% Mizar problem: t55_domain_1,domain_1,524,76 
fof(t55_domain_1,conjecture,(
    ! [A] : 
      ( ~ v1_xboole_0(A)
     => ! [B] : 
          ( ~ v1_xboole_0(B)
         => ! [C] : 
              ( ~ v1_xboole_0(C)
             => ! [D] : 
                  ( ~ v1_xboole_0(D)
                 => ! [E] : 
                      ( m1_subset_1(E,k1_zfmisc_1(A))
                     => ! [F] : 
                          ( m1_subset_1(F,k1_zfmisc_1(B))
                         => ! [G] : 
                              ( m1_subset_1(G,k1_zfmisc_1(C))
                             => ! [H] : 
                                  ( m1_subset_1(H,k1_zfmisc_1(D))
                                 => k14_mcart_1(A,B,C,D,E,F,G,H) = a_8_0_domain_1(A,B,C,D,E,F,G,H) ) ) ) ) ) ) ) ) ),
    inference(mizar_bg_added,[status(thm)],[antisymmetry_r2_hidden,d10_xboole_0,d3_tarski,dt_k10_mcart_1,dt_k11_mcart_1,dt_k14_mcart_1,dt_k1_xboole_0,dt_k1_zfmisc_1,dt_k4_mcart_1,dt_k4_zfmisc_1,dt_k5_domain_1,dt_k8_mcart_1,dt_k9_mcart_1,dt_m1_subset_1,existence_m1_subset_1,fc1_subset_1,fc1_xboole_0,fc6_subset_1,fraenkel_a_8_0_domain_1,rc1_subset_1,rc1_xboole_0,rc2_subset_1,rc2_xboole_0,redefinition_k14_mcart_1,redefinition_k5_domain_1,reflexivity_r1_tarski,t1_subset,t2_subset,t2_tarski,t3_subset,t4_subset,t5_subset,t60_mcart_1,t6_boole,t7_boole,t84_mcart_1,t87_mcart_1,t8_boole]),
    [file(domain_1,t55_domain_1)]).

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

fof(d10_xboole_0,axiom,(
    ! [A,B] : 
      ( A = B
    <=> ( r1_tarski(A,B)
        & r1_tarski(B,A) ) ) ),
    file(xboole_0,d10_xboole_0),
    []).

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

fof(dt_k10_mcart_1,axiom,(
    ! [A,B,C,D,E] : 
      ( m1_subset_1(E,k4_zfmisc_1(A,B,C,D))
     => m1_subset_1(k10_mcart_1(A,B,C,D,E),C) ) ),
    file(mcart_1,k10_mcart_1),
    []).

fof(dt_k11_mcart_1,axiom,(
    ! [A,B,C,D,E] : 
      ( m1_subset_1(E,k4_zfmisc_1(A,B,C,D))
     => m1_subset_1(k11_mcart_1(A,B,C,D,E),D) ) ),
    file(mcart_1,k11_mcart_1),
    []).

fof(dt_k14_mcart_1,axiom,(
    ! [A,B,C,D,E,F,G,H] : 
      ( ( m1_subset_1(E,k1_zfmisc_1(A))
        & m1_subset_1(F,k1_zfmisc_1(B))
        & m1_subset_1(G,k1_zfmisc_1(C))
        & m1_subset_1(H,k1_zfmisc_1(D)) )
     => m1_subset_1(k14_mcart_1(A,B,C,D,E,F,G,H),k1_zfmisc_1(k4_zfmisc_1(A,B,C,D))) ) ),
    file(mcart_1,k14_mcart_1),
    []).

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_k4_mcart_1,axiom,(
    $true ),
    file(mcart_1,k4_mcart_1),
    []).

fof(dt_k4_zfmisc_1,axiom,(
    $true ),
    file(zfmisc_1,k4_zfmisc_1),
    []).

fof(dt_k5_domain_1,axiom,(
    ! [A,B,C,D,E,F,G,H] : 
      ( ( ~ v1_xboole_0(A)
        & ~ v1_xboole_0(B)
        & ~ v1_xboole_0(C)
        & ~ v1_xboole_0(D)
        & m1_subset_1(E,A)
        & m1_subset_1(F,B)
        & m1_subset_1(G,C)
        & m1_subset_1(H,D) )
     => m1_subset_1(k5_domain_1(A,B,C,D,E,F,G,H),k4_zfmisc_1(A,B,C,D)) ) ),
    file(domain_1,k5_domain_1),
    []).

fof(dt_k8_mcart_1,axiom,(
    ! [A,B,C,D,E] : 
      ( m1_subset_1(E,k4_zfmisc_1(A,B,C,D))
     => m1_subset_1(k8_mcart_1(A,B,C,D,E),A) ) ),
    file(mcart_1,k8_mcart_1),
    []).

fof(dt_k9_mcart_1,axiom,(
    ! [A,B,C,D,E] : 
      ( m1_subset_1(E,k4_zfmisc_1(A,B,C,D))
     => m1_subset_1(k9_mcart_1(A,B,C,D,E),B) ) ),
    file(mcart_1,k9_mcart_1),
    []).

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_subset_1,axiom,(
    ! [A] : ~ v1_xboole_0(k1_zfmisc_1(A)) ),
    file(subset_1,fc1_subset_1),
    []).

