% Mizar ND problem: t3_topmetr,topmetr,69,33 fof(dh_c1_3__topmetr,definition, ( ( ( ~ v3_struct_0(c1_3__topmetr) & v2_pre_topc(c1_3__topmetr) & l1_pre_topc(c1_3__topmetr) ) => ! [A] : ( ( ~ v3_struct_0(A) & m1_pre_topc(A,c1_3__topmetr) ) => ( v3_compts_1(c1_3__topmetr) => v3_compts_1(A) ) ) ) => ! [B] : ( ( ~ v3_struct_0(B) & v2_pre_topc(B) & l1_pre_topc(B) ) => ! [C] : ( ( ~ v3_struct_0(C) & m1_pre_topc(C,B) ) => ( v3_compts_1(B) => v3_compts_1(C) ) ) ) ), introduced(definition,[new_symbol(c1_3__topmetr),file(topmetr,c1_3__topmetr)]), [interesting(0.8),axiom,file(topmetr,c1_3__topmetr)]). fof(dh_c2_3__topmetr,definition, ( ( ( ~ v3_struct_0(c2_3__topmetr) & m1_pre_topc(c2_3__topmetr,c1_3__topmetr) ) => ( v3_compts_1(c1_3__topmetr) => v3_compts_1(c2_3__topmetr) ) ) => ! [A] : ( ( ~ v3_struct_0(A) & m1_pre_topc(A,c1_3__topmetr) ) => ( v3_compts_1(c1_3__topmetr) => v3_compts_1(A) ) ) ), introduced(definition,[new_symbol(c2_3__topmetr),file(topmetr,c2_3__topmetr)]), [interesting(0.8),axiom,file(topmetr,c2_3__topmetr)]). fof(e1_3__topmetr,assumption,( v3_compts_1(c1_3__topmetr) ), introduced(assumption,[file(topmetr,e1_3__topmetr)]), [interesting(0.8),axiom,file(topmetr,e1_3__topmetr)]). fof(cc3_xreal_0,theorem,( ! [A] : ( ( v1_xreal_0(A) & v2_xreal_0(A) ) => ( ~ v1_xboole_0(A) & v1_xcmplx_0(A) & v1_xreal_0(A) & ~ v3_xreal_0(A) ) ) ), file(xreal_0,cc3_xreal_0), [interesting(0.9),axiom,file(xreal_0,cc3_xreal_0)]). fof(cc4_xreal_0,theorem,( ! [A] : ( ( ~ v1_xboole_0(A) & v1_xreal_0(A) & ~ v3_xreal_0(A) ) => ( v1_xcmplx_0(A) & v1_xreal_0(A) & v2_xreal_0(A) ) ) ), file(xreal_0,cc4_xreal_0), [interesting(0.9),axiom,file(xreal_0,cc4_xreal_0)]). fof(cc5_xreal_0,theorem,( ! [A] : ( ( v1_xreal_0(A) & v3_xreal_0(A) ) => ( ~ v1_xboole_0(A) & v1_xcmplx_0(A) & v1_xreal_0(A) & ~ v2_xreal_0(A) ) ) ), file(xreal_0,cc5_xreal_0), [interesting(0.9),axiom,file(xreal_0,cc5_xreal_0)]). fof(cc6_xreal_0,theorem,( ! [A] : ( ( ~ v1_xboole_0(A) & v1_xreal_0(A) & ~ v2_xreal_0(A) ) => ( v1_xcmplx_0(A) & v1_xreal_0(A) & v3_xreal_0(A) ) ) ), file(xreal_0,cc6_xreal_0), [interesting(0.9),axiom,file(xreal_0,cc6_xreal_0)]). fof(cc8_xreal_0,theorem,( ! [A] : ( ( v1_xreal_0(A) & ~ v2_xreal_0(A) & ~ v3_xreal_0(A) ) => ( v1_xboole_0(A) & v1_xcmplx_0(A) & v1_xreal_0(A) ) ) ), file(xreal_0,cc8_xreal_0), [interesting(0.9),axiom,file(xreal_0,cc8_xreal_0)]). fof(rc2_xreal_0,theorem,( ? [A] : ( ~ v1_xboole_0(A) & v1_xcmplx_0(A) & v1_xreal_0(A) & v2_xreal_0(A) & ~ v3_xreal_0(A) ) ), file(xreal_0,rc2_xreal_0), [interesting(0.9),axiom,file(xreal_0,rc2_xreal_0)]). fof(rc3_xreal_0,theorem,( ? [A] : ( ~ v1_xboole_0(A) & v1_xcmplx_0(A) & v1_xreal_0(A) & ~ v2_xreal_0(A) & v3_xreal_0(A) ) ), file(xreal_0,rc3_xreal_0), [interesting(0.9),axiom,file(xreal_0,rc3_xreal_0)]). fof(rc4_xreal_0,theorem,( ? [A] : ( v1_xboole_0(A) & v1_xcmplx_0(A) & v1_xreal_0(A) & ~ v2_xreal_0(A) & ~ v3_xreal_0(A) ) ), file(xreal_0,rc4_xreal_0), [interesting(0.9),axiom,file(xreal_0,rc4_xreal_0)]). fof(cc1_xreal_0,theorem,( ! [A] : ( v4_ordinal2(A) => v1_xreal_0(A) ) ), file(xreal_0,cc1_xreal_0), [interesting(0.9),axiom,file(xreal_0,cc1_xreal_0)]). fof(cc2_xreal_0,theorem,( ! [A] : ( v1_xreal_0(A) => v1_xcmplx_0(A) ) ), file(xreal_0,cc2_xreal_0), [interesting(0.9),axiom,file(xreal_0,cc2_xreal_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(cc7_xreal_0,theorem,( ! [A] : ( ( v1_xboole_0(A) & v1_xreal_0(A) ) => ( v1_xcmplx_0(A) & v1_xreal_0(A) & ~ v2_xreal_0(A) & ~ v3_xreal_0(A) ) ) ), file(xreal_0,cc7_xreal_0), [interesting(0.9),axiom,file(xreal_0,cc7_xreal_0)]). fof(rc1_xreal_0,theorem,( ? [A] : ( v1_xcmplx_0(A) & v1_xreal_0(A) ) ), file(xreal_0,rc1_xreal_0), [interesting(0.9),axiom,file(xreal_0,rc1_xreal_0)]). 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(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(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(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(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(dt_k1_xboole_0,axiom,( $true ), file(xboole_0,k1_xboole_0), [interesting(0.9),axiom,file(xboole_0,k1_xboole_0)]). fof(cc10_membered,theorem,( ! [A] : ( v1_membered(A) => ! [B] : ( m1_subset_1(B,A) => v1_xcmplx_0(B) ) ) ), file(membered,cc10_membered), [interesting(0.9),axiom,file(membered,cc10_membered)]). fof(cc11_membered,theorem,( ! [A] : ( v2_membered(A) => ! [B] : ( m1_subset_1(B,A) => ( v1_xcmplx_0(B) & v1_xreal_0(B) ) ) ) ), file(membered,cc11_membered), [interesting(0.9),axiom,file(membered,cc11_membered)]). fof(cc12_membered,theorem,( ! [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), [interesting(0.9),axiom,file(membered,cc12_membered)]). fof(cc13_membered,theorem,( ! [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), [interesting(0.9),axiom,file(membered,cc13_membered)]). fof(cc14_membered,theorem,( ! [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), [interesting(0.9),axiom,file(membered,cc14_membered)]). fof(cc16_membered,theorem,( ! [A] : ( v1_membered(A) => ! [B] : ( m1_subset_1(B,k1_zfmisc_1(A)) => v1_membered(B) ) ) ), file(membered,cc16_membered), [interesting(0.9),axiom,file(membered,cc16_membered)]). fof(cc17_membered,theorem,( ! [A] : ( v2_membered(A) => ! [B] : ( m1_subset_1(B,k1_zfmisc_1(A)) => ( v1_membered(B) & v2_membered(B) ) ) ) ), file(membered,cc17_membered), [interesting(0.9),axiom,file(membered,cc17_membered)]). fof(cc18_membered,theorem,( ! [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), [interesting(0.9),axiom,file(membered,cc18_membered)]). fof(cc19_membered,theorem,( ! [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), [interesting(0.9),axiom,file(membered,cc19_membered)]). fof(cc1_membered,theorem,( ! [A] : ( v5_membered(A) => v4_membered(A) ) ), file(membered,cc1_membered), [interesting(0.9),axiom,file(membered,cc1_membered)]). 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_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(cc20_membered,theorem,( ! [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), [interesting(0.9),axiom,file(membered,cc20_membered)]). fof(cc2_membered,theorem,( ! [A] : ( v4_membered(A) => v3_membered(A) ) ), file(membered,cc2_membered), [interesting(0.9),axiom,file(membered,cc2_membered)]). fof(cc3_membered,theorem,( ! [A] : ( v3_membered(A) => v2_membered(A) ) ), file(membered,cc3_membered), [interesting(0.9),axiom,file(membered,cc3_membered)]). 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(cc4_membered,theorem,( ! [A] : ( v2_membered(A) => v1_membered(A) ) ), file(membered,cc4_membered), [interesting(0.9),axiom,file(membered,cc4_membered)]). fof(fc6_membered,theorem, ( 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), [interesting(0.9),axiom,file(membered,fc6_membered)]). fof(rc1_membered,theorem,( ? [A] : ( ~ v1_xboole_0(A) & v1_membered(A) & v2_membered(A) & v3_membered(A) & v4_membered(A) & v5_membered(A) ) ), file(membered,rc1_membered), [interesting(0.9),axiom,file(membered,rc1_membered)]). 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(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(reflexivity_r1_tarski,theorem,( ! [A,B] : r1_tarski(A,A) ), file(tarski,r1_tarski), [interesting(0.9),axiom,file(tarski,r1_tarski)]). 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(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_c1_3__topmetr,assumption, ( ~ v3_struct_0(c1_3__topmetr) & v2_pre_topc(c1_3__topmetr) & l1_pre_topc(c1_3__topmetr) ), introduced(assumption,[file(topmetr,c1_3__topmetr)]), [interesting(0.8),axiom,file(topmetr,c1_3__topmetr)]). fof(cc15_membered,theorem,( ! [A] : ( v1_xboole_0(A) => ( v1_membered(A) & v2_membered(A) & v3_membered(A) & v4_membered(A) & v5_membered(A) ) ) ), file(membered,cc15_membered), [interesting(0.9),axiom,file(membered,cc15_membered)]). 