% Mizar problem: t49_hermitan,hermitan,1689,74 fof(t49_hermitan,conjecture,( ! [A] : ( ( ~ v3_struct_0(A) & v3_rlvect_1(A) & v4_rlvect_1(A) & v5_rlvect_1(A) & v6_rlvect_1(A) & v12_vectsp_1(A,k1_complfld) & l4_vectsp_1(A,k1_complfld) ) => ! [B] : ( ( v1_funct_1(B) & v1_funct_2(B,k2_zfmisc_1(u1_struct_0(A),u1_struct_0(A)),u1_struct_0(k1_complfld)) & v2_bilinear(B,k1_complfld,A,A) & v4_bilinear(B,k1_complfld,A,A) & v3_hermitan(B,A) & v5_hermitan(B,A) & m2_relset_1(B,k2_zfmisc_1(u1_struct_0(A),u1_struct_0(A)),u1_struct_0(k1_complfld)) ) => ! [C] : ( m1_subset_1(C,u1_struct_0(A)) => ! [D] : ( m1_subset_1(D,u1_struct_0(A)) => r1_xreal_0(k7_square_1(k3_complfld(k8_funct_2(k2_zfmisc_1(u1_struct_0(A),u1_struct_0(A)),u1_struct_0(k1_complfld),B,k1_domain_1(u1_struct_0(A),u1_struct_0(A),C,D)))),k4_real_1(k3_complfld(k8_funct_2(k2_zfmisc_1(u1_struct_0(A),u1_struct_0(A)),u1_struct_0(k1_complfld),B,k1_domain_1(u1_struct_0(A),u1_struct_0(A),C,C))),k3_complfld(k8_funct_2(k2_zfmisc_1(u1_struct_0(A),u1_struct_0(A)),u1_struct_0(k1_complfld),B,k1_domain_1(u1_struct_0(A),u1_struct_0(A),D,D))))) ) ) ) ) ), inference(mizar_bg_added,[status(thm)],[abstractness_v3_vectsp_1,antisymmetry_r2_hidden,cc10_membered,cc11_membered,cc12_membered,cc13_membered,cc14_membered,cc15_membered,cc16_membered,cc17_membered,cc18_membered,cc19_membered,cc1_complfld,cc1_membered,cc1_relset_1,cc1_xcmplx_0,cc1_xreal_0,cc20_membered,cc2_hermitan,cc2_membered,cc2_xcmplx_0,cc2_xreal_0,cc3_hermitan,cc3_membered,cc3_nat_1,cc3_xreal_0,cc4_hermitan,cc4_membered,cc4_xreal_0,cc5_hermitan,cc5_xreal_0,cc6_hermitan,cc6_membered,cc6_xreal_0,cc7_xreal_0,cc8_xreal_0,commutativity_k2_tarski,commutativity_k3_xcmplx_0,commutativity_k4_real_1,connectedness_r1_xreal_0,d3_square_1,d5_tarski,d9_hermitan,dt_g3_vectsp_1,dt_k16_complex1,dt_k1_complex1,dt_k1_complfld,dt_k1_domain_1,dt_k1_funct_1,dt_k1_numbers,dt_k1_tarski,dt_k1_xboole_0,dt_k1_zfmisc_1,dt_k2_tarski,dt_k2_zfmisc_1,dt_k3_complex1,dt_k3_complfld,dt_k3_xcmplx_0,dt_k4_hermitan,dt_k4_real_1,dt_k4_tarski,dt_k5_square_1,dt_k7_square_1,dt_k8_funct_2,dt_l1_group_1,dt_l1_rlvect_1,dt_l1_struct_0,dt_l1_vectsp_1,dt_l2_struct_0,dt_l2_vectsp_1,dt_l3_vectsp_1,dt_l4_vectsp_1,dt_m1_relset_1,dt_m1_subset_1,dt_m2_relset_1,dt_u1_group_1,dt_u1_rlvect_1,dt_u1_struct_0,dt_u1_vectsp_1,dt_u2_struct_0,existence_l1_group_1,existence_l1_rlvect_1,existence_l1_struct_0,existence_l1_vectsp_1,existence_l2_struct_0,existence_l2_vectsp_1,existence_l3_vectsp_1,existence_l4_vectsp_1,existence_m1_relset_1,existence_m1_subset_1,existence_m2_relset_1,fc10_membered,fc11_membered,fc12_membered,fc13_membered,fc14_membered,fc15_membered,fc16_membered,fc1_complex1,fc1_complfld,fc1_square_1,fc1_struct_0,fc1_subset_1,fc21_xreal_0,fc22_xreal_0,fc23_xreal_0,fc24_xreal_0,fc2_complfld,fc2_membered,fc2_nat_1,fc2_square_1,fc2_subset_1,fc3_complfld,fc3_subset_1,fc3_xcmplx_0,fc4_complex1,fc4_subset_1,fc4_xreal_0,fc6_membered,fc7_membered,fc8_membered,fc8_xcmplx_0,fc9_membered,free_g3_vectsp_1,projectivity_k16_complex1,projectivity_k3_complfld,rc1_bilinear,rc1_membered