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

fof(fc6_subset_1,axiom,(
    ! [A,B,C,D] : 
      ( ( ~ v1_xboole_0(A)
        & ~ v1_xboole_0(B)
        & ~ v1_xboole_0(C)
        & ~ v1_xboole_0(D) )
     => ~ v1_xboole_0(k4_zfmisc_1(A,B,C,D)) ) ),
    file(subset_1,fc6_subset_1),
    []).

fof(fraenkel_a_8_0_domain_1,axiom,(
    ! [A,B,C,D,E,F,G,H,I] : 
      ( ( ~ v1_xboole_0(B)
        & ~ v1_xboole_0(C)
        & ~ v1_xboole_0(D)
        & ~ v1_xboole_0(E)
        & m1_subset_1(F,k1_zfmisc_1(B))
        & m1_subset_1(G,k1_zfmisc_1(C))
        & m1_subset_1(H,k1_zfmisc_1(D))
        & m1_subset_1(I,k1_zfmisc_1(E)) )
     => ( r2_hidden(A,a_8_0_domain_1(B,C,D,E,F,G,H,I))
      <=> ? [J,K,L,M] : 
            ( m1_subset_1(J,B)
            & m1_subset_1(K,C)
            & m1_subset_1(L,D)
            & m1_subset_1(M,E)
            & A = k5_domain_1(B,C,D,E,J,K,L,M)
            & r2_hidden(J,F)
            & r2_hidden(K,G)
            & r2_hidden(L,H)
            & r2_hidden(M,I) ) ) ) ),
    file(domain_1,a_8_0_domain_1),
    []).

fof(rc1_subset_1,axiom,(
    ! [A] : 
      ( ~ v1_xboole_0(A)
     => ? [B] : 
          ( m1_subset_1(B,k1_zfmisc_1(A))
          & ~ v1_xboole_0(B) ) ) ),
    file(subset_1,rc1_subset_1),
    []).

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

fof(rc2_subset_1,axiom,(
    ! [A] : 
    ? [B] : 
      ( m1_subset_1(B,k1_zfmisc_1(A))
      & v1_xboole_0(B) ) ),
    file(subset_1,rc2_subset_1),
    []).

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

fof(redefinition_k14_mcart_1,axiom,(
    ! [A,B,C,D,E,F,G,H] : 
      ( ( m1_subset_1(E,k1_zfmisc_1(A))
        & m1_subset_1(F,k1_zfmisc_1(B))
        & m1_subset_1(G,k1_zfmisc_1(C))
        & m1_subset_1(H,k1_zfmisc_1(D)) )
     => k14_mcart_1(A,B,C,D,E,F,G,H) = k4_zfmisc_1(E,F,G,H) ) ),
    file(mcart_1,k14_mcart_1),
    []).

fof(redefinition_k5_domain_1,axiom,(
    ! [A,B,C,D,E,F,G,H] : 
      ( ( ~ v1_xboole_0(A)
        & ~ v1_xboole_0(B)
        & ~ v1_xboole_0(C)
        & ~ v1_xboole_0(D)
        & m1_subset_1(E,A)
        & m1_subset_1(F,B)
        & m1_subset_1(G,C)
        & m1_subset_1(H,D) )
     => k5_domain_1(A,B,C,D,E,F,G,H) = k4_mcart_1(E,F,G,H) ) ),
    file(domain_1,k5_domain_1),
    []).

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

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(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(t60_mcart_1,axiom,(
    ! [A,B,C,D] : ~ ( A != k1_xboole_0
      & B != k1_xboole_0
      & C != k1_xboole_0
      & D != k1_xboole_0
      & ~ ! [E] : 
            ( m1_subset_1(E,k4_zfmisc_1(A,B,C,D))
           => E = k4_mcart_1(k8_mcart_1(A,B,C,D,E),k9_mcart_1(A,B,C,D,E),k10_mcart_1(A,B,C,D,E),k11_mcart_1(A,B,C,D,E)) ) ) ),
    file(mcart_1,t60_mcart_1),
    []).

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(t84_mcart_1,axiom,(
    ! [A,B,C,D,E,F,G,H] : 
      ( r2_hidden(k4_mcart_1(A,B,C,D),k4_zfmisc_1(E,F,G,H))
    <=> ( r2_hidden(A,E)
        & r2_hidden(B,F)
        & r2_hidden(C,G)
        & r2_hidden(D,H) ) ) ),
    file(mcart_1,t84_mcart_1),
    []).

fof(t87_mcart_1,axiom,(
    ! [A,B,C,D,E] : 
      ( m1_subset_1(E,k1_zfmisc_1(A))
     => ! [F] : 
          ( m1_subset_1(F,k1_zfmisc_1(B))
         => ! [G] : 
              ( m1_subset_1(G,k1_zfmisc_1(C))
             => ! [H] : 
                  ( m1_subset_1(H,k1_zfmisc_1(D))
                 => ! [I] : 
                      ( m1_subset_1(I,k4_zfmisc_1(A,B,C,D))
                     => ( r2_hidden(I,k4_zfmisc_1(E,F,G,H))
                       => ( r2_hidden(k8_mcart_1(A,B,C,D,I),E)
                          & r2_hidden(k9_mcart_1(A,B,C,D,I),F)
                          & r2_hidden(k10_mcart_1(A,B,C,D,I),G)
                          & r2_hidden(k11_mcart_1(A,B,C,D,I),H) ) ) ) ) ) ) ) ),
    file(mcart_1,t87_mcart_1),
    []).

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