% Mizar problem: t1_vectmetr,vectmetr,72,25 fof(t1_vectmetr,conjecture,( ! [A] : ( ( ~ v3_struct_0(A) & v6_metric_1(A) & v7_metric_1(A) & v8_metric_1(A) & v9_metric_1(A) & l1_metric_1(A) ) => ( v1_vectmetr(A) => ! [B] : ( m1_subset_1(B,u1_struct_0(A)) => ! [C] : ( m1_subset_1(C,u1_struct_0(A)) => ! [D] : ( m1_subset_1(D,k1_numbers) => ~ ( ~ r1_xreal_0(D,0) & ! [E] : ( m2_finseq_1(E,u1_struct_0(A)) => ~ ( k4_finseq_4(k5_numbers,u1_struct_0(A),E,1) = B & k4_finseq_4(k5_numbers,u1_struct_0(A),E,k3_finseq_1(E)) = C & ! [F] : ( m2_subset_1(F,k1_numbers,k5_numbers) => ~ ( r1_xreal_0(1,F) & r1_xreal_0(F,k5_real_1(k3_finseq_1(E),1)) & r1_xreal_0(D,k4_metric_1(A,k4_finseq_4(k5_numbers,u1_struct_0(A),E,F),k4_finseq_4(k5_numbers,u1_struct_0(A),E,k1_nat_1(F,1)))) ) ) & ! [F] : ( m2_finseq_1(F,k1_numbers) => ( ( k3_finseq_1(F) = k5_real_1(k3_finseq_1(E),1) & ! [G] : ( m2_subset_1(G,k1_numbers,k5_numbers) => ( ( r1_xreal_0(1,G) & r1_xreal_0(G,k3_finseq_1(F)) ) => k4_finseq_4(k5_numbers,k1_numbers,F,G) = k4_metric_1(A,k4_finseq_4(k5_numbers,u1_struct_0(A),E,G),k4_finseq_4(k5_numbers,u1_struct_0(A),E,k1_nat_1(G,1))) ) ) ) => k4_metric_1(A,B,C) = k15_rvsum_1(F) ) ) ) ) ) ) ) ) ) ) ), inference(mizar_bg_added,[status(thm)],[antisymmetry_r2_hidden,cc10_membered,cc11_membered,cc12_membered,cc13_membered,cc14_membered,cc15_membered,cc16_membered,cc17_membered,cc18_membered,cc19_membered,cc1_bintree1,cc1_bintree2,cc1_funct_1,cc1_margrel1,cc1_membered,cc1_nat_1,cc1_relset_1,cc20_membered,cc2_funct_1,cc2_membered,cc2_nat_1,cc3_membered,cc3_nat_1,cc4_membered,cc6_membered,cc9_membered,commutativity_k11_binop_2,commutativity_k1_nat_1,commutativity_k23_binop_2,commutativity_k2_nat_1,commutativity_k2_tarski,commutativity_k2_xcmplx_0,commutativity_k3_real_1,commutativity_k3_xcmplx_0,commutativity_k4_metric_1,commutativity_k4_real_1,commutativity_k7_domain_1,connectedness_r1_xreal_0,d10_xboole_0,d11_finseq_1,d12_margrel1,d1_finseq_1,d1_tarski,d1_vectmetr,d2_bintree2,d2_finseq_2,d2_tarski,d2_zfmisc_1,d3_finseq_1,d3_tarski,d4_euclid,d4_finseq_4,d5_funct_1,d5_real_1,d5_tarski,d7_finseq_1,dt_k10_finseq_1,dt_k11_binop_2,dt_k11_margrel1,dt_k12_binarith,dt_k12_finseq_1,dt_k13_binarith,dt_k13_finseq_1,dt_k15_rvsum_1,dt_k18_group_2,dt_k1_card_1,dt_k1_domain_1,dt_k1_euclid,dt_k1_finseq_1,dt_k1_funct_1,dt_k1_int_1,dt_k1_mcart_1,dt_k1_nat_1,dt_k1_numbers,dt_k1_relat_1,dt_k1_tarski,dt_k1_xboole_0,dt_k1_zfmisc_1,dt_k23_binop_2,dt_k2_bintree2,dt_k2_domain_1,dt_k2_finseq_1,dt_k2_finseq_2,dt_k2_finseq_4,dt_k2_funcop_1,dt_k2_int_1,dt_k2_mcart_1,dt_k2_metric_1,dt_k2_nat_1,dt_k2_relat_1,dt_k2_tarski,dt_k2_trees_2,dt_k2_xcmplx_0,dt_k2_zfmisc_1,dt_k3_finseq_1,dt_k3_finseq_2,dt_k3_nat_1,dt_k3_power,dt_k3_real_1,dt_k3_series_1,dt_k3_trees_2,dt_k3_xcmplx_0,dt_k4_euclid,dt_k4_finseq_1,dt_k4_finseq_2,dt_k4_finseq_4,dt_k4_finseqop,dt_k4_metric_1,dt_k4_nat_1,dt_k4_real_1,dt_k4_tarski,dt_k5_binarith,dt_k5_euclid,dt_k5_finseq_1,dt_k5_finsub_1,dt_k5_numbers,dt_k5_ordinal2,dt_k5_power,dt_k5_real_1,dt_k6_binarith,dt_k6_finseq_1,dt_k6_margrel1,dt_k6_power,dt_k6_real_1,dt_k6_xcmplx_0,dt_k7_domain_1,dt_k7_finseq_1,dt_k7_margrel1,dt_k7_rlvect_1,dt_k7_xcmplx_0,dt_k8_finseq_1,dt_k8_margrel1,dt_k9_finseq_1,dt_k9_margrel1,dt_l1_metric_1,dt_l1_struct_0,dt_m1_bintree2,dt_m1_finseq_1,dt_m1_finseq_2,dt_m1_relset_1,dt_m1_subset_1,dt_m1_trees_1,dt_m1_trees_4,dt_m2_finseq_1,dt_m2_finseq_2,dt_m2_relset_1,dt_m2_subset_1,dt_m2_trees_2,dt_m3_trees_2,dt_u1_struct_0,existence_l1_metric_1,existence_l1_struct_0,existence_m1_bintree2,existence_m1_finseq_1,existence_m1_finseq_2,existence_m1_relset_1,existence_m1_subset_1,existence_m1_trees_1,existence_m1_trees_4,existence_m2_finseq_1,existence_m2_finseq_2,existence_m2_relset_1,existence_m2_subset_1,existence_m2_trees_2,existence_m3_trees_2,fc10_membered,fc11_membered,fc12_membered,fc13_membered,fc14_membered,fc15_membered,fc16_membered,fc1_margrel1,fc1_nat_1,fc1_struct_0,fc1_subset_1,fc2_bintree2,fc2_membered,fc2_nat_1,fc2_subset_1,fc2_trees_2,fc3_bintree2,fc3_margrel1,fc3_nat_1,fc3_subset_1,fc3_trees_2,fc4_margrel1,fc4_nat_1,fc4_subset_1,fc5_membered,fc5_trees_2,fc6_membered,fc7_membered,fc8_membered,fc9_membered,fraenkel_a_1_0_finseq_1,fraenkel_a_1_0_vectmetr,involutiveness_k11_margrel1,involutiveness_k9_margrel1,rc1_bintree1,rc1_bintree2,rc1_funct_1,rc1_margrel1,rc1_membered,rc1_nat_1,rc1_subset_1,rc1_trees_2,rc2_bintree1,rc2_funct_1,rc2_margrel1,rc2_nat_1,rc2_subset_1,rc3_bintree1,rc3_funct_1,rc3_nat_1,rc3_struct_0,rc5_struct_0,rc5_trees_2,rc6_trees_2,rc7_trees_2,rc8_trees_2,rc9_trees_2,redefinition_k11_binop_2,redefinition_k11_margrel1,redefinition_k12_binarith,redefinition_k12_finseq_1,redefinition_k13_binarith,redefinition_k18_group_2,redefinition_k1_domain_1,redefinition_k1_nat_1,redefinition_k23_binop_2,redefinition_k2_domain_1,redefinition_k2_finseq_1,redefinition_k2_finseq_4,redefinition_k2_nat_1,redefinition_k3_finseq_1,redefinition_k3_finseq_2,redefinition_k3_real_1,redefinition_k3_series_1,redefinition_k3_trees_2,redefinition_k4_finseq_1,redefinition_k4_finseqop,redefinition_k4_metric_1,redefinition_k4_real_1,redefinition_k5_euclid,redefinition_k5_numbers,redefinition_k5_real_1,redefinition_k6_power,redefinition_k6_real_1,redefinition_k7_domain_1,redefinition_k7_rlvect_1,redefinition_k8_finseq_1,redefinition_k9_finseq_1,redefinition_m1_bintree2,redefinition_m1_trees_1,redefinition_m1_trees_4,redefinition_m2_finseq_1,redefinition_m2_finseq_2,redefinition_m2_relset_1,redefinition_m2_subset_1,reflexivity_r1_nat_1,reflexivity_r1_tarski,reflexivity_r1_xreal_0,s1_binarith__e17_3_2_1_8__vectmetr,s1_binarith__e31_3_2_1__vectmetr,s1_binarith__e8_3_2_1_8_2_1_1__vectmetr,s1_bintree2__e4_3__vectmetr,s1_finseq_2__e19_3_2_1__vectmetr,s1_finseq_2__e22_3_2_1__vectmetr,s1_nat_1__e28_3_2_1__vectmetr,spc0_boole,spc1_boole,spc1_numerals,spc2_boole,spc2_numerals,t103_rvsum_1,t109_finseq_2,t10_bintree2,t110_finseq_2,t110_rvsum_1,t11_binari_2,t11_xreal_1,t12_pre_ff,t137_finseq_2,t13_finseq_2,t143_real_2,t14_nat_2,t15_binari_3,t15_bintree2,t15_finseq_5,t15_funcop_1,t15_nat_2,t16_bintree2,t17_nat_2,t19_bintree2,t19_nat_1,t1_arithm,t1_numerals,t1_real,t1_subset,t1_xreal_1,t20_bintree2,t20_finseq_2,t20_nat_2,t21_xreal_1,t22_bintree2,t23_bintree2,t23_mcart_1,t23_nat_2,t24_bintree2,t24_finseq_4,t24_nat_2,t26_finseq_4,t27_nat_2,t28_nat_2,t29_nat_1,t2_arithm,t2_real,t2_subset,t2_tarski,t2_xreal_1,t30_power,t32_power,t33_zfmisc_1,t35_finseq_1,t36_margrel1,t38_nat_1,t39_binarith,t39_nat_1,t39_power,t3_arithm,t3_finseq_1,t3_real,t3_subset,t41_margrel1,t47_finseq_1,t47_nat_1,t49_nat_1,t4_arithm,t4_binari_3,t4_finseq_1,t4_real,t4_subset,t50_binarith,t52_int_1,t53_binarith,t56_finseq_1,t59_finseq_1,t59_power,t5_arithm,t5_binari_3,t5_bintree2,t5_real,t5_subset,t60_power,t61_finseq_1,t62_nat_1,t63_power,t6_arithm,t6_boole,t6_finseq_1,t6_real,t70_finseq_2,t70_xreal_1,t72_finseq_2,t73_finseq_2,t73_nat_1,t73_real_1,t75_xcmplx_1,t79_xcmplx_1,t7_boole,t7_mcart_1,t7_real,t80_int_1,t83_finseq_4,t88_xcmplx_1,t8_boole,t8_jordan2b,t8_real,t8_xreal_1]), [file(vectmetr,t1_vectmetr)]). 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_bintree1,axiom,( ! [A] : ( ( ~ v1_xboole_0(A) & v1_trees_1(A) & v1_bintree1(A) ) => ( ~ v1_xboole_0(A) & v1_trees_1(A) & v1_trees_2(A) ) ) ), file(bintree1,cc1_bintree1), []). fof(cc1_bintree2,axiom,( ! [A] : ( ( ~ v1_xboole_0(A) & v1_trees_1(A) & v1_bintree2(A) ) => ( ~ v1_xboole_0(A) & v1_trees_1(A) & v1_trees_2(A) & v2_trees_9(A) & v1_bintree1(A) ) ) ), file(bintree2,cc1_bintree2), []). fof(cc1_funct_1,axiom,( ! [A] : ( v1_xboole_0(A) => v1_funct_1(A) ) ), file(funct_1,cc1_funct_1), []). fof(cc1_margrel1,axiom,( ! [A] : ( m1_subset_1(A,k6_margrel1) => v2_margrel1(A) ) ), file(margrel1,cc1_margrel1), []). 