% Mizar ND problem: t2_borsuk_3,borsuk_3,52,46 fof(dh_c1_4__borsuk_3,definition, ( ( ( ~ v3_struct_0(c1_4__borsuk_3) & v2_pre_topc(c1_4__borsuk_3) & l1_pre_topc(c1_4__borsuk_3) ) => ! [A] : ( ( ~ v3_struct_0(A) & v2_pre_topc(A) & l1_pre_topc(A) ) => ! [B] : ( m1_subset_1(B,u1_struct_0(c1_4__borsuk_3)) => ( v1_funct_1(k3_borsuk_1(A,c1_4__borsuk_3,B)) & v1_funct_2(k3_borsuk_1(A,c1_4__borsuk_3,B),u1_struct_0(A),u1_struct_0(k3_pre_topc(c1_4__borsuk_3,k1_struct_0(c1_4__borsuk_3,B)))) & v5_pre_topc(k3_borsuk_1(A,c1_4__borsuk_3,B),A,k3_pre_topc(c1_4__borsuk_3,k1_struct_0(c1_4__borsuk_3,B))) & m2_relset_1(k3_borsuk_1(A,c1_4__borsuk_3,B),u1_struct_0(A),u1_struct_0(k3_pre_topc(c1_4__borsuk_3,k1_struct_0(c1_4__borsuk_3,B)))) ) ) ) ) => ! [C] : ( ( ~ v3_struct_0(C) & v2_pre_topc(C) & l1_pre_topc(C) ) => ! [D] : ( ( ~ v3_struct_0(D) & v2_pre_topc(D) & l1_pre_topc(D) ) => ! [E] : ( m1_subset_1(E,u1_struct_0(C)) => ( v1_funct_1(k3_borsuk_1(D,C,E)) & v1_funct_2(k3_borsuk_1(D,C,E),u1_struct_0(D),u1_struct_0(k3_pre_topc(C,k1_struct_0(C,E)))) & v5_pre_topc(k3_borsuk_1(D,C,E),D,k3_pre_topc(C,k1_struct_0(C,E))) & m2_relset_1(k3_borsuk_1(D,C,E),u1_struct_0(D),u1_struct_0(k3_pre_topc(C,k1_struct_0(C,E)))) ) ) ) ) ), introduced(definition,[new_symbol(c1_4__borsuk_3),file(borsuk_3,c1_4__borsuk_3)]), [interesting(0.8),axiom,file(borsuk_3,c1_4__borsuk_3)]). fof(dh_c2_4__borsuk_3,definition, ( ( ( ~ v3_struct_0(c2_4__borsuk_3) & v2_pre_topc(c2_4__borsuk_3) & l1_pre_topc(c2_4__borsuk_3) ) => ! [A] : ( m1_subset_1(A,u1_struct_0(c1_4__borsuk_3)) => ( v1_funct_1(k3_borsuk_1(c2_4__borsuk_3,c1_4__borsuk_3,A)) & v1_funct_2(k3_borsuk_1(c2_4__borsuk_3,c1_4__borsuk_3,A),u1_struct_0(c2_4__borsuk_3),u1_struct_0(k3_pre_topc(c1_4__borsuk_3,k1_struct_0(c1_4__borsuk_3,A)))) & v5_pre_topc(k3_borsuk_1(c2_4__borsuk_3,c1_4__borsuk_3,A),c2_4__borsuk_3,k3_pre_topc(c1_4__borsuk_3,k1_struct_0(c1_4__borsuk_3,A))) & m2_relset_1(k3_borsuk_1(c2_4__borsuk_3,c1_4__borsuk_3,A),u1_struct_0(c2_4__borsuk_3),u1_struct_0(k3_pre_topc(c1_4__borsuk_3,k1_struct_0(c1_4__borsuk_3,A)))) ) ) ) => ! [B] : ( ( ~ v3_struct_0(B) & v2_pre_topc(B) & l1_pre_topc(B) ) => ! [C] : ( m1_subset_1(C,u1_struct_0(c1_4__borsuk_3)) => ( v1_funct_1(k3_borsuk_1(B,c1_4__borsuk_3,C)) & v1_funct_2(k3_borsuk_1(B,c1_4__borsuk_3,C),u1_struct_0(B),u1_struct_0(k3_pre_topc(c1_4__borsuk_3,k1_struct_0(c1_4__borsuk_3,C)))) & v5_pre_topc(k3_borsuk_1(B,c1_4__borsuk_3,C),B,k3_pre_topc(c1_4__borsuk_3,k1_struct_0(c1_4__borsuk_3,C))) & m2_relset_1(k3_borsuk_1(B,c1_4__borsuk_3,C),u1_struct_0(B),u1_struct_0(k3_pre_topc(c1_4__borsuk_3,k1_struct_0(c1_4__borsuk_3,C)))) ) ) ) ), introduced(definition,[new_symbol(c2_4__borsuk_3),file(borsuk_3,c2_4__borsuk_3)]), [interesting(0.8),axiom,file(borsuk_3,c2_4__borsuk_3)]). fof(dh_c3_4__borsuk_3,definition, ( ( m1_subset_1(c3_4__borsuk_3,u1_struct_0(c1_4__borsuk_3)) => ( v1_funct_1(k3_borsuk_1(c2_4__borsuk_3,c1_4__borsuk_3,c3_4__borsuk_3)) & v1_funct_2(k3_borsuk_1(c2_4__borsuk_3,c1_4__borsuk_3,c3_4__borsuk_3),u1_struct_0(c2_4__borsuk_3),u1_struct_0(k3_pre_topc(c1_4__borsuk_3,k1_struct_0(c1_4__borsuk_3,c3_4__borsuk_3)))) & v5_pre_topc(k3_borsuk_1(c2_4__borsuk_3,c1_4__borsuk_3,c3_4__borsuk_3),c2_4__borsuk_3,k3_pre_topc(c1_4__borsuk_3,k1_struct_0(c1_4__borsuk_3,c3_4__borsuk_3))) & m2_relset_1(k3_borsuk_1(c2_4__borsuk_3,c1_4__borsuk_3,c3_4__borsuk_3),u1_struct_0(c2_4__borsuk_3),u1_struct_0(k3_pre_topc(c1_4__borsuk_3,k1_struct_0(c1_4__borsuk_3,c3_4__borsuk_3)))) ) ) => ! [A] : ( m1_subset_1(A,u1_struct_0(c1_4__borsuk_3)) => ( v1_funct_1(k3_borsuk_1(c2_4__borsuk_3,c1_4__borsuk_3,A)) & v1_funct_2(k3_borsuk_1(c2_4__borsuk_3,c1_4__borsuk_3,A),u1_struct_0(c2_4__borsuk_3),u1_struct_0(k3_pre_topc(c1_4__borsuk_3,k1_struct_0(c1_4__borsuk_3,A)))) & v5_pre_topc(k3_borsuk_1(c2_4__borsuk_3,c1_4__borsuk_3,A),c2_4__borsuk_3,k3_pre_topc(c1_4__borsuk_3,k1_struct_0(c1_4__borsuk_3,A))) & m2_relset_1(k3_borsuk_1(c2_4__borsuk_3,c1_4__borsuk_3,A),u1_struct_0(c2_4__borsuk_3),u1_struct_0(k3_pre_topc(c1_4__borsuk_3,k1_struct_0(c1_4__borsuk_3,A)))) ) ) ), introduced(definition,[new_symbol(c3_4__borsuk_3),file(borsuk_3,c3_4__borsuk_3)]), [interesting(0.8),axiom,file(borsuk_3,c3_4__borsuk_3)]). fof(rc4_tops_1,theorem,( ! [A] : ( l1_pre_topc(A) => ? [B] : ( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) & v1_xboole_0(B) & v1_membered(B) & v2_membered(B) & v3_membered(B) & v4_membered(B) & v5_membered(B) & v2_tops_1(B,A) ) ) ), file(tops_1,rc4_tops_1), [interesting(0.9),axiom,file(tops_1,rc4_tops_1)]). fof(rc5_tops_1,theorem,( ! [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) & v3_pre_topc(B,A) & v4_pre_topc(B,A) & v1_membered(B) & v2_membered(B) & v3_membered(B) & v4_membered(B) & v5_membered(B) & v2_tops_1(B,A) & v3_tops_1(B,A) ) ) ), file(tops_1,rc5_tops_1), [interesting(0.9),axiom,file(tops_1,rc5_tops_1)]). fof(antisymmetry_r2_hidden,theorem,( ! [A,B] : ( r2_hidden(A,B) => ~ r2_hidden(B,A) ) ), file(hidden,r2_hidden), [interesting(0.9),axiom,file(hidden,r2_hidden)]). fof(dt_k1_xboole_0,axiom,( $true ), file(xboole_0,k1_xboole_0), [interesting(0.9),axiom,file(xboole_0,k1_xboole_0)]). fof(cc4_tops_1,theorem,( ! [A] : ( ( v2_pre_topc(A) & l1_pre_topc(A) ) => ! [B] : ( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) => ( v3_tops_1(B,A) => v2_tops_1(B,A) ) ) ) ), file(tops_1,cc4_tops_1), [interesting(0.9),axiom,file(tops_1,cc4_tops_1)]). fof(cc5_tops_1,theorem,( ! [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) & v2_tops_1(B,A) ) => ( v2_tops_1(B,A) & v3_tops_1(B,A) ) ) ) ) ), file(tops_1,cc5_tops_1), [interesting(0.9),axiom,file(tops_1,cc5_tops_1)]). fof(cc6_tops_1,theorem,( ! [A] : ( ( v2_pre_topc(A) & l1_pre_topc(A) ) => ! [B] : ( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) => ( ( v3_pre_topc(B,A) & v3_tops_1(B,A) ) => ( v1_xboole_0(B) & v3_pre_topc(B,A) & v4_pre_topc(B,A) & v1_membered(B) & v2_membered(B) & v3_membered(B) & v4_membered(B) & v5_membered(B) & v2_tops_1(B,A) & v3_tops_1(B,A) ) ) ) ) ), file(tops_1,cc6_tops_1), [interesting(0.9),axiom,file(tops_1,cc6_tops_1)]). fof(fc1_xboole_0,theorem,( v1_xboole_0(k1_xboole_0) ), file(xboole_0,fc1_xboole_0), [interesting(0.9),axiom,file(xboole_0,fc1_xboole_0)]). fof(rc1_tops_1,theorem,( ! [A] : ( ( v2_pre_topc(A) & l1_pre_topc(A) ) => ? [B] : ( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) & v3_pre_topc(B,A) ) ) ), file(tops_1,rc1_tops_1), [interesting(0.9),axiom,file(tops_1,rc1_tops_1)]). fof(rc2_tops_1,theorem,( ! [A] : ( ( v2_pre_topc(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(B,A) ) ) ), file(tops_1,rc2_tops_1), [interesting(0.9),axiom,file(tops_1,rc2_tops_1)]). fof(rc2_waybel12,theorem,( ? [A] : ( v1_xboole_0(A) & v1_finset_1(A) & v1_card_4(A) ) ), file(waybel12,rc2_waybel12), [interesting(0.9),axiom,file(waybel12,rc2_waybel12)]). fof(rc3_tops_1,theorem,( ! [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) & v3_pre_topc(B,A) & v4_pre_topc(B,A) ) ) ), file(tops_1,rc3_tops_1), [interesting(0.9),axiom,file(tops_1,rc3_tops_1)]). fof(rc3_waybel12,theorem,( ! [A] : ( l1_struct_0(A) => ? [B] : ( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) & v1_xboole_0(B) & v1_finset_1(B) & v1_card_4(B) ) ) ), file(waybel12,rc3_waybel12), [interesting(0.9),axiom,file(waybel12,rc3_waybel12)]). fof(rc4_waybel12,theorem,( ? [A] : ( ~ v1_xboole_0(A) & v1_finset_1(A) & v1_card_4(A) ) ), file(waybel12,rc4_waybel12), [interesting(0.9),axiom,file(waybel12,rc4_waybel12)]). fof(rc6_pre_topc,theorem,( ! [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), [interesting(0.9),axiom,file(pre_topc,rc6_pre_topc)]). fof(rc6_waybel12,theorem,( ? [A] : ( ~ v1_finset_1(A) & v1_card_4(A) ) ), file(waybel12,rc6_waybel12), [interesting(0.9),axiom,file(waybel12,rc6_waybel12)]). fof(rc7_pre_topc,theorem,( ! [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), [interesting(0.9),axiom,file(pre_topc,rc7_pre_topc)]). fof(rc7_waybel12,theorem,( ! [A] : ( l1_struct_0(A) => ? [B] : ( m1_subset_1(B,k1_zfmisc_1(k1_zfmisc_1(u1_struct_0(A)))) & v1_xboole_0(B) & v1_finset_1(B) & v1_card_4(B) ) ) ), file(waybel12,rc7_waybel12), [interesting(0.9),axiom,file(waybel12,rc7_waybel12)]). fof(t1_subset,theorem,( ! [A,B] : ( r2_hidden(A,B) => m1_subset_1(A,B) ) ), file(subset,t1_subset), [interesting(0.9),axiom,file(subset,t1_subset)]). fof(t4_subset,theorem,( ! [A,B,C] : ( ( r2_hidden(A,B) & m1_subset_1(B,k1_zfmisc_1(C)) ) => m1_subset_1(A,C) ) ), file(subset,t4_subset), [interesting(0.9),axiom,file(subset,t4_subset)]). fof(t5_subset,theorem,( ! [A,B,C] : ~ ( r2_hidden(A,B) & m1_subset_1(B,k1_zfmisc_1(C)) & v1_xboole_0(C) ) ), file(subset,t5_subset), [interesting(0.9),axiom,file(subset,t5_subset)]). fof(free_g1_pre_topc,definition,( ! [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), [interesting(0.9),axiom,file(pre_topc,g1_pre_topc)]). fof(reflexivity_r1_tarski,theorem,( ! [A,B] : r1_tarski(A,A) ), file(tarski,r1_tarski), [interesting(0.9),axiom,file(tarski,r1_tarski)]). 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), [interesting(0.9),axiom,file(pre_topc,g1_pre_topc)]). 