prime_exponent_entries_prime_divides

forall n pb pc eb ec vb vc l i q. (forall pvs_index_divisor_entries. (exists pvs_gap_divisor_entriesindex. pvs_gap_divisor_entriesindex + S (pvs_index_divisor_entries) = (l)) -> exists pvs_prime_divisor_entries pvs_exponent_divisor_entries pvs_power_divisor_entries. (((((exists ff_h_pvs_divisor_entriesprime. ff_h_pvs_divisor_entriesprime + S (pvs_prime_divisor_entries) = S ((S (pvs_index_divisor_entries)) * pc)) /\ exists ff_q_pvs_divisor_entriesprime. pb = ff_q_pvs_divisor_entriesprime * S ((S (pvs_index_divisor_entries)) * pc) + (pvs_prime_divisor_entries))) /\ (((((exists ff_h_pvs_divisor_entriesexponent. ff_h_pvs_divisor_entriesexponent + S (pvs_exponent_divisor_entries) = S ((S (pvs_index_divisor_entries)) * ec)) /\ exists ff_q_pvs_divisor_entriesexponent. eb = ff_q_pvs_divisor_entriesexponent * S ((S (pvs_index_divisor_entries)) * ec) + (pvs_exponent_divisor_entries))) /\ (((((exists ff_h_pvs_divisor_entriespower. ff_h_pvs_divisor_entriespower + S (pvs_power_divisor_entries) = S ((S (pvs_index_divisor_entries)) * vc)) /\ exists ff_q_pvs_divisor_entriespower. vb = ff_q_pvs_divisor_entriespower * S ((S (pvs_index_divisor_entries)) * vc) + (pvs_power_divisor_entries))) /\ (((~((pvs_prime_divisor_entries) = 1) /\ forall pvs_left_divisor_entriesdomain pvs_right_divisor_entriesdomain. (pvs_prime_divisor_entries) = pvs_left_divisor_entriesdomain * pvs_right_divisor_entriesdomain -> pvs_left_divisor_entriesdomain = 1 \/ pvs_right_divisor_entriesdomain = 1) /\ (((~(pvs_exponent_divisor_entries = 0)) /\ (((((exists bpd_gap_pvs_divisor_entriesvaluation_selected_bound. bpd_gap_pvs_divisor_entriesvaluation_selected_bound + (pvs_exponent_divisor_entries) = (n)) /\ (exists bpvi_result_pvs_divisor_entriesvaluation_selected. ((exists bpvi_b_pvs_divisor_entriesvaluation_selected_power bpvi_c_pvs_divisor_entriesvaluation_selected_power. ((forall bpvi_i_pvs_divisor_entriesvaluation_selected_power. (exists bpvi_repeat_gap_pvs_divisor_entriesvaluation_selected_power. bpvi_repeat_gap_pvs_divisor_entriesvaluation_selected_power + S bpvi_i_pvs_divisor_entriesvaluation_selected_power = pvs_exponent_divisor_entries) -> (((exists bpvi_h_pvs_divisor_entriesvaluation_selected_power_repeat. bpvi_h_pvs_divisor_entriesvaluation_selected_power_repeat + S (pvs_prime_divisor_entries) = S ((S (bpvi_i_pvs_divisor_entriesvaluation_selected_power)) * bpvi_c_pvs_divisor_entriesvaluation_selected_power)) /\ exists bpvi_q_pvs_divisor_entriesvaluation_selected_power_repeat. bpvi_b_pvs_divisor_entriesvaluation_selected_power = bpvi_q_pvs_divisor_entriesvaluation_selected_power_repeat * S ((S (bpvi_i_pvs_divisor_entriesvaluation_selected_power)) * bpvi_c_pvs_divisor_entriesvaluation_selected_power) + (pvs_prime_divisor_entries)))) /\ (exists bpvi_u_pvs_divisor_entriesvaluation_selected_power bpvi_v_pvs_divisor_entriesvaluation_selected_power. ((((exists bpvi_h_pvs_divisor_entriesvaluation_selected_power_start. bpvi_h_pvs_divisor_entriesvaluation_selected_power_start + S (1) = S ((S (0)) * bpvi_v_pvs_divisor_entriesvaluation_selected_power)) /\ exists bpvi_q_pvs_divisor_entriesvaluation_selected_power_start. bpvi_u_pvs_divisor_entriesvaluation_selected_power = bpvi_q_pvs_divisor_entriesvaluation_selected_power_start * S ((S (0)) * bpvi_v_pvs_divisor_entriesvaluation_selected_power) + (1))) /\ ((((exists bpvi_h_pvs_divisor_entriesvaluation_selected_power_terminal. bpvi_h_pvs_divisor_entriesvaluation_selected_power_terminal + S (bpvi_result_pvs_divisor_entriesvaluation_selected) = S ((S (pvs_exponent_divisor_entries)) * bpvi_v_pvs_divisor_entriesvaluation_selected_power)) /\ exists bpvi_q_pvs_divisor_entriesvaluation_selected_power_terminal. bpvi_u_pvs_divisor_entriesvaluation_selected_power = bpvi_q_pvs_divisor_entriesvaluation_selected_power_terminal * S ((S (pvs_exponent_divisor_entries)) * bpvi_v_pvs_divisor_entriesvaluation_selected_power) + (bpvi_result_pvs_divisor_entriesvaluation_selected))) /\ forall bpvi_j_pvs_divisor_entriesvaluation_selected_power. (exists bpvi_product_gap_pvs_divisor_entriesvaluation_selected_power. bpvi_product_gap_pvs_divisor_entriesvaluation_selected_power + S bpvi_j_pvs_divisor_entriesvaluation_selected_power = pvs_exponent_divisor_entries) -> exists bpvi_factor_pvs_divisor_entriesvaluation_selected_power bpvi_partial_pvs_divisor_entriesvaluation_selected_power bpvi_successor_pvs_divisor_entriesvaluation_selected_power. ((((exists bpvi_h_pvs_divisor_entriesvaluation_selected_power_factor. bpvi_h_pvs_divisor_entriesvaluation_selected_power_factor + S (bpvi_factor_pvs_divisor_entriesvaluation_selected_power) = S ((S (bpvi_j_pvs_divisor_entriesvaluation_selected_power)) * bpvi_c_pvs_divisor_entriesvaluation_selected_power)) /\ exists bpvi_q_pvs_divisor_entriesvaluation_selected_power_factor. bpvi_b_pvs_divisor_entriesvaluation_selected_power = bpvi_q_pvs_divisor_entriesvaluation_selected_power_factor * S ((S (bpvi_j_pvs_divisor_entriesvaluation_selected_power)) * bpvi_c_pvs_divisor_entriesvaluation_selected_power) + (bpvi_factor_pvs_divisor_entriesvaluation_selected_power))) /\ ((((exists bpvi_h_pvs_divisor_entriesvaluation_selected_power_partial. bpvi_h_pvs_divisor_entriesvaluation_selected_power_partial + S (bpvi_partial_pvs_divisor_entriesvaluation_selected_power) = S ((S (bpvi_j_pvs_divisor_entriesvaluation_selected_power)) * bpvi_v_pvs_divisor_entriesvaluation_selected_power)) /\ exists bpvi_q_pvs_divisor_entriesvaluation_selected_power_partial. bpvi_u_pvs_divisor_entriesvaluation_selected_power = bpvi_q_pvs_divisor_entriesvaluation_selected_power_partial * S ((S (bpvi_j_pvs_divisor_entriesvaluation_selected_power)) * bpvi_v_pvs_divisor_entriesvaluation_selected_power) + (bpvi_partial_pvs_divisor_entriesvaluation_selected_power))) /\ ((((exists bpvi_h_pvs_divisor_entriesvaluation_selected_power_successor. bpvi_h_pvs_divisor_entriesvaluation_selected_power_successor + S (bpvi_successor_pvs_divisor_entriesvaluation_selected_power) = S ((S (S bpvi_j_pvs_divisor_entriesvaluation_selected_power)) * bpvi_v_pvs_divisor_entriesvaluation_selected_power)) /\ exists bpvi_q_pvs_divisor_entriesvaluation_selected_power_successor. bpvi_u_pvs_divisor_entriesvaluation_selected_power = bpvi_q_pvs_divisor_entriesvaluation_selected_power_successor * S ((S (S bpvi_j_pvs_divisor_entriesvaluation_selected_power)) * bpvi_v_pvs_divisor_entriesvaluation_selected_power) + (bpvi_successor_pvs_divisor_entriesvaluation_selected_power))) /\ bpvi_successor_pvs_divisor_entriesvaluation_selected_power = bpvi_partial_pvs_divisor_entriesvaluation_selected_power * bpvi_factor_pvs_divisor_entriesvaluation_selected_power)))))))) /\ exists