,rc1_nat_1,rc1_subset_1,rc1_xcmplx_0,rc1_xreal_0,rc2_subset_1,rc2_xcmplx_0,rc2_xreal_0,rc3_hermitan,rc3_struct_0,rc3_xreal_0,rc4_hermitan,rc4_struct_0,rc4_xreal_0,rc5_struct_0,redefinition_k1_domain_1,redefinition_k3_complex1,redefinition_k3_complfld,redefinition_k4_real_1,redefinition_k7_square_1,redefinition_k8_funct_2,redefinition_m2_relset_1,reflexivity_r1_tarski,reflexivity_r1_xreal_0,t1_real,t1_subset,t2_real,t2_subset,t3_real,t3_subset,t44_hermitan,t48_hermitan,t4_real,t4_subset,t5_real,t5_subset,t6_boole,t6_real,t7_boole,t7_real,t8_boole,t8_real]), [file(hermitan,t49_hermitan)]). fof(abstractness_v3_vectsp_1,axiom,( ! [A] : ( l3_vectsp_1(A) => ( v3_vectsp_1(A) => A = g3_vectsp_1(u1_struct_0(A),u1_rlvect_1(A),u1_group_1(A),u1_vectsp_1(A),u2_struct_0(A)) ) ) ), file(vectsp_1,v3_vectsp_1), []). fof(antisymmetry_r2_hidden,axiom,( ! [A,B] : ( r2_hidden(A,B) => ~ r2_hidden(B,A) ) ), file(hidden,r2_hidden), []). fof(cc10_membered,axiom,( ! [A] : ( v1_membered(A) => ! [B] : ( m1_subset_1(B,A) => v1_xcmplx_0(B) ) ) ), file(membered,cc10_membered), []). fof(cc11_membered,axiom,( ! [A] : ( v2_membered(A) => ! [B] : ( m1_subset_1(B,A) => ( v1_xcmplx_0(B) & v1_xreal_0(B) ) ) ) ), file(membered,cc11_membered), []). fof(cc12_membered,axiom,( ! [A] : ( v3_membered(A) => ! [B] : ( m1_subset_1(B,A) => ( v1_xcmplx_0(B) & v1_xreal_0(B) & v1_rat_1(B) ) ) ) ), file(membered,cc12_membered), []). fof(cc13_membered,axiom,( ! [A] : ( v4_membered(A) => ! [B] : ( m1_subset_1(B,A) => ( v1_xcmplx_0(B) & v1_xreal_0(B) & v1_int_1(B) & v1_rat_1(B) ) ) ) ), file(membered,cc13_membered), []). fof(cc14_membered,axiom,( ! [A] : ( v5_membered(A) => ! [B] : ( m1_subset_1(B,A) => ( v1_xcmplx_0(B) & v4_ordinal2(B) & v1_xreal_0(B) & v1_int_1(B) & v1_rat_1(B) ) ) ) ), file(membered,cc14_membered), []). fof(cc15_membered,axiom,( ! [A] : ( v1_xboole_0(A) => ( v1_membered(A) & v2_membered(A) & v3_membered(A) & v4_membered(A) & v5_membered(A) ) ) ), file(membered,cc15_membered), []). fof(cc16_membered,axiom,( ! [A] : ( v1_membered(A) => ! [B] : ( m1_subset_1(B,k1_zfmisc_1(A)) => v1_membered(B) ) ) ), file(membered,cc16_membered), []). fof(cc17_membered,axiom,( ! [A] : ( v2_membered(A) => ! [B] : ( m1_subset_1(B,k1_zfmisc_1(A)) => ( v1_membered(B) & v2_membered(B) ) ) ) ), file(membered,cc17_membered), []). fof(cc18_membered,axiom,( ! [A] : ( v3_membered(A) => ! [B] : ( m1_subset_1(B,k1_zfmisc_1(A)) => ( v1_membered(B) & v2_membered(B) & v3_membered(B) ) ) ) ), file(membered,cc18_membered), []). fof(cc19_membered,axiom,( ! [A] : ( v4_membered(A) => ! [B] : ( m1_subset_1(B,k1_zfmisc_1(A)) => ( v1_membered(B) & v2_membered(B) & v3_membered(B) & v4_membered(B) ) ) ) ), file(membered,cc19_membered), []). fof(cc1_complfld,axiom,( ! [A] : ( m1_subset_1(A,u1_struct_0(k1_complfld)) => v1_xcmplx_0(A) ) ), file(complfld,cc1_complfld), []). 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_xcmplx_0,axiom,( ! [A] : ( m1_subset_1(A,k1_numbers) => v1_xcmplx_0(A) ) ), file(xcmplx_0,cc1_xcmplx_0), []). 