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), [interesting(0.9),axiom,file(pre_topc,u1_pre_topc)]). fof(cc1_tops_1,theorem,( ! [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) => ( v3_pre_topc(B,A) & v4_pre_topc(B,A) ) ) ) ) ), file(tops_1,cc1_tops_1), [interesting(0.9),axiom,file(tops_1,cc1_tops_1)]). fof(cc2_tops_1,theorem,( ! [A] : ( l1_pre_topc(A) => ! [B] : ( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) => ( v1_xboole_0(B) => v2_tops_1(B,A) ) ) ) ), file(tops_1,cc2_tops_1), [interesting(0.9),axiom,file(tops_1,cc2_tops_1)]). fof(cc2_waybel12,theorem,( ! [A] : ( v1_finset_1(A) => v1_card_4(A) ) ), file(waybel12,cc2_waybel12), [interesting(0.9),axiom,file(waybel12,cc2_waybel12)]). fof(cc3_tops_1,theorem,( ! [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) => v3_tops_1(B,A) ) ) ) ), file(tops_1,cc3_tops_1), [interesting(0.9),axiom,file(tops_1,cc3_tops_1)]). fof(fc1_borsuk_3,theorem,( ! [A,B] : ( v1_xboole_0(B) => v1_xboole_0(k2_zfmisc_1(A,B)) ) ), file(borsuk_3,fc1_borsuk_3), [interesting(0.9),axiom,file(borsuk_3,fc1_borsuk_3)]). fof(fc2_borsuk_3,theorem,( ! [A,B] : ( v1_xboole_0(A) => v1_xboole_0(k2_zfmisc_1(A,B)) ) ), file(borsuk_3,fc2_borsuk_3), [interesting(0.9),axiom,file(borsuk_3,fc2_borsuk_3)]). fof(fc3_pre_topc,theorem,( ! [A,B] : ( ( ~ v3_struct_0(A) & l1_pre_topc(A) & ~ v1_xboole_0(B) & m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) ) => ( ~ v3_struct_0(k3_pre_topc(A,B)) & v1_pre_topc(k3_pre_topc(A,B)) ) ) ), file(pre_topc,fc3_pre_topc), [interesting(0.9),axiom,file(pre_topc,fc3_pre_topc)]). fof(fc4_subset_1,theorem,( ! [A,B] : ( ( ~ v1_xboole_0(A) & ~ v1_xboole_0(B) ) => ~ v1_xboole_0(k2_zfmisc_1(A,B)) ) ), file(subset_1,fc4_subset_1), [interesting(0.9),axiom,file(subset_1,fc4_subset_1)]). fof(rc1_subset_1,theorem,( ! [A] : ( ~ v1_xboole_0(A) => ? [B] : ( m1_subset_1(B,k1_zfmisc_1(A)) & ~ v1_xboole_0(B) ) ) ), file(subset_1,rc1_subset_1), [interesting(0.9),axiom,file(subset_1,rc1_subset_1)]). fof(rc1_xboole_0,theorem,( ? [A] : v1_xboole_0(A) ), file(xboole_0,rc1_xboole_0), [interesting(0.9),axiom,file(xboole_0,rc1_xboole_0)]). fof(rc2_borsuk_1,theorem,( ! [A,B] : ( ( ~ v3_struct_0(A) & v2_pre_topc(A) & l1_pre_topc(A) & ~ v3_struct_0(B) & v2_pre_topc(B) & l1_pre_topc(B) ) => ? [C] : ( m1_relset_1(C,u1_struct_0(A),u1_struct_0(B)) & ~ v1_xboole_0(C) & v1_relat_1(C) & v1_funct_1(C) & v1_funct_2(C,u1_struct_0(A),u1_struct_0(B)) & v5_pre_topc(C,A,B) & v1_partfun1(C,u1_struct_0(A),u1_struct_0(B)) ) ) ), file(borsuk_1,rc2_borsuk_1), [interesting(0.9),axiom,file(borsuk_1,rc2_borsuk_1)]). fof(rc2_subset_1,theorem,( ! [A] : ? [B] : ( m1_subset_1(B,k1_zfmisc_1(A)) & v1_xboole_0(B) ) ), file(subset_1,rc2_subset_1), [interesting(0.9),axiom,file(subset_1,rc2_subset_1)]). fof(rc2_xboole_0,theorem,( ? [A] : ~ v1_xboole_0(A) ), file(xboole_0,rc2_xboole_0), [interesting(0.9),axiom,file(xboole_0,rc2_xboole_0)]). fof(rc5_struct_0,theorem,( ! [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), [interesting(0.9),axiom,file(struct_0,rc5_struct_0)]). fof(t2_subset,theorem,( ! [A,B] : ( m1_subset_1(A,B) => ( v1_xboole_0(B) | r2_hidden(A,B) ) ) ), file(subset,t2_subset), [interesting(0.9),axiom,file(subset,t2_subset)]). fof(t6_boole,theorem,( ! [A] : ( v1_xboole_0(A) => A = k1_xboole_0 ) ), file(boole,t6_boole), [interesting(0.9),axiom,file(boole,t6_boole)]). fof(t7_boole,theorem,( ! [A,B] : ~ ( r2_hidden(A,B) & v1_xboole_0(B) ) ), file(boole,t7_boole), [interesting(0.9),axiom,file(boole,t7_boole)]). fof(t8_boole,theorem,( ! [A,B] : ~ ( v1_xboole_0(A) & A != B & v1_xboole_0(B) ) ), file(boole,t8_boole), [interesting(0.9),axiom,file(boole,t8_boole)]). fof(abstractness_v1_pre_topc,theorem,( ! [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), [interesting(0.9),axiom,file(pre_topc,v1_pre_topc)]). fof(existence_l1_pre_topc,axiom,( ? [A] : l1_pre_topc(A) ), file(pre_topc,l1_pre_topc), [interesting(0.9),axiom,file(pre_topc,l1_pre_topc)]). fof(existence_l1_struct_0,axiom,( ? [A] : l1_struct_0(A) ), file(struct_0,l1_struct_0), [interesting(0.9),axiom,file(struct_0,l1_struct_0)]). fof(existence_m1_pre_topc,axiom,( ! [A] : ( l1_pre_topc(A) => ? [B] : m1_pre_topc(B,A) ) ), file(pre_topc,m1_pre_topc), [interesting(0.9),axiom,file(pre_topc,m1_pre_topc)]). fof(existence_m1_relset_1,axiom,( ! [A,B] : ? [C] : m1_relset_1(C,A,B) ), file(relset_1,m1_relset_1), [interesting(0.9),axiom,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), [interesting(0.9),axiom,file(subset_1,m1_subset_1)]). fof(dt_k1_tarski,axiom,( $true ), file(tarski,k1_tarski), [interesting(0.9),axiom,file(tarski,k1_tarski)]). fof(dt_k1_zfmisc_1,axiom,( $true ), file(zfmisc_1,k1_zfmisc_1), [interesting(0.9),axiom,file(zfmisc_1,k1_zfmisc_1)]). fof(dt_k2_zfmisc_1,axiom,( $true ), file(zfmisc_1,k2_zfmisc_1), [interesting(0.9),axiom,file(zfmisc_1,k2_zfmisc_1)]). fof(dt_l1_pre_topc,axiom,( ! [A] : ( l1_pre_topc(A) => l1_struct_0(A) ) ), file(pre_topc,l1_pre_topc), [interesting(0.9),axiom,file(pre_topc,l1_pre_topc)]). fof(dt_l1_struct_0,axiom,( $true ), file(struct_0,l1_struct_0), [interesting(0.9),axiom,file(struct_0,l1_struct_0)]). fof(dt_m1_pre_topc,axiom,( ! [A] : ( l1_pre_topc(A) => ! [B] : ( m1_pre_topc(B,A) => l1_pre_topc(B) ) ) ), file(pre_topc,m1_pre_topc), [interesting(0.9),axiom,file(pre_topc,m1_pre_topc)]). fof(dt_m1_relset_1,axiom,( $true ), file(relset_1,m1_relset_1), [interesting(0.9),axiom,file(relset_1,m1_relset_1)]). fof(dt_m1_subset_1,axiom,( $true ), file(subset_1,m1_subset_1), [interesting(0.9),axiom,file(subset_1,m1_subset_1)]). fof(cc1_pre_topc,theorem,( ! [A] : ( ( v2_pre_topc(A) & l1_pre_topc(A) ) => ! [B] : ( m1_pre_topc(B,A) => v2_pre_topc(B) ) ) ), file(pre_topc,cc1_pre_topc), [interesting(0.9),axiom,file(pre_topc,cc1_pre_topc)]). fof(cc1_relset_1,theorem,( ! [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), [interesting(0.9),axiom,file(relset_1,cc1_relset_1)]). fof(cc1_waybel12,theorem,( ! [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) & v1_finset_1(B) ) ) ) ), file(waybel12,cc1_waybel12), [interesting(0.9),axiom,file(waybel12,cc1_waybel12)]). fof(cc4_borsuk_2,theorem,( ! [A] : ( l1_pre_topc(A) => ( v3_struct_0(A) => v2_t_0topsp(A) ) ) ), file(borsuk_2,cc4_borsuk_2), [interesting(0.9),axiom,file(borsuk_2,cc4_borsuk_2)]). fof(fc1_borsuk_1,theorem,( ! [A,B,C] : ( ( ~ v3_struct_0(A) & v2_pre_topc(A) & l1_pre_topc(A) & ~ v3_struct_0(B) & v2_pre_topc(B) & l1_pre_topc(B) & m1_subset_1(C,u1_struct_0(B)) ) => ( ~ v1_xboole_0(k3_borsuk_1(A,B,C)) & v1_relat_1(k3_borsuk_1(A,B,C)) & v1_funct_1(k3_borsuk_1(A,B,C)) & v1_funct_2(k3_borsuk_1(A,B,C),u1_struct_0(A),u1_struct_0(B)) & v5_pre_topc(k3_borsuk_1(A,B,C),A,B) & v1_partfun1(k3_borsuk_1(A,B,C),u1_struct_0(A),u1_struct_0(B)) ) ) ), file(borsuk_1,fc1_borsuk_1), [interesting(0.9),axiom,file(borsuk_1,fc1_borsuk_1)]). fof(fc1_struct_0,theorem,( ! [A] : ( ( ~ v3_struct_0(A) & l1_struct_0(A) ) => ~ v1_xboole_0(u1_struct_0(A)) ) ), file(struct_0,fc1_struct_0), [interesting(0.9),axiom,file(struct_0,fc1_struct_0)]). fof(fc1_subset_1,theorem,( ! [A] : ~ v1_xboole_0(k1_zfmisc_1(A)) ), file(subset_1,fc1_subset_1), [interesting(0.9),axiom,file(subset_1,fc1_subset_1)]). fof(fc1_waybel12,theorem,( ! [A] : ( ( v3_struct_0(A) & l1_struct_0(A) ) => ( v1_xboole_0(u1_struct_0(A)) & v1_finset_1(u1_struct_0(A)) ) ) ), file(waybel12,fc1_waybel12), [interesting(0.9),axiom,file(waybel12,fc1_waybel12)]). fof(fc2_subset_1,theorem,( ! [A] : ~ v1_xboole_0(k1_tarski(A)) ), file(subset_1,fc2_subset_1), [interesting(0.9),axiom,file(subset_1,fc2_subset_1)]). fof(fc4_pre_topc,theorem,( ! [A,B] : ( ( v2_pre_topc(A) & l1_pre_topc(A) & m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) ) => ( v1_pre_topc(k3_pre_topc(A,B)) & v2_pre_topc(k3_pre_topc(A,B)) ) ) ), file(pre_topc,fc4_pre_topc), [interesting(0.9),axiom,file(pre_topc,fc4_pre_topc)]). fof(rc1_pre_topc,theorem,( ? [A] : ( l1_pre_topc(A) & v1_pre_topc(A) ) ), file(pre_topc,rc1_pre_topc), [interesting(0.9),axiom,file(pre_topc,rc1_pre_topc)]). fof(rc1_waybel12,theorem,( ? [A] : ( l1_struct_0(A) & v3_struct_0(A) ) ), file(waybel12,rc1_waybel12), [interesting(0.9),axiom,file(waybel12,rc1_waybel12)]). fof(rc2_pre_topc,theorem,( ? [A] : ( l1_pre_topc(A) & ~ v3_struct_0(A) & v1_pre_topc(A) & v2_pre_topc(A) ) ), file(pre_topc,rc2_pre_topc), [interesting(0.9),axiom,file(pre_topc,rc2_pre_topc)]). fof(rc3_pre_topc,theorem,( ! [A] : ( l1_pre_topc(A) => ? [B] : ( m1_pre_topc(B,A) & v1_pre_topc(B) ) ) ), file(pre_topc,rc3_pre_topc), [interesting(0.9),axiom,file(pre_topc,rc3_pre_topc)]). fof(rc3_struct_0,theorem,( ? [A] : ( l1_struct_0(A) & ~ v3_struct_0(A) ) ), file(struct_0,rc3_struct_0), [interesting(0.9),axiom,file(struct_0,rc3_struct_0)]). fof(rc4_pre_topc,theorem,( ! [A] : ( ( ~ v3_struct_0(A) & l1_pre_topc(A) ) => ? [B] : ( m1_pre_topc(B,A) & ~ v3_struct_0(B) & v1_pre_topc(B) ) ) ), file(pre_topc,rc4_pre_topc), [interesting(0.9),axiom,file(pre_topc,rc4_pre_topc)]). fof(rc5_pre_topc,theorem,( ! [A] : ( ( v2_pre_topc(A) & l1_pre_topc(A) ) => ? [B] : ( m1_pre_topc(B,A) & v1_pre_topc(B) & v2_pre_topc(B) ) ) ), file(pre_topc,rc5_pre_topc), [interesting(0.9),axiom,file(pre_topc,rc5_pre_topc)]). fof(t3_subset,theorem,( ! [A,B] : ( m1_subset_1(A,k1_zfmisc_1(B)) <=> r1_tarski(A,B) ) ), file(subset,t3_subset), [interesting(0.9),axiom,file(subset,t3_subset)]). fof(existence_m2_relset_1,axiom,( ! [A,B] : ? [C] : m2_relset_1(C,A,B) ), file(relset_1,m2_relset_1), [interesting(0.9),axiom,file(relset_1,m2_relset_1)]). fof(redefinition_k1_struct_0,definition,( ! [A,B] : ( ( ~ v3_struct_0(A) & l1_struct_0(A) & m1_subset_1(B,u1_struct_0(A)) ) => k1_struct_0(A,B) = k1_tarski(B) ) ), file(struct_0,k1_struct_0), [interesting(0.9),axiom,file(struct_0,k1_struct_0)]). fof(redefinition_m2_relset_1,definition,( ! [A,B,C] : ( m2_relset_1(C,A,B) <=> m1_relset_1(C,A,B) ) ), file(relset_1,m2_relset_1), [interesting(0.9),axiom,file(relset_1,m2_relset_1)]). fof(dt_k1_struct_0,axiom,( ! [A,B] : ( ( ~ v3_struct_0(A) & l1_struct_0(A) & m1_subset_1(B,u1_struct_0(A)) ) => m1_subset_1(k1_struct_0(A,B),k1_zfmisc_1(u1_struct_0(A))) ) ), file(struct_0,k1_struct_0), [interesting(0.9),axiom,file(struct_0,k1_struct_0)]). fof(dt_k3_borsuk_1,axiom,( ! [A,B,C] : ( ( l1_struct_0(A) & ~ v3_struct_0(B) & l1_struct_0(B) & m1_subset_1(C,u1_struct_0(B)) ) => ( v1_funct_1(k3_borsuk_1(A,B,C)) & v1_funct_2(k3_borsuk_1(A,B,C),u1_struct_0(A),u1_struct_0(B)) & m2_relset_1(k3_borsuk_1(A,B,C),u1_struct_0(A),u1_struct_0(B)) ) ) ), file(borsuk_1,k3_borsuk_1), [interesting(0.9),axiom,file(borsuk_1,k3_borsuk_1)]). fof(dt_k3_pre_topc,axiom,( ! [A,B] : ( ( l1_pre_topc(A) & m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) ) => ( v1_pre_topc(k3_pre_topc(A,B)) & m1_pre_topc(k3_pre_topc(A,B),A) ) ) ), file(pre_topc,k3_pre_topc), [interesting(0.9),axiom,file(pre_topc,k3_pre_topc)]). 