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_relset_1,axiom,( ! [A,B,C] : ( m1_subset_1(C,k1_zfmisc_1(k2_zfmisc_1(A,B))) => v1_relat_1(C) ) ), file(relset_1,cc1_relset_1), []). fof(cc20_membered,axiom,( ! [A] : ( v5_membered(A) => ! [B] : ( m1_subset_1(B,k1_zfmisc_1(A)) => ( v1_membered(B) & v2_membered(B) & v3_membered(B) & v4_membered(B) & v5_membered(B) ) ) ) ), file(membered,cc20_membered), []). fof(cc2_funct_1,axiom,( ! [A] : ( ( v1_relat_1(A) & v1_xboole_0(A) & v1_funct_1(A) ) => ( v1_relat_1(A) & v1_funct_1(A) & v2_funct_1(A) ) ) ), file(funct_1,cc2_funct_1), []). fof(cc2_membered,axiom,( ! [A] : ( v4_membered(A) => v3_membered(A) ) ), file(membered,cc2_membered), []). fof(cc2_nat_1,axiom,( ! [A] : ( m1_subset_1(A,k5_numbers) => ( v1_ordinal1(A) & v2_ordinal1(A) & v3_ordinal1(A) & v4_ordinal2(A) & v1_xcmplx_0(A) & v1_xreal_0(A) & ~ v3_xreal_0(A) ) ) ), file(nat_1,cc2_nat_1), []). fof(cc3_membered,axiom,( ! [A] : ( v3_membered(A) => v2_membered(A) ) ), file(membered,cc3_membered), []). fof(cc3_nat_1,axiom,( ! [A] : ( v4_ordinal2(A) => ( v4_ordinal2(A) & v1_xcmplx_0(A) & v1_xreal_0(A) & ~ v3_xreal_0(A) ) ) ), file(nat_1,cc3_nat_1), []). fof(cc4_membered,axiom,( ! [A] : ( v2_membered(A) => v1_membered(A) ) ), file(membered,cc4_membered), []). fof(cc6_membered,axiom,( ! [A] : ( m1_subset_1(A,k1_zfmisc_1(k1_numbers)) => ( v1_membered(A) & v2_membered(A) ) ) ), file(membered,cc6_membered), []). 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_k11_binop_2,axiom,( ! [A,B] : ( ( m1_subset_1(A,k1_numbers) & m1_subset_1(B,k1_numbers) ) => k11_binop_2(A,B) = k11_binop_2(B,A) ) ), file(binop_2,k11_binop_2), []). 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_k23_binop_2,axiom,( ! [A,B] : ( ( m1_subset_1(A,k5_numbers) & m1_subset_1(B,k5_numbers) ) => k23_binop_2(A,B) = k23_binop_2(B,A) ) ), file(binop_2,k23_binop_2), []). fof(commutativity_k2_nat_1,axiom,( ! [A,B] : ( ( m1_subset_1(A,k5_numbers) & m1_subset_1(B,k5_numbers) ) => k2_nat_1(A,B) = k2_nat_1(B,A) ) ), file(nat_1,k2_nat_1), []). fof(commutativity_k2_tarski,axiom,( ! [A,B] : k2_tarski(A,B) = k2_tarski(B,A) ), file(tarski,k2_tarski), []). 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_xcmplx_0,axiom,( ! [A,B] : ( ( v1_xcmplx_0(A) & v1_xcmplx_0(B) ) => k3_xcmplx_0(A,B) = k3_xcmplx_0(B,A) ) ), file(xcmplx_0,k3_xcmplx_0), []). fof(commutativity_k4_metric_1,axiom,( ! [A,B,C] : ( ( v8_metric_1(A) & l1_metric_1(A) & m1_subset_1(B,u1_struct_0(A)) & m1_subset_1(C,u1_struct_0(A)) ) => k4_metric_1(A,B,C) = k4_metric_1(A,C,B) ) ), file(metric_1,k4_metric_1), []). fof(commutativity_k4_real_1,axiom,( ! [A,B] : ( ( m1_subset_1(A,k1_numbers) & m1_subset_1(B,k1_numbers) ) => k4_real_1(A,B) = k4_real_1(B,A) ) ), file(real_1,k4_real_1), []). fof(commutativity_k7_domain_1,axiom,( ! [A,B,C] : ( ( ~ v1_xboole_0(A) & m1_subset_1(B,A) & m1_subset_1(C,A) ) => k7_domain_1(A,B,C) = k7_domain_1(A,C,B) ) ), file(domain_1,k7_domain_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(d10_xboole_0,axiom,( ! [A,B] : ( A = B <=> ( r1_tarski(A,B) & r1_tarski(B,A) ) ) ), file(xboole_0,d10_xboole_0), []). fof(d11_finseq_1,axiom,( ! [A,B] : ( B = k13_finseq_1(A) <=> ! [C] : ( r2_hidden(C,B) <=> m2_finseq_1(C,A) ) ) ), file(finseq_1,d11_finseq_1), []). fof(d12_margrel1,axiom,( k6_margrel1 = k2_tarski(0,1) ), file(margrel1,d12_margrel1), []). fof(d1_finseq_1,axiom,( ! [A] : ( v4_ordinal2(A) => k1_finseq_1(A) = a_1_0_finseq_1(A) ) ), file(finseq_1,d1_finseq_1), []). fof(d1_tarski,axiom,( ! [A,B] : ( B = k1_tarski(A) <=> ! [C] : ( r2_hidden(C,B) <=> C = A ) ) ), file(tarski,d1_tarski), []). fof(d1_vectmetr,axiom,( ! [A] : ( ( ~ v3_struct_0(A) & l1_metric_1(A) ) => ( v1_vectmetr(A) <=> ! [B] : ( m1_subset_1(B,u1_struct_0(A)) => ! [C] : ( m1_subset_1(C,u1_struct_0(A)) => ! [D] : ( m1_subset_1(D,k1_numbers) => ~ ( r1_xreal_0(0,D) & r1_xreal_0(D,1) & ! [E] : ( m1_subset_1(E,u1_struct_0(A)) => ~ ( k2_metric_1(A,B,E) = k4_real_1(D,k2_metric_1(A,B,C)) & k2_metric_1(A,E,C) = k4_real_1(k5_real_1(1,D),k2_metric_1(A,B,C)) ) ) ) ) ) ) ) ) ), file(vectmetr,d1_vectmetr), []). fof(d2_bintree2,axiom,( ! [A] : ( ( ~ v1_xboole_0(A) & v1_trees_1(A) ) => ( v1_bintree2(A) <=> A = k3_finseq_2(k7_domain_1(k5_numbers,0,1)) ) ) ), file(bintree2,d2_bintree2), []). fof(d2_finseq_2,axiom,( ! [A] : ( v4_ordinal2(A) => ! [B] : k2_finseq_2(A,B) = k2_funcop_1(k2_finseq_1(A),B) ) ), file(finseq_2,d2_finseq_2), []). fof(d2_tarski,axiom,( ! [A,B,C] : ( C = k2_tarski(A,B) <=> ! [D] : ( r2_hidden(D,C) <=> ( D = A | D = B ) ) ) ), file(tarski,d2_tarski), []). fof(d2_zfmisc_1,axiom,( ! [A,B,C] : ( C = k2_zfmisc_1(A,B) <=> ! [D] : ( r2_hidden(D,C) <=> ? [E,F] : ( r2_hidden(E,A) & r2_hidden(F,B) & D = k4_tarski(E,F) ) ) ) ), file(zfmisc_1,d2_zfmisc_1), []). 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_tarski,axiom,( ! [A,B] : ( r1_tarski(A,B) <=> ! [C] : ( r2_hidden(C,A) => r2_hidden(C,B) ) ) ), file(tarski,d3_tarski), []). fof(d4_euclid,axiom,( ! [A] : ( m2_subset_1(A,k1_numbers,k5_numbers) => k4_euclid(A) = k4_finseqop(k1_numbers,A,0) ) ), file(euclid,d4_euclid), []). 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(d5_funct_1,axiom,( ! [A] : ( ( v1_relat_1(A) & v1_funct_1(A) ) => ! [B] : ( B = k2_relat_1(A) <=> ! [C] : ( r2_hidden(C,B) <=> ? [D] : ( r2_hidden(D,k1_relat_1(A)) & C = k1_funct_1(A,D) ) ) ) ) ), file(funct_1,d5_funct_1), []). fof(d5_real_1,axiom,( ! [A] : ( v1_xreal_0(A) => ! [B] : ( v1_xreal_0(B) => ( r1_xreal_0(A,B) <=> ~ ( r1_xreal_0(B,A) & B != A ) ) ) ) ), file(real_1,d5_real_1), []). fof(d5_tarski,axiom,( ! [A,B] : k4_tarski(A,B) = k2_tarski(k2_tarski(A,B),k1_tarski(A)) ), file(tarski,d5_tarski), []). fof(d7_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) ) => ! [C] : ( ( v1_relat_1(C) & v1_funct_1(C) & v1_finseq_1(C) ) => ( C = k7_finseq_1(A,B) <=> ( k4_finseq_1(C) = k2_finseq_1(k1_nat_1(k3_finseq_1(A),k3_finseq_1(B))) & ! [D] : ( m2_subset_1(D,k1_numbers,k5_numbers) => ( r2_hidden(D,k4_finseq_1(A)) => k1_funct_1(C,D) = k1_funct_1(A,D) ) ) & ! [D] : ( m2_subset_1(D,k1_numbers,k5_numbers) => ( r2_hidden(D,k4_finseq_1(B)) => k1_funct_1(C,k1_nat_1(k3_finseq_1(A),D)) = k1_funct_1(B,D) ) ) ) ) ) ) ) ), file(finseq_1,d7_finseq_1), []). fof(dt_k10_finseq_1,axiom,( $true ), file(finseq_1,k10_finseq_1), []). fof(dt_k11_binop_2,axiom,( ! [A,B] : ( ( m1_subset_1(A,k1_numbers) & m1_subset_1(B,k1_numbers) ) => m1_subset_1(k11_binop_2(A,B),k1_numbers) ) ), file(binop_2,k11_binop_2), []). fof(dt_k11_margrel1,axiom,( ! [A] : ( m1_subset_1(A,k6_margrel1) => m1_subset_1(k11_margrel1(A),k6_margrel1) ) ), file(margrel1,k11_margrel1), []). fof(dt_k12_binarith,axiom,( ! [A,B,C,D,E] : ( ( ~ v1_xboole_0(A) & m1_subset_1(A,k5_numbers) & m1_subset_1(B,k5_numbers) & ~ v1_xboole_0(C) & m1_subset_1(D,k4_finseq_2(A,C)) & m1_subset_1(E,k4_finseq_2(B,C)) ) => m2_finseq_2(k12_binarith(A,B,C,D,E),C,k4_finseq_2(k23_binop_2(A,B),C)) ) ), file(binarith,k12_binarith), []). 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_binarith,axiom,( ! [A,B] : ( ( ~ v1_xboole_0(A) & m1_subset_1(B,A) ) => m2_finseq_2(k13_binarith(A,B),A,k4_finseq_2(1,A)) ) ), file(binarith,k13_binarith), []). fof(dt_k13_finseq_1,axiom,( $true ), file(finseq_1,k13_finseq_1), []). fof(dt_k15_rvsum_1,axiom,( ! [A] : ( m1_finseq_1(A,k1_numbers) => m1_subset_1(k15_rvsum_1(A),k1_numbers) ) ), file(rvsum_1,k15_rvsum_1), []). fof(dt_k18_group_2,axiom,( ! [A,B] : ( ( ~ v1_xboole_0(A) & m1_subset_1(B,A) ) => m1_subset_1(k18_group_2(A,B),k5_finsub_1(A)) ) ), file(group_2,k18_group_2), []). fof(dt_k1_card_1,axiom,( ! [A] : v1_card_1(k1_card_1(A)) ), file(card_1,k1_card_1), []). fof(dt_k1_domain_1,axiom,( ! [A,B,C,D] : ( ( ~ v1_xboole_0(A) & ~ v1_xboole_0(B) & m1_subset_1(C,A) & m1_subset_1(D,B) ) => m1_subset_1(k1_domain_1(A,B,C,D),k2_zfmisc_1(A,B)) ) ), file(domain_1,k1_domain_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_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_int_1,axiom,( ! [A] : ( v1_xreal_0(A) => v1_int_1(k1_int_1(A)) ) ), file(int_1,k1_int_1), []). fof(dt_k1_mcart_1,axiom,( $true ), file(mcart_1,k1_mcart_1), []). fof(dt_k1_nat_1,axiom,( ! [A,B] : ( ( m1_subset_1(A,k5_numbers) & m1_subset_1(B,k5_numbers) ) => m2_subset_1(k1_nat_1(A,B),k1_numbers,k5_numbers) ) ), file(nat_1,k1_nat_1), []). fof(dt_k1_numbers,axiom,( $true ), file(numbers,k1_numbers), []). fof(dt_k1_relat_1,axiom,( $true ), file(relat_1,k1_relat_1), []). fof(dt_k1_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_k23_binop_2,axiom,( ! [A,B] : ( ( m1_subset_1(A,k5_numbers) & m1_subset_1(B,k5_numbers) ) => m2_subset_1(k23_binop_2(A,B),k1_numbers,k5_numbers) ) ), file(binop_2,k23_binop_2), []). fof(dt_k2_bintree2,axiom,( ! [A,B] : ( ( ~ v1_xboole_0(A) & v1_trees_1(A) & v1_bintree2(A) & ~ v1_xboole_0(B) & m1_subset_1(B,k5_numbers) ) => m1_trees_4(k2_bintree2(A,B),A,k2_trees_2(A,B)) ) ), file(bintree2,k2_bintree2), []). fof(dt_k2_domain_1,axiom,( ! [A,B,C] : ( ( ~ v1_xboole_0(A) & ~ v1_xboole_0(B) & m1_subset_1(C,k2_zfmisc_1(A,B)) ) => m1_subset_1(k2_domain_1(A,B,C),A) ) ), file(domain_1,k2_domain_1), []). 