SET007 Axioms: SET007+92.ax


%------------------------------------------------------------------------------
% File     : SET007+92 : TPTP v7.5.0. Released v3.4.0.
% Domain   : Set Theory
% Axioms   : N-Tuples and Cartesian Products for n=5
% Version  : [Urb08] axioms.
% English  :

% Refs     : [Mat90] Matuszewski (1990), Formalized Mathematics
%          : [Urb07] Urban (2007), MPTP 0.2: Design, Implementation, and In
%          : [Urb08] Urban (2006), Email to G. Sutcliffe
% Source   : [Urb08]
% Names    : mcart_2 [Urb08]

% Status   : Satisfiable
% Syntax   : Number of formulae    :   54 (  13 unit)
%            Number of atoms       :  358 ( 202 equality)
%            Maximal formula depth :   32 (  15 average)
%            Number of connectives :  431 ( 127 ~  ;  32  |; 168  &)
%                                         (   8 <=>;  96 =>;   0 <=)
%                                         (   0 <~>;   0 ~|;   0 ~&)
%            Number of predicates  :    6 (   1 propositional; 0-2 arity)
%            Number of functors    :   19 (   1 constant; 0-10 arity)
%            Number of variables   :  423 (   0 singleton; 401 !;  22 ?)
%            Maximal term depth    :    5 (   1 average)
% SPC      : 

% Comments : The individual reference can be found in [Mat90] by looking for
%            the name provided by [Urb08].
%          : Translated by MPTP from the Mizar Mathematical Library 4.48.930.
%          : These set theory axioms are used in encodings of problems in
%            various domains, including ALG, CAT, GRP, LAT, SET, and TOP.
%------------------------------------------------------------------------------
fof(t1_mcart_2,axiom,(
    ! [A] :
      ~ ( A != k1_xboole_0
        & ! [B] :
            ~ ( r2_hidden(B,A)
              & ! [C,D,E,F,G,H] :
                  ( ( r2_hidden(C,D)
                    & r2_hidden(D,E)
                    & r2_hidden(E,F)
                    & r2_hidden(F,G)
                    & r2_hidden(G,H)
                    & r2_hidden(H,B) )
                 => r1_xboole_0(C,A) ) ) ) )).

fof(t2_mcart_2,axiom,(
    ! [A] :
      ~ ( A != k1_xboole_0
        & ! [B] :
            ~ ( r2_hidden(B,A)
              & ! [C,D,E,F,G,H,I] :
                  ( ( r2_hidden(C,D)
                    & r2_hidden(D,E)
                    & r2_hidden(E,F)
                    & r2_hidden(F,G)
                    & r2_hidden(G,H)
                    & r2_hidden(H,I)
                    & r2_hidden(I,B) )
                 => r1_xboole_0(C,A) ) ) ) )).

fof(d1_mcart_2,axiom,(
    ! [A,B,C,D,E] : k1_mcart_2(A,B,C,D,E) = k4_tarski(k4_mcart_1(A,B,C,D),E) )).

fof(t3_mcart_2,axiom,(
    ! [A,B,C,D,E] : k1_mcart_2(A,B,C,D,E) = k4_tarski(k4_tarski(k4_tarski(k4_tarski(A,B),C),D),E) )).

fof(t4_mcart_2,axiom,(
    $true )).

fof(t5_mcart_2,axiom,(
    ! [A,B,C,D,E] : k1_mcart_2(A,B,C,D,E) = k3_mcart_1(k3_mcart_1(A,B,C),D,E) )).

fof(t6_mcart_2,axiom,(
    ! [A,B,C,D,E] : k1_mcart_2(A,B,C,D,E) = k4_mcart_1(k4_tarski(A,B),C,D,E) )).

fof(t7_mcart_2,axiom,(
    ! [A,B,C,D,E,F,G,H,I,J] :
      ( k1_mcart_2(A,B,C,D,E) = k1_mcart_2(F,G,H,I,J)
     => ( A = F
        & B = G
        & C = H
        & D = I
        & E = J ) ) )).

