SET007 Axioms: SET007+507.ax


%------------------------------------------------------------------------------
% File     : SET007+507 : TPTP v7.5.0. Released v3.4.0.
% Domain   : Set Theory
% Axioms   : Basic Properties of Objects and Morphisms
% 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    : altcat_3 [Urb08]

% Status   : Satisfiable
% Syntax   : Number of formulae    :   52 (   0 unit)
%            Number of atoms       :  592 (  67 equality)
%            Maximal formula depth :   20 (  14 average)
%            Number of connectives :  651 ( 111 ~  ;  13  |; 302  &)
%                                         (  17 <=>; 208 =>;   0 <=)
%                                         (   0 <~>;   0 ~|;   0 ~&)
%            Number of predicates  :   28 (   0 propositional; 1-5 arity)
%            Number of functors    :    7 (   1 constant; 0-6 arity)
%            Number of variables   :  235 (   0 singleton; 208 !;  27 ?)
%            Maximal term depth    :    3 (   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(rc1_altcat_3,axiom,(
    ! [A,B] :
      ( ( ~ v3_struct_0(A)
        & v12_altcat_1(A)
        & l2_altcat_1(A)
        & m1_subset_1(B,u1_struct_0(A)) )
     => ? [C] :
          ( m1_subset_1(C,k1_altcat_1(A,B,B))
          & v1_altcat_3(C,A,B,B)
          & v2_altcat_3(C,A,B,B)
          & v4_altcat_3(C,A,B,B)
          & v5_altcat_3(C,A,B,B) ) ) )).

fof(rc2_altcat_3,axiom,(
    ! [A,B] :
      ( ( ~ v3_struct_0(A)
        & v2_altcat_1(A)
        & v11_altcat_1(A)
        & v12_altcat_1(A)
        & l2_altcat_1(A)
        & m1_subset_1(B,u1_struct_0(A)) )
     => ? [C] :
          ( m1_subset_1(C,k1_altcat_1(A,B,B))
          & v1_altcat_3(C,A,B,B)
          & v2_altcat_3(C,A,B,B)
          & v3_altcat_3(C,A,B,B)
          & v4_altcat_3(C,A,B,B)
          & v5_altcat_3(C,A,B,B) ) ) )).

fof(fc1_altcat_3,axiom,(
    ! [A,B,C,D] :
      ( ( ~ v3_struct_0(A)
        & v2_altcat_1(A)
        & v11_altcat_1(A)
        & v12_altcat_1(A)
        & l2_altcat_1(A)
        & m1_subset_1(B,u1_struct_0(A))
        & v4_altcat_3(C,A,B,B)
        & m1_subset_1(C,k1_altcat_1(A,B,B))
        & v4_altcat_3(D,A,B,B)
        & m1_subset_1(D,k1_altcat_1(A,B,B)) )
     => v4_altcat_3(k5_altcat_1(A,B,B,B,D,C),A,B,B) ) )).

fof(fc2_altcat_3,axiom,(
    ! [A,B,C,D] :
      ( ( ~ v3_struct_0(A)
        & v2_altcat_1(A)
        & v11_altcat_1(A)
        & v12_altcat_1(A)
        & l2_altcat_1(A)
        & m1_subset_1(B,u1_struct_0(A))
        & v5_altcat_3(C,A,B,B)
        & m1_subset_1(C,k1_altcat_1(A,B,B))
        & v5_altcat_3(D,A,B,B)
        & m1_subset_1(D,k1_altcat_1(A,B,B)) )
     => v5_altcat_3(k5_altcat_1(A,B,B,B,D,C),A,B,B) ) )).

fof(fc3_altcat_3,axiom,(
    ! [A,B,C,D] :
      ( ( ~ v3_struct_0(A)
        & v2_altcat_1(A)
        & v11_altcat_1(A)
        & v12_altcat_1(A)
        & l2_altcat_1(A)
        & m1_subset_1(B,u1_struct_0(A))
        & v3_altcat_3(C,A,B,B)
        & m1_subset_1(C,k1_altcat_1(A,B,B))
        & v3_altcat_3(D,A,B,B)
        & m1_subset_1(D,k1_altcat_1(A,B,B)) )
     => v3_altcat_3(k5_altcat_1(A,B,B,B,D,C),A,B,B) ) )).

