% Mizar problem: t16_lp_space,lp_space,1488,25 fof(t16_lp_space,conjecture,( ! [A] : ( m1_subset_1(A,k1_numbers) => ( r1_xreal_0(1,A) => ! [B] : ( ( ~ v3_struct_0(B) & v3_rlvect_1(B) & v4_rlvect_1(B) & v5_rlvect_1(B) & v6_rlvect_1(B) & v7_rlvect_1(B) & v2_normsp_1(B) & l1_normsp_1(B) ) => ( B = g1_normsp_1(k2_lp_space(A),k10_rsspace(k7_rsspace,k2_lp_space(A)),k8_rsspace(k7_rsspace,k2_lp_space(A)),k9_rsspace(k7_rsspace,k2_lp_space(A)),k3_lp_space(A)) => ! [C] : ( ( v1_funct_1(C) & v1_funct_2(C,k5_numbers,u1_struct_0(B)) & m2_relset_1(C,k5_numbers,u1_struct_0(B)) ) => ( v1_rsspace3(C,B) => v4_normsp_1(C,B) ) ) ) ) ) ) ), inference(mizar_bg_added,[status(thm)],[abstractness_v1_normsp_1,antisymmetry_r2_hidden,cc10_membered,cc11_membered,cc12_membered,cc13_membered,cc14_membered,cc15_membered,cc16_membered,cc17_membered,cc18_membered,cc19_membered,cc1_membered,cc1_relset_1,cc1_seq_1,cc1_xreal_0,cc20_membered,cc2_membered,cc2_xreal_0,cc3_membered,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_seq_1,commutativity_k3_xcmplx_0,commutativity_k4_real_1,commutativity_k4_rlvect_1,commutativity_k6_seq_1,commutativity_k9_seq_1,connectedness_r1_xreal_0,d11_rlvect_1,d1_lp_space,d1_normsp_1,d1_series_1,d1_xreal_0,d2_lp_space,d2_power,d2_series_1,d3_lp_space,d3_seq_2,d3_series_1,d9_normsp_1,dt_g1_normsp_1,dt_k10_rsspace,dt_k10_seq_1,dt_k12_prepower,dt_k12_seq_1,dt_k14_seq_1,dt_k15_seq_1,dt_k16_complex1,dt_k17_complex1,dt_k17_seq_1,dt_k18_complex1,dt_k1_funct_1,dt_k1_lp_space,dt_k1_nat_1,dt_k1_normsp_1,dt_k1_numbers,dt_k1_real_1,dt_k1_rlvect_1,dt_k1_rsspace,dt_k1_seq_2,dt_k1_series_1,dt_k1_xboole_0,dt_k1_zfmisc_1,dt_k20_seq_1,dt_k22_seq_1,dt_k2_lp_space,dt_k2_normsp_1,dt_k2_rlvect_1,dt_k2_rsspace,dt_k2_seq_1,dt_k2_seq_2,dt_k2_series_1,dt_k2_xcmplx_0,dt_k2_zfmisc_1,dt_k3_lp_space,dt_k3_power,dt_k3_real_1,dt_k3_rlvect_1,dt_k3_seq_1,dt_k3_xcmplx_0,dt_k4_power,dt_k4_real_1,dt_k4_rlvect_1,dt_k4_seq_1,dt_k4_xcmplx_0,dt_k5_numbers,dt_k5_ordinal2,dt_k5_real_1,dt_k5_rlvect_1,dt_k6_prepower,dt_k6_real_1,dt_k6_rlvect_1,dt_k6_rsspace,dt_k6_seq_1,dt_k6_xcmplx_0,dt_k7_rsspace,dt_k7_xcmplx_0,dt_k8_funct_2,dt_k8_rsspace,dt_k9_rsspace,dt_k9_seq_1,dt_l1_normsp_1,dt_l1_rlvect_1,dt_l1_struct_0,dt_l2_rlvect_1,dt_l2_struct_0,dt_m1_relset_1,dt_m1_subset_1,dt_m2_relset_1,dt_m2_subset_1,dt_u1_normsp_1,dt_u1_rlvect_1,dt_u1_struct_0,dt_u2_rlvect_1,dt_u2_struct_0,existence_l1_normsp_1,existence_l1_rlvect_1,existence_l1_struct_0,existence_l2_rlvect_1,existence_l2_struct_0,existence_m1_relset_1,existence_m1_subset_1,existence_m2_relset_1,existence_m2_subset_1,fc10_xreal_0,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_ordinal2,fc1_seq_1,fc1_struct_0,fc1_subset_1,fc1_xreal_0,fc20_xreal_0,fc21_xreal_0,fc22_xreal_0,fc23_xreal_0,fc24_xreal_0,fc27_xreal_0,fc28_xreal_0,fc29_xreal_0,fc2_membered,fc30_xreal_0,fc3_seq_1,fc3_xreal_0,fc4_rsspace3,fc4_seq_1,fc4_subset_1,fc4_xreal_0,fc5_membered,fc5_xreal_0,fc6_membered,fc6_seq_1,fc6_xreal_0,fc7_seq_1,fc7_xreal_0,fc8_seq_1,fc8_xreal_0,fc9_xreal_0,free_g1_normsp_1,involutiveness_k1_real_1,involutiveness_k4_xcmplx_0,l28_lp_space,l29_lp_space,l30_lp_space,l31_lp_space,l32_lp_space,l3_lp_space,projectivity_k16_complex1,projectivity_k17_complex1,projectivity_k18_complex1,rc1_membered,rc1_seq_1,rc1_subset_1,rc1_xreal_0,rc2_subset_1,rc2_xreal_0,rc3_rlvect_1,rc3_struct_0,rc3_xreal_0,rc4_struct_0,rc4_xreal_0,rc5_struct_0,redefinition_k10_seq_1,redefinition_k14_seq_1,redefinition_k17_complex1,redefinition_k17_seq_1,redefinition_k18_complex1,redefinition_k1_nat_1,redefinition_k1_real_1,redefinition_k22_seq_1,redefinition_k2_normsp_1,redefinition_k2_seq_1,redefinition_k2_seq_2,redefinition_k3_real_1,redefinition_k4_power,redefinition_k4_real_1,redefinition_k4_rlvect_1,redefinition_k5_numbers,redefinition_k5_real_1,redefinition_k6_real_1,redefinition_k6_seq_1,redefinition_k8_funct_2,redefinition_k9_seq_1,redefinition_m2_relset_1,redefinition_m2_subset_1,reflexivity_r1_tarski,reflexivity_r1_xreal_0,s1_funct_2__e9_28__lp_space,s1_nat_1__e4_28_4_2__lp_space,s1_seq_1__e2_28_4_2_3_1_1__lp_space,s1_seq_1__e2_28_4_3_1__lp_space,s1_seq_1__e3_28_2__lp_space,spc0_boole,spc1_boole,spc1_numerals,spc2_boole,spc2_numerals,spc4_boole,spc4_numerals,t107_xcmplx_1,t10_lp_space,t10_rsspace3,t11_seq_1,t122_real_2,t127_real_2,t132_complex1,t14_seq_1,t15_seq_1,t16_seq_1,t18_nat_1,t1_arithm,t1_lp_space,t1_numerals,t1_real,t1_subset,t20_seq_1,t20_series_1,t218_xreal_1,t21_series_1,t2_arithm,t2_holder_1,t2_real,t2_subset,t2_xreal_1,t30_power,t31_xreal_1,t38_power,t39_power,t3_arithm,t3_real,t3_rsspace,t3_seq_2,t3_subset,t47_rlvect_1,t4_arithm,t4_real,t4_rsspace2,t4_subset,t58_seq_4,t5_arithm,t5_real,t5_subset,t60_xcmplx_1,t6_arithm,t6_boole,t6_normsp_1,t6_real,t75_xcmplx_1,t7_boole,t7_real,t8_boole,t8_normsp_1,t8_real]), [file(lp_space,t16_lp_space)]). fof(abstractness_v1_normsp_1,axiom,( ! [A] : ( l1_normsp_1(A) => ( v1_normsp_1(A) => A = g1_normsp_1(u1_struct_0(A),u2_struct_0(A),u1_rlvect_1(A),u2_rlvect_1(A),u1_normsp_1(A)) ) ) ), file(normsp_1,v1_normsp_1), []). fof(antisymmetry_r2_hidden,axiom,( ! [A,B] : ( r2_hidden(A,B) => ~ r2_hidden(B,A) ) ), file(hidden,r2_hidden), []). fof(cc10_membered,axiom,( ! [A] : ( v1_membered(A) => ! [B] : ( m1_subset_1(B,A) => v1_xcmplx_0(B) ) ) ), file(membered,cc10_membered), []). fof(cc11_membered,axiom,( ! [A] : ( v2_membered(A) => ! [B] : ( m1_subset_1(B,A) => ( v1_xcmplx_0(B) & v1_xreal_0(B) ) ) ) ), file(membered,cc11_membered), []). fof(cc12_membered,axiom,( ! [A] : ( v3_membered(A) => ! [B] : ( m1_subset_1(B,A) => ( v1_xcmplx_0(B) & v1_xreal_0(B) & v1_rat_1(B) ) ) ) ), file(membered,cc12_membered), []). fof(cc13_membered,axiom,( ! [A] : ( v4_membered(A) => ! [B] : ( m1_subset_1(B,A) => ( v1_xcmplx_0(B) & v1_xreal_0(B) & v1_int_1(B) & v1_rat_1(B) ) ) ) ), file(membered,cc13_membered), []). fof(cc14_membered,axiom,( ! [A] : ( v5_membered(A) => ! [B] : ( m1_subset_1(B,A) => ( v1_xcmplx_0(B) & v4_ordinal2(B) & v1_xreal_0(B) & v1_int_1(B) & v1_rat_1(B) ) ) ) ), file(membered,cc14_membered), []). fof(cc15_membered,axiom,( ! [A] : ( v1_xboole_0(A) => ( v1_membered(A) & v2_membered(A) & v3_membered(A) & v4_membered(A) & v5_membered(A) ) ) ), file(membered,cc15_membered), []). fof(cc16_membered,axiom,( ! [A] : ( v1_membered(A) => ! [B] : ( m1_subset_1(B,k1_zfmisc_1(A)) => v1_membered(B) ) ) ), file(membered,cc16_membered), []). fof(cc17_membered,axiom,( ! [A] : ( v2_membered(A) => ! [B] : ( m1_subset_1(B,k1_zfmisc_1(A)) => ( v1_membered(B) & v2_membered(B) ) ) ) ), file(membered,cc17_membered), []). fof(cc18_membered,axiom,( ! [A] : ( v3_membered(A) => ! [B] : ( m1_subset_1(B,k1_zfmisc_1(A)) => ( v1_membered(B) & v2_membered(B) & v3_membered(B) ) ) ) ), file(membered,cc18_membered), []). fof(cc19_membered,axiom,( ! [A] : ( v4_membered(A) => ! [B] : ( m1_subset_1(B,k1_zfmisc_1(A)) => ( v1_membered(B) & v2_membered(B) & v3_membered(B) & v4_membered(B) ) ) ) ), file(membered,cc19_membered), []). fof(cc1_membered,axiom,( ! [A] : ( v5_membered(A) => v4_membered(A) ) ), file(membered,cc1_membered), []). fof(cc1_relset_1,axiom,( ! [A,B,C] : ( m1_subset_1(C,k1_zfmisc_1(k2_zfmisc_1(A,B))) => v1_relat_1(C) ) ), file(relset_1,cc1_relset_1), []). fof(cc1_seq_1,axiom,( ! [A,B] : ( v2_membered(B) => ! [C] : ( m1_relset_1(C,A,B) => ( v1_funct_1(C) => ( v1_funct_1(C) & v1_seq_1(C) ) ) ) ) ), file(seq_1,cc1_seq_1), []). fof(cc1_xreal_0,axiom,( ! [A] : ( v4_ordinal2(A) => v1_xreal_0(A) ) ), file(xreal_0,cc1_xreal_0), []). fof(cc20_membered,axiom,( ! [A] : ( v5_membered(A) => ! [B] : ( m1_subset_1(B,k1_zfmisc_1(A)) => ( v1_membered(B) & v2_membered(B) & v3_membered(B) & v4_membered(B) & v5_membered(B) ) ) ) ), file(membered,cc20_membered), []). fof(cc2_membered,axiom,( ! [A] : ( v4_membered(A) => v3_membered(A) ) ), file(membered,cc2_membered), []). 