% Mizar problem: t4_goboard9,goboard9,64,55 
fof(t4_goboard9,conjecture,(
    ! [A] : 
      ( ( v2_pre_topc(A)
        & l1_pre_topc(A) )
     => ! [B] : 
          ( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
         => ! [C] : 
              ( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A)))
             => ! [D] : 
                  ( m1_subset_1(D,k1_zfmisc_1(u1_struct_0(k3_pre_topc(A,C))))
                 => ( B = D
                   => k3_pre_topc(A,B) = k3_pre_topc(k3_pre_topc(A,C),D) ) ) ) ) ) ),
    inference(mizar_bg_added,[status(thm)],[abstractness_v1_pre_topc,cc1_pre_topc,d10_pre_topc,dt_g1_pre_topc,dt_k1_zfmisc_1,dt_k2_pre_topc,dt_k3_pre_topc,dt_l1_pre_topc,dt_l1_struct_0,dt_m1_pre_topc,dt_m1_subset_1,dt_u1_pre_topc,dt_u1_struct_0,existence_l1_pre_topc,existence_l1_struct_0,existence_m1_pre_topc,existence_m1_subset_1,fc4_pre_topc,fc5_pre_topc,free_g1_pre_topc,rc1_pre_topc,rc3_pre_topc,rc5_pre_topc,rc6_pre_topc,reflexivity_r1_tarski,t12_pre_topc,t3_subset,t4_tsep_1]),
    [file(goboard9,t4_goboard9)]).

fof(abstractness_v1_pre_topc,axiom,(
    ! [A] : 
      ( l1_pre_topc(A)
     => ( v1_pre_topc(A)
       => A = g1_pre_topc(u1_struct_0(A),u1_pre_topc(A)) ) ) ),
    file(pre_topc,v1_pre_topc),
    []).

fof(cc1_pre_topc,axiom,(
    ! [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),
    []).

fof(d10_pre_topc,axiom,(
    ! [A] : 
      ( l1_pre_topc(A)
     => ! [B] : 
          ( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
         => ! [C] : 
              ( ( v1_pre_topc(C)
                & m1_pre_topc(C,A) )
             => ( C = k3_pre_topc(A,B)
              <=> k2_pre_topc(C) = B ) ) ) ) ),
    file(pre_topc,d10_pre_topc),
    []).

fof(dt_g1_pre_topc,axiom,(
    ! [A,B] : 
      ( m1_subset_1(B,k1_zfmisc_1(k1_zfmisc_1(A)))
     => ( v1_pre_topc(g1_pre_topc(A,B))
        & l1_pre_topc(g1_pre_topc(A,B)) ) ) ),
    file(pre_topc,g1_pre_topc),
    []).

fof(dt_k1_zfmisc_1,axiom,(
    $true ),
    file(zfmisc_1,k1_zfmisc_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),
    []).

fof(dt_k3_pre_topc,axiom,(
    ! [A,B] : 
      ( ( l1_pre_topc(A)
        & m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) )
     => ( v1_pre_topc(k3_pre_topc(A,B))
        & m1_pre_topc(k3_pre_topc(A,B),A) ) ) ),
    file(pre_topc,k3_pre_topc),
    []).

fof(dt_l1_pre_topc,axiom,(
    ! [A] : 
      ( l1_pre_topc(A)
     => l1_struct_0(A) ) ),
    file(pre_topc,l1_pre_topc),
    []).

fof(dt_l1_struct_0,axiom,(
    $true ),
    file(struct_0,l1_struct_0),
    []).

fof(dt_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),
    []).

fof(dt_m1_subset_1,axiom,(
    $true ),
    file(subset_1,m1_subset_1),
    []).

fof(dt_u1_pre_topc,axiom,(
    ! [A] : 
      ( l1_pre_topc(A)
     => m1_subset_1(u1_pre_topc(A),k1_zfmisc_1(k1_zfmisc_1(u1_struct_0(A)))) ) ),
    file(pre_topc,u1_pre_topc),
    []).

fof(dt_u1_struct_0,axiom,(
    $true ),
    file(struct_0,u1_struct_0),
    []).

