% Mizar problem: t33_jordan20,jordan20,2702,38 fof(t33_jordan20,conjecture,( ! [A] : ( ( ~ v1_xboole_0(A) & m1_subset_1(A,k1_zfmisc_1(u1_struct_0(k15_euclid(2)))) ) => ! [B] : ( m1_subset_1(B,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))) => ! [E] : ( m1_subset_1(E,k1_numbers) => ~ ( r1_topreal4(A,B,C) & ~ r1_xreal_0(E,k21_euclid(B)) & ~ r1_xreal_0(k21_euclid(C),E) & r2_hidden(D,A) & k21_euclid(D) = E & ~ r1_jordan20(A,B,C,D,E) & ~ r2_jordan20(A,B,C,D,E) ) ) ) ) ) ) ), inference(mizar_bg_added,[status(thm)],[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_membered,cc1_nat_1,cc1_pre_topc,cc1_relset_1,cc1_sppol_2,cc1_xreal_0,cc20_membered,cc2_membered,cc2_nat_1,cc2_sppol_2,cc2_xreal_0,cc3_membered,cc3_nat_1,cc3_sppol_2,cc3_xreal_0,cc4_membered,cc4_xreal_0,cc5_xreal_0,cc6_membered,cc6_xreal_0,cc7_xreal_0,cc8_xreal_0,cc9_membered,commutativity_k1_nat_1,commutativity_k2_xcmplx_0,commutativity_k3_real_1,commutativity_k3_topreal1,connectedness_r1_xreal_0,d10_topreal1,d1_jordan20,d1_tarski,d1_topreal4,d2_jordan20,d3_binarith,d3_finseq_1,d3_jordan5c,d3_tarski,d4_finseq_4,d4_tarski,d5_topreal1,d6_topreal1,d7_topreal1,d8_funct_1,dt_g1_pre_topc,dt_k15_euclid,dt_k1_card_1,dt_k1_finseq_1,dt_k1_funct_1,dt_k1_jordan3,dt_k1_nat_1,dt_k1_numbers,dt_k1_relat_1,dt_k1_struct_0,dt_k1_tarski,dt_k1_topreal1,dt_k1_xboole_0,dt_k1_zfmisc_1,dt_k21_euclid,dt_k22_borsuk_1,dt_k22_euclid,dt_k2_finseq_1,dt_k2_xcmplx_0,dt_k2_zfmisc_1,dt_k3_finseq_1,dt_k3_pre_topc,dt_k3_real_1,dt_k3_tarski,dt_k3_topmetr,dt_k3_topreal1,dt_k4_finseq_1,dt_k4_finseq_4,dt_k4_topreal1,dt_k4_xcmplx_0,dt_k5_binarith,dt_k5_jordan6,dt_k5_numbers,dt_k5_ordinal2,dt_k5_real_1,dt_k5_topmetr,dt_k5_topreal1,dt_k6_xcmplx_0,dt_l1_pre_topc,dt_l1_struct_0,dt_m1_finseq_1,dt_m1_pre_topc,dt_m1_relset_1,dt_m1_subset_1,dt_m2_finseq_1,dt_m2_relset_1,dt_m2_subset_1,dt_u1_pre_topc,dt_u1_struct_0,existence_l1_pre_topc,existence_l1_struct_0,existence_m1_finseq_1,existence_m1_pre_topc,existence_m1_relset_1,existence_m1_subset_1,existence_m2_finseq_1,existence_m2_relset_1,existence_m2_subset_1,fc10_membered,fc10_xreal_0,fc11_membered,fc11_sppol_2,fc11_xreal_0,fc12_xreal_0,fc13_xreal_0,fc14_xreal_0,fc15_xreal_0,fc16_xreal_0,fc17_xreal_0,fc18_xreal_0,fc19_xreal_0,fc1_nat_1,fc1_struct_0,fc1_xreal_0,fc20_xreal_0,fc2_euclid,fc2_membered,fc2_topmetr,fc3_nat_1,fc3_pre_topc,fc3_topmetr,fc3_xreal_0,fc4_nat_1,fc4_pre_topc,fc5_borsuk_1,fc5_membered,fc5_xreal_0,fc6_membered,fc7_membered,fc7_xreal_0,fc8_membered,fc8_xreal_0,fc9_membered,fc9_xreal_0,fraenkel_a_1_1_jordan20,fraenkel_a_2_1_jordan20,fraenkel_a_2_2_topreal1,fraenkel_a_5_0_jordan6,fraenkel_a_6_0_jordan20,free_g1_pre_topc,involutiveness_k4_xcmplx_0,rc1_membered,rc1_nat_1,rc1_pre_topc,rc1_xreal_0,rc2_nat_1,rc2_pre_topc,rc2_xreal_0,rc3_nat_1,rc3_pre_topc,rc3_struct_0,rc3_xreal_0,rc4_pre_topc,rc4_xreal_0,rc5_pre_topc,rc5_struct_0,redefinition_k1_nat_1,redefinition_k1_struct_0,redefinition_k2_finseq_1,redefinition_k3_finseq_1,redefinition_k3_real_1,redefinition_k3_topreal1,redefinition_k4_finseq_1,redefinition_k5_numbers,redefinition_k5_real_1,redefinition_k5_topmetr,redefinition_m2_finseq_1,redefinition_m2_relset_1,redefinition_m2_subset_1,reflexivity_r1_tarski,reflexivity_r1_xreal_0,s5_nat_1__e21_33__jordan20,spc0_boole,spc1_boole,spc1_numerals,spc2_boole,spc2_numerals,t106_real_2,t10_xreal_1,t11_xreal_1,t12_jordan5c,t12_topreal1,t13_jordan3,t13_jordan5c,t14_jordan3,t15_topreal1,t19_jordan20,t1_arithm,t1_jordan20,t1_numerals,t1_real,t1_subset,t1_xboole_1,t1_xreal_1,t23_jordan5c,t24_finseq_4,t24_jordan5c,t25_jordan5c,t26_jordan5c,t27_finseq_3,t27_jordan3,t28_jordan20,t29_jordan6,t2_jordan20,t2_jordan3,t2_real,t2_subset,t2_tarski,t2_xreal_1,t31_jordan3,t33_sprect_3,t38_nat_1,t3_finseq_1,t3_jordan20,t3_real,t3_subset,t3_topreal4,t4_arithm,t4_real,t4_sprect_4,t4_subset,t50_xreal_1,t52_binarith,t52_xreal_1,t5_real,t5_subset,t6_boole,t6_jordan20,t6_real,t6_topreal1,t7_boole,t7_jct_misc,t7_jordan20,t7_real,t8_boole,t8_real,t9_topreal1,t9_xreal_1]), [file(jordan20,t33_jordan20)]). 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_membered,axiom,( ! [A] : ( v5_membered(A) => v4_membered(A) ) ), file(membered,cc1_membered), []). fof(cc1_nat_1,axiom,( ! [A] : ( m1_subset_1(A,k5_numbers) => ( v1_ordinal1(A) & v2_ordinal1(A) & v3_ordinal1(A) & v4_ordinal2(A) & v1_xcmplx_0(A) & v1_xreal_0(A) ) ) ), file(nat_1,cc1_nat_1), []). fof(cc1_pre_topc,axiom,( ! [A] : ( ( v2_pre_topc(A) & l1_pre_topc(A) ) => ! [B] : ( m1_pre_topc(B,A) => v2_pre_topc(B) ) ) ), file(pre_topc,cc1_pre_topc), []). 