% Mizar problem: t16_cat_4,cat_4,542,22 
fof(t16_cat_4,conjecture,(
    ! [A] : 
      ( ( v2_cat_1(A)
        & v3_cat_4(A)
        & l1_cat_4(A) )
     => ! [B] : 
          ( m1_subset_1(B,u1_cat_1(A))
         => k6_cat_4(A,B) = k10_cat_3(A,B,k1_cat_4(A)) ) ) ),
    inference(mizar_bg_added,[status(thm)],[antisymmetry_r2_hidden,cc1_finset_1,cc1_funct_1,cc1_relat_1,cc2_funct_1,d4_cat_4,dt_k10_cat_3,dt_k1_cat_4,dt_k1_xboole_0,dt_k6_cat_4,dt_l1_cat_1,dt_l1_cat_4,dt_m1_cat_1,dt_m1_subset_1,dt_u1_cat_1,dt_u1_cat_4,dt_u2_cat_1,existence_l1_cat_1,existence_l1_cat_4,existence_m1_cat_1,existence_m1_subset_1,fc12_relat_1,fc4_relat_1,rc1_finset_1,rc1_funct_1,rc1_relat_1,rc2_funct_1,rc2_relat_1,rc3_funct_1,rc3_relat_1,rc4_funct_1,t13_cat_4,t1_subset,t2_subset,t41_cat_3,t6_boole,t7_boole,t8_boole]),
    [file(cat_4,t16_cat_4)]).

fof(antisymmetry_r2_hidden,axiom,(
    ! [A,B] : 
      ( r2_hidden(A,B)
     => ~ r2_hidden(B,A) ) ),
    file(hidden,r2_hidden),
    []).

fof(cc1_finset_1,axiom,(
    ! [A] : 
      ( v1_xboole_0(A)
     => v1_finset_1(A) ) ),
    file(finset_1,cc1_finset_1),
    []).

fof(cc1_funct_1,axiom,(
    ! [A] : 
      ( v1_xboole_0(A)
     => v1_funct_1(A) ) ),
    file(funct_1,cc1_funct_1),
    []).

fof(cc1_relat_1,axiom,(
    ! [A] : 
      ( v1_xboole_0(A)
     => v1_relat_1(A) ) ),
    file(relat_1,cc1_relat_1),
    []).

fof(cc2_funct_1,axiom,(
    ! [A] : 
      ( ( v1_relat_1(A)
        & v1_xboole_0(A)
        & v1_funct_1(A) )
     => ( v1_relat_1(A)
        & v1_funct_1(A)
        & v2_funct_1(A) ) ) ),
    file(funct_1,cc2_funct_1),
    []).

fof(d4_cat_4,axiom,(
    ! [A] : 
      ( l1_cat_4(A)
     => k1_cat_4(A) = u1_cat_4(A) ) ),
    file(cat_4,d4_cat_4),
    []).

fof(dt_k10_cat_3,axiom,(
    ! [A,B,C] : 
      ( ( v2_cat_1(A)
        & l1_cat_1(A)
        & m1_subset_1(B,u1_cat_1(A))
        & m1_subset_1(C,u1_cat_1(A)) )
     => m1_cat_1(k10_cat_3(A,B,C),A,B,C) ) ),
    file(cat_3,k10_cat_3),
    []).

fof(dt_k1_cat_4,axiom,(
    ! [A] : 
      ( l1_cat_4(A)
     => m1_subset_1(k1_cat_4(A),u1_cat_1(A)) ) ),
    file(cat_4,k1_cat_4),
    []).

fof(dt_k1_xboole_0,axiom,(
    $true ),
    file(xboole_0,k1_xboole_0),
    []).

fof(dt_k6_cat_4,axiom,(
    ! [A,B] : 
      ( ( v2_cat_1(A)
        & v3_cat_4(A)
        & l1_cat_4(A)
        & m1_subset_1(B,u1_cat_1(A)) )
     => m1_cat_1(k6_cat_4(A,B),A,B,k1_cat_4(A)) ) ),
    file(cat_4,k6_cat_4),
    []).

fof(dt_l1_cat_1,axiom,(
    $true ),
    file(cat_1,l1_cat_1),
    []).

fof(dt_l1_cat_4,axiom,(
    ! [A] : 
      ( l1_cat_4(A)
     => l1_cat_1(A) ) ),
    file(cat_4,l1_cat_4),
    []).

fof(dt_m1_cat_1,axiom,(
    ! [A,B,C] : 
      ( ( l1_cat_1(A)
        & m1_subset_1(B,u1_cat_1(A))
        & m1_subset_1(C,u1_cat_1(A)) )
     => ! [D] : 
          ( m1_cat_1(D,A,B,C)
         => m1_subset_1(D,u2_cat_1(A)) ) ) ),
    file(cat_1,m1_cat_1),
    []).