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_hermitan,axiom,( ! [A] : ( ( ~ v3_struct_0(A) & l4_vectsp_1(A,k1_complfld) ) => ! [B] : ( m1_relset_1(B,k2_zfmisc_1(u1_struct_0(A),u1_struct_0(A)),u1_struct_0(k1_complfld)) => ( ( v1_funct_1(B) & v1_funct_2(B,k2_zfmisc_1(u1_struct_0(A),u1_struct_0(A)),u1_struct_0(k1_complfld)) & v3_hermitan(B,A) ) => ( v1_funct_1(B) & v1_funct_2(B,k2_zfmisc_1(u1_struct_0(A),u1_struct_0(A)),u1_struct_0(k1_complfld)) & v4_hermitan(B,A) ) ) ) ) ), file(hermitan,cc2_hermitan), []). fof(cc2_membered,axiom,( ! [A] : ( v4_membered(A) => v3_membered(A) ) ), file(membered,cc2_membered), []). fof(cc2_xcmplx_0,axiom,( ! [A] : ( v4_ordinal2(A) => v1_xcmplx_0(A) ) ), file(xcmplx_0,cc2_xcmplx_0), []). fof(cc2_xreal_0,axiom,( ! [A] : ( v1_xreal_0(A) => v1_xcmplx_0(A) ) ), file(xreal_0,cc2_xreal_0), []). fof(cc3_hermitan,axiom,( ! [A] : ( ( ~ v3_struct_0(A) & l4_vectsp_1(A,k1_complfld) ) => ! [B] : ( m1_relset_1(B,k2_zfmisc_1(u1_struct_0(A),u1_struct_0(A)),u1_struct_0(k1_complfld)) => ( ( v1_funct_1(B) & v1_funct_2(B,k2_zfmisc_1(u1_struct_0(A),u1_struct_0(A)),u1_struct_0(k1_complfld)) & v1_bilinear(B,k1_complfld,A,A) & v3_hermitan(B,A) ) => ( v1_funct_1(B) & v1_funct_2(B,k2_zfmisc_1(u1_struct_0(A),u1_struct_0(A)),u1_struct_0(k1_complfld)) & v2_bilinear(B,k1_complfld,A,A) ) ) ) ) ), file(hermitan,cc3_hermitan), []). fof(cc3_membered,axiom,( ! [A] : ( v3_membered(A) => v2_membered(A) ) ), file(membered,cc3_membered), []). fof(cc3_nat_1,axiom,( ! [A] : ( v4_ordinal2(A) => ( v4_ordinal2(A) & v1_xcmplx_0(A) & v1_xreal_0(A) & ~ v3_xreal_0(A) ) ) ), file(nat_1,cc3_nat_1), []). fof(cc3_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_hermitan,axiom,( ! [A] : ( ( ~ v3_struct_0(A) & l4_vectsp_1(A,k1_complfld) ) => ! [B] : ( m1_relset_1(B,k2_zfmisc_1(u1_struct_0(A),u1_struct_0(A)),u1_struct_0(k1_complfld)) => ( ( v1_funct_1(B) & v1_funct_2(B,k2_zfmisc_1(u1_struct_0(A),u1_struct_0(A)),u1_struct_0(k1_complfld)) & v2_bilinear(B,k1_complfld,A,A) & v3_hermitan(B,A) ) => ( v1_funct_1(B) & v1_funct_2(B,k2_zfmisc_1(u1_struct_0(A),u1_struct_0(A)),u1_struct_0(k1_complfld)) & v1_bilinear(B,k1_complfld,A,A) ) ) ) ) ), file(hermitan,cc4_hermitan), []). 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_hermitan,axiom,( ! [A] : ( ( ~ v3_struct_0(A) & l4_vectsp_1(A,k1_complfld) ) => ! [B] : ( m1_relset_1(B,k2_zfmisc_1(u1_struct_0(A),u1_struct_0(A)),u1_struct_0(k1_complfld)) => ( ( v1_funct_1(B) & v1_funct_2(B,k2_zfmisc_1(u1_struct_0(A),u1_struct_0(A)),u1_struct_0(k1_complfld)) & v4_bilinear(B,k1_complfld,A,A) & v3_hermitan(B,A) ) => ( v1_funct_1(B) & v1_funct_2(B,k2_zfmisc_1(u1_struct_0(A),u1_struct_0(A)),u1_struct_0(k1_complfld)) & v2_hermitan(B,A,A) ) ) ) ) ), file(hermitan,cc5_hermitan), []). 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_hermitan,axiom,( ! [A] : ( ( ~ v3_struct_0(A) & l4_vectsp_1(A,k1_complfld) ) => ! [B] : ( m1_relset_1(B,k2_zfmisc_1(u1_struct_0(A),u1_struct_0(A)),u1_struct_0(k1_complfld)) => ( ( v1_funct_1(B) & v1_funct_2(B,k2_zfmisc_1(u1_struct_0(A),u1_struct_0(A)),u1_struct_0(k1_complfld)) & v2_hermitan(B,A,A) & v3_hermitan(B,A) ) => ( v1_funct_1(B) & v1_funct_2(B,k2_zfmisc_1(u1_struct_0(A),u1_struct_0(A)),u1_struct_0(k1_complfld)) & v4_bilinear(B,k1_complfld,A,A) ) ) ) ) ), file(hermitan,cc6_hermitan), []). 