hensel_prime_nonsingular_root_is_simple

forall pb pc nb nc a l m p. ((~(p = 1) /\ forall frm_prime_left_hsc_root_prime frm_prime_right_hsc_root_prime. p = frm_prime_left_hsc_root_prime * frm_prime_right_hsc_root_prime -> frm_prime_left_hsc_root_prime = 1 \/ frm_prime_right_hsc_root_prime = 1)) -> (exists hsc_vp_nonsingular_source hsc_dp_nonsingular_source hsc_vn_nonsingular_source hsc_dn_nonsingular_source. ((((exists ff_u_hd_hpl_sph_hsc_nonsingular_source_pair_positive ff_v_hd_hpl_sph_hsc_nonsingular_source_pair_positive ff_d_hd_hpl_sph_hsc_nonsingular_source_pair_positive ff_e_hd_hpl_sph_hsc_nonsingular_source_pair_positive. ((((((exists fs_h_ph_hd_hpl_sph_hsc_nonsingular_source_pair_positive_body_value_start. fs_h_ph_hd_hpl_sph_hsc_nonsingular_source_pair_positive_body_value_start + S (0) = S ((S (0)) * ff_v_hd_hpl_sph_hsc_nonsingular_source_pair_positive)) /\ exists fs_q_ph_hd_hpl_sph_hsc_nonsingular_source_pair_positive_body_value_start. ff_u_hd_hpl_sph_hsc_nonsingular_source_pair_positive = fs_q_ph_hd_hpl_sph_hsc_nonsingular_source_pair_positive_body_value_start * S ((S (0)) * ff_v_hd_hpl_sph_hsc_nonsingular_source_pair_positive) + (0))) /\ ((((exists fs_h_ph_hd_hpl_sph_hsc_nonsingular_source_pair_positive_body_value_terminal. fs_h_ph_hd_hpl_sph_hsc_nonsingular_source_pair_positive_body_value_terminal + S (hsc_vp_nonsingular_source) = S ((S (l)) * ff_v_hd_hpl_sph_hsc_nonsingular_source_pair_positive)) /\ exists fs_q_ph_hd_hpl_sph_hsc_nonsingular_source_pair_positive_body_value_terminal. ff_u_hd_hpl_sph_hsc_nonsingular_source_pair_positive = fs_q_ph_hd_hpl_sph_hsc_nonsingular_source_pair_positive_body_value_terminal * S ((S (l)) * ff_v_hd_hpl_sph_hsc_nonsingular_source_pair_positive) + (hsc_vp_nonsingular_source))) /\ forall ff_i_ph_hd_hpl_sph_hsc_nonsingular_source_pair_positive_body_value_steps. (exists ph_bound_hd_hpl_sph_hsc_nonsingular_source_pair_positive_body_value_steps. ph_bound_hd_hpl_sph_hsc_nonsingular_source_pair_positive_body_value_steps + S ff_i_ph_hd_hpl_sph_hsc_nonsingular_source_pair_positive_body_value_steps = l) -> exists ff_coefficient_ph_hd_hpl_sph_hsc_nonsingular_source_pair_positive_body_value_steps ff_previous_ph_hd_hpl_sph_hsc_nonsingular_source_pair_positive_body_value_steps ff_current_ph_hd_hpl_sph_hsc_nonsingular_source_pair_positive_body_value_steps. ((((exists fs_h_ph_hd_hpl_sph_hsc_nonsingular_source_pair_positive_body_value_steps_coefficient. fs_h_ph_hd_hpl_sph_hsc_nonsingular_source_pair_positive_body_value_steps_coefficient + S (ff_coefficient_ph_hd_hpl_sph_hsc_nonsingular_source_pair_positive_body_value_steps) = S ((S (ff_i_ph_hd_hpl_sph_hsc_nonsingular_source_pair_positive_body_value_steps)) * pc)) /\ exists fs_q_ph_hd_hpl_sph_hsc_nonsingular_source_pair_positive_body_value_steps_coefficient. pb = fs_q_ph_hd_hpl_sph_hsc_nonsingular_source_pair_positive_body_value_steps_coefficient * S ((S (ff_i_ph_hd_hpl_sph_hsc_nonsingular_source_pair_positive_body_value_steps)) * pc) + (ff_coefficient_ph_hd_hpl_sph_hsc_nonsingular_source_pair_positive_body_value_steps))) /\ ((((exists fs_h_ph_hd_hpl_sph_hsc_nonsingular_source_pair_positive_body_value_steps_before. fs_h_ph_hd_hpl_sph_hsc_nonsingular_source_pair_positive_body_value_steps_before + S (ff_previous_ph_hd_hpl_sph_hsc_nonsingular_source_pair_positive_body_value_steps) = S ((S (ff_i_ph_hd_hpl_sph_hsc_nonsingular_source_pair_positive_body_value_steps)) * ff_v_hd_hpl_sph_hsc_nonsingular_source_pair_positive)) /\ exists fs_q_ph_hd_hpl_sph_hsc_nonsingular_source_pair_positive_body_value_steps_before. ff_u_hd_hpl_sph_hsc_nonsingular_source_pair_positive = fs_q_ph_hd_hpl_sph_hsc_nonsingular_source_pair_positive_body_value_steps_before * S ((S (ff_i_ph_hd_hpl_sph_hsc_nonsingular_source_pair_positive_body_value_steps)) * ff_v_hd_hpl_sph_hsc_nonsingular_source_pair_positive) + (ff_previous_ph_hd_hpl_sph_hsc_nonsingular_source_pair_positive_body_value_steps))) /\ ((((exists fs_h_ph_hd_hpl_sph_hsc_nonsingular_source_pair_positive_body_value_steps_after. fs_h_ph_hd_hpl_sph_hsc_nonsingular_source_pair_positive_body_value_steps_after + S (ff_current_ph_hd_hpl_sph_hsc_nonsingular_source_pair_positive_body_value_steps) = S ((S (S ff_i_ph_hd_hpl_sph_hsc_nonsingular_source_pair_positive_body_value_steps)) * ff_v_hd_hpl_sph_hsc_nonsingular_source_pair_positive)) /\ exists fs_q_ph_hd_hpl_sph_hsc_nonsingular_source_pair_positive_body_value_steps_after. ff_u_hd_hpl_sph_hsc_nonsingular_source_pair_positive = fs_q_ph_hd_hpl_sph_hsc_nonsingular_source_pair_positive_body_value_steps_after * S ((S (S