% Mizar problem: t31_hilbasis,hilbasis,2078,26 fof(t31_hilbasis,conjecture,( ! [A] : ( ( ~ v3_struct_0(A) & v2_group_1(A) & v4_group_1(A) & v7_group_1(A) & v3_rlvect_1(A) & v4_rlvect_1(A) & v5_rlvect_1(A) & v6_rlvect_1(A) & v7_vectsp_1(A) & ~ v3_realset2(A) & l3_vectsp_1(A) ) => ! [B] : ( m2_subset_1(B,k1_numbers,k5_numbers) => ? [C] : ( v1_funct_1(C) & v1_funct_2(C,u1_struct_0(k16_polynom3(k30_polynom1(B,A))),u1_struct_0(k30_polynom1(k1_nat_1(B,1),A))) & m2_relset_1(C,u1_struct_0(k16_polynom3(k30_polynom1(B,A))),u1_struct_0(k30_polynom1(k1_nat_1(B,1),A))) & v4_quofield(C,k16_polynom3(k30_polynom1(B,A)),k30_polynom1(k1_nat_1(B,1),A)) ) ) ) ), inference(mizar_bg_added,[status(thm)],[abstractness_v3_vectsp_1,antisymmetry_r2_hidden,cc10_membered,cc11_membered,cc12_membered,cc13_membered,cc14_membered,cc15_membered,cc16_membered,cc17_membered,cc18_membered,cc19_membered,cc1_algstr_1,cc1_binom,cc1_card_1,cc1_card_5,cc1_finseq_5,cc1_finset_1,cc1_funct_1,cc1_funct_7,cc1_gcd_1,cc1_membered,cc1_ordinal1,cc1_relat_1,cc1_relset_1,cc1_vectsp_1,cc1_xreal_0,cc20_membered,cc2_algstr_1,cc2_binom,cc2_card_1,cc2_card_5,cc2_finset_1,cc2_funct_1,cc2_funct_7,cc2_gcd_1,cc2_gobrd13,cc2_int_1,cc2_membered,cc2_ordinal1,cc2_vectsp_1,cc2_xreal_0,cc3_algstr_1,cc3_binom,cc3_card_1,cc3_gcd_1,cc3_int_1,cc3_membered,cc3_ordinal1,cc3_polynom1,cc3_vectsp_1,cc3_xreal_0,cc4_algstr_1,cc4_binom,cc4_gcd_1,cc4_int_1,cc4_membered,cc4_polynom1,cc4_vectsp_1,cc4_xreal_0,cc5_binom,cc5_polynom1,cc5_vectsp_1,cc5_xreal_0,cc6_membered,cc6_xreal_0,cc7_xreal_0,cc8_xreal_0,cc9_membered,commutativity_k1_nat_1,commutativity_k2_xcmplx_0,d10_xboole_0,d1_funct_2,d21_quofield,d22_quofield,d23_quofield,d24_quofield,d3_tarski,d5_funct_1,dt_g3_vectsp_1,dt_k16_polynom3,dt_k1_funct_1,dt_k1_nat_1,dt_k1_numbers,dt_k1_relat_1,dt_k1_xboole_0,dt_k1_zfmisc_1,dt_k2_relat_1,dt_k2_xcmplx_0,dt_k2_zfmisc_1,dt_k30_polynom1,dt_k4_relset_1,dt_k5_numbers,dt_k5_ordinal2,dt_k5_relset_1,dt_k6_hilbasis,dt_k7_hilbasis,dt_k8_funct_2,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,fc10_polynom3,fc10_xreal_0,fc11_polynom3,fc11_xreal_0,fc12_polynom3,fc12_relat_1,fc12_xreal_0,fc13_polynom3,fc14_finset_1,fc14_polynom1,fc14_polynom3,fc15_polynom3,fc16_polynom3,fc1_int_1,fc1_ordinal2,fc1_polynom1,fc1_struct_0,fc1_subset_1,fc1_xboole_0,fc20_polynom1,fc2_hilbasis,fc2_membered,fc2_ordinal1,fc38_polynom1,fc39_polynom1,fc3_funct_7,fc3_hilbasis,fc3_polynom1,fc3_polynom2,fc3_xreal_0,fc40_polynom1,fc41_polynom1,fc42_polynom1,fc43_polynom1,fc44_polynom1,fc4_hilbasis,fc4_relat_1,fc4_subset_1,fc5_hilbasis,fc5_membered,fc5_polynom1,fc5_relat_1,fc6_int_1,fc6_membered,fc6_relat_1,fc7_relat_1,fc7_xreal_0,fc8_relat_1,fc8_xreal_0,fc9_polynom3,fc9_xreal_0,free_g3_vectsp_1,rc11_polynom1,rc12_polynom1,rc1_binom,rc1_card_1,rc1_card_5,rc1_finseq_5,rc1_finset_1,rc1_funct_1,rc1_ideal_1,rc1_int_1,rc1_membered,rc1_ordinal1,rc1_polynom2,rc1_pre_circ,rc1_relat_1,rc1_subset_1,rc1_xboole_0,rc1_xreal_0,rc2_binom,rc2_card_1,rc2_card_5,rc2_finset_1,rc2_funct_1,rc2_ideal_1,rc2_int_1,rc2_ordinal1,rc2_polynom1,rc2_relat_1,rc2_subset_1,rc2_xboole_0,rc2_xreal_0,rc3_binom,rc3_finset_1,rc3_funct_1,rc3_ordinal1,rc3_relat_1,rc3_struct_0,rc3_xreal_0,rc4_binom,rc4_finset_1,rc4_funct_1,rc4_struct_0,rc4_xreal_0,rc5_struct_0,rc6_vectsp_1,rc7_polynom1,rc7_vectsp_1,rc8_vectsp_1,rc9_vectsp_1,redefinition_k1_nat_1,redefinition_k4_relset_1,redefinition_k5_numbers,redefinition_k5_relset_1,redefinition_k8_funct_2,redefinition_m2_relset_1,redefinition_m2_subset_1,reflexivity_r1_tarski,spc1_boole,spc1_numerals,t1_subset,t2_subset,t30_hilbasis,t3_subset,t4_subset,t5_subset,t6_boole,t7_boole,t8_boole]), [file(hilbasis,t31_hilbasis)]). 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(cc10_membered,axiom,( ! [A] : ( v1_membered(A) => ! [B] : ( m1_subset_1(B,A) => v1_xcmplx_0(B) ) ) ), file(membered,cc10_membered), []). 