fof(existence_l1_pre_topc,axiom,(
    ? [A] : l1_pre_topc(A) ),
    file(pre_topc,l1_pre_topc),
    []).

fof(existence_l1_struct_0,axiom,(
    ? [A] : l1_struct_0(A) ),
    file(struct_0,l1_struct_0),
    []).

fof(existence_m1_pre_topc,axiom,(
    ! [A] : 
      ( l1_pre_topc(A)
     => ? [B] : m1_pre_topc(B,A) ) ),
    file(pre_topc,m1_pre_topc),
    []).

fof(existence_m1_subset_1,axiom,(
    ! [A] : 
    ? [B] : m1_subset_1(B,A) ),
    file(subset_1,m1_subset_1),
    []).

fof(fc4_pre_topc,axiom,(
    ! [A,B] : 
      ( ( v2_pre_topc(A)
        & l1_pre_topc(A)
        & m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) )
     => ( v1_pre_topc(k3_pre_topc(A,B))
        & v2_pre_topc(k3_pre_topc(A,B)) ) ) ),
    file(pre_topc,fc4_pre_topc),
    []).

fof(fc5_pre_topc,axiom,(
    ! [A] : 
      ( ( v2_pre_topc(A)
        & l1_pre_topc(A) )
     => v4_pre_topc(k2_pre_topc(A),A) ) ),
    file(pre_topc,fc5_pre_topc),
    []).

fof(free_g1_pre_topc,axiom,(
    ! [A,B] : 
      ( m1_subset_1(B,k1_zfmisc_1(k1_zfmisc_1(A)))
     => ! [C,D] : 
          ( g1_pre_topc(A,B) = g1_pre_topc(C,D)
         => ( A = C
            & B = D ) ) ) ),
    file(pre_topc,g1_pre_topc),
    []).

fof(rc1_pre_topc,axiom,(
    ? [A] : 
      ( l1_pre_topc(A)
      & v1_pre_topc(A) ) ),
    file(pre_topc,rc1_pre_topc),
    []).

fof(rc3_pre_topc,axiom,(
    ! [A] : 
      ( l1_pre_topc(A)
     => ? [B] : 
          ( m1_pre_topc(B,A)
          & v1_pre_topc(B) ) ) ),
    file(pre_topc,rc3_pre_topc),
    []).

fof(rc5_pre_topc,axiom,(
    ! [A] : 
      ( ( v2_pre_topc(A)
        & l1_pre_topc(A) )
     => ? [B] : 
          ( m1_pre_topc(B,A)
          & v1_pre_topc(B)
          & v2_pre_topc(B) ) ) ),
    file(pre_topc,rc5_pre_topc),
    []).

fof(rc6_pre_topc,axiom,(
    ! [A] : 
      ( ( v2_pre_topc(A)
        & l1_pre_topc(A) )
     => ? [B] : 
          ( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
          & v4_pre_topc(B,A) ) ) ),
    file(pre_topc,rc6_pre_topc),
    []).

fof(reflexivity_r1_tarski,axiom,(
    ! [A,B] : r1_tarski(A,A) ),
    file(tarski,r1_tarski),
    []).

fof(t12_pre_topc,axiom,(
    ! [A] : 
      ( l1_struct_0(A)
     => k2_pre_topc(A) = u1_struct_0(A) ) ),
    file(pre_topc,t12_pre_topc),
    []).

fof(t3_subset,axiom,(
    ! [A,B] : 
      ( m1_subset_1(A,k1_zfmisc_1(B))
    <=> r1_tarski(A,B) ) ),
    file(subset,t3_subset),
    []).

fof(t4_tsep_1,axiom,(
    ! [A] : 
      ( ( v2_pre_topc(A)
        & l1_pre_topc(A) )
     => ! [B] : 
          ( m1_pre_topc(B,A)
         => ! [C] : 
              ( m1_pre_topc(C,A)
             => ( r1_tarski(u1_struct_0(B),u1_struct_0(C))
              <=> m1_pre_topc(B,C) ) ) ) ) ),
    file(tsep_1,t4_tsep_1),
    []).