% Mizar problem: t44_monoid_1,monoid_1,1237,18 fof(t44_monoid_1,conjecture,( ! [A,B] : ( ~ v1_xboole_0(B) => ~ ( m1_subset_1(A,u1_struct_0(k19_monoid_1(B))) & ! [C] : ( m2_finseq_1(C,B) => A != k20_monoid_1(B,C) ) ) ) ), inference(mizar_bg_added,[status(thm)],[abstractness_v1_group_1,abstractness_v1_vectsp_1,antisymmetry_r2_hidden,cc10_membered,cc10_monoid_0,cc11_membered,cc11_monoid_0,cc12_membered,cc12_monoid_0,cc13_membered,cc13_monoid_0,cc14_membered,cc14_monoid_0,cc15_membered,cc16_membered,cc16_monoid_0,cc17_membered,cc17_monoid_0,cc18_membered,cc18_monoid_0,cc19_membered,cc19_monoid_0,cc1_finseq_1,cc1_finset_1,cc1_funct_1,cc1_membered,cc1_monoid_0,cc1_nat_1,cc1_relat_1,cc1_relset_1,cc20_membered,cc20_monoid_0,cc21_monoid_0,cc22_monoid_0,cc23_monoid_0,cc26_monoid_0,cc27_monoid_0,cc28_monoid_0,cc29_monoid_0,cc2_finset_1,cc2_funct_1,cc2_membered,cc2_nat_1,cc30_monoid_0,cc33_monoid_0,cc34_monoid_0,cc35_monoid_0,cc36_monoid_0,cc3_membered,cc3_nat_1,cc4_membered,cc5_monoid_0,cc6_membered,cc6_monoid_0,cc7_monoid_0,cc8_monoid_0,cc9_membered,cc9_monoid_0,commutativity_k1_nat_1,commutativity_k2_xcmplx_0,commutativity_k3_xboole_0,d10_xboole_0,d13_funct_1,d1_funct_2,d1_tarski,d26_monoid_0,d3_tarski,d4_monoid_1,d4_xboole_0,d5_funct_1,d6_monoid_1,d7_xboole_0,dt_g1_group_1,dt_g1_vectsp_1,dt_k10_relat_1,dt_k12_monoid_1,dt_k13_finseq_1,dt_k13_monoid_1,dt_k14_monoid_1,dt_k15_monoid_1,dt_k17_monoid_1,dt_k19_monoid_1,dt_k1_card_1,dt_k1_funct_1,dt_k1_group_1,dt_k1_monoid_0,dt_k1_nat_1,dt_k1_numbers,dt_k1_relat_1,dt_k1_tarski,dt_k1_xboole_0,dt_k1_zfmisc_1,dt_k20_monoid_1,dt_k2_finseq_2,dt_k2_funcop_1,dt_k2_relat_1,dt_k2_xcmplx_0,dt_k2_zfmisc_1,dt_k33_binop_2,dt_k3_finseq_2,dt_k3_gr_cy_1,dt_k3_xboole_0,dt_k4_card_1,dt_k4_monoid_0,dt_k4_monoid_1,dt_k4_relset_1,dt_k4_xboole_0,dt_k5_monoid_0,dt_k5_numbers,dt_k5_ordinal2,dt_k5_real_1,dt_k6_domain_1,dt_k6_finseq_1,dt_k6_monoid_0,dt_k6_xcmplx_0,dt_k7_finseq_1,dt_k7_monoid_0,dt_k8_monoid_0,dt_k9_monoid_1,dt_l1_group_1,dt_l1_struct_0,dt_l1_vectsp_1,dt_m1_finseq_1,dt_m1_finseq_2,dt_m1_monoid_0,dt_m1_relset_1,dt_m1_subset_1,dt_m2_finseq_1,dt_m2_finseq_2,dt_m2_monoid_0,dt_m2_relset_1,dt_m2_subset_1,dt_m3_monoid_0,dt_m5_monoid_0,dt_u1_group_1,dt_u1_struct_0,dt_u1_vectsp_1,existence_l1_group_1,existence_l1_struct_0,existence_l1_vectsp_1,existence_m1_finseq_1,existence_m1_finseq_2,existence_m1_monoid_0,existence_m1_relset_1,existence_m1_subset_1,existence_m2_finseq_1,existence_m2_finseq_2,existence_m2_monoid_0,existence_m2_relset_1,existence_m2_subset_1,existence_m3_monoid_0,existence_m5_monoid_0,fc10_finset_1,fc10_membered,fc11_finseq_1,fc11_finset_1,fc11_membered,fc12_finset_1,fc12_relat_1,fc13_finseq_1,fc14_finseq_1,fc14_finset_1,fc16_finseq_1,fc17_finseq_1,fc18_finseq_1,fc1_finset_1,fc1_funcop_1,fc1_monoid_0,fc1_monoid_1,fc1_nat_1,fc1_ordinal2,fc1_relat_1,fc1_struct_0,fc1_subset_1,fc27_membered,fc28_membered,fc29_membered,fc2_finseq_1,fc2_funcop_1,fc2_membered,fc2_monoid_1,fc2_subset_1,fc30_membered,fc31_membered,fc32_membered,fc33_membered,fc34_membered,fc35_membered,fc36_membered,fc37_membered,fc38_membered,fc39_membered,fc3_funcop_1,fc3_monoid_1,fc3_nat_1,fc3_relat_1,fc40_membered,fc41_membered,fc4_nat_1,fc4_ordinal2,fc4_relat_1,fc4_subset_1,fc5_membered,fc5_relat_1,fc6_funct_1,fc6_membered,fc6_relat_1,fc7_membered,fc7_relat_1,fc8_membered,fc8_relat_1,fc9_finseq_1,fc9_funcop_1,fc9_membered,free_g1_group_1,free_g1_vectsp_1,idempotence_k3_xboole_0,rc10_monoid_0,rc11_monoid_0,rc1_finseq_1,rc1_finset_1,rc1_funct_1,rc1_membered,rc1_monoid_0,rc1_nat_1,rc1_relat_1,rc1_subset_1,rc2_finseq_1,rc2_finset_1,rc2_funct_1,rc2_nat_1,rc2_relat_1,rc2_subset_1,rc3_finseq_1,rc3_finset_1,rc3_funct_1,rc3_nat_1,rc3_relat_1,rc3_struct_0,rc4_finseq_1,rc4_finset_1,rc4_funct_1,rc5_funct_1,rc5_monoid_0,rc5_struct_0,rc6_finseq_1,rc6_monoid_0,rc7_finseq_1,rc7_monoid_0,rc8_finseq_1,rc8_monoid_0,redefinition_k12_monoid_1,redefinition_k14_monoid_1,redefinition_k15_monoid_1,redefinition_k17_monoid_1,redefinition_k1_monoid_0,redefinition_k1_nat_1,redefinition_k3_finseq_2,redefinition_k4_card_1,redefinition_k4_monoid_0,redefinition_k4_monoid_1,redefinition_k4_relset_1,redefinition_k5_numbers,redefinition_k5_real_1,redefinition_k6_domain_1,redefinition_k6_monoid_0,redefinition_m2_finseq_1,redefinition_m2_finseq_2,redefinition_m2_relset_1,redefinition_m2_subset_1,redefinition_m5_monoid_0,reflexivity_r1_tarski,s1_nat_1__e3_71__monoid_1,s1_partfun1__e10_71_2__monoid_1,spc0_boole,spc1_boole,spc1_numerals,symmetry_r1_xboole_0,t11_relset_1,t15_funcop_1,t16_finset_1,t173_relat_1,t1_arithm,t1_numerals,t1_subset,t25_monoid_0,t28_monoid_1,t28_xboole_1,t29_monoid_1,t2_boole,t2_subset,t30_monoid_1,t33_monoid_1,t37_xboole_1,t37_zfmisc_1,t38_monoid_1,t3_boole,t3_subset,t41_monoid_1,t42_monoid_1,t47_card_1,t49_xboole_1,t4_arithm,t4_boole,t4_subset,t59_card_2,t5_subset,t63_card_2,t6_boole,t79_card_1,t7_boole,t8_boole]), [file(monoid_1,t44_monoid_1)]). fof(abstractness_v1_group_1,axiom,( ! [A] : ( l1_group_1(A) => ( v1_group_1(A) => A = g1_group_1(u1_struct_0(A),u1_group_1(A)) ) ) ), file(group_1,v1_group_1), []). fof(abstractness_v1_vectsp_1,axiom,( ! [A] : ( l1_vectsp_1(A) => ( v1_vectsp_1(A) => A = g1_vectsp_1(u1_struct_0(A),u1_group_1(A),u1_vectsp_1(A)) ) ) ), file(vectsp_1,v1_vectsp_1), []). fof(antisymmetry_r2_hidden,axiom,( ! [A,B] : ( r2_hidden(A,B) => ~ r2_hidden(B,A) ) ), file(hidden,r2_hidden), []). fof(cc10_membered,axiom,( ! [A] : ( v1_membered(A) => ! [B] : ( m1_subset_1(B,A) => v1_xcmplx_0(B) ) ) ), file(membered,cc10_membered), []). fof(cc10_monoid_0,axiom,( ! [A] : ( l1_group_1(A) => ( ( ~ v3_struct_0(A) & v14_monoid_0(A) & v15_monoid_0(A) ) => ( ~ v3_struct_0(A) & v16_monoid_0(A) ) ) ) ), file(monoid_0,cc10_monoid_0), []). fof(cc11_membered,axiom,( ! [A] : ( v2_membered(A) => ! [B] : ( m1_subset_1(B,A) => ( v1_xcmplx_0(B) & v1_xreal_0(B) ) ) ) ), file(membered,cc11_membered), []). fof(cc11_monoid_0,axiom,( ! [A] : ( l1_group_1(A) => ( ( ~ v3_struct_0(A) & v4_group_1(A) & v13_monoid_0(A) ) => ( ~ v3_struct_0(A) & v2_group_1(A) & v16_monoid_0(A) ) ) ) ), file(monoid_0,cc11_monoid_0), []). fof(cc12_membered,axiom,( ! [A] : ( v3_membered(A) => ! [B] : ( m1_subset_1(B,A) => ( v1_xcmplx_0(B) & v1_xreal_0(B) & v1_rat_1(B) ) ) ) ), file(membered,cc12_membered), []). fof(cc12_monoid_0,axiom,( ! [A] : ( l1_vectsp_1(A) => ( ( ~ v3_struct_0(A) & v1_vectsp_2(A) ) => ( ~ v3_struct_0(A) & v2_group_1(A) & v6_vectsp_1(A) & v8_vectsp_1(A) ) ) ) ), file(monoid_0,cc12_monoid_0), []). fof(cc13_membered,axiom,( ! [A] : ( v4_membered(A) => ! [B] : ( m1_subset_1(B,A) => ( v1_xcmplx_0(B) & v1_xreal_0(B) & v1_int_1(B) & v1_rat_1(B) ) ) ) ), file(membered,cc13_membered), []). fof(cc13_monoid_0,axiom,( ! [A] : ( ( ~ v3_struct_0(A) & l1_group_1(A) ) => ! [B] : ( m1_monoid_0(B,A) => ~ v3_struct_0(B) ) ) ), file(monoid_0,cc13_monoid_0), []). fof(cc14_membered,axiom,( ! [A] : ( v5_membered(A) => ! [B] : ( m1_subset_1(B,A) => ( v1_xcmplx_0(B) & v4_ordinal2(B) & v1_xreal_0(B) & v1_int_1(B) & v1_rat_1(B) ) ) ) ), file(membered,cc14_membered), []). fof(cc14_monoid_0,axiom,( ! [A] : ( ( v1_monoid_0(A) & l1_group_1(A) ) => ! [B] : ( m1_monoid_0(B,A) => v1_monoid_0(B) ) ) ), file(monoid_0,cc14_monoid_0), []). fof(cc15_membered,axiom,( ! [A] : ( v1_xboole_0(A) => ( v1_membered(A) & v2_membered(A) & v3_membered(A) & v4_membered(A) & v5_membered(A) ) ) ), file(membered,cc15_membered), []). fof(cc16_membered,axiom,( ! [A] : ( v1_membered(A) => ! [B] : ( m1_subset_1(B,k1_zfmisc_1(A)) => v1_membered(B) ) ) ), file(membered,cc16_membered), []). fof(cc16_monoid_0,axiom,( ! [A] : ( ( ~ v3_struct_0(A) & v2_group_1(A) & l1_group_1(A) ) => ! [B] : ( m1_monoid_0(B,A) => ( ~ v3_struct_0(B) & v2_group_1(B) & v6_vectsp_1(B) & v8_vectsp_1(B) ) ) ) ), file(monoid_0,cc16_monoid_0), []). fof(cc17_membered,axiom,( ! [A] : ( v2_membered(A) => ! [B] : ( m1_subset_1(B,k1_zfmisc_1(A)) => ( v1_membered(B) & v2_membered(B) ) ) ) ), file(membered,cc17_membered), []). fof(cc17_monoid_0,axiom,( ! [A] : ( ( ~ v3_struct_0(A) & v4_group_1(A) & l1_group_1(A) ) => ! [B] : ( m1_monoid_0(B,A) => ( ~ v3_struct_0(B) & v4_group_1(B) ) ) ) ), file(monoid_0,cc17_monoid_0), []). fof(cc18_membered,axiom,( ! [A] : ( v3_membered(A) => ! [B] : ( m1_subset_1(B,k1_zfmisc_1(A)) => ( v1_membered(B) & v2_membered(B) & v3_membered(B) ) ) ) ), file(membered,cc18_membered), []). fof(cc18_monoid_0,axiom,( ! [A] : ( ( ~ v3_struct_0(A) & v7_group_1(A) & l1_group_1(A) ) => ! [B] : ( m1_monoid_0(B,A) => ( ~ v3_struct_0(B) & v7_group_1(B) ) ) ) ), file(monoid_0,cc18_monoid_0), []). fof(cc19_membered,axiom,( ! [A] : ( v4_membered(A) => ! [B] : ( m1_subset_1(B,k1_zfmisc_1(A)) => ( v1_membered(B) & v2_membered(B) & v3_membered(B) & v4_membered(B) ) ) ) ), file(membered,cc19_membered), []). fof(cc19_monoid_0,axiom,( ! [A] : ( ( ~ v3_struct_0(A) & v13_monoid_0(A) & l1_group_1(A) ) => ! [B] : ( m1_monoid_0(B,A) => ( ~ v3_struct_0(B) & v11_monoid_0(B) & v12_monoid_0(B) & v13_monoid_0(B) ) ) ) ), file(monoid_0,cc19_monoid_0), []). fof(cc1_finseq_1,axiom,( ! [A] : ( ( v1_relat_1(A) & v1_funct_1(A) & v1_finseq_1(A) ) => ( v1_relat_1(A) & v1_funct_1(A) & v1_finset_1(A) ) ) ), file(finseq_1,cc1_finseq_1), []). fof(cc1_finset_1,axiom,( ! [A] : ( v1_xboole_0(A) => v1_finset_1(A) ) ), file(finset_1,cc1_finset_1), []). fof(cc1_funct_1,axiom,( ! [A] : ( v1_xboole_0(A) => v1_funct_1(A) ) ), file(funct_1,cc1_funct_1), []). fof(cc1_membered,axiom,( ! [A] : ( v5_membered(A) => v4_membered(A) ) ), file(membered,cc1_membered), []). fof(cc1_monoid_0,axiom,( ! [A] : ( ( v1_monoid_0(A) & l1_struct_0(A) ) => ! [B] : ( m1_subset_1(B,u1_struct_0(A)) => ( v1_relat_1(B) & v1_funct_1(B) ) ) ) ), file(monoid_0,cc1_monoid_0), []). fof(cc1_nat_1,axiom,( ! [A] : ( m1_subset_1(A,k5_numbers) => ( v1_ordinal1(A) & v2_ordinal1(A) & v3_ordinal1(A) & v4_ordinal2(A) & v1_xcmplx_0(A) & v1_xreal_0(A) ) ) ), file(nat_1,cc1_nat_1), []). fof(cc1_relat_1,axiom,( ! [A] : ( v1_xboole_0(A) => v1_relat_1(A) ) ), file(relat_1,cc1_relat_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(cc20_membered,axiom,( ! [A] : ( v5_membered(A) => ! [B] : ( m1_subset_1(B,k1_zfmisc_1(A)) => ( v1_membered(B) & v2_membered(B) & v3_membered(B) & v4_membered(B) & v5_membered(B) ) ) ) ), file(membered,cc20_membered), []). fof(cc20_monoid_0,axiom,( ! [A] : ( ( ~ v3_struct_0(A) & v16_monoid_0(A) & l1_group_1(A) ) => ! [B] : ( m1_monoid_0(B,A) => ( ~ v3_struct_0(B) & v14_monoid_0(B) & v15_monoid_0(B) & v16_monoid_0(B) ) ) ) ), file(monoid_0,cc20_monoid_0), []). fof(cc21_monoid_0,axiom,( ! [A] : ( ( ~ v3_struct_0(A) & v17_monoid_0(A) & l1_group_1(A) ) => ! [B] : ( m1_monoid_0(B,A) => ( ~ v3_struct_0(B) & v17_monoid_0(B) ) ) ) ), file(monoid_0,cc21_monoid_0), []). fof(cc22_monoid_0,axiom,( ! [A] : ( ( ~ v3_struct_0(A) & v1_monoid_0(A) & l1_group_1(A) ) => ! [B] : ( m2_monoid_0(B,A) => ( ~ v3_struct_0(B) => ( ~ v3_struct_0(B) & v1_monoid_0(B) ) ) ) ) ), file(monoid_0,cc22_monoid_0), []). fof(cc23_monoid_0,axiom,( ! [A] : ( ( ~ v3_struct_0(A) & v1_monoid_0(A) & l1_group_1(A) ) => ! [B] : ( m3_monoid_0(B,A) => ( ~ v3_struct_0(B) => ( ~ v3_struct_0(B) & v1_monoid_0(B) ) ) ) ) ), file(monoid_0,cc23_monoid_0), []). fof(cc26_monoid_0,axiom,( ! [A] : ( ( ~ v3_struct_0(A) & v1_vectsp_2(A) & l1_vectsp_1(A) ) => ! [B] : ( m3_monoid_0(B,A) => ( ~ v3_struct_0(B) => ( ~ v3_struct_0(B) & v2_group_1(B) & v6_vectsp_1(B) & v8_vectsp_1(B) & v1_vectsp_2(B) ) ) ) ) ), file(monoid_0,cc26_monoid_0), []). fof(cc27_monoid_0,axiom,( ! [A] : ( ( ~ v3_struct_0(A) & v7_group_1(A) & l1_group_1(A) ) => ! [B] : ( m2_monoid_0(B,A) => ( ~ v3_struct_0(B) => ( ~ v3_struct_0(B) & v7_group_1(B) ) ) ) ) ), file(monoid_0,cc27_monoid_0), []). fof(cc28_monoid_0,axiom,( ! [A] : ( ( ~ v3_struct_0(A) & v7_group_1(A) & l1_group_1(A) ) => ! [B] : ( m3_monoid_0(B,A) => ( ~ v3_struct_0(B) => ( ~ v3_struct_0(B) & v7_group_1(B) ) ) ) ) ), file(monoid_0,cc28_monoid_0), []). fof(cc29_monoid_0,axiom,( ! [A] : ( ( ~ v3_struct_0(A) & v4_group_1(A) & l1_group_1(A) ) => ! [B] : ( m2_monoid_0(B,A) => ( ~ v3_struct_0(B) => ( ~ v3_struct_0(B) & v4_group_1(B) ) ) ) ) ), file(monoid_0,cc29_monoid_0), []). 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(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(cc2_membered,axiom,( ! [A] : ( v4_membered(A) => v3_membered(A) ) ), file(membered,cc2_membered), []). fof(cc2_nat_1,axiom,( ! [A] : ( m1_subset_1(A,k5_numbers) => ( v1_ordinal1(A) & v2_ordinal1(A) & v3_ordinal1(A) & v4_ordinal2(A) & v1_xcmplx_0(A) & v1_xreal_0(A) & ~ v3_xreal_0(A) ) ) ), file(nat_1,cc2_nat_1), []). fof(cc30_monoid_0,axiom,( ! [A] : ( ( ~ v3_struct_0(A) & v4_group_1(A) & l1_group_1(A) ) => ! [B] : ( m3_monoid_0(B,A) => ( ~ v3_struct_0(B) => ( ~ v3_struct_0(B) & v4_group_1(B) ) ) ) ) ), file(monoid_0,cc30_monoid_0), []). fof(cc33_monoid_0,axiom,( ! [A] : ( ( ~ v3_struct_0(A) & v16_monoid_0(A) & l1_group_1(A) ) => ! [B] : ( m2_monoid_0(B,A) => ( ~ v3_struct_0(B) => ( ~ v3_struct_0(B) & v14_monoid_0(B) & v15_monoid_0(B) & v16_monoid_0(B) ) ) ) ) ), file(monoid_0,cc33_monoid_0), []). fof(cc34_monoid_0,axiom,( ! [A] : ( ( ~ v3_struct_0(A) & v16_monoid_0(A) & l1_group_1(A) ) => ! [B] : ( m3_monoid_0(B,A) => ( ~ v3_struct_0(B) => ( ~ v3_struct_0(B) & v14_monoid_0(B) & v15_monoid_0(B) & v16_monoid_0(B) ) ) ) ) ), file(monoid_0,cc34_monoid_0), []). fof(cc35_monoid_0,axiom,( ! [A] : ( ( ~ v3_struct_0(A) & v1_vectsp_2(A) & v17_monoid_0(A) & l1_vectsp_1(A) ) => ! [B] : ( m3_monoid_0(B,A) => ( ~ v3_struct_0(B) => ( ~ v3_struct_0(B) & v2_group_1(B) & v6_vectsp_1(B) & v8_vectsp_1(B) & v1_vectsp_2(B) & v17_monoid_0(B) ) ) ) ) ), file(monoid_0,cc35_monoid_0), []). fof(cc36_monoid_0,axiom,( ! [A] : ( ( ~ v3_struct_0(A) & v2_group_1(A) & l1_group_1(A) ) => ! [B] : ( m2_monoid_0(B,A) => ( ( ~ v3_struct_0(B) & v4_group_1(B) & v13_monoid_0(B) ) => ( ~ v3_struct_0(B) & v2_group_1(B) & v3_group_1(B) & v14_monoid_0(B) & v15_monoid_0(B) & v16_monoid_0(B) ) ) ) ) ), file(monoid_0,cc36_monoid_0), []). fof(cc3_membered,axiom,( ! [A] : ( v3_membered(A) => v2_membered(A) ) ), file(membered,cc3_membered), []). fof(cc3_nat_1,axiom,( ! [A] : ( v4_ordinal2(A) => ( v4_ordinal2(A) & v1_xcmplx_0(A) & v1_xreal_0(A) & ~ v3_xreal_0(A) ) ) ), file(nat_1,cc3_nat_1), []). fof(cc4_membered,axiom,( ! [A] : ( v2_membered(A) => v1_membered(A) ) ), file(membered,cc4_membered), []). fof(cc5_monoid_0,axiom,( ! [A] : ( l1_group_1(A) => ( ( ~ v3_struct_0(A) & v3_group_1(A) & v4_group_1(A) ) => ( ~ v3_struct_0(A) & v13_monoid_0(A) ) ) ) ), file(monoid_0,cc5_monoid_0), []). fof(cc6_membered,axiom,( ! [A] : ( m1_subset_1(A,k1_zfmisc_1(k1_numbers)) => ( v1_membered(A) & v2_membered(A) ) ) ), file(membered,cc6_membered), []). fof(cc6_monoid_0,axiom,( ! [A] : ( l1_group_1(A) => ( ( ~ v3_struct_0(A) & v4_group_1(A) & v13_monoid_0(A) ) => ( ~ v3_struct_0(A) & v2_group_1(A) & v3_group_1(A) ) ) ) ), file(monoid_0,cc6_monoid_0), []). fof(cc7_monoid_0,axiom,( ! [A] : ( l1_group_1(A) => ( ( ~ v3_struct_0(A) & v13_monoid_0(A) ) => ( ~ v3_struct_0(A) & v11_monoid_0(A) & v12_monoid_0(A) ) ) ) ), file(monoid_0,cc7_monoid_0), []). fof(cc8_monoid_0,axiom,( ! [A] : ( l1_group_1(A) => ( ( ~ v3_struct_0(A) & v11_monoid_0(A) & v12_monoid_0(A) ) => ( ~ v3_struct_0(A) & v13_monoid_0(A) ) ) ) ), file(monoid_0,cc8_monoid_0), []). fof(cc9_membered,axiom,( ! [A] : ( m1_subset_1(A,k1_zfmisc_1(k5_numbers)) => ( v1_membered(A) & v2_membered(A) & v3_membered(A) & v4_membered(A) & v5_membered(A) ) ) ), file(membered,cc9_membered), []). fof(cc9_monoid_0,axiom,( ! [A] : ( l1_group_1(A) => ( ( ~ v3_struct_0(A) & v16_monoid_0(A) ) => ( ~ v3_struct_0(A) & v14_monoid_0(A) & v15_monoid_0(A) ) ) ) ), file(monoid_0,cc9_monoid_0), []). fof(commutativity_k1_nat_1,axiom,( ! [A,B] : ( ( m1_subset_1(A,k5_numbers) & m1_subset_1(B,k5_numbers) ) => k1_nat_1(A,B) = k1_nat_1(B,A) ) ), file(nat_1,k1_nat_1), []). fof(commutativity_k2_xcmplx_0,axiom,( ! [A,B] : ( ( v1_xcmplx_0(A) & v1_xcmplx_0(B) ) => k2_xcmplx_0(A,B) = k2_xcmplx_0(B,A) ) ), file(xcmplx_0,k2_xcmplx_0), []). fof(commutativity_k3_xboole_0,axiom,( ! [A,B] : k3_xboole_0(A,B) = k3_xboole_0(B,A) ), file(xboole_0,k3_xboole_0), []). fof(d10_xboole_0,axiom,( ! [A,B] : ( A = B <=> ( r1_tarski(A,B) & r1_tarski(B,A) ) ) ), file(xboole_0,d10_xboole_0), []). fof(d13_funct_1,axiom,( ! [A] : ( ( v1_relat_1(A) & v1_funct_1(A) ) => ! [B,C] : ( C = k10_relat_1(A,B) <=> ! [D] : ( r2_hidden(D,C) <=> ( r2_hidden(D,k1_relat_1(A)) & r2_hidden(k1_funct_1(A,D),B) ) ) ) ) ), file(funct_1,d13_funct_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_tarski,axiom,( ! [A,B] : ( B = k1_tarski(A) <=> ! [C] : ( r2_hidden(C,B) <=> C = A ) ) ), file(tarski,d1_tarski), []). fof(d26_monoid_0,axiom,( k5_monoid_0 = g1_group_1(k1_numbers,k33_binop_2) ), file(monoid_0,d26_monoid_0), []). fof(d3_tarski,axiom,( ! [A,B] : ( r1_tarski(A,B) <=> ! [C] : ( r2_hidden(C,A) => r2_hidden(C,B) ) ) ), file(tarski,d3_tarski), []). fof(d4_monoid_1,axiom,( ! [A] : k13_monoid_1(A) = k12_monoid_1(k8_monoid_0,A) ), file(monoid_1,d4_monoid_1), []). fof(d4_xboole_0,axiom,( ! [A,B,C] : ( C = k4_xboole_0(A,B) <=> ! [D] : ( r2_hidden(D,C) <=> ( r2_hidden(D,A) & ~ r2_hidden(D,B) ) ) ) ), file(xboole_0,d4_xboole_0), []). fof(d5_funct_1,axiom,( ! [A] : ( ( v1_relat_1(A) & v1_funct_1(A) ) => ! [B] : ( B = k2_relat_1(A) <=> ! [C] : ( r2_hidden(C,B) <=> ? [D] : ( r2_hidden(D,k1_relat_1(A)) & C = k1_funct_1(A,D) ) ) ) ) ), file(funct_1,d5_funct_1), []). fof(d6_monoid_1,axiom,( ! [A,B] : ( ( ~ v3_struct_0(B) & v1_vectsp_1(B) & m3_monoid_0(B,k13_monoid_1(A)) ) => ( B = k19_monoid_1(A) <=> ! [C] : ( m1_subset_1(C,u1_struct_0(k13_monoid_1(A))) => ( r2_hidden(C,u1_struct_0(B)) <=> v1_finset_1(k10_relat_1(C,k4_xboole_0(k5_numbers,k6_domain_1(k5_numbers,0)))) ) ) ) ) ), file(monoid_1,d6_monoid_1), []). fof(d7_xboole_0,axiom,( ! [A,B] : ( r1_xboole_0(A,B) <=> k3_xboole_0(A,B) = k1_xboole_0 ) ), file(xboole_0,d7_xboole_0), []). fof(dt_g1_group_1,axiom,( ! [A,B] : ( ( 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_group_1(g1_group_1(A,B)) & l1_group_1(g1_group_1(A,B)) ) ) ), file(group_1,g1_group_1), []). fof(dt_g1_vectsp_1,axiom,( ! [A,B,C] : ( ( v1_funct_1(B) & v1_funct_2(B,k2_zfmisc_1(A,A),A) & m1_relset_1(B,k2_zfmisc_1(A,A),A) & m1_subset_1(C,A) ) => ( v1_vectsp_1(g1_vectsp_1(A,B,C)) & l1_vectsp_1(g1_vectsp_1(A,B,C)) ) ) ), file(vectsp_1,g1_vectsp_1), []). fof(dt_k10_relat_1,axiom,( $true ), file(relat_1,k10_relat_1), []). fof(dt_k12_monoid_1,axiom,( ! [A,B] : ( ( ~ v3_struct_0(A) & v2_group_1(A) & v4_group_1(A) & v7_group_1(A) & v16_monoid_0(A) & v17_monoid_0(A) & l1_group_1(A) ) => ( ~ v3_struct_0(k12_monoid_1(A,B)) & v4_group_1(k12_monoid_1(A,B)) & v7_group_1(k12_monoid_1(A,B)) & v1_vectsp_1(k12_monoid_1(A,B)) & v1_vectsp_2(k12_monoid_1(A,B)) & v1_monoid_0(k12_monoid_1(A,B)) & v16_monoid_0(k12_monoid_1(A,B)) & v17_monoid_0(k12_monoid_1(A,B)) & l1_vectsp_1(k12_monoid_1(A,B)) ) ) ), file(monoid_1,k12_monoid_1), []). fof(dt_k13_finseq_1,axiom,( $true ), file(finseq_1,k13_finseq_1), []). fof(dt_k13_monoid_1,axiom,( ! [A] : ( ~ v3_struct_0(k13_monoid_1(A)) & v4_group_1(k13_monoid_1(A)) & v7_group_1(k13_monoid_1(A)) & v1_vectsp_1(k13_monoid_1(A)) & v1_vectsp_2(k13_monoid_1(A)) & v1_monoid_0(k13_monoid_1(A)) & v16_monoid_0(k13_monoid_1(A)) & v17_monoid_0(k13_monoid_1(A)) & l1_vectsp_1(k13_monoid_1(A)) ) ), file(monoid_1,k13_monoid_1), []). fof(dt_k14_monoid_1,axiom,( ! [A,B] : ( ( ~ v1_xboole_0(A) & m1_subset_1(B,u1_struct_0(k13_monoid_1(A))) ) => ( ~ v1_xboole_0(k14_monoid_1(A,B)) & m1_subset_1(k14_monoid_1(A,B),k1_zfmisc_1(k5_numbers)) ) ) ), file(monoid_1,k14_monoid_1), []). fof(dt_k15_monoid_1,axiom,( ! [A,B,C] : ( ( ~ v1_xboole_0(A) & m1_subset_1(B,u1_struct_0(k13_monoid_1(A))) & m1_subset_1(C,A) ) => m2_subset_1(k15_monoid_1(A,B,C),k1_numbers,k5_numbers) ) ), file(monoid_1,k15_monoid_1), []). fof(dt_k17_monoid_1,axiom,( ! [A,B] : ( m1_subset_1(B,k5_numbers) => m1_subset_1(k17_monoid_1(A,B),u1_struct_0(k13_monoid_1(A))) ) ), file(monoid_1,k17_monoid_1), []). fof(dt_k19_monoid_1,axiom,( ! [A] : ( ~ v3_struct_0(k19_monoid_1(A)) & v1_vectsp_1(k19_monoid_1(A)) & m3_monoid_0(k19_monoid_1(A),k13_monoid_1(A)) ) ), file(monoid_1,k19_monoid_1), []). fof(dt_k1_card_1,axiom,( ! [A] : v1_card_1(k1_card_1(A)) ), file(card_1,k1_card_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_monoid_0,axiom,( ! [A,B,C] : ( ( m1_finseq_1(B,A) & m1_finseq_1(C,A) ) => m2_finseq_2(k1_monoid_0(A,B,C),A,k3_finseq_2(A)) ) ), file(monoid_0,k1_monoid_0), []). fof(dt_k1_nat_1,axiom,( ! [A,B] : ( ( m1_subset_1(A,k5_numbers) & m1_subset_1(B,k5_numbers) ) => m2_subset_1(k1_nat_1(A,B),k1_numbers,k5_numbers) ) ), file(nat_1,k1_nat_1), []). fof(dt_k1_numbers,axiom,( $true ), file(numbers,k1_numbers), []). fof(dt_k1_relat_1,axiom,( $true ), file(relat_1,k1_relat_1), []). fof(dt_k1_tarski,axiom,( $true ), file(tarski,k1_tarski), []). 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_k20_monoid_1,axiom,( ! [A,B] : ( ( ~ v1_xboole_0(A) & m1_finseq_1(B,A) ) => m1_subset_1(k20_monoid_1(A,B),u1_struct_0(k13_monoid_1(A))) ) ), file(monoid_1,k20_monoid_1), []). fof(dt_k2_finseq_2,axiom,( ! [A,B] : ( v4_ordinal2(A) => ( v1_relat_1(k2_finseq_2(A,B)) & v1_funct_1(k2_finseq_2(A,B)) & v1_finseq_1(k2_finseq_2(A,B)) ) ) ), file(finseq_2,k2_finseq_2), []). fof(dt_k2_funcop_1,axiom,( $true ), file(funcop_1,k2_funcop_1), []). fof(dt_k2_relat_1,axiom,( $true ), file(relat_1,k2_relat_1), []). fof(dt_k2_xcmplx_0,axiom,( $true ), file(xcmplx_0,k2_xcmplx_0), []). fof(dt_k2_zfmisc_1,axiom,( $true ), file(zfmisc_1,k2_zfmisc_1), []). fof(dt_k33_binop_2,axiom, ( v1_funct_1(k33_binop_2) & v1_funct_2(k33_binop_2,k2_zfmisc_1(k1_numbers,k1_numbers),k1_numbers) & m2_relset_1(k33_binop_2,k2_zfmisc_1(k1_numbers,k1_numbers),k1_numbers) ), file(binop_2,k33_binop_2), []). fof(dt_k3_finseq_2,axiom,( ! [A] : ( ~ v1_xboole_0(k3_finseq_2(A)) & m1_finseq_2(k3_finseq_2(A),A) ) ), file(finseq_2,k3_finseq_2), []). fof(dt_k3_gr_cy_1,axiom, ( ~ v3_struct_0(k3_gr_cy_1) & v1_group_1(k3_gr_cy_1) & v3_group_1(k3_gr_cy_1) & v4_group_1(k3_gr_cy_1) & l1_group_1(k3_gr_cy_1) ), file(gr_cy_1,k3_gr_cy_1), []). fof(dt_k3_xboole_0,axiom,( $true ), file(xboole_0,k3_xboole_0), []). fof(dt_k4_card_1,axiom,( ! [A] : ( v1_finset_1(A) => m2_subset_1(k4_card_1(A),k1_numbers,k5_numbers) ) ), file(card_1,k4_card_1), []). fof(dt_k4_monoid_0,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(k4_monoid_0(A,B,C),u1_struct_0(A)) ) ), file(monoid_0,k4_monoid_0), []). fof(dt_k4_monoid_1,axiom,( ! [A,B,C] : ( ( ~ v1_xboole_0(A) & m1_subset_1(B,k5_numbers) & m1_subset_1(C,A) ) => m2_finseq_1(k4_monoid_1(A,B,C),A) ) ), file(monoid_1,k4_monoid_1), []). 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_xboole_0,axiom,( $true ), file(xboole_0,k4_xboole_0), []). fof(dt_k5_monoid_0,axiom, ( ~ v3_struct_0(k5_monoid_0) & v1_group_1(k5_monoid_0) & v2_group_1(k5_monoid_0) & v4_group_1(k5_monoid_0) & v7_group_1(k5_monoid_0) & v13_monoid_0(k5_monoid_0) & v16_monoid_0(k5_monoid_0) & l1_group_1(k5_monoid_0) ), file(monoid_0,k5_monoid_0), []). fof(dt_k5_numbers,axiom,( m1_subset_1(k5_numbers,k1_zfmisc_1(k1_numbers)) ), file(numbers,k5_numbers), []). fof(dt_k5_ordinal2,axiom,( $true ), file(ordinal2,k5_ordinal2), []). fof(dt_k5_real_1,axiom,( ! [A,B] : ( ( m1_subset_1(A,k1_numbers) & m1_subset_1(B,k1_numbers) ) => m1_subset_1(k5_real_1(A,B),k1_numbers) ) ), file(real_1,k5_real_1), []). 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_k6_finseq_1,axiom,( ! [A] : ( v1_xboole_0(k6_finseq_1(A)) & m2_finseq_1(k6_finseq_1(A),A) ) ), file(finseq_1,k6_finseq_1), []). fof(dt_k6_monoid_0,axiom, ( ~ v3_struct_0(k6_monoid_0) & v1_group_1(k6_monoid_0) & v2_group_1(k6_monoid_0) & v13_monoid_0(k6_monoid_0) & m2_monoid_0(k6_monoid_0,k5_monoid_0) ), file(monoid_0,k6_monoid_0), []). fof(dt_k6_xcmplx_0,axiom,( $true ), file(xcmplx_0,k6_xcmplx_0), []). fof(dt_k7_finseq_1,axiom,( ! [A,B] : ( ( v1_relat_1(A) & v1_funct_1(A) & v1_finseq_1(A) & v1_relat_1(B) & v1_funct_1(B) & v1_finseq_1(B) ) => ( v1_relat_1(k7_finseq_1(A,B)) & v1_funct_1(k7_finseq_1(A,B)) & v1_finseq_1(k7_finseq_1(A,B)) ) ) ), file(finseq_1,k7_finseq_1), []). fof(dt_k7_monoid_0,axiom, ( ~ v3_struct_0(k7_monoid_0) & v1_group_1(k7_monoid_0) & v2_group_1(k7_monoid_0) & v17_monoid_0(k7_monoid_0) & m5_monoid_0(k7_monoid_0,k5_monoid_0,k6_monoid_0) ), file(monoid_0,k7_monoid_0), []). fof(dt_k8_monoid_0,axiom, ( ~ v3_struct_0(k8_monoid_0) & v1_vectsp_1(k8_monoid_0) & v1_vectsp_2(k8_monoid_0) & m1_monoid_0(k8_monoid_0,k7_monoid_0) ), file(monoid_0,k8_monoid_0), []). fof(dt_k9_monoid_1,axiom,( ! [A,B] : ( ( ~ v3_struct_0(A) & l1_group_1(A) ) => l1_group_1(k9_monoid_1(A,B)) ) ), file(monoid_1,k9_monoid_1), []). fof(dt_l1_group_1,axiom,( ! [A] : ( l1_group_1(A) => l1_struct_0(A) ) ), file(group_1,l1_group_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_m1_finseq_1,axiom,( ! [A,B] : ( m1_finseq_1(B,A) => ( v1_relat_1(B) & v1_funct_1(B) & v1_finseq_1(B) ) ) ), file(finseq_1,m1_finseq_1), []). fof(dt_m1_finseq_2,axiom,( $true ), file(finseq_2,m1_finseq_2), []). fof(dt_m1_monoid_0,axiom,( ! [A] : ( l1_group_1(A) => ! [B] : ( m1_monoid_0(B,A) => l1_vectsp_1(B) ) ) ), file(monoid_0,m1_monoid_0), []). 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_finseq_1,axiom,( ! [A,B] : ( m2_finseq_1(B,A) => ( v1_funct_1(B) & v1_finseq_1(B) & m2_relset_1(B,k5_numbers,A) ) ) ), file(finseq_1,m2_finseq_1), []). fof(dt_m2_finseq_2,axiom,( ! [A,B] : ( ( ~ v1_xboole_0(B) & m1_finseq_2(B,A) ) => ! [C] : ( m2_finseq_2(C,A,B) => m2_finseq_1(C,A) ) ) ), file(finseq_2,m2_finseq_2), []). fof(dt_m2_monoid_0,axiom,( ! [A] : ( l1_group_1(A) => ! [B] : ( m2_monoid_0(B,A) => l1_group_1(B) ) ) ), file(monoid_0,m2_monoid_0), []). 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_m3_monoid_0,axiom,( ! [A] : ( l1_group_1(A) => ! [B] : ( m3_monoid_0(B,A) => l1_vectsp_1(B) ) ) ), file(monoid_0,m3_monoid_0), []). fof(dt_m5_monoid_0,axiom,( ! [A,B] : ( ( l1_group_1(A) & m2_monoid_0(B,A) ) => ! [C] : ( m5_monoid_0(C,A,B) => m2_monoid_0(C,A) ) ) ), file(monoid_0,m5_monoid_0), []). 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_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(existence_l1_group_1,axiom,( ? [A] : l1_group_1(A) ), file(group_1,l1_group_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_m1_finseq_1,axiom,( ! [A] : ? [B] : m1_finseq_1(B,A) ), file(finseq_1,m1_finseq_1), []). fof(existence_m1_finseq_2,axiom,( ! [A] : ? [B] : m1_finseq_2(B,A) ), file(finseq_2,m1_finseq_2), []). fof(existence_m1_monoid_0,axiom,( ! [A] : ( l1_group_1(A) => ? [B] : m1_monoid_0(B,A) ) ), file(monoid_0,m1_monoid_0), []). 