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_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_membered,axiom,( ! [A] : ( v4_membered(A) => v3_membered(A) ) ), file(membered,cc2_membered), []). fof(cc2_nat_1,axiom,( ! [A] : ( m1_subset_1(A,k5_numbers) => ( v1_ordinal1(A) & v2_ordinal1(A) & v3_ordinal1(A) & v4_ordinal2(A) & v1_xcmplx_0(A) & v1_xreal_0(A) & ~ v3_xreal_0(A) ) ) ), file(nat_1,cc2_nat_1), []). fof(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_xreal_0,axiom,( ! [A] : ( v1_xreal_0(A) => v1_xcmplx_0(A) ) ), file(xreal_0,cc2_xreal_0), []). fof(cc3_membered,axiom,( ! [A] : ( v3_membered(A) => v2_membered(A) ) ), file(membered,cc3_membered), []). fof(cc3_nat_1,axiom,( ! [A] : ( v4_ordinal2(A) => ( v4_ordinal2(A) & v1_xcmplx_0(A) & v1_xreal_0(A) & ~ v3_xreal_0(A) ) ) ), file(nat_1,cc3_nat_1), []). fof(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_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(commutativity_k3_real_1,axiom,( ! [A,B] : ( ( m1_subset_1(A,k1_numbers) & m1_subset_1(B,k1_numbers) ) => k3_real_1(A,B) = k3_real_1(B,A) ) ), file(real_1,k3_real_1), []). fof(commutativity_k3_topreal1,axiom,( ! [A,B,C] : ( ( m1_subset_1(A,k5_numbers) & m1_subset_1(B,u1_struct_0(k15_euclid(A))) & m1_subset_1(C,u1_struct_0(k15_euclid(A))) ) => k3_topreal1(A,B,C) = k3_topreal1(A,C,B) ) ), file(topreal1,k3_topreal1), []). 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(d10_topreal1,axiom,( ! [A] : ( m2_finseq_1(A,u1_struct_0(k15_euclid(2))) => ( v4_topreal1(A) <=> ( v2_funct_1(A) & r1_xreal_0(2,k3_finseq_1(A)) & v2_topreal1(A) & v3_topreal1(A) & v1_topreal1(A) ) ) ) ), file(topreal1,d10_topreal1), []). fof(d1_jordan20,axiom,( ! [A] : ( m1_subset_1(A,k1_zfmisc_1(u1_struct_0(k15_euclid(2)))) => ! [B] : ( m1_subset_1(B,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))) => ! [E] : ( m1_subset_1(E,k1_numbers) => ( r1_jordan20(A,B,C,D,E) <=> ( r1_topreal1(k15_euclid(2),B,C,A) & r2_hidden(D,A) & k21_euclid(D) = E & ? [F] : ( m1_subset_1(F,u1_struct_0(k15_euclid(2))) & ~ r1_xreal_0(E,k21_euclid(F)) & r1_jordan5c(A,B,C,F,D) & ! [G] : ( m1_subset_1(G,u1_struct_0(k15_euclid(2))) => ( ( r1_jordan5c(A,B,C,F,G) & r1_jordan5c(A,B,C,G,D) ) => r1_xreal_0(k21_euclid(G),E) ) ) ) ) ) ) ) ) ) ) ), file(jordan20,d1_jordan20), []). fof(d1_tarski,axiom,( ! [A,B] : ( B = k1_tarski(A) <=> ! [C] : ( r2_hidden(C,B) <=> C = A ) ) ), file(tarski,d1_tarski), []). fof(d1_topreal4,axiom,( ! [A] : ( m1_subset_1(A,k1_zfmisc_1(u1_struct_0(k15_euclid(2)))) => ! [B] : ( m1_subset_1(B,u1_struct_0(k15_euclid(2))) => ! [C] : ( m1_subset_1(C,u1_struct_0(k15_euclid(2))) => ( r1_topreal4(A,B,C) <=> ? [D] : ( m2_finseq_1(D,u1_struct_0(k15_euclid(2))) & v4_topreal1(D) & A = k5_topreal1(2,D) & B = k4_finseq_4(k5_numbers,u1_struct_0(k15_euclid(2)),D,1) & C = k4_finseq_4(k5_numbers,u1_struct_0(k15_euclid(2)),D,k3_finseq_1(D)) ) ) ) ) ) ), file(topreal4,d1_topreal4), []). fof(d2_jordan20,axiom,( ! [A] : ( m1_subset_1(A,k1_zfmisc_1(u1_struct_0(k15_euclid(2)))) => ! [B] : ( m1_subset_1(B,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))) => ! [E] : ( m1_subset_1(E,k1_numbers) => ( r2_jordan20(A,B,C,D,E) <=> ( r1_topreal1(k15_euclid(2),B,C,A) & r2_hidden(D,A) & k21_euclid(D) = E & ? [F] : ( m1_subset_1(F,u1_struct_0(k15_euclid(2))) & ~ r1_xreal_0(k21_euclid(F),E) & r1_jordan5c(A,B,C,F,D) & ! [G] : ( m1_subset_1(G,u1_struct_0(k15_euclid(2))) => ( ( r1_jordan5c(A,B,C,F,G) & r1_jordan5c(A,B,C,G,D) ) => r1_xreal_0(E,k21_euclid(G)) ) ) ) ) ) ) ) ) ) ) ), file(jordan20,d2_jordan20), []). fof(d3_binarith,axiom,( ! [A] : ( v4_ordinal2(A) => ! [B] : ( v4_ordinal2(B) => ( ( r1_xreal_0(0,k6_xcmplx_0(A,B)) => k5_binarith(A,B) = k6_xcmplx_0(A,B) ) & ( ~ r1_xreal_0(0,k6_xcmplx_0(A,B)) => k5_binarith(A,B) = 0 ) ) ) ) ), file(binarith,d3_binarith), []). fof(d3_finseq_1,axiom,( ! [A] : ( ( v1_relat_1(A) & v1_funct_1(A) & v1_finseq_1(A) ) => ! [B] : ( m2_subset_1(B,k1_numbers,k5_numbers) => ( B = k3_finseq_1(A) <=> k2_finseq_1(B) = k1_relat_1(A) ) ) ) ), file(finseq_1,d3_finseq_1), []). fof(d3_jordan5c,axiom,( ! [A] : ( m1_subset_1(A,k1_zfmisc_1(u1_struct_0(k15_euclid(2)))) => ! [B] : ( m1_subset_1(B,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))) => ! [E] : ( m1_subset_1(E,u1_struct_0(k15_euclid(2))) => ( r1_jordan5c(A,B,C,D,E) <=> ( r2_hidden(D,A) & r2_hidden(E,A) & ! [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) => ! [H] : ( m1_subset_1(H,k1_numbers) => ( ( v3_tops_2(F,k5_topmetr,k3_pre_topc(k15_euclid(2),A)) & k1_funct_1(F,0) = B & k1_funct_1(F,1) = C & k1_funct_1(F,G) = D & r1_xreal_0(0,G) & r1_xreal_0(G,1) & k1_funct_1(F,H) = E & r1_xreal_0(0,H) & r1_xreal_0(H,1) ) => r1_xreal_0(G,H) ) ) ) ) ) ) ) ) ) ) ) ), file(jordan5c,d3_jordan5c), []). fof(d3_tarski,axiom,( ! [A,B] : ( r1_tarski(A,B) <=> ! [C] : ( r2_hidden(C,A) => r2_hidden(C,B) ) ) ), file(tarski,d3_tarski), []). fof(d4_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_tarski,axiom,( ! [A,B] : ( B = k3_tarski(A) <=> ! [C] : ( r2_hidden(C,B) <=> ? [D] : ( r2_hidden(C,D) & r2_hidden(D,A) ) ) ) ), file(tarski,d4_tarski), []). fof(d5_topreal1,axiom,( ! [A] : ( m2_subset_1(A,k1_numbers,k5_numbers) => ! [B] : ( m2_finseq_1(B,u1_struct_0(k15_euclid(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(B)) ) => k4_topreal1(A,B,C) = k3_topreal1(A,k4_finseq_4(k5_numbers,u1_struct_0(k15_euclid(A)),B,C),k4_finseq_4(k5_numbers,u1_struct_0(k15_euclid(A)),B,k1_nat_1(C,1))) ) & ( ~ ( r1_xreal_0(1,C) & r1_xreal_0(k1_nat_1(C,1),k3_finseq_1(B)) ) => k4_topreal1(A,B,C) = k1_xboole_0 ) ) ) ) ) ), file(topreal1,d5_topreal1), []). fof(d6_topreal1,axiom,( ! [A] : ( m2_subset_1(A,k1_numbers,k5_numbers) => ! [B] : ( m2_finseq_1(B,u1_struct_0(k15_euclid(A))) => k5_topreal1(A,B) = k3_tarski(a_2_2_topreal1(A,B)) ) ) ), file(topreal1,d6_topreal1), []). fof(d7_topreal1,axiom,( ! [A] : ( m2_finseq_1(A,u1_struct_0(k15_euclid(2))) => ( v1_topreal1(A) <=> ! [B] : ( m2_subset_1(B,k1_numbers,k5_numbers) => ~ ( r1_xreal_0(1,B) & r1_xreal_0(k1_nat_1(B,1),k3_finseq_1(A)) & k21_euclid(k4_finseq_4(k5_numbers,u1_struct_0(k15_euclid(2)),A,B)) != k21_euclid(k4_finseq_4(k5_numbers,u1_struct_0(k15_euclid(2)),A,k1_nat_1(B,1))) & k22_euclid(k4_finseq_4(k5_numbers,u1_struct_0(k15_euclid(2)),A,B)) != k22_euclid(k4_finseq_4(k5_numbers,u1_struct_0(k15_euclid(2)),A,k1_nat_1(B,1))) ) ) ) ) ), file(topreal1,d7_topreal1), []). fof(d8_funct_1,axiom,( ! [A] : ( ( v1_relat_1(A) & v1_funct_1(A) ) => ( v2_funct_1(A) <=> ! [B,C] : ( ( r2_hidden(B,k1_relat_1(A)) & r2_hidden(C,k1_relat_1(A)) & k1_funct_1(A,B) = k1_funct_1(A,C) ) => B = C ) ) ) ), file(funct_1,d8_funct_1), []). fof(dt_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_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_k1_card_1,axiom,( ! [A] : v1_card_1(k1_card_1(A)) ), file(card_1,k1_card_1), []). fof(dt_k1_finseq_1,axiom,( $true ), file(finseq_1,k1_finseq_1), []). 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_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_topreal1,axiom,( ! [A,B,C] : ( ( m1_subset_1(A,k5_numbers) & m1_subset_1(B,u1_struct_0(k15_euclid(A))) & m1_subset_1(C,u1_struct_0(k15_euclid(A))) ) => m1_subset_1(k1_topreal1(A,B,C),k1_zfmisc_1(u1_struct_0(k15_euclid(A)))) ) ), file(topreal1,k1_topreal1), []). fof(dt_k1_xboole_0,axiom,( $true ), file(xboole_0,k1_xboole_0), []). fof(dt_k1_zfmisc_1,axiom,( $true ), file(zfmisc_1,k1_zfmisc_1), []). fof(dt_k21_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_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_finseq_1,axiom,( ! [A] : ( v4_ordinal2(A) => m1_subset_1(k2_finseq_1(A),k1_zfmisc_1(k5_numbers)) ) ), file(finseq_1,k2_finseq_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_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_real_1,axiom,( ! [A,B] : ( ( m1_subset_1(A,k1_numbers) & m1_subset_1(B,k1_numbers) ) => m1_subset_1(k3_real_1(A,B),k1_numbers) ) ), file(real_1,k3_real_1), []). fof(dt_k3_tarski,axiom,( $true ), file(tarski,k3_tarski), []). 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_topreal1,axiom,( ! [A,B,C] : ( ( m1_subset_1(A,k5_numbers) & m1_subset_1(B,u1_struct_0(k15_euclid(A))) & m1_subset_1(C,u1_struct_0(k15_euclid(A))) ) => m1_subset_1(k3_topreal1(A,B,C),k1_zfmisc_1(u1_struct_0(k15_euclid(A)))) ) ), file(topreal1,k3_topreal1), []). 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_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_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_k4_xcmplx_0,axiom,( ! [A] : ( v1_xcmplx_0(A) => v1_xcmplx_0(k4_xcmplx_0(A)) ) ), file(xcmplx_0,k4_xcmplx_0), []). 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_jordan6,axiom,( ! [A,B,C,D,E] : ( ( m1_subset_1(A,k1_zfmisc_1(u1_struct_0(k15_euclid(2)))) & m1_subset_1(B,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(E,u1_struct_0(k15_euclid(2))) ) => m1_subset_1(k5_jordan6(A,B,C,D,E),k1_zfmisc_1(u1_struct_0(k15_euclid(2)))) ) ), file(jordan6,k5_jordan6), []). fof(dt_k5_numbers,axiom,( m1_subset_1(k5_numbers,k1_zfmisc_1(k1_numbers)) ), file(numbers,k5_numbers), []). fof(dt_k5_ordinal2,axiom,( $true ), file(ordinal2,k5_ordinal2), []). fof(dt_k5_real_1,axiom,( ! [A,B] : ( ( m1_subset_1(A,k1_numbers) & m1_subset_1(B,k1_numbers) ) => m1_subset_1(k5_real_1(A,B),k1_numbers) ) ), file(real_1,k5_real_1), []). fof(dt_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_k6_xcmplx_0,axiom,( $true ), file(xcmplx_0,k6_xcmplx_0), []). 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_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_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_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_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_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_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_sppol_2,axiom,( ! [A,B,C] : ( ( m1_subset_1(A,k5_numbers) & m1_subset_1(B,u1_struct_0(k15_euclid(A))) & m1_subset_1(C,u1_struct_0(k15_euclid(A))) ) => ( ~ v1_xboole_0(k1_topreal1(A,B,C)) & v6_compts_1(k1_topreal1(A,B,C),k15_euclid(A)) ) ) ), file(sppol_2,fc11_sppol_2), []). 