fof(t8_mcart_2,axiom,(
    ! [A] :
      ~ ( A != k1_xboole_0
        & ! [B] :
            ~ ( r2_hidden(B,A)
              & ! [C,D,E,F,G] :
                  ~ ( ( r2_hidden(C,A)
                      | r2_hidden(D,A) )
                    & B = k1_mcart_2(C,D,E,F,G) ) ) ) )).

fof(d2_mcart_2,axiom,(
    ! [A,B,C,D,E] : k2_mcart_2(A,B,C,D,E) = k2_zfmisc_1(k4_zfmisc_1(A,B,C,D),E) )).

fof(t9_mcart_2,axiom,(
    ! [A,B,C,D,E] : k2_mcart_2(A,B,C,D,E) = k2_zfmisc_1(k2_zfmisc_1(k2_zfmisc_1(k2_zfmisc_1(A,B),C),D),E) )).

fof(t10_mcart_2,axiom,(
    $true )).

fof(t11_mcart_2,axiom,(
    ! [A,B,C,D,E] : k2_mcart_2(A,B,C,D,E) = k3_zfmisc_1(k3_zfmisc_1(A,B,C),D,E) )).

fof(t12_mcart_2,axiom,(
    ! [A,B,C,D,E] : k2_mcart_2(A,B,C,D,E) = k4_zfmisc_1(k2_zfmisc_1(A,B),C,D,E) )).

fof(t13_mcart_2,axiom,(
    ! [A,B,C,D,E] :
      ( ( A != k1_xboole_0
        & B != k1_xboole_0
        & C != k1_xboole_0
        & D != k1_xboole_0
        & E != k1_xboole_0 )
    <=> k2_mcart_2(A,B,C,D,E) != k1_xboole_0 ) )).

fof(t14_mcart_2,axiom,(
    ! [A,B,C,D,E,F,G,H,I,J] :
      ( k2_mcart_2(A,B,C,D,E) = k2_mcart_2(F,G,H,I,J)
     => ( A = k1_xboole_0
        | B = k1_xboole_0
        | C = k1_xboole_0
        | D = k1_xboole_0
        | E = k1_xboole_0
        | ( A = F
          & B = G
          & C = H
          & D = I
          & E = J ) ) ) )).

fof(t15_mcart_2,axiom,(
    ! [A,B,C,D,E,F,G,H,I,J] :
      ( k2_mcart_2(A,B,C,D,E) = k2_mcart_2(F,G,H,I,J)
     => ( k2_mcart_2(A,B,C,D,E) = k1_xboole_0
        | ( A = F
          & B = G
          & C = H
          & D = I
          & E = J ) ) ) )).

fof(t16_mcart_2,axiom,(
    ! [A,B] :
      ( k2_mcart_2(A,A,A,A,A) = k2_mcart_2(B,B,B,B,B)
     => A = B ) )).

fof(t17_mcart_2,axiom,(
    ! [A,B,C,D,E] :
      ~ ( A != k1_xboole_0
        & B != k1_xboole_0
        & C != k1_xboole_0
        & D != k1_xboole_0
        & E != k1_xboole_0
        & ? [F] :
            ( m1_subset_1(F,k2_mcart_2(A,B,C,D,E))
            & ! [G] :
                ( m1_subset_1(G,A)
               => ! [H] :
                    ( m1_subset_1(H,B)
                   => ! [I] :
                        ( m1_subset_1(I,C)
                       => ! [J] :
                            ( m1_subset_1(J,D)
                           => ! [K] :
                                ( m1_subset_1(K,E)
                               => F != k1_mcart_2(G,H,I,J,K) ) ) ) ) ) ) ) )).

fof(d3_mcart_2,axiom,(
    ! [A,B,C,D,E] :
      ~ ( A != k1_xboole_0
        & B != k1_xboole_0
        & C != k1_xboole_0
        & D != k1_xboole_0
        & E != k1_xboole_0
        & ~ ! [F] :
              ( m1_subset_1(F,k2_mcart_2(A,B,C,D,E))
             => ! [G] :
                  ( m1_subset_1(G,A)
                 => ( G = k3_mcart_2(A,B,C,D,E,F)
                  <=> ! [H,I,J,K,L] :
                        ( F = k1_mcart_2(H,I,J,K,L)
                       => G = H ) ) ) ) ) )).

