% Mizar problem: t24_substlat,substlat,813,17 
fof(t24_substlat,conjecture,(
    ! [A,B,C] : 
      ( m1_subset_1(C,k5_finsub_1(k4_partfun1(A,B)))
     => k3_substlat(A,B,k4_substlat(A,B,C,C)) = k3_substlat(A,B,C) ) ),
    inference(mizar_bg_added,[status(thm)],[antisymmetry_r2_hidden,cc1_finset_1,cc1_finsub_1,cc1_fraenkel,cc1_substlat,cc2_finset_1,cc2_finsub_1,cc2_substlat,cc3_finsub_1,commutativity_k1_finsub_1,commutativity_k2_xboole_0,d1_substlat,d2_substlat,d3_substlat,dt_k1_finsub_1,dt_k1_substlat,dt_k1_xboole_0,dt_k1_zfmisc_1,dt_k2_xboole_0,dt_k3_substlat,dt_k4_partfun1,dt_k4_substlat,dt_k5_finsub_1,dt_m1_subset_1,dt_m2_subset_1,existence_m1_subset_1,existence_m2_subset_1,fc1_finsub_1,fc1_substlat,fc1_wellfnd1,fc1_xboole_0,fc2_finsub_1,fc2_xboole_0,fc3_xboole_0,fc9_finset_1,fraenkel_a_2_0_substlat,fraenkel_a_3_0_substlat,fraenkel_a_4_0_substlat,idempotence_k1_finsub_1,idempotence_k2_xboole_0,l30_substlat,rc1_finset_1,rc1_fraenkel,rc1_substlat,rc1_xboole_0,rc2_substlat,rc2_xboole_0,rc3_finset_1,rc4_finset_1,redefinition_k1_finsub_1,redefinition_m2_subset_1,reflexivity_r1_partfun1,reflexivity_r1_tarski,symmetry_r1_partfun1,t12_xboole_1,t1_boole,t1_subset,t23_substlat,t2_subset,t2_tarski,t3_subset,t4_subset,t5_subset,t6_boole,t7_boole,t8_boole]),
    [file(substlat,t24_substlat)]).

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_finsub_1,axiom,(
    ! [A] : 
      ( v4_finsub_1(A)
     => ( v1_finsub_1(A)
        & v3_finsub_1(A) ) ) ),
    file(finsub_1,cc1_finsub_1),
    []).

fof(cc1_fraenkel,axiom,(
    ! [A] : 
      ( v1_fraenkel(A)
     => ! [B] : 
          ( m1_subset_1(B,A)
         => ( v1_relat_1(B)
            & v1_funct_1(B) ) ) ) ),
    file(fraenkel,cc1_fraenkel),
    []).

fof(cc1_substlat,axiom,(
    ! [A,B,C] : 
      ( m1_subset_1(C,k1_substlat(A,B))
     => v1_finset_1(C) ) ),
    file(substlat,cc1_substlat),
    []).

fof(cc2_finset_1,axiom,(
    ! [A] : 
      ( v1_finset_1(A)
     => ! [B] : 
          ( m1_subset_1(B,k1_zfmisc_1(A))
         => v1_finset_1(B) ) ) ),
    file(finset_1,cc2_finset_1),
    []).

fof(cc2_finsub_1,axiom,(
    ! [A] : 
      ( ( v1_finsub_1(A)
        & v3_finsub_1(A) )
     => v4_finsub_1(A) ) ),
    file(finsub_1,cc2_finsub_1),
    []).

fof(cc2_substlat,axiom,(
    ! [A,B,C] : 
      ( m1_subset_1(C,k1_substlat(A,B))
     => ( v1_finset_1(C)
        & v1_fraenkel(C) ) ) ),
    file(substlat,cc2_substlat),
    []).

fof(cc3_finsub_1,axiom,(
    ! [A,B] : 
      ( m1_subset_1(B,k5_finsub_1(A))
     => v1_finset_1(B) ) ),
    file(finsub_1,cc3_finsub_1),
    []).

fof(commutativity_k1_finsub_1,axiom,(
    ! [A,B,C] : 
      ( ( ~ v1_xboole_0(A)
        & v4_finsub_1(A)
        & m1_subset_1(B,A)
        & m1_subset_1(C,A) )
     => k1_finsub_1(A,B,C) = k1_finsub_1(A,C,B) ) ),
    file(finsub_1,k1_finsub_1),
    []).

fof(commutativity_k2_xboole_0,axiom,(
    ! [A,B] : k2_xboole_0(A,B) = k2_xboole_0(B,A) ),
    file(xboole_0,k2_xboole_0),
    []).

fof(d1_substlat,axiom,(
    ! [A,B] : k1_substlat(A,B) = a_2_0_substlat(A,B) ),
    file(substlat,d1_substlat),
    []).