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_finseq_2,axiom,( ! [A,B] : ( v4_ordinal2(A) => ( v1_relat_1(k2_finseq_2(A,B)) & v1_funct_1(k2_finseq_2(A,B)) & v1_finseq_1(k2_finseq_2(A,B)) ) ) ), file(finseq_2,k2_finseq_2), []). fof(dt_k2_finseq_4,axiom,( ! [A,B,C] : ( ( ~ v1_xboole_0(A) & m1_subset_1(B,A) & m1_subset_1(C,A) ) => m2_finseq_1(k2_finseq_4(A,B,C),A) ) ), file(finseq_4,k2_finseq_4), []). fof(dt_k2_funcop_1,axiom,( $true ), file(funcop_1,k2_funcop_1), []). fof(dt_k2_int_1,axiom,( ! [A] : ( v1_xreal_0(A) => v1_int_1(k2_int_1(A)) ) ), file(int_1,k2_int_1), []). fof(dt_k2_mcart_1,axiom,( $true ), file(mcart_1,k2_mcart_1), []). fof(dt_k2_metric_1,axiom,( ! [A,B,C] : ( ( l1_metric_1(A) & m1_subset_1(B,u1_struct_0(A)) & m1_subset_1(C,u1_struct_0(A)) ) => m1_subset_1(k2_metric_1(A,B,C),k1_numbers) ) ), file(metric_1,k2_metric_1), []). fof(dt_k2_nat_1,axiom,( ! [A,B] : ( ( m1_subset_1(A,k5_numbers) & m1_subset_1(B,k5_numbers) ) => m2_subset_1(k2_nat_1(A,B),k1_numbers,k5_numbers) ) ), file(nat_1,k2_nat_1), []). fof(dt_k2_relat_1,axiom,( $true ), file(relat_1,k2_relat_1), []). fof(dt_k2_tarski,axiom,( $true ), file(tarski,k2_tarski), []). fof(dt_k2_trees_2,axiom,( ! [A,B] : ( ( ~ v1_xboole_0(A) & v1_trees_1(A) & m1_subset_1(B,k5_numbers) ) => m2_trees_2(k2_trees_2(A,B),A) ) ), file(trees_2,k2_trees_2), []). 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_finseq_2,axiom,( ! [A] : ( ~ v1_xboole_0(k3_finseq_2(A)) & m1_finseq_2(k3_finseq_2(A),A) ) ), file(finseq_2,k3_finseq_2), []). fof(dt_k3_nat_1,axiom,( ! [A,B] : ( ( v4_ordinal2(A) & v4_ordinal2(B) ) => m2_subset_1(k3_nat_1(A,B),k1_numbers,k5_numbers) ) ), file(nat_1,k3_nat_1), []). fof(dt_k3_power,axiom,( ! [A,B] : ( ( v1_xreal_0(A) & v1_xreal_0(B) ) => v1_xreal_0(k3_power(A,B)) ) ), file(power,k3_power), []). 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_series_1,axiom,( ! [A,B] : ( ( m1_subset_1(A,k5_numbers) & m1_subset_1(B,k5_numbers) ) => m2_subset_1(k3_series_1(A,B),k1_numbers,k5_numbers) ) ), file(series_1,k3_series_1), []). fof(dt_k3_trees_2,axiom,( ! [A,B,C] : ( ( ~ v1_xboole_0(A) & v1_funct_1(B) & v3_trees_2(B) & m3_trees_2(B,A) & m1_subset_1(C,k1_relat_1(B)) ) => m1_subset_1(k3_trees_2(A,B,C),A) ) ), file(trees_2,k3_trees_2), []). fof(dt_k3_xcmplx_0,axiom,( $true ), file(xcmplx_0,k3_xcmplx_0), []). fof(dt_k4_euclid,axiom,( ! [A] : ( m1_subset_1(A,k5_numbers) => m2_finseq_1(k4_euclid(A),k1_numbers) ) ), file(euclid,k4_euclid), []). 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_finseqop,axiom,( ! [A,B,C] : ( ( ~ v1_xboole_0(A) & m1_subset_1(B,k5_numbers) & m1_subset_1(C,A) ) => m2_finseq_2(k4_finseqop(A,B,C),A,k4_finseq_2(B,A)) ) ), file(finseqop,k4_finseqop), []). fof(dt_k4_metric_1,axiom,( ! [A,B,C] : ( ( v8_metric_1(A) & l1_metric_1(A) & m1_subset_1(B,u1_struct_0(A)) & m1_subset_1(C,u1_struct_0(A)) ) => m1_subset_1(k4_metric_1(A,B,C),k1_numbers) ) ), file(metric_1,k4_metric_1), []). fof(dt_k4_nat_1,axiom,( ! [A,B] : ( ( v4_ordinal2(A) & v4_ordinal2(B) ) => m2_subset_1(k4_nat_1(A,B),k1_numbers,k5_numbers) ) ), file(nat_1,k4_nat_1), []). fof(dt_k4_real_1,axiom,( ! [A,B] : ( ( m1_subset_1(A,k1_numbers) & m1_subset_1(B,k1_numbers) ) => m1_subset_1(k4_real_1(A,B),k1_numbers) ) ), file(real_1,k4_real_1), []). fof(dt_k4_tarski,axiom,( $true ), file(tarski,k4_tarski), []). 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_euclid,axiom,( ! [A] : ( m1_subset_1(A,k5_numbers) => m2_finseq_2(k5_euclid(A),k1_numbers,k1_euclid(A)) ) ), file(euclid,k5_euclid), []). fof(dt_k5_finseq_1,axiom,( $true ), file(finseq_1,k5_finseq_1), []). fof(dt_k5_finsub_1,axiom,( ! [A] : v4_finsub_1(k5_finsub_1(A)) ), file(finsub_1,k5_finsub_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_power,axiom,( ! [A,B] : ( ( v1_xreal_0(A) & v1_xreal_0(B) ) => v1_xreal_0(k5_power(A,B)) ) ), file(power,k5_power), []). fof(dt_k5_real_1,axiom,( ! [A,B] : ( ( m1_subset_1(A,k1_numbers) & m1_subset_1(B,k1_numbers) ) => m1_subset_1(k5_real_1(A,B),k1_numbers) ) ), file(real_1,k5_real_1), []). fof(dt_k6_binarith,axiom,( ! [A,B] : ( ( m1_subset_1(A,k5_numbers) & m1_subset_1(B,k4_finseq_2(A,k6_margrel1)) ) => m2_finseq_2(k6_binarith(A,B),k6_margrel1,k4_finseq_2(A,k6_margrel1)) ) ), file(binarith,k6_binarith), []). fof(dt_k6_finseq_1,axiom,( ! [A] : ( v1_xboole_0(k6_finseq_1(A)) & m2_finseq_1(k6_finseq_1(A),A) ) ), file(finseq_1,k6_finseq_1), []). fof(dt_k6_margrel1,axiom,( $true ), file(margrel1,k6_margrel1), []). fof(dt_k6_power,axiom,( ! [A,B] : ( ( m1_subset_1(A,k1_numbers) & m1_subset_1(B,k1_numbers) ) => m1_subset_1(k6_power(A,B),k1_numbers) ) ), file(power,k6_power), []). fof(dt_k6_real_1,axiom,( ! [A,B] : ( ( m1_subset_1(A,k1_numbers) & m1_subset_1(B,k1_numbers) ) => m1_subset_1(k6_real_1(A,B),k1_numbers) ) ), file(real_1,k6_real_1), []). fof(dt_k6_xcmplx_0,axiom,( $true ), file(xcmplx_0,k6_xcmplx_0), []). fof(dt_k7_domain_1,axiom,( ! [A,B,C] : ( ( ~ v1_xboole_0(A) & m1_subset_1(B,A) & m1_subset_1(C,A) ) => m1_subset_1(k7_domain_1(A,B,C),k1_zfmisc_1(A)) ) ), file(domain_1,k7_domain_1), []). fof(dt_k7_finseq_1,axiom,( ! [A,B] : ( ( v1_relat_1(A) & v1_funct_1(A) & v1_finseq_1(A) & v1_relat_1(B) & v1_funct_1(B) & v1_finseq_1(B) ) => ( v1_relat_1(k7_finseq_1(A,B)) & v1_funct_1(k7_finseq_1(A,B)) & v1_finseq_1(k7_finseq_1(A,B)) ) ) ), file(finseq_1,k7_finseq_1), []). fof(dt_k7_margrel1,axiom,( m1_subset_1(k7_margrel1,k6_margrel1) ), file(margrel1,k7_margrel1), []). fof(dt_k7_rlvect_1,axiom,( ! [A,B,C] : ( ( ~ v3_struct_0(A) & l1_struct_0(A) & m1_subset_1(B,u1_struct_0(A)) & m1_subset_1(C,u1_struct_0(A)) ) => m2_finseq_1(k7_rlvect_1(A,B,C),u1_struct_0(A)) ) ), file(rlvect_1,k7_rlvect_1), []). fof(dt_k7_xcmplx_0,axiom,( $true ), file(xcmplx_0,k7_xcmplx_0), []). 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_k8_margrel1,axiom,( m1_subset_1(k8_margrel1,k6_margrel1) ), file(margrel1,k8_margrel1), []). 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_k9_margrel1,axiom,( ! [A] : ( v2_margrel1(A) => v2_margrel1(k9_margrel1(A)) ) ), file(margrel1,k9_margrel1), []). fof(dt_l1_metric_1,axiom,( ! [A] : ( l1_metric_1(A) => l1_struct_0(A) ) ), file(metric_1,l1_metric_1), []). fof(dt_l1_struct_0,axiom,( $true ), file(struct_0,l1_struct_0), []). fof(dt_m1_bintree2,axiom,( ! [A] : ( ( ~ v1_xboole_0(A) & v1_trees_1(A) & v1_bintree1(A) ) => ! [B] : ( m1_bintree2(B,A) => m2_finseq_1(B,k6_margrel1) ) ) ), file(bintree2,m1_bintree2), []). 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_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_m1_trees_1,axiom,( ! [A] : ( ( ~ v1_xboole_0(A) & v1_trees_1(A) ) => ! [B] : ( m1_trees_1(B,A) => m2_finseq_1(B,k5_numbers) ) ) ), file(trees_1,m1_trees_1), []). fof(dt_m1_trees_4,axiom,( ! [A,B] : ( ( ~ v1_xboole_0(A) & m1_subset_1(B,k1_zfmisc_1(A)) ) => ! [C] : ( m1_trees_4(C,A,B) => m2_finseq_1(C,A) ) ) ), file(trees_4,m1_trees_4), []). 