% Mizar problem: t71_sprect_3,sprect_3,2650,46 fof(t71_sprect_3,conjecture,( ! [A] : ( ( ~ v1_xboole_0(A) & v1_sprect_1(A) & v1_topreal1(A) & v2_topreal1(A) & v1_finseq_6(A,u1_struct_0(k15_euclid(2))) & v1_goboard5(A) & m2_finseq_1(A,u1_struct_0(k15_euclid(2))) ) => ! [B] : ( ( v4_topreal1(B) & m2_finseq_1(B,u1_struct_0(k15_euclid(2))) ) => ( ( r2_hidden(k4_finseq_4(k5_numbers,u1_struct_0(k15_euclid(2)),B,1),k2_goboard9(A)) & r2_hidden(k4_finseq_4(k5_numbers,u1_struct_0(k15_euclid(2)),B,k3_finseq_1(B)),k3_goboard9(A)) ) => r1_sprect_2(A,k3_jordan3(B,k2_jordan5c(k5_topreal1(2,B),k5_topreal1(2,A),k4_finseq_4(k5_numbers,u1_struct_0(k15_euclid(2)),B,1),k4_finseq_4(k5_numbers,u1_struct_0(k15_euclid(2)),B,k3_finseq_1(B))))) ) ) ) ), inference(mizar_bg_added,[status(thm)],[abstractness_v1_metric_1,abstractness_v1_pre_topc,antisymmetry_r2_hidden,cc10_membered,cc11_membered,cc12_membered,cc13_membered,cc14_membered,cc15_membered,cc16_membered,cc17_membered,cc18_membered,cc19_membered,cc1_arytm_3,cc1_finseq_5,cc1_membered,cc1_relset_1,cc1_sppol_2,cc1_sprect_1,cc1_sprect_3,cc1_xreal_0,cc20_membered,cc2_arytm_3,cc2_membered,cc2_sppol_2,cc2_sprect_1,cc2_xreal_0,cc3_arytm_3,cc3_membered,cc3_sppol_2,cc3_xreal_0,cc4_membered,cc4_xreal_0,cc5_sprect_1,cc5_xreal_0,cc6_membered,cc6_sprect_1,cc6_xreal_0,cc7_xreal_0,cc8_xreal_0,cc9_membered,commutativity_k1_nat_1,commutativity_k2_xcmplx_0,commutativity_k3_xboole_0,commutativity_k5_subset_1,connectedness_r1_xreal_0,d1_euclid,d1_sprect_2,d1_tarski,d2_jordan5c,d3_xboole_0,d4_finseq_4,d4_jordan3,d7_euclid,d8_euclid,d8_topreal1,d9_topreal1,dt_g1_metric_1,dt_g1_pre_topc,dt_k12_finseq_1,dt_k13_euclid,dt_k14_euclid,dt_k15_euclid,dt_k18_pscomp_1,dt_k19_pscomp_1,dt_k1_card_1,dt_k1_euclid,dt_k1_funct_1,dt_k1_jordan3,dt_k1_nat_1,dt_k1_numbers,dt_k1_relat_1,dt_k1_rfinseq,dt_k1_struct_0,dt_k1_tarski,dt_k1_xboole_0,dt_k1_zfmisc_1,dt_k20_pscomp_1,dt_k21_euclid,dt_k21_pscomp_1,dt_k22_borsuk_1,dt_k22_euclid,dt_k2_goboard9,dt_k2_jordan3,dt_k2_jordan5c,dt_k2_relat_1,dt_k2_xcmplx_0,dt_k2_zfmisc_1,dt_k3_finseq_1,dt_k3_goboard9,dt_k3_jordan3,dt_k3_pre_topc,dt_k3_topmetr,dt_k3_xboole_0,dt_k4_finseq_1,dt_k4_finseq_2,dt_k4_finseq_4,dt_k4_matrix_2,dt_k4_topreal1,dt_k5_binarith,dt_k5_finseq_1,dt_k5_numbers,dt_k5_ordinal2,dt_k5_pcomps_1,dt_k5_subset_1,dt_k5_topmetr,dt_k5_topreal1,dt_k7_finseq_1,dt_k8_finseq_1,dt_k9_finseq_1,dt_l1_metric_1,dt_l1_pre_topc,dt_l1_struct_0,dt_m1_finseq_1,dt_m1_finseq_2,dt_m1_pre_topc,dt_m1_relset_1,dt_m1_subset_1,dt_m2_finseq_1,dt_m2_finseq_2,dt_m2_relset_1,dt_m2_subset_1,dt_u1_metric_1,dt_u1_pre_topc,dt_u1_struct_0,existence_l1_metric_1,existence_l1_pre_topc,existence_l1_struct_0,existence_m1_finseq_1,existence_m1_finseq_2,existence_m1_pre_topc,existence_m1_relset_1,existence_m1_subset_1,existence_m2_finseq_1,existence_m2_finseq_2,existence_m2_relset_1,existence_m2_subset_1,fc10_membered,fc10_xreal_0,fc11_membered,fc11_sprect_1,fc11_xreal_0,fc12_sprect_1,fc12_xreal_0,fc13_sprect_1,fc1_euclid,fc1_finseq_5,fc1_sprect_1,fc1_sprect_3,fc1_struct_0,fc27_membered,fc28_membered,fc29_membered,fc2_euclid,fc2_membered,fc2_sppol_2,fc2_sprect_1,fc2_sprect_3,fc30_membered,fc31_membered,fc32_membered,fc33_membered,fc34_membered,fc35_membered,fc36_membered,fc3_sprect_1,fc3_sprect_2,fc3_sprect_3,fc3_xreal_0,fc4_finseq_5,fc5_membered,fc5_sppol_2,fc6_membered,fc7_membered,fc7_xreal_0,fc8_membered,fc8_xreal_0,fc9_membered,fc9_sppol_2,fc9_xreal_0,fraenkel_a_1_5_sprect_3,fraenkel_a_1_6_sprect_3,free_g1_metric_1,free_g1_pre_topc,idempotence_k3_xboole_0,idempotence_k5_subset_1,irreflexivity_r1_subset_1,irreflexivity_r2_subset_1,rc1_arytm_3,rc1_finseq_5,rc1_membered,rc1_sppol_2,rc1_sprect_1,rc1_sprect_2,rc1_sprect_3,rc1_xreal_0,rc2_finseq_5,rc2_sppol_2,rc2_sprect_1,rc2_sprect_3,rc2_xreal_0,rc3_struct_0,rc3_xreal_0,rc4_sprect_1,rc4_xreal_0,rc5_struct_0,redefinition_k12_finseq_1,redefinition_k1_nat_1,redefinition_k1_struct_0,redefinition_k3_finseq_1,redefinition_k4_finseq_1,redefinition_k4_matrix_2,redefinition_k5_numbers,redefinition_k5_subset_1,redefinition_k5_topmetr,redefinition_k8_finseq_1,redefinition_k9_finseq_1,redefinition_m2_finseq_1,redefinition_m2_finseq_2,redefinition_m2_relset_1,redefinition_m2_subset_1,redefinition_r1_subset_1,redefinition_r2_subset_1,reflexivity_r1_tarski,reflexivity_r1_xreal_0,spc0_boole,spc1_boole,spc1_numerals,spc2_boole,spc2_numerals,symmetry_r1_subset_1,symmetry_r1_xboole_0,symmetry_r2_subset_1,t10_xreal_1,t12_jordan3,t12_sprect_2,t13_sppol_2,t13_sprect_2,t16_finseq_5,t18_jordan4,t1_arithm,t1_numerals,t1_real,t1_subset,t1_xreal_1,t20_jordan4,t26_jordan3,t27_finseq_3,t27_jordan3,t28_jordan5c,t29_nat_1,t2_boole,t2_real,t2_subset,t2_tarski,t2_xreal_1,t31_jordan4,t31_topreal1,t31_xreal_1,t35_finseq_1,t35_tops_1,t38_nat_1,t39_binarith,t39_jordan3,t3_jordan4,t3_real,t3_subset,t3_xboole_0,t41_jordan3,t42_jordan3,t43_sprect_3,t49_sppol_1,t4_real,t4_subset,t4_xboole_0,t50_sprect_3,t53_binarith,t54_sprect_3,t56_finseq_1,t5_goboard2,t5_jordan3,t5_real,t5_subset,t6_boole,t6_finseq_5,t6_real,t6_sprect_3,t7_boole,t7_real,t7_sprect_2,t7_sprect_3,t8_boole,t8_real,t8_xreal_1,t96_sprect_1,t9_nat_2]), [file(sprect_3,t71_sprect_3)]). fof(abstractness_v1_metric_1,axiom,( ! [A] : ( l1_metric_1(A) => ( v1_metric_1(A) => A = g1_metric_1(u1_struct_0(A),u1_metric_1(A)) ) ) ), file(metric_1,v1_metric_1), []). fof(abstractness_v1_pre_topc,axiom,( ! [A] : ( l1_pre_topc(A) => ( v1_pre_topc(A) => A = g1_pre_topc(u1_struct_0(A),u1_pre_topc(A)) ) ) ), file(pre_topc,v1_pre_topc), []). 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_arytm_3,axiom,( ! [A] : ( v3_ordinal1(A) => ! [B] : ( m1_subset_1(B,A) => ( v1_ordinal1(B) & v2_ordinal1(B) & v3_ordinal1(B) ) ) ) ), file(arytm_3,cc1_arytm_3), []). fof(cc1_finseq_5,axiom,( ! [A] : ( v1_xboole_0(A) => v1_realset1(A) ) ), file(finseq_5,cc1_finseq_5), []). fof(cc1_membered,axiom,( ! [A] : ( v5_membered(A) => v4_membered(A) ) ), file(membered,cc1_membered), []). 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_sppol_2,axiom,( ! [A] : ( m1_finseq_1(A,u1_struct_0(k15_euclid(2))) => ( v4_topreal1(A) => ( v2_funct_1(A) & ~ v1_realset1(A) & v1_topreal1(A) & v2_topreal1(A) & v3_topreal1(A) ) ) ) ), file(sppol_2,cc1_sppol_2), []). fof(cc1_sprect_1,axiom,( ! [A] : ( ( v1_relat_1(A) & v1_funct_1(A) & ~ v5_seqm_3(A) ) => ( ~ v1_xboole_0(A) & v1_relat_1(A) & v1_funct_1(A) & ~ v1_realset1(A) ) ) ), file(sprect_1,cc1_sprect_1), []). fof(cc1_sprect_3,axiom,( ! [A] : ( m1_finseq_1(A,u1_struct_0(k15_euclid(2))) => ( ( ~ v1_xboole_0(A) & v1_sprect_1(A) & v1_topreal1(A) & v2_topreal1(A) & v1_finseq_6(A,u1_struct_0(k15_euclid(2))) & v1_goboard5(A) ) => ( ~ v1_xboole_0(A) & v1_sprect_2(A) & v1_sprect_1(A) & ~ v5_seqm_3(A) & v1_topreal1(A) & v2_topreal1(A) & v1_finseq_6(A,u1_struct_0(k15_euclid(2))) & v1_goboard5(A) & v2_goboard5(A) ) ) ) ), file(sprect_3,cc1_sprect_3), []). 