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

fof(dt_u1_cat_1,axiom,(
    ! [A] : 
      ( l1_cat_1(A)
     => ~ v1_xboole_0(u1_cat_1(A)) ) ),
    file(cat_1,u1_cat_1),
    []).

fof(dt_u1_cat_4,axiom,(
    ! [A] : 
      ( l1_cat_4(A)
     => m1_subset_1(u1_cat_4(A),u1_cat_1(A)) ) ),
    file(cat_4,u1_cat_4),
    []).

fof(dt_u2_cat_1,axiom,(
    ! [A] : 
      ( l1_cat_1(A)
     => ~ v1_xboole_0(u2_cat_1(A)) ) ),
    file(cat_1,u2_cat_1),
    []).

fof(existence_l1_cat_1,axiom,(
    ? [A] : l1_cat_1(A) ),
    file(cat_1,l1_cat_1),
    []).

fof(existence_l1_cat_4,axiom,(
    ? [A] : l1_cat_4(A) ),
    file(cat_4,l1_cat_4),
    []).

fof(existence_m1_cat_1,axiom,(
    ! [A,B,C] : 
      ( ( l1_cat_1(A)
        & m1_subset_1(B,u1_cat_1(A))
        & m1_subset_1(C,u1_cat_1(A)) )
     => ? [D] : m1_cat_1(D,A,B,C) ) ),
    file(cat_1,m1_cat_1),
    []).

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

fof(fc12_relat_1,axiom,
    ( v1_xboole_0(k1_xboole_0)
    & v1_relat_1(k1_xboole_0)
    & v3_relat_1(k1_xboole_0) ),
    file(relat_1,fc12_relat_1),
    []).

fof(fc4_relat_1,axiom,
    ( v1_xboole_0(k1_xboole_0)
    & v1_relat_1(k1_xboole_0) ),
    file(relat_1,fc4_relat_1),
    []).

fof(rc1_finset_1,axiom,(
    ? [A] : 
      ( ~ v1_xboole_0(A)
      & v1_finset_1(A) ) ),
    file(finset_1,rc1_finset_1),
    []).

fof(rc1_funct_1,axiom,(
    ? [A] : 
      ( v1_relat_1(A)
      & v1_funct_1(A) ) ),
    file(funct_1,rc1_funct_1),
    []).

fof(rc1_relat_1,axiom,(
    ? [A] : 
      ( v1_xboole_0(A)
      & v1_relat_1(A) ) ),
    file(relat_1,rc1_relat_1),
    []).

fof(rc2_funct_1,axiom,(
    ? [A] : 
      ( v1_relat_1(A)
      & v1_xboole_0(A)
      & v1_funct_1(A) ) ),
    file(funct_1,rc2_funct_1),
    []).

fof(rc2_relat_1,axiom,(
    ? [A] : 
      ( ~ v1_xboole_0(A)
      & v1_relat_1(A) ) ),
    file(relat_1,rc2_relat_1),
    []).

fof(rc3_funct_1,axiom,(
    ? [A] : 
      ( v1_relat_1(A)
      & v1_funct_1(A)
      & v2_funct_1(A) ) ),
    file(funct_1,rc3_funct_1),
    []).

fof(rc3_relat_1,axiom,(
    ? [A] : 
      ( v1_relat_1(A)
      & v3_relat_1(A) ) ),
    file(relat_1,rc3_relat_1),
    []).

fof(rc4_funct_1,axiom,(
    ? [A] : 
      ( v1_relat_1(A)
      & v3_relat_1(A)
      & v1_funct_1(A) ) ),
    file(funct_1,rc4_funct_1),
    []).

fof(t13_cat_4,axiom,(
    ! [A] : 
      ( ( v2_cat_1(A)
        & v3_cat_4(A)
        & l1_cat_4(A) )
     => v6_cat_1(k1_cat_4(A),A) ) ),
    file(cat_4,t13_cat_4),
    []).

fof(t1_subset,axiom,(
    ! [A,B] : 
      ( r2_hidden(A,B)
     => m1_subset_1(A,B) ) ),
    file(subset,t1_subset),
    []).

fof(t2_subset,axiom,(
    ! [A,B] : 
      ( m1_subset_1(A,B)
     => ( v1_xboole_0(B)
        | r2_hidden(A,B) ) ) ),
    file(subset,t2_subset),
    []).

fof(t41_cat_3,axiom,(
    ! [A] : 
      ( ( v2_cat_1(A)
        & l1_cat_1(A) )
     => ! [B] : 
          ( m1_subset_1(B,u1_cat_1(A))
         => ! [C] : 
              ( m1_subset_1(C,u1_cat_1(A))
             => ! [D] : 
                  ( m1_cat_1(D,A,B,C)
                 => ( v6_cat_1(C,A)
                   => k10_cat_3(A,B,C) = D ) ) ) ) ) ),
    file(cat_3,t41_cat_3),
    []).

fof(t6_boole,axiom,(
    ! [A] : 
      ( v1_xboole_0(A)
     => A = k1_xboole_0 ) ),
    file(boole,t6_boole),
    []).

fof(t7_boole,axiom,(
    ! [A,B] : ~ ( r2_hidden(A,B)
      & v1_xboole_0(B) ) ),
    file(boole,t7_boole),
    []).

fof(t8_boole,axiom,(
    ! [A,B] : ~ ( v1_xboole_0(A)
      & A != B
      & v1_xboole_0(B) ) ),
    file(boole,t8_boole),
    []).