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_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_xreal_0,axiom,( ! [A] : ( ( v1_xreal_0(A) & ~ v2_xreal_0(A) ) => ( v1_xcmplx_0(k4_xcmplx_0(A)) & v1_xreal_0(k4_xcmplx_0(A)) & ~ v3_xreal_0(k4_xcmplx_0(A)) ) ) ), file(xreal_0,fc13_xreal_0), []). fof(fc14_xreal_0,axiom,( ! [A] : ( ( v1_xreal_0(A) & ~ v3_xreal_0(A) ) => ( v1_xcmplx_0(k4_xcmplx_0(A)) & v1_xreal_0(k4_xcmplx_0(A)) & ~ v2_xreal_0(k4_xcmplx_0(A)) ) ) ), file(xreal_0,fc14_xreal_0), []). fof(fc15_xreal_0,axiom,( ! [A,B] : ( ( v1_xreal_0(A) & ~ v3_xreal_0(A) & v1_xreal_0(B) & ~ v2_xreal_0(B) ) => ( v1_xcmplx_0(k6_xcmplx_0(A,B)) & v1_xreal_0(k6_xcmplx_0(A,B)) & ~ v3_xreal_0(k6_xcmplx_0(A,B)) ) ) ), file(xreal_0,fc15_xreal_0), []). fof(fc16_xreal_0,axiom,( ! [A,B] : ( ( v1_xreal_0(A) & ~ v3_xreal_0(A) & v1_xreal_0(B) & ~ v2_xreal_0(B) ) => ( v1_xcmplx_0(k6_xcmplx_0(B,A)) & v1_xreal_0(k6_xcmplx_0(B,A)) & ~ v2_xreal_0(k6_xcmplx_0(B,A)) ) ) ), file(xreal_0,fc16_xreal_0), []). fof(fc17_xreal_0,axiom,( ! [A,B] : ( ( v1_xreal_0(A) & v2_xreal_0(A) & v1_xreal_0(B) & ~ v2_xreal_0(B) ) => ( ~ v1_xboole_0(k6_xcmplx_0(A,B)) & v1_xcmplx_0(k6_xcmplx_0(A,B)) & v1_xreal_0(k6_xcmplx_0(A,B)) & v2_xreal_0(k6_xcmplx_0(A,B)) & ~ v3_xreal_0(k6_xcmplx_0(A,B)) ) ) ), file(xreal_0,fc17_xreal_0), []). fof(fc18_xreal_0,axiom,( ! [A,B] : ( ( v1_xreal_0(A) & v2_xreal_0(A) & v1_xreal_0(B) & ~ v2_xreal_0(B) ) => ( ~ v1_xboole_0(k6_xcmplx_0(B,A)) & v1_xcmplx_0(k6_xcmplx_0(B,A)) & v1_xreal_0(k6_xcmplx_0(B,A)) & ~ v2_xreal_0(k6_xcmplx_0(B,A)) & v3_xreal_0(k6_xcmplx_0(B,A)) ) ) ), file(xreal_0,fc18_xreal_0), []). fof(fc19_xreal_0,axiom,( ! [A,B] : ( ( v1_xreal_0(A) & v3_xreal_0(A) & v1_xreal_0(B) & ~ v3_xreal_0(B) ) => ( ~ v1_xboole_0(k6_xcmplx_0(A,B)) & v1_xcmplx_0(k6_xcmplx_0(A,B)) & v1_xreal_0(k6_xcmplx_0(A,B)) & ~ v2_xreal_0(k6_xcmplx_0(A,B)) & v3_xreal_0(k6_xcmplx_0(A,B)) ) ) ), file(xreal_0,fc19_xreal_0), []). fof(fc1_nat_1,axiom,( ! [A,B] : ( ( v4_ordinal2(A) & v4_ordinal2(B) ) => ( v4_ordinal2(k2_xcmplx_0(A,B)) & v1_xcmplx_0(k2_xcmplx_0(A,B)) & v1_xreal_0(k2_xcmplx_0(A,B)) ) ) ), file(nat_1,fc1_nat_1), []). fof(fc1_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_xreal_0,axiom,( ! [A] : ( v1_xreal_0(A) => ( v1_xcmplx_0(k4_xcmplx_0(A)) & v1_xreal_0(k4_xcmplx_0(A)) ) ) ), file(xreal_0,fc1_xreal_0), []). fof(fc20_xreal_0,axiom,( ! [A,B] : ( ( v1_xreal_0(A) & v3_xreal_0(A) & v1_xreal_0(B) & ~ v3_xreal_0(B) ) => ( ~ v1_xboole_0(k6_xcmplx_0(B,A)) & v1_xcmplx_0(k6_xcmplx_0(B,A)) & v1_xreal_0(k6_xcmplx_0(B,A)) & v2_xreal_0(k6_xcmplx_0(B,A)) & ~ v3_xreal_0(k6_xcmplx_0(B,A)) ) ) ), file(xreal_0,fc20_xreal_0), []). 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_topmetr,axiom, ( ~ v3_struct_0(k3_topmetr) & v1_pre_topc(k3_topmetr) & v2_pre_topc(k3_topmetr) ), file(topmetr,fc2_topmetr), []). fof(fc3_nat_1,axiom,( ! [A,B] : ( ( v4_ordinal2(A) & ~ v1_xboole_0(B) & v4_ordinal2(B) ) => ( ~ v1_xboole_0(k2_xcmplx_0(A,B)) & v4_ordinal2(k2_xcmplx_0(A,B)) & v1_xcmplx_0(k2_xcmplx_0(A,B)) & v1_xreal_0(k2_xcmplx_0(A,B)) ) ) ), file(nat_1,fc3_nat_1), []). fof(fc3_pre_topc,axiom,( ! [A,B] : ( ( ~ v3_struct_0(A) & l1_pre_topc(A) & ~ v1_xboole_0(B) & m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) ) => ( ~ v3_struct_0(k3_pre_topc(A,B)) & v1_pre_topc(k3_pre_topc(A,B)) ) ) ), file(pre_topc,fc3_pre_topc), []). fof(fc3_topmetr,axiom,( ! [A,B,C] : ( ( v1_funct_1(B) & m1_relset_1(B,A,u1_struct_0(k3_topmetr)) ) => ( v1_xcmplx_0(k1_funct_1(B,C)) & v1_xreal_0(k1_funct_1(B,C)) ) ) ), file(topmetr,fc3_topmetr), []). 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_nat_1,axiom,( ! [A,B] : ( ( v4_ordinal2(A) & ~ v1_xboole_0(B) & v4_ordinal2(B) ) => ( ~ v1_xboole_0(k2_xcmplx_0(B,A)) & v4_ordinal2(k2_xcmplx_0(B,A)) & v1_xcmplx_0(k2_xcmplx_0(B,A)) & v1_xreal_0(k2_xcmplx_0(B,A)) ) ) ), file(nat_1,fc4_nat_1), []). fof(fc4_pre_topc,axiom,( ! [A,B] : ( ( v2_pre_topc(A) & l1_pre_topc(A) & m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) ) => ( v1_pre_topc(k3_pre_topc(A,B)) & v2_pre_topc(k3_pre_topc(A,B)) ) ) ), file(pre_topc,fc4_pre_topc), []). fof(fc5_borsuk_1,axiom, ( ~ v3_struct_0(k22_borsuk_1) & v1_pre_topc(k22_borsuk_1) & v2_pre_topc(k22_borsuk_1) ), file(borsuk_1,fc5_borsuk_1), []). fof(fc5_membered,axiom, ( v1_membered(k5_ordinal2) & v2_membered(k5_ordinal2) & v3_membered(k5_ordinal2) & v4_membered(k5_ordinal2) & v5_membered(k5_ordinal2) ), file(membered,fc5_membered), []). fof(fc5_xreal_0,axiom,( ! [A,B] : ( ( v1_xreal_0(A) & v1_xreal_0(B) ) => ( v1_xcmplx_0(k6_xcmplx_0(A,B)) & v1_xreal_0(k6_xcmplx_0(A,B)) ) ) ), file(xreal_0,fc5_xreal_0), []). 