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(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(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(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(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(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(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(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(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_binom,axiom,( ! [A] : ( l1_rlvect_1(A) => ( ( ~ v3_struct_0(A) & v2_algstr_1(A) & v3_algstr_1(A) ) => ( ~ v3_struct_0(A) & v1_binom(A) ) ) ) ), file(binom,cc1_binom), []). fof(cc1_card_1,axiom,( ! [A] : ( v1_card_1(A) => ( v1_ordinal1(A) & v2_ordinal1(A) & v3_ordinal1(A) ) ) ), file(card_1,cc1_card_1), []). fof(cc1_card_5,axiom,( ! [A] : ( ~ v1_finset_1(A) => ~ v1_xboole_0(A) ) ), file(card_5,cc1_card_5), []). fof(cc1_finseq_5,axiom,( ! [A] : ( v1_xboole_0(A) => v1_realset1(A) ) ), file(finseq_5,cc1_finseq_5), []). 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_funct_7,axiom,( ! [A] : ( ( v1_relat_1(A) & v1_funct_1(A) & v1_finseq_1(A) & v1_funct_7(A) ) => ( v1_relat_1(A) & v1_funct_1(A) & v1_finset_1(A) & v1_finseq_1(A) & v1_funcop_1(A) ) ) ), file(funct_7,cc1_funct_7), []). 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_membered,axiom,( ! [A] : ( v5_membered(A) => v4_membered(A) ) ), file(membered,cc1_membered), []). fof(cc1_ordinal1,axiom,( ! [A] : ( v3_ordinal1(A) => ( v1_ordinal1(A) & v2_ordinal1(A) ) ) ), file(ordinal1,cc1_ordinal1), []). 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(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(cc1_xreal_0,axiom,( ! [A] : ( v4_ordinal2(A) => v1_xreal_0(A) ) ), file(xreal_0,cc1_xreal_0), []). 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(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_binom,axiom,( ! [A] : ( l1_rlvect_1(A) => ( ( ~ v3_struct_0(A) & v1_binom(A) ) => ( ~ v3_struct_0(A) & v2_algstr_1(A) & v3_algstr_1(A) ) ) ) ), file(binom,cc2_binom), []). fof(cc2_card_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) & v1_card_1(A) ) ) ), file(card_1,cc2_card_1), []). fof(cc2_card_5,axiom,( ! [A] : ( ( ~ v1_finset_1(A) & v1_card_1(A) ) => ! [B] : ( m1_subset_1(B,A) => ( v1_ordinal1(B) & v2_ordinal1(B) & v3_ordinal1(B) ) ) ) ), file(card_5,cc2_card_5), []). 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_funct_7,axiom,( ! [A] : ( ( v1_xboole_0(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) & v1_finseq_1(A) & v1_funcop_1(A) & v1_funct_7(A) ) ) ), file(funct_7,cc2_funct_7), []). 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_gobrd13,axiom,( ! [A] : ( ( v1_relat_1(A) & v1_funct_1(A) & v1_matrlin(A) ) => ( v1_relat_1(A) & v1_funct_1(A) & v1_funcop_1(A) ) ) ), file(gobrd13,cc2_gobrd13), []). fof(cc2_int_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) & v1_int_1(A) ) ) ), file(int_1,cc2_int_1), []). fof(cc2_membered,axiom,( ! [A] : ( v4_membered(A) => v3_membered(A) ) ), file(membered,cc2_membered), []). fof(cc2_ordinal1,axiom,( ! [A] : ( ( v1_ordinal1(A) & v2_ordinal1(A) ) => v3_ordinal1(A) ) ), file(ordinal1,cc2_ordinal1), []). 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(cc2_xreal_0,axiom,( ! [A] : ( v1_xreal_0(A) => v1_xcmplx_0(A) ) ), file(xreal_0,cc2_xreal_0), []). 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_binom,axiom,( ! [A] : ( l1_rlvect_1(A) => ( ( ~ v3_struct_0(A) & v3_rlvect_1(A) & v3_algstr_1(A) ) => ( ~ v3_struct_0(A) & v2_algstr_1(A) ) ) ) ), file(binom,cc3_binom), []). fof(cc3_card_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_finset_1(A) & v1_xreal_0(A) & ~ v3_xreal_0(A) & v1_card_1(A) ) ) ), file(card_1,cc3_card_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_int_1,axiom,( ! [A] : ( v4_ordinal2(A) => v1_int_1(A) ) ), file(int_1,cc3_int_1), []). fof(cc3_membered,axiom,( ! [A] : ( v3_membered(A) => v2_membered(A) ) ), file(membered,cc3_membered), []). fof(cc3_ordinal1,axiom,( ! [A] : ( v1_xboole_0(A) => ( v1_ordinal1(A) & v2_ordinal1(A) & v3_ordinal1(A) ) ) ), file(ordinal1,cc3_ordinal1), []). fof(cc3_polynom1,axiom,( ! [A] : ( ( v1_relat_1(A) & v1_funct_1(A) & v7_seqm_3(A) ) => ( v1_relat_1(A) & v1_funct_1(A) & v1_seq_1(A) ) ) ), file(polynom1,cc3_polynom1), []). 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(cc3_xreal_0,axiom,( ! [A] : ( ( v1_xreal_0(A) & v2_xreal_0(A) ) => ( ~ v1_xboole_0(A) & v1_xcmplx_0(A) & v1_xreal_0(A) & ~ v3_xreal_0(A) ) ) ), file(xreal_0,cc3_xreal_0), []). 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_binom,axiom,( ! [A] : ( l1_rlvect_1(A) => ( ( ~ v3_struct_0(A) & v3_rlvect_1(A) & v2_algstr_1(A) ) => ( ~ v3_struct_0(A) & v3_algstr_1(A) ) ) ) ), file(binom,cc4_binom), []). 