% Mizar ND problem: t2_topmetr,topmetr,61,52 
fof(dh_c1_2__topmetr,definition,
    ( ( ( ~ v3_struct_0(c1_2__topmetr)
        & v2_pre_topc(c1_2__topmetr)
        & l1_pre_topc(c1_2__topmetr) )
     => ! [A] : 
          ( ( ~ v3_struct_0(A)
            & m1_pre_topc(A,c1_2__topmetr) )
         => ! [B] : 
              ( m1_subset_1(B,u1_struct_0(A))
             => m1_subset_1(B,u1_struct_0(c1_2__topmetr)) ) ) )
   => ! [C] : 
        ( ( ~ v3_struct_0(C)
          & v2_pre_topc(C)
          & l1_pre_topc(C) )
       => ! [D] : 
            ( ( ~ v3_struct_0(D)
              & m1_pre_topc(D,C) )
           => ! [E] : 
                ( m1_subset_1(E,u1_struct_0(D))
               => m1_subset_1(E,u1_struct_0(C)) ) ) ) ),
    introduced(definition,[new_symbol(c1_2__topmetr),file(topmetr,c1_2__topmetr)]),
    [interesting(0.8),axiom,file(topmetr,c1_2__topmetr)]).

fof(dh_c2_2__topmetr,definition,
    ( ( ( ~ v3_struct_0(c2_2__topmetr)
        & m1_pre_topc(c2_2__topmetr,c1_2__topmetr) )
     => ! [A] : 
          ( m1_subset_1(A,u1_struct_0(c2_2__topmetr))
         => m1_subset_1(A,u1_struct_0(c1_2__topmetr)) ) )
   => ! [B] : 
        ( ( ~ v3_struct_0(B)
          & m1_pre_topc(B,c1_2__topmetr) )
       => ! [C] : 
            ( m1_subset_1(C,u1_struct_0(B))
           => m1_subset_1(C,u1_struct_0(c1_2__topmetr)) ) ) ),
    introduced(definition,[new_symbol(c2_2__topmetr),file(topmetr,c2_2__topmetr)]),
    [interesting(0.8),axiom,file(topmetr,c2_2__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(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(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(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(cc4_membered,theorem,(
    ! [A] : 
      ( v2_membered(A)
     => v1_membered(A) ) ),
    file(membered,cc4_membered),
    [interesting(0.9),axiom,file(membered,cc4_membered)]).

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(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(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(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(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(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(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(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_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(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(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(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(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_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(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(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(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(reflexivity_r1_tarski,theorem,(
    ! [A,B] : r1_tarski(A,A) ),
    file(tarski,r1_tarski),
    [interesting(0.9),axiom,file(tarski,r1_tarski)]).

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_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_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_c1_2__topmetr,assumption,
    ( ~ v3_struct_0(c1_2__topmetr)
    & v2_pre_topc(c1_2__topmetr)
    & l1_pre_topc(c1_2__topmetr) ),
    introduced(assumption,[file(topmetr,c1_2__topmetr)]),
    [interesting(0.8),axiom,file(topmetr,c1_2__topmetr)]).

fof(dt_c2_2__topmetr,assumption,
    ( ~ v3_struct_0(c2_2__topmetr)
    & m1_pre_topc(c2_2__topmetr,c1_2__topmetr) ),
    introduced(assumption,[file(topmetr,c2_2__topmetr)]),
    [interesting(0.8),axiom,file(topmetr,c2_2__topmetr)]).

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(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(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(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(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(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(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(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(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_k3_xboole_0,axiom,(
    $true ),
    file(xboole_0,k3_xboole_0),
    [interesting(0.9),axiom,file(xboole_0,k3_xboole_0)]).