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(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(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(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(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(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(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(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(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(symmetry_r1_xboole_0,theorem,( ! [A,B] : ( r1_xboole_0(A,B) => r1_xboole_0(B,A) ) ), file(xboole_0,r1_xboole_0), [interesting(0.9),axiom,file(xboole_0,r1_xboole_0)]). 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(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_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_zfmisc_1,axiom,( $true ), file(zfmisc_1,k1_zfmisc_1), [interesting(0.9),axiom,file(zfmisc_1,k1_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_m1_subset_1,axiom,( $true ), file(subset_1,m1_subset_1), [interesting(0.9),axiom,file(subset_1,m1_subset_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_c2_3__topmetr,assumption, ( ~ v3_struct_0(c2_3__topmetr) & m1_pre_topc(c2_3__topmetr,c1_3__topmetr) ), introduced(assumption,[file(topmetr,c2_3__topmetr)]), [interesting(0.8),axiom,file(topmetr,c2_3__topmetr)]). 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(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(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(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(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(dh_c1_3_1__topmetr,definition, ( ( m1_subset_1(c1_3_1__topmetr,u1_struct_0(c2_3__topmetr)) => ! [A] : ( m1_subset_1(A,u1_struct_0(c2_3__topmetr)) => ~ ( c1_3_1__topmetr != A & ! [B] : ( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(c2_3__topmetr))) => ! [C] : ( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(c2_3__topmetr))) => ~ ( v3_pre_topc(B,c2_3__topmetr) & v3_pre_topc(C,c2_3__topmetr) & r2_hidden(c1_3_1__topmetr,B) & r2_hidden(A,C) & r1_xboole_0(B,C) ) ) ) ) ) ) => ! [D] : ( m1_subset_1(D,u1_struct_0(c2_3__topmetr)) => ! [E] : ( m1_subset_1(E,u1_struct_0(c2_3__topmetr)) => ~ ( D != E & ! [F] : ( m1_subset_1(F,k1_zfmisc_1(u1_struct_0(c2_3__topmetr))) => ! [G] : ( m1_subset_1(G,k1_zfmisc_1(u1_struct_0(c2_3__topmetr))) => ~ ( v3_pre_topc(F,c2_3__topmetr) & v3_pre_topc(G,c2_3__topmetr) & r2_hidden(D,F) & r2_hidden(E,G) & r1_xboole_0(F,G) ) ) ) ) ) ) ), introduced(definition,[new_symbol(c1_3_1__topmetr),file(topmetr,c1_3_1__topmetr)]), [interesting(0.65),axiom,file(topmetr,c1_3_1__topmetr)]). fof(dh_c2_3_1__topmetr,definition, ( ( m1_subset_1(c2_3_1__topmetr,u1_struct_0(c2_3__topmetr)) => ~ ( c1_3_1__topmetr != c2_3_1__topmetr & ! [A] : ( m1_subset_1(A,k1_zfmisc_1(u1_struct_0(c2_3__topmetr))) => ! [B] : ( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(c2_3__topmetr))) => ~ ( v3_pre_topc(A,c2_3__topmetr) & v3_pre_topc(B,c2_3__topmetr) & r2_hidden(c1_3_1__topmetr,A) & r2_hidden(c2_3_1__topmetr,B) & r1_xboole_0(A,B) ) ) ) ) ) => ! [C] : ( m1_subset_1(C,u1_struct_0(c2_3__topmetr)) => ~ ( c1_3_1__topmetr != C & ! [D] : ( m1_subset_1(D,k1_zfmisc_1(u1_struct_0(c2_3__topmetr))) => ! [E] : ( m1_subset_1(E,k1_zfmisc_1(u1_struct_0(c2_3__topmetr))) => ~ ( v3_pre_topc(D,c2_3__topmetr) & v3_pre_topc(E,c2_3__topmetr) & r2_hidden(c1_3_1__topmetr,D) & r2_hidden(C,E) & r1_xboole_0(D,E) ) ) ) ) ) ), introduced(definition,[new_symbol(c2_3_1__topmetr),file(topmetr,c2_3_1__topmetr)]), [interesting(0.65),axiom,file(topmetr,c2_3_1__topmetr)]). fof(e1_3_1__topmetr,assumption,( c1_3_1__topmetr != c2_3_1__topmetr ), introduced(assumption,[file(topmetr,e1_3_1__topmetr)]), [interesting(0.65),axiom,file(topmetr,e1_3_1__topmetr)]). fof(dt_c1_3_1__topmetr,assumption,( m1_subset_1(c1_3_1__topmetr,u1_struct_0(c2_3__topmetr)) ), introduced(assumption,[file(topmetr,c1_3_1__topmetr)]), [interesting(0.65),axiom,file(topmetr,c1_3_1__topmetr)]). fof(dt_c2_3_1__topmetr,assumption,( m1_subset_1(c2_3_1__topmetr,u1_struct_0(c2_3__topmetr)) ), introduced(assumption,[file(topmetr,c2_3_1__topmetr)]), [interesting(0.65),axiom,file(topmetr,c2_3_1__topmetr)]). fof(fc27_membered,theorem,( ! [A,B] : ( v1_membered(A) => v1_membered(k3_xboole_0(A,B)) ) ), file(membered,fc27_membered), [interesting(0.9),axiom,file(membered,fc27_membered)]). fof(fc28_membered,theorem,( ! [A,B] : ( v1_membered(A) => v1_membered(k3_xboole_0(B,A)) ) ), file(membered,fc28_membered), [interesting(0.9),axiom,file(membered,fc28_membered)]). fof(fc29_membered,theorem,( ! [A,B] : ( v2_membered(A) => ( v1_membered(k3_xboole_0(A,B)) & v2_membered(k3_xboole_0(A,B)) ) ) ), file(membered,fc29_membered), [interesting(0.9),axiom,file(membered,fc29_membered)]). fof(fc30_membered,theorem,( ! [A,B] : ( v2_membered(A) => ( v1_membered(k3_xboole_0(B,A)) & v2_membered(k3_xboole_0(B,A)) ) ) ), file(membered,fc30_membered), [interesting(0.9),axiom,file(membered,fc30_membered)]). fof(fc31_membered,theorem,( ! [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), [interesting(0.9),axiom,file(membered,fc31_membered)]). fof(fc32_membered,theorem,( ! [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), [interesting(0.9),axiom,file(membered,fc32_membered)]). fof(fc33_membered,theorem,( ! [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), [interesting(0.9),axiom,file(membered,fc33_membered)]). fof(fc34_membered,theorem,( ! [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), [interesting(0.9),axiom,file(membered,fc34_membered)]). fof(fc35_membered,theorem,( ! [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), [interesting(0.9),axiom,file(membered,fc35_membered)]). fof(fc36_membered,theorem,( ! [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), [interesting(0.9),axiom,file(membered,fc36_membered)]). fof(fc5_pre_topc,theorem,( ! [A] : ( ( v2_pre_topc(A) & l1_pre_topc(A) ) => v4_pre_topc(k2_pre_topc(A),A) ) ), file(pre_topc,fc5_pre_topc), [interesting(0.9),axiom,file(pre_topc,fc5_pre_topc)]). fof(fc8_tops_1,theorem,( ! [A] : ( ( v2_pre_topc(A) & l1_pre_topc(A) ) => ( v3_pre_topc(k2_pre_topc(A),A) & v4_pre_topc(k2_pre_topc(A),A) ) ) ), file(tops_1,fc8_tops_1), [interesting(0.9),axiom,file(tops_1,fc8_tops_1)]). fof(fc9_tops_1,theorem,( ! [A] : ( l1_pre_topc(A) => v1_tops_1(k2_pre_topc(A),A) ) ), file(tops_1,fc9_tops_1), [interesting(0.9),axiom,file(tops_1,fc9_tops_1)]). fof(t2_boole,theorem,( ! [A] : k3_xboole_0(A,k1_xboole_0) = k1_xboole_0 ), file(boole,t2_boole), [interesting(0.9),axiom,file(boole,t2_boole)]). fof(fc2_pre_topc,theorem,( ! [A] : ( ( ~ v3_struct_0(A) & l1_struct_0(A) ) => ~ v1_xboole_0(k2_pre_topc(A)) ) ), file(pre_topc,fc2_pre_topc), [interesting(0.9),axiom,file(pre_topc,fc2_pre_topc)]). fof(commutativity_k3_xboole_0,theorem,( ! [A,B] : k3_xboole_0(A,B) = k3_xboole_0(B,A) ), file(xboole_0,k3_xboole_0), [interesting(0.9),axiom,file(xboole_0,k3_xboole_0)]). fof(idempotence_k3_xboole_0,theorem,( ! [A,B] : k3_xboole_0(A,A) = A ), file(xboole_0,k3_xboole_0), [interesting(0.9),axiom,file(xboole_0,k3_xboole_0)]). 