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_int_1,axiom,( ! [A] : ( v1_int_1(A) => ( v1_xcmplx_0(A) & v1_xreal_0(A) ) ) ), file(int_1,cc4_int_1), []). fof(cc4_membered,axiom,( ! [A] : ( v2_membered(A) => v1_membered(A) ) ), file(membered,cc4_membered), []). fof(cc4_polynom1,axiom,( ! [A,B] : ( m1_relset_1(B,A,k5_numbers) => ( ( v1_funct_1(B) & v1_funct_2(B,A,k5_numbers) ) => ( v1_funct_1(B) & v1_funct_2(B,A,k5_numbers) & v7_seqm_3(B) ) ) ) ), file(polynom1,cc4_polynom1), []). 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(cc4_xreal_0,axiom,( ! [A] : ( ( ~ v1_xboole_0(A) & v1_xreal_0(A) & ~ v3_xreal_0(A) ) => ( v1_xcmplx_0(A) & v1_xreal_0(A) & v2_xreal_0(A) ) ) ), file(xreal_0,cc4_xreal_0), []). fof(cc5_binom,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) & v3_algstr_1(A) ) ) ) ), file(binom,cc5_binom), []). fof(cc5_polynom1,axiom,( ! [A] : ( ( v1_relat_1(A) & v1_funct_1(A) & v1_finset_1(A) ) => ( v1_relat_1(A) & v1_funct_1(A) & v1_polynom1(A) ) ) ), file(polynom1,cc5_polynom1), []). 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(cc5_xreal_0,axiom,( ! [A] : ( ( v1_xreal_0(A) & v3_xreal_0(A) ) => ( ~ v1_xboole_0(A) & v1_xcmplx_0(A) & v1_xreal_0(A) & ~ v2_xreal_0(A) ) ) ), file(xreal_0,cc5_xreal_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_xreal_0,axiom,( ! [A] : ( ( ~ v1_xboole_0(A) & v1_xreal_0(A) & ~ v2_xreal_0(A) ) => ( v1_xcmplx_0(A) & v1_xreal_0(A) & v3_xreal_0(A) ) ) ), file(xreal_0,cc6_xreal_0), []). fof(cc7_xreal_0,axiom,( ! [A] : ( ( v1_xboole_0(A) & v1_xreal_0(A) ) => ( v1_xcmplx_0(A) & v1_xreal_0(A) & ~ v2_xreal_0(A) & ~ v3_xreal_0(A) ) ) ), file(xreal_0,cc7_xreal_0), []). fof(cc8_xreal_0,axiom,( ! [A] : ( ( v1_xreal_0(A) & ~ v2_xreal_0(A) & ~ v3_xreal_0(A) ) => ( v1_xboole_0(A) & v1_xcmplx_0(A) & v1_xreal_0(A) ) ) ), file(xreal_0,cc8_xreal_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(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(d10_xboole_0,axiom,( ! [A,B] : ( A = B <=> ( r1_tarski(A,B) & r1_tarski(B,A) ) ) ), file(xboole_0,d10_xboole_0), []). 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(d21_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)) ) => ( v1_quofield(C,A,B) <=> ( v1_grcat_1(C,A,B) & v1_group_6(C,A,B) & v1_endalg(C,A,B) ) ) ) ) ) ), file(quofield,d21_quofield), []). fof(d22_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)) ) => ( v2_quofield(C,A,B) <=> ( v1_quofield(C,A,B) & k2_relat_1(C) = u1_struct_0(B) ) ) ) ) ) ), file(quofield,d22_quofield), []). 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(d24_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)) ) => ( v4_quofield(C,A,B) <=> ( v3_quofield(C,A,B) & v2_quofield(C,A,B) ) ) ) ) ) ), file(quofield,d24_quofield), []). fof(d3_tarski,axiom,( ! [A,B] : ( r1_tarski(A,B) <=> ! [C] : ( r2_hidden(C,A) => r2_hidden(C,B) ) ) ), file(tarski,d3_tarski), []). 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(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_k16_polynom3,axiom,( ! [A] : ( ( ~ v3_struct_0(A) & v4_rlvect_1(A) & v5_rlvect_1(A) & v6_rlvect_1(A) & v7_vectsp_1(A) & l3_vectsp_1(A) ) => ( ~ v3_struct_0(k16_polynom3(A)) & v3_vectsp_1(k16_polynom3(A)) & l3_vectsp_1(k16_polynom3(A)) ) ) ), file(polynom3,k16_polynom3), []). fof(dt_k1_funct_1,axiom,( $true ), file(funct_1,k1_funct_1), []). 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_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_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_k30_polynom1,axiom,( ! [A,B] : ( ( v3_ordinal1(A) & ~ v3_struct_0(B) & v4_rlvect_1(B) & v5_rlvect_1(B) & v6_rlvect_1(B) & v2_group_1(B) & v7_vectsp_1(B) & ~ v3_realset2(B) & l3_vectsp_1(B) ) => ( ~ v3_struct_0(k30_polynom1(A,B)) & v3_vectsp_1(k30_polynom1(A,B)) & l3_vectsp_1(k30_polynom1(A,B)) ) ) ), file(polynom1,k30_polynom1), []). 