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_arytm_3,axiom,( ! [A] : ( ( v1_xboole_0(A) & v3_ordinal1(A) ) => ( v1_ordinal1(A) & v2_ordinal1(A) & v3_ordinal1(A) & v4_ordinal2(A) ) ) ), file(arytm_3,cc2_arytm_3), []). fof(cc2_membered,axiom,( ! [A] : ( v4_membered(A) => v3_membered(A) ) ), file(membered,cc2_membered), []). fof(cc2_sppol_2,axiom,( ! [A] : ( m1_finseq_1(A,u1_struct_0(k15_euclid(2))) => ( ( v2_funct_1(A) & ~ v1_realset1(A) & v1_topreal1(A) & v2_topreal1(A) & v3_topreal1(A) ) => v4_topreal1(A) ) ) ), file(sppol_2,cc2_sppol_2), []). fof(cc2_sprect_1,axiom,( ! [A] : ( ( v1_relat_1(A) & v1_funct_1(A) & v1_realset1(A) ) => ( v1_relat_1(A) & v1_funct_1(A) & v5_seqm_3(A) ) ) ), file(sprect_1,cc2_sprect_1), []). fof(cc2_xreal_0,axiom,( ! [A] : ( v1_xreal_0(A) => v1_xcmplx_0(A) ) ), file(xreal_0,cc2_xreal_0), []). fof(cc3_arytm_3,axiom,( ! [A] : ( m1_subset_1(A,k5_ordinal2) => ( v1_ordinal1(A) & v2_ordinal1(A) & v3_ordinal1(A) & v4_ordinal2(A) ) ) ), file(arytm_3,cc3_arytm_3), []). fof(cc3_membered,axiom,( ! [A] : ( v3_membered(A) => v2_membered(A) ) ), file(membered,cc3_membered), []). fof(cc3_sppol_2,axiom,( ! [A] : ( m1_finseq_1(A,u1_struct_0(k15_euclid(2))) => ( v4_topreal1(A) => ~ v1_xboole_0(A) ) ) ), file(sppol_2,cc3_sppol_2), []). 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_membered,axiom,( ! [A] : ( v2_membered(A) => v1_membered(A) ) ), file(membered,cc4_membered), []). 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_sprect_1,axiom,( ! [A] : ( m1_finseq_1(A,u1_struct_0(k15_euclid(2))) => ( v1_sprect_1(A) => ~ v5_seqm_3(A) ) ) ), file(sprect_1,cc5_sprect_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_sprect_1,axiom,( ! [A] : ( m1_finseq_1(A,u1_struct_0(k15_euclid(2))) => ( ( ~ v1_xboole_0(A) & v1_sprect_1(A) ) => ( ~ v1_xboole_0(A) & v1_topreal1(A) & v2_topreal1(A) & v1_finseq_6(A,u1_struct_0(k15_euclid(2))) & v1_goboard5(A) & v2_goboard5(A) ) ) ) ), file(sprect_1,cc6_sprect_1), []). 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(commutativity_k3_xboole_0,axiom,( ! [A,B] : k3_xboole_0(A,B) = k3_xboole_0(B,A) ), file(xboole_0,k3_xboole_0), []). fof(commutativity_k5_subset_1,axiom,( ! [A,B,C] : ( ( m1_subset_1(B,k1_zfmisc_1(A)) & m1_subset_1(C,k1_zfmisc_1(A)) ) => k5_subset_1(A,B,C) = k5_subset_1(A,C,B) ) ), file(subset_1,k5_subset_1), []). fof(connectedness_r1_xreal_0,axiom,( ! [A,B] : ( ( v1_xreal_0(A) & v1_xreal_0(B) ) => ( r1_xreal_0(A,B) | r1_xreal_0(B,A) ) ) ), file(xreal_0,r1_xreal_0), []). fof(d1_euclid,axiom,( ! [A] : ( m2_subset_1(A,k1_numbers,k5_numbers) => k1_euclid(A) = k4_finseq_2(A,k1_numbers) ) ), file(euclid,d1_euclid), []). fof(d1_sprect_2,axiom,( ! [A] : ( m2_finseq_1(A,u1_struct_0(k15_euclid(2))) => ! [B] : ( m2_finseq_1(B,u1_struct_0(k15_euclid(2))) => ( r1_sprect_2(A,B) <=> ! [C] : ( m2_subset_1(C,k1_numbers,k5_numbers) => ( r2_hidden(C,k4_finseq_1(B)) => ( r1_xreal_0(k18_pscomp_1(k5_topreal1(2,A)),k21_euclid(k4_finseq_4(k5_numbers,u1_struct_0(k15_euclid(2)),B,C))) & r1_xreal_0(k21_euclid(k4_finseq_4(k5_numbers,u1_struct_0(k15_euclid(2)),B,C)),k20_pscomp_1(k5_topreal1(2,A))) & r1_xreal_0(k21_pscomp_1(k5_topreal1(2,A)),k22_euclid(k4_finseq_4(k5_numbers,u1_struct_0(k15_euclid(2)),B,C))) & r1_xreal_0(k22_euclid(k4_finseq_4(k5_numbers,u1_struct_0(k15_euclid(2)),B,C)),k19_pscomp_1(k5_topreal1(2,A))) ) ) ) ) ) ) ), file(sprect_2,d1_sprect_2), []). fof(d1_tarski,axiom,( ! [A,B] : ( B = k1_tarski(A) <=> ! [C] : ( r2_hidden(C,B) <=> C = A ) ) ), file(tarski,d1_tarski), []). fof(d2_jordan5c,axiom,( ! [A] : ( m1_subset_1(A,k1_zfmisc_1(u1_struct_0(k15_euclid(2)))) => ! [B] : ( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(k15_euclid(2)))) => ! [C] : ( m1_subset_1(C,u1_struct_0(k15_euclid(2))) => ! [D] : ( m1_subset_1(D,u1_struct_0(k15_euclid(2))) => ( ( v4_pre_topc(k5_subset_1(u1_struct_0(k15_euclid(2)),A,B),k15_euclid(2)) & r1_topreal1(k15_euclid(2),C,D,A) ) => ( r1_xboole_0(A,B) | ! [E] : ( m1_subset_1(E,u1_struct_0(k15_euclid(2))) => ( E = k2_jordan5c(A,B,C,D) <=> ( r2_hidden(E,k5_subset_1(u1_struct_0(k15_euclid(2)),A,B)) & ! [F] : ( ( v1_funct_1(F) & v1_funct_2(F,u1_struct_0(k5_topmetr),u1_struct_0(k3_pre_topc(k15_euclid(2),A))) & m2_relset_1(F,u1_struct_0(k5_topmetr),u1_struct_0(k3_pre_topc(k15_euclid(2),A))) ) => ! [G] : ( m1_subset_1(G,k1_numbers) => ( ( v3_tops_2(F,k5_topmetr,k3_pre_topc(k15_euclid(2),A)) & k1_funct_1(F,0) = C & k1_funct_1(F,1) = D & k1_funct_1(F,G) = E & r1_xreal_0(0,G) & r1_xreal_0(G,1) ) => ! [H] : ( m1_subset_1(H,k1_numbers) => ~ ( r1_xreal_0(H,1) & ~ r1_xreal_0(H,G) & r2_hidden(k1_funct_1(F,H),B) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ), file(jordan5c,d2_jordan5c), []). fof(d3_xboole_0,axiom,( ! [A,B,C] : ( C = k3_xboole_0(A,B) <=> ! [D] : ( r2_hidden(D,C) <=> ( r2_hidden(D,A) & r2_hidden(D,B) ) ) ) ), file(xboole_0,d3_xboole_0), []). fof(d4_finseq_4,axiom,( ! [A,B,C] : ( ( v1_funct_1(C) & m2_relset_1(C,A,B) ) => ! [D] : ( r2_hidden(D,k1_relat_1(C)) => k4_finseq_4(A,B,C,D) = k1_funct_1(C,D) ) ) ), file(finseq_4,d4_finseq_4), []). fof(d4_jordan3,axiom,( ! [A] : ( m2_finseq_1(A,u1_struct_0(k15_euclid(2))) => ! [B] : ( m1_subset_1(B,u1_struct_0(k15_euclid(2))) => ( ( B != k1_funct_1(A,k1_nat_1(k2_jordan3(A,B),1)) => k3_jordan3(A,B) = k8_finseq_1(u1_struct_0(k15_euclid(2)),k12_finseq_1(u1_struct_0(k15_euclid(2)),B),k1_jordan3(u1_struct_0(k15_euclid(2)),A,k1_nat_1(k2_jordan3(A,B),1),k3_finseq_1(A))) ) & ( B = k1_funct_1(A,k1_nat_1(k2_jordan3(A,B),1)) => k3_jordan3(A,B) = k1_jordan3(u1_struct_0(k15_euclid(2)),A,k1_nat_1(k2_jordan3(A,B),1),k3_finseq_1(A)) ) ) ) ) ), file(jordan3,d4_jordan3), []). fof(d7_euclid,axiom,( ! [A] : ( m2_subset_1(A,k1_numbers,k5_numbers) => k14_euclid(A) = g1_metric_1(k1_euclid(A),k13_euclid(A)) ) ), file(euclid,d7_euclid), []). fof(d8_euclid,axiom,( ! [A] : ( m2_subset_1(A,k1_numbers,k5_numbers) => k15_euclid(A) = k5_pcomps_1(k14_euclid(A)) ) ), file(euclid,d8_euclid), []). fof(d8_topreal1,axiom,( ! [A] : ( m2_finseq_1(A,u1_struct_0(k15_euclid(2))) => ( v2_topreal1(A) <=> ! [B] : ( m2_subset_1(B,k1_numbers,k5_numbers) => ( ( r1_xreal_0(1,B) & r1_xreal_0(k1_nat_1(B,2),k3_finseq_1(A)) ) => k5_subset_1(u1_struct_0(k15_euclid(2)),k4_topreal1(2,A,B),k4_topreal1(2,A,k1_nat_1(B,1))) = k1_struct_0(k15_euclid(2),k4_finseq_4(k5_numbers,u1_struct_0(k15_euclid(2)),A,k1_nat_1(B,1))) ) ) ) ) ), file(topreal1,d8_topreal1), []). fof(d9_topreal1,axiom,( ! [A] : ( m2_finseq_1(A,u1_struct_0(k15_euclid(2))) => ( v3_topreal1(A) <=> ! [B] : ( m2_subset_1(B,k1_numbers,k5_numbers) => ! [C] : ( m2_subset_1(C,k1_numbers,k5_numbers) => ( ~ r1_xreal_0(C,k1_nat_1(B,1)) => r1_xboole_0(k4_topreal1(2,A,B),k4_topreal1(2,A,C)) ) ) ) ) ) ), file(topreal1,d9_topreal1), []). fof(dt_g1_metric_1,axiom,( ! [A,B] : ( ( v1_funct_1(B) & v1_funct_2(B,k2_zfmisc_1(A,A),k1_numbers) & m1_relset_1(B,k2_zfmisc_1(A,A),k1_numbers) ) => ( v1_metric_1(g1_metric_1(A,B)) & l1_metric_1(g1_metric_1(A,B)) ) ) ), file(metric_1,g1_metric_1), []). fof(dt_g1_pre_topc,axiom,( ! [A,B] : ( m1_subset_1(B,k1_zfmisc_1(k1_zfmisc_1(A))) => ( v1_pre_topc(g1_pre_topc(A,B)) & l1_pre_topc(g1_pre_topc(A,B)) ) ) ), file(pre_topc,g1_pre_topc), []). fof(dt_k12_finseq_1,axiom,( ! [A,B] : ( ( ~ v1_xboole_0(A) & m1_subset_1(B,A) ) => m2_finseq_1(k12_finseq_1(A,B),A) ) ), file(finseq_1,k12_finseq_1), []). fof(dt_k13_euclid,axiom,( ! [A] : ( m1_subset_1(A,k5_numbers) => ( v1_funct_1(k13_euclid(A)) & v1_funct_2(k13_euclid(A),k2_zfmisc_1(k1_euclid(A),k1_euclid(A)),k1_numbers) & m2_relset_1(k13_euclid(A),k2_zfmisc_1(k1_euclid(A),k1_euclid(A)),k1_numbers) ) ) ), file(euclid,k13_euclid), []). fof(dt_k14_euclid,axiom,( ! [A] : ( m1_subset_1(A,k5_numbers) => ( v1_metric_1(k14_euclid(A)) & v6_metric_1(k14_euclid(A)) & v7_metric_1(k14_euclid(A)) & v8_metric_1(k14_euclid(A)) & v9_metric_1(k14_euclid(A)) & l1_metric_1(k14_euclid(A)) ) ) ), file(euclid,k14_euclid), []). fof(dt_k15_euclid,axiom,( ! [A] : ( m1_subset_1(A,k5_numbers) => ( v1_pre_topc(k15_euclid(A)) & v2_pre_topc(k15_euclid(A)) & l1_pre_topc(k15_euclid(A)) ) ) ), file(euclid,k15_euclid), []). fof(dt_k18_pscomp_1,axiom,( ! [A] : ( m1_subset_1(A,k1_zfmisc_1(u1_struct_0(k15_euclid(2)))) => m1_subset_1(k18_pscomp_1(A),k1_numbers) ) ), file(pscomp_1,k18_pscomp_1), []). fof(dt_k19_pscomp_1,axiom,( ! [A] : ( m1_subset_1(A,k1_zfmisc_1(u1_struct_0(k15_euclid(2)))) => m1_subset_1(k19_pscomp_1(A),k1_numbers) ) ), file(pscomp_1,k19_pscomp_1), []). fof(dt_k1_card_1,axiom,( ! [A] : v1_card_1(k1_card_1(A)) ), file(card_1,k1_card_1), []). fof(dt_k1_euclid,axiom,( ! [A] : ( m1_subset_1(A,k5_numbers) => ( ~ v1_xboole_0(k1_euclid(A)) & m1_finseq_2(k1_euclid(A),k1_numbers) ) ) ), file(euclid,k1_euclid), []). fof(dt_k1_funct_1,axiom,( $true ), file(funct_1,k1_funct_1), []). fof(dt_k1_jordan3,axiom,( ! [A,B,C,D] : ( ( ~ v1_xboole_0(A) & m1_finseq_1(B,A) & m1_subset_1(C,k5_numbers) & m1_subset_1(D,k5_numbers) ) => m2_finseq_1(k1_jordan3(A,B,C,D),A) ) ), file(jordan3,k1_jordan3), []). 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_rfinseq,axiom,( ! [A,B,C] : ( ( m1_finseq_1(B,A) & m1_subset_1(C,k5_numbers) ) => m2_finseq_1(k1_rfinseq(A,B,C),A) ) ), file(rfinseq,k1_rfinseq), []). fof(dt_k1_struct_0,axiom,( ! [A,B] : ( ( ~ v3_struct_0(A) & l1_struct_0(A) & m1_subset_1(B,u1_struct_0(A)) ) => m1_subset_1(k1_struct_0(A,B),k1_zfmisc_1(u1_struct_0(A))) ) ), file(struct_0,k1_struct_0), []). 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_pscomp_1,axiom,( ! [A] : ( m1_subset_1(A,k1_zfmisc_1(u1_struct_0(k15_euclid(2)))) => m1_subset_1(k20_pscomp_1(A),k1_numbers) ) ), file(pscomp_1,k20_pscomp_1), []). fof(dt_k21_euclid,axiom,( ! [A] : ( m1_subset_1(A,u1_struct_0(k15_euclid(2))) => m1_subset_1(k21_euclid(A),k1_numbers) ) ), file(euclid,k21_euclid), []). fof(dt_k21_pscomp_1,axiom,( ! [A] : ( m1_subset_1(A,k1_zfmisc_1(u1_struct_0(k15_euclid(2)))) => m1_subset_1(k21_pscomp_1(A),k1_numbers) ) ), file(pscomp_1,k21_pscomp_1), []). fof(dt_k22_borsuk_1,axiom,( l1_pre_topc(k22_borsuk_1) ), file(borsuk_1,k22_borsuk_1), []). fof(dt_k22_euclid,axiom,( ! [A] : ( m1_subset_1(A,u1_struct_0(k15_euclid(2))) => m1_subset_1(k22_euclid(A),k1_numbers) ) ), file(euclid,k22_euclid), []). fof(dt_k2_goboard9,axiom,( ! [A] : ( ( ~ v1_xboole_0(A) & ~ v5_seqm_3(A) & v1_topreal1(A) & v2_topreal1(A) & v1_finseq_6(A,u1_struct_0(k15_euclid(2))) & v1_goboard5(A) & v2_goboard5(A) & m1_finseq_1(A,u1_struct_0(k15_euclid(2))) ) => m1_subset_1(k2_goboard9(A),k1_zfmisc_1(u1_struct_0(k15_euclid(2)))) ) ), file(goboard9,k2_goboard9), []). fof(dt_k2_jordan3,axiom,( ! [A,B] : ( ( m1_finseq_1(A,u1_struct_0(k15_euclid(2))) & m1_subset_1(B,u1_struct_0(k15_euclid(2))) ) => m2_subset_1(k2_jordan3(A,B),k1_numbers,k5_numbers) ) ), file(jordan3,k2_jordan3), []). fof(dt_k2_jordan5c,axiom,( ! [A,B,C,D] : ( ( m1_subset_1(A,k1_zfmisc_1(u1_struct_0(k15_euclid(2)))) & m1_subset_1(B,k1_zfmisc_1(u1_struct_0(k15_euclid(2)))) & m1_subset_1(C,u1_struct_0(k15_euclid(2))) & m1_subset_1(D,u1_struct_0(k15_euclid(2))) ) => m1_subset_1(k2_jordan5c(A,B,C,D),u1_struct_0(k15_euclid(2))) ) ), file(jordan5c,k2_jordan5c), []). 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_k3_finseq_1,axiom,( ! [A] : ( ( v1_relat_1(A) & v1_funct_1(A) & v1_finseq_1(A) ) => m2_subset_1(k3_finseq_1(A),k1_numbers,k5_numbers) ) ), file(finseq_1,k3_finseq_1), []). fof(dt_k3_goboard9,axiom,( ! [A] : ( ( ~ v1_xboole_0(A) & ~ v5_seqm_3(A) & v1_topreal1(A) & v2_topreal1(A) & v1_finseq_6(A,u1_struct_0(k15_euclid(2))) & v1_goboard5(A) & v2_goboard5(A) & m1_finseq_1(A,u1_struct_0(k15_euclid(2))) ) => m1_subset_1(k3_goboard9(A),k1_zfmisc_1(u1_struct_0(k15_euclid(2)))) ) ), file(goboard9,k3_goboard9), []). fof(dt_k3_jordan3,axiom,( ! [A,B] : ( ( m1_finseq_1(A,u1_struct_0(k15_euclid(2))) & m1_subset_1(B,u1_struct_0(k15_euclid(2))) ) => m2_finseq_1(k3_jordan3(A,B),u1_struct_0(k15_euclid(2))) ) ), file(jordan3,k3_jordan3), []). fof(dt_k3_pre_topc,axiom,( ! [A,B] : ( ( l1_pre_topc(A) & m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) ) => ( v1_pre_topc(k3_pre_topc(A,B)) & m1_pre_topc(k3_pre_topc(A,B),A) ) ) ), file(pre_topc,k3_pre_topc), []). fof(dt_k3_topmetr,axiom, ( v1_pre_topc(k3_topmetr) & v2_pre_topc(k3_topmetr) & l1_pre_topc(k3_topmetr) ), file(topmetr,k3_topmetr), []). fof(dt_k3_xboole_0,axiom,( $true ), file(xboole_0,k3_xboole_0), []). fof(dt_k4_finseq_1,axiom,( ! [A] : ( ( v1_relat_1(A) & v1_funct_1(A) & v1_finseq_1(A) ) => m1_subset_1(k4_finseq_1(A),k1_zfmisc_1(k5_numbers)) ) ), file(finseq_1,k4_finseq_1), []). fof(dt_k4_finseq_2,axiom,( ! [A,B] : ( v4_ordinal2(A) => m1_finseq_2(k4_finseq_2(A,B),B) ) ), file(finseq_2,k4_finseq_2), []). fof(dt_k4_finseq_4,axiom,( ! [A,B,C,D] : ( ( v1_funct_1(C) & m1_relset_1(C,A,B) ) => m1_subset_1(k4_finseq_4(A,B,C,D),B) ) ), file(finseq_4,k4_finseq_4), []). fof(dt_k4_matrix_2,axiom,( ! [A,B] : ( ( ~ v1_xboole_0(A) & m1_subset_1(B,A) ) => m2_finseq_2(k4_matrix_2(A,B),A,k4_finseq_2(1,A)) ) ), file(matrix_2,k4_matrix_2), []). fof(dt_k4_topreal1,axiom,( ! [A,B,C] : ( ( m1_subset_1(A,k5_numbers) & m1_finseq_1(B,u1_struct_0(k15_euclid(A))) & m1_subset_1(C,k5_numbers) ) => m1_subset_1(k4_topreal1(A,B,C),k1_zfmisc_1(u1_struct_0(k15_euclid(A)))) ) ), file(topreal1,k4_topreal1), []). fof(dt_k5_binarith,axiom,( ! [A,B] : ( ( v4_ordinal2(A) & v4_ordinal2(B) ) => m2_subset_1(k5_binarith(A,B),k1_numbers,k5_numbers) ) ), file(binarith,k5_binarith), []). fof(dt_k5_finseq_1,axiom,( $true ), file(finseq_1,k5_finseq_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_pcomps_1,axiom,( ! [A] : ( l1_metric_1(A) => l1_pre_topc(k5_pcomps_1(A)) ) ), file(pcomps_1,k5_pcomps_1), []). fof(dt_k5_subset_1,axiom,( ! [A,B,C] : ( ( m1_subset_1(B,k1_zfmisc_1(A)) & m1_subset_1(C,k1_zfmisc_1(A)) ) => m1_subset_1(k5_subset_1(A,B,C),k1_zfmisc_1(A)) ) ), file(subset_1,k5_subset_1), []). fof(dt_k5_topmetr,axiom, ( v1_pre_topc(k5_topmetr) & m1_pre_topc(k5_topmetr,k3_topmetr) ), file(topmetr,k5_topmetr), []). fof(dt_k5_topreal1,axiom,( ! [A,B] : ( ( m1_subset_1(A,k5_numbers) & m1_finseq_1(B,u1_struct_0(k15_euclid(A))) ) => m1_subset_1(k5_topreal1(A,B),k1_zfmisc_1(u1_struct_0(k15_euclid(A)))) ) ), file(topreal1,k5_topreal1), []). 