fof(fc4_altcat_3,axiom,(
    ! [A,B,C,D] :
      ( ( ~ v3_struct_0(A)
        & v2_altcat_1(A)
        & v11_altcat_1(A)
        & v12_altcat_1(A)
        & l2_altcat_1(A)
        & m1_subset_1(B,u1_struct_0(A))
        & v1_altcat_3(C,A,B,B)
        & m1_subset_1(C,k1_altcat_1(A,B,B))
        & v1_altcat_3(D,A,B,B)
        & m1_subset_1(D,k1_altcat_1(A,B,B)) )
     => v1_altcat_3(k5_altcat_1(A,B,B,B,D,C),A,B,B) ) )).

fof(fc5_altcat_3,axiom,(
    ! [A,B,C,D] :
      ( ( ~ v3_struct_0(A)
        & v2_altcat_1(A)
        & v11_altcat_1(A)
        & v12_altcat_1(A)
        & l2_altcat_1(A)
        & m1_subset_1(B,u1_struct_0(A))
        & v2_altcat_3(C,A,B,B)
        & m1_subset_1(C,k1_altcat_1(A,B,B))
        & v2_altcat_3(D,A,B,B)
        & m1_subset_1(D,k1_altcat_1(A,B,B)) )
     => v2_altcat_3(k5_altcat_1(A,B,B,B,D,C),A,B,B) ) )).

fof(d1_altcat_3,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v12_altcat_1(A)
        & l2_altcat_1(A) )
     => ! [B] :
          ( m1_subset_1(B,u1_struct_0(A))
         => ! [C] :
              ( m1_subset_1(C,u1_struct_0(A))
             => ! [D] :
                  ( m1_subset_1(D,k1_altcat_1(A,B,C))
                 => ! [E] :
                      ( m1_subset_1(E,k1_altcat_1(A,C,B))
                     => ( r1_altcat_3(A,B,C,D,E)
                      <=> k5_altcat_1(A,C,B,C,E,D) = k8_altcat_1(A,C) ) ) ) ) ) ) )).

fof(d2_altcat_3,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v12_altcat_1(A)
        & l2_altcat_1(A) )
     => ! [B] :
          ( m1_subset_1(B,u1_struct_0(A))
         => ! [C] :
              ( m1_subset_1(C,u1_struct_0(A))
             => ! [D] :
                  ( m1_subset_1(D,k1_altcat_1(A,B,C))
                 => ( v1_altcat_3(D,A,B,C)
                  <=> ? [E] :
                        ( m1_subset_1(E,k1_altcat_1(A,C,B))
                        & r1_altcat_3(A,B,C,D,E) ) ) ) ) ) ) )).

fof(d3_altcat_3,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v12_altcat_1(A)
        & l2_altcat_1(A) )
     => ! [B] :
          ( m1_subset_1(B,u1_struct_0(A))
         => ! [C] :
              ( m1_subset_1(C,u1_struct_0(A))
             => ! [D] :
                  ( m1_subset_1(D,k1_altcat_1(A,B,C))
                 => ( v2_altcat_3(D,A,B,C)
                  <=> ? [E] :
                        ( m1_subset_1(E,k1_altcat_1(A,C,B))
                        & r1_altcat_3(A,C,B,E,D) ) ) ) ) ) ) )).

fof(t1_altcat_3,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v12_altcat_1(A)
        & l2_altcat_1(A) )
     => ! [B] :
          ( m1_subset_1(B,u1_struct_0(A))
         => ( v1_altcat_3(k8_altcat_1(A,B),A,B,B)
            & v2_altcat_3(k8_altcat_1(A,B),A,B,B) ) ) ) )).

fof(d4_altcat_3,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v2_altcat_1(A)
        & v11_altcat_1(A)
        & v12_altcat_1(A)
        & l2_altcat_1(A) )
     => ! [B] :
          ( m1_subset_1(B,u1_struct_0(A))
         => ! [C] :
              ( m1_subset_1(C,u1_struct_0(A))
             => ~ ( k1_altcat_1(A,B,C) != k1_xboole_0
                  & k1_altcat_1(A,C,B) != k1_xboole_0
                  & ? [D] :
                      ( m1_subset_1(D,k1_altcat_1(A,B,C))
                      & v1_altcat_3(D,A,B,C)
                      & v2_altcat_3(D,A,B,C)
                      & ~ ! [E] :
                            ( m1_subset_1(E,k1_altcat_1(A,C,B))
                           => ( E = k1_altcat_3(A,B,C,D)
                            <=> ( r1_altcat_3(A,C,B,E,D)
                                & r1_altcat_3(A,B,C,D,E) ) ) ) ) ) ) ) ) )).