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_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_relset_1,axiom,( ! [A,B,C] : ( m1_relset_1(C,A,B) => m1_subset_1(k5_relset_1(A,B,C),k1_zfmisc_1(B)) ) ), file(relset_1,k5_relset_1), []). fof(dt_k6_hilbasis,axiom,( ! [A,B] : ( ( ~ v3_struct_0(A) & v2_group_1(A) & v4_group_1(A) & v7_group_1(A) & v3_rlvect_1(A) & v4_rlvect_1(A) & v5_rlvect_1(A) & v6_rlvect_1(A) & v7_vectsp_1(A) & ~ v3_realset2(A) & l3_vectsp_1(A) & m1_subset_1(B,k5_numbers) ) => ( v1_funct_1(k6_hilbasis(A,B)) & v1_funct_2(k6_hilbasis(A,B),u1_struct_0(k16_polynom3(k30_polynom1(B,A))),u1_struct_0(k30_polynom1(k1_nat_1(B,1),A))) & m2_relset_1(k6_hilbasis(A,B),u1_struct_0(k16_polynom3(k30_polynom1(B,A))),u1_struct_0(k30_polynom1(k1_nat_1(B,1),A))) ) ) ), file(hilbasis,k6_hilbasis), []). fof(dt_k7_hilbasis,axiom,( ! [A,B] : ( ( ~ v3_struct_0(A) & v2_group_1(A) & v4_group_1(A) & v7_group_1(A) & v3_rlvect_1(A) & v4_rlvect_1(A) & v5_rlvect_1(A) & v6_rlvect_1(A) & v7_vectsp_1(A) & ~ v3_realset2(A) & l3_vectsp_1(A) & m1_subset_1(B,k5_numbers) ) => ( v1_funct_1(k7_hilbasis(A,B)) & v1_funct_2(k7_hilbasis(A,B),u1_struct_0(k30_polynom1(k1_nat_1(B,1),A)),u1_struct_0(k16_polynom3(k30_polynom1(B,A)))) & m2_relset_1(k7_hilbasis(A,B),u1_struct_0(k30_polynom1(k1_nat_1(B,1),A)),u1_struct_0(k16_polynom3(k30_polynom1(B,A)))) ) ) ), file(hilbasis,k7_hilbasis), []). 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_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(fc10_polynom3,axiom,( ! [A] : ( ( ~ v3_struct_0(A) & v4_rlvect_1(A) & v5_rlvect_1(A) & v6_rlvect_1(A) & v7_vectsp_1(A) & l3_vectsp_1(A) ) => ( ~ v3_struct_0(k16_polynom3(A)) & v4_rlvect_1(k16_polynom3(A)) & v3_vectsp_1(k16_polynom3(A)) ) ) ), file(polynom3,fc10_polynom3), []). fof(fc10_xreal_0,axiom,( ! [A,B] : ( ( v1_xreal_0(A) & v2_xreal_0(A) & v1_xreal_0(B) & ~ v3_xreal_0(B) ) => ( ~ v1_xboole_0(k2_xcmplx_0(B,A)) & v1_xcmplx_0(k2_xcmplx_0(B,A)) & v1_xreal_0(k2_xcmplx_0(B,A)) & v2_xreal_0(k2_xcmplx_0(B,A)) & ~ v3_xreal_0(k2_xcmplx_0(B,A)) ) ) ), file(xreal_0,fc10_xreal_0), []). fof(fc11_polynom3,axiom,( ! [A] : ( ( ~ v3_struct_0(A) & v4_rlvect_1(A) & v5_rlvect_1(A) & v6_rlvect_1(A) & v7_vectsp_1(A) & l3_vectsp_1(A) ) => ( ~ v3_struct_0(k16_polynom3(A)) & v5_rlvect_1(k16_polynom3(A)) & v3_vectsp_1(k16_polynom3(A)) ) ) ), file(polynom3,fc11_polynom3), []). fof(fc11_xreal_0,axiom,( ! [A,B] : ( ( v1_xreal_0(A) & v3_xreal_0(A) & v1_xreal_0(B) & ~ v2_xreal_0(B) ) => ( ~ v1_xboole_0(k2_xcmplx_0(A,B)) & v1_xcmplx_0(k2_xcmplx_0(A,B)) & v1_xreal_0(k2_xcmplx_0(A,B)) & ~ v2_xreal_0(k2_xcmplx_0(A,B)) & v3_xreal_0(k2_xcmplx_0(A,B)) ) ) ), file(xreal_0,fc11_xreal_0), []). fof(fc12_polynom3,axiom,( ! [A] : ( ( ~ v3_struct_0(A) & v4_rlvect_1(A) & v5_rlvect_1(A) & v6_rlvect_1(A) & v7_vectsp_1(A) & l3_vectsp_1(A) ) => ( ~ v3_struct_0(k16_polynom3(A)) & v6_rlvect_1(k16_polynom3(A)) & v3_vectsp_1(k16_polynom3(A)) ) ) ), file(polynom3,fc12_polynom3), []). 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(fc12_xreal_0,axiom,( ! [A,B] : ( ( v1_xreal_0(A) & v3_xreal_0(A) & v1_xreal_0(B) & ~ v2_xreal_0(B) ) => ( ~ v1_xboole_0(k2_xcmplx_0(B,A)) & v1_xcmplx_0(k2_xcmplx_0(B,A)) & v1_xreal_0(k2_xcmplx_0(B,A)) & ~ v2_xreal_0(k2_xcmplx_0(B,A)) & v3_xreal_0(k2_xcmplx_0(B,A)) ) ) ), file(xreal_0,fc12_xreal_0), []). fof(fc13_polynom3,axiom,( ! [A] : ( ( ~ v3_struct_0(A) & v3_rlvect_1(A) & v4_rlvect_1(A) & v5_rlvect_1(A) & v6_rlvect_1(A) & v7_group_1(A) & v7_vectsp_1(A) & l3_vectsp_1(A) ) => ( ~ v3_struct_0(k16_polynom3(A)) & v3_rlvect_1(k16_polynom3(A)) & v4_rlvect_1(k16_polynom3(A)) & v5_rlvect_1(k16_polynom3(A)) & v6_rlvect_1(k16_polynom3(A)) & v7_group_1(k16_polynom3(A)) & v3_vectsp_1(k16_polynom3(A)) ) ) ), file(polynom3,fc13_polynom3), []). 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(fc14_polynom1,axiom, ( 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_xboole_0(k1_xboole_0) & v1_finset_1(k1_xboole_0) & v1_finseq_1(k1_xboole_0) & v1_funcop_1(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) & v1_seq_1(k1_xboole_0) & v1_matrlin(k1_xboole_0) & v7_seqm_3(k1_xboole_0) ), file(polynom1,fc14_polynom1), []). fof(fc14_polynom3,axiom,( ! [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_vectsp_1(A) & l3_vectsp_1(A) ) => ( ~ v3_struct_0(k16_polynom3(A)) & v3_rlvect_1(k16_polynom3(A)) & v4_rlvect_1(k16_polynom3(A)) & v5_rlvect_1(k16_polynom3(A)) & v6_rlvect_1(k16_polynom3(A)) & v4_group_1(k16_polynom3(A)) & v3_vectsp_1(k16_polynom3(A)) ) ) ), file(polynom3,fc14_polynom3), []). fof(fc15_polynom3,axiom,( ! [A] : ( ( ~ v3_struct_0(A) & v3_rlvect_1(A) & v4_rlvect_1(A) & v5_rlvect_1(A) & v6_rlvect_1(A) & v7_group_1(A) & v6_vectsp_1(A) & v7_vectsp_1(A) & l3_vectsp_1(A) ) => ( ~ v3_struct_0(k16_polynom3(A)) & v3_rlvect_1(k16_polynom3(A)) & v4_rlvect_1(k16_polynom3(A)) & v5_rlvect_1(k16_polynom3(A)) & v6_rlvect_1(k16_polynom3(A)) & v2_group_1(k16_polynom3(A)) & v7_group_1(k16_polynom3(A)) & v3_vectsp_1(k16_polynom3(A)) & v6_vectsp_1(k16_polynom3(A)) & v8_vectsp_1(k16_polynom3(A)) ) ) ), file(polynom3,fc15_polynom3), []). fof(fc16_polynom3,axiom,( ! [A] : ( ( ~ v3_struct_0(A) & v3_rlvect_1(A) & v4_rlvect_1(A) & v5_rlvect_1(A) & v6_rlvect_1(A) & v7_vectsp_1(A) & l3_vectsp_1(A) ) => ( ~ v3_struct_0(k16_polynom3(A)) & v3_rlvect_1(k16_polynom3(A)) & v4_rlvect_1(k16_polynom3(A)) & v5_rlvect_1(k16_polynom3(A)) & v6_rlvect_1(k16_polynom3(A)) & v3_vectsp_1(k16_polynom3(A)) & v4_vectsp_1(k16_polynom3(A)) & v5_vectsp_1(k16_polynom3(A)) & v7_vectsp_1(k16_polynom3(A)) ) ) ), file(polynom3,fc16_polynom3), []). fof(fc1_int_1,axiom,( ! [A,B] : ( ( v1_int_1(A) & v1_int_1(B) ) => ( v1_xcmplx_0(k2_xcmplx_0(A,B)) & v1_xreal_0(k2_xcmplx_0(A,B)) & v1_int_1(k2_xcmplx_0(A,B)) ) ) ), file(int_1,fc1_int_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_polynom1,axiom,( ! [A,B,C,D,E] : ( ( ~ 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) & 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_struct_0(g3_vectsp_1(A,B,C,D,E)) & v3_vectsp_1(g3_vectsp_1(A,B,C,D,E)) ) ) ), file(polynom1,fc1_polynom1), []). 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(fc20_polynom1,axiom, ( 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_xboole_0(k1_xboole_0) & v1_finset_1(k1_xboole_0) & v1_finseq_1(k1_xboole_0) & v1_funcop_1(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) & v1_seq_1(k1_xboole_0) & v1_matrlin(k1_xboole_0) & v7_seqm_3(k1_xboole_0) & v1_polynom1(k1_xboole_0) ), file(polynom1,fc20_polynom1), []). fof(fc2_hilbasis,axiom,( ! [A,B] : ( ( ~ v3_struct_0(A) & v2_group_1(A) & v4_group_1(A) & v7_group_1(A) & v3_rlvect_1(A) & v4_rlvect_1(A) & v5_rlvect_1(A) & v6_rlvect_1(A) & v7_vectsp_1(A) & ~ v3_realset2(A) & l3_vectsp_1(A) & m1_subset_1(B,k5_numbers) ) => ( v1_relat_1(k6_hilbasis(A,B)) & v1_funct_1(k6_hilbasis(A,B)) & v1_funct_2(k6_hilbasis(A,B),u1_struct_0(k16_polynom3(k30_polynom1(B,A))),u1_struct_0(k30_polynom1(k1_nat_1(B,1),A))) & v1_grcat_1(k6_hilbasis(A,B),k16_polynom3(k30_polynom1(B,A)),k30_polynom1(k1_nat_1(B,1),A)) ) ) ), file(hilbasis,fc2_hilbasis), []). fof(fc2_membered,axiom, ( ~ v1_xboole_0(k1_numbers) & v1_membered(k1_numbers) & v2_membered(k1_numbers) ), file(membered,fc2_membered), []). fof(fc2_ordinal1,axiom, ( 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_xboole_0(k1_xboole_0) & v1_ordinal1(k1_xboole_0) & v2_ordinal1(k1_xboole_0) & v3_ordinal1(k1_xboole_0) ), file(ordinal1,fc2_ordinal1), []). fof(fc38_polynom1,axiom,( ! [A,B] : ( ( v3_ordinal1(A) & ~ v3_struct_0(B) & v3_rlvect_1(B) & v4_rlvect_1(B) & v5_rlvect_1(B) & v6_rlvect_1(B) & v2_group_1(B) & v7_vectsp_1(B) & ~ v3_realset2(B) & l3_vectsp_1(B) ) => ( ~ v3_struct_0(k30_polynom1(A,B)) & v3_rlvect_1(k30_polynom1(A,B)) & v3_vectsp_1(k30_polynom1(A,B)) ) ) ), file(polynom1,fc38_polynom1), []). fof(fc39_polynom1,axiom,( ! [A,B] : ( ( v3_ordinal1(A) & ~ v3_struct_0(B) & v4_rlvect_1(B) & v5_rlvect_1(B) & v6_rlvect_1(B) & v2_group_1(B) & v7_vectsp_1(B) & ~ v3_realset2(B) & l3_vectsp_1(B) ) => ( ~ v3_struct_0(k30_polynom1(A,B)) & v4_rlvect_1(k30_polynom1(A,B)) & v3_vectsp_1(k30_polynom1(A,B)) ) ) ), file(polynom1,fc39_polynom1), []). fof(fc3_funct_7,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) & v1_funcop_1(k1_xboole_0) ), file(funct_7,fc3_funct_7), []). fof(fc3_hilbasis,axiom,( ! [A,B] : ( ( ~ v3_struct_0(A) & v2_group_1(A) & v4_group_1(A) & v7_group_1(A) & v3_rlvect_1(A) & v4_rlvect_1(A) & v5_rlvect_1(A) & v6_rlvect_1(A) & v7_vectsp_1(A) & ~ v3_realset2(A) & l3_vectsp_1(A) & m1_subset_1(B,k5_numbers) ) => ( v1_relat_1(k6_hilbasis(A,B)) & v1_funct_1(k6_hilbasis(A,B)) & v1_funct_2(k6_hilbasis(A,B),u1_struct_0(k16_polynom3(k30_polynom1(B,A))),u1_struct_0(k30_polynom1(k1_nat_1(B,1),A))) & v1_group_6(k6_hilbasis(A,B),k16_polynom3(k30_polynom1(B,A)),k30_polynom1(k1_nat_1(B,1),A)) ) ) ), file(hilbasis,fc3_hilbasis), []). fof(fc3_polynom1,axiom, ( 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_xboole_0(k1_xboole_0) & v1_finset_1(k1_xboole_0) & v1_finseq_1(k1_xboole_0) & v1_funcop_1(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) & v1_matrlin(k1_xboole_0) ), file(polynom1,fc3_polynom1), []). fof(fc3_polynom2,axiom,( ! [A,B] : ( ( v3_ordinal1(A) & ~ v3_struct_0(B) & v3_rlvect_1(B) & v4_rlvect_1(B) & v5_rlvect_1(B) & v6_rlvect_1(B) & v2_group_1(B) & v4_group_1(B) & v7_vectsp_1(B) & ~ v3_realset2(B) & l3_vectsp_1(B) ) => ( ~ v3_struct_0(k30_polynom1(A,B)) & v3_rlvect_1(k30_polynom1(A,B)) & v4_rlvect_1(k30_polynom1(A,B)) & v5_rlvect_1(k30_polynom1(A,B)) & v6_rlvect_1(k30_polynom1(A,B)) & v2_group_1(k30_polynom1(A,B)) & v4_group_1(k30_polynom1(A,B)) & v3_vectsp_1(k30_polynom1(A,B)) & v4_vectsp_1(k30_polynom1(A,B)) & v6_vectsp_1(k30_polynom1(A,B)) & v8_vectsp_1(k30_polynom1(A,B)) & v1_algstr_1(k30_polynom1(A,B)) & v2_algstr_1(k30_polynom1(A,B)) & v3_algstr_1(k30_polynom1(A,B)) & v4_algstr_1(k30_polynom1(A,B)) & v5_algstr_1(k30_polynom1(A,B)) & v6_algstr_1(k30_polynom1(A,B)) ) ) ), file(polynom2,fc3_polynom2), []). fof(fc3_xreal_0,axiom,( ! [A,B] : ( ( v1_xreal_0(A) & v1_xreal_0(B) ) => ( v1_xcmplx_0(k2_xcmplx_0(A,B)) & v1_xreal_0(k2_xcmplx_0(A,B)) ) ) ), file(xreal_0,fc3_xreal_0), []). fof(fc40_polynom1,axiom,( ! [A,B] : ( ( v3_ordinal1(A) & ~ v3_struct_0(B) & v4_rlvect_1(B) & v5_rlvect_1(B) & v6_rlvect_1(B) & v2_group_1(B) & v7_vectsp_1(B) & ~ v3_realset2(B) & l3_vectsp_1(B) ) => ( ~ v3_struct_0(k30_polynom1(A,B)) & v4_rlvect_1(k30_polynom1(A,B)) & v5_rlvect_1(k30_polynom1(A,B)) & v3_vectsp_1(k30_polynom1(A,B)) ) ) ), file(polynom1,fc40_polynom1), []). fof(fc41_polynom1,axiom,( ! [A,B] : ( ( v3_ordinal1(A) & ~ v3_struct_0(B) & v4_rlvect_1(B) & v5_rlvect_1(B) & v6_rlvect_1(B) & v2_group_1(B) & v7_vectsp_1(B) & ~ v3_realset2(B) & l3_vectsp_1(B) ) => ( ~ v3_struct_0(k30_polynom1(A,B)) & v4_rlvect_1(k30_polynom1(A,B)) & v5_rlvect_1(k30_polynom1(A,B)) & v6_rlvect_1(k30_polynom1(A,B)) & v3_vectsp_1(k30_polynom1(A,B)) & v1_algstr_1(k30_polynom1(A,B)) & v2_algstr_1(k30_polynom1(A,B)) & v3_algstr_1(k30_polynom1(A,B)) & v4_algstr_1(k30_polynom1(A,B)) & v5_algstr_1(k30_polynom1(A,B)) & v6_algstr_1(k30_polynom1(A,B)) ) ) ), file(polynom1,fc41_polynom1), []). fof(fc42_polynom1,axiom,( ! [A,B] : ( ( v3_ordinal1(A) & ~ v3_struct_0(B) & v3_rlvect_1(B) & v4_rlvect_1(B) & v5_rlvect_1(B) & v6_rlvect_1(B) & v2_group_1(B) & v7_group_1(B) & v7_vectsp_1(B) & ~ v3_realset2(B) & l3_vectsp_1(B) ) => ( ~ v3_struct_0(k30_polynom1(A,B)) & v3_rlvect_1(k30_polynom1(A,B)) & v4_rlvect_1(k30_polynom1(A,B)) & v5_rlvect_1(k30_polynom1(A,B)) & v6_rlvect_1(k30_polynom1(A,B)) & v7_group_1(k30_polynom1(A,B)) & v3_vectsp_1(k30_polynom1(A,B)) & v1_algstr_1(k30_polynom1(A,B)) & v2_algstr_1(k30_polynom1(A,B)) & v3_algstr_1(k30_polynom1(A,B)) & v4_algstr_1(k30_polynom1(A,B)) & v5_algstr_1(k30_polynom1(A,B)) & v6_algstr_1(k30_polynom1(A,B)) ) ) ), file(polynom1,fc42_polynom1), []). fof(fc43_polynom1,axiom,( ! [A,B] : ( ( v3_ordinal1(A) & ~ v3_struct_0(B) & v3_rlvect_1(B) & v4_rlvect_1(B) & v5_rlvect_1(B) & v6_rlvect_1(B) & v2_group_1(B) & v4_group_1(B) & v7_vectsp_1(B) & ~ v3_realset2(B) & l3_vectsp_1(B) ) => ( ~ v3_struct_0(k30_polynom1(A,B)) & v3_rlvect_1(k30_polynom1(A,B)) & v4_rlvect_1(k30_polynom1(A,B)) & v5_rlvect_1(k30_polynom1(A,B)) & v6_rlvect_1(k30_polynom1(A,B)) & v4_group_1(k30_polynom1(A,B)) & v3_vectsp_1(k30_polynom1(A,B)) & v1_algstr_1(k30_polynom1(A,B)) & v2_algstr_1(k30_polynom1(A,B)) & v3_algstr_1(k30_polynom1(A,B)) & v4_algstr_1(k30_polynom1(A,B)) & v5_algstr_1(k30_polynom1(A,B)) & v6_algstr_1(k30_polynom1(A,B)) ) ) ), file(polynom1,fc43_polynom1), []). fof(fc44_polynom1,axiom,( ! [A,B] : ( ( v3_ordinal1(A) & ~ v3_struct_0(B) & v3_rlvect_1(B) & v4_rlvect_1(B) & v5_rlvect_1(B) & v6_rlvect_1(B) & v2_group_1(B) & v4_group_1(B) & v7_vectsp_1(B) & ~ v3_realset2(B) & l3_vectsp_1(B) ) => ( ~ v3_struct_0(k30_polynom1(A,B)) & v3_rlvect_1(k30_polynom1(A,B)) & v4_rlvect_1(k30_polynom1(A,B)) & v5_rlvect_1(k30_polynom1(A,B)) & v6_rlvect_1(k30_polynom1(A,B)) & v2_group_1(k30_polynom1(A,B)) & v4_group_1(k30_polynom1(A,B)) & v3_vectsp_1(k30_polynom1(A,B)) & v4_vectsp_1(k30_polynom1(A,B)) & v6_vectsp_1(k30_polynom1(A,B)) & v8_vectsp_1(k30_polynom1(A,B)) & v1_algstr_1(k30_polynom1(A,B)) & v2_algstr_1(k30_polynom1(A,B)) & v3_algstr_1(k30_polynom1(A,B)) & v4_algstr_1(k30_polynom1(A,B)) & v5_algstr_1(k30_polynom1(A,B)) & v6_algstr_1(k30_polynom1(A,B)) ) ) ), file(polynom1,fc44_polynom1), []). fof(fc4_hilbasis,axiom,( ! [A,B] : ( ( ~ v3_struct_0(A) & v2_group_1(A) & v4_group_1(A) & v7_group_1(A) & v3_rlvect_1(A) & v4_rlvect_1(A) & v5_rlvect_1(A) & v6_rlvect_1(A) & v7_vectsp_1(A) & ~ v3_realset2(A) & l3_vectsp_1(A) & m1_subset_1(B,k5_numbers) ) => ( v1_relat_1(k6_hilbasis(A,B)) & v1_funct_1(k6_hilbasis(A,B)) & v1_funct_2(k6_hilbasis(A,B),u1_struct_0(k16_polynom3(k30_polynom1(B,A))),u1_struct_0(k30_polynom1(k1_nat_1(B,1),A))) & v1_endalg(k6_hilbasis(A,B),k16_polynom3(k30_polynom1(B,A)),k30_polynom1(k1_nat_1(B,1),A)) ) ) ), file(hilbasis,fc4_hilbasis), []). 