fof(d2_substlat,axiom,(
    ! [A,B,C] : 
      ( m1_subset_1(C,k5_finsub_1(k4_partfun1(A,B)))
     => k3_substlat(A,B,C) = a_3_0_substlat(A,B,C) ) ),
    file(substlat,d2_substlat),
    []).

fof(d3_substlat,axiom,(
    ! [A,B,C] : 
      ( m1_subset_1(C,k5_finsub_1(k4_partfun1(A,B)))
     => ! [D] : 
          ( m1_subset_1(D,k5_finsub_1(k4_partfun1(A,B)))
         => k4_substlat(A,B,C,D) = a_4_0_substlat(A,B,C,D) ) ) ),
    file(substlat,d3_substlat),
    []).

fof(dt_k1_finsub_1,axiom,(
    ! [A,B,C] : 
      ( ( ~ v1_xboole_0(A)
        & v4_finsub_1(A)
        & m1_subset_1(B,A)
        & m1_subset_1(C,A) )
     => m1_subset_1(k1_finsub_1(A,B,C),A) ) ),
    file(finsub_1,k1_finsub_1),
    []).

fof(dt_k1_substlat,axiom,(
    ! [A,B] : m1_subset_1(k1_substlat(A,B),k1_zfmisc_1(k5_finsub_1(k4_partfun1(A,B)))) ),
    file(substlat,k1_substlat),
    []).

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

fof(dt_k1_zfmisc_1,axiom,(
    $true ),
    file(zfmisc_1,k1_zfmisc_1),
    []).

fof(dt_k2_xboole_0,axiom,(
    $true ),
    file(xboole_0,k2_xboole_0),
    []).

fof(dt_k3_substlat,axiom,(
    ! [A,B,C] : 
      ( m1_subset_1(C,k5_finsub_1(k4_partfun1(A,B)))
     => m2_subset_1(k3_substlat(A,B,C),k5_finsub_1(k4_partfun1(A,B)),k1_substlat(A,B)) ) ),
    file(substlat,k3_substlat),
    []).

fof(dt_k4_partfun1,axiom,(
    $true ),
    file(partfun1,k4_partfun1),
    []).

fof(dt_k4_substlat,axiom,(
    ! [A,B,C,D] : 
      ( ( m1_subset_1(C,k5_finsub_1(k4_partfun1(A,B)))
        & m1_subset_1(D,k5_finsub_1(k4_partfun1(A,B))) )
     => m1_subset_1(k4_substlat(A,B,C,D),k5_finsub_1(k4_partfun1(A,B))) ) ),
    file(substlat,k4_substlat),
    []).

fof(dt_k5_finsub_1,axiom,(
    ! [A] : v4_finsub_1(k5_finsub_1(A)) ),
    file(finsub_1,k5_finsub_1),
    []).

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

fof(dt_m2_subset_1,axiom,(
    ! [A,B] : 
      ( ( ~ v1_xboole_0(A)
        & ~ v1_xboole_0(B)
        & m1_subset_1(B,k1_zfmisc_1(A)) )
     => ! [C] : 
          ( m2_subset_1(C,A,B)
         => m1_subset_1(C,A) ) ) ),
    file(subset_1,m2_subset_1),
    []).

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

fof(existence_m2_subset_1,axiom,(
    ! [A,B] : 
      ( ( ~ v1_xboole_0(A)
        & ~ v1_xboole_0(B)
        & m1_subset_1(B,k1_zfmisc_1(A)) )
     => ? [C] : m2_subset_1(C,A,B) ) ),
    file(subset_1,m2_subset_1),
    []).

fof(fc1_finsub_1,axiom,(
    ! [A] : 
      ( ~ v1_xboole_0(k1_zfmisc_1(A))
      & v1_finsub_1(k1_zfmisc_1(A))
      & v3_finsub_1(k1_zfmisc_1(A))
      & v4_finsub_1(k1_zfmisc_1(A)) ) ),
    file(finsub_1,fc1_finsub_1),
    []).

fof(fc1_substlat,axiom,(
    ! [A,B] : ~ v1_xboole_0(k1_substlat(A,B)) ),
    file(substlat,fc1_substlat),
    []).

fof(fc1_wellfnd1,axiom,(
    ! [A,B] : 
      ( ~ v1_xboole_0(k4_partfun1(A,B))
      & v1_fraenkel(k4_partfun1(A,B)) ) ),
    file(wellfnd1,fc1_wellfnd1),
    []).

fof(fc1_xboole_0,axiom,(
    v1_xboole_0(k1_xboole_0) ),
    file(xboole_0,fc1_xboole_0),
    []).