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_m2_trees_2,axiom,( ! [A] : ( ( ~ v1_xboole_0(A) & v1_trees_1(A) ) => ! [B] : ( m2_trees_2(B,A) => m1_subset_1(B,k1_zfmisc_1(A)) ) ) ), file(trees_2,m2_trees_2), []). fof(dt_m3_trees_2,axiom,( ! [A,B] : ( m3_trees_2(B,A) => v1_relat_1(B) ) ), file(trees_2,m3_trees_2), []). 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_struct_0,axiom,( ? [A] : l1_struct_0(A) ), file(struct_0,l1_struct_0), []). fof(existence_m1_bintree2,axiom,( ! [A] : ( ( ~ v1_xboole_0(A) & v1_trees_1(A) & v1_bintree1(A) ) => ? [B] : m1_bintree2(B,A) ) ), file(bintree2,m1_bintree2), []). 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_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_m1_trees_1,axiom,( ! [A] : ( ( ~ v1_xboole_0(A) & v1_trees_1(A) ) => ? [B] : m1_trees_1(B,A) ) ), file(trees_1,m1_trees_1), []). fof(existence_m1_trees_4,axiom,( ! [A,B] : ( ( ~ v1_xboole_0(A) & m1_subset_1(B,k1_zfmisc_1(A)) ) => ? [C] : m1_trees_4(C,A,B) ) ), file(trees_4,m1_trees_4), []). 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(existence_m2_trees_2,axiom,( ! [A] : ( ( ~ v1_xboole_0(A) & v1_trees_1(A) ) => ? [B] : m2_trees_2(B,A) ) ), file(trees_2,m2_trees_2), []). fof(existence_m3_trees_2,axiom,( ! [A] : ? [B] : m3_trees_2(B,A) ), file(trees_2,m3_trees_2), []). fof(fc10_membered,axiom,( ! [A] : ( v1_int_1(A) => ( v1_membered(k1_tarski(A)) & v2_membered(k1_tarski(A)) & v3_membered(k1_tarski(A)) & v4_membered(k1_tarski(A)) ) ) ), file(membered,fc10_membered), []). fof(fc11_membered,axiom,( ! [A] : ( v4_ordinal2(A) => ( v1_membered(k1_tarski(A)) & v2_membered(k1_tarski(A)) & v3_membered(k1_tarski(A)) & v4_membered(k1_tarski(A)) & v5_membered(k1_tarski(A)) ) ) ), file(membered,fc11_membered), []). fof(fc12_membered,axiom,( ! [A,B] : ( ( v1_xcmplx_0(A) & v1_xcmplx_0(B) ) => v1_membered(k2_tarski(A,B)) ) ), file(membered,fc12_membered), []). fof(fc13_membered,axiom,( ! [A,B] : ( ( v1_xreal_0(A) & v1_xreal_0(B) ) => ( v1_membered(k2_tarski(A,B)) & v2_membered(k2_tarski(A,B)) ) ) ), file(membered,fc13_membered), []). fof(fc14_membered,axiom,( ! [A,B] : ( ( v1_rat_1(A) & v1_rat_1(B) ) => ( v1_membered(k2_tarski(A,B)) & v2_membered(k2_tarski(A,B)) & v3_membered(k2_tarski(A,B)) ) ) ), file(membered,fc14_membered), []). fof(fc15_membered,axiom,( ! [A,B] : ( ( v1_int_1(A) & v1_int_1(B) ) => ( v1_membered(k2_tarski(A,B)) & v2_membered(k2_tarski(A,B)) & v3_membered(k2_tarski(A,B)) & v4_membered(k2_tarski(A,B)) ) ) ), file(membered,fc15_membered), []). fof(fc16_membered,axiom,( ! [A,B] : ( ( v4_ordinal2(A) & v4_ordinal2(B) ) => ( v1_membered(k2_tarski(A,B)) & v2_membered(k2_tarski(A,B)) & v3_membered(k2_tarski(A,B)) & v4_membered(k2_tarski(A,B)) & v5_membered(k2_tarski(A,B)) ) ) ), file(membered,fc16_membered), []). fof(fc1_margrel1,axiom, ( v1_xboole_0(k1_xboole_0) & v1_margrel1(k1_xboole_0) ), file(margrel1,fc1_margrel1), []). 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_subset_1,axiom,( ! [A] : ~ v1_xboole_0(k1_zfmisc_1(A)) ), file(subset_1,fc1_subset_1), []). fof(fc2_bintree2,axiom,( ! [A,B] : ( ( ~ v1_xboole_0(A) & v1_trees_1(A) & v1_bintree1(A) & m1_subset_1(B,k5_numbers) ) => v1_finset_1(k2_trees_2(A,B)) ) ), file(bintree2,fc2_bintree2), []). fof(fc2_membered,axiom, ( ~ v1_xboole_0(k1_numbers) & v1_membered(k1_numbers) & v2_membered(k1_numbers) ), file(membered,fc2_membered), []). fof(fc2_nat_1,axiom,( ! [A,B] : ( ( v4_ordinal2(A) & v4_ordinal2(B) ) => ( v4_ordinal2(k3_xcmplx_0(A,B)) & v1_xcmplx_0(k3_xcmplx_0(A,B)) & v1_xreal_0(k3_xcmplx_0(A,B)) ) ) ), file(nat_1,fc2_nat_1), []). fof(fc2_subset_1,axiom,( ! [A] : ~ v1_xboole_0(k1_tarski(A)) ), file(subset_1,fc2_subset_1), []). fof(fc2_trees_2,axiom,( ! [A] : ( ( v1_relat_1(A) & v1_funct_1(A) & v3_trees_2(A) ) => ( ~ v1_xboole_0(k1_relat_1(A)) & v1_trees_1(k1_relat_1(A)) ) ) ), file(trees_2,fc2_trees_2), []). fof(fc3_bintree2,axiom,( ! [A,B] : ( ( ~ v1_xboole_0(A) & v1_trees_1(A) & v1_bintree2(A) & ~ v1_xboole_0(B) & m1_subset_1(B,k5_numbers) ) => ( v1_relat_1(k2_bintree2(A,B)) & v1_funct_1(k2_bintree2(A,B)) & v2_funct_1(k2_bintree2(A,B)) & v1_finset_1(k2_bintree2(A,B)) & v1_finseq_1(k2_bintree2(A,B)) ) ) ), file(bintree2,fc3_bintree2), []). fof(fc3_margrel1,axiom,( ~ v1_xboole_0(k6_margrel1) ), file(margrel1,fc3_margrel1), []). 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_subset_1,axiom,( ! [A,B] : ~ v1_xboole_0(k2_tarski(A,B)) ), file(subset_1,fc3_subset_1), []). fof(fc3_trees_2,axiom,( ! [A,B] : ( ( ~ v1_xboole_0(A) & v1_trees_1(A) ) => ( v1_relat_1(k2_funcop_1(A,B)) & v1_funct_1(k2_funcop_1(A,B)) & v3_trees_2(k2_funcop_1(A,B)) ) ) ), file(trees_2,fc3_trees_2), []). fof(fc4_margrel1,axiom,( ! [A] : ( v2_margrel1(A) => v2_margrel1(k9_margrel1(A)) ) ), file(margrel1,fc4_margrel1), []). 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_subset_1,axiom,( ! [A,B] : ( ( ~ v1_xboole_0(A) & ~ v1_xboole_0(B) ) => ~ v1_xboole_0(k2_zfmisc_1(A,B)) ) ), file(subset_1,fc4_subset_1), []). fof(fc5_membered,axiom, ( v1_membered(k5_ordinal2) & v2_membered(k5_ordinal2) & v3_membered(k5_ordinal2) & v4_membered(k5_ordinal2) & v5_membered(k5_ordinal2) ), file(membered,fc5_membered), []). fof(fc5_trees_2,axiom,( ! [A] : ( ( v1_relat_1(A) & v1_finset_1(A) ) => v1_finset_1(k1_relat_1(A)) ) ), file(trees_2,fc5_trees_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(fc8_membered,axiom,( ! [A] : ( v1_xreal_0(A) => ( v1_membered(k1_tarski(A)) & v2_membered(k1_tarski(A)) ) ) ), file(membered,fc8_membered), []). 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(fraenkel_a_1_0_finseq_1,axiom,( ! [A,B] : ( v4_ordinal2(B) => ( r2_hidden(A,a_1_0_finseq_1(B)) <=> ? [C] : ( m2_subset_1(C,k1_numbers,k5_numbers) & A = C & r1_xreal_0(1,C) & r1_xreal_0(C,B) ) ) ) ), file(finseq_1,a_1_0_finseq_1), []). fof(fraenkel_a_1_0_vectmetr,axiom,( ! [A,B] : ( m2_finseq_1(B,k1_numbers) => ( r2_hidden(A,a_1_0_vectmetr(B)) <=> ? [C] : ( m2_subset_1(C,k1_numbers,k5_numbers) & A = C & r1_xreal_0(1,C) & r1_xreal_0(C,k3_finseq_1(B)) ) ) ) ), file(vectmetr,a_1_0_vectmetr), []). fof(involutiveness_k11_margrel1,axiom,( ! [A] : ( m1_subset_1(A,k6_margrel1) => k11_margrel1(k11_margrel1(A)) = A ) ), file(margrel1,k11_margrel1), []). fof(involutiveness_k9_margrel1,axiom,( ! [A] : ( v2_margrel1(A) => k9_margrel1(k9_margrel1(A)) = A ) ), file(margrel1,k9_margrel1), []). fof(rc1_bintree1,axiom,( ? [A] : ( ~ v1_xboole_0(A) & v1_finset_1(A) & v1_trees_1(A) & v1_bintree1(A) ) ), file(bintree1,rc1_bintree1), []). fof(rc1_bintree2,axiom,( ? [A] : ( ~ v1_xboole_0(A) & v1_trees_1(A) & v1_trees_2(A) & v2_trees_9(A) & v1_bintree1(A) & v1_bintree2(A) ) ), file(bintree2,rc1_bintree2), []). fof(rc1_funct_1,axiom,( ? [A] : ( v1_relat_1(A) & v1_funct_1(A) ) ), file(funct_1,rc1_funct_1), []). fof(rc1_margrel1,axiom,( ? [A] : v1_margrel1(A) ), file(margrel1,rc1_margrel1), []). 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_subset_1,axiom,( ! [A] : ( ~ v1_xboole_0(A) => ? [B] : ( m1_subset_1(B,k1_zfmisc_1(A)) & ~ v1_xboole_0(B) ) ) ), file(subset_1,rc1_subset_1), []). fof(rc1_trees_2,axiom,( ? [A] : ( ~ v1_xboole_0(A) & v1_trees_1(A) & v1_trees_2(A) ) ), file(trees_2,rc1_trees_2), []). fof(rc2_bintree1,axiom,( ! [A] : ( ~ v1_xboole_0(A) => ? [B] : ( m3_trees_2(B,A) & v1_relat_1(B) & v1_funct_1(B) & v1_finset_1(B) & v3_trees_2(B) & v2_bintree1(B) ) ) ), file(bintree1,rc2_bintree1), []). fof(rc2_funct_1,axiom,( ? [A] : ( v1_relat_1(A) & v1_xboole_0(A) & v1_funct_1(A) ) ), file(funct_1,rc2_funct_1), []). fof(rc2_margrel1,axiom,( ? [A] : v2_margrel1(A) ), file(margrel1,rc2_margrel1), []). 