fof(t2_altcat_3,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v2_altcat_1(A)
        & v11_altcat_1(A)
        & v12_altcat_1(A)
        & l2_altcat_1(A) )
     => ! [B] :
          ( m1_subset_1(B,u1_struct_0(A))
         => ! [C] :
              ( m1_subset_1(C,u1_struct_0(A))
             => ~ ( k1_altcat_1(A,B,C) != k1_xboole_0
                  & k1_altcat_1(A,C,B) != k1_xboole_0
                  & ? [D] :
                      ( m1_subset_1(D,k1_altcat_1(A,B,C))
                      & v1_altcat_3(D,A,B,C)
                      & v2_altcat_3(D,A,B,C)
                      & ~ ( k5_altcat_1(A,B,C,B,D,k1_altcat_3(A,B,C,D)) = k8_altcat_1(A,B)
                          & k5_altcat_1(A,C,B,C,k1_altcat_3(A,B,C,D),D) = k8_altcat_1(A,C) ) ) ) ) ) ) )).

fof(t3_altcat_3,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v2_altcat_1(A)
        & v11_altcat_1(A)
        & v12_altcat_1(A)
        & l2_altcat_1(A) )
     => ! [B] :
          ( m1_subset_1(B,u1_struct_0(A))
         => ! [C] :
              ( m1_subset_1(C,u1_struct_0(A))
             => ~ ( k1_altcat_1(A,B,C) != k1_xboole_0
                  & k1_altcat_1(A,C,B) != k1_xboole_0
                  & ? [D] :
                      ( m1_subset_1(D,k1_altcat_1(A,B,C))
                      & v1_altcat_3(D,A,B,C)
                      & v2_altcat_3(D,A,B,C)
                      & k1_altcat_3(A,C,B,k1_altcat_3(A,B,C,D)) != D ) ) ) ) ) )).

fof(t4_altcat_3,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v2_altcat_1(A)
        & v11_altcat_1(A)
        & v12_altcat_1(A)
        & l2_altcat_1(A) )
     => ! [B] :
          ( m1_subset_1(B,u1_struct_0(A))
         => k1_altcat_3(A,B,B,k8_altcat_1(A,B)) = k8_altcat_1(A,B) ) ) )).

fof(d5_altcat_3,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v2_altcat_1(A)
        & v11_altcat_1(A)
        & v12_altcat_1(A)
        & l2_altcat_1(A) )
     => ! [B] :
          ( m1_subset_1(B,u1_struct_0(A))
         => ! [C] :
              ( m1_subset_1(C,u1_struct_0(A))
             => ! [D] :
                  ( m1_subset_1(D,k1_altcat_1(A,B,C))
                 => ( v3_altcat_3(D,A,B,C)
                  <=> ( k5_altcat_1(A,C,B,C,k1_altcat_3(A,B,C,D),D) = k8_altcat_1(A,C)
                      & k5_altcat_1(A,B,C,B,D,k1_altcat_3(A,B,C,D)) = k8_altcat_1(A,B) ) ) ) ) ) ) )).

fof(t5_altcat_3,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v2_altcat_1(A)
        & v11_altcat_1(A)
        & v12_altcat_1(A)
        & l2_altcat_1(A) )
     => ! [B] :
          ( m1_subset_1(B,u1_struct_0(A))
         => ! [C] :
              ( m1_subset_1(C,u1_struct_0(A))
             => ! [D] :
                  ( m1_subset_1(D,k1_altcat_1(A,B,C))
                 => ( v3_altcat_3(D,A,B,C)
                   => ( v1_altcat_3(D,A,B,C)
                      & v2_altcat_3(D,A,B,C) ) ) ) ) ) ) )).