bpvi_divisor_factor_pvs_divisor_entriesvaluation_selected. n = bpvi_result_pvs_divisor_entriesvaluation_selected * bpvi_divisor_factor_pvs_divisor_entriesvaluation_selected))) /\ forall bpd_candidate_pvs_divisor_entriesvaluation. (exists bpd_gap_pvs_divisor_entriesvaluation_candidate_bound. bpd_gap_pvs_divisor_entriesvaluation_candidate_bound + (bpd_candidate_pvs_divisor_entriesvaluation) = (n)) -> (exists bpvi_result_pvs_divisor_entriesvaluation_candidate. ((exists bpvi_b_pvs_divisor_entriesvaluation_candidate_power bpvi_c_pvs_divisor_entriesvaluation_candidate_power. ((forall bpvi_i_pvs_divisor_entriesvaluation_candidate_power. (exists bpvi_repeat_gap_pvs_divisor_entriesvaluation_candidate_power. bpvi_repeat_gap_pvs_divisor_entriesvaluation_candidate_power + S bpvi_i_pvs_divisor_entriesvaluation_candidate_power = bpd_candidate_pvs_divisor_entriesvaluation) -> (((exists bpvi_h_pvs_divisor_entriesvaluation_candidate_power_repeat. bpvi_h_pvs_divisor_entriesvaluation_candidate_power_repeat + S (pvs_prime_divisor_entries) = S ((S (bpvi_i_pvs_divisor_entriesvaluation_candidate_power)) * bpvi_c_pvs_divisor_entriesvaluation_candidate_power)) /\ exists bpvi_q_pvs_divisor_entriesvaluation_candidate_power_repeat. bpvi_b_pvs_divisor_entriesvaluation_candidate_power = bpvi_q_pvs_divisor_entriesvaluation_candidate_power_repeat * S ((S (bpvi_i_pvs_divisor_entriesvaluation_candidate_power)) * bpvi_c_pvs_divisor_entriesvaluation_candidate_power) + (pvs_prime_divisor_entries)))) /\ (exists bpvi_u_pvs_divisor_entriesvaluation_candidate_power bpvi_v_pvs_divisor_entriesvaluation_candidate_power. ((((exists bpvi_h_pvs_divisor_entriesvaluation_candidate_power_start. bpvi_h_pvs_divisor_entriesvaluation_candidate_power_start + S (1) = S ((S (0)) * bpvi_v_pvs_divisor_entriesvaluation_candidate_power)) /\ exists bpvi_q_pvs_divisor_entriesvaluation_candidate_power_start. bpvi_u_pvs_divisor_entriesvaluation_candidate_power = bpvi_q_pvs_divisor_entriesvaluation_candidate_power_start * S ((S (0)) * bpvi_v_pvs_divisor_entriesvaluation_candidate_power) + (1))) /\ ((((exists bpvi_h_pvs_divisor_entriesvaluation_candidate_power_terminal. bpvi_h_pvs_divisor_entriesvaluation_candidate_power_terminal + S (bpvi_result_pvs_divisor_entriesvaluation_candidate) = S ((S (bpd_candidate_pvs_divisor_entriesvaluation)) * bpvi_v_pvs_divisor_entriesvaluation_candidate_power)) /\ exists bpvi_q_pvs_divisor_entriesvaluation_candidate_power_terminal. bpvi_u_pvs_divisor_entriesvaluation_candidate_power = bpvi_q_pvs_divisor_entriesvaluation_candidate_power_terminal * S ((S (bpd_candidate_pvs_divisor_entriesvaluation)) * bpvi_v_pvs_divisor_entriesvaluation_candidate_power) + (bpvi_result_pvs_divisor_entriesvaluation_candidate))) /\ forall bpvi_j_pvs_divisor_entriesvaluation_candidate_power. (exists bpvi_product_gap_pvs_divisor_entriesvaluation_candidate_power. bpvi_product_gap_pvs_divisor_entriesvaluation_candidate_power + S bpvi_j_pvs_divisor_entriesvaluation_candidate_power = bpd_candidate_pvs_divisor_entriesvaluation) -> exists bpvi_factor_pvs_divisor_entriesvaluation_candidate_power bpvi_partial_pvs_divisor_entriesvaluation_candidate_power bpvi_successor_pvs_divisor_entriesvaluation_candidate_power. ((((exists bpvi_h_pvs_divisor_entriesvaluation_candidate_power_factor. bpvi_h_pvs_divisor_entriesvaluation_candidate_power_factor + S (bpvi_factor_pvs_divisor_entriesvaluation_candidate_power) = S ((S (bpvi_j_pvs_divisor_entriesvaluation_candidate_power)) * bpvi_c_pvs_divisor_entriesvaluation_candidate_power)) /\ exists bpvi_q_pvs_divisor_entriesvaluation_candidate_power_factor. bpvi_b_pvs_divisor_entriesvaluation_candidate_power = bpvi_q_pvs_divisor_entriesvaluation_candidate_power_factor * S ((S (bpvi_j_pvs_divisor_entriesvaluation_candidate_power)) * bpvi_c_pvs_divisor_entriesvaluation_candidate_power) + (bpvi_factor_pvs_divisor_entriesvaluation_candidate_power))) /\ ((((exists bpvi_h_pvs_divisor_entriesvaluation_candidate_power_partial. bpvi_h_pvs_divisor_entriesvaluation_candidate_power_partial + S (bpvi_partial_pvs_divisor_entriesvaluation_candidate_power) = S ((S (bpvi_j_pvs_divisor_entriesvaluation_candidate_power)) * bpvi_v_pvs_divisor_entriesvaluation_candidate_power)) /\ exists bpvi_q_pvs_divisor_entriesvaluation_candidate_power_partial. bpvi_u_pvs_divisor_entriesvaluation_candidate_power = bpvi_q_pvs_divisor_entriesvaluation_candidate_power_partial * S ((S (bpvi_j_pvs_divisor_entriesvaluation_candidate_power)) * bpvi_v_pvs_divisor_entriesvaluation_candidate_power) + (bpvi_partial_pvs_divisor_entriesvaluation_candidate_power))) /\ ((((exists bpvi_h_pvs_divisor_entriesvaluation_candidate_power_successor. bpvi_h_pvs_divisor_entriesvaluation_candidate_power_successor + S (bpvi_successor_pvs_divisor_entriesvaluation_candidate_power) = S ((S (S bpvi_j_pvs_divisor_entriesvaluation_candidate_power)) * bpvi_v_pvs_divisor_entriesvaluation_candidate_power)) /\ exists bpvi_q_pvs_divisor_entriesvaluation_candidate_power_successor. bpvi_u_pvs_divisor_entriesvaluation_candidate_power = bpvi_q_pvs_divisor_entriesvaluation_candidate_power_successor * S ((S (S bpvi_j_pvs_divisor_entriesvaluation_candidate_power)) * bpvi_v_pvs_divisor_entriesvaluation_candidate_power) + (bpvi_successor_pvs_divisor_entriesvaluation_candidate_power))) /\ bpvi_successor_pvs_divisor_entriesvaluation_candidate_power = bpvi_partial_pvs_divisor_entriesvaluation_candidate_power * bpvi_factor_pvs_divisor_entriesvaluation_candidate_power)))))))) /\ exists bpvi_divisor_factor_pvs_divisor_entriesvaluation_candidate. n = bpvi_result_pvs_divisor_entriesvaluation_candidate * bpvi_divisor_factor_pvs_divisor_entriesvaluation_candidate)) -> (exists bpd_gap_pvs_divisor_entriesvaluation_maximal. bpd_gap_pvs_divisor_entriesvaluation_maximal + (bpd_candidate_pvs_divisor_entriesvaluation) = (pvs_exponent_divisor_entries))) /\ (exists pa_b_pvs_divisor_entriesvalue pa_c_pvs_divisor_entriesvalue. ((forall pa_i_pvs_divisor_entriesvalue_repeat. (exists pa_lt_pvs_divisor_entriesvalue_repeat_bound. pa_lt_pvs_divisor_entriesvalue_repeat_bound + S pa_i_pvs_divisor_entriesvalue_repeat = pvs_exponent_divisor_entries) -> (((exists pa_h_pvs_divisor_entriesvalue_repeat_decoded. pa_h_pvs_divisor_entriesvalue_repeat_decoded + S (pvs_prime_divisor_entries) = S ((S (pa_i_pvs_divisor_entriesvalue_repeat)) * pa_c_pvs_divisor_entriesvalue)) /\ exists pa_q_pvs_divisor_entriesvalue_repeat_decoded. pa_b_pvs_divisor_entriesvalue = pa_q_pvs_divisor_entriesvalue_repeat_decoded * S ((S (pa_i_pvs_divisor_entriesvalue_repeat)) * pa_c_pvs_divisor_entriesvalue) + (pvs_prime_divisor_entries)))) /\ (exists pa_u_pvs_divisor_entriesvalue_product pa_v_pvs_divisor_entriesvalue_product. ((((exists