fof(d4_mcart_2,axiom,(
    ! [A,B,C,D,E] :
      ~ ( A != k1_xboole_0
        & B != k1_xboole_0
        & C != k1_xboole_0
        & D != k1_xboole_0
        & E != k1_xboole_0
        & ~ ! [F] :
              ( m1_subset_1(F,k2_mcart_2(A,B,C,D,E))
             => ! [G] :
                  ( m1_subset_1(G,B)
                 => ( G = k4_mcart_2(A,B,C,D,E,F)
                  <=> ! [H,I,J,K,L] :
                        ( F = k1_mcart_2(H,I,J,K,L)
                       => G = I ) ) ) ) ) )).

fof(d5_mcart_2,axiom,(
    ! [A,B,C,D,E] :
      ~ ( A != k1_xboole_0
        & B != k1_xboole_0
        & C != k1_xboole_0
        & D != k1_xboole_0
        & E != k1_xboole_0
        & ~ ! [F] :
              ( m1_subset_1(F,k2_mcart_2(A,B,C,D,E))
             => ! [G] :
                  ( m1_subset_1(G,C)
                 => ( G = k5_mcart_2(A,B,C,D,E,F)
                  <=> ! [H,I,J,K,L] :
                        ( F = k1_mcart_2(H,I,J,K,L)
                       => G = J ) ) ) ) ) )).

fof(d6_mcart_2,axiom,(
    ! [A,B,C,D,E] :
      ~ ( A != k1_xboole_0
        & B != k1_xboole_0
        & C != k1_xboole_0
        & D != k1_xboole_0
        & E != k1_xboole_0
        & ~ ! [F] :
              ( m1_subset_1(F,k2_mcart_2(A,B,C,D,E))
             => ! [G] :
                  ( m1_subset_1(G,D)
                 => ( G = k6_mcart_2(A,B,C,D,E,F)
                  <=> ! [H,I,J,K,L] :
                        ( F = k1_mcart_2(H,I,J,K,L)
                       => G = K ) ) ) ) ) )).

fof(d7_mcart_2,axiom,(
    ! [A,B,C,D,E] :
      ~ ( A != k1_xboole_0
        & B != k1_xboole_0
        & C != k1_xboole_0
        & D != k1_xboole_0
        & E != k1_xboole_0
        & ~ ! [F] :
              ( m1_subset_1(F,k2_mcart_2(A,B,C,D,E))
             => ! [G] :
                  ( m1_subset_1(G,E)
                 => ( G = k7_mcart_2(A,B,C,D,E,F)
                  <=> ! [H,I,J,K,L] :
                        ( F = k1_mcart_2(H,I,J,K,L)
                       => G = L ) ) ) ) ) )).

fof(t18_mcart_2,axiom,(
    ! [A,B,C,D,E] :
      ~ ( A != k1_xboole_0
        & B != k1_xboole_0
        & C != k1_xboole_0
        & D != k1_xboole_0
        & E != k1_xboole_0
        & ? [F] :
            ( m1_subset_1(F,k2_mcart_2(A,B,C,D,E))
            & ? [G,H,I,J,K] :
                ( F = k1_mcart_2(G,H,I,J,K)
                & ~ ( k3_mcart_2(A,B,C,D,E,F) = G
                    & k4_mcart_2(A,B,C,D,E,F) = H
                    & k5_mcart_2(A,B,C,D,E,F) = I
                    & k6_mcart_2(A,B,C,D,E,F) = J
                    & k7_mcart_2(A,B,C,D,E,F) = K ) ) ) ) )).

fof(t19_mcart_2,axiom,(
    ! [A,B,C,D,E] :
      ~ ( A != k1_xboole_0
        & B != k1_xboole_0
        & C != k1_xboole_0
        & D != k1_xboole_0
        & E != k1_xboole_0
        & ~ ! [F] :
              ( m1_subset_1(F,k2_mcart_2(A,B,C,D,E))
             => F = k1_mcart_2(k3_mcart_2(A,B,C,D,E,F),k4_mcart_2(A,B,C,D,E,F),k5_mcart_2(A,B,C,D,E,F),k6_mcart_2(A,B,C,D,E,F),k7_mcart_2(A,B,C,D,E,F)) ) ) )).