fof(t6_altcat_3,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v2_altcat_1(A)
        & v11_altcat_1(A)
        & v12_altcat_1(A)
        & l2_altcat_1(A) )
     => ! [B] :
          ( m1_subset_1(B,u1_struct_0(A))
         => ! [C] :
              ( m1_subset_1(C,u1_struct_0(A))
             => ~ ( k1_altcat_1(A,B,C) != k1_xboole_0
                  & k1_altcat_1(A,C,B) != k1_xboole_0
                  & ~ ! [D] :
                        ( m1_subset_1(D,k1_altcat_1(A,B,C))
                       => ( v3_altcat_3(D,A,B,C)
                        <=> ( v1_altcat_3(D,A,B,C)
                            & v2_altcat_3(D,A,B,C) ) ) ) ) ) ) ) )).

fof(t7_altcat_3,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v2_altcat_1(A)
        & v11_altcat_1(A)
        & v12_altcat_1(A)
        & l2_altcat_1(A) )
     => ! [B] :
          ( m1_subset_1(B,u1_struct_0(A))
         => ! [C] :
              ( m1_subset_1(C,u1_struct_0(A))
             => ! [D] :
                  ( m1_subset_1(D,u1_struct_0(A))
                 => ! [E] :
                      ( m1_subset_1(E,k1_altcat_1(A,B,C))
                     => ! [F] :
                          ( m1_subset_1(F,k1_altcat_1(A,C,D))
                         => ( ( v3_altcat_3(E,A,B,C)
                              & v3_altcat_3(F,A,C,D) )
                           => ( k1_altcat_1(A,B,C) = k1_xboole_0
                              | k1_altcat_1(A,C,D) = k1_xboole_0
                              | k1_altcat_1(A,D,B) = k1_xboole_0
                              | ( v3_altcat_3(k5_altcat_1(A,B,C,D,E,F),A,B,D)
                                & k1_altcat_3(A,B,D,k5_altcat_1(A,B,C,D,E,F)) = k5_altcat_1(A,D,C,B,k1_altcat_3(A,C,D,F),k1_altcat_3(A,B,C,E)) ) ) ) ) ) ) ) ) ) )).

fof(d6_altcat_3,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v2_altcat_1(A)
        & v11_altcat_1(A)
        & v12_altcat_1(A)
        & l2_altcat_1(A) )
     => ! [B] :
          ( m1_subset_1(B,u1_struct_0(A))
         => ! [C] :
              ( m1_subset_1(C,u1_struct_0(A))
             => ( r2_altcat_3(A,B,C)
              <=> ( k1_altcat_1(A,B,C) != k1_xboole_0
                  & k1_altcat_1(A,C,B) != k1_xboole_0
                  & ? [D] :
                      ( m1_subset_1(D,k1_altcat_1(A,B,C))
                      & v3_altcat_3(D,A,B,C) ) ) ) ) ) ) )).

fof(t8_altcat_3,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v2_altcat_1(A)
        & v11_altcat_1(A)
        & v12_altcat_1(A)
        & l2_altcat_1(A) )
     => ! [B] :
          ( m1_subset_1(B,u1_struct_0(A))
         => ! [C] :
              ( m1_subset_1(C,u1_struct_0(A))
             => ! [D] :
                  ( m1_subset_1(D,u1_struct_0(A))
                 => ( ( r2_altcat_3(A,B,C)
                      & r2_altcat_3(A,C,D) )
                   => r2_altcat_3(A,B,D) ) ) ) ) ) )).

fof(d7_altcat_3,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & l2_altcat_1(A) )
     => ! [B] :
          ( m1_subset_1(B,u1_struct_0(A))
         => ! [C] :
              ( m1_subset_1(C,u1_struct_0(A))
             => ! [D] :
                  ( m1_subset_1(D,k1_altcat_1(A,B,C))
                 => ( v4_altcat_3(D,A,B,C)
                  <=> ! [E] :
                        ( m1_subset_1(E,u1_struct_0(A))
                       => ( k1_altcat_1(A,E,B) != k1_xboole_0
                         => ! [F] :
                              ( m1_subset_1(F,k1_altcat_1(A,E,B))
                             => ! [G] :
                                  ( m1_subset_1(G,k1_altcat_1(A,E,B))
                                 => ( k5_altcat_1(A,E,B,C,F,D) = k5_altcat_1(A,E,B,C,G,D)
                                   => F = G ) ) ) ) ) ) ) ) ) ) )).