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_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_1_jordan20,axiom,( ! [A,B] : ( m2_finseq_1(B,u1_struct_0(k15_euclid(2))) => ( r2_hidden(A,a_1_1_jordan20(B)) <=> ? [C] : ( m2_subset_1(C,k1_numbers,k5_numbers) & A = k4_topreal1(2,B,C) & r1_xreal_0(1,C) & r1_xreal_0(k1_nat_1(C,1),k3_finseq_1(B)) ) ) ) ), file(jordan20,a_1_1_jordan20), []). fof(fraenkel_a_2_1_jordan20,axiom,( ! [A,B,C] : ( ( m2_finseq_1(B,u1_struct_0(k15_euclid(2))) & m2_subset_1(C,k1_numbers,k5_numbers) ) => ( r2_hidden(A,a_2_1_jordan20(B,C)) <=> ? [D] : ( m1_subset_1(D,u1_struct_0(k15_euclid(2))) & A = D & r1_jordan5c(k5_topreal1(2,B),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)),k4_finseq_4(k5_numbers,u1_struct_0(k15_euclid(2)),B,C),D) & r1_jordan5c(k5_topreal1(2,B),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)),D,k4_finseq_4(k5_numbers,u1_struct_0(k15_euclid(2)),B,k1_nat_1(C,1))) ) ) ) ), file(jordan20,a_2_1_jordan20), []). fof(fraenkel_a_2_2_topreal1,axiom,( ! [A,B,C] : ( ( m2_subset_1(B,k1_numbers,k5_numbers) & m2_finseq_1(C,u1_struct_0(k15_euclid(B))) ) => ( r2_hidden(A,a_2_2_topreal1(B,C)) <=> ? [D] : ( m2_subset_1(D,k1_numbers,k5_numbers) & A = k4_topreal1(B,C,D) & r1_xreal_0(1,D) & r1_xreal_0(k1_nat_1(D,1),k3_finseq_1(C)) ) ) ) ), file(topreal1,a_2_2_topreal1), []). fof(fraenkel_a_5_0_jordan6,axiom,( ! [A,B,C,D,E,F] : ( ( 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(E,u1_struct_0(k15_euclid(2))) & m1_subset_1(F,u1_struct_0(k15_euclid(2))) ) => ( r2_hidden(A,a_5_0_jordan6(B,C,D,E,F)) <=> ? [G] : ( m1_subset_1(G,u1_struct_0(k15_euclid(2))) & A = G & r1_jordan5c(B,C,D,E,G) & r1_jordan5c(B,C,D,G,F) ) ) ) ), file(jordan6,a_5_0_jordan6), []). fof(fraenkel_a_6_0_jordan20,axiom,( ! [A,B,C,D,E,F,G] : ( ( ~ v1_xboole_0(B) & 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(E,u1_struct_0(k15_euclid(2))) & m2_finseq_1(F,u1_struct_0(k15_euclid(2))) & m2_subset_1(G,k1_numbers,k5_numbers) ) => ( r2_hidden(A,a_6_0_jordan20(B,C,D,E,F,G)) <=> ? [H] : ( m1_subset_1(H,u1_struct_0(k15_euclid(2))) & A = H & r1_jordan5c(B,C,D,k4_finseq_4(k5_numbers,u1_struct_0(k15_euclid(2)),F,G),H) & r1_jordan5c(B,C,D,H,E) ) ) ) ), file(jordan20,a_6_0_jordan20), []). 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(involutiveness_k4_xcmplx_0,axiom,( ! [A] : ( v1_xcmplx_0(A) => k4_xcmplx_0(k4_xcmplx_0(A)) = A ) ), file(xcmplx_0,k4_xcmplx_0), []). 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_nat_1,axiom,( ? [A] : ( ~ v1_xboole_0(A) & v4_ordinal2(A) & v1_xcmplx_0(A) & v1_xreal_0(A) ) ), file(nat_1,rc1_nat_1), []). fof(rc1_pre_topc,axiom,( ? [A] : ( l1_pre_topc(A) & v1_pre_topc(A) ) ), file(pre_topc,rc1_pre_topc), []). fof(rc1_xreal_0,axiom,( ? [A] : ( v1_xcmplx_0(A) & v1_xreal_0(A) ) ), file(xreal_0,rc1_xreal_0), []). fof(rc2_nat_1,axiom,( ? [A] : ( m1_subset_1(A,k1_zfmisc_1(k1_numbers)) & ~ v1_xboole_0(A) & v3_ordinal1(A) ) ), file(nat_1,rc2_nat_1), []). fof(rc2_pre_topc,axiom,( ? [A] : ( l1_pre_topc(A) & ~ v3_struct_0(A) & v1_pre_topc(A) & v2_pre_topc(A) ) ), file(pre_topc,rc2_pre_topc), []). 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_nat_1,axiom,( ? [A] : ( m1_subset_1(A,k5_numbers) & ~ v1_xboole_0(A) & v1_ordinal1(A) & v2_ordinal1(A) & v3_ordinal1(A) & v4_ordinal2(A) & v1_xcmplx_0(A) & v1_xreal_0(A) ) ), file(nat_1,rc3_nat_1), []). fof(rc3_pre_topc,axiom,( ! [A] : ( l1_pre_topc(A) => ? [B] : ( m1_pre_topc(B,A) & v1_pre_topc(B) ) ) ), file(pre_topc,rc3_pre_topc), []). 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_pre_topc,axiom,( ! [A] : ( ( ~ v3_struct_0(A) & l1_pre_topc(A) ) => ? [B] : ( m1_pre_topc(B,A) & ~ v3_struct_0(B) & v1_pre_topc(B) ) ) ), file(pre_topc,rc4_pre_topc), []). 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_pre_topc,axiom,( ! [A] : ( ( v2_pre_topc(A) & l1_pre_topc(A) ) => ? [B] : ( m1_pre_topc(B,A) & v1_pre_topc(B) & v2_pre_topc(B) ) ) ), file(pre_topc,rc5_pre_topc), []). 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_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_k2_finseq_1,axiom,( ! [A] : ( v4_ordinal2(A) => k2_finseq_1(A) = k1_finseq_1(A) ) ), file(finseq_1,k2_finseq_1), []). 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_k3_real_1,axiom,( ! [A,B] : ( ( m1_subset_1(A,k1_numbers) & m1_subset_1(B,k1_numbers) ) => k3_real_1(A,B) = k2_xcmplx_0(A,B) ) ), file(real_1,k3_real_1), []). fof(redefinition_k3_topreal1,axiom,( ! [A,B,C] : ( ( m1_subset_1(A,k5_numbers) & m1_subset_1(B,u1_struct_0(k15_euclid(A))) & m1_subset_1(C,u1_struct_0(k15_euclid(A))) ) => k3_topreal1(A,B,C) = k1_topreal1(A,B,C) ) ), file(topreal1,k3_topreal1), []). 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_k5_numbers,axiom,( k5_numbers = k5_ordinal2 ), file(numbers,k5_numbers), []). fof(redefinition_k5_real_1,axiom,( ! [A,B] : ( ( m1_subset_1(A,k1_numbers) & m1_subset_1(B,k1_numbers) ) => k5_real_1(A,B) = k6_xcmplx_0(A,B) ) ), file(real_1,k5_real_1), []). fof(redefinition_k5_topmetr,axiom,( k5_topmetr = k22_borsuk_1 ), file(topmetr,k5_topmetr), []). 