FpPolynomialBezoutRepresentation(p,ab,ac,A,bb,bc,B,gb,gc,G,ub,uc,U,vb,vc,V)

exists pfbz_left_code_working_euclidean_definition pfbz_left_scale_working_euclidean_definition pfbz_left_length_working_euclidean_definition pfbz_right_code_working_euclidean_definition pfbz_right_scale_working_euclidean_definition pfbz_right_length_working_euclidean_definition. ((((forall fom_index_pfp_working_euclidean_definition_left_productleft. (exists fom_gap_pfp_working_euclidean_definition_left_productleft_index_bound. fom_gap_pfp_working_euclidean_definition_left_productleft_index_bound + S (fom_index_pfp_working_euclidean_definition_left_productleft) = (U)) -> exists fom_value_pfp_working_euclidean_definition_left_productleft. ((((exists fom_beta_height_pfp_working_euclidean_definition_left_productleft_entry. fom_beta_height_pfp_working_euclidean_definition_left_productleft_entry + S (fom_value_pfp_working_euclidean_definition_left_productleft) = S ((S (fom_index_pfp_working_euclidean_definition_left_productleft)) * (uc))) /\ exists fom_beta_quotient_pfp_working_euclidean_definition_left_productleft_entry. (ub) = fom_beta_quotient_pfp_working_euclidean_definition_left_productleft_entry * S ((S (fom_index_pfp_working_euclidean_definition_left_productleft)) * (uc)) + (fom_value_pfp_working_euclidean_definition_left_productleft))) /\ (exists fom_gap_pfp_working_euclidean_definition_left_productleft_value_bound. fom_gap_pfp_working_euclidean_definition_left_productleft_value_bound + S (fom_value_pfp_working_euclidean_definition_left_productleft) = (p)))) /\ (((forall fom_index_pfp_working_euclidean_definition_left_productright. (exists fom_gap_pfp_working_euclidean_definition_left_productright_index_bound. fom_gap_pfp_working_euclidean_definition_left_productright_index_bound + S (fom_index_pfp_working_euclidean_definition_left_productright) = (A)) -> exists fom_value_pfp_working_euclidean_definition_left_productright. ((((exists fom_beta_height_pfp_working_euclidean_definition_left_productright_entry. fom_beta_height_pfp_working_euclidean_definition_left_productright_entry + S (fom_value_pfp_working_euclidean_definition_left_productright) = S ((S (fom_index_pfp_working_euclidean_definition_left_productright)) * (ac))) /\ exists fom_beta_quotient_pfp_working_euclidean_definition_left_productright_entry. (ab) = fom_beta_quotient_pfp_working_euclidean_definition_left_productright_entry * S ((S (fom_index_pfp_working_euclidean_definition_left_productright)) * (ac)) + (fom_value_pfp_working_euclidean_definition_left_productright))) /\ (exists fom_gap_pfp_working_euclidean_definition_left_productright_value_bound. fom_gap_pfp_working_euclidean_definition_left_productright_value_bound + S (fom_value_pfp_working_euclidean_definition_left_productright) = (p)))) /\ ((((((((U))=0 \/ ((A))=0) /\ (((pfbz_left_length_working_euclidean_definition)=0)))) \/ (((~(((U))=0)) /\ (((~(((A))=0)) /\ ((((U))+((A))=S (pfbz_left_length_working_euclidean_definition)))))))) /\ ((forall pfc_index_working_euclidean_definition_left_productcoefficients. (exists pfa_gap_working_euclidean_definition_left_productcoefficientsbound. pfa_gap_working_euclidean_definition_left_productcoefficientsbound + S (pfc_index_working_euclidean_definition_left_productcoefficients) = (pfbz_left_length_working_euclidean_definition)) -> exists pfc_value_working_euclidean_definition_left_productcoefficients. ((((exists ff_h_pfp_working_euclidean_definition_left_productcoefficientsentry. ff_h_pfp_working_euclidean_definition_left_productcoefficientsentry + S (pfc_value_working_euclidean_definition_left_productcoefficients) = S ((S (pfc_index_working_euclidean_definition_left_productcoefficients)) * pfbz_left_scale_working_euclidean_definition)) /\ exists ff_q_pfp_working_euclidean_definition_left_productcoefficientsentry. pfbz_left_code_working_euclidean_definition = ff_q_pfp_working_euclidean_definition_left_productcoefficientsentry * S ((S (pfc_index_working_euclidean_definition_left_productcoefficients)) * pfbz_left_scale_working_euclidean_definition) + (pfc_value_working_euclidean_definition_left_productcoefficients))) /\ ((exists pfc_terms_code_working_euclidean_definition_left_productcoefficientscoefficient pfc_terms_scale_working_euclidean_definition_left_productcoefficientscoefficient pfc_natural_sum_working_euclidean_definition_left_productcoefficientscoefficient. ((forall pfc_index_working_euclidean_definition_left_productcoefficientscoefficientdiagonal. (exists pfa_gap_working_euclidean_definition_left_productcoefficientscoefficientdiagonalbound. pfa_gap_working_euclidean_definition_left_productcoefficientscoefficientdiagonalbound + S (pfc_index_working_euclidean_definition_left_productcoefficientscoefficientdiagonal) = (S (pfc_index_working_euclidean_definition_left_productcoefficients))) -> exists pfc_value_working_euclidean_definition_left_productcoefficientscoefficientdiagonal. ((((exists ff_h_pfp_working_euclidean_definition_left_productcoefficientscoefficientdiagonalentry. ff_h_pfp_working_euclidean_definition_left_productcoefficientscoefficientdiagonalentry + S (pfc_value_working_euclidean_definition_left_productcoefficientscoefficientdiagonal) = S ((S (pfc_index_working_euclidean_definition_left_productcoefficientscoefficientdiagonal)) * pfc_terms_scale_working_euclidean_definition_left_productcoefficientscoefficient)) /\ exists ff_q_pfp_working_euclidean_definition_left_productcoefficientscoefficientdiagonalentry. pfc_terms_code_working_euclidean_definition_left_productcoefficientscoefficient = ff_q_pfp_working_euclidean_definition_left_productcoefficientscoefficientdiagonalentry * S ((S (pfc_index_working_euclidean_definition_left_productcoefficientscoefficientdiagonal)) * pfc_terms_scale_working_euclidean_definition_left_productcoefficientscoefficient) + (pfc_value_working_euclidean_definition_left_productcoefficientscoefficientdiagonal))) /\ ((exists pfc_complement_working_euclidean_definition_left_productcoefficientscoefficientdiagonalterm pfc_left_working_euclidean_definition_left_productcoefficientscoefficientdiagonalterm pfc_right_working_euclidean_definition_left_productcoefficientscoefficientdiagonalterm. (((pfc_index_working_euclidean_definition_left_productcoefficientscoefficientdiagonal)+pfc_complement_working_euclidean_definition_left_productcoefficientscoefficientdiagonalterm=(pfc_index_working_euclidean_definition_left_productcoefficients)) /\ ((((((exists pfa_gap_working_euclidean_definition_left_productcoefficientscoefficientdiagonaltermleftinside. pfa_gap_working_euclidean_definition_left_productcoefficientscoefficientdiagonaltermleftinside + S (pfc_index_working_euclidean_definition_left_productcoefficientscoefficientdiagonal) = ((U))) /\ ((((exists ff_h_pfp_working_euclidean_definition_left_productcoefficientscoefficientdiagonaltermleftentry. ff_h_pfp_working_euclidean_definition_left_productcoefficientscoefficientdiagonaltermleftentry + S (pfc_left_working_euclidean_definition_left_productcoefficientscoefficientdiagonalterm) = S ((S (pfc_index_working_euclidean_definition_left_productcoefficientscoefficientdiagonal)) * (uc))) /\ exists ff_q_pfp_working_euclidean_definition_left_productcoefficientscoefficientdiagonaltermleftentry. (ub) = ff_q_pfp_working_euclidean_definition_left_productcoefficientscoefficientdiagonaltermleftentry * S ((S (pfc_index_working_euclidean_definition_left_productcoefficientscoefficientdiagonal)) * (uc)) + (pfc_left_working_euclidean_definition_left_productcoefficientscoefficientdiagonalterm)))))) \/ (((exists pfc_gap_working_euclidean_definition_left_productcoefficientscoefficientdiagonaltermleftoutside. pfc_gap_working_euclidean_definition_left_productcoefficientscoefficientdiagonaltermleftoutside+((U))=(pfc_index_working_euclidean_definition_left_productcoefficientscoefficientdiagonal)) /\ (((pfc_left_working_euclidean_definition_left_productcoefficientscoefficientdiagonalterm)=0))))) /\ ((((((exists pfa_gap_working_euclidean_definition_left_productcoefficientscoefficientdiagonaltermrightinside. pfa_gap_working_euclidean_definition_left_productcoefficientscoefficientdiagonaltermrightinside + S (pfc_complement_working_euclidean_definition_left_productcoefficientscoefficientdiagonalterm) = ((A))) /\ ((((exists ff_h_pfp_working_euclidean_definition_left_productcoefficientscoefficientdiagonaltermrightentry. ff_h_pfp_working_euclidean_definition_left_productcoefficientscoefficientdiagonaltermrightentry + S (pfc_right_working_euclidean_definition_left_productcoefficientscoefficientdiagonalterm) = S ((S (pfc_complement_working_euclidean_definition_left_productcoefficientscoefficientdiagonalterm)) * (ac))) /\ exists ff_q_pfp_working_euclidean_definition_left_productcoefficientscoefficientdiagonaltermrightentry. (ab) = ff_q_pfp_working_euclidean_definition_left_productcoefficientscoefficientdiagonaltermrightentry * S ((S (pfc_complement_working_euclidean_definition_left_productcoefficientscoefficientdiagonalterm)) * (ac)) + (pfc_right_working_euclidean_definition_left_productcoefficientscoefficientdiagonalterm)))))) \/ (((exists pfc_gap_working_euclidean_definition_left_productcoefficientscoefficientdiagonaltermrightoutside. pfc_gap_working_euclidean_definition_left_productcoefficientscoefficientdiagonaltermrightoutside+((A))=(pfc_complement_working_euclidean_definition_left_productcoefficientscoefficientdiagonalterm)) /\ (((pfc_right_working_euclidean_definition_left_productcoefficientscoefficientdiagonalterm)=0))))) /\ (((pfc_value_working_euclidean_definition_left_productcoefficientscoefficientdiagonal)=pfc_left_working_euclidean_definition_left_productcoefficientscoefficientdiagonalterm*pfc_right_working_euclidean_definition_left_productcoefficientscoefficientdiagonalterm))))))))))) /\ (((exists fs_u_pfc_working_euclidean_definition_left_productcoefficientscoefficientsum fs_v_pfc_working_euclidean_definition_left_productcoefficientscoefficientsum. ((((exists fs_h_pfc_working_euclidean_definition_left_productcoefficientscoefficientsum_body_start. fs_h_pfc_working_euclidean_definition_left_productcoefficientscoefficientsum_body_start + S (0) = S ((S (0)) * fs_v_pfc_working_euclidean_definition_left_productcoefficientscoefficientsum)) /\ exists fs_q_pfc_working_euclidean_definition_left_productcoefficientscoefficientsum_body_start. fs_u_pfc_working_euclidean_definition_left_productcoefficientscoefficientsum = fs_q_pfc_working_euclidean_definition_left_productcoefficientscoefficientsum_body_start * S ((S (0)) * fs_v_pfc_working_euclidean_definition_left_productcoefficientscoefficientsum) + (0))) /\ ((((exists fs_h_pfc_working_euclidean_definition_left_productcoefficientscoefficientsum_body_terminal. fs_h_pfc_working_euclidean_definition_left_productcoefficientscoefficientsum_body_terminal + S (pfc_natural_sum_working_euclidean_definition_left_productcoefficientscoefficient) = S ((S (S (pfc_index_working_euclidean_definition_left_productcoefficients))) * fs_v_pfc_working_euclidean_definition_left_productcoefficientscoefficientsum)) /\ exists fs_q_pfc_working_euclidean_definition_left_productcoefficientscoefficientsum_body_terminal. fs_u_pfc_working_euclidean_definition_left_productcoefficientscoefficientsum = fs_q_pfc_working_euclidean_definition_left_productcoefficientscoefficientsum_body_terminal * S ((S (S (pfc_index_working_euclidean_definition_left_productcoefficients))) * fs_v_pfc_working_euclidean_definition_left_productcoefficientscoefficientsum) + (pfc_natural_sum_working_euclidean_definition_left_productcoefficientscoefficient))) /\ forall fs_i_pfc_working_euclidean_definition_left_productcoefficientscoefficientsum_body_steps. (exists fs_lt_pfc_working_euclidean_definition_left_productcoefficientscoefficientsum_body_steps_bound. fs_lt_pfc_working_euclidean_definition_left_productcoefficientscoefficientsum_body_steps_bound + S fs_i_pfc_working_euclidean_definition_left_productcoefficientscoefficientsum_body_steps = S (pfc_index_working_euclidean_definition_left_productcoefficients)) -> exists fs_a_pfc_working_euclidean_definition_left_productcoefficientscoefficientsum_body_steps fs_r_pfc_working_euclidean_definition_left_productcoefficientscoefficientsum_body_steps fs_s_pfc_working_euclidean_definition_left_productcoefficientscoefficientsum_body_steps. ((((exists fs_h_pfc_working_euclidean_definition_left_productcoefficientscoefficientsum_body_steps_summand. fs_h_pfc_working_euclidean_definition_left_productcoefficientscoefficientsum_body_steps_summand + S (fs_a_pfc_working_euclidean_definition_left_productcoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_working_euclidean_definition_left_productcoefficientscoefficientsum_body_steps)) * pfc_terms_scale_working_euclidean_definition_left_productcoefficientscoefficient)) /\ exists fs_q_pfc_working_euclidean_definition_left_productcoefficientscoefficientsum_body_steps_summand. pfc_terms_code_working_euclidean_definition_left_productcoefficientscoefficient = fs_q_pfc_working_euclidean_definition_left_productcoefficientscoefficientsum_body_steps_summand * S ((S (fs_i_pfc_working_euclidean_definition_left_productcoefficientscoefficientsum_body_steps)) * pfc_terms_scale_working_euclidean_definition_left_productcoefficientscoefficient) + (fs_a_pfc_working_euclidean_definition_left_productcoefficientscoefficientsum_body_steps))) /\ ((((exists fs_h_pfc_working_euclidean_definition_left_productcoefficientscoefficientsum_body_steps_partial. fs_h_pfc_working_euclidean_definition_left_productcoefficientscoefficientsum_body_steps_partial + S (fs_r_pfc_working_euclidean_definition_left_productcoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_working_euclidean_definition_left_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_working_euclidean_definition_left_productcoefficientscoefficientsum)) /\ exists fs_q_pfc_working_euclidean_definition_left_productcoefficientscoefficientsum_body_steps_partial. fs_u_pfc_working_euclidean_definition_left_productcoefficientscoefficientsum = fs_q_pfc_working_euclidean_definition_left_productcoefficientscoefficientsum_body_steps_partial * S ((S (fs_i_pfc_working_euclidean_definition_left_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_working_euclidean_definition_left_productcoefficientscoefficientsum) + (fs_r_pfc_working_euclidean_definition_left_productcoefficientscoefficientsum_body_steps))) /\ ((((exists fs_h_pfc_working_euclidean_definition_left_productcoefficientscoefficientsum_body_steps_successor. fs_h_pfc_working_euclidean_definition_left_productcoefficientscoefficientsum_body_steps_successor + S (fs_s_pfc_working_euclidean_definition_left_productcoefficientscoefficientsum_body_steps) = S ((S (S fs_i_pfc_working_euclidean_definition_left_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_working_euclidean_definition_left_productcoefficientscoefficientsum)) /\ exists fs_q_pfc_working_euclidean_definition_left_productcoefficientscoefficientsum_body_steps_successor. fs_u_pfc_working_euclidean_definition_left_productcoefficientscoefficientsum = fs_q_pfc_working_euclidean_definition_left_productcoefficientscoefficientsum_body_steps_successor * S ((S (S fs_i_pfc_working_euclidean_definition_left_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_working_euclidean_definition_left_productcoefficientscoefficientsum) + (fs_s_pfc_working_euclidean_definition_left_productcoefficientscoefficientsum_body_steps))) /\ fs_s_pfc_working_euclidean_definition_left_productcoefficientscoefficientsum_body_steps = fs_r_pfc_working_euclidean_definition_left_productcoefficientscoefficientsum_body_steps + fs_a_pfc_working_euclidean_definition_left_productcoefficientscoefficientsum_body_steps)))))) /\ ((((exists pfa_gap_working_euclidean_definition_left_productcoefficientscoefficientresiduebound. pfa_gap_working_euclidean_definition_left_productcoefficientscoefficientresiduebound + S (pfc_value_working_euclidean_definition_left_productcoefficients) = ((p))) /\ ((exists pfa_offset_left_working_euclidean_definition_left_productcoefficientscoefficientresiduecongruence pfa_offset_right_working_euclidean_definition_left_productcoefficientscoefficientresiduecongruence. (pfc_natural_sum_working_euclidean_definition_left_productcoefficientscoefficient) + ((p)) * pfa_offset_left_working_euclidean_definition_left_productcoefficientscoefficientresiduecongruence = (pfc_value_working_euclidean_definition_left_productcoefficients) + ((p)) * pfa_offset_right_working_euclidean_definition_left_productcoefficientscoefficientresiduecongruence))))))))))))))))))) /\ (((((forall fom_index_pfp_working_euclidean_definition_right_productleft. (exists fom_gap_pfp_working_euclidean_definition_right_productleft_index_bound. fom_gap_pfp_working_euclidean_definition_right_productleft_index_bound + S (fom_index_pfp_working_euclidean_definition_right_productleft) = (V)) -> exists fom_value_pfp_working_euclidean_definition_right_productleft. ((((exists fom_beta_height_pfp_working_euclidean_definition_right_productleft_entry. fom_beta_height_pfp_working_euclidean_definition_right_productleft_entry + S (fom_value_pfp_working_euclidean_definition_right_productleft) = S ((S (fom_index_pfp_working_euclidean_definition_right_productleft)) * (vc))) /\ exists fom_beta_quotient_pfp_working_euclidean_definition_right_productleft_entry. (vb) = fom_beta_quotient_pfp_working_euclidean_definition_right_productleft_entry * S ((S (fom_index_pfp_working_euclidean_definition_right_productleft)) * (vc)) + (fom_value_pfp_working_euclidean_definition_right_productleft))) /\ (exists fom_gap_pfp_working_euclidean_definition_right_productleft_value_bound. fom_gap_pfp_working_euclidean_definition_right_productleft_value_bound + S (fom_value_pfp_working_euclidean_definition_right_productleft) = (p)))) /\ (((forall fom_index_pfp_working_euclidean_definition_right_productright. (exists fom_gap_pfp_working_euclidean_definition_right_productright_index_bound. fom_gap_pfp_working_euclidean_definition_right_productright_index_bound + S (fom_index_pfp_working_euclidean_definition_right_productright) = (B)) -> exists fom_value_pfp_working_euclidean_definition_right_productright. ((((exists fom_beta_height_pfp_working_euclidean_definition_right_productright_entry. fom_beta_height_pfp_working_euclidean_definition_right_productright_entry + S (fom_value_pfp_working_euclidean_definition_right_productright) = S ((S (fom_index_pfp_working_euclidean_definition_right_productright)) * (bc))) /\ exists fom_beta_quotient_pfp_working_euclidean_definition_right_productright_entry. (bb) = fom_beta_quotient_pfp_working_euclidean_definition_right_productright_entry * S ((S (fom_index_pfp_working_euclidean_definition_right_productright)) * (bc)) + (fom_value_pfp_working_euclidean_definition_right_productright))) /\ (exists fom_gap_pfp_working_euclidean_definition_right_productright_value_bound. fom_gap_pfp_working_euclidean_definition_right_productright_value_bound + S (fom_value_pfp_working_euclidean_definition_right_productright) = (p)))) /\ ((((((((V))=0 \/ ((B))=0) /\ (((pfbz_right_length_working_euclidean_definition)=0)))) \/ (((~(((V))=0)) /\ (((~(((B))=0)) /\ ((((V))+((B))=S (pfbz_right_length_working_euclidean_definition)))))))) /\ ((forall pfc_index_working_euclidean_definition_right_productcoefficients. (exists pfa_gap_working_euclidean_definition_right_productcoefficientsbound. pfa_gap_working_euclidean_definition_right_productcoefficientsbound + S (pfc_index_working_euclidean_definition_right_productcoefficients) = (pfbz_right_length_working_euclidean_definition)) -> exists pfc_value_working_euclidean_definition_right_productcoefficients. ((((exists ff_h_pfp_working_euclidean_definition_right_productcoefficientsentry. ff_h_pfp_working_euclidean_definition_right_productcoefficientsentry + S (pfc_value_working_euclidean_definition_right_productcoefficients) = S ((S (pfc_index_working_euclidean_definition_right_productcoefficients)) * pfbz_right_scale_working_euclidean_definition)) /\ exists ff_q_pfp_working_euclidean_definition_right_productcoefficientsentry. pfbz_right_code_working_euclidean_definition = ff_q_pfp_working_euclidean_definition_right_productcoefficientsentry * S ((S (pfc_index_working_euclidean_definition_right_productcoefficients)) * pfbz_right_scale_working_euclidean_definition) + (pfc_value_working_euclidean_definition_right_productcoefficients))) /\ ((exists pfc_terms_code_working_euclidean_definition_right_productcoefficientscoefficient pfc_terms_scale_working_euclidean_definition_right_productcoefficientscoefficient pfc_natural_sum_working_euclidean_definition_right_productcoefficientscoefficient. ((forall pfc_index_working_euclidean_definition_right_productcoefficientscoefficientdiagonal. (exists pfa_gap_working_euclidean_definition_right_productcoefficientscoefficientdiagonalbound. pfa_gap_working_euclidean_definition_right_productcoefficientscoefficientdiagonalbound + S (pfc_index_working_euclidean_definition_right_productcoefficientscoefficientdiagonal) = (S (pfc_index_working_euclidean_definition_right_productcoefficients))) -> exists pfc_value_working_euclidean_definition_right_productcoefficientscoefficientdiagonal. ((((exists ff_h_pfp_working_euclidean_definition_right_productcoefficientscoefficientdiagonalentry. ff_h_pfp_working_euclidean_definition_right_productcoefficientscoefficientdiagonalentry + S (pfc_value_working_euclidean_definition_right_productcoefficientscoefficientdiagonal) = S ((S (pfc_index_working_euclidean_definition_right_productcoefficientscoefficientdiagonal)) * pfc_terms_scale_working_euclidean_definition_right_productcoefficientscoefficient)) /\ exists ff_q_pfp_working_euclidean_definition_right_productcoefficientscoefficientdiagonalentry. pfc_terms_code_working_euclidean_definition_right_productcoefficientscoefficient = ff_q_pfp_working_euclidean_definition_right_productcoefficientscoefficientdiagonalentry * S ((S (pfc_index_working_euclidean_definition_right_productcoefficientscoefficientdiagonal)) * pfc_terms_scale_working_euclidean_definition_right_productcoefficientscoefficient) + (pfc_value_working_euclidean_definition_right_productcoefficientscoefficientdiagonal))) /\ ((exists pfc_complement_working_euclidean_definition_right_productcoefficientscoefficientdiagonalterm pfc_left_working_euclidean_definition_right_productcoefficientscoefficientdiagonalterm pfc_right_working_euclidean_definition_right_productcoefficientscoefficientdiagonalterm. (((pfc_index_working_euclidean_definition_right_productcoefficientscoefficientdiagonal)+pfc_complement_working_euclidean_definition_right_productcoefficientscoefficientdiagonalterm=(pfc_index_working_euclidean_definition_right_productcoefficients)) /\ ((((((exists pfa_gap_working_euclidean_definition_right_productcoefficientscoefficientdiagonaltermleftinside. pfa_gap_working_euclidean_definition_right_productcoefficientscoefficientdiagonaltermleftinside + S (pfc_index_working_euclidean_definition_right_productcoefficientscoefficientdiagonal) = ((V))) /\ ((((exists ff_h_pfp_working_euclidean_definition_right_productcoefficientscoefficientdiagonaltermleftentry. ff_h_pfp_working_euclidean_definition_right_productcoefficientscoefficientdiagonaltermleftentry + S (pfc_left_working_euclidean_definition_right_productcoefficientscoefficientdiagonalterm) = S ((S (pfc_index_working_euclidean_definition_right_productcoefficientscoefficientdiagonal)) * (vc))) /\ exists ff_q_pfp_working_euclidean_definition_right_productcoefficientscoefficientdiagonaltermleftentry. (vb) = ff_q_pfp_working_euclidean_definition_right_productcoefficientscoefficientdiagonaltermleftentry * S ((S (pfc_index_working_euclidean_definition_right_productcoefficientscoefficientdiagonal)) * (vc)) + (pfc_left_working_euclidean_definition_right_productcoefficientscoefficientdiagonalterm)))))) \/ (((exists pfc_gap_working_euclidean_definition_right_productcoefficientscoefficientdiagonaltermleftoutside. pfc_gap_working_euclidean_definition_right_productcoefficientscoefficientdiagonaltermleftoutside+((V))=(pfc_index_working_euclidean_definition_right_productcoefficientscoefficientdiagonal)) /\ (((pfc_left_working_euclidean_definition_right_productcoefficientscoefficientdiagonalterm)=0))))) /\ ((((((exists pfa_gap_working_euclidean_definition_right_productcoefficientscoefficientdiagonaltermrightinside. pfa_gap_working_euclidean_definition_right_productcoefficientscoefficientdiagonaltermrightinside + S (pfc_complement_working_euclidean_definition_right_productcoefficientscoefficientdiagonalterm) = ((B))) /\ ((((exists ff_h_pfp_working_euclidean_definition_right_productcoefficientscoefficientdiagonaltermrightentry. ff_h_pfp_working_euclidean_definition_right_productcoefficientscoefficientdiagonaltermrightentry + S (pfc_right_working_euclidean_definition_right_productcoefficientscoefficientdiagonalterm) = S ((S (pfc_complement_working_euclidean_definition_right_productcoefficientscoefficientdiagonalterm)) * (bc))) /\ exists ff_q_pfp_working_euclidean_definition_right_productcoefficientscoefficientdiagonaltermrightentry. (bb) = ff_q_pfp_working_euclidean_definition_right_productcoefficientscoefficientdiagonaltermrightentry * S ((S (pfc_complement_working_euclidean_definition_right_productcoefficientscoefficientdiagonalterm)) * (bc)) + (pfc_right_working_euclidean_definition_right_productcoefficientscoefficientdiagonalterm)))))) \/ (((exists pfc_gap_working_euclidean_definition_right_productcoefficientscoefficientdiagonaltermrightoutside. pfc_gap_working_euclidean_definition_right_productcoefficientscoefficientdiagonaltermrightoutside+((B))=(pfc_complement_working_euclidean_definition_right_productcoefficientscoefficientdiagonalterm)) /\ (((pfc_right_working_euclidean_definition_right_productcoefficientscoefficientdiagonalterm)=0))))) /\ (((pfc_value_working_euclidean_definition_right_productcoefficientscoefficientdiagonal)=pfc_left_working_euclidean_definition_right_productcoefficientscoefficientdiagonalterm*pfc_right_working_euclidean_definition_right_productcoefficientscoefficientdiagonalterm))))))))))) /\ (((exists fs_u_pfc_working_euclidean_definition_right_productcoefficientscoefficientsum fs_v_pfc_working_euclidean_definition_right_productcoefficientscoefficientsum. ((((exists fs_h_pfc_working_euclidean_definition_right_productcoefficientscoefficientsum_body_start. fs_h_pfc_working_euclidean_definition_right_productcoefficientscoefficientsum_body_start + S (0) = S ((S (0)) * fs_v_pfc_working_euclidean_definition_right_productcoefficientscoefficientsum)) /\ exists fs_q_pfc_working_euclidean_definition_right_productcoefficientscoefficientsum_body_start. fs_u_pfc_working_euclidean_definition_right_productcoefficientscoefficientsum = fs_q_pfc_working_euclidean_definition_right_productcoefficientscoefficientsum_body_start * S ((S (0)) * fs_v_pfc_working_euclidean_definition_right_productcoefficientscoefficientsum) + (0))) /\ ((((exists fs_h_pfc_working_euclidean_definition_right_productcoefficientscoefficientsum_body_terminal. fs_h_pfc_working_euclidean_definition_right_productcoefficientscoefficientsum_body_terminal + S (pfc_natural_sum_working_euclidean_definition_right_productcoefficientscoefficient) = S ((S (S (pfc_index_working_euclidean_definition_right_productcoefficients))) * fs_v_pfc_working_euclidean_definition_right_productcoefficientscoefficientsum)) /\ exists fs_q_pfc_working_euclidean_definition_right_productcoefficientscoefficientsum_body_terminal. fs_u_pfc_working_euclidean_definition_right_productcoefficientscoefficientsum = fs_q_pfc_working_euclidean_definition_right_productcoefficientscoefficientsum_body_terminal * S ((S (S (pfc_index_working_euclidean_definition_right_productcoefficients))) * fs_v_pfc_working_euclidean_definition_right_productcoefficientscoefficientsum) + (pfc_natural_sum_working_euclidean_definition_right_productcoefficientscoefficient))) /\ forall fs_i_pfc_working_euclidean_definition_right_productcoefficientscoefficientsum_body_steps. (exists fs_lt_pfc_working_euclidean_definition_right_productcoefficientscoefficientsum_body_steps_bound. fs_lt_pfc_working_euclidean_definition_right_productcoefficientscoefficientsum_body_steps_bound + S fs_i_pfc_working_euclidean_definition_right_productcoefficientscoefficientsum_body_steps = S (pfc_index_working_euclidean_definition_right_productcoefficients)) -> exists fs_a_pfc_working_euclidean_definition_right_productcoefficientscoefficientsum_body_steps fs_r_pfc_working_euclidean_definition_right_productcoefficientscoefficientsum_body_steps fs_s_pfc_working_euclidean_definition_right_productcoefficientscoefficientsum_body_steps. ((((exists fs_h_pfc_working_euclidean_definition_right_productcoefficientscoefficientsum_body_steps_summand. fs_h_pfc_working_euclidean_definition_right_productcoefficientscoefficientsum_body_steps_summand + S (fs_a_pfc_working_euclidean_definition_right_productcoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_working_euclidean_definition_right_productcoefficientscoefficientsum_body_steps)) * pfc_terms_scale_working_euclidean_definition_right_productcoefficientscoefficient)) /\ exists fs_q_pfc_working_euclidean_definition_right_productcoefficientscoefficientsum_body_steps_summand. pfc_terms_code_working_euclidean_definition_right_productcoefficientscoefficient = fs_q_pfc_working_euclidean_definition_right_productcoefficientscoefficientsum_body_steps_summand * S ((S (fs_i_pfc_working_euclidean_definition_right_productcoefficientscoefficientsum_body_steps)) * pfc_terms_scale_working_euclidean_definition_right_productcoefficientscoefficient) + (fs_a_pfc_working_euclidean_definition_right_productcoefficientscoefficientsum_body_steps))) /\ ((((exists fs_h_pfc_working_euclidean_definition_right_productcoefficientscoefficientsum_body_steps_partial. fs_h_pfc_working_euclidean_definition_right_productcoefficientscoefficientsum_body_steps_partial + S (fs_r_pfc_working_euclidean_definition_right_productcoefficientscoefficientsum_body_steps) = S ((S (fs_i_pfc_working_euclidean_definition_right_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_working_euclidean_definition_right_productcoefficientscoefficientsum)) /\ exists fs_q_pfc_working_euclidean_definition_right_productcoefficientscoefficientsum_body_steps_partial. fs_u_pfc_working_euclidean_definition_right_productcoefficientscoefficientsum = fs_q_pfc_working_euclidean_definition_right_productcoefficientscoefficientsum_body_steps_partial * S ((S (fs_i_pfc_working_euclidean_definition_right_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_working_euclidean_definition_right_productcoefficientscoefficientsum) + (fs_r_pfc_working_euclidean_definition_right_productcoefficientscoefficientsum_body_steps))) /\ ((((exists fs_h_pfc_working_euclidean_definition_right_productcoefficientscoefficientsum_body_steps_successor. fs_h_pfc_working_euclidean_definition_right_productcoefficientscoefficientsum_body_steps_successor + S (fs_s_pfc_working_euclidean_definition_right_productcoefficientscoefficientsum_body_steps) = S ((S (S fs_i_pfc_working_euclidean_definition_right_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_working_euclidean_definition_right_productcoefficientscoefficientsum)) /\ exists fs_q_pfc_working_euclidean_definition_right_productcoefficientscoefficientsum_body_steps_successor. fs_u_pfc_working_euclidean_definition_right_productcoefficientscoefficientsum = fs_q_pfc_working_euclidean_definition_right_productcoefficientscoefficientsum_body_steps_successor * S ((S (S