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), [interesting(0.9),axiom,file(pre_topc,k2_pre_topc)]). fof(dt_k3_xboole_0,axiom,( $true ), file(xboole_0,k3_xboole_0), [interesting(0.9),axiom,file(xboole_0,k3_xboole_0)]). fof(dh_c5_3_1__topmetr,definition, ( ? [A] : ( m1_subset_1(A,k1_zfmisc_1(u1_struct_0(c1_3__topmetr))) & ? [B] : ( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(c1_3__topmetr))) & v3_pre_topc(A,c1_3__topmetr) & v3_pre_topc(B,c1_3__topmetr) & r2_hidden(c3_3_1__topmetr,A) & r2_hidden(c4_3_1__topmetr,B) & r1_xboole_0(A,B) ) ) => ( m1_subset_1(c5_3_1__topmetr,k1_zfmisc_1(u1_struct_0(c1_3__topmetr))) & ? [C] : ( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(c1_3__topmetr))) & v3_pre_topc(c5_3_1__topmetr,c1_3__topmetr) & v3_pre_topc(C,c1_3__topmetr) & r2_hidden(c3_3_1__topmetr,c5_3_1__topmetr) & r2_hidden(c4_3_1__topmetr,C) & r1_xboole_0(c5_3_1__topmetr,C) ) ) ), introduced(definition,[new_symbol(c5_3_1__topmetr),file(topmetr,c5_3_1__topmetr)]), [interesting(0.65),axiom,file(topmetr,c5_3_1__topmetr)]). fof(de_c3_3_1__topmetr,definition,( c3_3_1__topmetr = c1_3_1__topmetr ), introduced(definition,[new_symbol(c3_3_1__topmetr),file(topmetr,c3_3_1__topmetr)]), [interesting(0.65),axiom,file(topmetr,c3_3_1__topmetr)]). fof(t2_topmetr,theorem,( ! [A] : ( ( ~ v3_struct_0(A) & v2_pre_topc(A) & l1_pre_topc(A) ) => ! [B] : ( ( ~ v3_struct_0(B) & m1_pre_topc(B,A) ) => ! [C] : ( m1_subset_1(C,u1_struct_0(B)) => m1_subset_1(C,u1_struct_0(A)) ) ) ) ), file(topmetr,t2_topmetr), [interesting(0.9),axiom,file(topmetr,t2_topmetr)]). fof(e2_3_1__topmetr,plain, ( m1_subset_1(c1_3_1__topmetr,u1_struct_0(c1_3__topmetr)) & m1_subset_1(c2_3_1__topmetr,u1_struct_0(c1_3__topmetr)) ), inference(mizar_by,[status(thm),assumptions([dt_c2_3__topmetr,dt_c1_3__topmetr,dt_c1_3_1__topmetr,dt_c2_3_1__topmetr])],[cc3_xreal_0,cc4_xreal_0,cc5_xreal_0,cc6_xreal_0,cc8_xreal_0,rc2_xreal_0,rc3_xreal_0,rc4_xreal_0,cc1_xreal_0,cc2_xreal_0,cc7_xreal_0,rc1_xreal_0,antisymmetry_r2_hidden,dt_k1_xboole_0,cc10_membered,cc11_membered,cc12_membered,cc13_membered,cc14_membered,cc1_membered,cc2_membered,cc3_membered,cc4_membered,fc6_membered,rc1_membered,t1_subset,t8_boole,cc15_membered,t2_subset,t6_boole,t7_boole,existence_l1_struct_0,dt_l1_struct_0,dt_c2_3__topmetr,fc1_struct_0,rc3_struct_0,existence_l1_pre_topc,existence_m1_pre_topc,existence_m1_subset_1,dt_l1_pre_topc,dt_m1_pre_topc,dt_m1_subset_1,dt_u1_struct_0,dt_c1_3__topmetr,dt_c1_3_1__topmetr,dt_c2_3_1__topmetr,cc1_pre_topc,t2_topmetr]), [interesting(0.65),file(topmetr,e2_3_1__topmetr),[file(topmetr,e2_3_1__topmetr)]]). fof(dt_c3_3_1__topmetr,plain,( m1_subset_1(c3_3_1__topmetr,u1_struct_0(c1_3__topmetr)) ), inference(mizar_by,[status(thm),assumptions([dt_c2_3__topmetr,dt_c1_3__topmetr,dt_c1_3_1__topmetr,dt_c2_3_1__topmetr])],[cc3_xreal_0,cc4_xreal_0,cc5_xreal_0,cc6_xreal_0,cc8_xreal_0,rc2_xreal_0,rc3_xreal_0,rc4_xreal_0,cc1_xreal_0,cc2_xreal_0,cc7_xreal_0,rc1_xreal_0,antisymmetry_r2_hidden,dt_k1_xboole_0,cc10_membered,cc11_membered,cc12_membered,cc13_membered,cc14_membered,cc1_membered,cc2_membered,cc3_membered,cc4_membered,fc6_membered,rc1_membered,t1_subset,existence_m1_pre_topc,dt_m1_pre_topc,cc15_membered,cc1_pre_topc,t2_subset,t6_boole,t7_boole,t8_boole,existence_l1_pre_topc,existence_l1_struct_0,dt_l1_pre_topc,dt_l1_struct_0,dt_c2_3__topmetr,fc1_struct_0,rc3_struct_0,existence_m1_subset_1,dt_m1_subset_1,dt_u1_struct_0,dt_c1_3__topmetr,dt_c1_3_1__topmetr,dt_c2_3_1__topmetr,de_c3_3_1__topmetr,e2_3_1__topmetr]), [interesting(0.65),file(topmetr,c3_3_1__topmetr),[file(topmetr,c3_3_1__topmetr)]]). fof(de_c4_3_1__topmetr,definition,( c4_3_1__topmetr = c2_3_1__topmetr ), introduced(definition,[new_symbol(c4_3_1__topmetr),file(topmetr,c4_3_1__topmetr)]), [interesting(0.65),axiom,file(topmetr,c4_3_1__topmetr)]). fof(dt_c4_3_1__topmetr,plain,( m1_subset_1(c4_3_1__topmetr,u1_struct_0(c1_3__topmetr)) ), inference(mizar_by,[status(thm),assumptions([dt_c2_3__topmetr,dt_c1_3__topmetr,dt_c1_3_1__topmetr,dt_c2_3_1__topmetr])],[cc3_xreal_0,cc4_xreal_0,cc5_xreal_0,cc6_xreal_0,cc8_xreal_0,rc2_xreal_0,rc3_xreal_0,rc4_xreal_0,cc1_xreal_0,cc2_xreal_0,cc7_xreal_0,rc1_xreal_0,antisymmetry_r2_hidden,dt_k1_xboole_0,cc10_membered,cc11_membered,cc12_membered,cc13_membered,cc14_membered,cc1_membered,cc2_membered,cc3_membered,cc4_membered,fc6_membered,rc1_membered,t1_subset,existence_m1_pre_topc,dt_m1_pre_topc,cc15_membered,cc1_pre_topc,t2_subset,t6_boole,t7_boole,t8_boole,existence_l1_pre_topc,existence_l1_struct_0,dt_l1_pre_topc,dt_l1_struct_0,dt_c2_3__topmetr,fc1_struct_0,rc3_struct_0,existence_m1_subset_1,dt_m1_subset_1,dt_u1_struct_0,dt_c1_3__topmetr,dt_c1_3_1__topmetr,dt_c2_3_1__topmetr,de_c4_3_1__topmetr,e2_3_1__topmetr]), [interesting(0.65),file(topmetr,c4_3_1__topmetr),[file(topmetr,c4_3_1__topmetr)]]). fof(d4_compts_1,definition,( ! [A] : ( l1_pre_topc(A) => ( v3_compts_1(A) <=> ! [B] : ( m1_subset_1(B,u1_struct_0(A)) => ! [C] : ( m1_subset_1(C,u1_struct_0(A)) => ~ ( B != C & ! [D] : ( m1_subset_1(D,k1_zfmisc_1(u1_struct_0(A))) => ! [E] : ( m1_subset_1(E,k1_zfmisc_1(u1_struct_0(A))) => ~ ( v3_pre_topc(D,A) & v3_pre_topc(E,A) & r2_hidden(B,D) & r2_hidden(C,E) & r1_xboole_0(D,E) ) ) ) ) ) ) ) ) ), file(compts_1,d4_compts_1), [interesting(0.9),axiom,file(compts_1,d4_compts_1)]). fof(e3_3_1__topmetr,plain,( ? [A] : ( m1_subset_1(A,k1_zfmisc_1(u1_struct_0(c1_3__topmetr))) & ? [B] : ( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(c1_3__topmetr))) & v3_pre_topc(A,c1_3__topmetr) & v3_pre_topc(B,c1_3__topmetr) & r2_hidden(c3_3_1__topmetr,A) & r2_hidden(c4_3_1__topmetr,B) & r1_xboole_0(A,B) ) ) ), inference(mizar_by,[status(thm),assumptions([dt_c2_3__topmetr,dt_c1_3__topmetr,dt_c1_3_1__topmetr,dt_c2_3_1__topmetr,e1_3__topmetr,e1_3_1__topmetr])],[cc3_xreal_0,cc4_xreal_0,cc5_xreal_0,cc6_xreal_0,cc8_xreal_0,rc2_xreal_0,rc3_xreal_0,rc4_xreal_0,cc1_xreal_0,cc2_xreal_0,cc7_xreal_0,rc1_xreal_0,existence_m1_pre_topc,dt_k1_xboole_0,dt_m1_pre_topc,cc10_membered,cc11_membered,cc12_membered,cc13_membered,cc14_membered,cc16_membered,cc17_membered,cc18_membered,cc19_membered,cc1_membered,cc1_pre_topc,cc20_membered,cc2_membered,cc3_membered,cc4_membered,cc4_tops_1,cc5_tops_1,cc6_tops_1,fc6_membered,rc1_membered,rc2_tops_1,rc3_tops_1,rc4_tops_1,rc5_tops_1,rc6_pre_topc,rc7_pre_topc,reflexivity_r1_tarski,existence_l1_struct_0,dt_l1_struct_0,dt_c2_3__topmetr,cc15_membered,cc1_tops_1,cc2_tops_1,cc3_tops_1,fc1_struct_0,rc1_subset_1,rc1_tops_1,rc2_subset_1,rc3_struct_0,rc5_struct_0,t2_subset,t5_subset,t6_boole,t8_boole,symmetry_r1_xboole_0,antisymmetry_r2_hidden,existence_l1_pre_topc,existence_m1_subset_1,dt_k1_zfmisc_1,dt_l1_pre_topc,dt_m1_subset_1,dt_u1_struct_0,dt_c1_3__topmetr,dt_c1_3_1__topmetr,dt_c2_3_1__topmetr,dt_c3_3_1__topmetr,dt_c4_3_1__topmetr,de_c3_3_1__topmetr,de_c4_3_1__topmetr,fc1_subset_1,t1_subset,t3_subset,t4_subset,t7_boole,e1_3__topmetr,e1_3_1__topmetr,d4_compts_1]), [interesting(0.65),file(topmetr,e3_3_1__topmetr),[file(topmetr,e3_3_1__topmetr)]]). fof(dt_c5_3_1__topmetr,plain,( m1_subset_1(c5_3_1__topmetr,k1_zfmisc_1(u1_struct_0(c1_3__topmetr))) ), inference(consider,[status(thm),assumptions([dt_c2_3__topmetr,dt_c1_3__topmetr,dt_c1_3_1__topmetr,dt_c2_3_1__topmetr,e1_3__topmetr,e1_3_1__topmetr])],[dh_c5_3_1__topmetr,e3_3_1__topmetr]), [interesting(0.65),file(topmetr,c5_3_1__topmetr),[file(topmetr,c5_3_1__topmetr)]]). fof(dh_c6_3_1__topmetr,definition, ( ? [A] : ( m1_subset_1(A,k1_zfmisc_1(u1_struct_0(c1_3__topmetr))) & v3_pre_topc(c5_3_1__topmetr,c1_3__topmetr) & v3_pre_topc(A,c1_3__topmetr) & r2_hidden(c3_3_1__topmetr,c5_3_1__topmetr) & r2_hidden(c4_3_1__topmetr,A) & r1_xboole_0(c5_3_1__topmetr,A) ) => ( m1_subset_1(c6_3_1__topmetr,k1_zfmisc_1(u1_struct_0(c1_3__topmetr))) & v3_pre_topc(c5_3_1__topmetr,c1_3__topmetr) & v3_pre_topc(c6_3_1__topmetr,c1_3__topmetr) & r2_hidden(c3_3_1__topmetr,c5_3_1__topmetr) & r2_hidden(c4_3_1__topmetr,c6_3_1__topmetr) & r1_xboole_0(c5_3_1__topmetr,c6_3_1__topmetr) ) ), introduced(definition,[new_symbol(c6_3_1__topmetr),file(topmetr,c6_3_1__topmetr)]), [interesting(0.65),axiom,file(topmetr,c6_3_1__topmetr)]). fof(dt_c6_3_1__topmetr,plain,( m1_subset_1(c6_3_1__topmetr,k1_zfmisc_1(u1_struct_0(c1_3__topmetr))) ), inference(consider,[status(thm),assumptions([dt_c2_3__topmetr,dt_c1_3__topmetr,dt_c1_3_1__topmetr,dt_c2_3_1__topmetr,e1_3__topmetr,e1_3_1__topmetr])],[dh_c5_3_1__topmetr,dh_c6_3_1__topmetr,e3_3_1__topmetr]), [interesting(0.65),file(topmetr,c6_3_1__topmetr),[file(topmetr,c6_3_1__topmetr)]]). fof(d3_pre_topc,definition,( ! [A] : ( l1_struct_0(A) => k2_pre_topc(A) = u1_struct_0(A) ) ), file(pre_topc,d3_pre_topc), [interesting(0.9),axiom,file(pre_topc,d3_pre_topc)]). fof(de_c7_3_1__topmetr,definition,( c7_3_1__topmetr = k3_xboole_0(c5_3_1__topmetr,k2_pre_topc(c2_3__topmetr)) ), introduced(definition,[new_symbol(c7_3_1__topmetr),file(topmetr,c7_3_1__topmetr)]), [interesting(0.65),axiom,file(topmetr,c7_3_1__topmetr)]). fof(t17_xboole_1,theorem,( ! [A,B] : r1_tarski(k3_xboole_0(A,B),A) ), file(xboole_1,t17_xboole_1), [interesting(0.9),axiom,file(xboole_1,t17_xboole_1)]). fof(e6_3_1__topmetr,plain, ( r1_tarski(k3_xboole_0(c5_3_1__topmetr,k2_pre_topc(c2_3__topmetr)),k2_pre_topc(c2_3__topmetr)) & r1_tarski(k3_xboole_0(c6_3_1__topmetr,k2_pre_topc(c2_3__topmetr)),k2_pre_topc(c2_3__topmetr)) ), inference(mizar_by,[status(thm),assumptions([dt_c2_3__topmetr,dt_c1_3__topmetr,dt_c1_3_1__topmetr,dt_c2_3_1__topmetr,e1_3__topmetr,e1_3_1__topmetr])],[cc3_xreal_0,cc4_xreal_0,cc5_xreal_0,cc6_xreal_0,cc8_xreal_0,rc2_xreal_0,rc3_xreal_0,rc4_xreal_0,cc1_xreal_0,cc2_xreal_0,cc7_xreal_0,rc1_xreal_0,antisymmetry_r2_hidden,dt_k1_xboole_0,cc10_membered,cc11_membered,cc12_membered,cc13_membered,cc14_membered,cc16_membered,cc17_membered,cc18_membered,cc19_membered,cc1_membered,cc20_membered,cc2_membered,cc3_membered,cc4_membered,cc4_tops_1,cc5_tops_1,cc6_tops_1,fc27_membered,fc28_membered,fc29_membered,fc30_membered,fc31_membered,fc32_membered,fc33_membered,fc34_membered,fc35_membered,fc36_membered,fc6_membered,rc1_membered,rc1_tops_1,rc2_tops_1,rc3_tops_1,rc4_tops_1,rc5_tops_1,rc6_pre_topc,rc7_pre_topc,t1_subset,t2_boole,t4_subset,t5_subset,existence_l1_pre_topc,dt_l1_pre_topc,cc15_membered,cc1_pre_topc,cc1_tops_1,cc2_tops_1,cc3_tops_1,fc5_pre_topc,fc8_tops_1,fc9_tops_1,rc1_subset_1,rc2_subset_1,rc5_struct_0,t2_subset,t6_boole,t7_boole,t8_boole,existence_l1_struct_0,existence_m1_pre_topc,existence_m1_subset_1,dt_k1_zfmisc_1,dt_l1_struct_0,dt_m1_pre_topc,dt_m1_subset_1,dt_u1_struct_0,dt_c1_3__topmetr,fc1_struct_0,fc1_subset_1,fc2_pre_topc,rc3_struct_0,commutativity_k3_xboole_0,idempotence_k3_xboole_0,reflexivity_r1_tarski,dt_k2_pre_topc,dt_k3_xboole_0,dt_c2_3__topmetr,dt_c5_3_1__topmetr,dt_c6_3_1__topmetr,t3_subset,d3_pre_topc,t17_xboole_1]), [interesting(0.65),file(topmetr,e6_3_1__topmetr),[file(topmetr,e6_3_1__topmetr)]]). fof(e7_3_1__topmetr,plain, ( m1_subset_1(k3_xboole_0(c5_3_1__topmetr,k2_pre_topc(c2_3__topmetr)),k1_zfmisc_1(u1_struct_0(c2_3__topmetr))) & m1_subset_1(k3_xboole_0(c6_3_1__topmetr,k2_pre_topc(c2_3__topmetr)),k1_zfmisc_1(u1_struct_0(c2_3__topmetr))) ), inference(mizar_by,[status(thm),assumptions([dt_c2_3__topmetr,dt_c1_3__topmetr,dt_c1_3_1__topmetr,dt_c2_3_1__topmetr,e1_3__topmetr,e1_3_1__topmetr])],[cc3_xreal_0,cc4_xreal_0,cc5_xreal_0,cc6_xreal_0,cc8_xreal_0,rc2_xreal_0,rc3_xreal_0,rc4_xreal_0,cc1_xreal_0,cc2_xreal_0,cc4_tops_1,cc5_tops_1,cc6_tops_1,cc7_xreal_0,rc1_tops_1,rc1_xreal_0,rc2_tops_1,rc3_tops_1,rc4_tops_1,rc5_tops_1,rc6_pre_topc,rc7_pre_topc,antisymmetry_r2_hidden,existence_l1_pre_topc,dt_k1_xboole_0,dt_l1_pre_topc,cc10_membered,cc11_membered,cc12_membered,cc13_membered,cc14_membered,cc16_membered,cc17_membered,cc18_membered,cc19_membered,cc1_membered,cc1_pre_topc,cc1_tops_1,cc20_membered,cc2_membered,cc2_tops_1,cc3_membered,cc3_tops_1,cc4_membered,fc27_membered,fc28_membered,fc29_membered,fc30_membered,fc31_membered,fc32_membered,fc33_membered,fc34_membered,fc35_membered,fc36_membered,fc5_pre_topc,fc6_membered,fc8_tops_1,fc9_tops_1,rc1_membered,t1_subset,t2_boole,t4_subset,t5_subset,existence_l1_struct_0,existence_m1_pre_topc,dt_l1_struct_0,dt_m1_pre_topc,dt_c1_3__topmetr,cc15_membered,fc1_struct_0,fc2_pre_topc,rc1_subset_1,rc2_subset_1,rc3_struct_0,rc5_struct_0,t2_subset,t6_boole,t7_boole,t8_boole,commutativity_k3_xboole_0,idempotence_k3_xboole_0,reflexivity_r1_tarski,existence_m1_subset_1,dt_k1_zfmisc_1,dt_k2_pre_topc,dt_k3_xboole_0,dt_m1_subset_1,dt_u1_struct_0,dt_c2_3__topmetr,dt_c5_3_1__topmetr,dt_c6_3_1__topmetr,fc1_subset_1,t3_subset,d3_pre_topc,e6_3_1__topmetr]), [interesting(0.65),file(topmetr,e7_3_1__topmetr),[file(topmetr,e7_3_1__topmetr)]]). fof(dt_c7_3_1__topmetr,plain,( m1_subset_1(c7_3_1__topmetr,k1_zfmisc_1(u1_struct_0(c2_3__topmetr))) ), inference(mizar_by,[status(thm),assumptions([dt_c2_3__topmetr,dt_c1_3__topmetr,dt_c1_3_1__topmetr,dt_c2_3_1__topmetr,e1_3__topmetr,e1_3_1__topmetr])],[cc3_xreal_0,cc4_xreal_0,cc5_xreal_0,cc6_xreal_0,cc8_xreal_0,rc2_xreal_0,rc3_xreal_0,rc4_xreal_0,cc1_xreal_0,cc2_xreal_0,cc4_tops_1,cc5_tops_1,cc6_tops_1,cc7_xreal_0,rc1_tops_1,rc1_xreal_0,rc2_tops_1,rc3_tops_1,rc4_tops_1,rc5_tops_1,rc6_pre_topc,rc7_pre_topc,antisymmetry_r2_hidden,existence_l1_pre_topc,dt_k1_xboole_0,dt_l1_pre_topc,cc10_membered,cc11_membered,cc12_membered,cc13_membered,cc14_membered,cc16_membered,cc17_membered,cc18_membered,cc19_membered,cc1_membered,cc1_pre_topc,cc1_tops_1,cc20_membered,cc2_membered,cc2_tops_1,cc3_membered,cc3_tops_1,cc4_membered,fc27_membered,fc28_membered,fc29_membered,fc30_membered,fc31_membered,fc32_membered,fc33_membered,fc34_membered,fc35_membered,fc36_membered,fc5_pre_topc,fc6_membered,fc8_tops_1,fc9_tops_1,rc1_membered,t1_subset,t2_boole,t4_subset,t5_subset,reflexivity_r1_tarski,existence_l1_struct_0,existence_m1_pre_topc,dt_l1_struct_0,dt_m1_pre_topc,dt_c1_3__topmetr,cc15_membered,fc1_struct_0,fc2_pre_topc,rc1_subset_1,rc2_subset_1,rc3_struct_0,rc5_struct_0,t2_subset,t6_boole,t7_boole,t8_boole,commutativity_k3_xboole_0,idempotence_k3_xboole_0,existence_m1_subset_1,dt_k1_zfmisc_1,dt_k2_pre_topc,dt_k3_xboole_0,dt_m1_subset_1,dt_u1_struct_0,dt_c2_3__topmetr,dt_c5_3_1__topmetr,dt_c6_3_1__topmetr,fc1_subset_1,t3_subset,d3_pre_topc,de_c7_3_1__topmetr,e7_3_1__topmetr]), [interesting(0.65),file(topmetr,c7_3_1__topmetr),[file(topmetr,c7_3_1__topmetr)]]). fof(de_c8_3_1__topmetr,definition,( c8_3_1__topmetr = k3_xboole_0(c6_3_1__topmetr,k2_pre_topc(c2_3__topmetr)) ), introduced(definition,[new_symbol(c8_3_1__topmetr),file(topmetr,c8_3_1__topmetr)]), [interesting(0.65),axiom,file(topmetr,c8_3_1__topmetr)]). fof(dt_c8_3_1__topmetr,plain,( m1_subset_1(c8_3_1__topmetr,k1_zfmisc_1(u1_struct_0(c2_3__topmetr))) ), inference(mizar_by,[status(thm),assumptions([dt_c2_3__topmetr,dt_c1_3__topmetr,dt_c1_3_1__topmetr,dt_c2_3_1__topmetr,e1_3__topmetr,e1_3_1__topmetr])],[cc3_xreal_0,cc4_xreal_0,cc5_xreal_0,cc6_xreal_0,cc8_xreal_0,rc2_xreal_0,rc3_xreal_0,rc4_xreal_0,cc1_xreal_0,cc2_xreal_0,cc4_tops_1,cc5_tops_1,cc6_tops_1,cc7_xreal_0,rc1_tops_1,rc1_xreal_0,rc2_tops_1,rc3_tops_1,rc4_tops_1,rc5_tops_1,rc6_pre_topc,rc7_pre_topc,antisymmetry_r2_hidden,existence_l1_pre_topc,dt_k1_xboole_0,dt_l1_pre_topc,cc10_membered,cc11_membered,cc12_membered,cc13_membered,cc14_membered,cc16_membered,cc17_membered,cc18_membered,cc19_membered,cc1_membered,cc1_pre_topc,cc1_tops_1,cc20_membered,cc2_membered,cc2_tops_1,cc3_membered,cc3_tops_1,cc4_membered,fc27_membered,fc28_membered,fc29_membered,fc30_membered,fc31_membered,fc32_membered,fc33_membered,fc34_membered,fc35_membered,fc36_membered,fc5_pre_topc,fc6_membered,fc8_tops_1,fc9_tops_1,rc1_membered,t1_subset,t2_boole,t4_subset,t5_subset,reflexivity_r1_tarski,existence_l1_struct_0,existence_m1_pre_topc,dt_l1_struct_0,dt_m1_pre_topc,dt_c1_3__topmetr,cc15_membered,fc1_struct_0,fc2_pre_topc,rc1_subset_1,rc2_subset_1,rc3_struct_0,rc5_struct_0,t2_subset,t6_boole,t7_boole,t8_boole,commutativity_k3_xboole_0,idempotence_k3_xboole_0,existence_m1_subset_1,dt_k1_zfmisc_1,dt_k2_pre_topc,dt_k3_xboole_0,dt_m1_subset_1,dt_u1_struct_0,dt_c2_3__topmetr,dt_c5_3_1__topmetr,dt_c6_3_1__topmetr,fc1_subset_1,t3_subset,d3_pre_topc,de_c8_3_1__topmetr,e7_3_1__topmetr]), [interesting(0.65),file(topmetr,c8_3_1__topmetr),[file(topmetr,c8_3_1__topmetr)]]). 