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_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_seq_1,axiom,( ! [A,B] : ( ( v1_relat_1(A) & v1_funct_1(A) & v1_seq_1(A) & v1_relat_1(B) & v1_funct_1(B) & v1_seq_1(B) ) => k3_seq_1(A,B) = k3_seq_1(B,A) ) ), file(seq_1,k3_seq_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_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_k4_rlvect_1,axiom,( ! [A,B,C] : ( ( ~ v3_struct_0(A) & v3_rlvect_1(A) & l1_rlvect_1(A) & m1_subset_1(B,u1_struct_0(A)) & m1_subset_1(C,u1_struct_0(A)) ) => k4_rlvect_1(A,B,C) = k4_rlvect_1(A,C,B) ) ), file(rlvect_1,k4_rlvect_1), []). fof(commutativity_k6_seq_1,axiom,( ! [A,B,C,D] : ( ( v2_membered(B) & v1_funct_1(C) & m1_relset_1(C,A,B) & v1_funct_1(D) & m1_relset_1(D,A,B) ) => k6_seq_1(A,B,C,D) = k6_seq_1(A,B,D,C) ) ), file(seq_1,k6_seq_1), []). fof(commutativity_k9_seq_1,axiom,( ! [A,B] : ( ( v1_funct_1(A) & v1_funct_2(A,k5_numbers,k1_numbers) & m1_relset_1(A,k5_numbers,k1_numbers) & v1_funct_1(B) & v1_funct_2(B,k5_numbers,k1_numbers) & m1_relset_1(B,k5_numbers,k1_numbers) ) => k9_seq_1(A,B) = k9_seq_1(B,A) ) ), file(seq_1,k9_seq_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(d11_rlvect_1,axiom,( ! [A] : ( ( ~ v3_struct_0(A) & l1_rlvect_1(A) ) => ! [B] : ( m1_subset_1(B,u1_struct_0(A)) => ! [C] : ( m1_subset_1(C,u1_struct_0(A)) => k6_rlvect_1(A,B,C) = k2_rlvect_1(A,B,k5_rlvect_1(A,C)) ) ) ) ), file(rlvect_1,d11_rlvect_1), []). fof(d1_lp_space,axiom,( ! [A] : ( ( v1_funct_1(A) & v1_funct_2(A,k5_numbers,k1_numbers) & m2_relset_1(A,k5_numbers,k1_numbers) ) => ! [B] : ( m1_subset_1(B,k1_numbers) => ! [C] : ( ( v1_funct_1(C) & v1_funct_2(C,k5_numbers,k1_numbers) & m2_relset_1(C,k5_numbers,k1_numbers) ) => ( C = k1_lp_space(A,B) <=> ! [D] : ( m2_subset_1(D,k1_numbers,k5_numbers) => k2_seq_1(k5_numbers,k1_numbers,C,D) = k4_power(k18_complex1(k2_seq_1(k5_numbers,k1_numbers,A,D)),B) ) ) ) ) ) ), file(lp_space,d1_lp_space), []). fof(d1_normsp_1,axiom,( ! [A] : ( ( ~ v3_struct_0(A) & l1_normsp_1(A) ) => ! [B] : ( m1_subset_1(B,u1_struct_0(A)) => k1_normsp_1(A,B) = k2_seq_1(u1_struct_0(A),k1_numbers,u1_normsp_1(A),B) ) ) ), file(normsp_1,d1_normsp_1), []). fof(d1_series_1,axiom,( ! [A] : ( ( v1_funct_1(A) & v1_funct_2(A,k5_numbers,k1_numbers) & m2_relset_1(A,k5_numbers,k1_numbers) ) => ! [B] : ( ( v1_funct_1(B) & v1_funct_2(B,k5_numbers,k1_numbers) & m2_relset_1(B,k5_numbers,k1_numbers) ) => ( B = k1_series_1(A) <=> ( k8_funct_2(k5_numbers,k1_numbers,B,0) = k8_funct_2(k5_numbers,k1_numbers,A,0) & ! [C] : ( m2_subset_1(C,k1_numbers,k5_numbers) => k8_funct_2(k5_numbers,k1_numbers,B,k1_nat_1(C,1)) = k3_real_1(k8_funct_2(k5_numbers,k1_numbers,B,C),k8_funct_2(k5_numbers,k1_numbers,A,k1_nat_1(C,1))) ) ) ) ) ) ), file(series_1,d1_series_1), []). fof(d1_xreal_0,axiom,( ! [A] : ( v1_xreal_0(A) <=> r2_hidden(A,k1_numbers) ) ), file(xreal_0,d1_xreal_0), []). fof(d2_lp_space,axiom,( ! [A] : ( m1_subset_1(A,k1_numbers) => ( r1_xreal_0(1,A) => ! [B] : ( ( ~ v1_xboole_0(B) & m1_subset_1(B,k1_zfmisc_1(u1_struct_0(k7_rsspace))) ) => ( B = k2_lp_space(A) <=> ! [C] : ( r2_hidden(C,B) <=> ( r2_hidden(C,k1_rsspace) & v1_series_1(k1_lp_space(k2_rsspace(C),A)) ) ) ) ) ) ) ), file(lp_space,d2_lp_space), []). fof(d2_power,axiom,( ! [A] : ( v1_xreal_0(A) => ! [B] : ( v1_xreal_0(B) => ! [C] : ( v1_xreal_0(C) => ( ( ~ r1_xreal_0(A,0) => ( C = k3_power(A,B) <=> C = k12_prepower(A,B) ) ) & ( A = 0 => ( r1_xreal_0(B,0) | ( C = k3_power(A,B) <=> C = 0 ) ) ) & ( ( A = 0 & B = 0 ) => ( C = k3_power(A,B) <=> C = 1 ) ) & ( v1_int_1(B) => ( r1_xreal_0(0,A) | ( C = k3_power(A,B) <=> ? [D] : ( v1_int_1(D) & D = B & C = k6_prepower(A,D) ) ) ) ) ) ) ) ) ), file(power,d2_power), []). fof(d2_series_1,axiom,( ! [A] : ( ( v1_funct_1(A) & v1_funct_2(A,k5_numbers,k1_numbers) & m2_relset_1(A,k5_numbers,k1_numbers) ) => ( v1_series_1(A) <=> v4_seq_2(k1_series_1(A)) ) ) ), file(series_1,d2_series_1), []). fof(d3_lp_space,axiom,( ! [A] : ( m1_subset_1(A,k1_numbers) => ! [B] : ( ( v1_funct_1(B) & v1_funct_2(B,k2_lp_space(A),k1_numbers) & m2_relset_1(B,k2_lp_space(A),k1_numbers) ) => ( B = k3_lp_space(A) <=> ! [C] : ( r2_hidden(C,k2_lp_space(A)) => k2_seq_1(k2_lp_space(A),k1_numbers,B,C) = k4_power(k2_series_1(k1_lp_space(k2_rsspace(C),A)),k6_real_1(1,A)) ) ) ) ) ), file(lp_space,d3_lp_space), []). fof(d3_seq_2,axiom,( ! [A] : ( ( v1_funct_1(A) & v1_funct_2(A,k5_numbers,k1_numbers) & m2_relset_1(A,k5_numbers,k1_numbers) ) => ( v1_seq_2(A) <=> ? [B] : ( v1_xreal_0(B) & ! [C] : ( m2_subset_1(C,k1_numbers,k5_numbers) => ~ r1_xreal_0(B,k2_seq_1(k5_numbers,k1_numbers,A,C)) ) ) ) ) ), file(seq_2,d3_seq_2), []). fof(d3_series_1,axiom,( ! [A] : ( ( v1_funct_1(A) & v1_funct_2(A,k5_numbers,k1_numbers) & m2_relset_1(A,k5_numbers,k1_numbers) ) => k2_series_1(A) = k2_seq_2(k1_series_1(A)) ) ), file(series_1,d3_series_1), []). fof(d9_normsp_1,axiom,( ! [A] : ( ( ~ v3_struct_0(A) & v3_rlvect_1(A) & v4_rlvect_1(A) & v5_rlvect_1(A) & v6_rlvect_1(A) & v7_rlvect_1(A) & v2_normsp_1(A) & l1_normsp_1(A) ) => ! [B] : ( ( v1_funct_1(B) & v1_funct_2(B,k5_numbers,u1_struct_0(A)) & m2_relset_1(B,k5_numbers,u1_struct_0(A)) ) => ( v4_normsp_1(B,A) <=> ? [C] : ( m1_subset_1(C,u1_struct_0(A)) & ! [D] : ( m1_subset_1(D,k1_numbers) => ~ ( ~ r1_xreal_0(D,0) & ! [E] : ( m2_subset_1(E,k1_numbers,k5_numbers) => ? [F] : ( m2_subset_1(F,k1_numbers,k5_numbers) & r1_xreal_0(E,F) & r1_xreal_0(D,k1_normsp_1(A,k6_rlvect_1(A,k2_normsp_1(A,B,F),C))) ) ) ) ) ) ) ) ) ), file(normsp_1,d9_normsp_1), []). fof(dt_g1_normsp_1,axiom,( ! [A,B,C,D,E] : ( ( m1_subset_1(B,A) & v1_funct_1(C) & v1_funct_2(C,k2_zfmisc_1(A,A),A) & m1_relset_1(C,k2_zfmisc_1(A,A),A) & v1_funct_1(D) & v1_funct_2(D,k2_zfmisc_1(k1_numbers,A),A) & m1_relset_1(D,k2_zfmisc_1(k1_numbers,A),A) & v1_funct_1(E) & v1_funct_2(E,A,k1_numbers) & m1_relset_1(E,A,k1_numbers) ) => ( v1_normsp_1(g1_normsp_1(A,B,C,D,E)) & l1_normsp_1(g1_normsp_1(A,B,C,D,E)) ) ) ), file(normsp_1,g1_normsp_1), []). fof(dt_k10_rsspace,axiom,( ! [A,B] : ( ( ~ v3_struct_0(A) & v3_rlvect_1(A) & v4_rlvect_1(A) & v5_rlvect_1(A) & v6_rlvect_1(A) & v7_rlvect_1(A) & l2_rlvect_1(A) & m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) ) => m1_subset_1(k10_rsspace(A,B),B) ) ), file(rsspace,k10_rsspace), []). fof(dt_k10_seq_1,axiom,( ! [A,B] : ( ( v1_funct_1(A) & v1_funct_2(A,k5_numbers,k1_numbers) & m1_relset_1(A,k5_numbers,k1_numbers) & v1_funct_1(B) & v1_funct_2(B,k5_numbers,k1_numbers) & m1_relset_1(B,k5_numbers,k1_numbers) ) => ( v1_funct_1(k10_seq_1(A,B)) & v1_funct_2(k10_seq_1(A,B),k5_numbers,k1_numbers) & m2_relset_1(k10_seq_1(A,B),k5_numbers,k1_numbers) ) ) ), file(seq_1,k10_seq_1), []). fof(dt_k12_prepower,axiom,( ! [A,B] : ( ( v1_xreal_0(A) & v1_xreal_0(B) ) => v1_xreal_0(k12_prepower(A,B)) ) ), file(prepower,k12_prepower), []). fof(dt_k12_seq_1,axiom,( ! [A,B] : ( ( v1_relat_1(A) & v1_funct_1(A) & v1_seq_1(A) & v1_xreal_0(B) ) => ( v1_relat_1(k12_seq_1(A,B)) & v1_funct_1(k12_seq_1(A,B)) ) ) ), file(seq_1,k12_seq_1), []). fof(dt_k14_seq_1,axiom,( ! [A,B] : ( ( v1_funct_1(A) & v1_funct_2(A,k5_numbers,k1_numbers) & m1_relset_1(A,k5_numbers,k1_numbers) & v1_xreal_0(B) ) => ( v1_funct_1(k14_seq_1(A,B)) & v1_funct_2(k14_seq_1(A,B),k5_numbers,k1_numbers) & m2_relset_1(k14_seq_1(A,B),k5_numbers,k1_numbers) ) ) ), file(seq_1,k14_seq_1), []). fof(dt_k15_seq_1,axiom,( ! [A] : ( ( v1_relat_1(A) & v1_funct_1(A) & v1_seq_1(A) ) => ( v1_relat_1(k15_seq_1(A)) & v1_funct_1(k15_seq_1(A)) ) ) ), file(seq_1,k15_seq_1), []). fof(dt_k16_complex1,axiom,( ! [A] : ( v1_xcmplx_0(A) => v1_xcmplx_0(k16_complex1(A)) ) ), file(complex1,k16_complex1), []). fof(dt_k17_complex1,axiom,( ! [A] : ( v1_xcmplx_0(A) => m1_subset_1(k17_complex1(A),k1_numbers) ) ), file(complex1,k17_complex1), []). fof(dt_k17_seq_1,axiom,( ! [A] : ( ( v1_funct_1(A) & v1_funct_2(A,k5_numbers,k1_numbers) & m1_relset_1(A,k5_numbers,k1_numbers) ) => ( v1_funct_1(k17_seq_1(A)) & v1_funct_2(k17_seq_1(A),k5_numbers,k1_numbers) & m2_relset_1(k17_seq_1(A),k5_numbers,k1_numbers) ) ) ), file(seq_1,k17_seq_1), []). fof(dt_k18_complex1,axiom,( ! [A] : ( v1_xcmplx_0(A) => m1_subset_1(k18_complex1(A),k1_numbers) ) ), file(complex1,k18_complex1), []). fof(dt_k1_funct_1,axiom,( $true ), file(funct_1,k1_funct_1), []). fof(dt_k1_lp_space,axiom,( ! [A,B] : ( ( v1_funct_1(A) & v1_funct_2(A,k5_numbers,k1_numbers) & m1_relset_1(A,k5_numbers,k1_numbers) & m1_subset_1(B,k1_numbers) ) => ( v1_funct_1(k1_lp_space(A,B)) & v1_funct_2(k1_lp_space(A,B),k5_numbers,k1_numbers) & m2_relset_1(k1_lp_space(A,B),k5_numbers,k1_numbers) ) ) ), file(lp_space,k1_lp_space), []). 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_normsp_1,axiom,( ! [A,B] : ( ( ~ v3_struct_0(A) & l1_normsp_1(A) & m1_subset_1(B,u1_struct_0(A)) ) => m1_subset_1(k1_normsp_1(A,B),k1_numbers) ) ), file(normsp_1,k1_normsp_1), []). fof(dt_k1_numbers,axiom,( $true ), file(numbers,k1_numbers), []). fof(dt_k1_real_1,axiom,( ! [A] : ( m1_subset_1(A,k1_numbers) => m1_subset_1(k1_real_1(A),k1_numbers) ) ), file(real_1,k1_real_1), []). fof(dt_k1_rlvect_1,axiom,( ! [A] : ( l2_struct_0(A) => m1_subset_1(k1_rlvect_1(A),u1_struct_0(A)) ) ), file(rlvect_1,k1_rlvect_1), []). fof(dt_k1_rsspace,axiom,( ~ v1_xboole_0(k1_rsspace) ), file(rsspace,k1_rsspace), []). fof(dt_k1_seq_2,axiom,( ! [A] : ( ( v1_funct_1(A) & v1_funct_2(A,k5_numbers,k1_numbers) & m1_relset_1(A,k5_numbers,k1_numbers) ) => v1_xreal_0(k1_seq_2(A)) ) ), file(seq_2,k1_seq_2), []). fof(dt_k1_series_1,axiom,( ! [A] : ( ( v1_funct_1(A) & v1_funct_2(A,k5_numbers,k1_numbers) & m1_relset_1(A,k5_numbers,k1_numbers) ) => ( v1_funct_1(k1_series_1(A)) & v1_funct_2(k1_series_1(A),k5_numbers,k1_numbers) & m2_relset_1(k1_series_1(A),k5_numbers,k1_numbers) ) ) ), file(series_1,k1_series_1), []). fof(dt_k1_xboole_0,axiom,( $true ), file(xboole_0,k1_xboole_0), []). fof(dt_k1_zfmisc_1,axiom,( $true ), file(zfmisc_1,k1_zfmisc_1), []). fof(dt_k20_seq_1,axiom,( ! [A] : ( ( v1_relat_1(A) & v1_funct_1(A) & v1_seq_1(A) ) => ( v1_relat_1(k20_seq_1(A)) & v1_funct_1(k20_seq_1(A)) ) ) ), file(seq_1,k20_seq_1), []). fof(dt_k22_seq_1,axiom,( ! [A] : ( ( v1_funct_1(A) & v1_funct_2(A,k5_numbers,k1_numbers) & m1_relset_1(A,k5_numbers,k1_numbers) ) => ( v1_funct_1(k22_seq_1(A)) & v1_funct_2(k22_seq_1(A),k5_numbers,k1_numbers) & m2_relset_1(k22_seq_1(A),k5_numbers,k1_numbers) ) ) ), file(seq_1,k22_seq_1), []). fof(dt_k2_lp_space,axiom,( ! [A] : ( m1_subset_1(A,k1_numbers) => ( ~ v1_xboole_0(k2_lp_space(A)) & m1_subset_1(k2_lp_space(A),k1_zfmisc_1(u1_struct_0(k7_rsspace))) ) ) ), file(lp_space,k2_lp_space), []). fof(dt_k2_normsp_1,axiom,( ! [A,B,C] : ( ( ~ v3_struct_0(A) & l1_struct_0(A) & v1_funct_1(B) & v1_funct_2(B,k5_numbers,u1_struct_0(A)) & m1_relset_1(B,k5_numbers,u1_struct_0(A)) & m1_subset_1(C,k5_numbers) ) => m1_subset_1(k2_normsp_1(A,B,C),u1_struct_0(A)) ) ), file(normsp_1,k2_normsp_1), []). fof(dt_k2_rlvect_1,axiom,( ! [A,B,C] : ( ( ~ v3_struct_0(A) & l1_rlvect_1(A) & m1_subset_1(B,u1_struct_0(A)) & m1_subset_1(C,u1_struct_0(A)) ) => m1_subset_1(k2_rlvect_1(A,B,C),u1_struct_0(A)) ) ), file(rlvect_1,k2_rlvect_1), []). fof(dt_k2_rsspace,axiom,( ! [A] : ( v1_funct_1(k2_rsspace(A)) & v1_funct_2(k2_rsspace(A),k5_numbers,k1_numbers) & m2_relset_1(k2_rsspace(A),k5_numbers,k1_numbers) ) ), file(rsspace,k2_rsspace), []). fof(dt_k2_seq_1,axiom,( ! [A,B,C,D] : ( ( v2_membered(B) & v1_funct_1(C) & m1_relset_1(C,A,B) ) => m1_subset_1(k2_seq_1(A,B,C,D),k1_numbers) ) ), file(seq_1,k2_seq_1), []). fof(dt_k2_seq_2,axiom,( ! [A] : ( ( v1_funct_1(A) & v1_funct_2(A,k5_numbers,k1_numbers) & m1_relset_1(A,k5_numbers,k1_numbers) ) => m1_subset_1(k2_seq_2(A),k1_numbers) ) ), file(seq_2,k2_seq_2), []). fof(dt_k2_series_1,axiom,( ! [A] : ( ( v1_funct_1(A) & v1_funct_2(A,k5_numbers,k1_numbers) & m1_relset_1(A,k5_numbers,k1_numbers) ) => m1_subset_1(k2_series_1(A),k1_numbers) ) ), file(series_1,k2_series_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_lp_space,axiom,( ! [A] : ( m1_subset_1(A,k1_numbers) => ( v1_funct_1(k3_lp_space(A)) & v1_funct_2(k3_lp_space(A),k2_lp_space(A),k1_numbers) & m2_relset_1(k3_lp_space(A),k2_lp_space(A),k1_numbers) ) ) ), file(lp_space,k3_lp_space), []). 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_rlvect_1,axiom,( ! [A,B,C] : ( ( ~ v3_struct_0(A) & l2_rlvect_1(A) & m1_subset_1(B,u1_struct_0(A)) & m1_subset_1(C,k1_numbers) ) => m1_subset_1(k3_rlvect_1(A,B,C),u1_struct_0(A)) ) ), file(rlvect_1,k3_rlvect_1), []). fof(dt_k3_seq_1,axiom,( ! [A,B] : ( ( v1_relat_1(A) & v1_funct_1(A) & v1_seq_1(A) & v1_relat_1(B) & v1_funct_1(B) & v1_seq_1(B) ) => ( v1_relat_1(k3_seq_1(A,B)) & v1_funct_1(k3_seq_1(A,B)) ) ) ), file(seq_1,k3_seq_1), []). fof(dt_k3_xcmplx_0,axiom,( $true ), file(xcmplx_0,k3_xcmplx_0), []). fof(dt_k4_power,axiom,( ! [A,B] : ( ( m1_subset_1(A,k1_numbers) & m1_subset_1(B,k1_numbers) ) => m1_subset_1(k4_power(A,B),k1_numbers) ) ), file(power,k4_power), []). 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_rlvect_1,axiom,( ! [A,B,C] : ( ( ~ v3_struct_0(A) & v3_rlvect_1(A) & l1_rlvect_1(A) & m1_subset_1(B,u1_struct_0(A)) & m1_subset_1(C,u1_struct_0(A)) ) => m1_subset_1(k4_rlvect_1(A,B,C),u1_struct_0(A)) ) ), file(rlvect_1,k4_rlvect_1), []). fof(dt_k4_seq_1,axiom,( ! [A,B] : ( ( v1_relat_1(A) & v1_funct_1(A) & v1_seq_1(A) & v1_relat_1(B) & v1_funct_1(B) & v1_seq_1(B) ) => ( v1_relat_1(k4_seq_1(A,B)) & v1_funct_1(k4_seq_1(A,B)) ) ) ), file(seq_1,k4_seq_1), []). 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_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_rlvect_1,axiom,( ! [A,B] : ( ( ~ v3_struct_0(A) & l1_rlvect_1(A) & m1_subset_1(B,u1_struct_0(A)) ) => m1_subset_1(k5_rlvect_1(A,B),u1_struct_0(A)) ) ), file(rlvect_1,k5_rlvect_1), []). fof(dt_k6_prepower,axiom,( $true ), file(prepower,k6_prepower), []). 