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_k8_finseq_1,axiom,( ! [A,B,C] : ( ( m1_finseq_1(B,A) & m1_finseq_1(C,A) ) => m2_finseq_1(k8_finseq_1(A,B,C),A) ) ), file(finseq_1,k8_finseq_1), []). fof(dt_k9_finseq_1,axiom,( ! [A] : ( v1_relat_1(k9_finseq_1(A)) & v1_funct_1(k9_finseq_1(A)) ) ), file(finseq_1,k9_finseq_1), []). fof(dt_l1_metric_1,axiom,( ! [A] : ( l1_metric_1(A) => l1_struct_0(A) ) ), file(metric_1,l1_metric_1), []). fof(dt_l1_pre_topc,axiom,( ! [A] : ( l1_pre_topc(A) => l1_struct_0(A) ) ), file(pre_topc,l1_pre_topc), []). fof(dt_l1_struct_0,axiom,( $true ), file(struct_0,l1_struct_0), []). 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_pre_topc,axiom,( ! [A] : ( l1_pre_topc(A) => ! [B] : ( m1_pre_topc(B,A) => l1_pre_topc(B) ) ) ), file(pre_topc,m1_pre_topc), []). 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_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_metric_1,axiom,( ! [A] : ( l1_metric_1(A) => ( v1_funct_1(u1_metric_1(A)) & v1_funct_2(u1_metric_1(A),k2_zfmisc_1(u1_struct_0(A),u1_struct_0(A)),k1_numbers) & m2_relset_1(u1_metric_1(A),k2_zfmisc_1(u1_struct_0(A),u1_struct_0(A)),k1_numbers) ) ) ), file(metric_1,u1_metric_1), []). fof(dt_u1_pre_topc,axiom,( ! [A] : ( l1_pre_topc(A) => m1_subset_1(u1_pre_topc(A),k1_zfmisc_1(k1_zfmisc_1(u1_struct_0(A)))) ) ), file(pre_topc,u1_pre_topc), []). fof(dt_u1_struct_0,axiom,( $true ), file(struct_0,u1_struct_0), []). fof(existence_l1_metric_1,axiom,( ? [A] : l1_metric_1(A) ), file(metric_1,l1_metric_1), []). fof(existence_l1_pre_topc,axiom,( ? [A] : l1_pre_topc(A) ), file(pre_topc,l1_pre_topc), []). fof(existence_l1_struct_0,axiom,( ? [A] : l1_struct_0(A) ), file(struct_0,l1_struct_0), []). 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_pre_topc,axiom,( ! [A] : ( l1_pre_topc(A) => ? [B] : m1_pre_topc(B,A) ) ), file(pre_topc,m1_pre_topc), []). 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_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_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(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_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(fc11_sprect_1,axiom,( ! [A] : ( ( v1_sprect_1(A) & m1_finseq_1(A,u1_struct_0(k15_euclid(2))) ) => ( ~ v1_xboole_0(k5_topreal1(2,A)) & v2_jordan1(k5_topreal1(2,A)) & v6_compts_1(k5_topreal1(2,A),k15_euclid(2)) ) ) ), file(sprect_1,fc11_sprect_1), []). 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_sprect_1,axiom,( ! [A] : ( ( ~ v1_xboole_0(A) & ~ v5_seqm_3(A) & v1_topreal1(A) & v2_topreal1(A) & v1_finseq_6(A,u1_struct_0(k15_euclid(2))) & v1_goboard5(A) & v2_goboard5(A) & m1_finseq_1(A,u1_struct_0(k15_euclid(2))) ) => ~ v1_xboole_0(k2_goboard9(A)) ) ), file(sprect_1,fc12_sprect_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_sprect_1,axiom,( ! [A] : ( ( ~ v1_xboole_0(A) & ~ v5_seqm_3(A) & v1_topreal1(A) & v2_topreal1(A) & v1_finseq_6(A,u1_struct_0(k15_euclid(2))) & v1_goboard5(A) & v2_goboard5(A) & m1_finseq_1(A,u1_struct_0(k15_euclid(2))) ) => ~ v1_xboole_0(k3_goboard9(A)) ) ), file(sprect_1,fc13_sprect_1), []). fof(fc1_euclid,axiom,( ! [A] : ( m1_subset_1(A,k5_numbers) => ( ~ v3_struct_0(k14_euclid(A)) & v1_metric_1(k14_euclid(A)) & v6_metric_1(k14_euclid(A)) & v7_metric_1(k14_euclid(A)) & v8_metric_1(k14_euclid(A)) & v9_metric_1(k14_euclid(A)) ) ) ), file(euclid,fc1_euclid), []). fof(fc1_finseq_5,axiom,( ! [A] : ( ~ v1_xboole_0(k5_finseq_1(A)) & v1_relat_1(k5_finseq_1(A)) & v1_funct_1(k5_finseq_1(A)) & v1_realset1(k5_finseq_1(A)) & v1_finset_1(k5_finseq_1(A)) & v1_finseq_1(k5_finseq_1(A)) ) ), file(finseq_5,fc1_finseq_5), []). fof(fc1_sprect_1,axiom,( ! [A] : ( ( v1_relat_1(A) & v1_funct_1(A) & v5_seqm_3(A) ) => v1_realset1(k2_relat_1(A)) ) ), file(sprect_1,fc1_sprect_1), []). fof(fc1_sprect_3,axiom,( ! [A] : ( ( ~ v1_xboole_0(A) & ~ v5_seqm_3(A) & v1_topreal1(A) & v2_topreal1(A) & v1_finseq_6(A,u1_struct_0(k15_euclid(2))) & v1_goboard5(A) & v2_goboard5(A) & m1_finseq_1(A,u1_struct_0(k15_euclid(2))) ) => ( ~ v1_xboole_0(k5_topreal1(2,A)) & ~ v1_sppol_1(k5_topreal1(2,A)) & ~ v2_sppol_1(k5_topreal1(2,A)) & v6_compts_1(k5_topreal1(2,A),k15_euclid(2)) & v1_topreal2(k5_topreal1(2,A)) ) ) ), file(sprect_3,fc1_sprect_3), []). 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(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_euclid,axiom,( ! [A] : ( m1_subset_1(A,k5_numbers) => ( ~ v3_struct_0(k15_euclid(A)) & v1_pre_topc(k15_euclid(A)) & v2_pre_topc(k15_euclid(A)) ) ) ), file(euclid,fc2_euclid), []). fof(fc2_membered,axiom, ( ~ v1_xboole_0(k1_numbers) & v1_membered(k1_numbers) & v2_membered(k1_numbers) ), file(membered,fc2_membered), []). fof(fc2_sppol_2,axiom,( ! [A,B] : ( ( v2_topreal1(A) & m1_finseq_1(A,u1_struct_0(k15_euclid(2))) & m1_subset_1(B,k5_numbers) ) => ( v1_relat_1(k1_rfinseq(u1_struct_0(k15_euclid(2)),A,B)) & v1_funct_1(k1_rfinseq(u1_struct_0(k15_euclid(2)),A,B)) & v1_finset_1(k1_rfinseq(u1_struct_0(k15_euclid(2)),A,B)) & v1_finseq_1(k1_rfinseq(u1_struct_0(k15_euclid(2)),A,B)) & v2_topreal1(k1_rfinseq(u1_struct_0(k15_euclid(2)),A,B)) ) ) ), file(sppol_2,fc2_sppol_2), []). fof(fc2_sprect_1,axiom,( ! [A,B] : ( ( m1_subset_1(A,k5_numbers) & ~ v1_realset1(B) & m1_finseq_1(B,u1_struct_0(k15_euclid(A))) ) => ~ v1_xboole_0(k5_topreal1(A,B)) ) ), file(sprect_1,fc2_sprect_1), []). fof(fc2_sprect_3,axiom,( ! [A] : ( ( ~ v1_xboole_0(A) & ~ v5_seqm_3(A) & v1_topreal1(A) & v2_topreal1(A) & v1_finseq_6(A,u1_struct_0(k15_euclid(2))) & v1_goboard5(A) & v2_goboard5(A) & m1_finseq_1(A,u1_struct_0(k15_euclid(2))) ) => ( ~ v1_xboole_0(k2_goboard9(A)) & v2_topreal4(k2_goboard9(A),k15_euclid(2)) ) ) ), file(sprect_3,fc2_sprect_3), []). 