fof(t20_mcart_2,axiom,(
    ! [A,B,C,D,E] :
      ~ ( A != k1_xboole_0
        & B != k1_xboole_0
        & C != k1_xboole_0
        & D != k1_xboole_0
        & E != k1_xboole_0
        & ~ ! [F] :
              ( m1_subset_1(F,k2_mcart_2(A,B,C,D,E))
             => ( k3_mcart_2(A,B,C,D,E,F) = k1_mcart_1(k1_mcart_1(k1_mcart_1(k1_mcart_1(F))))
                & k4_mcart_2(A,B,C,D,E,F) = k2_mcart_1(k1_mcart_1(k1_mcart_1(k1_mcart_1(F))))
                & k5_mcart_2(A,B,C,D,E,F) = k2_mcart_1(k1_mcart_1(k1_mcart_1(F)))
                & k6_mcart_2(A,B,C,D,E,F) = k2_mcart_1(k1_mcart_1(F))
                & k7_mcart_2(A,B,C,D,E,F) = k2_mcart_1(F) ) ) ) )).

fof(t21_mcart_2,axiom,(
    ! [A,B,C,D,E] :
      ( ~ ( ~ r1_tarski(A,k2_mcart_2(A,B,C,D,E))
          & ~ r1_tarski(A,k2_mcart_2(B,C,D,E,A))
          & ~ r1_tarski(A,k2_mcart_2(C,D,E,A,B))
          & ~ r1_tarski(A,k2_mcart_2(D,E,A,B,C))
          & ~ r1_tarski(A,k2_mcart_2(E,A,B,C,D)) )
     => A = k1_xboole_0 ) )).

fof(t22_mcart_2,axiom,(
    ! [A,B,C,D,E,F,G,H,I,J] :
      ( ~ r1_xboole_0(k2_mcart_2(A,B,C,D,E),k2_mcart_2(F,G,H,I,J))
     => ( ~ r1_xboole_0(A,F)
        & ~ r1_xboole_0(B,G)
        & ~ r1_xboole_0(C,H)
        & ~ r1_xboole_0(D,I)
        & ~ r1_xboole_0(E,J) ) ) )).

fof(t23_mcart_2,axiom,(
    ! [A,B,C,D,E] : k2_mcart_2(k1_tarski(A),k1_tarski(B),k1_tarski(C),k1_tarski(D),k1_tarski(E)) = k1_tarski(k1_mcart_2(A,B,C,D,E)) )).

fof(t24_mcart_2,axiom,(
    ! [A,B,C,D,E,F] :
      ( m1_subset_1(F,k2_mcart_2(A,B,C,D,E))
     => ~ ( A != k1_xboole_0
          & B != k1_xboole_0
          & C != k1_xboole_0
          & D != k1_xboole_0
          & E != k1_xboole_0
          & ? [G,H,I,J,K] :
              ( F = k1_mcart_2(G,H,I,J,K)
              & ~ ( k3_mcart_2(A,B,C,D,E,F) = G
                  & k4_mcart_2(A,B,C,D,E,F) = H
                  & k5_mcart_2(A,B,C,D,E,F) = I
                  & k6_mcart_2(A,B,C,D,E,F) = J
                  & k7_mcart_2(A,B,C,D,E,F) = K ) ) ) ) )).

fof(t25_mcart_2,axiom,(
    ! [A,B,C,D,E,F,G] :
      ( m1_subset_1(G,k2_mcart_2(A,B,C,D,E))
     => ( ! [H] :
            ( m1_subset_1(H,A)
           => ! [I] :
                ( m1_subset_1(I,B)
               => ! [J] :
                    ( m1_subset_1(J,C)
                   => ! [K] :
                        ( m1_subset_1(K,D)
                       => ! [L] :
                            ( m1_subset_1(L,E)
                           => ( G = k1_mcart_2(H,I,J,K,L)
                             => F = H ) ) ) ) ) )
       => ( A = k1_xboole_0
          | B = k1_xboole_0
          | C = k1_xboole_0
          | D = k1_xboole_0
          | E = k1_xboole_0
          | F = k3_mcart_2(A,B,C,D,E,G) ) ) ) )).