fof(d8_altcat_3,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & l2_altcat_1(A) )
     => ! [B] :
          ( m1_subset_1(B,u1_struct_0(A))
         => ! [C] :
              ( m1_subset_1(C,u1_struct_0(A))
             => ! [D] :
                  ( m1_subset_1(D,k1_altcat_1(A,B,C))
                 => ( v5_altcat_3(D,A,B,C)
                  <=> ! [E] :
                        ( m1_subset_1(E,u1_struct_0(A))
                       => ( k1_altcat_1(A,C,E) != k1_xboole_0
                         => ! [F] :
                              ( m1_subset_1(F,k1_altcat_1(A,C,E))
                             => ! [G] :
                                  ( m1_subset_1(G,k1_altcat_1(A,C,E))
                                 => ( k5_altcat_1(A,B,C,E,D,F) = k5_altcat_1(A,B,C,E,D,G)
                                   => F = G ) ) ) ) ) ) ) ) ) ) )).

fof(t9_altcat_3,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v2_altcat_1(A)
        & v11_altcat_1(A)
        & l2_altcat_1(A) )
     => ! [B] :
          ( m1_subset_1(B,u1_struct_0(A))
         => ! [C] :
              ( m1_subset_1(C,u1_struct_0(A))
             => ! [D] :
                  ( m1_subset_1(D,u1_struct_0(A))
                 => ~ ( k1_altcat_1(A,B,C) != k1_xboole_0
                      & k1_altcat_1(A,C,D) != k1_xboole_0
                      & ? [E] :
                          ( m1_subset_1(E,k1_altcat_1(A,B,C))
                          & ? [F] :
                              ( m1_subset_1(F,k1_altcat_1(A,C,D))
                              & v4_altcat_3(E,A,B,C)
                              & v4_altcat_3(F,A,C,D)
                              & ~ v4_altcat_3(k5_altcat_1(A,B,C,D,E,F),A,B,D) ) ) ) ) ) ) ) )).

fof(t10_altcat_3,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v2_altcat_1(A)
        & v11_altcat_1(A)
        & l2_altcat_1(A) )
     => ! [B] :
          ( m1_subset_1(B,u1_struct_0(A))
         => ! [C] :
              ( m1_subset_1(C,u1_struct_0(A))
             => ! [D] :
                  ( m1_subset_1(D,u1_struct_0(A))
                 => ~ ( k1_altcat_1(A,B,C) != k1_xboole_0
                      & k1_altcat_1(A,C,D) != k1_xboole_0
                      & ? [E] :
                          ( m1_subset_1(E,k1_altcat_1(A,B,C))
                          & ? [F] :
                              ( m1_subset_1(F,k1_altcat_1(A,C,D))
                              & v5_altcat_3(E,A,B,C)
                              & v5_altcat_3(F,A,C,D)
                              & ~ v5_altcat_3(k5_altcat_1(A,B,C,D,E,F),A,B,D) ) ) ) ) ) ) ) )).

fof(t11_altcat_3,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v2_altcat_1(A)
        & v11_altcat_1(A)
        & l2_altcat_1(A) )
     => ! [B] :
          ( m1_subset_1(B,u1_struct_0(A))
         => ! [C] :
              ( m1_subset_1(C,u1_struct_0(A))
             => ! [D] :
                  ( m1_subset_1(D,u1_struct_0(A))
                 => ~ ( k1_altcat_1(A,B,C) != k1_xboole_0
                      & k1_altcat_1(A,C,D) != k1_xboole_0
                      & ? [E] :
                          ( m1_subset_1(E,k1_altcat_1(A,B,C))
                          & ? [F] :
                              ( m1_subset_1(F,k1_altcat_1(A,C,D))
                              & v4_altcat_3(k5_altcat_1(A,B,C,D,E,F),A,B,D)
                              & ~ v4_altcat_3(E,A,B,C) ) ) ) ) ) ) ) )).