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(commutativity_k2_tarski,axiom,( ! [A,B] : k2_tarski(A,B) = k2_tarski(B,A) ), file(tarski,k2_tarski), []). 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(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(d3_square_1,axiom,( ! [A] : ( v1_xcmplx_0(A) => k5_square_1(A) = k3_xcmplx_0(A,A) ) ), file(square_1,d3_square_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(d9_hermitan,axiom,( ! [A] : ( ( ~ v3_struct_0(A) & l4_vectsp_1(A,k1_complfld) ) => ! [B] : ( ( v1_funct_1(B) & v1_funct_2(B,k2_zfmisc_1(u1_struct_0(A),u1_struct_0(A)),u1_struct_0(k1_complfld)) & m2_relset_1(B,k2_zfmisc_1(u1_struct_0(A),u1_struct_0(A)),u1_struct_0(k1_complfld)) ) => ! [C] : ( m1_subset_1(C,u1_struct_0(A)) => k4_hermitan(A,B,C) = k3_complex1(k8_funct_2(k2_zfmisc_1(u1_struct_0(A),u1_struct_0(A)),u1_struct_0(k1_complfld),B,k1_domain_1(u1_struct_0(A),u1_struct_0(A),C,C))) ) ) ) ), file(hermitan,d9_hermitan), []). fof(dt_g3_vectsp_1,axiom,( ! [A,B,C,D,E] : ( ( v1_funct_1(B) & v1_funct_2(B,k2_zfmisc_1(A,A),A) & m1_relset_1(B,k2_zfmisc_1(A,A),A) & v1_funct_1(C) & v1_funct_2(C,k2_zfmisc_1(A,A),A) & m1_relset_1(C,k2_zfmisc_1(A,A),A) & m1_subset_1(D,A) & m1_subset_1(E,A) ) => ( v3_vectsp_1(g3_vectsp_1(A,B,C,D,E)) & l3_vectsp_1(g3_vectsp_1(A,B,C,D,E)) ) ) ), file(vectsp_1,g3_vectsp_1), []). fof(dt_k16_complex1,axiom,( ! [A] : ( v1_xcmplx_0(A) => v1_xcmplx_0(k16_complex1(A)) ) ), file(complex1,k16_complex1), []). fof(dt_k1_complex1,axiom,( $true ), file(complex1,k1_complex1), []). fof(dt_k1_complfld,axiom, ( v3_vectsp_1(k1_complfld) & l3_vectsp_1(k1_complfld) ), file(complfld,k1_complfld), []). 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_funct_1,axiom,( $true ), file(funct_1,k1_funct_1), []). fof(dt_k1_numbers,axiom,( $true ), file(numbers,k1_numbers), []). 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_k2_tarski,axiom,( $true ), file(tarski,k2_tarski), []). fof(dt_k2_zfmisc_1,axiom,( $true ), file(zfmisc_1,k2_zfmisc_1), []). fof(dt_k3_complex1,axiom,( ! [A] : ( v1_xcmplx_0(A) => m1_subset_1(k3_complex1(A),k1_numbers) ) ), file(complex1,k3_complex1), []). fof(dt_k3_complfld,axiom,( ! [A] : ( m1_subset_1(A,u1_struct_0(k1_complfld)) => m1_subset_1(k3_complfld(A),k1_numbers) ) ), file(complfld,k3_complfld), []). fof(dt_k3_xcmplx_0,axiom,( $true ), file(xcmplx_0,k3_xcmplx_0), []). fof(dt_k4_hermitan,axiom,( ! [A,B,C] : ( ( ~ v3_struct_0(A) & l4_vectsp_1(A,k1_complfld) & v1_funct_1(B) & v1_funct_2(B,k2_zfmisc_1(u1_struct_0(A),u1_struct_0(A)),u1_struct_0(k1_complfld)) & m1_relset_1(B,k2_zfmisc_1(u1_struct_0(A),u1_struct_0(A)),u1_struct_0(k1_complfld)) & m1_subset_1(C,u1_struct_0(A)) ) => v1_xreal_0(k4_hermitan(A,B,C)) ) ), file(hermitan,k4_hermitan), []). 