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(fc3_sprect_1,axiom,( ! [A] : ( m1_finseq_1(A,u1_struct_0(k15_euclid(2))) => v6_compts_1(k5_topreal1(2,A),k15_euclid(2)) ) ), file(sprect_1,fc3_sprect_1), []). fof(fc3_sprect_2,axiom,( ! [A] : ( ( ~ v1_xboole_0(A) & v1_topreal1(A) & v2_topreal1(A) & v1_finseq_6(A,u1_struct_0(k15_euclid(2))) & v1_goboard5(A) & v2_goboard5(A) & ~ v5_seqm_3(A) & m1_finseq_1(A,u1_struct_0(k15_euclid(2))) ) => ( ~ v1_xboole_0(k5_topreal1(2,A)) & v1_topreal2(k5_topreal1(2,A)) & v6_compts_1(k5_topreal1(2,A),k15_euclid(2)) ) ) ), file(sprect_2,fc3_sprect_2), []). fof(fc3_sprect_3,axiom,( ! [A] : ( ( ~ v1_xboole_0(A) & ~ v5_seqm_3(A) & v1_topreal1(A) & v2_topreal1(A) & v1_finseq_6(A,u1_struct_0(k15_euclid(2))) & v1_goboard5(A) & v2_goboard5(A) & m1_finseq_1(A,u1_struct_0(k15_euclid(2))) ) => ( ~ v1_xboole_0(k3_goboard9(A)) & v2_topreal4(k3_goboard9(A),k15_euclid(2)) ) ) ), file(sprect_3,fc3_sprect_3), []). 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(fc4_finseq_5,axiom,( ! [A,B,C] : ( ( m1_subset_1(A,k5_numbers) & ~ v1_xboole_0(B) & v2_funct_1(C) & m1_finseq_1(C,B) ) => ( v1_relat_1(k1_rfinseq(B,C,A)) & v1_funct_1(k1_rfinseq(B,C,A)) & v2_funct_1(k1_rfinseq(B,C,A)) & v1_finset_1(k1_rfinseq(B,C,A)) & v1_finseq_1(k1_rfinseq(B,C,A)) ) ) ), file(finseq_5,fc4_finseq_5), []). 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_sppol_2,axiom,( ! [A,B] : ( ( v3_topreal1(A) & m1_finseq_1(A,u1_struct_0(k15_euclid(2))) & m1_subset_1(B,k5_numbers) ) => ( v1_relat_1(k1_rfinseq(u1_struct_0(k15_euclid(2)),A,B)) & v1_funct_1(k1_rfinseq(u1_struct_0(k15_euclid(2)),A,B)) & v1_finset_1(k1_rfinseq(u1_struct_0(k15_euclid(2)),A,B)) & v1_finseq_1(k1_rfinseq(u1_struct_0(k15_euclid(2)),A,B)) & v3_topreal1(k1_rfinseq(u1_struct_0(k15_euclid(2)),A,B)) ) ) ), file(sppol_2,fc5_sppol_2), []). 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(fc7_membered,axiom,( ! [A] : ( v1_xcmplx_0(A) => v1_membered(k1_tarski(A)) ) ), file(membered,fc7_membered), []). 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_membered,axiom,( ! [A] : ( v1_xreal_0(A) => ( v1_membered(k1_tarski(A)) & v2_membered(k1_tarski(A)) ) ) ), file(membered,fc8_membered), []). 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_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(fc9_sppol_2,axiom,( ! [A,B] : ( ( v1_topreal1(A) & m1_finseq_1(A,u1_struct_0(k15_euclid(2))) & m1_subset_1(B,k5_numbers) ) => ( v1_relat_1(k1_rfinseq(u1_struct_0(k15_euclid(2)),A,B)) & v1_funct_1(k1_rfinseq(u1_struct_0(k15_euclid(2)),A,B)) & v1_finset_1(k1_rfinseq(u1_struct_0(k15_euclid(2)),A,B)) & v1_finseq_1(k1_rfinseq(u1_struct_0(k15_euclid(2)),A,B)) & v1_topreal1(k1_rfinseq(u1_struct_0(k15_euclid(2)),A,B)) ) ) ), file(sppol_2,fc9_sppol_2), []). 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(fraenkel_a_1_5_sprect_3,axiom,( ! [A,B] : ( ( ~ v1_xboole_0(B) & v1_sprect_1(B) & v1_topreal1(B) & v2_topreal1(B) & v1_finseq_6(B,u1_struct_0(k15_euclid(2))) & v1_goboard5(B) & m2_finseq_1(B,u1_struct_0(k15_euclid(2))) ) => ( r2_hidden(A,a_1_5_sprect_3(B)) <=> ? [C] : ( m1_subset_1(C,u1_struct_0(k15_euclid(2))) & A = C & ~ ( r1_xreal_0(k18_pscomp_1(k5_topreal1(2,B)),k21_euclid(C)) & r1_xreal_0(k21_euclid(C),k20_pscomp_1(k5_topreal1(2,B))) & r1_xreal_0(k21_pscomp_1(k5_topreal1(2,B)),k22_euclid(C)) & r1_xreal_0(k22_euclid(C),k19_pscomp_1(k5_topreal1(2,B))) ) ) ) ) ), file(sprect_3,a_1_5_sprect_3), []). fof(fraenkel_a_1_6_sprect_3,axiom,( ! [A,B] : ( ( ~ v1_xboole_0(B) & v1_sprect_1(B) & v1_topreal1(B) & v2_topreal1(B) & v1_finseq_6(B,u1_struct_0(k15_euclid(2))) & v1_goboard5(B) & m2_finseq_1(B,u1_struct_0(k15_euclid(2))) ) => ( r2_hidden(A,a_1_6_sprect_3(B)) <=> ? [C] : ( m1_subset_1(C,u1_struct_0(k15_euclid(2))) & A = C & ~ r1_xreal_0(k21_euclid(C),k18_pscomp_1(k5_topreal1(2,B))) & ~ r1_xreal_0(k20_pscomp_1(k5_topreal1(2,B)),k21_euclid(C)) & ~ r1_xreal_0(k22_euclid(C),k21_pscomp_1(k5_topreal1(2,B))) & ~ r1_xreal_0(k19_pscomp_1(k5_topreal1(2,B)),k22_euclid(C)) ) ) ) ), file(sprect_3,a_1_6_sprect_3), []). fof(free_g1_metric_1,axiom,( ! [A,B] : ( ( v1_funct_1(B) & v1_funct_2(B,k2_zfmisc_1(A,A),k1_numbers) & m1_relset_1(B,k2_zfmisc_1(A,A),k1_numbers) ) => ! [C,D] : ( g1_metric_1(A,B) = g1_metric_1(C,D) => ( A = C & B = D ) ) ) ), file(metric_1,g1_metric_1), []). fof(free_g1_pre_topc,axiom,( ! [A,B] : ( m1_subset_1(B,k1_zfmisc_1(k1_zfmisc_1(A))) => ! [C,D] : ( g1_pre_topc(A,B) = g1_pre_topc(C,D) => ( A = C & B = D ) ) ) ), file(pre_topc,g1_pre_topc), []). fof(idempotence_k3_xboole_0,axiom,( ! [A,B] : k3_xboole_0(A,A) = A ), file(xboole_0,k3_xboole_0), []). fof(idempotence_k5_subset_1,axiom,( ! [A,B,C] : ( ( m1_subset_1(B,k1_zfmisc_1(A)) & m1_subset_1(C,k1_zfmisc_1(A)) ) => k5_subset_1(A,B,B) = B ) ), file(subset_1,k5_subset_1), []). fof(irreflexivity_r1_subset_1,axiom,( ! [A,B] : ( ( ~ v1_xboole_0(A) & ~ v1_xboole_0(B) ) => ~ r1_subset_1(A,A) ) ), file(subset_1,r1_subset_1), []). fof(irreflexivity_r2_subset_1,axiom,( ! [A,B] : ( ( ~ v1_xboole_0(A) & ~ v1_xboole_0(B) ) => ~ r2_subset_1(A,A) ) ), file(subset_1,r2_subset_1), []). fof(rc1_arytm_3,axiom,( ? [A] : ( ~ v1_xboole_0(A) & v1_ordinal1(A) & v2_ordinal1(A) & v3_ordinal1(A) & v4_ordinal2(A) ) ), file(arytm_3,rc1_arytm_3), []). 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_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_sppol_2,axiom,( ? [A] : ( m1_finseq_1(A,u1_struct_0(k15_euclid(2))) & v1_relat_1(A) & v1_funct_1(A) & v1_finset_1(A) & v1_finseq_1(A) & v4_topreal1(A) ) ), file(sppol_2,rc1_sppol_2), []). fof(rc1_sprect_1,axiom,( ? [A] : ( ~ v1_xboole_0(A) & v1_relat_1(A) & v1_funct_1(A) & v1_finset_1(A) & v1_finseq_1(A) & v5_seqm_3(A) ) ), file(sprect_1,rc1_sprect_1), []). fof(rc1_sprect_2,axiom,( ? [A] : ( m1_finseq_1(A,u1_struct_0(k15_euclid(2))) & ~ v1_xboole_0(A) & v1_relat_1(A) & v1_funct_1(A) & v1_finset_1(A) & v1_finseq_1(A) & v1_topreal1(A) & v2_topreal1(A) & v1_finseq_6(A,u1_struct_0(k15_euclid(2))) & v1_goboard5(A) & v2_goboard5(A) & ~ v1_realset1(A) & ~ v5_seqm_3(A) & v1_sprect_2(A) ) ), file(sprect_2,rc1_sprect_2), []). fof(rc1_sprect_3,axiom,( ? [A] : ( m1_finseq_1(A,u1_struct_0(k15_euclid(2))) & ~ v1_xboole_0(A) & v1_relat_1(A) & v1_funct_1(A) & v1_finseq_1(A) & ~ v1_realset1(A) & v1_sprect_1(A) & ~ v5_seqm_3(A) & v1_topreal1(A) & v2_topreal1(A) & v1_finseq_6(A,u1_struct_0(k15_euclid(2))) & v1_goboard5(A) & v2_goboard5(A) ) ), file(sprect_3,rc1_sprect_3), []). fof(rc1_xreal_0,axiom,( ? [A] : ( v1_xcmplx_0(A) & v1_xreal_0(A) ) ), file(xreal_0,rc1_xreal_0), []). fof(rc2_finseq_5,axiom,( ! [A] : ( ~ v1_xboole_0(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_5,rc2_finseq_5), []). fof(rc2_sppol_2,axiom,( ? [A] : ( m1_finseq_1(A,u1_struct_0(k15_euclid(2))) & ~ v1_xboole_0(A) & v1_relat_1(A) & v1_funct_1(A) & v2_funct_1(A) & ~ v1_realset1(A) & v1_finset_1(A) & v1_finseq_1(A) & v1_topreal1(A) & v2_topreal1(A) & v3_topreal1(A) & v4_topreal1(A) ) ), file(sppol_2,rc2_sppol_2), []). fof(rc2_sprect_1,axiom,( ? [A] : ( m1_subset_1(A,k1_zfmisc_1(u1_struct_0(k15_euclid(2)))) & ~ v1_xboole_0(A) & ~ v1_sppol_1(A) & ~ v2_sppol_1(A) & v6_compts_1(A,k15_euclid(2)) ) ), file(sprect_1,rc2_sprect_1), []). fof(rc2_sprect_3,axiom,( ? [A] : ( m1_finseq_1(A,u1_struct_0(k15_euclid(2))) & ~ v1_xboole_0(A) & v1_relat_1(A) & v1_funct_1(A) & v1_finseq_1(A) & v1_sprect_2(A) & ~ v1_realset1(A) & v1_sprect_1(A) & ~ v5_seqm_3(A) & v1_topreal1(A) & v2_topreal1(A) & v1_finseq_6(A,u1_struct_0(k15_euclid(2))) & v1_goboard5(A) & v2_goboard5(A) ) ), file(sprect_3,rc2_sprect_3), []). 