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_finseq_1,axiom,( ! [A] : ? [B] : m2_finseq_1(B,A) ), file(finseq_1,m2_finseq_1), []). fof(existence_m2_finseq_2,axiom,( ! [A,B] : ( ( ~ v1_xboole_0(B) & m1_finseq_2(B,A) ) => ? [C] : m2_finseq_2(C,A,B) ) ), file(finseq_2,m2_finseq_2), []). fof(existence_m2_monoid_0,axiom,( ! [A] : ( l1_group_1(A) => ? [B] : m2_monoid_0(B,A) ) ), file(monoid_0,m2_monoid_0), []). 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(existence_m3_monoid_0,axiom,( ! [A] : ( l1_group_1(A) => ? [B] : m3_monoid_0(B,A) ) ), file(monoid_0,m3_monoid_0), []). fof(existence_m5_monoid_0,axiom,( ! [A,B] : ( ( l1_group_1(A) & m2_monoid_0(B,A) ) => ? [C] : m5_monoid_0(C,A,B) ) ), file(monoid_0,m5_monoid_0), []). fof(fc10_finset_1,axiom,( ! [A,B] : ( v1_finset_1(B) => v1_finset_1(k3_xboole_0(A,B)) ) ), file(finset_1,fc10_finset_1), []). fof(fc10_membered,axiom,( ! [A] : ( v1_int_1(A) => ( v1_membered(k1_tarski(A)) & v2_membered(k1_tarski(A)) & v3_membered(k1_tarski(A)) & v4_membered(k1_tarski(A)) ) ) ), file(membered,fc10_membered), []). fof(fc11_finseq_1,axiom,( ! [A] : ( ( v1_relat_1(A) & v1_finset_1(A) ) => v1_finset_1(k2_relat_1(A)) ) ), file(finseq_1,fc11_finseq_1), []). fof(fc11_finset_1,axiom,( ! [A,B] : ( v1_finset_1(A) => v1_finset_1(k3_xboole_0(A,B)) ) ), file(finset_1,fc11_finset_1), []). fof(fc11_membered,axiom,( ! [A] : ( v4_ordinal2(A) => ( v1_membered(k1_tarski(A)) & v2_membered(k1_tarski(A)) & v3_membered(k1_tarski(A)) & v4_membered(k1_tarski(A)) & v5_membered(k1_tarski(A)) ) ) ), file(membered,fc11_membered), []). fof(fc12_finset_1,axiom,( ! [A,B] : ( v1_finset_1(A) => v1_finset_1(k4_xboole_0(A,B)) ) ), file(finset_1,fc12_finset_1), []). fof(fc12_relat_1,axiom, ( v1_xboole_0(k1_xboole_0) & v1_relat_1(k1_xboole_0) & v3_relat_1(k1_xboole_0) ), file(relat_1,fc12_relat_1), []). fof(fc13_finseq_1,axiom,( ! [A,B] : ( ( v1_relat_1(A) & v1_funct_1(A) & v1_finseq_1(A) & ~ v1_xboole_0(B) & v1_relat_1(B) & v1_funct_1(B) & v1_finseq_1(B) ) => ( ~ v1_xboole_0(k7_finseq_1(A,B)) & v1_relat_1(k7_finseq_1(A,B)) & v1_funct_1(k7_finseq_1(A,B)) & v1_finset_1(k7_finseq_1(A,B)) & v1_finseq_1(k7_finseq_1(A,B)) ) ) ), file(finseq_1,fc13_finseq_1), []). fof(fc14_finseq_1,axiom,( ! [A,B] : ( ( v1_relat_1(A) & v1_funct_1(A) & v1_finseq_1(A) & ~ v1_xboole_0(B) & v1_relat_1(B) & v1_funct_1(B) & v1_finseq_1(B) ) => ( ~ v1_xboole_0(k7_finseq_1(B,A)) & v1_relat_1(k7_finseq_1(B,A)) & v1_funct_1(k7_finseq_1(B,A)) & v1_finset_1(k7_finseq_1(B,A)) & v1_finseq_1(k7_finseq_1(B,A)) ) ) ), file(finseq_1,fc14_finseq_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(fc16_finseq_1,axiom,( ! [A] : ( ~ v1_xboole_0(k13_finseq_1(A)) & v1_fraenkel(k13_finseq_1(A)) ) ), file(finseq_1,fc16_finseq_1), []). fof(fc17_finseq_1,axiom,( ! [A] : ( ( v1_relat_1(A) & v1_funct_1(A) & v1_finset_1(A) ) => v1_finset_1(k1_relat_1(A)) ) ), file(finseq_1,fc17_finseq_1), []). fof(fc18_finseq_1,axiom,( ! [A,B] : ( ( v1_relat_1(A) & v1_funct_1(A) & v1_finset_1(A) ) => v1_finset_1(k10_relat_1(A,B)) ) ), file(finseq_1,fc18_finseq_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(fc1_funcop_1,axiom,( ! [A,B] : ( v1_relat_1(k2_funcop_1(A,B)) & v1_funct_1(k2_funcop_1(A,B)) ) ), file(funcop_1,fc1_funcop_1), []). fof(fc1_monoid_0,axiom,( ! [A,B,C] : ( ( ~ v1_xboole_0(A) & v1_funct_1(B) & v1_funct_2(B,k2_zfmisc_1(A,A),A) & m1_relset_1(B,k2_zfmisc_1(A,A),A) & m1_subset_1(C,A) ) => ( ~ v3_struct_0(g1_vectsp_1(A,B,C)) & v1_vectsp_1(g1_vectsp_1(A,B,C)) ) ) ), file(monoid_0,fc1_monoid_0), []). fof(fc1_monoid_1,axiom,( ! [A,B] : ( ( ~ v3_struct_0(A) & l1_group_1(A) ) => ~ v3_struct_0(k9_monoid_1(A,B)) ) ), file(monoid_1,fc1_monoid_1), []). fof(fc1_nat_1,axiom,( ! [A,B] : ( ( v4_ordinal2(A) & v4_ordinal2(B) ) => ( v4_ordinal2(k2_xcmplx_0(A,B)) & v1_xcmplx_0(k2_xcmplx_0(A,B)) & v1_xreal_0(k2_xcmplx_0(A,B)) ) ) ), file(nat_1,fc1_nat_1), []). fof(fc1_ordinal2,axiom, ( v1_ordinal1(k5_ordinal2) & v2_ordinal1(k5_ordinal2) & v3_ordinal1(k5_ordinal2) & ~ v1_xboole_0(k5_ordinal2) ), file(ordinal2,fc1_ordinal2), []). fof(fc1_relat_1,axiom,( ! [A,B] : ( ( v1_relat_1(A) & v1_relat_1(B) ) => v1_relat_1(k3_xboole_0(A,B)) ) ), file(relat_1,fc1_relat_1), []). 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(fc27_membered,axiom,( ! [A,B] : ( v1_membered(A) => v1_membered(k3_xboole_0(A,B)) ) ), file(membered,fc27_membered), []). fof(fc28_membered,axiom,( ! [A,B] : ( v1_membered(A) => v1_membered(k3_xboole_0(B,A)) ) ), file(membered,fc28_membered), []). fof(fc29_membered,axiom,( ! [A,B] : ( v2_membered(A) => ( v1_membered(k3_xboole_0(A,B)) & v2_membered(k3_xboole_0(A,B)) ) ) ), file(membered,fc29_membered), []). fof(fc2_finseq_1,axiom, ( v1_xboole_0(k1_xboole_0) & v1_relat_1(k1_xboole_0) & v3_relat_1(k1_xboole_0) & v1_funct_1(k1_xboole_0) & v2_funct_1(k1_xboole_0) & v1_finset_1(k1_xboole_0) & v1_finseq_1(k1_xboole_0) ), file(finseq_1,fc2_finseq_1), []). fof(fc2_funcop_1,axiom,( ! [A] : ( v1_xboole_0(k2_funcop_1(k1_xboole_0,A)) & v1_relat_1(k2_funcop_1(k1_xboole_0,A)) & v1_funct_1(k2_funcop_1(k1_xboole_0,A)) & v2_funct_1(k2_funcop_1(k1_xboole_0,A)) ) ), file(funcop_1,fc2_funcop_1), []). fof(fc2_membered,axiom, ( ~ v1_xboole_0(k1_numbers) & v1_membered(k1_numbers) & v2_membered(k1_numbers) ), file(membered,fc2_membered), []). fof(fc2_monoid_1,axiom,( ! [A,B] : ( ( ~ v3_struct_0(A) & l1_group_1(A) ) => ( ~ v3_struct_0(k9_monoid_1(A,B)) & v1_monoid_0(k9_monoid_1(A,B)) ) ) ), file(monoid_1,fc2_monoid_1), []). fof(fc2_subset_1,axiom,( ! [A] : ~ v1_xboole_0(k1_tarski(A)) ), file(subset_1,fc2_subset_1), []). fof(fc30_membered,axiom,( ! [A,B] : ( v2_membered(A) => ( v1_membered(k3_xboole_0(B,A)) & v2_membered(k3_xboole_0(B,A)) ) ) ), file(membered,fc30_membered), []). fof(fc31_membered,axiom,( ! [A,B] : ( v3_membered(A) => ( v1_membered(k3_xboole_0(A,B)) & v2_membered(k3_xboole_0(A,B)) & v3_membered(k3_xboole_0(A,B)) ) ) ), file(membered,fc31_membered), []). fof(fc32_membered,axiom,( ! [A,B] : ( v3_membered(A) => ( v1_membered(k3_xboole_0(B,A)) & v2_membered(k3_xboole_0(B,A)) & v3_membered(k3_xboole_0(B,A)) ) ) ), file(membered,fc32_membered), []). fof(fc33_membered,axiom,( ! [A,B] : ( v4_membered(A) => ( v1_membered(k3_xboole_0(A,B)) & v2_membered(k3_xboole_0(A,B)) & v3_membered(k3_xboole_0(A,B)) & v4_membered(k3_xboole_0(A,B)) ) ) ), file(membered,fc33_membered), []). fof(fc34_membered,axiom,( ! [A,B] : ( v4_membered(A) => ( v1_membered(k3_xboole_0(B,A)) & v2_membered(k3_xboole_0(B,A)) & v3_membered(k3_xboole_0(B,A)) & v4_membered(k3_xboole_0(B,A)) ) ) ), file(membered,fc34_membered), []). fof(fc35_membered,axiom,( ! [A,B] : ( v5_membered(A) => ( v1_membered(k3_xboole_0(A,B)) & v2_membered(k3_xboole_0(A,B)) & v3_membered(k3_xboole_0(A,B)) & v4_membered(k3_xboole_0(A,B)) & v5_membered(k3_xboole_0(A,B)) ) ) ), file(membered,fc35_membered), []). fof(fc36_membered,axiom,( ! [A,B] : ( v5_membered(A) => ( v1_membered(k3_xboole_0(B,A)) & v2_membered(k3_xboole_0(B,A)) & v3_membered(k3_xboole_0(B,A)) & v4_membered(k3_xboole_0(B,A)) & v5_membered(k3_xboole_0(B,A)) ) ) ), file(membered,fc36_membered), []). fof(fc37_membered,axiom,( ! [A,B] : ( v1_membered(A) => v1_membered(k4_xboole_0(A,B)) ) ), file(membered,fc37_membered), []). fof(fc38_membered,axiom,( ! [A,B] : ( v2_membered(A) => ( v1_membered(k4_xboole_0(A,B)) & v2_membered(k4_xboole_0(A,B)) ) ) ), file(membered,fc38_membered), []). fof(fc39_membered,axiom,( ! [A,B] : ( v3_membered(A) => ( v1_membered(k4_xboole_0(A,B)) & v2_membered(k4_xboole_0(A,B)) & v3_membered(k4_xboole_0(A,B)) ) ) ), file(membered,fc39_membered), []). fof(fc3_funcop_1,axiom,( ! [A,B] : ( v1_xboole_0(B) => ( v1_xboole_0(k2_funcop_1(B,A)) & v1_relat_1(k2_funcop_1(B,A)) & v1_funct_1(k2_funcop_1(B,A)) & v2_funct_1(k2_funcop_1(B,A)) ) ) ), file(funcop_1,fc3_funcop_1), []). fof(fc3_monoid_1,axiom,( ! [A,B,C] : ( ( ~ v3_struct_0(A) & l1_group_1(A) & ~ v1_xboole_0(B) & m1_subset_1(C,u1_struct_0(k9_monoid_1(A,B))) ) => ~ v1_xboole_0(k2_relat_1(C)) ) ), file(monoid_1,fc3_monoid_1), []). fof(fc3_nat_1,axiom,( ! [A,B] : ( ( v4_ordinal2(A) & ~ v1_xboole_0(B) & v4_ordinal2(B) ) => ( ~ v1_xboole_0(k2_xcmplx_0(A,B)) & v4_ordinal2(k2_xcmplx_0(A,B)) & v1_xcmplx_0(k2_xcmplx_0(A,B)) & v1_xreal_0(k2_xcmplx_0(A,B)) ) ) ), file(nat_1,fc3_nat_1), []). fof(fc3_relat_1,axiom,( ! [A,B] : ( ( v1_relat_1(A) & v1_relat_1(B) ) => v1_relat_1(k4_xboole_0(A,B)) ) ), file(relat_1,fc3_relat_1), []). fof(fc40_membered,axiom,( ! [A,B] : ( v4_membered(A) => ( v1_membered(k4_xboole_0(A,B)) & v2_membered(k4_xboole_0(A,B)) & v3_membered(k4_xboole_0(A,B)) & v4_membered(k4_xboole_0(A,B)) ) ) ), file(membered,fc40_membered), []). fof(fc41_membered,axiom,( ! [A,B] : ( v5_membered(A) => ( v1_membered(k4_xboole_0(A,B)) & v2_membered(k4_xboole_0(A,B)) & v3_membered(k4_xboole_0(A,B)) & v4_membered(k4_xboole_0(A,B)) & v5_membered(k4_xboole_0(A,B)) ) ) ), file(membered,fc41_membered), []). fof(fc4_nat_1,axiom,( ! [A,B] : ( ( v4_ordinal2(A) & ~ v1_xboole_0(B) & v4_ordinal2(B) ) => ( ~ v1_xboole_0(k2_xcmplx_0(B,A)) & v4_ordinal2(k2_xcmplx_0(B,A)) & v1_xcmplx_0(k2_xcmplx_0(B,A)) & v1_xreal_0(k2_xcmplx_0(B,A)) ) ) ), file(nat_1,fc4_nat_1), []). fof(fc4_ordinal2,axiom,( ! [A,B] : ( v3_ordinal1(B) => ( v1_relat_1(k2_funcop_1(A,B)) & v1_funct_1(k2_funcop_1(A,B)) & v1_ordinal2(k2_funcop_1(A,B)) ) ) ), file(ordinal2,fc4_ordinal2), []). fof(fc4_relat_1,axiom, ( v1_xboole_0(k1_xboole_0) & v1_relat_1(k1_xboole_0) ), file(relat_1,fc4_relat_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(fc5_membered,axiom, ( v1_membered(k5_ordinal2) & v2_membered(k5_ordinal2) & v3_membered(k5_ordinal2) & v4_membered(k5_ordinal2) & v5_membered(k5_ordinal2) ), file(membered,fc5_membered), []). fof(fc5_relat_1,axiom,( ! [A] : ( ( ~ v1_xboole_0(A) & v1_relat_1(A) ) => ~ v1_xboole_0(k1_relat_1(A)) ) ), file(relat_1,fc5_relat_1), []). fof(fc6_funct_1,axiom,( ! [A] : ( ( v1_relat_1(A) & v2_relat_1(A) & v1_funct_1(A) ) => v1_setfam_1(k2_relat_1(A)) ) ), file(funct_1,fc6_funct_1), []). fof(fc6_membered,axiom, ( v1_xboole_0(k1_xboole_0) & v1_membered(k1_xboole_0) & v2_membered(k1_xboole_0) & v3_membered(k1_xboole_0) & v4_membered(k1_xboole_0) & v5_membered(k1_xboole_0) ), file(membered,fc6_membered), []). fof(fc6_relat_1,axiom,( ! [A] : ( ( ~ v1_xboole_0(A) & v1_relat_1(A) ) => ~ v1_xboole_0(k2_relat_1(A)) ) ), file(relat_1,fc6_relat_1), []). fof(fc7_membered,axiom,( ! [A] : ( v1_xcmplx_0(A) => v1_membered(k1_tarski(A)) ) ), file(membered,fc7_membered), []). fof(fc7_relat_1,axiom,( ! [A] : ( v1_xboole_0(A) => ( v1_xboole_0(k1_relat_1(A)) & v1_relat_1(k1_relat_1(A)) ) ) ), file(relat_1,fc7_relat_1), []). fof(fc8_membered,axiom,( ! [A] : ( v1_xreal_0(A) => ( v1_membered(k1_tarski(A)) & v2_membered(k1_tarski(A)) ) ) ), file(membered,fc8_membered), []). fof(fc8_relat_1,axiom,( ! [A] : ( v1_xboole_0(A) => ( v1_xboole_0(k2_relat_1(A)) & v1_relat_1(k2_relat_1(A)) ) ) ), file(relat_1,fc8_relat_1), []). fof(fc9_finseq_1,axiom,( ! [A] : ~ v1_xboole_0(k13_finseq_1(A)) ), file(finseq_1,fc9_finseq_1), []). fof(fc9_funcop_1,axiom,( ! [A,B] : ( ~ v1_xboole_0(B) => ( v1_relat_1(k2_funcop_1(A,B)) & v2_relat_1(k2_funcop_1(A,B)) & v1_funct_1(k2_funcop_1(A,B)) ) ) ), file(funcop_1,fc9_funcop_1), []). fof(fc9_membered,axiom,( ! [A] : ( v1_rat_1(A) => ( v1_membered(k1_tarski(A)) & v2_membered(k1_tarski(A)) & v3_membered(k1_tarski(A)) ) ) ), file(membered,fc9_membered), []). fof(free_g1_group_1,axiom,( ! [A,B] : ( ( v1_funct_1(B) & v1_funct_2(B,k2_zfmisc_1(A,A),A) & m1_relset_1(B,k2_zfmisc_1(A,A),A) ) => ! [C,D] : ( g1_group_1(A,B) = g1_group_1(C,D) => ( A = C & B = D ) ) ) ), file(group_1,g1_group_1), []). fof(free_g1_vectsp_1,axiom,( ! [A,B,C] : ( ( v1_funct_1(B) & v1_funct_2(B,k2_zfmisc_1(A,A),A) & m1_relset_1(B,k2_zfmisc_1(A,A),A) & m1_subset_1(C,A) ) => ! [D,E,F] : ( g1_vectsp_1(A,B,C) = g1_vectsp_1(D,E,F) => ( A = D & B = E & C = F ) ) ) ), file(vectsp_1,g1_vectsp_1), []). fof(idempotence_k3_xboole_0,axiom,( ! [A,B] : k3_xboole_0(A,A) = A ), file(xboole_0,k3_xboole_0), []). fof(rc10_monoid_0,axiom,( ! [A] : ( l1_group_1(A) => ? [B] : ( m3_monoid_0(B,A) & v1_vectsp_1(B) ) ) ), file(monoid_0,rc10_monoid_0), []). fof(rc11_monoid_0,axiom,( ! [A] : ( ( ~ v3_struct_0(A) & l1_group_1(A) ) => ? [B] : ( m3_monoid_0(B,A) & ~ v3_struct_0(B) & v1_vectsp_1(B) ) ) ), file(monoid_0,rc11_monoid_0), []). fof(rc1_finseq_1,axiom,( ? [A] : ( v1_relat_1(A) & v1_funct_1(A) & v1_finseq_1(A) ) ), file(finseq_1,rc1_finseq_1), []). fof(rc1_finset_1,axiom,( ? [A] : ( ~ v1_xboole_0(A) & v1_finset_1(A) ) ), file(finset_1,rc1_finset_1), []). fof(rc1_funct_1,axiom,( ? [A] : ( v1_relat_1(A) & v1_funct_1(A) ) ), file(funct_1,rc1_funct_1), []). fof(rc1_membered,axiom,( ? [A] : ( ~ v1_xboole_0(A) & v1_membered(A) & v2_membered(A) & v3_membered(A) & v4_membered(A) & v5_membered(A) ) ), file(membered,rc1_membered), []). fof(rc1_monoid_0,axiom,( ? [A] : ( l1_struct_0(A) & v1_monoid_0(A) ) ), file(monoid_0,rc1_monoid_0), []). fof(rc1_nat_1,axiom,( ? [A] : ( ~ v1_xboole_0(A) & v4_ordinal2(A) & v1_xcmplx_0(A) & v1_xreal_0(A) ) ), file(nat_1,rc1_nat_1), []). fof(rc1_relat_1,axiom,( ? [A] : ( v1_xboole_0(A) & v1_relat_1(A) ) ), file(relat_1,rc1_relat_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_finseq_1,axiom,( ! [A] : ? [B] : ( m1_relset_1(B,k5_numbers,A) & v1_relat_1(B) & v1_funct_1(B) & v1_finset_1(B) & v1_finseq_1(B) ) ), file(finseq_1,rc2_finseq_1), []). fof(rc2_finset_1,axiom,( ! [A] : ? [B] : ( m1_subset_1(B,k1_zfmisc_1(A)) & v1_xboole_0(B) & v1_relat_1(B) & v1_funct_1(B) & v2_funct_1(B) & v1_ordinal1(B) & v2_ordinal1(B) & v3_ordinal1(B) & v4_ordinal2(B) & v1_finset_1(B) ) ), file(finset_1,rc2_finset_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(rc2_nat_1,axiom,( ? [A] : ( m1_subset_1(A,k1_zfmisc_1(k1_numbers)) & ~ v1_xboole_0(A) & v3_ordinal1(A) ) ), file(nat_1,rc2_nat_1), []). fof(rc2_relat_1,axiom,( ? [A] : ( ~ v1_xboole_0(A) & v1_relat_1(A) ) ), file(relat_1,rc2_relat_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(rc3_finseq_1,axiom,( ? [A] : ( v1_xboole_0(A) & v1_relat_1(A) & v1_funct_1(A) & v2_funct_1(A) & v1_finset_1(A) & v1_finseq_1(A) ) ), file(finseq_1,rc3_finseq_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(rc3_funct_1,axiom,( ? [A] : ( v1_relat_1(A) & v1_funct_1(A) & v2_funct_1(A) ) ), file(funct_1,rc3_funct_1), []). fof(rc3_nat_1,axiom,( ? [A] : ( m1_subset_1(A,k5_numbers) & ~ v1_xboole_0(A) & v1_ordinal1(A) & v2_ordinal1(A) & v3_ordinal1(A) & v4_ordinal2(A) & v1_xcmplx_0(A) & v1_xreal_0(A) ) ), file(nat_1,rc3_nat_1), []). fof(rc3_relat_1,axiom,( ? [A] : ( v1_relat_1(A) & v3_relat_1(A) ) ), file(relat_1,rc3_relat_1), []). fof(rc3_struct_0,axiom,( ? [A] : ( l1_struct_0(A) & ~ v3_struct_0(A) ) ), file(struct_0,rc3_struct_0), []). fof(rc4_finseq_1,axiom,( ! [A] : ? [B] : ( m1_finseq_1(B,A) & v1_xboole_0(B) & v1_relat_1(B) & v1_funct_1(B) & v2_funct_1(B) & v1_finset_1(B) & v1_finseq_1(B) ) ), file(finseq_1,rc4_finseq_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(rc4_funct_1,axiom,( ? [A] : ( v1_relat_1(A) & v3_relat_1(A) & v1_funct_1(A) ) ), file(funct_1,rc4_funct_1), []). fof(rc5_funct_1,axiom,( ? [A] : ( v1_relat_1(A) & v2_relat_1(A) & v1_funct_1(A) ) ), file(funct_1,rc5_funct_1), []). fof(rc5_monoid_0,axiom,( ! [A] : ( l1_group_1(A) => ? [B] : ( m1_monoid_0(B,A) & v1_vectsp_1(B) ) ) ), file(monoid_0,rc5_monoid_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(rc6_finseq_1,axiom,( ? [A] : ( v1_xboole_0(A) & v1_relat_1(A) & v1_funct_1(A) & v2_funct_1(A) & v1_finset_1(A) ) ), file(finseq_1,rc6_finseq_1), []). fof(rc6_monoid_0,axiom,( ! [A] : ( ( ~ v3_struct_0(A) & v2_group_1(A) & l1_group_1(A) ) => ? [B] : ( m1_monoid_0(B,A) & ~ v3_struct_0(B) & v2_group_1(B) & v1_vectsp_1(B) & v6_vectsp_1(B) & v8_vectsp_1(B) & v1_vectsp_2(B) ) ) ), file(monoid_0,rc6_monoid_0), []). fof(rc7_finseq_1,axiom,( ? [A] : ( ~ v1_xboole_0(A) & v1_relat_1(A) & v1_funct_1(A) & v1_finset_1(A) ) ), file(finseq_1,rc7_finseq_1), []). fof(rc7_monoid_0,axiom,( ! [A] : ( l1_group_1(A) => ? [B] : ( m2_monoid_0(B,A) & v1_group_1(B) ) ) ), file(monoid_0,rc7_monoid_0), []). fof(rc8_finseq_1,axiom,( ? [A] : ( ~ v1_xboole_0(A) & v1_relat_1(A) & v1_funct_1(A) & v1_finset_1(A) & v1_finseq_1(A) ) ), file(finseq_1,rc8_finseq_1), []). fof(rc8_monoid_0,axiom,( ! [A] : ( ( ~ v3_struct_0(A) & l1_group_1(A) ) => ? [B] : ( m2_monoid_0(B,A) & ~ v3_struct_0(B) & v1_group_1(B) ) ) ), file(monoid_0,rc8_monoid_0), []). fof(redefinition_k12_monoid_1,axiom,( ! [A,B] : ( ( ~ v3_struct_0(A) & v2_group_1(A) & v4_group_1(A) & v7_group_1(A) & v16_monoid_0(A) & v17_monoid_0(A) & l1_group_1(A) ) => k12_monoid_1(A,B) = k9_monoid_1(A,B) ) ), file(monoid_1,k12_monoid_1), []). fof(redefinition_k14_monoid_1,axiom,( ! [A,B] : ( ( ~ v1_xboole_0(A) & m1_subset_1(B,u1_struct_0(k13_monoid_1(A))) ) => k14_monoid_1(A,B) = k2_relat_1(B) ) ), file(monoid_1,k14_monoid_1), []). fof(redefinition_k15_monoid_1,axiom,( ! [A,B,C] : ( ( ~ v1_xboole_0(A) & m1_subset_1(B,u1_struct_0(k13_monoid_1(A))) & m1_subset_1(C,A) ) => k15_monoid_1(A,B,C) = k1_funct_1(B,C) ) ), file(monoid_1,k15_monoid_1), []). fof(redefinition_k17_monoid_1,axiom,( ! [A,B] : ( m1_subset_1(B,k5_numbers) => k17_monoid_1(A,B) = k2_funcop_1(A,B) ) ), file(monoid_1,k17_monoid_1), []). fof(redefinition_k1_monoid_0,axiom,( ! [A,B,C] : ( ( m1_finseq_1(B,A) & m1_finseq_1(C,A) ) => k1_monoid_0(A,B,C) = k7_finseq_1(B,C) ) ), file(monoid_0,k1_monoid_0), []). fof(redefinition_k1_nat_1,axiom,( ! [A,B] : ( ( m1_subset_1(A,k5_numbers) & m1_subset_1(B,k5_numbers) ) => k1_nat_1(A,B) = k2_xcmplx_0(A,B) ) ), file(nat_1,k1_nat_1), []). fof(redefinition_k3_finseq_2,axiom,( ! [A] : k3_finseq_2(A) = k13_finseq_1(A) ), file(finseq_2,k3_finseq_2), []). fof(redefinition_k4_card_1,axiom,( ! [A] : ( v1_finset_1(A) => k4_card_1(A) = k1_card_1(A) ) ), file(card_1,k4_card_1), []). fof(redefinition_k4_monoid_0,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)) ) => k4_monoid_0(A,B,C) = k1_group_1(A,B,C) ) ), file(monoid_0,k4_monoid_0), []). fof(redefinition_k4_monoid_1,axiom,( ! [A,B,C] : ( ( ~ v1_xboole_0(A) & m1_subset_1(B,k5_numbers) & m1_subset_1(C,A) ) => k4_monoid_1(A,B,C) = k2_finseq_2(B,C) ) ), file(monoid_1,k4_monoid_1), []). 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_k5_numbers,axiom,( k5_numbers = k5_ordinal2 ), file(numbers,k5_numbers), []). fof(redefinition_k5_real_1,axiom,( ! [A,B] : ( ( m1_subset_1(A,k1_numbers) & m1_subset_1(B,k1_numbers) ) => k5_real_1(A,B) = k6_xcmplx_0(A,B) ) ), file(real_1,k5_real_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(redefinition_k6_monoid_0,axiom,( k6_monoid_0 = k3_gr_cy_1 ), file(monoid_0,k6_monoid_0), []). fof(redefinition_m2_finseq_1,axiom,( ! [A,B] : ( m2_finseq_1(B,A) <=> m1_finseq_1(B,A) ) ), file(finseq_1,m2_finseq_1), []). fof(redefinition_m2_finseq_2,axiom,( ! [A,B] : ( ( ~ v1_xboole_0(B) & m1_finseq_2(B,A) ) => ! [C] : ( m2_finseq_2(C,A,B) <=> m1_subset_1(C,B) ) ) ), file(finseq_2,m2_finseq_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(redefinition_m5_monoid_0,axiom,( ! [A,B] : ( ( l1_group_1(A) & m2_monoid_0(B,A) ) => ! [C] : ( m5_monoid_0(C,A,B) <=> m2_monoid_0(C,B) ) ) ), file(monoid_0,m5_monoid_0), []). fof(reflexivity_r1_tarski,axiom,( ! [A,B] : r1_tarski(A,A) ), file(tarski,r1_tarski), []). fof(s1_nat_1__e3_71__monoid_1,axiom,( ! [A] : ( ~ v1_xboole_0(A) => ( ( ! [B] : ( m1_subset_1(B,u1_struct_0(k19_monoid_1(A))) => ~ ( ! [C] : ( v1_finset_1(C) => ( C = k10_relat_1(B,k4_xboole_0(k5_numbers,k6_domain_1(k5_numbers,0))) => k4_card_1(C) = 0 ) ) & ! [C] : ( m2_finseq_1(C,A) => B != k20_monoid_1(A,C) ) ) ) & ! [D] : ( m2_subset_1(D,k1_numbers,k5_numbers) => ( ! [E] : ( m1_subset_1(E,u1_struct_0(k19_monoid_1(A))) => ~ ( ! [F] : ( v1_finset_1(F) => ( F = k10_relat_1(E,k4_xboole_0(k5_numbers,k6_domain_1(k5_numbers,0))) => k4_card_1(F) = D ) ) & ! [F] : ( m2_finseq_1(F,A) => E != k20_monoid_1(A,F) ) ) ) => ! [G] : ( m1_subset_1(G,u1_struct_0(k19_monoid_1(A))) => ~ ( ! [H] : ( v1_finset_1(H) => ( H = k10_relat_1(G,k4_xboole_0(k5_numbers,k6_domain_1(k5_numbers,0))) => k4_card_1(H) = k1_nat_1(D,1) ) ) & ! [H] : ( m2_finseq_1(H,A) => G != k20_monoid_1(A,H) ) ) ) ) ) ) => ! [D] : ( m2_subset_1(D,k1_numbers,k5_numbers) => ! [I] : ( m1_subset_1(I,u1_struct_0(k19_monoid_1(A))) => ~ ( ! [J] : ( v1_finset_1(J) => ( J = k10_relat_1(I,k4_xboole_0(k5_numbers,k6_domain_1(k5_numbers,0))) => k4_card_1(J) = D ) ) & ! [J] : ( m2_finseq_1(J,A) => I != k20_monoid_1(A,J) ) ) ) ) ) ) ), file(monoid_1,s1_nat_1__e3_71__monoid_1), []). fof(s1_partfun1__e10_71_2__monoid_1,axiom,( ! [A,B,C] : ( ( ~ v1_xboole_0(A) & m1_subset_1(B,u1_struct_0(k19_monoid_1(A))) & m1_subset_1(C,A) ) => ? [D] : ( v1_relat_1(D) & v1_funct_1(D) & k1_relat_1(D) = A & ! [E] : ( r2_hidden(E,A) => ( ( C = E => k1_funct_1(D,E) = 0 ) & ( C != E => k1_funct_1(D,E) = k1_funct_1(B,E) ) ) ) ) ) ), file(monoid_1,s1_partfun1__e10_71_2__monoid_1), []). fof(spc0_boole,axiom,( v1_xboole_0(0) ), file(boole,spc0_boole), []). fof(spc1_boole,axiom,( ~ v1_xboole_0(1) ), file(boole,spc1_boole), []). fof(spc1_numerals,axiom, ( v2_xreal_0(1) & m1_subset_1(1,k5_numbers) ), file(numerals,spc1_numerals), []). fof(symmetry_r1_xboole_0,axiom,( ! [A,B] : ( r1_xboole_0(A,B) => r1_xboole_0(B,A) ) ), file(xboole_0,r1_xboole_0), []). fof(t11_relset_1,axiom,( ! [A,B,C] : ( v1_relat_1(C) => ( ( r1_tarski(k1_relat_1(C),A) & r1_tarski(k2_relat_1(C),B) ) => m2_relset_1(C,A,B) ) ) ), file(relset_1,t11_relset_1), []). fof(t15_funcop_1,axiom,( ! [A] : ( ( v1_relat_1(A) & v1_funct_1(A) ) => ! [B] : ( k2_relat_1(A) = k1_tarski(B) => A = k2_funcop_1(k1_relat_1(A),B) ) ) ), file(funcop_1,t15_funcop_1), []). fof(t16_finset_1,axiom,( ! [A,B] : ( v1_finset_1(A) => v1_finset_1(k4_xboole_0(A,B)) ) ), file(finset_1,t16_finset_1), []). fof(t173_relat_1,axiom,( ! [A,B] : ( v1_relat_1(B) => ( k10_relat_1(B,A) = k1_xboole_0 <=> r1_xboole_0(k2_relat_1(B),A) ) ) ), file(relat_1,t173_relat_1), []). fof(t1_arithm,axiom,( ! [A] : ( v1_xcmplx_0(A) => k2_xcmplx_0(A,0) = A ) ), file(arithm,t1_arithm), []). fof(t1_numerals,axiom,( m1_subset_1(0,k5_numbers) ), file(numerals,t1_numerals), []). fof(t1_subset,axiom,( ! [A,B] : ( r2_hidden(A,B) => m1_subset_1(A,B) ) ), file(subset,t1_subset), []). fof(t25_monoid_0,axiom,( ! [A] : ( ( ~ v3_struct_0(A) & l1_group_1(A) ) => ! [B] : ( ( ~ v3_struct_0(B) & m2_monoid_0(B,A) ) => ! [C] : ( ( ~ v3_struct_0(C) & m3_monoid_0(C,A) ) => ( r1_tarski(u1_struct_0(B),u1_struct_0(A)) & r1_tarski(u1_struct_0(C),u1_struct_0(A)) ) ) ) ) ), file(monoid_0,t25_monoid_0), []). fof(t28_monoid_1,axiom,( ! [A,B] : ( m1_subset_1(A,u1_struct_0(k13_monoid_1(B))) <=> ( v1_funct_1(A) & v1_funct_2(A,B,k5_numbers) & m2_relset_1(A,B,k5_numbers) ) ) ), file(monoid_1,t28_monoid_1), []). fof(t28_xboole_1,axiom,( ! [A,B] : ( r1_tarski(A,B) => k3_xboole_0(A,B) = A ) ), file(xboole_1,t28_xboole_1), []). fof(t29_monoid_1,axiom,( ! [A,B] : ( m1_subset_1(B,u1_struct_0(k13_monoid_1(A))) => ( k1_relat_1(B) = A & r1_tarski(k2_relat_1(B),k5_numbers) ) ) ), file(monoid_1,t29_monoid_1), []). fof(t2_boole,axiom,( ! [A] : k3_xboole_0(A,k1_xboole_0) = k1_xboole_0 ), file(boole,t2_boole), []). fof(t2_subset,axiom,( ! [A,B] : ( m1_subset_1(A,B) => ( v1_xboole_0(B) | r2_hidden(A,B) ) ) ), file(subset,t2_subset), []). fof(t30_monoid_1,axiom,( ! [A] : ( ~ v1_xboole_0(A) => ! [B] : ( m1_subset_1(B,u1_struct_0(k13_monoid_1(A))) => ! [C] : ( m1_subset_1(C,u1_struct_0(k13_monoid_1(A))) => ! [D] : ( m1_subset_1(D,A) => k15_monoid_1(A,k4_monoid_0(k13_monoid_1(A),B,C),D) = k1_nat_1(k15_monoid_1(A,B,D),k15_monoid_1(A,C,D)) ) ) ) ) ), file(monoid_1,t30_monoid_1), []). fof(t33_monoid_1,axiom,( ! [A] : ( ~ v1_xboole_0(A) => ! [B] : ( m1_subset_1(B,u1_struct_0(k13_monoid_1(A))) => ! [C] : ( m1_subset_1(C,u1_struct_0(k13_monoid_1(A))) => ( ! [D] : ( m1_subset_1(D,A) => k15_monoid_1(A,B,D) = k15_monoid_1(A,C,D) ) => B = C ) ) ) ) ), file(monoid_1,t33_monoid_1), []). fof(t37_xboole_1,axiom,( ! [A,B] : ( k4_xboole_0(A,B) = k1_xboole_0 <=> r1_tarski(A,B) ) ), file(xboole_1,t37_xboole_1), []). fof(t37_zfmisc_1,axiom,( ! [A,B] : ( r1_tarski(k1_tarski(A),B) <=> r2_hidden(A,B) ) ), file(zfmisc_1,t37_zfmisc_1), []). fof(t38_monoid_1,axiom,( ! [A] : ( ~ v1_xboole_0(A) => k20_monoid_1(A,k6_finseq_1(A)) = k17_monoid_1(A,0) ) ), file(monoid_1,t38_monoid_1), []). fof(t3_boole,axiom,( ! [A] : k4_xboole_0(A,k1_xboole_0) = A ), file(boole,t3_boole), []). fof(t3_subset,axiom,( ! [A,B] : ( m1_subset_1(A,k1_zfmisc_1(B)) <=> r1_tarski(A,B) ) ), file(subset,t3_subset), []). fof(t41_monoid_1,axiom,( ! [A] : ( ~ v1_xboole_0(A) => ! [B] : ( m2_finseq_1(B,A) => ! [C] : ( m2_finseq_1(C,A) => k20_monoid_1(A,k1_monoid_0(A,B,C)) = k4_monoid_0(k13_monoid_1(A),k20_monoid_1(A,B),k20_monoid_1(A,C)) ) ) ) ), file(monoid_1,t41_monoid_1), []). fof(t42_monoid_1,axiom,( ! [A] : ( m2_subset_1(A,k1_numbers,k5_numbers) => ! [B] : ( ~ v1_xboole_0(B) => ! [C] : ( m1_subset_1(C,B) => ( k15_monoid_1(B,k20_monoid_1(B,k4_monoid_1(B,A,C)),C) = A & ! [D] : ( m1_subset_1(D,B) => ( D != C => k15_monoid_1(B,k20_monoid_1(B,k4_monoid_1(B,A,C)),D) = 0 ) ) ) ) ) ) ), file(monoid_1,t42_monoid_1), []). fof(t47_card_1,axiom,( k1_card_1(k1_xboole_0) = k1_xboole_0 ), file(card_1,t47_card_1), []). fof(t49_xboole_1,axiom,( ! [A,B,C] : k3_xboole_0(A,k4_xboole_0(B,C)) = k4_xboole_0(k3_xboole_0(A,B),C) ), file(xboole_1,t49_xboole_1), []). fof(t4_arithm,axiom,( ! [A] : ( v1_xcmplx_0(A) => k6_xcmplx_0(A,0) = A ) ), file(arithm,t4_arithm), []). fof(t4_boole,axiom,( ! [A] : k4_xboole_0(k1_xboole_0,A) = k1_xboole_0 ), file(boole,t4_boole), []). 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_card_2,axiom,( ! [A] : ( k1_card_1(A) = 0 => A = k1_xboole_0 ) ), file(card_2,t59_card_2), []). 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(t63_card_2,axiom,( ! [A] : ( v1_finset_1(A) => ! [B] : ( v1_finset_1(B) => ( r1_tarski(B,A) => k4_card_1(k4_xboole_0(A,B)) = k5_real_1(k4_card_1(A),k4_card_1(B)) ) ) ) ), file(card_2,t63_card_2), []). fof(t6_boole,axiom,( ! [A] : ( v1_xboole_0(A) => A = k1_xboole_0 ) ), file(boole,t6_boole), []). fof(t79_card_1,axiom,( ! [A] : k4_card_1(k1_tarski(A)) = 1 ), file(card_1,t79_card_1), []). 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), []).