ff_i_ph_hd_hpl_sph_hsc_nonsingular_source_pair_positive_body_value_steps)) * ff_v_hd_hpl_sph_hsc_nonsingular_source_pair_positive) + (ff_current_ph_hd_hpl_sph_hsc_nonsingular_source_pair_positive_body_value_steps))) /\ ff_current_ph_hd_hpl_sph_hsc_nonsingular_source_pair_positive_body_value_steps = ff_previous_ph_hd_hpl_sph_hsc_nonsingular_source_pair_positive_body_value_steps * a + ff_coefficient_ph_hd_hpl_sph_hsc_nonsingular_source_pair_positive_body_value_steps)))))) /\ (((((exists fs_h_ph_hd_hpl_sph_hsc_nonsingular_source_pair_positive_body_derivative_start. fs_h_ph_hd_hpl_sph_hsc_nonsingular_source_pair_positive_body_derivative_start + S (0) = S ((S (0)) * ff_e_hd_hpl_sph_hsc_nonsingular_source_pair_positive)) /\ exists fs_q_ph_hd_hpl_sph_hsc_nonsingular_source_pair_positive_body_derivative_start. ff_d_hd_hpl_sph_hsc_nonsingular_source_pair_positive = fs_q_ph_hd_hpl_sph_hsc_nonsingular_source_pair_positive_body_derivative_start * S ((S (0)) * ff_e_hd_hpl_sph_hsc_nonsingular_source_pair_positive) + (0))) /\ ((((exists fs_h_ph_hd_hpl_sph_hsc_nonsingular_source_pair_positive_body_derivative_terminal. fs_h_ph_hd_hpl_sph_hsc_nonsingular_source_pair_positive_body_derivative_terminal + S (hsc_dp_nonsingular_source) = S ((S (l)) * ff_e_hd_hpl_sph_hsc_nonsingular_source_pair_positive)) /\ exists fs_q_ph_hd_hpl_sph_hsc_nonsingular_source_pair_positive_body_derivative_terminal. ff_d_hd_hpl_sph_hsc_nonsingular_source_pair_positive = fs_q_ph_hd_hpl_sph_hsc_nonsingular_source_pair_positive_body_derivative_terminal * S ((S (l)) * ff_e_hd_hpl_sph_hsc_nonsingular_source_pair_positive) + (hsc_dp_nonsingular_source))) /\ forall ff_i_ph_hd_hpl_sph_hsc_nonsingular_source_pair_positive_body_derivative_steps. (exists ph_bound_hd_hpl_sph_hsc_nonsingular_source_pair_positive_body_derivative_steps. ph_bound_hd_hpl_sph_hsc_nonsingular_source_pair_positive_body_derivative_steps + S ff_i_ph_hd_hpl_sph_hsc_nonsingular_source_pair_positive_body_derivative_steps = l) -> exists ff_coefficient_ph_hd_hpl_sph_hsc_nonsingular_source_pair_positive_body_derivative_steps ff_previous_ph_hd_hpl_sph_hsc_nonsingular_source_pair_positive_body_derivative_steps ff_current_ph_hd_hpl_sph_hsc_nonsingular_source_pair_positive_body_derivative_steps. ((((exists fs_h_ph_hd_hpl_sph_hsc_nonsingular_source_pair_positive_body_derivative_steps_coefficient. fs_h_ph_hd_hpl_sph_hsc_nonsingular_source_pair_positive_body_derivative_steps_coefficient + S (ff_coefficient_ph_hd_hpl_sph_hsc_nonsingular_source_pair_positive_body_derivative_steps) = S ((S (ff_i_ph_hd_hpl_sph_hsc_nonsingular_source_pair_positive_body_derivative_steps)) * ff_v_hd_hpl_sph_hsc_nonsingular_source_pair_positive)) /\ exists fs_q_ph_hd_hpl_sph_hsc_nonsingular_source_pair_positive_body_derivative_steps_coefficient. ff_u_hd_hpl_sph_hsc_nonsingular_source_pair_positive = fs_q_ph_hd_hpl_sph_hsc_nonsingular_source_pair_positive_body_derivative_steps_coefficient * S ((S (ff_i_ph_hd_hpl_sph_hsc_nonsingular_source_pair_positive_body_derivative_steps)) * ff_v_hd_hpl_sph_hsc_nonsingular_source_pair_positive) + (ff_coefficient_ph_hd_hpl_sph_hsc_nonsingular_source_pair_positive_body_derivative_steps))) /\ ((((exists fs_h_ph_hd_hpl_sph_hsc_nonsingular_source_pair_positive_body_derivative_steps_before. fs_h_ph_hd_hpl_sph_hsc_nonsingular_source_pair_positive_body_derivative_steps_before + S (ff_previous_ph_hd_hpl_sph_hsc_nonsingular_source_pair_positive_body_derivative_steps) = S ((S (ff_i_ph_hd_hpl_sph_hsc_nonsingular_source_pair_positive_body_derivative_steps)) * ff_e_hd_hpl_sph_hsc_nonsingular_source_pair_positive)) /\ exists fs_q_ph_hd_hpl_sph_hsc_nonsingular_source_pair_positive_body_derivative_steps_before. ff_d_hd_hpl_sph_hsc_nonsingular_source_pair_positive = fs_q_ph_hd_hpl_sph_hsc_nonsingular_source_pair_positive_body_derivative_steps_before * S ((S (ff_i_ph_hd_hpl_sph_hsc_nonsingular_source_pair_positive_body_derivative_steps)) * ff_e_hd_hpl_sph_hsc_nonsingular_source_pair_positive) + (ff_previous_ph_hd_hpl_sph_hsc_nonsingular_source_pair_positive_body_derivative_steps))) /\ ((((exists fs_h_ph_hd_hpl_sph_hsc_nonsingular_source_pair_positive_body_derivative_steps_after. fs_h_ph_hd_hpl_sph_hsc_nonsingular_source_pair_positive_body_derivative_steps_after + S (ff_current_ph_hd_hpl_sph_hsc_nonsingular_source_pair_positive_body_derivative_steps) = S ((S (S ff_i_ph_hd_hpl_sph_hsc_nonsingular_source_pair_positive_body_derivative_steps)) * ff_e_hd_hpl_sph_hsc_nonsingular_source_pair_positive)) /\ exists fs_q_ph_hd_hpl_sph_hsc_nonsingular_source_pair_positive_body_derivative_steps_after. ff_d_hd_hpl_sph_hsc_nonsingular_source_pair_positive = fs_q_ph_hd_hpl_sph_hsc_nonsingular_source_pair_positive_body_derivative_steps_after * S ((S (S ff_i_ph_hd_hpl_sph_hsc_nonsingular_source_pair_positive_body_derivative_steps)) * ff_e_hd_hpl_sph_hsc_nonsingular_source_pair_positive) + (ff_current_ph_hd_hpl_sph_hsc_nonsingular_source_pair_positive_body_derivative_steps))) /\ ff_current_ph_hd_hpl_sph_hsc_nonsingular_source_pair_positive_body_derivative_steps = ff_previous_ph_hd_hpl_sph_hsc_nonsingular_source_pair_positive_body_derivative_steps * a + ff_coefficient_ph_hd_hpl_sph_hsc_nonsingular_source_pair_positive_body_derivative_steps)))))))) /\ (exists ff_u_hd_hpl_sph_hsc_nonsingular_source_pair_negative ff_v_hd_hpl_sph_hsc_nonsingular_source_pair_negative ff_d_hd_hpl_sph_hsc_nonsingular_source_pair_negative ff_e_hd_hpl_sph_hsc_nonsingular_source_pair_negative. ((((((exists fs_h_ph_hd_hpl_sph_hsc_nonsingular_source_pair_negative_body_value_start. fs_h_ph_hd_hpl_sph_hsc_nonsingular_source_pair_negative_body_value_start + S (0) = S ((S (0)) * ff_v_hd_hpl_sph_hsc_nonsingular_source_pair_negative)) /\ exists fs_q_ph_hd_hpl_sph_hsc_nonsingular_source_pair_negative_body_value_start. ff_u_hd_hpl_sph_hsc_nonsingular_source_pair_negative = fs_q_ph_hd_hpl_sph_hsc_nonsingular_source_pair_negative_body_value_start * S ((S (0)) * ff_v_hd_hpl_sph_hsc_nonsingular_source_pair_negative) + (0))) /\ ((((exists fs_h_ph_hd_hpl_sph_hsc_nonsingular_source_pair_negative_body_value_terminal. fs_h_ph_hd_hpl_sph_hsc_nonsingular_source_pair_negative_body_value_terminal + S (hsc_vn_nonsingular_source) = S ((S (l)) * ff_v_hd_hpl_sph_hsc_nonsingular_source_pair_negative)) /\ exists fs_q_ph_hd_hpl_sph_hsc_nonsingular_source_pair_negative_body_value_terminal. ff_u_hd_hpl_sph_hsc_nonsingular_source_pair_negative = fs_q_ph_hd_hpl_sph_hsc_nonsingular_source_pair_negative_body_value_terminal * S ((S (l)) * ff_v_hd_hpl_sph_hsc_nonsingular_source_pair_negative) + (hsc_vn_nonsingular_source))) /\ forall ff_i_ph_hd_hpl_sph_hsc_nonsingular_source_pair_negative_body_value_steps. (exists ph_bound_hd_hpl_sph_hsc_nonsingular_source_pair_negative_body_value_steps. ph_bound_hd_hpl_sph_hsc_nonsingular_source_pair_negative_body_value_steps + S ff_i_ph_hd_hpl_sph_hsc_nonsingular_source_pair_negative_body_value_steps = l) -> exists ff_coefficient_ph_hd_hpl_sph_hsc_nonsingular_source_pair_negative_body_value_steps ff_previous_ph_hd_hpl_sph_hsc_nonsingular_source_pair_negative_body_value_steps ff_current_ph_hd_hpl_sph_hsc_nonsingular_source_pair_negative_body_value_steps. ((((exists fs_h_ph_hd_hpl_sph_hsc_nonsingular_source_pair_negative_body_value_steps_coefficient. fs_h_ph_hd_hpl_sph_hsc_nonsingular_source_pair_negative_body_value_steps_coefficient + S (ff_coefficient_ph_hd_hpl_sph_hsc_nonsingular_source_pair_negative_body_value_steps) = S ((S (ff_i_ph_hd_hpl_sph_hsc_nonsingular_source_pair_negative_body_value_steps)) * nc)) /\ exists fs_q_ph_hd_hpl_sph_hsc_nonsingular_source_pair_negative_body_value_steps_coefficient. nb = fs_q_ph_hd_hpl_sph_hsc_nonsingular_source_pair_negative_body_value_steps_coefficient * S ((S (ff_i_ph_hd_hpl_sph_hsc_nonsingular_source_pair_negative_body_value_steps)) * nc) + (ff_coefficient_ph_hd_hpl_sph_hsc_nonsingular_source_pair_negative_body_value_steps))) /\ ((((exists fs_h_ph_hd_hpl_sph_hsc_nonsingular_source_pair_negative_body_value_steps_before. fs_h_ph_hd_hpl_sph_hsc_nonsingular_source_pair_negative_body_value_steps_before + S (ff_previous_ph_hd_hpl_sph_hsc_nonsingular_source_pair_negative_body_value_steps) = S ((S (ff_i_ph_hd_hpl_sph_hsc_nonsingular_source_pair_negative_body_value_steps)) * ff_v_hd_hpl_sph_hsc_nonsingular_source_pair_negative)) /\ exists fs_q_ph_hd_hpl_sph_hsc_nonsingular_source_pair_negative_body_value_steps_before. ff_u_hd_hpl_sph_hsc_nonsingular_source_pair_negative = fs_q_ph_hd_hpl_sph_hsc_nonsingular_source_pair_negative_body_value_steps_before * S ((S (ff_i_ph_hd_hpl_sph_hsc_nonsingular_source_pair_negative_body_value_steps)) * ff_v_hd_hpl_sph_hsc_nonsingular_source_pair_negative) + (ff_previous_ph_hd_hpl_sph_hsc_nonsingular_source_pair_negative_body_value_steps))) /\ ((((exists fs_h_ph_hd_hpl_sph_hsc_nonsingular_source_pair_negative_body_value_steps_after. fs_h_ph_hd_hpl_sph_hsc_nonsingular_source_pair_negative_body_value_steps_after + S (ff_current_ph_hd_hpl_sph_hsc_nonsingular_source_pair_negative_body_value_steps) = S ((S (S ff_i_ph_hd_hpl_sph_hsc_nonsingular_source_pair_negative_body_value_steps)) * ff_v_hd_hpl_sph_hsc_nonsingular_source_pair_negative)) /\ exists fs_q_ph_hd_hpl_sph_hsc_nonsingular_source_pair_negative_body_value_steps_after. ff_u_hd_hpl_sph_hsc_nonsingular_source_pair_negative = fs_q_ph_hd_hpl_sph_hsc_nonsingular_source_pair_negative_body_value_steps_after * S ((S (S ff_i_ph_hd_hpl_sph_hsc_nonsingular_source_pair_negative_body_value_steps)) * ff_v_hd_hpl_sph_hsc_nonsingular_source_pair_negative) + (ff_current_ph_hd_hpl_sph_hsc_nonsingular_source_pair_negative_body_value_steps))) /\ ff_current_ph_hd_hpl_sph_hsc_nonsingular_source_pair_negative_body_value_steps = ff_previous_ph_hd_hpl_sph_hsc_nonsingular_source_pair_negative_body_value_steps * a + ff_coefficient_ph_hd_hpl_sph_hsc_nonsingular_source_pair_negative_body_value_steps)))))) /\ (((((exists fs_h_ph_hd_hpl_sph_hsc_nonsingular_source_pair_negative_body_derivative_start. fs_h_ph_hd_hpl_sph_hsc_nonsingular_source_pair_negative_body_derivative_start + S (0) = S ((S (0)) * ff_e_hd_hpl_sph_hsc_nonsingular_source_pair_negative)) /\ exists fs_q_ph_hd_hpl_sph_hsc_nonsingular_source_pair_negative_body_derivative_start. ff_d_hd_hpl_sph_hsc_nonsingular_source_pair_negative = fs_q_ph_hd_hpl_sph_hsc_nonsingular_source_pair_negative_body_derivative_start * S ((S (0)) * ff_e_hd_hpl_sph_hsc_nonsingular_source_pair_negative) + (0))) /\ ((((exists fs_h_ph_hd_hpl_sph_hsc_nonsingular_source_pair_negative_body_derivative_terminal. fs_h_ph_hd_hpl_sph_hsc_nonsingular_source_pair_negative_body_derivative_terminal + S (hsc_dn_nonsingular_source) = S ((S (l)) * ff_e_hd_hpl_sph_hsc_nonsingular_source_pair_negative)) /\ exists fs_q_ph_hd_hpl_sph_hsc_nonsingular_source_pair_negative_body_derivative_terminal. ff_d_hd_hpl_sph_hsc_nonsingular_source_pair_negative = fs_q_ph_hd_hpl_sph_hsc_nonsingular_source_pair_negative_body_derivative_terminal * S ((S (l)) * ff_e_hd_hpl_sph_hsc_nonsingular_source_pair_negative) + (hsc_dn_nonsingular_source))) /\ forall ff_i_ph_hd_hpl_sph_hsc_nonsingular_source_pair_negative_body_derivative_steps. (exists ph_bound_hd_hpl_sph_hsc_nonsingular_source_pair_negative_body_derivative_steps. ph_bound_hd_hpl_sph_hsc_nonsingular_source_pair_negative_body_derivative_steps + S ff_i_ph_hd_hpl_sph_hsc_nonsingular_source_pair_negative_body_derivative_steps = l) -> exists ff_coefficient_ph_hd_hpl_sph_hsc_nonsingular_source_pair_negative_body_derivative_steps ff_previous_ph_hd_hpl_sph_hsc_nonsingular_source_pair_negative_body_derivative_steps ff_current_ph_hd_hpl_sph_hsc_nonsingular_source_pair_negative_body_derivative_steps. ((((exists fs_h_ph_hd_hpl_sph_hsc_nonsingular_source_pair_negative_body_derivative_steps_coefficient. fs_h_ph_hd_hpl_sph_hsc_nonsingular_source_pair_negative_body_derivative_steps_coefficient + S (ff_coefficient_ph_hd_hpl_sph_hsc_nonsingular_source_pair_negative_body_derivative_steps) = S ((S (ff_i_ph_hd_hpl_sph_hsc_nonsingular_source_pair_negative_body_derivative_steps)) * ff_v_hd_hpl_sph_hsc_nonsingular_source_pair_negative)) /\ exists fs_q_ph_hd_hpl_sph_hsc_nonsingular_source_pair_negative_body_derivative_steps_coefficient. ff_u_hd_hpl_sph_hsc_nonsingular_source_pair_negative = fs_q_ph_hd_hpl_sph_hsc_nonsingular_source_pair_negative_body_derivative_steps_coefficient * S ((S (ff_i_ph_hd_hpl_sph_hsc_nonsingular_source_pair_negative_body_derivative_steps)) * ff_v_hd_hpl_sph_hsc_nonsingular_source_pair_negative) + (ff_coefficient_ph_hd_hpl_sph_hsc_nonsingular_source_pair_negative_body_derivative_steps))) /\ ((((exists fs_h_ph_hd_hpl_sph_hsc_nonsingular_source_pair_negative_body_derivative_steps_before. fs_h_ph_hd_hpl_sph_hsc_nonsingular_source_pair_negative_body_derivative_steps_before + S (ff_previous_ph_hd_hpl_sph_hsc_nonsingular_source_pair_negative_body_derivative_steps) = S ((S (ff_i_ph_hd_hpl_sph_hsc_nonsingular_source_pair_negative_body_derivative_steps)) * ff_e_hd_hpl_sph_hsc_nonsingular_source_pair_negative)) /\ exists fs_q_ph_hd_hpl_sph_hsc_nonsingular_source_pair_negative_body_derivative_steps_before. ff_d_hd_hpl_sph_hsc_nonsingular_source_pair_negative = fs_q_ph_hd_hpl_sph_hsc_nonsingular_source_pair_negative_body_derivative_steps_before * S ((S (ff_i_ph_hd_hpl_sph_hsc_nonsingular_source_pair_negative_body_derivative_steps)) * ff_e_hd_hpl_sph_hsc_nonsingular_source_pair_negative) + (ff_previous_ph_hd_hpl_sph_hsc_nonsingular_source_pair_negative_body_derivative_steps))) /\ ((((exists fs_h_ph_hd_hpl_sph_hsc_nonsingular_source_pair_negative_body_derivative_steps_after. fs_h_ph_hd_hpl_sph_hsc_nonsingular_source_pair_negative_body_derivative_steps_after + S (ff_current_ph_hd_hpl_sph_hsc_nonsingular_source_pair_negative_body_derivative_steps) = S ((S (S ff_i_ph_hd_hpl_sph_hsc_nonsingular_source_pair_negative_body_derivative_steps)) * ff_e_hd_hpl_sph_hsc_nonsingular_source_pair_negative)) /\ exists