% Mizar problem: t40_lopclset,lopclset,1336,50 fof(t40_lopclset,conjecture,( ! [A] : ( ( ~ v3_struct_0(A) & v10_lattices(A) & v17_lattices(A) & ~ v3_realset2(A) & l3_lattices(A) ) => k10_lopclset(A) = k1_lopclset(k11_lopclset(A)) ) ), inference(mizar_bg_added,[status(thm)],[abstractness_v1_pre_topc,antisymmetry_r2_hidden,cc10_membered,cc11_membered,cc12_membered,cc13_membered,cc14_membered,cc15_membered,cc16_membered,cc17_membered,cc18_membered,cc19_membered,cc1_finset_1,cc1_finsub_1,cc1_funct_2,cc1_lattices,cc1_membered,cc1_partfun1,cc1_relset_1,cc20_membered,cc2_finset_1,cc2_finsub_1,cc2_funct_2,cc2_lattices,cc2_membered,cc3_finsub_1,cc3_funct_2,cc3_lattices,cc3_membered,cc4_funct_2,cc4_lattices,cc4_membered,cc5_funct_2,cc5_lattices,cc6_funct_2,cc6_lattices,cc7_lattices,commutativity_k1_finsub_1,commutativity_k2_tarski,commutativity_k2_xboole_0,commutativity_k3_finsub_1,commutativity_k3_lattices,commutativity_k3_xboole_0,commutativity_k4_subset_1,commutativity_k5_setwiseo,d10_xboole_0,d11_grcat_1,d12_lattice4,d13_binop_1,d14_binop_1,d15_binop_1,d1_binop_1,d1_funct_2,d1_lattices,d1_lopclset,d3_lattice2,d3_pre_topc,d3_tarski,d3_xboole_0,d5_finsub_1,d5_lopclset,d5_pre_topc,d5_tarski,d7_lopclset,d7_xboole_0,d8_lopclset,dt_g1_pre_topc,dt_k10_lopclset,dt_k10_relat_1,dt_k11_lopclset,dt_k1_binop_1,dt_k1_card_1,dt_k1_finsub_1,dt_k1_funct_1,dt_k1_lattice4,dt_k1_lattices,dt_k1_lopclset,dt_k1_partfun1,dt_k1_relat_1,dt_k1_setwiseo,dt_k1_tarski,dt_k1_xboole_0,dt_k1_zfmisc_1,dt_k2_binop_1,dt_k2_lattice2,dt_k2_pre_topc,dt_k2_relat_1,dt_k2_setwiseo,dt_k2_tarski,dt_k2_xboole_0,dt_k2_zfmisc_1,dt_k3_finsub_1,dt_k3_funct_2,dt_k3_lattices,dt_k3_subset_1,dt_k3_tarski,dt_k3_xboole_0,dt_k4_relset_1,dt_k4_subset_1,dt_k4_tarski,dt_k5_finsub_1,dt_k5_lattices,dt_k5_relat_1,dt_k5_setfam_1,dt_k5_setwiseo,dt_k6_partfun1,dt_k6_relat_1,dt_k7_funct_2,dt_k7_grcat_1,dt_k7_lopclset,dt_k7_relset_1,dt_k7_setwiseo,dt_k8_funct_2,dt_k8_lopclset,dt_k8_setwiseo,dt_k9_lopclset,dt_k9_relat_1,dt_l1_lattices,dt_l1_pre_topc,dt_l1_struct_0,dt_l2_lattices,dt_l3_lattices,dt_m1_filter_0,dt_m1_relset_1,dt_m1_subset_1,dt_m2_relset_1,dt_u1_pre_topc,dt_u1_struct_0,dt_u2_lattices,existence_l1_lattices,existence_l1_pre_topc,existence_l1_struct_0,existence_l2_lattices,existence_l3_lattices,existence_m1_filter_0,existence_m1_relset_1,existence_m1_subset_1,existence_m2_relset_1,fc10_finset_1,fc10_membered,fc11_finset_1,fc11_membered,fc12_membered,fc13_finset_1,fc13_membered,fc14_finset_1,fc14_membered,fc15_membered,fc16_membered,fc1_finset_1,fc1_finsub_1,fc1_lopclset,fc1_struct_0,fc22_membered,fc23_membered,fc24_membered,fc25_membered,fc26_membered,fc27_membered,fc28_membered,fc29_membered,fc2_finset_1,fc2_finsub_1,fc2_lopclset,fc2_partfun1,fc2_pre_topc,fc30_membered,fc31_membered,fc32_membered,fc33_membered,fc34_membered,fc35_membered,fc36_membered,fc3_lopclset,fc4_lopclset,fc5_pre_topc,fc6_membered,fc7_membered,fc8_membered,fc9_finset_1,fc9_membered,fraenkel_a_1_0_lopclset,fraenkel_a_1_2_lopclset,fraenkel_a_1_3_lopclset,free_g1_pre_topc,idempotence_k1_finsub_1,idempotence_k2_xboole_0,idempotence_k3_finsub_1,idempotence_k3_xboole_0,idempotence_k4_subset_1,idempotence_k5_setwiseo,involutiveness_k3_subset_1,l49_lopclset,rc1_finset_1,rc1_funct_2,rc1_lopclset,rc1_membered,rc1_partfun1,rc1_pre_topc,rc2_funct_2,rc2_lopclset,rc2_partfun1,rc2_pre_topc,rc3_finset_1,rc3_partfun1,rc3_struct_0,rc4_finset_1,rc5_struct_0,rc6_pre_topc,rc7_pre_topc,redefinition_k1_finsub_1,redefinition_k1_partfun1,redefinition_k2_binop_1,redefinition_k2_setwiseo,redefinition_k3_finsub_1,redefinition_k3_funct_2,redefinition_k3_lattices,redefinition_k4_relset_1,redefinition_k4_subset_1,redefinition_k5_setfam_1,redefinition_k5_setwiseo,redefinition_k6_partfun1,redefinition_k7_funct_2,redefinition_k7_relset_1,redefinition_k8_funct_2,redefinition_k8_setwiseo,redefinition_k9_lopclset,redefinition_m2_relset_1,reflexivity_r1_rfinseq,reflexivity_r1_tarski,s4_setwiseo__e29_42_1_4_2__lopclset,s8_funct_2__e18_42_1_4_2__lopclset,symmetry_r1_rfinseq,symmetry_r1_xboole_0,t147_funct_1,t149_relat_1,t17_xboole_1,t18_pre_topc,t1_boole,t1_card_4,t1_subset,t1_xboole_1,t23_funct_2,t23_xboole_1,t24_lopclset,t26_lattice2,t26_setwiseo,t26_subset_1,t26_xboole_1,t27_lattice2,t28_xboole_1,t29_lattice2,t29_lopclset,t29_setwiseo,t29_tops_1,t2_boole,t2_lopclset,t2_subset,t2_tarski,t2_zfmisc_1,t30_tops_1,t31_zfmisc_1,t35_funct_1,t36_lopclset,t38_lopclset,t38_setwiseo,t39_setwiseo,t3_lopclset,t3_rfinseq,t3_subset,t3_xboole_1,t42_funct_1,t4_borsuk_1,t4_rfinseq,t4_subset,t4_xboole_1,t57_setfam_1,t5_subset,t6_boole,t6_funct_2,t7_boole,t7_xboole_1,t8_boole,t8_xboole_1,t92_zfmisc_1,t96_zfmisc_1,t97_zfmisc_1]), [file(lopclset,t40_lopclset)]). fof(abstractness_v1_pre_topc,axiom,( ! [A] : ( l1_pre_topc(A) => ( v1_pre_topc(A) => A = g1_pre_topc(u1_struct_0(A),u1_pre_topc(A)) ) ) ), file(pre_topc,v1_pre_topc), []). fof(antisymmetry_r2_hidden,axiom,( ! [A,B] : ( r2_hidden(A,B) => ~ r2_hidden(B,A) ) ), file(hidden,r2_hidden), []). fof(cc10_membered,axiom,( ! [A] : ( v1_membered(A) => ! [B] : ( m1_subset_1(B,A) => v1_xcmplx_0(B) ) ) ), file(membered,cc10_membered), []). fof(cc11_membered,axiom,( ! [A] : ( v2_membered(A) => ! [B] : ( m1_subset_1(B,A) => ( v1_xcmplx_0(B) & v1_xreal_0(B) ) ) ) ), file(membered,cc11_membered), []). fof(cc12_membered,axiom,( ! [A] : ( v3_membered(A) => ! [B] : ( m1_subset_1(B,A) => ( v1_xcmplx_0(B) & v1_xreal_0(B) & v1_rat_1(B) ) ) ) ), file(membered,cc12_membered), []). fof(cc13_membered,axiom,( ! [A] : ( v4_membered(A) => ! [B] : ( m1_subset_1(B,A) => ( v1_xcmplx_0(B) & v1_xreal_0(B) & v1_int_1(B) & v1_rat_1(B) ) ) ) ), file(membered,cc13_membered), []). fof(cc14_membered,axiom,( ! [A] : ( v5_membered(A) => ! [B] : ( m1_subset_1(B,A) => ( v1_xcmplx_0(B) & v4_ordinal2(B) & v1_xreal_0(B) & v1_int_1(B) & v1_rat_1(B) ) ) ) ), file(membered,cc14_membered), []). fof(cc15_membered,axiom,( ! [A] : ( v1_xboole_0(A) => ( v1_membered(A) & v2_membered(A) & v3_membered(A) & v4_membered(A) & v5_membered(A) ) ) ), file(membered,cc15_membered), []). fof(cc16_membered,axiom,( ! [A] : ( v1_membered(A) => ! [B] : ( m1_subset_1(B,k1_zfmisc_1(A)) => v1_membered(B) ) ) ), file(membered,cc16_membered), []). fof(cc17_membered,axiom,( ! [A] : ( v2_membered(A) => ! [B] : ( m1_subset_1(B,k1_zfmisc_1(A)) => ( v1_membered(B) & v2_membered(B) ) ) ) ), file(membered,cc17_membered), []). fof(cc18_membered,axiom,( ! [A] : ( v3_membered(A) => ! [B] : ( m1_subset_1(B,k1_zfmisc_1(A)) => ( v1_membered(B) & v2_membered(B) & v3_membered(B) ) ) ) ), file(membered,cc18_membered), []). fof(cc19_membered,axiom,( ! [A] : ( v4_membered(A) => ! [B] : ( m1_subset_1(B,k1_zfmisc_1(A)) => ( v1_membered(B) & v2_membered(B) & v3_membered(B) & v4_membered(B) ) ) ) ), file(membered,cc19_membered), []). fof(cc1_finset_1,axiom,( ! [A] : ( v1_xboole_0(A) => v1_finset_1(A) ) ), file(finset_1,cc1_finset_1), []). fof(cc1_finsub_1,axiom,( ! [A] : ( v4_finsub_1(A) => ( v1_finsub_1(A) & v3_finsub_1(A) ) ) ), file(finsub_1,cc1_finsub_1), []). fof(cc1_funct_2,axiom,( ! [A,B,C] : ( m1_relset_1(C,A,B) => ( ( v1_funct_1(C) & v1_partfun1(C,A,B) ) => ( v1_funct_1(C) & v1_funct_2(C,A,B) ) ) ) ), file(funct_2,cc1_funct_2), []). fof(cc1_lattices,axiom,( ! [A] : ( l3_lattices(A) => ( ( ~ v3_struct_0(A) & v10_lattices(A) ) => ( ~ v3_struct_0(A) & v4_lattices(A) & v5_lattices(A) & v6_lattices(A) & v7_lattices(A) & v8_lattices(A) & v9_lattices(A) ) ) ) ), file(lattices,cc1_lattices), []). fof(cc1_membered,axiom,( ! [A] : ( v5_membered(A) => v4_membered(A) ) ), file(membered,cc1_membered), []). fof(cc1_partfun1,axiom,( ! [A] : ( ( v1_relat_1(A) & v3_relat_2(A) & v8_relat_2(A) ) => ( v1_relat_1(A) & v1_relat_2(A) ) ) ), file(partfun1,cc1_partfun1), []). fof(cc1_relset_1,axiom,( ! [A,B,C] : ( m1_subset_1(C,k1_zfmisc_1(k2_zfmisc_1(A,B))) => v1_relat_1(C) ) ), file(relset_1,cc1_relset_1), []). fof(cc20_membered,axiom,( ! [A] : ( v5_membered(A) => ! [B] : ( m1_subset_1(B,k1_zfmisc_1(A)) => ( v1_membered(B) & v2_membered(B) & v3_membered(B) & v4_membered(B) & v5_membered(B) ) ) ) ), file(membered,cc20_membered), []). fof(cc2_finset_1,axiom,( ! [A] : ( v1_finset_1(A) => ! [B] : ( m1_subset_1(B,k1_zfmisc_1(A)) => v1_finset_1(B) ) ) ), file(finset_1,cc2_finset_1), []). fof(cc2_finsub_1,axiom,( ! [A] : ( ( v1_finsub_1(A) & v3_finsub_1(A) ) => v4_finsub_1(A) ) ), file(finsub_1,cc2_finsub_1), []). fof(cc2_funct_2,axiom,( ! [A,B,C] : ( m1_relset_1(C,A,B) => ( ( v1_funct_1(C) & v1_funct_2(C,A,B) & v3_funct_2(C,A,B) ) => ( v1_funct_1(C) & v2_funct_1(C) & v1_funct_2(C,A,B) & v2_funct_2(C,A,B) ) ) ) ), file(funct_2,cc2_funct_2), []). fof(cc2_lattices,axiom,( ! [A] : ( l3_lattices(A) => ( ( ~ v3_struct_0(A) & v4_lattices(A) & v5_lattices(A) & v6_lattices(A) & v7_lattices(A) & v8_lattices(A) & v9_lattices(A) ) => ( ~ v3_struct_0(A) & v10_lattices(A) ) ) ) ), file(lattices,cc2_lattices), []). fof(cc2_membered,axiom,( ! [A] : ( v4_membered(A) => v3_membered(A) ) ), file(membered,cc2_membered), []). fof(cc3_finsub_1,axiom,( ! [A,B] : ( m1_subset_1(B,k5_finsub_1(A)) => v1_finset_1(B) ) ), file(finsub_1,cc3_finsub_1), []). fof(cc3_funct_2,axiom,( ! [A,B,C] : ( m1_relset_1(C,A,B) => ( ( v1_funct_1(C) & v2_funct_1(C) & v1_funct_2(C,A,B) & v2_funct_2(C,A,B) ) => ( v1_funct_1(C) & v1_funct_2(C,A,B) & v3_funct_2(C,A,B) ) ) ) ), file(funct_2,cc3_funct_2), []). fof(cc3_lattices,axiom,( ! [A] : ( l3_lattices(A) => ( ( ~ v3_struct_0(A) & v13_lattices(A) & v14_lattices(A) ) => ( ~ v3_struct_0(A) & v15_lattices(A) ) ) ) ), file(lattices,cc3_lattices), []). fof(cc3_membered,axiom,( ! [A] : ( v3_membered(A) => v2_membered(A) ) ), file(membered,cc3_membered), []). fof(cc4_funct_2,axiom,( ! [A,B] : ( m1_relset_1(B,A,A) => ( ( v1_funct_1(B) & v1_partfun1(B,A,A) & v1_relat_2(B) & v1_funct_2(B,A,A) ) => ( v1_funct_1(B) & v2_funct_1(B) & v1_funct_2(B,A,A) & v2_funct_2(B,A,A) & v3_funct_2(B,A,A) ) ) ) ), file(funct_2,cc4_funct_2), []). fof(cc4_lattices,axiom,( ! [A] : ( l3_lattices(A) => ( ( ~ v3_struct_0(A) & v15_lattices(A) ) => ( ~ v3_struct_0(A) & v13_lattices(A) & v14_lattices(A) ) ) ) ), file(lattices,cc4_lattices), []). fof(cc4_membered,axiom,( ! [A] : ( v2_membered(A) => v1_membered(A) ) ), file(membered,cc4_membered), []). fof(cc5_funct_2,axiom,( ! [A,B] : ( ~ v1_xboole_0(B) => ! [C] : ( m1_relset_1(C,A,B) => ( ( v1_funct_1(C) & v1_funct_2(C,A,B) ) => ( v1_funct_1(C) & v1_partfun1(C,A,B) & v1_funct_2(C,A,B) ) ) ) ) ), file(funct_2,cc5_funct_2), []). fof(cc5_lattices,axiom,( ! [A] : ( l3_lattices(A) => ( ( ~ v3_struct_0(A) & v17_lattices(A) ) => ( ~ v3_struct_0(A) & v11_lattices(A) & v13_lattices(A) & v14_lattices(A) & v15_lattices(A) & v16_lattices(A) ) ) ) ), file(lattices,cc5_lattices), []). fof(cc6_funct_2,axiom,( ! [A,B] : ( ( ~ v1_xboole_0(A) & ~ v1_xboole_0(B) ) => ! [C] : ( m1_relset_1(C,A,B) => ( ( v1_funct_1(C) & v1_funct_2(C,A,B) ) => ( v1_funct_1(C) & ~ v1_xboole_0(C) & v1_partfun1(C,A,B) & v1_funct_2(C,A,B) ) ) ) ) ), file(funct_2,cc6_funct_2), []). fof(cc6_lattices,axiom,( ! [A] : ( l3_lattices(A) => ( ( ~ v3_struct_0(A) & v11_lattices(A) & v15_lattices(A) & v16_lattices(A) ) => ( ~ v3_struct_0(A) & v17_lattices(A) ) ) ) ), file(lattices,cc6_lattices), []). fof(cc7_lattices,axiom,( ! [A] : ( l3_lattices(A) => ( ( ~ v3_struct_0(A) & v10_lattices(A) & v11_lattices(A) ) => ( ~ v3_struct_0(A) & v4_lattices(A) & v5_lattices(A) & v6_lattices(A) & v7_lattices(A) & v8_lattices(A) & v9_lattices(A) & v10_lattices(A) & v12_lattices(A) ) ) ) ), file(lattices,cc7_lattices), []). fof(commutativity_k1_finsub_1,axiom,( ! [A,B,C] : ( ( ~ v1_xboole_0(A) & v4_finsub_1(A) & m1_subset_1(B,A) & m1_subset_1(C,A) ) => k1_finsub_1(A,B,C) = k1_finsub_1(A,C,B) ) ), file(finsub_1,k1_finsub_1), []). fof(commutativity_k2_tarski,axiom,( ! [A,B] : k2_tarski(A,B) = k2_tarski(B,A) ), file(tarski,k2_tarski), []). fof(commutativity_k2_xboole_0,axiom,( ! [A,B] : k2_xboole_0(A,B) = k2_xboole_0(B,A) ), file(xboole_0,k2_xboole_0), []). fof(commutativity_k3_finsub_1,axiom,( ! [A,B,C] : ( ( ~ v1_xboole_0(A) & v4_finsub_1(A) & m1_subset_1(B,A) & m1_subset_1(C,A) ) => k3_finsub_1(A,B,C) = k3_finsub_1(A,C,B) ) ), file(finsub_1,k3_finsub_1), []). fof(commutativity_k3_lattices,axiom,( ! [A,B,C] : ( ( ~ v3_struct_0(A) & v4_lattices(A) & l2_lattices(A) & m1_subset_1(B,u1_struct_0(A)) & m1_subset_1(C,u1_struct_0(A)) ) => k3_lattices(A,B,C) = k3_lattices(A,C,B) ) ), file(lattices,k3_lattices), []). fof(commutativity_k3_xboole_0,axiom,( ! [A,B] : k3_xboole_0(A,B) = k3_xboole_0(B,A) ), file(xboole_0,k3_xboole_0), []). fof(commutativity_k4_subset_1,axiom,( ! [A,B,C] : ( ( m1_subset_1(B,k1_zfmisc_1(A)) & m1_subset_1(C,k1_zfmisc_1(A)) ) => k4_subset_1(A,B,C) = k4_subset_1(A,C,B) ) ), file(subset_1,k4_subset_1), []). fof(commutativity_k5_setwiseo,axiom,( ! [A,B,C] : ( ( m1_subset_1(B,k5_finsub_1(A)) & m1_subset_1(C,k5_finsub_1(A)) ) => k5_setwiseo(A,B,C) = k5_setwiseo(A,C,B) ) ), file(setwiseo,k5_setwiseo), []). fof(d10_xboole_0,axiom,( ! [A,B] : ( A = B <=> ( r1_tarski(A,B) & r1_tarski(B,A) ) ) ), file(xboole_0,d10_xboole_0), []). fof(d11_grcat_1,axiom,( ! [A] : ( l1_struct_0(A) => k7_grcat_1(A) = k6_partfun1(u1_struct_0(A)) ) ), file(grcat_1,d11_grcat_1), []). fof(d12_lattice4,axiom,( ! [A] : ( ( ~ v3_struct_0(A) & v10_lattices(A) & l3_lattices(A) ) => ! [B] : ( m1_subset_1(B,k5_finsub_1(u1_struct_0(A))) => k1_lattice4(A,B) = k2_lattice2(u1_struct_0(A),A,B,k7_grcat_1(A)) ) ) ), file(lattice4,d12_lattice4), []). fof(d13_binop_1,axiom,( ! [A] : ( ~ v1_xboole_0(A) => ! [B] : ( ( v1_funct_1(B) & v1_funct_2(B,k2_zfmisc_1(A,A),A) & m2_relset_1(B,k2_zfmisc_1(A,A),A) ) => ( v1_binop_1(B,A) <=> ! [C] : ( m1_subset_1(C,A) => ! [D] : ( m1_subset_1(D,A) => k2_binop_1(A,A,A,B,C,D) = k2_binop_1(A,A,A,B,D,C) ) ) ) ) ) ), file(binop_1,d13_binop_1), []). fof(d14_binop_1,axiom,( ! [A] : ( ~ v1_xboole_0(A) => ! [B] : ( ( v1_funct_1(B) & v1_funct_2(B,k2_zfmisc_1(A,A),A) & m2_relset_1(B,k2_zfmisc_1(A,A),A) ) => ( v2_binop_1(B,A) <=> ! [C] : ( m1_subset_1(C,A) => ! [D] : ( m1_subset_1(D,A) => ! [E] : ( m1_subset_1(E,A) => k2_binop_1(A,A,A,B,C,k2_binop_1(A,A,A,B,D,E)) = k2_binop_1(A,A,A,B,k2_binop_1(A,A,A,B,C,D),E) ) ) ) ) ) ) ), file(binop_1,d14_binop_1), []). fof(d15_binop_1,axiom,( ! [A] : ( ~ v1_xboole_0(A) => ! [B] : ( ( v1_funct_1(B) & v1_funct_2(B,k2_zfmisc_1(A,A),A) & m2_relset_1(B,k2_zfmisc_1(A,A),A) ) => ( v3_binop_1(B,A) <=> ! [C] : ( m1_subset_1(C,A) => k2_binop_1(A,A,A,B,C,C) = C ) ) ) ) ), file(binop_1,d15_binop_1), []). fof(d1_binop_1,axiom,( ! [A] : ( ( v1_relat_1(A) & v1_funct_1(A) ) => ! [B,C] : k1_binop_1(A,B,C) = k1_funct_1(A,k4_tarski(B,C)) ) ), file(binop_1,d1_binop_1), []). fof(d1_funct_2,axiom,( ! [A,B,C] : ( m2_relset_1(C,A,B) => ( ( ( B = k1_xboole_0 => A = k1_xboole_0 ) => ( v1_funct_2(C,A,B) <=> A = k4_relset_1(A,B,C) ) ) & ( B = k1_xboole_0 => ( A = k1_xboole_0 | ( v1_funct_2(C,A,B) <=> C = k1_xboole_0 ) ) ) ) ) ), file(funct_2,d1_funct_2), []). fof(d1_lattices,axiom,( ! [A] : ( ( ~ v3_struct_0(A) & l2_lattices(A) ) => ! [B] : ( m1_subset_1(B,u1_struct_0(A)) => ! [C] : ( m1_subset_1(C,u1_struct_0(A)) => k1_lattices(A,B,C) = k2_binop_1(u1_struct_0(A),u1_struct_0(A),u1_struct_0(A),u2_lattices(A),B,C) ) ) ) ), file(lattices,d1_lattices), []). fof(d1_lopclset,axiom,( ! [A] : ( ( ~ v3_struct_0(A) & l1_pre_topc(A) ) => k1_lopclset(A) = a_1_0_lopclset(A) ) ), file(lopclset,d1_lopclset), []). fof(d3_lattice2,axiom,( ! [A] : ( ~ v1_xboole_0(A) => ! [B] : ( ( ~ v3_struct_0(B) & v10_lattices(B) & l3_lattices(B) ) => ! [C] : ( m1_subset_1(C,k5_finsub_1(A)) => ! [D] : ( ( v1_funct_1(D) & v1_funct_2(D,A,u1_struct_0(B)) & m2_relset_1(D,A,u1_struct_0(B)) ) => k2_lattice2(A,B,C,D) = k7_setwiseo(A,u1_struct_0(B),u2_lattices(B),C,D) ) ) ) ) ), file(lattice2,d3_lattice2), []). fof(d3_pre_topc,axiom,( ! [A] : ( l1_struct_0(A) => k2_pre_topc(A) = u1_struct_0(A) ) ), file(pre_topc,d3_pre_topc), []). fof(d3_tarski,axiom,( ! [A,B] : ( r1_tarski(A,B) <=> ! [C] : ( r2_hidden(C,A) => r2_hidden(C,B) ) ) ), file(tarski,d3_tarski), []). fof(d3_xboole_0,axiom,( ! [A,B,C] : ( C = k3_xboole_0(A,B) <=> ! [D] : ( r2_hidden(D,C) <=> ( r2_hidden(D,A) & r2_hidden(D,B) ) ) ) ), file(xboole_0,d3_xboole_0), []). fof(d5_finsub_1,axiom,( ! [A,B] : ( v4_finsub_1(B) => ( B = k5_finsub_1(A) <=> ! [C] : ( r2_hidden(C,B) <=> ( r1_tarski(C,A) & v1_finset_1(C) ) ) ) ) ), file(finsub_1,d5_finsub_1), []). fof(d5_lopclset,axiom,( ! [A] : ( ( ~ v3_struct_0(A) & v10_lattices(A) & v17_lattices(A) & ~ v3_realset2(A) & l3_lattices(A) ) => k7_lopclset(A) = a_1_2_lopclset(A) ) ), file(lopclset,d5_lopclset), []). fof(d5_pre_topc,axiom,( ! [A] : ( l1_pre_topc(A) => ! [B] : ( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) => ( v3_pre_topc(B,A) <=> r2_hidden(B,u1_pre_topc(A)) ) ) ) ), file(pre_topc,d5_pre_topc), []). fof(d5_tarski,axiom,( ! [A,B] : k4_tarski(A,B) = k2_tarski(k2_tarski(A,B),k1_tarski(A)) ), file(tarski,d5_tarski), []). fof(d7_lopclset,axiom,( ! [A] : ( ( ~ v3_struct_0(A) & v10_lattices(A) & v17_lattices(A) & ~ v3_realset2(A) & l3_lattices(A) ) => k10_lopclset(A) = k2_relat_1(k9_lopclset(A)) ) ), file(lopclset,d7_lopclset), []). fof(d7_xboole_0,axiom,( ! [A,B] : ( r1_xboole_0(A,B) <=> k3_xboole_0(A,B) = k1_xboole_0 ) ), file(xboole_0,d7_xboole_0), []). fof(d8_lopclset,axiom,( ! [A] : ( ( ~ v3_struct_0(A) & v10_lattices(A) & v17_lattices(A) & ~ v3_realset2(A) & l3_lattices(A) ) => ! [B] : ( ( v1_pre_topc(B) & v2_pre_topc(B) & l1_pre_topc(B) ) => ( B = k11_lopclset(A) <=> ( u1_struct_0(B) = k7_lopclset(A) & u1_pre_topc(B) = a_1_3_lopclset(A) ) ) ) ) ), file(lopclset,d8_lopclset), []). fof(dt_g1_pre_topc,axiom,( ! [A,B] : ( m1_subset_1(B,k1_zfmisc_1(k1_zfmisc_1(A))) => ( v1_pre_topc(g1_pre_topc(A,B)) & l1_pre_topc(g1_pre_topc(A,B)) ) ) ), file(pre_topc,g1_pre_topc), []). fof(dt_k10_lopclset,axiom,( $true ), file(lopclset,k10_lopclset), []). fof(dt_k10_relat_1,axiom,( $true ), file(relat_1,k10_relat_1), []). fof(dt_k11_lopclset,axiom,( ! [A] : ( ( ~ v3_struct_0(A) & v10_lattices(A) & v17_lattices(A) & ~ v3_realset2(A) & l3_lattices(A) ) => ( v1_pre_topc(k11_lopclset(A)) & v2_pre_topc(k11_lopclset(A)) & l1_pre_topc(k11_lopclset(A)) ) ) ), file(lopclset,k11_lopclset), []). fof(dt_k1_binop_1,axiom,( $true ), file(binop_1,k1_binop_1), []). fof(dt_k1_card_1,axiom,( ! [A] : v1_card_1(k1_card_1(A)) ), file(card_1,k1_card_1), []). fof(dt_k1_finsub_1,axiom,( ! [A,B,C] : ( ( ~ v1_xboole_0(A) & v4_finsub_1(A) & m1_subset_1(B,A) & m1_subset_1(C,A) ) => m1_subset_1(k1_finsub_1(A,B,C),A) ) ), file(finsub_1,k1_finsub_1), []). fof(dt_k1_funct_1,axiom,( $true ), file(funct_1,k1_funct_1), []). fof(dt_k1_lattice4,axiom,( ! [A,B] : ( ( ~ v3_struct_0(A) & v10_lattices(A) & l3_lattices(A) & m1_subset_1(B,k5_finsub_1(u1_struct_0(A))) ) => m1_subset_1(k1_lattice4(A,B),u1_struct_0(A)) ) ), file(lattice4,k1_lattice4), []). fof(dt_k1_lattices,axiom,( ! [A,B,C] : ( ( ~ v3_struct_0(A) & l2_lattices(A) & m1_subset_1(B,u1_struct_0(A)) & m1_subset_1(C,u1_struct_0(A)) ) => m1_subset_1(k1_lattices(A,B,C),u1_struct_0(A)) ) ), file(lattices,k1_lattices), []). fof(dt_k1_lopclset,axiom,( ! [A] : ( ( ~ v3_struct_0(A) & l1_pre_topc(A) ) => m1_subset_1(k1_lopclset(A),k1_zfmisc_1(k1_zfmisc_1(u1_struct_0(A)))) ) ), file(lopclset,k1_lopclset), []). fof(dt_k1_partfun1,axiom,( ! [A,B,C,D,E,F] : ( ( v1_funct_1(E) & m1_relset_1(E,A,B) & v1_funct_1(F) & m1_relset_1(F,C,D) ) => ( v1_funct_1(k1_partfun1(A,B,C,D,E,F)) & m2_relset_1(k1_partfun1(A,B,C,D,E,F),A,D) ) ) ), file(partfun1,k1_partfun1), []). fof(dt_k1_relat_1,axiom,( $true ), file(relat_1,k1_relat_1), []). fof(dt_k1_setwiseo,axiom,( ! [A] : ( v1_xboole_0(k1_setwiseo(A)) & m1_subset_1(k1_setwiseo(A),k5_finsub_1(A)) ) ), file(setwiseo,k1_setwiseo), []). 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_binop_1,axiom,( ! [A,B,C,D,E,F] : ( ( ~ v1_xboole_0(A) & ~ v1_xboole_0(B) & v1_funct_1(D) & v1_funct_2(D,k2_zfmisc_1(A,B),C) & m1_relset_1(D,k2_zfmisc_1(A,B),C) & m1_subset_1(E,A) & m1_subset_1(F,B) ) => m1_subset_1(k2_binop_1(A,B,C,D,E,F),C) ) ), file(binop_1,k2_binop_1), []). fof(dt_k2_lattice2,axiom,( ! [A,B,C,D] : ( ( ~ v1_xboole_0(A) & ~ v3_struct_0(B) & v10_lattices(B) & l3_lattices(B) & m1_subset_1(C,k5_finsub_1(A)) & v1_funct_1(D) & v1_funct_2(D,A,u1_struct_0(B)) & m1_relset_1(D,A,u1_struct_0(B)) ) => m1_subset_1(k2_lattice2(A,B,C,D),u1_struct_0(B)) ) ), file(lattice2,k2_lattice2), []). fof(dt_k2_pre_topc,axiom,( ! [A] : ( l1_struct_0(A) => m1_subset_1(k2_pre_topc(A),k1_zfmisc_1(u1_struct_0(A))) ) ), file(pre_topc,k2_pre_topc), []). fof(dt_k2_relat_1,axiom,( $true ), file(relat_1,k2_relat_1), []). fof(dt_k2_setwiseo,axiom,( ! [A,B] : ( ( ~ v1_xboole_0(A) & m1_subset_1(B,A) ) => m1_subset_1(k2_setwiseo(A,B),k5_finsub_1(A)) ) ), file(setwiseo,k2_setwiseo), []). fof(dt_k2_tarski,axiom,( $true ), file(tarski,k2_tarski), []). fof(dt_k2_xboole_0,axiom,( $true ), file(xboole_0,k2_xboole_0), []). fof(dt_k2_zfmisc_1,axiom,( $true ), file(zfmisc_1,k2_zfmisc_1), []). fof(dt_k3_finsub_1,axiom,( ! [A,B,C] : ( ( ~ v1_xboole_0(A) & v4_finsub_1(A) & m1_subset_1(B,A) & m1_subset_1(C,A) ) => m1_subset_1(k3_finsub_1(A,B,C),A) ) ), file(finsub_1,k3_finsub_1), []). fof(dt_k3_funct_2,axiom,( ! [A,B,C,D] : ( ( v1_funct_1(C) & m1_relset_1(C,A,B) ) => m1_subset_1(k3_funct_2(A,B,C,D),k1_zfmisc_1(A)) ) ), file(funct_2,k3_funct_2), []). fof(dt_k3_lattices,axiom,( ! [A,B,C] : ( ( ~ v3_struct_0(A) & v4_lattices(A) & l2_lattices(A) & m1_subset_1(B,u1_struct_0(A)) & m1_subset_1(C,u1_struct_0(A)) ) => m1_subset_1(k3_lattices(A,B,C),u1_struct_0(A)) ) ), file(lattices,k3_lattices), []). fof(dt_k3_subset_1,axiom,( ! [A,B] : ( m1_subset_1(B,k1_zfmisc_1(A)) => m1_subset_1(k3_subset_1(A,B),k1_zfmisc_1(A)) ) ), file(subset_1,k3_subset_1), []). fof(dt_k3_tarski,axiom,( $true ), file(tarski,k3_tarski), []). fof(dt_k3_xboole_0,axiom,( $true ), file(xboole_0,k3_xboole_0), []). fof(dt_k4_relset_1,axiom,( ! [A,B,C] : ( m1_relset_1(C,A,B) => m1_subset_1(k4_relset_1(A,B,C),k1_zfmisc_1(A)) ) ), file(relset_1,k4_relset_1), []). fof(dt_k4_subset_1,axiom,( ! [A,B,C] : ( ( m1_subset_1(B,k1_zfmisc_1(A)) & m1_subset_1(C,k1_zfmisc_1(A)) ) => m1_subset_1(k4_subset_1(A,B,C),k1_zfmisc_1(A)) ) ), file(subset_1,k4_subset_1), []). fof(dt_k4_tarski,axiom,( $true ), file(tarski,k4_tarski), []). fof(dt_k5_finsub_1,axiom,( ! [A] : v4_finsub_1(k5_finsub_1(A)) ), file(finsub_1,k5_finsub_1), []). fof(dt_k5_lattices,axiom,( ! [A] : ( ( ~ v3_struct_0(A) & l1_lattices(A) ) => m1_subset_1(k5_lattices(A),u1_struct_0(A)) ) ), file(lattices,k5_lattices), []). fof(dt_k5_relat_1,axiom,( ! [A,B] : ( ( v1_relat_1(A) & v1_relat_1(B) ) => v1_relat_1(k5_relat_1(A,B)) ) ), file(relat_1,k5_relat_1), []). fof(dt_k5_setfam_1,axiom,( ! [A,B] : ( m1_subset_1(B,k1_zfmisc_1(k1_zfmisc_1(A))) => m1_subset_1(k5_setfam_1(A,B),k1_zfmisc_1(A)) ) ), file(setfam_1,k5_setfam_1), []). fof(dt_k5_setwiseo,axiom,( ! [A,B,C] : ( ( m1_subset_1(B,k5_finsub_1(A)) & m1_subset_1(C,k5_finsub_1(A)) ) => m1_subset_1(k5_setwiseo(A,B,C),k5_finsub_1(A)) ) ), file(setwiseo,k5_setwiseo), []). fof(dt_k6_partfun1,axiom,( ! [A] : ( v1_partfun1(k6_partfun1(A),A,A) & m2_relset_1(k6_partfun1(A),A,A) ) ), file(partfun1,k6_partfun1), []). fof(dt_k6_relat_1,axiom,( ! [A] : v1_relat_1(k6_relat_1(A)) ), file(relat_1,k6_relat_1), []). fof(dt_k7_funct_2,axiom,( ! [A,B,C,D,E] : ( ( ~ v1_xboole_0(B) & v1_funct_1(D) & v1_funct_2(D,A,B) & m1_relset_1(D,A,B) & v1_funct_1(E) & v1_funct_2(E,B,C) & m1_relset_1(E,B,C) ) => ( v1_funct_1(k7_funct_2(A,B,C,D,E)) & v1_funct_2(k7_funct_2(A,B,C,D,E),A,C) & m2_relset_1(k7_funct_2(A,B,C,D,E),A,C) ) ) ), file(funct_2,k7_funct_2), []). fof(dt_k7_grcat_1,axiom,( ! [A] : ( l1_struct_0(A) => ( v1_funct_1(k7_grcat_1(A)) & v1_funct_2(k7_grcat_1(A),u1_struct_0(A),u1_struct_0(A)) & m2_relset_1(k7_grcat_1(A),u1_struct_0(A),u1_struct_0(A)) ) ) ), file(grcat_1,k7_grcat_1), []). fof(dt_k7_lopclset,axiom,( ! [A] : ( ( ~ v3_struct_0(A) & v10_lattices(A) & v17_lattices(A) & ~ v3_realset2(A) & l3_lattices(A) ) => m1_subset_1(k7_lopclset(A),k1_zfmisc_1(k1_zfmisc_1(u1_struct_0(A)))) ) ), file(lopclset,k7_lopclset), []). fof(dt_k7_relset_1,axiom,( ! [A,B,C,D,E,F] : ( ( m1_relset_1(E,A,B) & m1_relset_1(F,C,D) ) => m2_relset_1(k7_relset_1(A,B,C,D,E,F),A,D) ) ), file(relset_1,k7_relset_1), []). fof(dt_k7_setwiseo,axiom,( ! [A,B,C,D,E] : ( ( ~ v1_xboole_0(A) & ~ v1_xboole_0(B) & v1_funct_1(C) & v1_funct_2(C,k2_zfmisc_1(B,B),B) & m1_relset_1(C,k2_zfmisc_1(B,B),B) & m1_subset_1(D,k5_finsub_1(A)) & v1_funct_1(E) & v1_funct_2(E,A,B) & m1_relset_1(E,A,B) ) => m1_subset_1(k7_setwiseo(A,B,C,D,E),B) ) ), file(setwiseo,k7_setwiseo), []). 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_lopclset,axiom,( ! [A] : ( ( ~ v3_struct_0(A) & v10_lattices(A) & v17_lattices(A) & ~ v3_realset2(A) & l3_lattices(A) ) => ( v1_relat_1(k8_lopclset(A)) & v1_funct_1(k8_lopclset(A)) ) ) ), file(lopclset,k8_lopclset), []). fof(dt_k8_setwiseo,axiom,( ! [A,B,C,D] : ( ( ~ v1_xboole_0(A) & ~ v1_xboole_0(B) & v1_funct_1(C) & v1_funct_2(C,A,B) & m1_relset_1(C,A,B) & m1_subset_1(D,k5_finsub_1(A)) ) => m1_subset_1(k8_setwiseo(A,B,C,D),k5_finsub_1(B)) ) ), file(setwiseo,k8_setwiseo), []). fof(dt_k9_lopclset,axiom,( ! [A] : ( ( ~ v3_struct_0(A) & v10_lattices(A) & v17_lattices(A) & ~ v3_realset2(A) & l3_lattices(A) ) => ( v1_funct_1(k9_lopclset(A)) & v1_funct_2(k9_lopclset(A),u1_struct_0(A),k1_zfmisc_1(k7_lopclset(A))) & m2_relset_1(k9_lopclset(A),u1_struct_0(A),k1_zfmisc_1(k7_lopclset(A))) ) ) ), file(lopclset,k9_lopclset), []). fof(dt_k9_relat_1,axiom,( $true ), file(relat_1,k9_relat_1), []). fof(dt_l1_lattices,axiom,( ! [A] : ( l1_lattices(A) => l1_struct_0(A) ) ), file(lattices,l1_lattices), []). fof(dt_l1_pre_topc,axiom,( ! [A] : ( l1_pre_topc(A) => l1_struct_0(A) ) ), file(pre_topc,l1_pre_topc), []). fof(dt_l1_struct_0,axiom,( $true ), file(struct_0,l1_struct_0), []). fof(dt_l2_lattices,axiom,( ! [A] : ( l2_lattices(A) => l1_struct_0(A) ) ), file(lattices,l2_lattices), []). fof(dt_l3_lattices,axiom,( ! [A] : ( l3_lattices(A) => ( l1_lattices(A) & l2_lattices(A) ) ) ), file(lattices,l3_lattices), []). fof(dt_m1_filter_0,axiom,( ! [A] : ( ( ~ v3_struct_0(A) & v10_lattices(A) & l3_lattices(A) ) => ! [B] : ( m1_filter_0(B,A) => ( ~ v1_xboole_0(B) & m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) ) ) ) ), file(filter_0,m1_filter_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_u1_pre_topc,axiom,( ! [A] : ( l1_pre_topc(A) => m1_subset_1(u1_pre_topc(A),k1_zfmisc_1(k1_zfmisc_1(u1_struct_0(A)))) ) ), file(pre_topc,u1_pre_topc), []). fof(dt_u1_struct_0,axiom,( $true ), file(struct_0,u1_struct_0), []). fof(dt_u2_lattices,axiom,( ! [A] : ( l2_lattices(A) => ( v1_funct_1(u2_lattices(A)) & v1_funct_2(u2_lattices(A),k2_zfmisc_1(u1_struct_0(A),u1_struct_0(A)),u1_struct_0(A)) & m2_relset_1(u2_lattices(A),k2_zfmisc_1(u1_struct_0(A),u1_struct_0(A)),u1_struct_0(A)) ) ) ), file(lattices,u2_lattices), []). fof(existence_l1_lattices,axiom,( ? [A] : l1_lattices(A) ), file(lattices,l1_lattices), []). fof(existence_l1_pre_topc,axiom,( ? [A] : l1_pre_topc(A) ), file(pre_topc,l1_pre_topc), []). fof(existence_l1_struct_0,axiom,( ? [A] : l1_struct_0(A) ), file(struct_0,l1_struct_0), []). fof(existence_l2_lattices,axiom,( ? [A] : l2_lattices(A) ), file(lattices,l2_lattices), []). fof(existence_l3_lattices,axiom,( ? [A] : l3_lattices(A) ), file(lattices,l3_lattices), []). fof(existence_m1_filter_0,axiom,( ! [A] : ( ( ~ v3_struct_0(A) & v10_lattices(A) & l3_lattices(A) ) => ? [B] : m1_filter_0(B,A) ) ), file(filter_0,m1_filter_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(fc10_finset_1,axiom,( ! [A,B] : ( v1_finset_1(B) => v1_finset_1(k3_xboole_0(A,B)) ) ), file(finset_1,fc10_finset_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_finset_1,axiom,( ! [A,B] : ( v1_finset_1(A) => v1_finset_1(k3_xboole_0(A,B)) ) ), file(finset_1,fc11_finset_1), []). 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_finset_1,axiom,( ! [A,B] : ( ( v1_relat_1(A) & v1_funct_1(A) & v1_finset_1(B) ) => v1_finset_1(k9_relat_1(A,B)) ) ), file(finset_1,fc13_finset_1), []). 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_finset_1,axiom,( ! [A,B] : ( ( v1_finset_1(A) & v1_finset_1(B) ) => v1_finset_1(k2_zfmisc_1(A,B)) ) ), file(finset_1,fc14_finset_1), []). 