fof(t12_altcat_3,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v2_altcat_1(A)
        & v11_altcat_1(A)
        & l2_altcat_1(A) )
     => ! [B] :
          ( m1_subset_1(B,u1_struct_0(A))
         => ! [C] :
              ( m1_subset_1(C,u1_struct_0(A))
             => ! [D] :
                  ( m1_subset_1(D,u1_struct_0(A))
                 => ~ ( k1_altcat_1(A,B,C) != k1_xboole_0
                      & k1_altcat_1(A,C,D) != k1_xboole_0
                      & ? [E] :
                          ( m1_subset_1(E,k1_altcat_1(A,B,C))
                          & ? [F] :
                              ( m1_subset_1(F,k1_altcat_1(A,C,D))
                              & v5_altcat_3(k5_altcat_1(A,B,C,D,E,F),A,B,D)
                              & ~ v5_altcat_3(F,A,C,D) ) ) ) ) ) ) ) )).

fof(t13_altcat_3,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( m1_subset_1(B,u1_struct_0(k7_altcat_1(A)))
         => ! [C] :
              ( m1_subset_1(C,u1_struct_0(k7_altcat_1(A)))
             => ( k1_altcat_1(k7_altcat_1(A),B,C) != k1_xboole_0
               => ! [D] :
                    ( m1_subset_1(D,k1_altcat_1(k7_altcat_1(A),B,C))
                   => ! [E] :
                        ( ( v1_funct_1(E)
                          & v1_funct_2(E,B,C)
                          & m2_relset_1(E,B,C) )
                       => ( E = D
                         => ( v4_altcat_3(D,k7_altcat_1(A),B,C)
                          <=> v2_funct_1(E) ) ) ) ) ) ) ) ) )).

fof(t14_altcat_3,axiom,(
    ! [A] :
      ( ( ~ v1_xboole_0(A)
        & v1_setfam_1(A) )
     => ! [B] :
          ( m1_subset_1(B,u1_struct_0(k7_altcat_1(A)))
         => ! [C] :
              ( m1_subset_1(C,u1_struct_0(k7_altcat_1(A)))
             => ( k1_altcat_1(k7_altcat_1(A),B,C) != k1_xboole_0
               => ! [D] :
                    ( m1_subset_1(D,k1_altcat_1(k7_altcat_1(A),B,C))
                   => ! [E] :
                        ( ( v1_funct_1(E)
                          & v1_funct_2(E,B,C)
                          & m2_relset_1(E,B,C) )
                       => ( E = D
                         => ( v5_altcat_3(D,k7_altcat_1(A),B,C)
                          <=> v2_funct_2(E,B,C) ) ) ) ) ) ) ) ) )).

fof(t15_altcat_3,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v2_altcat_1(A)
        & v11_altcat_1(A)
        & v12_altcat_1(A)
        & l2_altcat_1(A) )
     => ! [B] :
          ( m1_subset_1(B,u1_struct_0(A))
         => ! [C] :
              ( m1_subset_1(C,u1_struct_0(A))
             => ~ ( k1_altcat_1(A,B,C) != k1_xboole_0
                  & k1_altcat_1(A,C,B) != k1_xboole_0
                  & ? [D] :
                      ( m1_subset_1(D,k1_altcat_1(A,B,C))
                      & v1_altcat_3(D,A,B,C)
                      & ~ v5_altcat_3(D,A,B,C) ) ) ) ) ) )).

fof(t16_altcat_3,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v2_altcat_1(A)
        & v11_altcat_1(A)
        & v12_altcat_1(A)
        & l2_altcat_1(A) )
     => ! [B] :
          ( m1_subset_1(B,u1_struct_0(A))
         => ! [C] :
              ( m1_subset_1(C,u1_struct_0(A))
             => ~ ( k1_altcat_1(A,B,C) != k1_xboole_0
                  & k1_altcat_1(A,C,B) != k1_xboole_0
                  & ? [D] :
                      ( m1_subset_1(D,k1_altcat_1(A,B,C))
                      & v2_altcat_3(D,A,B,C)
                      & ~ v4_altcat_3(D,A,B,C) ) ) ) ) ) )).

fof(t17_altcat_3,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v2_altcat_1(A)
        & v11_altcat_1(A)
        & v12_altcat_1(A)
        & l2_altcat_1(A) )
     => ! [B] :
          ( m1_subset_1(B,u1_struct_0(A))
         => ! [C] :
              ( m1_subset_1(C,u1_struct_0(A))
             => ~ ( k1_altcat_1(A,B,C) != k1_xboole_0
                  & k1_altcat_1(A,C,B) != k1_xboole_0
                  & ? [D] :
                      ( m1_subset_1(D,k1_altcat_1(A,B,C))
                      & v3_altcat_3(D,A,B,C)
                      & ~ ( v4_altcat_3(D,A,B,C)
                          & v5_altcat_3(D,A,B,C) ) ) ) ) ) ) )).