pa_h_pvs_divisor_entriesvalue_product_start. pa_h_pvs_divisor_entriesvalue_product_start + S (1) = S ((S (0)) * pa_v_pvs_divisor_entriesvalue_product)) /\ exists pa_q_pvs_divisor_entriesvalue_product_start. pa_u_pvs_divisor_entriesvalue_product = pa_q_pvs_divisor_entriesvalue_product_start * S ((S (0)) * pa_v_pvs_divisor_entriesvalue_product) + (1))) /\ ((((exists pa_h_pvs_divisor_entriesvalue_product_terminal. pa_h_pvs_divisor_entriesvalue_product_terminal + S (pvs_power_divisor_entries) = S ((S (pvs_exponent_divisor_entries)) * pa_v_pvs_divisor_entriesvalue_product)) /\ exists pa_q_pvs_divisor_entriesvalue_product_terminal. pa_u_pvs_divisor_entriesvalue_product = pa_q_pvs_divisor_entriesvalue_product_terminal * S ((S (pvs_exponent_divisor_entries)) * pa_v_pvs_divisor_entriesvalue_product) + (pvs_power_divisor_entries))) /\ forall pa_i_pvs_divisor_entriesvalue_product. (exists pa_lt_pvs_divisor_entriesvalue_product_bound. pa_lt_pvs_divisor_entriesvalue_product_bound + S pa_i_pvs_divisor_entriesvalue_product = pvs_exponent_divisor_entries) -> exists pa_p_pvs_divisor_entriesvalue_product pa_r_pvs_divisor_entriesvalue_product pa_s_pvs_divisor_entriesvalue_product. ((((exists pa_h_pvs_divisor_entriesvalue_product_factor. pa_h_pvs_divisor_entriesvalue_product_factor + S (pa_p_pvs_divisor_entriesvalue_product) = S ((S (pa_i_pvs_divisor_entriesvalue_product)) * pa_c_pvs_divisor_entriesvalue)) /\ exists pa_q_pvs_divisor_entriesvalue_product_factor. pa_b_pvs_divisor_entriesvalue = pa_q_pvs_divisor_entriesvalue_product_factor * S ((S (pa_i_pvs_divisor_entriesvalue_product)) * pa_c_pvs_divisor_entriesvalue) + (pa_p_pvs_divisor_entriesvalue_product))) /\ ((((exists pa_h_pvs_divisor_entriesvalue_product_partial. pa_h_pvs_divisor_entriesvalue_product_partial + S (pa_r_pvs_divisor_entriesvalue_product) = S ((S (pa_i_pvs_divisor_entriesvalue_product)) * pa_v_pvs_divisor_entriesvalue_product)) /\ exists pa_q_pvs_divisor_entriesvalue_product_partial. pa_u_pvs_divisor_entriesvalue_product = pa_q_pvs_divisor_entriesvalue_product_partial * S ((S (pa_i_pvs_divisor_entriesvalue_product)) * pa_v_pvs_divisor_entriesvalue_product) + (pa_r_pvs_divisor_entriesvalue_product))) /\ ((((exists pa_h_pvs_divisor_entriesvalue_product_successor. pa_h_pvs_divisor_entriesvalue_product_successor + S (pa_s_pvs_divisor_entriesvalue_product) = S ((S (S pa_i_pvs_divisor_entriesvalue_product)) * pa_v_pvs_divisor_entriesvalue_product)) /\ exists pa_q_pvs_divisor_entriesvalue_product_successor. pa_u_pvs_divisor_entriesvalue_product = pa_q_pvs_divisor_entriesvalue_product_successor * S ((S (S pa_i_pvs_divisor_entriesvalue_product)) * pa_v_pvs_divisor_entriesvalue_product) + (pa_s_pvs_divisor_entriesvalue_product))) /\ pa_s_pvs_divisor_entriesvalue_product = pa_r_pvs_divisor_entriesvalue_product * pa_p_pvs_divisor_entriesvalue_product))))))))))))))))))))) -> (exists pvs_gap_divisor_index. pvs_gap_divisor_index + S (i) = (l)) -> (((exists ff_h_pvs_divisor_at. ff_h_pvs_divisor_at + S (q) = S ((S (i)) * pc)) /\ exists ff_q_pvs_divisor_at. pb = ff_q_pvs_divisor_at * S ((S (i)) * pc) + (q))) -> (~((q) = 1) /\ forall pvs_left_entry_prime pvs_right_entry_prime. (q) = pvs_left_entry_prime * pvs_right_entry_prime -> pvs_left_entry_prime = 1 \/ pvs_right_entry_prime = 1) /\ (exists pvs_factor_entry_divisor. (n) = (q) * pvs_factor_entry_divisor)