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_finset_1,axiom,( ! [A] : ( ~ v1_xboole_0(k1_tarski(A)) & v1_finset_1(k1_tarski(A)) ) ), file(finset_1,fc1_finset_1), []). fof(fc1_finsub_1,axiom,( ! [A] : ( ~ v1_xboole_0(k1_zfmisc_1(A)) & v1_finsub_1(k1_zfmisc_1(A)) & v3_finsub_1(k1_zfmisc_1(A)) & v4_finsub_1(k1_zfmisc_1(A)) ) ), file(finsub_1,fc1_finsub_1), []). fof(fc1_lopclset,axiom,( ! [A] : ( ( ~ v3_struct_0(A) & v2_pre_topc(A) & l1_pre_topc(A) ) => ~ v1_xboole_0(k1_lopclset(A)) ) ), file(lopclset,fc1_lopclset), []). 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(fc22_membered,axiom,( ! [A,B] : ( ( v1_membered(A) & v1_membered(B) ) => v1_membered(k2_xboole_0(A,B)) ) ), file(membered,fc22_membered), []). fof(fc23_membered,axiom,( ! [A,B] : ( ( v2_membered(A) & v2_membered(B) ) => ( v1_membered(k2_xboole_0(A,B)) & v2_membered(k2_xboole_0(A,B)) ) ) ), file(membered,fc23_membered), []). fof(fc24_membered,axiom,( ! [A,B] : ( ( v3_membered(A) & v3_membered(B) ) => ( v1_membered(k2_xboole_0(A,B)) & v2_membered(k2_xboole_0(A,B)) & v3_membered(k2_xboole_0(A,B)) ) ) ), file(membered,fc24_membered), []). fof(fc25_membered,axiom,( ! [A,B] : ( ( v4_membered(A) & v4_membered(B) ) => ( v1_membered(k2_xboole_0(A,B)) & v2_membered(k2_xboole_0(A,B)) & v3_membered(k2_xboole_0(A,B)) & v4_membered(k2_xboole_0(A,B)) ) ) ), file(membered,fc25_membered), []). fof(fc26_membered,axiom,( ! [A,B] : ( ( v5_membered(A) & v5_membered(B) ) => ( v1_membered(k2_xboole_0(A,B)) & v2_membered(k2_xboole_0(A,B)) & v3_membered(k2_xboole_0(A,B)) & v4_membered(k2_xboole_0(A,B)) & v5_membered(k2_xboole_0(A,B)) ) ) ), file(membered,fc26_membered), []). fof(fc27_membered,axiom,( ! [A,B] : ( v1_membered(A) => v1_membered(k3_xboole_0(A,B)) ) ), file(membered,fc27_membered), []). fof(fc28_membered,axiom,( ! [A,B] : ( v1_membered(A) => v1_membered(k3_xboole_0(B,A)) ) ), file(membered,fc28_membered), []). fof(fc29_membered,axiom,( ! [A,B] : ( v2_membered(A) => ( v1_membered(k3_xboole_0(A,B)) & v2_membered(k3_xboole_0(A,B)) ) ) ), file(membered,fc29_membered), []). fof(fc2_finset_1,axiom,( ! [A,B] : ( ~ v1_xboole_0(k2_tarski(A,B)) & v1_finset_1(k2_tarski(A,B)) ) ), file(finset_1,fc2_finset_1), []). fof(fc2_finsub_1,axiom,( ! [A] : ( ~ v1_xboole_0(k5_finsub_1(A)) & v1_finsub_1(k5_finsub_1(A)) & v3_finsub_1(k5_finsub_1(A)) & v4_finsub_1(k5_finsub_1(A)) ) ), file(finsub_1,fc2_finsub_1), []). fof(fc2_lopclset,axiom,( ! [A] : ( ( ~ v3_struct_0(A) & v10_lattices(A) & v17_lattices(A) & ~ v3_realset2(A) & l3_lattices(A) ) => ~ v1_xboole_0(k7_lopclset(A)) ) ), file(lopclset,fc2_lopclset), []). fof(fc2_partfun1,axiom,( ! [A] : ( v1_relat_1(k6_relat_1(A)) & v1_funct_1(k6_relat_1(A)) & v1_relat_2(k6_relat_1(A)) & v3_relat_2(k6_relat_1(A)) & v4_relat_2(k6_relat_1(A)) & v8_relat_2(k6_relat_1(A)) ) ), file(partfun1,fc2_partfun1), []). fof(fc2_pre_topc,axiom,( ! [A] : ( ( ~ v3_struct_0(A) & l1_struct_0(A) ) => ~ v1_xboole_0(k2_pre_topc(A)) ) ), file(pre_topc,fc2_pre_topc), []). fof(fc30_membered,axiom,( ! [A,B] : ( v2_membered(A) => ( v1_membered(k3_xboole_0(B,A)) & v2_membered(k3_xboole_0(B,A)) ) ) ), file(membered,fc30_membered), []). fof(fc31_membered,axiom,( ! [A,B] : ( v3_membered(A) => ( v1_membered(k3_xboole_0(A,B)) & v2_membered(k3_xboole_0(A,B)) & v3_membered(k3_xboole_0(A,B)) ) ) ), file(membered,fc31_membered), []). fof(fc32_membered,axiom,( ! [A,B] : ( v3_membered(A) => ( v1_membered(k3_xboole_0(B,A)) & v2_membered(k3_xboole_0(B,A)) & v3_membered(k3_xboole_0(B,A)) ) ) ), file(membered,fc32_membered), []). fof(fc33_membered,axiom,( ! [A,B] : ( v4_membered(A) => ( v1_membered(k3_xboole_0(A,B)) & v2_membered(k3_xboole_0(A,B)) & v3_membered(k3_xboole_0(A,B)) & v4_membered(k3_xboole_0(A,B)) ) ) ), file(membered,fc33_membered), []). fof(fc34_membered,axiom,( ! [A,B] : ( v4_membered(A) => ( v1_membered(k3_xboole_0(B,A)) & v2_membered(k3_xboole_0(B,A)) & v3_membered(k3_xboole_0(B,A)) & v4_membered(k3_xboole_0(B,A)) ) ) ), file(membered,fc34_membered), []). fof(fc35_membered,axiom,( ! [A,B] : ( v5_membered(A) => ( v1_membered(k3_xboole_0(A,B)) & v2_membered(k3_xboole_0(A,B)) & v3_membered(k3_xboole_0(A,B)) & v4_membered(k3_xboole_0(A,B)) & v5_membered(k3_xboole_0(A,B)) ) ) ), file(membered,fc35_membered), []). fof(fc36_membered,axiom,( ! [A,B] : ( v5_membered(A) => ( v1_membered(k3_xboole_0(B,A)) & v2_membered(k3_xboole_0(B,A)) & v3_membered(k3_xboole_0(B,A)) & v4_membered(k3_xboole_0(B,A)) & v5_membered(k3_xboole_0(B,A)) ) ) ), file(membered,fc36_membered), []). fof(fc3_lopclset,axiom,( ! [A] : ( ( ~ v3_struct_0(A) & v10_lattices(A) & v17_lattices(A) & ~ v3_realset2(A) & l3_lattices(A) ) => ~ v1_xboole_0(k10_lopclset(A)) ) ), file(lopclset,fc3_lopclset), []). fof(fc4_lopclset,axiom,( ! [A] : ( ( ~ v3_struct_0(A) & v10_lattices(A) & v17_lattices(A) & ~ v3_realset2(A) & l3_lattices(A) ) => ( ~ v3_struct_0(k11_lopclset(A)) & v1_pre_topc(k11_lopclset(A)) & v2_pre_topc(k11_lopclset(A)) ) ) ), file(lopclset,fc4_lopclset), []). fof(fc5_pre_topc,axiom,( ! [A] : ( ( v2_pre_topc(A) & l1_pre_topc(A) ) => v4_pre_topc(k2_pre_topc(A),A) ) ), file(pre_topc,fc5_pre_topc), []). fof(fc6_membered,axiom, ( v1_xboole_0(k1_xboole_0) & v1_membered(k1_xboole_0) & v2_membered(k1_xboole_0) & v3_membered(k1_xboole_0) & v4_membered(k1_xboole_0) & v5_membered(k1_xboole_0) ), file(membered,fc6_membered), []). fof(fc7_membered,axiom,( ! [A] : ( v1_xcmplx_0(A) => v1_membered(k1_tarski(A)) ) ), file(membered,fc7_membered), []). fof(fc8_membered,axiom,( ! [A] : ( v1_xreal_0(A) => ( v1_membered(k1_tarski(A)) & v2_membered(k1_tarski(A)) ) ) ), file(membered,fc8_membered), []). fof(fc9_finset_1,axiom,( ! [A,B] : ( ( v1_finset_1(A) & v1_finset_1(B) ) => v1_finset_1(k2_xboole_0(A,B)) ) ), file(finset_1,fc9_finset_1), []). fof(fc9_membered,axiom,( ! [A] : ( v1_rat_1(A) => ( v1_membered(k1_tarski(A)) & v2_membered(k1_tarski(A)) & v3_membered(k1_tarski(A)) ) ) ), file(membered,fc9_membered), []). fof(fraenkel_a_1_0_lopclset,axiom,( ! [A,B] : ( ( ~ v3_struct_0(B) & l1_pre_topc(B) ) => ( r2_hidden(A,a_1_0_lopclset(B)) <=> ? [C] : ( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(B))) & A = C & v3_pre_topc(C,B) & v4_pre_topc(C,B) ) ) ) ), file(lopclset,a_1_0_lopclset), []). fof(fraenkel_a_1_2_lopclset,axiom,( ! [A,B] : ( ( ~ v3_struct_0(B) & v10_lattices(B) & v17_lattices(B) & ~ v3_realset2(B) & l3_lattices(B) ) => ( r2_hidden(A,a_1_2_lopclset(B)) <=> ? [C] : ( m1_filter_0(C,B) & A = C & v1_filter_0(C,B) ) ) ) ), file(lopclset,a_1_2_lopclset), []). fof(fraenkel_a_1_3_lopclset,axiom,( ! [A,B] : ( ( ~ v3_struct_0(B) & v10_lattices(B) & v17_lattices(B) & ~ v3_realset2(B) & l3_lattices(B) ) => ( r2_hidden(A,a_1_3_lopclset(B)) <=> ? [C] : ( m1_subset_1(C,k1_zfmisc_1(k1_zfmisc_1(k7_lopclset(B)))) & A = k5_setfam_1(k7_lopclset(B),C) & r1_tarski(C,k10_lopclset(B)) ) ) ) ), file(lopclset,a_1_3_lopclset), []). fof(free_g1_pre_topc,axiom,( ! [A,B] : ( m1_subset_1(B,k1_zfmisc_1(k1_zfmisc_1(A))) => ! [C,D] : ( g1_pre_topc(A,B) = g1_pre_topc(C,D) => ( A = C & B = D ) ) ) ), file(pre_topc,g1_pre_topc), []). fof(idempotence_k1_finsub_1,axiom,( ! [A,B,C] : ( ( ~ v1_xboole_0(A) & v4_finsub_1(A) & m1_subset_1(B,A) & m1_subset_1(C,A) ) => k1_finsub_1(A,B,B) = B ) ), file(finsub_1,k1_finsub_1), []). fof(idempotence_k2_xboole_0,axiom,( ! [A,B] : k2_xboole_0(A,A) = A ), file(xboole_0,k2_xboole_0), []). fof(idempotence_k3_finsub_1,axiom,( ! [A,B,C] : ( ( ~ v1_xboole_0(A) & v4_finsub_1(A) & m1_subset_1(B,A) & m1_subset_1(C,A) ) => k3_finsub_1(A,B,B) = B ) ), file(finsub_1,k3_finsub_1), []). fof(idempotence_k3_xboole_0,axiom,( ! [A,B] : k3_xboole_0(A,A) = A ), file(xboole_0,k3_xboole_0), []). fof(idempotence_k4_subset_1,axiom,( ! [A,B,C] : ( ( m1_subset_1(B,k1_zfmisc_1(A)) & m1_subset_1(C,k1_zfmisc_1(A)) ) => k4_subset_1(A,B,B) = B ) ), file(subset_1,k4_subset_1), []). fof(idempotence_k5_setwiseo,axiom,( ! [A,B,C] : ( ( m1_subset_1(B,k5_finsub_1(A)) & m1_subset_1(C,k5_finsub_1(A)) ) => k5_setwiseo(A,B,B) = B ) ), file(setwiseo,k5_setwiseo), []). fof(involutiveness_k3_subset_1,axiom,( ! [A,B] : ( m1_subset_1(B,k1_zfmisc_1(A)) => k3_subset_1(A,k3_subset_1(A,B)) = B ) ), file(subset_1,k3_subset_1), []). fof(l49_lopclset,axiom,( ! [A] : ( ( ~ v3_struct_0(A) & v10_lattices(A) & v17_lattices(A) & ~ v3_realset2(A) & l3_lattices(A) ) => r1_tarski(k10_lopclset(A),k1_lopclset(k11_lopclset(A))) ) ), file(lopclset,l49_lopclset), []). fof(rc1_finset_1,axiom,( ? [A] : ( ~ v1_xboole_0(A) & v1_finset_1(A) ) ), file(finset_1,rc1_finset_1), []). fof(rc1_funct_2,axiom,( ! [A,B] : ? [C] : ( m1_relset_1(C,A,B) & v1_relat_1(C) & v1_funct_1(C) & v1_funct_2(C,A,B) ) ), file(funct_2,rc1_funct_2), []). fof(rc1_lopclset,axiom,( ? [A] : ( l3_lattices(A) & ~ v3_struct_0(A) & v4_lattices(A) & v5_lattices(A) & v6_lattices(A) & v7_lattices(A) & v8_lattices(A) & v9_lattices(A) & v10_lattices(A) & v11_lattices(A) & v12_lattices(A) & v13_lattices(A) & v14_lattices(A) & v15_lattices(A) & v16_lattices(A) & v17_lattices(A) & ~ v3_realset2(A) ) ), file(lopclset,rc1_lopclset), []). 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_partfun1,axiom,( ? [A] : ( v1_relat_1(A) & v1_funct_1(A) & v2_funct_1(A) & v1_xboole_0(A) ) ), file(partfun1,rc1_partfun1), []). fof(rc1_pre_topc,axiom,( ? [A] : ( l1_pre_topc(A) & v1_pre_topc(A) ) ), file(pre_topc,rc1_pre_topc), []). fof(rc2_funct_2,axiom,( ! [A] : ? [B] : ( m1_relset_1(B,A,A) & v1_relat_1(B) & v1_funct_1(B) & v2_funct_1(B) & v1_funct_2(B,A,A) & v2_funct_2(B,A,A) & v3_funct_2(B,A,A) ) ), file(funct_2,rc2_funct_2), []). fof(rc2_lopclset,axiom,( ! [A] : ( ~ v1_xboole_0(A) => ? [B] : ( m1_subset_1(B,k5_finsub_1(A)) & ~ v1_xboole_0(B) & v1_finset_1(B) ) ) ), file(lopclset,rc2_lopclset), []). fof(rc2_partfun1,axiom,( ! [A,B] : ? [C] : ( m1_relset_1(C,A,B) & v1_relat_1(C) & v1_funct_1(C) ) ), file(partfun1,rc2_partfun1), []). fof(rc2_pre_topc,axiom,( ? [A] : ( l1_pre_topc(A) & ~ v3_struct_0(A) & v1_pre_topc(A) & v2_pre_topc(A) ) ), file(pre_topc,rc2_pre_topc), []). fof(rc3_finset_1,axiom,( ! [A] : ( ~ v1_xboole_0(A) => ? [B] : ( m1_subset_1(B,k1_zfmisc_1(A)) & ~ v1_xboole_0(B) & v1_finset_1(B) ) ) ), file(finset_1,rc3_finset_1), []). fof(rc3_partfun1,axiom,( ! [A] : ? [B] : ( m1_relset_1(B,A,A) & v1_relat_1(B) & v1_relat_2(B) & v3_relat_2(B) & v4_relat_2(B) & v8_relat_2(B) & v1_partfun1(B,A,A) ) ), file(partfun1,rc3_partfun1), []). fof(rc3_struct_0,axiom,( ? [A] : ( l1_struct_0(A) & ~ v3_struct_0(A) ) ), file(struct_0,rc3_struct_0), []). fof(rc4_finset_1,axiom,( ! [A] : ( ~ v1_xboole_0(A) => ? [B] : ( m1_subset_1(B,k1_zfmisc_1(A)) & ~ v1_xboole_0(B) & v1_finset_1(B) ) ) ), file(finset_1,rc4_finset_1), []). 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(rc6_pre_topc,axiom,( ! [A] : ( ( v2_pre_topc(A) & l1_pre_topc(A) ) => ? [B] : ( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) & v4_pre_topc(B,A) ) ) ), file(pre_topc,rc6_pre_topc), []). fof(rc7_pre_topc,axiom,( ! [A] : ( ( ~ v3_struct_0(A) & v2_pre_topc(A) & l1_pre_topc(A) ) => ? [B] : ( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) & ~ v1_xboole_0(B) & v4_pre_topc(B,A) ) ) ), file(pre_topc,rc7_pre_topc), []). fof(redefinition_k1_finsub_1,axiom,( ! [A,B,C] : ( ( ~ v1_xboole_0(A) & v4_finsub_1(A) & m1_subset_1(B,A) & m1_subset_1(C,A) ) => k1_finsub_1(A,B,C) = k2_xboole_0(B,C) ) ), file(finsub_1,k1_finsub_1), []). fof(redefinition_k1_partfun1,axiom,( ! [A,B,C,D,E,F] : ( ( v1_funct_1(E) & m1_relset_1(E,A,B) & v1_funct_1(F) & m1_relset_1(F,C,D) ) => k1_partfun1(A,B,C,D,E,F) = k5_relat_1(E,F) ) ), file(partfun1,k1_partfun1), []). fof(redefinition_k2_binop_1,axiom,( ! [A,B,C,D,E,F] : ( ( ~ v1_xboole_0(A) & ~ v1_xboole_0(B) & v1_funct_1(D) & v1_funct_2(D,k2_zfmisc_1(A,B),C) & m1_relset_1(D,k2_zfmisc_1(A,B),C) & m1_subset_1(E,A) & m1_subset_1(F,B) ) => k2_binop_1(A,B,C,D,E,F) = k1_binop_1(D,E,F) ) ), file(binop_1,k2_binop_1), []). fof(redefinition_k2_setwiseo,axiom,( ! [A,B] : ( ( ~ v1_xboole_0(A) & m1_subset_1(B,A) ) => k2_setwiseo(A,B) = k1_tarski(B) ) ), file(setwiseo,k2_setwiseo), []). fof(redefinition_k3_finsub_1,axiom,( ! [A,B,C] : ( ( ~ v1_xboole_0(A) & v4_finsub_1(A) & m1_subset_1(B,A) & m1_subset_1(C,A) ) => k3_finsub_1(A,B,C) = k3_xboole_0(B,C) ) ), file(finsub_1,k3_finsub_1), []). fof(redefinition_k3_funct_2,axiom,( ! [A,B,C,D] : ( ( v1_funct_1(C) & m1_relset_1(C,A,B) ) => k3_funct_2(A,B,C,D) = k10_relat_1(C,D) ) ), file(funct_2,k3_funct_2), []). fof(redefinition_k3_lattices,axiom,( ! [A,B,C] : ( ( ~ v3_struct_0(A) & v4_lattices(A) & l2_lattices(A) & m1_subset_1(B,u1_struct_0(A)) & m1_subset_1(C,u1_struct_0(A)) ) => k3_lattices(A,B,C) = k1_lattices(A,B,C) ) ), file(lattices,k3_lattices), []). fof(redefinition_k4_relset_1,axiom,( ! [A,B,C] : ( m1_relset_1(C,A,B) => k4_relset_1(A,B,C) = k1_relat_1(C) ) ), file(relset_1,k4_relset_1), []). fof(redefinition_k4_subset_1,axiom,( ! [A,B,C] : ( ( m1_subset_1(B,k1_zfmisc_1(A)) & m1_subset_1(C,k1_zfmisc_1(A)) ) => k4_subset_1(A,B,C) = k2_xboole_0(B,C) ) ), file(subset_1,k4_subset_1), []). fof(redefinition_k5_setfam_1,axiom,( ! [A,B] : ( m1_subset_1(B,k1_zfmisc_1(k1_zfmisc_1(A))) => k5_setfam_1(A,B) = k3_tarski(B) ) ), file(setfam_1,k5_setfam_1), []). fof(redefinition_k5_setwiseo,axiom,( ! [A,B,C] : ( ( m1_subset_1(B,k5_finsub_1(A)) & m1_subset_1(C,k5_finsub_1(A)) ) => k5_setwiseo(A,B,C) = k2_xboole_0(B,C) ) ), file(setwiseo,k5_setwiseo), []). fof(redefinition_k6_partfun1,axiom,( ! [A] : k6_partfun1(A) = k6_relat_1(A) ), file(partfun1,k6_partfun1), []). fof(redefinition_k7_funct_2,axiom,( ! [A,B,C,D,E] : ( ( ~ v1_xboole_0(B) & v1_funct_1(D) & v1_funct_2(D,A,B) & m1_relset_1(D,A,B) & v1_funct_1(E) & v1_funct_2(E,B,C) & m1_relset_1(E,B,C) ) => k7_funct_2(A,B,C,D,E) = k5_relat_1(D,E) ) ), file(funct_2,k7_funct_2), []). fof(redefinition_k7_relset_1,axiom,( ! [A,B,C,D,E,F] : ( ( m1_relset_1(E,A,B) & m1_relset_1(F,C,D) ) => k7_relset_1(A,B,C,D,E,F) = k5_relat_1(E,F) ) ), file(relset_1,k7_relset_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_k8_setwiseo,axiom,( ! [A,B,C,D] : ( ( ~ v1_xboole_0(A) & ~ v1_xboole_0(B) & v1_funct_1(C) & v1_funct_2(C,A,B) & m1_relset_1(C,A,B) & m1_subset_1(D,k5_finsub_1(A)) ) => k8_setwiseo(A,B,C,D) = k9_relat_1(C,D) ) ), file(setwiseo,k8_setwiseo), []). fof(redefinition_k9_lopclset,axiom,( ! [A] : ( ( ~ v3_struct_0(A) & v10_lattices(A) & v17_lattices(A) & ~ v3_realset2(A) & l3_lattices(A) ) => k9_lopclset(A) = k8_lopclset(A) ) ), file(lopclset,k9_lopclset), []). 