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_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_sprect_1,axiom,( ? [A] : ( m1_finseq_1(A,u1_struct_0(k15_euclid(2))) & v1_relat_1(A) & v1_funct_1(A) & v1_finset_1(A) & v1_finseq_1(A) & v1_sprect_1(A) ) ), file(sprect_1,rc4_sprect_1), []). 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(redefinition_k12_finseq_1,axiom,( ! [A,B] : ( ( ~ v1_xboole_0(A) & m1_subset_1(B,A) ) => k12_finseq_1(A,B) = k5_finseq_1(B) ) ), file(finseq_1,k12_finseq_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_k1_struct_0,axiom,( ! [A,B] : ( ( ~ v3_struct_0(A) & l1_struct_0(A) & m1_subset_1(B,u1_struct_0(A)) ) => k1_struct_0(A,B) = k1_tarski(B) ) ), file(struct_0,k1_struct_0), []). fof(redefinition_k3_finseq_1,axiom,( ! [A] : ( ( v1_relat_1(A) & v1_funct_1(A) & v1_finseq_1(A) ) => k3_finseq_1(A) = k1_card_1(A) ) ), file(finseq_1,k3_finseq_1), []). fof(redefinition_k4_finseq_1,axiom,( ! [A] : ( ( v1_relat_1(A) & v1_funct_1(A) & v1_finseq_1(A) ) => k4_finseq_1(A) = k1_relat_1(A) ) ), file(finseq_1,k4_finseq_1), []). fof(redefinition_k4_matrix_2,axiom,( ! [A,B] : ( ( ~ v1_xboole_0(A) & m1_subset_1(B,A) ) => k4_matrix_2(A,B) = k5_finseq_1(B) ) ), file(matrix_2,k4_matrix_2), []). fof(redefinition_k5_numbers,axiom,( k5_numbers = k5_ordinal2 ), file(numbers,k5_numbers), []). fof(redefinition_k5_subset_1,axiom,( ! [A,B,C] : ( ( m1_subset_1(B,k1_zfmisc_1(A)) & m1_subset_1(C,k1_zfmisc_1(A)) ) => k5_subset_1(A,B,C) = k3_xboole_0(B,C) ) ), file(subset_1,k5_subset_1), []). fof(redefinition_k5_topmetr,axiom,( k5_topmetr = k22_borsuk_1 ), file(topmetr,k5_topmetr), []). fof(redefinition_k8_finseq_1,axiom,( ! [A,B,C] : ( ( m1_finseq_1(B,A) & m1_finseq_1(C,A) ) => k8_finseq_1(A,B,C) = k7_finseq_1(B,C) ) ), file(finseq_1,k8_finseq_1), []). fof(redefinition_k9_finseq_1,axiom,( ! [A] : k9_finseq_1(A) = k5_finseq_1(A) ), file(finseq_1,k9_finseq_1), []). 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_r1_subset_1,axiom,( ! [A,B] : ( ( ~ v1_xboole_0(A) & ~ v1_xboole_0(B) ) => ( r1_subset_1(A,B) <=> r1_xboole_0(A,B) ) ) ), file(subset_1,r1_subset_1), []). fof(redefinition_r2_subset_1,axiom,( ! [A,B] : ( ( ~ v1_xboole_0(A) & ~ v1_xboole_0(B) ) => ( r2_subset_1(A,B) <=> r1_xboole_0(A,B) ) ) ), file(subset_1,r2_subset_1), []). fof(reflexivity_r1_tarski,axiom,( ! [A,B] : r1_tarski(A,A) ), file(tarski,r1_tarski), []). fof(reflexivity_r1_xreal_0,axiom,( ! [A,B] : ( ( v1_xreal_0(A) & v1_xreal_0(B) ) => r1_xreal_0(A,A) ) ), file(xreal_0,r1_xreal_0), []). 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(spc2_boole,axiom,( ~ v1_xboole_0(2) ), file(boole,spc2_boole), []). fof(spc2_numerals,axiom, ( v2_xreal_0(2) & m1_subset_1(2,k5_numbers) ), file(numerals,spc2_numerals), []). fof(symmetry_r1_subset_1,axiom,( ! [A,B] : ( ( ~ v1_xboole_0(A) & ~ v1_xboole_0(B) ) => ( r1_subset_1(A,B) => r1_subset_1(B,A) ) ) ), file(subset_1,r1_subset_1), []). 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(symmetry_r2_subset_1,axiom,( ! [A,B] : ( ( ~ v1_xboole_0(A) & ~ v1_xboole_0(B) ) => ( r2_subset_1(A,B) => r2_subset_1(B,A) ) ) ), file(subset_1,r2_subset_1), []). fof(t10_xreal_1,axiom,( ! [A] : ( v1_xreal_0(A) => ! [B] : ( v1_xreal_0(B) => ! [C] : ( v1_xreal_0(C) => ! [D] : ( v1_xreal_0(D) => ~ ( ~ r1_xreal_0(B,A) & r1_xreal_0(C,D) & r1_xreal_0(k2_xcmplx_0(B,D),k2_xcmplx_0(A,C)) ) ) ) ) ) ), file(xreal_1,t10_xreal_1), []). fof(t12_jordan3,axiom,( ! [A] : ( v4_ordinal2(A) => ! [B] : ( v4_ordinal2(B) => ( ~ ( r1_xreal_0(B,A) & ~ r1_xreal_0(k2_xcmplx_0(A,1),B) ) => r1_xreal_0(A,k5_binarith(B,1)) ) ) ) ), file(jordan3,t12_jordan3), []). fof(t12_sprect_2,axiom,( ! [A] : ( m2_subset_1(A,k1_numbers,k5_numbers) => ! [B] : ( m2_subset_1(B,k1_numbers,k5_numbers) => ! [C] : ( ~ v1_xboole_0(C) => ! [D] : ( m2_finseq_1(D,C) => ( ( r2_hidden(A,k4_finseq_1(D)) & r2_hidden(B,k4_finseq_1(D)) ) => k4_finseq_4(k5_numbers,C,k1_jordan3(C,D,A,B),1) = k4_finseq_4(k5_numbers,C,D,A) ) ) ) ) ) ), file(sprect_2,t12_sprect_2), []). fof(t13_sppol_2,axiom,( ! [A] : ( m2_finseq_1(A,u1_struct_0(k15_euclid(2))) => ! [B] : ( m1_subset_1(B,u1_struct_0(k15_euclid(2))) => ~ ( r2_hidden(B,k5_topreal1(2,A)) & ! [C] : ( m2_subset_1(C,k1_numbers,k5_numbers) => ~ ( r1_xreal_0(1,C) & r1_xreal_0(k1_nat_1(C,1),k3_finseq_1(A)) & r2_hidden(B,k4_topreal1(2,A,C)) ) ) ) ) ) ), file(sppol_2,t13_sppol_2), []). fof(t13_sprect_2,axiom,( ! [A] : ( m2_subset_1(A,k1_numbers,k5_numbers) => ! [B] : ( m2_subset_1(B,k1_numbers,k5_numbers) => ! [C] : ( ~ v1_xboole_0(C) => ! [D] : ( m2_finseq_1(D,C) => ( ( r2_hidden(A,k4_finseq_1(D)) & r2_hidden(B,k4_finseq_1(D)) ) => k4_finseq_4(k5_numbers,C,k1_jordan3(C,D,A,B),k3_finseq_1(k1_jordan3(C,D,A,B))) = k4_finseq_4(k5_numbers,C,D,B) ) ) ) ) ) ), file(sprect_2,t13_sprect_2), []). fof(t16_finseq_5,axiom,( ! [A] : ( ~ v1_xboole_0(A) => ! [B] : ( m1_subset_1(B,A) => ! [C] : ( m2_finseq_1(C,A) => k4_finseq_4(k5_numbers,A,k8_finseq_1(A,k12_finseq_1(A,B),C),1) = B ) ) ) ), file(finseq_5,t16_finseq_5), []). fof(t18_jordan4,axiom,( ! [A] : ( ~ v1_xboole_0(A) => ! [B] : ( m2_finseq_1(B,A) => ! [C] : ( m2_subset_1(C,k1_numbers,k5_numbers) => ( ~ r1_xreal_0(k3_finseq_1(B),C) => ( k1_funct_1(k1_rfinseq(A,B,C),k3_finseq_1(k1_rfinseq(A,B,C))) = k1_funct_1(B,k3_finseq_1(B)) & k4_finseq_4(k5_numbers,A,k1_rfinseq(A,B,C),k3_finseq_1(k1_rfinseq(A,B,C))) = k4_finseq_4(k5_numbers,A,B,k3_finseq_1(B)) ) ) ) ) ) ), file(jordan4,t18_jordan4), []). 