fs_i_pfc_working_euclidean_definition_right_productcoefficientscoefficientsum_body_steps)) * fs_v_pfc_working_euclidean_definition_right_productcoefficientscoefficientsum) + (fs_s_pfc_working_euclidean_definition_right_productcoefficientscoefficientsum_body_steps))) /\ fs_s_pfc_working_euclidean_definition_right_productcoefficientscoefficientsum_body_steps = fs_r_pfc_working_euclidean_definition_right_productcoefficientscoefficientsum_body_steps + fs_a_pfc_working_euclidean_definition_right_productcoefficientscoefficientsum_body_steps)))))) /\ ((((exists pfa_gap_working_euclidean_definition_right_productcoefficientscoefficientresiduebound. pfa_gap_working_euclidean_definition_right_productcoefficientscoefficientresiduebound + S (pfc_value_working_euclidean_definition_right_productcoefficients) = ((p))) /\ ((exists pfa_offset_left_working_euclidean_definition_right_productcoefficientscoefficientresiduecongruence pfa_offset_right_working_euclidean_definition_right_productcoefficientscoefficientresiduecongruence. (pfc_natural_sum_working_euclidean_definition_right_productcoefficientscoefficient) + ((p)) * pfa_offset_left_working_euclidean_definition_right_productcoefficientscoefficientresiduecongruence = (pfc_value_working_euclidean_definition_right_productcoefficients) + ((p)) * pfa_offset_right_working_euclidean_definition_right_productcoefficientscoefficientresiduecongruence))))))))))))))))))) /\ ((((forall fom_index_pfp_working_euclidean_definition_sum_left_bounded. (exists fom_gap_pfp_working_euclidean_definition_sum_left_bounded_index_bound. fom_gap_pfp_working_euclidean_definition_sum_left_bounded_index_bound + S (fom_index_pfp_working_euclidean_definition_sum_left_bounded) = pfbz_left_length_working_euclidean_definition) -> exists fom_value_pfp_working_euclidean_definition_sum_left_bounded. ((((exists fom_beta_height_pfp_working_euclidean_definition_sum_left_bounded_entry. fom_beta_height_pfp_working_euclidean_definition_sum_left_bounded_entry + S (fom_value_pfp_working_euclidean_definition_sum_left_bounded) = S ((S (fom_index_pfp_working_euclidean_definition_sum_left_bounded)) * pfbz_left_scale_working_euclidean_definition)) /\ exists fom_beta_quotient_pfp_working_euclidean_definition_sum_left_bounded_entry. pfbz_left_code_working_euclidean_definition = fom_beta_quotient_pfp_working_euclidean_definition_sum_left_bounded_entry * S ((S (fom_index_pfp_working_euclidean_definition_sum_left_bounded)) * pfbz_left_scale_working_euclidean_definition) + (fom_value_pfp_working_euclidean_definition_sum_left_bounded))) /\ (exists fom_gap_pfp_working_euclidean_definition_sum_left_bounded_value_bound. fom_gap_pfp_working_euclidean_definition_sum_left_bounded_value_bound + S (fom_value_pfp_working_euclidean_definition_sum_left_bounded) = (p)))) /\ (((forall fom_index_pfp_working_euclidean_definition_sum_right_bounded. (exists fom_gap_pfp_working_euclidean_definition_sum_right_bounded_index_bound. fom_gap_pfp_working_euclidean_definition_sum_right_bounded_index_bound + S (fom_index_pfp_working_euclidean_definition_sum_right_bounded) = pfbz_right_length_working_euclidean_definition) -> exists fom_value_pfp_working_euclidean_definition_sum_right_bounded. ((((exists fom_beta_height_pfp_working_euclidean_definition_sum_right_bounded_entry. fom_beta_height_pfp_working_euclidean_definition_sum_right_bounded_entry + S (fom_value_pfp_working_euclidean_definition_sum_right_bounded) = S ((S (fom_index_pfp_working_euclidean_definition_sum_right_bounded)) * pfbz_right_scale_working_euclidean_definition)) /\ exists fom_beta_quotient_pfp_working_euclidean_definition_sum_right_bounded_entry. pfbz_right_code_working_euclidean_definition = fom_beta_quotient_pfp_working_euclidean_definition_sum_right_bounded_entry * S ((S (fom_index_pfp_working_euclidean_definition_sum_right_bounded)) * pfbz_right_scale_working_euclidean_definition) + (fom_value_pfp_working_euclidean_definition_sum_right_bounded))) /\ (exists fom_gap_pfp_working_euclidean_definition_sum_right_bounded_value_bound. fom_gap_pfp_working_euclidean_definition_sum_right_bounded_value_bound + S (fom_value_pfp_working_euclidean_definition_sum_right_bounded) = (p)))) /\ (((forall fom_index_pfp_working_euclidean_definition_sum_result_bounded. (exists fom_gap_pfp_working_euclidean_definition_sum_result_bounded_index_bound. fom_gap_pfp_working_euclidean_definition_sum_result_bounded_index_bound + S (fom_index_pfp_working_euclidean_definition_sum_result_bounded) = (G)) -> exists fom_value_pfp_working_euclidean_definition_sum_result_bounded. ((((exists fom_beta_height_pfp_working_euclidean_definition_sum_result_bounded_entry. fom_beta_height_pfp_working_euclidean_definition_sum_result_bounded_entry + S (fom_value_pfp_working_euclidean_definition_sum_result_bounded) = S ((S (fom_index_pfp_working_euclidean_definition_sum_result_bounded)) * (gc))) /\ exists fom_beta_quotient_pfp_working_euclidean_definition_sum_result_bounded_entry. (gb) = fom_beta_quotient_pfp_working_euclidean_definition_sum_result_bounded_entry * S ((S (fom_index_pfp_working_euclidean_definition_sum_result_bounded)) * (gc)) + (fom_value_pfp_working_euclidean_definition_sum_result_bounded))) /\ (exists fom_gap_pfp_working_euclidean_definition_sum_result_bounded_value_bound. fom_gap_pfp_working_euclidean_definition_sum_result_bounded_value_bound + S (fom_value_pfp_working_euclidean_definition_sum_result_bounded) = (p)))) /\ ((exists pfaa_left_b_working_euclidean_definition_sum pfaa_left_c_working_euclidean_definition_sum pfaa_right_b_working_euclidean_definition_sum pfaa_right_c_working_euclidean_definition_sum pfaa_sum_b_working_euclidean_definition_sum pfaa_sum_c_working_euclidean_definition_sum pfaa_length_working_euclidean_definition_sum. ((((forall pfrep_power_working_euclidean_definition_sum_witness_common_left pfrep_left_working_euclidean_definition_sum_witness_common_left pfrep_right_working_euclidean_definition_sum_witness_common_left. ((exists pfrep_position_working_euclidean_definition_sum_witness_common_leftfirst. ((pfrep_position_working_euclidean_definition_sum_witness_common_leftfirst+S (pfrep_power_working_euclidean_definition_sum_witness_common_left)=(pfbz_left_length_working_euclidean_definition)) /\ ((((exists ff_h_pfp_working_euclidean_definition_sum_witness_common_leftfirstentry. ff_h_pfp_working_euclidean_definition_sum_witness_common_leftfirstentry + S (pfrep_left_working_euclidean_definition_sum_witness_common_left) = S ((S (pfrep_position_working_euclidean_definition_sum_witness_common_leftfirst)) * pfbz_left_scale_working_euclidean_definition)) /\ exists ff_q_pfp_working_euclidean_definition_sum_witness_common_leftfirstentry. pfbz_left_code_working_euclidean_definition = ff_q_pfp_working_euclidean_definition_sum_witness_common_leftfirstentry * S ((S (pfrep_position_working_euclidean_definition_sum_witness_common_leftfirst)) * pfbz_left_scale_working_euclidean_definition) + (pfrep_left_working_euclidean_definition_sum_witness_common_left)))))) \/ (((exists pfrep_gap_working_euclidean_definition_sum_witness_common_leftfirstoutside. pfrep_gap_working_euclidean_definition_sum_witness_common_leftfirstoutside+(pfbz_left_length_working_euclidean_definition)=(pfrep_power_working_euclidean_definition_sum_witness_common_left)) /\ (((pfrep_left_working_euclidean_definition_sum_witness_common_left)=0))))) -> ((exists pfrep_position_working_euclidean_definition_sum_witness_common_leftsecond. ((pfrep_position_working_euclidean_definition_sum_witness_common_leftsecond+S (pfrep_power_working_euclidean_definition_sum_witness_common_left)=(pfaa_length_working_euclidean_definition_sum)) /\ ((((exists ff_h_pfp_working_euclidean_definition_sum_witness_common_leftsecondentry. ff_h_pfp_working_euclidean_definition_sum_witness_common_leftsecondentry + S (pfrep_right_working_euclidean_definition_sum_witness_common_left) = S ((S (pfrep_position_working_euclidean_definition_sum_witness_common_leftsecond)) * pfaa_left_c_working_euclidean_definition_sum)) /\ exists ff_q_pfp_working_euclidean_definition_sum_witness_common_leftsecondentry. pfaa_left_b_working_euclidean_definition_sum = ff_q_pfp_working_euclidean_definition_sum_witness_common_leftsecondentry * S ((S (pfrep_position_working_euclidean_definition_sum_witness_common_leftsecond)) * pfaa_left_c_working_euclidean_definition_sum) + (pfrep_right_working_euclidean_definition_sum_witness_common_left)))))) \/ (((exists pfrep_gap_working_euclidean_definition_sum_witness_common_leftsecondoutside. pfrep_gap_working_euclidean_definition_sum_witness_common_leftsecondoutside+(pfaa_length_working_euclidean_definition_sum)=(pfrep_power_working_euclidean_definition_sum_witness_common_left)) /\ (((pfrep_right_working_euclidean_definition_sum_witness_common_left)=0))))) -> pfrep_left_working_euclidean_definition_sum_witness_common_left=pfrep_right_working_euclidean_definition_sum_witness_common_left) /\ ((forall pfrep_power_working_euclidean_definition_sum_witness_common_right pfrep_left_working_euclidean_definition_sum_witness_common_right pfrep_right_working_euclidean_definition_sum_witness_common_right. ((exists pfrep_position_working_euclidean_definition_sum_witness_common_rightfirst. ((pfrep_position_working_euclidean_definition_sum_witness_common_rightfirst+S (pfrep_power_working_euclidean_definition_sum_witness_common_right)=(pfbz_right_length_working_euclidean_definition)) /\ ((((exists ff_h_pfp_working_euclidean_definition_sum_witness_common_rightfirstentry. ff_h_pfp_working_euclidean_definition_sum_witness_common_rightfirstentry + S (pfrep_left_working_euclidean_definition_sum_witness_common_right) = S ((S (pfrep_position_working_euclidean_definition_sum_witness_common_rightfirst)) * pfbz_right_scale_working_euclidean_definition)) /\ exists ff_q_pfp_working_euclidean_definition_sum_witness_common_rightfirstentry. pfbz_right_code_working_euclidean_definition = ff_q_pfp_working_euclidean_definition_sum_witness_common_rightfirstentry * S ((S (pfrep_position_working_euclidean_definition_sum_witness_common_rightfirst)) * pfbz_right_scale_working_euclidean_definition) + (pfrep_left_working_euclidean_definition_sum_witness_common_right)))))) \/ (((exists pfrep_gap_working_euclidean_definition_sum_witness_common_rightfirstoutside. pfrep_gap_working_euclidean_definition_sum_witness_common_rightfirstoutside+(pfbz_right_length_working_euclidean_definition)=(pfrep_power_working_euclidean_definition_sum_witness_common_right)) /\ (((pfrep_left_working_euclidean_definition_sum_witness_common_right)=0))))) -> ((exists pfrep_position_working_euclidean_definition_sum_witness_common_rightsecond. ((pfrep_position_working_euclidean_definition_sum_witness_common_rightsecond+S (pfrep_power_working_euclidean_definition_sum_witness_common_right)=(pfaa_length_working_euclidean_definition_sum)) /\ ((((exists ff_h_pfp_working_euclidean_definition_sum_witness_common_rightsecondentry. ff_h_pfp_working_euclidean_definition_sum_witness_common_rightsecondentry + S (pfrep_right_working_euclidean_definition_sum_witness_common_right) = S ((S (pfrep_position_working_euclidean_definition_sum_witness_common_rightsecond)) * pfaa_right_c_working_euclidean_definition_sum)) /\ exists ff_q_pfp_working_euclidean_definition_sum_witness_common_rightsecondentry. pfaa_right_b_working_euclidean_definition_sum = ff_q_pfp_working_euclidean_definition_sum_witness_common_rightsecondentry * S ((S (pfrep_position_working_euclidean_definition_sum_witness_common_rightsecond)) * pfaa_right_c_working_euclidean_definition_sum) + (pfrep_right_working_euclidean_definition_sum_witness_common_right)))))) \/ (((exists pfrep_gap_working_euclidean_definition_sum_witness_common_rightsecondoutside. pfrep_gap_working_euclidean_definition_sum_witness_common_rightsecondoutside+(pfaa_length_working_euclidean_definition_sum)=(pfrep_power_working_euclidean_definition_sum_witness_common_right)) /\ (((pfrep_right_working_euclidean_definition_sum_witness_common_right)=0))))) -> pfrep_left_working_euclidean_definition_sum_witness_common_right=pfrep_right_working_euclidean_definition_sum_witness_common_right)))) /\ (((forall pfp_index_working_euclidean_definition_sum_witness_operation. (exists pfa_gap_working_euclidean_definition_sum_witness_operationindex. pfa_gap_working_euclidean_definition_sum_witness_operationindex + S (pfp_index_working_euclidean_definition_sum_witness_operation) = (pfaa_length_working_euclidean_definition_sum)) -> exists pfp_left_working_euclidean_definition_sum_witness_operation pfp_right_working_euclidean_definition_sum_witness_operation pfp_value_working_euclidean_definition_sum_witness_operation. ((((exists