fof(t26_mcart_2,axiom,(
    ! [A,B,C,D,E,F,G] :
      ( m1_subset_1(G,k2_mcart_2(A,B,C,D,E))
     => ( ! [H] :
            ( m1_subset_1(H,A)
           => ! [I] :
                ( m1_subset_1(I,B)
               => ! [J] :
                    ( m1_subset_1(J,C)
                   => ! [K] :
                        ( m1_subset_1(K,D)
                       => ! [L] :
                            ( m1_subset_1(L,E)
                           => ( G = k1_mcart_2(H,I,J,K,L)
                             => F = I ) ) ) ) ) )
       => ( A = k1_xboole_0
          | B = k1_xboole_0
          | C = k1_xboole_0
          | D = k1_xboole_0
          | E = k1_xboole_0
          | F = k4_mcart_2(A,B,C,D,E,G) ) ) ) )).

fof(t27_mcart_2,axiom,(
    ! [A,B,C,D,E,F,G] :
      ( m1_subset_1(G,k2_mcart_2(A,B,C,D,E))
     => ( ! [H] :
            ( m1_subset_1(H,A)
           => ! [I] :
                ( m1_subset_1(I,B)
               => ! [J] :
                    ( m1_subset_1(J,C)
                   => ! [K] :
                        ( m1_subset_1(K,D)
                       => ! [L] :
                            ( m1_subset_1(L,E)
                           => ( G = k1_mcart_2(H,I,J,K,L)
                             => F = J ) ) ) ) ) )
       => ( A = k1_xboole_0
          | B = k1_xboole_0
          | C = k1_xboole_0
          | D = k1_xboole_0
          | E = k1_xboole_0
          | F = k5_mcart_2(A,B,C,D,E,G) ) ) ) )).

fof(t28_mcart_2,axiom,(
    ! [A,B,C,D,E,F,G] :
      ( m1_subset_1(G,k2_mcart_2(A,B,C,D,E))
     => ( ! [H] :
            ( m1_subset_1(H,A)
           => ! [I] :
                ( m1_subset_1(I,B)
               => ! [J] :
                    ( m1_subset_1(J,C)
                   => ! [K] :
                        ( m1_subset_1(K,D)
                       => ! [L] :
                            ( m1_subset_1(L,E)
                           => ( G = k1_mcart_2(H,I,J,K,L)
                             => F = K ) ) ) ) ) )
       => ( A = k1_xboole_0
          | B = k1_xboole_0
          | C = k1_xboole_0
          | D = k1_xboole_0
          | E = k1_xboole_0
          | F = k6_mcart_2(A,B,C,D,E,G) ) ) ) )).

fof(t29_mcart_2,axiom,(
    ! [A,B,C,D,E,F,G] :
      ( m1_subset_1(G,k2_mcart_2(A,B,C,D,E))
     => ( ! [H] :
            ( m1_subset_1(H,A)
           => ! [I] :
                ( m1_subset_1(I,B)
               => ! [J] :
                    ( m1_subset_1(J,C)
                   => ! [K] :
                        ( m1_subset_1(K,D)
                       => ! [L] :
                            ( m1_subset_1(L,E)
                           => ( G = k1_mcart_2(H,I,J,K,L)
                             => F = L ) ) ) ) ) )
       => ( A = k1_xboole_0
          | B = k1_xboole_0
          | C = k1_xboole_0
          | D = k1_xboole_0
          | E = k1_xboole_0
          | F = k7_mcart_2(A,B,C,D,E,G) ) ) ) )).

fof(t30_mcart_2,axiom,(
    ! [A,B,C,D,E,F] :
      ~ ( r2_hidden(A,k2_mcart_2(B,C,D,E,F))
        & ! [G,H,I,J,K] :
            ~ ( r2_hidden(G,B)
              & r2_hidden(H,C)
              & r2_hidden(I,D)
              & r2_hidden(J,E)
              & r2_hidden(K,F)
              & A = k1_mcart_2(G,H,I,J,K) ) ) )).