fof(fc2_finsub_1,axiom,(
    ! [A] : 
      ( ~ v1_xboole_0(k5_finsub_1(A))
      & v1_finsub_1(k5_finsub_1(A))
      & v3_finsub_1(k5_finsub_1(A))
      & v4_finsub_1(k5_finsub_1(A)) ) ),
    file(finsub_1,fc2_finsub_1),
    []).

fof(fc2_xboole_0,axiom,(
    ! [A,B] : 
      ( ~ v1_xboole_0(A)
     => ~ v1_xboole_0(k2_xboole_0(A,B)) ) ),
    file(xboole_0,fc2_xboole_0),
    []).

fof(fc3_xboole_0,axiom,(
    ! [A,B] : 
      ( ~ v1_xboole_0(A)
     => ~ v1_xboole_0(k2_xboole_0(B,A)) ) ),
    file(xboole_0,fc3_xboole_0),
    []).

fof(fc9_finset_1,axiom,(
    ! [A,B] : 
      ( ( v1_finset_1(A)
        & v1_finset_1(B) )
     => v1_finset_1(k2_xboole_0(A,B)) ) ),
    file(finset_1,fc9_finset_1),
    []).

fof(fraenkel_a_2_0_substlat,axiom,(
    ! [A,B,C] : 
      ( r2_hidden(A,a_2_0_substlat(B,C))
    <=> ? [D] : 
          ( m1_subset_1(D,k5_finsub_1(k4_partfun1(B,C)))
          & A = D
          & ! [E] : 
              ( r2_hidden(E,D)
             => v1_finset_1(E) )
          & ! [E] : 
              ( m1_subset_1(E,k4_partfun1(B,C))
             => ! [F] : 
                  ( m1_subset_1(F,k4_partfun1(B,C))
                 => ( ( r2_hidden(E,D)
                      & r2_hidden(F,D)
                      & r1_tarski(E,F) )
                   => E = F ) ) ) ) ) ),
    file(substlat,a_2_0_substlat),
    []).

fof(fraenkel_a_3_0_substlat,axiom,(
    ! [A,B,C,D] : 
      ( m1_subset_1(D,k5_finsub_1(k4_partfun1(B,C)))
     => ( r2_hidden(A,a_3_0_substlat(B,C,D))
      <=> ? [E] : 
            ( m1_subset_1(E,k4_partfun1(B,C))
            & A = E
            & v1_finset_1(E)
            & ! [F] : 
                ( m1_subset_1(F,k4_partfun1(B,C))
               => ( ( r2_hidden(F,D)
                    & r1_tarski(F,E) )
                <=> F = E ) ) ) ) ) ),
    file(substlat,a_3_0_substlat),
    []).

fof(fraenkel_a_4_0_substlat,axiom,(
    ! [A,B,C,D,E] : 
      ( ( m1_subset_1(D,k5_finsub_1(k4_partfun1(B,C)))
        & m1_subset_1(E,k5_finsub_1(k4_partfun1(B,C))) )
     => ( r2_hidden(A,a_4_0_substlat(B,C,D,E))
      <=> ? [F,G] : 
            ( m1_subset_1(F,k4_partfun1(B,C))
            & m1_subset_1(G,k4_partfun1(B,C))
            & A = k2_xboole_0(F,G)
            & r2_hidden(F,D)
            & r2_hidden(G,E)
            & r1_partfun1(F,G) ) ) ) ),
    file(substlat,a_4_0_substlat),
    []).

fof(idempotence_k1_finsub_1,axiom,(
    ! [A,B,C] : 
      ( ( ~ v1_xboole_0(A)
        & v4_finsub_1(A)
        & m1_subset_1(B,A)
        & m1_subset_1(C,A) )
     => k1_finsub_1(A,B,B) = B ) ),
    file(finsub_1,k1_finsub_1),
    []).

fof(idempotence_k2_xboole_0,axiom,(
    ! [A,B] : k2_xboole_0(A,A) = A ),
    file(xboole_0,k2_xboole_0),
    []).

fof(l30_substlat,axiom,(
    ! [A,B,C] : 
      ( m1_subset_1(C,k5_finsub_1(k4_partfun1(A,B)))
     => ! [D] : 
          ( m1_subset_1(D,k5_finsub_1(k4_partfun1(A,B)))
         => k3_substlat(A,B,k1_finsub_1(k5_finsub_1(k4_partfun1(A,B)),k4_substlat(A,B,C,D),D)) = k3_substlat(A,B,D) ) ) ),
    file(substlat,l30_substlat),
    []).