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(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(e1_2__topmetr,plain,(
    r1_tarski(k2_pre_topc(c2_2__topmetr),k2_pre_topc(c1_2__topmetr)) ),
    inference(mizar_by,[status(thm),assumptions([dt_c1_2__topmetr,dt_c2_2__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_2__topmetr,dt_c2_2__topmetr,fc1_subset_1,fc9_tops_1,t1_subset,t3_subset,t4_subset,t7_boole,d3_pre_topc,d9_pre_topc]),
    [interesting(0.8),file(topmetr,e1_2__topmetr),[file(topmetr,e1_2__topmetr)]]).

fof(e2_2__topmetr,plain,(
    r1_tarski(u1_struct_0(c2_2__topmetr),k2_pre_topc(c1_2__topmetr)) ),
    inference(mizar_by,[status(thm),assumptions([dt_c1_2__topmetr,dt_c2_2__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,fc6_membered,rc1_membered,rc4_tops_1,rc5_tops_1,t1_subset,t4_subset,t5_subset,cc15_membered,cc1_tops_1,cc2_tops_1,cc3_tops_1,rc1_subset_1,rc1_tops_1,rc2_subset_1,rc2_tops_1,rc3_tops_1,rc5_struct_0,rc6_pre_topc,rc7_pre_topc,t2_subset,t6_boole,t7_boole,t8_boole,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,cc1_pre_topc,fc1_struct_0,fc1_subset_1,fc2_pre_topc,fc5_pre_topc,fc8_tops_1,fc9_tops_1,rc3_struct_0,reflexivity_r1_tarski,dt_k2_pre_topc,dt_u1_struct_0,dt_c1_2__topmetr,dt_c2_2__topmetr,t3_subset,d3_pre_topc,e1_2__topmetr]),
    [interesting(0.8),file(topmetr,e2_2__topmetr),[file(topmetr,e2_2__topmetr)]]).

fof(e3_2__topmetr,plain,(
    r1_tarski(u1_struct_0(c2_2__topmetr),u1_struct_0(c1_2__topmetr)) ),
    inference(mizar_by,[status(thm),assumptions([dt_c1_2__topmetr,dt_c2_2__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,fc6_membered,rc1_membered,rc4_tops_1,rc5_tops_1,t1_subset,t4_subset,t5_subset,cc15_membered,cc1_tops_1,cc2_tops_1,cc3_tops_1,rc1_subset_1,rc1_tops_1,rc2_subset_1,rc2_tops_1,rc3_tops_1,rc5_struct_0,rc6_pre_topc,rc7_pre_topc,t2_subset,t6_boole,t7_boole,t8_boole,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,cc1_pre_topc,fc1_struct_0,fc1_subset_1,fc2_pre_topc,fc5_pre_topc,fc8_tops_1,fc9_tops_1,rc3_struct_0,reflexivity_r1_tarski,dt_k2_pre_topc,dt_u1_struct_0,dt_c1_2__topmetr,dt_c2_2__topmetr,t3_subset,d3_pre_topc,e2_2__topmetr]),
    [interesting(0.8),file(topmetr,e3_2__topmetr),[file(topmetr,e3_2__topmetr)]]).

fof(d3_tarski,definition,(
    ! [A,B] : 
      ( r1_tarski(A,B)
    <=> ! [C] : 
          ( r2_hidden(C,A)
         => r2_hidden(C,B) ) ) ),
    file(tarski,d3_tarski),
    [interesting(0.9),axiom,file(tarski,d3_tarski)]).