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_square_1,axiom,( $true ), file(square_1,k5_square_1), []). fof(dt_k7_square_1,axiom,( ! [A] : ( m1_subset_1(A,k1_numbers) => m1_subset_1(k7_square_1(A),k1_numbers) ) ), file(square_1,k7_square_1), []). fof(dt_k8_funct_2,axiom,( ! [A,B,C,D] : ( ( ~ v1_xboole_0(A) & v1_funct_1(C) & v1_funct_2(C,A,B) & m1_relset_1(C,A,B) & m1_subset_1(D,A) ) => m1_subset_1(k8_funct_2(A,B,C,D),B) ) ), file(funct_2,k8_funct_2), []). fof(dt_l1_group_1,axiom,( ! [A] : ( l1_group_1(A) => l1_struct_0(A) ) ), file(group_1,l1_group_1), []). fof(dt_l1_rlvect_1,axiom,( ! [A] : ( l1_rlvect_1(A) => l2_struct_0(A) ) ), file(rlvect_1,l1_rlvect_1), []). fof(dt_l1_struct_0,axiom,( $true ), file(struct_0,l1_struct_0), []). fof(dt_l1_vectsp_1,axiom,( ! [A] : ( l1_vectsp_1(A) => l1_group_1(A) ) ), file(vectsp_1,l1_vectsp_1), []). fof(dt_l2_struct_0,axiom,( ! [A] : ( l2_struct_0(A) => l1_struct_0(A) ) ), file(struct_0,l2_struct_0), []). fof(dt_l2_vectsp_1,axiom,( ! [A] : ( l2_vectsp_1(A) => ( l1_vectsp_1(A) & l2_struct_0(A) ) ) ), file(vectsp_1,l2_vectsp_1), []). fof(dt_l3_vectsp_1,axiom,( ! [A] : ( l3_vectsp_1(A) => ( l1_rlvect_1(A) & l2_vectsp_1(A) ) ) ), file(vectsp_1,l3_vectsp_1), []). fof(dt_l4_vectsp_1,axiom,( ! [A] : ( l1_struct_0(A) => ! [B] : ( l4_vectsp_1(B,A) => l1_rlvect_1(B) ) ) ), file(vectsp_1,l4_vectsp_1), []). fof(dt_m1_relset_1,axiom,( $true ), file(relset_1,m1_relset_1), []). fof(dt_m1_subset_1,axiom,( $true ), file(subset_1,m1_subset_1), []). fof(dt_m2_relset_1,axiom,( ! [A,B,C] : ( m2_relset_1(C,A,B) => m1_subset_1(C,k1_zfmisc_1(k2_zfmisc_1(A,B))) ) ), file(relset_1,m2_relset_1), []). fof(dt_u1_group_1,axiom,( ! [A] : ( l1_group_1(A) => ( v1_funct_1(u1_group_1(A)) & v1_funct_2(u1_group_1(A),k2_zfmisc_1(u1_struct_0(A),u1_struct_0(A)),u1_struct_0(A)) & m2_relset_1(u1_group_1(A),k2_zfmisc_1(u1_struct_0(A),u1_struct_0(A)),u1_struct_0(A)) ) ) ), file(group_1,u1_group_1), []). fof(dt_u1_rlvect_1,axiom,( ! [A] : ( l1_rlvect_1(A) => ( v1_funct_1(u1_rlvect_1(A)) & v1_funct_2(u1_rlvect_1(A),k2_zfmisc_1(u1_struct_0(A),u1_struct_0(A)),u1_struct_0(A)) & m2_relset_1(u1_rlvect_1(A),k2_zfmisc_1(u1_struct_0(A),u1_struct_0(A)),u1_struct_0(A)) ) ) ), file(rlvect_1,u1_rlvect_1), []). fof(dt_u1_struct_0,axiom,( $true ), file(struct_0,u1_struct_0), []). fof(dt_u1_vectsp_1,axiom,( ! [A] : ( l1_vectsp_1(A) => m1_subset_1(u1_vectsp_1(A),u1_struct_0(A)) ) ), file(vectsp_1,u1_vectsp_1), []). fof(dt_u2_struct_0,axiom,( ! [A] : ( l2_struct_0(A) => m1_subset_1(u2_struct_0(A),u1_struct_0(A)) ) ), file(struct_0,u2_struct_0), []). fof(existence_l1_group_1,axiom,( ? [A] : l1_group_1(A) ), file(group_1,l1_group_1), []). fof(existence_l1_rlvect_1,axiom,( ? [A] : l1_rlvect_1(A) ), file(rlvect_1,l1_rlvect_1), []). fof(existence_l1_struct_0,axiom,( ? [A] : l1_struct_0(A) ), file(struct_0,l1_struct_0), []). fof(existence_l1_vectsp_1,axiom,( ? [A] : l1_vectsp_1(A) ), file(vectsp_1,l1_vectsp_1), []). fof(existence_l2_struct_0,axiom,( ? [A] : l2_struct_0(A) ), file(struct_0,l2_struct_0), []). fof(existence_l2_vectsp_1,axiom,( ? [A] : l2_vectsp_1(A) ), file(vectsp_1,l2_vectsp_1), []). fof(existence_l3_vectsp_1,axiom,( ? [A] : l3_vectsp_1(A) ), file(vectsp_1,l3_vectsp_1), []). fof(existence_l4_vectsp_1,axiom,( ! [A] : ( l1_struct_0(A) => ? [B] : l4_vectsp_1(B,A) ) ), file(vectsp_1,l4_vectsp_1), []). fof(existence_m1_relset_1,axiom,( ! [A,B] : ? [C] : m1_relset_1(C,A,B) ), file(relset_1,m1_relset_1), []). fof(existence_m1_subset_1,axiom,( ! [A] : ? [B] : m1_subset_1(B,A) ), file(subset_1,m1_subset_1), []). fof(existence_m2_relset_1,axiom,( ! [A,B] : ? [C] : m2_relset_1(C,A,B) ), file(relset_1,m2_relset_1), []). fof(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_complex1,axiom,( ! [A] : ( v1_xcmplx_0(A) => ( v1_xcmplx_0(k1_complex1(A)) & v1_xreal_0(k1_complex1(A)) ) ) ), file(complex1,fc1_complex1), []). fof(fc1_complfld,axiom, ( ~ v3_struct_0(k1_complfld) & v3_vectsp_1(k1_complfld) ), file(complfld,fc1_complfld), []). fof(fc1_square_1,axiom,( ! [A] : ( v1_xcmplx_0(A) => v1_xcmplx_0(k5_square_1(A)) ) ), file(square_1,fc1_square_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(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(fc2_complfld,axiom, ( ~ v3_struct_0(k1_complfld) & v2_group_1(k1_complfld) & v3_vectsp_1(k1_complfld) & v6_vectsp_1(k1_complfld) & v8_vectsp_1(k1_complfld) ), file(complfld,fc2_complfld), []). 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_square_1,axiom,( ! [A] : ( v1_xreal_0(A) => ( v1_xcmplx_0(k5_square_1(A)) & v1_xreal_0(k5_square_1(A)) ) ) ), file(square_1,fc2_square_1), []). fof(fc2_subset_1,axiom,( ! [A] : ~ v1_xboole_0(k1_tarski(A)) ), file(subset_1,fc2_subset_1), []). fof(fc3_complfld,axiom, ( ~ v3_struct_0(k1_complfld) & v3_rlvect_1(k1_complfld) & v4_rlvect_1(k1_complfld) & v5_rlvect_1(k1_complfld) & v6_rlvect_1(k1_complfld) & v2_group_1(k1_complfld) & v4_group_1(k1_complfld) & v7_group_1(k1_complfld) & v3_vectsp_1(k1_complfld) & v4_vectsp_1(k1_complfld) & v5_vectsp_1(k1_complfld) & v6_vectsp_1(k1_complfld) & v7_vectsp_1(k1_complfld) & v8_vectsp_1(k1_complfld) & v9_vectsp_1(k1_complfld) & ~ v10_vectsp_1(k1_complfld) ), file(complfld,fc3_complfld), []). fof(fc3_subset_1,axiom,( ! [A,B] : ~ v1_xboole_0(k2_tarski(A,B)) ), file(subset_1,fc3_subset_1), []). fof(fc3_xcmplx_0,axiom,( ! [A,B] : ( ( v1_xcmplx_0(A) & v1_xcmplx_0(B) ) => v1_xcmplx_0(k3_xcmplx_0(A,B)) ) ), file(xcmplx_0,fc3_xcmplx_0), []). fof(fc4_complex1,axiom,( ! [A] : ( v1_xcmplx_0(A) => ( v1_xcmplx_0(k16_complex1(A)) & v1_xreal_0(k16_complex1(A)) ) ) ), file(complex1,fc4_complex1), []). 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(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(fc8_xcmplx_0,axiom,( ! [A,B] : ( ( ~ v1_xboole_0(A) & v1_xcmplx_0(A) & ~ v1_xboole_0(B) & v1_xcmplx_0(B) ) => ( ~ v1_xboole_0(k3_xcmplx_0(A,B)) & v1_xcmplx_0(k3_xcmplx_0(A,B)) ) ) ), file(xcmplx_0,fc8_xcmplx_0), []). fof(fc9_membered,axiom,( ! [A] : ( v1_rat_1(A) => ( v1_membered(k1_tarski(A)) & v2_membered(k1_tarski(A)) & v3_membered(k1_tarski(A)) ) ) ), file(membered,fc9_membered), []). fof(free_g3_vectsp_1,axiom,( ! [A,B,C,D,E] : ( ( v1_funct_1(B) & v1_funct_2(B,k2_zfmisc_1(A,A),A) & m1_relset_1(B,k2_zfmisc_1(A,A),A) & v1_funct_1(C) & v1_funct_2(C,k2_zfmisc_1(A,A),A) & m1_relset_1(C,k2_zfmisc_1(A,A),A) & m1_subset_1(D,A) & m1_subset_1(E,A) ) => ! [F,G,H,I,J] : ( g3_vectsp_1(A,B,C,D,E) = g3_vectsp_1(F,G,H,I,J) => ( A = F & B = G & C = H & D = I & E = J ) ) ) ), file(vectsp_1,g3_vectsp_1), []). fof(projectivity_k16_complex1,axiom,( ! [A] : ( v1_xcmplx_0(A) => k16_complex1(k16_complex1(A)) = k16_complex1(A) ) ), file(complex1,k16_complex1), []). fof(projectivity_k3_complfld,axiom,( ! [A] : ( m1_subset_1(A,u1_struct_0(k1_complfld)) => k3_complfld(k3_complfld(A)) = k3_complfld(A) ) ), file(complfld,k3_complfld), []). fof(rc1_bilinear,axiom,( ! [A,B,C] : ( ( ~ v3_struct_0(A) & v5_rlvect_1(A) & l1_rlvect_1(A) & ~ v3_struct_0(B) & l4_vectsp_1(B,A) & ~ v3_struct_0(C) & l4_vectsp_1(C,A) ) => ? [D] : ( m1_relset_1(D,k2_zfmisc_1(u1_struct_0(B),u1_struct_0(C)),u1_struct_0(A)) & v1_relat_1(D) & v1_funct_1(D) & v1_funct_2(D,k2_zfmisc_1(u1_struct_0(B),u1_struct_0(C)),u1_struct_0(A)) & v1_bilinear(D,A,B,C) & v2_bilinear(D,A,B,C) ) ) ), file(bilinear,rc1_bilinear), []). 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_xcmplx_0,axiom,( ? [A] : v1_xcmplx_0(A) ), file(xcmplx_0,rc1_xcmplx_0), []). 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_xcmplx_0,axiom,( ? [A] : ( ~ v1_xboole_0(A) & v1_xcmplx_0(A) ) ), file(xcmplx_0,rc2_xcmplx_0), []). fof(rc2_xreal_0,axiom,( ? [A] : ( ~ v1_xboole_0(A) & v1_xcmplx_0(A) & v1_xreal_0(A) & v2_xreal_0(A) & ~ v3_xreal_0(A) ) ), file(xreal_0,rc2_xreal_0), []). fof(rc3_hermitan,axiom,( ! [A] : ( ( ~ v3_struct_0(A) & l4_vectsp_1(A,k1_complfld) ) => ? [B] : ( m1_relset_1(B,k2_zfmisc_1(u1_struct_0(A),u1_struct_0(A)),u1_struct_0(k1_complfld)) & v1_relat_1(B) & v1_funct_1(B) & v1_funct_2(B,k2_zfmisc_1(u1_struct_0(A),u1_struct_0(A)),u1_struct_0(k1_complfld)) & v1_bilinear(B,k1_complfld,A,A) & v2_bilinear(B,k1_complfld,A,A) & v4_bilinear(B,k1_complfld,A,A) & v2_hermitan(B,A,A) & v3_hermitan(B,A) & v4_hermitan(B,A) & v5_hermitan(B,A) ) ) ), file(hermitan,rc3_hermitan), []). 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_hermitan,axiom,( ! [A,B] : ( ( ~ v3_struct_0(A) & l4_vectsp_1(A,k1_complfld) & ~ v3_struct_0(B) & l4_vectsp_1(B,k1_complfld) ) => ? [C] : ( m1_relset_1(C,k2_zfmisc_1(u1_struct_0(A),u1_struct_0(B)),u1_struct_0(k1_complfld)) & v1_relat_1(C) & v1_funct_1(C) & v1_funct_2(C,k2_zfmisc_1(u1_struct_0(A),u1_struct_0(B)),u1_struct_0(k1_complfld)) & v1_bilinear(C,k1_complfld,A,B) & v2_bilinear(C,k1_complfld,A,B) & v4_bilinear(C,k1_complfld,A,B) & v2_hermitan(C,A,B) ) ) ), file(hermitan,rc4_hermitan), []). 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_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_k3_complex1,axiom,( ! [A] : ( v1_xcmplx_0(A) => k3_complex1(A) = k1_complex1(A) ) ), file(complex1,k3_complex1), []). fof(redefinition_k3_complfld,axiom,( ! [A] : ( m1_subset_1(A,u1_struct_0(k1_complfld)) => k3_complfld(A) = k16_complex1(A) ) ), file(complfld,k3_complfld), []). 