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(e4_3_1__topmetr,plain, ( v3_pre_topc(c5_3_1__topmetr,c1_3__topmetr) & v3_pre_topc(c6_3_1__topmetr,c1_3__topmetr) & r2_hidden(c3_3_1__topmetr,c5_3_1__topmetr) & r2_hidden(c4_3_1__topmetr,c6_3_1__topmetr) & r1_xboole_0(c5_3_1__topmetr,c6_3_1__topmetr) ), inference(consider,[status(thm),assumptions([dt_c2_3__topmetr,dt_c1_3__topmetr,dt_c1_3_1__topmetr,dt_c2_3_1__topmetr,e1_3__topmetr,e1_3_1__topmetr])],[dh_c5_3_1__topmetr,dh_c6_3_1__topmetr,e3_3_1__topmetr]), [interesting(0.65),file(topmetr,e4_3_1__topmetr),[file(topmetr,e4_3_1__topmetr)]]). fof(d5_pre_topc,definition,( ! [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), [interesting(0.9),axiom,file(pre_topc,d5_pre_topc)]). fof(e8_3_1__topmetr,plain, ( r2_hidden(c5_3_1__topmetr,u1_pre_topc(c1_3__topmetr)) & r2_hidden(c6_3_1__topmetr,u1_pre_topc(c1_3__topmetr)) ), inference(mizar_by,[status(thm),assumptions([dt_c2_3__topmetr,dt_c1_3__topmetr,dt_c1_3_1__topmetr,dt_c2_3_1__topmetr,e1_3__topmetr,e1_3_1__topmetr])],[cc3_xreal_0,cc4_xreal_0,cc5_xreal_0,cc6_xreal_0,cc8_xreal_0,rc2_xreal_0,rc3_xreal_0,rc4_xreal_0,existence_m1_pre_topc,dt_m1_pre_topc,cc1_pre_topc,cc1_xreal_0,cc2_xreal_0,cc7_xreal_0,rc1_xreal_0,dt_k1_xboole_0,dt_c2_3__topmetr,cc10_membered,cc11_membered,cc12_membered,cc13_membered,cc14_membered,cc16_membered,cc17_membered,cc18_membered,cc19_membered,cc1_membered,cc20_membered,cc2_membered,cc3_membered,cc4_membered,cc4_tops_1,cc5_tops_1,cc6_tops_1,fc6_membered,rc1_membered,rc2_tops_1,rc3_tops_1,rc4_tops_1,rc5_tops_1,rc6_pre_topc,rc7_pre_topc,reflexivity_r1_tarski,existence_l1_struct_0,dt_l1_struct_0,dt_c1_3_1__topmetr,dt_c2_3_1__topmetr,cc15_membered,cc1_tops_1,cc2_tops_1,cc3_tops_1,fc1_struct_0,rc1_subset_1,rc1_tops_1,rc2_subset_1,rc3_struct_0,rc5_struct_0,t2_subset,t5_subset,t6_boole,t8_boole,symmetry_r1_xboole_0,antisymmetry_r2_hidden,existence_l1_pre_topc,existence_m1_subset_1,dt_k1_zfmisc_1,dt_l1_pre_topc,dt_m1_subset_1,dt_u1_pre_topc,dt_u1_struct_0,dt_c1_3__topmetr,dt_c3_3_1__topmetr,dt_c4_3_1__topmetr,dt_c5_3_1__topmetr,dt_c6_3_1__topmetr,de_c3_3_1__topmetr,de_c4_3_1__topmetr,fc1_subset_1,t1_subset,t3_subset,t4_subset,t7_boole,e4_3_1__topmetr,d5_pre_topc]), [interesting(0.65),file(topmetr,e8_3_1__topmetr),[file(topmetr,e8_3_1__topmetr)]]). fof(d9_pre_topc,definition,( ! [A] : ( l1_pre_topc(A) => ! [B] : ( l1_pre_topc(B) => ( m1_pre_topc(B,A) <=> ( r1_tarski(k2_pre_topc(B),k2_pre_topc(A)) & ! [C] : ( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(B))) => ( r2_hidden(C,u1_pre_topc(B)) <=> ? [D] : ( m1_subset_1(D,k1_zfmisc_1(u1_struct_0(A))) & r2_hidden(D,u1_pre_topc(A)) & C = k3_xboole_0(D,k2_pre_topc(B)) ) ) ) ) ) ) ) ), file(pre_topc,d9_pre_topc), [interesting(0.9),axiom,file(pre_topc,d9_pre_topc)]). fof(e9_3_1__topmetr,plain, ( r2_hidden(c7_3_1__topmetr,u1_pre_topc(c2_3__topmetr)) & r2_hidden(c8_3_1__topmetr,u1_pre_topc(c2_3__topmetr)) ), inference(mizar_by,[status(thm),assumptions([dt_c2_3__topmetr,dt_c1_3__topmetr,dt_c1_3_1__topmetr,dt_c2_3_1__topmetr,e1_3__topmetr,e1_3_1__topmetr])],[cc3_xreal_0,cc4_xreal_0,cc5_xreal_0,cc6_xreal_0,cc8_xreal_0,rc2_xreal_0,rc3_xreal_0,rc4_xreal_0,cc1_xreal_0,cc2_xreal_0,cc7_xreal_0,rc1_xreal_0,dt_k1_xboole_0,cc10_membered,cc11_membered,cc12_membered,cc13_membered,cc14_membered,cc16_membered,cc17_membered,cc18_membered,cc19_membered,cc1_membered,cc20_membered,cc2_membered,cc3_membered,cc4_membered,cc4_tops_1,cc5_tops_1,cc6_tops_1,fc27_membered,fc28_membered,fc29_membered,fc30_membered,fc31_membered,fc32_membered,fc33_membered,fc34_membered,fc35_membered,fc36_membered,fc6_membered,rc1_membered,rc1_tops_1,rc2_tops_1,rc3_tops_1,rc4_tops_1,rc5_tops_1,rc6_pre_topc,rc7_pre_topc,t2_boole,existence_l1_struct_0,dt_l1_struct_0,cc15_membered,cc1_pre_topc,cc1_tops_1,cc2_tops_1,cc3_tops_1,fc1_struct_0,fc2_pre_topc,fc5_pre_topc,fc8_tops_1,rc1_subset_1,rc2_subset_1,rc3_struct_0,rc5_struct_0,t2_subset,t5_subset,t6_boole,t8_boole,commutativity_k3_xboole_0,idempotence_k3_xboole_0,reflexivity_r1_tarski,antisymmetry_r2_hidden,existence_l1_pre_topc,existence_m1_pre_topc,existence_m1_subset_1,dt_k1_zfmisc_1,dt_k2_pre_topc,dt_k3_xboole_0,dt_l1_pre_topc,dt_m1_pre_topc,dt_m1_subset_1,dt_u1_pre_topc,dt_u1_struct_0,dt_c1_3__topmetr,dt_c2_3__topmetr,dt_c5_3_1__topmetr,dt_c6_3_1__topmetr,dt_c7_3_1__topmetr,dt_c8_3_1__topmetr,de_c7_3_1__topmetr,de_c8_3_1__topmetr,fc1_subset_1,fc9_tops_1,t1_subset,t3_subset,t4_subset,t7_boole,d3_pre_topc,e8_3_1__topmetr,d9_pre_topc]), [interesting(0.65),file(topmetr,e9_3_1__topmetr),[file(topmetr,e9_3_1__topmetr)]]). fof(e10_3_1__topmetr,plain, ( v3_pre_topc(c7_3_1__topmetr,c2_3__topmetr) & v3_pre_topc(c8_3_1__topmetr,c2_3__topmetr) ), inference(mizar_by,[status(thm),assumptions([dt_c2_3__topmetr,dt_c1_3__topmetr,dt_c1_3_1__topmetr,dt_c2_3_1__topmetr,e1_3__topmetr,e1_3_1__topmetr])],[cc3_xreal_0,cc4_xreal_0,cc5_xreal_0,cc6_xreal_0,cc8_xreal_0,rc2_xreal_0,rc3_xreal_0,rc4_xreal_0,cc1_xreal_0,cc2_xreal_0,cc4_tops_1,cc5_tops_1,cc6_tops_1,cc7_xreal_0,rc1_xreal_0,rc2_tops_1,rc3_tops_1,rc5_tops_1,rc6_pre_topc,rc7_pre_topc,dt_k1_xboole_0,cc10_membered,cc11_membered,cc12_membered,cc13_membered,cc14_membered,cc16_membered,cc17_membered,cc18_membered,cc19_membered,cc1_membered,cc1_pre_topc,cc1_tops_1,cc20_membered,cc2_membered,cc3_membered,cc3_tops_1,cc4_membered,fc27_membered,fc28_membered,fc29_membered,fc30_membered,fc31_membered,fc32_membered,fc33_membered,fc34_membered,fc35_membered,fc36_membered,fc5_pre_topc,fc6_membered,fc8_tops_1,rc1_membered,rc1_tops_1,rc4_tops_1,t2_boole,commutativity_k3_xboole_0,idempotence_k3_xboole_0,reflexivity_r1_tarski,existence_l1_struct_0,existence_m1_pre_topc,dt_k2_pre_topc,dt_k3_xboole_0,dt_l1_struct_0,dt_m1_pre_topc,dt_c1_3__topmetr,dt_c5_3_1__topmetr,dt_c6_3_1__topmetr,cc15_membered,cc2_tops_1,fc1_struct_0,fc2_pre_topc,fc9_tops_1,rc1_subset_1,rc2_subset_1,rc3_struct_0,rc5_struct_0,t2_subset,t5_subset,t6_boole,t8_boole,d3_pre_topc,antisymmetry_r2_hidden,existence_l1_pre_topc,existence_m1_subset_1,dt_k1_zfmisc_1,dt_l1_pre_topc,dt_m1_subset_1,dt_u1_pre_topc,dt_u1_struct_0,dt_c2_3__topmetr,dt_c7_3_1__topmetr,dt_c8_3_1__topmetr,de_c7_3_1__topmetr,de_c8_3_1__topmetr,fc1_subset_1,t1_subset,t3_subset,t4_subset,t7_boole,e9_3_1__topmetr,d5_pre_topc]), [interesting(0.65),file(topmetr,e10_3_1__topmetr),[file(topmetr,e10_3_1__topmetr)]]). fof(d3_xboole_0,definition,( ! [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), [interesting(0.9),axiom,file(xboole_0,d3_xboole_0)]). fof(e11_3_1__topmetr,plain, ( r2_hidden(c1_3_1__topmetr,c7_3_1__topmetr) & r2_hidden(c2_3_1__topmetr,c8_3_1__topmetr) ), inference(mizar_by,[status(thm),assumptions([dt_c2_3__topmetr,dt_c1_3__topmetr,dt_c1_3_1__topmetr,dt_c2_3_1__topmetr,e1_3__topmetr,e1_3_1__topmetr])],[cc3_xreal_0,cc4_xreal_0,cc5_xreal_0,cc6_xreal_0,cc8_xreal_0,rc2_xreal_0,rc3_xreal_0,rc4_xreal_0,cc1_xreal_0,cc2_xreal_0,cc7_xreal_0,rc1_xreal_0,reflexivity_r1_tarski,existence_l1_struct_0,existence_m1_pre_topc,dt_k1_xboole_0,dt_l1_struct_0,dt_m1_pre_topc,cc10_membered,cc11_membered,cc12