Current library: Alpha v34, 4,223 checked-use theorems; Stable remains 432. Historical first admissions, original proof editions, and non-admitted aliases are preserved. Exact original first-admission records.
Exact expanded first-order arithmetic statement
forall z. (exists ge_real_positive_unique_factorization_domain ge_real_negative_unique_factorization_domain ge_imaginary_positive_unique_factorization_domain ge_imaginary_negative_unique_factorization_domain. (exists ge_real_code_unique_factorization_domaindecode ge_imaginary_code_unique_factorization_domaindecode. (((z) = ((ge_real_code_unique_factorization_domaindecode) + (ge_imaginary_code_unique_factorization_domaindecode)) * S ((ge_real_code_unique_factorization_domaindecode) + (ge_imaginary_code_unique_factorization_domaindecode)) + ((ge_imaginary_code_unique_factorization_domaindecode) + (ge_imaginary_code_unique_factorization_domaindecode))) /\ (((((ge_real_code_unique_factorization_domaindecode) = 2 * (ge_real_positive_unique_factorization_domain) /\ (ge_real_negative_unique_factorization_domain) = 0) \/ exists ge_signed_half_ge_unique_factorization_domaindecode_real. (((ge_real_code_unique_factorization_domaindecode) = 2 * ge_signed_half_ge_unique_factorization_domaindecode_real + 1 /\ (ge_real_positive_unique_factorization_domain) = 0) /\ (ge_real_negative_unique_factorization_domain) = S ge_signed_half_ge_unique_factorization_domaindecode_real))) /\ ((((ge_imaginary_code_unique_factorization_domaindecode) = 2 * (ge_imaginary_positive_unique_factorization_domain) /\ (ge_imaginary_negative_unique_factorization_domain) = 0) \/ exists ge_signed_half_ge_unique_factorization_domaindecode_imaginary. (((ge_imaginary_code_unique_factorization_domaindecode) = 2 * ge_signed_half_ge_unique_factorization_domaindecode_imaginary + 1 /\ (ge_imaginary_positive_unique_factorization_domain) = 0) /\ (ge_imaginary_negative_unique_factorization_domain) = S ge_signed_half_ge_unique_factorization_domaindecode_imaginary))))))) -> ~(z=0) -> (exists gr_chosen_unit_unique_factorization_result gr_chosen_code_unique_factorization_result gr_chosen_scale_unique_factorization_result gr_chosen_length_unique_factorization_result. ((((exists gr_inverse_unique_factorization_resultchosen_factorizationunit. (exists ge_first_rp_unique_factorization_resultchosen_factorizationunitidentity ge_first_rn_unique_factorization_resultchosen_factorizationunitidentity ge_first_ip_unique_factorization_resultchosen_factorizationunitidentity ge_first_in_unique_factorization_resultchosen_factorizationunitidentity ge_second_rp_unique_factorization_resultchosen_factorizationunitidentity ge_second_rn_unique_factorization_resultchosen_factorizationunitidentity ge_second_ip_unique_factorization_resultchosen_factorizationunitidentity ge_second_in_unique_factorization_resultchosen_factorizationunitidentity. ((exists ge_representation_real_code_unique_factorization_resultchosen_factorizationunitidentityfirst ge_representation_imaginary_code_unique_factorization_resultchosen_factorizationunitidentityfirst. (((gr_chosen_unit_unique_factorization_result) = ((ge_representation_real_code_unique_factorization_resultchosen_factorizationunitidentityfirst) + (ge_representation_imaginary_code_unique_factorization_resultchosen_factorizationunitidentityfirst)) * S ((ge_representation_real_code_unique_factorization_resultchosen_factorizationunitidentityfirst) + (ge_representation_imaginary_code_unique_factorization_resultchosen_factorizationunitidentityfirst)) + ((ge_representation_imaginary_code_unique_factorization_resultchosen_factorizationunitidentityfirst) + (ge_representation_imaginary_code_unique_factorization_resultchosen_factorizationunitidentityfirst))) /\ ((exists ge_balance_positive_unique_factorization_resultchosen_factorizationunitidentityfirstreal ge_balance_negative_unique_factorization_resultchosen_factorizationunitidentityfirstreal. (((((ge_representation_real_code_unique_factorization_resultchosen_factorizationunitidentityfirst) = 2 * (ge_balance_positive_unique_factorization_resultchosen_factorizationunitidentityfirstreal) /\ (ge_balance_negative_unique_factorization_resultchosen_factorizationunitidentityfirstreal) = 0) \/ exists ge_signed_half_unique_factorization_resultchosen_factorizationunitidentityfirstrealdecode. (((ge_representation_real_code_unique_factorization_resultchosen_factorizationunitidentityfirst) = 2 * ge_signed_half_unique_factorization_resultchosen_factorizationunitidentityfirstrealdecode + 1 /\ (ge_balance_positive_unique_factorization_resultchosen_factorizationunitidentityfirstreal) = 0) /\ (ge_balance_negative_unique_factorization_resultchosen_factorizationunitidentityfirstreal) = S ge_signed_half_unique_factorization_resultchosen_factorizationunitidentityfirstrealdecode))) /\ ((ge_first_rp_unique_factorization_resultchosen_factorizationunitidentity) + ge_balance_negative_unique_factorization_resultchosen_factorizationunitidentityfirstreal = (ge_first_rn_unique_factorization_resultchosen_factorizationunitidentity) + ge_balance_positive_unique_factorization_resultchosen_factorizationunitidentityfirstreal))) /\ (exists ge_balance_positive_unique_factorization_resultchosen_factorizationunitidentityfirstimaginary ge_balance_negative_unique_factorization_resultchosen_factorizationunitidentityfirstimaginary. (((((ge_representation_imaginary_code_unique_factorization_resultchosen_factorizationunitidentityfirst) = 2 * (ge_balance_positive_unique_factorization_resultchosen_factorizationunitidentityfirstimaginary) /\ (ge_balance_negative_unique_factorization_resultchosen_factorizationunitidentityfirstimaginary) = 0) \/ exists ge_signed_half_unique_factorization_resultchosen_factorizationunitidentityfirstimaginarydecode. (((ge_representation_imaginary_code_unique_factorization_resultchosen_factorizationunitidentityfirst) = 2 * ge_signed_half_unique_factorization_resultchosen_factorizationunitidentityfirstimaginarydecode + 1 /\ (ge_balance_positive_unique_factorization_resultchosen_factorizationunitidentityfirstimaginary) = 0) /\ (ge_balance_negative_unique_factorization_resultchosen_factorizationunitidentityfirstimaginary) = S ge_signed_half_unique_factorization_resultchosen_factorizationunitidentityfirstimaginarydecode))) /\ ((ge_first_ip_unique_factorization_resultchosen_factorizationunitidentity) + ge_balance_negative_unique_factorization_resultchosen_factorizationunitidentityfirstimaginary = (ge_first_in_unique_factorization_resultchosen_factorizationunitidentity) + ge_balance_positive_unique_factorization_resultchosen_factorizationunitidentityfirstimaginary)))))) /\ ((exists ge_representation_real_code_unique_factorization_resultchosen_factorizationunitidentitysecond ge_representation_imaginary_code_unique_factorization_resultchosen_factorizationunitidentitysecond. (((gr_inverse_unique_factorization_resultchosen_factorizationunit) = ((ge_representation_real_code_unique_factorization_resultchosen_factorizationunitidentitysecond) + (ge_representation_imaginary_code_unique_factorization_resultchosen_factorizationunitidentitysecond)) * S ((ge_representation_real_code_unique_factorization_resultchosen_factorizationunitidentitysecond) + (ge_representation_imaginary_code_unique_factorization_resultchosen_factorizationunitidentitysecond)) + ((ge_representation_imaginary_code_unique_factorization_resultchosen_factorizationunitidentitysecond) + (ge_representation_imaginary_code_unique_factorization_resultchosen_factorizationunitidentitysecond))) /\ ((exists ge_balance_positive_unique_factorization_resultchosen_factorizationunitidentitysecondreal ge_balance_negative_unique_factorization_resultchosen_factorizationunitidentitysecondreal. (((((ge_representation_real_code_unique_factorization_resultchosen_factorizationunitidentitysecond) = 2 * (ge_balance_positive_unique_factorization_resultchosen_factorizationunitidentitysecondreal) /\ (ge_balance_negative_unique_factorization_resultchosen_factorizationunitidentitysecondreal) = 0) \/ exists ge_signed_half_unique_factorization_resultchosen_factorizationunitidentitysecondrealdecode. (((ge_representation_real_code_unique_factorization_resultchosen_factorizationunitidentitysecond) = 2 * ge_signed_half_unique_factorization_resultchosen_factorizationunitidentitysecondrealdecode + 1 /\ (ge_balance_positive_unique_factorization_resultchosen_factorizationunitidentitysecondreal) = 0) /\ (ge_balance_negative_unique_factorization_resultchosen_factorizationunitidentitysecondreal) = S ge_signed_half_unique_factorization_resultchosen_factorizationunitidentitysecondrealdecode))) /\ ((ge_second_rp_unique_factorization_resultchosen_factorizationunitidentity) + ge_balance_negative_unique_factorization_resultchosen_factorizationunitidentitysecondreal = (ge_second_rn_unique_factorization_resultchosen_factorizationunitidentity) + ge_balance_positive_unique_factorization_resultchosen_factorizationunitidentitysecondreal))) /\ (exists ge_balance_positive_unique_factorization_resultchosen_factorizationunitidentitysecondimaginary ge_balance_negative_unique_factorization_resultchosen_factorizationunitidentitysecondimaginary. (((((ge_representation_imaginary_code_unique_factorization_resultchosen_factorizationunitidentitysecond) = 2 * (ge_balance_positive_unique_factorization_resultchosen_factorizationunitidentitysecondimaginary) /\ (ge_balance_negative_unique_factorization_resultchosen_factorizationunitidentitysecondimaginary) = 0) \/ exists ge_signed_half_unique_factorization_resultchosen_factorizationunitidentitysecondimaginarydecode. (((ge_representation_imaginary_code_unique_factorization_resultchosen_factorizationunitidentitysecond) = 2 * ge_signed_half_unique_factorization_resultchosen_factorizationunitidentitysecondimaginarydecode + 1 /\ (ge_balance_positive_unique_factorization_resultchosen_factorizationunitidentitysecondimaginary) = 0) /\ (ge_balance_negative_unique_factorization_resultchosen_factorizationunitidentitysecondimaginary) = S ge_signed_half_unique_factorization_resultchosen_factorizationunitidentitysecondimaginarydecode))) /\ ((ge_second_ip_unique_factorization_resultchosen_factorizationunitidentity) + ge_balance_negative_unique_factorization_resultchosen_factorizationunitidentitysecondimaginary = (ge_second_in_unique_factorization_resultchosen_factorizationunitidentity) + ge_balance_positive_unique_factorization_resultchosen_factorizationunitidentitysecondimaginary)))))) /\ (exists ge_representation_real_code_unique_factorization_resultchosen_factorizationunitidentityoutput ge_representation_imaginary_code_unique_factorization_resultchosen_factorizationunitidentityoutput. (((6) = ((ge_representation_real_code_unique_factorization_resultchosen_factorizationunitidentityoutput) + (ge_representation_imaginary_code_unique_factorization_resultchosen_factorizationunitidentityoutput)) * S ((ge_representation_real_code_unique_factorization_resultchosen_factorizationunitidentityoutput) + (ge_representation_imaginary_code_unique_factorization_resultchosen_factorizationunitidentityoutput)) + ((ge_representation_imaginary_code_unique_factorization_resultchosen_factorizationunitidentityoutput) + (ge_representation_imaginary_code_unique_factorization_resultchosen_factorizationunitidentityoutput))) /\ ((exists ge_balance_positive_unique_factorization_resultchosen_factorizationunitidentityoutputreal ge_balance_negative_unique_factorization_resultchosen_factorizationunitidentityoutputreal. (((((ge_representation_real_code_unique_factorization_resultchosen_factorizationunitidentityoutput) = 2 * (ge_balance_positive_unique_factorization_resultchosen_factorizationunitidentityoutputreal) /\ (ge_balance_negative_unique_factorization_resultchosen_factorizationunitidentityoutputreal) = 0) \/ exists ge_signed_half_unique_factorization_resultchosen_factorizationunitidentityoutputrealdecode. (((ge_representation_real_code_unique_factorization_resultchosen_factorizationunitidentityoutput) = 2 * ge_signed_half_unique_factorization_resultchosen_factorizationunitidentityoutputrealdecode + 1 /\ (ge_balance_positive_unique_factorization_resultchosen_factorizationunitidentityoutputreal) = 0) /\ (ge_balance_negative_unique_factorization_resultchosen_factorizationunitidentityoutputreal) = S ge_signed_half_unique_factorization_resultchosen_factorizationunitidentityoutputrealdecode))) /\ ((((((((ge_first_rp_unique_factorization_resultchosen_factorizationunitidentity) * (ge_second_rp_unique_factorization_resultchosen_factorizationunitidentity))) + (((ge_first_rn_unique_factorization_resultchosen_factorizationunitidentity) * (ge_second_rn_unique_factorization_resultchosen_factorizationunitidentity))))) + (((((ge_first_ip_unique_factorization_resultchosen_factorizationunitidentity) * (ge_second_in_unique_factorization_resultchosen_factorizationunitidentity))) + (((ge_first_in_unique_factorization_resultchosen_factorizationunitidentity) * (ge_second_ip_unique_factorization_resultchosen_factorizationunitidentity))))))) + ge_balance_negative_unique_factorization_resultchosen_factorizationunitidentityoutputreal = (((((((ge_first_rp_unique_factorization_resultchosen_factorizationunitidentity) * (ge_second_rn_unique_factorization_resultchosen_factorizationunitidentity))) + (((ge_first_rn_unique_factorization_resultchosen_factorizationunitidentity) * (ge_second_rp_unique_factorization_resultchosen_factorizationunitidentity))))) + (((((ge_first_ip_unique_factorization_resultchosen_factorizationunitidentity) * (ge_second_ip_unique_factorization_resultchosen_factorizationunitidentity))) + (((ge_first_in_unique_factorization_resultchosen_factorizationunitidentity) * (ge_second_in_unique_factorization_resultchosen_factorizationunitidentity))))))) + ge_balance_positive_unique_factorization_resultchosen_factorizationunitidentityoutputreal))) /\ (exists ge_balance_positive_unique_factorization_resultchosen_factorizationunitidentityoutputimaginary ge_balance_negative_unique_factorization_resultchosen_factorizationunitidentityoutputimaginary. (((((ge_representation_imaginary_code_unique_factorization_resultchosen_factorizationunitidentityoutput) = 2 * (ge_balance_positive_unique_factorization_resultchosen_factorizationunitidentityoutputimaginary) /\ (ge_balance_negative_unique_factorization_resultchosen_factorizationunitidentityoutputimaginary) = 0) \/ exists ge_signed_half_unique_factorization_resultchosen_factorizationunitidentityoutputimaginarydecode. (((ge_representation_imaginary_code_unique_factorization_resultchosen_factorizationunitidentityoutput) = 2 * ge_signed_half_unique_factorization_resultchosen_factorizationunitidentityoutputimaginarydecode + 1 /\ (ge_balance_positive_unique_factorization_resultchosen_factorizationunitidentityoutputimaginary) = 0) /\ (ge_balance_negative_unique_factorization_resultchosen_factorizationunitidentityoutputimaginary) = S ge_signed_half_unique_factorization_resultchosen_factorizationunitidentityoutputimaginarydecode))) /\ ((((((((ge_first_rp_unique_factorization_resultchosen_factorizationunitidentity) * (ge_second_ip_unique_factorization_resultchosen_factorizationunitidentity))) + (((ge_first_rn_unique_factorization_resultchosen_factorizationunitidentity) * (ge_second_in_unique_factorization_resultchosen_factorizationunitidentity))))) + (((((ge_first_ip_unique_factorization_resultchosen_factorizationunitidentity) * (ge_second_rp_unique_factorization_resultchosen_factorizationunitidentity))) + (((ge_first_in_unique_factorization_resultchosen_factorizationunitidentity) * (ge_second_rn_unique_factorization_resultchosen_factorizationunitidentity))))))) + ge_balance_negative_unique_factorization_resultchosen_factorizationunitidentityoutputimaginary = (((((((ge_first_rp_unique_factorization_resultchosen_factorizationunitidentity) * (ge_second_in_unique_factorization_resultchosen_factorizationunitidentity))) + (((ge_first_rn_unique_factorization_resultchosen_factorizationunitidentity) * (ge_second_ip_unique_factorization_resultchosen_factorizationunitidentity))))) + (((((ge_first_ip_unique_factorization_resultchosen_factorizationunitidentity) * (ge_second_rn_unique_factorization_resultchosen_factorizationunitidentity))) + (((ge_first_in_unique_factorization_resultchosen_factorizationunitidentity) * (ge_second_rp_unique_factorization_resultchosen_factorizationunitidentity))))))) + ge_balance_positive_unique_factorization_resultchosen_factorizationunitidentityoutputimaginary)))))))))) /\ ((forall gr_prime_factor_index_unique_factorization_resultchosen_factorizationprimes gr_prime_factor_value_unique_factorization_resultchosen_factorizationprimes. (exists ge_gap_unique_factorization_resultchosen_factorizationprimesindex. ge_gap_unique_factorization_resultchosen_factorizationprimesindex + S (gr_prime_factor_index_unique_factorization_resultchosen_factorizationprimes) = (gr_chosen_length_unique_factorization_result)) -> (((exists ff_h_gprod_unique_factorization_resultchosen_factorizationprimesentry. ff_h_gprod_unique_factorization_resultchosen_factorizationprimesentry + S (gr_prime_factor_value_unique_factorization_resultchosen_factorizationprimes) = S ((S (gr_prime_factor_index_unique_factorization_resultchosen_factorizationprimes)) * gr_chosen_scale_unique_factorization_result)) /\ exists ff_q_gprod_unique_factorization_resultchosen_factorizationprimesentry. gr_chosen_code_unique_factorization_result = ff_q_gprod_unique_factorization_resultchosen_factorizationprimesentry * S ((S (gr_prime_factor_index_unique_factorization_resultchosen_factorizationprimes)) * gr_chosen_scale_unique_factorization_result) + (gr_prime_factor_value_unique_factorization_resultchosen_factorizationprimes))) -> (((exists ge_real_positive_unique_factorization_resultchosen_factorizationprimesprimecarrier ge_real_negative_unique_factorization_resultchosen_factorizationprimesprimecarrier ge_imaginary_positive_unique_factorization_resultchosen_factorizationprimesprimecarrier ge_imaginary_negative_unique_factorization_resultchosen_factorizationprimesprimecarrier. (exists ge_real_code_unique_factorization_resultchosen_factorizationprimesprimecarrierdecode ge_imaginary_code_unique_factorization_resultchosen_factorizationprimesprimecarrierdecode. (((gr_prime_factor_value_unique_factorization_resultchosen_factorizationprimes) = ((ge_real_code_unique_factorization_resultchosen_factorizationprimesprimecarrierdecode) + (ge_imaginary_code_unique_factorization_resultchosen_factorizationprimesprimecarrierdecode)) * S ((ge_real_code_unique_factorization_resultchosen_factorizationprimesprimecarrierdecode) + (ge_imaginary_code_unique_factorization_resultchosen_factorizationprimesprimecarrierdecode)) + ((ge_imaginary_code_unique_factorization_resultchosen_factorizationprimesprimecarrierdecode) + (ge_imaginary_code_unique_factorization_resultchosen_factorizationprimesprimecarrierdecode))) /\ (((((ge_real_code_unique_factorization_resultchosen_factorizationprimesprimecarrierdecode) = 2 * (ge_real_positive_unique_factorization_resultchosen_factorizationprimesprimecarrier) /\ (ge_real_negative_unique_factorization_resultchosen_factorizationprimesprimecarrier) = 0) \/ exists ge_signed_half_ge_unique_factorization_resultchosen_factorizationprimesprimecarrierdecode_real. (((ge_real_code_unique_factorization_resultchosen_factorizationprimesprimecarrierdecode) = 2 * ge_signed_half_ge_unique_factorization_resultchosen_factorizationprimesprimecarrierdecode_real + 1 /\ (ge_real_positive_unique_factorization_resultchosen_factorizationprimesprimecarrier) = 0) /\ (ge_real_negative_unique_factorization_resultchosen_factorizationprimesprimecarrier) = S ge_signed_half_ge_unique_factorization_resultchosen_factorizationprimesprimecarrierdecode_real))) /\ ((((ge_imaginary_code_unique_factorization_resultchosen_factorizationprimesprimecarrierdecode) = 2 * (ge_imaginary_positive_unique_factorization_resultchosen_factorizationprimesprimecarrier) /\ (ge_imaginary_negative_unique_factorization_resultchosen_factorizationprimesprimecarrier) = 0) \/ exists ge_signed_half_ge_unique_factorization_resultchosen_factorizationprimesprimecarrierdecode_imaginary. (((ge_imaginary_code_unique_factorization_resultchosen_factorizationprimesprimecarrierdecode) = 2 * ge_signed_half_ge_unique_factorization_resultchosen_factorizationprimesprimecarrierdecode_imaginary + 1 /\ (ge_imaginary_positive_unique_factorization_resultchosen_factorizationprimesprimecarrier) = 0) /\ (ge_imaginary_negative_unique_factorization_resultchosen_factorizationprimesprimecarrier) = S ge_signed_half_ge_unique_factorization_resultchosen_factorizationprimesprimecarrierdecode_imaginary))))))) /\ ((~((gr_prime_factor_value_unique_factorization_resultchosen_factorizationprimes)=0)) /\ ((~(exists gr_inverse_unique_factorization_resultchosen_factorizationprimesprimenonunit. (exists ge_first_rp_unique_factorization_resultchosen_factorizationprimesprimenonunitidentity ge_first_rn_unique_factorization_resultchosen_factorizationprimesprimenonunitidentity ge_first_ip_unique_factorization_resultchosen_factorizationprimesprimenonunitidentity ge_first_in_unique_factorization_resultchosen_factorizationprimesprimenonunitidentity ge_second_rp_unique_factorization_resultchosen_factorizationprimesprimenonunitidentity ge_second_rn_unique_factorization_resultchosen_factorizationprimesprimenonunitidentity ge_second_ip_unique_factorization_resultchosen_factorizationprimesprimenonunitidentity ge_second_in_unique_factorization_resultchosen_factorizationprimesprimenonunitidentity. ((exists ge_representation_real_code_unique_factorization_resultchosen_factorizationprimesprimenonunitidentityfirst ge_representation_imaginary_code_unique_factorization_resultchosen_factorizationprimesprimenonunitidentityfirst. (((gr_prime_factor_value_unique_factorization_resultchosen_factorizationprimes) = ((ge_representation_real_code_unique_factorization_resultchosen_factorizationprimesprimenonunitidentityfirst) + (ge_representation_imaginary_code_unique_factorization_resultchosen_factorizationprimesprimenonunitidentityfirst)) * S ((ge_representation_real_code_unique_factorization_resultchosen_factorizationprimesprimenonunitidentityfirst) + (ge_representation_imaginary_code_unique_factorization_resultchosen_factorizationprimesprimenonunitidentityfirst)) + ((ge_representation_imaginary_code_unique_factorization_resultchosen_factorizationprimesprimenonunitidentityfirst) + (ge_representation_imaginary_code_unique_factorization_resultchosen_factorizationprimesprimenonunitidentityfirst))) /\ ((exists ge_balance_positive_unique_factorization_resultchosen_factorizationprimesprimenonunitidentityfirstreal ge_balance_negative_unique_factorization_resultchosen_factorizationprimesprimenonunitidentityfirstreal. (((((ge_representation_real_code_unique_factorization_resultchosen_factorizationprimesprimenonunitidentityfirst) = 2 * (ge_balance_positive_unique_factorization_resultchosen_factorizationprimesprimenonunitidentityfirstreal) /\ (ge_balance_negative_unique_factorization_resultchosen_factorizationprimesprimenonunitidentityfirstreal) = 0) \/ exists ge_signed_half_unique_factorization_resultchosen_factorizationprimesprimenonunitidentityfirstrealdecode. (((ge_representation_real_code_unique_factorization_resultchosen_factorizationprimesprimenonunitidentityfirst) = 2 * ge_signed_half_unique_factorization_resultchosen_factorizationprimesprimenonunitidentityfirstrealdecode + 1 /\ (ge_balance_positive_unique_factorization_resultchosen_factorizationprimesprimenonunitidentityfirstreal) = 0) /\ (ge_balance_negative_unique_factorization_resultchosen_factorizationprimesprimenonunitidentityfirstreal) = S ge_signed_half_unique_factorization_resultchosen_factorizationprimesprimenonunitidentityfirstrealdecode))) /\ ((ge_first_rp_unique_factorization_resultchosen_factorizationprimesprimenonunitidentity) + ge_balance_negative_unique_factorization_resultchosen_factorizationprimesprimenonunitidentityfirstreal = (ge_first_rn_unique_factorization_resultchosen_factorizationprimesprimenonunitidentity) + ge_balance_positive_unique_factorization_resultchosen_factorizationprimesprimenonunitidentityfirstreal))) /\ (exists ge_balance_positive_unique_factorization_resultchosen_factorizationprimesprimenonunitidentityfirstimaginary ge_balance_negative_unique_factorization_resultchosen_factorizationprimesprimenonunitidentityfirstimaginary. (((((ge_representation_imaginary_code_unique_factorization_resultchosen_factorizationprimesprimenonunitidentityfirst) = 2 * (ge_balance_positive_unique_factorization_resultchosen_factorizationprimesprimenonunitidentityfirstimaginary) /\ (ge_balance_negative_unique_factorization_resultchosen_factorizationprimesprimenonunitidentityfirstimaginary) = 0) \/ exists ge_signed_half_unique_factorization_resultchosen_factorizationprimesprimenonunitidentityfirstimaginarydecode. (((ge_representation_imaginary_code_unique_factorization_resultchosen_factorizationprimesprimenonunitidentityfirst) = 2 * ge_signed_half_unique_factorization_resultchosen_factorizationprimesprimenonunitidentityfirstimaginarydecode + 1 /\ (ge_balance_positive_unique_factorization_resultchosen_factorizationprimesprimenonunitidentityfirstimaginary) = 0) /\ (ge_balance_negative_unique_factorization_resultchosen_factorizationprimesprimenonunitidentityfirstimaginary) = S ge_signed_half_unique_factorization_resultchosen_factorizationprimesprimenonunitidentityfirstimaginarydecode))) /\ ((ge_first_ip_unique_factorization_resultchosen_factorizationprimesprimenonunitidentity) + ge_balance_negative_unique_factorization_resultchosen_factorizationprimesprimenonunitidentityfirstimaginary = (ge_first_in_unique_factorization_resultchosen_factorizationprimesprimenonunitidentity) + ge_balance_positive_unique_factorization_resultchosen_factorizationprimesprimenonunitidentityfirstimaginary)))))) /\ ((exists ge_representation_real_code_unique_factorization_resultchosen_factorizationprimesprimenonunitidentitysecond ge_representation_imaginary_code_unique_factorization_resultchosen_factorizationprimesprimenonunitidentitysecond. (((gr_inverse_unique_factorization_resultchosen_factorizationprimesprimenonunit) = ((ge_representation_real_code_unique_factorization_resultchosen_factorizationprimesprimenonunitidentitysecond) + (ge_representation_imaginary_code_unique_factorization_resultchosen_factorizationprimesprimenonunitidentitysecond)) * S ((ge_representation_real_code_unique_factorization_resultchosen_factorizationprimesprimenonunitidentitysecond) + (ge_representation_imaginary_code_unique_factorization_resultchosen_factorizationprimesprimenonunitidentitysecond)) + ((ge_representation_imaginary_code_unique_factorization_resultchosen_factorizationprimesprimenonunitidentitysecond) + (ge_representation_imaginary_code_unique_factorization_resultchosen_factorizationprimesprimenonunitidentitysecond))) /\ ((exists ge_balance_positive_unique_factorization_resultchosen_factorizationprimesprimenonunitidentitysecondreal ge_balance_negative_unique_factorization_resultchosen_factorizationprimesprimenonunitidentitysecondreal. (((((ge_representation_real_code_unique_factorization_resultchosen_factorizationprimesprimenonunitidentitysecond) = 2 * (ge_balance_positive_unique_factorization_resultchosen_factorizationprimesprimenonunitidentitysecondreal) /\ (ge_balance_negative_unique_factorization_resultchosen_factorizationprimesprimenonunitidentitysecondreal) = 0) \/ exists ge_signed_half_unique_factorization_resultchosen_factorizationprimesprimenonunitidentitysecondrealdecode. (((ge_representation_real_code_unique_factorization_resultchosen_factorizationprimesprimenonunitidentitysecond) = 2 * ge_signed_half_unique_factorization_resultchosen_factorizationprimesprimenonunitidentitysecondrealdecode + 1 /\ (ge_balance_positive_unique_factorization_resultchosen_factorizationprimesprimenonunitidentitysecondreal) = 0) /\ (ge_balance_negative_unique_factorization_resultchosen_factorizationprimesprimenonunitidentitysecondreal) = S ge_signed_half_unique_factorization_resultchosen_factorizationprimesprimenonunitidentitysecondrealdecode))) /\ ((ge_second_rp_unique_factorization_resultchosen_factorizationprimesprimenonunitidentity) + ge_balance_negative_unique_factorization_resultchosen_factorizationprimesprimenonunitidentitysecondreal = (ge_second_rn_unique_factorization_resultchosen_factorizationprimesprimenonunitidentity) + ge_balance_positive_unique_factorization_resultchosen_factorizationprimesprimenonunitidentitysecondreal))) /\ (exists ge_balance_positive_unique_factorization_resultchosen_factorizationprimesprimenonunitidentitysecondimaginary ge_balance_negative_unique_factorization_resultchosen_factorizationprimesprimenonunitidentitysecondimaginary. (((((ge_representation_imaginary_code_unique_factorization_resultchosen_factorizationprimesprimenonunitidentitysecond) = 2 * (ge_balance_positive_unique_factorization_resultchosen_factorizationprimesprimenonunitidentitysecondimaginary) /\ (ge_balance_negative_unique_factorization_resultchosen_factorizationprimesprimenonunitidentitysecondimaginary) = 0) \/ exists ge_signed_half_unique_factorization_resultchosen_factorizationprimesprimenonunitidentitysecondimaginarydecode. (((ge_representation_imaginary_code_unique_factorization_resultchosen_factorizationprimesprimenonunitidentitysecond) = 2 * ge_signed_half_unique_factorization_resultchosen_factorizationprimesprimenonunitidentitysecondimaginarydecode + 1 /\ (ge_balance_positive_unique_factorization_resultchosen_factorizationprimesprimenonunitidentitysecondimaginary) = 0) /\ (ge_balance_negative_unique_factorization_resultchosen_factorizationprimesprimenonunitidentitysecondimaginary) = S ge_signed_half_unique_factorization_resultchosen_factorizationprimesprimenonunitidentitysecondimaginarydecode))) /\ ((ge_second_ip_unique_factorization_resultchosen_factorizationprimesprimenonunitidentity) + ge_balance_negative_unique_factorization_resultchosen_factorizationprimesprimenonunitidentitysecondimaginary = (ge_second_in_unique_factorization_resultchosen_factorizationprimesprimenonunitidentity) + ge_balance_positive_unique_factorization_resultchosen_factorizationprimesprimenonunitidentitysecondimaginary)))))) /\ (exists ge_representation_real_code_unique_factorization_resultchosen_factorizationprimesprimenonunitidentityoutput ge_representation_imaginary_code_unique_factorization_resultchosen_factorizationprimesprimenonunitidentityoutput. (((6) = ((ge_representation_real_code_unique_factorization_resultchosen_factorizationprimesprimenonunitidentityoutput) + (ge_representation_imaginary_code_unique_factorization_resultchosen_factorizationprimesprimenonunitidentityoutput)) * S ((ge_representation_real_code_unique_factorization_resultchosen_factorizationprimesprimenonunitidentityoutput) + (ge_representation_imaginary_code_unique_factorization_resultchosen_factorizationprimesprimenonunitidentityoutput)) + ((ge_representation_imaginary_code_unique_factorization_resultchosen_factorizationprimesprimenonunitidentityoutput) + (ge_representation_imaginary_code_unique_factorization_resultchosen_factorizationprimesprimenonunitidentityoutput))) /\ ((exists ge_balance_positive_unique_factorization_resultchosen_factorizationprimesprimenonunitidentityoutputreal ge_balance_negative_unique_factorization_resultchosen_factorizationprimesprimenonunitidentityoutputreal. (((((ge_representation_real_code_unique_factorization_resultchosen_factorizationprimesprimenonunitidentityoutput) = 2 * (ge_balance_positive_unique_factorization_resultchosen_factorizationprimesprimenonunitidentityoutputreal) /\ (ge_balance_negative_unique_factorization_resultchosen_factorizationprimesprimenonunitidentityoutputreal) = 0) \/ exists ge_signed_half_unique_factorization_resultchosen_factorizationprimesprimenonunitidentityoutputrealdecode. (((ge_representation_real_code_unique_factorization_resultchosen_factorizationprimesprimenonunitidentityoutput) = 2 * ge_signed_half_unique_factorization_resultchosen_factorizationprimesprimenonunitidentityoutputrealdecode + 1 /\ (ge_balance_positive_unique_factorization_resultchosen_factorizationprimesprimenonunitidentityoutputreal) = 0) /\ (ge_balance_negative_unique_factorization_resultchosen_factorizationprimesprimenonunitidentityoutputreal) = S ge_signed_half_unique_factorization_resultchosen_factorizationprimesprimenonunitidentityoutputrealdecode))) /\ ((((((((ge_first_rp_unique_factorization_resultchosen_factorizationprimesprimenonunitidentity) * (ge_second_rp_unique_factorization_resultchosen_factorizationprimesprimenonunitidentity))) + (((ge_first_rn_unique_factorization_resultchosen_factorizationprimesprimenonunitidentity) * (ge_second_rn_unique_factorization_resultchosen_factorizationprimesprimenonunitidentity))))) + (((((ge_first_ip_unique_factorization_resultchosen_factorizationprimesprimenonunitidentity) * (ge_second_in_unique_factorization_resultchosen_factorizationprimesprimenonunitidentity))) + (((ge_first_in_unique_factorization_resultchosen_factorizationprimesprimenonunitidentity) * (ge_second_ip_unique_factorization_resultchosen_factorizationprimesprimenonunitidentity))))))) + ge_balance_negative_unique_factorization_resultchosen_factorizationprimesprimenonunitidentityoutputreal = (((((((ge_first_rp_unique_factorization_resultchosen_factorizationprimesprimenonunitidentity) * (ge_second_rn_unique_factorization_resultchosen_factorizationprimesprimenonunitidentity))) + (((ge_first_rn_unique_factorization_resultchosen_factorizationprimesprimenonunitidentity) * (ge_second_rp_unique_factorization_resultchosen_factorizationprimesprimenonunitidentity))))) + (((((ge_first_ip_unique_factorization_resultchosen_factorizationprimesprimenonunitidentity) * (ge_second_ip_unique_factorization_resultchosen_factorizationprimesprimenonunitidentity))) + (((ge_first_in_unique_factorization_resultchosen_factorizationprimesprimenonunitidentity) * (ge_second_in_unique_factorization_resultchosen_factorizationprimesprimenonunitidentity))))))) + ge_balance_positive_unique_factorization_resultchosen_factorizationprimesprimenonunitidentityoutputreal))) /\ (exists ge_balance_positive_unique_factorization_resultchosen_factorizationprimesprimenonunitidentityoutputimaginary ge_balance_negative_unique_factorization_resultchosen_factorizationprimesprimenonunitidentityoutputimaginary. (((((ge_representation_imaginary_code_unique_factorization_resultchosen_factorizationprimesprimenonunitidentityoutput) = 2 * (ge_balance_positive_unique_factorization_resultchosen_factorizationprimesprimenonunitidentityoutputimaginary) /\ (ge_balance_negative_unique_factorization_resultchosen_factorizationprimesprimenonunitidentityoutputimaginary) = 0) \/ exists ge_signed_half_unique_factorization_resultchosen_factorizationprimesprimenonunitidentityoutputimaginarydecode. (((ge_representation_imaginary_code_unique_factorization_resultchosen_factorizationprimesprimenonunitidentityoutput) = 2 * ge_signed_half_unique_factorization_resultchosen_factorizationprimesprimenonunitidentityoutputimaginarydecode + 1 /\ (ge_balance_positive_unique_factorization_resultchosen_factorizationprimesprimenonunitidentityoutputimaginary) = 0) /\ (ge_balance_negative_unique_factorization_resultchosen_factorizationprimesprimenonunitidentityoutputimaginary) = S ge_signed_half_unique_factorization_resultchosen_factorizationprimesprimenonunitidentityoutputimaginarydecode))) /\ ((((((((ge_first_rp_unique_factorization_resultchosen_factorizationprimesprimenonunitidentity) * (ge_second_ip_unique_factorization_resultchosen_factorizationprimesprimenonunitidentity))) + (((ge_first_rn_unique_factorization_resultchosen_factorizationprimesprimenonunitidentity) * (ge_second_in_unique_factorization_resultchosen_factorizationprimesprimenonunitidentity))))) + (((((ge_first_ip_unique_factorization_resultchosen_factorizationprimesprimenonunitidentity) * (ge_second_rp_unique_factorization_resultchosen_factorizationprimesprimenonunitidentity))) + (((ge_first_in_unique_factorization_resultchosen_factorizationprimesprimenonunitidentity) * (ge_second_rn_unique_factorization_resultchosen_factorizationprimesprimenonunitidentity))))))) + ge_balance_negative_unique_factorization_resultchosen_factorizationprimesprimenonunitidentityoutputimaginary = (((((((ge_first_rp_unique_factorization_resultchosen_factorizationprimesprimenonunitidentity) * (ge_second_in_unique_factorization_resultchosen_factorizationprimesprimenonunitidentity))) + (((ge_first_rn_unique_factorization_resultchosen_factorizationprimesprimenonunitidentity) * (ge_second_ip_unique_factorization_resultchosen_factorizationprimesprimenonunitidentity))))) + (((((ge_first_ip_unique_factorization_resultchosen_factorizationprimesprimenonunitidentity) * (ge_second_rn_unique_factorization_resultchosen_factorizationprimesprimenonunitidentity))) + (((ge_first_in_unique_factorization_resultchosen_factorizationprimesprimenonunitidentity) * (ge_second_rp_unique_factorization_resultchosen_factorizationprimesprimenonunitidentity))))))) + ge_balance_positive_unique_factorization_resultchosen_factorizationprimesprimenonunitidentityoutputimaginary))))))))))) /\ (forall gr_first_factor_unique_factorization_resultchosen_factorizationprimesprime gr_second_factor_unique_factorization_resultchosen_factorizationprimesprime gr_product_unique_factorization_resultchosen_factorizationprimesprime. (exists ge_first_rp_unique_factorization_resultchosen_factorizationprimesprimeproduct ge_first_rn_unique_factorization_resultchosen_factorizationprimesprimeproduct ge_first_ip_unique_factorization_resultchosen_factorizationprimesprimeproduct ge_first_in_unique_factorization_resultchosen_factorizationprimesprimeproduct ge_second_rp_unique_factorization_resultchosen_factorizationprimesprimeproduct ge_second_rn_unique_factorization_resultchosen_factorizationprimesprimeproduct ge_second_ip_unique_factorization_resultchosen_factorizationprimesprimeproduct ge_second_in_unique_factorization_resultchosen_factorizationprimesprimeproduct. ((exists ge_representation_real_code_unique_factorization_resultchosen_factorizationprimesprimeproductfirst ge_representation_imaginary_code_unique_factorization_resultchosen_factorizationprimesprimeproductfirst. (((gr_first_factor_unique_factorization_resultchosen_factorizationprimesprime) = ((ge_representation_real_code_unique_factorization_resultchosen_factorizationprimesprimeproductfirst) + (ge_representation_imaginary_code_unique_factorization_resultchosen_factorizationprimesprimeproductfirst)) * S ((ge_representation_real_code_unique_factorization_resultchosen_factorizationprimesprimeproductfirst) + (ge_representation_imaginary_code_unique_factorization_resultchosen_factorizationprimesprimeproductfirst)) + ((ge_representation_imaginary_code_unique_factorization_resultchosen_factorizationprimesprimeproductfirst) + (ge_representation_imaginary_code_unique_factorization_resultchosen_factorizationprimesprimeproductfirst))) /\ ((exists ge_balance_positive_unique_factorization_resultchosen_factorizationprimesprimeproductfirstreal ge_balance_negative_unique_factorization_resultchosen_factorizationprimesprimeproductfirstreal. (((((ge_representation_real_code_unique_factorization_resultchosen_factorizationprimesprimeproductfirst) = 2 * (ge_balance_positive_unique_factorization_resultchosen_factorizationprimesprimeproductfirstreal) /\ (ge_balance_negative_unique_factorization_resultchosen_factorizationprimesprimeproductfirstreal) = 0) \/ exists ge_signed_half_unique_factorization_resultchosen_factorizationprimesprimeproductfirstrealdecode. (((ge_representation_real_code_unique_factorization_resultchosen_factorizationprimesprimeproductfirst) = 2 * ge_signed_half_unique_factorization_resultchosen_factorizationprimesprimeproductfirstrealdecode + 1 /\ (ge_balance_positive_unique_factorization_resultchosen_factorizationprimesprimeproductfirstreal) = 0) /\ (ge_balance_negative_unique_factorization_resultchosen_factorizationprimesprimeproductfirstreal) = S ge_signed_half_unique_factorization_resultchosen_factorizationprimesprimeproductfirstrealdecode))) /\ ((ge_first_rp_unique_factorization_resultchosen_factorizationprimesprimeproduct) + ge_balance_negative_unique_factorization_resultchosen_factorizationprimesprimeproductfirstreal = (ge_first_rn_unique_factorization_resultchosen_factorizationprimesprimeproduct) + ge_balance_positive_unique_factorization_resultchosen_factorizationprimesprimeproductfirstreal))) /\ (exists ge_balance_positive_unique_factorization_resultchosen_factorizationprimesprimeproductfirstimaginary ge_balance_negative_unique_factorization_resultchosen_factorizationprimesprimeproductfirstimaginary. (((((ge_representation_imaginary_code_unique_factorization_resultchosen_factorizationprimesprimeproductfirst) = 2 * (ge_balance_positive_unique_factorization_resultchosen_factorizationprimesprimeproductfirstimaginary) /\ (ge_balance_negative_unique_factorization_resultchosen_factorizationprimesprimeproductfirstimaginary) = 0) \/ exists ge_signed_half_unique_factorization_resultchosen_factorizationprimesprimeproductfirstimaginarydecode. (((ge_representation_imaginary_code_unique_factorization_resultchosen_factorizationprimesprimeproductfirst) = 2 * ge_signed_half_unique_factorization_resultchosen_factorizationprimesprimeproductfirstimaginarydecode + 1 /\ (ge_balance_positive_unique_factorization_resultchosen_factorizationprimesprimeproductfirstimaginary) = 0) /\ (ge_balance_negative_unique_factorization_resultchosen_factorizationprimesprimeproductfirstimaginary) = S ge_signed_half_unique_factorization_resultchosen_factorizationprimesprimeproductfirstimaginarydecode))) /\ ((ge_first_ip_unique_factorization_resultchosen_factorizationprimesprimeproduct) + ge_balance_negative_unique_factorization_resultchosen_factorizationprimesprimeproductfirstimaginary = (ge_first_in_unique_factorization_resultchosen_factorizationprimesprimeproduct) + ge_balance_positive_unique_factorization_resultchosen_factorizationprimesprimeproductfirstimaginary)))))) /\ ((exists ge_representation_real_code_unique_factorization_resultchosen_factorizationprimesprimeproductsecond ge_representation_imaginary_code_unique_factorization_resultchosen_factorizationprimesprimeproductsecond. (((gr_second_factor_unique_factorization_resultchosen_factorizationprimesprime) = ((ge_representation_real_code_unique_factorization_resultchosen_factorizationprimesprimeproductsecond) + (ge_representation_imaginary_code_unique_factorization_resultchosen_factorizationprimesprimeproductsecond)) * S ((ge_representation_real_code_unique_factorization_resultchosen_factorizationprimesprimeproductsecond) + (ge_representation_imaginary_code_unique_factorization_resultchosen_factorizationprimesprimeproductsecond)) + ((ge_representation_imaginary_code_unique_factorization_resultchosen_factorizationprimesprimeproductsecond) + (ge_representation_imaginary_code_unique_factorization_resultchosen_factorizationprimesprimeproductsecond))) /\ ((exists ge_balance_positive_unique_factorization_resultchosen_factorizationprimesprimeproductsecondreal ge_balance_negative_unique_factorization_resultchosen_factorizationprimesprimeproductsecondreal. (((((ge_representation_real_code_unique_factorization_resultchosen_factorizationprimesprimeproductsecond) = 2 * (ge_balance_positive_unique_factorization_resultchosen_factorizationprimesprimeproductsecondreal) /\ (ge_balance_negative_unique_factorization_resultchosen_factorizationprimesprimeproductsecondreal) = 0) \/ exists ge_signed_half_unique_factorization_resultchosen_factorizationprimesprimeproductsecondrealdecode. (((ge_representation_real_code_unique_factorization_resultchosen_factorizationprimesprimeproductsecond) = 2 * ge_signed_half_unique_factorization_resultchosen_factorizationprimesprimeproductsecondrealdecode + 1 /\ (ge_balance_positive_unique_factorization_resultchosen_factorizationprimesprimeproductsecondreal) = 0) /\ (ge_balance_negative_unique_factorization_resultchosen_factorizationprimesprimeproductsecondreal) = S ge_signed_half_unique_factorization_resultchosen_factorizationprimesprimeproductsecondrealdecode))) /\ ((ge_second_rp_unique_factorization_resultchosen_factorizationprimesprimeproduct) + ge_balance_negative_unique_factorization_resultchosen_factorizationprimesprimeproductsecondreal = (ge_second_rn_unique_factorization_resultchosen_factorizationprimesprimeproduct) + ge_balance_positive_unique_factorization_resultchosen_factorizationprimesprimeproductsecondreal))) /\ (exists ge_balance_positive_unique_factorization_resultchosen_factorizationprimesprimeproductsecondimaginary ge_balance_negative_unique_factorization_resultchosen_factorizationprimesprimeproductsecondimaginary. (((((ge_representation_imaginary_code_unique_factorization_resultchosen_factorizationprimesprimeproductsecond) = 2 * (ge_balance_positive_unique_factorization_resultchosen_factorizationprimesprimeproductsecondimaginary) /\ (ge_balance_negative_unique_factorization_resultchosen_factorizationprimesprimeproductsecondimaginary) = 0) \/ exists ge_signed_half_unique_factorization_resultchosen_factorizationprimesprimeproductsecondimaginarydecode. (((ge_representation_imaginary_code_unique_factorization_resultchosen_factorizationprimesprimeproductsecond) = 2 * ge_signed_half_unique_factorization_resultchosen_factorizationprimesprimeproductsecondimaginarydecode + 1 /\ (ge_balance_positive_unique_factorization_resultchosen_factorizationprimesprimeproductsecondimaginary) = 0) /\ (ge_balance_negative_unique_factorization_resultchosen_factorizationprimesprimeproductsecondimaginary) = S ge_signed_half_unique_factorization_resultchosen_factorizationprimesprimeproductsecondimaginarydecode))) /\ ((ge_second_ip_unique_factorization_resultchosen_factorizationprimesprimeproduct) + ge_balance_negative_unique_factorization_resultchosen_factorizationprimesprimeproductsecondimaginary = (ge_second_in_unique_factorization_resultchosen_factorizationprimesprimeproduct) + ge_balance_positive_unique_factorization_resultchosen_factorizationprimesprimeproductsecondimaginary)))))) /\ (exists ge_representation_real_code_unique_factorization_resultchosen_factorizationprimesprimeproductoutput ge_representation_imaginary_code_unique_factorization_resultchosen_factorizationprimesprimeproductoutput. (((gr_product_unique_factorization_resultchosen_factorizationprimesprime) = ((ge_representation_real_code_unique_factorization_resultchosen_factorizationprimesprimeproductoutput) + (ge_representation_imaginary_code_unique_factorization_resultchosen_factorizationprimesprimeproductoutput)) * S ((ge_representation_real_code_unique_factorization_resultchosen_factorizationprimesprimeproductoutput) + (ge_representation_imaginary_code_unique_factorization_resultchosen_factorizationprimesprimeproductoutput)) + ((ge_representation_imaginary_code_unique_factorization_resultchosen_factorizationprimesprimeproductoutput) + (ge_representation_imaginary_code_unique_factorization_resultchosen_factorizationprimesprimeproductoutput))) /\ ((exists ge_balance_positive_unique_factorization_resultchosen_factorizationprimesprimeproductoutputreal ge_balance_negative_unique_factorization_resultchosen_factorizationprimesprimeproductoutputreal. (((((ge_representation_real_code_unique_factorization_resultchosen_factorizationprimesprimeproductoutput) = 2 * (ge_balance_positive_unique_factorization_resultchosen_factorizationprimesprimeproductoutputreal) /\ (ge_balance_negative_unique_factorization_resultchosen_factorizationprimesprimeproductoutputreal) = 0) \/ exists ge_signed_half_unique_factorization_resultchosen_factorizationprimesprimeproductoutputrealdecode. (((ge_representation_real_code_unique_factorization_resultchosen_factorizationprimesprimeproductoutput) = 2 * ge_signed_half_unique_factorization_resultchosen_factorizationprimesprimeproductoutputrealdecode + 1 /\ (ge_balance_positive_unique_factorization_resultchosen_factorizationprimesprimeproductoutputreal) = 0) /\ (ge_balance_negative_unique_factorization_resultchosen_factorizationprimesprimeproductoutputreal) = S ge_signed_half_unique_factorization_resultchosen_factorizationprimesprimeproductoutputrealdecode))) /\ ((((((((ge_first_rp_unique_factorization_resultchosen_factorizationprimesprimeproduct) * (ge_second_rp_unique_factorization_resultchosen_factorizationprimesprimeproduct))) + (((ge_first_rn_unique_factorization_resultchosen_factorizationprimesprimeproduct) * (ge_second_rn_unique_factorization_resultchosen_factorizationprimesprimeproduct))))) + (((((ge_first_ip_unique_factorization_resultchosen_factorizationprimesprimeproduct) * (ge_second_in_unique_factorization_resultchosen_factorizationprimesprimeproduct))) + (((ge_first_in_unique_factorization_resultchosen_factorizationprimesprimeproduct) * (ge_second_ip_unique_factorization_resultchosen_factorizationprimesprimeproduct))))))) + ge_balance_negative_unique_factorization_resultchosen_factorizationprimesprimeproductoutputreal = (((((((ge_first_rp_unique_factorization_resultchosen_factorizationprimesprimeproduct) * (ge_second_rn_unique_factorization_resultchosen_factorizationprimesprimeproduct))) + (((ge_first_rn_unique_factorization_resultchosen_factorizationprimesprimeproduct) * (ge_second_rp_unique_factorization_resultchosen_factorizationprimesprimeproduct))))) + (((((ge_first_ip_unique_factorization_resultchosen_factorizationprimesprimeproduct) * (ge_second_ip_unique_factorization_resultchosen_factorizationprimesprimeproduct))) + (((ge_first_in_unique_factorization_resultchosen_factorizationprimesprimeproduct) * (ge_second_in_unique_factorization_resultchosen_factorizationprimesprimeproduct))))))) + ge_balance_positive_unique_factorization_resultchosen_factorizationprimesprimeproductoutputreal))) /\ (exists ge_balance_positive_unique_factorization_resultchosen_factorizationprimesprimeproductoutputimaginary ge_balance_negative_unique_factorization_resultchosen_factorizationprimesprimeproductoutputimaginary. (((((ge_representation_imaginary_code_unique_factorization_resultchosen_factorizationprimesprimeproductoutput) = 2 * (ge_balance_positive_unique_factorization_resultchosen_factorizationprimesprimeproductoutputimaginary) /\ (ge_balance_negative_unique_factorization_resultchosen_factorizationprimesprimeproductoutputimaginary) = 0) \/ exists ge_signed_half_unique_factorization_resultchosen_factorizationprimesprimeproductoutputimaginarydecode. (((ge_representation_imaginary_code_unique_factorization_resultchosen_factorizationprimesprimeproductoutput) = 2 * ge_signed_half_unique_factorization_resultchosen_factorizationprimesprimeproductoutputimaginarydecode + 1 /\ (ge_balance_positive_unique_factorization_resultchosen_factorizationprimesprimeproductoutputimaginary) = 0) /\ (ge_balance_negative_unique_factorization_resultchosen_factorizationprimesprimeproductoutputimaginary) = S ge_signed_half_unique_factorization_resultchosen_factorizationprimesprimeproductoutputimaginarydecode))) /\ ((((((((ge_first_rp_unique_factorization_resultchosen_factorizationprimesprimeproduct) * (ge_second_ip_unique_factorization_resultchosen_factorizationprimesprimeproduct))) + (((ge_first_rn_unique_factorization_resultchosen_factorizationprimesprimeproduct) * (ge_second_in_unique_factorization_resultchosen_factorizationprimesprimeproduct))))) + (((((ge_first_ip_unique_factorization_resultchosen_factorizationprimesprimeproduct) * (ge_second_rp_unique_factorization_resultchosen_factorizationprimesprimeproduct))) + (((ge_first_in_unique_factorization_resultchosen_factorizationprimesprimeproduct) * (ge_second_rn_unique_factorization_resultchosen_factorizationprimesprimeproduct))))))) + ge_balance_negative_unique_factorization_resultchosen_factorizationprimesprimeproductoutputimaginary = (((((((ge_first_rp_unique_factorization_resultchosen_factorizationprimesprimeproduct) * (ge_second_in_unique_factorization_resultchosen_factorizationprimesprimeproduct))) + (((ge_first_rn_unique_factorization_resultchosen_factorizationprimesprimeproduct) * (ge_second_ip_unique_factorization_resultchosen_factorizationprimesprimeproduct))))) + (((((ge_first_ip_unique_factorization_resultchosen_factorizationprimesprimeproduct) * (ge_second_rn_unique_factorization_resultchosen_factorizationprimesprimeproduct))) + (((ge_first_in_unique_factorization_resultchosen_factorizationprimesprimeproduct) * (ge_second_rp_unique_factorization_resultchosen_factorizationprimesprimeproduct))))))) + ge_balance_positive_unique_factorization_resultchosen_factorizationprimesprimeproductoutputimaginary))))))))) -> (exists