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_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(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(s5_nat_1__e21_33__jordan20,axiom,( ! [A,B,C] : ( ( m1_subset_1(A,k1_numbers) & m2_finseq_1(B,u1_struct_0(k15_euclid(2))) & m2_subset_1(C,k1_numbers,k5_numbers) ) => ( ? [D] : ( m2_subset_1(D,k1_numbers,k5_numbers) & ! [E] : ( m1_subset_1(E,u1_struct_0(k15_euclid(2))) => ~ ( E = k1_funct_1(B,k5_binarith(C,D)) & k21_euclid(E) = A ) ) ) => ? [D] : ( m2_subset_1(D,k1_numbers,k5_numbers) & ! [F] : ( m1_subset_1(F,u1_struct_0(k15_euclid(2))) => ~ ( F = k1_funct_1(B,k5_binarith(C,D)) & k21_euclid(F) = A ) ) & ! [G] : ( m2_subset_1(G,k1_numbers,k5_numbers) => ( ! [H] : ( m1_subset_1(H,u1_struct_0(k15_euclid(2))) => ~ ( H = k1_funct_1(B,k5_binarith(C,G)) & k21_euclid(H) = A ) ) => r1_xreal_0(D,G) ) ) ) ) ) ), file(jordan20,s5_nat_1__e21_33__jordan20), []). 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(t106_real_2,axiom,( ! [A] : ( v1_xreal_0(A) => ! [B] : ( v1_xreal_0(B) => ! [C] : ( v1_xreal_0(C) => ~ ( ~ ( r1_xreal_0(0,k2_xcmplx_0(A,k4_xcmplx_0(B))) & r1_xreal_0(k6_xcmplx_0(B,A),0) & r1_xreal_0(k2_xcmplx_0(k4_xcmplx_0(A),B),0) & r1_xreal_0(k2_xcmplx_0(B,k4_xcmplx_0(C)),k6_xcmplx_0(A,C)) & r1_xreal_0(k6_xcmplx_0(B,C),k2_xcmplx_0(A,k4_xcmplx_0(C))) & r1_xreal_0(k6_xcmplx_0(C,A),k6_xcmplx_0(C,B)) ) & r1_xreal_0(B,A) ) ) ) ) ), file(real_2,t106_real_2), []). 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(t11_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(k6_xcmplx_0(A,C),k6_xcmplx_0(B,C)) ) ) ) ) ), file(xreal_1,t11_xreal_1), []). fof(t12_jordan5c,axiom,( ! [A] : ( m1_subset_1(A,k1_zfmisc_1(u1_struct_0(k15_euclid(2)))) => ! [B] : ( m1_subset_1(B,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))) => ! [E] : ( m1_subset_1(E,u1_struct_0(k15_euclid(2))) => ( ( r1_topreal1(k15_euclid(2),B,C,A) & r1_jordan5c(A,B,C,D,E) & r1_jordan5c(A,B,C,E,D) ) => D = E ) ) ) ) ) ) ), file(jordan5c,t12_jordan5c), []). fof(t12_topreal1,axiom,( ! [A] : ( m2_subset_1(A,k1_numbers,k5_numbers) => ! [B] : ( m1_subset_1(B,u1_struct_0(k15_euclid(A))) => ! [C] : ( m1_subset_1(C,u1_struct_0(k15_euclid(A))) => ! [D] : ( m1_subset_1(D,u1_struct_0(k15_euclid(A))) => ! [E] : ( m1_subset_1(E,u1_struct_0(k15_euclid(A))) => ( ( r2_hidden(D,k3_topreal1(A,B,C)) & r2_hidden(E,k3_topreal1(A,B,C)) ) => r1_tarski(k3_topreal1(A,D,E),k3_topreal1(A,B,C)) ) ) ) ) ) ) ), file(topreal1,t12_topreal1), []). fof(t13_jordan3,axiom,( ! [A] : ( v4_ordinal2(A) => ! [B] : ( v4_ordinal2(B) => ! [C] : ( v4_ordinal2(C) => ( r1_xreal_0(B,A) => r1_xreal_0(k5_binarith(B,C),A) ) ) ) ) ), file(jordan3,t13_jordan3), []). fof(t13_jordan5c,axiom,( ! [A] : ( m1_subset_1(A,k1_zfmisc_1(u1_struct_0(k15_euclid(2)))) => ! [B] : ( m1_subset_1(B,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))) => ! [E] : ( m1_subset_1(E,u1_struct_0(k15_euclid(2))) => ! [F] : ( m1_subset_1(F,u1_struct_0(k15_euclid(2))) => ( ( r1_jordan5c(A,B,C,D,E) & r1_jordan5c(A,B,C,E,F) ) => r1_jordan5c(A,B,C,D,F) ) ) ) ) ) ) ) ), file(jordan5c,t13_jordan5c), []). fof(t14_jordan3,axiom,( ! [A] : ( v4_ordinal2(A) => ! [B] : ( v4_ordinal2(B) => ~ ( r1_xreal_0(1,A) & r1_xreal_0(1,k5_binarith(B,A)) & r1_xreal_0(B,k5_binarith(B,A)) ) ) ) ), file(jordan3,t14_jordan3), []). fof(t15_topreal1,axiom,( ! [A] : ( m2_subset_1(A,k1_numbers,k5_numbers) => ! [B] : ( m1_subset_1(B,u1_struct_0(k15_euclid(A))) => ! [C] : ( m1_subset_1(C,u1_struct_0(k15_euclid(A))) => ( B != C => r1_topreal1(k15_euclid(A),B,C,k3_topreal1(A,B,C)) ) ) ) ) ), file(topreal1,t15_topreal1), []). fof(t19_jordan20,axiom,( ! [A] : ( ( ~ v1_xboole_0(A) & m1_subset_1(A,k1_zfmisc_1(u1_struct_0(k15_euclid(2)))) ) => ! [B] : ( m1_subset_1(B,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))) => ! [E] : ( m1_subset_1(E,u1_struct_0(k15_euclid(2))) => ~ ( r1_topreal1(k15_euclid(2),B,C,A) & r2_hidden(D,A) & r2_hidden(E,A) & ~ r1_jordan5c(A,B,C,D,E) & ~ r1_jordan5c(A,B,C,E,D) ) ) ) ) ) ) ), file(jordan20,t19_jordan20), []). fof(t1_arithm,axiom,( ! [A] : ( v1_xcmplx_0(A) => k2_xcmplx_0(A,0) = A ) ), file(arithm,t1_arithm), []). fof(t1_jordan20,axiom,( ! [A] : ( m1_subset_1(A,k1_zfmisc_1(u1_struct_0(k15_euclid(2)))) => ! [B] : ( m1_subset_1(B,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))) => ( ( r1_topreal1(k15_euclid(2),B,C,A) & r2_hidden(D,A) ) => k5_jordan6(A,B,C,D,D) = k1_struct_0(k15_euclid(2),D) ) ) ) ) ) ), file(jordan20,t1_jordan20), []). 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_xboole_1,axiom,( ! [A,B,C] : ( ( r1_tarski(A,B) & r1_tarski(B,C) ) => r1_tarski(A,C) ) ), file(xboole_1,t1_xboole_1), []). 