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_rfinseq,axiom,( ! [A,B] : ( ( v1_relat_1(A) & v1_relat_1(B) ) => r1_rfinseq(A,A) ) ), file(rfinseq,r1_rfinseq), []). fof(reflexivity_r1_tarski,axiom,( ! [A,B] : r1_tarski(A,A) ), file(tarski,r1_tarski), []). fof(s4_setwiseo__e29_42_1_4_2__lopclset,axiom,( ! [A,B] : ( ( ~ v3_struct_0(A) & v10_lattices(A) & v17_lattices(A) & ~ v3_realset2(A) & l3_lattices(A) & v1_funct_1(B) & v1_funct_2(B,k2_zfmisc_1(k1_zfmisc_1(k7_lopclset(A)),k1_zfmisc_1(k7_lopclset(A))),k1_zfmisc_1(k7_lopclset(A))) & m2_relset_1(B,k2_zfmisc_1(k1_zfmisc_1(k7_lopclset(A)),k1_zfmisc_1(k7_lopclset(A))),k1_zfmisc_1(k7_lopclset(A))) ) => ( ( ! [C] : ( ( ~ v1_xboole_0(C) & m1_subset_1(C,k1_zfmisc_1(k1_zfmisc_1(k7_lopclset(A)))) ) => ( C = k1_setwiseo(k1_zfmisc_1(k7_lopclset(A))) => k7_setwiseo(k1_zfmisc_1(k7_lopclset(A)),k1_zfmisc_1(k7_lopclset(A)),B,k1_setwiseo(k1_zfmisc_1(k7_lopclset(A))),k6_partfun1(k1_zfmisc_1(k7_lopclset(A)))) = k5_setfam_1(k7_lopclset(A),C) ) ) & ! [D] : ( m1_subset_1(D,k5_finsub_1(k1_zfmisc_1(k7_lopclset(A)))) => ! [E] : ( m1_subset_1(E,k1_zfmisc_1(k7_lopclset(A))) => ( ! [F] : ( ( ~ v1_xboole_0(F) & m1_subset_1(F,k1_zfmisc_1(k1_zfmisc_1(k7_lopclset(A)))) ) => ( F = D => k7_setwiseo(k1_zfmisc_1(k7_lopclset(A)),k1_zfmisc_1(k7_lopclset(A)),B,D,k6_partfun1(k1_zfmisc_1(k7_lopclset(A)))) = k5_setfam_1(k7_lopclset(A),F) ) ) => ! [G] : ( ( ~ v1_xboole_0(G) & m1_subset_1(G,k1_zfmisc_1(k1_zfmisc_1(k7_lopclset(A)))) ) => ( G = k5_setwiseo(k1_zfmisc_1(k7_lopclset(A)),D,k2_setwiseo(k1_zfmisc_1(k7_lopclset(A)),E)) => k7_setwiseo(k1_zfmisc_1(k7_lopclset(A)),k1_zfmisc_1(k7_lopclset(A)),B,k5_setwiseo(k1_zfmisc_1(k7_lopclset(A)),D,k2_setwiseo(k1_zfmisc_1(k7_lopclset(A)),E)),k6_partfun1(k1_zfmisc_1(k7_lopclset(A)))) = k5_setfam_1(k7_lopclset(A),G) ) ) ) ) ) ) => ! [D] : ( m1_subset_1(D,k5_finsub_1(k1_zfmisc_1(k7_lopclset(A)))) => ! [H] : ( ( ~ v1_xboole_0(H) & m1_subset_1(H,k1_zfmisc_1(k1_zfmisc_1(k7_lopclset(A)))) ) => ( H = D => k7_setwiseo(k1_zfmisc_1(k7_lopclset(A)),k1_zfmisc_1(k7_lopclset(A)),B,D,k6_partfun1(k1_zfmisc_1(k7_lopclset(A)))) = k5_setfam_1(k7_lopclset(A),H) ) ) ) ) ) ), file(lopclset,s4_setwiseo__e29_42_1_4_2__lopclset), []). fof(s8_funct_2__e18_42_1_4_2__lopclset,axiom,( ! [A] : ( ( ~ v3_struct_0(A) & v10_lattices(A) & v17_lattices(A) & ~ v3_realset2(A) & l3_lattices(A) ) => ? [B] : ( v1_funct_1(B) & v1_funct_2(B,k2_zfmisc_1(k1_zfmisc_1(k7_lopclset(A)),k1_zfmisc_1(k7_lopclset(A))),k1_zfmisc_1(k7_lopclset(A))) & m2_relset_1(B,k2_zfmisc_1(k1_zfmisc_1(k7_lopclset(A)),k1_zfmisc_1(k7_lopclset(A))),k1_zfmisc_1(k7_lopclset(A))) & ! [C] : ( m1_subset_1(C,k1_zfmisc_1(k7_lopclset(A))) => ! [D] : ( m1_subset_1(D,k1_zfmisc_1(k7_lopclset(A))) => k1_funct_1(B,k4_tarski(C,D)) = k1_finsub_1(k1_zfmisc_1(k7_lopclset(A)),C,D) ) ) ) ) ), file(lopclset,s8_funct_2__e18_42_1_4_2__lopclset), []). fof(symmetry_r1_rfinseq,axiom,( ! [A,B] : ( ( v1_relat_1(A) & v1_relat_1(B) ) => ( r1_rfinseq(A,B) => r1_rfinseq(B,A) ) ) ), file(rfinseq,r1_rfinseq), []). fof(symmetry_r1_xboole_0,axiom,( ! [A,B] : ( r1_xboole_0(A,B) => r1_xboole_0(B,A) ) ), file(xboole_0,r1_xboole_0), []). fof(t147_funct_1,axiom,( ! [A,B] : ( ( v1_relat_1(B) & v1_funct_1(B) ) => ( r1_tarski(A,k2_relat_1(B)) => k9_relat_1(B,k10_relat_1(B,A)) = A ) ) ), file(funct_1,t147_funct_1), []). fof(t149_relat_1,axiom,( ! [A] : ( v1_relat_1(A) => k9_relat_1(A,k1_xboole_0) = k1_xboole_0 ) ), file(relat_1,t149_relat_1), []). fof(t17_xboole_1,axiom,( ! [A,B] : r1_tarski(k3_xboole_0(A,B),A) ), file(xboole_1,t17_xboole_1), []). fof(t18_pre_topc,axiom,( ! [A] : ( l1_struct_0(A) => ! [B] : ( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) => k4_subset_1(u1_struct_0(A),B,k3_subset_1(u1_struct_0(A),B)) = k2_pre_topc(A) ) ) ), file(pre_topc,t18_pre_topc), []). fof(t1_boole,axiom,( ! [A] : k2_xboole_0(A,k1_xboole_0) = A ), file(boole,t1_boole), []). fof(t1_card_4,axiom,( ! [A] : ( v1_finset_1(A) <=> v1_finset_1(k1_card_1(A)) ) ), file(card_4,t1_card_4), []). fof(t1_subset,axiom,( ! [A,B] : ( r2_hidden(A,B) => m1_subset_1(A,B) ) ), file(subset,t1_subset), []). fof(t1_xboole_1,axiom,( ! [A,B,C] : ( ( r1_tarski(A,B) & r1_tarski(B,C) ) => r1_tarski(A,C) ) ), file(xboole_1,t1_xboole_1), []). fof(t23_funct_2,axiom,( ! [A,B,C] : ( m2_relset_1(C,A,B) => ( k7_relset_1(A,A,A,B,k6_partfun1(A),C) = C & k7_relset_1(A,B,B,B,C,k6_partfun1(B)) = C ) ) ), file(funct_2,t23_funct_2), []). fof(t23_xboole_1,axiom,( ! [A,B,C] : k3_xboole_0(A,k2_xboole_0(B,C)) = k2_xboole_0(k3_xboole_0(A,B),k3_xboole_0(A,C)) ), file(xboole_1,t23_xboole_1), []). fof(t24_lopclset,axiom,( ! [A] : ( ( ~ v3_struct_0(A) & v10_lattices(A) & v17_lattices(A) & ~ v3_realset2(A) & l3_lattices(A) ) => ! [B] : ( m1_subset_1(B,u1_struct_0(A)) => ! [C] : ( m1_subset_1(C,u1_struct_0(A)) => k1_funct_1(k8_lopclset(A),k3_lattices(A,B,C)) = k2_xboole_0(k1_funct_1(k8_lopclset(A),B),k1_funct_1(k8_lopclset(A),C)) ) ) ) ), file(lopclset,t24_lopclset), []). fof(t26_lattice2,axiom,( ! [A] : ( ( ~ v3_struct_0(A) & v10_lattices(A) & l3_lattices(A) ) => v3_binop_1(u2_lattices(A),u1_struct_0(A)) ) ), file(lattice2,t26_lattice2), []). fof(t26_setwiseo,axiom,( ! [A] : ( ~ v1_xboole_0(A) => ! [B] : ( ~ v1_xboole_0(B) => ! [C] : ( ( v1_funct_1(C) & v1_funct_2(C,k2_zfmisc_1(A,A),A) & m2_relset_1(C,k2_zfmisc_1(A,A),A) ) => ! [D] : ( ( v1_funct_1(D) & v1_funct_2(D,B,A) & m2_relset_1(D,B,A) ) => ( ( v1_binop_1(C,A) & v2_binop_1(C,A) ) => ! [E] : ( m1_subset_1(E,B) => k7_setwiseo(B,A,C,k2_setwiseo(B,E),D) = k8_funct_2(B,A,D,E) ) ) ) ) ) ) ), file(setwiseo,t26_setwiseo), []). fof(t26_subset_1,axiom,( ! [A,B] : ( m1_subset_1(B,k1_zfmisc_1(A)) => r1_xboole_0(B,k3_subset_1(A,B)) ) ), file(subset_1,t26_subset_1), []). fof(t26_xboole_1,axiom,( ! [A,B,C] : ( r1_tarski(A,B) => r1_tarski(k3_xboole_0(A,C),k3_xboole_0(B,C)) ) ), file(xboole_1,t26_xboole_1), []). fof(t27_lattice2,axiom,( ! [A] : ( ( ~ v3_struct_0(A) & v4_lattices(A) & l2_lattices(A) ) => v1_binop_1(u2_lattices(A),u1_struct_0(A)) ) ), file(lattice2,t27_lattice2), []). fof(t28_xboole_1,axiom,( ! [A,B] : ( r1_tarski(A,B) => k3_xboole_0(A,B) = A ) ), file(xboole_1,t28_xboole_1), []). fof(t29_lattice2,axiom,( ! [A] : ( ( ~ v3_struct_0(A) & v5_lattices(A) & l2_lattices(A) ) => v2_binop_1(u2_lattices(A),u1_struct_0(A)) ) ), file(lattice2,t29_lattice2), []). fof(t29_lopclset,axiom,( ! [A] : ( ( ~ v3_struct_0(A) & v10_lattices(A) & v17_lattices(A) & ~ v3_realset2(A) & l3_lattices(A) ) => v2_funct_1(k9_lopclset(A)) ) ), file(lopclset,t29_lopclset), []). fof(t29_setwiseo,axiom,( ! [A] : ( ~ v1_xboole_0(A) => ! [B] : ( ~ v1_xboole_0(B) => ! [C] : ( ( v1_funct_1(C) & v1_funct_2(C,k2_zfmisc_1(A,A),A) & m2_relset_1(C,k2_zfmisc_1(A,A),A) ) => ! [D] : ( m1_subset_1(D,k5_finsub_1(B)) => ! [E] : ( ( v1_funct_1(E) & v1_funct_2(E,B,A) & m2_relset_1(E,B,A) ) => ( ( v3_binop_1(C,A) & v1_binop_1(C,A) & v2_binop_1(C,A) ) => ( D = k1_xboole_0 | ! [F] : ( m1_subset_1(F,B) => k7_setwiseo(B,A,C,k5_setwiseo(B,D,k2_setwiseo(B,F)),E) = k2_binop_1(A,A,A,C,k7_setwiseo(B,A,C,D,E),k8_funct_2(B,A,E,F)) ) ) ) ) ) ) ) ) ), file(setwiseo,t29_setwiseo), []). fof(t29_tops_1,axiom,( ! [A] : ( l1_pre_topc(A) => ! [B] : ( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) => ( v4_pre_topc(B,A) <=> v3_pre_topc(k3_subset_1(u1_struct_0(A),B),A) ) ) ) ), file(tops_1,t29_tops_1), []). fof(t2_boole,axiom,( ! [A] : k3_xboole_0(A,k1_xboole_0) = k1_xboole_0 ), file(boole,t2_boole), []). fof(t2_lopclset,axiom,( ! [A] : ( ( ~ v3_struct_0(A) & v2_pre_topc(A) & l1_pre_topc(A) ) => ! [B] : ( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) => ( r2_hidden(B,k1_lopclset(A)) => v3_pre_topc(B,A) ) ) ) ), file(lopclset,t2_lopclset), []). fof(t2_subset,axiom,( ! [A,B] : ( m1_subset_1(A,B) => ( v1_xboole_0(B) | r2_hidden(A,B) ) ) ), file(subset,t2_subset), []). fof(t2_tarski,axiom,( ! [A,B] : ( ! [C] : ( r2_hidden(C,A) <=> r2_hidden(C,B) ) => A = B ) ), file(tarski,t2_tarski), []). fof(t2_zfmisc_1,axiom,( k3_tarski(k1_xboole_0) = k1_xboole_0 ), file(zfmisc_1,t2_zfmisc_1), []). fof(t30_tops_1,axiom,( ! [A] : ( l1_pre_topc(A) => ! [B] : ( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) => ( v3_pre_topc(B,A) <=> v4_pre_topc(k3_subset_1(u1_struct_0(A),B),A) ) ) ) ), file(tops_1,t30_tops_1), []). fof(t31_zfmisc_1,axiom,( ! [A] : k3_tarski(k1_tarski(A)) = A ), file(zfmisc_1,t31_zfmisc_1), []). fof(t35_funct_1,axiom,( ! [A,B] : ( r2_hidden(B,A) => k1_funct_1(k6_relat_1(A),B) = B ) ), file(funct_1,t35_funct_1), []). fof(t36_lopclset,axiom,( ! [A] : ( ( ~ v3_struct_0(A) & v10_lattices(A) & v17_lattices(A) & ~ v3_realset2(A) & l3_lattices(A) ) => k8_funct_2(u1_struct_0(A),k1_zfmisc_1(k7_lopclset(A)),k9_lopclset(A),k5_lattices(A)) = k1_xboole_0 ) ), file(lopclset,t36_lopclset), []). fof(t38_lopclset,axiom,( ! [A,B] : ( ( ~ v3_struct_0(B) & v10_lattices(B) & v17_lattices(B) & ~ v3_realset2(B) & l3_lattices(B) ) => ~ ( k7_lopclset(B) = k3_tarski(A) & m1_subset_1(A,k1_zfmisc_1(k10_lopclset(B))) & ! [C] : ( m1_subset_1(C,k5_finsub_1(A)) => k7_lopclset(B) != k3_tarski(C) ) ) ) ), file(lopclset,t38_lopclset), []). fof(t38_setwiseo,axiom,( ! [A] : ( ~ v1_xboole_0(A) => ! [B] : ( ~ v1_xboole_0(B) => ! [C] : ( ~ v1_xboole_0(C) => ! [D] : ( ( v1_funct_1(D) & v1_funct_2(D,k2_zfmisc_1(A,A),A) & m2_relset_1(D,k2_zfmisc_1(A,A),A) ) => ( ( v3_binop_1(D,A) & v1_binop_1(D,A) & v2_binop_1(D,A) ) => ! [E] : ( m1_subset_1(E,k5_finsub_1(B)) => ( E != k1_xboole_0 => ! [F] : ( ( v1_funct_1(F) & v1_funct_2(F,B,C) & m2_relset_1(F,B,C) ) => ! [G] : ( ( v1_funct_1(G) & v1_funct_2(G,C,A) & m2_relset_1(G,C,A) ) => k7_setwiseo(C,A,D,k8_setwiseo(B,C,F,E),G) = k7_setwiseo(B,A,D,E,k7_funct_2(B,C,A,F,G)) ) ) ) ) ) ) ) ) ) ), file(setwiseo,t38_setwiseo), []). fof(t39_setwiseo,axiom,( ! [A] : ( ~ v1_xboole_0(A) => ! [B] : ( ~ v1_xboole_0(B) => ! [C] : ( ( v1_funct_1(C) & v1_funct_2(C,k2_zfmisc_1(B,B),B) & m2_relset_1(C,k2_zfmisc_1(B,B),B) ) => ( ( v1_binop_1(C,B) & v2_binop_1(C,B) & v3_binop_1(C,B) ) => ! [D] : ( ~ v1_xboole_0(D) => ! [E] : ( ( v1_funct_1(E) & v1_funct_2(E,k2_zfmisc_1(D,D),D) & m2_relset_1(E,k2_zfmisc_1(D,D),D) ) => ( ( v1_binop_1(E,D) & v2_binop_1(E,D) & v3_binop_1(E,D) ) => ! [F] : ( ( v1_funct_1(F) & v1_funct_2(F,A,B) & m2_relset_1(F,A,B) ) => ! [G] : ( ( v1_funct_1(G) & v1_funct_2(G,B,D) & m2_relset_1(G,B,D) ) => ( ! [H] : ( m1_subset_1(H,B) => ! [I] : ( m1_subset_1(I,B) => k8_funct_2(B,D,G,k2_binop_1(B,B,B,C,H,I)) = k2_binop_1(D,D,D,E,k8_funct_2(B,D,G,H),k8_funct_2(B,D,G,I)) ) ) => ! [H] : ( m1_subset_1(H,k5_finsub_1(A)) => ( H != k1_xboole_0 => k8_funct_2(B,D,G,k7_setwiseo(A,B,C,H,F)) = k7_setwiseo(A,D,E,H,k7_funct_2(A,B,D,F,G)) ) ) ) ) ) ) ) ) ) ) ) ) ), file(setwiseo,t39_setwiseo), []). fof(t3_lopclset,axiom,( ! [A] : ( ( ~ v3_struct_0(A) & v2_pre_topc(A) & l1_pre_topc(A) ) => ! [B] : ( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) => ( r2_hidden(B,k1_lopclset(A)) => v4_pre_topc(B,A) ) ) ) ), file(lopclset,t3_lopclset), []). fof(t3_rfinseq,axiom,( ! [A] : ( ( v1_relat_1(A) & v1_funct_1(A) ) => ! [B] : ( ( v1_relat_1(B) & v1_funct_1(B) ) => ( r1_rfinseq(A,B) <=> ? [C] : ( v1_relat_1(C) & v1_funct_1(C) & k1_relat_1(C) = k1_relat_1(A) & k2_relat_1(C) = k1_relat_1(B) & v2_funct_1(C) & A = k5_relat_1(C,B) ) ) ) ) ), file(rfinseq,t3_rfinseq), []). fof(t3_subset,axiom,( ! [A,B] : ( m1_subset_1(A,k1_zfmisc_1(B)) <=> r1_tarski(A,B) ) ), file(subset,t3_subset), []). fof(t3_xboole_1,axiom,( ! [A] : ( r1_tarski(A,k1_xboole_0) => A = k1_xboole_0 ) ), file(xboole_1,t3_xboole_1), []). fof(t42_funct_1,axiom,( ! [A] : ( ( v1_relat_1(A) & v1_funct_1(A) ) => ( k5_relat_1(k6_relat_1(k1_relat_1(A)),A) = A & k5_relat_1(A,k6_relat_1(k2_relat_1(A))) = A ) ) ), file(funct_1,t42_funct_1), []). fof(t4_borsuk_1,axiom,( ! [A,B] : ( m1_subset_1(B,k1_zfmisc_1(A)) => k3_funct_2(A,A,k6_partfun1(A),B) = B ) ), file(borsuk_1,t4_borsuk_1), []). fof(t4_rfinseq,axiom,( ! [A] : ( ( v1_relat_1(A) & v1_funct_1(A) ) => ! [B] : ( ( v1_relat_1(B) & v1_funct_1(B) ) => ( r1_rfinseq(A,B) <=> ! [C] : k1_card_1(k10_relat_1(A,C)) = k1_card_1(k10_relat_1(B,C)) ) ) ) ), file(rfinseq,t4_rfinseq), []). fof(t4_subset,axiom,( ! [A,B,C] : ( ( r2_hidden(A,B) & m1_subset_1(B,k1_zfmisc_1(C)) ) => m1_subset_1(A,C) ) ), file(subset,t4_subset), []). fof(t4_xboole_1,axiom,( ! [A,B,C] : k2_xboole_0(k2_xboole_0(A,B),C) = k2_xboole_0(A,k2_xboole_0(B,C)) ), file(xboole_1,t4_xboole_1), []). fof(t57_setfam_1,axiom,( ! [A,B,C] : ( ( r1_tarski(C,k3_tarski(k2_xboole_0(A,B))) & ! [D] : ( r2_hidden(D,B) => r1_xboole_0(D,C) ) ) => r1_tarski(C,k3_tarski(A)) ) ), file(setfam_1,t57_setfam_1), []). 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_funct_2,axiom,( ! [A,B,C,D] : ( ( v1_funct_1(D) & v1_funct_2(D,A,B) & m2_relset_1(D,A,B) ) => ( r2_hidden(C,A) => ( B = k1_xboole_0 | r2_hidden(k1_funct_1(D,C),k2_relat_1(D)) ) ) ) ), file(funct_2,t6_funct_2), []). fof(t7_boole,axiom,( ! [A,B] : ~ ( r2_hidden(A,B) & v1_xboole_0(B) ) ), file(boole,t7_boole), []). fof(t7_xboole_1,axiom,( ! [A,B] : r1_tarski(A,k2_xboole_0(A,B)) ), file(xboole_1,t7_xboole_1), []). fof(t8_boole,axiom,( ! [A,B] : ~ ( v1_xboole_0(A) & A != B & v1_xboole_0(B) ) ), file(boole,t8_boole), []). fof(t8_xboole_1,axiom,( ! [A,B,C] : ( ( r1_tarski(A,B) & r1_tarski(C,B) ) => r1_tarski(k2_xboole_0(A,C),B) ) ), file(xboole_1,t8_xboole_1), []). fof(t92_zfmisc_1,axiom,( ! [A,B] : ( r2_hidden(A,B) => r1_tarski(A,k3_tarski(B)) ) ), file(zfmisc_1,t92_zfmisc_1), []). fof(t96_zfmisc_1,axiom,( ! [A,B] : k3_tarski(k2_xboole_0(A,B)) = k2_xboole_0(k3_tarski(A),k3_tarski(B)) ), file(zfmisc_1,t96_zfmisc_1), []). fof(t97_zfmisc_1,axiom,( ! [A,B] : r1_tarski(k3_tarski(k3_xboole_0(A,B)),k3_xboole_0(k3_tarski(A),k3_tarski(B))) ), file(zfmisc_1,t97_zfmisc_1), []).