gr_quotient_unique_factorization_resultchosen_factorizationprimesprimedivisor. (exists ge_first_rp_unique_factorization_resultchosen_factorizationprimesprimedivisorproduct ge_first_rn_unique_factorization_resultchosen_factorizationprimesprimedivisorproduct ge_first_ip_unique_factorization_resultchosen_factorizationprimesprimedivisorproduct ge_first_in_unique_factorization_resultchosen_factorizationprimesprimedivisorproduct ge_second_rp_unique_factorization_resultchosen_factorizationprimesprimedivisorproduct ge_second_rn_unique_factorization_resultchosen_factorizationprimesprimedivisorproduct ge_second_ip_unique_factorization_resultchosen_factorizationprimesprimedivisorproduct ge_second_in_unique_factorization_resultchosen_factorizationprimesprimedivisorproduct. ((exists ge_representation_real_code_unique_factorization_resultchosen_factorizationprimesprimedivisorproductfirst ge_representation_imaginary_code_unique_factorization_resultchosen_factorizationprimesprimedivisorproductfirst. (((gr_prime_factor_value_unique_factorization_resultchosen_factorizationprimes) = ((ge_representation_real_code_unique_factorization_resultchosen_factorizationprimesprimedivisorproductfirst) + (ge_representation_imaginary_code_unique_factorization_resultchosen_factorizationprimesprimedivisorproductfirst)) * S ((ge_representation_real_code_unique_factorization_resultchosen_factorizationprimesprimedivisorproductfirst) + (ge_representation_imaginary_code_unique_factorization_resultchosen_factorizationprimesprimedivisorproductfirst)) + ((ge_representation_imaginary_code_unique_factorization_resultchosen_factorizationprimesprimedivisorproductfirst) + (ge_representation_imaginary_code_unique_factorization_resultchosen_factorizationprimesprimedivisorproductfirst))) /\ ((exists ge_balance_positive_unique_factorization_resultchosen_factorizationprimesprimedivisorproductfirstreal ge_balance_negative_unique_factorization_resultchosen_factorizationprimesprimedivisorproductfirstreal. (((((ge_representation_real_code_unique_factorization_resultchosen_factorizationprimesprimedivisorproductfirst) = 2 * (ge_balance_positive_unique_factorization_resultchosen_factorizationprimesprimedivisorproductfirstreal) /\ (ge_balance_negative_unique_factorization_resultchosen_factorizationprimesprimedivisorproductfirstreal) = 0) \/ exists ge_signed_half_unique_factorization_resultchosen_factorizationprimesprimedivisorproductfirstrealdecode. (((ge_representation_real_code_unique_factorization_resultchosen_factorizationprimesprimedivisorproductfirst) = 2 * ge_signed_half_unique_factorization_resultchosen_factorizationprimesprimedivisorproductfirstrealdecode + 1 /\ (ge_balance_positive_unique_factorization_resultchosen_factorizationprimesprimedivisorproductfirstreal) = 0) /\ (ge_balance_negative_unique_factorization_resultchosen_factorizationprimesprimedivisorproductfirstreal) = S ge_signed_half_unique_factorization_resultchosen_factorizationprimesprimedivisorproductfirstrealdecode))) /\ ((ge_first_rp_unique_factorization_resultchosen_factorizationprimesprimedivisorproduct) + ge_balance_negative_unique_factorization_resultchosen_factorizationprimesprimedivisorproductfirstreal = (ge_first_rn_unique_factorization_resultchosen_factorizationprimesprimedivisorproduct) + ge_balance_positive_unique_factorization_resultchosen_factorizationprimesprimedivisorproductfirstreal))) /\ (exists ge_balance_positive_unique_factorization_resultchosen_factorizationprimesprimedivisorproductfirstimaginary ge_balance_negative_unique_factorization_resultchosen_factorizationprimesprimedivisorproductfirstimaginary. (((((ge_representation_imaginary_code_unique_factorization_resultchosen_factorizationprimesprimedivisorproductfirst) = 2 * (ge_balance_positive_unique_factorization_resultchosen_factorizationprimesprimedivisorproductfirstimaginary) /\ (ge_balance_negative_unique_factorization_resultchosen_factorizationprimesprimedivisorproductfirstimaginary) = 0) \/ exists ge_signed_half_unique_factorization_resultchosen_factorizationprimesprimedivisorproductfirstimaginarydecode. (((ge_representation_imaginary_code_unique_factorization_resultchosen_factorizationprimesprimedivisorproductfirst) = 2 * ge_signed_half_unique_factorization_resultchosen_factorizationprimesprimedivisorproductfirstimaginarydecode + 1 /\ (ge_balance_positive_unique_factorization_resultchosen_factorizationprimesprimedivisorproductfirstimaginary) = 0) /\ (ge_balance_negative_unique_factorization_resultchosen_factorizationprimesprimedivisorproductfirstimaginary) = S ge_signed_half_unique_factorization_resultchosen_factorizationprimesprimedivisorproductfirstimaginarydecode))) /\ ((ge_first_ip_unique_factorization_resultchosen_factorizationprimesprimedivisorproduct) + ge_balance_negative_unique_factorization_resultchosen_factorizationprimesprimedivisorproductfirstimaginary = (ge_first_in_unique_factorization_resultchosen_factorizationprimesprimedivisorproduct) + ge_balance_positive_unique_factorization_resultchosen_factorizationprimesprimedivisorproductfirstimaginary)))))) /\ ((exists ge_representation_real_code_unique_factorization_resultchosen_factorizationprimesprimedivisorproductsecond ge_representation_imaginary_code_unique_factorization_resultchosen_factorizationprimesprimedivisorproductsecond. (((gr_quotient_unique_factorization_resultchosen_factorizationprimesprimedivisor) = ((ge_representation_real_code_unique_factorization_resultchosen_factorizationprimesprimedivisorproductsecond) + (ge_representation_imaginary_code_unique_factorization_resultchosen_factorizationprimesprimedivisorproductsecond)) * S ((ge_representation_real_code_unique_factorization_resultchosen_factorizationprimesprimedivisorproductsecond) + (ge_representation_imaginary_code_unique_factorization_resultchosen_factorizationprimesprimedivisorproductsecond)) + ((ge_representation_imaginary_code_unique_factorization_resultchosen_factorizationprimesprimedivisorproductsecond) + (ge_representation_imaginary_code_unique_factorization_resultchosen_factorizationprimesprimedivisorproductsecond))) /\ ((exists ge_balance_positive_unique_factorization_resultchosen_factorizationprimesprimedivisorproductsecondreal ge_balance_negative_unique_factorization_resultchosen_factorizationprimesprimedivisorproductsecondreal. (((((ge_representation_real_code_unique_factorization_resultchosen_factorizationprimesprimedivisorproductsecond) = 2 * (ge_balance_positive_unique_factorization_resultchosen_factorizationprimesprimedivisorproductsecondreal) /\ (ge_balance_negative_unique_factorization_resultchosen_factorizationprimesprimedivisorproductsecondreal) = 0) \/ exists ge_signed_half_unique_factorization_resultchosen_factorizationprimesprimedivisorproductsecondrealdecode. (((ge_representation_real_code_unique_factorization_resultchosen_factorizationprimesprimedivisorproductsecond) = 2 * ge_signed_half_unique_factorization_resultchosen_factorizationprimesprimedivisorproductsecondrealdecode + 1 /\ (ge_balance_positive_unique_factorization_resultchosen_factorizationprimesprimedivisorproductsecondreal) = 0) /\ (ge_balance_negative_unique_factorization_resultchosen_factorizationprimesprimedivisorproductsecondreal) = S ge_signed_half_unique_factorization_resultchosen_factorizationprimesprimedivisorproductsecondrealdecode))) /\ ((ge_second_rp_unique_factorization_resultchosen_factorizationprimesprimedivisorproduct) + ge_balance_negative_unique_factorization_resultchosen_factorizationprimesprimedivisorproductsecondreal = (ge_second_rn_unique_factorization_resultchosen_factorizationprimesprimedivisorproduct) + ge_balance_positive_unique_factorization_resultchosen_factorizationprimesprimedivisorproductsecondreal))) /\ (exists ge_balance_positive_unique_factorization_resultchosen_factorizationprimesprimedivisorproductsecondimaginary ge_balance_negative_unique_factorization_resultchosen_factorizationprimesprimedivisorproductsecondimaginary. (((((ge_representation_imaginary_code_unique_factorization_resultchosen_factorizationprimesprimedivisorproductsecond) = 2 * (ge_balance_positive_unique_factorization_resultchosen_factorizationprimesprimedivisorproductsecondimaginary) /\ (ge_balance_negative_unique_factorization_resultchosen_factorizationprimesprimedivisorproductsecondimaginary) = 0) \/ exists ge_signed_half_unique_factorization_resultchosen_factorizationprimesprimedivisorproductsecondimaginarydecode. (((ge_representation_imaginary_code_unique_factorization_resultchosen_factorizationprimesprimedivisorproductsecond) = 2 * ge_signed_half_unique_factorization_resultchosen_factorizationprimesprimedivisorproductsecondimaginarydecode + 1 /\ (ge_balance_positive_unique_factorization_resultchosen_factorizationprimesprimedivisorproductsecondimaginary) = 0) /\ (ge_balance_negative_unique_factorization_resultchosen_factorizationprimesprimedivisorproductsecondimaginary) = S ge_signed_half_unique_factorization_resultchosen_factorizationprimesprimedivisorproductsecondimaginarydecode))) /\ ((ge_second_ip_unique_factorization_resultchosen_factorizationprimesprimedivisorproduct) + ge_balance_negative_unique_factorization_resultchosen_factorizationprimesprimedivisorproductsecondimaginary = (ge_second_in_unique_factorization_resultchosen_factorizationprimesprimedivisorproduct) + ge_balance_positive_unique_factorization_resultchosen_factorizationprimesprimedivisorproductsecondimaginary)))))) /\ (exists ge_representation_real_code_unique_factorization_resultchosen_factorizationprimesprimedivisorproductoutput ge_representation_imaginary_code_unique_factorization_resultchosen_factorizationprimesprimedivisorproductoutput. (((gr_product_unique_factorization_resultchosen_factorizationprimesprime) = ((ge_representation_real_code_unique_factorization_resultchosen_factorizationprimesprimedivisorproductoutput) + (ge_representation_imaginary_code_unique_factorization_resultchosen_factorizationprimesprimedivisorproductoutput)) * S ((ge_representation_real_code_unique_factorization_resultchosen_factorizationprimesprimedivisorproductoutput) + (ge_representation_imaginary_code_unique_factorization_resultchosen_factorizationprimesprimedivisorproductoutput)) + ((ge_representation_imaginary_code_unique_factorization_resultchosen_factorizationprimesprimedivisorproductoutput) + (ge_representation_imaginary_code_unique_factorization_resultchosen_factorizationprimesprimedivisorproductoutput))) /\ ((exists ge_balance_positive_unique_factorization_resultchosen_factorizationprimesprimedivisorproductoutputreal ge_balance_negative_unique_factorization_resultchosen_factorizationprimesprimedivisorproductoutputreal. (((((ge_representation_real_code_unique_factorization_resultchosen_factorizationprimesprimedivisorproductoutput) = 2 * (ge_balance_positive_unique_factorization_resultchosen_factorizationprimesprimedivisorproductoutputreal) /\ (ge_balance_negative_unique_factorization_resultchosen_factorizationprimesprimedivisorproductoutputreal) = 0) \/ exists ge_signed_half_unique_factorization_resultchosen_factorizationprimesprimedivisorproductoutputrealdecode. (((ge_representation_real_code_unique_factorization_resultchosen_factorizationprimesprimedivisorproductoutput) = 2 * ge_signed_half_unique_factorization_resultchosen_factorizationprimesprimedivisorproductoutputrealdecode + 1 /\ (ge_balance_positive_unique_factorization_resultchosen_factorizationprimesprimedivisorproductoutputreal) = 0) /\ (ge_balance_negative_unique_factorization_resultchosen_factorizationprimesprimedivisorproductoutputreal) = S ge_signed_half_unique_factorization_resultchosen_factorizationprimesprimedivisorproductoutputrealdecode))) /\ ((((((((ge_first_rp_unique_factorization_resultchosen_factorizationprimesprimedivisorproduct) * (ge_second_rp_unique_factorization_resultchosen_factorizationprimesprimedivisorproduct))) + (((ge_first_rn_unique_factorization_resultchosen_factorizationprimesprimedivisorproduct) * (ge_second_rn_unique_factorization_resultchosen_factorizationprimesprimedivisorproduct))))) + (((((ge_first_ip_unique_factorization_resultchosen_factorizationprimesprimedivisorproduct) * (ge_second_in_unique_factorization_resultchosen_factorizationprimesprimedivisorproduct))) + (((ge_first_in_unique_factorization_resultchosen_factorizationprimesprimedivisorproduct) * (ge_second_ip_unique_factorization_resultchosen_factorizationprimesprimedivisorproduct))))))) + ge_balance_negative_unique_factorization_resultchosen_factorizationprimesprimedivisorproductoutputreal = (((((((ge_first_rp_unique_factorization_resultchosen_factorizationprimesprimedivisorproduct) * (ge_second_rn_unique_factorization_resultchosen_factorizationprimesprimedivisorproduct))) + (((ge_first_rn_unique_factorization_resultchosen_factorizationprimesprimedivisorproduct) * (ge_second_rp_unique_factorization_resultchosen_factorizationprimesprimedivisorproduct))))) + (((((ge_first_ip_unique_factorization_resultchosen_factorizationprimesprimedivisorproduct) * (ge_second_ip_unique_factorization_resultchosen_factorizationprimesprimedivisorproduct))) + (((ge_first_in_unique_factorization_resultchosen_factorizationprimesprimedivisorproduct) * (ge_second_in_unique_factorization_resultchosen_factorizationprimesprimedivisorproduct))))))) + ge_balance_positive_unique_factorization_resultchosen_factorizationprimesprimedivisorproductoutputreal))) /\ (exists ge_balance_positive_unique_factorization_resultchosen_factorizationprimesprimedivisorproductoutputimaginary ge_balance_negative_unique_factorization_resultchosen_factorizationprimesprimedivisorproductoutputimaginary. (((((ge_representation_imaginary_code_unique_factorization_resultchosen_factorizationprimesprimedivisorproductoutput) = 2 * (ge_balance_positive_unique_factorization_resultchosen_factorizationprimesprimedivisorproductoutputimaginary) /\ (ge_balance_negative_unique_factorization_resultchosen_factorizationprimesprimedivisorproductoutputimaginary) = 0) \/ exists ge_signed_half_unique_factorization_resultchosen_factorizationprimesprimedivisorproductoutputimaginarydecode. (((ge_representation_imaginary_code_unique_factorization_resultchosen_factorizationprimesprimedivisorproductoutput) = 2 * ge_signed_half_unique_factorization_resultchosen_factorizationprimesprimedivisorproductoutputimaginarydecode + 1 /\ (ge_balance_positive_unique_factorization_resultchosen_factorizationprimesprimedivisorproductoutputimaginary) = 0) /\ (ge_balance_negative_unique_factorization_resultchosen_factorizationprimesprimedivisorproductoutputimaginary) = S ge_signed_half_unique_factorization_resultchosen_factorizationprimesprimedivisorproductoutputimaginarydecode))) /\ ((((((((ge_first_rp_unique_factorization_resultchosen_factorizationprimesprimedivisorproduct) * (ge_second_ip_unique_factorization_resultchosen_factorizationprimesprimedivisorproduct))) + (((ge_first_rn_unique_factorization_resultchosen_factorizationprimesprimedivisorproduct) * (ge_second_in_unique_factorization_resultchosen_factorizationprimesprimedivisorproduct))))) + (((((ge_first_ip_unique_factorization_resultchosen_factorizationprimesprimedivisorproduct) * (ge_second_rp_unique_factorization_resultchosen_factorizationprimesprimedivisorproduct))) + (((ge_first_in_unique_factorization_resultchosen_factorizationprimesprimedivisorproduct) * (ge_second_rn_unique_factorization_resultchosen_factorizationprimesprimedivisorproduct))))))) + ge_balance_negative_unique_factorization_resultchosen_factorizationprimesprimedivisorproductoutputimaginary = (((((((ge_first_rp_unique_factorization_resultchosen_factorizationprimesprimedivisorproduct) * (ge_second_in_unique_factorization_resultchosen_factorizationprimesprimedivisorproduct))) + (((ge_first_rn_unique_factorization_resultchosen_factorizationprimesprimedivisorproduct) * (ge_second_ip_unique_factorization_resultchosen_factorizationprimesprimedivisorproduct))))) + (((((ge_first_ip_unique_factorization_resultchosen_factorizationprimesprimedivisorproduct) * (ge_second_rn_unique_factorization_resultchosen_factorizationprimesprimedivisorproduct))) + (((ge_first_in_unique_factorization_resultchosen_factorizationprimesprimedivisorproduct) * (ge_second_rp_unique_factorization_resultchosen_factorizationprimesprimedivisorproduct))))))) + ge_balance_positive_unique_factorization_resultchosen_factorizationprimesprimedivisorproductoutputimaginary)))))))))) -> (exists gr_quotient_unique_factorization_resultchosen_factorizationprimesprimefirst_divisor. (exists ge_first_rp_unique_factorization_resultchosen_factorizationprimesprimefirst_divisorproduct ge_first_rn_unique_factorization_resultchosen_factorizationprimesprimefirst_divisorproduct ge_first_ip_unique_factorization_resultchosen_factorizationprimesprimefirst_divisorproduct ge_first_in_unique_factorization_resultchosen_factorizationprimesprimefirst_divisorproduct ge_second_rp_unique_factorization_resultchosen_factorizationprimesprimefirst_divisorproduct ge_second_rn_unique_factorization_resultchosen_factorizationprimesprimefirst_divisorproduct ge_second_ip_unique_factorization_resultchosen_factorizationprimesprimefirst_divisorproduct ge_second_in_unique_factorization_resultchosen_factorizationprimesprimefirst_divisorproduct. ((exists ge_representation_real_code_unique_factorization_resultchosen_factorizationprimesprimefirst_divisorproductfirst ge_representation_imaginary_code_unique_factorization_resultchosen_factorizationprimesprimefirst_divisorproductfirst. (((gr_prime_factor_value_unique_factorization_resultchosen_factorizationprimes) = ((ge_representation_real_code_unique_factorization_resultchosen_factorizationprimesprimefirst_divisorproductfirst) + (ge_representation_imaginary_code_unique_factorization_resultchosen_factorizationprimesprimefirst_divisorproductfirst)) * S ((ge_representation_real_code_unique_factorization_resultchosen_factorizationprimesprimefirst_divisorproductfirst) + (ge_representation_imaginary_code_unique_factorization_resultchosen_factorizationprimesprimefirst_divisorproductfirst)) + ((ge_representation_imaginary_code_unique_factorization_resultchosen_factorizationprimesprimefirst_divisorproductfirst) + (ge_representation_imaginary_code_unique_factorization_resultchosen_factorizationprimesprimefirst_divisorproductfirst))) /\ ((exists ge_balance_positive_unique_factorization_resultchosen_factorizationprimesprimefirst_divisorproductfirstreal ge_balance_negative_unique_factorization_resultchosen_factorizationprimesprimefirst_divisorproductfirstreal. (((((ge_representation_real_code_unique_factorization_resultchosen_factorizationprimesprimefirst_divisorproductfirst) = 2 * (ge_balance_positive_unique_factorization_resultchosen_factorizationprimesprimefirst_divisorproductfirstreal) /\ (ge_balance_negative_unique_factorization_resultchosen_factorizationprimesprimefirst_divisorproductfirstreal) = 0) \/ exists ge_signed_half_unique_factorization_resultchosen_factorizationprimesprimefirst_divisorproductfirstrealdecode. (((ge_representation_real_code_unique_factorization_resultchosen_factorizationprimesprimefirst_divisorproductfirst) = 2 * ge_signed_half_unique_factorization_resultchosen_factorizationprimesprimefirst_divisorproductfirstrealdecode + 1 /\ (ge_balance_positive_unique_factorization_resultchosen_factorizationprimesprimefirst_divisorproductfirstreal) = 0) /\ (ge_balance_negative_unique_factorization_resultchosen_factorizationprimesprimefirst_divisorproductfirstreal) = S ge_signed_half_unique_factorization_resultchosen_factorizationprimesprimefirst_divisorproductfirstrealdecode))) /\ ((ge_first_rp_unique_factorization_resultchosen_factorizationprimesprimefirst_divisorproduct) + ge_balance_negative_unique_factorization_resultchosen_factorizationprimesprimefirst_divisorproductfirstreal = (ge_first_rn_unique_factorization_resultchosen_factorizationprimesprimefirst_divisorproduct) + ge_balance_positive_unique_factorization_resultchosen_factorizationprimesprimefirst_divisorproductfirstreal))) /\ (exists ge_balance_positive_unique_factorization_resultchosen_factorizationprimesprimefirst_divisorproductfirstimaginary ge_balance_negative_unique_factorization_resultchosen_factorizationprimesprimefirst_divisorproductfirstimaginary. (((((ge_representation_imaginary_code_unique_factorization_resultchosen_factorizationprimesprimefirst_divisorproductfirst) = 2 * (ge_balance_positive_unique_factorization_resultchosen_factorizationprimesprimefirst_divisorproductfirstimaginary) /\ (ge_balance_negative_unique_factorization_resultchosen_factorizationprimesprimefirst_divisorproductfirstimaginary) = 0) \/ exists ge_signed_half_unique_factorization_resultchosen_factorizationprimesprimefirst_divisorproductfirstimaginarydecode. (((ge_representation_imaginary_code_unique_factorization_resultchosen_factorizationprimesprimefirst_divisorproductfirst) = 2 * ge_signed_half_unique_factorization_resultchosen_factorizationprimesprimefirst_divisorproductfirstimaginarydecode + 1 /\ (ge_balance_positive_unique_factorization_resultchosen_factorizationprimesprimefirst_divisorproductfirstimaginary) = 0) /\ (ge_balance_negative_unique_factorization_resultchosen_factorizationprimesprimefirst_divisorproductfirstimaginary) = S ge_signed_half_unique_factorization_resultchosen_factorizationprimesprimefirst_divisorproductfirstimaginarydecode))) /\ ((ge_first_ip_unique_factorization_resultchosen_factorizationprimesprimefirst_divisorproduct) + ge_balance_negative_unique_factorization_resultchosen_factorizationprimesprimefirst_divisorproductfirstimaginary = (ge_first_in_unique_factorization_resultchosen_factorizationprimesprimefirst_divisorproduct) + ge_balance_positive_unique_factorization_resultchosen_factorizationprimesprimefirst_divisorproductfirstimaginary)))))) /\ ((exists ge_representation_real_code_unique_factorization_resultchosen_factorizationprimesprimefirst_divisorproductsecond ge_representation_imaginary_code_unique_factorization_resultchosen_factorizationprimesprimefirst_divisorproductsecond. (((gr_quotient_unique_factorization_resultchosen_factorizationprimesprimefirst_divisor) = ((ge_representation_real_code_unique_factorization_resultchosen_factorizationprimesprimefirst_divisorproductsecond) + (ge_representation_imaginary_code_unique_factorization_resultchosen_factorizationprimesprimefirst_divisorproductsecond)) * S ((ge_representation_real_code_unique_factorization_resultchosen_factorizationprimesprimefirst_divisorproductsecond) + (ge_representation_imaginary_code_unique_factorization_resultchosen_factorizationprimesprimefirst_divisorproductsecond)) + ((ge_representation_imaginary_code_unique_factorization_resultchosen_factorizationprimesprimefirst_divisorproductsecond) + (ge_representation_imaginary_code_unique_factorization_resultchosen_factorizationprimesprimefirst_divisorproductsecond))) /\ ((exists ge_balance_positive_unique_factorization_resultchosen_factorizationprimesprimefirst_divisorproductsecondreal ge_balance_negative_unique_factorization_resultchosen_factorizationprimesprimefirst_divisorproductsecondreal. (((((ge_representation_real_code_unique_factorization_resultchosen_factorizationprimesprimefirst_divisorproductsecond) = 2 * (ge_balance_positive_unique_factorization_resultchosen_factorizationprimesprimefirst_divisorproductsecondreal) /\ (ge_balance_negative_unique_factorization_resultchosen_factorizationprimesprimefirst_divisorproductsecondreal) = 0) \/ exists ge_signed_half_unique_factorization_resultchosen_factorizationprimesprimefirst_divisorproductsecondrealdecode. (((ge_representation_real_code_unique_factorization_resultchosen_factorizationprimesprimefirst_divisorproductsecond) = 2 * ge_signed_half_unique_factorization_resultchosen_factorizationprimesprimefirst_divisorproductsecondrealdecode + 1 /\ (ge_balance_positive_unique_factorization_resultchosen_factorizationprimesprimefirst_divisorproductsecondreal) = 0) /\ (ge_balance_negative_unique_factorization_resultchosen_factorizationprimesprimefirst_divisorproductsecondreal) = S ge_signed_half_unique_factorization_resultchosen_factorizationprimesprimefirst_divisorproductsecondrealdecode))) /\ ((ge_second_rp_unique_factorization_resultchosen_factorizationprimesprimefirst_divisorproduct) + ge_balance_negative_unique_factorization_resultchosen_factorizationprimesprimefirst_divisorproductsecondreal = (ge_second_rn_unique_factorization_resultchosen_factorizationprimesprimefirst_divisorproduct) + ge_balance_positive_unique_factorization_resultchosen_factorizationprimesprimefirst_divisorproductsecondreal))) /\ (exists ge_balance_positive_unique_factorization_resultchosen_factorizationprimesprimefirst_divisorproductsecondimaginary ge_balance_negative_unique_factorization_resultchosen_factorizationprimesprimefirst_divisorproductsecondimaginary. (((((ge_representation_imaginary_code_unique_factorization_resultchosen_factorizationprimesprimefirst_divisorproductsecond) = 2 * (ge_balance_positive_unique_factorization_resultchosen_factorizationprimesprimefirst_divisorproductsecondimaginary) /\ (ge_balance_negative_unique_factorization_resultchosen_factorizationprimesprimefirst_divisorproductsecondimaginary) = 0) \/ exists ge_signed_half_unique_factorization_resultchosen_factorizationprimesprimefirst_divisorproductsecondimaginarydecode. (((ge_representation_imaginary_code_unique_factorization_resultchosen_factorizationprimesprimefirst_divisorproductsecond) = 2 * ge_signed_half_unique_factorization_resultchosen_factorizationprimesprimefirst_divisorproductsecondimaginarydecode + 1 /\ (ge_balance_positive_unique_factorization_resultchosen_factorizationprimesprimefirst_divisorproductsecondimaginary) = 0) /\ (ge_balance_negative_unique_factorization_resultchosen_factorizationprimesprimefirst_divisorproductsecondimaginary) = S ge_signed_half_unique_factorization_resultchosen_factorizationprimesprimefirst_divisorproductsecondimaginarydecode))) /\ ((ge_second_ip_unique_factorization_resultchosen_factorizationprimesprimefirst_divisorproduct) + ge_balance_negative_unique_factorization_resultchosen_factorizationprimesprimefirst_divisorproductsecondimaginary = (ge_second_in_unique_factorization_resultchosen_factorizationprimesprimefirst_divisorproduct) + ge_balance_positive_unique_factorization_resultchosen_factorizationprimesprimefirst_divisorproductsecondimaginary)))))) /\ (exists ge_representation_real_code_unique_factorization_resultchosen_factorizationprimesprimefirst_divisorproductoutput ge_representation_imaginary_code_unique_factorization_resultchosen_factorizationprimesprimefirst_divisorproductoutput. (((gr_first_factor_unique_factorization_resultchosen_factorizationprimesprime) = ((ge_representation_real_code_unique_factorization_resultchosen_factorizationprimesprimefirst_divisorproductoutput) + (ge_representation_imaginary_code_unique_factorization_resultchosen_factorizationprimesprimefirst_divisorproductoutput)) * S ((ge_representation_real_code_unique_factorization_resultchosen_factorizationprimesprimefirst_divisorproductoutput) + (ge_representation_imaginary_code_unique_factorization_resultchosen_factorizationprimesprimefirst_divisorproductoutput)) + ((ge_representation_imaginary_code_unique_factorization_resultchosen_factorizationprimesprimefirst_divisorproductoutput) + (ge_representation_imaginary_code_unique_factorization_resultchosen_factorizationprimesprimefirst_divisorproductoutput))) /\ ((exists ge_balance_positive_unique_factorization_resultchosen_factorizationprimesprimefirst_divisorproductoutputreal ge_balance_negative_unique_factorization_resultchosen_factorizationprimesprimefirst_divisorproductoutputreal. (((((ge_representation_real_code_unique_factorization_resultchosen_factorizationprimesprimefirst_divisorproductoutput) = 2 * (ge_balance_positive_unique_factorization_resultchosen_factorizationprimesprimefirst_divisorproductoutputreal) /\ (ge_balance_negative_unique_factorization_resultchosen_factorizationprimesprimefirst_divisorproductoutputreal) = 0) \/ exists ge_signed_half_unique_factorization_resultchosen_factorizationprimesprimefirst_divisorproductoutputrealdecode. (((ge_representation_real_code_unique_factorization_resultchosen_factorizationprimesprimefirst_divisorproductoutput) = 2 * ge_signed_half_unique_factorization_resultchosen_factorizationprimesprimefirst_divisorproductoutputrealdecode + 1 /\ (ge_balance_positive_unique_factorization_resultchosen_factorizationprimesprimefirst_divisorproductoutputreal) = 0) /\ (ge_balance_negative_unique_factorization_resultchosen_factorizationprimesprimefirst_divisorproductoutputreal) = S ge_signed_half_unique_factorization_resultchosen_factorizationprimesprimefirst_divisorproductoutputrealdecode))) /\ ((((((((ge_first_rp_unique_factorization_resultchosen_factorizationprimesprimefirst_divisorproduct) * (ge_second_rp_unique_factorization_resultchosen_factorizationprimesprimefirst_divisorproduct))) + (((ge_first_rn_unique_factorization_resultchosen_factorizationprimesprimefirst_divisorproduct) * (ge_second_rn_unique_factorization_resultchosen_factorizationprimesprimefirst_divisorproduct))))) + (((((ge_first_ip_unique_factorization_resultchosen_factorizationprimesprimefirst_divisorproduct) * (ge_second_in_unique_factorization_resultchosen_factorizationprimesprimefirst_divisorproduct))) + (((ge_first_in_unique_factorization_resultchosen_factorizationprimesprimefirst_divisorproduct) * (ge_second_ip_unique_factorization_resultchosen_factorizationprimesprimefirst_divisorproduct))))))) + ge_balance_negative_unique_factorization_resultchosen_factorizationprimesprimefirst_divisorproductoutputreal = (((((((ge_first_rp_unique_factorization_resultchosen_factorizationprimesprimefirst_divisorproduct) * (ge_second_rn_unique_factorization_resultchosen_factorizationprimesprimefirst_divisorproduct))) + (((ge_first_rn_unique_factorization_resultchosen_factorizationprimesprimefirst_divisorproduct) * (ge_second_rp_unique_factorization_resultchosen_factorizationprimesprimefirst_divisorproduct))))) + (((((ge_first_ip_unique_factorization_resultchosen_factorizationprimesprimefirst_divisorproduct) * (ge_second_ip_unique_factorization_resultchosen_factorizationprimesprimefirst_divisorproduct))) + (((ge_first_in_unique_factorization_resultchosen_factorizationprimesprimefirst_divisorproduct) * (ge_second_in_unique_factorization_resultchosen_factorizationprimesprimefirst_divisorproduct))))))) + ge_balance_positive_unique_factorization_resultchosen_factorizationprimesprimefirst_divisorproductoutputreal))) /\ (exists ge_balance_positive_unique_factorization_resultchosen_factorizationprimesprimefirst_divisorproductoutputimaginary ge_balance_negative_unique_factorization_resultchosen_factorizationprimesprimefirst_divisorproductoutputimaginary. (((((ge_representation_imaginary_code_unique_factorization_resultchosen_factorizationprimesprimefirst_divisorproductoutput) = 2 * (ge_balance_positive_unique_factorization_resultchosen_factorizationprimesprimefirst_divisorproductoutputimaginary) /\ (ge_balance_negative_unique_factorization_resultchosen_factorizationprimesprimefirst_divisorproductoutputimaginary) = 0) \/ exists ge_signed_half_unique_factorization_resultchosen_factorizationprimesprimefirst_divisorproductoutputimaginarydecode. (((ge_representation_imaginary_code_unique_factorization_resultchosen_factorizationprimesprimefirst_divisorproductoutput) = 2 * ge_signed_half_unique_factorization_resultchosen_factorizationprimesprimefirst_divisorproductoutputimaginarydecode + 1 /\ (ge_balance_positive_unique_factorization_resultchosen_factorizationprimesprimefirst_divisorproductoutputimaginary) = 0) /\ (ge_balance_negative_unique_factorization_resultchosen_factorizationprimesprimefirst_divisorproductoutputimaginary) = S ge_signed_half_unique_factorization_resultchosen_factorizationprimesprimefirst_divisorproductoutputimaginarydecode))) /\ ((((((((ge_first_rp_unique_factorization_resultchosen_factorizationprimesprimefirst_divisorproduct) * (ge_second_ip_unique_factorization_resultchosen_factorizationprimesprimefirst_divisorproduct))) + (((ge_first_rn_unique_factorization_resultchosen_factorizationprimesprimefirst_divisorproduct) * (ge_second_in_unique_factorization_resultchosen_factorizationprimesprimefirst_divisorproduct))))) + (((((ge_first_ip_unique_factorization_resultchosen_factorizationprimesprimefirst_divisorproduct) * (ge_second_rp_unique_factorization_resultchosen_factorizationprimesprimefirst_divisorproduct))) + (((ge_first_in_unique_factorization_resultchosen_factorizationprimesprimefirst_divisorproduct) * (ge_second_rn_unique_factorization_resultchosen_factorizationprimesprimefirst_divisorproduct))))))) + ge_balance_negative_unique_factorization_resultchosen_factorizationprimesprimefirst_divisorproductoutputimaginary = (((((((ge_first_rp_unique_factorization_resultchosen_factorizationprimesprimefirst_divisorproduct) * (ge_second_in_unique_factorization_resultchosen_factorizationprimesprimefirst_divisorproduct))) + (((ge_first_rn_unique_factorization_resultchosen_factorizationprimesprimefirst_divisorproduct) * (ge_second_ip_unique_factorization_resultchosen_factorizationprimesprimefirst_divisorproduct))))) + (((((ge_first_ip_unique_factorization_resultchosen_factorizationprimesprimefirst_divisorproduct) * (ge_second_rn_unique_factorization_resultchosen_factorizationprimesprimefirst_divisorproduct))) + (((ge_first_in_unique_factorization_resultchosen_factorizationprimesprimefirst_divisorproduct) * (ge_second_rp_unique_factorization_resultchosen_factorizationprimesprimefirst_divisorproduct))))))) + ge_balance_positive_unique_factorization_resultchosen_factorizationprimesprimefirst_divisorproductoutputimaginary)))))))))) \/ (exists gr_quotient_unique_factorization_resultchosen_factorizationprimesprimesecond_divisor. (exists ge_first_rp_unique_factorization_resultchosen_factorizationprimesprimesecond_divisorproduct ge_first_rn_unique_factorization_resultchosen_factorizationprimesprimesecond_divisorproduct ge_first_ip_unique_factorization_resultchosen_factorizationprimesprimesecond_divisorproduct ge_first_in_unique_factorization_resultchosen_factorizationprimesprimesecond_divisorproduct ge_second_rp_unique_factorization_resultchosen_factorizationprimesprimesecond_divisorproduct ge_second_rn_unique_factorization_resultchosen_factorizationprimesprimesecond_divisorproduct ge_second_ip_unique_factorization_resultchosen_factorizationprimesprimesecond_divisorproduct ge_second_in_unique_factorization_resultchosen_factorizationprimesprimesecond_divisorproduct. ((exists ge_representation_real_code_unique_factorization_resultchosen_factorizationprimesprimesecond_divisorproductfirst ge_representation_imaginary_code_unique_factorization_resultchosen_factorizationprimesprimesecond_divisorproductfirst. (((gr_prime_factor_value_unique_factorization_resultchosen_factorizationprimes) = ((ge_representation_real_code_unique_factorization_resultchosen_factorizationprimesprimesecond_divisorproductfirst) + (ge_representation_imaginary_code_unique_factorization_resultchosen_factorizationprimesprimesecond_divisorproductfirst)) * S ((ge_representation_real_code_unique_factorization_resultchosen_factorizationprimesprimesecond_divisorproductfirst) + (ge_representation_imaginary_code_unique_factorization_resultchosen_factorizationprimesprimesecond_divisorproductfirst)) + ((ge_representation_imaginary_code_unique_factorization_resultchosen_factorizationprimesprimesecond_divisorproductfirst) + (ge_representation_imaginary_code_unique_factorization_resultchosen_factorizationprimesprimesecond_divisorproductfirst))) /\ ((exists ge_balance_positive_unique_factorization_resultchosen_factorizationprimesprimesecond_divisorproductfirstreal ge_balance_negative_unique_factorization_resultchosen_factorizationprimesprimesecond_divisorproductfirstreal. (((((ge_representation_real_code_unique_factorization_resultchosen_factorizationprimesprimesecond_divisorproductfirst) = 2 * (ge_balance_positive_unique_factorization_resultchosen_factorizationprimesprimesecond_divisorproductfirstreal) /\ (ge_balance_negative_unique_factorization_resultchosen_factorizationprimesprimesecond_divisorproductfirstreal) = 0) \/ exists ge_signed_half_unique_factorization_resultchosen_factorizationprimesprimesecond_divisorproductfirstrealdecode. (((ge_representation_real_code_unique_factorization_resultchosen_factorizationprimesprimesecond_divisorproductfirst) = 2 * ge_signed_half_unique_factorization_resultchosen_factorizationprimesprimesecond_divisorproductfirstrealdecode + 1 /\ (ge_balance_positive_unique_factorization_resultchosen_factorizationprimesprimesecond_divisorproductfirstreal) = 0) /\ (ge_balance_negative_unique_factorization_resultchosen_factorizationprimesprimesecond_divisorproductfirstreal) = S ge_signed_half_unique_factorization_resultchosen_factorizationprimesprimesecond_divisorproductfirstrealdecode))) /\ ((ge_first_rp_unique_factorization_resultchosen_factorizationprimesprimesecond_divisorproduct) + ge_balance_negative_unique_factorization_resultchosen_factorizationprimesprimesecond_divisorproductfirstreal = (ge_first_rn_unique_factorization_resultchosen_factorizationprimesprimesecond_divisorproduct) + ge_balance_positive_unique_factorization_resultchosen_factorizationprimesprimesecond_divisorproductfirstreal))) /\ (exists ge_balance_positive_unique_factorization_resultchosen_factorizationprimesprimesecond_divisorproductfirstimaginary ge_balance_negative_unique_factorization_resultchosen_factorizationprimesprimesecond_divisorproductfirstimaginary. (((((ge_representation_imaginary_code_unique_factorization_resultchosen_factorizationprimesprimesecond_divisorproductfirst) = 2 * (ge_balance_positive_unique_factorization_resultchosen_factorizationprimesprimesecond_divisorproductfirstimaginary) /\ (ge_balance_negative_unique_factorization_resultchosen_factorizationprimesprimesecond_divisorproductfirstimaginary) = 0) \/ exists ge_signed_half_unique_factorization_resultchosen_factorizationprimesprimesecond_divisorproductfirstimaginarydecode. (((ge_representation_imaginary_code_unique_factorization_resultchosen_factorizationprimesprimesecond_divisorproductfirst) = 2 * ge_signed_half_unique_factorization_resultchosen_factorizationprimesprimesecond_divisorproductfirstimaginarydecode + 1 /\ (ge_balance_positive_unique_factorization_resultchosen_factorizationprimesprimesecond_divisorproductfirstimaginary) = 0) /\ (ge_balance_negative_unique_factorization_resultchosen_factorizationprimesprimesecond_divisorproductfirstimaginary) = S ge_signed_half_unique_factorization_resultchosen_factorizationprimesprimesecond_divisorproductfirstimaginarydecode))) /\ ((ge_first_ip_unique_factorization_resultchosen_factorizationprimesprimesecond_divisorproduct) + ge_balance_negative_unique_factorization_resultchosen_factorizationprimesprimesecond_divisorproductfirstimaginary = (ge_first_in_unique_factorization_resultchosen_factorizationprimesprimesecond_divisorproduct) + ge_balance_positive_unique_factorization_resultchosen_factorizationprimesprimesecond_divisorproductfirstimaginary)))))) /\ ((exists ge_representation_real_code_unique_factorization_resultchosen_factorizationprimesprimesecond_divisorproductsecond ge_representation_imaginary_code_unique_factorization_resultchosen_factorizationprimesprimesecond_divisorproductsecond. (((gr_quotient_unique_factorization_resultchosen_factorizationprimesprimesecond_divisor) = ((ge_representation_real_code_unique_factorization_resultchosen_factorizationprimesprimesecond_divisorproductsecond) + (ge_representation_imaginary_code_unique_factorization_resultchosen_factorizationprimesprimesecond_divisorproductsecond)) * S ((ge_representation_real_code_unique_factorization_resultchosen_factorizationprimesprimesecond_divisorproductsecond) + (ge_representation_imaginary_code_unique_factorization_resultchosen_factorizationprimesprimesecond_divisorproductsecond)) + ((ge_representation_imaginary_code_unique_factorization_resultchosen_factorizationprimesprimesecond_divisorproductsecond) + (ge_representation_imaginary_code_unique_factorization_resultchosen_factorizationprimesprimesecond_divisorproductsecond))) /\ ((exists ge_balance_positive_unique_factorization_resultchosen_factorizationprimesprimesecond_divisorproductsecondreal ge_balance_negative_unique_factorization_resultchosen_factorizationprimesprimesecond_divisorproductsecondreal. (((((ge_representation_real_code_unique_factorization_resultchosen_factorizationprimesprimesecond_divisorproductsecond) = 2 * (ge_balance_positive_unique_factorization_resultchosen_factorizationprimesprimesecond_divisorproductsecondreal) /\ (ge_balance_negative_unique_factorization_resultchosen_factorizationprimesprimesecond_divisorproductsecondreal) = 0) \/ exists ge_signed_half_unique_factorization_resultchosen_factorizationprimesprimesecond_divisorproductsecondrealdecode. (((ge_representation_real_code_unique_factorization_resultchosen_factorizationprimesprimesecond_divisorproductsecond) = 2 * ge_signed_half_unique_factorization_resultchosen_factorizationprimesprimesecond_divisorproductsecondrealdecode + 1 /\ (ge_balance_positive_unique_factorization_resultchosen_factorizationprimesprimesecond_divisorproductsecondreal) = 0) /\ (ge_balance_negative_unique_factorization_resultchosen_factorizationprimesprimesecond_divisorproductsecondreal) = S ge_signed_half_unique_factorization_resultchosen_factorizationprimesprimesecond_divisorproductsecondrealdecode))) /\ ((ge_second_rp_unique_factorization_resultchosen_factorizationprimesprimesecond_divisorproduct) + ge_balance_negative_unique_factorization_resultchosen_factorizationprimesprimesecond_divisorproductsecondreal = (ge_second_rn_unique_factorization_resultchosen_factorizationprimesprimesecond_divisorproduct) + ge_balance_positive_unique_factorization_resultchosen_factorizationprimesprimesecond_divisorproductsecondreal))) /\ (exists ge_balance_positive_unique_factorization_resultchosen_factorizationprimesprimesecond_divisorproductsecondimaginary ge_balance_negative_unique_factorization_resultchosen_factorizationprimesprimesecond_divisorproductsecondimaginary. (((((ge_representation_imaginary_code_unique_factorization_resultchosen_factorizationprimesprimesecond_divisorproductsecond) = 2 * (ge_balance_positive_unique_factorization_resultchosen_factorizationprimesprimesecond_divisorproductsecondimaginary) /\ (ge_balance_negative_unique_factorization_resultchosen_factorizationprimesprimesecond_divisorproductsecondimaginary) = 0) \/ exists ge_signed_half_unique_factorization_resultchosen_factorizationprimesprimesecond_divisorproductsecondimaginarydecode. (((ge_representation_imaginary_code_unique_factorization_resultchosen_factorizationprimesprimesecond_divisorproductsecond) = 2 * ge_signed_half_unique_factorization_resultchosen_factorizationprimesprimesecond_divisorproductsecondimaginarydecode + 1 /\ (ge_balance_positive_unique_factorization_resultchosen_factorizationprimesprimesecond_divisorproductsecondimaginary) = 0) /\ (ge_balance_negative_unique_factorization_resultchosen_factorizationprimesprimesecond_divisorproductsecondimaginary) = S ge_signed_half_unique_factorization_resultchosen_factorizationprimesprimesecond_divisorproductsecondimaginarydecode))) /\ ((ge_second_ip_unique_factorization_resultchosen_factorizationprimesprimesecond_divisorproduct) + ge_balance_negative_unique_factorization_resultchosen_factorizationprimesprimesecond_divisorproductsecondimaginary = (ge_second_in_unique_factorization_resultchosen_factorizationprimesprimesecond_divisorproduct) + ge_balance_positive_unique_factorization_resultchosen_factorizationprimesprimesecond_divisorproductsecondimaginary)))))) /\ (exists ge_representation_real_code_unique_factorization_resultchosen_factorizationprimesprimesecond_divisorproductoutput ge_representation_imaginary_code_unique_factorization_resultchosen_factorizationprimesprimesecond_divisorproductoutput. (((gr_second_factor_unique_factorization_resultchosen_factorizationprimesprime) = ((ge_representation_real_code_unique_factorization_resultchosen_factorizationprimesprimesecond_divisorproductoutput) + (ge_representation_imaginary_code_unique_factorization_resultchosen_factorizationprimesprimesecond_divisorproductoutput)) * S ((ge_representation_real_code_unique_factorization_resultchosen_factorizationprimesprimesecond_divisorproductoutput) + (ge_representation_imaginary_code_unique_factorization_resultchosen_factorizationprimesprimesecond_divisorproductoutput)) + ((ge_representation_imaginary_code_unique_factorization_resultchosen_factorizationprimesprimesecond_divisorproductoutput) + (ge_representation_imaginary_code_unique_factorization_resultchosen_factorizationprimesprimesecond_divisorproductoutput))) /\ ((exists ge_balance_positive_unique_factorization_resultchosen_factorizationprimesprimesecond_divisorproductoutputreal ge_balance_negative_unique_factorization_resultchosen_factorizationprimesprimesecond_divisorproductoutputreal. (((((ge_representation_real_code_unique_factorization_resultchosen_factorizationprimesprimesecond_divisorproductoutput) = 2 * (ge_balance_positive_unique_factorization_resultchosen_factorizationprimesprimesecond_divisorproductoutputreal) /\ (ge_balance_negative_unique_factorization_resultchosen_factorizationprimesprimesecond_divisorproductoutputreal) = 0) \/ exists ge_signed_half_unique_factorization_resultchosen_factorizationprimesprimesecond_divisorproductoutputrealdecode. (((ge_representation_real_code_unique_factorization_resultchosen_factorizationprimesprimesecond_divisorproductoutput) = 2 * ge_signed_half_unique_factorization_resultchosen_factorizationprimesprimesecond_divisorproductoutputrealdecode + 1 /\ (ge_balance_positive_unique_factorization_resultchosen_factorizationprimesprimesecond_divisorproductoutputreal) = 0) /\ (ge_balance_negative_unique_factorization_resultchosen_factorizationprimesprimesecond_divisorproductoutputreal) = S ge_signed_half_unique_factorization_resultchosen_factorizationprimesprimesecond_divisorproductoutputrealdecode))) /\ ((((((((ge_first_rp_unique_factorization_resultchosen_factorizationprimesprimesecond_divisorproduct) * (ge_second_rp_unique_factorization_resultchosen_factorizationprimesprimesecond_divisorproduct))) + (((ge_first_rn_unique_factorization_resultchosen_factorizationprimesprimesecond_divisorproduct) * (ge_second_rn_unique_factorization_resultchosen_factorizationprimesprimesecond_divisorproduct))))) + (((((ge_first_ip_unique_factorization_resultchosen_factorizationprimesprimesecond_divisorproduct) * (ge_second_in_unique_factorization_resultchosen_factorizationprimesprimesecond_divisorproduct))) + (((ge_first_in_unique_factorization_resultchosen_factorizationprimesprimesecond_divisorproduct) * (ge_second_ip_unique_factorization_resultchosen_factorizationprimesprimesecond_divisorproduct))))))) + ge_balance_negative_unique_factorization_resultchosen_factorizationprimesprimesecond_divisorproductoutputreal = (((((((ge_first_rp_unique_factorization_resultchosen_factorizationprimesprimesecond_divisorproduct) * (ge_second_rn_unique_factorization_resultchosen_factorizationprimesprimesecond_divisorproduct))) + (((ge_first_rn_unique_factorization_resultchosen_factorizationprimesprimesecond_divisorproduct) * (ge_second_rp_unique_factorization_resultchosen_factorizationprimesprimesecond_divisorproduct))))) + (((((ge_first_ip_unique_factorization_resultchosen_factorizationprimesprimesecond_divisorproduct) * (ge_second_ip_unique_factorization_resultchosen_factorizationprimesprimesecond_divisorproduct))) + (((ge_first_in_unique_factorization_resultchosen_factorizationprimesprimesecond_divisorproduct) * (ge_second_in_unique_factorization_resultchosen_factorizationprimesprimesecond_divisorproduct))))))) + ge_balance_positive_unique_factorization_resultchosen_factorizationprimesprimesecond_divisorproductoutputreal))) /\ (exists ge_balance_positive_unique_factorization_resultchosen_factorizationprimesprimesecond_divisorproductoutputimaginary ge_balance_negative_unique_factorization_resultchosen_factorizationprimesprimesecond_divisorproductoutputimaginary. (((((ge_representation_imaginary_code_unique_factorization_resultchosen_factorizationprimesprimesecond_divisorproductoutput) = 2 * (ge_balance_positive_unique_factorization_resultchosen_factorizationprimesprimesecond_divisorproductoutputimaginary) /\ (ge_balance_negative_unique_factorization_resultchosen_factorizationprimesprimesecond_divisorproductoutputimaginary) = 0) \/ exists ge_signed_half_unique_factorization_resultchosen_factorizationprimesprimesecond_divisorproductoutputimaginarydecode. (((ge_representation_imaginary_code_unique_factorization_resultchosen_factorizationprimesprimesecond_divisorproductoutput) = 2 * ge_signed_half_unique_factorization_resultchosen_factorizationprimesprimesecond_divisorproductoutputimaginarydecode + 1 /\ (ge_balance_positive_unique_factorization_resultchosen_factorizationprimesprimesecond_divisorproductoutputimaginary) = 0) /\ (ge_balance_negative_unique_factorization_resultchosen_factorizationprimesprimesecond_divisorproductoutputimaginary) = S ge_signed_half_unique_factorization_resultchosen_factorizationprimesprimesecond_divisorproductoutputimaginarydecode))) /\ ((((((((ge_first_rp_unique_factorization_resultchosen_factorizationprimesprimesecond_divisorproduct) * (ge_second_ip_unique_factorization_resultchosen_factorizationprimesprimesecond_divisorproduct))) + (((ge_first_rn_unique_factorization_resultchosen_factorizationprimesprimesecond_divisorproduct) * (ge_second_in_unique_factorization_resultchosen_factorizationprimesprimesecond_divisorproduct))))) + (((((ge_first_ip_unique_factorization_resultchosen_factorizationprimesprimesecond_divisorproduct) * (ge_second_rp_unique_factorization_resultchosen_factorizationprimesprimesecond_divisorproduct))) + (((ge_first_in_unique_factorization_resultchosen_factorizationprimesprimesecond_divisorproduct) * (ge_second_rn_unique_factorization_resultchosen_factorizationprimesprimesecond_divisorproduct))))))) + ge_balance_negative_unique_factorization_resultchosen_factorizationprimesprimesecond_divisorproductoutputimaginary = (((((((ge_first_rp_unique_factorization_resultchosen_factorizationprimesprimesecond_divisorproduct) * (ge_second_in_unique_factorization_resultchosen_factorizationprimesprimesecond_divisorproduct))) + (((ge_first_rn_unique_factorization_resultchosen_factorizationprimesprimesecond_divisorproduct) * (ge_second_ip_unique_factorization_resultchosen_factorizationprimesprimesecond_divisorproduct))))) + (((((ge_first_ip_unique_factorization_resultchosen_factorizationprimesprimesecond_divisorproduct) * (ge_second_rn_unique_factorization_resultchosen_factorizationprimesprimesecond_divisorproduct))) + (((ge_first_in_unique_factorization_resultchosen_factorizationprimesprimesecond_divisorproduct) * (ge_second_rp_unique_factorization_resultchosen_factorizationprimesprimesecond_divisorproduct))))))) + ge_balance_positive_unique_factorization_resultchosen_factorizationprimesprimesecond_divisorproductoutputimaginary)))))))))))))))) /\ (exists gr_prime_factor_product_unique_factorization_resultchosen_factorization. ((exists gr_product_trace_unique_factorization_resultchosen_factorizationtrace gr_product_scale_unique_factorization_resultchosen_factorizationtrace. ((((exists ff_h_gprod_unique_factorization_resultchosen_factorizationtracestart. ff_h_gprod_unique_factorization_resultchosen_factorizationtracestart + S (6) = S ((S (0)) * gr_product_scale_unique_factorization_resultchosen_factorizationtrace)) /\ exists ff_q_gprod_unique_factorization_resultchosen_factorizationtracestart. gr_product_trace_unique_factorization_resultchosen_factorizationtrace = ff_q_gprod_unique_factorization_resultchosen_factorizationtracestart * S ((S (0)) * gr_product_scale_unique_factorization_resultchosen_factorizationtrace) + (6))) /\ ((((exists ff_h_gprod_unique_factorization_resultchosen_factorizationtraceend. ff_h_gprod_unique_factorization_resultchosen_factorizationtraceend + S (gr_prime_factor_product_unique_factorization_resultchosen_factorization) = S ((S (gr_chosen_length_unique_factorization_result)) * gr_product_scale_unique_factorization_resultchosen_factorizationtrace)) /\ exists