ff_h_pfp_working_euclidean_definition_sum_witness_operationleft. ff_h_pfp_working_euclidean_definition_sum_witness_operationleft + S (pfp_left_working_euclidean_definition_sum_witness_operation) = S ((S (pfp_index_working_euclidean_definition_sum_witness_operation)) * pfaa_left_c_working_euclidean_definition_sum)) /\ exists ff_q_pfp_working_euclidean_definition_sum_witness_operationleft. pfaa_left_b_working_euclidean_definition_sum = ff_q_pfp_working_euclidean_definition_sum_witness_operationleft * S ((S (pfp_index_working_euclidean_definition_sum_witness_operation)) * pfaa_left_c_working_euclidean_definition_sum) + (pfp_left_working_euclidean_definition_sum_witness_operation))) /\ (((((exists ff_h_pfp_working_euclidean_definition_sum_witness_operationright. ff_h_pfp_working_euclidean_definition_sum_witness_operationright + S (pfp_right_working_euclidean_definition_sum_witness_operation) = S ((S (pfp_index_working_euclidean_definition_sum_witness_operation)) * pfaa_right_c_working_euclidean_definition_sum)) /\ exists ff_q_pfp_working_euclidean_definition_sum_witness_operationright. pfaa_right_b_working_euclidean_definition_sum = ff_q_pfp_working_euclidean_definition_sum_witness_operationright * S ((S (pfp_index_working_euclidean_definition_sum_witness_operation)) * pfaa_right_c_working_euclidean_definition_sum) + (pfp_right_working_euclidean_definition_sum_witness_operation))) /\ (((((exists ff_h_pfp_working_euclidean_definition_sum_witness_operationtarget. ff_h_pfp_working_euclidean_definition_sum_witness_operationtarget + S (pfp_value_working_euclidean_definition_sum_witness_operation) = S ((S (pfp_index_working_euclidean_definition_sum_witness_operation)) * pfaa_sum_c_working_euclidean_definition_sum)) /\ exists ff_q_pfp_working_euclidean_definition_sum_witness_operationtarget. pfaa_sum_b_working_euclidean_definition_sum = ff_q_pfp_working_euclidean_definition_sum_witness_operationtarget * S ((S (pfp_index_working_euclidean_definition_sum_witness_operation)) * pfaa_sum_c_working_euclidean_definition_sum) + (pfp_value_working_euclidean_definition_sum_witness_operation))) /\ ((((exists pfa_gap_working_euclidean_definition_sum_witness_operationoperationleft. pfa_gap_working_euclidean_definition_sum_witness_operationoperationleft + S (pfp_left_working_euclidean_definition_sum_witness_operation) = ((p))) /\ (((exists pfa_gap_working_euclidean_definition_sum_witness_operationoperationright. pfa_gap_working_euclidean_definition_sum_witness_operationoperationright + S (pfp_right_working_euclidean_definition_sum_witness_operation) = ((p))) /\ ((((exists pfa_gap_working_euclidean_definition_sum_witness_operationoperationresultbound. pfa_gap_working_euclidean_definition_sum_witness_operationoperationresultbound + S (pfp_value_working_euclidean_definition_sum_witness_operation) = ((p))) /\ ((exists pfa_offset_left_working_euclidean_definition_sum_witness_operationoperationresultcongruence pfa_offset_right_working_euclidean_definition_sum_witness_operationoperationresultcongruence. ((pfp_left_working_euclidean_definition_sum_witness_operation) + (pfp_right_working_euclidean_definition_sum_witness_operation)) + ((p)) * pfa_offset_left_working_euclidean_definition_sum_witness_operationoperationresultcongruence = (pfp_value_working_euclidean_definition_sum_witness_operation) + ((p)) * pfa_offset_right_working_euclidean_definition_sum_witness_operationoperationresultcongruence)))))))))))))))) /\ ((forall pfrep_power_working_euclidean_definition_sum_witness_output pfrep_left_working_euclidean_definition_sum_witness_output pfrep_right_working_euclidean_definition_sum_witness_output. ((exists pfrep_position_working_euclidean_definition_sum_witness_outputfirst. ((pfrep_position_working_euclidean_definition_sum_witness_outputfirst+S (pfrep_power_working_euclidean_definition_sum_witness_output)=(pfaa_length_working_euclidean_definition_sum)) /\ ((((exists ff_h_pfp_working_euclidean_definition_sum_witness_outputfirstentry. ff_h_pfp_working_euclidean_definition_sum_witness_outputfirstentry + S (pfrep_left_working_euclidean_definition_sum_witness_output) = S ((S (pfrep_position_working_euclidean_definition_sum_witness_outputfirst)) * pfaa_sum_c_working_euclidean_definition_sum)) /\ exists ff_q_pfp_working_euclidean_definition_sum_witness_outputfirstentry. pfaa_sum_b_working_euclidean_definition_sum = ff_q_pfp_working_euclidean_definition_sum_witness_outputfirstentry * S ((S (pfrep_position_working_euclidean_definition_sum_witness_outputfirst)) * pfaa_sum_c_working_euclidean_definition_sum) + (pfrep_left_working_euclidean_definition_sum_witness_output)))))) \/ (((exists pfrep_gap_working_euclidean_definition_sum_witness_outputfirstoutside. pfrep_gap_working_euclidean_definition_sum_witness_outputfirstoutside+(pfaa_length_working_euclidean_definition_sum)=(pfrep_power_working_euclidean_definition_sum_witness_output)) /\ (((pfrep_left_working_euclidean_definition_sum_witness_output)=0))))) -> ((exists pfrep_position_working_euclidean_definition_sum_witness_outputsecond. ((pfrep_position_working_euclidean_definition_sum_witness_outputsecond+S (pfrep_power_working_euclidean_definition_sum_witness_output)=((G))) /\ ((((exists ff_h_pfp_working_euclidean_definition_sum_witness_outputsecondentry. ff_h_pfp_working_euclidean_definition_sum_witness_outputsecondentry + S (pfrep_right_working_euclidean_definition_sum_witness_output) = S ((S (pfrep_position_working_euclidean_definition_sum_witness_outputsecond)) * (gc))) /\ exists ff_q_pfp_working_euclidean_definition_sum_witness_outputsecondentry. (gb) = ff_q_pfp_working_euclidean_definition_sum_witness_outputsecondentry * S ((S (pfrep_position_working_euclidean_definition_sum_witness_outputsecond)) * (gc)) + (pfrep_right_working_euclidean_definition_sum_witness_output)))))) \/ (((exists pfrep_gap_working_euclidean_definition_sum_witness_outputsecondoutside. pfrep_gap_working_euclidean_definition_sum_witness_outputsecondoutside+((G))=(pfrep_power_working_euclidean_definition_sum_witness_output)) /\ (((pfrep_right_working_euclidean_definition_sum_witness_output)=0))))) -> pfrep_left_working_euclidean_definition_sum_witness_output=pfrep_right_working_euclidean_definition_sum_witness_output)))))))))))))))))

The unchanged native kernel never receives this surface symbol. Binder-safe expansion produces only its existing first-order syntax.