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_hilbasis,axiom,( ! [A,B] : ( ( ~ v3_struct_0(A) & v2_group_1(A) & v4_group_1(A) & v7_group_1(A) & v3_rlvect_1(A) & v4_rlvect_1(A) & v5_rlvect_1(A) & v6_rlvect_1(A) & v7_vectsp_1(A) & ~ v3_realset2(A) & l3_vectsp_1(A) & m1_subset_1(B,k5_numbers) ) => ( v1_relat_1(k6_hilbasis(A,B)) & v1_funct_1(k6_hilbasis(A,B)) & v2_funct_1(k6_hilbasis(A,B)) & v1_funct_2(k6_hilbasis(A,B),u1_struct_0(k16_polynom3(k30_polynom1(B,A))),u1_struct_0(k30_polynom1(k1_nat_1(B,1),A))) ) ) ), file(hilbasis,fc5_hilbasis), []). 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_polynom1,axiom,( ! [A,B] : ( ( v1_relat_1(A) & v1_funct_1(A) & v1_finseq_1(A) & v1_matrlin(A) ) => ( v1_relat_1(k1_funct_1(A,B)) & v1_funct_1(k1_funct_1(A,B)) & v1_finset_1(k1_funct_1(A,B)) & v1_finseq_1(k1_funct_1(A,B)) ) ) ), file(polynom1,fc5_polynom1), []). 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_int_1,axiom,( ! [A,B] : ( ( m1_subset_1(A,k5_numbers) & v1_int_1(B) ) => ( v1_xcmplx_0(k2_xcmplx_0(B,A)) & v1_xreal_0(k2_xcmplx_0(B,A)) & v1_int_1(k2_xcmplx_0(B,A)) ) ) ), file(int_1,fc6_int_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_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(fc7_xreal_0,axiom,( ! [A,B] : ( ( v1_xreal_0(A) & ~ v3_xreal_0(A) & v1_xreal_0(B) & ~ v3_xreal_0(B) ) => ( v1_xcmplx_0(k2_xcmplx_0(A,B)) & v1_xreal_0(k2_xcmplx_0(A,B)) & ~ v3_xreal_0(k2_xcmplx_0(A,B)) ) ) ), file(xreal_0,fc7_xreal_0), []). 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(fc8_xreal_0,axiom,( ! [A,B] : ( ( v1_xreal_0(A) & ~ v2_xreal_0(A) & v1_xreal_0(B) & ~ v2_xreal_0(B) ) => ( v1_xcmplx_0(k2_xcmplx_0(A,B)) & v1_xreal_0(k2_xcmplx_0(A,B)) & ~ v2_xreal_0(k2_xcmplx_0(A,B)) ) ) ), file(xreal_0,fc8_xreal_0), []). fof(fc9_polynom3,axiom,( ! [A] : ( ( ~ v3_struct_0(A) & v3_rlvect_1(A) & v4_rlvect_1(A) & v5_rlvect_1(A) & v6_rlvect_1(A) & v7_vectsp_1(A) & l3_vectsp_1(A) ) => ( ~ v3_struct_0(k16_polynom3(A)) & v3_rlvect_1(k16_polynom3(A)) & v3_vectsp_1(k16_polynom3(A)) ) ) ), file(polynom3,fc9_polynom3), []). fof(fc9_xreal_0,axiom,( ! [A,B] : ( ( v1_xreal_0(A) & v2_xreal_0(A) & v1_xreal_0(B) & ~ v3_xreal_0(B) ) => ( ~ v1_xboole_0(k2_xcmplx_0(A,B)) & v1_xcmplx_0(k2_xcmplx_0(A,B)) & v1_xreal_0(k2_xcmplx_0(A,B)) & v2_xreal_0(k2_xcmplx_0(A,B)) & ~ v3_xreal_0(k2_xcmplx_0(A,B)) ) ) ), file(xreal_0,fc9_xreal_0), []). 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(rc11_polynom1,axiom,( ? [A] : ( v1_relat_1(A) & v1_funct_1(A) & ~ v1_xboole_0(A) & v1_seq_1(A) & v7_seqm_3(A) & v1_polynom1(A) ) ), file(polynom1,rc11_polynom1), []). fof(rc12_polynom1,axiom,( ! [A] : ? [B] : ( m1_relset_1(B,A,k5_numbers) & v1_relat_1(B) & v1_funct_1(B) & v1_funct_2(B,A,k5_numbers) & v1_seq_1(B) & v7_seqm_3(B) & v1_polynom1(B) ) ), file(polynom1,rc12_polynom1), []). fof(rc1_binom,axiom,( ? [A] : ( l1_rlvect_1(A) & ~ v3_struct_0(A) & v2_algstr_1(A) ) ), file(binom,rc1_binom), []). fof(rc1_card_1,axiom,( ? [A] : v1_card_1(A) ), file(card_1,rc1_card_1), []). fof(rc1_card_5,axiom,( ? [A] : ~ v1_finset_1(A) ), file(card_5,rc1_card_5), []). fof(rc1_finseq_5,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_5,rc1_finseq_5), []). 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_ideal_1,axiom,( ? [A] : ( l1_rlvect_1(A) & ~ v3_struct_0(A) & v4_rlvect_1(A) & v5_rlvect_1(A) & v1_algstr_1(A) ) ), file(ideal_1,rc1_ideal_1), []). fof(rc1_int_1,axiom,( ? [A] : ( m1_subset_1(A,k1_numbers) & v1_xcmplx_0(A) & v1_xreal_0(A) & v1_int_1(A) ) ), file(int_1,rc1_int_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_ordinal1,axiom,( ? [A] : ( v1_ordinal1(A) & v2_ordinal1(A) & v3_ordinal1(A) ) ), file(ordinal1,rc1_ordinal1), []). fof(rc1_polynom2,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) & v1_algstr_1(A) & v2_algstr_1(A) & v3_algstr_1(A) & v4_algstr_1(A) & v5_algstr_1(A) & v6_algstr_1(A) & ~ v3_realset2(A) ) ), file(polynom2,rc1_polynom2), []). fof(rc1_pre_circ,axiom,( ? [A] : ( ~ v1_xboole_0(A) & v1_finset_1(A) & v1_membered(A) & v2_membered(A) & v3_membered(A) & v4_membered(A) & v5_membered(A) ) ), file(pre_circ,rc1_pre_circ), []). 