ff_q_gprod_unique_factorization_resultchosen_factorizationtraceend. gr_product_trace_unique_factorization_resultchosen_factorizationtrace = ff_q_gprod_unique_factorization_resultchosen_factorizationtraceend * S ((S (gr_chosen_length_unique_factorization_result)) * gr_product_scale_unique_factorization_resultchosen_factorizationtrace) + (gr_prime_factor_product_unique_factorization_resultchosen_factorization))) /\ (forall gr_product_index_unique_factorization_resultchosen_factorizationtracesteps. (exists ge_gap_unique_factorization_resultchosen_factorizationtracestepsindex_bound. ge_gap_unique_factorization_resultchosen_factorizationtracestepsindex_bound + S (gr_product_index_unique_factorization_resultchosen_factorizationtracesteps) = (gr_chosen_length_unique_factorization_result)) -> exists gr_product_factor_unique_factorization_resultchosen_factorizationtracesteps gr_product_before_unique_factorization_resultchosen_factorizationtracesteps gr_product_after_unique_factorization_resultchosen_factorizationtracesteps. ((((exists ff_h_gprod_unique_factorization_resultchosen_factorizationtracestepsfactor. ff_h_gprod_unique_factorization_resultchosen_factorizationtracestepsfactor + S (gr_product_factor_unique_factorization_resultchosen_factorizationtracesteps) = S ((S (gr_product_index_unique_factorization_resultchosen_factorizationtracesteps)) * gr_chosen_scale_unique_factorization_result)) /\ exists ff_q_gprod_unique_factorization_resultchosen_factorizationtracestepsfactor. gr_chosen_code_unique_factorization_result = ff_q_gprod_unique_factorization_resultchosen_factorizationtracestepsfactor * S ((S (gr_product_index_unique_factorization_resultchosen_factorizationtracesteps)) * gr_chosen_scale_unique_factorization_result) + (gr_product_factor_unique_factorization_resultchosen_factorizationtracesteps))) /\ ((((exists ff_h_gprod_unique_factorization_resultchosen_factorizationtracestepsbefore. ff_h_gprod_unique_factorization_resultchosen_factorizationtracestepsbefore + S (gr_product_before_unique_factorization_resultchosen_factorizationtracesteps) = S ((S (gr_product_index_unique_factorization_resultchosen_factorizationtracesteps)) * gr_product_scale_unique_factorization_resultchosen_factorizationtrace)) /\ exists ff_q_gprod_unique_factorization_resultchosen_factorizationtracestepsbefore. gr_product_trace_unique_factorization_resultchosen_factorizationtrace = ff_q_gprod_unique_factorization_resultchosen_factorizationtracestepsbefore * S ((S (gr_product_index_unique_factorization_resultchosen_factorizationtracesteps)) * gr_product_scale_unique_factorization_resultchosen_factorizationtrace) + (gr_product_before_unique_factorization_resultchosen_factorizationtracesteps))) /\ ((((exists ff_h_gprod_unique_factorization_resultchosen_factorizationtracestepsafter. ff_h_gprod_unique_factorization_resultchosen_factorizationtracestepsafter + S (gr_product_after_unique_factorization_resultchosen_factorizationtracesteps) = S ((S (S (gr_product_index_unique_factorization_resultchosen_factorizationtracesteps))) * gr_product_scale_unique_factorization_resultchosen_factorizationtrace)) /\ exists ff_q_gprod_unique_factorization_resultchosen_factorizationtracestepsafter. gr_product_trace_unique_factorization_resultchosen_factorizationtrace = ff_q_gprod_unique_factorization_resultchosen_factorizationtracestepsafter * S ((S (S (gr_product_index_unique_factorization_resultchosen_factorizationtracesteps))) * gr_product_scale_unique_factorization_resultchosen_factorizationtrace) + (gr_product_after_unique_factorization_resultchosen_factorizationtracesteps))) /\ (exists ge_first_rp_unique_factorization_resultchosen_factorizationtracestepsmultiply ge_first_rn_unique_factorization_resultchosen_factorizationtracestepsmultiply ge_first_ip_unique_factorization_resultchosen_factorizationtracestepsmultiply ge_first_in_unique_factorization_resultchosen_factorizationtracestepsmultiply ge_second_rp_unique_factorization_resultchosen_factorizationtracestepsmultiply ge_second_rn_unique_factorization_resultchosen_factorizationtracestepsmultiply ge_second_ip_unique_factorization_resultchosen_factorizationtracestepsmultiply ge_second_in_unique_factorization_resultchosen_factorizationtracestepsmultiply. ((exists ge_representation_real_code_unique_factorization_resultchosen_factorizationtracestepsmultiplyfirst ge_representation_imaginary_code_unique_factorization_resultchosen_factorizationtracestepsmultiplyfirst. (((gr_product_before_unique_factorization_resultchosen_factorizationtracesteps) = ((ge_representation_real_code_unique_factorization_resultchosen_factorizationtracestepsmultiplyfirst) + (ge_representation_imaginary_code_unique_factorization_resultchosen_factorizationtracestepsmultiplyfirst)) * S ((ge_representation_real_code_unique_factorization_resultchosen_factorizationtracestepsmultiplyfirst) + (ge_representation_imaginary_code_unique_factorization_resultchosen_factorizationtracestepsmultiplyfirst)) + ((ge_representation_imaginary_code_unique_factorization_resultchosen_factorizationtracestepsmultiplyfirst) + (ge_representation_imaginary_code_unique_factorization_resultchosen_factorizationtracestepsmultiplyfirst))) /\ ((exists ge_balance_positive_unique_factorization_resultchosen_factorizationtracestepsmultiplyfirstreal ge_balance_negative_unique_factorization_resultchosen_factorizationtracestepsmultiplyfirstreal. (((((ge_representation_real_code_unique_factorization_resultchosen_factorizationtracestepsmultiplyfirst) = 2 * (ge_balance_positive_unique_factorization_resultchosen_factorizationtracestepsmultiplyfirstreal) /\ (ge_balance_negative_unique_factorization_resultchosen_factorizationtracestepsmultiplyfirstreal) = 0) \/ exists ge_signed_half_unique_factorization_resultchosen_factorizationtracestepsmultiplyfirstrealdecode. (((ge_representation_real_code_unique_factorization_resultchosen_factorizationtracestepsmultiplyfirst) = 2 * ge_signed_half_unique_factorization_resultchosen_factorizationtracestepsmultiplyfirstrealdecode + 1 /\ (ge_balance_positive_unique_factorization_resultchosen_factorizationtracestepsmultiplyfirstreal) = 0) /\ (ge_balance_negative_unique_factorization_resultchosen_factorizationtracestepsmultiplyfirstreal) = S ge_signed_half_unique_factorization_resultchosen_factorizationtracestepsmultiplyfirstrealdecode))) /\ ((ge_first_rp_unique_factorization_resultchosen_factorizationtracestepsmultiply) + ge_balance_negative_unique_factorization_resultchosen_factorizationtracestepsmultiplyfirstreal = (ge_first_rn_unique_factorization_resultchosen_factorizationtracestepsmultiply) + ge_balance_positive_unique_factorization_resultchosen_factorizationtracestepsmultiplyfirstreal))) /\ (exists ge_balance_positive_unique_factorization_resultchosen_factorizationtracestepsmultiplyfirstimaginary ge_balance_negative_unique_factorization_resultchosen_factorizationtracestepsmultiplyfirstimaginary. (((((ge_representation_imaginary_code_unique_factorization_resultchosen_factorizationtracestepsmultiplyfirst) = 2 * (ge_balance_positive_unique_factorization_resultchosen_factorizationtracestepsmultiplyfirstimaginary) /\ (ge_balance_negative_unique_factorization_resultchosen_factorizationtracestepsmultiplyfirstimaginary) = 0) \/ exists ge_signed_half_unique_factorization_resultchosen_factorizationtracestepsmultiplyfirstimaginarydecode. (((ge_representation_imaginary_code_unique_factorization_resultchosen_factorizationtracestepsmultiplyfirst) = 2 * ge_signed_half_unique_factorization_resultchosen_factorizationtracestepsmultiplyfirstimaginarydecode + 1 /\ (ge_balance_positive_unique_factorization_resultchosen_factorizationtracestepsmultiplyfirstimaginary) = 0) /\ (ge_balance_negative_unique_factorization_resultchosen_factorizationtracestepsmultiplyfirstimaginary) = S ge_signed_half_unique_factorization_resultchosen_factorizationtracestepsmultiplyfirstimaginarydecode))) /\ ((ge_first_ip_unique_factorization_resultchosen_factorizationtracestepsmultiply) + ge_balance_negative_unique_factorization_resultchosen_factorizationtracestepsmultiplyfirstimaginary = (ge_first_in_unique_factorization_resultchosen_factorizationtracestepsmultiply) + ge_balance_positive_unique_factorization_resultchosen_factorizationtracestepsmultiplyfirstimaginary)))))) /\ ((exists ge_representation_real_code_unique_factorization_resultchosen_factorizationtracestepsmultiplysecond ge_representation_imaginary_code_unique_factorization_resultchosen_factorizationtracestepsmultiplysecond. (((gr_product_factor_unique_factorization_resultchosen_factorizationtracesteps) = ((ge_representation_real_code_unique_factorization_resultchosen_factorizationtracestepsmultiplysecond) + (ge_representation_imaginary_code_unique_factorization_resultchosen_factorizationtracestepsmultiplysecond)) * S ((ge_representation_real_code_unique_factorization_resultchosen_factorizationtracestepsmultiplysecond) + (ge_representation_imaginary_code_unique_factorization_resultchosen_factorizationtracestepsmultiplysecond)) + ((ge_representation_imaginary_code_unique_factorization_resultchosen_factorizationtracestepsmultiplysecond) + (ge_representation_imaginary_code_unique_factorization_resultchosen_factorizationtracestepsmultiplysecond))) /\ ((exists ge_balance_positive_unique_factorization_resultchosen_factorizationtracestepsmultiplysecondreal ge_balance_negative_unique_factorization_resultchosen_factorizationtracestepsmultiplysecondreal. (((((ge_representation_real_code_unique_factorization_resultchosen_factorizationtracestepsmultiplysecond) = 2 * (ge_balance_positive_unique_factorization_resultchosen_factorizationtracestepsmultiplysecondreal) /\ (ge_balance_negative_unique_factorization_resultchosen_factorizationtracestepsmultiplysecondreal) = 0) \/ exists ge_signed_half_unique_factorization_resultchosen_factorizationtracestepsmultiplysecondrealdecode. (((ge_representation_real_code_unique_factorization_resultchosen_factorizationtracestepsmultiplysecond) = 2 * ge_signed_half_unique_factorization_resultchosen_factorizationtracestepsmultiplysecondrealdecode + 1 /\ (ge_balance_positive_unique_factorization_resultchosen_factorizationtracestepsmultiplysecondreal) = 0) /\ (ge_balance_negative_unique_factorization_resultchosen_factorizationtracestepsmultiplysecondreal) = S ge_signed_half_unique_factorization_resultchosen_factorizationtracestepsmultiplysecondrealdecode))) /\ ((ge_second_rp_unique_factorization_resultchosen_factorizationtracestepsmultiply) + ge_balance_negative_unique_factorization_resultchosen_factorizationtracestepsmultiplysecondreal = (ge_second_rn_unique_factorization_resultchosen_factorizationtracestepsmultiply) + ge_balance_positive_unique_factorization_resultchosen_factorizationtracestepsmultiplysecondreal))) /\ (exists ge_balance_positive_unique_factorization_resultchosen_factorizationtracestepsmultiplysecondimaginary ge_balance_negative_unique_factorization_resultchosen_factorizationtracestepsmultiplysecondimaginary. (((((ge_representation_imaginary_code_unique_factorization_resultchosen_factorizationtracestepsmultiplysecond) = 2 * (ge_balance_positive_unique_factorization_resultchosen_factorizationtracestepsmultiplysecondimaginary) /\ (ge_balance_negative_unique_factorization_resultchosen_factorizationtracestepsmultiplysecondimaginary) = 0) \/ exists ge_signed_half_unique_factorization_resultchosen_factorizationtracestepsmultiplysecondimaginarydecode. (((ge_representation_imaginary_code_unique_factorization_resultchosen_factorizationtracestepsmultiplysecond) = 2 * ge_signed_half_unique_factorization_resultchosen_factorizationtracestepsmultiplysecondimaginarydecode + 1 /\ (ge_balance_positive_unique_factorization_resultchosen_factorizationtracestepsmultiplysecondimaginary) = 0) /\ (ge_balance_negative_unique_factorization_resultchosen_factorizationtracestepsmultiplysecondimaginary) = S ge_signed_half_unique_factorization_resultchosen_factorizationtracestepsmultiplysecondimaginarydecode))) /\ ((ge_second_ip_unique_factorization_resultchosen_factorizationtracestepsmultiply) + ge_balance_negative_unique_factorization_resultchosen_factorizationtracestepsmultiplysecondimaginary = (ge_second_in_unique_factorization_resultchosen_factorizationtracestepsmultiply) + ge_balance_positive_unique_factorization_resultchosen_factorizationtracestepsmultiplysecondimaginary)))))) /\ (exists ge_representation_real_code_unique_factorization_resultchosen_factorizationtracestepsmultiplyoutput ge_representation_imaginary_code_unique_factorization_resultchosen_factorizationtracestepsmultiplyoutput. (((gr_product_after_unique_factorization_resultchosen_factorizationtracesteps) = ((ge_representation_real_code_unique_factorization_resultchosen_factorizationtracestepsmultiplyoutput) + (ge_representation_imaginary_code_unique_factorization_resultchosen_factorizationtracestepsmultiplyoutput)) * S ((ge_representation_real_code_unique_factorization_resultchosen_factorizationtracestepsmultiplyoutput) + (ge_representation_imaginary_code_unique_factorization_resultchosen_factorizationtracestepsmultiplyoutput)) + ((ge_representation_imaginary_code_unique_factorization_resultchosen_factorizationtracestepsmultiplyoutput) + (ge_representation_imaginary_code_unique_factorization_resultchosen_factorizationtracestepsmultiplyoutput))) /\ ((exists ge_balance_positive_unique_factorization_resultchosen_factorizationtracestepsmultiplyoutputreal ge_balance_negative_unique_factorization_resultchosen_factorizationtracestepsmultiplyoutputreal. (((((ge_representation_real_code_unique_factorization_resultchosen_factorizationtracestepsmultiplyoutput) = 2 * (ge_balance_positive_unique_factorization_resultchosen_factorizationtracestepsmultiplyoutputreal) /\ (ge_balance_negative_unique_factorization_resultchosen_factorizationtracestepsmultiplyoutputreal) = 0) \/ exists ge_signed_half_unique_factorization_resultchosen_factorizationtracestepsmultiplyoutputrealdecode. (((ge_representation_real_code_unique_factorization_resultchosen_factorizationtracestepsmultiplyoutput) = 2 * ge_signed_half_unique_factorization_resultchosen_factorizationtracestepsmultiplyoutputrealdecode + 1 /\ (ge_balance_positive_unique_factorization_resultchosen_factorizationtracestepsmultiplyoutputreal) = 0) /\ (ge_balance_negative_unique_factorization_resultchosen_factorizationtracestepsmultiplyoutputreal) = S ge_signed_half_unique_factorization_resultchosen_factorizationtracestepsmultiplyoutputrealdecode))) /\ ((((((((ge_first_rp_unique_factorization_resultchosen_factorizationtracestepsmultiply) * (ge_second_rp_unique_factorization_resultchosen_factorizationtracestepsmultiply))) + (((ge_first_rn_unique_factorization_resultchosen_factorizationtracestepsmultiply) * (ge_second_rn_unique_factorization_resultchosen_factorizationtracestepsmultiply))))) + (((((ge_first_ip_unique_factorization_resultchosen_factorizationtracestepsmultiply) * (ge_second_in_unique_factorization_resultchosen_factorizationtracestepsmultiply))) + (((ge_first_in_unique_factorization_resultchosen_factorizationtracestepsmultiply) * (ge_second_ip_unique_factorization_resultchosen_factorizationtracestepsmultiply))))))) + ge_balance_negative_unique_factorization_resultchosen_factorizationtracestepsmultiplyoutputreal = (((((((ge_first_rp_unique_factorization_resultchosen_factorizationtracestepsmultiply) * (ge_second_rn_unique_factorization_resultchosen_factorizationtracestepsmultiply))) + (((ge_first_rn_unique_factorization_resultchosen_factorizationtracestepsmultiply) * (ge_second_rp_unique_factorization_resultchosen_factorizationtracestepsmultiply))))) + (((((ge_first_ip_unique_factorization_resultchosen_factorizationtracestepsmultiply) * (ge_second_ip_unique_factorization_resultchosen_factorizationtracestepsmultiply))) + (((ge_first_in_unique_factorization_resultchosen_factorizationtracestepsmultiply) * (ge_second_in_unique_factorization_resultchosen_factorizationtracestepsmultiply))))))) + ge_balance_positive_unique_factorization_resultchosen_factorizationtracestepsmultiplyoutputreal))) /\ (exists ge_balance_positive_unique_factorization_resultchosen_factorizationtracestepsmultiplyoutputimaginary ge_balance_negative_unique_factorization_resultchosen_factorizationtracestepsmultiplyoutputimaginary. (((((ge_representation_imaginary_code_unique_factorization_resultchosen_factorizationtracestepsmultiplyoutput) = 2 * (ge_balance_positive_unique_factorization_resultchosen_factorizationtracestepsmultiplyoutputimaginary) /\ (ge_balance_negative_unique_factorization_resultchosen_factorizationtracestepsmultiplyoutputimaginary) = 0) \/ exists ge_signed_half_unique_factorization_resultchosen_factorizationtracestepsmultiplyoutputimaginarydecode. (((ge_representation_imaginary_code_unique_factorization_resultchosen_factorizationtracestepsmultiplyoutput) = 2 * ge_signed_half_unique_factorization_resultchosen_factorizationtracestepsmultiplyoutputimaginarydecode + 1 /\ (ge_balance_positive_unique_factorization_resultchosen_factorizationtracestepsmultiplyoutputimaginary) = 0) /\ (ge_balance_negative_unique_factorization_resultchosen_factorizationtracestepsmultiplyoutputimaginary) = S ge_signed_half_unique_factorization_resultchosen_factorizationtracestepsmultiplyoutputimaginarydecode))) /\ ((((((((ge_first_rp_unique_factorization_resultchosen_factorizationtracestepsmultiply) * (ge_second_ip_unique_factorization_resultchosen_factorizationtracestepsmultiply))) + (((ge_first_rn_unique_factorization_resultchosen_factorizationtracestepsmultiply) * (ge_second_in_unique_factorization_resultchosen_factorizationtracestepsmultiply))))) + (((((ge_first_ip_unique_factorization_resultchosen_factorizationtracestepsmultiply) * (ge_second_rp_unique_factorization_resultchosen_factorizationtracestepsmultiply))) + (((ge_first_in_unique_factorization_resultchosen_factorizationtracestepsmultiply) * (ge_second_rn_unique_factorization_resultchosen_factorizationtracestepsmultiply))))))) + ge_balance_negative_unique_factorization_resultchosen_factorizationtracestepsmultiplyoutputimaginary = (((((((ge_first_rp_unique_factorization_resultchosen_factorizationtracestepsmultiply) * (ge_second_in_unique_factorization_resultchosen_factorizationtracestepsmultiply))) + (((ge_first_rn_unique_factorization_resultchosen_factorizationtracestepsmultiply) * (ge_second_ip_unique_factorization_resultchosen_factorizationtracestepsmultiply))))) + (((((ge_first_ip_unique_factorization_resultchosen_factorizationtracestepsmultiply) * (ge_second_rn_unique_factorization_resultchosen_factorizationtracestepsmultiply))) + (((ge_first_in_unique_factorization_resultchosen_factorizationtracestepsmultiply) * (ge_second_rp_unique_factorization_resultchosen_factorizationtracestepsmultiply))))))) + ge_balance_positive_unique_factorization_resultchosen_factorizationtracestepsmultiplyoutputimaginary)))))))))))))))) /\ (exists ge_first_rp_unique_factorization_resultchosen_factorizationreconstruct ge_first_rn_unique_factorization_resultchosen_factorizationreconstruct ge_first_ip_unique_factorization_resultchosen_factorizationreconstruct ge_first_in_unique_factorization_resultchosen_factorizationreconstruct ge_second_rp_unique_factorization_resultchosen_factorizationreconstruct ge_second_rn_unique_factorization_resultchosen_factorizationreconstruct ge_second_ip_unique_factorization_resultchosen_factorizationreconstruct ge_second_in_unique_factorization_resultchosen_factorizationreconstruct. ((exists ge_representation_real_code_unique_factorization_resultchosen_factorizationreconstructfirst ge_representation_imaginary_code_unique_factorization_resultchosen_factorizationreconstructfirst. (((gr_chosen_unit_unique_factorization_result) = ((ge_representation_real_code_unique_factorization_resultchosen_factorizationreconstructfirst) + (ge_representation_imaginary_code_unique_factorization_resultchosen_factorizationreconstructfirst)) * S ((ge_representation_real_code_unique_factorization_resultchosen_factorizationreconstructfirst) + (ge_representation_imaginary_code_unique_factorization_resultchosen_factorizationreconstructfirst)) + ((ge_representation_imaginary_code_unique_factorization_resultchosen_factorizationreconstructfirst) + (ge_representation_imaginary_code_unique_factorization_resultchosen_factorizationreconstructfirst))) /\ ((exists ge_balance_positive_unique_factorization_resultchosen_factorizationreconstructfirstreal ge_balance_negative_unique_factorization_resultchosen_factorizationreconstructfirstreal. (((((ge_representation_real_code_unique_factorization_resultchosen_factorizationreconstructfirst) = 2 * (ge_balance_positive_unique_factorization_resultchosen_factorizationreconstructfirstreal) /\ (ge_balance_negative_unique_factorization_resultchosen_factorizationreconstructfirstreal) = 0) \/ exists ge_signed_half_unique_factorization_resultchosen_factorizationreconstructfirstrealdecode. (((ge_representation_real_code_unique_factorization_resultchosen_factorizationreconstructfirst) = 2 * ge_signed_half_unique_factorization_resultchosen_factorizationreconstructfirstrealdecode + 1 /\ (ge_balance_positive_unique_factorization_resultchosen_factorizationreconstructfirstreal) = 0) /\ (ge_balance_negative_unique_factorization_resultchosen_factorizationreconstructfirstreal) = S ge_signed_half_unique_factorization_resultchosen_factorizationreconstructfirstrealdecode))) /\ ((ge_first_rp_unique_factorization_resultchosen_factorizationreconstruct) + ge_balance_negative_unique_factorization_resultchosen_factorizationreconstructfirstreal = (ge_first_rn_unique_factorization_resultchosen_factorizationreconstruct) + ge_balance_positive_unique_factorization_resultchosen_factorizationreconstructfirstreal))) /\ (exists ge_balance_positive_unique_factorization_resultchosen_factorizationreconstructfirstimaginary ge_balance_negative_unique_factorization_resultchosen_factorizationreconstructfirstimaginary. (((((ge_representation_imaginary_code_unique_factorization_resultchosen_factorizationreconstructfirst) = 2 * (ge_balance_positive_unique_factorization_resultchosen_factorizationreconstructfirstimaginary) /\ (ge_balance_negative_unique_factorization_resultchosen_factorizationreconstructfirstimaginary) = 0) \/ exists ge_signed_half_unique_factorization_resultchosen_factorizationreconstructfirstimaginarydecode. (((ge_representation_imaginary_code_unique_factorization_resultchosen_factorizationreconstructfirst) = 2 * ge_signed_half_unique_factorization_resultchosen_factorizationreconstructfirstimaginarydecode + 1 /\ (ge_balance_positive_unique_factorization_resultchosen_factorizationreconstructfirstimaginary) = 0) /\ (ge_balance_negative_unique_factorization_resultchosen_factorizationreconstructfirstimaginary) = S ge_signed_half_unique_factorization_resultchosen_factorizationreconstructfirstimaginarydecode))) /\ ((ge_first_ip_unique_factorization_resultchosen_factorizationreconstruct) + ge_balance_negative_unique_factorization_resultchosen_factorizationreconstructfirstimaginary = (ge_first_in_unique_factorization_resultchosen_factorizationreconstruct) + ge_balance_positive_unique_factorization_resultchosen_factorizationreconstructfirstimaginary)))))) /\ ((exists ge_representation_real_code_unique_factorization_resultchosen_factorizationreconstructsecond ge_representation_imaginary_code_unique_factorization_resultchosen_factorizationreconstructsecond. (((gr_prime_factor_product_unique_factorization_resultchosen_factorization) = ((ge_representation_real_code_unique_factorization_resultchosen_factorizationreconstructsecond) + (ge_representation_imaginary_code_unique_factorization_resultchosen_factorizationreconstructsecond)) * S ((ge_representation_real_code_unique_factorization_resultchosen_factorizationreconstructsecond) + (ge_representation_imaginary_code_unique_factorization_resultchosen_factorizationreconstructsecond)) + ((ge_representation_imaginary_code_unique_factorization_resultchosen_factorizationreconstructsecond) + (ge_representation_imaginary_code_unique_factorization_resultchosen_factorizationreconstructsecond))) /\ ((exists ge_balance_positive_unique_factorization_resultchosen_factorizationreconstructsecondreal ge_balance_negative_unique_factorization_resultchosen_factorizationreconstructsecondreal. (((((ge_representation_real_code_unique_factorization_resultchosen_factorizationreconstructsecond) = 2 * (ge_balance_positive_unique_factorization_resultchosen_factorizationreconstructsecondreal) /\ (ge_balance_negative_unique_factorization_resultchosen_factorizationreconstructsecondreal) = 0) \/ exists ge_signed_half_unique_factorization_resultchosen_factorizationreconstructsecondrealdecode. (((ge_representation_real_code_unique_factorization_resultchosen_factorizationreconstructsecond) = 2 * ge_signed_half_unique_factorization_resultchosen_factorizationreconstructsecondrealdecode + 1 /\ (ge_balance_positive_unique_factorization_resultchosen_factorizationreconstructsecondreal) = 0) /\ (ge_balance_negative_unique_factorization_resultchosen_factorizationreconstructsecondreal) = S ge_signed_half_unique_factorization_resultchosen_factorizationreconstructsecondrealdecode))) /\ ((ge_second_rp_unique_factorization_resultchosen_factorizationreconstruct) + ge_balance_negative_unique_factorization_resultchosen_factorizationreconstructsecondreal = (ge_second_rn_unique_factorization_resultchosen_factorizationreconstruct) + ge_balance_positive_unique_factorization_resultchosen_factorizationreconstructsecondreal))) /\ (exists ge_balance_positive_unique_factorization_resultchosen_factorizationreconstructsecondimaginary ge_balance_negative_unique_factorization_resultchosen_factorizationreconstructsecondimaginary. (((((ge_representation_imaginary_code_unique_factorization_resultchosen_factorizationreconstructsecond) = 2 * (ge_balance_positive_unique_factorization_resultchosen_factorizationreconstructsecondimaginary) /\ (ge_balance_negative_unique_factorization_resultchosen_factorizationreconstructsecondimaginary) = 0) \/ exists ge_signed_half_unique_factorization_resultchosen_factorizationreconstructsecondimaginarydecode. (((ge_representation_imaginary_code_unique_factorization_resultchosen_factorizationreconstructsecond) = 2 * ge_signed_half_unique_factorization_resultchosen_factorizationreconstructsecondimaginarydecode + 1 /\ (ge_balance_positive_unique_factorization_resultchosen_factorizationreconstructsecondimaginary) = 0) /\ (ge_balance_negative_unique_factorization_resultchosen_factorizationreconstructsecondimaginary) = S ge_signed_half_unique_factorization_resultchosen_factorizationreconstructsecondimaginarydecode))) /\ ((ge_second_ip_unique_factorization_resultchosen_factorizationreconstruct) + ge_balance_negative_unique_factorization_resultchosen_factorizationreconstructsecondimaginary = (ge_second_in_unique_factorization_resultchosen_factorizationreconstruct) + ge_balance_positive_unique_factorization_resultchosen_factorizationreconstructsecondimaginary)))))) /\ (exists ge_representation_real_code_unique_factorization_resultchosen_factorizationreconstructoutput ge_representation_imaginary_code_unique_factorization_resultchosen_factorizationreconstructoutput. (((z) = ((ge_representation_real_code_unique_factorization_resultchosen_factorizationreconstructoutput) + (ge_representation_imaginary_code_unique_factorization_resultchosen_factorizationreconstructoutput)) * S ((ge_representation_real_code_unique_factorization_resultchosen_factorizationreconstructoutput) + (ge_representation_imaginary_code_unique_factorization_resultchosen_factorizationreconstructoutput)) + ((ge_representation_imaginary_code_unique_factorization_resultchosen_factorizationreconstructoutput) + (ge_representation_imaginary_code_unique_factorization_resultchosen_factorizationreconstructoutput))) /\ ((exists ge_balance_positive_unique_factorization_resultchosen_factorizationreconstructoutputreal ge_balance_negative_unique_factorization_resultchosen_factorizationreconstructoutputreal. (((((ge_representation_real_code_unique_factorization_resultchosen_factorizationreconstructoutput) = 2 * (ge_balance_positive_unique_factorization_resultchosen_factorizationreconstructoutputreal) /\ (ge_balance_negative_unique_factorization_resultchosen_factorizationreconstructoutputreal) = 0) \/ exists ge_signed_half_unique_factorization_resultchosen_factorizationreconstructoutputrealdecode. (((ge_representation_real_code_unique_factorization_resultchosen_factorizationreconstructoutput) = 2 * ge_signed_half_unique_factorization_resultchosen_factorizationreconstructoutputrealdecode + 1 /\ (ge_balance_positive_unique_factorization_resultchosen_factorizationreconstructoutputreal) = 0) /\ (ge_balance_negative_unique_factorization_resultchosen_factorizationreconstructoutputreal) = S ge_signed_half_unique_factorization_resultchosen_factorizationreconstructoutputrealdecode))) /\ ((((((((ge_first_rp_unique_factorization_resultchosen_factorizationreconstruct) * (ge_second_rp_unique_factorization_resultchosen_factorizationreconstruct))) + (((ge_first_rn_unique_factorization_resultchosen_factorizationreconstruct) * (ge_second_rn_unique_factorization_resultchosen_factorizationreconstruct))))) + (((((ge_first_ip_unique_factorization_resultchosen_factorizationreconstruct) * (ge_second_in_unique_factorization_resultchosen_factorizationreconstruct))) + (((ge_first_in_unique_factorization_resultchosen_factorizationreconstruct) * (ge_second_ip_unique_factorization_resultchosen_factorizationreconstruct))))))) + ge_balance_negative_unique_factorization_resultchosen_factorizationreconstructoutputreal = (((((((ge_first_rp_unique_factorization_resultchosen_factorizationreconstruct) * (ge_second_rn_unique_factorization_resultchosen_factorizationreconstruct))) + (((ge_first_rn_unique_factorization_resultchosen_factorizationreconstruct) * (ge_second_rp_unique_factorization_resultchosen_factorizationreconstruct))))) + (((((ge_first_ip_unique_factorization_resultchosen_factorizationreconstruct) * (ge_second_ip_unique_factorization_resultchosen_factorizationreconstruct))) + (((ge_first_in_unique_factorization_resultchosen_factorizationreconstruct) * (ge_second_in_unique_factorization_resultchosen_factorizationreconstruct))))))) + ge_balance_positive_unique_factorization_resultchosen_factorizationreconstructoutputreal))) /\ (exists ge_balance_positive_unique_factorization_resultchosen_factorizationreconstructoutputimaginary ge_balance_negative_unique_factorization_resultchosen_factorizationreconstructoutputimaginary. (((((ge_representation_imaginary_code_unique_factorization_resultchosen_factorizationreconstructoutput) = 2 * (ge_balance_positive_unique_factorization_resultchosen_factorizationreconstructoutputimaginary) /\ (ge_balance_negative_unique_factorization_resultchosen_factorizationreconstructoutputimaginary) = 0) \/ exists ge_signed_half_unique_factorization_resultchosen_factorizationreconstructoutputimaginarydecode. (((ge_representation_imaginary_code_unique_factorization_resultchosen_factorizationreconstructoutput) = 2 * ge_signed_half_unique_factorization_resultchosen_factorizationreconstructoutputimaginarydecode + 1 /\ (ge_balance_positive_unique_factorization_resultchosen_factorizationreconstructoutputimaginary) = 0) /\ (ge_balance_negative_unique_factorization_resultchosen_factorizationreconstructoutputimaginary) = S ge_signed_half_unique_factorization_resultchosen_factorizationreconstructoutputimaginarydecode))) /\ ((((((((ge_first_rp_unique_factorization_resultchosen_factorizationreconstruct) * (ge_second_ip_unique_factorization_resultchosen_factorizationreconstruct))) + (((ge_first_rn_unique_factorization_resultchosen_factorizationreconstruct) * (ge_second_in_unique_factorization_resultchosen_factorizationreconstruct))))) + (((((ge_first_ip_unique_factorization_resultchosen_factorizationreconstruct) * (ge_second_rp_unique_factorization_resultchosen_factorizationreconstruct))) + (((ge_first_in_unique_factorization_resultchosen_factorizationreconstruct) * (ge_second_rn_unique_factorization_resultchosen_factorizationreconstruct))))))) + ge_balance_negative_unique_factorization_resultchosen_factorizationreconstructoutputimaginary = (((((((ge_first_rp_unique_factorization_resultchosen_factorizationreconstruct) * (ge_second_in_unique_factorization_resultchosen_factorizationreconstruct))) + (((ge_first_rn_unique_factorization_resultchosen_factorizationreconstruct) * (ge_second_ip_unique_factorization_resultchosen_factorizationreconstruct))))) + (((((ge_first_ip_unique_factorization_resultchosen_factorizationreconstruct) * (ge_second_rn_unique_factorization_resultchosen_factorizationreconstruct))) + (((ge_first_in_unique_factorization_resultchosen_factorizationreconstruct) * (ge_second_rp_unique_factorization_resultchosen_factorizationreconstruct))))))) + ge_balance_positive_unique_factorization_resultchosen_factorizationreconstructoutputimaginary)))))))))))))) /\ (forall gr_other_unit_unique_factorization_result gr_other_code_unique_factorization_result gr_other_scale_unique_factorization_result gr_other_length_unique_factorization_result. (((exists gr_inverse_unique_factorization_resultother_factorizationunit. (exists ge_first_rp_unique_factorization_resultother_factorizationunitidentity ge_first_rn_unique_factorization_resultother_factorizationunitidentity ge_first_ip_unique_factorization_resultother_factorizationunitidentity ge_first_in_unique_factorization_resultother_factorizationunitidentity ge_second_rp_unique_factorization_resultother_factorizationunitidentity ge_second_rn_unique_factorization_resultother_factorizationunitidentity ge_second_ip_unique_factorization_resultother_factorizationunitidentity ge_second_in_unique_factorization_resultother_factorizationunitidentity. ((exists ge_representation_real_code_unique_factorization_resultother_factorizationunitidentityfirst ge_representation_imaginary_code_unique_factorization_resultother_factorizationunitidentityfirst. (((gr_other_unit_unique_factorization_result) = ((ge_representation_real_code_unique_factorization_resultother_factorizationunitidentityfirst) + (ge_representation_imaginary_code_unique_factorization_resultother_factorizationunitidentityfirst)) * S ((ge_representation_real_code_unique_factorization_resultother_factorizationunitidentityfirst) + (ge_representation_imaginary_code_unique_factorization_resultother_factorizationunitidentityfirst)) + ((ge_representation_imaginary_code_unique_factorization_resultother_factorizationunitidentityfirst) + (ge_representation_imaginary_code_unique_factorization_resultother_factorizationunitidentityfirst))) /\ ((exists ge_balance_positive_unique_factorization_resultother_factorizationunitidentityfirstreal ge_balance_negative_unique_factorization_resultother_factorizationunitidentityfirstreal. (((((ge_representation_real_code_unique_factorization_resultother_factorizationunitidentityfirst) = 2 * (ge_balance_positive_unique_factorization_resultother_factorizationunitidentityfirstreal) /\ (ge_balance_negative_unique_factorization_resultother_factorizationunitidentityfirstreal) = 0) \/ exists ge_signed_half_unique_factorization_resultother_factorizationunitidentityfirstrealdecode. (((ge_representation_real_code_unique_factorization_resultother_factorizationunitidentityfirst) = 2 * ge_signed_half_unique_factorization_resultother_factorizationunitidentityfirstrealdecode + 1 /\ (ge_balance_positive_unique_factorization_resultother_factorizationunitidentityfirstreal) = 0) /\ (ge_balance_negative_unique_factorization_resultother_factorizationunitidentityfirstreal) = S ge_signed_half_unique_factorization_resultother_factorizationunitidentityfirstrealdecode))) /\ ((ge_first_rp_unique_factorization_resultother_factorizationunitidentity) + ge_balance_negative_unique_factorization_resultother_factorizationunitidentityfirstreal = (ge_first_rn_unique_factorization_resultother_factorizationunitidentity) + ge_balance_positive_unique_factorization_resultother_factorizationunitidentityfirstreal))) /\ (exists ge_balance_positive_unique_factorization_resultother_factorizationunitidentityfirstimaginary ge_balance_negative_unique_factorization_resultother_factorizationunitidentityfirstimaginary. (((((ge_representation_imaginary_code_unique_factorization_resultother_factorizationunitidentityfirst) = 2 * (ge_balance_positive_unique_factorization_resultother_factorizationunitidentityfirstimaginary) /\ (ge_balance_negative_unique_factorization_resultother_factorizationunitidentityfirstimaginary) = 0) \/ exists ge_signed_half_unique_factorization_resultother_factorizationunitidentityfirstimaginarydecode. (((ge_representation_imaginary_code_unique_factorization_resultother_factorizationunitidentityfirst) = 2 * ge_signed_half_unique_factorization_resultother_factorizationunitidentityfirstimaginarydecode + 1 /\ (ge_balance_positive_unique_factorization_resultother_factorizationunitidentityfirstimaginary) = 0) /\ (ge_balance_negative_unique_factorization_resultother_factorizationunitidentityfirstimaginary) = S ge_signed_half_unique_factorization_resultother_factorizationunitidentityfirstimaginarydecode))) /\ ((ge_first_ip_unique_factorization_resultother_factorizationunitidentity) + ge_balance_negative_unique_factorization_resultother_factorizationunitidentityfirstimaginary = (ge_first_in_unique_factorization_resultother_factorizationunitidentity) + ge_balance_positive_unique_factorization_resultother_factorizationunitidentityfirstimaginary)))))) /\ ((exists ge_representation_real_code_unique_factorization_resultother_factorizationunitidentitysecond ge_representation_imaginary_code_unique_factorization_resultother_factorizationunitidentitysecond. (((gr_inverse_unique_factorization_resultother_factorizationunit) = ((ge_representation_real_code_unique_factorization_resultother_factorizationunitidentitysecond) + (ge_representation_imaginary_code_unique_factorization_resultother_factorizationunitidentitysecond)) * S ((ge_representation_real_code_unique_factorization_resultother_factorizationunitidentitysecond) + (ge_representation_imaginary_code_unique_factorization_resultother_factorizationunitidentitysecond)) + ((ge_representation_imaginary_code_unique_factorization_resultother_factorizationunitidentitysecond) + (ge_representation_imaginary_code_unique_factorization_resultother_factorizationunitidentitysecond))) /\ ((exists ge_balance_positive_unique_factorization_resultother_factorizationunitidentitysecondreal ge_balance_negative_unique_factorization_resultother_factorizationunitidentitysecondreal. (((((ge_representation_real_code_unique_factorization_resultother_factorizationunitidentitysecond) = 2 * (ge_balance_positive_unique_factorization_resultother_factorizationunitidentitysecondreal) /\ (ge_balance_negative_unique_factorization_resultother_factorizationunitidentitysecondreal) = 0) \/ exists ge_signed_half_unique_factorization_resultother_factorizationunitidentitysecondrealdecode. (((ge_representation_real_code_unique_factorization_resultother_factorizationunitidentitysecond) = 2 * ge_signed_half_unique_factorization_resultother_factorizationunitidentitysecondrealdecode + 1 /\ (ge_balance_positive_unique_factorization_resultother_factorizationunitidentitysecondreal) = 0) /\ (ge_balance_negative_unique_factorization_resultother_factorizationunitidentitysecondreal) = S ge_signed_half_unique_factorization_resultother_factorizationunitidentitysecondrealdecode))) /\ ((ge_second_rp_unique_factorization_resultother_factorizationunitidentity) + ge_balance_negative_unique_factorization_resultother_factorizationunitidentitysecondreal = (ge_second_rn_unique_factorization_resultother_factorizationunitidentity) + ge_balance_positive_unique_factorization_resultother_factorizationunitidentitysecondreal))) /\ (exists ge_balance_positive_unique_factorization_resultother_factorizationunitidentitysecondimaginary ge_balance_negative_unique_factorization_resultother_factorizationunitidentitysecondimaginary. (((((ge_representation_imaginary_code_unique_factorization_resultother_factorizationunitidentitysecond) = 2 * (ge_balance_positive_unique_factorization_resultother_factorizationunitidentitysecondimaginary) /\ (ge_balance_negative_unique_factorization_resultother_factorizationunitidentitysecondimaginary) = 0) \/ exists ge_signed_half_unique_factorization_resultother_factorizationunitidentitysecondimaginarydecode. (((ge_representation_imaginary_code_unique_factorization_resultother_factorizationunitidentitysecond) = 2 * ge_signed_half_unique_factorization_resultother_factorizationunitidentitysecondimaginarydecode + 1 /\ (ge_balance_positive_unique_factorization_resultother_factorizationunitidentitysecondimaginary) = 0) /\ (ge_balance_negative_unique_factorization_resultother_factorizationunitidentitysecondimaginary) = S ge_signed_half_unique_factorization_resultother_factorizationunitidentitysecondimaginarydecode))) /\ ((ge_second_ip_unique_factorization_resultother_factorizationunitidentity) + ge_balance_negative_unique_factorization_resultother_factorizationunitidentitysecondimaginary = (ge_second_in_unique_factorization_resultother_factorizationunitidentity) + ge_balance_positive_unique_factorization_resultother_factorizationunitidentitysecondimaginary)))))) /\ (exists ge_representation_real_code_unique_factorization_resultother_factorizationunitidentityoutput ge_representation_imaginary_code_unique_factorization_resultother_factorizationunitidentityoutput. (((6) = ((ge_representation_real_code_unique_factorization_resultother_factorizationunitidentityoutput) + (ge_representation_imaginary_code_unique_factorization_resultother_factorizationunitidentityoutput)) * S ((ge_representation_real_code_unique_factorization_resultother_factorizationunitidentityoutput) + (ge_representation_imaginary_code_unique_factorization_resultother_factorizationunitidentityoutput)) + ((ge_representation_imaginary_code_unique_factorization_resultother_factorizationunitidentityoutput) + (ge_representation_imaginary_code_unique_factorization_resultother_factorizationunitidentityoutput))) /\ ((exists ge_balance_positive_unique_factorization_resultother_factorizationunitidentityoutputreal ge_balance_negative_unique_factorization_resultother_factorizationunitidentityoutputreal. (((((ge_representation_real_code_unique_factorization_resultother_factorizationunitidentityoutput) = 2 * (ge_balance_positive_unique_factorization_resultother_factorizationunitidentityoutputreal) /\ (ge_balance_negative_unique_factorization_resultother_factorizationunitidentityoutputreal) = 0) \/ exists ge_signed_half_unique_factorization_resultother_factorizationunitidentityoutputrealdecode. (((ge_representation_real_code_unique_factorization_resultother_factorizationunitidentityoutput) = 2 * ge_signed_half_unique_factorization_resultother_factorizationunitidentityoutputrealdecode + 1 /\ (ge_balance_positive_unique_factorization_resultother_factorizationunitidentityoutputreal) = 0) /\ (ge_balance_negative_unique_factorization_resultother_factorizationunitidentityoutputreal) = S ge_signed_half_unique_factorization_resultother_factorizationunitidentityoutputrealdecode))) /\ ((((((((ge_first_rp_unique_factorization_resultother_factorizationunitidentity) * (ge_second_rp_unique_factorization_resultother_factorizationunitidentity))) + (((ge_first_rn_unique_factorization_resultother_factorizationunitidentity) * (ge_second_rn_unique_factorization_resultother_factorizationunitidentity))))) + (((((ge_first_ip_unique_factorization_resultother_factorizationunitidentity) * (ge_second_in_unique_factorization_resultother_factorizationunitidentity))) + (((ge_first_in_unique_factorization_resultother_factorizationunitidentity) * (ge_second_ip_unique_factorization_resultother_factorizationunitidentity))))))) + ge_balance_negative_unique_factorization_resultother_factorizationunitidentityoutputreal = (((((((ge_first_rp_unique_factorization_resultother_factorizationunitidentity) * (ge_second_rn_unique_factorization_resultother_factorizationunitidentity))) + (((ge_first_rn_unique_factorization_resultother_factorizationunitidentity) * (ge_second_rp_unique_factorization_resultother_factorizationunitidentity))))) + (((((ge_first_ip_unique_factorization_resultother_factorizationunitidentity) * (ge_second_ip_unique_factorization_resultother_factorizationunitidentity))) + (((ge_first_in_unique_factorization_resultother_factorizationunitidentity) * (ge_second_in_unique_factorization_resultother_factorizationunitidentity))))))) + ge_balance_positive_unique_factorization_resultother_factorizationunitidentityoutputreal))) /\ (exists ge_balance_positive_unique_factorization_resultother_factorizationunitidentityoutputimaginary ge_balance_negative_unique_factorization_resultother_factorizationunitidentityoutputimaginary. (((((ge_representation_imaginary_code_unique_factorization_resultother_factorizationunitidentityoutput) = 2 * (ge_balance_positive_unique_factorization_resultother_factorizationunitidentityoutputimaginary) /\ (ge_balance_negative_unique_factorization_resultother_factorizationunitidentityoutputimaginary) = 0) \/ exists ge_signed_half_unique_factorization_resultother_factorizationunitidentityoutputimaginarydecode. (((ge_representation_imaginary_code_unique_factorization_resultother_factorizationunitidentityoutput) = 2 * ge_signed_half_unique_factorization_resultother_factorizationunitidentityoutputimaginarydecode + 1 /\ (ge_balance_positive_unique_factorization_resultother_factorizationunitidentityoutputimaginary) = 0) /\ (ge_balance_negative_unique_factorization_resultother_factorizationunitidentityoutputimaginary) = S ge_signed_half_unique_factorization_resultother_factorizationunitidentityoutputimaginarydecode))) /\ ((((((((ge_first_rp_unique_factorization_resultother_factorizationunitidentity) * (ge_second_ip_unique_factorization_resultother_factorizationunitidentity))) + (((ge_first_rn_unique_factorization_resultother_factorizationunitidentity) * (ge_second_in_unique_factorization_resultother_factorizationunitidentity))))) + (((((ge_first_ip_unique_factorization_resultother_factorizationunitidentity) * (ge_second_rp_unique_factorization_resultother_factorizationunitidentity))) + (((ge_first_in_unique_factorization_resultother_factorizationunitidentity) * (ge_second_rn_unique_factorization_resultother_factorizationunitidentity))))))) + ge_balance_negative_unique_factorization_resultother_factorizationunitidentityoutputimaginary = (((((((ge_first_rp_unique_factorization_resultother_factorizationunitidentity) * (ge_second_in_unique_factorization_resultother_factorizationunitidentity))) + (((ge_first_rn_unique_factorization_resultother_factorizationunitidentity) * (ge_second_ip_unique_factorization_resultother_factorizationunitidentity))))) + (((((ge_first_ip_unique_factorization_resultother_factorizationunitidentity) * (ge_second_rn_unique_factorization_resultother_factorizationunitidentity))) + (((ge_first_in_unique_factorization_resultother_factorizationunitidentity) * (ge_second_rp_unique_factorization_resultother_factorizationunitidentity))))))) + ge_balance_positive_unique_factorization_resultother_factorizationunitidentityoutputimaginary)))))))))) /\ ((forall gr_prime_factor_index_unique_factorization_resultother_factorizationprimes gr_prime_factor_value_unique_factorization_resultother_factorizationprimes. (exists ge_gap_unique_factorization_resultother_factorizationprimesindex. ge_gap_unique_factorization_resultother_factorizationprimesindex + S (gr_prime_factor_index_unique_factorization_resultother_factorizationprimes) = (gr_other_length_unique_factorization_result)) -> (((exists ff_h_gprod_unique_factorization_resultother_factorizationprimesentry. ff_h_gprod_unique_factorization_resultother_factorizationprimesentry + S (gr_prime_factor_value_unique_factorization_resultother_factorizationprimes) = S ((S (gr_prime_factor_index_unique_factorization_resultother_factorizationprimes)) * gr_other_scale_unique_factorization_result)) /\ exists ff_q_gprod_unique_factorization_resultother_factorizationprimesentry. gr_other_code_unique_factorization_result = ff_q_gprod_unique_factorization_resultother_factorizationprimesentry * S ((S (gr_prime_factor_index_unique_factorization_resultother_factorizationprimes)) * gr_other_scale_unique_factorization_result) + (gr_prime_factor_value_unique_factorization_resultother_factorizationprimes))) -> (((exists ge_real_positive_unique_factorization_resultother_factorizationprimesprimecarrier ge_real_negative_unique_factorization_resultother_factorizationprimesprimecarrier ge_imaginary_positive_unique_factorization_resultother_factorizationprimesprimecarrier ge_imaginary_negative_unique_factorization_resultother_factorizationprimesprimecarrier. (exists ge_real_code_unique_factorization_resultother_factorizationprimesprimecarrierdecode ge_imaginary_code_unique_factorization_resultother_factorizationprimesprimecarrierdecode. (((gr_prime_factor_value_unique_factorization_resultother_factorizationprimes) = ((ge_real_code_unique_factorization_resultother_factorizationprimesprimecarrierdecode) + (ge_imaginary_code_unique_factorization_resultother_factorizationprimesprimecarrierdecode)) * S ((ge_real_code_unique_factorization_resultother_factorizationprimesprimecarrierdecode) + (ge_imaginary_code_unique_factorization_resultother_factorizationprimesprimecarrierdecode)) + ((ge_imaginary_code_unique_factorization_resultother_factorizationprimesprimecarrierdecode) + (ge_imaginary_code_unique_factorization_resultother_factorizationprimesprimecarrierdecode))) /\ (((((ge_real_code_unique_factorization_resultother_factorizationprimesprimecarrierdecode) = 2 * (ge_real_positive_unique_factorization_resultother_factorizationprimesprimecarrier) /\ (ge_real_negative_unique_factorization_resultother_factorizationprimesprimecarrier) = 0) \/ exists ge_signed_half_ge_unique_factorization_resultother_factorizationprimesprimecarrierdecode_real. (((ge_real_code_unique_factorization_resultother_factorizationprimesprimecarrierdecode) = 2 * ge_signed_half_ge_unique_factorization_resultother_factorizationprimesprimecarrierdecode_real + 1 /\ (ge_real_positive_unique_factorization_resultother_factorizationprimesprimecarrier) = 0) /\ (ge_real_negative_unique_factorization_resultother_factorizationprimesprimecarrier) = S ge_signed_half_ge_unique_factorization_resultother_factorizationprimesprimecarrierdecode_real))) /\ ((((ge_imaginary_code_unique_factorization_resultother_factorizationprimesprimecarrierdecode) = 2 * (ge_imaginary_positive_unique_factorization_resultother_factorizationprimesprimecarrier) /\ (ge_imaginary_negative_unique_factorization_resultother_factorizationprimesprimecarrier) = 0) \/ exists ge_signed_half_ge_unique_factorization_resultother_factorizationprimesprimecarrierdecode_imaginary. (((ge_imaginary_code_unique_factorization_resultother_factorizationprimesprimecarrierdecode) = 2 * ge_signed_half_ge_unique_factorization_resultother_factorizationprimesprimecarrierdecode_imaginary + 1 /\ (ge_imaginary_positive_unique_factorization_resultother_factorizationprimesprimecarrier) = 0) /\ (ge_imaginary_negative_unique_factorization_resultother_factorizationprimesprimecarrier) = S ge_signed_half_ge_unique_factorization_resultother_factorizationprimesprimecarrierdecode_imaginary))))))) /\ ((~((gr_prime_factor_value_unique_factorization_resultother_factorizationprimes)=0)) /\ ((~(exists gr_inverse_unique_factorization_resultother_factorizationprimesprimenonunit. (exists ge_first_rp_unique_factorization_resultother_factorizationprimesprimenonunitidentity ge_first_rn_unique_factorization_resultother_factorizationprimesprimenonunitidentity ge_first_ip_unique_factorization_resultother_factorizationprimesprimenonunitidentity ge_first_in_unique_factorization_resultother_factorizationprimesprimenonunitidentity ge_second_rp_unique_factorization_resultother_factorizationprimesprimenonunitidentity ge_second_rn_unique_factorization_resultother_factorizationprimesprimenonunitidentity ge_second_ip_unique_factorization_resultother_factorizationprimesprimenonunitidentity ge_second_in_unique_factorization_resultother_factorizationprimesprimenonunitidentity. ((exists ge_representation_real_code_unique_factorization_resultother_factorizationprimesprimenonunitidentityfirst ge_representation_imaginary_code_unique_factorization_resultother_factorizationprimesprimenonunitidentityfirst. (((gr_prime_factor_value_unique_factorization_resultother_factorizationprimes) = ((ge_representation_real_code_unique_factorization_resultother_factorizationprimesprimenonunitidentityfirst) + (ge_representation_imaginary_code_unique_factorization_resultother_factorizationprimesprimenonunitidentityfirst)) * S ((ge_representation_real_code_unique_factorization_resultother_factorizationprimesprimenonunitidentityfirst) + (ge_representation_imaginary_code_unique_factorization_resultother_factorizationprimesprimenonunitidentityfirst)) + ((ge_representation_imaginary_code_unique_factorization_resultother_factorizationprimesprimenonunitidentityfirst) + (ge_representation_imaginary_code_unique_factorization_resultother_factorizationprimesprimenonunitidentityfirst))) /\ ((exists ge_balance_positive_unique_factorization_resultother_factorizationprimesprimenonunitidentityfirstreal ge_balance_negative_unique_factorization_resultother_factorizationprimesprimenonunitidentityfirstreal. (((((ge_representation_real_code_unique_factorization_resultother_factorizationprimesprimenonunitidentityfirst) = 2 * (ge_balance_positive_unique_factorization_resultother_factorizationprimesprimenonunitidentityfirstreal) /\ (ge_balance_negative_unique_factorization_resultother_factorizationprimesprimenonunitidentityfirstreal) = 0) \/ exists ge_signed_half_unique_factorization_resultother_factorizationprimesprimenonunitidentityfirstrealdecode. (((ge_representation_real_code_unique_factorization_resultother_factorizationprimesprimenonunitidentityfirst) = 2 * ge_signed_half_unique_factorization_resultother_factorizationprimesprimenonunitidentityfirstrealdecode + 1 /\ (ge_balance_positive_unique_factorization_resultother_factorizationprimesprimenonunitidentityfirstreal) = 0) /\ (ge_balance_negative_unique_factorization_resultother_factorizationprimesprimenonunitidentityfirstreal) = S ge_signed_half_unique_factorization_resultother_factorizationprimesprimenonunitidentityfirstrealdecode))) /\ ((ge_first_rp_unique_factorization_resultother_factorizationprimesprimenonunitidentity) + ge_balance_negative_unique_factorization_resultother_factorizationprimesprimenonunitidentityfirstreal = (ge_first_rn_unique_factorization_resultother_factorizationprimesprimenonunitidentity) + ge_balance_positive_unique_factorization_resultother_factorizationprimesprimenonunitidentityfirstreal))) /\ (exists ge_balance_positive_unique_factorization_resultother_factorizationprimesprimenonunitidentityfirstimaginary ge_balance_negative_unique_factorization_resultother_factorizationprimesprimenonunitidentityfirstimaginary. (((((ge_representation_imaginary_code_unique_factorization_resultother_factorizationprimesprimenonunitidentityfirst) = 2 * (ge_balance_positive_unique_factorization_resultother_factorizationprimesprimenonunitidentityfirstimaginary) /\ (ge_balance_negative_unique_factorization_resultother_factorizationprimesprimenonunitidentityfirstimaginary) = 0) \/ exists ge_signed_half_unique_factorization_resultother_factorizationprimesprimenonunitidentityfirstimaginarydecode. (((ge_representation_imaginary_code_unique_factorization_resultother_factorizationprimesprimenonunitidentityfirst) = 2 * ge_signed_half_unique_factorization_resultother_factorizationprimesprimenonunitidentityfirstimaginarydecode + 1 /\ (ge_balance_positive_unique_factorization_resultother_factorizationprimesprimenonunitidentityfirstimaginary) = 0) /\ (ge_balance_negative_unique_factorization_resultother_factorizationprimesprimenonunitidentityfirstimaginary) = S ge_signed_half_unique_factorization_resultother_factorizationprimesprimenonunitidentityfirstimaginarydecode))) /\ ((ge_first_ip_unique_factorization_resultother_factorizationprimesprimenonunitidentity) + ge_balance_negative_unique_factorization_resultother_factorizationprimesprimenonunitidentityfirstimaginary = (ge_first_in_unique_factorization_resultother_factorizationprimesprimenonunitidentity) + ge_balance_positive_unique_factorization_resultother_factorizationprimesprimenonunitidentityfirstimaginary)))))) /\ ((exists ge_representation_real_code_unique_factorization_resultother_factorizationprimesprimenonunitidentitysecond ge_representation_imaginary_code_unique_factorization_resultother_factorizationprimesprimenonunitidentitysecond. (((gr_inverse_unique_factorization_resultother_factorizationprimesprimenonunit) = ((ge_representation_real_code_unique_factorization_resultother_factorizationprimesprimenonunitidentitysecond) + (ge_representation_imaginary_code_unique_factorization_resultother_factorizationprimesprimenonunitidentitysecond)) * S ((ge_representation_real_code_unique_factorization_resultother_factorizationprimesprimenonunitidentitysecond) + (ge_representation_imaginary_code_unique_factorization_resultother_factorizationprimesprimenonunitidentitysecond)) + ((ge_representation_imaginary_code_unique_factorization_resultother_factorizationprimesprimenonunitidentitysecond) + (ge_representation_imaginary_code_unique_factorization_resultother_factorizationprimesprimenonunitidentitysecond))) /\ ((exists ge_balance_positive_unique_factorization_resultother_factorizationprimesprimenonunitidentitysecondreal ge_balance_negative_unique_factorization_resultother_factorizationprimesprimenonunitidentitysecondreal. (((((ge_representation_real_code_unique_factorization_resultother_factorizationprimesprimenonunitidentitysecond) = 2 * (ge_balance_positive_unique_factorization_resultother_factorizationprimesprimenonunitidentitysecondreal) /\ (ge_balance_negative_unique_factorization_resultother_factorizationprimesprimenonunitidentitysecondreal) = 0) \/ exists ge_signed_half_unique_factorization_resultother_factorizationprimesprimenonunitidentitysecondrealdecode. (((ge_representation_real_code_unique_factorization_resultother_factorizationprimesprimenonunitidentitysecond) = 2 * ge_signed_half_unique_factorization_resultother_factorizationprimesprimenonunitidentitysecondrealdecode + 1 /\ (ge_balance_positive_unique_factorization_resultother_factorizationprimesprimenonunitidentitysecondreal) = 0) /\ (ge_balance_negative_unique_factorization_resultother_factorizationprimesprimenonunitidentitysecondreal) = S ge_signed_half_unique_factorization_resultother_factorizationprimesprimenonunitidentitysecondrealdecode))) /\ ((ge_second_rp_unique_factorization_resultother_factorizationprimesprimenonunitidentity) + ge_balance_negative_unique_factorization_resultother_factorizationprimesprimenonunitidentitysecondreal = (ge_second_rn_unique_factorization_resultother_factorizationprimesprimenonunitidentity) + ge_balance_positive_unique_factorization_resultother_factorizationprimesprimenonunitidentitysecondreal))) /\ (exists ge_balance_positive_unique_factorization_resultother_factorizationprimesprimenonunitidentitysecondimaginary ge_balance_negative_unique_factorization_resultother_factorizationprimesprimenonunitidentitysecondimaginary. (((((ge_representation_imaginary_code_unique_factorization_resultother_factorizationprimesprimenonunitidentitysecond) = 2 * (ge_balance_positive_unique_factorization_resultother_factorizationprimesprimenonunitidentitysecondimaginary) /\ (ge_balance_negative_unique_factorization_resultother_factorizationprimesprimenonunitidentitysecondimaginary) = 0) \/ exists ge_signed_half_unique_factorization_resultother_factorizationprimesprimenonunitidentitysecondimaginarydecode. (((ge_representation_imaginary_code_unique_factorization_resultother_factorizationprimesprimenonunitidentitysecond) = 2 * ge_signed_half_unique_factorization_resultother_factorizationprimesprimenonunitidentitysecondimaginarydecode + 1 /\ (ge_balance_positive_unique_factorization_resultother_factorizationprimesprimenonunitidentitysecondimaginary) = 0) /\ (ge_balance_negative_unique_factorization_resultother_factorizationprimesprimenonunitidentitysecondimaginary) = S ge_signed_half_unique_factorization_resultother_factorizationprimesprimenonunitidentitysecondimaginarydecode))) /\ ((ge_second_ip_unique_factorization_resultother_factorizationprimesprimenonunitidentity) + ge_balance_negative_unique_factorization_resultother_factorizationprimesprimenonunitidentitysecondimaginary = (ge_second_in_unique_factorization_resultother_factorizationprimesprimenonunitidentity) + ge_balance_positive_unique_factorization_resultother_factorizationprimesprimenonunitidentitysecondimaginary)))))) /\ (exists ge_representation_real_code_unique_factorization_resultother_factorizationprimesprimenonunitidentityoutput ge_representation_imaginary_code_unique_factorization_resultother_factorizationprimesprimenonunitidentityoutput. (((6) = ((ge_representation_real_code_unique_factorization_resultother_factorizationprimesprimenonunitidentityoutput) + (ge_representation_imaginary_code_unique_factorization_resultother_factorizationprimesprimenonunitidentityoutput)) * S ((ge_representation_real_code_unique_factorization_resultother_factorizationprimesprimenonunitidentityoutput) + (ge_representation_imaginary_code_unique_factorization_resultother_factorizationprimesprimenonunitidentityoutput)) + ((ge_representation_imaginary_code_unique_factorization_resultother_factorizationprimesprimenonunitidentityoutput) + (ge_representation_imaginary_code_unique_factorization_resultother_factorizationprimesprimenonunitidentityoutput))) /\ ((exists ge_balance_positive_unique_factorization_resultother_factorizationprimesprimenonunitidentityoutputreal ge_balance_negative_unique_factorization_resultother_factorizationprimesprimenonunitidentityoutputreal. (((((ge_representation_real_code_unique_factorization_resultother_factorizationprimesprimenonunitidentityoutput) = 2 * (ge_balance_positive_unique_factorization_resultother_factorizationprimesprimenonunitidentityoutputreal) /\ (ge_balance_negative_unique_factorization_resultother_factorizationprimesprimenonunitidentityoutputreal) = 0) \/ exists ge_signed_half_unique_factorization_resultother_factorizationprimesprimenonunitidentityoutputrealdecode. (((ge_representation_real_code_unique_factorization_resultother_factorizationprimesprimenonunitidentityoutput) = 2 * ge_signed_half_unique_factorization_resultother_factorizationprimesprimenonunitidentityoutputrealdecode + 1 /\ (ge_balance_positive_unique_factorization_resultother_factorizationprimesprimenonunitidentityoutputreal) = 0) /\ (ge_balance_negative_unique_factorization_resultother_factorizationprimesprimenonunitidentityoutputreal) = S ge_signed_half_unique_factorization_resultother_factorizationprimesprimenonunitidentityoutputrealdecode))) /\ ((((((((ge_first_rp_unique_factorization_resultother_factorizationprimesprimenonunitidentity) * (ge_second_rp_unique_factorization_resultother_factorizationprimesprimenonunitidentity))) + (((ge_first_rn_unique_factorization_resultother_factorizationprimesprimenonunitidentity) * (ge_second_rn_unique_factorization_resultother_factorizationprimesprimenonunitidentity))))) + (((((ge_first_ip_unique_factorization_resultother_factorizationprimesprimenonunitidentity) * (ge_second_in_unique_factorization_resultother_factorizationprimesprimenonunitidentity))) + (((ge_first_in_unique_factorization_resultother_factorizationprimesprimenonunitidentity) * (ge_second_ip_unique_factorization_resultother_factorizationprimesprimenonunitidentity))))))) + ge_balance_negative_unique_factorization_resultother_factorizationprimesprimenonunitidentityoutputreal = (((((((ge_first_rp_unique_factorization_resultother_factorizationprimesprimenonunitidentity) * (ge_second_rn_unique_factorization_resultother_factorizationprimesprimenonunitidentity))) + (((ge_first_rn_unique_factorization_resultother_factorizationprimesprimenonunitidentity) * (ge_second_rp_unique_factorization_resultother_factorizationprimesprimenonunitidentity))))) + (((((ge_first_ip_unique_factorization_resultother_factorizationprimesprimenonunitidentity) * (ge_second_ip_unique_factorization_resultother_factorizationprimesprimenonunitidentity))) + (((ge_first_in_unique_factorization_resultother_factorizationprimesprimenonunitidentity) * (ge_second_in_unique_factorization_resultother_factorizationprimesprimenonunitidentity))))))) + ge_balance_positive_unique_factorization_resultother_factorizationprimesprimenonunitidentityoutputreal))) /\ (exists ge_balance_positive_unique_factorization_resultother_factorizationprimesprimenonunitidentityoutputimaginary ge_balance_negative_unique_factorization_resultother_factorizationprimesprimenonunitidentityoutputimaginary. (((((ge_representation_imaginary_code_unique_factorization_resultother_factorizationprimesprimenonunitidentityoutput) = 2 * (ge_balance_positive_unique_factorization_resultother_factorizationprimesprimenonunitidentityoutputimaginary) /\ (ge_balance_negative_unique_factorization_resultother_factorizationprimesprimenonunitidentityoutputimaginary) = 0) \/ exists ge_signed_half_unique_factorization_resultother_factorizationprimesprimenonunitidentityoutputimaginarydecode. (((ge_representation_imaginary_code_unique_factorization_resultother_factorizationprimesprimenonunitidentityoutput) = 2 * ge_signed_half_unique_factorization_resultother_factorizationprimesprimenonunitidentityoutputimaginarydecode + 1 /\ (ge_balance_positive_unique_factorization_resultother_factorizationprimesprimenonunitidentityoutputimaginary) = 0) /\ (ge_balance_negative_unique_factorization_resultother_factorizationprimesprimenonunitidentityoutputimaginary) = S ge_signed_half_unique_factorization_resultother_factorizationprimesprimenonunitidentityoutputimaginarydecode))) /\ ((((((((ge_first_rp_unique_factorization_resultother_factorizationprimesprimenonunitidentity) * (ge_second_ip_unique_factorization_resultother_factorizationprimesprimenonunitidentity))) + (((ge_first_rn_unique_factorization_resultother_factorizationprimesprimenonunitidentity) * (ge_second_in_unique_factorization_resultother_factorizationprimesprimenonunitidentity))))) + (((((ge_first_ip_unique_factorization_resultother_factorizationprimesprimenonunitidentity) * (ge_second_rp_unique_factorization_resultother_factorizationprimesprimenonunitidentity))) + (((ge_first_in_unique_factorization_resultother_factorizationprimesprimenonunitidentity) * (ge_second_rn_unique_factorization_resultother_factorizationprimesprimenonunitidentity))))))) + ge_balance_negative_unique_factorization_resultother_factorizationprimesprimenonunitidentityoutputimaginary = (((((((ge_first_rp_unique_factorization_resultother_factorizationprimesprimenonunitidentity) * (ge_second_in_unique_factorization_resultother_factorizationprimesprimenonunitidentity))) + (((ge_first_rn_unique_factorization_resultother_factorizationprimesprimenonunitidentity) * (ge_second_ip_unique_factorization_resultother_factorizationprimesprimenonunitidentity))))) + (((((ge_first_ip_unique_factorization_resultother_factorizationprimesprimenonunitidentity) * (ge_second_rn_unique_factorization_resultother_factorizationprimesprimenonunitidentity))) + (((ge_first_in_unique_factorization_resultother_factorizationprimesprimenonunitidentity) * (ge_second_rp_unique_factorization_resultother_factorizationprimesprimenonunitidentity))))))) + ge_balance_positive_unique_factorization_resultother_factorizationprimesprimenonunitidentityoutputimaginary))))))))))) /\ (forall gr_first_factor_unique_factorization_resultother_factorizationprimesprime gr_second_factor_unique_factorization_resultother_factorizationprimesprime gr_product_unique_factorization_resultother_factorizationprimesprime. (exists ge_first_rp_unique_factorization_resultother_factorizationprimesprimeproduct ge_first_rn_unique_factorization_resultother_factorizationprimesprimeproduct ge_first_ip_unique_factorization_resultother_factorizationprimesprimeproduct ge_first_in_unique_factorization_resultother_factorizationprimesprimeproduct ge_second_rp_unique_factorization_resultother_factorizationprimesprimeproduct ge_second_rn_unique_factorization_resultother_factorizationprimesprimeproduct ge_second_ip_unique_factorization_resultother_factorizationprimesprimeproduct ge_second_in_unique_factorization_resultother_factorizationprimesprimeproduct. ((exists ge_representation_real_code_unique_factorization_resultother_factorizationprimesprimeproductfirst ge_representation_imaginary_code_unique_factorization_resultother_factorizationprimesprimeproductfirst. (((gr_first_factor_unique_factorization_resultother_factorizationprimesprime) = ((ge_representation_real_code_unique_factorization_resultother_factorizationprimesprimeproductfirst) + (ge_representation_imaginary_code_unique_factorization_resultother_factorizationprimesprimeproductfirst)) * S ((ge_representation_real_code_unique_factorization_resultother_factorizationprimesprimeproductfirst) + (ge_representation_imaginary_code_unique_factorization_resultother_factorizationprimesprimeproductfirst)) + ((ge_representation_imaginary_code_unique_factorization_resultother_factorizationprimesprimeproductfirst) + (ge_representation_imaginary_code_unique_factorization_resultother_factorizationprimesprimeproductfirst))) /\ ((exists ge_balance_positive_unique_factorization_resultother_factorizationprimesprimeproductfirstreal ge_balance_negative_unique_factorization_resultother_factorizationprimesprimeproductfirstreal. (((((ge_representation_real_code_unique_factorization_resultother_factorizationprimesprimeproductfirst) = 2 * (ge_balance_positive_unique_factorization_resultother_factorizationprimesprimeproductfirstreal) /\ (ge_balance_negative_unique_factorization_resultother_factorizationprimesprimeproductfirstreal) = 0) \/ exists ge_signed_half_unique_factorization_resultother_factorizationprimesprimeproductfirstrealdecode. (((ge_representation_real_code_unique_factorization_resultother_factorizationprimesprimeproductfirst) = 2 * ge_signed_half_unique_factorization_resultother_factorizationprimesprimeproductfirstrealdecode + 1 /\ (ge_balance_positive_unique_factorization_resultother_factorizationprimesprimeproductfirstreal) = 0) /\ (ge_balance_negative_unique_factorization_resultother_factorizationprimesprimeproductfirstreal) = S ge_signed_half_unique_factorization_resultother_factorizationprimesprimeproductfirstrealdecode))) /\ ((ge_first_rp_unique_factorization_resultother_factorizationprimesprimeproduct) + ge_balance_negative_unique_factorization_resultother_factorizationprimesprimeproductfirstreal = (ge_first_rn_unique_factorization_resultother_factorizationprimesprimeproduct) + ge_balance_positive_unique_factorization_resultother_factorizationprimesprimeproductfirstreal))) /\ (exists ge_balance_positive_unique_factorization_resultother_factorizationprimesprimeproductfirstimaginary ge_balance_negative_unique_factorization_resultother_factorizationprimesprimeproductfirstimaginary. (((((ge_representation_imaginary_code_unique_factorization_resultother_factorizationprimesprimeproductfirst) = 2 * (ge_balance_positive_unique_factorization_resultother_factorizationprimesprimeproductfirstimaginary) /\ (ge_balance_negative_unique_factorization_resultother_factorizationprimesprimeproductfirstimaginary) = 0) \/ exists ge_signed_half_unique_factorization_resultother_factorizationprimesprimeproductfirstimaginarydecode. (((ge_representation_imaginary_code_unique_factorization_resultother_factorizationprimesprimeproductfirst) = 2 * ge_signed_half_unique_factorization_resultother_factorizationprimesprimeproductfirstimaginarydecode + 1 /\ (ge_balance_positive_unique_factorization_resultother_factorizationprimesprimeproductfirstimaginary) = 0) /\ (ge_balance_negative_unique_factorization_resultother_factorizationprimesprimeproductfirstimaginary) = S ge_signed_half_unique_factorization_resultother_factorizationprimesprimeproductfirstimaginarydecode))) /\ ((ge_first_ip_unique_factorization_resultother_factorizationprimesprimeproduct) + ge_balance_negative_unique_factorization_resultother_factorizationprimesprimeproductfirstimaginary = (ge_first_in_unique_factorization_resultother_factorizationprimesprimeproduct) + ge_balance_positive_unique_factorization_resultother_factorizationprimesprimeproductfirstimaginary)))))) /\ ((exists ge_representation_real_code_unique_factorization_resultother_factorizationprimesprimeproductsecond ge_representation_imaginary_code_unique_factorization_resultother_factorizationprimesprimeproductsecond. (((gr_second_factor_unique_factorization_resultother_factorizationprimesprime) = ((ge_representation_real_code_unique_factorization_resultother_factorizationprimesprimeproductsecond) + (ge_representation_imaginary_code_unique_factorization_resultother_factorizationprimesprimeproductsecond)) * S ((ge_representation_real_code_unique_factorization_resultother_factorizationprimesprimeproductsecond) + (ge_representation_imaginary_code_unique_factorization_resultother_factorizationprimesprimeproductsecond)) + ((ge_representation_imaginary_code_unique_factorization_resultother_factorizationprimesprimeproductsecond) + (ge_representation_imaginary_code_unique_factorization_resultother_factorizationprimesprimeproductsecond))) /\ ((exists ge_balance_positive_unique_factorization_resultother_factorizationprimesprimeproductsecondreal ge_balance_negative_unique_factorization_resultother_factorizationprimesprimeproductsecondreal. (((((ge_representation_real_code_unique_factorization_resultother_factorizationprimesprimeproductsecond) = 2 * (ge_balance_positive_unique_factorization_resultother_factorizationprimesprimeproductsecondreal) /\ (ge_balance_negative_unique_factorization_resultother_factorizationprimesprimeproductsecondreal) = 0) \/ exists ge_signed_half_unique_factorization_resultother_factorizationprimesprimeproductsecondrealdecode. (((ge_representation_real_code_unique_factorization_resultother_factorizationprimesprimeproductsecond) = 2 * ge_signed_half_unique_factorization_resultother_factorizationprimesprimeproductsecondrealdecode + 1 /\ (ge_balance_positive_unique_factorization_resultother_factorizationprimesprimeproductsecondreal) = 0) /\ (ge_balance_negative_unique_factorization_resultother_factorizationprimesprimeproductsecondreal) = S ge_signed_half_unique_factorization_resultother_factorizationprimesprimeproductsecondrealdecode))) /\ ((ge_second_rp_unique_factorization_resultother_factorizationprimesprimeproduct) + ge_balance_negative_unique_factorization_resultother_factorizationprimesprimeproductsecondreal = (ge_second_rn_unique_factorization_resultother_factorizationprimesprimeproduct) + ge_balance_positive_unique_factorization_resultother_factorizationprimesprimeproductsecondreal))) /\ (exists ge_balance_positive_unique_factorization_resultother_factorizationprimesprimeproductsecondimaginary ge_balance_negative_unique_factorization_resultother_factorizationprimesprimeproductsecondimaginary. (((((ge_representation_imaginary_code_unique_factorization_resultother_factorizationprimesprimeproductsecond) = 2 * (ge_balance_positive_unique_factorization_resultother_factorizationprimesprimeproductsecondimaginary) /\ (ge_balance_negative_unique_factorization_resultother_factorizationprimesprimeproductsecondimaginary) = 0) \/ exists ge_signed_half_unique_factorization_resultother_factorizationprimesprimeproductsecondimaginarydecode. (((ge_representation_imaginary_code_unique_factorization_resultother_factorizationprimesprimeproductsecond) = 2 * ge_signed_half_unique_factorization_resultother_factorizationprimesprimeproductsecondimaginarydecode + 1 /\ (ge_balance_positive_unique_factorization_resultother_factorizationprimesprimeproductsecondimaginary) = 0) /\ (ge_balance_negative_unique_factorization_resultother_factorizationprimesprimeproductsecondimaginary) = S ge_signed_half_unique_factorization_resultother_factorizationprimesprimeproductsecondimaginarydecode))) /\ ((ge_second_ip_unique_factorization_resultother_factorizationprimesprimeproduct) + ge_balance_negative_unique_factorization_resultother_factorizationprimesprimeproductsecondimaginary = (ge_second_in_unique_factorization_resultother_factorizationprimesprimeproduct) + ge_balance_positive_unique_factorization_resultother_factorizationprimesprimeproductsecondimaginary)))))) /\ (exists ge_representation_real_code_unique_factorization_resultother_factorizationprimesprimeproductoutput ge_representation_imaginary_code_unique_factorization_resultother_factorizationprimesprimeproductoutput. (((gr_product_unique_factorization_resultother_factorizationprimesprime) = ((ge_representation_real_code_unique_factorization_resultother_factorizationprimesprimeproductoutput) + (ge_representation_imaginary_code_unique_factorization_resultother_factorizationprimesprimeproductoutput)) * S ((ge_representation_real_code_unique_factorization_resultother_factorizationprimesprimeproductoutput) + (ge_representation_imaginary_code_unique_factorization_resultother_factorizationprimesprimeproductoutput)) + ((ge_representation_imaginary_code_unique_factorization_resultother_factorizationprimesprimeproductoutput) + (ge_representation_imaginary_code_unique_factorization_resultother_factorizationprimesprimeproductoutput))) /\ ((exists ge_balance_positive_unique_factorization_resultother_factorizationprimesprimeproductoutputreal ge_balance_negative_unique_factorization_resultother_factorizationprimesprimeproductoutputreal. (((((ge_representation_real_code_unique_factorization_resultother_factorizationprimesprimeproductoutput) = 2 * (ge_balance_positive_unique_factorization_resultother_factorizationprimesprimeproductoutputreal) /\ (ge_balance_negative_unique_factorization_resultother_factorizationprimesprimeproductoutputreal) = 0) \/ exists ge_signed_half_unique_factorization_resultother_factorizationprimesprimeproductoutputrealdecode. (((ge_representation_real_code_unique_factorization_resultother_factorizationprimesprimeproductoutput) = 2 * ge_signed_half_unique_factorization_resultother_factorizationprimesprimeproductoutputrealdecode + 1 /\ (ge_balance_positive_unique_factorization_resultother_factorizationprimesprimeproductoutputreal) = 0) /\ (ge_balance_negative_unique_factorization_resultother_factorizationprimesprimeproductoutputreal) = S ge_signed_half_unique_factorization_resultother_factorizationprimesprimeproductoutputrealdecode))) /\ ((((((((ge_first_rp_unique_factorization_resultother_factorizationprimesprimeproduct) * (ge_second_rp_unique_factorization_resultother_factorizationprimesprimeproduct))) + (((ge_first_rn_unique_factorization_resultother_factorizationprimesprimeproduct) * (ge_second_rn_unique_factorization_resultother_factorizationprimesprimeproduct))))) + (((((ge_first_ip_unique_factorization_resultother_factorizationprimesprimeproduct) * (ge_second_in_unique_factorization_resultother_factorizationprimesprimeproduct))) + (((ge_first_in_unique_factorization_resultother_factorizationprimesprimeproduct) * (ge_second_ip_unique_factorization_resultother_factorizationprimesprimeproduct))))))) + ge_balance_negative_unique_factorization_resultother_factorizationprimesprimeproductoutputreal = (((((((ge_first_rp_unique_factorization_resultother_factorizationprimesprimeproduct) * (ge_second_rn_unique_factorization_resultother_factorizationprimesprimeproduct))) + (((ge_first_rn_unique_factorization_resultother_factorizationprimesprimeproduct) * (ge_second_rp_unique_factorization_resultother_factorizationprimesprimeproduct))))) + (((((ge_first_ip_unique_factorization_resultother_factorizationprimesprimeproduct) * (ge_second_ip_unique_factorization_resultother_factorizationprimesprimeproduct))) + (((ge_first_in_unique_factorization_resultother_factorizationprimesprimeproduct) * (ge_second_in_unique_factorization_resultother_factorizationprimesprimeproduct))))))) + ge_balance_positive_unique_factorization_resultother_factorizationprimesprimeproductoutputreal))) /\ (exists ge_balance_positive_unique_factorization_resultother_factorizationprimesprimeproductoutputimaginary ge_balance_negative_unique_factorization_resultother_factorizationprimesprimeproductoutputimaginary. (((((ge_representation_imaginary_code_unique_factorization_resultother_factorizationprimesprimeproductoutput) = 2 * (ge_balance_positive_unique_factorization_resultother_factorizationprimesprimeproductoutputimaginary) /\ (ge_balance_negative_unique_factorization_resultother_factorizationprimesprimeproductoutputimaginary) = 0) \/ exists ge_signed_half_unique_factorization_resultother_factorizationprimesprimeproductoutputimaginarydecode. (((ge_representation_imaginary_code_unique_factorization_resultother_factorizationprimesprimeproductoutput) = 2 * ge_signed_half_unique_factorization_resultother_factorizationprimesprimeproductoutputimaginarydecode + 1 /\ (ge_balance_positive_unique_factorization_resultother_factorizationprimesprimeproductoutputimaginary) = 0) /\ (ge_balance_negative_unique_factorization_resultother_factorizationprimesprimeproductoutputimaginary) = S ge_signed_half_unique_factorization_resultother_factorizationprimesprimeproductoutputimaginarydecode))) /\ ((((((((ge_first_rp_unique_factorization_resultother_factorizationprimesprimeproduct) * (ge_second_ip_unique_factorization_resultother_factorizationprimesprimeproduct))) + (((ge_first_rn_unique_factorization_resultother_factorizationprimesprimeproduct) * (ge_second_in_unique_factorization_resultother_factorizationprimesprimeproduct))))) + (((((ge_first_ip_unique_factorization_resultother_factorizationprimesprimeproduct) * (ge_second_rp_unique_factorization_resultother_factorizationprimesprimeproduct))) + (((ge_first_in_unique_factorization_resultother_factorizationprimesprimeproduct) * (ge_second_rn_unique_factorization_resultother_factorizationprimesprimeproduct))))))) + ge_balance_negative_unique_factorization_resultother_factorizationprimesprimeproductoutputimaginary = (((((((ge_first_rp_unique_factorization_resultother_factorizationprimesprimeproduct) * (ge_second_in_unique_factorization_resultother_factorizationprimesprimeproduct))) + (((ge_first_rn_unique_factorization_resultother_factorizationprimesprimeproduct) * (ge_second_ip_unique_factorization_resultother_factorizationprimesprimeproduct))))) + (((((ge_first_ip_unique_factorization_resultother_factorizationprimesprimeproduct) * (ge_second_rn_unique_factorization_resultother_factorizationprimesprimeproduct))) + (((ge_first_in_unique_factorization_resultother_factorizationprimesprimeproduct) * (ge_second_rp_unique_factorization_resultother_factorizationprimesprimeproduct))))))) + ge_balance_positive_unique_factorization_resultother_factorizationprimesprimeproductoutputimaginary))))))))) -> (exists gr_quotient_unique_factorization_resultother_factorizationprimesprimedivisor. (exists ge_first_rp_unique_factorization_resultother_factorizationprimesprimedivisorproduct ge_first_rn_unique_factorization_resultother_factorizationprimesprimedivisorproduct ge_first_ip_unique_factorization_resultother_factorizationprimesprimedivisorproduct ge_first_in_unique_factorization_resultother_factorizationprimesprimedivisorproduct ge_second_rp_unique_factorization_resultother_factorizationprimesprimedivisorproduct ge_second_rn_unique_factorization_resultother_factorizationprimesprimedivisorproduct ge_second_ip_unique_factorization_resultother_factorizationprimesprimedivisorproduct ge_second_in_unique_factorization_resultother_factorizationprimesprimedivisorproduct. ((exists ge_representation_real_code_unique_factorization_resultother_factorizationprimesprimedivisorproductfirst ge_representation_imaginary_code_unique_factorization_resultother_factorizationprimesprimedivisorproductfirst. (((gr_prime_factor_value_unique_factorization_resultother_factorizationprimes) = ((ge_representation_real_code_unique_factorization_resultother_factorizationprimesprimedivisorproductfirst) + (ge_representation_imaginary_code_unique_factorization_resultother_factorizationprimesprimedivisorproductfirst)) * S ((ge_representation_real_code_unique_factorization_resultother_factorizationprimesprimedivisorproductfirst) + (ge_representation_imaginary_code_unique_factorization_resultother_factorizationprimesprimedivisorproductfirst)) + ((ge_representation_imaginary_code_unique_factorization_resultother_factorizationprimesprimedivisorproductfirst) + (ge_representation_imaginary_code_unique_factorization_resultother_factorizationprimesprimedivisorproductfirst))) /\ ((exists ge_balance_positive_unique_factorization_resultother_factorizationprimesprimedivisorproductfirstreal ge_balance_negative_unique_factorization_resultother_factorizationprimesprimedivisorproductfirstreal. (((((ge_representation_real_code_unique_factorization_resultother_factorizationprimesprimedivisorproductfirst) = 2 * (ge_balance_positive_unique_factorization_resultother_factorizationprimesprimedivisorproductfirstreal) /\ (ge_balance_negative_unique_factorization_resultother_factorizationprimesprimedivisorproductfirstreal) = 0) \/ exists ge_signed_half_unique_factorization_resultother_factorizationprimesprimedivisorproductfirstrealdecode. (((ge_representation_real_code_unique_factorization_resultother_factorizationprimesprimedivisorproductfirst) = 2 * ge_signed_half_unique_factorization_resultother_factorizationprimesprimedivisorproductfirstrealdecode + 1 /\ (ge_balance_positive_unique_factorization_resultother_factorizationprimesprimedivisorproductfirstreal) = 0) /\ (ge_balance_negative_unique_factorization_resultother_factorizationprimesprimedivisorproductfirstreal) = S ge_signed_half_unique_factorization_resultother_factorizationprimesprimedivisorproductfirstrealdecode))) /\ ((ge_first_rp_unique_factorization_resultother_factorizationprimesprimedivisorproduct) + ge_balance_negative_unique_factorization_resultother_factorizationprimesprimedivisorproductfirstreal = (ge_first_rn_unique_factorization_resultother_factorizationprimesprimedivisorproduct) + ge_balance_positive_unique_factorization_resultother_factorizationprimesprimedivisorproductfirstreal))) /\ (exists ge_balance_positive_unique_factorization_resultother_factorizationprimesprimedivisorproductfirstimaginary ge_balance_negative_unique_factorization_resultother_factorizationprimesprimedivisorproductfirstimaginary. (((((ge_representation_imaginary_code_unique_factorization_resultother_factorizationprimesprimedivisorproductfirst) = 2 * (ge_balance_positive_unique_factorization_resultother_factorizationprimesprimedivisorproductfirstimaginary) /\ (ge_balance_negative_unique_factorization_resultother_factorizationprimesprimedivisorproductfirstimaginary) = 0) \/ exists ge_signed_half_unique_factorization_resultother_factorizationprimesprimedivisorproductfirstimaginarydecode. (((ge_representation_imaginary_code_unique_factorization_resultother_factorizationprimesprimedivisorproductfirst) = 2 * ge_signed_half_unique_factorization_resultother_factorizationprimesprimedivisorproductfirstimaginarydecode + 1 /\ (ge_balance_positive_unique_factorization_resultother_factorizationprimesprimedivisorproductfirstimaginary) = 0) /\ (ge_balance_negative_unique_factorization_resultother_factorizationprimesprimedivisorproductfirstimaginary) = S ge_signed_half_unique_factorization_resultother_factorizationprimesprimedivisorproductfirstimaginarydecode))) /\ ((ge_first_ip_unique_factorization_resultother_factorizationprimesprimedivisorproduct) + ge_balance_negative_unique_factorization_resultother_factorizationprimesprimedivisorproductfirstimaginary = (ge_first_in_unique_factorization_resultother_factorizationprimesprimedivisorproduct) + ge_balance_positive_unique_factorization_resultother_factorizationprimesprimedivisorproductfirstimaginary)))))) /\ ((exists ge_representation_real_code_unique_factorization_resultother_factorizationprimesprimedivisorproductsecond ge_representation_imaginary_code_unique_factorization_resultother_factorizationprimesprimedivisorproductsecond. (((gr_quotient_unique_factorization_resultother_factorizationprimesprimedivisor) = ((ge_representation_real_code_unique_factorization_resultother_factorizationprimesprimedivisorproductsecond) + (ge_representation_imaginary_code_unique_factorization_resultother_factorizationprimesprimedivisorproductsecond)) * S ((ge_representation_real_code_unique_factorization_resultother_factorizationprimesprimedivisorproductsecond) + (ge_representation_imaginary_code_unique_factorization_resultother_factorizationprimesprimedivisorproductsecond)) + ((ge_representation_imaginary_code_unique_factorization_resultother_factorizationprimesprimedivisorproductsecond) + (ge_representation_imaginary_code_unique_factorization_resultother_factorizationprimesprimedivisorproductsecond))) /\ ((exists ge_balance_positive_unique_factorization_resultother_factorizationprimesprimedivisorproductsecondreal ge_balance_negative_unique_factorization_resultother_factorizationprimesprimedivisorproductsecondreal. (((((ge_representation_real_code_unique_factorization_resultother_factorizationprimesprimedivisorproductsecond) = 2 * (ge_balance_positive_unique_factorization_resultother_factorizationprimesprimedivisorproductsecondreal) /\ (ge_balance_negative_unique_factorization_resultother_factorizationprimesprimedivisorproductsecondreal) = 0) \/ exists ge_signed_half_unique_factorization_resultother_factorizationprimesprimedivisorproductsecondrealdecode. (((ge_representation_real_code_unique_factorization_resultother_factorizationprimesprimedivisorproductsecond) = 2 * ge_signed_half_unique_factorization_resultother_factorizationprimesprimedivisorproductsecondrealdecode + 1 /\ (ge_balance_positive_unique_factorization_resultother_factorizationprimesprimedivisorproductsecondreal) = 0) /\ (ge_balance_negative_unique_factorization_resultother_factorizationprimesprimedivisorproductsecondreal) = S ge_signed_half_unique_factorization_resultother_factorizationprimesprimedivisorproductsecondrealdecode))) /\ ((ge_second_rp_unique_factorization_resultother_factorizationprimesprimedivisorproduct) + ge_balance_negative_unique_factorization_resultother_factorizationprimesprimedivisorproductsecondreal = (ge_second_rn_unique_factorization_resultother_factorizationprimesprimedivisorproduct) + ge_balance_positive_unique_factorization_resultother_factorizationprimesprimedivisorproductsecondreal))) /\ (exists ge_balance_positive_unique_factorization_resultother_factorizationprimesprimedivisorproductsecondimaginary ge_balance_negative_unique_factorization_resultother_factorizationprimesprimedivisorproductsecondimaginary. (((((ge_representation_imaginary_code_unique_factorization_resultother_factorizationprimesprimedivisorproductsecond) = 2 * (ge_balance_positive_unique_factorization_resultother_factorizationprimesprimedivisorproductsecondimaginary) /\ (ge_balance_negative_unique_factorization_resultother_factorizationprimesprimedivisorproductsecondimaginary) = 0) \/ exists ge_signed_half_unique_factorization_resultother_factorizationprimesprimedivisorproductsecondimaginarydecode. (((ge_representation_imaginary_code_unique_factorization_resultother_factorizationprimesprimedivisorproductsecond) = 2 * ge_signed_half_unique_factorization_resultother_factorizationprimesprimedivisorproductsecondimaginarydecode + 1 /\ (ge_balance_positive_unique_factorization_resultother_factorizationprimesprimedivisorproductsecondimaginary) = 0) /\ (ge_balance_negative_unique_factorization_resultother_factorizationprimesprimedivisorproductsecondimaginary) = S ge_signed_half_unique_factorization_resultother_factorizationprimesprimedivisorproductsecondimaginarydecode))) /\ ((ge_second_ip_unique_factorization_resultother_factorizationprimesprimedivisorproduct) + ge_balance_negative_unique_factorization_resultother_factorizationprimesprimedivisorproductsecondimaginary = (ge_second_in_unique_factorization_resultother_factorizationprimesprimedivisorproduct) + ge_balance_positive_unique_factorization_resultother_factorizationprimesprimedivisorproductsecondimaginary)))))) /\ (exists ge_representation_real_code_unique_factorization_resultother_factorizationprimesprimedivisorproductoutput ge_representation_imaginary_code_unique_factorization_resultother_factorizationprimesprimedivisorproductoutput. (((gr_product_unique_factorization_resultother_factorizationprimesprime) = ((ge_representation_real_code_unique_factorization_resultother_factorizationprimesprimedivisorproductoutput) + (ge_representation_imaginary_code_unique_factorization_resultother_factorizationprimesprimedivisorproductoutput)) * S ((ge_representation_real_code_unique_factorization_resultother_factorizationprimesprimedivisorproductoutput) + (ge_representation_imaginary_code_unique_factorization_resultother_factorizationprimesprimedivisorproductoutput)) + ((ge_representation_imaginary_code_unique_factorization_resultother_factorizationprimesprimedivisorproductoutput) + (ge_representation_imaginary_code_unique_factorization_resultother_factorizationprimesprimedivisorproductoutput))) /\ ((exists ge_balance_positive_unique_factorization_resultother_factorizationprimesprimedivisorproductoutputreal ge_balance_negative_unique_factorization_resultother_factorizationprimesprimedivisorproductoutputreal. (((((ge_representation_real_code_unique_factorization_resultother_factorizationprimesprimedivisorproductoutput) = 2 * (ge_balance_positive_unique_factorization_resultother_factorizationprimesprimedivisorproductoutputreal) /\ (ge_balance_negative_unique_factorization_resultother_factorizationprimesprimedivisorproductoutputreal) = 0) \/ exists ge_signed_half_unique_factorization_resultother_factorizationprimesprimedivisorproductoutputrealdecode. (((ge_representation_real_code_unique_factorization_resultother_factorizationprimesprimedivisorproductoutput) = 2 * ge_signed_half_unique_factorization_resultother_factorizationprimesprimedivisorproductoutputrealdecode + 1 /\ (ge_balance_positive_unique_factorization_resultother_factorizationprimesprimedivisorproductoutputreal) = 0) /\ (ge_balance_negative_unique_factorization_resultother_factorizationprimesprimedivisorproductoutputreal) = S ge_signed_half_unique_factorization_resultother_factorizationprimesprimedivisorproductoutputrealdecode))) /\ ((((((((ge_first_rp_unique_factorization_resultother_factorizationprimesprimedivisorproduct) * (ge_second_rp_unique_factorization_resultother_factorizationprimesprimedivisorproduct))) + (((ge_first_rn_unique_factorization_resultother_factorizationprimesprimedivisorproduct) * (ge_second_rn_unique_factorization_resultother_factorizationprimesprimedivisorproduct))))) + (((((ge_first_ip_unique_factorization_resultother_factorizationprimesprimedivisorproduct) * (ge_second_in_unique_factorization_resultother_factorizationprimesprimedivisorproduct))) + (((ge_first_in_unique_factorization_resultother_factorizationprimesprimedivisorproduct) * (ge_second_ip_unique_factorization_resultother_factorizationprimesprimedivisorproduct))))))) + ge_balance_negative_unique_factorization_resultother_factorizationprimesprimedivisorproductoutputreal = (((((((ge_first_rp_unique_factorization_resultother_factorizationprimesprimedivisorproduct) * (ge_second_rn_unique_factorization_resultother_factorizationprimesprimedivisorproduct))) + (((ge_first_rn_unique_factorization_resultother_factorizationprimesprimedivisorproduct) * (ge_second_rp_unique_factorization_resultother_factorizationprimesprimedivisorproduct))))) + (((((ge_first_ip_unique_factorization_resultother_factorizationprimesprimedivisorproduct) * (ge_second_ip_unique_factorization_resultother_factorizationprimesprimedivisorproduct))) + (((ge_first_in_unique_factorization_resultother_factorizationprimesprimedivisorproduct) * (ge_second_in_unique_factorization_resultother_factorizationprimesprimedivisorproduct))))))) + ge_balance_positive_unique_factorization_resultother_factorizationprimesprimedivisorproductoutputreal))) /\ (exists ge_balance_positive_unique_factorization_resultother_factorizationprimesprimedivisorproductoutputimaginary ge_balance_negative_unique_factorization_resultother_factorizationprimesprimedivisorproductoutputimaginary. (((((ge_representation_imaginary_code_unique_factorization_resultother_factorizationprimesprimedivisorproductoutput) = 2 * (ge_balance_positive_unique_factorization_resultother_factorizationprimesprimedivisorproductoutputimaginary) /\ (ge_balance_negative_unique_factorization_resultother_factorizationprimesprimedivisorproductoutputimaginary) = 0) \/ exists ge_signed_half_unique_factorization_resultother_factorizationprimesprimedivisorproductoutputimaginarydecode. (((ge_representation_imaginary_code_unique_factorization_resultother_factorizationprimesprimedivisorproductoutput) = 2 * ge_signed_half_unique_factorization_resultother_factorizationprimesprimedivisorproductoutputimaginarydecode + 1 /\ (ge_balance_positive_unique_factorization_resultother_factorizationprimesprimedivisorproductoutputimaginary) = 0) /\ (ge_balance_negative_unique_factorization_resultother_factorizationprimesprimedivisorproductoutputimaginary) = S ge_signed_half_unique_factorization_resultother_factorizationprimesprimedivisorproductoutputimaginarydecode))) /\ ((((((((ge_first_rp_unique_factorization_resultother_factorizationprimesprimedivisorproduct) * (ge_second_ip_unique_factorization_resultother_factorizationprimesprimedivisorproduct))) + (((ge_first_rn_unique_factorization_resultother_factorizationprimesprimedivisorproduct) * (ge_second_in_unique_factorization_resultother_factorizationprimesprimedivisorproduct))))) + (((((ge_first_ip_unique_factorization_resultother_factorizationprimesprimedivisorproduct) * (ge_second_rp_unique_factorization_resultother_factorizationprimesprimedivisorproduct))) + (((ge_first_in_unique_factorization_resultother_factorizationprimesprimedivisorproduct) * (ge_second_rn_unique_factorization_resultother_factorizationprimesprimedivisorproduct))))))) + ge_balance_negative_unique_factorization_resultother_factorizationprimesprimedivisorproductoutputimaginary = (((((((ge_first_rp_unique_factorization_resultother_factorizationprimesprimedivisorproduct) * (ge_second_in_unique_factorization_resultother_factorizationprimesprimedivisorproduct))) + (((ge_first_rn_unique_factorization_resultother_factorizationprimesprimedivisorproduct) * (ge_second_ip_unique_factorization_resultother_factorizationprimesprimedivisorproduct))))) + (((((ge_first_ip_unique_factorization_resultother_factorizationprimesprimedivisorproduct) * (ge_second_rn_unique_factorization_resultother_factorizationprimesprimedivisorproduct))) + (((ge_first_in_unique_factorization_resultother_factorizationprimesprimedivisorproduct) * (ge_second_rp_unique_factorization_resultother_factorizationprimesprimedivisorproduct))))))) + ge_balance_positive_unique_factorization_resultother_factorizationprimesprimedivisorproductoutputimaginary)))))))))) -> (exists gr_quotient_unique_factorization_resultother_factorizationprimesprimefirst_divisor. (exists ge_first_rp_unique_factorization_resultother_factorizationprimesprimefirst_divisorproduct ge_first_rn_unique_factorization_resultother_factorizationprimesprimefirst_divisorproduct ge_first_ip_unique_factorization_resultother_factorizationprimesprimefirst_divisorproduct ge_first_in_unique_factorization_resultother_factorizationprimesprimefirst_divisorproduct ge_second_rp_unique_factorization_resultother_factorizationprimesprimefirst_divisorproduct ge_second_rn_unique_factorization_resultother_factorizationprimesprimefirst_divisorproduct ge_second_ip_unique_factorization_resultother_factorizationprimesprimefirst_divisorproduct ge_second_in_unique_factorization_resultother_factorizationprimesprimefirst_divisorproduct. ((exists ge_representation_real_code_unique_factorization_resultother_factorizationprimesprimefirst_divisorproductfirst ge_representation_imaginary_code_unique_factorization_resultother_factorizationprimesprimefirst_divisorproductfirst. (((gr_prime_factor_value_unique_factorization_resultother_factorizationprimes) = ((ge_representation_real_code_unique_factorization_resultother_factorizationprimesprimefirst_divisorproductfirst) + (ge_representation_imaginary_code_unique_factorization_resultother_factorizationprimesprimefirst_divisorproductfirst)) * S ((ge_representation_real_code_unique_factorization_resultother_factorizationprimesprimefirst_divisorproductfirst) + (ge_representation_imaginary_code_unique_factorization_resultother_factorizationprimesprimefirst_divisorproductfirst)) + ((ge_representation_imaginary_code_unique_factorization_resultother_factorizationprimesprimefirst_divisorproductfirst) + (ge_representation_imaginary_code_unique_factorization_resultother_factorizationprimesprimefirst_divisorproductfirst))) /\ ((exists ge_balance_positive_unique_factorization_resultother_factorizationprimesprimefirst_divisorproductfirstreal ge_balance_negative_unique_factorization_resultother_factorizationprimesprimefirst_divisorproductfirstreal. (((((ge_representation_real_code_unique_factorization_resultother_factorizationprimesprimefirst_divisorproductfirst) = 2 * (ge_balance_positive_unique_factorization_resultother_factorizationprimesprimefirst_divisorproductfirstreal) /\ (ge_balance_negative_unique_factorization_resultother_factorizationprimesprimefirst_divisorproductfirstreal) = 0) \/ exists ge_signed_half_unique_factorization_resultother_factorizationprimesprimefirst_divisorproductfirstrealdecode. (((ge_representation_real_code_unique_factorization_resultother_factorizationprimesprimefirst_divisorproductfirst) = 2 * ge_signed_half_unique_factorization_resultother_factorizationprimesprimefirst_divisorproductfirstrealdecode + 1 /\ (ge_balance_positive_unique_factorization_resultother_factorizationprimesprimefirst_divisorproductfirstreal) = 0) /\ (ge_balance_negative_unique_factorization_resultother_factorizationprimesprimefirst_divisorproductfirstreal) = S ge_signed_half_unique_factorization_resultother_factorizationprimesprimefirst_divisorproductfirstrealdecode))) /\ ((ge_first_rp_unique_factorization_resultother_factorizationprimesprimefirst_divisorproduct) + ge_balance_negative_unique_factorization_resultother_factorizationprimesprimefirst_divisorproductfirstreal = (ge_first_rn_unique_factorization_resultother_factorizationprimesprimefirst_divisorproduct) + ge_balance_positive_unique_factorization_resultother_factorizationprimesprimefirst_divisorproductfirstreal))) /\ (exists ge_balance_positive_unique_factorization_resultother_factorizationprimesprimefirst_divisorproductfirstimaginary ge_balance_negative_unique_factorization_resultother_factorizationprimesprimefirst_divisorproductfirstimaginary. (((((ge_representation_imaginary_code_unique_factorization_resultother_factorizationprimesprimefirst_divisorproductfirst) = 2 * (ge_balance_positive_unique_factorization_resultother_factorizationprimesprimefirst_divisorproductfirstimaginary) /\ (ge_balance_negative_unique_factorization_resultother_factorizationprimesprimefirst_divisorproductfirstimaginary) = 0) \/ exists ge_signed_half_unique_factorization_resultother_factorizationprimesprimefirst_divisorproductfirstimaginarydecode. (((ge_representation_imaginary_code_unique_factorization_resultother_factorizationprimesprimefirst_divisorproductfirst) = 2 * ge_signed_half_unique_factorization_resultother_factorizationprimesprimefirst_divisorproductfirstimaginarydecode + 1 /\ (ge_balance_positive_unique_factorization_resultother_factorizationprimesprimefirst_divisorproductfirstimaginary) = 0) /\ (ge_balance_negative_unique_factorization_resultother_factorizationprimesprimefirst_divisorproductfirstimaginary) = S ge_signed_half_unique_factorization_resultother_factorizationprimesprimefirst_divisorproductfirstimaginarydecode))) /\ ((ge_first_ip_unique_factorization_resultother_factorizationprimesprimefirst_divisorproduct) + ge_balance_negative_unique_factorization_resultother_factorizationprimesprimefirst_divisorproductfirstimaginary = (ge_first_in_unique_factorization_resultother_factorizationprimesprimefirst_divisorproduct) + ge_balance_positive_unique_factorization_resultother_factorizationprimesprimefirst_divisorproductfirstimaginary)))))) /\ ((exists ge_representation_real_code_unique_factorization_resultother_factorizationprimesprimefirst_divisorproductsecond ge_representation_imaginary_code_unique_factorization_resultother_factorizationprimesprimefirst_divisorproductsecond. (((gr_quotient_unique_factorization_resultother_factorizationprimesprimefirst_divisor) = ((ge_representation_real_code_unique_factorization_resultother_factorizationprimesprimefirst_divisorproductsecond) + (ge_representation_imaginary_code_unique_factorization_resultother_factorizationprimesprimefirst_divisorproductsecond)) * S ((ge_representation_real_code_unique_factorization_resultother_factorizationprimesprimefirst_divisorproductsecond) + (ge_representation_imaginary_code_unique_factorization_resultother_factorizationprimesprimefirst_divisorproductsecond)) + ((ge_representation_imaginary_code_unique_factorization_resultother_factorizationprimesprimefirst_divisorproductsecond) + (ge_representation_imaginary_code_unique_factorization_resultother_factorizationprimesprimefirst_divisorproductsecond))) /\ ((exists ge_balance_positive_unique_factorization_resultother_factorizationprimesprimefirst_divisorproductsecondreal ge_balance_negative_unique_factorization_resultother_factorizationprimesprimefirst_divisorproductsecondreal. (((((ge_representation_real_code_unique_factorization_resultother_factorizationprimesprimefirst_divisorproductsecond) = 2 * (ge_balance_positive_unique_factorization_resultother_factorizationprimesprimefirst_divisorproductsecondreal) /\ (ge_balance_negative_unique_factorization_resultother_factorizationprimesprimefirst_divisorproductsecondreal) = 0) \/ exists ge_signed_half_unique_factorization_resultother_factorizationprimesprimefirst_divisorproductsecondrealdecode. (((ge_representation_real_code_unique_factorization_resultother_factorizationprimesprimefirst_divisorproductsecond) = 2 * ge_signed_half_unique_factorization_resultother_factorizationprimesprimefirst_divisorproductsecondrealdecode + 1 /\ (ge_balance_positive_unique_factorization_resultother_factorizationprimesprimefirst_divisorproductsecondreal) = 0) /\ (ge_balance_negative_unique_factorization_resultother_factorizationprimesprimefirst_divisorproductsecondreal) = S ge_signed_half_unique_factorization_resultother_factorizationprimesprimefirst_divisorproductsecondrealdecode))) /\ ((ge_second_rp_unique_factorization_resultother_factorizationprimesprimefirst_divisorproduct) + ge_balance_negative_unique_factorization_resultother_factorizationprimesprimefirst_divisorproductsecondreal = (ge_second_rn_unique_factorization_resultother_factorizationprimesprimefirst_divisorproduct) + ge_balance_positive_unique_factorization_resultother_factorizationprimesprimefirst_divisorproductsecondreal))) /\ (exists ge_balance_positive_unique_factorization_resultother_factorizationprimesprimefirst_divisorproductsecondimaginary ge_balance_negative_unique_factorization_resultother_factorizationprimesprimefirst_divisorproductsecondimaginary. (((((ge_representation_imaginary_code_unique_factorization_resultother_factorizationprimesprimefirst_divisorproductsecond) = 2 * (ge_balance_positive_unique_factorization_resultother_factorizationprimesprimefirst_divisorproductsecondimaginary) /\ (ge_balance_negative_unique_factorization_resultother_factorizationprimesprimefirst_divisorproductsecondimaginary) = 0) \/ exists ge_signed_half_unique_factorization_resultother_factorizationprimesprimefirst_divisorproductsecondimaginarydecode. (((ge_representation_imaginary_code_unique_factorization_resultother_factorizationprimesprimefirst_divisorproductsecond) = 2 * ge_signed_half_unique_factorization_resultother_factorizationprimesprimefirst_divisorproductsecondimaginarydecode + 1 /\ (ge_balance_positive_unique_factorization_resultother_factorizationprimesprimefirst_divisorproductsecondimaginary) = 0) /\ (ge_balance_negative_unique_factorization_resultother_factorizationprimesprimefirst_divisorproductsecondimaginary) = S ge_signed_half_unique_factorization_resultother_factorizationprimesprimefirst_divisorproductsecondimaginarydecode))) /\ ((ge_second_ip_unique_factorization_resultother_factorizationprimesprimefirst_divisorproduct) + ge_balance_negative_unique_factorization_resultother_factorizationprimesprimefirst_divisorproductsecondimaginary = (ge_second_in_unique_factorization_resultother_factorizationprimesprimefirst_divisorproduct) + ge_balance_positive_unique_factorization_resultother_factorizationprimesprimefirst_divisorproductsecondimaginary)))))) /\ (exists ge_representation_real_code_unique_factorization_resultother_factorizationprimesprimefirst_divisorproductoutput ge_representation_imaginary_code_unique_factorization_resultother_factorizationprimesprimefirst_divisorproductoutput. (((gr_first_factor_unique_factorization_resultother_factorizationprimesprime) = ((ge_representation_real_code_unique_factorization_resultother_factorizationprimesprimefirst_divisorproductoutput) + (ge_representation_imaginary_code_unique_factorization_resultother_factorizationprimesprimefirst_divisorproductoutput)) * S ((ge_representation_real_code_unique_factorization_resultother_factorizationprimesprimefirst_divisorproductoutput) + (ge_representation_imaginary_code_unique_factorization_resultother_factorizationprimesprimefirst_divisorproductoutput)) + ((ge_representation_imaginary_code_unique_factorization_resultother_factorizationprimesprimefirst_divisorproductoutput) + (ge_representation_imaginary_code_unique_factorization_resultother_factorizationprimesprimefirst_divisorproductoutput))) /\ ((exists ge_balance_positive_unique_factorization_resultother_factorizationprimesprimefirst_divisorproductoutputreal ge_balance_negative_unique_factorization_resultother_factorizationprimesprimefirst_divisorproductoutputreal. (((((ge_representation_real_code_unique_factorization_resultother_factorizationprimesprimefirst_divisorproductoutput) = 2 * (ge_balance_positive_unique_factorization_resultother_factorizationprimesprimefirst_divisorproductoutputreal) /\ (ge_balance_negative_unique_factorization_resultother_factorizationprimesprimefirst_divisorproductoutputreal) = 0) \/ exists ge_signed_half_unique_factorization_resultother_factorizationprimesprimefirst_divisorproductoutputrealdecode. (((ge_representation_real_code_unique_factorization_resultother_factorizationprimesprimefirst_divisorproductoutput) = 2 * ge_signed_half_unique_factorization_resultother_factorizationprimesprimefirst_divisorproductoutputrealdecode + 1 /\ (ge_balance_positive_unique_factorization_resultother_factorizationprimesprimefirst_divisorproductoutputreal) = 0) /\ (ge_balance_negative_unique_factorization_resultother_factorizationprimesprimefirst_divisorproductoutputreal) = S ge_signed_half_unique_factorization_resultother_factorizationprimesprimefirst_divisorproductoutputrealdecode))) /\ ((((((((ge_first_rp_unique_factorization_resultother_factorizationprimesprimefirst_divisorproduct) * (ge_second_rp_unique_factorization_resultother_factorizationprimesprimefirst_divisorproduct))) + (((ge_first_rn_unique_factorization_resultother_factorizationprimesprimefirst_divisorproduct) * (ge_second_rn_unique_factorization_resultother_factorizationprimesprimefirst_divisorproduct))))) + (((((ge_first_ip_unique_factorization_resultother_factorizationprimesprimefirst_divisorproduct) * (ge_second_in_unique_factorization_resultother_factorizationprimesprimefirst_divisorproduct))) + (((ge_first_in_unique_factorization_resultother_factorizationprimesprimefirst_divisorproduct) * (ge_second_ip_unique_factorization_resultother_factorizationprimesprimefirst_divisorproduct))))))) + ge_balance_negative_unique_factorization_resultother_factorizationprimesprimefirst_divisorproductoutputreal = (((((((ge_first_rp_unique_factorization_resultother_factorizationprimesprimefirst_divisorproduct) * (ge_second_rn_unique_factorization_resultother_factorizationprimesprimefirst_divisorproduct))) + (((ge_first_rn_unique_factorization_resultother_factorizationprimesprimefirst_divisorproduct) * (ge_second_rp_unique_factorization_resultother_factorizationprimesprimefirst_divisorproduct))))) + (((((ge_first_ip_unique_factorization_resultother_factorizationprimesprimefirst_divisorproduct) * (ge_second_ip_unique_factorization_resultother_factorizationprimesprimefirst_divisorproduct))) + (((ge_first_in_unique_factorization_resultother_factorizationprimesprimefirst_divisorproduct) * (ge_second_in_unique_factorization_resultother_factorizationprimesprimefirst_divisorproduct))))))) + ge_balance_positive_unique_factorization_resultother_factorizationprimesprimefirst_divisorproductoutputreal))) /\ (exists ge_balance_positive_unique_factorization_resultother_factorizationprimesprimefirst_divisorproductoutputimaginary ge_balance_negative_unique_factorization_resultother_factorizationprimesprimefirst_divisorproductoutputimaginary. (((((ge_representation_imaginary_code_unique_factorization_resultother_factorizationprimesprimefirst_divisorproductoutput) = 2 * (ge_balance_positive_unique_factorization_resultother_factorizationprimesprimefirst_divisorproductoutputimaginary) /\ (ge_balance_negative_unique_factorization_resultother_factorizationprimesprimefirst_divisorproductoutputimaginary) = 0) \/ exists ge_signed_half_unique_factorization_resultother_factorizationprimesprimefirst_divisorproductoutputimaginarydecode. (((ge_representation_imaginary_code_unique_factorization_resultother_factorizationprimesprimefirst_divisorproductoutput) = 2 * ge_signed_half_unique_factorization_resultother_factorizationprimesprimefirst_divisorproductoutputimaginarydecode + 1 /\ (ge_balance_positive_unique_factorization_resultother_factorizationprimesprimefirst_divisorproductoutputimaginary) = 0) /\ (ge_balance_negative_unique_factorization_resultother_factorizationprimesprimefirst_divisorproductoutputimaginary) = S ge_signed_half_unique_factorization_resultother_factorizationprimesprimefirst_divisorproductoutputimaginarydecode))) /\ ((((((((ge_first_rp_unique_factorization_resultother_factorizationprimesprimefirst_divisorproduct) * (ge_second_ip_unique_factorization_resultother_factorizationprimesprimefirst_divisorproduct))) + (((ge_first_rn_unique_factorization_resultother_factorizationprimesprimefirst_divisorproduct) * (ge_second_in_unique_factorization_resultother_factorizationprimesprimefirst_divisorproduct))))) + (((((ge_first_ip_unique_factorization_resultother_factorizationprimesprimefirst_divisorproduct) * (ge_second_rp_unique_factorization_resultother_factorizationprimesprimefirst_divisorproduct))) + (((ge_first_in_unique_factorization_resultother_factorizationprimesprimefirst_divisorproduct) * (ge_second_rn_unique_factorization_resultother_factorizationprimesprimefirst_divisorproduct))))))) + ge_balance_negative_unique_factorization_resultother_factorizationprimesprimefirst_divisorproductoutputimaginary = (((((((ge_first_rp_unique_factorization_resultother_factorizationprimesprimefirst_divisorproduct) * (ge_second_in_unique_factorization_resultother_factorizationprimesprimefirst_divisorproduct))) + (((ge_first_rn_unique_factorization_resultother_factorizationprimesprimefirst_divisorproduct) * (ge_second_ip_unique_factorization_resultother_factorizationprimesprimefirst_divisorproduct))))) + (((((ge_first_ip_unique_factorization_resultother_factorizationprimesprimefirst_divisorproduct) * (ge_second_rn_unique_factorization_resultother_factorizationprimesprimefirst_divisorproduct))) + (((ge_first_in_unique_factorization_resultother_factorizationprimesprimefirst_divisorproduct) * (ge_second_rp_unique_factorization_resultother_factorizationprimesprimefirst_divisorproduct))))))) + ge_balance_positive_unique_factorization_resultother_factorizationprimesprimefirst_divisorproductoutputimaginary)))))))))) \/ (exists gr_quotient_unique_factorization_resultother_factorizationprimesprimesecond_divisor. (exists ge_first_rp_unique_factorization_resultother_factorizationprimesprimesecond_divisorproduct ge_first_rn_unique_factorization_resultother_factorizationprimesprimesecond_divisorproduct ge_first_ip_unique_factorization_resultother_factorizationprimesprimesecond_divisorproduct ge_first_in_unique_factorization_resultother_factorizationprimesprimesecond_divisorproduct ge_second_rp_unique_factorization_resultother_factorizationprimesprimesecond_divisorproduct ge_second_rn_unique_factorization_resultother_factorizationprimesprimesecond_divisorproduct ge_second_ip_unique_factorization_resultother_factorizationprimesprimesecond_divisorproduct ge_second_in_unique_factorization_resultother_factorizationprimesprimesecond_divisorproduct. ((exists ge_representation_real_code_unique_factorization_resultother_factorizationprimesprimesecond_divisorproductfirst ge_representation_imaginary_code_unique_factorization_resultother_factorizationprimesprimesecond_divisorproductfirst. (((gr_prime_factor_value_unique_factorization_resultother_factorizationprimes) = ((ge_representation_real_code_unique_factorization_resultother_factorizationprimesprimesecond_divisorproductfirst) + (ge_representation_imaginary_code_unique_factorization_resultother_factorizationprimesprimesecond_divisorproductfirst)) * S ((ge_representation_real_code_unique_factorization_resultother_factorizationprimesprimesecond_divisorproductfirst) + (ge_representation_imaginary_code_unique_factorization_resultother_factorizationprimesprimesecond_divisorproductfirst)) + ((ge_representation_imaginary_code_unique_factorization_resultother_factorizationprimesprimesecond_divisorproductfirst) + (ge_representation_imaginary_code_unique_factorization_resultother_factorizationprimesprimesecond_divisorproductfirst))) /\ ((exists ge_balance_positive_unique_factorization_resultother_factorizationprimesprimesecond_divisorproductfirstreal ge_balance_negative_unique_factorization_resultother_factorizationprimesprimesecond_divisorproductfirstreal. (((((ge_representation_real_code_unique_factorization_resultother_factorizationprimesprimesecond_divisorproductfirst) = 2 * (ge_balance_positive_unique_factorization_resultother_factorizationprimesprimesecond_divisorproductfirstreal) /\ (ge_balance_negative_unique_factorization_resultother_factorizationprimesprimesecond_divisorproductfirstreal) = 0) \/ exists ge_signed_half_unique_factorization_resultother_factorizationprimesprimesecond_divisorproductfirstrealdecode. (((ge_representation_real_code_unique_factorization_resultother_factorizationprimesprimesecond_divisorproductfirst) = 2 * ge_signed_half_unique_factorization_resultother_factorizationprimesprimesecond_divisorproductfirstrealdecode + 1 /\ (ge_balance_positive_unique_factorization_resultother_factorizationprimesprimesecond_divisorproductfirstreal) = 0) /\ (ge_balance_negative_unique_factorization_resultother_factorizationprimesprimesecond_divisorproductfirstreal) = S ge_signed_half_unique_factorization_resultother_factorizationprimesprimesecond_divisorproductfirstrealdecode))) /\ ((ge_first_rp_unique_factorization_resultother_factorizationprimesprimesecond_divisorproduct) + ge_balance_negative_unique_factorization_resultother_factorizationprimesprimesecond_divisorproductfirstreal = (ge_first_rn_unique_factorization_resultother_factorizationprimesprimesecond_divisorproduct) + ge_balance_positive_unique_factorization_resultother_factorizationprimesprimesecond_divisorproductfirstreal))) /\ (exists ge_balance_positive_unique_factorization_resultother_factorizationprimesprimesecond_divisorproductfirstimaginary ge_balance_negative_unique_factorization_resultother_factorizationprimesprimesecond_divisorproductfirstimaginary. (((((ge_representation_imaginary_code_unique_factorization_resultother_factorizationprimesprimesecond_divisorproductfirst) = 2 * (ge_balance_positive_unique_factorization_resultother_factorizationprimesprimesecond_divisorproductfirstimaginary) /\ (ge_balance_negative_unique_factorization_resultother_factorizationprimesprimesecond_divisorproductfirstimaginary) = 0) \/ exists ge_signed_half_unique_factorization_resultother_factorizationprimesprimesecond_divisorproductfirstimaginarydecode. (((ge_representation_imaginary_code_unique_factorization_resultother_factorizationprimesprimesecond_divisorproductfirst) = 2 * ge_signed_half_unique_factorization_resultother_factorizationprimesprimesecond_divisorproductfirstimaginarydecode + 1 /\ (ge_balance_positive_unique_factorization_resultother_factorizationprimesprimesecond_divisorproductfirstimaginary) = 0) /\ (ge_balance_negative_unique_factorization_resultother_factorizationprimesprimesecond_divisorproductfirstimaginary) = S ge_signed_half_unique_factorization_resultother_factorizationprimesprimesecond_divisorproductfirstimaginarydecode))) /\ ((ge_first_ip_unique_factorization_resultother_factorizationprimesprimesecond_divisorproduct) + ge_balance_negative_unique_factorization_resultother_factorizationprimesprimesecond_divisorproductfirstimaginary = (ge_first_in_unique_factorization_resultother_factorizationprimesprimesecond_divisorproduct) + ge_balance_positive_unique_factorization_resultother_factorizationprimesprimesecond_divisorproductfirstimaginary)))))) /\ ((exists ge_representation_real_code_unique_factorization_resultother_factorizationprimesprimesecond_divisorproductsecond ge_representation_imaginary_code_unique_factorization_resultother_factorizationprimesprimesecond_divisorproductsecond. (((gr_quotient_unique_factorization_resultother_factorizationprimesprimesecond_divisor) = ((ge_representation_real_code_unique_factorization_resultother_factorizationprimesprimesecond_divisorproductsecond) + (ge_representation_imaginary_code_unique_factorization_resultother_factorizationprimesprimesecond_divisorproductsecond)) * S ((ge_representation_real_code_unique_factorization_resultother_factorizationprimesprimesecond_divisorproductsecond) + (ge_representation_imaginary_code_unique_factorization_resultother_factorizationprimesprimesecond_divisorproductsecond)) + ((ge_representation_imaginary_code_unique_factorization_resultother_factorizationprimesprimesecond_divisorproductsecond) + (ge_representation_imaginary_code_unique_factorization_resultother_factorizationprimesprimesecond_divisorproductsecond))) /\ ((exists ge_balance_positive_unique_factorization_resultother_factorizationprimesprimesecond_divisorproductsecondreal ge_balance_negative_unique_factorization_resultother_factorizationprimesprimesecond_divisorproductsecondreal. (((((ge_representation_real_code_unique_factorization_resultother_factorizationprimesprimesecond_divisorproductsecond) = 2 * (ge_balance_positive_unique_factorization_resultother_factorizationprimesprimesecond_divisorproductsecondreal) /\ (ge_balance_negative_unique_factorization_resultother_factorizationprimesprimesecond_divisorproductsecondreal) = 0) \/ exists ge_signed_half_unique_factorization_resultother_factorizationprimesprimesecond_divisorproductsecondrealdecode. (((ge_representation_real_code_unique_factorization_resultother_factorizationprimesprimesecond_divisorproductsecond) = 2 * ge_signed_half_unique_factorization_resultother_factorizationprimesprimesecond_divisorproductsecondrealdecode + 1 /\ (ge_balance_positive_unique_factorization_resultother_factorizationprimesprimesecond_divisorproductsecondreal) = 0) /\ (ge_balance_negative_unique_factorization_resultother_factorizationprimesprimesecond_divisorproductsecondreal) = S ge_signed_half_unique_factorization_resultother_factorizationprimesprimesecond_divisorproductsecondrealdecode))) /\ ((ge_second_rp_unique_factorization_resultother_factorizationprimesprimesecond_divisorproduct) + ge_balance_negative_unique_factorization_resultother_factorizationprimesprimesecond_divisorproductsecondreal = (ge_second_rn_unique_factorization_resultother_factorizationprimesprimesecond_divisorproduct) + ge_balance_positive_unique_factorization_resultother_factorizationprimesprimesecond_divisorproductsecondreal))) /\ (exists ge_balance_positive_unique_factorization_resultother_factorizationprimesprimesecond_divisorproductsecondimaginary ge_balance_negative_unique_factorization_resultother_factorizationprimesprimesecond_divisorproductsecondimaginary. (((((ge_representation_imaginary_code_unique_factorization_resultother_factorizationprimesprimesecond_divisorproductsecond) = 2 * (ge_balance_positive_unique_factorization_resultother_factorizationprimesprimesecond_divisorproductsecondimaginary) /\ (ge_balance_negative_unique_factorization_resultother_factorizationprimesprimesecond_divisorproductsecondimaginary) = 0) \/ exists ge_signed_half_unique_factorization_resultother_factorizationprimesprimesecond_divisorproductsecondimaginarydecode. (((ge_representation_imaginary_code_unique_factorization_resultother_factorizationprimesprimesecond_divisorproductsecond) = 2 * ge_signed_half_unique_factorization_resultother_factorizationprimesprimesecond_divisorproductsecondimaginarydecode + 1 /\ (ge_balance_positive_unique_factorization_resultother_factorizationprimesprimesecond_divisorproductsecondimaginary) = 0) /\ (ge_balance_negative_unique_factorization_resultother_factorizationprimesprimesecond_divisorproductsecondimaginary) = S ge_signed_half_unique_factorization_resultother_factorizationprimesprimesecond_divisorproductsecondimaginarydecode))) /\ ((ge_second_ip_unique_factorization_resultother_factorizationprimesprimesecond_divisorproduct) + ge_balance_negative_unique_factorization_resultother_factorizationprimesprimesecond_divisorproductsecondimaginary = (ge_second_in_unique_factorization_resultother_factorizationprimesprimesecond_divisorproduct) + ge_balance_positive_unique_factorization_resultother_factorizationprimesprimesecond_divisorproductsecondimaginary)))))) /\ (exists ge_representation_real_code_unique_factorization_resultother_factorizationprimesprimesecond_divisorproductoutput ge_representation_imaginary_code_unique_factorization_resultother_factorizationprimesprimesecond_divisorproductoutput. (((gr_second_factor_unique_factorization_resultother_factorizationprimesprime) = ((ge_representation_real_code_unique_factorization_resultother_factorizationprimesprimesecond_divisorproductoutput) + (ge_representation_imaginary_code_unique_factorization_resultother_factorizationprimesprimesecond_divisorproductoutput)) * S ((ge_representation_real_code_unique_factorization_resultother_factorizationprimesprimesecond_divisorproductoutput) + (ge_representation_imaginary_code_unique_factorization_resultother_factorizationprimesprimesecond_divisorproductoutput)) + ((ge_representation_imaginary_code_unique_factorization_resultother_factorizationprimesprimesecond_divisorproductoutput) + (ge_representation_imaginary_code_unique_factorization_resultother_factorizationprimesprimesecond_divisorproductoutput))) /\ ((exists ge_balance_positive_unique_factorization_resultother_factorizationprimesprimesecond_divisorproductoutputreal ge_balance_negative_unique_factorization_resultother_factorizationprimesprimesecond_divisorproductoutputreal. (((((ge_representation_real_code_unique_factorization_resultother_factorizationprimesprimesecond_divisorproductoutput) = 2 * (ge_balance_positive_unique_factorization_resultother_factorizationprimesprimesecond_divisorproductoutputreal) /\ (ge_balance_negative_unique_factorization_resultother_factorizationprimesprimesecond_divisorproductoutputreal) = 0) \/ exists ge_signed_half_unique_factorization_resultother_factorizationprimesprimesecond_divisorproductoutputrealdecode. (((ge_representation_real_code_unique_factorization_resultother_factorizationprimesprimesecond_divisorproductoutput) = 2 * ge_signed_half_unique_factorization_resultother_factorizationprimesprimesecond_divisorproductoutputrealdecode + 1 /\ (ge_balance_positive_unique_factorization_resultother_factorizationprimesprimesecond_divisorproductoutputreal) = 0) /\ (ge_balance_negative_unique_factorization_resultother_factorizationprimesprimesecond_divisorproductoutputreal) = S ge_signed_half_unique_factorization_resultother_factorizationprimesprimesecond_divisorproductoutputrealdecode))) /\ ((((((((ge_first_rp_unique_factorization_resultother_factorizationprimesprimesecond_divisorproduct) * (ge_second_rp_unique_factorization_resultother_factorizationprimesprimesecond_divisorproduct))) + (((ge_first_rn_unique_factorization_resultother_factorizationprimesprimesecond_divisorproduct) * (ge_second_rn_unique_factorization_resultother_factorizationprimesprimesecond_divisorproduct))))) + (((((ge_first_ip_unique_factorization_resultother_factorizationprimesprimesecond_divisorproduct) * (ge_second_in_unique_factorization_resultother_factorizationprimesprimesecond_divisorproduct))) + (((ge_first_in_unique_factorization_resultother_factorizationprimesprimesecond_divisorproduct) * (ge_second_ip_unique_factorization_resultother_factorizationprimesprimesecond_divisorproduct))))))) + ge_balance_negative_unique_factorization_resultother_factorizationprimesprimesecond_divisorproductoutputreal = (((((((ge_first_rp_unique_factorization_resultother_factorizationprimesprimesecond_divisorproduct) * (ge_second_rn_unique_factorization_resultother_factorizationprimesprimesecond_divisorproduct))) + (((ge_first_rn_unique_factorization_resultother_factorizationprimesprimesecond_divisorproduct) * (ge_second_rp_unique_factorization_resultother_factorizationprimesprimesecond_divisorproduct))))) + (((((ge_first_ip_unique_factorization_resultother_factorizationprimesprimesecond_divisorproduct) * (ge_second_ip_unique_factorization_resultother_factorizationprimesprimesecond_divisorproduct))) + (((ge_first_in_unique_factorization_resultother_factorizationprimesprimesecond_divisorproduct) * (ge_second_in_unique_factorization_resultother_factorizationprimesprimesecond_divisorproduct))))))) + ge_balance_positive_unique_factorization_resultother_factorizationprimesprimesecond_divisorproductoutputreal))) /\ (exists ge_balance_positive_unique_factorization_resultother_factorizationprimesprimesecond_divisorproductoutputimaginary ge_balance_negative_unique_factorization_resultother_factorizationprimesprimesecond_divisorproductoutputimaginary. (((((ge_representation_imaginary_code_unique_factorization_resultother_factorizationprimesprimesecond_divisorproductoutput) = 2 * (ge_balance_positive_unique_factorization_resultother_factorizationprimesprimesecond_divisorproductoutputimaginary) /\ (ge_balance_negative_unique_factorization_resultother_factorizationprimesprimesecond_divisorproductoutputimaginary) = 0) \/ exists ge_signed_half_unique_factorization_resultother_factorizationprimesprimesecond_divisorproductoutputimaginarydecode. (((ge_representation_imaginary_code_unique_factorization_resultother_factorizationprimesprimesecond_divisorproductoutput) = 2 * ge_signed_half_unique_factorization_resultother_factorizationprimesprimesecond_divisorproductoutputimaginarydecode + 1 /\ (ge_balance_positive_unique_factorization_resultother_factorizationprimesprimesecond_divisorproductoutputimaginary) = 0) /\ (ge_balance_negative_unique_factorization_resultother_factorizationprimesprimesecond_divisorproductoutputimaginary) = S ge_signed_half_unique_factorization_resultother_factorizationprimesprimesecond_divisorproductoutputimaginarydecode))) /\ ((((((((ge_first_rp_unique_factorization_resultother_factorizationprimesprimesecond_divisorproduct) * (ge_second_ip_unique_factorization_resultother_factorizationprimesprimesecond_divisorproduct))) + (((ge_first_rn_unique_factorization_resultother_factorizationprimesprimesecond_divisorproduct) * (ge_second_in_unique_factorization_resultother_factorizationprimesprimesecond_divisorproduct))))) + (((((ge_first_ip_unique_factorization_resultother_factorizationprimesprimesecond_divisorproduct) * (ge_second_rp_unique_factorization_resultother_factorizationprimesprimesecond_divisorproduct))) + (((ge_first_in_unique_factorization_resultother_factorizationprimesprimesecond_divisorproduct) * (ge_second_rn_unique_factorization_resultother_factorizationprimesprimesecond_divisorproduct))))))) + ge_balance_negative_unique_factorization_resultother_factorizationprimesprimesecond_divisorproductoutputimaginary = (((((((ge_first_rp_unique_factorization_resultother_factorizationprimesprimesecond_divisorproduct) * (ge_second_in_unique_factorization_resultother_factorizationprimesprimesecond_divisorproduct))) + (((ge_first_rn_unique_factorization_resultother_factorizationprimesprimesecond_divisorproduct) * (ge_second_ip_unique_factorization_resultother_factorizationprimesprimesecond_divisorproduct))))) + (((((ge_first_ip_unique_factorization_resultother_factorizationprimesprimesecond_divisorproduct) * (ge_second_rn_unique_factorization_resultother_factorizationprimesprimesecond_divisorproduct))) + (((ge_first_in_unique_factorization_resultother_factorizationprimesprimesecond_divisorproduct) * (ge_second_rp_unique_factorization_resultother_factorizationprimesprimesecond_divisorproduct))))))) + ge_balance_positive_unique_factorization_resultother_factorizationprimesprimesecond_divisorproductoutputimaginary)))))))))))))))) /\ (exists gr_prime_factor_product_unique_factorization_resultother_factorization. ((exists gr_product_trace_unique_factorization_resultother_factorizationtrace gr_product_scale_unique_factorization_resultother_factorizationtrace. ((((exists ff_h_gprod_unique_factorization_resultother_factorizationtracestart. ff_h_gprod_unique_factorization_resultother_factorizationtracestart + S (6) = S ((S (0)) * gr_product_scale_unique_factorization_resultother_factorizationtrace)) /\ exists ff_q_gprod_unique_factorization_resultother_factorizationtracestart. gr_product_trace_unique_factorization_resultother_factorizationtrace = ff_q_gprod_unique_factorization_resultother_factorizationtracestart * S ((S (0)) * gr_product_scale_unique_factorization_resultother_factorizationtrace) + (6))) /\ ((((exists ff_h_gprod_unique_factorization_resultother_factorizationtraceend. ff_h_gprod_unique_factorization_resultother_factorizationtraceend + S (gr_prime_factor_product_unique_factorization_resultother_factorization) = S ((S (gr_other_length_unique_factorization_result)) * gr_product_scale_unique_factorization_resultother_factorizationtrace)) /\ exists ff_q_gprod_unique_factorization_resultother_factorizationtraceend. gr_product_trace_unique_factorization_resultother_factorizationtrace = ff_q_gprod_unique_factorization_resultother_factorizationtraceend * S ((S (gr_other_length_unique_factorization_result)) * gr_product_scale_unique_factorization_resultother_factorizationtrace) + (gr_prime_factor_product_unique_factorization_resultother_factorization))) /\ (forall gr_product_index_unique_factorization_resultother_factorizationtracesteps. (exists ge_gap_unique_factorization_resultother_factorizationtracestepsindex_bound. ge_gap_unique_factorization_resultother_factorizationtracestepsindex_bound + S (gr_product_index_unique_factorization_resultother_factorizationtracesteps) = (gr_other_length_unique_factorization_result)) -> exists gr_product_factor_unique_factorization_resultother_factorizationtracesteps gr_product_before_unique_factorization_resultother_factorizationtracesteps gr_product_after_unique_factorization_resultother_factorizationtracesteps. ((((exists ff_h_gprod_unique_factorization_resultother_factorizationtracestepsfactor. ff_h_gprod_unique_factorization_resultother_factorizationtracestepsfactor + S (gr_product_factor_unique_factorization_resultother_factorizationtracesteps) = S ((S (gr_product_index_unique_factorization_resultother_factorizationtracesteps)) * gr_other_scale_unique_factorization_result)) /\ exists ff_q_gprod_unique_factorization_resultother_factorizationtracestepsfactor. gr_other_code_unique_factorization_result = ff_q_gprod_unique_factorization_resultother_factorizationtracestepsfactor * S ((S (gr_product_index_unique_factorization_resultother_factorizationtracesteps)) * gr_other_scale_unique_factorization_result) + (gr_product_factor_unique_factorization_resultother_factorizationtracesteps))) /\ ((((exists ff_h_gprod_unique_factorization_resultother_factorizationtracestepsbefore. ff_h_gprod_unique_factorization_resultother_factorizationtracestepsbefore + S (gr_product_before_unique_factorization_resultother_factorizationtracesteps) = S ((S (gr_product_index_unique_factorization_resultother_factorizationtracesteps)) * gr_product_scale_unique_factorization_resultother_factorizationtrace)) /\ exists ff_q_gprod_unique_factorization_resultother_factorizationtracestepsbefore. gr_product_trace_unique_factorization_resultother_factorizationtrace = ff_q_gprod_unique_factorization_resultother_factorizationtracestepsbefore * S ((S (gr_product_index_unique_factorization_resultother_factorizationtracesteps)) * gr_product_scale_unique_factorization_resultother_factorizationtrace) + (gr_product_before_unique_factorization_resultother_factorizationtracesteps))) /\ ((((exists ff_h_gprod_unique_factorization_resultother_factorizationtracestepsafter. ff_h_gprod_unique_factorization_resultother_factorizationtracestepsafter + S (gr_product_after_unique_factorization_resultother_factorizationtracesteps) = S ((S (S (gr_product_index_unique_factorization_resultother_factorizationtracesteps))) * gr_product_scale_unique_factorization_resultother_factorizationtrace)) /\ exists ff_q_gprod_unique_factorization_resultother_factorizationtracestepsafter. gr_product_trace_unique_factorization_resultother_factorizationtrace = ff_q_gprod_unique_factorization_resultother_factorizationtracestepsafter * S ((S (S (gr_product_index_unique_factorization_resultother_factorizationtracesteps))) * gr_product_scale_unique_factorization_resultother_factorizationtrace) + (gr_product_after_unique_factorization_resultother_factorizationtracesteps))) /\ (exists ge_first_rp_unique_factorization_resultother_factorizationtracestepsmultiply ge_first_rn_unique_factorization_resultother_factorizationtracestepsmultiply ge_first_ip_unique_factorization_resultother_factorizationtracestepsmultiply ge_first_in_unique_factorization_resultother_factorizationtracestepsmultiply ge_second_rp_unique_factorization_resultother_factorizationtracestepsmultiply ge_second_rn_unique_factorization_resultother_factorizationtracestepsmultiply ge_second_ip_unique_factorization_resultother_factorizationtracestepsmultiply ge_second_in_unique_factorization_resultother_factorizationtracestepsmultiply. ((exists ge_representation_real_code_unique_factorization_resultother_factorizationtracestepsmultiplyfirst ge_representation_imaginary_code_unique_factorization_resultother_factorizationtracestepsmultiplyfirst. (((gr_product_before_unique_factorization_resultother_factorizationtracesteps) = ((ge_representation_real_code_unique_factorization_resultother_factorizationtracestepsmultiplyfirst) + (ge_representation_imaginary_code_unique_factorization_resultother_factorizationtracestepsmultiplyfirst)) * S ((ge_representation_real_code_unique_factorization_resultother_factorizationtracestepsmultiplyfirst) + (ge_representation_imaginary_code_unique_factorization_resultother_factorizationtracestepsmultiplyfirst)) + ((ge_representation_imaginary_code_unique_factorization_resultother_factorizationtracestepsmultiplyfirst) + (ge_representation_imaginary_code_unique_factorization_resultother_factorizationtracestepsmultiplyfirst))) /\ ((exists ge_balance_positive_unique_factorization_resultother_factorizationtracestepsmultiplyfirstreal ge_balance_negative_unique_factorization_resultother_factorizationtracestepsmultiplyfirstreal. (((((ge_representation_real_code_unique_factorization_resultother_factorizationtracestepsmultiplyfirst) = 2 * (ge_balance_positive_unique_factorization_resultother_factorizationtracestepsmultiplyfirstreal) /\ (ge_balance_negative_unique_factorization_resultother_factorizationtracestepsmultiplyfirstreal) = 0) \/ exists ge_signed_half_unique_factorization_resultother_factorizationtracestepsmultiplyfirstrealdecode. (((ge_representation_real_code_unique_factorization_resultother_factorizationtracestepsmultiplyfirst) = 2 * ge_signed_half_unique_factorization_resultother_factorizationtracestepsmultiplyfirstrealdecode + 1 /\ (ge_balance_positive_unique_factorization_resultother_factorizationtracestepsmultiplyfirstreal) = 0) /\ (ge_balance_negative_unique_factorization_resultother_factorizationtracestepsmultiplyfirstreal) = S ge_signed_half_unique_factorization_resultother_factorizationtracestepsmultiplyfirstrealdecode))) /\ ((ge_first_rp_unique_factorization_resultother_factorizationtracestepsmultiply) + ge_balance_negative_unique_factorization_resultother_factorizationtracestepsmultiplyfirstreal = (ge_first_rn_unique_factorization_resultother_factorizationtracestepsmultiply) + ge_balance_positive_unique_factorization_resultother_factorizationtracestepsmultiplyfirstreal))) /\ (exists ge_balance_positive_unique_factorization_resultother_factorizationtracestepsmultiplyfirstimaginary ge_balance_negative_unique_factorization_resultother_factorizationtracestepsmultiplyfirstimaginary. (((((ge_representation_imaginary_code_unique_factorization_resultother_factorizationtracestepsmultiplyfirst) = 2 * (ge_balance_positive_unique_factorization_resultother_factorizationtracestepsmultiplyfirstimaginary) /\ (ge_balance_negative_unique_factorization_resultother_factorizationtracestepsmultiplyfirstimaginary) = 0) \/ exists ge_signed_half_unique_factorization_resultother_factorizationtracestepsmultiplyfirstimaginarydecode. (((ge_representation_imaginary_code_unique_factorization_resultother_factorizationtracestepsmultiplyfirst) = 2 * ge_signed_half_unique_factorization_resultother_factorizationtracestepsmultiplyfirstimaginarydecode + 1 /\ (ge_balance_positive_unique_factorization_resultother_factorizationtracestepsmultiplyfirstimaginary) = 0) /\ (ge_balance_negative_unique_factorization_resultother_factorizationtracestepsmultiplyfirstimaginary) = S ge_signed_half_unique_factorization_resultother_factorizationtracestepsmultiplyfirstimaginarydecode))) /\ ((ge_first_ip_unique_factorization_resultother_factorizationtracestepsmultiply) + ge_balance_negative_unique_factorization_resultother_factorizationtracestepsmultiplyfirstimaginary = (ge_first_in_unique_factorization_resultother_factorizationtracestepsmultiply) + ge_balance_positive_unique_factorization_resultother_factorizationtracestepsmultiplyfirstimaginary)))))) /\ ((exists ge_representation_real_code_unique_factorization_resultother_factorizationtracestepsmultiplysecond ge_representation_imaginary_code_unique_factorization_resultother_factorizationtracestepsmultiplysecond. (((gr_product_factor_unique_factorization_resultother_factorizationtracesteps) = ((ge_representation_real_code_unique_factorization_resultother_factorizationtracestepsmultiplysecond) + (ge_representation_imaginary_code_unique_factorization_resultother_factorizationtracestepsmultiplysecond)) * S ((ge_representation_real_code_unique_factorization_resultother_factorizationtracestepsmultiplysecond) + (ge_representation_imaginary_code_unique_factorization_resultother_factorizationtracestepsmultiplysecond)) + ((ge_representation_imaginary_code_unique_factorization_resultother_factorizationtracestepsmultiplysecond) + (ge_representation_imaginary_code_unique_factorization_resultother_factorizationtracestepsmultiplysecond))) /\ ((exists ge_balance_positive_unique_factorization_resultother_factorizationtracestepsmultiplysecondreal ge_balance_negative_unique_factorization_resultother_factorizationtracestepsmultiplysecondreal. (((((ge_representation_real_code_unique_factorization_resultother_factorizationtracestepsmultiplysecond) = 2 * (ge_balance_positive_unique_factorization_resultother_factorizationtracestepsmultiplysecondreal) /\ (ge_balance_negative_unique_factorization_resultother_factorizationtracestepsmultiplysecondreal) = 0) \/ exists ge_signed_half_unique_factorization_resultother_factorizationtracestepsmultiplysecondrealdecode. (((ge_representation_real_code_unique_factorization_resultother_factorizationtracestepsmultiplysecond) = 2 * ge_signed_half_unique_factorization_resultother_factorizationtracestepsmultiplysecondrealdecode + 1 /\ (ge_balance_positive_unique_factorization_resultother_factorizationtracestepsmultiplysecondreal) = 0) /\ (ge_balance_negative_unique_factorization_resultother_factorizationtracestepsmultiplysecondreal) = S ge_signed_half_unique_factorization_resultother_factorizationtracestepsmultiplysecondrealdecode))) /\ ((ge_second_rp_unique_factorization_resultother_factorizationtracestepsmultiply) + ge_balance_negative_unique_factorization_resultother_factorizationtracestepsmultiplysecondreal = (ge_second_rn_unique_factorization_resultother_factorizationtracestepsmultiply) + ge_balance_positive_unique_factorization_resultother_factorizationtracestepsmultiplysecondreal))) /\ (exists ge_balance_positive_unique_factorization_resultother_factorizationtracestepsmultiplysecondimaginary ge_balance_negative_unique_factorization_resultother_factorizationtracestepsmultiplysecondimaginary. (((((ge_representation_imaginary_code_unique_factorization_resultother_factorizationtracestepsmultiplysecond) = 2 * (ge_balance_positive_unique_factorization_resultother_factorizationtracestepsmultiplysecondimaginary) /\ (ge_balance_negative_unique_factorization_resultother_factorizationtracestepsmultiplysecondimaginary) = 0) \/ exists ge_signed_half_unique_factorization_resultother_factorizationtracestepsmultiplysecondimaginarydecode. (((ge_representation_imaginary_code_unique_factorization_resultother_factorizationtracestepsmultiplysecond) = 2 * ge_signed_half_unique_factorization_resultother_factorizationtracestepsmultiplysecondimaginarydecode + 1 /\ (ge_balance_positive_unique_factorization_resultother_factorizationtracestepsmultiplysecondimaginary) = 0) /\ (ge_balance_negative_unique_factorization_resultother_factorizationtracestepsmultiplysecondimaginary) = S ge_signed_half_unique_factorization_resultother_factorizationtracestepsmultiplysecondimaginarydecode))) /\ ((ge_second_ip_unique_factorization_resultother_factorizationtracestepsmultiply) + ge_balance_negative_unique_factorization_resultother_factorizationtracestepsmultiplysecondimaginary = (ge_second_in_unique_factorization_resultother_factorizationtracestepsmultiply) + ge_balance_positive_unique_factorization_resultother_factorizationtracestepsmultiplysecondimaginary)))))) /\ (exists ge_representation_real_code_unique_factorization_resultother_factorizationtracestepsmultiplyoutput ge_representation_imaginary_code_unique_factorization_resultother_factorizationtracestepsmultiplyoutput. (((gr_product_after_unique_factorization_resultother_factorizationtracesteps) = ((ge_representation_real_code_unique_factorization_resultother_factorizationtracestepsmultiplyoutput) + (ge_representation_imaginary_code_unique_factorization_resultother_factorizationtracestepsmultiplyoutput)) * S ((ge_representation_real_code_unique_factorization_resultother_factorizationtracestepsmultiplyoutput) + (ge_representation_imaginary_code_unique_factorization_resultother_factorizationtracestepsmultiplyoutput)) + ((ge_representation_imaginary_code_unique_factorization_resultother_factorizationtracestepsmultiplyoutput) + (ge_representation_imaginary_code_unique_factorization_resultother_factorizationtracestepsmultiplyoutput))) /\ ((exists ge_balance_positive_unique_factorization_resultother_factorizationtracestepsmultiplyoutputreal ge_balance_negative_unique_factorization_resultother_factorizationtracestepsmultiplyoutputreal. (((((ge_representation_real_code_unique_factorization_resultother_factorizationtracestepsmultiplyoutput) = 2 * (ge_balance_positive_unique_factorization_resultother_factorizationtracestepsmultiplyoutputreal) /\ (ge_balance_negative_unique_factorization_resultother_factorizationtracestepsmultiplyoutputreal) = 0) \/ exists ge_signed_half_unique_factorization_resultother_factorizationtracestepsmultiplyoutputrealdecode. (((ge_representation_real_code_unique_factorization_resultother_factorizationtracestepsmultiplyoutput) = 2 * ge_signed_half_unique_factorization_resultother_factorizationtracestepsmultiplyoutputrealdecode + 1 /\ (ge_balance_positive_unique_factorization_resultother_factorizationtracestepsmultiplyoutputreal) = 0) /\ (ge_balance_negative_unique_factorization_resultother_factorizationtracestepsmultiplyoutputreal) = S ge_signed_half_unique_factorization_resultother_factorizationtracestepsmultiplyoutputrealdecode))) /\ ((((((((ge_first_rp_unique_factorization_resultother_factorizationtracestepsmultiply) * (ge_second_rp_unique_factorization_resultother_factorizationtracestepsmultiply))) + (((ge_first_rn_unique_factorization_resultother_factorizationtracestepsmultiply) * (ge_second_rn_unique_factorization_resultother_factorizationtracestepsmultiply))))) + (((((ge_first_ip_unique_factorization_resultother_factorizationtracestepsmultiply) * (ge_second_in_unique_factorization_resultother_factorizationtracestepsmultiply))) + (((ge_first_in_unique_factorization_resultother_factorizationtracestepsmultiply) * (ge_second_ip_unique_factorization_resultother_factorizationtracestepsmultiply))))))) + ge_balance_negative_unique_factorization_resultother_factorizationtracestepsmultiplyoutputreal = (((((((ge_first_rp_unique_factorization_resultother_factorizationtracestepsmultiply) * (ge_second_rn_unique_factorization_resultother_factorizationtracestepsmultiply))) + (((ge_first_rn_unique_factorization_resultother_factorizationtracestepsmultiply) * (ge_second_rp_unique_factorization_resultother_factorizationtracestepsmultiply))))) + (((((ge_first_ip_unique_factorization_resultother_factorizationtracestepsmultiply) * (ge_second_ip_unique_factorization_resultother_factorizationtracestepsmultiply))) + (((ge_first_in_unique_factorization_resultother_factorizationtracestepsmultiply) * (ge_second_in_unique_factorization_resultother_factorizationtracestepsmultiply))))))) + ge_balance_positive_unique_factorization_resultother_factorizationtracestepsmultiplyoutputreal))) /\ (exists ge_balance_positive_unique_factorization_resultother_factorizationtracestepsmultiplyoutputimaginary ge_balance_negative_unique_factorization_resultother_factorizationtracestepsmultiplyoutputimaginary. (((((ge_representation_imaginary_code_unique_factorization_resultother_factorizationtracestepsmultiplyoutput) = 2 * (ge_balance_positive_unique_factorization_resultother_factorizationtracestepsmultiplyoutputimaginary) /\ (ge_balance_negative_unique_factorization_resultother_factorizationtracestepsmultiplyoutputimaginary) = 0) \/ exists ge_signed_half_unique_factorization_resultother_factorizationtracestepsmultiplyoutputimaginarydecode. (((ge_representation_imaginary_code_unique_factorization_resultother_factorizationtracestepsmultiplyoutput) = 2 * ge_signed_half_unique_factorization_resultother_factorizationtracestepsmultiplyoutputimaginarydecode + 1 /\ (ge_balance_positive_unique_factorization_resultother_factorizationtracestepsmultiplyoutputimaginary) = 0) /\ (ge_balance_negative_unique_factorization_resultother_factorizationtracestepsmultiplyoutputimaginary) = S ge_signed_half_unique_factorization_resultother_factorizationtracestepsmultiplyoutputimaginarydecode))) /\ ((((((((ge_first_rp_unique_factorization_resultother_factorizationtracestepsmultiply) * (ge_second_ip_unique_factorization_resultother_factorizationtracestepsmultiply))) + (((ge_first_rn_unique_factorization_resultother_factorizationtracestepsmultiply) * (ge_second_in_unique_factorization_resultother_factorizationtracestepsmultiply))))) + (((((ge_first_ip_unique_factorization_resultother_factorizationtracestepsmultiply) * (ge_second_rp_unique_factorization_resultother_factorizationtracestepsmultiply))) + (((ge_first_in_unique_factorization_resultother_factorizationtracestepsmultiply) * (ge_second_rn_unique_factorization_resultother_factorizationtracestepsmultiply))))))) + ge_balance_negative_unique_factorization_resultother_factorizationtracestepsmultiplyoutputimaginary = (((((((ge_first_rp_unique_factorization_resultother_factorizationtracestepsmultiply) * (ge_second_in_unique_factorization_resultother_factorizationtracestepsmultiply))) + (((ge_first_rn_unique_factorization_resultother_factorizationtracestepsmultiply) * (ge_second_ip_unique_factorization_resultother_factorizationtracestepsmultiply))))) + (((((ge_first_ip_unique_factorization_resultother_factorizationtracestepsmultiply) * (ge_second_rn_unique_factorization_resultother_factorizationtracestepsmultiply))) + (((ge_first_in_unique_factorization_resultother_factorizationtracestepsmultiply) * (ge_second_rp_unique_factorization_resultother_factorizationtracestepsmultiply))))))) + ge_balance_positive_unique_factorization_resultother_factorizationtracestepsmultiplyoutputimaginary)))))))))))))))) /\ (exists ge_first_rp_unique_factorization_resultother_factorizationreconstruct ge_first_rn_unique_factorization_resultother_factorizationreconstruct ge_first_ip_unique_factorization_resultother_factorizationreconstruct ge_first_in_unique_factorization_resultother_factorizationreconstruct ge_second_rp_unique_factorization_resultother_factorizationreconstruct ge_second_rn_unique_factorization_resultother_factorizationreconstruct ge_second_ip_unique_factorization_resultother_factorizationreconstruct ge_second_in_unique_factorization_resultother_factorizationreconstruct. ((exists ge_representation_real_code_unique_factorization_resultother_factorizationreconstructfirst ge_representation_imaginary_code_unique_factorization_resultother_factorizationreconstructfirst. (((gr_other_unit_unique_factorization_result) = ((ge_representation_real_code_unique_factorization_resultother_factorizationreconstructfirst) + (ge_representation_imaginary_code_unique_factorization_resultother_factorizationreconstructfirst)) * S ((ge_representation_real_code_unique_factorization_resultother_factorizationreconstructfirst) + (ge_representation_imaginary_code_unique_factorization_resultother_factorizationreconstructfirst)) + ((ge_representation_imaginary_code_unique_factorization_resultother_factorizationreconstructfirst) + (ge_representation_imaginary_code_unique_factorization_resultother_factorizationreconstructfirst))) /\ ((exists ge_balance_positive_unique_factorization_resultother_factorizationreconstructfirstreal ge_balance_negative_unique_factorization_resultother_factorizationreconstructfirstreal. (((((ge_representation_real_code_unique_factorization_resultother_factorizationreconstructfirst) = 2 * (ge_balance_positive_unique_factorization_resultother_factorizationreconstructfirstreal) /\ (ge_balance_negative_unique_factorization_resultother_factorizationreconstructfirstreal) = 0) \/ exists ge_signed_half_unique_factorization_resultother_factorizationreconstructfirstrealdecode. (((ge_representation_real_code_unique_factorization_resultother_factorizationreconstructfirst) = 2 * ge_signed_half_unique_factorization_resultother_factorizationreconstructfirstrealdecode + 1 /\ (ge_balance_positive_unique_factorization_resultother_factorizationreconstructfirstreal) = 0) /\ (ge_balance_negative_unique_factorization_resultother_factorizationreconstructfirstreal) = S ge_signed_half_unique_factorization_resultother_factorizationreconstructfirstrealdecode))) /\ ((ge_first_rp_unique_factorization_resultother_factorizationreconstruct) + ge_balance_negative_unique_factorization_resultother_factorizationreconstructfirstreal = (ge_first_rn_unique_factorization_resultother_factorizationreconstruct) + ge_balance_positive_unique_factorization_resultother_factorizationreconstructfirstreal))) /\ (exists ge_balance_positive_unique_factorization_resultother_factorizationreconstructfirstimaginary ge_balance_negative_unique_factorization_resultother_factorizationreconstructfirstimaginary. (((((ge_representation_imaginary_code_unique_factorization_resultother_factorizationreconstructfirst) = 2 * (ge_balance_positive_unique_factorization_resultother_factorizationreconstructfirstimaginary) /\ (ge_balance_negative_unique_factorization_resultother_factorizationreconstructfirstimaginary) = 0) \/ exists ge_signed_half_unique_factorization_resultother_factorizationreconstructfirstimaginarydecode. (((ge_representation_imaginary_code_unique_factorization_resultother_factorizationreconstructfirst) = 2 * ge_signed_half_unique_factorization_resultother_factorizationreconstructfirstimaginarydecode + 1 /\ (ge_balance_positive_unique_factorization_resultother_factorizationreconstructfirstimaginary) = 0) /\ (ge_balance_negative_unique_factorization_resultother_factorizationreconstructfirstimaginary) = S ge_signed_half_unique_factorization_resultother_factorizationreconstructfirstimaginarydecode))) /\ ((ge_first_ip_unique_factorization_resultother_factorizationreconstruct) + ge_balance_negative_unique_factorization_resultother_factorizationreconstructfirstimaginary = (ge_first_in_unique_factorization_resultother_factorizationreconstruct) + ge_balance_positive_unique_factorization_resultother_factorizationreconstructfirstimaginary)))))) /\ ((exists ge_representation_real_code_unique_factorization_resultother_factorizationreconstructsecond ge_representation_imaginary_code_unique_factorization_resultother_factorizationreconstructsecond. (((gr_prime_factor_product_unique_factorization_resultother_factorization) = ((ge_representation_real_code_unique_factorization_resultother_factorizationreconstructsecond) + (ge_representation_imaginary_code_unique_factorization_resultother_factorizationreconstructsecond)) * S ((ge_representation_real_code_unique_factorization_resultother_factorizationreconstructsecond) + (ge_representation_imaginary_code_unique_factorization_resultother_factorizationreconstructsecond)) + ((ge_representation_imaginary_code_unique_factorization_resultother_factorizationreconstructsecond) + (ge_representation_imaginary_code_unique_factorization_resultother_factorizationreconstructsecond))) /\ ((exists ge_balance_positive_unique_factorization_resultother_factorizationreconstructsecondreal ge_balance_negative_unique_factorization_resultother_factorizationreconstructsecondreal. (((((ge_representation_real_code_unique_factorization_resultother_factorizationreconstructsecond) = 2 * (ge_balance_positive_unique_factorization_resultother_factorizationreconstructsecondreal) /\ (ge_balance_negative_unique_factorization_resultother_factorizationreconstructsecondreal) = 0) \/ exists ge_signed_half_unique_factorization_resultother_factorizationreconstructsecondrealdecode. (((ge_representation_real_code_unique_factorization_resultother_factorizationreconstructsecond) = 2 * ge_signed_half_unique_factorization_resultother_factorizationreconstructsecondrealdecode + 1 /\ (ge_balance_positive_unique_factorization_resultother_factorizationreconstructsecondreal) = 0) /\ (ge_balance_negative_unique_factorization_resultother_factorizationreconstructsecondreal) = S ge_signed_half_unique_factorization_resultother_factorizationreconstructsecondrealdecode))) /\ ((ge_second_rp_unique_factorization_resultother_factorizationreconstruct) + ge_balance_negative_unique_factorization_resultother_factorizationreconstructsecondreal = (ge_second_rn_unique_factorization_resultother_factorizationreconstruct) + ge_balance_positive_unique_factorization_resultother_factorizationreconstructsecondreal))) /\ (exists ge_balance_positive_unique_factorization_resultother_factorizationreconstructsecondimaginary ge_balance_negative_unique_factorization_resultother_factorizationreconstructsecondimaginary. (((((ge_representation_imaginary_code_unique_factorization_resultother_factorizationreconstructsecond) = 2 * (ge_balance_positive_unique_factorization_resultother_factorizationreconstructsecondimaginary) /\ (ge_balance_negative_unique_factorization_resultother_factorizationreconstructsecondimaginary) = 0) \/ exists ge_signed_half_unique_factorization_resultother_factorizationreconstructsecondimaginarydecode. (((ge_representation_imaginary_code_unique_factorization_resultother_factorizationreconstructsecond) = 2 * ge_signed_half_unique_factorization_resultother_factorizationreconstructsecondimaginarydecode + 1 /\ (ge_balance_positive_unique_factorization_resultother_factorizationreconstructsecondimaginary) = 0) /\ (ge_balance_negative_unique_factorization_resultother_factorizationreconstructsecondimaginary) = S ge_signed_half_unique_factorization_resultother_factorizationreconstructsecondimaginarydecode))) /\ ((ge_second_ip_unique_factorization_resultother_factorizationreconstruct) + ge_balance_negative_unique_factorization_resultother_factorizationreconstructsecondimaginary = (ge_second_in_unique_factorization_resultother_factorizationreconstruct) + ge_balance_positive_unique_factorization_resultother_factorizationreconstructsecondimaginary)))))) /\ (exists ge_representation_real_code_unique_factorization_resultother_factorizationreconstructoutput ge_representation_imaginary_code_unique_factorization_resultother_factorizationreconstructoutput. (((z) = ((ge_representation_real_code_unique_factorization_resultother_factorizationreconstructoutput) + (ge_representation_imaginary_code_unique_factorization_resultother_factorizationreconstructoutput)) * S ((ge_representation_real_code_unique_factorization_resultother_factorizationreconstructoutput) + (ge_representation_imaginary_code_unique_factorization_resultother_factorizationreconstructoutput)) + ((ge_representation_imaginary_code_unique_factorization_resultother_factorizationreconstructoutput) + (ge_representation_imaginary_code_unique_factorization_resultother_factorizationreconstructoutput))) /\ ((exists ge_balance_positive_unique_factorization_resultother_factorizationreconstructoutputreal ge_balance_negative_unique_factorization_resultother_factorizationreconstructoutputreal. (((((ge_representation_real_code_unique_factorization_resultother_factorizationreconstructoutput) = 2 * (ge_balance_positive_unique_factorization_resultother_factorizationreconstructoutputreal) /\ (ge_balance_negative_unique_factorization_resultother_factorizationreconstructoutputreal) = 0) \/ exists ge_signed_half_unique_factorization_resultother_factorizationreconstructoutputrealdecode. (((ge_representation_real_code_unique_factorization_resultother_factorizationreconstructoutput) = 2 * ge_signed_half_unique_factorization_resultother_factorizationreconstructoutputrealdecode + 1 /\ (ge_balance_positive_unique_factorization_resultother_factorizationreconstructoutputreal) = 0) /\ (ge_balance_negative_unique_factorization_resultother_factorizationreconstructoutputreal) = S ge_signed_half_unique_factorization_resultother_factorizationreconstructoutputrealdecode))) /\ ((((((((ge_first_rp_unique_factorization_resultother_factorizationreconstruct) * (ge_second_rp_unique_factorization_resultother_factorizationreconstruct))) + (((ge_first_rn_unique_factorization_resultother_factorizationreconstruct) * (ge_second_rn_unique_factorization_resultother_factorizationreconstruct))))) + (((((ge_first_ip_unique_factorization_resultother_factorizationreconstruct) * (ge_second_in_unique_factorization_resultother_factorizationreconstruct))) + (((ge_first_in_unique_factorization_resultother_factorizationreconstruct) * (ge_second_ip_unique_factorization_resultother_factorizationreconstruct))))))) + ge_balance_negative_unique_factorization_resultother_factorizationreconstructoutputreal = (((((((ge_first_rp_unique_factorization_resultother_factorizationreconstruct) * (ge_second_rn_unique_factorization_resultother_factorizationreconstruct))) + (((ge_first_rn_unique_factorization_resultother_factorizationreconstruct) * (ge_second_rp_unique_factorization_resultother_factorizationreconstruct))))) + (((((ge_first_ip_unique_factorization_resultother_factorizationreconstruct) * (ge_second_ip_unique_factorization_resultother_factorizationreconstruct))) + (((ge_first_in_unique_factorization_resultother_factorizationreconstruct) * (ge_second_in_unique_factorization_resultother_factorizationreconstruct))))))) + ge_balance_positive_unique_factorization_resultother_factorizationreconstructoutputreal))) /\ (exists ge_balance_positive_unique_factorization_resultother_factorizationreconstructoutputimaginary ge_balance_negative_unique_factorization_resultother_factorizationreconstructoutputimaginary. (((((ge_representation_imaginary_code_unique_factorization_resultother_factorizationreconstructoutput) = 2 * (ge_balance_positive_unique_factorization_resultother_factorizationreconstructoutputimaginary) /\ (ge_balance_negative_unique_factorization_resultother_factorizationreconstructoutputimaginary) = 0) \/ exists ge_signed_half_unique_factorization_resultother_factorizationreconstructoutputimaginarydecode. (((ge_representation_imaginary_code_unique_factorization_resultother_factorizationreconstructoutput) = 2 * ge_signed_half_unique_factorization_resultother_factorizationreconstructoutputimaginarydecode + 1 /\ (ge_balance_positive_unique_factorization_resultother_factorizationreconstructoutputimaginary) = 0) /\ (ge_balance_negative_unique_factorization_resultother_factorizationreconstructoutputimaginary) = S ge_signed_half_unique_factorization_resultother_factorizationreconstructoutputimaginarydecode))) /\ ((((((((ge_first_rp_unique_factorization_resultother_factorizationreconstruct) * (ge_second_ip_unique_factorization_resultother_factorizationreconstruct))) + (((ge_first_rn_unique_factorization_resultother_factorizationreconstruct) * (ge_second_in_unique_factorization_resultother_factorizationreconstruct))))) + (((((ge_first_ip_unique_factorization_resultother_factorizationreconstruct) * (ge_second_rp_unique_factorization_resultother_factorizationreconstruct))) + (((ge_first_in_unique_factorization_resultother_factorizationreconstruct) * (ge_second_rn_unique_factorization_resultother_factorizationreconstruct))))))) + ge_balance_negative_unique_factorization_resultother_factorizationreconstructoutputimaginary = (((((((ge_first_rp_unique_factorization_resultother_factorizationreconstruct) * (ge_second_in_unique_factorization_resultother_factorizationreconstruct))) + (((ge_first_rn_unique_factorization_resultother_factorizationreconstruct) * (ge_second_ip_unique_factorization_resultother_factorizationreconstruct))))) + (((((ge_first_ip_unique_factorization_resultother_factorizationreconstruct) * (ge_second_rn_unique_factorization_resultother_factorizationreconstruct))) + (((ge_first_in_unique_factorization_resultother_factorizationreconstruct) * (ge_second_rp_unique_factorization_resultother_factorizationreconstruct))))))) + ge_balance_positive_unique_factorization_resultother_factorizationreconstructoutputimaginary)))))))))))))) -> ((((gr_chosen_length_unique_factorization_result)=(gr_other_length_unique_factorization_result)) /\ (exists gr_unique_map_unique_factorization_resultunique gr_unique_scale_unique_factorization_resultunique. (((((forall pfp_i_unique_factorization_resultuniquematchingbijectionbounded. (exists pfp_gap_unique_factorization_resultuniquematchingbijectionboundedindex. pfp_gap_unique_factorization_resultuniquematchingbijectionboundedindex + S (pfp_i_unique_factorization_resultuniquematchingbijectionbounded) = (gr_chosen_length_unique_factorization_result)) -> exists pfp_a_unique_factorization_resultuniquematchingbijectionbounded. (((exists ff_h_pfp_unique_factorization_resultuniquematchingbijectionboundedentry. ff_h_pfp_unique_factorization_resultuniquematchingbijectionboundedentry + S (pfp_a_unique_factorization_resultuniquematchingbijectionbounded) = S ((S (pfp_i_unique_factorization_resultuniquematchingbijectionbounded)) * gr_unique_scale_unique_factorization_resultunique)) /\ exists ff_q_pfp_unique_factorization_resultuniquematchingbijectionboundedentry. gr_unique_map_unique_factorization_resultunique = ff_q_pfp_unique_factorization_resultuniquematchingbijectionboundedentry * S ((S (pfp_i_unique_factorization_resultuniquematchingbijectionbounded)) * gr_unique_scale_unique_factorization_resultunique) + (pfp_a_unique_factorization_resultuniquematchingbijectionbounded))) /\ (exists pfp_gap_unique_factorization_resultuniquematchingbijectionboundedvalue. pfp_gap_unique_factorization_resultuniquematchingbijectionboundedvalue + S (pfp_a_unique_factorization_resultuniquematchingbijectionbounded) = (gr_chosen_length_unique_factorization_result))) /\ (((forall pfp_i_unique_factorization_resultuniquematchingbijectioninjective pfp_j_unique_factorization_resultuniquematchingbijectioninjective pfp_a_unique_factorization_resultuniquematchingbijectioninjective. (exists pfp_gap_unique_factorization_resultuniquematchingbijectioninjectivefirst. pfp_gap_unique_factorization_resultuniquematchingbijectioninjectivefirst + S (pfp_i_unique_factorization_resultuniquematchingbijectioninjective) = (gr_chosen_length_unique_factorization_result)) -> (exists pfp_gap_unique_factorization_resultuniquematchingbijectioninjectivesecond. pfp_gap_unique_factorization_resultuniquematchingbijectioninjectivesecond + S (pfp_j_unique_factorization_resultuniquematchingbijectioninjective) = (gr_chosen_length_unique_factorization_result)) -> (((exists ff_h_pfp_unique_factorization_resultuniquematchingbijectioninjectiveleft. ff_h_pfp_unique_factorization_resultuniquematchingbijectioninjectiveleft + S (pfp_a_unique_factorization_resultuniquematchingbijectioninjective) = S ((S (pfp_i_unique_factorization_resultuniquematchingbijectioninjective)) * gr_unique_scale_unique_factorization_resultunique)) /\ exists ff_q_pfp_unique_factorization_resultuniquematchingbijectioninjectiveleft. gr_unique_map_unique_factorization_resultunique = ff_q_pfp_unique_factorization_resultuniquematchingbijectioninjectiveleft * S ((S (pfp_i_unique_factorization_resultuniquematchingbijectioninjective)) * gr_unique_scale_unique_factorization_resultunique) + (pfp_a_unique_factorization_resultuniquematchingbijectioninjective))) -> (((exists ff_h_pfp_unique_factorization_resultuniquematchingbijectioninjectiveright. ff_h_pfp_unique_factorization_resultuniquematchingbijectioninjectiveright + S (pfp_a_unique_factorization_resultuniquematchingbijectioninjective) = S ((S (pfp_j_unique_factorization_resultuniquematchingbijectioninjective)) * gr_unique_scale_unique_factorization_resultunique)) /\ exists ff_q_pfp_unique_factorization_resultuniquematchingbijectioninjectiveright. gr_unique_map_unique_factorization_resultunique = ff_q_pfp_unique_factorization_resultuniquematchingbijectioninjectiveright * S ((S (pfp_j_unique_factorization_resultuniquematchingbijectioninjective)) * gr_unique_scale_unique_factorization_resultunique) + (pfp_a_unique_factorization_resultuniquematchingbijectioninjective))) -> pfp_i_unique_factorization_resultuniquematchingbijectioninjective = pfp_j_unique_factorization_resultuniquematchingbijectioninjective) /\ (forall pfp_a_unique_factorization_resultuniquematchingbijectionsurjective. (exists pfp_gap_unique_factorization_resultuniquematchingbijectionsurjectivevalue. pfp_gap_unique_factorization_resultuniquematchingbijectionsurjectivevalue + S (pfp_a_unique_factorization_resultuniquematchingbijectionsurjective) = (gr_chosen_length_unique_factorization_result)) -> exists pfp_i_unique_factorization_resultuniquematchingbijectionsurjective. (exists pfp_gap_unique_factorization_resultuniquematchingbijectionsurjectiveindex. pfp_gap_unique_factorization_resultuniquematchingbijectionsurjectiveindex + S (pfp_i_unique_factorization_resultuniquematchingbijectionsurjective) = (gr_chosen_length_unique_factorization_result)) /\ (((exists ff_h_pfp_unique_factorization_resultuniquematchingbijectionsurjectiveentry. ff_h_pfp_unique_factorization_resultuniquematchingbijectionsurjectiveentry + S (pfp_a_unique_factorization_resultuniquematchingbijectionsurjective) = S ((S (pfp_i_unique_factorization_resultuniquematchingbijectionsurjective)) * gr_unique_scale_unique_factorization_resultunique)) /\ exists ff_q_pfp_unique_factorization_resultuniquematchingbijectionsurjectiveentry. gr_unique_map_unique_factorization_resultunique = ff_q_pfp_unique_factorization_resultuniquematchingbijectionsurjectiveentry * S ((S (pfp_i_unique_factorization_resultuniquematchingbijectionsurjective)) * gr_unique_scale_unique_factorization_resultunique) + (pfp_a_unique_factorization_resultuniquematchingbijectionsurjective)))))))) /\ (forall gr_match_index_unique_factorization_resultuniquematchingmatching gr_match_image_unique_factorization_resultuniquematchingmatching gr_match_source_unique_factorization_resultuniquematchingmatching gr_match_target_unique_factorization_resultuniquematchingmatching. (exists ge_gap_unique_factorization_resultuniquematchingmatchingindex. ge_gap_unique_factorization_resultuniquematchingmatchingindex + S (gr_match_index_unique_factorization_resultuniquematchingmatching) = (gr_chosen_length_unique_factorization_result)) -> (((exists ff_h_gprod_unique_factorization_resultuniquematchingmatchingmap. ff_h_gprod_unique_factorization_resultuniquematchingmatchingmap + S (gr_match_image_unique_factorization_resultuniquematchingmatching) = S ((S (gr_match_index_unique_factorization_resultuniquematchingmatching)) * gr_unique_scale_unique_factorization_resultunique)) /\ exists ff_q_gprod_unique_factorization_resultuniquematchingmatchingmap. gr_unique_map_unique_factorization_resultunique = ff_q_gprod_unique_factorization_resultuniquematchingmatchingmap * S ((S (gr_match_index_unique_factorization_resultuniquematchingmatching)) * gr_unique_scale_unique_factorization_resultunique) + (gr_match_image_unique_factorization_resultuniquematchingmatching))) -> (((exists ff_h_gprod_unique_factorization_resultuniquematchingmatchingsource. ff_h_gprod_unique_factorization_resultuniquematchingmatchingsource + S (gr_match_source_unique_factorization_resultuniquematchingmatching) = S ((S (gr_match_index_unique_factorization_resultuniquematchingmatching)) * gr_chosen_scale_unique_factorization_result)) /\ exists ff_q_gprod_unique_factorization_resultuniquematchingmatchingsource. gr_chosen_code_unique_factorization_result = ff_q_gprod_unique_factorization_resultuniquematchingmatchingsource * S ((S (gr_match_index_unique_factorization_resultuniquematchingmatching)) * gr_chosen_scale_unique_factorization_result) + (gr_match_source_unique_factorization_resultuniquematchingmatching))) -> (((exists ff_h_gprod_unique_factorization_resultuniquematchingmatchingtarget. ff_h_gprod_unique_factorization_resultuniquematchingmatchingtarget + S (gr_match_target_unique_factorization_resultuniquematchingmatching) = S ((S (gr_match_image_unique_factorization_resultuniquematchingmatching)) * gr_other_scale_unique_factorization_result)) /\ exists ff_q_gprod_unique_factorization_resultuniquematchingmatchingtarget. gr_other_code_unique_factorization_result = ff_q_gprod_unique_factorization_resultuniquematchingmatchingtarget * S ((S (gr_match_image_unique_factorization_resultuniquematchingmatching)) * gr_other_scale_unique_factorization_result) + (gr_match_target_unique_factorization_resultuniquematchingmatching))) -> (exists gr_unit_unique_factorization_resultuniquematchingmatchingunit_witness. ((exists gr_inverse_unique_factorization_resultuniquematchingmatchingunit_witnessunit. (exists ge_first_rp_unique_factorization_resultuniquematchingmatchingunit_witnessunitidentity ge_first_rn_unique_factorization_resultuniquematchingmatchingunit_witnessunitidentity ge_first_ip_unique_factorization_resultuniquematchingmatchingunit_witnessunitidentity ge_first_in_unique_factorization_resultuniquematchingmatchingunit_witnessunitidentity ge_second_rp_unique_factorization_resultuniquematchingmatchingunit_witnessunitidentity ge_second_rn_unique_factorization_resultuniquematchingmatchingunit_witnessunitidentity ge_second_ip_unique_factorization_resultuniquematchingmatchingunit_witnessunitidentity ge_second_in_unique_factorization_resultuniquematchingmatchingunit_witnessunitidentity. ((exists ge_representation_real_code_unique_factorization_resultuniquematchingmatchingunit_witnessunitidentityfirst ge_representation_imaginary_code_unique_factorization_resultuniquematchingmatchingunit_witnessunitidentityfirst. (((gr_unit_unique_factorization_resultuniquematchingmatchingunit_witness) = ((ge_representation_real_code_unique_factorization_resultuniquematchingmatchingunit_witnessunitidentityfirst) + (ge_representation_imaginary_code_unique_factorization_resultuniquematchingmatchingunit_witnessunitidentityfirst)) * S ((ge_representation_real_code_unique_factorization_resultuniquematchingmatchingunit_witnessunitidentityfirst) + (ge_representation_imaginary_code_unique_factorization_resultuniquematchingmatchingunit_witnessunitidentityfirst)) + ((ge_representation_imaginary_code_unique_factorization_resultuniquematchingmatchingunit_witnessunitidentityfirst) + (ge_representation_imaginary_code_unique_factorization_resultuniquematchingmatchingunit_witnessunitidentityfirst))) /\ ((exists ge_balance_positive_unique_factorization_resultuniquematchingmatchingunit_witnessunitidentityfirstreal ge_balance_negative_unique_factorization_resultuniquematchingmatchingunit_witnessunitidentityfirstreal. (((((ge_representation_real_code_unique_factorization_resultuniquematchingmatchingunit_witnessunitidentityfirst) = 2 * (ge_balance_positive_unique_factorization_resultuniquematchingmatchingunit_witnessunitidentityfirstreal) /\ (ge_balance_negative_unique_factorization_resultuniquematchingmatchingunit_witnessunitidentityfirstreal) = 0) \/ exists ge_signed_half_unique_factorization_resultuniquematchingmatchingunit_witnessunitidentityfirstrealdecode. (((ge_representation_real_code_unique_factorization_resultuniquematchingmatchingunit_witnessunitidentityfirst) = 2 * ge_signed_half_unique_factorization_resultuniquematchingmatchingunit_witnessunitidentityfirstrealdecode + 1 /\ (ge_balance_positive_unique_factorization_resultuniquematchingmatchingunit_witnessunitidentityfirstreal) = 0) /\ (ge_balance_negative_unique_factorization_resultuniquematchingmatchingunit_witnessunitidentityfirstreal) = S ge_signed_half_unique_factorization_resultuniquematchingmatchingunit_witnessunitidentityfirstrealdecode))) /\ ((ge_first_rp_unique_factorization_resultuniquematchingmatchingunit_witnessunitidentity) + ge_balance_negative_unique_factorization_resultuniquematchingmatchingunit_witnessunitidentityfirstreal = (ge_first_rn_unique_factorization_resultuniquematchingmatchingunit_witnessunitidentity) + ge_balance_positive_unique_factorization_resultuniquematchingmatchingunit_witnessunitidentityfirstreal))) /\ (exists ge_balance_positive_unique_factorization_resultuniquematchingmatchingunit_witnessunitidentityfirstimaginary ge_balance_negative_unique_factorization_resultuniquematchingmatchingunit_witnessunitidentityfirstimaginary. (((((ge_representation_imaginary_code_unique_factorization_resultuniquematchingmatchingunit_witnessunitidentityfirst) = 2 * (ge_balance_positive_unique_factorization_resultuniquematchingmatchingunit_witnessunitidentityfirstimaginary) /\ (ge_balance_negative_unique_factorization_resultuniquematchingmatchingunit_witnessunitidentityfirstimaginary) = 0) \/ exists ge_signed_half_unique_factorization_resultuniquematchingmatchingunit_witnessunitidentityfirstimaginarydecode. (((ge_representation_imaginary_code_unique_factorization_resultuniquematchingmatchingunit_witnessunitidentityfirst) = 2 * ge_signed_half_unique_factorization_resultuniquematchingmatchingunit_witnessunitidentityfirstimaginarydecode + 1 /\ (ge_balance_positive_unique_factorization_resultuniquematchingmatchingunit_witnessunitidentityfirstimaginary) = 0) /\ (ge_balance_negative_unique_factorization_resultuniquematchingmatchingunit_witnessunitidentityfirstimaginary) = S ge_signed_half_unique_factorization_resultuniquematchingmatchingunit_witnessunitidentityfirstimaginarydecode))) /\ ((ge_first_ip_unique_factorization_resultuniquematchingmatchingunit_witnessunitidentity) + ge_balance_negative_unique_factorization_resultuniquematchingmatchingunit_witnessunitidentityfirstimaginary = (ge_first_in_unique_factorization_resultuniquematchingmatchingunit_witnessunitidentity) + ge_balance_positive_unique_factorization_resultuniquematchingmatchingunit_witnessunitidentityfirstimaginary)))))) /\ ((exists ge_representation_real_code_unique_factorization_resultuniquematchingmatchingunit_witnessunitidentitysecond ge_representation_imaginary_code_unique_factorization_resultuniquematchingmatchingunit_witnessunitidentitysecond. (((gr_inverse_unique_factorization_resultuniquematchingmatchingunit_witnessunit) = ((ge_representation_real_code_unique_factorization_resultuniquematchingmatchingunit_witnessunitidentitysecond) + (ge_representation_imaginary_code_unique_factorization_resultuniquematchingmatchingunit_witnessunitidentitysecond)) * S ((ge_representation_real_code_unique_factorization_resultuniquematchingmatchingunit_witnessunitidentitysecond) + (ge_representation_imaginary_code_unique_factorization_resultuniquematchingmatchingunit_witnessunitidentitysecond)) + ((ge_representation_imaginary_code_unique_factorization_resultuniquematchingmatchingunit_witnessunitidentitysecond) + (ge_representation_imaginary_code_unique_factorization_resultuniquematchingmatchingunit_witnessunitidentitysecond))) /\ ((exists ge_balance_positive_unique_factorization_resultuniquematchingmatchingunit_witnessunitidentitysecondreal ge_balance_negative_unique_factorization_resultuniquematchingmatchingunit_witnessunitidentitysecondreal. (((((ge_representation_real_code_unique_factorization_resultuniquematchingmatchingunit_witnessunitidentitysecond) = 2 * (ge_balance_positive_unique_factorization_resultuniquematchingmatchingunit_witnessunitidentitysecondreal) /\ (ge_balance_negative_unique_factorization_resultuniquematchingmatchingunit_witnessunitidentitysecondreal) = 0) \/ exists ge_signed_half_unique_factorization_resultuniquematchingmatchingunit_witnessunitidentitysecondrealdecode. (((ge_representation_real_code_unique_factorization_resultuniquematchingmatchingunit_witnessunitidentitysecond) = 2 * ge_signed_half_unique_factorization_resultuniquematchingmatchingunit_witnessunitidentitysecondrealdecode + 1 /\ (ge_balance_positive_unique_factorization_resultuniquematchingmatchingunit_witnessunitidentitysecondreal) = 0) /\ (ge_balance_negative_unique_factorization_resultuniquematchingmatchingunit_witnessunitidentitysecondreal) = S ge_signed_half_unique_factorization_resultuniquematchingmatchingunit_witnessunitidentitysecondrealdecode))) /\ ((ge_second_rp_unique_factorization_resultuniquematchingmatchingunit_witnessunitidentity) + ge_balance_negative_unique_factorization_resultuniquematchingmatchingunit_witnessunitidentitysecondreal = (ge_second_rn_unique_factorization_resultuniquematchingmatchingunit_witnessunitidentity) + ge_balance_positive_unique_factorization_resultuniquematchingmatchingunit_witnessunitidentitysecondreal))) /\ (exists ge_balance_positive_unique_factorization_resultuniquematchingmatchingunit_witnessunitidentitysecondimaginary ge_balance_negative_unique_factorization_resultuniquematchingmatchingunit_witnessunitidentitysecondimaginary. (((((ge_representation_imaginary_code_unique_factorization_resultuniquematchingmatchingunit_witnessunitidentitysecond) = 2 * (ge_balance_positive_unique_factorization_resultuniquematchingmatchingunit_witnessunitidentitysecondimaginary) /\ (ge_balance_negative_unique_factorization_resultuniquematchingmatchingunit_witnessunitidentitysecondimaginary) = 0) \/ exists ge_signed_half_unique_factorization_resultuniquematchingmatchingunit_witnessunitidentitysecondimaginarydecode. (((ge_representation_imaginary_code_unique_factorization_resultuniquematchingmatchingunit_witnessunitidentitysecond) = 2 * ge_signed_half_unique_factorization_resultuniquematchingmatchingunit_witnessunitidentitysecondimaginarydecode + 1 /\ (ge_balance_positive_unique_factorization_resultuniquematchingmatchingunit_witnessunitidentitysecondimaginary) = 0) /\ (ge_balance_negative_unique_factorization_resultuniquematchingmatchingunit_witnessunitidentitysecondimaginary) = S ge_signed_half_unique_factorization_resultuniquematchingmatchingunit_witnessunitidentitysecondimaginarydecode))) /\ ((ge_second_ip_unique_factorization_resultuniquematchingmatchingunit_witnessunitidentity) + ge_balance_negative_unique_factorization_resultuniquematchingmatchingunit_witnessunitidentitysecondimaginary = (ge_second_in_unique_factorization_resultuniquematchingmatchingunit_witnessunitidentity) + ge_balance_positive_unique_factorization_resultuniquematchingmatchingunit_witnessunitidentitysecondimaginary)))))) /\ (exists ge_representation_real_code_unique_factorization_resultuniquematchingmatchingunit_witnessunitidentityoutput ge_representation_imaginary_code_unique_factorization_resultuniquematchingmatchingunit_witnessunitidentityoutput. (((6) = ((ge_representation_real_code_unique_factorization_resultuniquematchingmatchingunit_witnessunitidentityoutput) + (ge_representation_imaginary_code_unique_factorization_resultuniquematchingmatchingunit_witnessunitidentityoutput)) * S ((ge_representation_real_code_unique_factorization_resultuniquematchingmatchingunit_witnessunitidentityoutput) + (ge_representation_imaginary_code_unique_factorization_resultuniquematchingmatchingunit_witnessunitidentityoutput)) + ((ge_representation_imaginary_code_unique_factorization_resultuniquematchingmatchingunit_witnessunitidentityoutput) + (ge_representation_imaginary_code_unique_factorization_resultuniquematchingmatchingunit_witnessunitidentityoutput))) /\ ((exists ge_balance_positive_unique_factorization_resultuniquematchingmatchingunit_witnessunitidentityoutputreal ge_balance_negative_unique_factorization_resultuniquematchingmatchingunit_witnessunitidentityoutputreal. (((((ge_representation_real_code_unique_factorization_resultuniquematchingmatchingunit_witnessunitidentityoutput) = 2 * (ge_balance_positive_unique_factorization_resultuniquematchingmatchingunit_witnessunitidentityoutputreal) /\ (ge_balance_negative_unique_factorization_resultuniquematchingmatchingunit_witnessunitidentityoutputreal) = 0) \/ exists ge_signed_half_unique_factorization_resultuniquematchingmatchingunit_witnessunitidentityoutputrealdecode. (((ge_representation_real_code_unique_factorization_resultuniquematchingmatchingunit_witnessunitidentityoutput) = 2 * ge_signed_half_unique_factorization_resultuniquematchingmatchingunit_witnessunitidentityoutputrealdecode + 1 /\ (ge_balance_positive_unique_factorization_resultuniquematchingmatchingunit_witnessunitidentityoutputreal) = 0) /\ (ge_balance_negative_unique_factorization_resultuniquematchingmatchingunit_witnessunitidentityoutputreal) = S ge_signed_half_unique_factorization_resultuniquematchingmatchingunit_witnessunitidentityoutputrealdecode))) /\ ((((((((ge_first_rp_unique_factorization_resultuniquematchingmatchingunit_witnessunitidentity) * (ge_second_rp_unique_factorization_resultuniquematchingmatchingunit_witnessunitidentity))) + (((ge_first_rn_unique_factorization_resultuniquematchingmatchingunit_witnessunitidentity) * (ge_second_rn_unique_factorization_resultuniquematchingmatchingunit_witnessunitidentity))))) + (((((ge_first_ip_unique_factorization_resultuniquematchingmatchingunit_witnessunitidentity) * (ge_second_in_unique_factorization_resultuniquematchingmatchingunit_witnessunitidentity))) + (((ge_first_in_unique_factorization_resultuniquematchingmatchingunit_witnessunitidentity) * (ge_second_ip_unique_factorization_resultuniquematchingmatchingunit_witnessunitidentity))))))) + ge_balance_negative_unique_factorization_resultuniquematchingmatchingunit_witnessunitidentityoutputreal = (((((((ge_first_rp_unique_factorization_resultuniquematchingmatchingunit_witnessunitidentity) * (ge_second_rn_unique_factorization_resultuniquematchingmatchingunit_witnessunitidentity))) + (((ge_first_rn_unique_factorization_resultuniquematchingmatchingunit_witnessunitidentity) * (ge_second_rp_unique_factorization_resultuniquematchingmatchingunit_witnessunitidentity))))) + (((((ge_first_ip_unique_factorization_resultuniquematchingmatchingunit_witnessunitidentity) * (ge_second_ip_unique_factorization_resultuniquematchingmatchingunit_witnessunitidentity))) + (((ge_first_in_unique_factorization_resultuniquematchingmatchingunit_witnessunitidentity) * (ge_second_in_unique_factorization_resultuniquematchingmatchingunit_witnessunitidentity))))))) + ge_balance_positive_unique_factorization_resultuniquematchingmatchingunit_witnessunitidentityoutputreal))) /\ (exists ge_balance_positive_unique_factorization_resultuniquematchingmatchingunit_witnessunitidentityoutputimaginary ge_balance_negative_unique_factorization_resultuniquematchingmatchingunit_witnessunitidentityoutputimaginary. (((((ge_representation_imaginary_code_unique_factorization_resultuniquematchingmatchingunit_witnessunitidentityoutput) = 2 * (ge_balance_positive_unique_factorization_resultuniquematchingmatchingunit_witnessunitidentityoutputimaginary) /\ (ge_balance_negative_unique_factorization_resultuniquematchingmatchingunit_witnessunitidentityoutputimaginary) = 0) \/ exists ge_signed_half_unique_factorization_resultuniquematchingmatchingunit_witnessunitidentityoutputimaginarydecode. (((ge_representation_imaginary_code_unique_factorization_resultuniquematchingmatchingunit_witnessunitidentityoutput) = 2 * ge_signed_half_unique_factorization_resultuniquematchingmatchingunit_witnessunitidentityoutputimaginarydecode + 1 /\ (ge_balance_positive_unique_factorization_resultuniquematchingmatchingunit_witnessunitidentityoutputimaginary) = 0) /\ (ge_balance_negative_unique_factorization_resultuniquematchingmatchingunit_witnessunitidentityoutputimaginary) = S ge_signed_half_unique_factorization_resultuniquematchingmatchingunit_witnessunitidentityoutputimaginarydecode))) /\ ((((((((ge_first_rp_unique_factorization_resultuniquematchingmatchingunit_witnessunitidentity) * (ge_second_ip_unique_factorization_resultuniquematchingmatchingunit_witnessunitidentity))) + (((ge_first_rn_unique_factorization_resultuniquematchingmatchingunit_witnessunitidentity) * (ge_second_in_unique_factorization_resultuniquematchingmatchingunit_witnessunitidentity))))) + (((((ge_first_ip_unique_factorization_resultuniquematchingmatchingunit_witnessunitidentity) * (ge_second_rp_unique_factorization_resultuniquematchingmatchingunit_witnessunitidentity))) + (((ge_first_in_unique_factorization_resultuniquematchingmatchingunit_witnessunitidentity) * (ge_second_rn_unique_factorization_resultuniquematchingmatchingunit_witnessunitidentity))))))) + ge_balance_negative_unique_factorization_resultuniquematchingmatchingunit_witnessunitidentityoutputimaginary = (((((((ge_first_rp_unique_factorization_resultuniquematchingmatchingunit_witnessunitidentity) * (ge_second_in_unique_factorization_resultuniquematchingmatchingunit_witnessunitidentity))) + (((ge_first_rn_unique_factorization_resultuniquematchingmatchingunit_witnessunitidentity) * (ge_second_ip_unique_factorization_resultuniquematchingmatchingunit_witnessunitidentity))))) + (((((ge_first_ip_unique_factorization_resultuniquematchingmatchingunit_witnessunitidentity) * (ge_second_rn_unique_factorization_resultuniquematchingmatchingunit_witnessunitidentity))) + (((ge_first_in_unique_factorization_resultuniquematchingmatchingunit_witnessunitidentity) * (ge_second_rp_unique_factorization_resultuniquematchingmatchingunit_witnessunitidentity))))))) + ge_balance_positive_unique_factorization_resultuniquematchingmatchingunit_witnessunitidentityoutputimaginary)))))))))) /\ (exists ge_first_rp_unique_factorization_resultuniquematchingmatchingunit_witnesstransport ge_first_rn_unique_factorization_resultuniquematchingmatchingunit_witnesstransport ge_first_ip_unique_factorization_resultuniquematchingmatchingunit_witnesstransport ge_first_in_unique_factorization_resultuniquematchingmatchingunit_witnesstransport ge_second_rp_unique_factorization_resultuniquematchingmatchingunit_witnesstransport ge_second_rn_unique_factorization_resultuniquematchingmatchingunit_witnesstransport ge_second_ip_unique_factorization_resultuniquematchingmatchingunit_witnesstransport ge_second_in_unique_factorization_resultuniquematchingmatchingunit_witnesstransport. ((exists ge_representation_real_code_unique_factorization_resultuniquematchingmatchingunit_witnesstransportfirst ge_representation_imaginary_code_unique_factorization_resultuniquematchingmatchingunit_witnesstransportfirst. (((gr_unit_unique_factorization_resultuniquematchingmatchingunit_witness) = ((ge_representation_real_code_unique_factorization_resultuniquematchingmatchingunit_witnesstransportfirst) + (ge_representation_imaginary_code_unique_factorization_resultuniquematchingmatchingunit_witnesstransportfirst)) * S ((ge_representation_real_code_unique_factorization_resultuniquematchingmatchingunit_witnesstransportfirst) + (ge_representation_imaginary_code_unique_factorization_resultuniquematchingmatchingunit_witnesstransportfirst)) + ((ge_representation_imaginary_code_unique_factorization_resultuniquematchingmatchingunit_witnesstransportfirst) + (ge_representation_imaginary_code_unique_factorization_resultuniquematchingmatchingunit_witnesstransportfirst))) /\ ((exists ge_balance_positive_unique_factorization_resultuniquematchingmatchingunit_witnesstransportfirstreal ge_balance_negative_unique_factorization_resultuniquematchingmatchingunit_witnesstransportfirstreal. (((((ge_representation_real_code_unique_factorization_resultuniquematchingmatchingunit_witnesstransportfirst) = 2 * (ge_balance_positive_unique_factorization_resultuniquematchingmatchingunit_witnesstransportfirstreal) /\ (ge_balance_negative_unique_factorization_resultuniquematchingmatchingunit_witnesstransportfirstreal) = 0) \/ exists ge_signed_half_unique_factorization_resultuniquematchingmatchingunit_witnesstransportfirstrealdecode. (((ge_representation_real_code_unique_factorization_resultuniquematchingmatchingunit_witnesstransportfirst) = 2 * ge_signed_half_unique_factorization_resultuniquematchingmatchingunit_witnesstransportfirstrealdecode + 1 /\ (ge_balance_positive_unique_factorization_resultuniquematchingmatchingunit_witnesstransportfirstreal) = 0) /\ (ge_balance_negative_unique_factorization_resultuniquematchingmatchingunit_witnesstransportfirstreal) = S ge_signed_half_unique_factorization_resultuniquematchingmatchingunit_witnesstransportfirstrealdecode))) /\ ((ge_first_rp_unique_factorization_resultuniquematchingmatchingunit_witnesstransport) + ge_balance_negative_unique_factorization_resultuniquematchingmatchingunit_witnesstransportfirstreal = (ge_first_rn_unique_factorization_resultuniquematchingmatchingunit_witnesstransport) + ge_balance_positive_unique_factorization_resultuniquematchingmatchingunit_witnesstransportfirstreal))) /\ (exists ge_balance_positive_unique_factorization_resultuniquematchingmatchingunit_witnesstransportfirstimaginary ge_balance_negative_unique_factorization_resultuniquematchingmatchingunit_witnesstransportfirstimaginary. (((((ge_representation_imaginary_code_unique_factorization_resultuniquematchingmatchingunit_witnesstransportfirst) = 2 * (ge_balance_positive_unique_factorization_resultuniquematchingmatchingunit_witnesstransportfirstimaginary) /\ (ge_balance_negative_unique_factorization_resultuniquematchingmatchingunit_witnesstransportfirstimaginary) = 0) \/ exists ge_signed_half_unique_factorization_resultuniquematchingmatchingunit_witnesstransportfirstimaginarydecode. (((ge_representation_imaginary_code_unique_factorization_resultuniquematchingmatchingunit_witnesstransportfirst) = 2 * ge_signed_half_unique_factorization_resultuniquematchingmatchingunit_witnesstransportfirstimaginarydecode + 1 /\ (ge_balance_positive_unique_factorization_resultuniquematchingmatchingunit_witnesstransportfirstimaginary) = 0) /\ (ge_balance_negative_unique_factorization_resultuniquematchingmatchingunit_witnesstransportfirstimaginary) = S ge_signed_half_unique_factorization_resultuniquematchingmatchingunit_witnesstransportfirstimaginarydecode))) /\ ((ge_first_ip_unique_factorization_resultuniquematchingmatchingunit_witnesstransport) + ge_balance_negative_unique_factorization_resultuniquematchingmatchingunit_witnesstransportfirstimaginary = (ge_first_in_unique_factorization_resultuniquematchingmatchingunit_witnesstransport) + ge_balance_positive_unique_factorization_resultuniquematchingmatchingunit_witnesstransportfirstimaginary)))))) /\ ((exists ge_representation_real_code_unique_factorization_resultuniquematchingmatchingunit_witnesstransportsecond ge_representation_imaginary_code_unique_factorization_resultuniquematchingmatchingunit_witnesstransportsecond. (((gr_match_source_unique_factorization_resultuniquematchingmatching) = ((ge_representation_real_code_unique_factorization_resultuniquematchingmatchingunit_witnesstransportsecond) + (ge_representation_imaginary_code_unique_factorization_resultuniquematchingmatchingunit_witnesstransportsecond)) * S ((ge_representation_real_code_unique_factorization_resultuniquematchingmatchingunit_witnesstransportsecond) + (ge_representation_imaginary_code_unique_factorization_resultuniquematchingmatchingunit_witnesstransportsecond)) + ((ge_representation_imaginary_code_unique_factorization_resultuniquematchingmatchingunit_witnesstransportsecond) + (ge_representation_imaginary_code_unique_factorization_resultuniquematchingmatchingunit_witnesstransportsecond))) /\ ((exists ge_balance_positive_unique_factorization_resultuniquematchingmatchingunit_witnesstransportsecondreal ge_balance_negative_unique_factorization_resultuniquematchingmatchingunit_witnesstransportsecondreal. (((((ge_representation_real_code_unique_factorization_resultuniquematchingmatchingunit_witnesstransportsecond) = 2 * (ge_balance_positive_unique_factorization_resultuniquematchingmatchingunit_witnesstransportsecondreal) /\ (ge_balance_negative_unique_factorization_resultuniquematchingmatchingunit_witnesstransportsecondreal) = 0) \/ exists ge_signed_half_unique_factorization_resultuniquematchingmatchingunit_witnesstransportsecondrealdecode. (((ge_representation_real_code_unique_factorization_resultuniquematchingmatchingunit_witnesstransportsecond) = 2 * ge_signed_half_unique_factorization_resultuniquematchingmatchingunit_witnesstransportsecondrealdecode + 1 /\ (ge_balance_positive_unique_factorization_resultuniquematchingmatchingunit_witnesstransportsecondreal) = 0) /\ (ge_balance_negative_unique_factorization_resultuniquematchingmatchingunit_witnesstransportsecondreal) = S ge_signed_half_unique_factorization_resultuniquematchingmatchingunit_witnesstransportsecondrealdecode))) /\ ((ge_second_rp_unique_factorization_resultuniquematchingmatchingunit_witnesstransport) + ge_balance_negative_unique_factorization_resultuniquematchingmatchingunit_witnesstransportsecondreal = (ge_second_rn_unique_factorization_resultuniquematchingmatchingunit_witnesstransport) + ge_balance_positive_unique_factorization_resultuniquematchingmatchingunit_witnesstransportsecondreal))) /\ (exists ge_balance_positive_unique_factorization_resultuniquematchingmatchingunit_witnesstransportsecondimaginary ge_balance_negative_unique_factorization_resultuniquematchingmatchingunit_witnesstransportsecondimaginary. (((((ge_representation_imaginary_code_unique_factorization_resultuniquematchingmatchingunit_witnesstransportsecond) = 2 * (ge_balance_positive_unique_factorization_resultuniquematchingmatchingunit_witnesstransportsecondimaginary) /\ (ge_balance_negative_unique_factorization_resultuniquematchingmatchingunit_witnesstransportsecondimaginary) = 0) \/ exists ge_signed_half_unique_factorization_resultuniquematchingmatchingunit_witnesstransportsecondimaginarydecode. (((ge_representation_imaginary_code_unique_factorization_resultuniquematchingmatchingunit_witnesstransportsecond) = 2 * ge_signed_half_unique_factorization_resultuniquematchingmatchingunit_witnesstransportsecondimaginarydecode + 1 /\ (ge_balance_positive_unique_factorization_resultuniquematchingmatchingunit_witnesstransportsecondimaginary) = 0) /\ (ge_balance_negative_unique_factorization_resultuniquematchingmatchingunit_witnesstransportsecondimaginary) = S ge_signed_half_unique_factorization_resultuniquematchingmatchingunit_witnesstransportsecondimaginarydecode))) /\ ((ge_second_ip_unique_factorization_resultuniquematchingmatchingunit_witnesstransport) + ge_balance_negative_unique_factorization_resultuniquematchingmatchingunit_witnesstransportsecondimaginary = (ge_second_in_unique_factorization_resultuniquematchingmatchingunit_witnesstransport) + ge_balance_positive_unique_factorization_resultuniquematchingmatchingunit_witnesstransportsecondimaginary)))))) /\ (exists ge_representation_real_code_unique_factorization_resultuniquematchingmatchingunit_witnesstransportoutput ge_representation_imaginary_code_unique_factorization_resultuniquematchingmatchingunit_witnesstransportoutput. (((gr_match_target_unique_factorization_resultuniquematchingmatching) = ((ge_representation_real_code_unique_factorization_resultuniquematchingmatchingunit_witnesstransportoutput) + (ge_representation_imaginary_code_unique_factorization_resultuniquematchingmatchingunit_witnesstransportoutput)) * S ((ge_representation_real_code_unique_factorization_resultuniquematchingmatchingunit_witnesstransportoutput) + (ge_representation_imaginary_code_unique_factorization_resultuniquematchingmatchingunit_witnesstransportoutput)) + ((ge_representation_imaginary_code_unique_factorization_resultuniquematchingmatchingunit_witnesstransportoutput) + (ge_representation_imaginary_code_unique_factorization_resultuniquematchingmatchingunit_witnesstransportoutput))) /\ ((exists ge_balance_positive_unique_factorization_resultuniquematchingmatchingunit_witnesstransportoutputreal ge_balance_negative_unique_factorization_resultuniquematchingmatchingunit_witnesstransportoutputreal. (((((ge_representation_real_code_unique_factorization_resultuniquematchingmatchingunit_witnesstransportoutput) = 2 * (ge_balance_positive_unique_factorization_resultuniquematchingmatchingunit_witnesstransportoutputreal) /\ (ge_balance_negative_unique_factorization_resultuniquematchingmatchingunit_witnesstransportoutputreal) = 0) \/ exists ge_signed_half_unique_factorization_resultuniquematchingmatchingunit_witnesstransportoutputrealdecode. (((ge_representation_real_code_unique_factorization_resultuniquematchingmatchingunit_witnesstransportoutput) = 2 * ge_signed_half_unique_factorization_resultuniquematchingmatchingunit_witnesstransportoutputrealdecode + 1 /\ (ge_balance_positive_unique_factorization_resultuniquematchingmatchingunit_witnesstransportoutputreal) = 0) /\ (ge_balance_negative_unique_factorization_resultuniquematchingmatchingunit_witnesstransportoutputreal) = S ge_signed_half_unique_factorization_resultuniquematchingmatchingunit_witnesstransportoutputrealdecode))) /\ ((((((((ge_first_rp_unique_factorization_resultuniquematchingmatchingunit_witnesstransport) * (ge_second_rp_unique_factorization_resultuniquematchingmatchingunit_witnesstransport))) + (((ge_first_rn_unique_factorization_resultuniquematchingmatchingunit_witnesstransport) * (ge_second_rn_unique_factorization_resultuniquematchingmatchingunit_witnesstransport))))) + (((((ge_first_ip_unique_factorization_resultuniquematchingmatchingunit_witnesstransport) * (ge_second_in_unique_factorization_resultuniquematchingmatchingunit_witnesstransport))) + (((ge_first_in_unique_factorization_resultuniquematchingmatchingunit_witnesstransport) * (ge_second_ip_unique_factorization_resultuniquematchingmatchingunit_witnesstransport))))))) + ge_balance_negative_unique_factorization_resultuniquematchingmatchingunit_witnesstransportoutputreal = (((((((ge_first_rp_unique_factorization_resultuniquematchingmatchingunit_witnesstransport) * (ge_second_rn_unique_factorization_resultuniquematchingmatchingunit_witnesstransport))) + (((ge_first_rn_unique_factorization_resultuniquematchingmatchingunit_witnesstransport) * (ge_second_rp_unique_factorization_resultuniquematchingmatchingunit_witnesstransport))))) + (((((ge_first_ip_unique_factorization_resultuniquematchingmatchingunit_witnesstransport) * (ge_second_ip_unique_factorization_resultuniquematchingmatchingunit_witnesstransport))) + (((ge_first_in_unique_factorization_resultuniquematchingmatchingunit_witnesstransport) * (ge_second_in_unique_factorization_resultuniquematchingmatchingunit_witnesstransport))))))) + ge_balance_positive_unique_factorization_resultuniquematchingmatchingunit_witnesstransportoutputreal))) /\ (exists ge_balance_positive_unique_factorization_resultuniquematchingmatchingunit_witnesstransportoutputimaginary ge_balance_negative_unique_factorization_resultuniquematchingmatchingunit_witnesstransportoutputimaginary. (((((ge_representation_imaginary_code_unique_factorization_resultuniquematchingmatchingunit_witnesstransportoutput) = 2 * (ge_balance_positive_unique_factorization_resultuniquematchingmatchingunit_witnesstransportoutputimaginary) /\ (ge_balance_negative_unique_factorization_resultuniquematchingmatchingunit_witnesstransportoutputimaginary) = 0) \/ exists ge_signed_half_unique_factorization_resultuniquematchingmatchingunit_witnesstransportoutputimaginarydecode. (((ge_representation_imaginary_code_unique_factorization_resultuniquematchingmatchingunit_witnesstransportoutput) = 2 * ge_signed_half_unique_factorization_resultuniquematchingmatchingunit_witnesstransportoutputimaginarydecode + 1 /\ (ge_balance_positive_unique_factorization_resultuniquematchingmatchingunit_witnesstransportoutputimaginary) = 0) /\ (ge_balance_negative_unique_factorization_resultuniquematchingmatchingunit_witnesstransportoutputimaginary) = S ge_signed_half_unique_factorization_resultuniquematchingmatchingunit_witnesstransportoutputimaginarydecode))) /\ ((((((((ge_first_rp_unique_factorization_resultuniquematchingmatchingunit_witnesstransport) * (ge_second_ip_unique_factorization_resultuniquematchingmatchingunit_witnesstransport))) + (((ge_first_rn_unique_factorization_resultuniquematchingmatchingunit_witnesstransport) * (ge_second_in_unique_factorization_resultuniquematchingmatchingunit_witnesstransport))))) + (((((ge_first_ip_unique_factorization_resultuniquematchingmatchingunit_witnesstransport) * (ge_second_rp_unique_factorization_resultuniquematchingmatchingunit_witnesstransport))) + (((ge_first_in_unique_factorization_resultuniquematchingmatchingunit_witnesstransport) * (ge_second_rn_unique_factorization_resultuniquematchingmatchingunit_witnesstransport))))))) + ge_balance_negative_unique_factorization_resultuniquematchingmatchingunit_witnesstransportoutputimaginary = (((((((ge_first_rp_unique_factorization_resultuniquematchingmatchingunit_witnesstransport) * (ge_second_in_unique_factorization_resultuniquematchingmatchingunit_witnesstransport))) + (((ge_first_rn_unique_factorization_resultuniquematchingmatchingunit_witnesstransport) * (ge_second_ip_unique_factorization_resultuniquematchingmatchingunit_witnesstransport))))) + (((((ge_first_ip_unique_factorization_resultuniquematchingmatchingunit_witnesstransport) * (ge_second_rn_unique_factorization_resultuniquematchingmatchingunit_witnesstransport))) + (((ge_first_in_unique_factorization_resultuniquematchingmatchingunit_witnesstransport) * (ge_second_rp_unique_factorization_resultuniquematchingmatchingunit_witnesstransport))))))) + ge_balance_positive_unique_factorization_resultuniquematchingmatchingunit_witnesstransportoutputimaginary))))))))))))))))))))Constructive proof overview
Generated structural guide
Full constructive G082: every actual nonzero Gaussian integer has a finite actual RingPrime factorization, and every other such factorization differs by a constructed beta permutation and witnessed multiplicative units.
The unchanged tactic script uses 2 declared prerequisites and contains 35 exact native proof lines.
Alpha v34 checked-use · first admitted v30 · independently kernel and Lean verified; not Stable
Proof neighborhood
Direct dependencies
Direct dependents
none
Formal native tactic body
Dependencies are introduced as named hypotheses before line 1. Local theorem links identify exact declared prerequisites. This exact body belongs to a complete independently kernel-checked constructive proof bundle and has Alpha checked-use authority; it does not imply Stable membership.
Read the argument
Proof checkpoints
This is a reading aid, not a new proof or a proof-tree certificate. Checkpoint groups are consecutive commands, not inferred branch boundaries. Every step links to the preserved script.
Named ingredients (2)
01Fix variables and assumptionsL1–3
02Establish hfL4–8
Establish this local claim before using it. It is not an additional assumption. The following proof commands apply gaussian prime factorization exists.
03Separate the logical casesL9–12
04Construct an explicit witnessL13–16
05Separate the logical casesL17–17
Follow the explicit conjunction, disjunction, witness, or contradiction step recorded below.
- L17
split
06Use earlier factsL18–18
Instantiate or apply named facts and discharge the corresponding proof obligations.
- L18
exact hf_witness_witness_witness_witness
07Fix variables and assumptionsL19–23
08Use earlier factsL24–33
Instantiate or apply named facts and discharge the corresponding proof obligations.