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_subset_1,axiom,( ! [A] : ? [B] : ( m1_subset_1(B,k1_zfmisc_1(A)) & v1_xboole_0(B) ) ), file(subset_1,rc2_subset_1), []). fof(rc3_bintree1,axiom,( ? [A] : ( v1_relat_1(A) & v1_funct_1(A) & v1_finset_1(A) & v3_trees_2(A) & v2_bintree1(A) ) ), file(bintree1,rc3_bintree1), []). fof(rc3_funct_1,axiom,( ? [A] : ( v1_relat_1(A) & v1_funct_1(A) & v2_funct_1(A) ) ), file(funct_1,rc3_funct_1), []). fof(rc3_nat_1,axiom,( ? [A] : ( m1_subset_1(A,k5_numbers) & ~ v1_xboole_0(A) & v1_ordinal1(A) & v2_ordinal1(A) & v3_ordinal1(A) & v4_ordinal2(A) & v1_xcmplx_0(A) & v1_xreal_0(A) ) ), file(nat_1,rc3_nat_1), []). fof(rc3_struct_0,axiom,( ? [A] : ( l1_struct_0(A) & ~ v3_struct_0(A) ) ), file(struct_0,rc3_struct_0), []). fof(rc5_struct_0,axiom,( ! [A] : ( ( ~ v3_struct_0(A) & l1_struct_0(A) ) => ? [B] : ( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) & ~ v1_xboole_0(B) ) ) ), file(struct_0,rc5_struct_0), []). fof(rc5_trees_2,axiom,( ? [A] : ( v1_relat_1(A) & v1_funct_1(A) & v3_trees_2(A) ) ), file(trees_2,rc5_trees_2), []). fof(rc6_trees_2,axiom,( ! [A] : ( ~ v1_xboole_0(A) => ? [B] : ( m3_trees_2(B,A) & v1_relat_1(B) & v1_funct_1(B) & v3_trees_2(B) ) ) ), file(trees_2,rc6_trees_2), []). fof(rc7_trees_2,axiom,( ? [A] : ( v1_relat_1(A) & v1_funct_1(A) & v1_finset_1(A) & v3_trees_2(A) ) ), file(trees_2,rc7_trees_2), []). fof(rc8_trees_2,axiom,( ! [A] : ( ~ v1_xboole_0(A) => ? [B] : ( m3_trees_2(B,A) & v1_relat_1(B) & v1_funct_1(B) & v1_finset_1(B) & v3_trees_2(B) ) ) ), file(trees_2,rc8_trees_2), []). fof(rc9_trees_2,axiom,( ! [A,B] : ( ( ~ v1_xboole_0(A) & ~ v1_xboole_0(B) ) => ? [C] : ( m1_relset_1(C,A,B) & v1_relat_1(C) & ~ v1_xboole_0(C) ) ) ), file(trees_2,rc9_trees_2), []). fof(redefinition_k11_binop_2,axiom,( ! [A,B] : ( ( m1_subset_1(A,k1_numbers) & m1_subset_1(B,k1_numbers) ) => k11_binop_2(A,B) = k3_xcmplx_0(A,B) ) ), file(binop_2,k11_binop_2), []). fof(redefinition_k11_margrel1,axiom,( ! [A] : ( m1_subset_1(A,k6_margrel1) => k11_margrel1(A) = k9_margrel1(A) ) ), file(margrel1,k11_margrel1), []). fof(redefinition_k12_binarith,axiom,( ! [A,B,C,D,E] : ( ( ~ v1_xboole_0(A) & m1_subset_1(A,k5_numbers) & m1_subset_1(B,k5_numbers) & ~ v1_xboole_0(C) & m1_subset_1(D,k4_finseq_2(A,C)) & m1_subset_1(E,k4_finseq_2(B,C)) ) => k12_binarith(A,B,C,D,E) = k7_finseq_1(D,E) ) ), file(binarith,k12_binarith), []). 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_k13_binarith,axiom,( ! [A,B] : ( ( ~ v1_xboole_0(A) & m1_subset_1(B,A) ) => k13_binarith(A,B) = k5_finseq_1(B) ) ), file(binarith,k13_binarith), []). fof(redefinition_k18_group_2,axiom,( ! [A,B] : ( ( ~ v1_xboole_0(A) & m1_subset_1(B,A) ) => k18_group_2(A,B) = k1_tarski(B) ) ), file(group_2,k18_group_2), []). fof(redefinition_k1_domain_1,axiom,( ! [A,B,C,D] : ( ( ~ v1_xboole_0(A) & ~ v1_xboole_0(B) & m1_subset_1(C,A) & m1_subset_1(D,B) ) => k1_domain_1(A,B,C,D) = k4_tarski(C,D) ) ), file(domain_1,k1_domain_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_k23_binop_2,axiom,( ! [A,B] : ( ( m1_subset_1(A,k5_numbers) & m1_subset_1(B,k5_numbers) ) => k23_binop_2(A,B) = k2_xcmplx_0(A,B) ) ), file(binop_2,k23_binop_2), []). fof(redefinition_k2_domain_1,axiom,( ! [A,B,C] : ( ( ~ v1_xboole_0(A) & ~ v1_xboole_0(B) & m1_subset_1(C,k2_zfmisc_1(A,B)) ) => k2_domain_1(A,B,C) = k1_mcart_1(C) ) ), file(domain_1,k2_domain_1), []). 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_k2_finseq_4,axiom,( ! [A,B,C] : ( ( ~ v1_xboole_0(A) & m1_subset_1(B,A) & m1_subset_1(C,A) ) => k2_finseq_4(A,B,C) = k10_finseq_1(B,C) ) ), file(finseq_4,k2_finseq_4), []). fof(redefinition_k2_nat_1,axiom,( ! [A,B] : ( ( m1_subset_1(A,k5_numbers) & m1_subset_1(B,k5_numbers) ) => k2_nat_1(A,B) = k3_xcmplx_0(A,B) ) ), file(nat_1,k2_nat_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_finseq_2,axiom,( ! [A] : k3_finseq_2(A) = k13_finseq_1(A) ), file(finseq_2,k3_finseq_2), []). 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_series_1,axiom,( ! [A,B] : ( ( m1_subset_1(A,k5_numbers) & m1_subset_1(B,k5_numbers) ) => k3_series_1(A,B) = k3_power(A,B) ) ), file(series_1,k3_series_1), []). fof(redefinition_k3_trees_2,axiom,( ! [A,B,C] : ( ( ~ v1_xboole_0(A) & v1_funct_1(B) & v3_trees_2(B) & m3_trees_2(B,A) & m1_subset_1(C,k1_relat_1(B)) ) => k3_trees_2(A,B,C) = k1_funct_1(B,C) ) ), file(trees_2,k3_trees_2), []). 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_finseqop,axiom,( ! [A,B,C] : ( ( ~ v1_xboole_0(A) & m1_subset_1(B,k5_numbers) & m1_subset_1(C,A) ) => k4_finseqop(A,B,C) = k2_finseq_2(B,C) ) ), file(finseqop,k4_finseqop), []). fof(redefinition_k4_metric_1,axiom,( ! [A,B,C] : ( ( v8_metric_1(A) & l1_metric_1(A) & m1_subset_1(B,u1_struct_0(A)) & m1_subset_1(C,u1_struct_0(A)) ) => k4_metric_1(A,B,C) = k2_metric_1(A,B,C) ) ), file(metric_1,k4_metric_1), []). fof(redefinition_k4_real_1,axiom,( ! [A,B] : ( ( m1_subset_1(A,k1_numbers) & m1_subset_1(B,k1_numbers) ) => k4_real_1(A,B) = k3_xcmplx_0(A,B) ) ), file(real_1,k4_real_1), []). fof(redefinition_k5_euclid,axiom,( ! [A] : ( m1_subset_1(A,k5_numbers) => k5_euclid(A) = k4_euclid(A) ) ), file(euclid,k5_euclid), []). fof(redefinition_k5_numbers,axiom,( k5_numbers = k5_ordinal2 ), file(numbers,k5_numbers), []). fof(redefinition_k5_real_1,axiom,( ! [A,B] : ( ( m1_subset_1(A,k1_numbers) & m1_subset_1(B,k1_numbers) ) => k5_real_1(A,B) = k6_xcmplx_0(A,B) ) ), file(real_1,k5_real_1), []). fof(redefinition_k6_power,axiom,( ! [A,B] : ( ( m1_subset_1(A,k1_numbers) & m1_subset_1(B,k1_numbers) ) => k6_power(A,B) = k5_power(A,B) ) ), file(power,k6_power), []). fof(redefinition_k6_real_1,axiom,( ! [A,B] : ( ( m1_subset_1(A,k1_numbers) & m1_subset_1(B,k1_numbers) ) => k6_real_1(A,B) = k7_xcmplx_0(A,B) ) ), file(real_1,k6_real_1), []). fof(redefinition_k7_domain_1,axiom,( ! [A,B,C] : ( ( ~ v1_xboole_0(A) & m1_subset_1(B,A) & m1_subset_1(C,A) ) => k7_domain_1(A,B,C) = k2_tarski(B,C) ) ), file(domain_1,k7_domain_1), []). fof(redefinition_k7_rlvect_1,axiom,( ! [A,B,C] : ( ( ~ v3_struct_0(A) & l1_struct_0(A) & m1_subset_1(B,u1_struct_0(A)) & m1_subset_1(C,u1_struct_0(A)) ) => k7_rlvect_1(A,B,C) = k10_finseq_1(B,C) ) ), file(rlvect_1,k7_rlvect_1), []). 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_m1_bintree2,axiom,( ! [A] : ( ( ~ v1_xboole_0(A) & v1_trees_1(A) & v1_bintree1(A) ) => ! [B] : ( m1_bintree2(B,A) <=> m1_subset_1(B,A) ) ) ), file(bintree2,m1_bintree2), []). fof(redefinition_m1_trees_1,axiom,( ! [A] : ( ( ~ v1_xboole_0(A) & v1_trees_1(A) ) => ! [B] : ( m1_trees_1(B,A) <=> m1_subset_1(B,A) ) ) ), file(trees_1,m1_trees_1), []). fof(redefinition_m1_trees_4,axiom,( ! [A,B] : ( ( ~ v1_xboole_0(A) & m1_subset_1(B,k1_zfmisc_1(A)) ) => ! [C] : ( m1_trees_4(C,A,B) <=> m1_finseq_1(C,B) ) ) ), file(trees_4,m1_trees_4), []). 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(reflexivity_r1_nat_1,axiom,( ! [A,B] : ( ( v4_ordinal2(A) & v4_ordinal2(B) ) => r1_nat_1(A,A) ) ), file(nat_1,r1_nat_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(s1_binarith__e17_3_2_1_8__vectmetr,axiom,( ! [A,B,C,D,E] : ( ( ~ v3_struct_0(A) & v6_metric_1(A) & v7_metric_1(A) & v8_metric_1(A) & v9_metric_1(A) & l1_metric_1(A) & m1_subset_1(B,u1_struct_0(A)) & m1_subset_1(C,u1_struct_0(A)) & v1_funct_1(D) & v3_trees_2(D) & v2_bintree1(D) & m3_trees_2(D,k2_zfmisc_1(u1_struct_0(A),u1_struct_0(A))) & ~ v1_xboole_0(E) & v1_trees_1(E) & v1_bintree2(E) ) => ( ( ! [F] : ( ( ~ v1_xboole_0(F) & m2_subset_1(F,k1_numbers,k5_numbers) ) => ( r1_xreal_0(F,k3_series_1(2,1)) => ! [G] : ( m1_subset_1(G,u1_struct_0(A)) => ! [H] : ( m1_subset_1(H,u1_struct_0(A)) => ( ( G = k1_mcart_1(k1_funct_1(D,k1_funct_1(k2_bintree2(E,1),F))) & H = k2_mcart_1(k1_funct_1(D,k1_funct_1(k2_bintree2(E,1),F))) ) => k4_metric_1(A,G,H) = k6_real_1(k4_metric_1(A,B,C),k3_series_1(2,1)) ) ) ) ) ) & ! [I] : ( ( ~ v1_xboole_0(I) & m2_subset_1(I,k1_numbers,k5_numbers) ) => ( ! [J] : ( ( ~ v1_xboole_0(J) & m2_subset_1(J,k1_numbers,k5_numbers) ) => ( r1_xreal_0(J,k3_series_1(2,I)) => ! [K] : ( m1_subset_1(K,u1_struct_0(A)) => ! [L] : ( m1_subset_1(L,u1_struct_0(A)) => ( ( K = k1_mcart_1(k1_funct_1(D,k1_funct_1(k2_bintree2(E,I),J))) & L = k2_mcart_1(k1_funct_1(D,k1_funct_1(k2_bintree2(E,I),J))) ) => k4_metric_1(A,K,L) = k6_real_1(k4_metric_1(A,B,C),k3_series_1(2,I)) ) ) ) ) ) => ! [M] : ( ( ~ v1_xboole_0(M) & m2_subset_1(M,k1_numbers,k5_numbers) ) => ( r1_xreal_0(M,k3_series_1(2,k23_binop_2(I,1))) => ! [N] : ( m1_subset_1(N,u1_struct_0(A)) => ! [O] : ( m1_subset_1(O,u1_struct_0(A)) => ( ( N = k1_mcart_1(k1_funct_1(D,k1_funct_1(k2_bintree2(E,k23_binop_2(I,1)),M))) & O = k2_mcart_1(k1_funct_1(D,k1_funct_1(k2_bintree2(E,k23_binop_2(I,1)),M))) ) => k4_metric_1(A,N,O) = k6_real_1(k4_metric_1(A,B,C),k3_series_1(2,k23_binop_2(I,1))) ) ) ) ) ) ) ) ) => ! [I] : ( ( ~ v1_xboole_0(I) & m2_subset_1(I,k1_numbers,k5_numbers) ) => ! [P] : ( ( ~ v1_xboole_0(P) & m2_subset_1(P,k1_numbers,k5_numbers) ) => ( r1_xreal_0(P,k3_series_1(2,I)) => ! [Q] : ( m1_subset_1(Q,u1_struct_0(A)) => ! [R] : ( m1_subset_1(R,u1_struct_0(A)) => ( ( Q = k1_mcart_1(k1_funct_1(D,k1_funct_1(k2_bintree2(E,I),P))) & R = k2_mcart_1(k1_funct_1(D,k1_funct_1(k2_bintree2(E,I),P))) ) => k4_metric_1(A,Q,R) = k6_real_1(k4_metric_1(A,B,C),k3_series_1(2,I)) ) ) ) ) ) ) ) ) ), file(vectmetr,s1_binarith__e17_3_2_1_8__vectmetr), []). fof(s1_binarith__e31_3_2_1__vectmetr,axiom,( ! [A,B,C] : ( ( ~ v3_struct_0(A) & v6_metric_1(A) & v7_metric_1(A) & v8_metric_1(A) & v9_metric_1(A) & l1_metric_1(A) & m1_subset_1(B,u1_struct_0(A)) & v1_funct_1(C) & v3_trees_2(C) & v2_bintree1(C) & m3_trees_2(C,k2_zfmisc_1(u1_struct_0(A),u1_struct_0(A))) ) => ( ( ! [D] : ( m2_finseq_2(D,k6_margrel1,k4_finseq_2(1,k6_margrel1)) => ( D = k5_euclid(1) => k2_mcart_1(k1_funct_1(C,k6_binarith(1,D))) = B ) ) & ! [E] : ( ( ~ v1_xboole_0(E) & m2_subset_1(E,k1_numbers,k5_numbers) ) => ( ! [F] : ( m2_finseq_2(F,k6_margrel1,k4_finseq_2(E,k6_margrel1)) => ( F = k5_euclid(E) => k2_mcart_1(k1_funct_1(C,k6_binarith(E,F))) = B ) ) => ! [G] : ( m2_finseq_2(G,k6_margrel1,k4_finseq_2(k23_binop_2(E,1),k6_margrel1)) => ( G = k5_euclid(k23_binop_2(E,1)) => k2_mcart_1(k1_funct_1(C,k6_binarith(k23_binop_2(E,1),G))) = B ) ) ) ) ) => ! [E] : ( ( ~ v1_xboole_0(E) & m2_subset_1(E,k1_numbers,k5_numbers) ) => ! [H] : ( m2_finseq_2(H,k6_margrel1,k4_finseq_2(E,k6_margrel1)) => ( H = k5_euclid(E) => k2_mcart_1(k1_funct_1(C,k6_binarith(E,H))) = B ) ) ) ) ) ), file(vectmetr,s1_binarith__e31_3_2_1__vectmetr), []). fof(s1_binarith__e8_3_2_1_8_2_1_1__vectmetr,axiom,( ! [A,B,C] : ( ( ~ v3_struct_0(A) & v6_metric_1(A) & v7_metric_1(A) & v8_metric_1(A) & v9_metric_1(A) & l1_metric_1(A) & v1_funct_1(B) & v3_trees_2(B) & v2_bintree1(B) & m3_trees_2(B,k2_zfmisc_1(u1_struct_0(A),u1_struct_0(A))) & ~ v1_xboole_0(C) & v1_trees_1(C) & v1_bintree2(C) ) => ( ( ! [D] : ( m2_subset_1(D,k1_numbers,k5_numbers) => ( ( r1_xreal_0(1,D) & r1_xreal_0(D,k5_real_1(k3_series_1(2,1),1)) ) => k1_mcart_1(k1_funct_1(B,k1_funct_1(k2_bintree2(C,1),k1_nat_1(D,1)))) = k2_mcart_1(k1_funct_1(B,k1_funct_1(k2_bintree2(C,1),D))) ) ) & ! [E] : ( ( ~ v1_xboole_0(E) & m2_subset_1(E,k1_numbers,k5_numbers) ) => ( ! [F] : ( m2_subset_1(F,k1_numbers,k5_numbers) => ( ( r1_xreal_0(1,F) & r1_xreal_0(F,k5_real_1(k3_series_1(2,E),1)) ) => k1_mcart_1(k1_funct_1(B,k1_funct_1(k2_bintree2(C,E),k1_nat_1(F,1)))) = k2_mcart_1(k1_funct_1(B,k1_funct_1(k2_bintree2(C,E),F))) ) ) => ! [G] : ( m2_subset_1(G,k1_numbers,k5_numbers) => ( ( r1_xreal_0(1,G) & r1_xreal_0(G,k5_real_1(k3_series_1(2,k23_binop_2(E,1)),1)) ) => k1_mcart_1(k1_funct_1(B,k1_funct_1(k2_bintree2(C,k23_binop_2(E,1)),k1_nat_1(G,1)))) = k2_mcart_1(k1_funct_1(B,k1_funct_1(k2_bintree2(C,k23_binop_2(E,1)),G))) ) ) ) ) ) => ! [E] : ( ( ~ v1_xboole_0(E) & m2_subset_1(E,k1_numbers,k5_numbers) ) => ! [H] : ( m2_subset_1(H,k1_numbers,k5_numbers) => ( ( r1_xreal_0(1,H) & r1_xreal_0(H,k5_real_1(k3_series_1(2,E),1)) ) => k1_mcart_1(k1_funct_1(B,k1_funct_1(k2_bintree2(C,E),k1_nat_1(H,1)))) = k2_mcart_1(k1_funct_1(B,k1_funct_1(k2_bintree2(C,E),H))) ) ) ) ) ) ), file(vectmetr,s1_binarith__e8_3_2_1_8_2_1_1__vectmetr), []). fof(s1_bintree2__e4_3__vectmetr,axiom,( ! [A,B,C] : ( ( ~ v3_struct_0(A) & v6_metric_1(A) & v7_metric_1(A) & v8_metric_1(A) & v9_metric_1(A) & l1_metric_1(A) & m1_subset_1(B,u1_struct_0(A)) & m1_subset_1(C,u1_struct_0(A)) ) => ( ! [D] : ( m1_subset_1(D,k2_zfmisc_1(u1_struct_0(A),u1_struct_0(A))) => ? [E] : ( m1_subset_1(E,k2_zfmisc_1(u1_struct_0(A),u1_struct_0(A))) & ? [F] : ( m1_subset_1(F,k2_zfmisc_1(u1_struct_0(A),u1_struct_0(A))) & ! [G] : ( m1_subset_1(G,u1_struct_0(A)) => ! [H] : ( m1_subset_1(H,u1_struct_0(A)) => ~ ( D = k1_domain_1(u1_struct_0(A),u1_struct_0(A),G,H) & ! [I] : ( m1_subset_1(I,u1_struct_0(A)) => ~ ( E = k1_domain_1(u1_struct_0(A),u1_struct_0(A),G,I) & F = k1_domain_1(u1_struct_0(A),u1_struct_0(A),I,H) & k4_metric_1(A,G,H) = k4_real_1(2,k4_metric_1(A,G,I)) & k4_metric_1(A,G,H) = k4_real_1(2,k4_metric_1(A,I,H)) ) ) ) ) ) ) ) ) => ? [D] : ( v1_funct_1(D) & v3_trees_2(D) & v2_bintree1(D) & m3_trees_2(D,k2_zfmisc_1(u1_struct_0(A),u1_struct_0(A))) & k1_relat_1(D) = k3_finseq_2(k7_domain_1(k5_numbers,0,1)) & k1_funct_1(D,k1_xboole_0) = k1_domain_1(u1_struct_0(A),u1_struct_0(A),B,C) & ! [E] : ( m1_trees_1(E,k1_relat_1(D)) => ! [J] : ( m1_subset_1(J,u1_struct_0(A)) => ! [K] : ( m1_subset_1(K,u1_struct_0(A)) => ~ ( k3_trees_2(k2_zfmisc_1(u1_struct_0(A),u1_struct_0(A)),D,E) = k1_domain_1(u1_struct_0(A),u1_struct_0(A),J,K) & ! [L] : ( m1_subset_1(L,u1_struct_0(A)) => ~ ( k1_funct_1(D,k8_finseq_1(k5_numbers,E,k13_binarith(k5_numbers,0))) = k1_domain_1(u1_struct_0(A),u1_struct_0(A),J,L) & k1_funct_1(D,k8_finseq_1(k5_numbers,E,k13_binarith(k5_numbers,1))) = k1_domain_1(u1_struct_0(A),u1_struct_0(A),L,K) & k4_metric_1(A,J,K) = k4_real_1(2,k4_metric_1(A,J,L)) & k4_metric_1(A,J,K) = k4_real_1(2,k4_metric_1(A,L,K)) ) ) ) ) ) ) ) ) ) ), file(vectmetr,s1_bintree2__e4_3__vectmetr), []). fof(s1_finseq_2__e19_3_2_1__vectmetr,axiom,( ! [A,B,C,D] : ( ( ~ v3_struct_0(A) & v6_metric_1(A) & v7_metric_1(A) & v8_metric_1(A) & v9_metric_1(A) & l1_metric_1(A) & v1_funct_1(B) & v3_trees_2(B) & v2_bintree1(B) & m3_trees_2(B,k2_zfmisc_1(u1_struct_0(A),u1_struct_0(A))) & ~ v1_xboole_0(C) & m2_subset_1(C,k1_numbers,k5_numbers) & m2_finseq_1(D,k1_relat_1(B)) ) => ? [E] : ( m2_finseq_1(E,k2_zfmisc_1(u1_struct_0(A),u1_struct_0(A))) & k3_finseq_1(E) = C & ! [F] : ( m2_subset_1(F,k1_numbers,k5_numbers) => ( r2_hidden(F,k2_finseq_1(C)) => k1_funct_1(E,F) = k3_trees_2(k2_zfmisc_1(u1_struct_0(A),u1_struct_0(A)),B,k4_finseq_4(k5_numbers,k1_relat_1(B),D,F)) ) ) ) ) ), file(vectmetr,s1_finseq_2__e19_3_2_1__vectmetr), []). fof(s1_finseq_2__e22_3_2_1__vectmetr,axiom,( ! [A,B,C] : ( ( ~ v3_struct_0(A) & v6_metric_1(A) & v7_metric_1(A) & v8_metric_1(A) & v9_metric_1(A) & l1_metric_1(A) & ~ v1_xboole_0(B) & m2_subset_1(B,k1_numbers,k5_numbers) & m2_finseq_1(C,k2_zfmisc_1(u1_struct_0(A),u1_struct_0(A))) ) => ? [D] : ( m2_finseq_1(D,u1_struct_0(A)) & k3_finseq_1(D) = B & ! [E] : ( m2_subset_1(E,k1_numbers,k5_numbers) => ( r2_hidden(E,k2_finseq_1(B)) => k1_funct_1(D,E) = k2_domain_1(u1_struct_0(A),u1_struct_0(A),k4_finseq_4(k5_numbers,k2_zfmisc_1(u1_struct_0(A),u1_struct_0(A)),C,E)) ) ) ) ) ), file(vectmetr,s1_finseq_2__e22_3_2_1__vectmetr), []). fof(s1_nat_1__e28_3_2_1__vectmetr,axiom,( ! [A,B,C] : ( ( ~ v3_struct_0(A) & v6_metric_1(A) & v7_metric_1(A) & v8_metric_1(A) & v9_metric_1(A) & l1_metric_1(A) & m1_subset_1(B,u1_struct_0(A)) & v1_funct_1(C) & v3_trees_2(C) & v2_bintree1(C) & m3_trees_2(C,k2_zfmisc_1(u1_struct_0(A),u1_struct_0(A))) ) => ( ( k1_mcart_1(k1_funct_1(C,k5_euclid(0))) = B & ! [D] : ( m2_subset_1(D,k1_numbers,k5_numbers) => ( k1_mcart_1(k1_funct_1(C,k5_euclid(D))) = B => k1_mcart_1(k1_funct_1(C,k5_euclid(k1_nat_1(D,1)))) = B ) ) ) => ! [D] : ( m2_subset_1(D,k1_numbers,k5_numbers) => k1_mcart_1(k1_funct_1(C,k5_euclid(D))) = B ) ) ) ), file(vectmetr,s1_nat_1__e28_3_2_1__vectmetr), []). 