fof(t18_altcat_3,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v2_altcat_1(A)
        & v11_altcat_1(A)
        & v12_altcat_1(A)
        & l2_altcat_1(A) )
     => ! [B] :
          ( m1_subset_1(B,u1_struct_0(A))
         => ! [C] :
              ( m1_subset_1(C,u1_struct_0(A))
             => ! [D] :
                  ( m1_subset_1(D,u1_struct_0(A))
                 => ~ ( k1_altcat_1(A,B,C) != k1_xboole_0
                      & k1_altcat_1(A,C,D) != k1_xboole_0
                      & k1_altcat_1(A,D,B) != k1_xboole_0
                      & ? [E] :
                          ( m1_subset_1(E,k1_altcat_1(A,B,C))
                          & ? [F] :
                              ( m1_subset_1(F,k1_altcat_1(A,C,D))
                              & v1_altcat_3(E,A,B,C)
                              & v1_altcat_3(F,A,C,D)
                              & ~ v1_altcat_3(k5_altcat_1(A,B,C,D,E,F),A,B,D) ) ) ) ) ) ) ) )).

fof(t19_altcat_3,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v2_altcat_1(A)
        & v11_altcat_1(A)
        & v12_altcat_1(A)
        & l2_altcat_1(A) )
     => ! [B] :
          ( m1_subset_1(B,u1_struct_0(A))
         => ! [C] :
              ( m1_subset_1(C,u1_struct_0(A))
             => ! [D] :
                  ( m1_subset_1(D,u1_struct_0(A))
                 => ~ ( k1_altcat_1(A,B,C) != k1_xboole_0
                      & k1_altcat_1(A,C,D) != k1_xboole_0
                      & k1_altcat_1(A,D,B) != k1_xboole_0
                      & ? [E] :
                          ( m1_subset_1(E,k1_altcat_1(A,B,C))
                          & ? [F] :
                              ( m1_subset_1(F,k1_altcat_1(A,C,D))
                              & v2_altcat_3(E,A,B,C)
                              & v2_altcat_3(F,A,C,D)
                              & ~ v2_altcat_3(k5_altcat_1(A,B,C,D,E,F),A,B,D) ) ) ) ) ) ) ) )).

fof(t20_altcat_3,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v2_altcat_1(A)
        & v11_altcat_1(A)
        & v12_altcat_1(A)
        & l2_altcat_1(A) )
     => ! [B] :
          ( m1_subset_1(B,u1_struct_0(A))
         => ! [C] :
              ( m1_subset_1(C,u1_struct_0(A))
             => ! [D] :
                  ( m1_subset_1(D,k1_altcat_1(A,B,C))
                 => ( ( v1_altcat_3(D,A,B,C)
                      & v4_altcat_3(D,A,B,C) )
                   => ( k1_altcat_1(A,B,C) = k1_xboole_0
                      | k1_altcat_1(A,C,B) = k1_xboole_0
                      | v3_altcat_3(D,A,B,C) ) ) ) ) ) ) )).

fof(t21_altcat_3,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v2_altcat_1(A)
        & v11_altcat_1(A)
        & v12_altcat_1(A)
        & l2_altcat_1(A) )
     => ! [B] :
          ( m1_subset_1(B,u1_struct_0(A))
         => ! [C] :
              ( m1_subset_1(C,u1_struct_0(A))
             => ! [D] :
                  ( m1_subset_1(D,k1_altcat_1(A,B,C))
                 => ( ( v2_altcat_3(D,A,B,C)
                      & v5_altcat_3(D,A,B,C) )
                   => ( k1_altcat_1(A,B,C) = k1_xboole_0
                      | k1_altcat_1(A,C,B) = k1_xboole_0
                      | v3_altcat_3(D,A,B,C) ) ) ) ) ) ) )).