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(rc1_xboole_0,axiom,( ? [A] : v1_xboole_0(A) ), file(xboole_0,rc1_xboole_0), []). fof(rc1_xreal_0,axiom,( ? [A] : ( v1_xcmplx_0(A) & v1_xreal_0(A) ) ), file(xreal_0,rc1_xreal_0), []). fof(rc2_binom,axiom,( ? [A] : ( l1_rlvect_1(A) & ~ v3_struct_0(A) & v3_algstr_1(A) ) ), file(binom,rc2_binom), []). fof(rc2_card_1,axiom,( ? [A] : ( v1_ordinal1(A) & v2_ordinal1(A) & v3_ordinal1(A) & v1_finset_1(A) & v1_card_1(A) ) ), file(card_1,rc2_card_1), []). fof(rc2_card_5,axiom,( ? [A] : ( v1_ordinal1(A) & v2_ordinal1(A) & v3_ordinal1(A) & ~ v1_finset_1(A) & v1_card_1(A) ) ), file(card_5,rc2_card_5), []). 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_ideal_1,axiom,( ? [A] : ( l3_vectsp_1(A) & ~ v3_struct_0(A) & v2_group_1(A) & v4_group_1(A) & v7_group_1(A) & v3_rlvect_1(A) & v4_rlvect_1(A) & v5_rlvect_1(A) & v4_vectsp_1(A) & v5_vectsp_1(A) & v6_vectsp_1(A) & v7_vectsp_1(A) & v8_vectsp_1(A) & v1_binom(A) & v1_algstr_1(A) & v2_algstr_1(A) & v3_algstr_1(A) & ~ v3_realset2(A) ) ), file(ideal_1,rc2_ideal_1), []). fof(rc2_int_1,axiom,( ? [A] : v1_int_1(A) ), file(int_1,rc2_int_1), []). fof(rc2_ordinal1,axiom,( ? [A] : ( v1_relat_1(A) & v1_funct_1(A) & v2_funct_1(A) & v1_xboole_0(A) & v1_ordinal1(A) & v2_ordinal1(A) & v3_ordinal1(A) ) ), file(ordinal1,rc2_ordinal1), []). fof(rc2_polynom1,axiom,( ? [A] : ( v1_relat_1(A) & v1_funct_1(A) & v1_finset_1(A) & v1_finseq_1(A) & v1_funcop_1(A) & v1_matrlin(A) ) ), file(polynom1,rc2_polynom1), []). 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(rc2_xboole_0,axiom,( ? [A] : ~ v1_xboole_0(A) ), file(xboole_0,rc2_xboole_0), []). fof(rc2_xreal_0,axiom,( ? [A] : ( ~ v1_xboole_0(A) & v1_xcmplx_0(A) & v1_xreal_0(A) & v2_xreal_0(A) & ~ v3_xreal_0(A) ) ), file(xreal_0,rc2_xreal_0), []). fof(rc3_binom,axiom,( ? [A] : ( l1_rlvect_1(A) & ~ v3_struct_0(A) & v1_binom(A) ) ), file(binom,rc3_binom), []). 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_ordinal1,axiom,( ? [A] : ( ~ v1_xboole_0(A) & v1_ordinal1(A) & v2_ordinal1(A) & v3_ordinal1(A) ) ), file(ordinal1,rc3_ordinal1), []). 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(rc3_xreal_0,axiom,( ? [A] : ( ~ v1_xboole_0(A) & v1_xcmplx_0(A) & v1_xreal_0(A) & ~ v2_xreal_0(A) & v3_xreal_0(A) ) ), file(xreal_0,rc3_xreal_0), []). fof(rc4_binom,axiom,( ? [A] : ( l3_vectsp_1(A) & ~ v3_struct_0(A) & v3_rlvect_1(A) & v4_rlvect_1(A) & v5_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) & v1_algstr_1(A) & v2_algstr_1(A) & v3_algstr_1(A) & v1_binom(A) ) ), file(binom,rc4_binom), []). 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(rc4_struct_0,axiom,( ? [A] : ( l2_struct_0(A) & ~ v3_struct_0(A) ) ), file(struct_0,rc4_struct_0), []). fof(rc4_xreal_0,axiom,( ? [A] : ( v1_xboole_0(A) & v1_xcmplx_0(A) & v1_xreal_0(A) & ~ v2_xreal_0(A) & ~ v3_xreal_0(A) ) ), file(xreal_0,rc4_xreal_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_vectsp_1,axiom,( ? [A] : ( l3_vectsp_1(A) & v3_vectsp_1(A) ) ), file(vectsp_1,rc6_vectsp_1), []). fof(rc7_polynom1,axiom,( ? [A] : ( v1_relat_1(A) & v1_funct_1(A) & v1_seq_1(A) & v7_seqm_3(A) ) ), file(polynom1,rc7_polynom1), []). 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_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_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_relset_1,axiom,( ! [A,B,C] : ( m1_relset_1(C,A,B) => k5_relset_1(A,B,C) = k2_relat_1(C) ) ), file(relset_1,k5_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(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(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(t30_hilbasis,axiom,( ! [A] : ( ( ~ v3_struct_0(A) & v2_group_1(A) & v4_group_1(A) & v7_group_1(A) & v3_rlvect_1(A) & v4_rlvect_1(A) & v5_rlvect_1(A) & v6_rlvect_1(A) & v7_vectsp_1(A) & ~ v3_realset2(A) & l3_vectsp_1(A) ) => ! [B] : ( m2_subset_1(B,k1_numbers,k5_numbers) => ! [C] : ( m1_subset_1(C,u1_struct_0(k30_polynom1(k1_nat_1(B,1),A))) => k8_funct_2(u1_struct_0(k16_polynom3(k30_polynom1(B,A))),u1_struct_0(k30_polynom1(k1_nat_1(B,1),A)),k6_hilbasis(A,B),k8_funct_2(u1_struct_0(k30_polynom1(k1_nat_1(B,1),A)),u1_struct_0(k16_polynom3(k30_polynom1(B,A))),k7_hilbasis(A,B),C)) = C ) ) ) ), file(hilbasis,t30_hilbasis), []). 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(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(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), []).