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_rlvect_1,axiom,( ! [A,B,C] : ( ( ~ v3_struct_0(A) & l1_rlvect_1(A) & m1_subset_1(B,u1_struct_0(A)) & m1_subset_1(C,u1_struct_0(A)) ) => m1_subset_1(k6_rlvect_1(A,B,C),u1_struct_0(A)) ) ), file(rlvect_1,k6_rlvect_1), []). fof(dt_k6_rsspace,axiom,( m1_subset_1(k6_rsspace,k1_rsspace) ), file(rsspace,k6_rsspace), []). fof(dt_k6_seq_1,axiom,( ! [A,B,C,D] : ( ( v2_membered(B) & v1_funct_1(C) & m1_relset_1(C,A,B) & v1_funct_1(D) & m1_relset_1(D,A,B) ) => ( v1_funct_1(k6_seq_1(A,B,C,D)) & m2_relset_1(k6_seq_1(A,B,C,D),A,k1_numbers) ) ) ), file(seq_1,k6_seq_1), []). fof(dt_k6_xcmplx_0,axiom,( $true ), file(xcmplx_0,k6_xcmplx_0), []). fof(dt_k7_rsspace,axiom, ( ~ v3_struct_0(k7_rsspace) & v3_rlvect_1(k7_rsspace) & v4_rlvect_1(k7_rsspace) & v5_rlvect_1(k7_rsspace) & v6_rlvect_1(k7_rsspace) & v7_rlvect_1(k7_rsspace) & l2_rlvect_1(k7_rsspace) ), file(rsspace,k7_rsspace), []). fof(dt_k7_xcmplx_0,axiom,( $true ), file(xcmplx_0,k7_xcmplx_0), []). fof(dt_k8_funct_2,axiom,( ! [A,B,C,D] : ( ( ~ v1_xboole_0(A) & v1_funct_1(C) & v1_funct_2(C,A,B) & m1_relset_1(C,A,B) & m1_subset_1(D,A) ) => m1_subset_1(k8_funct_2(A,B,C,D),B) ) ), file(funct_2,k8_funct_2), []). fof(dt_k8_rsspace,axiom,( ! [A,B] : ( ( ~ v3_struct_0(A) & v3_rlvect_1(A) & v4_rlvect_1(A) & v5_rlvect_1(A) & v6_rlvect_1(A) & v7_rlvect_1(A) & l2_rlvect_1(A) & m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) ) => ( v1_funct_1(k8_rsspace(A,B)) & v1_funct_2(k8_rsspace(A,B),k2_zfmisc_1(B,B),B) & m2_relset_1(k8_rsspace(A,B),k2_zfmisc_1(B,B),B) ) ) ), file(rsspace,k8_rsspace), []). fof(dt_k9_rsspace,axiom,( ! [A,B] : ( ( ~ v3_struct_0(A) & v3_rlvect_1(A) & v4_rlvect_1(A) & v5_rlvect_1(A) & v6_rlvect_1(A) & v7_rlvect_1(A) & l2_rlvect_1(A) & m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) ) => ( v1_funct_1(k9_rsspace(A,B)) & v1_funct_2(k9_rsspace(A,B),k2_zfmisc_1(k1_numbers,B),B) & m2_relset_1(k9_rsspace(A,B),k2_zfmisc_1(k1_numbers,B),B) ) ) ), file(rsspace,k9_rsspace), []). fof(dt_k9_seq_1,axiom,( ! [A,B] : ( ( v1_funct_1(A) & v1_funct_2(A,k5_numbers,k1_numbers) & m1_relset_1(A,k5_numbers,k1_numbers) & v1_funct_1(B) & v1_funct_2(B,k5_numbers,k1_numbers) & m1_relset_1(B,k5_numbers,k1_numbers) ) => ( v1_funct_1(k9_seq_1(A,B)) & v1_funct_2(k9_seq_1(A,B),k5_numbers,k1_numbers) & m2_relset_1(k9_seq_1(A,B),k5_numbers,k1_numbers) ) ) ), file(seq_1,k9_seq_1), []). fof(dt_l1_normsp_1,axiom,( ! [A] : ( l1_normsp_1(A) => l2_rlvect_1(A) ) ), file(normsp_1,l1_normsp_1), []). fof(dt_l1_rlvect_1,axiom,( ! [A] : ( l1_rlvect_1(A) => l2_struct_0(A) ) ), file(rlvect_1,l1_rlvect_1), []). fof(dt_l1_struct_0,axiom,( $true ), file(struct_0,l1_struct_0), []). fof(dt_l2_rlvect_1,axiom,( ! [A] : ( l2_rlvect_1(A) => l1_rlvect_1(A) ) ), file(rlvect_1,l2_rlvect_1), []). fof(dt_l2_struct_0,axiom,( ! [A] : ( l2_struct_0(A) => l1_struct_0(A) ) ), file(struct_0,l2_struct_0), []). fof(dt_m1_relset_1,axiom,( $true ), file(relset_1,m1_relset_1), []). fof(dt_m1_subset_1,axiom,( $true ), file(subset_1,m1_subset_1), []). fof(dt_m2_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_normsp_1,axiom,( ! [A] : ( l1_normsp_1(A) => ( v1_funct_1(u1_normsp_1(A)) & v1_funct_2(u1_normsp_1(A),u1_struct_0(A),k1_numbers) & m2_relset_1(u1_normsp_1(A),u1_struct_0(A),k1_numbers) ) ) ), file(normsp_1,u1_normsp_1), []). fof(dt_u1_rlvect_1,axiom,( ! [A] : ( l1_rlvect_1(A) => ( v1_funct_1(u1_rlvect_1(A)) & v1_funct_2(u1_rlvect_1(A),k2_zfmisc_1(u1_struct_0(A),u1_struct_0(A)),u1_struct_0(A)) & m2_relset_1(u1_rlvect_1(A),k2_zfmisc_1(u1_struct_0(A),u1_struct_0(A)),u1_struct_0(A)) ) ) ), file(rlvect_1,u1_rlvect_1), []). fof(dt_u1_struct_0,axiom,( $true ), file(struct_0,u1_struct_0), []). fof(dt_u2_rlvect_1,axiom,( ! [A] : ( l2_rlvect_1(A) => ( v1_funct_1(u2_rlvect_1(A)) & v1_funct_2(u2_rlvect_1(A),k2_zfmisc_1(k1_numbers,u1_struct_0(A)),u1_struct_0(A)) & m2_relset_1(u2_rlvect_1(A),k2_zfmisc_1(k1_numbers,u1_struct_0(A)),u1_struct_0(A)) ) ) ), file(rlvect_1,u2_rlvect_1), []). fof(dt_u2_struct_0,axiom,( ! [A] : ( l2_struct_0(A) => m1_subset_1(u2_struct_0(A),u1_struct_0(A)) ) ), file(struct_0,u2_struct_0), []). fof(existence_l1_normsp_1,axiom,( ? [A] : l1_normsp_1(A) ), file(normsp_1,l1_normsp_1), []). fof(existence_l1_rlvect_1,axiom,( ? [A] : l1_rlvect_1(A) ), file(rlvect_1,l1_rlvect_1), []). fof(existence_l1_struct_0,axiom,( ? [A] : l1_struct_0(A) ), file(struct_0,l1_struct_0), []). fof(existence_l2_rlvect_1,axiom,( ? [A] : l2_rlvect_1(A) ), file(rlvect_1,l2_rlvect_1), []). fof(existence_l2_struct_0,axiom,( ? [A] : l2_struct_0(A) ), file(struct_0,l2_struct_0), []). fof(existence_m1_relset_1,axiom,( ! [A,B] : ? [C] : m1_relset_1(C,A,B) ), file(relset_1,m1_relset_1), []). fof(existence_m1_subset_1,axiom,( ! [A] : ? [B] : m1_subset_1(B,A) ), file(subset_1,m1_subset_1), []). fof(existence_m2_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_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_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_ordinal2,axiom, ( v1_ordinal1(k5_ordinal2) & v2_ordinal1(k5_ordinal2) & v3_ordinal1(k5_ordinal2) & ~ v1_xboole_0(k5_ordinal2) ), file(ordinal2,fc1_ordinal2), []). fof(fc1_seq_1,axiom,( ! [A,B] : ( ( v1_relat_1(A) & v1_funct_1(A) & v1_seq_1(A) ) => ( v1_xcmplx_0(k1_funct_1(A,B)) & v1_xreal_0(k1_funct_1(A,B)) ) ) ), file(seq_1,fc1_seq_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(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(fc21_xreal_0,axiom,( ! [A,B] : ( ( v1_xreal_0(A) & ~ v2_xreal_0(A) & v1_xreal_0(B) & ~ v3_xreal_0(B) ) => ( v1_xcmplx_0(k3_xcmplx_0(A,B)) & v1_xreal_0(k3_xcmplx_0(A,B)) & ~ v2_xreal_0(k3_xcmplx_0(A,B)) ) ) ), file(xreal_0,fc21_xreal_0), []). fof(fc22_xreal_0,axiom,( ! [A,B] : ( ( v1_xreal_0(A) & ~ v2_xreal_0(A) & v1_xreal_0(B) & ~ v3_xreal_0(B) ) => ( v1_xcmplx_0(k3_xcmplx_0(B,A)) & v1_xreal_0(k3_xcmplx_0(B,A)) & ~ v2_xreal_0(k3_xcmplx_0(B,A)) ) ) ), file(xreal_0,fc22_xreal_0), []). fof(fc23_xreal_0,axiom,( ! [A,B] : ( ( v1_xreal_0(A) & ~ v2_xreal_0(A) & v1_xreal_0(B) & ~ v2_xreal_0(B) ) => ( v1_xcmplx_0(k3_xcmplx_0(A,B)) & v1_xreal_0(k3_xcmplx_0(A,B)) & ~ v3_xreal_0(k3_xcmplx_0(A,B)) ) ) ), file(xreal_0,fc23_xreal_0), []). fof(fc24_xreal_0,axiom,( ! [A,B] : ( ( v1_xreal_0(A) & ~ v3_xreal_0(A) & v1_xreal_0(B) & ~ v3_xreal_0(B) ) => ( v1_xcmplx_0(k3_xcmplx_0(A,B)) & v1_xreal_0(k3_xcmplx_0(A,B)) & ~ v3_xreal_0(k3_xcmplx_0(A,B)) ) ) ), file(xreal_0,fc24_xreal_0), []). fof(fc27_xreal_0,axiom,( ! [A,B] : ( ( v1_xreal_0(A) & ~ v3_xreal_0(A) & v1_xreal_0(B) & ~ v2_xreal_0(B) ) => ( v1_xcmplx_0(k7_xcmplx_0(A,B)) & v1_xreal_0(k7_xcmplx_0(A,B)) & ~ v2_xreal_0(k7_xcmplx_0(A,B)) ) ) ), file(xreal_0,fc27_xreal_0), []). fof(fc28_xreal_0,axiom,( ! [A,B] : ( ( v1_xreal_0(A) & ~ v3_xreal_0(A) & v1_xreal_0(B) & ~ v2_xreal_0(B) ) => ( v1_xcmplx_0(k7_xcmplx_0(B,A)) & v1_xreal_0(k7_xcmplx_0(B,A)) & ~ v2_xreal_0(k7_xcmplx_0(B,A)) ) ) ), file(xreal_0,fc28_xreal_0), []). fof(fc29_xreal_0,axiom,( ! [A,B] : ( ( v1_xreal_0(A) & ~ v3_xreal_0(A) & v1_xreal_0(B) & ~ v3_xreal_0(B) ) => ( v1_xcmplx_0(k7_xcmplx_0(A,B)) & v1_xreal_0(k7_xcmplx_0(A,B)) & ~ v3_xreal_0(k7_xcmplx_0(A,B)) ) ) ), file(xreal_0,fc29_xreal_0), []). fof(fc2_membered,axiom, ( ~ v1_xboole_0(k1_numbers) & v1_membered(k1_numbers) & v2_membered(k1_numbers) ), file(membered,fc2_membered), []). fof(fc30_xreal_0,axiom,( ! [A,B] : ( ( v1_xreal_0(A) & ~ v2_xreal_0(A) & v1_xreal_0(B) & ~ v2_xreal_0(B) ) => ( v1_xcmplx_0(k7_xcmplx_0(A,B)) & v1_xreal_0(k7_xcmplx_0(A,B)) & ~ v3_xreal_0(k7_xcmplx_0(A,B)) ) ) ), file(xreal_0,fc30_xreal_0), []). fof(fc3_seq_1,axiom,( ! [A,B] : ( ( v1_relat_1(A) & v1_funct_1(A) & v1_seq_1(A) & v1_relat_1(B) & v1_funct_1(B) & v1_seq_1(B) ) => ( v1_relat_1(k3_seq_1(A,B)) & v1_funct_1(k3_seq_1(A,B)) & v1_seq_1(k3_seq_1(A,B)) ) ) ), file(seq_1,fc3_seq_1), []). 