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), [interesting(0.9),axiom,file(relset_1,m2_relset_1)]). fof(dt_u1_struct_0,axiom,( $true ), file(struct_0,u1_struct_0), [interesting(0.9),axiom,file(struct_0,u1_struct_0)]). fof(dt_c1_4__borsuk_3,assumption, ( ~ v3_struct_0(c1_4__borsuk_3) & v2_pre_topc(c1_4__borsuk_3) & l1_pre_topc(c1_4__borsuk_3) ), introduced(assumption,[file(borsuk_3,c1_4__borsuk_3)]), [interesting(0.8),axiom,file(borsuk_3,c1_4__borsuk_3)]). fof(dt_c2_4__borsuk_3,assumption, ( ~ v3_struct_0(c2_4__borsuk_3) & v2_pre_topc(c2_4__borsuk_3) & l1_pre_topc(c2_4__borsuk_3) ), introduced(assumption,[file(borsuk_3,c2_4__borsuk_3)]), [interesting(0.8),axiom,file(borsuk_3,c2_4__borsuk_3)]). fof(dt_c3_4__borsuk_3,assumption,( m1_subset_1(c3_4__borsuk_3,u1_struct_0(c1_4__borsuk_3)) ), introduced(assumption,[file(borsuk_3,c3_4__borsuk_3)]), [interesting(0.8),axiom,file(borsuk_3,c3_4__borsuk_3)]). fof(de_c6_4__borsuk_3,definition,( c6_4__borsuk_3 = k3_borsuk_1(c2_4__borsuk_3,c1_4__borsuk_3,c3_4__borsuk_3) ), introduced(definition,[new_symbol(c6_4__borsuk_3),file(borsuk_3,c6_4__borsuk_3)]), [interesting(0.8),axiom,file(borsuk_3,c6_4__borsuk_3)]). fof(redefinition_k1_borsuk_1,definition,( ! [A,B,C] : ( ( ~ v1_xboole_0(A) & m1_subset_1(C,A) ) => k1_borsuk_1(A,B,C) = k2_funcop_1(B,C) ) ), file(borsuk_1,k1_borsuk_1), [interesting(0.9),axiom,file(borsuk_1,k1_borsuk_1)]). fof(dt_k1_borsuk_1,axiom,( ! [A,B,C] : ( ( ~ v1_xboole_0(A) & m1_subset_1(C,A) ) => ( v1_funct_1(k1_borsuk_1(A,B,C)) & v1_funct_2(k1_borsuk_1(A,B,C),B,A) & m2_relset_1(k1_borsuk_1(A,B,C),B,A) ) ) ), file(borsuk_1,k1_borsuk_1), [interesting(0.9),axiom,file(borsuk_1,k1_borsuk_1)]). fof(dt_k2_funcop_1,axiom,( $true ), file(funcop_1,k2_funcop_1), [interesting(0.9),axiom,file(funcop_1,k2_funcop_1)]). fof(t1_jordan1,theorem,( ! [A] : ( l1_pre_topc(A) => ! [B] : ( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) => u1_struct_0(k3_pre_topc(A,B)) = B ) ) ), file(jordan1,t1_jordan1), [interesting(0.9),axiom,file(jordan1,t1_jordan1)]). fof(e3_4__borsuk_3,plain,( u1_struct_0(k3_pre_topc(c1_4__borsuk_3,k1_struct_0(c1_4__borsuk_3,c3_4__borsuk_3))) = k1_struct_0(c1_4__borsuk_3,c3_4__borsuk_3) ), inference(mizar_by,[status(thm),assumptions([dt_c1_4__borsuk_3,dt_c3_4__borsuk_3])],[rc2_waybel12,rc3_waybel12,rc4_tops_1,rc4_waybel12,rc5_tops_1,rc6_waybel12,rc7_waybel12,free_g1_pre_topc,antisymmetry_r2_hidden,dt_g1_pre_topc,dt_k1_xboole_0,dt_u1_pre_topc,cc2_waybel12,cc4_tops_1,cc5_tops_1,cc6_tops_1,fc1_xboole_0,rc1_tops_1,rc2_tops_1,rc3_tops_1,rc6_pre_topc,rc7_pre_topc,t1_subset,t4_subset,t5_subset,reflexivity_r1_tarski,abstractness_v1_pre_topc,existence_l1_struct_0,existence_m1_pre_topc,dt_k1_tarski,dt_l1_struct_0,dt_m1_pre_topc,cc1_pre_topc,cc1_tops_1,cc1_waybel12,cc2_tops_1,cc3_tops_1,cc4_borsuk_2,fc1_struct_0,fc1_waybel12,fc2_subset_1,fc3_pre_topc,fc4_pre_topc,rc1_pre_topc,rc1_subset_1,rc1_waybel12,rc1_xboole_0,rc2_pre_topc,rc2_subset_1,rc2_xboole_0,rc3_pre_topc,rc3_struct_0,rc4_pre_topc,rc5_pre_topc,rc5_struct_0,t2_subset,t6_boole,t7_boole,t8_boole,existence_l1_pre_topc,existence_m1_subset_1,redefinition_k1_struct_0,dt_k1_struct_0,dt_k1_zfmisc_1,dt_k3_pre_topc,dt_l1_pre_topc,dt_m1_subset_1,dt_u1_struct_0,dt_c1_4__borsuk_3,dt_c3_4__borsuk_3,fc1_subset_1,t3_subset,t1_jordan1]), [interesting(0.8),file(borsuk_3,e3_4__borsuk_3),[file(borsuk_3,e3_4__borsuk_3)]]). fof(d3_borsuk_1,definition,( ! [A] : ( l1_struct_0(A) => ! [B] : ( ( ~ v3_struct_0(B) & l1_struct_0(B) ) => ! [C] : ( m1_subset_1(C,u1_struct_0(B)) => k3_borsuk_1(A,B,C) = k1_borsuk_1(u1_struct_0(B),u1_struct_0(A),C) ) ) ) ), file(borsuk_1,d3_borsuk_1), [interesting(0.9),axiom,file(borsuk_1,d3_borsuk_1)]). fof(e1_4__borsuk_3,plain,( k3_borsuk_1(c2_4__borsuk_3,c1_4__borsuk_3,c3_4__borsuk_3) = k1_borsuk_1(u1_struct_0(c1_4__borsuk_3),u1_struct_0(c2_4__borsuk_3),c3_4__borsuk_3) ), inference(mizar_by,[status(thm),assumptions([dt_c1_4__borsuk_3,dt_c2_4__borsuk_3,dt_c3_4__borsuk_3])],[rc4_tops_1,rc5_tops_1,reflexivity_r1_tarski,cc4_tops_1,cc5_tops_1,cc6_tops_1,rc1_tops_1,rc2_tops_1,rc3_tops_1,rc6_pre_topc,rc7_pre_topc,antisymmetry_r2_hidden,existence_m1_relset_1,dt_k1_xboole_0,dt_k1_zfmisc_1,dt_k2_zfmisc_1,dt_m1_relset_1,cc1_relset_1,cc1_tops_1,cc1_waybel12,cc2_tops_1,cc3_tops_1,fc1_borsuk_3,fc1_subset_1,fc1_xboole_0,fc2_borsuk_3,fc4_subset_1,rc1_subset_1,rc2_borsuk_1,rc2_subset_1,rc2_waybel12,rc3_waybel12,rc4_waybel12,rc5_struct_0,rc6_waybel12,rc7_waybel12,t1_subset,t3_subset,t4_subset,t5_subset,existence_l1_pre_topc,existence_m2_relset_1,redefinition_m2_relset_1,dt_k2_funcop_1,dt_l1_pre_topc,dt_m2_relset_1,cc2_waybel12,cc4_borsuk_2,fc1_borsuk_1,rc1_xboole_0,rc2_xboole_0,t2_subset,t6_boole,t7_boole,t8_boole,existence_l1_struct_0,existence_m1_subset_1,redefinition_k1_borsuk_1,dt_k1_borsuk_1,dt_k3_borsuk_1,dt_l1_struct_0,dt_m1_subset_1,dt_u1_struct_0,dt_c1_4__borsuk_3,dt_c2_4__borsuk_3,dt_c3_4__borsuk_3,fc1_struct_0,fc1_waybel12,rc1_waybel12,rc3_struct_0,d3_borsuk_1]), [interesting(0.8),file(borsuk_3,e1_4__borsuk_3),[file(borsuk_3,e1_4__borsuk_3)]]). fof(d1_tarski,definition,( ! [A,B] : ( B = k1_tarski(A) <=> ! [C] : ( r2_hidden(C,B) <=> C = A ) ) ), file(tarski,d1_tarski), [interesting(0.9),axiom,file(tarski,d1_tarski)]). fof(e2_4__borsuk_3,plain,( r2_hidden(c3_4__borsuk_3,k1_struct_0(c1_4__borsuk_3,c3_4__borsuk_3)) ), inference(mizar_by,[status(thm),assumptions([dt_c1_4__borsuk_3,dt_c3_4__borsuk_3])],[rc2_waybel12,rc3_waybel12,rc4_tops_1,rc4_waybel12,rc5_tops_1,rc6_waybel12,rc7_waybel12,reflexivity_r1_tarski,dt_k1_xboole_0,cc2_waybel12,cc4_tops_1,cc5_tops_1,cc6_tops_1,fc1_xboole_0,rc1_tops_1,rc2_tops_1,rc3_tops_1,rc6_pre_topc,rc7_pre_topc,existence_l1_pre_topc,existence_l1_struct_0,existence_m1_subset_1,dt_k1_zfmisc_1,dt_l1_pre_topc,dt_l1_struct_0,dt_m1_subset_1,dt_u1_struct_0,cc1_tops_1,cc1_waybel12,cc2_tops_1,cc3_tops_1,cc4_borsuk_2,fc1_struct_0,fc1_subset_1,fc1_waybel12,rc1_subset_1,rc1_waybel12,rc1_xboole_0,rc2_subset_1,rc2_xboole_0,rc3_struct_0,rc5_struct_0,t2_subset,t3_subset,t4_subset,t5_subset,t6_boole,t8_boole,antisymmetry_r2_hidden,redefinition_k1_struct_0,dt_k1_struct_0,dt_k1_tarski,dt_c1_4__borsuk_3,dt_c3_4__borsuk_3,fc2_subset_1,t1_subset,t7_boole,d1_tarski]), [interesting(0.8),file(borsuk_3,e2_4__borsuk_3),[file(borsuk_3,e2_4__borsuk_3)]]). fof(t57_funcop_1,theorem,( ! [A,B,C] : ( r2_hidden(C,B) => ( v1_funct_1(k2_funcop_1(A,C)) & v1_funct_2(k2_funcop_1(A,C),A,B) & m2_relset_1(k2_funcop_1(A,C),A,B) ) ) ), file(funcop_1,t57_funcop_1), [interesting(0.9),axiom,file(funcop_1,t57_funcop_1)]). fof(e4_4__borsuk_3,plain, ( v1_funct_1(k3_borsuk_1(c2_4__borsuk_3,c1_4__borsuk_3,c3_4__borsuk_3)) & v1_funct_2(k3_borsuk_1(c2_4__borsuk_3,c1_4__borsuk_3,c3_4__borsuk_3),u1_struct_0(c2_4__borsuk_3),u1_struct_0(k3_pre_topc(c1_4__borsuk_3,k1_struct_0(c1_4__borsuk_3,c3_4__borsuk_3)))) & m2_relset_1(k3_borsuk_1(c2_4__borsuk_3,c1_4__borsuk_3,c3_4__borsuk_3),u1_struct_0(c2_4__borsuk_3),u1_struct_0(k3_pre_topc(c1_4__borsuk_3,k1_struct_0(c1_4__borsuk_3,c3_4__borsuk_3)))) ), inference(mizar_by,[status(thm),assumptions([dt_c2_4__borsuk_3,dt_c1_4__borsuk_3,dt_c3_4__borsuk_3])],[rc2_waybel12,rc3_waybel12,rc4_tops_1,rc4_waybel12,rc5_tops_1,rc6_waybel12,rc7_waybel12,free_g1_pre_topc,reflexivity_r1_tarski,dt_g1_pre_topc,dt_k1_xboole_0,dt_u1_pre_topc,cc2_waybel12,cc4_tops_1,cc5_tops_1,cc6_tops_1,fc1_xboole_0,rc1_tops_1,rc2_borsuk_1,rc2_tops_1,rc3_tops_1,rc6_pre_topc,rc7_pre_topc,abstractness_v1_pre_topc,existence_l1_pre_topc,existence_l1_struct_0,existence_m1_pre_topc,existence_m1_relset_1,existence_m1_subset_1,dt_k1_tarski,dt_k1_zfmisc_1,dt_k2_zfmisc_1,dt_l1_pre_topc,dt_l1_struct_0,dt_m1_pre_topc,dt_m1_relset_1,dt_m1_subset_1,cc1_pre_topc,cc1_relset_1,cc1_tops_1,cc1_waybel12,cc2_tops_1,cc3_tops_1,cc4_borsuk_2,fc1_borsuk_1,fc1_borsuk_3,fc1_struct_0,fc1_subset_1,fc1_waybel12,fc2_borsuk_3,fc2_subset_1,fc3_pre_topc,fc4_pre_topc,fc4_subset_1,rc1_pre_topc,rc1_subset_1,rc1_waybel12,rc1_xboole_0,rc2_pre_topc,rc2_subset_1,rc2_xboole_0,rc3_pre_topc,rc3_struct_0,rc4_pre_topc,rc5_pre_topc,rc5_struct_0,t2_subset,t3_subset,t4_subset,t5_subset,t6_boole,t8_boole,antisymmetry_r2_hidden,existence_m2_relset_1,redefinition_k1_borsuk_1,redefinition_k1_struct_0,redefinition_m2_relset_1,dt_k1_borsuk_1,dt_k1_struct_0,dt_k2_funcop_1,dt_k3_borsuk_1,dt_k3_pre_topc,dt_m2_relset_1,dt_u1_struct_0,dt_c1_4__borsuk_3,dt_c2_4__borsuk_3,dt_c3_4__borsuk_3,t1_subset,t7_boole,e3_4__borsuk_3,e1_4__borsuk_3,e2_4__borsuk_3,t57_funcop_1]), [interesting(0.8),file(borsuk_3,e4_4__borsuk_3),[file(borsuk_3,e4_4__borsuk_3)]]). fof(dt_c6_4__borsuk_3,plain, ( v1_funct_1(c6_4__borsuk_3) & v1_funct_2(c6_4__borsuk_3,u1_struct_0(c2_4__borsuk_3),u1_struct_0(k3_pre_topc(c1_4__borsuk_3,k1_struct_0(c1_4__borsuk_3,c3_4__borsuk_3)))) & m2_relset_1(c6_4__borsuk_3,u1_struct_0(c2_4__borsuk_3),u1_struct_0(k3_pre_topc(c1_4__borsuk_3,k1_struct_0(c1_4__borsuk_3,c3_4__borsuk_3)))) ), inference(mizar_by,[status(thm),assumptions([dt_c2_4__borsuk_3,dt_c1_4__borsuk_3,dt_c3_4__borsuk_3])],[rc4_tops_1,rc5_tops_1,antisymmetry_r2_hidden,dt_k1_xboole_0,cc4_tops_1,cc5_tops_1,cc6_tops_1,fc1_xboole_0,rc1_tops_1,rc2_tops_1,rc2_waybel12,rc3_tops_1,rc3_waybel12,rc4_waybel12,rc6_pre_topc,rc6_waybel12,rc7_pre_topc,rc7_waybel12,t1_subset,t4_subset,t5_subset,free_g1_pre_topc,reflexivity_r1_tarski,dt_g1_pre_topc,dt_u1_pre_topc,cc1_tops_1