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(t23_jordan5c,axiom,( ! [A] : ( m2_finseq_1(A,u1_struct_0(k15_euclid(2))) => ! [B] : ( m2_subset_1(B,k1_numbers,k5_numbers) => ( ( v4_topreal1(A) & r1_xreal_0(1,B) & r1_xreal_0(k1_nat_1(B,1),k3_finseq_1(A)) ) => r1_jordan5c(k5_topreal1(2,A),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_finseq_4(k5_numbers,u1_struct_0(k15_euclid(2)),A,B),k4_finseq_4(k5_numbers,u1_struct_0(k15_euclid(2)),A,k1_nat_1(B,1))) ) ) ) ), file(jordan5c,t23_jordan5c), []). fof(t24_finseq_4,axiom,( ! [A,B] : ( m2_finseq_1(B,A) => ! [C] : ( m2_subset_1(C,k1_numbers,k5_numbers) => ( ( r1_xreal_0(1,C) & r1_xreal_0(C,k3_finseq_1(B)) ) => k4_finseq_4(k5_numbers,A,B,C) = k1_funct_1(B,C) ) ) ) ), file(finseq_4,t24_finseq_4), []). fof(t24_jordan5c,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,C) & r1_xreal_0(C,k3_finseq_1(A)) ) => r1_jordan5c(k5_topreal1(2,A),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_finseq_4(k5_numbers,u1_struct_0(k15_euclid(2)),A,B),k4_finseq_4(k5_numbers,u1_struct_0(k15_euclid(2)),A,C)) ) ) ) ) ), file(jordan5c,t24_jordan5c), []). fof(t25_jordan5c,axiom,( ! [A] : ( m2_finseq_1(A,u1_struct_0(k15_euclid(2))) => ! [B] : ( m1_subset_1(B,u1_struct_0(k15_euclid(2))) => ! [C] : ( m2_subset_1(C,k1_numbers,k5_numbers) => ( ( v4_topreal1(A) & 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)) ) => r1_jordan5c(k5_topreal1(2,A),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_finseq_4(k5_numbers,u1_struct_0(k15_euclid(2)),A,C),B) ) ) ) ) ), file(jordan5c,t25_jordan5c), []). fof(t26_jordan5c,axiom,( ! [A] : ( m2_finseq_1(A,u1_struct_0(k15_euclid(2))) => ! [B] : ( m1_subset_1(B,u1_struct_0(k15_euclid(2))) => ! [C] : ( m2_subset_1(C,k1_numbers,k5_numbers) => ( ( v4_topreal1(A) & 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)) ) => r1_jordan5c(k5_topreal1(2,A),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)),B,k4_finseq_4(k5_numbers,u1_struct_0(k15_euclid(2)),A,k1_nat_1(C,1))) ) ) ) ) ), file(jordan5c,t26_jordan5c), []). 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_jordan20,axiom,( ! [A] : ( ( ~ v1_xboole_0(A) & m1_subset_1(A,k1_zfmisc_1(u1_struct_0(k15_euclid(2)))) ) => ! [B] : ( ( ~ v1_xboole_0(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))) => ! [E] : ( m1_subset_1(E,u1_struct_0(k15_euclid(2))) => ! [F] : ( m1_subset_1(F,u1_struct_0(k15_euclid(2))) => ( ( r1_topreal1(k15_euclid(2),C,D,A) & r1_topreal1(k15_euclid(2),E,F,B) & r1_jordan5c(A,C,D,E,F) & r1_tarski(B,A) ) => B = k5_jordan6(A,C,D,E,F) ) ) ) ) ) ) ) ), file(jordan20,t28_jordan20), []). fof(t29_jordan6,axiom,( ! [A] : ( m1_subset_1(A,k1_zfmisc_1(u1_struct_0(k15_euclid(2)))) => ! [B] : ( m1_subset_1(B,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))) => ! [E] : ( m1_subset_1(E,u1_struct_0(k15_euclid(2))) => k5_jordan6(A,B,C,D,E) = a_5_0_jordan6(A,B,C,D,E) ) ) ) ) ) ), file(jordan6,t29_jordan6), []). fof(t2_jordan20,axiom,( ! [A] : ( m1_subset_1(A,u1_struct_0(k15_euclid(2))) => ! [B] : ( m1_subset_1(B,u1_struct_0(k15_euclid(2))) => ! [C] : ( m1_subset_1(C,u1_struct_0(k15_euclid(2))) => ! [D] : ( m1_subset_1(D,k1_numbers) => ( ( r2_hidden(C,k3_topreal1(2,A,B)) & r1_xreal_0(k21_euclid(A),D) & r1_xreal_0(k21_euclid(B),D) ) => r1_xreal_0(k21_euclid(C),D) ) ) ) ) ) ), file(jordan20,t2_jordan20), []). fof(t2_jordan3,axiom,( ! [A] : ( v4_ordinal2(A) => k5_binarith(A,0) = A ) ), file(jordan3,t2_jordan3), []). 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_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) => ! [E] : ( m2_subset_1(E,k1_numbers,k5_numbers) => ( ( r1_xreal_0(1,C) & r1_xreal_0(C,D) & r1_xreal_0(D,k3_finseq_1(B)) & r1_xreal_0(1,E) ) => ( ( ~ r1_xreal_0(E,k1_nat_1(k5_binarith(D,C),1)) & ~ r1_xreal_0(E,k3_real_1(k5_real_1(D,C),1)) & ~ r1_xreal_0(E,k5_real_1(k1_nat_1(D,1),C)) ) | ( k1_funct_1(k1_jordan3(A,B,C,D),E) = k1_funct_1(B,k5_binarith(k1_nat_1(E,C),1)) & k1_funct_1(k1_jordan3(A,B,C,D),E) = k1_funct_1(B,k1_nat_1(k5_binarith(E,1),C)) & k1_funct_1(k1_jordan3(A,B,C,D),E) = k1_funct_1(B,k5_real_1(k1_nat_1(E,C),1)) & k1_funct_1(k1_jordan3(A,B,C,D),E) = k1_funct_1(B,k3_real_1(k5_real_1(E,1),C)) ) ) ) ) ) ) ) ) ), file(jordan3,t31_jordan3), []). fof(t33_sprect_3,axiom,( ! [A] : ( m2_subset_1(A,k1_numbers,k5_numbers) => ! [B] : ( ( v4_topreal1(B) & m2_finseq_1(B,u1_struct_0(k15_euclid(2))) ) => ! [C] : ( m1_subset_1(C,u1_struct_0(k15_euclid(2))) => ( ( r1_xreal_0(A,k3_finseq_1(B)) & r2_hidden(C,k5_topreal1(2,k1_jordan3(u1_struct_0(k15_euclid(2)),B,1,A))) ) => ( r1_xreal_0(A,1) | r1_jordan5c(k5_topreal1(2,B),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)),C,k4_finseq_4(k5_numbers,u1_struct_0(k15_euclid(2)),B,A)) ) ) ) ) ) ), file(sprect_3,t33_sprect_3), []). 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(t3_finseq_1,axiom,( ! [A] : ( v4_ordinal2(A) => ! [B] : ( v4_ordinal2(B) => ( r2_hidden(A,k2_finseq_1(B)) <=> ( r1_xreal_0(1,A) & r1_xreal_0(A,B) ) ) ) ) ), file(finseq_1,t3_finseq_1), []). fof(t3_jordan20,axiom,( ! [A] : ( m1_subset_1(A,u1_struct_0(k15_euclid(2))) => ! [B] : ( m1_subset_1(B,u1_struct_0(k15_euclid(2))) => ! [C] : ( m1_subset_1(C,u1_struct_0(k15_euclid(2))) => ! [D] : ( m1_subset_1(D,k1_numbers) => ( ( r2_hidden(C,k3_topreal1(2,A,B)) & r1_xreal_0(D,k21_euclid(A)) & r1_xreal_0(D,k21_euclid(B)) ) => r1_xreal_0(D,k21_euclid(C)) ) ) ) ) ) ), file(jordan20,t3_jordan20), []). 