% Mizar problem: t7_group_3,group_3,111,12 fof(t7_group_3,conjecture,( ! [A] : ( ( ~ v3_struct_0(A) & v3_group_1(A) & v4_group_1(A) & l1_group_1(A) ) => ! [B] : ( m1_subset_1(B,u1_struct_0(A)) => ! [C] : ( m1_group_2(C,A) => k12_group_2(A,C,B) = k10_group_2(A,C,k6_domain_1(u1_struct_0(A),B)) ) ) ) ), inference(mizar_bg_added,[status(thm)],[commutativity_k2_tarski,dt_k2_tarski,fc3_subset_1,rc3_funct_1,fc2_finset_1,dt_k1_funct_1,dt_k4_tarski,rc1_funct_1,rc2_funct_1,cc2_funct_1,d5_tarski,existence_m1_relset_1,existence_m2_relset_1,redefinition_m2_relset_1,dt_k1_binop_1,dt_k2_zfmisc_1,dt_m1_relset_1,dt_m2_relset_1,fc4_subset_1,fc14_finset_1,cc1_relset_1,d1_binop_1,redefinition_k2_binop_1,dt_k2_binop_1,dt_u1_group_1,dt_k1_group_1,d1_group_1,reflexivity_r1_tarski,antisymmetry_r2_hidden,dt_k1_xboole_0,rc1_finset_1,rc3_finset_1,cc2_finset_1,rc4_finset_1,fc1_xboole_0,t1_subset,t4_subset,t5_subset,t2_tarski,fraenkel_a_3_0_group_2,existence_l1_struct_0,dt_k1_tarski,dt_k1_zfmisc_1,dt_k2_group_2,dt_k3_group_2,dt_k7_group_2,dt_l1_struct_0,fc1_subset_1,fc2_subset_1,rc1_subset_1,rc2_subset_1,cc1_funct_1,cc1_finset_1,fc1_finset_1,rc3_struct_0,fc1_struct_0,rc5_struct_0,rc1_xboole_0,rc2_xboole_0,t2_subset,t3_subset,t6_boole,t7_boole,t8_boole,d2_group_2,d3_group_2,d9_group_2,existence_l1_group_1,existence_m1_group_2,existence_m1_subset_1,redefinition_k6_domain_1,dt_k10_group_2,dt_k12_group_2,dt_k6_domain_1,dt_l1_group_1,dt_m1_group_2,dt_m1_subset_1,dt_u1_struct_0,cc1_group_2,cc1_group_1,d11_group_2,d13_group_2]), [file(group_3,t7_group_3)]). fof(commutativity_k2_tarski,axiom,( ! [A,B] : k2_tarski(A,B) = k2_tarski(B,A) ), file(tarski,k2_tarski), []). fof(dt_k2_tarski,axiom,( $true ), file(tarski,k2_tarski), []). fof(fc3_subset_1,axiom,( ! [A,B] : ~ v1_xboole_0(k2_tarski(A,B)) ), file(subset_1,fc3_subset_1), []). fof(rc3_funct_1,axiom,( ? [A] : ( v1_relat_1(A) & v1_funct_1(A) & v2_funct_1(A) ) ), file(funct_1,rc3_funct_1), []). fof(fc2_finset_1,axiom,( ! [A,B] : ( ~ v1_xboole_0(k2_tarski(A,B)) & v1_finset_1(k2_tarski(A,B)) ) ), file(finset_1,fc2_finset_1), []). fof(dt_k1_funct_1,axiom,( $true ), file(funct_1,k1_funct_1), []). fof(dt_k4_tarski,axiom,( $true ), file(tarski,k4_tarski), []). fof(rc1_funct_1,axiom,( ? [A] : ( v1_relat_1(A) & v1_funct_1(A) ) ), file(funct_1,rc1_funct_1), []). fof(rc2_funct_1,axiom,( ? [A] : ( v1_relat_1(A) & v1_xboole_0(A) & v1_funct_1(A) ) ), file(funct_1,rc2_funct_1), []). fof(cc2_funct_1,axiom,( ! [A] : ( ( v1_relat_1(A) & v1_xboole_0(A) & v1_funct_1(A) ) => ( v1_relat_1(A) & v1_funct_1(A) & v2_funct_1(A) ) ) ), file(funct_1,cc2_funct_1), []). fof(d5_tarski,axiom,( ! [A,B] : k4_tarski(A,B) = k2_tarski(k2_tarski(A,B),k1_tarski(A)) ), file(tarski,d5_tarski), []). fof(existence_m1_relset_1,axiom,( ! [A,B] : ? [C] : m1_relset_1(C,A,B) ), file(relset_1,m1_relset_1), []). fof(existence_m2_relset_1,axiom,( ! [A,B] : ? [C] : m2_relset_1(C,A,B) ), file(relset_1,m2_relset_1), []). 