- L24
specialize gaussian_prime_factorizations_unique (z) - L25
specialize gaussian_prime_factorizations_unique (x) - L26
specialize gaussian_prime_factorizations_unique (x1) - L27
specialize gaussian_prime_factorizations_unique (x2) - L28
specialize gaussian_prime_factorizations_unique (x3) - L29
specialize gaussian_prime_factorizations_unique (v) - L30
specialize gaussian_prime_factorizations_unique (d) - L31
specialize gaussian_prime_factorizations_unique (e) - L32
specialize gaussian_prime_factorizations_unique (m) - L33
apply gaussian_prime_factorizations_unique
Original exact command ledger · 35 lines
- 0001
intro z - 0002
intro hv - 0003
intro hz - 0004
have hf : exists u b c l. (((exists gr_inverse_unique_factorization_constructedunit. (exists ge_first_rp_unique_factorization_constructedunitidentity ge_first_rn_unique_factorization_constructedunitidentity ge_first_ip_unique_factorization_constructedunitidentity ge_first_in_unique_factorization_constructedunitidentity ge_second_rp_unique_factorization_constructedunitidentity ge_second_rn_unique_factorization_constructedunitidentity ge_second_ip_unique_factorization_constructedunitidentity ge_second_in_unique_factorization_constructedunitidentity. ((exists ge_representation_real_code_unique_factorization_constructedunitidentityfirst ge_representation_imaginary_code_unique_factorization_constructedunitidentityfirst. (((u) = ((ge_representation_real_code_unique_factorization_constructedunitidentityfirst) + (ge_representation_imaginary_code_unique_factorization_constructedunitidentityfirst)) * S ((ge_representation_real_code_unique_factorization_constructedunitidentityfirst) + (ge_representation_imaginary_code_unique_factorization_constructedunitidentityfirst)) + ((ge_representation_imaginary_code_unique_factorization_constructedunitidentityfirst) + (ge_representation_imaginary_code_unique_factorization_constructedunitidentityfirst))) /\ ((exists ge_balance_positive_unique_factorization_constructedunitidentityfirstreal ge_balance_negative_unique_factorization_constructedunitidentityfirstreal. (((((ge_representation_real_code_unique_factorization_constructedunitidentityfirst) = 2 * (ge_balance_positive_unique_factorization_constructedunitidentityfirstreal) /\ (ge_balance_negative_unique_factorization_constructedunitidentityfirstreal) = 0) \/ exists ge_signed_half_unique_factorization_constructedunitidentityfirstrealdecode. (((ge_representation_real_code_unique_factorization_constructedunitidentityfirst) = 2 * ge_signed_half_unique_factorization_constructedunitidentityfirstrealdecode + 1 /\ (ge_balance_positive_unique_factorization_constructedunitidentityfirstreal) = 0) /\ (ge_balance_negative_unique_factorization_constructedunitidentityfirstreal) = S ge_signed_half_unique_factorization_constructedunitidentityfirstrealdecode))) /\ ((ge_first_rp_unique_factorization_constructedunitidentity) + ge_balance_negative_unique_factorization_constructedunitidentityfirstreal = (ge_first_rn_unique_factorization_constructedunitidentity) + ge_balance_positive_unique_factorization_constructedunitidentityfirstreal))) /\ (exists ge_balance_positive_unique_factorization_constructedunitidentityfirstimaginary ge_balance_negative_unique_factorization_constructedunitidentityfirstimaginary. (((((ge_representation_imaginary_code_unique_factorization_constructedunitidentityfirst) = 2 * (ge_balance_positive_unique_factorization_constructedunitidentityfirstimaginary) /\ (ge_balance_negative_unique_factorization_constructedunitidentityfirstimaginary) = 0) \/ exists ge_signed_half_unique_factorization_constructedunitidentityfirstimaginarydecode. (((ge_representation_imaginary_code_unique_factorization_constructedunitidentityfirst) = 2 * ge_signed_half_unique_factorization_constructedunitidentityfirstimaginarydecode + 1 /\ (ge_balance_positive_unique_factorization_constructedunitidentityfirstimaginary) = 0) /\ (ge_balance_negative_unique_factorization_constructedunitidentityfirstimaginary) = S ge_signed_half_unique_factorization_constructedunitidentityfirstimaginarydecode))) /\ ((ge_first_ip_unique_factorization_constructedunitidentity) + ge_balance_negative_unique_factorization_constructedunitidentityfirstimaginary = (ge_first_in_unique_factorization_constructedunitidentity) + ge_balance_positive_unique_factorization_constructedunitidentityfirstimaginary)))))) /\ ((exists ge_representation_real_code_unique_factorization_constructedunitidentitysecond ge_representation_imaginary_code_unique_factorization_constructedunitidentitysecond. (((gr_inverse_unique_factorization_constructedunit) = ((ge_representation_real_code_unique_factorization_constructedunitidentitysecond) + (ge_representation_imaginary_code_unique_factorization_constructedunitidentitysecond)) * S ((ge_representation_real_code_unique_factorization_constructedunitidentitysecond) + (ge_representation_imaginary_code_unique_factorization_constructedunitidentitysecond)) + ((ge_representation_imaginary_code_unique_factorization_constructedunitidentitysecond) + (ge_representation_imaginary_code_unique_factorization_constructedunitidentitysecond))) /\ ((exists ge_balance_positive_unique_factorization_constructedunitidentitysecondreal ge_balance_negative_unique_factorization_constructedunitidentitysecondreal. (((((ge_representation_real_code_unique_factorization_constructedunitidentitysecond) = 2 * (ge_balance_positive_unique_factorization_constructedunitidentitysecondreal) /\ (ge_balance_negative_unique_factorization_constructedunitidentitysecondreal) = 0) \/ exists ge_signed_half_unique_factorization_constructedunitidentitysecondrealdecode. (((ge_representation_real_code_unique_factorization_constructedunitidentitysecond) = 2 * ge_signed_half_unique_factorization_constructedunitidentitysecondrealdecode + 1 /\ (ge_balance_positive_unique_factorization_constructedunitidentitysecondreal) = 0) /\ (ge_balance_negative_unique_factorization_constructedunitidentitysecondreal) = S ge_signed_half_unique_factorization_constructedunitidentitysecondrealdecode))) /\ ((ge_second_rp_unique_factorization_constructedunitidentity) + ge_balance_negative_unique_factorization_constructedunitidentitysecondreal = (ge_second_rn_unique_factorization_constructedunitidentity) + ge_balance_positive_unique_factorization_constructedunitidentitysecondreal))) /\ (exists ge_balance_positive_unique_factorization_constructedunitidentitysecondimaginary ge_balance_negative_unique_factorization_constructedunitidentitysecondimaginary. (((((ge_representation_imaginary_code_unique_factorization_constructedunitidentitysecond) = 2 * (ge_balance_positive_unique_factorization_constructedunitidentitysecondimaginary) /\ (ge_balance_negative_unique_factorization_constructedunitidentitysecondimaginary) = 0) \/ exists ge_signed_half_unique_factorization_constructedunitidentitysecondimaginarydecode. (((ge_representation_imaginary_code_unique_factorization_constructedunitidentitysecond) = 2 * ge_signed_half_unique_factorization_constructedunitidentitysecondimaginarydecode + 1 /\ (ge_balance_positive_unique_factorization_constructedunitidentitysecondimaginary) = 0) /\ (ge_balance_negative_unique_factorization_constructedunitidentitysecondimaginary) = S ge_signed_half_unique_factorization_constructedunitidentitysecondimaginarydecode))) /\ ((ge_second_ip_unique_factorization_constructedunitidentity) + ge_balance_negative_unique_factorization_constructedunitidentitysecondimaginary = (ge_second_in_unique_factorization_constructedunitidentity) + ge_balance_positive_unique_factorization_constructedunitidentitysecondimaginary)))))) /\ (exists ge_representation_real_code_unique_factorization_constructedunitidentityoutput ge_representation_imaginary_code_unique_factorization_constructedunitidentityoutput. (((6) = ((ge_representation_real_code_unique_factorization_constructedunitidentityoutput) + (ge_representation_imaginary_code_unique_factorization_constructedunitidentityoutput)) * S ((ge_representation_real_code_unique_factorization_constructedunitidentityoutput) + (ge_representation_imaginary_code_unique_factorization_constructedunitidentityoutput)) + ((ge_representation_imaginary_code_unique_factorization_constructedunitidentityoutput) + (ge_representation_imaginary_code_unique_factorization_constructedunitidentityoutput))) /\ ((exists ge_balance_positive_unique_factorization_constructedunitidentityoutputreal ge_balance_negative_unique_factorization_constructedunitidentityoutputreal. (((((ge_representation_real_code_unique_factorization_constructedunitidentityoutput) = 2 * (ge_balance_positive_unique_factorization_constructedunitidentityoutputreal) /\ (ge_balance_negative_unique_factorization_constructedunitidentityoutputreal) = 0) \/ exists ge_signed_half_unique_factorization_constructedunitidentityoutputrealdecode. (((ge_representation_real_code_unique_factorization_constructedunitidentityoutput) = 2 * ge_signed_half_unique_factorization_constructedunitidentityoutputrealdecode + 1 /\ (ge_balance_positive_unique_factorization_constructedunitidentityoutputreal) = 0) /\ (ge_balance_negative_unique_factorization_constructedunitidentityoutputreal) = S ge_signed_half_unique_factorization_constructedunitidentityoutputrealdecode))) /\ ((((((((ge_first_rp_unique_factorization_constructedunitidentity) * (ge_second_rp_unique_factorization_constructedunitidentity))) + (((ge_first_rn_unique_factorization_constructedunitidentity) * (ge_second_rn_unique_factorization_constructedunitidentity))))) + (((((ge_first_ip_unique_factorization_constructedunitidentity) * (ge_second_in_unique_factorization_constructedunitidentity))) + (((ge_first_in_unique_factorization_constructedunitidentity) * (ge_second_ip_unique_factorization_constructedunitidentity))))))) + ge_balance_negative_unique_factorization_constructedunitidentityoutputreal = (((((((ge_first_rp_unique_factorization_constructedunitidentity) * (ge_second_rn_unique_factorization_constructedunitidentity))) + (((ge_first_rn_unique_factorization_constructedunitidentity) * (ge_second_rp_unique_factorization_constructedunitidentity))))) + (((((ge_first_ip_unique_factorization_constructedunitidentity) * (ge_second_ip_unique_factorization_constructedunitidentity))) + (((ge_first_in_unique_factorization_constructedunitidentity) * (ge_second_in_unique_factorization_constructedunitidentity))))))) + ge_balance_positive_unique_factorization_constructedunitidentityoutputreal))) /\ (exists ge_balance_positive_unique_factorization_constructedunitidentityoutputimaginary ge_balance_negative_unique_factorization_constructedunitidentityoutputimaginary. (((((ge_representation_imaginary_code_unique_factorization_constructedunitidentityoutput) = 2 * (ge_balance_positive_unique_factorization_constructedunitidentityoutputimaginary) /\ (ge_balance_negative_unique_factorization_constructedunitidentityoutputimaginary) = 0) \/ exists ge_signed_half_unique_factorization_constructedunitidentityoutputimaginarydecode. (((ge_representation_imaginary_code_unique_factorization_constructedunitidentityoutput) = 2 * ge_signed_half_unique_factorization_constructedunitidentityoutputimaginarydecode + 1 /\ (ge_balance_positive_unique_factorization_constructedunitidentityoutputimaginary) = 0) /\ (ge_balance_negative_unique_factorization_constructedunitidentityoutputimaginary) = S ge_signed_half_unique_factorization_constructedunitidentityoutputimaginarydecode))) /\ ((((((((ge_first_rp_unique_factorization_constructedunitidentity) * (ge_second_ip_unique_factorization_constructedunitidentity))) + (((ge_first_rn_unique_factorization_constructedunitidentity) * (ge_second_in_unique_factorization_constructedunitidentity))))) + (((((ge_first_ip_unique_factorization_constructedunitidentity) * (ge_second_rp_unique_factorization_constructedunitidentity))) + (((ge_first_in_unique_factorization_constructedunitidentity) * (ge_second_rn_unique_factorization_constructedunitidentity))))))) + ge_balance_negative_unique_factorization_constructedunitidentityoutputimaginary = (((((((ge_first_rp_unique_factorization_constructedunitidentity) * (ge_second_in_unique_factorization_constructedunitidentity))) + (((ge_first_rn_unique_factorization_constructedunitidentity) * (ge_second_ip_unique_factorization_constructedunitidentity))))) + (((((ge_first_ip_unique_factorization_constructedunitidentity) * (ge_second_rn_unique_factorization_constructedunitidentity))) + (((ge_first_in_unique_factorization_constructedunitidentity) * (ge_second_rp_unique_factorization_constructedunitidentity))))))) + ge_balance_positive_unique_factorization_constructedunitidentityoutputimaginary)))))))))) /\ ((forall gr_prime_factor_index_unique_factorization_constructedprimes gr_prime_factor_value_unique_factorization_constructedprimes. (exists ge_gap_unique_factorization_constructedprimesindex. ge_gap_unique_factorization_constructedprimesindex + S (gr_prime_factor_index_unique_factorization_constructedprimes) = (l)) -> (((exists ff_h_gprod_unique_factorization_constructedprimesentry. ff_h_gprod_unique_factorization_constructedprimesentry + S (gr_prime_factor_value_unique_factorization_constructedprimes) = S ((S (gr_prime_factor_index_unique_factorization_constructedprimes)) * c)) /\ exists ff_q_gprod_unique_factorization_constructedprimesentry. b = ff_q_gprod_unique_factorization_constructedprimesentry * S ((S (gr_prime_factor_index_unique_factorization_constructedprimes)) * c) + (gr_prime_factor_value_unique_factorization_constructedprimes))) -> (((exists ge_real_positive_unique_factorization_constructedprimesprimecarrier ge_real_negative_unique_factorization_constructedprimesprimecarrier ge_imaginary_positive_unique_factorization_constructedprimesprimecarrier ge_imaginary_negative_unique_factorization_constructedprimesprimecarrier. (exists ge_real_code_unique_factorization_constructedprimesprimecarrierdecode ge_imaginary_code_unique_factorization_constructedprimesprimecarrierdecode. (((gr_prime_factor_value_unique_factorization_constructedprimes) = ((ge_real_code_unique_factorization_constructedprimesprimecarrierdecode) + (ge_imaginary_code_unique_factorization_constructedprimesprimecarrierdecode)) * S ((ge_real_code_unique_factorization_constructedprimesprimecarrierdecode) + (ge_imaginary_code_unique_factorization_constructedprimesprimecarrierdecode)) + ((ge_imaginary_code_unique_factorization_constructedprimesprimecarrierdecode) + (ge_imaginary_code_unique_factorization_constructedprimesprimecarrierdecode))) /\ (((((ge_real_code_unique_factorization_constructedprimesprimecarrierdecode) = 2 * (ge_real_positive_unique_factorization_constructedprimesprimecarrier) /\ (ge_real_negative_unique_factorization_constructedprimesprimecarrier) = 0) \/ exists ge_signed_half_ge_unique_factorization_constructedprimesprimecarrierdecode_real. (((ge_real_code_unique_factorization_constructedprimesprimecarrierdecode) = 2 * ge_signed_half_ge_unique_factorization_constructedprimesprimecarrierdecode_real + 1 /\ (ge_real_positive_unique_factorization_constructedprimesprimecarrier) = 0) /\ (ge_real_negative_unique_factorization_constructedprimesprimecarrier) = S ge_signed_half_ge_unique_factorization_constructedprimesprimecarrierdecode_real))) /\ ((((ge_imaginary_code_unique_factorization_constructedprimesprimecarrierdecode) = 2 * (ge_imaginary_positive_unique_factorization_constructedprimesprimecarrier) /\ (ge_imaginary_negative_unique_factorization_constructedprimesprimecarrier) = 0) \/ exists ge_signed_half_ge_unique_factorization_constructedprimesprimecarrierdecode_imaginary. (((ge_imaginary_code_unique_factorization_constructedprimesprimecarrierdecode) = 2 * ge_signed_half_ge_unique_factorization_constructedprimesprimecarrierdecode_imaginary + 1 /\ (ge_imaginary_positive_unique_factorization_constructedprimesprimecarrier) = 0) /\ (ge_imaginary_negative_unique_factorization_constructedprimesprimecarrier) = S ge_signed_half_ge_unique_factorization_constructedprimesprimecarrierdecode_imaginary))))))) /\ ((~((gr_prime_factor_value_unique_factorization_constructedprimes)=0)) /\ ((~(exists gr_inverse_unique_factorization_constructedprimesprimenonunit. (exists ge_first_rp_unique_factorization_constructedprimesprimenonunitidentity ge_first_rn_unique_factorization_constructedprimesprimenonunitidentity ge_first_ip_unique_factorization_constructedprimesprimenonunitidentity ge_first_in_unique_factorization_constructedprimesprimenonunitidentity ge_second_rp_unique_factorization_constructedprimesprimenonunitidentity ge_second_rn_unique_factorization_constructedprimesprimenonunitidentity ge_second_ip_unique_factorization_constructedprimesprimenonunitidentity ge_second_in_unique_factorization_constructedprimesprimenonunitidentity. ((exists ge_representation_real_code_unique_factorization_constructedprimesprimenonunitidentityfirst ge_representation_imaginary_code_unique_factorization_constructedprimesprimenonunitidentityfirst. (((gr_prime_factor_value_unique_factorization_constructedprimes) = ((ge_representation_real_code_unique_factorization_constructedprimesprimenonunitidentityfirst) + (ge_representation_imaginary_code_unique_factorization_constructedprimesprimenonunitidentityfirst)) * S ((ge_representation_real_code_unique_factorization_constructedprimesprimenonunitidentityfirst) + (ge_representation_imaginary_code_unique_factorization_constructedprimesprimenonunitidentityfirst)) + ((ge_representation_imaginary_code_unique_factorization_constructedprimesprimenonunitidentityfirst) + (ge_representation_imaginary_code_unique_factorization_constructedprimesprimenonunitidentityfirst))) /\ ((exists ge_balance_positive_unique_factorization_constructedprimesprimenonunitidentityfirstreal ge_balance_negative_unique_factorization_constructedprimesprimenonunitidentityfirstreal. (((((ge_representation_real_code_unique_factorization_constructedprimesprimenonunitidentityfirst) = 2 * (ge_balance_positive_unique_factorization_constructedprimesprimenonunitidentityfirstreal) /\ (ge_balance_negative_unique_factorization_constructedprimesprimenonunitidentityfirstreal) = 0) \/ exists ge_signed_half_unique_factorization_constructedprimesprimenonunitidentityfirstrealdecode. (((ge_representation_real_code_unique_factorization_constructedprimesprimenonunitidentityfirst) = 2 * ge_signed_half_unique_factorization_constructedprimesprimenonunitidentityfirstrealdecode + 1 /\ (ge_balance_positive_unique_factorization_constructedprimesprimenonunitidentityfirstreal) = 0) /\ (ge_balance_negative_unique_factorization_constructedprimesprimenonunitidentityfirstreal) = S ge_signed_half_unique_factorization_constructedprimesprimenonunitidentityfirstrealdecode))) /\ ((ge_first_rp_unique_factorization_constructedprimesprimenonunitidentity) + ge_balance_negative_unique_factorization_constructedprimesprimenonunitidentityfirstreal = (ge_first_rn_unique_factorization_constructedprimesprimenonunitidentity) + ge_balance_positive_unique_factorization_constructedprimesprimenonunitidentityfirstreal))) /\ (exists ge_balance_positive_unique_factorization_constructedprimesprimenonunitidentityfirstimaginary ge_balance_negative_unique_factorization_constructedprimesprimenonunitidentityfirstimaginary. (((((ge_representation_imaginary_code_unique_factorization_constructedprimesprimenonunitidentityfirst) = 2 * (ge_balance_positive_unique_factorization_constructedprimesprimenonunitidentityfirstimaginary) /\ (ge_balance_negative_unique_factorization_constructedprimesprimenonunitidentityfirstimaginary) = 0) \/ exists ge_signed_half_unique_factorization_constructedprimesprimenonunitidentityfirstimaginarydecode. (((ge_representation_imaginary_code_unique_factorization_constructedprimesprimenonunitidentityfirst) = 2 * ge_signed_half_unique_factorization_constructedprimesprimenonunitidentityfirstimaginarydecode + 1 /\ (ge_balance_positive_unique_factorization_constructedprimesprimenonunitidentityfirstimaginary) = 0) /\ (ge_balance_negative_unique_factorization_constructedprimesprimenonunitidentityfirstimaginary) = S ge_signed_half_unique_factorization_constructedprimesprimenonunitidentityfirstimaginarydecode))) /\ ((ge_first_ip_unique_factorization_constructedprimesprimenonunitidentity) + ge_balance_negative_unique_factorization_constructedprimesprimenonunitidentityfirstimaginary = (ge_first_in_unique_factorization_constructedprimesprimenonunitidentity) + ge_balance_positive_unique_factorization_constructedprimesprimenonunitidentityfirstimaginary)))))) /\ ((exists ge_representation_real_code_unique_factorization_constructedprimesprimenonunitidentitysecond ge_representation_imaginary_code_unique_factorization_constructedprimesprimenonunitidentitysecond. (((gr_inverse_unique_factorization_constructedprimesprimenonunit) = ((ge_representation_real_code_unique_factorization_constructedprimesprimenonunitidentitysecond) + (ge_representation_imaginary_code_unique_factorization_constructedprimesprimenonunitidentitysecond)) * S ((ge_representation_real_code_unique_factorization_constructedprimesprimenonunitidentitysecond) + (ge_representation_imaginary_code_unique_factorization_constructedprimesprimenonunitidentitysecond)) + ((ge_representation_imaginary_code_unique_factorization_constructedprimesprimenonunitidentitysecond) + (ge_representation_imaginary_code_unique_factorization_constructedprimesprimenonunitidentitysecond))) /\ ((exists ge_balance_positive_unique_factorization_constructedprimesprimenonunitidentitysecondreal ge_balance_negative_unique_factorization_constructedprimesprimenonunitidentitysecondreal. (((((ge_representation_real_code_unique_factorization_constructedprimesprimenonunitidentitysecond) = 2 * (ge_balance_positive_unique_factorization_constructedprimesprimenonunitidentitysecondreal) /\ (ge_balance_negative_unique_factorization_constructedprimesprimenonunitidentitysecondreal) = 0) \/ exists ge_signed_half_unique_factorization_constructedprimesprimenonunitidentitysecondrealdecode. (((ge_representation_real_code_unique_factorization_constructedprimesprimenonunitidentitysecond) = 2 * ge_signed_half_unique_factorization_constructedprimesprimenonunitidentitysecondrealdecode + 1 /\ (ge_balance_positive_unique_factorization_constructedprimesprimenonunitidentitysecondreal) = 0) /\ (ge_balance_negative_unique_factorization_constructedprimesprimenonunitidentitysecondreal) = S ge_signed_half_unique_factorization_constructedprimesprimenonunitidentitysecondrealdecode))) /\ ((ge_second_rp_unique_factorization_constructedprimesprimenonunitidentity) + ge_balance_negative_unique_factorization_constructedprimesprimenonunitidentitysecondreal = (ge_second_rn_unique_factorization_constructedprimesprimenonunitidentity) + ge_balance_positive_unique_factorization_constructedprimesprimenonunitidentitysecondreal))) /\ (exists ge_balance_positive_unique_factorization_constructedprimesprimenonunitidentitysecondimaginary ge_balance_negative_unique_factorization_constructedprimesprimenonunitidentitysecondimaginary. (((((ge_representation_imaginary_code_unique_factorization_constructedprimesprimenonunitidentitysecond) = 2 * (ge_balance_positive_unique_factorization_constructedprimesprimenonunitidentitysecondimaginary) /\ (ge_balance_negative_unique_factorization_constructedprimesprimenonunitidentitysecondimaginary) = 0) \/ exists ge_signed_half_unique_factorization_constructedprimesprimenonunitidentitysecondimaginarydecode. (((ge_representation_imaginary_code_unique_factorization_constructedprimesprimenonunitidentitysecond) = 2 * ge_signed_half_unique_factorization_constructedprimesprimenonunitidentitysecondimaginarydecode + 1 /\ (ge_balance_positive_unique_factorization_constructedprimesprimenonunitidentitysecondimaginary) = 0) /\ (ge_balance_negative_unique_factorization_constructedprimesprimenonunitidentitysecondimaginary) = S ge_signed_half_unique_factorization_constructedprimesprimenonunitidentitysecondimaginarydecode))) /\ ((ge_second_ip_unique_factorization_constructedprimesprimenonunitidentity) + ge_balance_negative_unique_factorization_constructedprimesprimenonunitidentitysecondimaginary = (ge_second_in_unique_factorization_constructedprimesprimenonunitidentity) + ge_balance_positive_unique_factorization_constructedprimesprimenonunitidentitysecondimaginary)))))) /\ (exists ge_representation_real_code_unique_factorization_constructedprimesprimenonunitidentityoutput ge_representation_imaginary_code_unique_factorization_constructedprimesprimenonunitidentityoutput. (((6) = ((ge_representation_real_code_unique_factorization_constructedprimesprimenonunitidentityoutput) + (ge_representation_imaginary_code_unique_factorization_constructedprimesprimenonunitidentityoutput)) * S ((ge_representation_real_code_unique_factorization_constructedprimesprimenonunitidentityoutput) + (ge_representation_imaginary_code_unique_factorization_constructedprimesprimenonunitidentityoutput)) + ((ge_representation_imaginary_code_unique_factorization_constructedprimesprimenonunitidentityoutput) + (ge_representation_imaginary_code_unique_factorization_constructedprimesprimenonunitidentityoutput))) /\ ((exists ge_balance_positive_unique_factorization_constructedprimesprimenonunitidentityoutputreal ge_balance_negative_unique_factorization_constructedprimesprimenonunitidentityoutputreal. (((((ge_representation_real_code_unique_factorization_constructedprimesprimenonunitidentityoutput) = 2 * (ge_balance_positive_unique_factorization_constructedprimesprimenonunitidentityoutputreal) /\ (ge_balance_negative_unique_factorization_constructedprimesprimenonunitidentityoutputreal) = 0) \/ exists ge_signed_half_unique_factorization_constructedprimesprimenonunitidentityoutputrealdecode. (((ge_representation_real_code_unique_factorization_constructedprimesprimenonunitidentityoutput) = 2 * ge_signed_half_unique_factorization_constructedprimesprimenonunitidentityoutputrealdecode + 1 /\ (ge_balance_positive_unique_factorization_constructedprimesprimenonunitidentityoutputreal) = 0) /\ (ge_balance_negative_unique_factorization_constructedprimesprimenonunitidentityoutputreal) = S ge_signed_half_unique_factorization_constructedprimesprimenonunitidentityoutputrealdecode))) /\ ((((((((ge_first_rp_unique_factorization_constructedprimesprimenonunitidentity) * (ge_second_rp_unique_factorization_constructedprimesprimenonunitidentity))) + (((ge_first_rn_unique_factorization_constructedprimesprimenonunitidentity) * (ge_second_rn_unique_factorization_constructedprimesprimenonunitidentity))))) + (((((ge_first_ip_unique_factorization_constructedprimesprimenonunitidentity) * (ge_second_in_unique_factorization_constructedprimesprimenonunitidentity))) + (((ge_first_in_unique_factorization_constructedprimesprimenonunitidentity) * (ge_second_ip_unique_factorization_constructedprimesprimenonunitidentity))))))) + ge_balance_negative_unique_factorization_constructedprimesprimenonunitidentityoutputreal = (((((((ge_first_rp_unique_factorization_constructedprimesprimenonunitidentity) * (ge_second_rn_unique_factorization_constructedprimesprimenonunitidentity))) + (((ge_first_rn_unique_factorization_constructedprimesprimenonunitidentity) * (ge_second_rp_unique_factorization_constructedprimesprimenonunitidentity))))) + (((((ge_first_ip_unique_factorization_constructedprimesprimenonunitidentity) * (ge_second_ip_unique_factorization_constructedprimesprimenonunitidentity))) + (((ge_first_in_unique_factorization_constructedprimesprimenonunitidentity) * (ge_second_in_unique_factorization_constructedprimesprimenonunitidentity))))))) + ge_balance_positive_unique_factorization_constructedprimesprimenonunitidentityoutputreal))) /\ (exists ge_balance_positive_unique_factorization_constructedprimesprimenonunitidentityoutputimaginary ge_balance_negative_unique_factorization_constructedprimesprimenonunitidentityoutputimaginary. (((((ge_representation_imaginary_code_unique_factorization_constructedprimesprimenonunitidentityoutput) = 2 * (ge_balance_positive_unique_factorization_constructedprimesprimenonunitidentityoutputimaginary) /\ (ge_balance_negative_unique_factorization_constructedprimesprimenonunitidentityoutputimaginary) = 0) \/ exists ge_signed_half_unique_factorization_constructedprimesprimenonunitidentityoutputimaginarydecode. (((ge_representation_imaginary_code_unique_factorization_constructedprimesprimenonunitidentityoutput) = 2 * ge_signed_half_unique_factorization_constructedprimesprimenonunitidentityoutputimaginarydecode + 1 /\ (ge_balance_positive_unique_factorization_constructedprimesprimenonunitidentityoutputimaginary) = 0) /\ (ge_balance_negative_unique_factorization_constructedprimesprimenonunitidentityoutputimaginary) = S ge_signed_half_unique_factorization_constructedprimesprimenonunitidentityoutputimaginarydecode))) /\ ((((((((ge_first_rp_unique_factorization_constructedprimesprimenonunitidentity) * (ge_second_ip_unique_factorization_constructedprimesprimenonunitidentity))) + (((ge_first_rn_unique_factorization_constructedprimesprimenonunitidentity) * (ge_second_in_unique_factorization_constructedprimesprimenonunitidentity))))) + (((((ge_first_ip_unique_factorization_constructedprimesprimenonunitidentity) * (ge_second_rp_unique_factorization_constructedprimesprimenonunitidentity))) + (((ge_first_in_unique_factorization_constructedprimesprimenonunitidentity) * (ge_second_rn_unique_factorization_constructedprimesprimenonunitidentity))))))) + ge_balance_negative_unique_factorization_constructedprimesprimenonunitidentityoutputimaginary = (((((((ge_first_rp_unique_factorization_constructedprimesprimenonunitidentity) * (ge_second_in_unique_factorization_constructedprimesprimenonunitidentity))) + (((ge_first_rn_unique_factorization_constructedprimesprimenonunitidentity) * (ge_second_ip_unique_factorization_constructedprimesprimenonunitidentity))))) + (((((ge_first_ip_unique_factorization_constructedprimesprimenonunitidentity) * (ge_second_rn_unique_factorization_constructedprimesprimenonunitidentity))) + (((ge_first_in_unique_factorization_constructedprimesprimenonunitidentity) * (ge_second_rp_unique_factorization_constructedprimesprimenonunitidentity))))))) + ge_balance_positive_unique_factorization_constructedprimesprimenonunitidentityoutputimaginary))))))))))) /\ (forall gr_first_factor_unique_factorization_constructedprimesprime gr_second_factor_unique_factorization_constructedprimesprime gr_product_unique_factorization_constructedprimesprime. (exists ge_first_rp_unique_factorization_constructedprimesprimeproduct ge_first_rn_unique_factorization_constructedprimesprimeproduct ge_first_ip_unique_factorization_constructedprimesprimeproduct ge_first_in_unique_factorization_constructedprimesprimeproduct ge_second_rp_unique_factorization_constructedprimesprimeproduct ge_second_rn_unique_factorization_constructedprimesprimeproduct ge_second_ip_unique_factorization_constructedprimesprimeproduct ge_second_in_unique_factorization_constructedprimesprimeproduct. ((exists ge_representation_real_code_unique_factorization_constructedprimesprimeproductfirst ge_representation_imaginary_code_unique_factorization_constructedprimesprimeproductfirst. (((gr_first_factor_unique_factorization_constructedprimesprime) = ((ge_representation_real_code_unique_factorization_constructedprimesprimeproductfirst) + (ge_representation_imaginary_code_unique_factorization_constructedprimesprimeproductfirst)) * S ((ge_representation_real_code_unique_factorization_constructedprimesprimeproductfirst) + (ge_representation_imaginary_code_unique_factorization_constructedprimesprimeproductfirst)) + ((ge_representation_imaginary_code_unique_factorization_constructedprimesprimeproductfirst) + (ge_representation_imaginary_code_unique_factorization_constructedprimesprimeproductfirst))) /\ ((exists ge_balance_positive_unique_factorization_constructedprimesprimeproductfirstreal ge_balance_negative_unique_factorization_constructedprimesprimeproductfirstreal. (((((ge_representation_real_code_unique_factorization_constructedprimesprimeproductfirst) = 2 * (ge_balance_positive_unique_factorization_constructedprimesprimeproductfirstreal) /\ (ge_balance_negative_unique_factorization_constructedprimesprimeproductfirstreal) = 0) \/ exists ge_signed_half_unique_factorization_constructedprimesprimeproductfirstrealdecode. (((ge_representation_real_code_unique_factorization_constructedprimesprimeproductfirst) = 2 * ge_signed_half_unique_factorization_constructedprimesprimeproductfirstrealdecode + 1 /\ (ge_balance_positive_unique_factorization_constructedprimesprimeproductfirstreal) = 0) /\ (ge_balance_negative_unique_factorization_constructedprimesprimeproductfirstreal) = S ge_signed_half_unique_factorization_constructedprimesprimeproductfirstrealdecode))) /\ ((ge_first_rp_unique_factorization_constructedprimesprimeproduct) + ge_balance_negative_unique_factorization_constructedprimesprimeproductfirstreal = (ge_first_rn_unique_factorization_constructedprimesprimeproduct) + ge_balance_positive_unique_factorization_constructedprimesprimeproductfirstreal))) /\ (exists ge_balance_positive_unique_factorization_constructedprimesprimeproductfirstimaginary ge_balance_negative_unique_factorization_constructedprimesprimeproductfirstimaginary. (((((ge_representation_imaginary_code_unique_factorization_constructedprimesprimeproductfirst) = 2 * (ge_balance_positive_unique_factorization_constructedprimesprimeproductfirstimaginary) /\ (ge_balance_negative_unique_factorization_constructedprimesprimeproductfirstimaginary) = 0) \/ exists ge_signed_half_unique_factorization_constructedprimesprimeproductfirstimaginarydecode. (((ge_representation_imaginary_code_unique_factorization_constructedprimesprimeproductfirst) = 2 * ge_signed_half_unique_factorization_constructedprimesprimeproductfirstimaginarydecode + 1 /\ (ge_balance_positive_unique_factorization_constructedprimesprimeproductfirstimaginary) = 0) /\ (ge_balance_negative_unique_factorization_constructedprimesprimeproductfirstimaginary) = S ge_signed_half_unique_factorization_constructedprimesprimeproductfirstimaginarydecode))) /\ ((ge_first_ip_unique_factorization_constructedprimesprimeproduct) + ge_balance_negative_unique_factorization_constructedprimesprimeproductfirstimaginary = (ge_first_in_unique_factorization_constructedprimesprimeproduct) + ge_balance_positive_unique_factorization_constructedprimesprimeproductfirstimaginary)))))) /\ ((exists ge_representation_real_code_unique_factorization_constructedprimesprimeproductsecond ge_representation_imaginary_code_unique_factorization_constructedprimesprimeproductsecond. (((gr_second_factor_unique_factorization_constructedprimesprime) = ((ge_representation_real_code_unique_factorization_constructedprimesprimeproductsecond) + (ge_representation_imaginary_code_unique_factorization_constructedprimesprimeproductsecond)) * S ((ge_representation_real_code_unique_factorization_constructedprimesprimeproductsecond) + (ge_representation_imaginary_code_unique_factorization_constructedprimesprimeproductsecond)) + ((ge_representation_imaginary_code_unique_factorization_constructedprimesprimeproductsecond) + (ge_representation_imaginary_code_unique_factorization_constructedprimesprimeproductsecond))) /\ ((exists ge_balance_positive_unique_factorization_constructedprimesprimeproductsecondreal ge_balance_negative_unique_factorization_constructedprimesprimeproductsecondreal. (((((ge_representation_real_code_unique_factorization_constructedprimesprimeproductsecond) = 2 * (ge_balance_positive_unique_factorization_constructedprimesprimeproductsecondreal) /\ (ge_balance_negative_unique_factorization_constructedprimesprimeproductsecondreal) = 0) \/ exists ge_signed_half_unique_factorization_constructedprimesprimeproductsecondrealdecode. (((ge_representation_real_code_unique_factorization_constructedprimesprimeproductsecond) = 2 * ge_signed_half_unique_factorization_constructedprimesprimeproductsecondrealdecode + 1 /\ (ge_balance_positive_unique_factorization_constructedprimesprimeproductsecondreal) = 0) /\ (ge_balance_negative_unique_factorization_constructedprimesprimeproductsecondreal) = S ge_signed_half_unique_factorization_constructedprimesprimeproductsecondrealdecode))) /\ ((ge_second_rp_unique_factorization_constructedprimesprimeproduct) + ge_balance_negative_unique_factorization_constructedprimesprimeproductsecondreal = (ge_second_rn_unique_factorization_constructedprimesprimeproduct) + ge_balance_positive_unique_factorization_constructedprimesprimeproductsecondreal))) /\ (exists ge_balance_positive_unique_factorization_constructedprimesprimeproductsecondimaginary ge_balance_negative_unique_factorization_constructedprimesprimeproductsecondimaginary. (((((ge_representation_imaginary_code_unique_factorization_constructedprimesprimeproductsecond) = 2 * (ge_balance_positive_unique_factorization_constructedprimesprimeproductsecondimaginary) /\ (ge_balance_negative_unique_factorization_constructedprimesprimeproductsecondimaginary) = 0) \/ exists ge_signed_half_unique_factorization_constructedprimesprimeproductsecondimaginarydecode. (((ge_representation_imaginary_code_unique_factorization_constructedprimesprimeproductsecond) = 2 * ge_signed_half_unique_factorization_constructedprimesprimeproductsecondimaginarydecode + 1 /\ (ge_balance_positive_unique_factorization_constructedprimesprimeproductsecondimaginary) = 0) /\ (ge_balance_negative_unique_factorization_constructedprimesprimeproductsecondimaginary) = S ge_signed_half_unique_factorization_constructedprimesprimeproductsecondimaginarydecode))) /\ ((ge_second_ip_unique_factorization_constructedprimesprimeproduct) + ge_balance_negative_unique_factorization_constructedprimesprimeproductsecondimaginary = (ge_second_in_unique_factorization_constructedprimesprimeproduct) + ge_balance_positive_unique_factorization_constructedprimesprimeproductsecondimaginary)))))) /\ (exists ge_representation_real_code_unique_factorization_constructedprimesprimeproductoutput ge_representation_imaginary_code_unique_factorization_constructedprimesprimeproductoutput. (((gr_product_unique_factorization_constructedprimesprime) = ((ge_representation_real_code_unique_factorization_constructedprimesprimeproductoutput) + (ge_representation_imaginary_code_unique_factorization_constructedprimesprimeproductoutput)) * S ((ge_representation_real_code_unique_factorization_constructedprimesprimeproductoutput) + (ge_representation_imaginary_code_unique_factorization_constructedprimesprimeproductoutput)) + ((ge_representation_imaginary_code_unique_factorization_constructedprimesprimeproductoutput) + (ge_representation_imaginary_code_unique_factorization_constructedprimesprimeproductoutput))) /\ ((exists ge_balance_positive_unique_factorization_constructedprimesprimeproductoutputreal ge_balance_negative_unique_factorization_constructedprimesprimeproductoutputreal. (((((ge_representation_real_code_unique_factorization_constructedprimesprimeproductoutput) = 2 * (ge_balance_positive_unique_factorization_constructedprimesprimeproductoutputreal) /\ (ge_balance_negative_unique_factorization_constructedprimesprimeproductoutputreal) = 0) \/ exists ge_signed_half_unique_factorization_constructedprimesprimeproductoutputrealdecode. (((ge_representation_real_code_unique_factorization_constructedprimesprimeproductoutput) = 2 * ge_signed_half_unique_factorization_constructedprimesprimeproductoutputrealdecode + 1 /\ (ge_balance_positive_unique_factorization_constructedprimesprimeproductoutputreal) = 0) /\ (ge_balance_negative_unique_factorization_constructedprimesprimeproductoutputreal) = S ge_signed_half_unique_factorization_constructedprimesprimeproductoutputrealdecode))) /\ ((((((((ge_first_rp_unique_factorization_constructedprimesprimeproduct) * (ge_second_rp_unique_factorization_constructedprimesprimeproduct))) + (((ge_first_rn_unique_factorization_constructedprimesprimeproduct) * (ge_second_rn_unique_factorization_constructedprimesprimeproduct))))) + (((((ge_first_ip_unique_factorization_constructedprimesprimeproduct) * (ge_second_in_unique_factorization_constructedprimesprimeproduct))) + (((ge_first_in_unique_factorization_constructedprimesprimeproduct) * (ge_second_ip_unique_factorization_constructedprimesprimeproduct))))))) + ge_balance_negative_unique_factorization_constructedprimesprimeproductoutputreal = (((((((ge_first_rp_unique_factorization_constructedprimesprimeproduct) * (ge_second_rn_unique_factorization_constructedprimesprimeproduct))) + (((ge_first_rn_unique_factorization_constructedprimesprimeproduct) * (ge_second_rp_unique_factorization_constructedprimesprimeproduct))))) + (((((ge_first_ip_unique_factorization_constructedprimesprimeproduct) * (ge_second_ip_unique_factorization_constructedprimesprimeproduct))) + (((ge_first_in_unique_factorization_constructedprimesprimeproduct) * (ge_second_in_unique_factorization_constructedprimesprimeproduct))))))) + ge_balance_positive_unique_factorization_constructedprimesprimeproductoutputreal))) /\ (exists ge_balance_positive_unique_factorization_constructedprimesprimeproductoutputimaginary ge_balance_negative_unique_factorization_constructedprimesprimeproductoutputimaginary. (((((ge_representation_imaginary_code_unique_factorization_constructedprimesprimeproductoutput) = 2 * (ge_balance_positive_unique_factorization_constructedprimesprimeproductoutputimaginary) /\ (ge_balance_negative_unique_factorization_constructedprimesprimeproductoutputimaginary) = 0) \/ exists ge_signed_half_unique_factorization_constructedprimesprimeproductoutputimaginarydecode. (((ge_representation_imaginary_code_unique_factorization_constructedprimesprimeproductoutput) = 2 * ge_signed_half_unique_factorization_constructedprimesprimeproductoutputimaginarydecode + 1 /\ (ge_balance_positive_unique_factorization_constructedprimesprimeproductoutputimaginary) = 0) /\ (ge_balance_negative_unique_factorization_constructedprimesprimeproductoutputimaginary) = S ge_signed_half_unique_factorization_constructedprimesprimeproductoutputimaginarydecode))) /\ ((((((((ge_first_rp_unique_factorization_constructedprimesprimeproduct) * (ge_second_ip_unique_factorization_constructedprimesprimeproduct))) + (((ge_first_rn_unique_factorization_constructedprimesprimeproduct) * (ge_second_in_unique_factorization_constructedprimesprimeproduct))))) + (((((ge_first_ip_unique_factorization_constructedprimesprimeproduct) * (ge_second_rp_unique_factorization_constructedprimesprimeproduct))) + (((ge_first_in_unique_factorization_constructedprimesprimeproduct) * (ge_second_rn_unique_factorization_constructedprimesprimeproduct))))))) + ge_balance_negative_unique_factorization_constructedprimesprimeproductoutputimaginary = (((((((ge_first_rp_unique_factorization_constructedprimesprimeproduct) * (ge_second_in_unique_factorization_constructedprimesprimeproduct))) + (((ge_first_rn_unique_factorization_constructedprimesprimeproduct) * (ge_second_ip_unique_factorization_constructedprimesprimeproduct))))) + (((((ge_first_ip_unique_factorization_constructedprimesprimeproduct) * (ge_second_rn_unique_factorization_constructedprimesprimeproduct))) + (((ge_first_in_unique_factorization_constructedprimesprimeproduct) * (ge_second_rp_unique_factorization_constructedprimesprimeproduct))))))) + ge_balance_positive_unique_factorization_constructedprimesprimeproductoutputimaginary))))))))) -> (exists gr_quotient_unique_factorization_constructedprimesprimedivisor. (exists ge_first_rp_unique_factorization_constructedprimesprimedivisorproduct ge_first_rn_unique_factorization_constructedprimesprimedivisorproduct ge_first_ip_unique_factorization_constructedprimesprimedivisorproduct ge_first_in_unique_factorization_constructedprimesprimedivisorproduct ge_second_rp_unique_factorization_constructedprimesprimedivisorproduct ge_second_rn_unique_factorization_constructedprimesprimedivisorproduct ge_second_ip_unique_factorization_constructedprimesprimedivisorproduct ge_second_in_unique_factorization_constructedprimesprimedivisorproduct. ((exists ge_representation_real_code_unique_factorization_constructedprimesprimedivisorproductfirst ge_representation_imaginary_code_unique_factorization_constructedprimesprimedivisorproductfirst. (((gr_prime_factor_value_unique_factorization_constructedprimes) = ((ge_representation_real_code_unique_factorization_constructedprimesprimedivisorproductfirst) + (ge_representation_imaginary_code_unique_factorization_constructedprimesprimedivisorproductfirst)) * S ((ge_representation_real_code_unique_factorization_constructedprimesprimedivisorproductfirst) + (ge_representation_imaginary_code_unique_factorization_constructedprimesprimedivisorproductfirst)) + ((ge_representation_imaginary_code_unique_factorization_constructedprimesprimedivisorproductfirst) + (ge_representation_imaginary_code_unique_factorization_constructedprimesprimedivisorproductfirst))) /\ ((exists ge_balance_positive_unique_factorization_constructedprimesprimedivisorproductfirstreal ge_balance_negative_unique_factorization_constructedprimesprimedivisorproductfirstreal. (((((ge_representation_real_code_unique_factorization_constructedprimesprimedivisorproductfirst) = 2 * (ge_balance_positive_unique_factorization_constructedprimesprimedivisorproductfirstreal) /\ (ge_balance_negative_unique_factorization_constructedprimesprimedivisorproductfirstreal) = 0) \/ exists ge_signed_half_unique_factorization_constructedprimesprimedivisorproductfirstrealdecode. (((ge_representation_real_code_unique_factorization_constructedprimesprimedivisorproductfirst) = 2 * ge_signed_half_unique_factorization_constructedprimesprimedivisorproductfirstrealdecode + 1 /\ (ge_balance_positive_unique_factorization_constructedprimesprimedivisorproductfirstreal) = 0) /\ (ge_balance_negative_unique_factorization_constructedprimesprimedivisorproductfirstreal) = S ge_signed_half_unique_factorization_constructedprimesprimedivisorproductfirstrealdecode))) /\ ((ge_first_rp_unique_factorization_constructedprimesprimedivisorproduct) + ge_balance_negative_unique_factorization_constructedprimesprimedivisorproductfirstreal = (ge_first_rn_unique_factorization_constructedprimesprimedivisorproduct) + ge_balance_positive_unique_factorization_constructedprimesprimedivisorproductfirstreal))) /\ (exists ge_balance_positive_unique_factorization_constructedprimesprimedivisorproductfirstimaginary ge_balance_negative_unique_factorization_constructedprimesprimedivisorproductfirstimaginary. (((((ge_representation_imaginary_code_unique_factorization_constructedprimesprimedivisorproductfirst) = 2 * (ge_balance_positive_unique_factorization_constructedprimesprimedivisorproductfirstimaginary) /\ (ge_balance_negative_unique_factorization_constructedprimesprimedivisorproductfirstimaginary) = 0) \/ exists ge_signed_half_unique_factorization_constructedprimesprimedivisorproductfirstimaginarydecode. (((ge_representation_imaginary_code_unique_factorization_constructedprimesprimedivisorproductfirst) = 2 * ge_signed_half_unique_factorization_constructedprimesprimedivisorproductfirstimaginarydecode + 1 /\ (ge_balance_positive_unique_factorization_constructedprimesprimedivisorproductfirstimaginary) = 0) /\ (ge_balance_negative_unique_factorization_constructedprimesprimedivisorproductfirstimaginary) = S ge_signed_half_unique_factorization_constructedprimesprimedivisorproductfirstimaginarydecode))) /\ ((ge_first_ip_unique_factorization_constructedprimesprimedivisorproduct) + ge_balance_negative_unique_factorization_constructedprimesprimedivisorproductfirstimaginary = (ge_first_in_unique_factorization_constructedprimesprimedivisorproduct) + ge_balance_positive_unique_factorization_constructedprimesprimedivisorproductfirstimaginary)))))) /\ ((exists ge_representation_real_code_unique_factorization_constructedprimesprimedivisorproductsecond ge_representation_imaginary_code_unique_factorization_constructedprimesprimedivisorproductsecond. (((gr_quotient_unique_factorization_constructedprimesprimedivisor) = ((ge_representation_real_code_unique_factorization_constructedprimesprimedivisorproductsecond) + (ge_representation_imaginary_code_unique_factorization_constructedprimesprimedivisorproductsecond)) * S ((ge_representation_real_code_unique_factorization_constructedprimesprimedivisorproductsecond) + (ge_representation_imaginary_code_unique_factorization_constructedprimesprimedivisorproductsecond)) + ((ge_representation_imaginary_code_unique_factorization_constructedprimesprimedivisorproductsecond) + (ge_representation_imaginary_code_unique_factorization_constructedprimesprimedivisorproductsecond))) /\ ((exists ge_balance_positive_unique_factorization_constructedprimesprimedivisorproductsecondreal ge_balance_negative_unique_factorization_constructedprimesprimedivisorproductsecondreal. (((((ge_representation_real_code_unique_factorization_constructedprimesprimedivisorproductsecond) = 2 * (ge_balance_positive_unique_factorization_constructedprimesprimedivisorproductsecondreal) /\ (ge_balance_negative_unique_factorization_constructedprimesprimedivisorproductsecondreal) = 0) \/ exists ge_signed_half_unique_factorization_constructedprimesprimedivisorproductsecondrealdecode. (((ge_representation_real_code_unique_factorization_constructedprimesprimedivisorproductsecond) = 2 * ge_signed_half_unique_factorization_constructedprimesprimedivisorproductsecondrealdecode + 1 /\ (ge_balance_positive_unique_factorization_constructedprimesprimedivisorproductsecondreal) = 0) /\ (ge_balance_negative_unique_factorization_constructedprimesprimedivisorproductsecondreal) = S ge_signed_half_unique_factorization_constructedprimesprimedivisorproductsecondrealdecode))) /\ ((ge_second_rp_unique_factorization_constructedprimesprimedivisorproduct) + ge_balance_negative_unique_factorization_constructedprimesprimedivisorproductsecondreal = (ge_second_rn_unique_factorization_constructedprimesprimedivisorproduct) + ge_balance_positive_unique_factorization_constructedprimesprimedivisorproductsecondreal))) /\ (exists ge_balance_positive_unique_factorization_constructedprimesprimedivisorproductsecondimaginary ge_balance_negative_unique_factorization_constructedprimesprimedivisorproductsecondimaginary. (((((ge_representation_imaginary_code_unique_factorization_constructedprimesprimedivisorproductsecond) = 2 * (ge_balance_positive_unique_factorization_constructedprimesprimedivisorproductsecondimaginary) /\ (ge_balance_negative_unique_factorization_constructedprimesprimedivisorproductsecondimaginary) = 0) \/ exists ge_signed_half_unique_factorization_constructedprimesprimedivisorproductsecondimaginarydecode. (((ge_representation_imaginary_code_unique_factorization_constructedprimesprimedivisorproductsecond) = 2 * ge_signed_half_unique_factorization_constructedprimesprimedivisorproductsecondimaginarydecode + 1 /\ (ge_balance_positive_unique_factorization_constructedprimesprimedivisorproductsecondimaginary) = 0) /\ (ge_balance_negative_unique_factorization_constructedprimesprimedivisorproductsecondimaginary) = S ge_signed_half_unique_factorization_constructedprimesprimedivisorproductsecondimaginarydecode))) /\ ((ge_second_ip_unique_factorization_constructedprimesprimedivisorproduct) + ge_balance_negative_unique_factorization_constructedprimesprimedivisorproductsecondimaginary = (ge_second_in_unique_factorization_constructedprimesprimedivisorproduct) + ge_balance_positive_unique_factorization_constructedprimesprimedivisorproductsecondimaginary)))))) /\ (exists ge_representation_real_code_unique_factorization_constructedprimesprimedivisorproductoutput ge_representation_imaginary_code_unique_factorization_constructedprimesprimedivisorproductoutput. (((gr_product_unique_factorization_constructedprimesprime) = ((ge_representation_real_code_unique_factorization_constructedprimesprimedivisorproductoutput) + (ge_representation_imaginary_code_unique_factorization_constructedprimesprimedivisorproductoutput)) * S ((ge_representation_real_code_unique_factorization_constructedprimesprimedivisorproductoutput) + (ge_representation_imaginary_code_unique_factorization_constructedprimesprimedivisorproductoutput)) + ((ge_representation_imaginary_code_unique_factorization_constructedprimesprimedivisorproductoutput) + (ge_representation_imaginary_code_unique_factorization_constructedprimesprimedivisorproductoutput))) /\ ((exists ge_balance_positive_unique_factorization_constructedprimesprimedivisorproductoutputreal ge_balance_negative_unique_factorization_constructedprimesprimedivisorproductoutputreal. (((((ge_representation_real_code_unique_factorization_constructedprimesprimedivisorproductoutput) = 2 * (ge_balance_positive_unique_factorization_constructedprimesprimedivisorproductoutputreal) /\ (ge_balance_negative_unique_factorization_constructedprimesprimedivisorproductoutputreal) = 0) \/ exists ge_signed_half_unique_factorization_constructedprimesprimedivisorproductoutputrealdecode. (((ge_representation_real_code_unique_factorization_constructedprimesprimedivisorproductoutput) = 2 * ge_signed_half_unique_factorization_constructedprimesprimedivisorproductoutputrealdecode + 1 /\ (ge_balance_positive_unique_factorization_constructedprimesprimedivisorproductoutputreal) = 0) /\ (ge_balance_negative_unique_factorization_constructedprimesprimedivisorproductoutputreal) = S ge_signed_half_unique_factorization_constructedprimesprimedivisorproductoutputrealdecode))) /\ ((((((((ge_first_rp_unique_factorization_constructedprimesprimedivisorproduct) * (ge_second_rp_unique_factorization_constructedprimesprimedivisorproduct))) + (((ge_first_rn_unique_factorization_constructedprimesprimedivisorproduct) * (ge_second_rn_unique_factorization_constructedprimesprimedivisorproduct))))) + (((((ge_first_ip_unique_factorization_constructedprimesprimedivisorproduct) * (ge_second_in_unique_factorization_constructedprimesprimedivisorproduct))) + (((ge_first_in_unique_factorization_constructedprimesprimedivisorproduct) * (ge_second_ip_unique_factorization_constructedprimesprimedivisorproduct))))))) + ge_balance_negative_unique_factorization_constructedprimesprimedivisorproductoutputreal = (((((((ge_first_rp_unique_factorization_constructedprimesprimedivisorproduct) * (ge_second_rn_unique_factorization_constructedprimesprimedivisorproduct))) + (((ge_first_rn_unique_factorization_constructedprimesprimedivisorproduct) * (ge_second_rp_unique_factorization_constructedprimesprimedivisorproduct))))) + (((((ge_first_ip_unique_factorization_constructedprimesprimedivisorproduct) * (ge_second_ip_unique_factorization_constructedprimesprimedivisorproduct))) + (((ge_first_in_unique_factorization_constructedprimesprimedivisorproduct) * (ge_second_in_unique_factorization_constructedprimesprimedivisorproduct))))))) + ge_balance_positive_unique_factorization_constructedprimesprimedivisorproductoutputreal))) /\ (exists ge_balance_positive_unique_factorization_constructedprimesprimedivisorproductoutputimaginary ge_balance_negative_unique_factorization_constructedprimesprimedivisorproductoutputimaginary. (((((ge_representation_imaginary_code_unique_factorization_constructedprimesprimedivisorproductoutput) = 2 * (ge_balance_positive_unique_factorization_constructedprimesprimedivisorproductoutputimaginary) /\ (ge_balance_negative_unique_factorization_constructedprimesprimedivisorproductoutputimaginary) = 0) \/ exists ge_signed_half_unique_factorization_constructedprimesprimedivisorproductoutputimaginarydecode. (((ge_representation_imaginary_code_unique_factorization_constructedprimesprimedivisorproductoutput) = 2 * ge_signed_half_unique_factorization_constructedprimesprimedivisorproductoutputimaginarydecode + 1 /\ (ge_balance_positive_unique_factorization_constructedprimesprimedivisorproductoutputimaginary) = 0) /\ (ge_balance_negative_unique_factorization_constructedprimesprimedivisorproductoutputimaginary) = S ge_signed_half_unique_factorization_constructedprimesprimedivisorproductoutputimaginarydecode))) /\ ((((((((ge_first_rp_unique_factorization_constructedprimesprimedivisorproduct) * (ge_second_ip_unique_factorization_constructedprimesprimedivisorproduct))) + (((ge_first_rn_unique_factorization_constructedprimesprimedivisorproduct) * (ge_second_in_unique_factorization_constructedprimesprimedivisorproduct))))) + (((((ge_first_ip_unique_factorization_constructedprimesprimedivisorproduct) * (ge_second_rp_unique_factorization_constructedprimesprimedivisorproduct))) + (((ge_first_in_unique_factorization_constructedprimesprimedivisorproduct) * (ge_second_rn_unique_factorization_constructedprimesprimedivisorproduct))))))) + ge_balance_negative_unique_factorization_constructedprimesprimedivisorproductoutputimaginary = (((((((ge_first_rp_unique_factorization_constructedprimesprimedivisorproduct) * (ge_second_in_unique_factorization_constructedprimesprimedivisorproduct))) + (((ge_first_rn_unique_factorization_constructedprimesprimedivisorproduct) * (ge_second_ip_unique_factorization_constructedprimesprimedivisorproduct))))) + (((((ge_first_ip_unique_factorization_constructedprimesprimedivisorproduct) * (ge_second_rn_unique_factorization_constructedprimesprimedivisorproduct))) + (((ge_first_in_unique_factorization_constructedprimesprimedivisorproduct) * (ge_second_rp_unique_factorization_constructedprimesprimedivisorproduct))))))) + ge_balance_positive_unique_factorization_constructedprimesprimedivisorproductoutputimaginary)))))))))) -> (exists gr_quotient_unique_factorization_constructedprimesprimefirst_divisor. (exists ge_first_rp_unique_factorization_constructedprimesprimefirst_divisorproduct ge_first_rn_unique_factorization_constructedprimesprimefirst_divisorproduct ge_first_ip_unique_factorization_constructedprimesprimefirst_divisorproduct ge_first_in_unique_factorization_constructedprimesprimefirst_divisorproduct ge_second_rp_unique_factorization_constructedprimesprimefirst_divisorproduct ge_second_rn_unique_factorization_constructedprimesprimefirst_divisorproduct ge_second_ip_unique_factorization_constructedprimesprimefirst_divisorproduct ge_second_in_unique_factorization_constructedprimesprimefirst_divisorproduct. ((exists ge_representation_real_code_unique_factorization_constructedprimesprimefirst_divisorproductfirst ge_representation_imaginary_code_unique_factorization_constructedprimesprimefirst_divisorproductfirst. (((gr_prime_factor_value_unique_factorization_constructedprimes) = ((ge_representation_real_code_unique_factorization_constructedprimesprimefirst_divisorproductfirst) + (ge_representation_imaginary_code_unique_factorization_constructedprimesprimefirst_divisorproductfirst)) * S ((ge_representation_real_code_unique_factorization_constructedprimesprimefirst_divisorproductfirst) + (ge_representation_imaginary_code_unique_factorization_constructedprimesprimefirst_divisorproductfirst)) + ((ge_representation_imaginary_code_unique_factorization_constructedprimesprimefirst_divisorproductfirst) + (ge_representation_imaginary_code_unique_factorization_constructedprimesprimefirst_divisorproductfirst))) /\ ((exists ge_balance_positive_unique_factorization_constructedprimesprimefirst_divisorproductfirstreal ge_balance_negative_unique_factorization_constructedprimesprimefirst_divisorproductfirstreal. (((((ge_representation_real_code_unique_factorization_constructedprimesprimefirst_divisorproductfirst) = 2 * (ge_balance_positive_unique_factorization_constructedprimesprimefirst_divisorproductfirstreal) /\ (ge_balance_negative_unique_factorization_constructedprimesprimefirst_divisorproductfirstreal) = 0) \/ exists ge_signed_half_unique_factorization_constructedprimesprimefirst_divisorproductfirstrealdecode. (((ge_representation_real_code_unique_factorization_constructedprimesprimefirst_divisorproductfirst) = 2 * ge_signed_half_unique_factorization_constructedprimesprimefirst_divisorproductfirstrealdecode + 1 /\ (ge_balance_positive_unique_factorization_constructedprimesprimefirst_divisorproductfirstreal) = 0) /\ (ge_balance_negative_unique_factorization_constructedprimesprimefirst_divisorproductfirstreal) = S ge_signed_half_unique_factorization_constructedprimesprimefirst_divisorproductfirstrealdecode))) /\ ((ge_first_rp_unique_factorization_constructedprimesprimefirst_divisorproduct) + ge_balance_negative_unique_factorization_constructedprimesprimefirst_divisorproductfirstreal = (ge_first_rn_unique_factorization_constructedprimesprimefirst_divisorproduct) + ge_balance_positive_unique_factorization_constructedprimesprimefirst_divisorproductfirstreal))) /\ (exists ge_balance_positive_unique_factorization_constructedprimesprimefirst_divisorproductfirstimaginary ge_balance_negative_unique_factorization_constructedprimesprimefirst_divisorproductfirstimaginary. (((((ge_representation_imaginary_code_unique_factorization_constructedprimesprimefirst_divisorproductfirst) = 2 * (ge_balance_positive_unique_factorization_constructedprimesprimefirst_divisorproductfirstimaginary) /\ (ge_balance_negative_unique_factorization_constructedprimesprimefirst_divisorproductfirstimaginary) = 0) \/ exists ge_signed_half_unique_factorization_constructedprimesprimefirst_divisorproductfirstimaginarydecode. (((ge_representation_imaginary_code_unique_factorization_constructedprimesprimefirst_divisorproductfirst) = 2 * ge_signed_half_unique_factorization_constructedprimesprimefirst_divisorproductfirstimaginarydecode + 1 /\ (ge_balance_positive_unique_factorization_constructedprimesprimefirst_divisorproductfirstimaginary) = 0) /\ (ge_balance_negative_unique_factorization_constructedprimesprimefirst_divisorproductfirstimaginary) = S ge_signed_half_unique_factorization_constructedprimesprimefirst_divisorproductfirstimaginarydecode))) /\ ((ge_first_ip_unique_factorization_constructedprimesprimefirst_divisorproduct) + ge_balance_negative_unique_factorization_constructedprimesprimefirst_divisorproductfirstimaginary = (ge_first_in_unique_factorization_constructedprimesprimefirst_divisorproduct) + ge_balance_positive_unique_factorization_constructedprimesprimefirst_divisorproductfirstimaginary)))))) /\ ((exists ge_representation_real_code_unique_factorization_constructedprimesprimefirst_divisorproductsecond ge_representation_imaginary_code_unique_factorization_constructedprimesprimefirst_divisorproductsecond. (((gr_quotient_unique_factorization_constructedprimesprimefirst_divisor) = ((ge_representation_real_code_unique_factorization_constructedprimesprimefirst_divisorproductsecond) + (ge_representation_imaginary_code_unique_factorization_constructedprimesprimefirst_divisorproductsecond)) * S ((ge_representation_real_code_unique_factorization_constructedprimesprimefirst_divisorproductsecond) + (ge_representation_imaginary_code_unique_factorization_constructedprimesprimefirst_divisorproductsecond)) + ((ge_representation_imaginary_code_unique_factorization_constructedprimesprimefirst_divisorproductsecond) + (ge_representation_imaginary_code_unique_factorization_constructedprimesprimefirst_divisorproductsecond))) /\ ((exists ge_balance_positive_unique_factorization_constructedprimesprimefirst_divisorproductsecondreal ge_balance_negative_unique_factorization_constructedprimesprimefirst_divisorproductsecondreal. (((((ge_representation_real_code_unique_factorization_constructedprimesprimefirst_divisorproductsecond) = 2 * (ge_balance_positive_unique_factorization_constructedprimesprimefirst_divisorproductsecondreal) /\ (ge_balance_negative_unique_factorization_constructedprimesprimefirst_divisorproductsecondreal) = 0) \/ exists ge_signed_half_unique_factorization_constructedprimesprimefirst_divisorproductsecondrealdecode. (((ge_representation_real_code_unique_factorization_constructedprimesprimefirst_divisorproductsecond) = 2 * ge_signed_half_unique_factorization_constructedprimesprimefirst_divisorproductsecondrealdecode + 1 /\ (ge_balance_positive_unique_factorization_constructedprimesprimefirst_divisorproductsecondreal) = 0) /\ (ge_balance_negative_unique_factorization_constructedprimesprimefirst_divisorproductsecondreal) = S ge_signed_half_unique_factorization_constructedprimesprimefirst_divisorproductsecondrealdecode))) /\ ((ge_second_rp_unique_factorization_constructedprimesprimefirst_divisorproduct) + ge_balance_negative_unique_factorization_constructedprimesprimefirst_divisorproductsecondreal = (ge_second_rn_unique_factorization_constructedprimesprimefirst_divisorproduct) + ge_balance_positive_unique_factorization_constructedprimesprimefirst_divisorproductsecondreal))) /\ (exists ge_balance_positive_unique_factorization_constructedprimesprimefirst_divisorproductsecondimaginary ge_balance_negative_unique_factorization_constructedprimesprimefirst_divisorproductsecondimaginary. (((((ge_representation_imaginary_code_unique_factorization_constructedprimesprimefirst_divisorproductsecond) = 2 * (ge_balance_positive_unique_factorization_constructedprimesprimefirst_divisorproductsecondimaginary) /\ (ge_balance_negative_unique_factorization_constructedprimesprimefirst_divisorproductsecondimaginary) = 0) \/ exists ge_signed_half_unique_factorization_constructedprimesprimefirst_divisorproductsecondimaginarydecode. (((ge_representation_imaginary_code_unique_factorization_constructedprimesprimefirst_divisorproductsecond) = 2 * ge_signed_half_unique_factorization_constructedprimesprimefirst_divisorproductsecondimaginarydecode + 1 /\ (ge_balance_positive_unique_factorization_constructedprimesprimefirst_divisorproductsecondimaginary) = 0) /\ (ge_balance_negative_unique_factorization_constructedprimesprimefirst_divisorproductsecondimaginary) = S ge_signed_half_unique_factorization_constructedprimesprimefirst_divisorproductsecondimaginarydecode))) /\ ((ge_second_ip_unique_factorization_constructedprimesprimefirst_divisorproduct) + ge_balance_negative_unique_factorization_constructedprimesprimefirst_divisorproductsecondimaginary = (ge_second_in_unique_factorization_constructedprimesprimefirst_divisorproduct) + ge_balance_positive_unique_factorization_constructedprimesprimefirst_divisorproductsecondimaginary)))))) /\ (exists ge_representation_real_code_unique_factorization_constructedprimesprimefirst_divisorproductoutput ge_representation_imaginary_code_unique_factorization_constructedprimesprimefirst_divisorproductoutput. (((gr_first_factor_unique_factorization_constructedprimesprime) = ((ge_representation_real_code_unique_factorization_constructedprimesprimefirst_divisorproductoutput) + (ge_representation_imaginary_code_unique_factorization_constructedprimesprimefirst_divisorproductoutput)) * S ((ge_representation_real_code_unique_factorization_constructedprimesprimefirst_divisorproductoutput) + (ge_representation_imaginary_code_unique_factorization_constructedprimesprimefirst_divisorproductoutput)) + ((ge_representation_imaginary_code_unique_factorization_constructedprimesprimefirst_divisorproductoutput) + (ge_representation_imaginary_code_unique_factorization_constructedprimesprimefirst_divisorproductoutput))) /\ ((exists ge_balance_positive_unique_factorization_constructedprimesprimefirst_divisorproductoutputreal ge_balance_negative_unique_factorization_constructedprimesprimefirst_divisorproductoutputreal. (((((ge_representation_real_code_unique_factorization_constructedprimesprimefirst_divisorproductoutput) = 2 * (ge_balance_positive_unique_factorization_constructedprimesprimefirst_divisorproductoutputreal) /\ (ge_balance_negative_unique_factorization_constructedprimesprimefirst_divisorproductoutputreal) = 0) \/ exists ge_signed_half_unique_factorization_constructedprimesprimefirst_divisorproductoutputrealdecode. (((ge_representation_real_code_unique_factorization_constructedprimesprimefirst_divisorproductoutput) = 2 * ge_signed_half_unique_factorization_constructedprimesprimefirst_divisorproductoutputrealdecode + 1 /\ (ge_balance_positive_unique_factorization_constructedprimesprimefirst_divisorproductoutputreal) = 0) /\ (ge_balance_negative_unique_factorization_constructedprimesprimefirst_divisorproductoutputreal) = S ge_signed_half_unique_factorization_constructedprimesprimefirst_divisorproductoutputrealdecode))) /\ ((((((((ge_first_rp_unique_factorization_constructedprimesprimefirst_divisorproduct) * (ge_second_rp_unique_factorization_constructedprimesprimefirst_divisorproduct))) + (((ge_first_rn_unique_factorization_constructedprimesprimefirst_divisorproduct) * (ge_second_rn_unique_factorization_constructedprimesprimefirst_divisorproduct))))) + (((((ge_first_ip_unique_factorization_constructedprimesprimefirst_divisorproduct) * (ge_second_in_unique_factorization_constructedprimesprimefirst_divisorproduct))) + (((ge_first_in_unique_factorization_constructedprimesprimefirst_divisorproduct) * (ge_second_ip_unique_factorization_constructedprimesprimefirst_divisorproduct))))))) + ge_balance_negative_unique_factorization_constructedprimesprimefirst_divisorproductoutputreal = (((((((ge_first_rp_unique_factorization_constructedprimesprimefirst_divisorproduct) * (ge_second_rn_unique_factorization_constructedprimesprimefirst_divisorproduct))) + (((ge_first_rn_unique_factorization_constructedprimesprimefirst_divisorproduct) * (ge_second_rp_unique_factorization_constructedprimesprimefirst_divisorproduct))))) + (((((ge_first_ip_unique_factorization_constructedprimesprimefirst_divisorproduct) * (ge_second_ip_unique_factorization_constructedprimesprimefirst_divisorproduct))) + (((ge_first_in_unique_factorization_constructedprimesprimefirst_divisorproduct) * (ge_second_in_unique_factorization_constructedprimesprimefirst_divisorproduct))))))) + ge_balance_positive_unique_factorization_constructedprimesprimefirst_divisorproductoutputreal))) /\ (exists ge_balance_positive_unique_factorization_constructedprimesprimefirst_divisorproductoutputimaginary ge_balance_negative_unique_factorization_constructedprimesprimefirst_divisorproductoutputimaginary. (((((ge_representation_imaginary_code_unique_factorization_constructedprimesprimefirst_divisorproductoutput) = 2 * (ge_balance_positive_unique_factorization_constructedprimesprimefirst_divisorproductoutputimaginary) /\ (ge_balance_negative_unique_factorization_constructedprimesprimefirst_divisorproductoutputimaginary) = 0) \/ exists ge_signed_half_unique_factorization_constructedprimesprimefirst_divisorproductoutputimaginarydecode. (((ge_representation_imaginary_code_unique_factorization_constructedprimesprimefirst_divisorproductoutput) = 2 * ge_signed_half_unique_factorization_constructedprimesprimefirst_divisorproductoutputimaginarydecode + 1 /\ (ge_balance_positive_unique_factorization_constructedprimesprimefirst_divisorproductoutputimaginary) = 0) /\ (ge_balance_negative_unique_factorization_constructedprimesprimefirst_divisorproductoutputimaginary) = S ge_signed_half_unique_factorization_constructedprimesprimefirst_divisorproductoutputimaginarydecode))) /\ ((((((((ge_first_rp_unique_factorization_constructedprimesprimefirst_divisorproduct) * (ge_second_ip_unique_factorization_constructedprimesprimefirst_divisorproduct))) + (((ge_first_rn_unique_factorization_constructedprimesprimefirst_divisorproduct) * (ge_second_in_unique_factorization_constructedprimesprimefirst_divisorproduct))))) + (((((ge_first_ip_unique_factorization_constructedprimesprimefirst_divisorproduct) * (ge_second_rp_unique_factorization_constructedprimesprimefirst_divisorproduct))) + (((ge_first_in_unique_factorization_constructedprimesprimefirst_divisorproduct) * (ge_second_rn_unique_factorization_constructedprimesprimefirst_divisorproduct))))))) + ge_balance_negative_unique_factorization_constructedprimesprimefirst_divisorproductoutputimaginary = (((((((ge_first_rp_unique_factorization_constructedprimesprimefirst_divisorproduct) * (ge_second_in_unique_factorization_constructedprimesprimefirst_divisorproduct))) + (((ge_first_rn_unique_factorization_constructedprimesprimefirst_divisorproduct) * (ge_second_ip_unique_factorization_constructedprimesprimefirst_divisorproduct))))) + (((((ge_first_ip_unique_factorization_constructedprimesprimefirst_divisorproduct) * (ge_second_rn_unique_factorization_constructedprimesprimefirst_divisorproduct))) + (((ge_first_in_unique_factorization_constructedprimesprimefirst_divisorproduct) * (ge_second_rp_unique_factorization_constructedprimesprimefirst_divisorproduct))))))) + ge_balance_positive_unique_factorization_constructedprimesprimefirst_divisorproductoutputimaginary)))))))))) \/ (exists gr_quotient_unique_factorization_constructedprimesprimesecond_divisor. (exists ge_first_rp_unique_factorization_constructedprimesprimesecond_divisorproduct ge_first_rn_unique_factorization_constructedprimesprimesecond_divisorproduct ge_first_ip_unique_factorization_constructedprimesprimesecond_divisorproduct ge_first_in_unique_factorization_constructedprimesprimesecond_divisorproduct ge_second_rp_unique_factorization_constructedprimesprimesecond_divisorproduct ge_second_rn_unique_factorization_constructedprimesprimesecond_divisorproduct ge_second_ip_unique_factorization_constructedprimesprimesecond_divisorproduct ge_second_in_unique_factorization_constructedprimesprimesecond_divisorproduct. ((exists ge_representation_real_code_unique_factorization_constructedprimesprimesecond_divisorproductfirst ge_representation_imaginary_code_unique_factorization_constructedprimesprimesecond_divisorproductfirst. (((gr_prime_factor_value_unique_factorization_constructedprimes) = ((ge_representation_real_code_unique_factorization_constructedprimesprimesecond_divisorproductfirst) + (ge_representation_imaginary_code_unique_factorization_constructedprimesprimesecond_divisorproductfirst)) * S ((ge_representation_real_code_unique_factorization_constructedprimesprimesecond_divisorproductfirst) + (ge_representation_imaginary_code_unique_factorization_constructedprimesprimesecond_divisorproductfirst)) + ((ge_representation_imaginary_code_unique_factorization_constructedprimesprimesecond_divisorproductfirst) + (ge_representation_imaginary_code_unique_factorization_constructedprimesprimesecond_divisorproductfirst))) /\ ((exists ge_balance_positive_unique_factorization_constructedprimesprimesecond_divisorproductfirstreal ge_balance_negative_unique_factorization_constructedprimesprimesecond_divisorproductfirstreal. (((((ge_representation_real_code_unique_factorization_constructedprimesprimesecond_divisorproductfirst) = 2 * (ge_balance_positive_unique_factorization_constructedprimesprimesecond_divisorproductfirstreal) /\ (ge_balance_negative_unique_factorization_constructedprimesprimesecond_divisorproductfirstreal) = 0) \/ exists ge_signed_half_unique_factorization_constructedprimesprimesecond_divisorproductfirstrealdecode. (((ge_representation_real_code_unique_factorization_constructedprimesprimesecond_divisorproductfirst) = 2 * ge_signed_half_unique_factorization_constructedprimesprimesecond_divisorproductfirstrealdecode + 1 /\ (ge_balance_positive_unique_factorization_constructedprimesprimesecond_divisorproductfirstreal) = 0) /\ (ge_balance_negative_unique_factorization_constructedprimesprimesecond_divisorproductfirstreal) = S ge_signed_half_unique_factorization_constructedprimesprimesecond_divisorproductfirstrealdecode))) /\ ((ge_first_rp_unique_factorization_constructedprimesprimesecond_divisorproduct) + ge_balance_negative_unique_factorization_constructedprimesprimesecond_divisorproductfirstreal = (ge_first_rn_unique_factorization_constructedprimesprimesecond_divisorproduct) + ge_balance_positive_unique_factorization_constructedprimesprimesecond_divisorproductfirstreal))) /\ (exists ge_balance_positive_unique_factorization_constructedprimesprimesecond_divisorproductfirstimaginary ge_balance_negative_unique_factorization_constructedprimesprimesecond_divisorproductfirstimaginary. (((((ge_representation_imaginary_code_unique_factorization_constructedprimesprimesecond_divisorproductfirst) = 2 * (ge_balance_positive_unique_factorization_constructedprimesprimesecond_divisorproductfirstimaginary) /\ (ge_balance_negative_unique_factorization_constructedprimesprimesecond_divisorproductfirstimaginary) = 0) \/ exists ge_signed_half_unique_factorization_constructedprimesprimesecond_divisorproductfirstimaginarydecode. (((ge_representation_imaginary_code_unique_factorization_constructedprimesprimesecond_divisorproductfirst) = 2 * ge_signed_half_unique_factorization_constructedprimesprimesecond_divisorproductfirstimaginarydecode + 1 /\ (ge_balance_positive_unique_factorization_constructedprimesprimesecond_divisorproductfirstimaginary) = 0) /\ (ge_balance_negative_unique_factorization_constructedprimesprimesecond_divisorproductfirstimaginary) = S ge_signed_half_unique_factorization_constructedprimesprimesecond_divisorproductfirstimaginarydecode))) /\ ((ge_first_ip_unique_factorization_constructedprimesprimesecond_divisorproduct) + ge_balance_negative_unique_factorization_constructedprimesprimesecond_divisorproductfirstimaginary = (ge_first_in_unique_factorization_constructedprimesprimesecond_divisorproduct) + ge_balance_positive_unique_factorization_constructedprimesprimesecond_divisorproductfirstimaginary)))))) /\ ((exists ge_representation_real_code_unique_factorization_constructedprimesprimesecond_divisorproductsecond ge_representation_imaginary_code_unique_factorization_constructedprimesprimesecond_divisorproductsecond. (((gr_quotient_unique_factorization_constructedprimesprimesecond_divisor) = ((ge_representation_real_code_unique_factorization_constructedprimesprimesecond_divisorproductsecond) + (ge_representation_imaginary_code_unique_factorization_constructedprimesprimesecond_divisorproductsecond)) * S ((ge_representation_real_code_unique_factorization_constructedprimesprimesecond_divisorproductsecond) + (ge_representation_imaginary_code_unique_factorization_constructedprimesprimesecond_divisorproductsecond)) + ((ge_representation_imaginary_code_unique_factorization_constructedprimesprimesecond_divisorproductsecond) + (ge_representation_imaginary_code_unique_factorization_constructedprimesprimesecond_divisorproductsecond))) /\ ((exists ge_balance_positive_unique_factorization_constructedprimesprimesecond_divisorproductsecondreal ge_balance_negative_unique_factorization_constructedprimesprimesecond_divisorproductsecondreal. (((((ge_representation_real_code_unique_factorization_constructedprimesprimesecond_divisorproductsecond) = 2 * (ge_balance_positive_unique_factorization_constructedprimesprimesecond_divisorproductsecondreal) /\ (ge_balance_negative_unique_factorization_constructedprimesprimesecond_divisorproductsecondreal) = 0) \/ exists ge_signed_half_unique_factorization_constructedprimesprimesecond_divisorproductsecondrealdecode. (((ge_representation_real_code_unique_factorization_constructedprimesprimesecond_divisorproductsecond) = 2 * ge_signed_half_unique_factorization_constructedprimesprimesecond_divisorproductsecondrealdecode + 1 /\ (ge_balance_positive_unique_factorization_constructedprimesprimesecond_divisorproductsecondreal) = 0) /\ (ge_balance_negative_unique_factorization_constructedprimesprimesecond_divisorproductsecondreal) = S ge_signed_half_unique_factorization_constructedprimesprimesecond_divisorproductsecondrealdecode))) /\ ((ge_second_rp_unique_factorization_constructedprimesprimesecond_divisorproduct) + ge_balance_negative_unique_factorization_constructedprimesprimesecond_divisorproductsecondreal = (ge_second_rn_unique_factorization_constructedprimesprimesecond_divisorproduct) + ge_balance_positive_unique_factorization_constructedprimesprimesecond_divisorproductsecondreal))) /\ (exists ge_balance_positive_unique_factorization_constructedprimesprimesecond_divisorproductsecondimaginary ge_balance_negative_unique_factorization_constructedprimesprimesecond_divisorproductsecondimaginary. (((((ge_representation_imaginary_code_unique_factorization_constructedprimesprimesecond_divisorproductsecond) = 2 * (ge_balance_positive_unique_factorization_constructedprimesprimesecond_divisorproductsecondimaginary) /\ (ge_balance_negative_unique_factorization_constructedprimesprimesecond_divisorproductsecondimaginary) = 0) \/ exists ge_signed_half_unique_factorization_constructedprimesprimesecond_divisorproductsecondimaginarydecode. (((ge_representation_imaginary_code_unique_factorization_constructedprimesprimesecond_divisorproductsecond) = 2 * ge_signed_half_unique_factorization_constructedprimesprimesecond_divisorproductsecondimaginarydecode + 1 /\ (ge_balance_positive_unique_factorization_constructedprimesprimesecond_divisorproductsecondimaginary) = 0) /\ (ge_balance_negative_unique_factorization_constructedprimesprimesecond_divisorproductsecondimaginary) = S ge_signed_half_unique_factorization_constructedprimesprimesecond_divisorproductsecondimaginarydecode))) /\ ((ge_second_ip_unique_factorization_constructedprimesprimesecond_divisorproduct) + ge_balance_negative_unique_factorization_constructedprimesprimesecond_divisorproductsecondimaginary = (ge_second_in_unique_factorization_constructedprimesprimesecond_divisorproduct) + ge_balance_positive_unique_factorization_constructedprimesprimesecond_divisorproductsecondimaginary)))))) /\ (exists ge_representation_real_code_unique_factorization_constructedprimesprimesecond_divisorproductoutput ge_representation_imaginary_code_unique_factorization_constructedprimesprimesecond_divisorproductoutput. (((gr_second_factor_unique_factorization_constructedprimesprime) = ((ge_representation_real_code_unique_factorization_constructedprimesprimesecond_divisorproductoutput) + (ge_representation_imaginary_code_unique_factorization_constructedprimesprimesecond_divisorproductoutput)) * S ((ge_representation_real_code_unique_factorization_constructedprimesprimesecond_divisorproductoutput) + (ge_representation_imaginary_code_unique_factorization_constructedprimesprimesecond_divisorproductoutput)) + ((ge_representation_imaginary_code_unique_factorization_constructedprimesprimesecond_divisorproductoutput) + (ge_representation_imaginary_code_unique_factorization_constructedprimesprimesecond_divisorproductoutput))) /\ ((exists ge_balance_positive_unique_factorization_constructedprimesprimesecond_divisorproductoutputreal ge_balance_negative_unique_factorization_constructedprimesprimesecond_divisorproductoutputreal. (((((ge_representation_real_code_unique_factorization_constructedprimesprimesecond_divisorproductoutput) = 2 * (ge_balance_positive_unique_factorization_constructedprimesprimesecond_divisorproductoutputreal) /\ (ge_balance_negative_unique_factorization_constructedprimesprimesecond_divisorproductoutputreal) = 0) \/ exists ge_signed_half_unique_factorization_constructedprimesprimesecond_divisorproductoutputrealdecode. (((ge_representation_real_code_unique_factorization_constructedprimesprimesecond_divisorproductoutput) = 2 * ge_signed_half_unique_factorization_constructedprimesprimesecond_divisorproductoutputrealdecode + 1 /\ (ge_balance_positive_unique_factorization_constructedprimesprimesecond_divisorproductoutputreal) = 0) /\ (ge_balance_negative_unique_factorization_constructedprimesprimesecond_divisorproductoutputreal) = S ge_signed_half_unique_factorization_constructedprimesprimesecond_divisorproductoutputrealdecode))) /\ ((((((((ge_first_rp_unique_factorization_constructedprimesprimesecond_divisorproduct) * (ge_second_rp_unique_factorization_constructedprimesprimesecond_divisorproduct))) + (((ge_first_rn_unique_factorization_constructedprimesprimesecond_divisorproduct) * (ge_second_rn_unique_factorization_constructedprimesprimesecond_divisorproduct))))) + (((((ge_first_ip_unique_factorization_constructedprimesprimesecond_divisorproduct) * (ge_second_in_unique_factorization_constructedprimesprimesecond_divisorproduct))) + (((ge_first_in_unique_factorization_constructedprimesprimesecond_divisorproduct) * (ge_second_ip_unique_factorization_constructedprimesprimesecond_divisorproduct))))))) + ge_balance_negative_unique_factorization_constructedprimesprimesecond_divisorproductoutputreal = (((((((ge_first_rp_unique_factorization_constructedprimesprimesecond_divisorproduct) * (ge_second_rn_unique_factorization_constructedprimesprimesecond_divisorproduct))) + (((ge_first_rn_unique_factorization_constructedprimesprimesecond_divisorproduct) * (ge_second_rp_unique_factorization_constructedprimesprimesecond_divisorproduct))))) + (((((ge_first_ip_unique_factorization_constructedprimesprimesecond_divisorproduct) * (ge_second_ip_unique_factorization_constructedprimesprimesecond_divisorproduct))) + (((ge_first_in_unique_factorization_constructedprimesprimesecond_divisorproduct) * (ge_second_in_unique_factorization_constructedprimesprimesecond_divisorproduct))))))) + ge_balance_positive_unique_factorization_constructedprimesprimesecond_divisorproductoutputreal))) /\ (exists ge_balance_positive_unique_factorization_constructedprimesprimesecond_divisorproductoutputimaginary ge_balance_negative_unique_factorization_constructedprimesprimesecond_divisorproductoutputimaginary. (((((ge_representation_imaginary_code_unique_factorization_constructedprimesprimesecond_divisorproductoutput) = 2 * (ge_balance_positive_unique_factorization_constructedprimesprimesecond_divisorproductoutputimaginary) /\ (ge_balance_negative_unique_factorization_constructedprimesprimesecond_divisorproductoutputimaginary) = 0) \/ exists ge_signed_half_unique_factorization_constructedprimesprimesecond_divisorproductoutputimaginarydecode. (((ge_representation_imaginary_code_unique_factorization_constructedprimesprimesecond_divisorproductoutput) = 2 * ge_signed_half_unique_factorization_constructedprimesprimesecond_divisorproductoutputimaginarydecode + 1 /\ (ge_balance_positive_unique_factorization_constructedprimesprimesecond_divisorproductoutputimaginary) = 0) /\ (ge_balance_negative_unique_factorization_constructedprimesprimesecond_divisorproductoutputimaginary) = S ge_signed_half_unique_factorization_constructedprimesprimesecond_divisorproductoutputimaginarydecode))) /\ ((((((((ge_first_rp_unique_factorization_constructedprimesprimesecond_divisorproduct) * (ge_second_ip_unique_factorization_constructedprimesprimesecond_divisorproduct))) + (((ge_first_rn_unique_factorization_constructedprimesprimesecond_divisorproduct) * (ge_second_in_unique_factorization_constructedprimesprimesecond_divisorproduct))))) + (((((ge_first_ip_unique_factorization_constructedprimesprimesecond_divisorproduct) * (ge_second_rp_unique_factorization_constructedprimesprimesecond_divisorproduct))) + (((ge_first_in_unique_factorization_constructedprimesprimesecond_divisorproduct) * (ge_second_rn_unique_factorization_constructedprimesprimesecond_divisorproduct))))))) + ge_balance_negative_unique_factorization_constructedprimesprimesecond_divisorproductoutputimaginary = (((((((ge_first_rp_unique_factorization_constructedprimesprimesecond_divisorproduct) * (ge_second_in_unique_factorization_constructedprimesprimesecond_divisorproduct))) + (((ge_first_rn_unique_factorization_constructedprimesprimesecond_divisorproduct) * (ge_second_ip_unique_factorization_constructedprimesprimesecond_divisorproduct))))) + (((((ge_first_ip_unique_factorization_constructedprimesprimesecond_divisorproduct) * (ge_second_rn_unique_factorization_constructedprimesprimesecond_divisorproduct))) + (((ge_first_in_unique_factorization_constructedprimesprimesecond_divisorproduct) * (ge_second_rp_unique_factorization_constructedprimesprimesecond_divisorproduct))))))) + ge_balance_positive_unique_factorization_constructedprimesprimesecond_divisorproductoutputimaginary)))))))))))))))) /\ (exists gr_prime_factor_product_unique_factorization_constructed. ((exists gr_product_trace_unique_factorization_constructedtrace gr_product_scale_unique_factorization_constructedtrace. ((((exists ff_h_gprod_unique_factorization_constructedtracestart. ff_h_gprod_unique_factorization_constructedtracestart + S (6) = S ((S (0)) * gr_product_scale_unique_factorization_constructedtrace)) /\ exists ff_q_gprod_unique_factorization_constructedtracestart. gr_product_trace_unique_factorization_constructedtrace = ff_q_gprod_unique_factorization_constructedtracestart * S ((S (0)) * gr_product_scale_unique_factorization_constructedtrace) + (6))) /\ ((((exists ff_h_gprod_unique_factorization_constructedtraceend. ff_h_gprod_unique_factorization_constructedtraceend + S (gr_prime_factor_product_unique_factorization_constructed) = S ((S (l)) * gr_product_scale_unique_factorization_constructedtrace)) /\ exists ff_q_gprod_unique_factorization_constructedtraceend. gr_product_trace_unique_factorization_constructedtrace = ff_q_gprod_unique_factorization_constructedtraceend * S ((S (l)) * gr_product_scale_unique_factorization_constructedtrace) + (gr_prime_factor_product_unique_factorization_constructed))) /\ (forall gr_product_index_unique_factorization_constructedtracesteps. (exists ge_gap_unique_factorization_constructedtracestepsindex_bound. ge_gap_unique_factorization_constructedtracestepsindex_bound + S (gr_product_index_unique_factorization_constructedtracesteps) = (l)) -> exists gr_product_factor_unique_factorization_constructedtracesteps gr_product_before_unique_factorization_constructedtracesteps gr_product_after_unique_factorization_constructedtracesteps. ((((exists ff_h_gprod_unique_factorization_constructedtracestepsfactor. ff_h_gprod_unique_factorization_constructedtracestepsfactor + S (gr_product_factor_unique_factorization_constructedtracesteps) = S ((S (gr_product_index_unique_factorization_constructedtracesteps)) * c)) /\ exists ff_q_gprod_unique_factorization_constructedtracestepsfactor. b = ff_q_gprod_unique_factorization_constructedtracestepsfactor * S ((S (gr_product_index_unique_factorization_constructedtracesteps)) * c) + (gr_product_factor_unique_factorization_constructedtracesteps))) /\ ((((exists ff_h_gprod_unique_factorization_constructedtracestepsbefore. ff_h_gprod_unique_factorization_constructedtracestepsbefore + S (gr_product_before_unique_factorization_constructedtracesteps) = S ((S (gr_product_index_unique_factorization_constructedtracesteps)) * gr_product_scale_unique_factorization_constructedtrace)) /\ exists ff_q_gprod_unique_factorization_constructedtracestepsbefore. gr_product_trace_unique_factorization_constructedtrace = ff_q_gprod_unique_factorization_constructedtracestepsbefore * S ((S (gr_product_index_unique_factorization_constructedtracesteps)) * gr_product_scale_unique_factorization_constructedtrace) + (gr_product_before_unique_factorization_constructedtracesteps))) /\ ((((exists ff_h_gprod_unique_factorization_constructedtracestepsafter. ff_h_gprod_unique_factorization_constructedtracestepsafter + S (gr_product_after_unique_factorization_constructedtracesteps) = S ((S (S (gr_product_index_unique_factorization_constructedtracesteps))) * gr_product_scale_unique_factorization_constructedtrace)) /\ exists ff_q_gprod_unique_factorization_constructedtracestepsafter. gr_product_trace_unique_factorization_constructedtrace = ff_q_gprod_unique_factorization_constructedtracestepsafter * S ((S (S (gr_product_index_unique_factorization_constructedtracesteps))) * gr_product_scale_unique_factorization_constructedtrace) + (gr_product_after_unique_factorization_constructedtracesteps))) /\ (exists ge_first_rp_unique_factorization_constructedtracestepsmultiply ge_first_rn_unique_factorization_constructedtracestepsmultiply ge_first_ip_unique_factorization_constructedtracestepsmultiply ge_first_in_unique_factorization_constructedtracestepsmultiply ge_second_rp_unique_factorization_constructedtracestepsmultiply ge_second_rn_unique_factorization_constructedtracestepsmultiply ge_second_ip_unique_factorization_constructedtracestepsmultiply ge_second_in_unique_factorization_constructedtracestepsmultiply. ((exists ge_representation_real_code_unique_factorization_constructedtracestepsmultiplyfirst ge_representation_imaginary_code_unique_factorization_constructedtracestepsmultiplyfirst. (((gr_product_before_unique_factorization_constructedtracesteps) = ((ge_representation_real_code_unique_factorization_constructedtracestepsmultiplyfirst) + (ge_representation_imaginary_code_unique_factorization_constructedtracestepsmultiplyfirst)) * S ((ge_representation_real_code_unique_factorization_constructedtracestepsmultiplyfirst) + (ge_representation_imaginary_code_unique_factorization_constructedtracestepsmultiplyfirst)) + ((ge_representation_imaginary_code_unique_factorization_constructedtracestepsmultiplyfirst) + (ge_representation_imaginary_code_unique_factorization_constructedtracestepsmultiplyfirst))) /\ ((exists ge_balance_positive_unique_factorization_constructedtracestepsmultiplyfirstreal ge_balance_negative_unique_factorization_constructedtracestepsmultiplyfirstreal. (((((ge_representation_real_code_unique_factorization_constructedtracestepsmultiplyfirst) = 2 * (ge_balance_positive_unique_factorization_constructedtracestepsmultiplyfirstreal) /\ (ge_balance_negative_unique_factorization_constructedtracestepsmultiplyfirstreal) = 0) \/ exists ge_signed_half_unique_factorization_constructedtracestepsmultiplyfirstrealdecode. (((ge_representation_real_code_unique_factorization_constructedtracestepsmultiplyfirst) = 2 * ge_signed_half_unique_factorization_constructedtracestepsmultiplyfirstrealdecode + 1 /\ (ge_balance_positive_unique_factorization_constructedtracestepsmultiplyfirstreal) = 0) /\ (ge_balance_negative_unique_factorization_constructedtracestepsmultiplyfirstreal) = S ge_signed_half_unique_factorization_constructedtracestepsmultiplyfirstrealdecode))) /\ ((ge_first_rp_unique_factorization_constructedtracestepsmultiply) + ge_balance_negative_unique_factorization_constructedtracestepsmultiplyfirstreal = (ge_first_rn_unique_factorization_constructedtracestepsmultiply) + ge_balance_positive_unique_factorization_constructedtracestepsmultiplyfirstreal))) /\ (exists ge_balance_positive_unique_factorization_constructedtracestepsmultiplyfirstimaginary ge_balance_negative_unique_factorization_constructedtracestepsmultiplyfirstimaginary. (((((ge_representation_imaginary_code_unique_factorization_constructedtracestepsmultiplyfirst) = 2 * (ge_balance_positive_unique_factorization_constructedtracestepsmultiplyfirstimaginary) /\ (ge_balance_negative_unique_factorization_constructedtracestepsmultiplyfirstimaginary) = 0) \/ exists ge_signed_half_unique_factorization_constructedtracestepsmultiplyfirstimaginarydecode. (((ge_representation_imaginary_code_unique_factorization_constructedtracestepsmultiplyfirst) = 2 * ge_signed_half_unique_factorization_constructedtracestepsmultiplyfirstimaginarydecode + 1 /\ (ge_balance_positive_unique_factorization_constructedtracestepsmultiplyfirstimaginary) = 0) /\ (ge_balance_negative_unique_factorization_constructedtracestepsmultiplyfirstimaginary) = S ge_signed_half_unique_factorization_constructedtracestepsmultiplyfirstimaginarydecode))) /\ ((ge_first_ip_unique_factorization_constructedtracestepsmultiply) + ge_balance_negative_unique_factorization_constructedtracestepsmultiplyfirstimaginary = (ge_first_in_unique_factorization_constructedtracestepsmultiply) + ge_balance_positive_unique_factorization_constructedtracestepsmultiplyfirstimaginary)))))) /\ ((exists ge_representation_real_code_unique_factorization_constructedtracestepsmultiplysecond ge_representation_imaginary_code_unique_factorization_constructedtracestepsmultiplysecond. (((gr_product_factor_unique_factorization_constructedtracesteps) = ((ge_representation_real_code_unique_factorization_constructedtracestepsmultiplysecond) + (ge_representation_imaginary_code_unique_factorization_constructedtracestepsmultiplysecond)) * S ((ge_representation_real_code_unique_factorization_constructedtracestepsmultiplysecond) + (ge_representation_imaginary_code_unique_factorization_constructedtracestepsmultiplysecond)) + ((ge_representation_imaginary_code_unique_factorization_constructedtracestepsmultiplysecond) + (ge_representation_imaginary_code_unique_factorization_constructedtracestepsmultiplysecond))) /\ ((exists ge_balance_positive_unique_factorization_constructedtracestepsmultiplysecondreal ge_balance_negative_unique_factorization_constructedtracestepsmultiplysecondreal. (((((ge_representation_real_code_unique_factorization_constructedtracestepsmultiplysecond) = 2 * (ge_balance_positive_unique_factorization_constructedtracestepsmultiplysecondreal) /\ (ge_balance_negative_unique_factorization_constructedtracestepsmultiplysecondreal) = 0) \/ exists ge_signed_half_unique_factorization_constructedtracestepsmultiplysecondrealdecode. (((ge_representation_real_code_unique_factorization_constructedtracestepsmultiplysecond) = 2 * ge_signed_half_unique_factorization_constructedtracestepsmultiplysecondrealdecode + 1 /\ (ge_balance_positive_unique_factorization_constructedtracestepsmultiplysecondreal) = 0) /\ (ge_balance_negative_unique_factorization_constructedtracestepsmultiplysecondreal) = S ge_signed_half_unique_factorization_constructedtracestepsmultiplysecondrealdecode))) /\ ((ge_second_rp_unique_factorization_constructedtracestepsmultiply) + ge_balance_negative_unique_factorization_constructedtracestepsmultiplysecondreal = (ge_second_rn_unique_factorization_constructedtracestepsmultiply) + ge_balance_positive_unique_factorization_constructedtracestepsmultiplysecondreal))) /\ (exists ge_balance_positive_unique_factorization_constructedtracestepsmultiplysecondimaginary ge_balance_negative_unique_factorization_constructedtracestepsmultiplysecondimaginary. (((((ge_representation_imaginary_code_unique_factorization_constructedtracestepsmultiplysecond) = 2 * (ge_balance_positive_unique_factorization_constructedtracestepsmultiplysecondimaginary) /\ (ge_balance_negative_unique_factorization_constructedtracestepsmultiplysecondimaginary) = 0) \/ exists ge_signed_half_unique_factorization_constructedtracestepsmultiplysecondimaginarydecode. (((ge_representation_imaginary_code_unique_factorization_constructedtracestepsmultiplysecond) = 2 * ge_signed_half_unique_factorization_constructedtracestepsmultiplysecondimaginarydecode + 1 /\ (ge_balance_positive_unique_factorization_constructedtracestepsmultiplysecondimaginary) = 0) /\ (ge_balance_negative_unique_factorization_constructedtracestepsmultiplysecondimaginary) = S ge_signed_half_unique_factorization_constructedtracestepsmultiplysecondimaginarydecode))) /\ ((ge_second_ip_unique_factorization_constructedtracestepsmultiply) + ge_balance_negative_unique_factorization_constructedtracestepsmultiplysecondimaginary = (ge_second_in_unique_factorization_constructedtracestepsmultiply) + ge_balance_positive_unique_factorization_constructedtracestepsmultiplysecondimaginary)))))) /\ (exists ge_representation_real_code_unique_factorization_constructedtracestepsmultiplyoutput ge_representation_imaginary_code_unique_factorization_constructedtracestepsmultiplyoutput. (((gr_product_after_unique_factorization_constructedtracesteps) = ((ge_representation_real_code_unique_factorization_constructedtracestepsmultiplyoutput) + (ge_representation_imaginary_code_unique_factorization_constructedtracestepsmultiplyoutput)) * S ((ge_representation_real_code_unique_factorization_constructedtracestepsmultiplyoutput) + (ge_representation_imaginary_code_unique_factorization_constructedtracestepsmultiplyoutput)) + ((ge_representation_imaginary_code_unique_factorization_constructedtracestepsmultiplyoutput) + (ge_representation_imaginary_code_unique_factorization_constructedtracestepsmultiplyoutput))) /\ ((exists ge_balance_positive_unique_factorization_constructedtracestepsmultiplyoutputreal ge_balance_negative_unique_factorization_constructedtracestepsmultiplyoutputreal. (((((ge_representation_real_code_unique_factorization_constructedtracestepsmultiplyoutput) = 2 * (ge_balance_positive_unique_factorization_constructedtracestepsmultiplyoutputreal) /\ (ge_balance_negative_unique_factorization_constructedtracestepsmultiplyoutputreal) = 0) \/ exists ge_signed_half_unique_factorization_constructedtracestepsmultiplyoutputrealdecode. (((ge_representation_real_code_unique_factorization_constructedtracestepsmultiplyoutput) = 2 * ge_signed_half_unique_factorization_constructedtracestepsmultiplyoutputrealdecode + 1 /\ (ge_balance_positive_unique_factorization_constructedtracestepsmultiplyoutputreal) = 0) /\ (ge_balance_negative_unique_factorization_constructedtracestepsmultiplyoutputreal) = S ge_signed_half_unique_factorization_constructedtracestepsmultiplyoutputrealdecode))) /\ ((((((((ge_first_rp_unique_factorization_constructedtracestepsmultiply) * (ge_second_rp_unique_factorization_constructedtracestepsmultiply))) + (((ge_first_rn_unique_factorization_constructedtracestepsmultiply) * (ge_second_rn_unique_factorization_constructedtracestepsmultiply))))) + (((((ge_first_ip_unique_factorization_constructedtracestepsmultiply) * (ge_second_in_unique_factorization_constructedtracestepsmultiply))) + (((ge_first_in_unique_factorization_constructedtracestepsmultiply) * (ge_second_ip_unique_factorization_constructedtracestepsmultiply))))))) + ge_balance_negative_unique_factorization_constructedtracestepsmultiplyoutputreal = (((((((ge_first_rp_unique_factorization_constructedtracestepsmultiply) * (ge_second_rn_unique_factorization_constructedtracestepsmultiply))) + (((ge_first_rn_unique_factorization_constructedtracestepsmultiply) * (ge_second_rp_unique_factorization_constructedtracestepsmultiply))))) + (((((ge_first_ip_unique_factorization_constructedtracestepsmultiply) * (ge_second_ip_unique_factorization_constructedtracestepsmultiply))) + (((ge_first_in_unique_factorization_constructedtracestepsmultiply) * (ge_second_in_unique_factorization_constructedtracestepsmultiply))))))) + ge_balance_positive_unique_factorization_constructedtracestepsmultiplyoutputreal))) /\ (exists ge_balance_positive_unique_factorization_constructedtracestepsmultiplyoutputimaginary ge_balance_negative_unique_factorization_constructedtracestepsmultiplyoutputimaginary. (((((ge_representation_imaginary_code_unique_factorization_constructedtracestepsmultiplyoutput) = 2 * (ge_balance_positive_unique_factorization_constructedtracestepsmultiplyoutputimaginary) /\ (ge_balance_negative_unique_factorization_constructedtracestepsmultiplyoutputimaginary) = 0) \/ exists ge_signed_half_unique_factorization_constructedtracestepsmultiplyoutputimaginarydecode. (((ge_representation_imaginary_code_unique_factorization_constructedtracestepsmultiplyoutput) = 2 * ge_signed_half_unique_factorization_constructedtracestepsmultiplyoutputimaginarydecode + 1 /\ (ge_balance_positive_unique_factorization_constructedtracestepsmultiplyoutputimaginary) = 0) /\ (ge_balance_negative_unique_factorization_constructedtracestepsmultiplyoutputimaginary) = S ge_signed_half_unique_factorization_constructedtracestepsmultiplyoutputimaginarydecode))) /\ ((((((((ge_first_rp_unique_factorization_constructedtracestepsmultiply) * (ge_second_ip_unique_factorization_constructedtracestepsmultiply))) + (((ge_first_rn_unique_factorization_constructedtracestepsmultiply) * (ge_second_in_unique_factorization_constructedtracestepsmultiply))))) + (((((ge_first_ip_unique_factorization_constructedtracestepsmultiply) * (ge_second_rp_unique_factorization_constructedtracestepsmultiply))) + (((ge_first_in_unique_factorization_constructedtracestepsmultiply) * (ge_second_rn_unique_factorization_constructedtracestepsmultiply))))))) + ge_balance_negative_unique_factorization_constructedtracestepsmultiplyoutputimaginary = (((((((ge_first_rp_unique_factorization_constructedtracestepsmultiply) * (ge_second_in_unique_factorization_constructedtracestepsmultiply))) + (((ge_first_rn_unique_factorization_constructedtracestepsmultiply) * (ge_second_ip_unique_factorization_constructedtracestepsmultiply))))) + (((((ge_first_ip_unique_factorization_constructedtracestepsmultiply) * (ge_second_rn_unique_factorization_constructedtracestepsmultiply))) + (((ge_first_in_unique_factorization_constructedtracestepsmultiply) * (ge_second_rp_unique_factorization_constructedtracestepsmultiply))))))) + ge_balance_positive_unique_factorization_constructedtracestepsmultiplyoutputimaginary)))))))))))))))) /\ (exists ge_first_rp_unique_factorization_constructedreconstruct ge_first_rn_unique_factorization_constructedreconstruct ge_first_ip_unique_factorization_constructedreconstruct ge_first_in_unique_factorization_constructedreconstruct ge_second_rp_unique_factorization_constructedreconstruct ge_second_rn_unique_factorization_constructedreconstruct ge_second_ip_unique_factorization_constructedreconstruct ge_second_in_unique_factorization_constructedreconstruct. ((exists ge_representation_real_code_unique_factorization_constructedreconstructfirst ge_representation_imaginary_code_unique_factorization_constructedreconstructfirst. (((u) = ((ge_representation_real_code_unique_factorization_constructedreconstructfirst) + (ge_representation_imaginary_code_unique_factorization_constructedreconstructfirst)) * S ((ge_representation_real_code_unique_factorization_constructedreconstructfirst) + (ge_representation_imaginary_code_unique_factorization_constructedreconstructfirst)) + ((ge_representation_imaginary_code_unique_factorization_constructedreconstructfirst) + (ge_representation_imaginary_code_unique_factorization_constructedreconstructfirst))) /\ ((exists ge_balance_positive_unique_factorization_constructedreconstructfirstreal ge_balance_negative_unique_factorization_constructedreconstructfirstreal. (((((ge_representation_real_code_unique_factorization_constructedreconstructfirst) = 2 * (ge_balance_positive_unique_factorization_constructedreconstructfirstreal) /\ (ge_balance_negative_unique_factorization_constructedreconstructfirstreal) = 0) \/ exists ge_signed_half_unique_factorization_constructedreconstructfirstrealdecode. (((ge_representation_real_code_unique_factorization_constructedreconstructfirst) = 2 * ge_signed_half_unique_factorization_constructedreconstructfirstrealdecode + 1 /\ (ge_balance_positive_unique_factorization_constructedreconstructfirstreal) = 0) /\ (ge_balance_negative_unique_factorization_constructedreconstructfirstreal) = S ge_signed_half_unique_factorization_constructedreconstructfirstrealdecode))) /\ ((ge_first_rp_unique_factorization_constructedreconstruct) + ge_balance_negative_unique_factorization_constructedreconstructfirstreal = (ge_first_rn_unique_factorization_constructedreconstruct) + ge_balance_positive_unique_factorization_constructedreconstructfirstreal))) /\ (exists ge_balance_positive_unique_factorization_constructedreconstructfirstimaginary ge_balance_negative_unique_factorization_constructedreconstructfirstimaginary. (((((ge_representation_imaginary_code_unique_factorization_constructedreconstructfirst) = 2 * (ge_balance_positive_unique_factorization_constructedreconstructfirstimaginary) /\ (ge_balance_negative_unique_factorization_constructedreconstructfirstimaginary) = 0) \/ exists ge_signed_half_unique_factorization_constructedreconstructfirstimaginarydecode. (((ge_representation_imaginary_code_unique_factorization_constructedreconstructfirst) = 2 * ge_signed_half_unique_factorization_constructedreconstructfirstimaginarydecode + 1 /\ (ge_balance_positive_unique_factorization_constructedreconstructfirstimaginary) = 0) /\ (ge_balance_negative_unique_factorization_constructedreconstructfirstimaginary) = S ge_signed_half_unique_factorization_constructedreconstructfirstimaginarydecode))) /\ ((ge_first_ip_unique_factorization_constructedreconstruct) + ge_balance_negative_unique_factorization_constructedreconstructfirstimaginary = (ge_first_in_unique_factorization_constructedreconstruct) + ge_balance_positive_unique_factorization_constructedreconstructfirstimaginary)))))) /\ ((exists ge_representation_real_code_unique_factorization_constructedreconstructsecond ge_representation_imaginary_code_unique_factorization_constructedreconstructsecond. (((gr_prime_factor_product_unique_factorization_constructed) = ((ge_representation_real_code_unique_factorization_constructedreconstructsecond) + (ge_representation_imaginary_code_unique_factorization_constructedreconstructsecond)) * S ((ge_representation_real_code_unique_factorization_constructedreconstructsecond) + (ge_representation_imaginary_code_unique_factorization_constructedreconstructsecond)) + ((ge_representation_imaginary_code_unique_factorization_constructedreconstructsecond) + (ge_representation_imaginary_code_unique_factorization_constructedreconstructsecond))) /\ ((exists ge_balance_positive_unique_factorization_constructedreconstructsecondreal ge_balance_negative_unique_factorization_constructedreconstructsecondreal. (((((ge_representation_real_code_unique_factorization_constructedreconstructsecond) = 2 * (ge_balance_positive_unique_factorization_constructedreconstructsecondreal) /\ (ge_balance_negative_unique_factorization_constructedreconstructsecondreal) = 0) \/ exists ge_signed_half_unique_factorization_constructedreconstructsecondrealdecode. (((ge_representation_real_code_unique_factorization_constructedreconstructsecond) = 2 * ge_signed_half_unique_factorization_constructedreconstructsecondrealdecode + 1 /\ (ge_balance_positive_unique_factorization_constructedreconstructsecondreal) = 0) /\ (ge_balance_negative_unique_factorization_constructedreconstructsecondreal) = S ge_signed_half_unique_factorization_constructedreconstructsecondrealdecode))) /\ ((ge_second_rp_unique_factorization_constructedreconstruct) + ge_balance_negative_unique_factorization_constructedreconstructsecondreal = (ge_second_rn_unique_factorization_constructedreconstruct) + ge_balance_positive_unique_factorization_constructedreconstructsecondreal))) /\ (exists ge_balance_positive_unique_factorization_constructedreconstructsecondimaginary ge_balance_negative_unique_factorization_constructedreconstructsecondimaginary. (((((ge_representation_imaginary_code_unique_factorization_constructedreconstructsecond) = 2 * (ge_balance_positive_unique_factorization_constructedreconstructsecondimaginary) /\ (ge_balance_negative_unique_factorization_constructedreconstructsecondimaginary) = 0) \/ exists ge_signed_half_unique_factorization_constructedreconstructsecondimaginarydecode. (((ge_representation_imaginary_code_unique_factorization_constructedreconstructsecond) = 2 * ge_signed_half_unique_factorization_constructedreconstructsecondimaginarydecode + 1 /\ (ge_balance_positive_unique_factorization_constructedreconstructsecondimaginary) = 0) /\ (ge_balance_negative_unique_factorization_constructedreconstructsecondimaginary) = S ge_signed_half_unique_factorization_constructedreconstructsecondimaginarydecode))) /\ ((ge_second_ip_unique_factorization_constructedreconstruct) + ge_balance_negative_unique_factorization_constructedreconstructsecondimaginary = (ge_second_in_unique_factorization_constructedreconstruct) + ge_balance_positive_unique_factorization_constructedreconstructsecondimaginary)))))) /\ (exists ge_representation_real_code_unique_factorization_constructedreconstructoutput ge_representation_imaginary_code_unique_factorization_constructedreconstructoutput. (((z) = ((ge_representation_real_code_unique_factorization_constructedreconstructoutput) + (ge_representation_imaginary_code_unique_factorization_constructedreconstructoutput)) * S ((ge_representation_real_code_unique_factorization_constructedreconstructoutput) + (ge_representation_imaginary_code_unique_factorization_constructedreconstructoutput)) + ((ge_representation_imaginary_code_unique_factorization_constructedreconstructoutput) + (ge_representation_imaginary_code_unique_factorization_constructedreconstructoutput))) /\ ((exists ge_balance_positive_unique_factorization_constructedreconstructoutputreal ge_balance_negative_unique_factorization_constructedreconstructoutputreal. (((((ge_representation_real_code_unique_factorization_constructedreconstructoutput) = 2 * (ge_balance_positive_unique_factorization_constructedreconstructoutputreal) /\ (ge_balance_negative_unique_factorization_constructedreconstructoutputreal) = 0) \/ exists ge_signed_half_unique_factorization_constructedreconstructoutputrealdecode. (((ge_representation_real_code_unique_factorization_constructedreconstructoutput) = 2 * ge_signed_half_unique_factorization_constructedreconstructoutputrealdecode + 1 /\ (ge_balance_positive_unique_factorization_constructedreconstructoutputreal) = 0) /\ (ge_balance_negative_unique_factorization_constructedreconstructoutputreal) = S ge_signed_half_unique_factorization_constructedreconstructoutputrealdecode))) /\ ((((((((ge_first_rp_unique_factorization_constructedreconstruct) * (ge_second_rp_unique_factorization_constructedreconstruct))) + (((ge_first_rn_unique_factorization_constructedreconstruct) * (ge_second_rn_unique_factorization_constructedreconstruct))))) + (((((ge_first_ip_unique_factorization_constructedreconstruct) * (ge_second_in_unique_factorization_constructedreconstruct))) + (((ge_first_in_unique_factorization_constructedreconstruct) * (ge_second_ip_unique_factorization_constructedreconstruct))))))) + ge_balance_negative_unique_factorization_constructedreconstructoutputreal = (((((((ge_first_rp_unique_factorization_constructedreconstruct) * (ge_second_rn_unique_factorization_constructedreconstruct))) + (((ge_first_rn_unique_factorization_constructedreconstruct) * (ge_second_rp_unique_factorization_constructedreconstruct))))) + (((((ge_first_ip_unique_factorization_constructedreconstruct) * (ge_second_ip_unique_factorization_constructedreconstruct))) + (((ge_first_in_unique_factorization_constructedreconstruct) * (ge_second_in_unique_factorization_constructedreconstruct))))))) + ge_balance_positive_unique_factorization_constructedreconstructoutputreal))) /\ (exists ge_balance_positive_unique_factorization_constructedreconstructoutputimaginary ge_balance_negative_unique_factorization_constructedreconstructoutputimaginary. (((((ge_representation_imaginary_code_unique_factorization_constructedreconstructoutput) = 2 * (ge_balance_positive_unique_factorization_constructedreconstructoutputimaginary) /\ (ge_balance_negative_unique_factorization_constructedreconstructoutputimaginary) = 0) \/ exists ge_signed_half_unique_factorization_constructedreconstructoutputimaginarydecode. (((ge_representation_imaginary_code_unique_factorization_constructedreconstructoutput) = 2 * ge_signed_half_unique_factorization_constructedreconstructoutputimaginarydecode + 1 /\ (ge_balance_positive_unique_factorization_constructedreconstructoutputimaginary) = 0) /\ (ge_balance_negative_unique_factorization_constructedreconstructoutputimaginary) = S ge_signed_half_unique_factorization_constructedreconstructoutputimaginarydecode))) /\ ((((((((ge_first_rp_unique_factorization_constructedreconstruct) * (ge_second_ip_unique_factorization_constructedreconstruct))) + (((ge_first_rn_unique_factorization_constructedreconstruct) * (ge_second_in_unique_factorization_constructedreconstruct))))) + (((((ge_first_ip_unique_factorization_constructedreconstruct) * (ge_second_rp_unique_factorization_constructedreconstruct))) + (((ge_first_in_unique_factorization_constructedreconstruct) * (ge_second_rn_unique_factorization_constructedreconstruct))))))) + ge_balance_negative_unique_factorization_constructedreconstructoutputimaginary = (((((((ge_first_rp_unique_factorization_constructedreconstruct) * (ge_second_in_unique_factorization_constructedreconstruct))) + (((ge_first_rn_unique_factorization_constructedreconstruct) * (ge_second_ip_unique_factorization_constructedreconstruct))))) + (((((ge_first_ip_unique_factorization_constructedreconstruct) * (ge_second_rn_unique_factorization_constructedreconstruct))) + (((ge_first_in_unique_factorization_constructedreconstruct) * (ge_second_rp_unique_factorization_constructedreconstruct))))))) + ge_balance_positive_unique_factorization_constructedreconstructoutputimaginary)))))))))))))) - 0005
specialize gaussian_prime_factorization_exists (z) - 0006
apply gaussian_prime_factorization_exists - 0007
exact hv - 0008
exact hz - 0009
cases hf - 0010
cases hf_witness - 0011
cases hf_witness_witness - 0012
cases hf_witness_witness_witness - 0013
exists (x) - 0014
exists (x1) - 0015
exists (x2) - 0016
exists (x3) - 0017
split - 0018
exact hf_witness_witness_witness_witness - 0019
intro v - 0020
intro d - 0021
intro e - 0022
intro m - 0023
intro hg - 0024
specialize gaussian_prime_factorizations_unique (z) - 0025
specialize gaussian_prime_factorizations_unique (x) - 0026
specialize gaussian_prime_factorizations_unique (x1) - 0027
specialize gaussian_prime_factorizations_unique (x2) - 0028
specialize gaussian_prime_factorizations_unique (x3) - 0029
specialize gaussian_prime_factorizations_unique (v) - 0030
specialize gaussian_prime_factorizations_unique (d) - 0031
specialize gaussian_prime_factorizations_unique (e) - 0032
specialize gaussian_prime_factorizations_unique (m) - 0033
apply gaussian_prime_factorizations_unique - 0034
exact hf_witness_witness_witness_witness - 0035
exact hg