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_k7_square_1,axiom,( ! [A] : ( m1_subset_1(A,k1_numbers) => k7_square_1(A) = k5_square_1(A) ) ), file(square_1,k7_square_1), []). fof(redefinition_k8_funct_2,axiom,( ! [A,B,C,D] : ( ( ~ v1_xboole_0(A) & v1_funct_1(C) & v1_funct_2(C,A,B) & m1_relset_1(C,A,B) & m1_subset_1(D,A) ) => k8_funct_2(A,B,C,D) = k1_funct_1(C,D) ) ), file(funct_2,k8_funct_2), []). fof(redefinition_m2_relset_1,axiom,( ! [A,B,C] : ( m2_relset_1(C,A,B) <=> m1_relset_1(C,A,B) ) ), file(relset_1,m2_relset_1), []). fof(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(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(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(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(t44_hermitan,axiom,( ! [A] : ( ( ~ v3_struct_0(A) & v4_rlvect_1(A) & v5_rlvect_1(A) & v6_rlvect_1(A) & v12_vectsp_1(A,k1_complfld) & l4_vectsp_1(A,k1_complfld) ) => ! [B] : ( ( v1_funct_1(B) & v1_funct_2(B,k2_zfmisc_1(u1_struct_0(A),u1_struct_0(A)),u1_struct_0(k1_complfld)) & v4_hermitan(B,A) & v5_hermitan(B,A) & m2_relset_1(B,k2_zfmisc_1(u1_struct_0(A),u1_struct_0(A)),u1_struct_0(k1_complfld)) ) => ! [C] : ( m1_subset_1(C,u1_struct_0(A)) => ( k3_complfld(k8_funct_2(k2_zfmisc_1(u1_struct_0(A),u1_struct_0(A)),u1_struct_0(k1_complfld),B,k1_domain_1(u1_struct_0(A),u1_struct_0(A),C,C))) = k3_complex1(k8_funct_2(k2_zfmisc_1(u1_struct_0(A),u1_struct_0(A)),u1_struct_0(k1_complfld),B,k1_domain_1(u1_struct_0(A),u1_struct_0(A),C,C))) & k4_hermitan(A,B,C) = k3_complfld(k8_funct_2(k2_zfmisc_1(u1_struct_0(A),u1_struct_0(A)),u1_struct_0(k1_complfld),B,k1_domain_1(u1_struct_0(A),u1_struct_0(A),C,C))) ) ) ) ) ), file(hermitan,t44_hermitan), []). fof(t48_hermitan,axiom,( ! [A] : ( ( ~ v3_struct_0(A) & v3_rlvect_1(A) & v4_rlvect_1(A) & v5_rlvect_1(A) & v6_rlvect_1(A) & v12_vectsp_1(A,k1_complfld) & l4_vectsp_1(A,k1_complfld) ) => ! [B] : ( m1_subset_1(B,u1_struct_0(A)) => ! [C] : ( m1_subset_1(C,u1_struct_0(A)) => ! [D] : ( ( v1_funct_1(D) & v1_funct_2(D,k2_zfmisc_1(u1_struct_0(A),u1_struct_0(A)),u1_struct_0(k1_complfld)) & v2_bilinear(D,k1_complfld,A,A) & v4_bilinear(D,k1_complfld,A,A) & v3_hermitan(D,A) & v5_hermitan(D,A) & m2_relset_1(D,k2_zfmisc_1(u1_struct_0(A),u1_struct_0(A)),u1_struct_0(k1_complfld)) ) => r1_xreal_0(k7_square_1(k3_complfld(k8_funct_2(k2_zfmisc_1(u1_struct_0(A),u1_struct_0(A)),u1_struct_0(k1_complfld),D,k1_domain_1(u1_struct_0(A),u1_struct_0(A),B,C)))),k3_xcmplx_0(k4_hermitan(A,D,B),k4_hermitan(A,D,C))) ) ) ) ) ), file(hermitan,t48_hermitan), []). 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(t5_real,axiom,( ! [A] : ( v1_xreal_0(A) => ! [B] : ( v1_xreal_0(B) => ( r1_xreal_0(A,B) => ( v1_xboole_0(B) | v3_xreal_0(A) | v2_xreal_0(B) ) ) ) ) ), file(real,t5_real), []). fof(t5_subset,axiom,( ! [A,B,C] : ~ ( r2_hidden(A,B) & m1_subset_1(B,k1_zfmisc_1(C)) & v1_xboole_0(C) ) ), file(subset,t5_subset), []). fof(t6_boole,axiom,( ! [A] : ( v1_xboole_0(A) => A = k1_xboole_0 ) ), file(boole,t6_boole), []). fof(t6_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(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_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), []).