fof(t31_mcart_2,axiom,(
    ! [A,B,C,D,E,F,G,H,I,J] :
      ( r2_hidden(k1_mcart_2(A,B,C,D,E),k2_mcart_2(F,G,H,I,J))
    <=> ( r2_hidden(A,F)
        & r2_hidden(B,G)
        & r2_hidden(C,H)
        & r2_hidden(D,I)
        & r2_hidden(E,J) ) ) )).

fof(t32_mcart_2,axiom,(
    ! [A,B,C,D,E,F] :
      ( ! [G] :
          ( r2_hidden(G,A)
        <=> ? [H,I,J,K,L] :
              ( r2_hidden(H,B)
              & r2_hidden(I,C)
              & r2_hidden(J,D)
              & r2_hidden(K,E)
              & r2_hidden(L,F)
              & G = k1_mcart_2(H,I,J,K,L) ) )
     => A = k2_mcart_2(B,C,D,E,F) ) )).

fof(t33_mcart_2,axiom,(
    ! [A,B,C,D,E,F,G,H,I,J] :
      ~ ( A != k1_xboole_0
        & B != k1_xboole_0
        & C != k1_xboole_0
        & D != k1_xboole_0
        & E != k1_xboole_0
        & F != k1_xboole_0
        & G != k1_xboole_0
        & H != k1_xboole_0
        & I != k1_xboole_0
        & J != k1_xboole_0
        & ? [K] :
            ( m1_subset_1(K,k2_mcart_2(A,B,C,D,E))
            & ? [L] :
                ( m1_subset_1(L,k2_mcart_2(F,G,H,I,J))
                & K = L
                & ~ ( k3_mcart_2(A,B,C,D,E,K) = k3_mcart_2(F,G,H,I,J,L)
                    & k4_mcart_2(A,B,C,D,E,K) = k4_mcart_2(F,G,H,I,J,L)
                    & k5_mcart_2(A,B,C,D,E,K) = k5_mcart_2(F,G,H,I,J,L)
                    & k6_mcart_2(A,B,C,D,E,K) = k6_mcart_2(F,G,H,I,J,L)
                    & k7_mcart_2(A,B,C,D,E,K) = k7_mcart_2(F,G,H,I,J,L) ) ) ) ) )).

fof(t34_mcart_2,axiom,(
    ! [A,B,C,D,E,F] :
      ( m1_subset_1(F,k1_zfmisc_1(A))
     => ! [G] :
          ( m1_subset_1(G,k1_zfmisc_1(B))
         => ! [H] :
              ( m1_subset_1(H,k1_zfmisc_1(C))
             => ! [I] :
                  ( m1_subset_1(I,k1_zfmisc_1(D))
                 => ! [J] :
                      ( m1_subset_1(J,k1_zfmisc_1(E))
                     => ! [K] :
                          ( m1_subset_1(K,k2_mcart_2(A,B,C,D,E))
                         => ( r2_hidden(K,k2_mcart_2(F,G,H,I,J))
                           => ( r2_hidden(k3_mcart_2(A,B,C,D,E,K),F)
                              & r2_hidden(k4_mcart_2(A,B,C,D,E,K),G)
                              & r2_hidden(k5_mcart_2(A,B,C,D,E,K),H)
                              & r2_hidden(k6_mcart_2(A,B,C,D,E,K),I)
                              & r2_hidden(k7_mcart_2(A,B,C,D,E,K),J) ) ) ) ) ) ) ) ) )).

fof(t35_mcart_2,axiom,(
    ! [A,B,C,D,E,F,G,H,I,J] :
      ( ( r1_tarski(A,B)
        & r1_tarski(C,D)
        & r1_tarski(E,F)
        & r1_tarski(G,H)
        & r1_tarski(I,J) )
     => r1_tarski(k2_mcart_2(A,C,E,G,I),k2_mcart_2(B,D,F,H,J)) ) )).