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(t103_rvsum_1,axiom,( ! [A] : ( m1_subset_1(A,k1_numbers) => k15_rvsum_1(k12_finseq_1(k1_numbers,A)) = A ) ), file(rvsum_1,t103_rvsum_1), []). fof(t109_finseq_2,axiom,( ! [A] : ( m2_subset_1(A,k1_numbers,k5_numbers) => ! [B] : ( ~ v1_xboole_0(B) => ! [C] : ( m2_finseq_2(C,B,k4_finseq_2(A,B)) => k3_finseq_1(C) = A ) ) ) ), file(finseq_2,t109_finseq_2), []). fof(t10_bintree2,axiom,( ! [A] : ( ( ~ v1_xboole_0(A) & v1_trees_1(A) ) => ( A = k3_finseq_2(k7_domain_1(k5_numbers,0,1)) => ! [B] : ( m2_subset_1(B,k1_numbers,k5_numbers) => r2_hidden(k5_euclid(B),k2_trees_2(A,B)) ) ) ) ), file(bintree2,t10_bintree2), []). fof(t110_finseq_2,axiom,( ! [A,B] : ( m2_finseq_1(B,A) => m1_subset_1(B,k4_finseq_2(k3_finseq_1(B),A)) ) ), file(finseq_2,t110_finseq_2), []). fof(t110_rvsum_1,axiom,( ! [A] : ( m2_subset_1(A,k1_numbers,k5_numbers) => ! [B] : ( m1_subset_1(B,k1_numbers) => k15_rvsum_1(k4_finseqop(k1_numbers,A,B)) = k11_binop_2(A,B) ) ) ), file(rvsum_1,t110_rvsum_1), []). fof(t11_binari_2,axiom,( ! [A] : ( ( ~ v1_xboole_0(A) & m2_subset_1(A,k1_numbers,k5_numbers) ) => ! [B] : ( m2_finseq_2(B,k6_margrel1,k4_finseq_2(A,k6_margrel1)) => ! [C] : ( m1_subset_1(C,k6_margrel1) => k6_binarith(k23_binop_2(A,1),k12_binarith(A,1,k6_margrel1,B,k13_binarith(k6_margrel1,C))) = k12_binarith(A,1,k6_margrel1,k6_binarith(A,B),k13_binarith(k6_margrel1,k11_margrel1(C))) ) ) ) ), file(binari_2,t11_binari_2), []). 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_pre_ff,axiom,( ! [A] : ( v1_xreal_0(A) => ! [B] : ( v1_xreal_0(B) => ! [C] : ( v1_xreal_0(C) => ( r1_xreal_0(B,C) => ( r1_xreal_0(A,1) | r1_xreal_0(B,0) | r1_xreal_0(k5_power(A,B),k5_power(A,C)) ) ) ) ) ) ), file(pre_ff,t12_pre_ff), []). fof(t137_finseq_2,axiom,( ! [A] : ( m2_subset_1(A,k1_numbers,k5_numbers) => ! [B] : ( ~ v1_xboole_0(B) => ! [C] : ( m2_finseq_2(C,B,k4_finseq_2(k1_nat_1(A,1),B)) => ? [D] : ( m2_finseq_2(D,B,k4_finseq_2(A,B)) & ? [E] : ( m1_subset_1(E,B) & C = k8_finseq_1(B,D,k12_finseq_1(B,E)) ) ) ) ) ) ), file(finseq_2,t137_finseq_2), []). fof(t13_finseq_2,axiom,( ! [A] : ( m2_subset_1(A,k1_numbers,k5_numbers) => ! [B,C] : ( m2_finseq_1(C,B) => ( r2_hidden(A,k4_finseq_1(C)) => r2_hidden(k1_funct_1(C,A),B) ) ) ) ), file(finseq_2,t13_finseq_2), []). fof(t143_real_2,axiom,( ! [A] : ( v1_xreal_0(A) => ! [B] : ( v1_xreal_0(B) => ( ( ( ~ r1_xreal_0(A,0) & r1_xreal_0(A,B) ) | ( r1_xreal_0(B,A) & ~ r1_xreal_0(0,A) ) ) => ( r1_xreal_0(k7_xcmplx_0(A,B),1) & r1_xreal_0(1,k7_xcmplx_0(B,A)) ) ) ) ) ), file(real_2,t143_real_2), []). fof(t14_nat_2,axiom,( ! [A] : ( v4_ordinal2(A) => ! [B] : ( v4_ordinal2(B) => ~ ( ~ r1_xreal_0(B,0) & k3_nat_1(A,B) = 0 & r1_xreal_0(B,A) ) ) ) ), file(nat_2,t14_nat_2), []). fof(t15_binari_3,axiom,( k6_binarith(1,k13_binarith(k6_margrel1,k7_margrel1)) = k13_binarith(k6_margrel1,k8_margrel1) ), file(binari_3,t15_binari_3), []). fof(t15_bintree2,axiom,( ! [A] : ( ( ~ v1_xboole_0(A) & v1_trees_1(A) & v1_bintree2(A) ) => ! [B] : ( ( ~ v1_xboole_0(B) & m2_subset_1(B,k1_numbers,k5_numbers) ) => k1_funct_1(k2_bintree2(A,B),1) = k5_euclid(B) ) ) ), file(bintree2,t15_bintree2), []). fof(t15_finseq_5,axiom,( ! [A] : ( ~ v1_xboole_0(A) => ! [B] : ( m2_finseq_1(B,A) => ( k3_finseq_1(B) = 1 => B = k12_finseq_1(A,k4_finseq_4(k5_numbers,A,B,1)) ) ) ) ), file(finseq_5,t15_finseq_5), []). fof(t15_funcop_1,axiom,( ! [A] : ( ( v1_relat_1(A) & v1_funct_1(A) ) => ! [B] : ( k2_relat_1(A) = k1_tarski(B) => A = k2_funcop_1(k1_relat_1(A),B) ) ) ), file(funcop_1,t15_funcop_1), []). fof(t15_nat_2,axiom,( ! [A] : ( v4_ordinal2(A) => ! [B] : ( v4_ordinal2(B) => ( r1_xreal_0(A,B) => ( r1_xreal_0(A,0) | r1_xreal_0(1,k3_nat_1(B,A)) ) ) ) ) ), file(nat_2,t15_nat_2), []). fof(t16_bintree2,axiom,( ! [A] : ( ( ~ v1_xboole_0(A) & v1_trees_1(A) & v1_bintree2(A) ) => ! [B] : ( ( ~ v1_xboole_0(B) & m2_subset_1(B,k1_numbers,k5_numbers) ) => ! [C] : ( m2_finseq_2(C,k6_margrel1,k4_finseq_2(B,k6_margrel1)) => ( C = k5_euclid(B) => k1_funct_1(k2_bintree2(A,B),k3_series_1(2,B)) = k6_binarith(B,C) ) ) ) ) ), file(bintree2,t16_bintree2), []). fof(t17_nat_2,axiom,( ! [A] : ( v4_ordinal2(A) => ! [B] : ( v4_ordinal2(B) => ! [C] : ( v4_ordinal2(C) => ( ( r1_nat_1(A,B) & r1_xreal_0(1,B) & r1_xreal_0(1,C) & r1_xreal_0(C,A) ) => k3_nat_1(k5_binarith(B,C),A) = k6_xcmplx_0(k3_nat_1(B,A),1) ) ) ) ) ), file(nat_2,t17_nat_2), []). fof(t19_bintree2,axiom,( ! [A] : ( ( ~ v1_xboole_0(A) & v1_trees_1(A) & v1_bintree2(A) ) => ! [B] : ( ( ~ v1_xboole_0(B) & m2_subset_1(B,k1_numbers,k5_numbers) ) => k3_finseq_1(k2_bintree2(A,B)) = k3_series_1(2,B) ) ) ), file(bintree2,t19_bintree2), []). fof(t19_nat_1,axiom,( ! [A] : ( v4_ordinal2(A) => ~ ( 0 != A & r1_xreal_0(A,0) ) ) ), file(nat_1,t19_nat_1), []). fof(t1_arithm,axiom,( ! [A] : ( v1_xcmplx_0(A) => k2_xcmplx_0(A,0) = A ) ), file(arithm,t1_arithm), []). fof(t1_numerals,axiom,( m1_subset_1(0,k5_numbers) ), file(numerals,t1_numerals), []). fof(t1_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_bintree2,axiom,( ! [A] : ( ( ~ v1_xboole_0(A) & v1_trees_1(A) & v1_bintree2(A) ) => ! [B] : ( ( ~ v1_xboole_0(B) & m2_subset_1(B,k1_numbers,k5_numbers) ) => k4_finseq_1(k2_bintree2(A,B)) = k2_finseq_1(k3_series_1(2,B)) ) ) ), file(bintree2,t20_bintree2), []). fof(t20_finseq_2,axiom,( ! [A,B,C] : ( ( v1_relat_1(C) & v1_funct_1(C) & v1_finseq_1(C) ) => ! [D] : ( ( v1_relat_1(D) & v1_funct_1(D) & v1_finseq_1(D) ) => ( k7_finseq_1(C,k9_finseq_1(A)) = k7_finseq_1(D,k9_finseq_1(B)) => ( C = D & A = B ) ) ) ) ), file(finseq_2,t20_finseq_2), []). fof(t20_nat_2,axiom,( ! [A] : ( m2_subset_1(A,k1_numbers,k5_numbers) => ( ~ r1_xreal_0(A,0) => ( k4_nat_1(A,2) = 0 <=> k4_nat_1(k5_binarith(A,1),2) = 1 ) ) ) ), file(nat_2,t20_nat_2), []). fof(t21_xreal_1,axiom,( ! [A] : ( v1_xreal_0(A) => ! [B] : ( v1_xreal_0(B) => ! [C] : ( v1_xreal_0(C) => ( r1_xreal_0(k2_xcmplx_0(A,B),C) <=> r1_xreal_0(A,k6_xcmplx_0(C,B)) ) ) ) ) ), file(xreal_1,t21_xreal_1), []). fof(t22_bintree2,axiom,( ! [A] : ( ( ~ v1_xboole_0(A) & v1_trees_1(A) & v1_bintree2(A) ) => k1_funct_1(k2_bintree2(A,1),1) = k13_binarith(k5_numbers,0) ) ), file(bintree2,t22_bintree2), []). fof(t23_bintree2,axiom,( ! [A] : ( ( ~ v1_xboole_0(A) & v1_trees_1(A) & v1_bintree2(A) ) => k1_funct_1(k2_bintree2(A,1),2) = k13_binarith(k5_numbers,1) ) ), file(bintree2,t23_bintree2), []). fof(t23_mcart_1,axiom,( ! [A,B,C] : ( r2_hidden(A,k2_zfmisc_1(B,C)) => A = k4_tarski(k1_mcart_1(A),k2_mcart_1(A)) ) ), file(mcart_1,t23_mcart_1), []). fof(t23_nat_2,axiom,( ! [A] : ( m2_subset_1(A,k1_numbers,k5_numbers) => ( v1_abian(A) <=> k4_nat_1(A,2) = 0 ) ) ), file(nat_2,t23_nat_2), []). fof(t24_bintree2,axiom,( ! [A] : ( ( ~ v1_xboole_0(A) & v1_trees_1(A) & v1_bintree2(A) ) => ! [B] : ( ( ~ v1_xboole_0(B) & m2_subset_1(B,k1_numbers,k5_numbers) ) => ! [C] : ( ( ~ v1_xboole_0(C) & m2_subset_1(C,k1_numbers,k5_numbers) ) => ( r1_xreal_0(C,k3_series_1(2,k1_nat_1(B,1))) => ! [D] : ( m2_finseq_2(D,k6_margrel1,k4_finseq_2(B,k6_margrel1)) => ( D = k1_funct_1(k2_bintree2(A,B),k3_nat_1(k1_nat_1(C,1),2)) => k1_funct_1(k2_bintree2(A,k1_nat_1(B,1)),C) = k7_finseq_1(D,k13_binarith(k5_numbers,k4_nat_1(k1_nat_1(C,1),2))) ) ) ) ) ) ) ), file(bintree2,t24_bintree2), []). 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_nat_2,axiom,( ! [A] : ( m2_subset_1(A,k1_numbers,k5_numbers) => ( ~ v1_abian(A) <=> k4_nat_1(A,2) = 1 ) ) ), file(nat_2,t24_nat_2), []). fof(t26_finseq_4,axiom,( ! [A] : ( ~ v1_xboole_0(A) => ! [B] : ( m1_subset_1(B,A) => ! [C] : ( m1_subset_1(C,A) => ( k4_finseq_4(k5_numbers,A,k2_finseq_4(A,B,C),1) = B & k4_finseq_4(k5_numbers,A,k2_finseq_4(A,B,C),2) = C ) ) ) ) ), file(finseq_4,t26_finseq_4), []). fof(t27_nat_2,axiom,( ! [A] : ( m2_subset_1(A,k1_numbers,k5_numbers) => ! [B] : ( m2_subset_1(B,k1_numbers,k5_numbers) => ( r1_xreal_0(B,k2_nat_1(2,A)) => r1_xreal_0(k3_nat_1(k1_nat_1(B,1),2),A) ) ) ) ), file(nat_2,t27_nat_2), []). fof(t28_nat_2,axiom,( ! [A] : ( ( v1_abian(A) & m2_subset_1(A,k1_numbers,k5_numbers) ) => k3_nat_1(A,2) = k3_nat_1(k1_nat_1(A,1),2) ) ), file(nat_2,t28_nat_2), []). 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_arithm,axiom,( ! [A] : ( v1_xcmplx_0(A) => k3_xcmplx_0(A,0) = 0 ) ), file(arithm,t2_arithm), []). 