,cc2_tops_1,cc2_waybel12,cc3_tops_1,fc1_borsuk_3,fc2_borsuk_3,fc3_pre_topc,fc4_subset_1,rc1_subset_1,rc1_xboole_0,rc2_borsuk_1,rc2_subset_1,rc2_xboole_0,rc5_struct_0,t2_subset,t6_boole,t7_boole,t8_boole,abstractness_v1_pre_topc,existence_l1_pre_topc,existence_l1_struct_0,existence_m1_pre_topc,existence_m1_relset_1,existence_m1_subset_1,dt_k1_tarski,dt_k1_zfmisc_1,dt_k2_zfmisc_1,dt_l1_pre_topc,dt_l1_struct_0,dt_m1_pre_topc,dt_m1_relset_1,dt_m1_subset_1,cc1_pre_topc,cc1_relset_1,cc1_waybel12,cc4_borsuk_2,fc1_borsuk_1,fc1_struct_0,fc1_subset_1,fc1_waybel12,fc2_subset_1,fc4_pre_topc,rc1_pre_topc,rc1_waybel12,rc2_pre_topc,rc3_pre_topc,rc3_struct_0,rc4_pre_topc,rc5_pre_topc,t3_subset,existence_m2_relset_1,redefinition_k1_struct_0,redefinition_m2_relset_1,dt_k1_struct_0,dt_k3_borsuk_1,dt_k3_pre_topc,dt_m2_relset_1,dt_u1_struct_0,dt_c1_4__borsuk_3,dt_c2_4__borsuk_3,dt_c3_4__borsuk_3,de_c6_4__borsuk_3,e4_4__borsuk_3]), [interesting(0.8),file(borsuk_3,c6_4__borsuk_3),[file(borsuk_3,c6_4__borsuk_3)]]). fof(t9_topmetr,theorem,( ! [A] : ( ( v2_pre_topc(A) & l1_pre_topc(A) ) => ! [B] : ( ( v2_pre_topc(B) & l1_pre_topc(B) ) => ! [C] : ( ( v1_funct_1(C) & v1_funct_2(C,u1_struct_0(A),u1_struct_0(B)) & m2_relset_1(C,u1_struct_0(A),u1_struct_0(B)) ) => ! [D] : ( m1_subset_1(D,k1_zfmisc_1(u1_struct_0(B))) => ( v5_pre_topc(C,A,B) => ! [E] : ( ( v1_funct_1(E) & v1_funct_2(E,u1_struct_0(A),u1_struct_0(k3_pre_topc(B,D))) & m2_relset_1(E,u1_struct_0(A),u1_struct_0(k3_pre_topc(B,D))) ) => ( E = C => v5_pre_topc(E,A,k3_pre_topc(B,D)) ) ) ) ) ) ) ) ), file(topmetr,t9_topmetr), [interesting(0.9),axiom,file(topmetr,t9_topmetr)]). fof(e5_4__borsuk_3,plain,( v5_pre_topc(c6_4__borsuk_3,c2_4__borsuk_3,k3_pre_topc(c1_4__borsuk_3,k1_struct_0(c1_4__borsuk_3,c3_4__borsuk_3))) ), inference(mizar_by,[status(thm),assumptions([dt_c2_4__borsuk_3,dt_c1_4__borsuk_3,dt_c3_4__borsuk_3])],[rc2_waybel12,rc3_waybel12,rc4_tops_1,rc4_waybel12,rc5_tops_1,rc6_waybel12,rc7_waybel12,free_g1_pre_topc,antisymmetry_r2_hidden,dt_g1_pre_topc,dt_k1_xboole_0,dt_u1_pre_topc,cc2_waybel12,cc4_tops_1,cc5_tops_1,cc6_tops_1,fc1_xboole_0,rc1_tops_1,rc2_borsuk_1,rc2_tops_1,rc3_tops_1,rc6_pre_topc,rc7_pre_topc,t1_subset,t4_subset,t5_subset,reflexivity_r1_tarski,abstractness_v1_pre_topc,existence_l1_struct_0,existence_m1_pre_topc,existence_m1_relset_1,dt_k1_tarski,dt_k2_zfmisc_1,dt_k3_borsuk_1,dt_l1_struct_0,dt_m1_pre_topc,dt_m1_relset_1,cc1_pre_topc,cc1_relset_1,cc1_tops_1,cc1_waybel12,cc2_tops_1,cc3_tops_1,cc4_borsuk_2,fc1_borsuk_1,fc1_borsuk_3,fc1_struct_0,fc1_waybel12,fc2_borsuk_3,fc2_subset_1,fc3_pre_topc,fc4_subset_1,rc1_pre_topc,rc1_subset_1,rc1_waybel12,rc1_xboole_0,rc2_pre_topc,rc2_subset_1,rc2_xboole_0,rc3_pre_topc,rc3_struct_0,rc4_pre_topc,rc5_pre_topc,rc5_struct_0,t2_subset,t6_boole,t7_boole,t8_boole,existence_l1_pre_topc,existence_m1_subset_1,existence_m2_relset_1,redefinition_k1_struct_0,redefinition_m2_relset_1,dt_k1_struct_0,dt_k1_zfmisc_1,dt_k3_pre_topc,dt_l1_pre_topc,dt_m1_subset_1,dt_m2_relset_1,dt_u1_struct_0,dt_c1_4__borsuk_3,dt_c2_4__borsuk_3,dt_c3_4__borsuk_3,dt_c6_4__borsuk_3,de_c6_4__borsuk_3,fc1_subset_1,fc4_pre_topc,t3_subset,t9_topmetr]), [interesting(0.8),file(borsuk_3,e5_4__borsuk_3),[file(borsuk_3,e5_4__borsuk_3)]]). fof(e6_4__borsuk_3,plain, ( v1_funct_1(k3_borsuk_1(c2_4__borsuk_3,c1_4__borsuk_3,c3_4__borsuk_3)) & v1_funct_2(k3_borsuk_1(c2_4__borsuk_3,c1_4__borsuk_3,c3_4__borsuk_3),u1_struct_0(c2_4__borsuk_3),u1_struct_0(k3_pre_topc(c1_4__borsuk_3,k1_struct_0(c1_4__borsuk_3,c3_4__borsuk_3)))) & v5_pre_topc(k3_borsuk_1(c2_4__borsuk_3,c1_4__borsuk_3,c3_4__borsuk_3),c2_4__borsuk_3,k3_pre_topc(c1_4__borsuk_3,k1_struct_0(c1_4__borsuk_3,c3_4__borsuk_3))) & m2_relset_1(k3_borsuk_1(c2_4__borsuk_3,c1_4__borsuk_3,c3_4__borsuk_3),u1_struct_0(c2_4__borsuk_3),u1_struct_0(k3_pre_topc(c1_4__borsuk_3,k1_struct_0(c1_4__borsuk_3,c3_4__borsuk_3)))) ), inference(mizar_by,[status(thm),assumptions([dt_c2_4__borsuk_3,dt_c1_4__borsuk_3,dt_c3_4__borsuk_3])],[rc4_tops_1,rc5_tops_1,antisymmetry_r2_hidden,dt_k1_xboole_0,cc4_tops_1,cc5_tops_1,cc6_tops_1,fc1_xboole_0,rc1_tops_1,rc2_tops_1,rc2_waybel12,rc3_tops_1,rc3_waybel12,rc4_waybel12,rc6_pre_topc,rc6_waybel12,rc7_pre_topc,rc7_waybel12,t1_subset,t4_subset,t5_subset,free_g1_pre_topc,reflexivity_r1_tarski,dt_g1_pre_topc,dt_u1_pre_topc,cc1_tops_1,cc2_tops_1,cc2_waybel12,cc3_tops_1,fc1_borsuk_3,fc2_borsuk_3,fc3_pre_topc,fc4_subset_1,rc1_subset_1,rc1_xboole_0,rc2_borsuk_1,rc2_subset_1,rc2_xboole_0,rc5_struct_0,t2_subset,t6_boole,t7_boole,t8_boole,abstractness_v1_pre_topc,existence_l1_pre_topc,existence_l1_struct_0,existence_m1_pre_topc,existence_m1_relset_1,existence_m1_subset_1,dt_k1_tarski,dt_k1_zfmisc_1,dt_k2_zfmisc_1,dt_l1_pre_topc,dt_l1_struct_0,dt_m1_pre_topc,dt_m1_relset_1,dt_m1_subset_1,cc1_pre_topc,cc1_relset_1,cc1_waybel12,cc4_borsuk_2,fc1_borsuk_1,fc1_struct_0,fc1_subset_1,fc1_waybel12,fc2_subset_1,fc4_pre_topc,rc1_pre_topc,rc1_waybel12,rc2_pre_topc,rc3_pre_topc,rc3_struct_0,rc4_pre_topc,rc5_pre_topc,t3_subset,existence_m2_relset_1,redefinition_k1_struct_0,redefinition_m2_relset_1,dt_k1_struct_0,dt_k3_borsuk_1,dt_k3_pre_topc,dt_m2_relset_1,dt_u1_struct_0,dt_c1_4__borsuk_3,dt_c2_4__borsuk_3,dt_c3_4__borsuk_3,dt_c6_4__borsuk_3,de_c6_4__borsuk_3,e5_4__borsuk_3]), [interesting(0.8),file(borsuk_3,e6_4__borsuk_3),[file(borsuk_3,e6_4__borsuk_3)]]). fof(i3_4__borsuk_3,theorem,( $true ), introduced(tautology,[file(borsuk_3,i3_4__borsuk_3)]), [interesting(0.8),trivial,file(borsuk_3,i3_4__borsuk_3)]). fof(i2_4__borsuk_3,plain, ( v1_funct_1(k3_borsuk_1(c2_4__borsuk_3,c1_4__borsuk_3,c3_4__borsuk_3)) & v1_funct_2(k3_borsuk_1(c2_4__borsuk_3,c1_4__borsuk_3,c3_4__borsuk_3),u1_struct_0(c2_4__borsuk_3),u1_struct_0(k3_pre_topc(c1_4__borsuk_3,k1_struct_0(c1_4__borsuk_3,c3_4__borsuk_3)))) & v5_pre_topc(k3_borsuk_1(c2_4__borsuk_3,c1_4__borsuk_3,c3_4__borsuk_3),c2_4__borsuk_3,k3_pre_topc(c1_4__borsuk_3,k1_struct_0(c1_4__borsuk_3,c3_4__borsuk_3))) & m2_relset_1(k3_borsuk_1(c2_4__borsuk_3,c1_4__borsuk_3,c3_4__borsuk_3),u1_struct_0(c2_4__borsuk_3),u1_struct_0(k3_pre_topc(c1_4__borsuk_3,k1_struct_0(c1_4__borsuk_3,c3_4__borsuk_3)))) ), inference(conclusion,[status(thm),assumptions([dt_c2_4__borsuk_3,dt_c1_4__borsuk_3,dt_c3_4__borsuk_3])],[e6_4__borsuk_3,i3_4__borsuk_3]), [interesting(0.8),file(borsuk_3,i2_4__borsuk_3),[file(borsuk_3,i2_4__borsuk_3)]]). fof(i2_4_tmp__borsuk_3,plain, ( m1_subset_1(c3_4__borsuk_3,u1_struct_0(c1_4__borsuk_3)) => ( v1_funct_1(k3_borsuk_1(c2_4__borsuk_3,c1_4__borsuk_3,c3_4__borsuk_3)) & v1_funct_2(k3_borsuk_1(c2_4__borsuk_3,c1_4__borsuk_3,c3_4__borsuk_3),u1_struct_0(c2_4__borsuk_3),u1_struct_0(k3_pre_topc(c1_4__borsuk_3,k1_struct_0(c1_4__borsuk_3,c3_4__borsuk_3)))) & v5_pre_topc(k3_borsuk_1(c2_4__borsuk_3,c1_4__borsuk_3,c3_4__borsuk_3),c2_4__borsuk_3,k3_pre_topc(c1_4__borsuk_3,k1_struct_0(c1_4__borsuk_3,c3_4__borsuk_3))) & m2_relset_1(k3_borsuk_1(c2_4__borsuk_3,c1_4__borsuk_3,c3_4__borsuk_3),u1_struct_0(c2_4__borsuk_3),u1_struct_0(k3_pre_topc(c1_4__borsuk_3,k1_struct_0(c1_4__borsuk_3,c3_4__borsuk_3)))) ) ), inference(discharge_asm,[status(thm),assumptions([dt_c2_4__borsuk_3,dt_c1_4__borsuk_3]),discharge_asm(discharge,[dt_c3_4__borsuk_3])],[dt_c3_4__borsuk_3,i2_4__borsuk_3]), [interesting(0.8),i1_4__borsuk_3]). fof(i1_4__borsuk_3,plain,( ! [A] : ( m1_subset_1(A,u1_struct_0(c1_4__borsuk_3)) => ( v1_funct_1(k3_borsuk_1(c2_4__borsuk_3,c1_4__borsuk_3,A)) & v1_funct_2(k3_borsuk_1(c2_4__borsuk_3,c1_4__borsuk_3,A),u1_struct_0(c2_4__borsuk_3),u1_struct_0(k3_pre_topc(c1_4__borsuk_3,k1_struct_0(c1_4__borsuk_3,A)))) & v5_pre_topc(k3_borsuk_1(c2_4__borsuk_3,c1_4__borsuk_3,A),c2_4__borsuk_3,k3_pre_topc(c1_4__borsuk_3,k1_struct_0(c1_4__borsuk_3,A))) & m2_relset_1(k3_borsuk_1(c2_4__borsuk_3,c1_4__borsuk_3,A),u1_struct_0(c2_4__borsuk_3),u1_struct_0(k3_pre_topc(c1_4__borsuk_3,k1_struct_0(c1_4__borsuk_3,A)))) ) ) ), inference(let,[status(thm),assumptions([dt_c2_4__borsuk_3,dt_c1_4__borsuk_3])],[i2_4_tmp__borsuk_3,dh_c3_4__borsuk_3]), [interesting(0.8),file(borsuk_3,i1_4__borsuk_3),[file(borsuk_3,i1_4__borsuk_3)]]). fof(i1_4_tmp__borsuk_3,plain, ( ( ~ v3_struct_0(c1_4__borsuk_3) & v2_pre_topc(c1_4__borsuk_3) & l1_pre_topc(c1_4__borsuk_3) & ~ v3_struct_0(c2_4__borsuk_3) & v2_pre_topc(c2_4__borsuk_3) & l1_pre_topc(c2_4__borsuk_3) ) => ! [A] : ( m1_subset_1(A,u1_struct_0(c1_4__borsuk_3)) => ( v1_funct_1(k3_borsuk_1(c2_4__borsuk_3,c1_4__borsuk_3,A)) & v1_funct_2(k3_borsuk_1(c2_4__borsuk_3,c1_4__borsuk_3,A),u1_struct_0(c2_4__borsuk_3),u1_struct_0(k3_pre_topc(c1_4__borsuk_3,k1_struct_0(c1_4__borsuk_3,A)))) & v5_pre_topc(k3_borsuk_1(c2_4__borsuk_3,c1_4__borsuk_3,A),c2_4__borsuk_3,k3_pre_topc(c1_4__borsuk_3,k1_struct_0(c1_4__borsuk_3,A))) & m2_relset_1(k3_borsuk_1(c2_4__borsuk_3,c1_4__borsuk_3,A),u1_struct_0(c2_4__borsuk_3),u1_struct_0(k3_pre_topc(c1_4__borsuk_3,k1_struct_0(c1_4__borsuk_3,A)))) ) ) ), inference(discharge_asm,[status(thm),assumptions([]),discharge_asm(discharge,[dt_c1_4__borsuk_3,dt_c2_4__borsuk_3])],[dt_c1_4__borsuk_3,dt_c2_4__borsuk_3,i1_4__borsuk_3]), [interesting(1),t2_borsuk_3]). fof(t2_borsuk_3,theorem,( ! [A] : ( ( ~ v3_struct_0(A) & v2_pre_topc(A) & l1_pre_topc(A) ) => ! [B] : ( ( ~ v3_struct_0(B) & v2_pre_topc(B) & l1_pre_topc(B) ) => ! [C] : ( m1_subset_1(C,u1_struct_0(A)) => ( v1_funct_1(k3_borsuk_1(B,A,C)) & v1_funct_2(k3_borsuk_1(B,A,C),u1_struct_0(B),u1_struct_0(k3_pre_topc(A,k1_struct_0(A,C)))) & v5_pre_topc(k3_borsuk_1(B,A,C),B,k3_pre_topc(A,k1_struct_0(A,C))) & m2_relset_1(k3_borsuk_1(B,A,C),u1_struct_0(B),u1_struct_0(k3_pre_topc(A,k1_struct_0(A,C)))) ) ) ) ) ), inference(let,[status(thm),assumptions([])],[i1_4_tmp__borsuk_3,dh_c1_4__borsuk_3,dh_c2_4__borsuk_3]), [interesting(1),file(borsuk_3,t2_borsuk_3),[file(borsuk_3,t2_borsuk_3)]]).