fof(t36_mcart_2,axiom,(
    ! [A,B] :
      ~ ( A != k1_xboole_0
        & B != k1_xboole_0
        & ? [C] :
            ( m1_subset_1(C,k2_zfmisc_1(A,B))
            & ! [D] :
                ( m1_subset_1(D,A)
               => ! [E] :
                    ( m1_subset_1(E,B)
                   => C != k4_tarski(D,E) ) ) ) ) )).

fof(t37_mcart_2,axiom,(
    ! [A,B,C] :
      ~ ( A != k1_xboole_0
        & B != k1_xboole_0
        & C != k1_xboole_0
        & ? [D] :
            ( m1_subset_1(D,k3_zfmisc_1(A,B,C))
            & ! [E] :
                ( m1_subset_1(E,A)
               => ! [F] :
                    ( m1_subset_1(F,B)
                   => ! [G] :
                        ( m1_subset_1(G,C)
                       => D != k3_mcart_1(E,F,G) ) ) ) ) ) )).

fof(t38_mcart_2,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))
            & ! [F] :
                ( m1_subset_1(F,A)
               => ! [G] :
                    ( m1_subset_1(G,B)
                   => ! [H] :
                        ( m1_subset_1(H,C)
                       => ! [I] :
                            ( m1_subset_1(I,D)
                           => E != k4_mcart_1(F,G,H,I) ) ) ) ) ) ) )).

fof(dt_k1_mcart_2,axiom,(
    $true )).

fof(dt_k2_mcart_2,axiom,(
    $true )).

fof(dt_k3_mcart_2,axiom,(
    ! [A,B,C,D,E,F] :
      ( m1_subset_1(F,k2_mcart_2(A,B,C,D,E))
     => m1_subset_1(k3_mcart_2(A,B,C,D,E,F),A) ) )).

fof(dt_k4_mcart_2,axiom,(
    ! [A,B,C,D,E,F] :
      ( m1_subset_1(F,k2_mcart_2(A,B,C,D,E))
     => m1_subset_1(k4_mcart_2(A,B,C,D,E,F),B) ) )).

fof(dt_k5_mcart_2,axiom,(
    ! [A,B,C,D,E,F] :
      ( m1_subset_1(F,k2_mcart_2(A,B,C,D,E))
     => m1_subset_1(k5_mcart_2(A,B,C,D,E,F),C) ) )).

fof(dt_k6_mcart_2,axiom,(
    ! [A,B,C,D,E,F] :
      ( m1_subset_1(F,k2_mcart_2(A,B,C,D,E))
     => m1_subset_1(k6_mcart_2(A,B,C,D,E,F),D) ) )).

fof(dt_k7_mcart_2,axiom,(
    ! [A,B,C,D,E,F] :
      ( m1_subset_1(F,k2_mcart_2(A,B,C,D,E))
     => m1_subset_1(k7_mcart_2(A,B,C,D,E,F),E) ) )).

fof(dt_k8_mcart_2,axiom,(
    ! [A,B,C,D,E,F,G,H,I,J] :
      ( ( m1_subset_1(F,k1_zfmisc_1(A))
        & m1_subset_1(G,k1_zfmisc_1(B))
        & m1_subset_1(H,k1_zfmisc_1(C))
        & m1_subset_1(I,k1_zfmisc_1(D))
        & m1_subset_1(J,k1_zfmisc_1(E)) )
     => m1_subset_1(k8_mcart_2(A,B,C,D,E,F,G,H,I,J),k1_zfmisc_1(k2_mcart_2(A,B,C,D,E))) ) )).

fof(redefinition_k8_mcart_2,axiom,(
    ! [A,B,C,D,E,F,G,H,I,J] :
      ( ( m1_subset_1(F,k1_zfmisc_1(A))
        & m1_subset_1(G,k1_zfmisc_1(B))
        & m1_subset_1(H,k1_zfmisc_1(C))
        & m1_subset_1(I,k1_zfmisc_1(D))
        & m1_subset_1(J,k1_zfmisc_1(E)) )
     => k8_mcart_2(A,B,C,D,E,F,G,H,I,J) = k2_mcart_2(F,G,H,I,J) ) )).
%------------------------------------------------------------------------------