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(t30_power,axiom,( ! [A] : ( v1_xreal_0(A) => k3_power(A,1) = A ) ), file(power,t30_power), []). fof(t32_power,axiom,( ! [A] : ( v1_xreal_0(A) => ! [B] : ( v1_xreal_0(B) => ! [C] : ( v1_xreal_0(C) => ( ~ r1_xreal_0(A,0) => k3_power(A,k2_xcmplx_0(B,C)) = k3_xcmplx_0(k3_power(A,B),k3_power(A,C)) ) ) ) ) ), file(power,t32_power), []). fof(t33_zfmisc_1,axiom,( ! [A,B,C,D] : ( k4_tarski(A,B) = k4_tarski(C,D) => ( A = C & B = D ) ) ), file(zfmisc_1,t33_zfmisc_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(t36_margrel1,axiom, ( k7_margrel1 = 0 & k8_margrel1 = 1 ), file(margrel1,t36_margrel1), []). 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_nat_1,axiom,( ! [A] : ( v4_ordinal2(A) => ( ~ r1_xreal_0(1,A) => A = 0 ) ) ), file(nat_1,t39_nat_1), []). fof(t39_power,axiom,( ! [A] : ( v1_xreal_0(A) => ! [B] : ( v1_xreal_0(B) => ~ ( ~ r1_xreal_0(A,0) & r1_xreal_0(k3_power(A,B),0) ) ) ) ), file(power,t39_power), []). fof(t3_arithm,axiom,( ! [A] : ( v1_xcmplx_0(A) => k3_xcmplx_0(1,A) = A ) ), file(arithm,t3_arithm), []). 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_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(t41_margrel1,axiom,( ! [A] : ( v2_margrel1(A) => ( ( A = k7_margrel1 => k9_margrel1(A) = k8_margrel1 ) & ( k9_margrel1(A) = k8_margrel1 => A = k7_margrel1 ) & ( A = k8_margrel1 => k9_margrel1(A) = k7_margrel1 ) & ( k9_margrel1(A) = k7_margrel1 => A = k8_margrel1 ) ) ) ), file(margrel1,t41_margrel1), []). fof(t47_finseq_1,axiom,( ! [A] : ( ( v1_relat_1(A) & v1_funct_1(A) & v1_finseq_1(A) ) => ( k7_finseq_1(A,k1_xboole_0) = A & k7_finseq_1(k1_xboole_0,A) = A ) ) ), file(finseq_1,t47_finseq_1), []). fof(t47_nat_1,axiom,( ! [A] : ( v4_ordinal2(A) => ! [B] : ( v4_ordinal2(B) => ( ~ r1_xreal_0(A,0) => B = k2_xcmplx_0(k3_xcmplx_0(A,k3_nat_1(B,A)),k4_nat_1(B,A)) ) ) ) ), file(nat_1,t47_nat_1), []). fof(t49_nat_1,axiom,( ! [A] : ( v4_ordinal2(A) => ! [B] : ( v4_ordinal2(B) => ( r1_nat_1(A,B) <=> B = k3_xcmplx_0(A,k3_nat_1(B,A)) ) ) ) ), file(nat_1,t49_nat_1), []). fof(t4_arithm,axiom,( ! [A] : ( v1_xcmplx_0(A) => k6_xcmplx_0(A,0) = A ) ), file(arithm,t4_arithm), []). fof(t4_binari_3,axiom,( ! [A] : ( m2_subset_1(A,k1_numbers,k5_numbers) => k5_euclid(k1_nat_1(A,1)) = k8_finseq_1(k1_numbers,k5_euclid(A),k13_binarith(k1_numbers,0)) ) ), file(binari_3,t4_binari_3), []). fof(t4_finseq_1,axiom, ( k2_finseq_1(0) = k1_xboole_0 & k2_finseq_1(1) = k1_tarski(1) & k2_finseq_1(2) = k2_tarski(1,2) ), file(finseq_1,t4_finseq_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(t50_binarith,axiom,( ! [A] : ( v4_ordinal2(A) => ! [B] : ( v4_ordinal2(B) => ( r1_xreal_0(A,B) => k5_binarith(B,A) = k6_xcmplx_0(B,A) ) ) ) ), file(binarith,t50_binarith), []). fof(t52_int_1,axiom,( ! [A] : ( v1_xreal_0(A) => ~ r1_xreal_0(k2_xcmplx_0(k1_int_1(A),1),A) ) ), file(int_1,t52_int_1), []). 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(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(t59_finseq_1,axiom,( ! [A,B] : ( ( v1_relat_1(B) & v1_funct_1(B) & v1_finseq_1(B) ) => k1_funct_1(k7_finseq_1(B,k9_finseq_1(A)),k1_nat_1(k3_finseq_1(B),1)) = A ) ), file(finseq_1,t59_finseq_1), []). fof(t59_power,axiom,( ! [A] : ( v1_xreal_0(A) => ~ ( ~ r1_xreal_0(A,0) & A != 1 & k5_power(A,1) != 0 ) ) ), file(power,t59_power), []). fof(t5_arithm,axiom,( ! [A] : ( v1_xcmplx_0(A) => k7_xcmplx_0(0,A) = 0 ) ), file(arithm,t5_arithm), []). fof(t5_binari_3,axiom,( ! [A] : ( m2_subset_1(A,k1_numbers,k5_numbers) => r2_hidden(k5_euclid(A),k13_finseq_1(k6_margrel1)) ) ), file(binari_3,t5_binari_3), []). fof(t5_bintree2,axiom,( ! [A] : ( ( ~ v1_xboole_0(A) & v1_trees_1(A) & v1_bintree1(A) ) => ! [B] : ( m2_subset_1(B,k1_numbers,k5_numbers) => ! [C] : ( m1_bintree2(C,A) => ( r2_hidden(C,k2_trees_2(A,B)) => m2_finseq_2(C,k6_margrel1,k4_finseq_2(B,k6_margrel1)) ) ) ) ) ), file(bintree2,t5_bintree2), []). 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(t60_power,axiom,( ! [A] : ( v1_xreal_0(A) => ~ ( ~ r1_xreal_0(A,0) & A != 1 & k5_power(A,A) != 1 ) ) ), file(power,t60_power), []). fof(t61_finseq_1,axiom,( ! [A,B,C] : ( ( v1_relat_1(C) & v1_funct_1(C) & v1_finseq_1(C) ) => ( C = k10_finseq_1(A,B) <=> ( k3_finseq_1(C) = 2 & k1_funct_1(C,1) = A & k1_funct_1(C,2) = B ) ) ) ), file(finseq_1,t61_finseq_1), []). fof(t62_nat_1,axiom,( ! [A] : ( v4_ordinal2(A) => ( k4_nat_1(A,2) = 0 | k4_nat_1(A,2) = 1 ) ) ), file(nat_1,t62_nat_1), []). fof(t63_power,axiom,( ! [A] : ( v1_xreal_0(A) => ! [B] : ( v1_xreal_0(B) => ! [C] : ( v1_xreal_0(C) => ~ ( ~ r1_xreal_0(A,0) & A != 1 & ~ r1_xreal_0(B,0) & k5_power(A,k3_power(B,C)) != k3_xcmplx_0(C,k5_power(A,B)) ) ) ) ) ), file(power,t63_power), []). fof(t6_arithm,axiom,( ! [A] : ( v1_xcmplx_0(A) => k7_xcmplx_0(A,1) = A ) ), file(arithm,t6_arithm), []). fof(t6_boole,axiom,( ! [A] : ( v1_xboole_0(A) => A = k1_xboole_0 ) ), file(boole,t6_boole), []). fof(t6_finseq_1,axiom,( ! [A] : ( v4_ordinal2(A) => r2_hidden(k2_xcmplx_0(A,1),k2_finseq_1(k2_xcmplx_0(A,1))) ) ), file(finseq_1,t6_finseq_1), []). 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(t70_finseq_2,axiom,( ! [A] : ( v4_ordinal2(A) => ! [B,C] : ( r2_hidden(B,k2_finseq_1(A)) => k1_funct_1(k2_finseq_2(A,C),B) = C ) ) ), file(finseq_2,t70_finseq_2), []). fof(t70_xreal_1,axiom,( ! [A] : ( v1_xreal_0(A) => ! [B] : ( v1_xreal_0(B) => ! [C] : ( v1_xreal_0(C) => ~ ( ~ r1_xreal_0(A,0) & ~ r1_xreal_0(C,B) & r1_xreal_0(k3_xcmplx_0(C,A),k3_xcmplx_0(B,A)) ) ) ) ) ), file(xreal_1,t70_xreal_1), []). fof(t72_finseq_2,axiom,( ! [A] : k2_finseq_2(0,A) = k1_xboole_0 ), file(finseq_2,t72_finseq_2), []). fof(t73_finseq_2,axiom,( ! [A] : k2_finseq_2(1,A) = k9_finseq_1(A) ), file(finseq_2,t73_finseq_2), []). fof(t73_nat_1,axiom,( ! [A] : ( v4_ordinal2(A) => ! [B] : ( v4_ordinal2(B) => ! [C] : ( v4_ordinal2(C) => k4_nat_1(A,B) = k4_nat_1(k2_xcmplx_0(k3_xcmplx_0(B,C),A),B) ) ) ) ), file(nat_1,t73_nat_1), []). fof(t73_real_1,axiom,( ! [A] : ( v1_xreal_0(A) => ! [B] : ( v1_xreal_0(B) => ! [C] : ( v1_xreal_0(C) => ( ~ r1_xreal_0(A,0) => ( ~ ( ~ r1_xreal_0(C,B) & r1_xreal_0(k7_xcmplx_0(C,A),k7_xcmplx_0(B,A)) ) & ~ ( ~ r1_xreal_0(k7_xcmplx_0(C,A),k7_xcmplx_0(B,A)) & r1_xreal_0(C,B) ) ) ) ) ) ) ), file(real_1,t73_real_1), []). fof(t75_xcmplx_1,axiom,( ! [A] : ( v1_xcmplx_0(A) => ! [B] : ( v1_xcmplx_0(B) => ! [C] : ( v1_xcmplx_0(C) => k3_xcmplx_0(A,k7_xcmplx_0(B,C)) = k7_xcmplx_0(k3_xcmplx_0(A,B),C) ) ) ) ), file(xcmplx_1,t75_xcmplx_1), []). fof(t79_xcmplx_1,axiom,( ! [A] : ( v1_xcmplx_0(A) => ! [B] : ( v1_xcmplx_0(B) => ! [C] : ( v1_xcmplx_0(C) => k7_xcmplx_0(A,k3_xcmplx_0(B,C)) = k7_xcmplx_0(k7_xcmplx_0(A,B),C) ) ) ) ), file(xcmplx_1,t79_xcmplx_1), []). fof(t7_boole,axiom,( ! [A,B] : ~ ( r2_hidden(A,B) & v1_xboole_0(B) ) ), file(boole,t7_boole), []). fof(t7_mcart_1,axiom,( ! [A,B] : ( k1_mcart_1(k4_tarski(A,B)) = A & k2_mcart_1(k4_tarski(A,B)) = B ) ), file(mcart_1,t7_mcart_1), []). 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(t80_int_1,axiom,( ! [A] : ( v1_xreal_0(A) => ( r1_xreal_0(0,A) => ( r1_xreal_0(0,k2_int_1(A)) & r1_xreal_0(0,k1_int_1(A)) & m2_subset_1(k2_int_1(A),k1_numbers,k5_numbers) & m2_subset_1(k1_int_1(A),k1_numbers,k5_numbers) ) ) ) ), file(int_1,t80_int_1), []). fof(t83_finseq_4,axiom,( ! [A] : ( m2_subset_1(A,k1_numbers,k5_numbers) => ! [B] : ( ~ v1_xboole_0(B) => ! [C] : ( m2_finseq_1(C,B) => ! [D] : ( m2_finseq_1(D,B) => ( r2_hidden(A,k4_finseq_1(C)) => k4_finseq_4(k5_numbers,B,k8_finseq_1(B,C,D),A) = k4_finseq_4(k5_numbers,B,C,A) ) ) ) ) ) ), file(finseq_4,t83_finseq_4), []). fof(t88_xcmplx_1,axiom,( ! [A] : ( v1_xcmplx_0(A) => ! [B] : ( v1_xcmplx_0(B) => ( A != 0 => k3_xcmplx_0(k7_xcmplx_0(B,A),A) = B ) ) ) ), file(xcmplx_1,t88_xcmplx_1), []). fof(t8_boole,axiom,( ! [A,B] : ~ ( v1_xboole_0(A) & A != B & v1_xboole_0(B) ) ), file(boole,t8_boole), []). fof(t8_jordan2b,axiom,( ! [A] : ( m2_subset_1(A,k1_numbers,k5_numbers) => k3_finseq_1(k5_euclid(A)) = A ) ), file(jordan2b,t8_jordan2b), []). 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), []).