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_topreal4,axiom,( ! [A] : ( m1_subset_1(A,k1_zfmisc_1(u1_struct_0(k15_euclid(2)))) => ! [B] : ( m1_subset_1(B,u1_struct_0(k15_euclid(2))) => ! [C] : ( m1_subset_1(C,u1_struct_0(k15_euclid(2))) => ( r1_topreal4(A,B,C) => r1_topreal1(k15_euclid(2),B,C,A) ) ) ) ) ), file(topreal4,t3_topreal4), []). fof(t4_arithm,axiom,( ! [A] : ( v1_xcmplx_0(A) => k6_xcmplx_0(A,0) = A ) ), file(arithm,t4_arithm), []). 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_sprect_4,axiom,( ! [A] : ( ( v4_topreal1(A) & m2_finseq_1(A,u1_struct_0(k15_euclid(2))) ) => ! [B] : ( m1_subset_1(B,u1_struct_0(k15_euclid(2))) => ! [C] : ( m2_subset_1(C,k1_numbers,k5_numbers) => ( ( r1_xreal_0(1,C) & r2_hidden(B,k5_topreal1(2,k1_jordan3(u1_struct_0(k15_euclid(2)),A,C,k3_finseq_1(A)))) ) => ( r1_xreal_0(k3_finseq_1(A),C) | r1_jordan5c(k5_topreal1(2,A),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_finseq_4(k5_numbers,u1_struct_0(k15_euclid(2)),A,C),B) ) ) ) ) ) ), file(sprect_4,t4_sprect_4), []). 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(t50_xreal_1,axiom,( ! [A] : ( v1_xreal_0(A) => ! [B] : ( v1_xreal_0(B) => ( r1_xreal_0(A,B) => r1_xreal_0(0,k6_xcmplx_0(B,A)) ) ) ) ), file(xreal_1,t50_xreal_1), []). fof(t52_binarith,axiom,( ! [A] : ( v4_ordinal2(A) => ! [B] : ( v4_ordinal2(B) => r1_xreal_0(k5_binarith(A,B),A) ) ) ), file(binarith,t52_binarith), []). fof(t52_xreal_1,axiom,( ! [A] : ( v1_xreal_0(A) => ! [B] : ( v1_xreal_0(B) => ~ ( ~ r1_xreal_0(B,A) & r1_xreal_0(k6_xcmplx_0(B,A),0) ) ) ) ), file(xreal_1,t52_xreal_1), []). 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_jordan20,axiom,( ! [A] : ( m2_subset_1(A,k1_numbers,k5_numbers) => ! [B] : ( ( v4_topreal1(B) & m2_finseq_1(B,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))) => ( ( r1_xreal_0(1,A) & r2_hidden(C,k4_topreal1(2,B,A)) & r2_hidden(D,k4_topreal1(2,B,A)) & k22_euclid(k4_finseq_4(k5_numbers,u1_struct_0(k15_euclid(2)),B,A)) = k22_euclid(k4_finseq_4(k5_numbers,u1_struct_0(k15_euclid(2)),B,k1_nat_1(A,1))) & r1_jordan5c(k5_topreal1(2,B),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)),C,D) ) => ( r1_xreal_0(k3_finseq_1(B),A) | r1_xreal_0(k21_euclid(k4_finseq_4(k5_numbers,u1_struct_0(k15_euclid(2)),B,A)),k21_euclid(k4_finseq_4(k5_numbers,u1_struct_0(k15_euclid(2)),B,k1_nat_1(A,1)))) | r1_xreal_0(k21_euclid(D),k21_euclid(C)) ) ) ) ) ) ) ), file(jordan20,t6_jordan20), []). 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_topreal1,axiom,( ! [A] : ( m2_subset_1(A,k1_numbers,k5_numbers) => ! [B] : ( m1_subset_1(B,u1_struct_0(k15_euclid(A))) => ! [C] : ( m1_subset_1(C,u1_struct_0(k15_euclid(A))) => ( r2_hidden(B,k1_topreal1(A,B,C)) & r2_hidden(C,k1_topreal1(A,B,C)) ) ) ) ) ), file(topreal1,t6_topreal1), []). fof(t7_boole,axiom,( ! [A,B] : ~ ( r2_hidden(A,B) & v1_xboole_0(B) ) ), file(boole,t7_boole), []). fof(t7_jct_misc,axiom,( ! [A] : ( m2_subset_1(A,k1_numbers,k5_numbers) => ! [B] : ( m2_subset_1(B,k1_numbers,k5_numbers) => ( r1_xreal_0(A,B) => k5_binarith(B,k5_binarith(B,A)) = A ) ) ) ), file(jct_misc,t7_jct_misc), []). fof(t7_jordan20,axiom,( ! [A] : ( m2_subset_1(A,k1_numbers,k5_numbers) => ! [B] : ( ( v4_topreal1(B) & m2_finseq_1(B,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))) => ( ( r1_xreal_0(1,A) & r2_hidden(C,k4_topreal1(2,B,A)) & r2_hidden(D,k4_topreal1(2,B,A)) & k22_euclid(k4_finseq_4(k5_numbers,u1_struct_0(k15_euclid(2)),B,A)) = k22_euclid(k4_finseq_4(k5_numbers,u1_struct_0(k15_euclid(2)),B,k1_nat_1(A,1))) & r1_jordan5c(k5_topreal1(2,B),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)),C,D) ) => ( r1_xreal_0(k3_finseq_1(B),A) | r1_xreal_0(k21_euclid(k4_finseq_4(k5_numbers,u1_struct_0(k15_euclid(2)),B,k1_nat_1(A,1))),k21_euclid(k4_finseq_4(k5_numbers,u1_struct_0(k15_euclid(2)),B,A))) | r1_xreal_0(k21_euclid(C),k21_euclid(D)) ) ) ) ) ) ) ), file(jordan20,t7_jordan20), []). 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(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(t9_topreal1,axiom,( ! [A] : ( m1_subset_1(A,u1_struct_0(k15_euclid(2))) => ! [B] : ( m1_subset_1(B,u1_struct_0(k15_euclid(2))) => ! [C] : ( m1_subset_1(C,u1_struct_0(k15_euclid(2))) => ( ( r1_xreal_0(k21_euclid(A),k21_euclid(B)) & r2_hidden(C,k3_topreal1(2,A,B)) ) => ( r1_xreal_0(k21_euclid(A),k21_euclid(C)) & r1_xreal_0(k21_euclid(C),k21_euclid(B)) ) ) ) ) ) ), file(topreal1,t9_topreal1), []). fof(t9_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(A,B) & r1_xreal_0(C,D) ) => r1_xreal_0(k2_xcmplx_0(A,C),k2_xcmplx_0(B,D)) ) ) ) ) ) ), file(xreal_1,t9_xreal_1), []).