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

fof(rc1_fraenkel,axiom,(
    ? [A] : 
      ( ~ v1_xboole_0(A)
      & v1_fraenkel(A) ) ),
    file(fraenkel,rc1_fraenkel),
    []).

fof(rc1_substlat,axiom,(
    ! [A,B] : 
    ? [C] : 
      ( m1_subset_1(C,k1_substlat(A,B))
      & ~ v1_xboole_0(C) ) ),
    file(substlat,rc1_substlat),
    []).

fof(rc1_xboole_0,axiom,(
    ? [A] : v1_xboole_0(A) ),
    file(xboole_0,rc1_xboole_0),
    []).

fof(rc2_substlat,axiom,(
    ! [A,B] : 
    ? [C] : 
      ( m1_subset_1(C,k4_partfun1(A,B))
      & v1_relat_1(C)
      & v1_funct_1(C)
      & v1_finset_1(C) ) ),
    file(substlat,rc2_substlat),
    []).

fof(rc2_xboole_0,axiom,(
    ? [A] : ~ v1_xboole_0(A) ),
    file(xboole_0,rc2_xboole_0),
    []).

fof(rc3_finset_1,axiom,(
    ! [A] : 
      ( ~ v1_xboole_0(A)
     => ? [B] : 
          ( m1_subset_1(B,k1_zfmisc_1(A))
          & ~ v1_xboole_0(B)
          & v1_finset_1(B) ) ) ),
    file(finset_1,rc3_finset_1),
    []).

fof(rc4_finset_1,axiom,(
    ! [A] : 
      ( ~ v1_xboole_0(A)
     => ? [B] : 
          ( m1_subset_1(B,k1_zfmisc_1(A))
          & ~ v1_xboole_0(B)
          & v1_finset_1(B) ) ) ),
    file(finset_1,rc4_finset_1),
    []).

fof(redefinition_k1_finsub_1,axiom,(
    ! [A,B,C] : 
      ( ( ~ v1_xboole_0(A)
        & v4_finsub_1(A)
        & m1_subset_1(B,A)
        & m1_subset_1(C,A) )
     => k1_finsub_1(A,B,C) = k2_xboole_0(B,C) ) ),
    file(finsub_1,k1_finsub_1),
    []).

fof(redefinition_m2_subset_1,axiom,(
    ! [A,B] : 
      ( ( ~ v1_xboole_0(A)
        & ~ v1_xboole_0(B)
        & m1_subset_1(B,k1_zfmisc_1(A)) )
     => ! [C] : 
          ( m2_subset_1(C,A,B)
        <=> m1_subset_1(C,B) ) ) ),
    file(subset_1,m2_subset_1),
    []).

fof(reflexivity_r1_partfun1,axiom,(
    ! [A,B] : 
      ( ( v1_relat_1(A)
        & v1_funct_1(A)
        & v1_relat_1(B)
        & v1_funct_1(B) )
     => r1_partfun1(A,A) ) ),
    file(partfun1,r1_partfun1),
    []).

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

fof(symmetry_r1_partfun1,axiom,(
    ! [A,B] : 
      ( ( v1_relat_1(A)
        & v1_funct_1(A)
        & v1_relat_1(B)
        & v1_funct_1(B) )
     => ( r1_partfun1(A,B)
       => r1_partfun1(B,A) ) ) ),
    file(partfun1,r1_partfun1),
    []).

fof(t12_xboole_1,axiom,(
    ! [A,B] : 
      ( r1_tarski(A,B)
     => k2_xboole_0(A,B) = B ) ),
    file(xboole_1,t12_xboole_1),
    []).

fof(t1_boole,axiom,(
    ! [A] : k2_xboole_0(A,k1_xboole_0) = A ),
    file(boole,t1_boole),
    []).

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

fof(t23_substlat,axiom,(
    ! [A,B,C] : 
      ( m1_subset_1(C,k5_finsub_1(k4_partfun1(A,B)))
     => r1_tarski(C,k4_substlat(A,B,C,C)) ) ),
    file(substlat,t23_substlat),
    []).

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

fof(t2_tarski,axiom,(
    ! [A,B] : 
      ( ! [C] : 
          ( r2_hidden(C,A)
        <=> r2_hidden(C,B) )
     => A = B ) ),
    file(tarski,t2_tarski),
    []).

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

fof(t4_subset,axiom,(
    ! [A,B,C] : 
      ( ( r2_hidden(A,B)
        & m1_subset_1(B,k1_zfmisc_1(C)) )
     => m1_subset_1(A,C) ) ),
    file(subset,t4_subset),
    []).

fof(t5_subset,axiom,(
    ! [A,B,C] : ~ ( r2_hidden(A,B)
      & m1_subset_1(B,k1_zfmisc_1(C))
      & v1_xboole_0(C) ) ),
    file(subset,t5_subset),
    []).

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),
    []).