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_real,axiom,( ! [A] : ( v1_xreal_0(A) => ! [B] : ( v1_xreal_0(B) => ( ( r1_xreal_0(A,B) & v2_xreal_0(A) ) => v2_xreal_0(B) ) ) ) ), file(real,t1_real), []). fof(t1_subset,axiom,( ! [A,B] : ( r2_hidden(A,B) => m1_subset_1(A,B) ) ), file(subset,t1_subset), []). fof(t1_xreal_1,axiom,( ! [A] : ( v1_xreal_0(A) => ! [B] : ( v1_xreal_0(B) => ( ( r1_xreal_0(A,B) & r1_xreal_0(B,A) ) => A = B ) ) ) ), file(xreal_1,t1_xreal_1), []). fof(t20_jordan4,axiom,( ! [A] : ( ~ v1_xboole_0(A) => ! [B] : ( m2_finseq_1(B,A) => ! [C] : ( m2_subset_1(C,k1_numbers,k5_numbers) => ! [D] : ( m2_subset_1(D,k1_numbers,k5_numbers) => ( ( r1_xreal_0(1,D) & r1_xreal_0(D,C) & r1_xreal_0(C,k3_finseq_1(B)) ) => k3_finseq_1(k1_jordan3(A,B,D,C)) = k1_nat_1(k5_binarith(C,D),1) ) ) ) ) ) ), file(jordan4,t20_jordan4), []). fof(t26_jordan3,axiom,( ! [A] : ( ~ v1_xboole_0(A) => ! [B] : ( m2_finseq_1(B,A) => ! [C] : ( m2_subset_1(C,k1_numbers,k5_numbers) => ( r1_xreal_0(C,k3_finseq_1(B)) => k1_jordan3(A,B,C,k3_finseq_1(B)) = k1_rfinseq(A,B,k5_binarith(C,1)) ) ) ) ) ), file(jordan3,t26_jordan3), []). fof(t27_finseq_3,axiom,( ! [A] : ( ( v1_relat_1(A) & v1_funct_1(A) & v1_finseq_1(A) ) => ! [B] : ( v4_ordinal2(B) => ( r2_hidden(B,k4_finseq_1(A)) <=> ( r1_xreal_0(1,B) & r1_xreal_0(B,k3_finseq_1(A)) ) ) ) ) ), file(finseq_3,t27_finseq_3), []). fof(t27_jordan3,axiom,( ! [A] : ( ~ v1_xboole_0(A) => ! [B] : ( m2_finseq_1(B,A) => ! [C] : ( m2_subset_1(C,k1_numbers,k5_numbers) => ! [D] : ( m2_subset_1(D,k1_numbers,k5_numbers) => ( ( r1_xreal_0(1,C) & r1_xreal_0(C,k3_finseq_1(B)) & r1_xreal_0(1,D) & r1_xreal_0(D,k3_finseq_1(B)) ) => ( k1_funct_1(k1_jordan3(A,B,C,D),1) = k1_funct_1(B,C) & ( r1_xreal_0(C,D) => ( k3_finseq_1(k1_jordan3(A,B,C,D)) = k1_nat_1(k5_binarith(D,C),1) & ! [E] : ( m2_subset_1(E,k1_numbers,k5_numbers) => ( ( r1_xreal_0(1,E) & r1_xreal_0(E,k3_finseq_1(k1_jordan3(A,B,C,D))) ) => k1_funct_1(k1_jordan3(A,B,C,D),E) = k1_funct_1(B,k5_binarith(k1_nat_1(E,C),1)) ) ) ) ) & ( ~ r1_xreal_0(C,D) => ( k3_finseq_1(k1_jordan3(A,B,C,D)) = k1_nat_1(k5_binarith(C,D),1) & ! [E] : ( m2_subset_1(E,k1_numbers,k5_numbers) => ( ( r1_xreal_0(1,E) & r1_xreal_0(E,k3_finseq_1(k1_jordan3(A,B,C,D))) ) => k1_funct_1(k1_jordan3(A,B,C,D),E) = k1_funct_1(B,k1_nat_1(k5_binarith(C,E),1)) ) ) ) ) ) ) ) ) ) ) ), file(jordan3,t27_jordan3), []). fof(t28_jordan5c,axiom,( ! [A] : ( m2_finseq_1(A,u1_struct_0(k15_euclid(2))) => ! [B] : ( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(k15_euclid(2)))) => ! [C] : ( m1_subset_1(C,u1_struct_0(k15_euclid(2))) => ! [D] : ( m2_subset_1(D,k1_numbers,k5_numbers) => ! [E] : ( m2_subset_1(E,k1_numbers,k5_numbers) => ( ( v4_topreal1(A) & v4_pre_topc(B,k15_euclid(2)) & r2_hidden(k2_jordan5c(k5_topreal1(2,A),B,k4_finseq_4(k5_numbers,u1_struct_0(k15_euclid(2)),A,1),k4_finseq_4(k5_numbers,u1_struct_0(k15_euclid(2)),A,k3_finseq_1(A))),k4_topreal1(2,A,D)) & r1_xreal_0(1,D) & r1_xreal_0(k1_nat_1(D,1),k3_finseq_1(A)) & r2_hidden(C,k4_topreal1(2,A,E)) & r1_xreal_0(1,E) & r1_xreal_0(k1_nat_1(E,1),k3_finseq_1(A)) & r2_hidden(C,B) ) => ( r1_xboole_0(k5_topreal1(2,A),B) | k2_jordan5c(k5_topreal1(2,A),B,k4_finseq_4(k5_numbers,u1_struct_0(k15_euclid(2)),A,1),k4_finseq_4(k5_numbers,u1_struct_0(k15_euclid(2)),A,k3_finseq_1(A))) = C | ( r1_xreal_0(E,D) & ( D = E => r2_jordan3(k4_finseq_4(k5_numbers,u1_struct_0(k15_euclid(2)),A,D),k4_finseq_4(k5_numbers,u1_struct_0(k15_euclid(2)),A,k1_nat_1(D,1)),C,k2_jordan5c(k5_topreal1(2,A),B,k4_finseq_4(k5_numbers,u1_struct_0(k15_euclid(2)),A,1),k4_finseq_4(k5_numbers,u1_struct_0(k15_euclid(2)),A,k3_finseq_1(A)))) ) ) ) ) ) ) ) ) ) ), file(jordan5c,t28_jordan5c), []). fof(t29_nat_1,axiom,( ! [A] : ( v4_ordinal2(A) => ! [B] : ( v4_ordinal2(B) => r1_xreal_0(A,k2_xcmplx_0(A,B)) ) ) ), file(nat_1,t29_nat_1), []). fof(t2_boole,axiom,( ! [A] : k3_xboole_0(A,k1_xboole_0) = k1_xboole_0 ), file(boole,t2_boole), []). fof(t2_real,axiom,( ! [A] : ( v1_xreal_0(A) => ! [B] : ( v1_xreal_0(B) => ( ( r1_xreal_0(A,B) & v3_xreal_0(B) ) => v3_xreal_0(A) ) ) ) ), file(real,t2_real), []). fof(t2_subset,axiom,( ! [A,B] : ( m1_subset_1(A,B) => ( v1_xboole_0(B) | r2_hidden(A,B) ) ) ), file(subset,t2_subset), []). fof(t2_tarski,axiom,( ! [A,B] : ( ! [C] : ( r2_hidden(C,A) <=> r2_hidden(C,B) ) => A = B ) ), file(tarski,t2_tarski), []). fof(t2_xreal_1,axiom,( ! [A] : ( v1_xreal_0(A) => ! [B] : ( v1_xreal_0(B) => ! [C] : ( v1_xreal_0(C) => ( ( r1_xreal_0(A,B) & r1_xreal_0(B,C) ) => r1_xreal_0(A,C) ) ) ) ) ), file(xreal_1,t2_xreal_1), []). fof(t31_jordan4,axiom,( ! [A] : ( m2_finseq_1(A,u1_struct_0(k15_euclid(2))) => ! [B] : ( m2_subset_1(B,k1_numbers,k5_numbers) => ! [C] : ( m2_subset_1(C,k1_numbers,k5_numbers) => ! [D] : ( m2_subset_1(D,k1_numbers,k5_numbers) => ( ( r1_xreal_0(1,B) & r1_xreal_0(C,k3_finseq_1(A)) & r1_xreal_0(1,D) ) => ( r1_xreal_0(C,B) | r1_xreal_0(k1_nat_1(k5_binarith(C,B),1),D) | k4_topreal1(2,k1_jordan3(u1_struct_0(k15_euclid(2)),A,B,C),D) = k4_topreal1(2,A,k5_binarith(k1_nat_1(D,B),1)) ) ) ) ) ) ) ), file(jordan4,t31_jordan4), []). fof(t31_topreal1,axiom,( ! [A] : ( m2_finseq_1(A,u1_struct_0(k15_euclid(2))) => ( v4_topreal1(A) => r1_topreal1(k15_euclid(2),k4_finseq_4(k5_numbers,u1_struct_0(k15_euclid(2)),A,1),k4_finseq_4(k5_numbers,u1_struct_0(k15_euclid(2)),A,k3_finseq_1(A)),k5_topreal1(2,A)) ) ) ), file(topreal1,t31_topreal1), []). fof(t31_xreal_1,axiom,( ! [A] : ( v1_xreal_0(A) => ! [B] : ( v1_xreal_0(B) => ~ ( ~ r1_xreal_0(A,0) & r1_xreal_0(k2_xcmplx_0(B,A),B) ) ) ) ), file(xreal_1,t31_xreal_1), []). fof(t35_finseq_1,axiom,( ! [A] : ( ( v1_relat_1(A) & v1_funct_1(A) & v1_finseq_1(A) ) => ! [B] : ( ( v1_relat_1(B) & v1_funct_1(B) & v1_finseq_1(B) ) => k3_finseq_1(k7_finseq_1(A,B)) = k1_nat_1(k3_finseq_1(A),k3_finseq_1(B)) ) ) ), file(finseq_1,t35_finseq_1), []). fof(t35_tops_1,axiom,( ! [A] : ( ( v2_pre_topc(A) & l1_pre_topc(A) ) => ! [B] : ( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) => ! [C] : ( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A))) => ( ( v4_pre_topc(B,A) & v4_pre_topc(C,A) ) => v4_pre_topc(k5_subset_1(u1_struct_0(A),B,C),A) ) ) ) ) ), file(tops_1,t35_tops_1), []). fof(t38_nat_1,axiom,( ! [A] : ( v4_ordinal2(A) => ! [B] : ( v4_ordinal2(B) => ( ~ r1_xreal_0(k2_xcmplx_0(B,1),A) <=> r1_xreal_0(A,B) ) ) ) ), file(nat_1,t38_nat_1), []). fof(t39_binarith,axiom,( ! [A] : ( v4_ordinal2(A) => ! [B] : ( v4_ordinal2(B) => k5_binarith(k2_xcmplx_0(A,B),B) = A ) ) ), file(binarith,t39_binarith), []). fof(t39_jordan3,axiom,( ! [A] : ( m2_finseq_1(A,u1_struct_0(k15_euclid(2))) => ! [B] : ( m2_subset_1(B,k1_numbers,k5_numbers) => ! [C] : ( m2_subset_1(C,k1_numbers,k5_numbers) => ( ( v4_topreal1(A) & r1_xreal_0(1,B) & r1_xreal_0(B,k3_finseq_1(A)) & r1_xreal_0(1,C) & r1_xreal_0(C,k3_finseq_1(A)) ) => ( B = C | v4_topreal1(k1_jordan3(u1_struct_0(k15_euclid(2)),A,B,C)) ) ) ) ) ) ), file(jordan3,t39_jordan3), []). fof(t3_jordan4,axiom,( ! [A] : ( m2_subset_1(A,k1_numbers,k5_numbers) => ! [B] : ( m2_subset_1(B,k1_numbers,k5_numbers) => ! [C] : ( m2_subset_1(C,k1_numbers,k5_numbers) => ( r1_xreal_0(A,B) => k5_binarith(k1_nat_1(B,C),A) = k1_nat_1(k5_binarith(B,A),C) ) ) ) ) ), file(jordan4,t3_jordan4), []). fof(t3_real,axiom,( ! [A] : ( v1_xreal_0(A) => ! [B] : ( v1_xreal_0(B) => ~ ( r1_xreal_0(A,B) & ~ v3_xreal_0(A) & v3_xreal_0(B) ) ) ) ), file(real,t3_real), []). fof(t3_subset,axiom,( ! [A,B] : ( m1_subset_1(A,k1_zfmisc_1(B)) <=> r1_tarski(A,B) ) ), file(subset,t3_subset), []). fof(t3_xboole_0,axiom,( ! [A,B] : ( ~ ( ~ r1_xboole_0(A,B) & ! [C] : ~ ( r2_hidden(C,A) & r2_hidden(C,B) ) ) & ~ ( ? [C] : ( r2_hidden(C,A) & r2_hidden(C,B) ) & r1_xboole_0(A,B) ) ) ), file(xboole_0,t3_xboole_0), []). fof(t41_jordan3,axiom,( ! [A] : ( m2_finseq_1(A,u1_struct_0(k15_euclid(2))) => ! [B] : ( m1_subset_1(B,u1_struct_0(k15_euclid(2))) => ( r2_hidden(B,k5_topreal1(2,A)) => ( r1_xreal_0(1,k2_jordan3(A,B)) & ~ r1_xreal_0(k3_finseq_1(A),k2_jordan3(A,B)) ) ) ) ) ), file(jordan3,t41_jordan3), []). fof(t42_jordan3,axiom,( ! [A] : ( m2_finseq_1(A,u1_struct_0(k15_euclid(2))) => ! [B] : ( m1_subset_1(B,u1_struct_0(k15_euclid(2))) => ( r2_hidden(B,k5_topreal1(2,A)) => r2_hidden(B,k4_topreal1(2,A,k2_jordan3(A,B))) ) ) ) ), file(jordan3,t42_jordan3), []). fof(t43_sprect_3,axiom,( ! [A] : ( ( ~ v1_xboole_0(A) & ~ v5_seqm_3(A) & v1_topreal1(A) & v2_topreal1(A) & v1_finseq_6(A,u1_struct_0(k15_euclid(2))) & v1_goboard5(A) & v2_goboard5(A) & m2_finseq_1(A,u1_struct_0(k15_euclid(2))) ) => r1_subset_1(k2_goboard9(A),k5_topreal1(2,A)) ) ), file(sprect_3,t43_sprect_3), []). fof(t49_sppol_1,axiom,( ! [A] : ( m2_finseq_1(A,u1_struct_0(k15_euclid(2))) => v4_pre_topc(k5_topreal1(2,A),k15_euclid(2)) ) ), file(sppol_1,t49_sppol_1), []). fof(t4_real,axiom,( ! [A] : ( v1_xreal_0(A) => ! [B] : ( v1_xreal_0(B) => ~ ( r1_xreal_0(A,B) & ~ v2_xreal_0(B) & v2_xreal_0(A) ) ) ) ), file(real,t4_real), []). 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(t4_xboole_0,axiom,( ! [A,B] : ( ~ ( ~ r1_xboole_0(A,B) & ! [C] : ~ r2_hidden(C,k3_xboole_0(A,B)) ) & ~ ( ? [C] : r2_hidden(C,k3_xboole_0(A,B)) & r1_xboole_0(A,B) ) ) ), file(xboole_0,t4_xboole_0), []). fof(t50_sprect_3,axiom,( ! [A] : ( ( ~ v1_xboole_0(A) & v1_sprect_1(A) & v1_topreal1(A) & v2_topreal1(A) & v1_finseq_6(A,u1_struct_0(k15_euclid(2))) & v1_goboard5(A) & m2_finseq_1(A,u1_struct_0(k15_euclid(2))) ) => ! [B] : ( ( v4_topreal1(B) & m2_finseq_1(B,u1_struct_0(k15_euclid(2))) ) => ~ ( r2_hidden(k4_finseq_4(k5_numbers,u1_struct_0(k15_euclid(2)),B,1),k2_goboard9(A)) & r2_hidden(k4_finseq_4(k5_numbers,u1_struct_0(k15_euclid(2)),B,k3_finseq_1(B)),k3_goboard9(A)) & r2_subset_1(k5_topreal1(2,A),k5_topreal1(2,B)) ) ) ) ), file(sprect_3,t50_sprect_3), []). fof(t53_binarith,axiom,( ! [A] : ( v4_ordinal2(A) => ! [B] : ( v4_ordinal2(B) => ( r1_xreal_0(B,A) => k2_xcmplx_0(k5_binarith(A,B),B) = A ) ) ) ), file(binarith,t53_binarith), []). fof(t54_sprect_3,axiom,( ! [A] : ( ( ~ v1_xboole_0(A) & v1_sprect_1(A) & v1_topreal1(A) & v2_topreal1(A) & v1_finseq_6(A,u1_struct_0(k15_euclid(2))) & v1_goboard5(A) & m2_finseq_1(A,u1_struct_0(k15_euclid(2))) ) => ( k2_goboard9(A) = a_1_5_sprect_3(A) & k3_goboard9(A) = a_1_6_sprect_3(A) ) ) ), file(sprect_3,t54_sprect_3), []). fof(t56_finseq_1,axiom,( ! [A,B] : ( ( v1_relat_1(B) & v1_funct_1(B) & v1_finseq_1(B) ) => ( B = k9_finseq_1(A) <=> ( k3_finseq_1(B) = 1 & k2_relat_1(B) = k1_tarski(A) ) ) ) ), file(finseq_1,t56_finseq_1), []). fof(t5_goboard2,axiom,( ! [A] : ( ~ v1_xboole_0(A) => ! [B] : ( m2_finseq_1(B,A) => ! [C] : ( m2_finseq_1(C,A) => ! [D] : ( m2_subset_1(D,k1_numbers,k5_numbers) => ( ( r1_xreal_0(1,D) & r1_xreal_0(D,k3_finseq_1(C)) ) => k4_finseq_4(k5_numbers,A,k8_finseq_1(A,B,C),k1_nat_1(D,k3_finseq_1(B))) = k4_finseq_4(k5_numbers,A,C,D) ) ) ) ) ) ), file(goboard2,t5_goboard2), []). fof(t5_jordan3,axiom,( ! [A] : ( v4_ordinal2(A) => ! [B] : ( v4_ordinal2(B) => ! [C] : ( v4_ordinal2(C) => ( r1_xreal_0(A,B) => r1_xreal_0(k5_binarith(A,C),k5_binarith(B,C)) ) ) ) ) ), file(jordan3,t5_jordan3), []). fof(t5_real,axiom,( ! [A] : ( v1_xreal_0(A) => ! [B] : ( v1_xreal_0(B) => ( r1_xreal_0(A,B) => ( v1_xboole_0(B) | v3_xreal_0(A) | v2_xreal_0(B) ) ) ) ) ), file(real,t5_real), []). fof(t5_subset,axiom,( ! [A,B,C] : ~ ( r2_hidden(A,B) & m1_subset_1(B,k1_zfmisc_1(C)) & v1_xboole_0(C) ) ), file(subset,t5_subset), []). fof(t6_boole,axiom,( ! [A] : ( v1_xboole_0(A) => A = k1_xboole_0 ) ), file(boole,t6_boole), []). fof(t6_finseq_5,axiom,( ! [A] : ( ( ~ v1_xboole_0(A) & v1_relat_1(A) & v1_funct_1(A) & v1_finseq_1(A) ) => ( r2_hidden(1,k4_finseq_1(A)) & r2_hidden(k3_finseq_1(A),k4_finseq_1(A)) ) ) ), file(finseq_5,t6_finseq_5), []). fof(t6_real,axiom,( ! [A] : ( v1_xreal_0(A) => ! [B] : ( v1_xreal_0(B) => ( r1_xreal_0(A,B) => ( v1_xboole_0(A) | v2_xreal_0(B) | v3_xreal_0(A) ) ) ) ) ), file(real,t6_real), []). fof(t6_sprect_3,axiom,( ! [A] : ( m2_subset_1(A,k1_numbers,k5_numbers) => ! [B] : ( m2_subset_1(B,k1_numbers,k5_numbers) => ! [C] : ( m2_subset_1(C,k1_numbers,k5_numbers) => ( r1_xreal_0(A,k1_nat_1(B,C)) => r1_xreal_0(k5_binarith(A,C),B) ) ) ) ) ), file(sprect_3,t6_sprect_3), []). fof(t7_boole,axiom,( ! [A,B] : ~ ( r2_hidden(A,B) & v1_xboole_0(B) ) ), file(boole,t7_boole), []). fof(t7_real,axiom,( ! [A] : ( v1_xreal_0(A) => ! [B] : ( v1_xreal_0(B) => ~ ( ~ r1_xreal_0(A,B) & ~ v2_xreal_0(A) & ~ v3_xreal_0(B) ) ) ) ), file(real,t7_real), []). fof(t7_sprect_2,axiom,( ! [A] : ( m2_subset_1(A,k1_numbers,k5_numbers) => ! [B] : ( m2_subset_1(B,k1_numbers,k5_numbers) => ! [C] : ( m2_subset_1(C,k1_numbers,k5_numbers) => ! [D] : ( ~ v1_xboole_0(D) => ! [E] : ( m2_finseq_1(E,D) => ( ( r1_xreal_0(A,B) & r2_hidden(A,k4_finseq_1(E)) & r2_hidden(B,k4_finseq_1(E)) & r2_hidden(C,k4_finseq_1(k1_jordan3(D,E,A,B))) ) => k4_finseq_4(k5_numbers,D,k1_jordan3(D,E,A,B),C) = k4_finseq_4(k5_numbers,D,E,k5_binarith(k1_nat_1(C,A),1)) ) ) ) ) ) ) ), file(sprect_2,t7_sprect_2), []). fof(t7_sprect_3,axiom,( ! [A] : ( m2_subset_1(A,k1_numbers,k5_numbers) => ! [B] : ( m2_subset_1(B,k1_numbers,k5_numbers) => ! [C] : ( m2_subset_1(C,k1_numbers,k5_numbers) => ( ( r1_xreal_0(A,k5_binarith(B,C)) & r1_xreal_0(C,B) ) => r1_xreal_0(k1_nat_1(A,C),B) ) ) ) ) ), file(sprect_3,t7_sprect_3), []). fof(t8_boole,axiom,( ! [A,B] : ~ ( v1_xboole_0(A) & A != B & v1_xboole_0(B) ) ), file(boole,t8_boole), []). fof(t8_real,axiom,( ! [A] : ( v1_xreal_0(A) => ! [B] : ( v1_xreal_0(B) => ~ ( ~ r1_xreal_0(A,B) & ~ v3_xreal_0(B) & ~ v2_xreal_0(A) ) ) ) ), file(real,t8_real), []). fof(t8_xreal_1,axiom,( ! [A] : ( v1_xreal_0(A) => ! [B] : ( v1_xreal_0(B) => ! [C] : ( v1_xreal_0(C) => ( r1_xreal_0(A,B) <=> r1_xreal_0(k2_xcmplx_0(A,C),k2_xcmplx_0(B,C)) ) ) ) ) ), file(xreal_1,t8_xreal_1), []). fof(t96_sprect_1,axiom,( ! [A] : ( ( v1_sprect_1(A) & m2_finseq_1(A,u1_struct_0(k15_euclid(2))) ) => r1_xboole_0(k2_goboard9(A),k3_goboard9(A)) ) ), file(sprect_1,t96_sprect_1), []). fof(t9_nat_2,axiom,( ! [A] : ( v4_ordinal2(A) => ! [B] : ( v4_ordinal2(B) => ( ~ r1_xreal_0(A,B) => k1_nat_1(k5_binarith(A,k2_xcmplx_0(B,1)),1) = k5_binarith(A,B) ) ) ) ), file(nat_2,t9_nat_2), []).