_membered,cc13_membered,cc14_membered,cc16_membered,cc17_membered,cc18_membered,cc19_membered,cc1_membered,cc1_pre_topc,cc20_membered,cc2_membered,cc3_membered,cc4_membered,cc4_tops_1,cc5_tops_1,cc6_tops_1,fc1_struct_0,fc27_membered,fc28_membered,fc29_membered,fc2_pre_topc,fc30_membered,fc31_membered,fc32_membered,fc33_membered,fc34_membered,fc35_membered,fc36_membered,fc6_membered,rc1_membered,rc2_tops_1,rc3_struct_0,rc3_tops_1,rc4_tops_1,rc5_struct_0,rc5_tops_1,rc6_pre_topc,rc7_pre_topc,t2_boole,existence_l1_pre_topc,existence_m1_subset_1,dt_k1_zfmisc_1,dt_k2_pre_topc,dt_l1_pre_topc,dt_m1_subset_1,dt_u1_struct_0,dt_c2_3__topmetr,cc15_membered,cc1_tops_1,cc2_tops_1,cc3_tops_1,fc1_subset_1,fc5_pre_topc,fc8_tops_1,fc9_tops_1,rc1_subset_1,rc1_tops_1,rc2_subset_1,t2_subset,t3_subset,t4_subset,t5_subset,t6_boole,t8_boole,d3_pre_topc,commutativity_k3_xboole_0,idempotence_k3_xboole_0,symmetry_r1_xboole_0,antisymmetry_r2_hidden,dt_k3_xboole_0,dt_c1_3__topmetr,dt_c1_3_1__topmetr,dt_c2_3_1__topmetr,dt_c3_3_1__topmetr,dt_c4_3_1__topmetr,dt_c5_3_1__topmetr,dt_c6_3_1__topmetr,dt_c7_3_1__topmetr,dt_c8_3_1__topmetr,de_c3_3_1__topmetr,de_c4_3_1__topmetr,de_c7_3_1__topmetr,de_c8_3_1__topmetr,t1_subset,t7_boole,e4_3_1__topmetr,d3_xboole_0]), [interesting(0.65),file(topmetr,e11_3_1__topmetr),[file(topmetr,e11_3_1__topmetr)]]). fof(commutativity_k5_subset_1,theorem,( ! [A,B,C] : ( ( m1_subset_1(B,k1_zfmisc_1(A)) & m1_subset_1(C,k1_zfmisc_1(A)) ) => k5_subset_1(A,B,C) = k5_subset_1(A,C,B) ) ), file(subset_1,k5_subset_1), [interesting(0.9),axiom,file(subset_1,k5_subset_1)]). fof(idempotence_k5_subset_1,theorem,( ! [A,B,C] : ( ( m1_subset_1(B,k1_zfmisc_1(A)) & m1_subset_1(C,k1_zfmisc_1(A)) ) => k5_subset_1(A,B,B) = B ) ), file(subset_1,k5_subset_1), [interesting(0.9),axiom,file(subset_1,k5_subset_1)]). fof(redefinition_k5_subset_1,definition,( ! [A,B,C] : ( ( m1_subset_1(B,k1_zfmisc_1(A)) & m1_subset_1(C,k1_zfmisc_1(A)) ) => k5_subset_1(A,B,C) = k3_xboole_0(B,C) ) ), file(subset_1,k5_subset_1), [interesting(0.9),axiom,file(subset_1,k5_subset_1)]). fof(dt_k5_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(k5_subset_1(A,B,C),k1_zfmisc_1(A)) ) ), file(subset_1,k5_subset_1), [interesting(0.9),axiom,file(subset_1,k5_subset_1)]). fof(t16_xboole_1,theorem,( ! [A,B,C] : k3_xboole_0(k3_xboole_0(A,B),C) = k3_xboole_0(A,k3_xboole_0(B,C)) ), file(xboole_1,t16_xboole_1), [interesting(0.9),axiom,file(xboole_1,t16_xboole_1)]). fof(e1_3_1_1__topmetr,plain,( k5_subset_1(u1_struct_0(c2_3__topmetr),c7_3_1__topmetr,c8_3_1__topmetr) = k3_xboole_0(k3_xboole_0(c5_3_1__topmetr,k3_xboole_0(c6_3_1__topmetr,k2_pre_topc(c2_3__topmetr))),k2_pre_topc(c2_3__topmetr)) ), inference(mizar_by,[status(thm),assumptions([dt_c2_3__topmetr,dt_c1_3__topmetr,dt_c1_3_1__topmetr,dt_c2_3_1__topmetr,e1_3__topmetr,e1_3_1__topmetr])],[cc3_xreal_0,cc4_xreal_0,cc5_xreal_0,cc6_xreal_0,cc8_xreal_0,rc2_xreal_0,rc3_xreal_0,rc4_xreal_0,cc1_xreal_0,cc2_xreal_0,cc7_xreal_0,rc1_xreal_0,antisymmetry_r2_hidden,dt_k1_xboole_0,cc10_membered,cc11_membered,cc12_membered,cc13_membered,cc14_membered,cc16_membered,cc17_membered,cc18_membered,cc19_membered,cc1_membered,cc20_membered,cc2_membered,cc3_membered,cc4_membered,cc4_tops_1,cc5_tops_1,cc6_tops_1,fc27_membered,fc28_membered,fc29_membered,fc30_membered,fc31_membered,fc32_membered,fc33_membered,fc34_membered,fc35_membered,fc36_membered,fc6_membered,rc1_membered,rc1_tops_1,rc2_tops_1,rc3_tops_1,rc4_tops_1,rc5_tops_1,rc6_pre_topc,rc7_pre_topc,t1_subset,t2_boole,t4_subset,t5_subset,reflexivity_r1_tarski,existence_l1_pre_topc,dt_l1_pre_topc,cc15_membered,cc1_pre_topc,cc1_tops_1,cc2_tops_1,cc3_tops_1,fc5_pre_topc,fc8_tops_1,fc9_tops_1,rc1_subset_1,rc2_subset_1,rc5_struct_0,t2_subset,t6_boole,t7_boole,t8_boole,existence_l1_struct_0,existence_m1_pre_topc,existence_m1_subset_1,dt_k1_zfmisc_1,dt_l1_struct_0,dt_m1_pre_topc,dt_m1_subset_1,dt_c1_3__topmetr,fc1_struct_0,fc1_subset_1,fc2_pre_topc,rc3_struct_0,t3_subset,commutativity_k3_xboole_0,idempotence_k3_xboole_0,commutativity_k5_subset_1,idempotence_k5_subset_1,redefinition_k5_subset_1,dt_k2_pre_topc,dt_k3_xboole_0,dt_k5_subset_1,dt_u1_struct_0,dt_c2_3__topmetr,dt_c5_3_1__topmetr,dt_c6_3_1__topmetr,dt_c7_3_1__topmetr,dt_c8_3_1__topmetr,de_c7_3_1__topmetr,de_c8_3_1__topmetr,d3_pre_topc,t16_xboole_1]), [interesting(0.5),file(topmetr,e1_3_1_1__topmetr),[file(topmetr,e1_3_1_1__topmetr)]]). fof(d7_xboole_0,definition,( ! [A,B] : ( r1_xboole_0(A,B) <=> k3_xboole_0(A,B) = k1_xboole_0 ) ), file(xboole_0,d7_xboole_0), [interesting(0.9),axiom,file(xboole_0,d7_xboole_0)]). fof(e5_3_1__topmetr,plain,( k5_subset_1(u1_struct_0(c1_3__topmetr),c5_3_1__topmetr,c6_3_1__topmetr) = k1_xboole_0 ), inference(mizar_by,[status(thm),assumptions([dt_c2_3__topmetr,dt_c1_3__topmetr,dt_c1_3_1__topmetr,dt_c2_3_1__topmetr,e1_3__topmetr,e1_3_1__topmetr])],[existence_m1_pre_topc,dt_m1_pre_topc,cc1_pre_topc,cc3_xreal_0,cc4_xreal_0,cc5_xreal_0,cc6_xreal_0,cc8_xreal_0,rc2_xreal_0,rc3_xreal_0,rc4_xreal_0,reflexivity_r1_tarski,dt_c2_3__topmetr,cc1_xreal_0,cc2_xreal_0,cc4_tops_1,cc5_tops_1,cc6_tops_1,cc7_xreal_0,rc1_xreal_0,rc2_tops_1,rc3_tops_1,rc4_tops_1,rc5_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_c1_3_1__topmetr,dt_c2_3_1__topmetr,cc10_membered,cc11_membered,cc12_membered,cc13_membered,cc14_membered,cc15_membered,cc16_membered,cc17_membered,cc18_membered,cc19_membered,cc1_membered,cc1_tops_1,cc20_membered,cc2_membered,cc2_tops_1,cc3_membered,cc3_tops_1,cc4_membered,fc1_struct_0,fc1_subset_1,fc27_membered,fc28_membered,fc29_membered,fc30_membered,fc31_membered,fc32_membered,fc33_membered,fc34_membered,fc35_membered,fc36_membered,rc1_membered,rc1_subset_1,rc1_tops_1,rc2_subset_1,rc3_struct_0,rc5_struct_0,t2_subset,t3_subset,t4_subset,t5_subset,t8_boole,commutativity_k3_xboole_0,idempotence_k3_xboole_0,commutativity_k5_subset_1,idempotence_k5_subset_1,symmetry_r1_xboole_0,antisymmetry_r2_hidden,redefinition_k5_subset_1,dt_k1_xboole_0,dt_k3_xboole_0,dt_k5_subset_1,dt_u1_struct_0,dt_c1_3__topmetr,dt_c3_3_1__topmetr,dt_c4_3_1__topmetr,dt_c5_3_1__topmetr,dt_c6_3_1__topmetr,de_c3_3_1__topmetr,de_c4_3_1__topmetr,fc6_membered,t1_subset,t2_boole,t6_boole,t7_boole,e4_3_1__topmetr,d7_xboole_0]), [interesting(0.65),file(topmetr,e5_3_1__topmetr),[file(topmetr,e5_3_1__topmetr)]]). fof(e2_3_1_1__topmetr,plain,( k3_xboole_0(k3_xboole_0(c5_3_1__topmetr,k3_xboole_0(c6_3_1__topmetr,k2_pre_topc(c2_3__topmetr))),k2_pre_topc(c2_3__topmetr)) = k3_xboole_0(k1_xboole_0,k2_pre_topc(c2_3__topmetr)) ), inference(mizar_by,[status(thm),assumptions([dt_c2_3__topmetr,dt_c1_3__topmetr,dt_c1_3_1__topmetr,dt_c2_3_1__topmetr,e1_3__topmetr,e1_3_1__topmetr])],[cc3_xreal_0,cc4_xreal_0,cc5_xreal_0,cc6_xreal_0,cc8_xreal_0,rc2_xreal_0,rc3_xreal_0,rc4_xreal_0,reflexivity_r1_tarski,antisymmetry_r2_hidden,cc1_xreal_0,cc2_xreal_0,cc4_tops_1,cc5_tops_1,cc6_tops_1,cc7_xreal_0,rc1_tops_1,rc1_xreal_0,rc2_tops_1,rc3_tops_1,rc4_tops_1,rc5_tops_1,rc6_pre_topc,rc7_pre_topc,t1_subset,t4_subset,t5_subset,existence_l1_pre_topc,existence_l1_struct_0,existence_m1_pre_topc,existence_m1_subset_1,dt_k1_zfmisc_1,dt_l1_pre_topc,dt_l1_struct_0,dt_m1_pre_topc,dt_m1_subset_1,cc10_membered,cc11_membered,cc12_membered,cc13_membered,cc14_membered,cc15_membered,cc16_membered,cc17_membered,cc18_membered,cc19_membered,cc1_membered,cc1_pre_topc,cc1_tops_1,cc20_membered,cc2_membered,cc2_tops_1,cc3_membered,cc3_tops_1,cc4_membered,fc1_struct_0,fc1_subset_1,fc27_membered,fc28_membered,fc29_membered,fc2_pre_topc,fc30_membered,fc31_membered,fc32_membered,fc33_membered,fc34_membered,fc35_membered,fc36_membered,fc5_pre_topc,fc8_tops_1,fc9_tops_1,rc1_membered,rc1_subset_1,rc2_subset_1,rc3_struct_0,rc5_struct_0,t2_subset,t3_subset,t7_boole,t8_boole,commutativity_k3_xboole_0,idempotence_k3_xboole_0,commutativity_k5_subset_1,idempotence_k5_subset_1,redefinition_k5_subset_1,dt_k1_xboole_0,dt_k2_pre_topc,dt_k3_xboole_0,dt_k5_subset_1,dt_u1_struct_0,dt_c1_3__topmetr,dt_c2_3__topmetr,dt_c5_3_1__topmetr,dt_c6_3_1__topmetr,fc6_membered,t2_boole,t6_boole,d3_pre_topc,e5_3_1__topmetr,t16_xboole_1]), [interesting(0.5),file(topmetr,e2_3_1_1__topmetr),[file(topmetr,e2_3_1_1__topmetr)]]). fof(e3_3_1_1__topmetr,plain,( k3_xboole_0(k1_xboole_0,k2_pre_topc(c2_3__topmetr)) = k1_xboole_0 ), inference(mizar_by,[status(thm),assumptions([dt_c1_3__topmetr,dt_c2_3__topmetr])],[cc3_xreal_0,cc4_tops_1,cc4_xreal_0,cc5_tops_1,cc5_xreal_0,cc6_tops_1,cc6_xreal_0,cc8_xreal_0,rc1_tops_1,rc2_tops_1,rc2_xreal_0,rc3_tops_1,rc3_xreal_0,rc4_tops_1,rc4_xreal_0,rc5_tops_1,rc6_pre_topc,rc7_pre_topc,reflexivity_r1_tarski,antisymmetry_r2_hidden,existence_l1_pre_topc,dt_l1_pre_topc,cc1_pre_topc,cc1_tops_1,cc1_xreal_0,cc2_tops_1,cc2_xreal_0,cc3_tops_1,cc7_xreal_0,fc5_pre_topc,fc8_tops_1,fc9_tops_1,rc1_xreal_0,t1_subset,t4_subset,t5_subset,existence_l1_struct_0,existence_m1_pre_topc,existence_m1_subset_1,dt_k1_zfmisc_1,dt_l1_struct_0,dt_m1_pre_topc,dt_m1_subset_1,dt_u1_struct_0,dt_c1_3__topmetr,cc10_membered,cc11_membered,cc12_membered,cc13_membered,cc14_membered,cc15_membered,cc16_membered,cc17_membered,cc18_membered,cc19_membered,cc1_membered,cc20_membered,cc2_membered,cc3_membered,cc4_membered,fc1_struct_0,fc1_subset_1,fc27_membered,fc28_membered,fc29_membered,fc2_pre_topc,fc30_membered,fc31_membered,fc32_membered,fc33_membered,fc34_membered,fc35_membered,fc36_membered,rc1_membered,rc1_subset_1,rc2_subset_1,rc3_struct_0,rc5_struct_0,t2_subset,t3_subset,t7_boole,t8_boole,commutativity_k3_xboole_0,idempotence_k3_xboole_0,dt_k1_xboole_0,dt_k2_pre_topc,dt_k3_xboole_0,dt_c2_3__topmetr,fc6_membered,t2_boole,t6_boole,d3_pre_topc]), [interesting(0.5),file(topmetr,e3_3_1_1__topmetr),[file(topmetr,e3_3_1_1__topmetr)]]). fof(e12_3_1__topmetr,plain,( k5_subset_1(u1_struct_0(c2_3__topmetr),c7_3_1__topmetr,c8_3_1__topmetr) = k1_xboole_0 ), inference(iterative_eq,[status(thm),assumptions([dt_c1_3_1__topmetr,dt_c2_3_1__topmetr,e1_3__topmetr,e1_3_1__topmetr,dt_c1_3__topmetr,dt_c2_3__topmetr])],[cc3_xreal_0,cc4_tops_1,cc4_xreal_0,cc5_tops_1,cc5_xreal_0,cc6_tops_1,cc6_xreal_0,cc8_xreal_0,rc1_tops_1,rc2_tops_1,rc2_xreal_0,rc3_tops_1,rc3_xreal_0,rc4_tops_1,rc4_xreal_0,rc5_tops_1,rc6_pre_topc,rc7_pre_topc,dt_l1_pre_topc,cc1_pre_topc,cc1_tops_1,cc1_xreal_0,cc2_tops_1,cc2_xreal_0,cc3_tops_1,cc7_xreal_0,fc5_pre_topc,fc8_tops_1,fc9_tops_1,rc1_xreal_0,dt_k1_zfmisc_1,dt_l1_struct_0,dt_m1_pre_topc,dt_m1_subset_1,dt_c1_3__topmetr,cc10_membered,cc11_membered,cc12_membered,cc13_membered,cc14_membered,cc15_membered,cc16_membered,cc17_membered,cc18_membered,cc19_membered,cc1_membered,cc20_membered,cc2_membered,cc3_membered,cc4_membered,fc1_struct_0,fc1_subset_1,fc27_membered,fc28_membered,fc29_membered,fc2_pre_topc,fc30_membered,fc31_membered,fc32_membered,fc33_membered,fc34_membered,fc35_membered,fc36_membered,rc1_membered,rc1_subset_1,rc2_subset_1,rc3_struct_0,rc5_struct_0,commutativity_k3_xboole_0,idempotence_k3_xboole_0,commutativity_k5_subset_1,idempotence_k5_subset_1,redefinition_k5_subset_1,dt_k1_xboole_0,dt_k2_pre_topc,dt_k3_xboole_0,dt_k5_subset_1,dt_u1_struct_0,dt_c2_3__topmetr,dt_c5_3_1__topmetr,dt_c6_3_1__topmetr,dt_c7_3_1__topmetr,dt_c8_3_1__topmetr,fc6_membered,e1_3_1_1__topmetr,e2_3_1_1__topmetr,e3_3_1_1__topmetr]), [interesting(0.65),file(topmetr,e12_3_1__topmetr),[file(topmetr,e12_3_1__topmetr)]]). fof(e13_3_1__topmetr,plain,( r1_xboole_0(c7_3_1__topmetr,c8_3_1__topmetr) ), inference(mizar_by,[status(thm),assumptions([dt_c1_3_1__topmetr,dt_c2_3_1__topmetr,e1_3__topmetr,e1_3_1__topmetr,dt_c1_3__topmetr,dt_c2_3__topmetr])],[cc3_xreal_0,cc4_tops_1,cc4_xreal_0,cc5_tops_1,cc5_xreal_0,cc6_tops_1,cc6_xreal_0,cc8_xreal_0,rc1_tops_1,rc2_tops_1,rc2_xreal_0,rc3_tops_1,rc3_xreal_0,rc4_tops_1,rc4_xreal_0,rc5_tops_1,rc6_pre_topc,rc7_pre_topc,reflexivity_r1_tarski,antisymmetry_r2_hidden,existence_l1_pre_topc,dt_l1_pre_topc,cc1_pre_topc,cc1_tops_1,cc1_xreal_0,cc2_tops_1,cc2_xreal_0,cc3_tops_1,cc7_xreal_0,fc5_pre_topc,fc8_tops_1,fc9_tops_1,rc1_xreal_0,t1_subset,t4_subset,t5_subset,existence_l1_struct_0,existence_m1_pre_topc,existence_m1_subset_1,dt_k1_zfmisc_1,dt_k2_pre_topc,dt_l1_struct_0,dt_m1_pre_topc,dt_m1_subset_1,dt_c1_3__topmetr,dt_c5_3_1__topmetr,dt_c6_3_1__topmetr,cc10_membered,cc11_membered,cc12_membered,cc13_membered,cc14_membered,cc15_membered,cc16_membered,cc17_membered,cc18_membered,cc19_membered,cc1_membered,cc20_membered,cc2_membered,cc3_membered,cc4_membered,fc1_struct_0,fc1_subset_1,fc27_membered,fc28_membered,fc29_membered,fc2_pre_topc,fc30_membered,fc31_membered,fc32_membered,fc33_membered,fc34_membered,fc35_membered,fc36_membered,rc1_membered,rc1_subset_1,rc2_subset_1,rc3_struct_0,rc5_struct_0,t2_subset,t3_subset,t7_boole,t8_boole,d3_pre_topc,commutativity_k3_xboole_0,idempotence_k3_xboole_0,commutativity_k5_subset_1,idempotence_k5_subset_1,symmetry_r1_xboole_0,redefinition_k5_subset_1,dt_k1_xboole_0,dt_k3_xboole_0,dt_k5_subset_1,dt_u1_struct_0,dt_c2_3__topmetr,dt_c7_3_1__topmetr,dt_c8_3_1__topmetr,de_c7_3_1__topmetr,de_c8_3_1__topmetr,fc6_membered,t2_boole,t6_boole,e12_3_1__topmetr,d7_xboole_0]), [interesting(0.65),file(topmetr,e13_3_1__topmetr),[file(topmetr,e13_3_1__topmetr)]]). fof(i7_3_1__topmetr,theorem,( $true ), introduced(tautology,[file(topmetr,i7_3_1__topmetr)]), [interesting(0.65),trivial,file(topmetr,i7_3_1__topmetr)]). fof(i6_3_1__topmetr,plain,( r1_xboole_0(c7_3_1__topmetr,c8_3_1__topmetr) ), inference(conclusion,[status(thm),assumptions([dt_c1_3_1__topmetr,dt_c2_3_1__topmetr,e1_3__topmetr,e1_3_1__topmetr,dt_c1_3__topmetr,dt_c2_3__topmetr])],[e13_3_1__topmetr,i7_3_1__topmetr]), [interesting(0.65),file(topmetr,i6_3_1__topmetr),[file(topmetr,i6_3_1__topmetr)]]). fof(i5_3_1__topmetr,plain, ( r2_hidden(c1_3_1__topmetr,c7_3_1__topmetr) & r2_hidden(c2_3_1__topmetr,c8_3_1__topmetr) & r1_xboole_0(c7_3_1__topmetr,c8_3_1__topmetr) ), inference(conclusion,[status(thm),assumptions([dt_c1_3_1__topmetr,dt_c2_3_1__topmetr,e1_3__topmetr,e1_3_1__topmetr,dt_c1_3__topmetr,dt_c2_3__topmetr])],[e11_3_1__topmetr,i6_3_1__topmetr]), [interesting(0.65),file(topmetr,i5_3_1__topmetr),[file(topmetr,i5_3_1__topmetr)]]). fof(i4_3_1__topmetr,plain, ( v3_pre_topc(c7_3_1__topmetr,c2_3__topmetr) & v3_pre_topc(c8_3_1__topmetr,c2_3__topmetr) & r2_hidden(c1_3_1__topmetr,c7_3_1__topmetr) & r2_hidden(c2_3_1__topmetr,c8_3_1__topmetr) & r1_xboole_0(c7_3_1__topmetr,c8_3_1__topmetr) ), inference(conclusion,[status(thm),assumptions([dt_c1_3_1__topmetr,dt_c2_3_1__topmetr,e1_3__topmetr,e1_3_1__topmetr,dt_c1_3__topmetr,dt_c2_3__topmetr])],[e10_3_1__topmetr,i5_3_1__topmetr]), [interesting(0.65),file(topmetr,i4_3_1__topmetr),[file(topmetr,i4_3_1__topmetr)]]). fof(i3_3_1__topmetr,plain,( ? [A] : ( m1_subset_1(A,k1_zfmisc_1(u1_struct_0(c2_3__topmetr))) & v3_pre_topc(c7_3_1__topmetr,c2_3__topmetr) & v3_pre_topc(A,c2_3__topmetr) & r2_hidden(c1_3_1__topmetr,c7_3_1__topmetr) & r2_hidden(c2_3_1__topmetr,A) & r1_xboole_0(c7_3_1__topmetr,A) ) ), inference(take,[status(thm),assumptions([dt_c1_3_1__topmetr,dt_c2_3_1__topmetr,e1_3__topmetr,e1_3_1__topmetr,dt_c1_3__topmetr,dt_c2_3__topmetr])],[cc3_xreal_0,cc4_xreal_0,cc5_xreal_0,cc6_xreal_0,cc8_xreal_0,rc2_xreal_0,rc3_xreal_0,rc4_xreal_0,cc1_xreal_0,cc2_xreal_0,cc4_tops_1,cc5_tops_1,cc6_tops_1,cc7_xreal_0,rc1_xreal_0,rc2_tops_1,rc3_tops_1,rc4_tops_1,rc5_tops_1,rc6_pre_topc,rc7_pre_topc,