fs_q_ph_hd_hpl_sph_hsc_nonsingular_source_pair_negative_body_derivative_steps_after. ff_d_hd_hpl_sph_hsc_nonsingular_source_pair_negative = fs_q_ph_hd_hpl_sph_hsc_nonsingular_source_pair_negative_body_derivative_steps_after * S ((S (S ff_i_ph_hd_hpl_sph_hsc_nonsingular_source_pair_negative_body_derivative_steps)) * ff_e_hd_hpl_sph_hsc_nonsingular_source_pair_negative) + (ff_current_ph_hd_hpl_sph_hsc_nonsingular_source_pair_negative_body_derivative_steps))) /\ ff_current_ph_hd_hpl_sph_hsc_nonsingular_source_pair_negative_body_derivative_steps = ff_previous_ph_hd_hpl_sph_hsc_nonsingular_source_pair_negative_body_derivative_steps * a + ff_coefficient_ph_hd_hpl_sph_hsc_nonsingular_source_pair_negative_body_derivative_steps)))))))))) /\ ((exists hgcrt_mod_left_hpl_hsc_nonsingular_source_root hgcrt_mod_right_hpl_hsc_nonsingular_source_root. hsc_vp_nonsingular_source + m * hgcrt_mod_left_hpl_hsc_nonsingular_source_root = hsc_vn_nonsingular_source + m * hgcrt_mod_right_hpl_hsc_nonsingular_source_root) /\ (~(exists hgcrt_mod_left_hpl_hsc_nonsingular_source hgcrt_mod_right_hpl_hsc_nonsingular_source. hsc_dp_nonsingular_source + p * hgcrt_mod_left_hpl_hsc_nonsingular_source = hsc_dn_nonsingular_source + p * hgcrt_mod_right_hpl_hsc_nonsingular_source))))) -> (exists sph_vp_hsc_simple_result sph_dp_hsc_simple_result sph_vn_hsc_simple_result sph_dn_hsc_simple_result. ((((exists ff_u_hd_hpl_sph_hsc_simple_result_positive ff_v_hd_hpl_sph_hsc_simple_result_positive ff_d_hd_hpl_sph_hsc_simple_result_positive ff_e_hd_hpl_sph_hsc_simple_result_positive. ((((((exists fs_h_ph_hd_hpl_sph_hsc_simple_result_positive_body_value_start. fs_h_ph_hd_hpl_sph_hsc_simple_result_positive_body_value_start + S (0) = S ((S (0)) * ff_v_hd_hpl_sph_hsc_simple_result_positive)) /\ exists fs_q_ph_hd_hpl_sph_hsc_simple_result_positive_body_value_start. ff_u_hd_hpl_sph_hsc_simple_result_positive = fs_q_ph_hd_hpl_sph_hsc_simple_result_positive_body_value_start * S ((S (0)) * ff_v_hd_hpl_sph_hsc_simple_result_positive) + (0))) /\ ((((exists fs_h_ph_hd_hpl_sph_hsc_simple_result_positive_body_value_terminal. fs_h_ph_hd_hpl_sph_hsc_simple_result_positive_body_value_terminal + S (sph_vp_hsc_simple_result) = S ((S (l)) * ff_v_hd_hpl_sph_hsc_simple_result_positive)) /\ exists fs_q_ph_hd_hpl_sph_hsc_simple_result_positive_body_value_terminal. ff_u_hd_hpl_sph_hsc_simple_result_positive = fs_q_ph_hd_hpl_sph_hsc_simple_result_positive_body_value_terminal * S ((S (l)) * ff_v_hd_hpl_sph_hsc_simple_result_positive) + (sph_vp_hsc_simple_result))) /\ forall ff_i_ph_hd_hpl_sph_hsc_simple_result_positive_body_value_steps. (exists ph_bound_hd_hpl_sph_hsc_simple_result_positive_body_value_steps. ph_bound_hd_hpl_sph_hsc_simple_result_positive_body_value_steps + S ff_i_ph_hd_hpl_sph_hsc_simple_result_positive_body_value_steps = l) -> exists ff_coefficient_ph_hd_hpl_sph_hsc_simple_result_positive_body_value_steps ff_previous_ph_hd_hpl_sph_hsc_simple_result_positive_body_value_steps ff_current_ph_hd_hpl_sph_hsc_simple_result_positive_body_value_steps. ((((exists fs_h_ph_hd_hpl_sph_hsc_simple_result_positive_body_value_steps_coefficient. fs_h_ph_hd_hpl_sph_hsc_simple_result_positive_body_value_steps_coefficient + S (ff_coefficient_ph_hd_hpl_sph_hsc_simple_result_positive_body_value_steps) = S ((S (ff_i_ph_hd_hpl_sph_hsc_simple_result_positive_body_value_steps)) * pc)) /\ exists fs_q_ph_hd_hpl_sph_hsc_simple_result_positive_body_value_steps_coefficient. pb = fs_q_ph_hd_hpl_sph_hsc_simple_result_positive_body_value_steps_coefficient * S ((S (ff_i_ph_hd_hpl_sph_hsc_simple_result_positive_body_value_steps)) * pc) + (ff_coefficient_ph_hd_hpl_sph_hsc_simple_result_positive_body_value_steps))) /\ ((((exists fs_h_ph_hd_hpl_sph_hsc_simple_result_positive_body_value_steps_before. fs_h_ph_hd_hpl_sph_hsc_simple_result_positive_body_value_steps_before + S (ff_previous_ph_hd_hpl_sph_hsc_simple_result_positive_body_value_steps) = S ((S (ff_i_ph_hd_hpl_sph_hsc_simple_result_positive_body_value_steps)) * ff_v_hd_hpl_sph_hsc_simple_result_positive)) /\ exists fs_q_ph_hd_hpl_sph_hsc_simple_result_positive_body_value_steps_before. ff_u_hd_hpl_sph_hsc_simple_result_positive = fs_q_ph_hd_hpl_sph_hsc_simple_result_positive_body_value_steps_before * S ((S (ff_i_ph_hd_hpl_sph_hsc_simple_result_positive_body_value_steps)) * ff_v_hd_hpl_sph_hsc_simple_result_positive) + (ff_previous_ph_hd_hpl_sph_hsc_simple_result_positive_body_value_steps))) /\ ((((exists fs_h_ph_hd_hpl_sph_hsc_simple_result_positive_body_value_steps_after. fs_h_ph_hd_hpl_sph_hsc_simple_result_positive_body_value_steps_after + S (ff_current_ph_hd_hpl_sph_hsc_simple_result_positive_body_value_steps) = S ((S (S ff_i_ph_hd_hpl_sph_hsc_simple_result_positive_body_value_steps)) * ff_v_hd_hpl_sph_hsc_simple_result_positive)) /\ exists fs_q_ph_hd_hpl_sph_hsc_simple_result_positive_body_value_steps_after. ff_u_hd_hpl_sph_hsc_simple_result_positive = fs_q_ph_hd_hpl_sph_hsc_simple_result_positive_body_value_steps_after * S ((S (S ff_i_ph_hd_hpl_sph_hsc_simple_result_positive_body_value_steps)) * ff_v_hd_hpl_sph_hsc_simple_result_positive) + (ff_current_ph_hd_hpl_sph_hsc_simple_result_positive_body_value_steps))) /\ ff_current_ph_hd_hpl_sph_hsc_simple_result_positive_body_value_steps = ff_previous_ph_hd_hpl_sph_hsc_simple_result_positive_body_value_steps * a + ff_coefficient_ph_hd_hpl_sph_hsc_simple_result_positive_body_value_steps)))))) /\ (((((exists fs_h_ph_hd_hpl_sph_hsc_simple_result_positive_body_derivative_start. fs_h_ph_hd_hpl_sph_hsc_simple_result_positive_body_derivative_start + S (0) = S ((S (0)) * ff_e_hd_hpl_sph_hsc_simple_result_positive)) /\ exists fs_q_ph_hd_hpl_sph_hsc_simple_result_positive_body_derivative_start. ff_d_hd_hpl_sph_hsc_simple_result_positive = fs_q_ph_hd_hpl_sph_hsc_simple_result_positive_body_derivative_start * S ((S (0)) * ff_e_hd_hpl_sph_hsc_simple_result_positive) + (0))) /\ ((((exists fs_h_ph_hd_hpl_sph_hsc_simple_result_positive_body_derivative_terminal. fs_h_ph_hd_hpl_sph_hsc_simple_result_positive_body_derivative_terminal + S (sph_dp_hsc_simple_result) = S ((S (l)) * ff_e_hd_hpl_sph_hsc_simple_result_positive)) /\ exists fs_q_ph_hd_hpl_sph_hsc_simple_result_positive_body_derivative_terminal. ff_d_hd_hpl_sph_hsc_simple_result_positive = fs_q_ph_hd_hpl_sph_hsc_simple_result_positive_body_derivative_terminal * S ((S (l)) * ff_e_hd_hpl_sph_hsc_simple_result_positive) + (sph_dp_hsc_simple_result))) /\ forall ff_i_ph_hd_hpl_sph_hsc_simple_result_positive_body_derivative_steps. (exists ph_bound_hd_hpl_sph_hsc_simple_result_positive_body_derivative_steps. ph_bound_hd_hpl_sph_hsc_simple_result_positive_body_derivative_steps + S ff_i_ph_hd_hpl_sph_hsc_simple_result_positive_body_derivative_steps = l) -> exists ff_coefficient_ph_hd_hpl_sph_hsc_simple_result_positive_body_derivative_steps ff_previous_ph_hd_hpl_sph_hsc_simple_result_positive_body_derivative_steps ff_current_ph_hd_hpl_sph_hsc_simple_result_positive_body_derivative_steps. ((((exists fs_h_ph_hd_hpl_sph_hsc_simple_result_positive_body_derivative_steps_coefficient. fs_h_ph_hd_hpl_sph_hsc_simple_result_positive_body_derivative_steps_coefficient + S (ff_coefficient_ph_hd_hpl_sph_hsc_simple_result_positive_body_derivative_steps) = S ((S (ff_i_ph_hd_hpl_sph_hsc_simple_result_positive_body_derivative_steps)) * ff_v_hd_hpl_sph_hsc_simple_result_positive)) /\ exists fs_q_ph_hd_hpl_sph_hsc_simple_result_positive_body_derivative_steps_coefficient. ff_u_hd_hpl_sph_hsc_simple_result_positive = fs_q_ph_hd_hpl_sph_hsc_simple_result_positive_body_derivative_steps_coefficient * S ((S (ff_i_ph_hd_hpl_sph_hsc_simple_result_positive_body_derivative_steps)) * ff_v_hd_hpl_sph_hsc_simple_result_positive) + (ff_coefficient_ph_hd_hpl_sph_hsc_simple_result_positive_body_derivative_steps))) /\ ((((exists fs_h_ph_hd_hpl_sph_hsc_simple_result_positive_body_derivative_steps_before. fs_h_ph_hd_hpl_sph_hsc_simple_result_positive_body_derivative_steps_before + S (ff_previous_ph_hd_hpl_sph_hsc_simple_result_positive_body_derivative_steps) = S ((S (ff_i_ph_hd_hpl_sph_hsc_simple_result_positive_body_derivative_steps)) * ff_e_hd_hpl_sph_hsc_simple_result_positive)) /\ exists fs_q_ph_hd_hpl_sph_hsc_simple_result_positive_body_derivative_steps_before. ff_d_hd_hpl_sph_hsc_simple_result_positive = fs_q_ph_hd_hpl_sph_hsc_simple_result_positive_body_derivative_steps_before * S ((S (ff_i_ph_hd_hpl_sph_hsc_simple_result_positive_body_derivative_steps)) * ff_e_hd_hpl_sph_hsc_simple_result_positive) + (ff_previous_ph_hd_hpl_sph_hsc_simple_result_positive_body_derivative_steps))) /\ ((((exists fs_h_ph_hd_hpl_sph_hsc_simple_result_positive_body_derivative_steps_after. fs_h_ph_hd_hpl_sph_hsc_simple_result_positive_body_derivative_steps_after + S (ff_current_ph_hd_hpl_sph_hsc_simple_result_positive_body_derivative_steps) = S ((S (S ff_i_ph_hd_hpl_sph_hsc_simple_result_positive_body_derivative_steps)) * ff_e_hd_hpl_sph_hsc_simple_result_positive)) /\ exists fs_q_ph_hd_hpl_sph_hsc_simple_result_positive_body_derivative_steps_after. ff_d_hd_hpl_sph_hsc_simple_result_positive = fs_q_ph_hd_hpl_sph_hsc_simple_result_positive_body_derivative_steps_after * S ((S (S ff_i_ph_hd_hpl_sph_hsc_simple_result_positive_body_derivative_steps)) * ff_e_hd_hpl_sph_hsc_simple_result_positive) + (ff_current_ph_hd_hpl_sph_hsc_simple_result_positive_body_derivative_steps))) /\ ff_current_ph_hd_hpl_sph_hsc_simple_result_positive_body_derivative_steps = ff_previous_ph_hd_hpl_sph_hsc_simple_result_positive_body_derivative_steps * a + ff_coefficient_ph_hd_hpl_sph_hsc_simple_result_positive_body_derivative_steps)))))))) /\ (exists ff_u_hd_hpl_sph_hsc_simple_result_negative ff_v_hd_hpl_sph_hsc_simple_result_negative ff_d_hd_hpl_sph_hsc_simple_result_negative ff_e_hd_hpl_sph_hsc_simple_result_negative. ((((((exists fs_h_ph_hd_hpl_sph_hsc_simple_result_negative_body_value_start. fs_h_ph_hd_hpl_sph_hsc_simple_result_negative_body_value_start + S (0) = S ((S (0)) * ff_v_hd_hpl_sph_hsc_simple_result_negative)) /\ exists fs_q_ph_hd_hpl_sph_hsc_simple_result_negative_body_value_start. ff_u_hd_hpl_sph_hsc_simple_result_negative = fs_q_ph_hd_hpl_sph_hsc_simple_result_negative_body_value_start * S ((S (0)) * ff_v_hd_hpl_sph_hsc_simple_result_negative) + (0))) /\ ((((exists fs_h_ph_hd_hpl_sph_hsc_simple_result_negative_body_value_terminal. fs_h_ph_hd_hpl_sph_hsc_simple_result_negative_body_value_terminal + S (sph_vn_hsc_simple_result) = S ((S (l)) * ff_v_hd_hpl_sph_hsc_simple_result_negative)) /\ exists fs_q_ph_hd_hpl_sph_hsc_simple_result_negative_body_value_terminal. ff_u_hd_hpl_sph_hsc_simple_result_negative = fs_q_ph_hd_hpl_sph_hsc_simple_result_negative_body_value_terminal * S ((S (l)) * ff_v_hd_hpl_sph_hsc_simple_result_negative) + (sph_vn_hsc_simple_result))) /\ forall ff_i_ph_hd_hpl_sph_hsc_simple_result_negative_body_value_steps. (exists ph_bound_hd_hpl_sph_hsc_simple_result_negative_body_value_steps. ph_bound_hd_hpl_sph_hsc_simple_result_negative_body_value_steps + S ff_i_ph_hd_hpl_sph_hsc_simple_result_negative_body_value_steps = l) -> exists ff_coefficient_ph_hd_hpl_sph_hsc_simple_result_negative_body_value_steps ff_previous_ph_hd_hpl_sph_hsc_simple_result_negative_body_value_steps ff_current_ph_hd_hpl_sph_hsc_simple_result_negative_body_value_steps. ((((exists fs_h_ph_hd_hpl_sph_hsc_simple_result_negative_body_value_steps_coefficient. fs_h_ph_hd_hpl_sph_hsc_simple_result_negative_body_value_steps_coefficient + S (ff_coefficient_ph_hd_hpl_sph_hsc_simple_result_negative_body_value_steps) = S ((S (ff_i_ph_hd_hpl_sph_hsc_simple_result_negative_body_value_steps)) * nc)) /\ exists fs_q_ph_hd_hpl_sph_hsc_simple_result_negative_body_value_steps_coefficient. nb = fs_q_ph_hd_hpl_sph_hsc_simple_result_negative_body_value_steps_coefficient * S ((S (ff_i_ph_hd_hpl_sph_hsc_simple_result_negative_body_value_steps)) * nc) + (ff_coefficient_ph_hd_hpl_sph_hsc_simple_result_negative_body_value_steps))) /\ ((((exists fs_h_ph_hd_hpl_sph_hsc_simple_result_negative_body_value_steps_before. fs_h_ph_hd_hpl_sph_hsc_simple_result_negative_body_value_steps_before + S (ff_previous_ph_hd_hpl_sph_hsc_simple_result_negative_body_value_steps) = S ((S (ff_i_ph_hd_hpl_sph_hsc_simple_result_negative_body_value_steps)) * ff_v_hd_hpl_sph_hsc_simple_result_negative)) /\ exists fs_q_ph_hd_hpl_sph_hsc_simple_result_negative_body_value_steps_before. ff_u_hd_hpl_sph_hsc_simple_result_negative = fs_q_ph_hd_hpl_sph_hsc_simple_result_negative_body_value_steps_before * S ((S (ff_i_ph_hd_hpl_sph_hsc_simple_result_negative_body_value_steps)) * ff_v_hd_hpl_sph_hsc_simple_result_negative) + (ff_previous_ph_hd_hpl_sph_hsc_simple_result_negative_body_value_steps))) /\ ((((exists fs_h_ph_hd_hpl_sph_hsc_simple_result_negative_body_value_steps_after. fs_h_ph_hd_hpl_sph_hsc_simple_result_negative_body_value_steps_after + S (ff_current_ph_hd_hpl_sph_hsc_simple_result_negative_body_value_steps) = S ((S (S ff_i_ph_hd_hpl_sph_hsc_simple_result_negative_body_value_steps)) * ff_v_hd_hpl_sph_hsc_simple_result_negative)) /\ exists fs_q_ph_hd_hpl_sph_hsc_simple_result_negative_body_value_steps_after. ff_u_hd_hpl_sph_hsc_simple_result_negative = fs_q_ph_hd_hpl_sph_hsc_simple_result_negative_body_value_steps_after * S ((S (S ff_i_ph_hd_hpl_sph_hsc_simple_result_negative_body_value_steps)) * ff_v_hd_hpl_sph_hsc_simple_result_negative) + (ff_current_ph_hd_hpl_sph_hsc_simple_result_negative_body_value_steps))) /\ ff_current_ph_hd_hpl_sph_hsc_simple_result_negative_body_value_steps = ff_previous_ph_hd_hpl_sph_hsc_simple_result_negative_body_value_steps * a + ff_coefficient_ph_hd_hpl_sph_hsc_simple_result_negative_body_value_steps)))))) /\ (((((exists fs_h_ph_hd_hpl_sph_hsc_simple_result_negative_body_derivative_start. fs_h_ph_hd_hpl_sph_hsc_simple_result_negative_body_derivative_start + S (0) = S ((S (0)) * ff_e_hd_hpl_sph_hsc_simple_result_negative)) /\ exists