fof(e4_2__topmetr,plain,(
    ! [A] : 
      ( m1_subset_1(A,u1_struct_0(c2_2__topmetr))
     => m1_subset_1(A,u1_struct_0(c1_2__topmetr)) ) ),
    inference(mizar_by,[status(thm),assumptions([dt_c1_2__topmetr,dt_c2_2__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,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,t8_boole,existence_l1_pre_topc,existence_l1_struct_0,existence_m1_pre_topc,dt_k1_zfmisc_1,dt_l1_pre_topc,dt_l1_struct_0,dt_m1_pre_topc,cc15_membered,cc1_pre_topc,cc1_tops_1,cc2_tops_1,cc3_tops_1,fc1_struct_0,fc1_subset_1,rc1_subset_1,rc2_subset_1,rc3_struct_0,rc5_struct_0,t2_subset,t4_subset,t5_subset,t6_boole,reflexivity_r1_tarski,antisymmetry_r2_hidden,existence_m1_subset_1,dt_m1_subset_1,dt_u1_struct_0,dt_c1_2__topmetr,dt_c2_2__topmetr,t1_subset,t3_subset,t7_boole,e3_2__topmetr,d3_tarski]),
    [interesting(0.8),file(topmetr,e4_2__topmetr),[file(topmetr,e4_2__topmetr)]]).

fof(i3_2__topmetr,theorem,(
    $true ),
    introduced(tautology,[file(topmetr,i3_2__topmetr)]),
    [interesting(0.8),trivial,file(topmetr,i3_2__topmetr)]).

fof(i2_2__topmetr,plain,(
    ! [A] : 
      ( m1_subset_1(A,u1_struct_0(c2_2__topmetr))
     => m1_subset_1(A,u1_struct_0(c1_2__topmetr)) ) ),
    inference(conclusion,[status(thm),assumptions([dt_c1_2__topmetr,dt_c2_2__topmetr])],[e4_2__topmetr,i3_2__topmetr]),
    [interesting(0.8),file(topmetr,i2_2__topmetr),[file(topmetr,i2_2__topmetr)]]).

fof(i2_2_tmp__topmetr,plain,
    ( ( ~ v3_struct_0(c2_2__topmetr)
      & m1_pre_topc(c2_2__topmetr,c1_2__topmetr) )
   => ! [A] : 
        ( m1_subset_1(A,u1_struct_0(c2_2__topmetr))
       => m1_subset_1(A,u1_struct_0(c1_2__topmetr)) ) ),
    inference(discharge_asm,[status(thm),assumptions([dt_c1_2__topmetr]),discharge_asm(discharge,[dt_c2_2__topmetr])],[dt_c2_2__topmetr,i2_2__topmetr]),
    [interesting(0.8),i1_2__topmetr]).

fof(i1_2__topmetr,plain,(
    ! [A] : 
      ( ( ~ v3_struct_0(A)
        & m1_pre_topc(A,c1_2__topmetr) )
     => ! [B] : 
          ( m1_subset_1(B,u1_struct_0(A))
         => m1_subset_1(B,u1_struct_0(c1_2__topmetr)) ) ) ),
    inference(let,[status(thm),assumptions([dt_c1_2__topmetr])],[i2_2_tmp__topmetr,dh_c2_2__topmetr]),
    [interesting(0.8),file(topmetr,i1_2__topmetr),[file(topmetr,i1_2__topmetr)]]).

fof(i1_2_tmp__topmetr,plain,
    ( ( ~ v3_struct_0(c1_2__topmetr)
      & v2_pre_topc(c1_2__topmetr)
      & l1_pre_topc(c1_2__topmetr) )
   => ! [A] : 
        ( ( ~ v3_struct_0(A)
          & m1_pre_topc(A,c1_2__topmetr) )
       => ! [B] : 
            ( m1_subset_1(B,u1_struct_0(A))
           => m1_subset_1(B,u1_struct_0(c1_2__topmetr)) ) ) ),
    inference(discharge_asm,[status(thm),assumptions([]),discharge_asm(discharge,[dt_c1_2__topmetr])],[dt_c1_2__topmetr,i1_2__topmetr]),
    [interesting(1),t2_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)) ) ) ) ),
    inference(let,[status(thm),assumptions([])],[i1_2_tmp__topmetr,dh_c1_2__topmetr]),
    [interesting(1),file(topmetr,t2_topmetr),[file(topmetr,t2_topmetr)]]).