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_rsspace3,axiom,( ! [A,B,C,D,E] : ( ( ~ v1_xboole_0(A) & m1_subset_1(B,A) & v1_funct_1(C) & v1_funct_2(C,k2_zfmisc_1(A,A),A) & m1_relset_1(C,k2_zfmisc_1(A,A),A) & v1_funct_1(D) & v1_funct_2(D,k2_zfmisc_1(k1_numbers,A),A) & m1_relset_1(D,k2_zfmisc_1(k1_numbers,A),A) & v1_funct_1(E) & v1_funct_2(E,A,k1_numbers) & m1_relset_1(E,A,k1_numbers) ) => ( ~ v3_struct_0(g1_normsp_1(A,B,C,D,E)) & v1_normsp_1(g1_normsp_1(A,B,C,D,E)) ) ) ), file(rsspace3,fc4_rsspace3), []). fof(fc4_seq_1,axiom,( ! [A,B] : ( ( v1_relat_1(A) & v1_funct_1(A) & v1_seq_1(A) & v1_relat_1(B) & v1_funct_1(B) & v1_seq_1(B) ) => ( v1_relat_1(k4_seq_1(A,B)) & v1_funct_1(k4_seq_1(A,B)) & v1_seq_1(k4_seq_1(A,B)) ) ) ), file(seq_1,fc4_seq_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(fc4_xreal_0,axiom,( ! [A,B] : ( ( v1_xreal_0(A) & v1_xreal_0(B) ) => ( v1_xcmplx_0(k3_xcmplx_0(A,B)) & v1_xreal_0(k3_xcmplx_0(A,B)) ) ) ), file(xreal_0,fc4_xreal_0), []). 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(fc6_seq_1,axiom,( ! [A,B] : ( ( v1_xreal_0(A) & v1_relat_1(B) & v1_funct_1(B) & v1_seq_1(B) ) => ( v1_relat_1(k12_seq_1(B,A)) & v1_funct_1(k12_seq_1(B,A)) & v1_seq_1(k12_seq_1(B,A)) ) ) ), file(seq_1,fc6_seq_1), []). fof(fc6_xreal_0,axiom,( ! [A,B] : ( ( v1_xreal_0(A) & v1_xreal_0(B) ) => ( v1_xcmplx_0(k7_xcmplx_0(A,B)) & v1_xreal_0(k7_xcmplx_0(A,B)) ) ) ), file(xreal_0,fc6_xreal_0), []). fof(fc7_seq_1,axiom,( ! [A] : ( ( v1_relat_1(A) & v1_funct_1(A) & v1_seq_1(A) ) => ( v1_relat_1(k15_seq_1(A)) & v1_funct_1(k15_seq_1(A)) & v1_seq_1(k15_seq_1(A)) ) ) ), file(seq_1,fc7_seq_1), []). fof(fc7_xreal_0,axiom,( ! [A,B] : ( ( v1_xreal_0(A) & ~ v3_xreal_0(A) & v1_xreal_0(B) & ~ v3_xreal_0(B) ) => ( v1_xcmplx_0(k2_xcmplx_0(A,B)) & v1_xreal_0(k2_xcmplx_0(A,B)) & ~ v3_xreal_0(k2_xcmplx_0(A,B)) ) ) ), file(xreal_0,fc7_xreal_0), []). fof(fc8_seq_1,axiom,( ! [A] : ( ( v1_relat_1(A) & v1_funct_1(A) & v1_seq_1(A) ) => ( v1_relat_1(k20_seq_1(A)) & v1_funct_1(k20_seq_1(A)) & v1_seq_1(k20_seq_1(A)) ) ) ), file(seq_1,fc8_seq_1), []). fof(fc8_xreal_0,axiom,( ! [A,B] : ( ( v1_xreal_0(A) & ~ v2_xreal_0(A) & v1_xreal_0(B) & ~ v2_xreal_0(B) ) => ( v1_xcmplx_0(k2_xcmplx_0(A,B)) & v1_xreal_0(k2_xcmplx_0(A,B)) & ~ v2_xreal_0(k2_xcmplx_0(A,B)) ) ) ), file(xreal_0,fc8_xreal_0), []). fof(fc9_xreal_0,axiom,( ! [A,B] : ( ( v1_xreal_0(A) & v2_xreal_0(A) & v1_xreal_0(B) & ~ v3_xreal_0(B) ) => ( ~ v1_xboole_0(k2_xcmplx_0(A,B)) & v1_xcmplx_0(k2_xcmplx_0(A,B)) & v1_xreal_0(k2_xcmplx_0(A,B)) & v2_xreal_0(k2_xcmplx_0(A,B)) & ~ v3_xreal_0(k2_xcmplx_0(A,B)) ) ) ), file(xreal_0,fc9_xreal_0), []). fof(free_g1_normsp_1,axiom,( ! [A,B,C,D,E] : ( ( m1_subset_1(B,A) & v1_funct_1(C) & v1_funct_2(C,k2_zfmisc_1(A,A),A) & m1_relset_1(C,k2_zfmisc_1(A,A),A) & v1_funct_1(D) & v1_funct_2(D,k2_zfmisc_1(k1_numbers,A),A) & m1_relset_1(D,k2_zfmisc_1(k1_numbers,A),A) & v1_funct_1(E) & v1_funct_2(E,A,k1_numbers) & m1_relset_1(E,A,k1_numbers) ) => ! [F,G,H,I,J] : ( g1_normsp_1(A,B,C,D,E) = g1_normsp_1(F,G,H,I,J) => ( A = F & B = G & C = H & D = I & E = J ) ) ) ), file(normsp_1,g1_normsp_1), []). fof(involutiveness_k1_real_1,axiom,( ! [A] : ( m1_subset_1(A,k1_numbers) => k1_real_1(k1_real_1(A)) = A ) ), file(real_1,k1_real_1), []). 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(l28_lp_space,axiom,( ! [A] : ( m1_subset_1(A,k1_numbers) => ( ~ r1_xreal_0(A,0) => ! [B] : ( m1_subset_1(B,k1_numbers) => ! [C] : ( ( v1_funct_1(C) & v1_funct_2(C,k5_numbers,k1_numbers) & m2_relset_1(C,k5_numbers,k1_numbers) ) => ( v4_seq_2(C) => ! [D] : ( ( v1_funct_1(D) & v1_funct_2(D,k5_numbers,k1_numbers) & m2_relset_1(D,k5_numbers,k1_numbers) ) => ( ! [E] : ( m2_subset_1(E,k1_numbers,k5_numbers) => k2_seq_1(k5_numbers,k1_numbers,D,E) = k4_power(k18_complex1(k5_real_1(k2_seq_1(k5_numbers,k1_numbers,C,E),B)),A) ) => ( v4_seq_2(D) & k2_seq_2(D) = k4_power(k18_complex1(k5_real_1(k2_seq_2(C),B)),A) ) ) ) ) ) ) ) ) ), file(lp_space,l28_lp_space), []). fof(l29_lp_space,axiom,( ! [A] : ( m1_subset_1(A,k1_numbers) => ( ~ r1_xreal_0(A,0) => ! [B] : ( m1_subset_1(B,k1_numbers) => ! [C] : ( ( v1_funct_1(C) & v1_funct_2(C,k5_numbers,k1_numbers) & m2_relset_1(C,k5_numbers,k1_numbers) ) => ! [D] : ( ( v1_funct_1(D) & v1_funct_2(D,k5_numbers,k1_numbers) & m2_relset_1(D,k5_numbers,k1_numbers) ) => ( ( v4_seq_2(C) & v4_seq_2(D) ) => ! [E] : ( ( v1_funct_1(E) & v1_funct_2(E,k5_numbers,k1_numbers) & m2_relset_1(E,k5_numbers,k1_numbers) ) => ( ! [F] : ( m2_subset_1(F,k1_numbers,k5_numbers) => k2_seq_1(k5_numbers,k1_numbers,E,F) = k3_real_1(k4_power(k18_complex1(k5_real_1(k2_seq_1(k5_numbers,k1_numbers,C,F),B)),A),k2_seq_1(k5_numbers,k1_numbers,D,F)) ) => ( v4_seq_2(E) & k2_seq_2(E) = k3_real_1(k4_power(k18_complex1(k5_real_1(k2_seq_2(C),B)),A),k2_seq_2(D)) ) ) ) ) ) ) ) ) ) ), file(lp_space,l29_lp_space), []). fof(l30_lp_space,axiom,( ! [A] : ( m1_subset_1(A,k1_numbers) => ! [B] : ( ( v1_funct_1(B) & v1_funct_2(B,k5_numbers,k1_numbers) & m2_relset_1(B,k5_numbers,k1_numbers) ) => ( v4_seq_2(B) => ( ! [C] : ( m2_subset_1(C,k1_numbers,k5_numbers) => ? [D] : ( m2_subset_1(D,k1_numbers,k5_numbers) & r1_xreal_0(C,D) & ~ r1_xreal_0(k2_seq_1(k5_numbers,k1_numbers,B,D),A) ) ) | r1_xreal_0(k2_seq_2(B),A) ) ) ) ) ), file(lp_space,l30_lp_space), []). fof(l31_lp_space,axiom,( ! [A] : ( ( v1_funct_1(A) & v1_funct_2(A,k5_numbers,k1_numbers) & m2_relset_1(A,k5_numbers,k1_numbers) ) => ! [B] : ( ( v1_funct_1(B) & v1_funct_2(B,k5_numbers,k1_numbers) & m2_relset_1(B,k5_numbers,k1_numbers) ) => A = k10_seq_1(k9_seq_1(A,B),B) ) ) ), file(lp_space,l31_lp_space), []). fof(l32_lp_space,axiom,( ! [A] : ( m1_subset_1(A,k1_numbers) => ( r1_xreal_0(1,A) => ! [B] : ( ( v1_funct_1(B) & v1_funct_2(B,k5_numbers,k1_numbers) & m2_relset_1(B,k5_numbers,k1_numbers) ) => ! [C] : ( ( v1_funct_1(C) & v1_funct_2(C,k5_numbers,k1_numbers) & m2_relset_1(C,k5_numbers,k1_numbers) ) => ( ( v1_series_1(k1_lp_space(B,A)) & v1_series_1(k1_lp_space(C,A)) ) => v1_series_1(k1_lp_space(k9_seq_1(B,C),A)) ) ) ) ) ) ), file(lp_space,l32_lp_space), []). fof(l3_lp_space,axiom,( ! [A] : ( m1_subset_1(A,k1_numbers) => ! [B] : ( m1_subset_1(B,k1_numbers) => ( r1_xreal_0(0,A) => ( r1_xreal_0(B,0) | r1_xreal_0(0,k4_power(A,B)) ) ) ) ) ), file(lp_space,l3_lp_space), []). fof(projectivity_k16_complex1,axiom,( ! [A] : ( v1_xcmplx_0(A) => k16_complex1(k16_complex1(A)) = k16_complex1(A) ) ), file(complex1,k16_complex1), []). fof(projectivity_k17_complex1,axiom,( ! [A] : ( v1_xcmplx_0(A) => k17_complex1(k17_complex1(A)) = k17_complex1(A) ) ), file(complex1,k17_complex1), []). fof(projectivity_k18_complex1,axiom,( ! [A] : ( v1_xcmplx_0(A) => k18_complex1(k18_complex1(A)) = k18_complex1(A) ) ), file(complex1,k18_complex1), []). 