dt_l1_pre_topc,cc10_membered,cc11_membered,cc12_membered,cc13_membered,cc14_membered,cc16_membered,cc17_membered,cc18_membered,cc19_membered,cc1_membered,cc1_pre_topc,cc1_tops_1,cc20_membered,cc2_membered,cc2_tops_1,cc3_membered,cc3_tops_1,cc4_membered,rc1_membered,rc1_tops_1,dt_l1_struct_0,dt_m1_pre_topc,dt_c1_3__topmetr,cc15_membered,fc1_struct_0,rc1_subset_1,rc2_subset_1,rc3_struct_0,rc5_struct_0,symmetry_r1_xboole_0,antisymmetry_r2_hidden,dt_k1_zfmisc_1,dt_m1_subset_1,dt_u1_struct_0,dt_c1_3_1__topmetr,dt_c2_3__topmetr,dt_c2_3_1__topmetr,dt_c7_3_1__topmetr,dt_c8_3_1__topmetr,fc1_subset_1,i4_3_1__topmetr]), [interesting(0.65),file(topmetr,i3_3_1__topmetr),[file(topmetr,i3_3_1__topmetr)]]). fof(i2_3_1__topmetr,plain,( ? [A] : ( m1_subset_1(A,k1_zfmisc_1(u1_struct_0(c2_3__topmetr))) & ? [B] : ( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(c2_3__topmetr))) & v3_pre_topc(A,c2_3__topmetr) & v3_pre_topc(B,c2_3__topmetr) & r2_hidden(c1_3_1__topmetr,A) & r2_hidden(c2_3_1__topmetr,B) & r1_xboole_0(A,B) ) ) ), inference(take,[status(thm),assumptions([dt_c1_3_1__topmetr,dt_c2_3_1__topmetr,e1_3__topmetr,e1_3_1__topmetr,dt_c1_3__topmetr,dt_c2_3__topmetr])],[cc3_xreal_0,cc4_xreal_0,cc5_xreal_0,cc6_xreal_0,cc8_xreal_0,rc2_xreal_0,rc3_xreal_0,rc4_xreal_0,cc1_xreal_0,cc2_xreal_0,cc4_tops_1,cc5_tops_1,cc6_tops_1,cc7_xreal_0,rc1_xreal_0,rc2_tops_1,rc3_tops_1,rc4_tops_1,rc5_tops_1,rc6_pre_topc,rc7_pre_topc,dt_l1_pre_topc,cc10_membered,cc11_membered,cc12_membered,cc13_membered,cc14_membered,cc16_membered,cc17_membered,cc18_membered,cc19_membered,cc1_membered,cc1_pre_topc,cc1_tops_1,cc20_membered,cc2_membered,cc2_tops_1,cc3_membered,cc3_tops_1,cc4_membered,rc1_membered,rc1_tops_1,dt_l1_struct_0,dt_m1_pre_topc,dt_c1_3__topmetr,cc15_membered,fc1_struct_0,rc1_subset_1,rc2_subset_1,rc3_struct_0,rc5_struct_0,symmetry_r1_xboole_0,antisymmetry_r2_hidden,dt_k1_zfmisc_1,dt_m1_subset_1,dt_u1_struct_0,dt_c1_3_1__topmetr,dt_c2_3__topmetr,dt_c2_3_1__topmetr,dt_c7_3_1__topmetr,fc1_subset_1,i3_3_1__topmetr]), [interesting(0.65),file(topmetr,i2_3_1__topmetr),[file(topmetr,i2_3_1__topmetr)]]). fof(i1_3_1__topmetr,plain,( ~ ( c1_3_1__topmetr != c2_3_1__topmetr & ! [A] : ( m1_subset_1(A,k1_zfmisc_1(u1_struct_0(c2_3__topmetr))) => ! [B] : ( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(c2_3__topmetr))) => ~ ( v3_pre_topc(A,c2_3__topmetr) & v3_pre_topc(B,c2_3__topmetr) & r2_hidden(c1_3_1__topmetr,A) & r2_hidden(c2_3_1__topmetr,B) & r1_xboole_0(A,B) ) ) ) ) ), inference(discharge_asm,[status(thm),assumptions([dt_c1_3_1__topmetr,dt_c2_3_1__topmetr,e1_3__topmetr,dt_c1_3__topmetr,dt_c2_3__topmetr]),discharge_asm(discharge,[e1_3_1__topmetr])],[e1_3_1__topmetr,i2_3_1__topmetr]), [interesting(0.65),file(topmetr,i1_3_1__topmetr),[file(topmetr,i1_3_1__topmetr)]]). fof(i1_3_1_tmp__topmetr,plain, ( ( m1_subset_1(c1_3_1__topmetr,u1_struct_0(c2_3__topmetr)) & m1_subset_1(c2_3_1__topmetr,u1_struct_0(c2_3__topmetr)) ) => ~ ( c1_3_1__topmetr != c2_3_1__topmetr & ! [A] : ( m1_subset_1(A,k1_zfmisc_1(u1_struct_0(c2_3__topmetr))) => ! [B] : ( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(c2_3__topmetr))) => ~ ( v3_pre_topc(A,c2_3__topmetr) & v3_pre_topc(B,c2_3__topmetr) & r2_hidden(c1_3_1__topmetr,A) & r2_hidden(c2_3_1__topmetr,B) & r1_xboole_0(A,B) ) ) ) ) ), inference(discharge_asm,[status(thm),assumptions([e1_3__topmetr,dt_c1_3__topmetr,dt_c2_3__topmetr]),discharge_asm(discharge,[dt_c1_3_1__topmetr,dt_c2_3_1__topmetr])],[dt_c1_3_1__topmetr,dt_c2_3_1__topmetr,i1_3_1__topmetr]), [interesting(0.8),e2_3__topmetr]). fof(e2_3__topmetr,plain,( ! [A] : ( m1_subset_1(A,u1_struct_0(c2_3__topmetr)) => ! [B] : ( m1_subset_1(B,u1_struct_0(c2_3__topmetr)) => ~ ( A != B & ! [C] : ( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(c2_3__topmetr))) => ! [D] : ( m1_subset_1(D,k1_zfmisc_1(u1_struct_0(c2_3__topmetr))) => ~ ( v3_pre_topc(C,c2_3__topmetr) & v3_pre_topc(D,c2_3__topmetr) & r2_hidden(A,C) & r2_hidden(B,D) & r1_xboole_0(C,D) ) ) ) ) ) ) ), inference(let,[status(thm),assumptions([e1_3__topmetr,dt_c1_3__topmetr,dt_c2_3__topmetr])],[i1_3_1_tmp__topmetr,dh_c1_3_1__topmetr,dh_c2_3_1__topmetr]), [interesting(0.8),file(topmetr,e2_3__topmetr),[file(topmetr,e2_3__topmetr)]]). fof(e3_3__topmetr,plain,( v3_compts_1(c2_3__topmetr) ), inference(mizar_by,[status(thm),assumptions([e1_3__topmetr,dt_c1_3__topmetr,dt_c2_3__topmetr])],[cc3_xreal_0,cc4_xreal_0,cc5_xreal_0,cc6_xreal_0,cc8_xreal_0,rc2_xreal_0,rc3_xreal_0,rc4_xreal_0,cc1_xreal_0,cc2_xreal_0,cc4_tops_1,cc5_tops_1,cc6_tops_1,cc7_xreal_0,rc1_xreal_0,rc2_tops_1,rc3_tops_1,rc5_tops_1,rc6_pre_topc,rc7_pre_topc,dt_k1_xboole_0,cc10_membered,cc11_membered,cc12_membered,cc13_membered,cc14_membered,cc16_membered,cc17_membered,cc18_membered,cc19_membered,cc1_membered,cc1_pre_topc,cc1_tops_1,cc20_membered,cc2_membered,cc3_membered,cc3_tops_1,cc4_membered,fc6_membered,rc1_membered,rc1_tops_1,rc4_tops_1,reflexivity_r1_tarski,existence_l1_struct_0,existence_m1_pre_topc,dt_l1_struct_0,dt_m1_pre_topc,dt_c1_3__topmetr,cc15_membered,cc2_tops_1,fc1_struct_0,rc1_subset_1,rc2_subset_1,rc3_struct_0,rc5_struct_0,t2_subset,t5_subset,t6_boole,t8_boole,symmetry_r1_xboole_0,antisymmetry_r2_hidden,existence_l1_pre_topc,existence_m1_subset_1,dt_k1_zfmisc_1,dt_l1_pre_topc,dt_m1_subset_1,dt_u1_struct_0,dt_c2_3__topmetr,fc1_subset_1,t1_subset,t3_subset,t4_subset,t7_boole,e2_3__topmetr,d4_compts_1]), [interesting(0.8),file(topmetr,e3_3__topmetr),[file(topmetr,e3_3__topmetr)]]). fof(i4_3__topmetr,theorem,( $true ), introduced(tautology,[file(topmetr,i4_3__topmetr)]), [interesting(0.8),trivial,file(topmetr,i4_3__topmetr)]). fof(i3_3__topmetr,plain,( v3_compts_1(c2_3__topmetr) ), inference(conclusion,[status(thm),assumptions([e1_3__topmetr,dt_c1_3__topmetr,dt_c2_3__topmetr])],[e3_3__topmetr,i4_3__topmetr]), [interesting(0.8),file(topmetr,i3_3__topmetr),[file(topmetr,i3_3__topmetr)]]). fof(i2_3__topmetr,plain, ( v3_compts_1(c1_3__topmetr) => v3_compts_1(c2_3__topmetr) ), inference(discharge_asm,[status(thm),assumptions([dt_c1_3__topmetr,dt_c2_3__topmetr]),discharge_asm(discharge,[e1_3__topmetr])],[e1_3__topmetr,i3_3__topmetr]), [interesting(0.8),file(topmetr,i2_3__topmetr),[file(topmetr,i2_3__topmetr)]]). fof(i2_3_tmp__topmetr,plain, ( ( ~ v3_struct_0(c2_3__topmetr) & m1_pre_topc(c2_3__topmetr,c1_3__topmetr) ) => ( v3_compts_1(c1_3__topmetr) => v3_compts_1(c2_3__topmetr) ) ), inference(discharge_asm,[status(thm),assumptions([dt_c1_3__topmetr]),discharge_asm(discharge,[dt_c2_3__topmetr])],[dt_c2_3__topmetr,i2_3__topmetr]), [interesting(0.8),i1_3__topmetr]). fof(i1_3__topmetr,plain,( ! [A] : ( ( ~ v3_struct_0(A) & m1_pre_topc(A,c1_3__topmetr) ) => ( v3_compts_1(c1_3__topmetr) => v3_compts_1(A) ) ) ), inference(let,[status(thm),assumptions([dt_c1_3__topmetr])],[i2_3_tmp__topmetr,dh_c2_3__topmetr]), [interesting(0.8),file(topmetr,i1_3__topmetr),[file(topmetr,i1_3__topmetr)]]). fof(i1_3_tmp__topmetr,plain, ( ( ~ v3_struct_0(c1_3__topmetr) & v2_pre_topc(c1_3__topmetr) & l1_pre_topc(c1_3__topmetr) ) => ! [A] : ( ( ~ v3_struct_0(A) & m1_pre_topc(A,c1_3__topmetr) ) => ( v3_compts_1(c1_3__topmetr) => v3_compts_1(A) ) ) ), inference(discharge_asm,[status(thm),assumptions([]),discharge_asm(discharge,[dt_c1_3__topmetr])],[dt_c1_3__topmetr,i1_3__topmetr]), [interesting(1),t3_topmetr]). fof(t3_topmetr,theorem,( ! [A] : ( ( ~ v3_struct_0(A) & v2_pre_topc(A) & l1_pre_topc(A) ) => ! [B] : ( ( ~ v3_struct_0(B) & m1_pre_topc(B,A) ) => ( v3_compts_1(A) => v3_compts_1(B) ) ) ) ), inference(let,[status(thm),assumptions([])],[i1_3_tmp__topmetr,dh_c1_3__topmetr]), [interesting(1),file(topmetr,t3_topmetr),[file(topmetr,t3_topmetr)]]).