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(dt_k1_binop_1,axiom,( $true ), file(binop_1,k1_binop_1), []). fof(dt_k2_zfmisc_1,axiom,( $true ), file(zfmisc_1,k2_zfmisc_1), []). fof(dt_m1_relset_1,axiom,( $true ), file(relset_1,m1_relset_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(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(fc14_finset_1,axiom,( ! [A,B] : ( ( v1_finset_1(A) & v1_finset_1(B) ) => v1_finset_1(k2_zfmisc_1(A,B)) ) ), file(finset_1,fc14_finset_1), []). 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(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(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(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_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_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(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(reflexivity_r1_tarski,axiom,( ! [A,B] : r1_tarski(A,A) ), file(tarski,r1_tarski), []). fof(antisymmetry_r2_hidden,axiom,( ! [A,B] : ( r2_hidden(A,B) => ~ r2_hidden(B,A) ) ), file(hidden,r2_hidden), []). fof(dt_k1_xboole_0,axiom,( $true ), file(xboole_0,k1_xboole_0), []). fof(rc1_finset_1,axiom,( ? [A] : ( ~ v1_xboole_0(A) & v1_finset_1(A) ) ), file(finset_1,rc1_finset_1), []). fof(rc3_finset_1,axiom,( ! [A] : ( ~ v1_xboole_0(A) => ? [B] : ( m1_subset_1(B,k1_zfmisc_1(A)) & ~ v1_xboole_0(B) & v1_finset_1(B) ) ) ), file(finset_1,rc3_finset_1), []). fof(cc2_finset_1,axiom,( ! [A] : ( v1_finset_1(A) => ! [B] : ( m1_subset_1(B,k1_zfmisc_1(A)) => v1_finset_1(B) ) ) ), file(finset_1,cc2_finset_1), []). fof(rc4_finset_1,axiom,( ! [A] : ( ~ v1_xboole_0(A) => ? [B] : ( m1_subset_1(B,k1_zfmisc_1(A)) & ~ v1_xboole_0(B) & v1_finset_1(B) ) ) ), file(finset_1,rc4_finset_1), []). fof(fc1_xboole_0,axiom,( v1_xboole_0(k1_xboole_0) ), file(xboole_0,fc1_xboole_0), []). fof(t1_subset,axiom,( ! [A,B] : ( r2_hidden(A,B) => m1_subset_1(A,B) ) ), file(subset,t1_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(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(t2_tarski,axiom,( ! [A,B] : ( ! [C] : ( r2_hidden(C,A) <=> r2_hidden(C,B) ) => A = B ) ), file(tarski,t2_tarski), []). fof(fraenkel_a_3_0_group_2,axiom,( ! [A,B,C,D] : ( ( ~ v3_struct_0(B) & l1_group_1(B) & m1_subset_1(C,k1_zfmisc_1(u1_struct_0(B))) & m1_subset_1(D,k1_zfmisc_1(u1_struct_0(B))) ) => ( r2_hidden(A,a_3_0_group_2(B,C,D)) <=> ? [E,F] : ( m1_subset_1(E,u1_struct_0(B)) & m1_subset_1(F,u1_struct_0(B)) & A = k1_group_1(B,E,F) & r2_hidden(E,C) & r2_hidden(F,D) ) ) ) ), file(group_2,a_3_0_group_2), []). fof(existence_l1_struct_0,axiom,( ? [A] : l1_struct_0(A) ), file(struct_0,l1_struct_0), []). fof(dt_k1_tarski,axiom,( $true ), file(tarski,k1_tarski), []). fof(dt_k1_zfmisc_1,axiom,( $true ), file(zfmisc_1,k1_zfmisc_1), []). fof(dt_k2_group_2,axiom,( ! [A,B,C] : ( ( ~ v3_struct_0(A) & l1_group_1(A) & m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) & m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A))) ) => m1_subset_1(k2_group_2(A,B,C),k1_zfmisc_1(u1_struct_0(A))) ) ), file(group_2,k2_group_2), []). fof(dt_k3_group_2,axiom,( ! [A,B,C] : ( ( ~ v3_struct_0(A) & l1_group_1(A) & m1_subset_1(B,u1_struct_0(A)) & m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A))) ) => m1_subset_1(k3_group_2(A,B,C),k1_zfmisc_1(u1_struct_0(A))) ) ), file(group_2,k3_group_2), []). fof(dt_k7_group_2,axiom,( ! [A,B] : ( ( ~ v3_struct_0(A) & v3_group_1(A) & v4_group_1(A) & l1_group_1(A) & m1_group_2(B,A) ) => m1_subset_1(k7_group_2(A,B),k1_zfmisc_1(u1_struct_0(A))) ) ), file(group_2,k7_group_2), []). fof(dt_l1_struct_0,axiom,( $true ), file(struct_0,l1_struct_0), []). fof(fc1_subset_1,axiom,( ! [A] : ~ v1_xboole_0(k1_zfmisc_1(A)) ), file(subset_1,fc1_subset_1), []). fof(fc2_subset_1,axiom,( ! [A] : ~ v1_xboole_0(k1_tarski(A)) ), file(subset_1,fc2_subset_1), []). 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(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(cc1_funct_1,axiom,( ! [A] : ( v1_xboole_0(A) => v1_funct_1(A) ) ), file(funct_1,cc1_funct_1), []). fof(cc1_finset_1,axiom,( ! [A] : ( v1_xboole_0(A) => v1_finset_1(A) ) ), file(finset_1,cc1_finset_1), []). fof(fc1_finset_1,axiom,( ! [A] : ( ~ v1_xboole_0(k1_tarski(A)) & v1_finset_1(k1_tarski(A)) ) ), file(finset_1,fc1_finset_1), []). fof(rc3_struct_0,axiom,( ? [A] : ( l1_struct_0(A) & ~ v3_struct_0(A) ) ), file(struct_0,rc3_struct_0), []). 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(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(rc1_xboole_0,axiom,( ? [A] : v1_xboole_0(A) ), file(xboole_0,rc1_xboole_0), []). fof(rc2_xboole_0,axiom,( ? [A] : ~ v1_xboole_0(A) ), file(xboole_0,rc2_xboole_0), []). 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(t6_boole,axiom,( ! [A] : ( v1_xboole_0(A) => A = k1_xboole_0 ) ), file(boole,t6_boole), []). 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), []). fof(d2_group_2,axiom,( ! [A] : ( ( ~ v3_struct_0(A) & l1_group_1(A) ) => ! [B] : ( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) => ! [C] : ( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A))) => k2_group_2(A,B,C) = a_3_0_group_2(A,B,C) ) ) ) ), file(group_2,d2_group_2), []). fof(d3_group_2,axiom,( ! [A] : ( ( ~ v3_struct_0(A) & l1_group_1(A) ) => ! [B] : ( m1_subset_1(B,u1_struct_0(A)) => ! [C] : ( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A))) => k3_group_2(A,B,C) = k2_group_2(A,k6_domain_1(u1_struct_0(A),B),C) ) ) ) ), file(group_2,d3_group_2), []). fof(d9_group_2,axiom,( ! [A] : ( ( ~ v3_struct_0(A) & v3_group_1(A) & v4_group_1(A) & l1_group_1(A) ) => ! [B] : ( m1_group_2(B,A) => k7_group_2(A,B) = u1_struct_0(B) ) ) ), file(group_2,d9_group_2), []). fof(existence_l1_group_1,axiom,( ? [A] : l1_group_1(A) ), file(group_1,l1_group_1), []). fof(existence_m1_group_2,axiom,( ! [A] : ( ( ~ v3_struct_0(A) & v3_group_1(A) & l1_group_1(A) ) => ? [B] : m1_group_2(B,A) ) ), file(group_2,m1_group_2), []). fof(existence_m1_subset_1,axiom,( ! [A] : ? [B] : m1_subset_1(B,A) ), file(subset_1,m1_subset_1), []). fof(redefinition_k6_domain_1,axiom,( ! [A,B] : ( ( ~ v1_xboole_0(A) & m1_subset_1(B,A) ) => k6_domain_1(A,B) = k1_tarski(B) ) ), file(domain_1,k6_domain_1), []). fof(dt_k10_group_2,axiom,( ! [A,B,C] : ( ( ~ v3_struct_0(A) & v3_group_1(A) & v4_group_1(A) & l1_group_1(A) & m1_group_2(B,A) & m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A))) ) => m1_subset_1(k10_group_2(A,B,C),k1_zfmisc_1(u1_struct_0(A))) ) ), file(group_2,k10_group_2), []). fof(dt_k12_group_2,axiom,( ! [A,B,C] : ( ( ~ v3_struct_0(A) & v3_group_1(A) & v4_group_1(A) & l1_group_1(A) & m1_group_2(B,A) & m1_subset_1(C,u1_struct_0(A)) ) => m1_subset_1(k12_group_2(A,B,C),k1_zfmisc_1(u1_struct_0(A))) ) ), file(group_2,k12_group_2), []). fof(dt_k6_domain_1,axiom,( ! [A,B] : ( ( ~ v1_xboole_0(A) & m1_subset_1(B,A) ) => m1_subset_1(k6_domain_1(A,B),k1_zfmisc_1(A)) ) ), file(domain_1,k6_domain_1), []). fof(dt_l1_group_1,axiom,( ! [A] : ( l1_group_1(A) => l1_struct_0(A) ) ), file(group_1,l1_group_1), []). fof(dt_m1_group_2,axiom,( ! [A] : ( ( ~ v3_struct_0(A) & v3_group_1(A) & l1_group_1(A) ) => ! [B] : ( m1_group_2(B,A) => ( ~ v3_struct_0(B) & v3_group_1(B) & l1_group_1(B) ) ) ) ), file(group_2,m1_group_2), []). fof(dt_m1_subset_1,axiom,( $true ), file(subset_1,m1_subset_1), []). fof(dt_u1_struct_0,axiom,( $true ), file(struct_0,u1_struct_0), []). fof(cc1_group_2,axiom,( ! [A] : ( ( ~ v3_struct_0(A) & v3_group_1(A) & v4_group_1(A) & l1_group_1(A) ) => ! [B] : ( m1_group_2(B,A) => v4_group_1(B) ) ) ), file(group_2,cc1_group_2), []). fof(cc1_group_1,axiom,( ! [A] : ( l1_group_1(A) => ( ( ~ v3_struct_0(A) & v3_group_1(A) ) => ( ~ v3_struct_0(A) & v2_group_1(A) ) ) ) ), file(group_1,cc1_group_1), []). fof(d11_group_2,axiom,( ! [A] : ( ( ~ v3_struct_0(A) & v3_group_1(A) & v4_group_1(A) & l1_group_1(A) ) => ! [B] : ( m1_group_2(B,A) => ! [C] : ( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A))) => k10_group_2(A,B,C) = k2_group_2(A,C,k7_group_2(A,B)) ) ) ) ), file(group_2,d11_group_2), []). fof(d13_group_2,axiom,( ! [A] : ( ( ~ v3_struct_0(A) & v3_group_1(A) & v4_group_1(A) & l1_group_1(A) ) => ! [B] : ( m1_group_2(B,A) => ! [C] : ( m1_subset_1(C,u1_struct_0(A)) => k12_group_2(A,B,C) = k3_group_2(A,C,k7_group_2(A,B)) ) ) ) ), file(group_2,d13_group_2), []).