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_seq_1,axiom,( ? [A] : ( v1_relat_1(A) & v1_funct_1(A) & v1_seq_1(A) ) ), file(seq_1,rc1_seq_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_xreal_0,axiom,( ? [A] : ( v1_xcmplx_0(A) & v1_xreal_0(A) ) ), file(xreal_0,rc1_xreal_0), []). fof(rc2_subset_1,axiom,( ! [A] : ? [B] : ( m1_subset_1(B,k1_zfmisc_1(A)) & v1_xboole_0(B) ) ), file(subset_1,rc2_subset_1), []). fof(rc2_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_rlvect_1,axiom,( ? [A] : ( l2_rlvect_1(A) & ~ v3_struct_0(A) ) ), file(rlvect_1,rc3_rlvect_1), []). fof(rc3_struct_0,axiom,( ? [A] : ( l1_struct_0(A) & ~ v3_struct_0(A) ) ), file(struct_0,rc3_struct_0), []). fof(rc3_xreal_0,axiom,( ? [A] : ( ~ v1_xboole_0(A) & v1_xcmplx_0(A) & v1_xreal_0(A) & ~ v2_xreal_0(A) & v3_xreal_0(A) ) ), file(xreal_0,rc3_xreal_0), []). fof(rc4_struct_0,axiom,( ? [A] : ( l2_struct_0(A) & ~ v3_struct_0(A) ) ), file(struct_0,rc4_struct_0), []). fof(rc4_xreal_0,axiom,( ? [A] : ( v1_xboole_0(A) & v1_xcmplx_0(A) & v1_xreal_0(A) & ~ v2_xreal_0(A) & ~ v3_xreal_0(A) ) ), file(xreal_0,rc4_xreal_0), []). fof(rc5_struct_0,axiom,( ! [A] : ( ( ~ v3_struct_0(A) & l1_struct_0(A) ) => ? [B] : ( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) & ~ v1_xboole_0(B) ) ) ), file(struct_0,rc5_struct_0), []). fof(redefinition_k10_seq_1,axiom,( ! [A,B] : ( ( v1_funct_1(A) & v1_funct_2(A,k5_numbers,k1_numbers) & m1_relset_1(A,k5_numbers,k1_numbers) & v1_funct_1(B) & v1_funct_2(B,k5_numbers,k1_numbers) & m1_relset_1(B,k5_numbers,k1_numbers) ) => k10_seq_1(A,B) = k4_seq_1(A,B) ) ), file(seq_1,k10_seq_1), []). fof(redefinition_k14_seq_1,axiom,( ! [A,B] : ( ( v1_funct_1(A) & v1_funct_2(A,k5_numbers,k1_numbers) & m1_relset_1(A,k5_numbers,k1_numbers) & v1_xreal_0(B) ) => k14_seq_1(A,B) = k12_seq_1(A,B) ) ), file(seq_1,k14_seq_1), []). fof(redefinition_k17_complex1,axiom,( ! [A] : ( v1_xcmplx_0(A) => k17_complex1(A) = k16_complex1(A) ) ), file(complex1,k17_complex1), []). fof(redefinition_k17_seq_1,axiom,( ! [A] : ( ( v1_funct_1(A) & v1_funct_2(A,k5_numbers,k1_numbers) & m1_relset_1(A,k5_numbers,k1_numbers) ) => k17_seq_1(A) = k15_seq_1(A) ) ), file(seq_1,k17_seq_1), []). fof(redefinition_k18_complex1,axiom,( ! [A] : ( v1_xcmplx_0(A) => k18_complex1(A) = k16_complex1(A) ) ), file(complex1,k18_complex1), []). 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_real_1,axiom,( ! [A] : ( m1_subset_1(A,k1_numbers) => k1_real_1(A) = k4_xcmplx_0(A) ) ), file(real_1,k1_real_1), []). fof(redefinition_k22_seq_1,axiom,( ! [A] : ( ( v1_funct_1(A) & v1_funct_2(A,k5_numbers,k1_numbers) & m1_relset_1(A,k5_numbers,k1_numbers) ) => k22_seq_1(A) = k20_seq_1(A) ) ), file(seq_1,k22_seq_1), []). fof(redefinition_k2_normsp_1,axiom,( ! [A,B,C] : ( ( ~ v3_struct_0(A) & l1_struct_0(A) & v1_funct_1(B) & v1_funct_2(B,k5_numbers,u1_struct_0(A)) & m1_relset_1(B,k5_numbers,u1_struct_0(A)) & m1_subset_1(C,k5_numbers) ) => k2_normsp_1(A,B,C) = k1_funct_1(B,C) ) ), file(normsp_1,k2_normsp_1), []). fof(redefinition_k2_seq_1,axiom,( ! [A,B,C,D] : ( ( v2_membered(B) & v1_funct_1(C) & m1_relset_1(C,A,B) ) => k2_seq_1(A,B,C,D) = k1_funct_1(C,D) ) ), file(seq_1,k2_seq_1), []). fof(redefinition_k2_seq_2,axiom,( ! [A] : ( ( v1_funct_1(A) & v1_funct_2(A,k5_numbers,k1_numbers) & m1_relset_1(A,k5_numbers,k1_numbers) ) => k2_seq_2(A) = k1_seq_2(A) ) ), file(seq_2,k2_seq_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_k4_power,axiom,( ! [A,B] : ( ( m1_subset_1(A,k1_numbers) & m1_subset_1(B,k1_numbers) ) => k4_power(A,B) = k3_power(A,B) ) ), file(power,k4_power), []). 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_k4_rlvect_1,axiom,( ! [A,B,C] : ( ( ~ v3_struct_0(A) & v3_rlvect_1(A) & l1_rlvect_1(A) & m1_subset_1(B,u1_struct_0(A)) & m1_subset_1(C,u1_struct_0(A)) ) => k4_rlvect_1(A,B,C) = k2_rlvect_1(A,B,C) ) ), file(rlvect_1,k4_rlvect_1), []). fof(redefinition_k5_numbers,axiom,( k5_numbers = k5_ordinal2 ), file(numbers,k5_numbers), []). fof(redefinition_k5_real_1,axiom,( ! [A,B] : ( ( m1_subset_1(A,k1_numbers) & m1_subset_1(B,k1_numbers) ) => k5_real_1(A,B) = k6_xcmplx_0(A,B) ) ), file(real_1,k5_real_1), []). fof(redefinition_k6_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_k6_seq_1,axiom,( ! [A,B,C,D] : ( ( v2_membered(B) & v1_funct_1(C) & m1_relset_1(C,A,B) & v1_funct_1(D) & m1_relset_1(D,A,B) ) => k6_seq_1(A,B,C,D) = k3_seq_1(C,D) ) ), file(seq_1,k6_seq_1), []). fof(redefinition_k8_funct_2,axiom,( ! [A,B,C,D] : ( ( ~ v1_xboole_0(A) & v1_funct_1(C) & v1_funct_2(C,A,B) & m1_relset_1(C,A,B) & m1_subset_1(D,A) ) => k8_funct_2(A,B,C,D) = k1_funct_1(C,D) ) ), file(funct_2,k8_funct_2), []). fof(redefinition_k9_seq_1,axiom,( ! [A,B] : ( ( v1_funct_1(A) & v1_funct_2(A,k5_numbers,k1_numbers) & m1_relset_1(A,k5_numbers,k1_numbers) & v1_funct_1(B) & v1_funct_2(B,k5_numbers,k1_numbers) & m1_relset_1(B,k5_numbers,k1_numbers) ) => k9_seq_1(A,B) = k3_seq_1(A,B) ) ), file(seq_1,k9_seq_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(s1_funct_2__e9_28__lp_space,axiom,( ! [A,B] : ( ( ~ v3_struct_0(A) & v3_rlvect_1(A) & v4_rlvect_1(A) & v5_rlvect_1(A) & v6_rlvect_1(A) & v7_rlvect_1(A) & v2_normsp_1(A) & l1_normsp_1(A) & v1_funct_1(B) & v1_funct_2(B,k5_numbers,u1_struct_0(A)) & m2_relset_1(B,k5_numbers,u1_struct_0(A)) ) => ( ! [C] : ~ ( r2_hidden(C,k5_numbers) & ! [D] : ~ ( r2_hidden(D,k1_numbers) & ? [E] : ( m2_subset_1(E,k1_numbers,k5_numbers) & C = E & ? [F] : ( v1_funct_1(F) & v1_funct_2(F,k5_numbers,k1_numbers) & m2_relset_1(F,k5_numbers,k1_numbers) & ! [G] : ( m2_subset_1(G,k1_numbers,k5_numbers) => k2_seq_1(k5_numbers,k1_numbers,F,G) = k2_seq_1(k5_numbers,k1_numbers,k2_rsspace(k2_normsp_1(A,B,G)),E) ) & v4_seq_2(F) & D = k2_seq_2(F) ) ) ) ) => ? [C] : ( v1_funct_1(C) & v1_funct_2(C,k5_numbers,k1_numbers) & m2_relset_1(C,k5_numbers,k1_numbers) & ! [D] : ( r2_hidden(D,k5_numbers) => ? [H] : ( m2_subset_1(H,k1_numbers,k5_numbers) & D = H & ? [I] : ( v1_funct_1(I) & v1_funct_2(I,k5_numbers,k1_numbers) & m2_relset_1(I,k5_numbers,k1_numbers) & ! [J] : ( m2_subset_1(J,k1_numbers,k5_numbers) => k2_seq_1(k5_numbers,k1_numbers,I,J) = k2_seq_1(k5_numbers,k1_numbers,k2_rsspace(k2_normsp_1(A,B,J)),H) ) & v4_seq_2(I) & k1_funct_1(C,D) = k2_seq_2(I) ) ) ) ) ) ) ), file(lp_space,s1_funct_2__e9_28__lp_space), []). fof(s1_nat_1__e4_28_4_2__lp_space,axiom,( ! [A,B,C,D,E] : ( ( m1_subset_1(A,k1_numbers) & ~ v3_struct_0(B) & v3_rlvect_1(B) & v4_rlvect_1(B) & v5_rlvect_1(B) & v6_rlvect_1(B) & v7_rlvect_1(B) & v2_normsp_1(B) & l1_normsp_1(B) & v1_funct_1(C) & v1_funct_2(C,k5_numbers,u1_struct_0(B)) & m2_relset_1(C,k5_numbers,u1_struct_0(B)) & v1_funct_1(D) & v1_funct_2(D,k5_numbers,k1_numbers) & m2_relset_1(D,k5_numbers,k1_numbers) & m2_subset_1(E,k1_numbers,k5_numbers) ) => ( ( ! [F] : ( ( v1_funct_1(F) & v1_funct_2(F,k5_numbers,k1_numbers) & m2_relset_1(F,k5_numbers,k1_numbers) ) => ( ! [G] : ( m2_subset_1(G,k1_numbers,k5_numbers) => k2_seq_1(k5_numbers,k1_numbers,F,G) = k2_seq_1(k5_numbers,k1_numbers,k1_series_1(k1_lp_space(k2_rsspace(k6_rlvect_1(B,k2_normsp_1(B,C,G),k2_normsp_1(B,C,E))),A)),0) ) => ( v4_seq_2(F) & k2_seq_2(F) = k2_seq_1(k5_numbers,k1_numbers,k1_series_1(k1_lp_space(k10_seq_1(k2_rsspace(D),k2_rsspace(k2_normsp_1(B,C,E))),A)),0) ) ) ) & ! [H] : ( m2_subset_1(H,k1_numbers,k5_numbers) => ( ! [I] : ( ( v1_funct_1(I) & v1_funct_2(I,k5_numbers,k1_numbers) & m2_relset_1(I,k5_numbers,k1_numbers) ) => ( ! [J] : ( m2_subset_1(J,k1_numbers,k5_numbers) => k2_seq_1(k5_numbers,k1_numbers,I,J) = k2_seq_1(k5_numbers,k1_numbers,k1_series_1(k1_lp_space(k2_rsspace(k6_rlvect_1(B,k2_normsp_1(B,C,J),k2_normsp_1(B,C,E))),A)),H) ) => ( v4_seq_2(I) & k2_seq_2(I) = k2_seq_1(k5_numbers,k1_numbers,k1_series_1(k1_lp_space(k10_seq_1(k2_rsspace(D),k2_rsspace(k2_normsp_1(B,C,E))),A)),H) ) ) ) => ! [K] : ( ( v1_funct_1(K) & v1_funct_2(K,k5_numbers,k1_numbers) & m2_relset_1(K,k5_numbers,k1_numbers) ) => ( ! [L] : ( m2_subset_1(L,k1_numbers,k5_numbers) => k2_seq_1(k5_numbers,k1_numbers,K,L) = k2_seq_1(k5_numbers,k1_numbers,k1_series_1(k1_lp_space(k2_rsspace(k6_rlvect_1(B,k2_normsp_1(B,C,L),k2_normsp_1(B,C,E))),A)),k1_nat_1(H,1)) ) => ( v4_seq_2(K) & k2_seq_2(K) = k2_seq_1(k5_numbers,k1_numbers,k1_series_1(k1_lp_space(k10_seq_1(k2_rsspace(D),k2_rsspace(k2_normsp_1(B,C,E))),A)),k1_nat_1(H,1)) ) ) ) ) ) ) => ! [H] : ( m2_subset_1(H,k1_numbers,k5_numbers) => ! [M] : ( ( v1_funct_1(M) & v1_funct_2(M,k5_numbers,k1_numbers) & m2_relset_1(M,k5_numbers,k1_numbers) ) => ( ! [N] : ( m2_subset_1(N,k1_numbers,k5_numbers) => k2_seq_1(k5_numbers,k1_numbers,M,N) = k2_seq_1(k5_numbers,k1_numbers,k1_series_1(k1_lp_space(k2_rsspace(k6_rlvect_1(B,k2_normsp_1(B,C,N),k2_normsp_1(B,C,E))),A)),H) ) => ( v4_seq_2(M) & k2_seq_2(M) = k2_seq_1(k5_numbers,k1_numbers,k1_series_1(k1_lp_space(k10_seq_1(k2_rsspace(D),k2_rsspace(k2_normsp_1(B,C,E))),A)),H) ) ) ) ) ) ) ), file(lp_space,s1_nat_1__e4_28_4_2__lp_space), []). fof(s1_seq_1__e2_28_4_2_3_1_1__lp_space,axiom,( ! [A,B,C,D,E] : ( ( m1_subset_1(A,k1_numbers) & ~ v3_struct_0(B) & v3_rlvect_1(B) & v4_rlvect_1(B) & v5_rlvect_1(B) & v6_rlvect_1(B) & v7_rlvect_1(B) & v2_normsp_1(B) & l1_normsp_1(B) & v1_funct_1(C) & v1_funct_2(C,k5_numbers,u1_struct_0(B)) & m2_relset_1(C,k5_numbers,u1_struct_0(B)) & m2_subset_1(D,k1_numbers,k5_numbers) & m2_subset_1(E,k1_numbers,k5_numbers) ) => ? [F] : ( v1_funct_1(F) & v1_funct_2(F,k5_numbers,k1_numbers) & m2_relset_1(F,k5_numbers,k1_numbers) & ! [G] : ( m2_subset_1(G,k1_numbers,k5_numbers) => k2_seq_1(k5_numbers,k1_numbers,F,G) = k2_seq_1(k5_numbers,k1_numbers,k1_series_1(k1_lp_space(k2_rsspace(k6_rlvect_1(B,k2_normsp_1(B,C,G),k2_normsp_1(B,C,D))),A)),E) ) ) ) ), file(lp_space,s1_seq_1__e2_28_4_2_3_1_1__lp_space), []). fof(s1_seq_1__e2_28_4_3_1__lp_space,axiom,( ! [A,B,C,D,E] : ( ( m1_subset_1(A,k1_numbers) & ~ v3_struct_0(B) & v3_rlvect_1(B) & v4_rlvect_1(B) & v5_rlvect_1(B) & v6_rlvect_1(B) & v7_rlvect_1(B) & v2_normsp_1(B) & l1_normsp_1(B) & v1_funct_1(C) & v1_funct_2(C,k5_numbers,u1_struct_0(B)) & m2_relset_1(C,k5_numbers,u1_struct_0(B)) & m2_subset_1(D,k1_numbers,k5_numbers) & m2_subset_1(E,k1_numbers,k5_numbers) ) => ? [F] : ( v1_funct_1(F) & v1_funct_2(F,k5_numbers,k1_numbers) & m2_relset_1(F,k5_numbers,k1_numbers) & ! [G] : ( m2_subset_1(G,k1_numbers,k5_numbers) => k2_seq_1(k5_numbers,k1_numbers,F,G) = k2_seq_1(k5_numbers,k1_numbers,k1_series_1(k1_lp_space(k2_rsspace(k6_rlvect_1(B,k2_normsp_1(B,C,G),k2_normsp_1(B,C,D))),A)),E) ) ) ) ), file(lp_space,s1_seq_1__e2_28_4_3_1__lp_space), []). fof(s1_seq_1__e3_28_2__lp_space,axiom,( ! [A,B,C] : ( ( ~ v3_struct_0(A) & v3_rlvect_1(A) & v4_rlvect_1(A) & v5_rlvect_1(A) & v6_rlvect_1(A) & v7_rlvect_1(A) & v2_normsp_1(A) & l1_normsp_1(A) & v1_funct_1(B) & v1_funct_2(B,k5_numbers,u1_struct_0(A)) & m2_relset_1(B,k5_numbers,u1_struct_0(A)) & m2_subset_1(C,k1_numbers,k5_numbers) ) => ? [D] : ( v1_funct_1(D) & v1_funct_2(D,k5_numbers,k1_numbers) & m2_relset_1(D,k5_numbers,k1_numbers) & ! [E] : ( m2_subset_1(E,k1_numbers,k5_numbers) => k2_seq_1(k5_numbers,k1_numbers,D,E) = k2_seq_1(k5_numbers,k1_numbers,k2_rsspace(k2_normsp_1(A,B,E)),C) ) ) ) ), file(lp_space,s1_seq_1__e3_28_2__lp_space), []). 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(spc4_boole,axiom,( ~ v1_xboole_0(4) ), file(boole,spc4_boole), []). fof(spc4_numerals,axiom, ( v2_xreal_0(4) & m1_subset_1(4,k5_numbers) ), file(numerals,spc4_numerals), []). fof(t107_xcmplx_1,axiom,( ! [A] : ( v1_xcmplx_0(A) => ( A != 0 => k3_xcmplx_0(A,k7_xcmplx_0(1,A)) = 1 ) ) ), file(xcmplx_1,t107_xcmplx_1), []). fof(t10_lp_space,axiom,( ! [A] : ( m1_subset_1(A,k1_numbers) => ( r1_xreal_0(1,A) => ! [B] : ( ( ~ v3_struct_0(B) & l1_normsp_1(B) ) => ( B = g1_normsp_1(k2_lp_space(A),k10_rsspace(k7_rsspace,k2_lp_space(A)),k8_rsspace(k7_rsspace,k2_lp_space(A)),k9_rsspace(k7_rsspace,k2_lp_space(A)),k3_lp_space(A)) => ( u1_struct_0(B) = k2_lp_space(A) & ! [C] : ( m1_subset_1(C,u1_struct_0(B)) <=> ( v1_funct_1(C) & v1_funct_2(C,k5_numbers,k1_numbers) & m2_relset_1(C,k5_numbers,k1_numbers) & v1_series_1(k1_lp_space(k2_rsspace(C),A)) ) ) & k1_rlvect_1(B) = k6_rsspace & ! [C] : ( m1_subset_1(C,u1_struct_0(B)) => C = k2_rsspace(C) ) & ! [C] : ( m1_subset_1(C,u1_struct_0(B)) => ! [D] : ( m1_subset_1(D,u1_struct_0(B)) => k2_rlvect_1(B,C,D) = k9_seq_1(k2_rsspace(C),k2_rsspace(D)) ) ) & ! [C] : ( m1_subset_1(C,k1_numbers) => ! [D] : ( m1_subset_1(D,u1_struct_0(B)) => k3_rlvect_1(B,D,C) = k14_seq_1(k2_rsspace(D),C) ) ) & ! [C] : ( m1_subset_1(C,u1_struct_0(B)) => ( k5_rlvect_1(B,C) = k17_seq_1(k2_rsspace(C)) & k2_rsspace(k5_rlvect_1(B,C)) = k17_seq_1(k2_rsspace(C)) ) ) & ! [C] : ( m1_subset_1(C,u1_struct_0(B)) => ! [D] : ( m1_subset_1(D,u1_struct_0(B)) => k6_rlvect_1(B,C,D) = k10_seq_1(k2_rsspace(C),k2_rsspace(D)) ) ) & ! [C] : ( m1_subset_1(C,u1_struct_0(B)) => v1_series_1(k1_lp_space(k2_rsspace(C),A)) ) & ! [C] : ( m1_subset_1(C,u1_struct_0(B)) => k1_normsp_1(B,C) = k4_power(k2_series_1(k1_lp_space(k2_rsspace(C),A)),k6_real_1(1,A)) ) ) ) ) ) ) ), file(lp_space,t10_lp_space), []). fof(t10_rsspace3,axiom,( ! [A] : ( ( ~ v3_struct_0(A) & v3_rlvect_1(A) & v4_rlvect_1(A) & v5_rlvect_1(A) & v6_rlvect_1(A) & v7_rlvect_1(A) & v2_normsp_1(A) & l1_normsp_1(A) ) => ! [B] : ( ( v1_funct_1(B) & v1_funct_2(B,k5_numbers,u1_struct_0(A)) & m2_relset_1(B,k5_numbers,u1_struct_0(A)) ) => ( v1_rsspace3(B,A) <=> ! [C] : ( m1_subset_1(C,k1_numbers) => ~ ( ~ r1_xreal_0(C,0) & ! [D] : ( m2_subset_1(D,k1_numbers,k5_numbers) => ? [E] : ( m2_subset_1(E,k1_numbers,k5_numbers) & ? [F] : ( m2_subset_1(F,k1_numbers,k5_numbers) & r1_xreal_0(D,E) & r1_xreal_0(D,F) & r1_xreal_0(C,k1_normsp_1(A,k6_rlvect_1(A,k2_normsp_1(A,B,E),k2_normsp_1(A,B,F)))) ) ) ) ) ) ) ) ) ), file(rsspace3,t10_rsspace3), []). fof(t11_seq_1,axiom,( ! [A] : ( ( v1_funct_1(A) & v1_funct_2(A,k5_numbers,k1_numbers) & m2_relset_1(A,k5_numbers,k1_numbers) ) => ! [B] : ( ( v1_funct_1(B) & v1_funct_2(B,k5_numbers,k1_numbers) & m2_relset_1(B,k5_numbers,k1_numbers) ) => ! [C] : ( ( v1_funct_1(C) & v1_funct_2(C,k5_numbers,k1_numbers) & m2_relset_1(C,k5_numbers,k1_numbers) ) => ( A = k6_seq_1(k5_numbers,k1_numbers,B,C) <=> ! [D] : ( m2_subset_1(D,k1_numbers,k5_numbers) => k2_seq_1(k5_numbers,k1_numbers,A,D) = k3_real_1(k2_seq_1(k5_numbers,k1_numbers,B,D),k2_seq_1(k5_numbers,k1_numbers,C,D)) ) ) ) ) ) ), file(seq_1,t11_seq_1), []). fof(t122_real_2,axiom,( ! [A] : ( v1_xreal_0(A) => ! [B] : ( v1_xreal_0(B) => ~ ( ( ( ~ r1_xreal_0(0,A) & ~ r1_xreal_0(0,B) ) | ( ~ r1_xreal_0(A,0) & ~ r1_xreal_0(B,0) ) ) & r1_xreal_0(k3_xcmplx_0(A,B),0) ) ) ) ), file(real_2,t122_real_2), []). fof(t127_real_2,axiom,( ! [A] : ( v1_xreal_0(A) => ! [B] : ( v1_xreal_0(B) => ~ ( ( ( ~ r1_xreal_0(A,0) & ~ r1_xreal_0(B,0) ) | ( ~ r1_xreal_0(0,A) & ~ r1_xreal_0(0,B) ) ) & r1_xreal_0(k7_xcmplx_0(A,B),0) ) ) ) ), file(real_2,t127_real_2), []). fof(t132_complex1,axiom,( ! [A] : ( v1_xcmplx_0(A) => r1_xreal_0(0,k17_complex1(A)) ) ), file(complex1,t132_complex1), []). fof(t14_seq_1,axiom,( ! [A] : ( ( v1_funct_1(A) & v1_funct_2(A,k5_numbers,k1_numbers) & m2_relset_1(A,k5_numbers,k1_numbers) ) => ! [B] : ( ( v1_funct_1(B) & v1_funct_2(B,k5_numbers,k1_numbers) & m2_relset_1(B,k5_numbers,k1_numbers) ) => ( A = k17_seq_1(B) <=> ! [C] : ( m2_subset_1(C,k1_numbers,k5_numbers) => k2_seq_1(k5_numbers,k1_numbers,A,C) = k1_real_1(k2_seq_1(k5_numbers,k1_numbers,B,C)) ) ) ) ) ), file(seq_1,t14_seq_1), []). fof(t15_seq_1,axiom,( ! [A] : ( ( v1_funct_1(A) & v1_funct_2(A,k5_numbers,k1_numbers) & m2_relset_1(A,k5_numbers,k1_numbers) ) => ! [B] : ( ( v1_funct_1(B) & v1_funct_2(B,k5_numbers,k1_numbers) & m2_relset_1(B,k5_numbers,k1_numbers) ) => k10_seq_1(A,B) = k9_seq_1(A,k17_seq_1(B)) ) ) ), file(seq_1,t15_seq_1), []). fof(t16_seq_1,axiom,( ! [A] : ( ( v1_funct_1(A) & v1_funct_2(A,k5_numbers,k1_numbers) & m2_relset_1(A,k5_numbers,k1_numbers) ) => ! [B] : ( ( v1_funct_1(B) & v1_funct_2(B,k5_numbers,k1_numbers) & m2_relset_1(B,k5_numbers,k1_numbers) ) => ( A = k22_seq_1(B) <=> ! [C] : ( m2_subset_1(C,k1_numbers,k5_numbers) => k2_seq_1(k5_numbers,k1_numbers,A,C) = k18_complex1(k2_seq_1(k5_numbers,k1_numbers,B,C)) ) ) ) ) ), file(seq_1,t16_seq_1), []). fof(t18_nat_1,axiom,( ! [A] : ( v4_ordinal2(A) => r1_xreal_0(0,A) ) ), file(nat_1,t18_nat_1), []). fof(t1_arithm,axiom,( ! [A] : ( v1_xcmplx_0(A) => k2_xcmplx_0(A,0) = A ) ), file(arithm,t1_arithm), []). fof(t1_lp_space,axiom,( ! [A] : ( m1_subset_1(A,k1_numbers) => ! [B] : ( m1_subset_1(B,k1_numbers) => ! [C] : ( m1_subset_1(C,k1_numbers) => ~ ( r1_xreal_0(0,A) & ~ r1_xreal_0(B,A) & ~ r1_xreal_0(C,0) & r1_xreal_0(k4_power(B,C),k4_power(A,C)) ) ) ) ) ), file(lp_space,t1_lp_space), []). 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(t20_seq_1,axiom,( ! [A] : ( ( v1_funct_1(A) & v1_funct_2(A,k5_numbers,k1_numbers) & m2_relset_1(A,k5_numbers,k1_numbers) ) => ! [B] : ( ( v1_funct_1(B) & v1_funct_2(B,k5_numbers,k1_numbers) & m2_relset_1(B,k5_numbers,k1_numbers) ) => ! [C] : ( ( v1_funct_1(C) & v1_funct_2(C,k5_numbers,k1_numbers) & m2_relset_1(C,k5_numbers,k1_numbers) ) => k9_seq_1(k9_seq_1(A,B),C) = k9_seq_1(A,k9_seq_1(B,C)) ) ) ) ), file(seq_1,t20_seq_1), []). fof(t20_series_1,axiom,( ! [A] : ( ( v1_funct_1(A) & v1_funct_2(A,k5_numbers,k1_numbers) & m2_relset_1(A,k5_numbers,k1_numbers) ) => ( ! [B] : ( m2_subset_1(B,k1_numbers,k5_numbers) => r1_xreal_0(0,k8_funct_2(k5_numbers,k1_numbers,A,B)) ) => ( v1_seq_2(k1_series_1(A)) <=> v1_series_1(A) ) ) ) ), file(series_1,t20_series_1), []). fof(t218_xreal_1,axiom,( ! [A] : ( v1_xreal_0(A) => ~ ( ~ r1_xreal_0(A,0) & r1_xreal_0(A,k7_xcmplx_0(A,2)) ) ) ), file(xreal_1,t218_xreal_1), []). fof(t21_series_1,axiom,( ! [A] : ( ( v1_funct_1(A) & v1_funct_2(A,k5_numbers,k1_numbers) & m2_relset_1(A,k5_numbers,k1_numbers) ) => ( ( v1_series_1(A) & ! [B] : ( m2_subset_1(B,k1_numbers,k5_numbers) => r1_xreal_0(0,k8_funct_2(k5_numbers,k1_numbers,A,B)) ) ) => r1_xreal_0(0,k2_series_1(A)) ) ) ), file(series_1,t21_series_1), []). fof(t2_arithm,axiom,( ! [A] : ( v1_xcmplx_0(A) => k3_xcmplx_0(A,0) = 0 ) ), file(arithm,t2_arithm), []). fof(t2_holder_1,axiom,( ! [A] : ( m1_subset_1(A,k1_numbers) => ! [B] : ( m1_subset_1(B,k1_numbers) => ~ ( ~ r1_xreal_0(A,0) & ~ r1_xreal_0(B,0) & ? [C] : ( m1_subset_1(C,k1_numbers) & r1_xreal_0(0,C) & k4_power(k4_power(C,A),B) != k4_power(C,k4_real_1(A,B)) ) ) ) ) ), file(holder_1,t2_holder_1), []). 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_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(t31_xreal_1,axiom,( ! [A] : ( v1_xreal_0(A) => ! [B] : ( v1_xreal_0(B) => ~ ( ~ r1_xreal_0(A,0) & r1_xreal_0(k2_xcmplx_0(B,A),B) ) ) ) ), file(xreal_1,t31_xreal_1), []). fof(t38_power,axiom,( ! [A] : ( v1_xreal_0(A) => ! [B] : ( v1_xreal_0(B) => ! [C] : ( v1_xreal_0(C) => ( ~ r1_xreal_0(A,0) => k3_power(k3_power(A,B),C) = k3_power(A,k3_xcmplx_0(B,C)) ) ) ) ) ), file(power,t38_power), []). 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_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_rsspace,axiom,( ! [A] : ( ( v1_funct_1(A) & v1_funct_2(A,k5_numbers,k1_numbers) & m2_relset_1(A,k5_numbers,k1_numbers) ) => k2_rsspace(A) = A ) ), file(rsspace,t3_rsspace), []). fof(t3_seq_2,axiom,( ! [A] : ( v1_xreal_0(A) => ( ~ r1_xreal_0(A,0) => ( ~ r1_xreal_0(k7_xcmplx_0(A,2),0) & ~ r1_xreal_0(k7_xcmplx_0(A,4),0) ) ) ) ), file(seq_2,t3_seq_2), []). fof(t3_subset,axiom,( ! [A,B] : ( m1_subset_1(A,k1_zfmisc_1(B)) <=> r1_tarski(A,B) ) ), file(subset,t3_subset), []). fof(t47_rlvect_1,axiom,( ! [A] : ( ( ~ v3_struct_0(A) & v4_rlvect_1(A) & v5_rlvect_1(A) & v6_rlvect_1(A) & l1_rlvect_1(A) ) => ! [B] : ( m1_subset_1(B,u1_struct_0(A)) => ! [C] : ( m1_subset_1(C,u1_struct_0(A)) => k5_rlvect_1(A,k6_rlvect_1(A,B,C)) = k2_rlvect_1(A,C,k5_rlvect_1(A,B)) ) ) ) ), file(rlvect_1,t47_rlvect_1), []). 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_rsspace2,axiom,( ! [A] : ( ( v1_funct_1(A) & v1_funct_2(A,k5_numbers,k1_numbers) & m2_relset_1(A,k5_numbers,k1_numbers) ) => ( ! [B] : ( m2_subset_1(B,k1_numbers,k5_numbers) => r1_xreal_0(0,k2_seq_1(k5_numbers,k1_numbers,A,B)) ) => ( ! [B] : ( m2_subset_1(B,k1_numbers,k5_numbers) => r1_xreal_0(0,k2_seq_1(k5_numbers,k1_numbers,k1_series_1(A),B)) ) & ! [B] : ( m2_subset_1(B,k1_numbers,k5_numbers) => r1_xreal_0(k2_seq_1(k5_numbers,k1_numbers,A,B),k2_seq_1(k5_numbers,k1_numbers,k1_series_1(A),B)) ) & ( v1_series_1(A) => ( ! [B] : ( m2_subset_1(B,k1_numbers,k5_numbers) => r1_xreal_0(k2_seq_1(k5_numbers,k1_numbers,k1_series_1(A),B),k2_series_1(A)) ) & ! [B] : ( m2_subset_1(B,k1_numbers,k5_numbers) => r1_xreal_0(k2_seq_1(k5_numbers,k1_numbers,A,B),k2_series_1(A)) ) ) ) ) ) ) ), file(rsspace2,t4_rsspace2), []). 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(t58_seq_4,axiom,( ! [A] : ( ( v1_funct_1(A) & v1_funct_2(A,k5_numbers,k1_numbers) & m2_relset_1(A,k5_numbers,k1_numbers) ) => ( v4_seq_2(A) <=> ! [B] : ( v1_xreal_0(B) => ~ ( ~ r1_xreal_0(B,0) & ! [C] : ( m2_subset_1(C,k1_numbers,k5_numbers) => ? [D] : ( m2_subset_1(D,k1_numbers,k5_numbers) & r1_xreal_0(C,D) & r1_xreal_0(B,k18_complex1(k5_real_1(k8_funct_2(k5_numbers,k1_numbers,A,D),k8_funct_2(k5_numbers,k1_numbers,A,C)))) ) ) ) ) ) ) ), file(seq_4,t58_seq_4), []). fof(t5_arithm,axiom,( ! [A] : ( v1_xcmplx_0(A) => k7_xcmplx_0(0,A) = 0 ) ), file(arithm,t5_arithm), []). 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_xcmplx_1,axiom,( ! [A] : ( v1_xcmplx_0(A) => ( A != 0 => k7_xcmplx_0(A,A) = 1 ) ) ), file(xcmplx_1,t60_xcmplx_1), []). 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_normsp_1,axiom,( ! [A] : ( ( ~ v3_struct_0(A) & v3_rlvect_1(A) & v4_rlvect_1(A) & v5_rlvect_1(A) & v6_rlvect_1(A) & v7_rlvect_1(A) & v2_normsp_1(A) & l1_normsp_1(A) ) => ! [B] : ( m1_subset_1(B,u1_struct_0(A)) => k1_normsp_1(A,k5_rlvect_1(A,B)) = k1_normsp_1(A,B) ) ) ), file(normsp_1,t6_normsp_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(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(t7_boole,axiom,( ! [A,B] : ~ ( r2_hidden(A,B) & v1_xboole_0(B) ) ), file(boole,t7_boole), []). fof(t7_real,axiom,( ! [A] : ( v1_xreal_0(A) => ! [B] : ( v1_xreal_0(B) => ~ ( ~ r1_xreal_0(A,B) & ~ v2_xreal_0(A) & ~ v3_xreal_0(B) ) ) ) ), file(real,t7_real), []). fof(t8_boole,axiom,( ! [A,B] : ~ ( v1_xboole_0(A) & A != B & v1_xboole_0(B) ) ), file(boole,t8_boole), []). fof(t8_normsp_1,axiom,( ! [A] : ( ( ~ v3_struct_0(A) & v3_rlvect_1(A) & v4_rlvect_1(A) & v5_rlvect_1(A) & v6_rlvect_1(A) & v7_rlvect_1(A) & v2_normsp_1(A) & l1_normsp_1(A) ) => ! [B] : ( m1_subset_1(B,u1_struct_0(A)) => r1_xreal_0(0,k1_normsp_1(A,B)) ) ) ), file(normsp_1,t8_normsp_1), []). 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), []).