% Mizar problem: t4_mcart_1,mcart_1,117,65 
fof(t4_mcart_1,conjecture,(
    ! [A] : ~ ( A != k1_xboole_0
      & ! [B] : ~ ( r2_hidden(B,A)
          & ! [C,D,E] : 
              ( ( r2_hidden(C,D)
                & r2_hidden(D,E)
                & r2_hidden(E,B) )
             => r1_xboole_0(C,A) ) ) ) ),
    inference(mizar_bg_added,[status(thm)],[antisymmetry_r2_hidden,commutativity_k2_xboole_0,d2_xboole_0,d4_tarski,dt_k1_xboole_0,dt_k2_xboole_0,dt_k3_tarski,dt_m1_subset_1,existence_m1_subset_1,fc1_xboole_0,fc2_xboole_0,fc3_xboole_0,idempotence_k2_xboole_0,rc1_xboole_0,rc2_xboole_0,s1_xboole_0__e1_4__mcart_1,s1_xboole_0__e3_4__mcart_1,s1_xboole_0__e5_4__mcart_1,symmetry_r1_xboole_0,t15_xboole_1,t1_boole,t1_mcart_1,t1_subset,t2_subset,t3_xboole_0,t4_xboole_1,t6_boole,t70_xboole_1,t7_boole,t8_boole]),
    [file(mcart_1,t4_mcart_1)]).

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

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(d2_xboole_0,axiom,(
    ! [A,B,C] : 
      ( C = k2_xboole_0(A,B)
    <=> ! [D] : 
          ( r2_hidden(D,C)
        <=> ( r2_hidden(D,A)
            | r2_hidden(D,B) ) ) ) ),
    file(xboole_0,d2_xboole_0),
    []).

fof(d4_tarski,axiom,(
    ! [A,B] : 
      ( B = k3_tarski(A)
    <=> ! [C] : 
          ( r2_hidden(C,B)
        <=> ? [D] : 
              ( r2_hidden(C,D)
              & r2_hidden(D,A) ) ) ) ),
    file(tarski,d4_tarski),
    []).

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

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

fof(dt_k3_tarski,axiom,(
    $true ),
    file(tarski,k3_tarski),
    []).

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

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

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

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

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

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

fof(s1_xboole_0__e1_4__mcart_1,axiom,(
    ! [A] : 
    ? [B] : 
    ! [C] : 
      ( r2_hidden(C,B)
    <=> ( r2_hidden(C,k3_tarski(A))
        & ? [D,E] : 
            ( r2_hidden(D,E)
            & r2_hidden(E,C)
            & ~ r1_xboole_0(D,A) ) ) ) ),
    file(mcart_1,s1_xboole_0__e1_4__mcart_1),
    []).

fof(s1_xboole_0__e3_4__mcart_1,axiom,(
    ! [A] : 
    ? [B] : 
    ! [C] : 
      ( r2_hidden(C,B)
    <=> ( r2_hidden(C,k3_tarski(k3_tarski(A)))
        & ? [D] : 
            ( r2_hidden(D,C)
            & ~ r1_xboole_0(D,A) ) ) ) ),
    file(mcart_1,s1_xboole_0__e3_4__mcart_1),
    []).

fof(s1_xboole_0__e5_4__mcart_1,axiom,(
    ! [A] : 
    ? [B] : 
    ! [C] : 
      ( r2_hidden(C,B)
    <=> ( r2_hidden(C,k3_tarski(k3_tarski(k3_tarski(A))))
        & ~ r1_xboole_0(C,A) ) ) ),
    file(mcart_1,s1_xboole_0__e5_4__mcart_1),
    []).

fof(symmetry_r1_xboole_0,axiom,(
    ! [A,B] : 
      ( r1_xboole_0(A,B)
     => r1_xboole_0(B,A) ) ),
    file(xboole_0,r1_xboole_0),
    []).

fof(t15_xboole_1,axiom,(
    ! [A,B] : 
      ( k2_xboole_0(A,B) = k1_xboole_0
     => A = k1_xboole_0 ) ),
    file(xboole_1,t15_xboole_1),
    []).

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

fof(t1_mcart_1,axiom,(
    ! [A] : ~ ( A != k1_xboole_0
      & ! [B] : ~ ( r2_hidden(B,A)
          & r1_xboole_0(B,A) ) ) ),
    file(mcart_1,t1_mcart_1),
    []).

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(t3_xboole_0,axiom,(
    ! [A,B] : 
      ( ~ ( ~ r1_xboole_0(A,B)
          & ! [C] : ~ ( r2_hidden(C,A)
              & r2_hidden(C,B) ) )
      & ~ ( ? [C] : 
              ( r2_hidden(C,A)
              & r2_hidden(C,B) )
          & r1_xboole_0(A,B) ) ) ),
    file(xboole_0,t3_xboole_0),
    []).

fof(t4_xboole_1,axiom,(
    ! [A,B,C] : k2_xboole_0(k2_xboole_0(A,B),C) = k2_xboole_0(A,k2_xboole_0(B,C)) ),
    file(xboole_1,t4_xboole_1),
    []).

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

fof(t70_xboole_1,axiom,(
    ! [A,B,C] : 
      ( ~ ( ~ r1_xboole_0(A,k2_xboole_0(B,C))
          & r1_xboole_0(A,B)
          & r1_xboole_0(A,C) )
      & ~ ( ~ ( r1_xboole_0(A,B)
              & r1_xboole_0(A,C) )
          & r1_xboole_0(A,k2_xboole_0(B,C)) ) ) ),
    file(xboole_1,t70_xboole_1),
    []).

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),
    []).