fs_q_ph_hd_hpl_sph_hsc_simple_result_negative_body_derivative_start. ff_d_hd_hpl_sph_hsc_simple_result_negative = fs_q_ph_hd_hpl_sph_hsc_simple_result_negative_body_derivative_start * S ((S (0)) * ff_e_hd_hpl_sph_hsc_simple_result_negative) + (0))) /\ ((((exists fs_h_ph_hd_hpl_sph_hsc_simple_result_negative_body_derivative_terminal. fs_h_ph_hd_hpl_sph_hsc_simple_result_negative_body_derivative_terminal + S (sph_dn_hsc_simple_result) = S ((S (l)) * ff_e_hd_hpl_sph_hsc_simple_result_negative)) /\ exists fs_q_ph_hd_hpl_sph_hsc_simple_result_negative_body_derivative_terminal. ff_d_hd_hpl_sph_hsc_simple_result_negative = fs_q_ph_hd_hpl_sph_hsc_simple_result_negative_body_derivative_terminal * S ((S (l)) * ff_e_hd_hpl_sph_hsc_simple_result_negative) + (sph_dn_hsc_simple_result))) /\ forall ff_i_ph_hd_hpl_sph_hsc_simple_result_negative_body_derivative_steps. (exists ph_bound_hd_hpl_sph_hsc_simple_result_negative_body_derivative_steps. ph_bound_hd_hpl_sph_hsc_simple_result_negative_body_derivative_steps + S ff_i_ph_hd_hpl_sph_hsc_simple_result_negative_body_derivative_steps = l) -> exists ff_coefficient_ph_hd_hpl_sph_hsc_simple_result_negative_body_derivative_steps ff_previous_ph_hd_hpl_sph_hsc_simple_result_negative_body_derivative_steps ff_current_ph_hd_hpl_sph_hsc_simple_result_negative_body_derivative_steps. ((((exists fs_h_ph_hd_hpl_sph_hsc_simple_result_negative_body_derivative_steps_coefficient. fs_h_ph_hd_hpl_sph_hsc_simple_result_negative_body_derivative_steps_coefficient + S (ff_coefficient_ph_hd_hpl_sph_hsc_simple_result_negative_body_derivative_steps) = S ((S (ff_i_ph_hd_hpl_sph_hsc_simple_result_negative_body_derivative_steps)) * ff_v_hd_hpl_sph_hsc_simple_result_negative)) /\ exists fs_q_ph_hd_hpl_sph_hsc_simple_result_negative_body_derivative_steps_coefficient. ff_u_hd_hpl_sph_hsc_simple_result_negative = fs_q_ph_hd_hpl_sph_hsc_simple_result_negative_body_derivative_steps_coefficient * S ((S (ff_i_ph_hd_hpl_sph_hsc_simple_result_negative_body_derivative_steps)) * ff_v_hd_hpl_sph_hsc_simple_result_negative) + (ff_coefficient_ph_hd_hpl_sph_hsc_simple_result_negative_body_derivative_steps))) /\ ((((exists fs_h_ph_hd_hpl_sph_hsc_simple_result_negative_body_derivative_steps_before. fs_h_ph_hd_hpl_sph_hsc_simple_result_negative_body_derivative_steps_before + S (ff_previous_ph_hd_hpl_sph_hsc_simple_result_negative_body_derivative_steps) = S ((S (ff_i_ph_hd_hpl_sph_hsc_simple_result_negative_body_derivative_steps)) * ff_e_hd_hpl_sph_hsc_simple_result_negative)) /\ exists fs_q_ph_hd_hpl_sph_hsc_simple_result_negative_body_derivative_steps_before. ff_d_hd_hpl_sph_hsc_simple_result_negative = fs_q_ph_hd_hpl_sph_hsc_simple_result_negative_body_derivative_steps_before * S ((S (ff_i_ph_hd_hpl_sph_hsc_simple_result_negative_body_derivative_steps)) * ff_e_hd_hpl_sph_hsc_simple_result_negative) + (ff_previous_ph_hd_hpl_sph_hsc_simple_result_negative_body_derivative_steps))) /\ ((((exists fs_h_ph_hd_hpl_sph_hsc_simple_result_negative_body_derivative_steps_after. fs_h_ph_hd_hpl_sph_hsc_simple_result_negative_body_derivative_steps_after + S (ff_current_ph_hd_hpl_sph_hsc_simple_result_negative_body_derivative_steps) = S ((S (S ff_i_ph_hd_hpl_sph_hsc_simple_result_negative_body_derivative_steps)) * ff_e_hd_hpl_sph_hsc_simple_result_negative)) /\ exists fs_q_ph_hd_hpl_sph_hsc_simple_result_negative_body_derivative_steps_after. ff_d_hd_hpl_sph_hsc_simple_result_negative = fs_q_ph_hd_hpl_sph_hsc_simple_result_negative_body_derivative_steps_after * S ((S (S ff_i_ph_hd_hpl_sph_hsc_simple_result_negative_body_derivative_steps)) * ff_e_hd_hpl_sph_hsc_simple_result_negative) + (ff_current_ph_hd_hpl_sph_hsc_simple_result_negative_body_derivative_steps))) /\ ff_current_ph_hd_hpl_sph_hsc_simple_result_negative_body_derivative_steps = ff_previous_ph_hd_hpl_sph_hsc_simple_result_negative_body_derivative_steps * a + ff_coefficient_ph_hd_hpl_sph_hsc_simple_result_negative_body_derivative_steps)))))))))) /\ ((exists hgcrt_mod_left_hpl_hsc_simple_result hgcrt_mod_right_hpl_hsc_simple_result. sph_vp_hsc_simple_result + m * hgcrt_mod_left_hpl_hsc_simple_result = sph_vn_hsc_simple_result + m * hgcrt_mod_right_hpl_hsc_simple_result) /\ (exists sph_inverse_hsc_simple_result. ((exists hpl_gap_hsc_simple_result. hpl_gap_hsc_simple_result + S (sph_inverse_hsc_simple_result) = (p)) /\ (exists hgcrt_mod_left_hpl_hsc_simple_result hgcrt_mod_right_hpl_hsc_simple_result. (sph_dp_hsc_simple_result * sph_inverse_hsc_simple_result) + p * hgcrt_mod_left_hpl_hsc_simple_result = (1 + sph_dn_hsc_simple_result * sph_inverse_hsc_simple_result) + p * hgcrt_mod_right_hpl_hsc_simple_result))))))