% Mizar problem: t60_quofield,quofield,3195,22 fof(t60_quofield,conjecture,( ! [A] : ( ( ~ v3_struct_0(A) & v3_rlvect_1(A) & v4_rlvect_1(A) & v5_rlvect_1(A) & v6_rlvect_1(A) & v4_group_1(A) & v7_group_1(A) & v6_vectsp_1(A) & v7_vectsp_1(A) & v8_vectsp_1(A) & ~ v10_vectsp_1(A) & v2_vectsp_2(A) & l3_vectsp_1(A) ) => v3_quofield(k24_quofield(A),A,k21_quofield(A)) ) ), inference(mizar_bg_added,[status(thm)],[abstractness_v3_vectsp_1,antisymmetry_r2_hidden,cc1_algstr_1,cc1_gcd_1,cc1_quofield,cc1_relset_1,cc1_vectsp_1,cc2_algstr_1,cc2_gcd_1,cc2_quofield,cc2_vectsp_1,cc3_algstr_1,cc3_gcd_1,cc3_quofield,cc3_vectsp_1,cc4_algstr_1,cc4_gcd_1,cc4_vectsp_1,cc5_gcd_1,cc5_vectsp_1,commutativity_k10_group_1,d13_vectsp_1,d16_quofield,d19_vectsp_1,d1_binop_1,d1_funct_2,d1_group_1,d1_mcart_1,d21_vectsp_1,d23_quofield,d27_quofield,d28_quofield,d2_mcart_1,d2_rlvect_1,d4_quofield,d8_funct_1,dt_g3_vectsp_1,dt_k10_group_1,dt_k12_quofield,dt_k13_quofield,dt_k14_quofield,dt_k17_quofield,dt_k18_quofield,dt_k1_binop_1,dt_k1_domain_1,dt_k1_funct_1,dt_k1_group_1,dt_k1_mcart_1,dt_k1_quofield,dt_k1_relat_1,dt_k1_rlvect_1,dt_k1_xboole_0,dt_k1_zfmisc_1,dt_k21_quofield,dt_k23_quofield,dt_k24_quofield,dt_k2_binop_1,dt_k2_domain_1,dt_k2_group_1,dt_k2_mcart_1,dt_k2_quofield,dt_k2_zfmisc_1,dt_k3_domain_1,dt_k3_quofield,dt_k4_relset_1,dt_k4_tarski,dt_k8_funct_2,dt_k8_quofield,dt_k9_quofield,dt_l1_group_1,dt_l1_rlvect_1,dt_l1_struct_0,dt_l1_vectsp_1,dt_l2_struct_0,dt_l2_vectsp_1,dt_l3_vectsp_1,dt_m1_relset_1,dt_m1_subset_1,dt_m2_relset_1,dt_m2_subset_1,dt_u1_group_1,dt_u1_rlvect_1,dt_u1_struct_0,dt_u1_vectsp_1,dt_u2_struct_0,existence_l1_group_1,existence_l1_rlvect_1,existence_l1_struct_0,existence_l1_vectsp_1,existence_l2_struct_0,existence_l2_vectsp_1,existence_l3_vectsp_1,existence_m1_relset_1,existence_m1_subset_1,existence_m2_relset_1,existence_m2_subset_1,fc1_quofield,fc1_struct_0,fc1_subset_1,fc1_xboole_0,fc2_quofield,fc3_quofield,fc4_quofield,fc4_subset_1,fc5_quofield,fc6_quofield,fc7_quofield,fc8_quofield,free_g3_vectsp_1,rc10_vectsp_1,rc1_gcd_1,rc1_quofield,rc1_subset_1,rc1_xboole_0,rc2_gcd_1,rc2_subset_1,rc2_xboole_0,rc3_gcd_1,rc3_struct_0,rc4_gcd_1,rc4_struct_0,rc5_gcd_1,rc5_struct_0,rc6_vectsp_1,rc7_vectsp_1,rc8_vectsp_1,rc9_vectsp_1,redefinition_k10_group_1,redefinition_k12_quofield,redefinition_k1_domain_1,redefinition_k2_binop_1,redefinition_k2_domain_1,redefinition_k2_quofield,redefinition_k3_domain_1,redefinition_k3_quofield,redefinition_k4_relset_1,redefinition_k8_funct_2,redefinition_m2_relset_1,redefinition_m2_subset_1,reflexivity_r1_tarski,t1_subset,t2_subset,t3_subset,t4_subset,t59_quofield,t5_subset,t6_boole,t6_quofield,t7_boole,t8_boole]), [file(quofield,t60_quofield)]). fof(abstractness_v3_vectsp_1,axiom,( ! [A] : ( l3_vectsp_1(A) => ( v3_vectsp_1(A) => A = g3_vectsp_1(u1_struct_0(A),u1_rlvect_1(A),u1_group_1(A),u1_vectsp_1(A),u2_struct_0(A)) ) ) ), file(vectsp_1,v3_vectsp_1), []). fof(antisymmetry_r2_hidden,axiom,( ! [A,B] : ( r2_hidden(A,B) => ~ r2_hidden(B,A) ) ), file(hidden,r2_hidden), []). fof(cc1_algstr_1,axiom,( ! [A] : ( l1_rlvect_1(A) => ( ( ~ v3_struct_0(A) & v6_algstr_1(A) ) => ( ~ v3_struct_0(A) & v2_algstr_1(A) & v3_algstr_1(A) & v4_algstr_1(A) & v5_algstr_1(A) ) ) ) ), file(algstr_1,cc1_algstr_1), []). fof(cc1_gcd_1,axiom,( ! [A] : ( l1_vectsp_1(A) => ( ( ~ v3_struct_0(A) & v7_group_1(A) & v6_vectsp_1(A) ) => ( ~ v3_struct_0(A) & v8_vectsp_1(A) ) ) ) ), file(gcd_1,cc1_gcd_1), []). fof(cc1_quofield,axiom,( ! [A] : ( l3_vectsp_1(A) => ( ( ~ v3_struct_0(A) & v4_rlvect_1(A) & v5_rlvect_1(A) & v6_rlvect_1(A) & v4_group_1(A) & v7_group_1(A) & v7_vectsp_1(A) & v8_vectsp_1(A) & v9_vectsp_1(A) & ~ v10_vectsp_1(A) ) => ( ~ v3_struct_0(A) & v4_rlvect_1(A) & v5_rlvect_1(A) & v6_rlvect_1(A) & v2_group_1(A) & v4_group_1(A) & v7_group_1(A) & v4_vectsp_1(A) & v5_vectsp_1(A) & v6_vectsp_1(A) & v7_vectsp_1(A) & v8_vectsp_1(A) & v9_vectsp_1(A) & ~ v10_vectsp_1(A) & v2_vectsp_2(A) & v3_gcd_1(A) & v1_algstr_1(A) & v2_algstr_1(A) & v3_algstr_1(A) & v4_algstr_1(A) & v5_algstr_1(A) & v6_algstr_1(A) ) ) ) ), file(quofield,cc1_quofield), []). fof(cc1_relset_1,axiom,( ! [A,B,C] : ( m1_subset_1(C,k1_zfmisc_1(k2_zfmisc_1(A,B))) => v1_relat_1(C) ) ), file(relset_1,cc1_relset_1), []). fof(cc1_vectsp_1,axiom,( ! [A] : ( l3_vectsp_1(A) => ( ( ~ v3_struct_0(A) & v7_vectsp_1(A) ) => ( ~ v3_struct_0(A) & v4_vectsp_1(A) & v5_vectsp_1(A) ) ) ) ), file(vectsp_1,cc1_vectsp_1), []). fof(cc2_algstr_1,axiom,( ! [A] : ( l1_rlvect_1(A) => ( ( ~ v3_struct_0(A) & v2_algstr_1(A) & v3_algstr_1(A) & v4_algstr_1(A) & v5_algstr_1(A) ) => ( ~ v3_struct_0(A) & v6_algstr_1(A) ) ) ) ), file(algstr_1,cc2_algstr_1), []). fof(cc2_gcd_1,axiom,( ! [A] : ( l3_vectsp_1(A) => ( ( ~ v3_struct_0(A) & v7_group_1(A) & v4_vectsp_1(A) ) => ( ~ v3_struct_0(A) & v4_vectsp_1(A) & v5_vectsp_1(A) & v7_vectsp_1(A) ) ) ) ), file(gcd_1,cc2_gcd_1), []). fof(cc2_quofield,axiom,( ! [A,B] : ( ( ~ v3_struct_0(A) & l3_vectsp_1(A) & ~ v3_struct_0(B) & l3_vectsp_1(B) ) => ! [C] : ( m1_relset_1(C,u1_struct_0(A),u1_struct_0(B)) => ( ( v1_funct_1(C) & v1_funct_2(C,u1_struct_0(A),u1_struct_0(B)) & v1_quofield(C,A,B) ) => ( v1_funct_1(C) & v1_funct_2(C,u1_struct_0(A),u1_struct_0(B)) & v1_grcat_1(C,A,B) & v1_endalg(C,A,B) & v1_group_6(C,A,B) ) ) ) ) ), file(quofield,cc2_quofield), []). fof(cc2_vectsp_1,axiom,( ! [A] : ( l3_vectsp_1(A) => ( ( ~ v3_struct_0(A) & v4_vectsp_1(A) & v5_vectsp_1(A) ) => ( ~ v3_struct_0(A) & v7_vectsp_1(A) ) ) ) ), file(vectsp_1,cc2_vectsp_1), []). fof(cc3_algstr_1,axiom,( ! [A] : ( l1_rlvect_1(A) => ( ( ~ v3_struct_0(A) & v6_algstr_1(A) ) => ( ~ v3_struct_0(A) & v6_rlvect_1(A) ) ) ) ), file(algstr_1,cc3_algstr_1), []). fof(cc3_gcd_1,axiom,( ! [A] : ( l3_vectsp_1(A) => ( ( ~ v3_struct_0(A) & v7_group_1(A) & v5_vectsp_1(A) ) => ( ~ v3_struct_0(A) & v4_vectsp_1(A) & v5_vectsp_1(A) & v7_vectsp_1(A) ) ) ) ), file(gcd_1,cc3_gcd_1), []). fof(cc3_quofield,axiom,( ! [A,B] : ( ( ~ v3_struct_0(A) & l3_vectsp_1(A) & ~ v3_struct_0(B) & l3_vectsp_1(B) ) => ! [C] : ( m1_relset_1(C,u1_struct_0(A),u1_struct_0(B)) => ( ( v1_funct_1(C) & v1_funct_2(C,u1_struct_0(A),u1_struct_0(B)) & v1_grcat_1(C,A,B) & v1_endalg(C,A,B) & v1_group_6(C,A,B) ) => ( v1_funct_1(C) & v1_funct_2(C,u1_struct_0(A),u1_struct_0(B)) & v1_quofield(C,A,B) ) ) ) ) ), file(quofield,cc3_quofield), []). fof(cc3_vectsp_1,axiom,( ! [A] : ( l1_vectsp_1(A) => ( ( ~ v3_struct_0(A) & v2_group_1(A) ) => ( ~ v3_struct_0(A) & v6_vectsp_1(A) & v8_vectsp_1(A) ) ) ) ), file(vectsp_1,cc3_vectsp_1), []). fof(cc4_algstr_1,axiom,( ! [A] : ( l1_rlvect_1(A) => ( ( ~ v3_struct_0(A) & v4_rlvect_1(A) & v5_rlvect_1(A) & v6_rlvect_1(A) ) => ( ~ v3_struct_0(A) & v1_algstr_1(A) & v2_algstr_1(A) & v3_algstr_1(A) & v4_algstr_1(A) & v5_algstr_1(A) & v6_algstr_1(A) ) ) ) ), file(algstr_1,cc4_algstr_1), []). fof(cc4_gcd_1,axiom,( ! [A] : ( l3_vectsp_1(A) => ( ( ~ v3_struct_0(A) & v3_rlvect_1(A) & v4_rlvect_1(A) & v5_rlvect_1(A) & v6_rlvect_1(A) & v4_group_1(A) & v6_vectsp_1(A) & v7_vectsp_1(A) & v8_vectsp_1(A) ) => ( ~ v3_struct_0(A) & v3_rlvect_1(A) & v4_rlvect_1(A) & v5_rlvect_1(A) & v6_rlvect_1(A) & v2_group_1(A) & v4_group_1(A) & v4_vectsp_1(A) & v5_vectsp_1(A) & v6_vectsp_1(A) & v7_vectsp_1(A) & v8_vectsp_1(A) ) ) ) ), file(gcd_1,cc4_gcd_1), []). fof(cc4_vectsp_1,axiom,( ! [A] : ( l1_vectsp_1(A) => ( ( ~ v3_struct_0(A) & v6_vectsp_1(A) & v8_vectsp_1(A) ) => ( ~ v3_struct_0(A) & v2_group_1(A) ) ) ) ), file(vectsp_1,cc4_vectsp_1), []). fof(cc5_gcd_1,axiom,( ! [A] : ( l3_vectsp_1(A) => ( ( ~ v3_struct_0(A) & v4_rlvect_1(A) & v5_rlvect_1(A) & v6_rlvect_1(A) & v7_group_1(A) & v4_vectsp_1(A) & v5_vectsp_1(A) & v6_vectsp_1(A) & v8_vectsp_1(A) & v9_vectsp_1(A) ) => ( ~ v3_struct_0(A) & v4_rlvect_1(A) & v5_rlvect_1(A) & v6_rlvect_1(A) & v2_group_1(A) & v7_group_1(A) & v4_vectsp_1(A) & v5_vectsp_1(A) & v6_vectsp_1(A) & v7_vectsp_1(A) & v8_vectsp_1(A) & v9_vectsp_1(A) & v3_gcd_1(A) ) ) ) ), file(gcd_1,cc5_gcd_1), []). fof(cc5_vectsp_1,axiom,( ! [A] : ( l1_vectsp_1(A) => ( ( ~ v3_struct_0(A) & v7_group_1(A) & v8_vectsp_1(A) ) => ( ~ v3_struct_0(A) & v6_vectsp_1(A) ) ) ) ), file(vectsp_1,cc5_vectsp_1), []). fof(commutativity_k10_group_1,axiom,( ! [A,B,C] : ( ( ~ v3_struct_0(A) & v7_group_1(A) & l1_group_1(A) & m1_subset_1(B,u1_struct_0(A)) & m1_subset_1(C,u1_struct_0(A)) ) => k10_group_1(A,B,C) = k10_group_1(A,C,B) ) ), file(group_1,k10_group_1), []). fof(d13_vectsp_1,axiom,( ! [A] : ( ( ~ v3_struct_0(A) & l1_vectsp_1(A) ) => ( v6_vectsp_1(A) <=> ! [B] : ( m1_subset_1(B,u1_struct_0(A)) => k1_group_1(A,B,k2_group_1(A)) = B ) ) ) ), file(vectsp_1,d13_vectsp_1), []). fof(d16_quofield,axiom,( ! [A] : ( ( ~ v3_struct_0(A) & v3_rlvect_1(A) & v4_rlvect_1(A) & v5_rlvect_1(A) & v6_rlvect_1(A) & v4_group_1(A) & v7_group_1(A) & v6_vectsp_1(A) & v7_vectsp_1(A) & v8_vectsp_1(A) & ~ v10_vectsp_1(A) & v2_vectsp_2(A) & l3_vectsp_1(A) ) => k21_quofield(A) = g3_vectsp_1(k9_quofield(A),k17_quofield(A),k18_quofield(A),k14_quofield(A),k13_quofield(A)) ) ), file(quofield,d16_quofield), []). fof(d19_vectsp_1,axiom,( ! [A] : ( ( ~ v3_struct_0(A) & l1_vectsp_1(A) ) => ( v8_vectsp_1(A) <=> ! [B] : ( m1_subset_1(B,u1_struct_0(A)) => k1_group_1(A,k2_group_1(A),B) = B ) ) ) ), file(vectsp_1,d19_vectsp_1), []). fof(d1_binop_1,axiom,( ! [A] : ( ( v1_relat_1(A) & v1_funct_1(A) ) => ! [B,C] : k1_binop_1(A,B,C) = k1_funct_1(A,k4_tarski(B,C)) ) ), file(binop_1,d1_binop_1), []). fof(d1_funct_2,axiom,( ! [A,B,C] : ( m2_relset_1(C,A,B) => ( ( ( B = k1_xboole_0 => A = k1_xboole_0 ) => ( v1_funct_2(C,A,B) <=> A = k4_relset_1(A,B,C) ) ) & ( B = k1_xboole_0 => ( A = k1_xboole_0 | ( v1_funct_2(C,A,B) <=> C = k1_xboole_0 ) ) ) ) ) ), file(funct_2,d1_funct_2), []). fof(d1_group_1,axiom,( ! [A] : ( ( ~ v3_struct_0(A) & l1_group_1(A) ) => ! [B] : ( m1_subset_1(B,u1_struct_0(A)) => ! [C] : ( m1_subset_1(C,u1_struct_0(A)) => k1_group_1(A,B,C) = k2_binop_1(u1_struct_0(A),u1_struct_0(A),u1_struct_0(A),u1_group_1(A),B,C) ) ) ) ), file(group_1,d1_group_1), []). fof(d1_mcart_1,axiom,( ! [A] : ( ? [B,C] : A = k4_tarski(B,C) => ! [B] : ( B = k1_mcart_1(A) <=> ! [C,D] : ( A = k4_tarski(C,D) => B = C ) ) ) ), file(mcart_1,d1_mcart_1), []). fof(d21_vectsp_1,axiom,( ! [A] : ( ( ~ v3_struct_0(A) & l2_vectsp_1(A) ) => ( v10_vectsp_1(A) <=> k1_rlvect_1(A) = k2_group_1(A) ) ) ), file(vectsp_1,d21_vectsp_1), []). fof(d23_quofield,axiom,( ! [A] : ( ( ~ v3_struct_0(A) & l3_vectsp_1(A) ) => ! [B] : ( ( ~ v3_struct_0(B) & l3_vectsp_1(B) ) => ! [C] : ( ( v1_funct_1(C) & v1_funct_2(C,u1_struct_0(A),u1_struct_0(B)) & m2_relset_1(C,u1_struct_0(A),u1_struct_0(B)) ) => ( v3_quofield(C,A,B) <=> ( v1_quofield(C,A,B) & v2_funct_1(C) ) ) ) ) ) ), file(quofield,d23_quofield), []). fof(d27_quofield,axiom,( ! [A] : ( ( ~ v3_struct_0(A) & l2_struct_0(A) ) => ! [B] : ( m1_subset_1(B,u1_struct_0(A)) => ! [C] : ( m1_subset_1(C,u1_struct_0(A)) => ( C != k1_rlvect_1(A) => k23_quofield(A,B,C) = k1_domain_1(u1_struct_0(A),u1_struct_0(A),B,C) ) ) ) ) ), file(quofield,d27_quofield), []). fof(d28_quofield,axiom,( ! [A] : ( ( ~ v3_struct_0(A) & v3_rlvect_1(A) & v4_rlvect_1(A) & v5_rlvect_1(A) & v6_rlvect_1(A) & v4_group_1(A) & v7_group_1(A) & v6_vectsp_1(A) & v7_vectsp_1(A) & v8_vectsp_1(A) & ~ v10_vectsp_1(A) & v2_vectsp_2(A) & l3_vectsp_1(A) ) => ! [B] : ( ( v1_funct_1(B) & v1_funct_2(B,u1_struct_0(A),u1_struct_0(k21_quofield(A))) & m2_relset_1(B,u1_struct_0(A),u1_struct_0(k21_quofield(A))) ) => ( B = k24_quofield(A) <=> ! [C] : ( m1_subset_1(C,u1_struct_0(A)) => k8_funct_2(u1_struct_0(A),u1_struct_0(k21_quofield(A)),B,C) = k12_quofield(A,k23_quofield(A,C,k2_group_1(A))) ) ) ) ) ), file(quofield,d28_quofield), []). fof(d2_mcart_1,axiom,( ! [A] : ( ? [B,C] : A = k4_tarski(B,C) => ! [B] : ( B = k2_mcart_1(A) <=> ! [C,D] : ( A = k4_tarski(C,D) => B = D ) ) ) ), file(mcart_1,d2_mcart_1), []). fof(d2_rlvect_1,axiom,( ! [A] : ( l2_struct_0(A) => k1_rlvect_1(A) = u2_struct_0(A) ) ), file(rlvect_1,d2_rlvect_1), []). fof(d4_quofield,axiom,( ! [A] : ( ( ~ v3_struct_0(A) & ~ v10_vectsp_1(A) & l2_vectsp_1(A) ) => ! [B] : ( m2_subset_1(B,k2_zfmisc_1(u1_struct_0(A),u1_struct_0(A)),k1_quofield(A)) => ! [C] : ( m1_subset_1(C,k1_zfmisc_1(k1_quofield(A))) => ( C = k8_quofield(A,B) <=> ! [D] : ( m2_subset_1(D,k2_zfmisc_1(u1_struct_0(A),u1_struct_0(A)),k1_quofield(A)) => ( r2_hidden(D,C) <=> k1_group_1(A,k2_quofield(A,D),k3_quofield(A,B)) = k1_group_1(A,k3_quofield(A,D),k2_quofield(A,B)) ) ) ) ) ) ) ), file(quofield,d4_quofield), []). fof(d8_funct_1,axiom,( ! [A] : ( ( v1_relat_1(A) & v1_funct_1(A) ) => ( v2_funct_1(A) <=> ! [B,C] : ( ( r2_hidden(B,k1_relat_1(A)) & r2_hidden(C,k1_relat_1(A)) & k1_funct_1(A,B) = k1_funct_1(A,C) ) => B = C ) ) ) ), file(funct_1,d8_funct_1), []). fof(dt_g3_vectsp_1,axiom,( ! [A,B,C,D,E] : ( ( v1_funct_1(B) & v1_funct_2(B,k2_zfmisc_1(A,A),A) & m1_relset_1(B,k2_zfmisc_1(A,A),A) & v1_funct_1(C) & v1_funct_2(C,k2_zfmisc_1(A,A),A) & m1_relset_1(C,k2_zfmisc_1(A,A),A) & m1_subset_1(D,A) & m1_subset_1(E,A) ) => ( v3_vectsp_1(g3_vectsp_1(A,B,C,D,E)) & l3_vectsp_1(g3_vectsp_1(A,B,C,D,E)) ) ) ), file(vectsp_1,g3_vectsp_1), []). fof(dt_k10_group_1,axiom,( ! [A,B,C] : ( ( ~ v3_struct_0(A) & v7_group_1(A) & l1_group_1(A) & m1_subset_1(B,u1_struct_0(A)) & m1_subset_1(C,u1_struct_0(A)) ) => m1_subset_1(k10_group_1(A,B,C),u1_struct_0(A)) ) ), file(group_1,k10_group_1), []). fof(dt_k12_quofield,axiom,( ! [A,B] : ( ( ~ v3_struct_0(A) & ~ v10_vectsp_1(A) & l2_vectsp_1(A) & m1_subset_1(B,k1_quofield(A)) ) => m2_subset_1(k12_quofield(A,B),k1_zfmisc_1(k1_quofield(A)),k9_quofield(A)) ) ), file(quofield,k12_quofield), []). fof(dt_k13_quofield,axiom,( ! [A] : ( ( ~ v3_struct_0(A) & v3_rlvect_1(A) & v4_rlvect_1(A) & v5_rlvect_1(A) & v6_rlvect_1(A) & v4_group_1(A) & v7_group_1(A) & v6_vectsp_1(A) & v7_vectsp_1(A) & v8_vectsp_1(A) & ~ v10_vectsp_1(A) & v2_vectsp_2(A) & l3_vectsp_1(A) ) => m2_subset_1(k13_quofield(A),k1_zfmisc_1(k1_quofield(A)),k9_quofield(A)) ) ), file(quofield,k13_quofield), []). fof(dt_k14_quofield,axiom,( ! [A] : ( ( ~ v3_struct_0(A) & v3_rlvect_1(A) & v4_rlvect_1(A) & v5_rlvect_1(A) & v6_rlvect_1(A) & v4_group_1(A) & v7_group_1(A) & v6_vectsp_1(A) & v7_vectsp_1(A) & v8_vectsp_1(A) & ~ v10_vectsp_1(A) & v2_vectsp_2(A) & l3_vectsp_1(A) ) => m2_subset_1(k14_quofield(A),k1_zfmisc_1(k1_quofield(A)),k9_quofield(A)) ) ), file(quofield,k14_quofield), []). fof(dt_k17_quofield,axiom,( ! [A] : ( ( ~ v3_struct_0(A) & v3_rlvect_1(A) & v4_rlvect_1(A) & v5_rlvect_1(A) & v6_rlvect_1(A) & v4_group_1(A) & v7_group_1(A) & v6_vectsp_1(A) & v7_vectsp_1(A) & v8_vectsp_1(A) & ~ v10_vectsp_1(A) & v2_vectsp_2(A) & l3_vectsp_1(A) ) => ( v1_funct_1(k17_quofield(A)) & v1_funct_2(k17_quofield(A),k2_zfmisc_1(k9_quofield(A),k9_quofield(A)),k9_quofield(A)) & m2_relset_1(k17_quofield(A),k2_zfmisc_1(k9_quofield(A),k9_quofield(A)),k9_quofield(A)) ) ) ), file(quofield,k17_quofield), []). fof(dt_k18_quofield,axiom,( ! [A] : ( ( ~ v3_struct_0(A) & v3_rlvect_1(A) & v4_rlvect_1(A) & v5_rlvect_1(A) & v6_rlvect_1(A) & v4_group_1(A) & v7_group_1(A) & v6_vectsp_1(A) & v7_vectsp_1(A) & v8_vectsp_1(A) & ~ v10_vectsp_1(A) & v2_vectsp_2(A) & l3_vectsp_1(A) ) => ( v1_funct_1(k18_quofield(A)) & v1_funct_2(k18_quofield(A),k2_zfmisc_1(k9_quofield(A),k9_quofield(A)),k9_quofield(A)) & m2_relset_1(k18_quofield(A),k2_zfmisc_1(k9_quofield(A),k9_quofield(A)),k9_quofield(A)) ) ) ), file(quofield,k18_quofield), []). fof(dt_k1_binop_1,axiom,( $true ), file(binop_1,k1_binop_1), []). fof(dt_k1_domain_1,axiom,( ! [A,B,C,D] : ( ( ~ v1_xboole_0(A) & ~ v1_xboole_0(B) & m1_subset_1(C,A) & m1_subset_1(D,B) ) => m1_subset_1(k1_domain_1(A,B,C,D),k2_zfmisc_1(A,B)) ) ), file(domain_1,k1_domain_1), []). fof(dt_k1_funct_1,axiom,( $true ), file(funct_1,k1_funct_1), []). fof(dt_k1_group_1,axiom,( ! [A,B,C] : ( ( ~ v3_struct_0(A) & l1_group_1(A) & m1_subset_1(B,u1_struct_0(A)) & m1_subset_1(C,u1_struct_0(A)) ) => m1_subset_1(k1_group_1(A,B,C),u1_struct_0(A)) ) ), file(group_1,k1_group_1), []). fof(dt_k1_mcart_1,axiom,( $true ), file(mcart_1,k1_mcart_1), []). fof(dt_k1_quofield,axiom,( ! [A] : ( ( ~ v3_struct_0(A) & l2_struct_0(A) ) => m1_subset_1(k1_quofield(A),k1_zfmisc_1(k2_zfmisc_1(u1_struct_0(A),u1_struct_0(A)))) ) ), file(quofield,k1_quofield), []). fof(dt_k1_relat_1,axiom,( $true ), file(relat_1,k1_relat_1), []). fof(dt_k1_rlvect_1,axiom,( ! [A] : ( l2_struct_0(A) => m1_subset_1(k1_rlvect_1(A),u1_struct_0(A)) ) ), file(rlvect_1,k1_rlvect_1), []). fof(dt_k1_xboole_0,axiom,( $true ), file(xboole_0,k1_xboole_0), []). fof(dt_k1_zfmisc_1,axiom,( $true ), file(zfmisc_1,k1_zfmisc_1), []). fof(dt_k21_quofield,axiom,( ! [A] : ( ( ~ v3_struct_0(A) & v3_rlvect_1(A) & v4_rlvect_1(A) & v5_rlvect_1(A) & v6_rlvect_1(A) & v4_group_1(A) & v7_group_1(A) & v6_vectsp_1(A) & v7_vectsp_1(A) & v8_vectsp_1(A) & ~ v10_vectsp_1(A) & v2_vectsp_2(A) & l3_vectsp_1(A) ) => ( v3_vectsp_1(k21_quofield(A)) & l3_vectsp_1(k21_quofield(A)) ) ) ), file(quofield,k21_quofield), []). fof(dt_k23_quofield,axiom,( ! [A,B,C] : ( ( ~ v3_struct_0(A) & l2_struct_0(A) & m1_subset_1(B,u1_struct_0(A)) & m1_subset_1(C,u1_struct_0(A)) ) => m1_subset_1(k23_quofield(A,B,C),k1_quofield(A)) ) ), file(quofield,k23_quofield), []). fof(dt_k24_quofield,axiom,( ! [A] : ( ( ~ v3_struct_0(A) & v3_rlvect_1(A) & v4_rlvect_1(A) & v5_rlvect_1(A) & v6_rlvect_1(A) & v4_group_1(A) & v7_group_1(A) & v6_vectsp_1(A) & v7_vectsp_1(A) & v8_vectsp_1(A) & ~ v10_vectsp_1(A) & v2_vectsp_2(A) & l3_vectsp_1(A) ) => ( v1_funct_1(k24_quofield(A)) & v1_funct_2(k24_quofield(A),u1_struct_0(A),u1_struct_0(k21_quofield(A))) & m2_relset_1(k24_quofield(A),u1_struct_0(A),u1_struct_0(k21_quofield(A))) ) ) ), file(quofield,k24_quofield), []). fof(dt_k2_binop_1,axiom,( ! [A,B,C,D,E,F] : ( ( ~ v1_xboole_0(A) & ~ v1_xboole_0(B) & v1_funct_1(D) & v1_funct_2(D,k2_zfmisc_1(A,B),C) & m1_relset_1(D,k2_zfmisc_1(A,B),C) & m1_subset_1(E,A) & m1_subset_1(F,B) ) => m1_subset_1(k2_binop_1(A,B,C,D,E,F),C) ) ), file(binop_1,k2_binop_1), []). fof(dt_k2_domain_1,axiom,( ! [A,B,C] : ( ( ~ v1_xboole_0(A) & ~ v1_xboole_0(B) & m1_subset_1(C,k2_zfmisc_1(A,B)) ) => m1_subset_1(k2_domain_1(A,B,C),A) ) ), file(domain_1,k2_domain_1), []). fof(dt_k2_group_1,axiom,( ! [A] : ( ( ~ v3_struct_0(A) & l1_group_1(A) ) => m1_subset_1(k2_group_1(A),u1_struct_0(A)) ) ), file(group_1,k2_group_1), []). fof(dt_k2_mcart_1,axiom,( $true ), file(mcart_1,k2_mcart_1), []). fof(dt_k2_quofield,axiom,( ! [A,B] : ( ( ~ v3_struct_0(A) & ~ v10_vectsp_1(A) & l2_vectsp_1(A) & m1_subset_1(B,k1_quofield(A)) ) => m1_subset_1(k2_quofield(A,B),u1_struct_0(A)) ) ), file(quofield,k2_quofield), []). fof(dt_k2_zfmisc_1,axiom,( $true ), file(zfmisc_1,k2_zfmisc_1), []). fof(dt_k3_domain_1,axiom,( ! [A,B,C] : ( ( ~ v1_xboole_0(A) & ~ v1_xboole_0(B) & m1_subset_1(C,k2_zfmisc_1(A,B)) ) => m1_subset_1(k3_domain_1(A,B,C),B) ) ), file(domain_1,k3_domain_1), []). fof(dt_k3_quofield,axiom,( ! [A,B] : ( ( ~ v3_struct_0(A) & ~ v10_vectsp_1(A) & l2_vectsp_1(A) & m1_subset_1(B,k1_quofield(A)) ) => m1_subset_1(k3_quofield(A,B),u1_struct_0(A)) ) ), file(quofield,k3_quofield), []). fof(dt_k4_relset_1,axiom,( ! [A,B,C] : ( m1_relset_1(C,A,B) => m1_subset_1(k4_relset_1(A,B,C),k1_zfmisc_1(A)) ) ), file(relset_1,k4_relset_1), []). fof(dt_k4_tarski,axiom,( $true ), file(tarski,k4_tarski), []). fof(dt_k8_funct_2,axiom,( ! [A,B,C,D] : ( ( ~ v1_xboole_0(A) & v1_funct_1(C) & v1_funct_2(C,A,B) & m1_relset_1(C,A,B) & m1_subset_1(D,A) ) => m1_subset_1(k8_funct_2(A,B,C,D),B) ) ), file(funct_2,k8_funct_2), []). fof(dt_k8_quofield,axiom,( ! [A,B] : ( ( ~ v3_struct_0(A) & ~ v10_vectsp_1(A) & l2_vectsp_1(A) & m1_subset_1(B,k1_quofield(A)) ) => m1_subset_1(k8_quofield(A,B),k1_zfmisc_1(k1_quofield(A))) ) ), file(quofield,k8_quofield), []). fof(dt_k9_quofield,axiom,( ! [A] : ( ( ~ v3_struct_0(A) & ~ v10_vectsp_1(A) & l2_vectsp_1(A) ) => m1_subset_1(k9_quofield(A),k1_zfmisc_1(k1_zfmisc_1(k1_quofield(A)))) ) ), file(quofield,k9_quofield), []). fof(dt_l1_group_1,axiom,( ! [A] : ( l1_group_1(A) => l1_struct_0(A) ) ), file(group_1,l1_group_1), []). fof(dt_l1_rlvect_1,axiom,( ! [A] : ( l1_rlvect_1(A) => l2_struct_0(A) ) ), file(rlvect_1,l1_rlvect_1), []). fof(dt_l1_struct_0,axiom,( $true ), file(struct_0,l1_struct_0), []). fof(dt_l1_vectsp_1,axiom,( ! [A] : ( l1_vectsp_1(A) => l1_group_1(A) ) ), file(vectsp_1,l1_vectsp_1), []). fof(dt_l2_struct_0,axiom,( ! [A] : ( l2_struct_0(A) => l1_struct_0(A) ) ), file(struct_0,l2_struct_0), []). fof(dt_l2_vectsp_1,axiom,( ! [A] : ( l2_vectsp_1(A) => ( l1_vectsp_1(A) & l2_struct_0(A) ) ) ), file(vectsp_1,l2_vectsp_1), []). fof(dt_l3_vectsp_1,axiom,( ! [A] : ( l3_vectsp_1(A) => ( l1_rlvect_1(A) & l2_vectsp_1(A) ) ) ), file(vectsp_1,l3_vectsp_1), []). fof(dt_m1_relset_1,axiom,( $true ), file(relset_1,m1_relset_1), []). fof(dt_m1_subset_1,axiom,( $true ), file(subset_1,m1_subset_1), []). fof(dt_m2_relset_1,axiom,( ! [A,B,C] : ( m2_relset_1(C,A,B) => m1_subset_1(C,k1_zfmisc_1(k2_zfmisc_1(A,B))) ) ), file(relset_1,m2_relset_1), []). fof(dt_m2_subset_1,axiom,( ! [A,B] : ( ( ~ v1_xboole_0(A) & ~ v1_xboole_0(B) & m1_subset_1(B,k1_zfmisc_1(A)) ) => ! [C] : ( m2_subset_1(C,A,B) => m1_subset_1(C,A) ) ) ), file(subset_1,m2_subset_1), []). fof(dt_u1_group_1,axiom,( ! [A] : ( l1_group_1(A) => ( v1_funct_1(u1_group_1(A)) & v1_funct_2(u1_group_1(A),k2_zfmisc_1(u1_struct_0(A),u1_struct_0(A)),u1_struct_0(A)) & m2_relset_1(u1_group_1(A),k2_zfmisc_1(u1_struct_0(A),u1_struct_0(A)),u1_struct_0(A)) ) ) ), file(group_1,u1_group_1), []). fof(dt_u1_rlvect_1,axiom,( ! [A] : ( l1_rlvect_1(A) => ( v1_funct_1(u1_rlvect_1(A)) & v1_funct_2(u1_rlvect_1(A),k2_zfmisc_1(u1_struct_0(A),u1_struct_0(A)),u1_struct_0(A)) & m2_relset_1(u1_rlvect_1(A),k2_zfmisc_1(u1_struct_0(A),u1_struct_0(A)),u1_struct_0(A)) ) ) ), file(rlvect_1,u1_rlvect_1), []). fof(dt_u1_struct_0,axiom,( $true ), file(struct_0,u1_struct_0), []). fof(dt_u1_vectsp_1,axiom,( ! [A] : ( l1_vectsp_1(A) => m1_subset_1(u1_vectsp_1(A),u1_struct_0(A)) ) ), file(vectsp_1,u1_vectsp_1), []). fof(dt_u2_struct_0,axiom,( ! [A] : ( l2_struct_0(A) => m1_subset_1(u2_struct_0(A),u1_struct_0(A)) ) ), file(struct_0,u2_struct_0), []). fof(existence_l1_group_1,axiom,( ? [A] : l1_group_1(A) ), file(group_1,l1_group_1), []). fof(existence_l1_rlvect_1,axiom,( ? [A] : l1_rlvect_1(A) ), file(rlvect_1,l1_rlvect_1), []). fof(existence_l1_struct_0,axiom,( ? [A] : l1_struct_0(A) ), file(struct_0,l1_struct_0), []). fof(existence_l1_vectsp_1,axiom,( ? [A] : l1_vectsp_1(A) ), file(vectsp_1,l1_vectsp_1), []). fof(existence_l2_struct_0,axiom,( ? [A] : l2_struct_0(A) ), file(struct_0,l2_struct_0), []). fof(existence_l2_vectsp_1,axiom,( ? [A] : l2_vectsp_1(A) ), file(vectsp_1,l2_vectsp_1), []). fof(existence_l3_vectsp_1,axiom,( ? [A] : l3_vectsp_1(A) ), file(vectsp_1,l3_vectsp_1), []). fof(existence_m1_relset_1,axiom,( ! [A,B] : ? [C] : m1_relset_1(C,A,B) ), file(relset_1,m1_relset_1), []). fof(existence_m1_subset_1,axiom,( ! [A] : ? [B] : m1_subset_1(B,A) ), file(subset_1,m1_subset_1), []). fof(existence_m2_relset_1,axiom,( ! [A,B] : ? [C] : m2_relset_1(C,A,B) ), file(relset_1,m2_relset_1), []). fof(existence_m2_subset_1,axiom,( ! [A,B] : ( ( ~ v1_xboole_0(A) & ~ v1_xboole_0(B) & m1_subset_1(B,k1_zfmisc_1(A)) ) => ? [C] : m2_subset_1(C,A,B) ) ), file(subset_1,m2_subset_1), []). fof(fc1_quofield,axiom,( ! [A] : ( ( ~ v3_struct_0(A) & ~ v10_vectsp_1(A) & l2_vectsp_1(A) ) => ( ~ v1_xboole_0(k1_quofield(A)) & v1_relat_1(k1_quofield(A)) ) ) ), file(quofield,fc1_quofield), []). fof(fc1_struct_0,axiom,( ! [A] : ( ( ~ v3_struct_0(A) & l1_struct_0(A) ) => ~ v1_xboole_0(u1_struct_0(A)) ) ), file(struct_0,fc1_struct_0), []). fof(fc1_subset_1,axiom,( ! [A] : ~ v1_xboole_0(k1_zfmisc_1(A)) ), file(subset_1,fc1_subset_1), []). fof(fc1_xboole_0,axiom,( v1_xboole_0(k1_xboole_0) ), file(xboole_0,fc1_xboole_0), []). fof(fc2_quofield,axiom,( ! [A,B] : ( ( ~ v3_struct_0(A) & v7_group_1(A) & ~ v10_vectsp_1(A) & l2_vectsp_1(A) & m1_subset_1(B,k1_quofield(A)) ) => ~ v1_xboole_0(k8_quofield(A,B)) ) ), file(quofield,fc2_quofield), []). fof(fc3_quofield,axiom,( ! [A] : ( ( ~ v3_struct_0(A) & ~ v10_vectsp_1(A) & l2_vectsp_1(A) ) => ~ v1_xboole_0(k9_quofield(A)) ) ), file(quofield,fc3_quofield), []). fof(fc4_quofield,axiom,( ! [A] : ( ( ~ v3_struct_0(A) & v3_rlvect_1(A) & v4_rlvect_1(A) & v5_rlvect_1(A) & v6_rlvect_1(A) & v4_group_1(A) & v7_group_1(A) & v6_vectsp_1(A) & v7_vectsp_1(A) & v8_vectsp_1(A) & ~ v10_vectsp_1(A) & v2_vectsp_2(A) & l3_vectsp_1(A) ) => ( ~ v3_struct_0(k21_quofield(A)) & v3_vectsp_1(k21_quofield(A)) ) ) ), file(quofield,fc4_quofield), []). fof(fc4_subset_1,axiom,( ! [A,B] : ( ( ~ v1_xboole_0(A) & ~ v1_xboole_0(B) ) => ~ v1_xboole_0(k2_zfmisc_1(A,B)) ) ), file(subset_1,fc4_subset_1), []). fof(fc5_quofield,axiom,( ! [A] : ( ( ~ v3_struct_0(A) & v3_rlvect_1(A) & v4_rlvect_1(A) & v5_rlvect_1(A) & v6_rlvect_1(A) & v4_group_1(A) & v7_group_1(A) & v6_vectsp_1(A) & v7_vectsp_1(A) & v8_vectsp_1(A) & ~ v10_vectsp_1(A) & v2_vectsp_2(A) & l3_vectsp_1(A) ) => ( ~ v3_struct_0(k21_quofield(A)) & v4_rlvect_1(k21_quofield(A)) & v5_rlvect_1(k21_quofield(A)) & v6_rlvect_1(k21_quofield(A)) & v3_vectsp_1(k21_quofield(A)) & v1_algstr_1(k21_quofield(A)) & v2_algstr_1(k21_quofield(A)) & v3_algstr_1(k21_quofield(A)) & v4_algstr_1(k21_quofield(A)) & v5_algstr_1(k21_quofield(A)) & v6_algstr_1(k21_quofield(A)) ) ) ), file(quofield,fc5_quofield), []). fof(fc6_quofield,axiom,( ! [A] : ( ( ~ v3_struct_0(A) & v3_rlvect_1(A) & v4_rlvect_1(A) & v5_rlvect_1(A) & v6_rlvect_1(A) & v4_group_1(A) & v7_group_1(A) & v6_vectsp_1(A) & v7_vectsp_1(A) & v8_vectsp_1(A) & ~ v10_vectsp_1(A) & v2_vectsp_2(A) & l3_vectsp_1(A) ) => ( ~ v3_struct_0(k21_quofield(A)) & v4_rlvect_1(k21_quofield(A)) & v5_rlvect_1(k21_quofield(A)) & v6_rlvect_1(k21_quofield(A)) & v7_group_1(k21_quofield(A)) & v3_vectsp_1(k21_quofield(A)) & v1_algstr_1(k21_quofield(A)) & v2_algstr_1(k21_quofield(A)) & v3_algstr_1(k21_quofield(A)) & v4_algstr_1(k21_quofield(A)) & v5_algstr_1(k21_quofield(A)) & v6_algstr_1(k21_quofield(A)) ) ) ), file(quofield,fc6_quofield), []). fof(fc7_quofield,axiom,( ! [A] : ( ( ~ v3_struct_0(A) & v3_rlvect_1(A) & v4_rlvect_1(A) & v5_rlvect_1(A) & v6_rlvect_1(A) & v4_group_1(A) & v7_group_1(A) & v6_vectsp_1(A) & v7_vectsp_1(A) & v8_vectsp_1(A) & ~ v10_vectsp_1(A) & v2_vectsp_2(A) & l3_vectsp_1(A) ) => ( ~ v3_struct_0(k21_quofield(A)) & v4_rlvect_1(k21_quofield(A)) & v5_rlvect_1(k21_quofield(A)) & v6_rlvect_1(k21_quofield(A)) & v2_group_1(k21_quofield(A)) & v7_group_1(k21_quofield(A)) & v3_vectsp_1(k21_quofield(A)) & v6_vectsp_1(k21_quofield(A)) & v8_vectsp_1(k21_quofield(A)) & v1_algstr_1(k21_quofield(A)) & v2_algstr_1(k21_quofield(A)) & v3_algstr_1(k21_quofield(A)) & v4_algstr_1(k21_quofield(A)) & v5_algstr_1(k21_quofield(A)) & v6_algstr_1(k21_quofield(A)) ) ) ), file(quofield,fc7_quofield), []). fof(fc8_quofield,axiom,( ! [A] : ( ( ~ v3_struct_0(A) & v3_rlvect_1(A) & v4_rlvect_1(A) & v5_rlvect_1(A) & v6_rlvect_1(A) & v4_group_1(A) & v7_group_1(A) & v6_vectsp_1(A) & v7_vectsp_1(A) & v8_vectsp_1(A) & ~ v10_vectsp_1(A) & v2_vectsp_2(A) & l3_vectsp_1(A) ) => ( ~ v3_struct_0(k21_quofield(A)) & v3_rlvect_1(k21_quofield(A)) & v4_rlvect_1(k21_quofield(A)) & v5_rlvect_1(k21_quofield(A)) & v6_rlvect_1(k21_quofield(A)) & v2_group_1(k21_quofield(A)) & v4_group_1(k21_quofield(A)) & v7_group_1(k21_quofield(A)) & v3_vectsp_1(k21_quofield(A)) & v4_vectsp_1(k21_quofield(A)) & v5_vectsp_1(k21_quofield(A)) & v6_vectsp_1(k21_quofield(A)) & v7_vectsp_1(k21_quofield(A)) & v8_vectsp_1(k21_quofield(A)) & v9_vectsp_1(k21_quofield(A)) & ~ v10_vectsp_1(k21_quofield(A)) & v3_gcd_1(k21_quofield(A)) & v1_algstr_1(k21_quofield(A)) & v2_algstr_1(k21_quofield(A)) & v3_algstr_1(k21_quofield(A)) & v4_algstr_1(k21_quofield(A)) & v5_algstr_1(k21_quofield(A)) & v6_algstr_1(k21_quofield(A)) ) ) ), file(quofield,fc8_quofield), []). fof(free_g3_vectsp_1,axiom,( ! [A,B,C,D,E] : ( ( v1_funct_1(B) & v1_funct_2(B,k2_zfmisc_1(A,A),A) & m1_relset_1(B,k2_zfmisc_1(A,A),A) & v1_funct_1(C) & v1_funct_2(C,k2_zfmisc_1(A,A),A) & m1_relset_1(C,k2_zfmisc_1(A,A),A) & m1_subset_1(D,A) & m1_subset_1(E,A) ) => ! [F,G,H,I,J] : ( g3_vectsp_1(A,B,C,D,E) = g3_vectsp_1(F,G,H,I,J) => ( A = F & B = G & C = H & D = I & E = J ) ) ) ), file(vectsp_1,g3_vectsp_1), []). fof(rc10_vectsp_1,axiom,( ? [A] : ( l3_vectsp_1(A) & ~ v3_struct_0(A) & v3_rlvect_1(A) & v4_rlvect_1(A) & v5_rlvect_1(A) & v6_rlvect_1(A) & v2_group_1(A) & v4_group_1(A) & v7_group_1(A) & v3_vectsp_1(A) & v4_vectsp_1(A) & v5_vectsp_1(A) & v6_vectsp_1(A) & v7_vectsp_1(A) & v8_vectsp_1(A) & v9_vectsp_1(A) & ~ v10_vectsp_1(A) ) ), file(vectsp_1,rc10_vectsp_1), []). fof(rc1_gcd_1,axiom,( ? [A] : ( l3_vectsp_1(A) & ~ v3_struct_0(A) & v3_rlvect_1(A) & v4_rlvect_1(A) & v5_rlvect_1(A) & v6_rlvect_1(A) & v2_group_1(A) & v4_group_1(A) & v7_group_1(A) & v3_vectsp_1(A) & v4_vectsp_1(A) & v5_vectsp_1(A) & v6_vectsp_1(A) & v7_vectsp_1(A) & v8_vectsp_1(A) & v9_vectsp_1(A) & ~ v10_vectsp_1(A) & v2_vectsp_2(A) ) ), file(gcd_1,rc1_gcd_1), []). fof(rc1_quofield,axiom,( ? [A] : ( l3_vectsp_1(A) & ~ v3_struct_0(A) & v3_rlvect_1(A) & v4_rlvect_1(A) & v5_rlvect_1(A) & v6_rlvect_1(A) & v2_group_1(A) & v4_group_1(A) & v7_group_1(A) & v4_vectsp_1(A) & v5_vectsp_1(A) & v6_vectsp_1(A) & v7_vectsp_1(A) & v8_vectsp_1(A) & v9_vectsp_1(A) & ~ v10_vectsp_1(A) & v2_vectsp_2(A) & v3_gcd_1(A) & v1_algstr_1(A) & v2_algstr_1(A) & v3_algstr_1(A) & v4_algstr_1(A) & v5_algstr_1(A) & v6_algstr_1(A) ) ), file(quofield,rc1_quofield), []). fof(rc1_subset_1,axiom,( ! [A] : ( ~ v1_xboole_0(A) => ? [B] : ( m1_subset_1(B,k1_zfmisc_1(A)) & ~ v1_xboole_0(B) ) ) ), file(subset_1,rc1_subset_1), []). fof(rc1_xboole_0,axiom,( ? [A] : v1_xboole_0(A) ), file(xboole_0,rc1_xboole_0), []). fof(rc2_gcd_1,axiom,( ? [A] : ( l3_vectsp_1(A) & ~ v3_struct_0(A) & v3_rlvect_1(A) & v4_rlvect_1(A) & v5_rlvect_1(A) & v6_rlvect_1(A) & v2_group_1(A) & v4_group_1(A) & v7_group_1(A) & v4_vectsp_1(A) & v5_vectsp_1(A) & v6_vectsp_1(A) & v7_vectsp_1(A) & v8_vectsp_1(A) & ~ v10_vectsp_1(A) & v2_vectsp_2(A) & v3_gcd_1(A) ) ), file(gcd_1,rc2_gcd_1), []). fof(rc2_subset_1,axiom,( ! [A] : ? [B] : ( m1_subset_1(B,k1_zfmisc_1(A)) & v1_xboole_0(B) ) ), file(subset_1,rc2_subset_1), []). fof(rc2_xboole_0,axiom,( ? [A] : ~ v1_xboole_0(A) ), file(xboole_0,rc2_xboole_0), []). fof(rc3_gcd_1,axiom,( ? [A] : ( l1_vectsp_1(A) & ~ v3_struct_0(A) & v2_group_1(A) & v4_group_1(A) & v7_group_1(A) & v6_vectsp_1(A) & v8_vectsp_1(A) & v3_gcd_1(A) ) ), file(gcd_1,rc3_gcd_1), []). fof(rc3_struct_0,axiom,( ? [A] : ( l1_struct_0(A) & ~ v3_struct_0(A) ) ), file(struct_0,rc3_struct_0), []). fof(rc4_gcd_1,axiom,( ? [A] : ( l2_vectsp_1(A) & ~ v3_struct_0(A) & v2_group_1(A) & v4_group_1(A) & v7_group_1(A) & v6_vectsp_1(A) & v8_vectsp_1(A) & v3_gcd_1(A) ) ), file(gcd_1,rc4_gcd_1), []). fof(rc4_struct_0,axiom,( ? [A] : ( l2_struct_0(A) & ~ v3_struct_0(A) ) ), file(struct_0,rc4_struct_0), []). fof(rc5_gcd_1,axiom,( ? [A] : ( l3_vectsp_1(A) & ~ v3_struct_0(A) & v3_rlvect_1(A) & v4_rlvect_1(A) & v5_rlvect_1(A) & v6_rlvect_1(A) & v2_group_1(A) & v4_group_1(A) & v7_group_1(A) & v4_vectsp_1(A) & v5_vectsp_1(A) & v6_vectsp_1(A) & v7_vectsp_1(A) & v8_vectsp_1(A) & ~ v10_vectsp_1(A) & v2_vectsp_2(A) & v3_gcd_1(A) ) ), file(gcd_1,rc5_gcd_1), []). fof(rc5_struct_0,axiom,( ! [A] : ( ( ~ v3_struct_0(A) & l1_struct_0(A) ) => ? [B] : ( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) & ~ v1_xboole_0(B) ) ) ), file(struct_0,rc5_struct_0), []). fof(rc6_vectsp_1,axiom,( ? [A] : ( l3_vectsp_1(A) & v3_vectsp_1(A) ) ), file(vectsp_1,rc6_vectsp_1), []). fof(rc7_vectsp_1,axiom,( ? [A] : ( l3_vectsp_1(A) & ~ v3_struct_0(A) & v3_vectsp_1(A) ) ), file(vectsp_1,rc7_vectsp_1), []). fof(rc8_vectsp_1,axiom,( ? [A] : ( l1_group_1(A) & ~ v3_struct_0(A) & v4_group_1(A) & v7_group_1(A) ) ), file(vectsp_1,rc8_vectsp_1), []). fof(rc9_vectsp_1,axiom,( ? [A] : ( l1_vectsp_1(A) & ~ v3_struct_0(A) & v2_group_1(A) & v4_group_1(A) & v7_group_1(A) & v6_vectsp_1(A) & v8_vectsp_1(A) ) ), file(vectsp_1,rc9_vectsp_1), []). fof(redefinition_k10_group_1,axiom,( ! [A,B,C] : ( ( ~ v3_struct_0(A) & v7_group_1(A) & l1_group_1(A) & m1_subset_1(B,u1_struct_0(A)) & m1_subset_1(C,u1_struct_0(A)) ) => k10_group_1(A,B,C) = k1_group_1(A,B,C) ) ), file(group_1,k10_group_1), []). fof(redefinition_k12_quofield,axiom,( ! [A,B] : ( ( ~ v3_struct_0(A) & ~ v10_vectsp_1(A) & l2_vectsp_1(A) & m1_subset_1(B,k1_quofield(A)) ) => k12_quofield(A,B) = k8_quofield(A,B) ) ), file(quofield,k12_quofield), []). fof(redefinition_k1_domain_1,axiom,( ! [A,B,C,D] : ( ( ~ v1_xboole_0(A) & ~ v1_xboole_0(B) & m1_subset_1(C,A) & m1_subset_1(D,B) ) => k1_domain_1(A,B,C,D) = k4_tarski(C,D) ) ), file(domain_1,k1_domain_1), []). fof(redefinition_k2_binop_1,axiom,( ! [A,B,C,D,E,F] : ( ( ~ v1_xboole_0(A) & ~ v1_xboole_0(B) & v1_funct_1(D) & v1_funct_2(D,k2_zfmisc_1(A,B),C) & m1_relset_1(D,k2_zfmisc_1(A,B),C) & m1_subset_1(E,A) & m1_subset_1(F,B) ) => k2_binop_1(A,B,C,D,E,F) = k1_binop_1(D,E,F) ) ), file(binop_1,k2_binop_1), []). fof(redefinition_k2_domain_1,axiom,( ! [A,B,C] : ( ( ~ v1_xboole_0(A) & ~ v1_xboole_0(B) & m1_subset_1(C,k2_zfmisc_1(A,B)) ) => k2_domain_1(A,B,C) = k1_mcart_1(C) ) ), file(domain_1,k2_domain_1), []). fof(redefinition_k2_quofield,axiom,( ! [A,B] : ( ( ~ v3_struct_0(A) & ~ v10_vectsp_1(A) & l2_vectsp_1(A) & m1_subset_1(B,k1_quofield(A)) ) => k2_quofield(A,B) = k1_mcart_1(B) ) ), file(quofield,k2_quofield), []). fof(redefinition_k3_domain_1,axiom,( ! [A,B,C] : ( ( ~ v1_xboole_0(A) & ~ v1_xboole_0(B) & m1_subset_1(C,k2_zfmisc_1(A,B)) ) => k3_domain_1(A,B,C) = k2_mcart_1(C) ) ), file(domain_1,k3_domain_1), []). fof(redefinition_k3_quofield,axiom,( ! [A,B] : ( ( ~ v3_struct_0(A) & ~ v10_vectsp_1(A) & l2_vectsp_1(A) & m1_subset_1(B,k1_quofield(A)) ) => k3_quofield(A,B) = k2_mcart_1(B) ) ), file(quofield,k3_quofield), []). fof(redefinition_k4_relset_1,axiom,( ! [A,B,C] : ( m1_relset_1(C,A,B) => k4_relset_1(A,B,C) = k1_relat_1(C) ) ), file(relset_1,k4_relset_1), []). fof(redefinition_k8_funct_2,axiom,( ! [A,B,C,D] : ( ( ~ v1_xboole_0(A) & v1_funct_1(C) & v1_funct_2(C,A,B) & m1_relset_1(C,A,B) & m1_subset_1(D,A) ) => k8_funct_2(A,B,C,D) = k1_funct_1(C,D) ) ), file(funct_2,k8_funct_2), []). fof(redefinition_m2_relset_1,axiom,( ! [A,B,C] : ( m2_relset_1(C,A,B) <=> m1_relset_1(C,A,B) ) ), file(relset_1,m2_relset_1), []). fof(redefinition_m2_subset_1,axiom,( ! [A,B] : ( ( ~ v1_xboole_0(A) & ~ v1_xboole_0(B) & m1_subset_1(B,k1_zfmisc_1(A)) ) => ! [C] : ( m2_subset_1(C,A,B) <=> m1_subset_1(C,B) ) ) ), file(subset_1,m2_subset_1), []). fof(reflexivity_r1_tarski,axiom,( ! [A,B] : r1_tarski(A,A) ), file(tarski,r1_tarski), []). fof(t1_subset,axiom,( ! [A,B] : ( r2_hidden(A,B) => m1_subset_1(A,B) ) ), file(subset,t1_subset), []). fof(t2_subset,axiom,( ! [A,B] : ( m1_subset_1(A,B) => ( v1_xboole_0(B) | r2_hidden(A,B) ) ) ), file(subset,t2_subset), []). fof(t3_subset,axiom,( ! [A,B] : ( m1_subset_1(A,k1_zfmisc_1(B)) <=> r1_tarski(A,B) ) ), file(subset,t3_subset), []). fof(t4_subset,axiom,( ! [A,B,C] : ( ( r2_hidden(A,B) & m1_subset_1(B,k1_zfmisc_1(C)) ) => m1_subset_1(A,C) ) ), file(subset,t4_subset), []). fof(t59_quofield,axiom,( ! [A] : ( ( ~ v3_struct_0(A) & v3_rlvect_1(A) & v4_rlvect_1(A) & v5_rlvect_1(A) & v6_rlvect_1(A) & v4_group_1(A) & v7_group_1(A) & v6_vectsp_1(A) & v7_vectsp_1(A) & v8_vectsp_1(A) & ~ v10_vectsp_1(A) & v2_vectsp_2(A) & l3_vectsp_1(A) ) => v1_quofield(k24_quofield(A),A,k21_quofield(A)) ) ), file(quofield,t59_quofield), []). fof(t5_subset,axiom,( ! [A,B,C] : ~ ( r2_hidden(A,B) & m1_subset_1(B,k1_zfmisc_1(C)) & v1_xboole_0(C) ) ), file(subset,t5_subset), []). fof(t6_boole,axiom,( ! [A] : ( v1_xboole_0(A) => A = k1_xboole_0 ) ), file(boole,t6_boole), []). fof(t6_quofield,axiom,( ! [A] : ( ( ~ v3_struct_0(A) & v7_group_1(A) & ~ v10_vectsp_1(A) & l2_vectsp_1(A) ) => ! [B] : ( m2_subset_1(B,k2_zfmisc_1(u1_struct_0(A),u1_struct_0(A)),k1_quofield(A)) => r2_hidden(B,k8_quofield(A,B)) ) ) ), file(quofield,t6_quofield), []). fof(t7_boole,axiom,( ! [A,B] : ~ ( r2_hidden(A,B) & v1_xboole_0(B) ) ), file(boole,t7_boole), []). fof(t8_boole,axiom,( ! [A,B] : ~ ( v1_xboole_0(A) & A != B & v1_xboole_0(B) ) ), file(boole,t8_boole), []).