fof(t22_altcat_3,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v2_altcat_1(A)
        & v11_altcat_1(A)
        & v12_altcat_1(A)
        & l2_altcat_1(A) )
     => ! [B] :
          ( m1_subset_1(B,u1_struct_0(A))
         => ! [C] :
              ( m1_subset_1(C,u1_struct_0(A))
             => ! [D] :
                  ( m1_subset_1(D,u1_struct_0(A))
                 => ! [E] :
                      ( m1_subset_1(E,k1_altcat_1(A,B,C))
                     => ! [F] :
                          ( m1_subset_1(F,k1_altcat_1(A,C,D))
                         => ( v1_altcat_3(k5_altcat_1(A,B,C,D,E,F),A,B,D)
                           => ( k1_altcat_1(A,B,C) = k1_xboole_0
                              | k1_altcat_1(A,C,D) = k1_xboole_0
                              | k1_altcat_1(A,D,B) = k1_xboole_0
                              | v1_altcat_3(F,A,C,D) ) ) ) ) ) ) ) ) )).

fof(t23_altcat_3,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v2_altcat_1(A)
        & v11_altcat_1(A)
        & v12_altcat_1(A)
        & l2_altcat_1(A) )
     => ! [B] :
          ( m1_subset_1(B,u1_struct_0(A))
         => ! [C] :
              ( m1_subset_1(C,u1_struct_0(A))
             => ! [D] :
                  ( m1_subset_1(D,u1_struct_0(A))
                 => ! [E] :
                      ( m1_subset_1(E,k1_altcat_1(A,B,C))
                     => ! [F] :
                          ( m1_subset_1(F,k1_altcat_1(A,C,D))
                         => ( v2_altcat_3(k5_altcat_1(A,B,C,D,E,F),A,B,D)
                           => ( k1_altcat_1(A,B,C) = k1_xboole_0
                              | k1_altcat_1(A,C,D) = k1_xboole_0
                              | k1_altcat_1(A,D,B) = k1_xboole_0
                              | v2_altcat_3(E,A,B,C) ) ) ) ) ) ) ) ) )).

fof(t24_altcat_3,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v2_altcat_1(A)
        & v11_altcat_1(A)
        & v12_altcat_1(A)
        & l2_altcat_1(A) )
     => ( ! [B] :
            ( m1_subset_1(B,u1_struct_0(A))
           => ! [C] :
                ( m1_subset_1(C,u1_struct_0(A))
               => ! [D] :
                    ( m1_subset_1(D,k1_altcat_1(A,B,C))
                   => v1_altcat_3(D,A,B,C) ) ) )
       => ! [B] :
            ( m1_subset_1(B,u1_struct_0(A))
           => ! [C] :
                ( m1_subset_1(C,u1_struct_0(A))
               => ! [D] :
                    ( m1_subset_1(D,k1_altcat_1(A,B,C))
                   => ~ ( k1_altcat_1(A,B,C) != k1_xboole_0
                        & k1_altcat_1(A,C,B) != k1_xboole_0
                        & ~ v3_altcat_3(D,A,B,C) ) ) ) ) ) ) )).

fof(d9_altcat_3,axiom,(
    ! [A] :
      ( l1_altcat_1(A)
     => ! [B] :
          ( m1_subset_1(B,u1_struct_0(A))
         => ( v6_altcat_3(B,A)
          <=> ! [C] :
                ( m1_subset_1(C,u1_struct_0(A))
               => ? [D] :
                    ( m1_subset_1(D,k1_altcat_1(A,B,C))
                    & r2_hidden(D,k1_altcat_1(A,B,C))
                    & v1_realset1(k1_altcat_1(A,B,C)) ) ) ) ) ) )).

fof(t25_altcat_3,axiom,(
    ! [A] :
      ( l1_altcat_1(A)
     => ! [B] :
          ( m1_subset_1(B,u1_struct_0(A))
         => ( v6_altcat_3(B,A)
          <=> ! [C] :
                ( m1_subset_1(C,u1_struct_0(A))
               => ? [D] :
                    ( m1_subset_1(D,k1_altcat_1(A,B,C))
                    & r2_hidden(D,k1_altcat_1(A,B,C))
                    & ! [E] :
                        ( m1_subset_1(E,k1_altcat_1(A,B,C))
                       => ( r2_hidden(E,k1_altcat_1(A,B,C))
                         => D = E ) ) ) ) ) ) ) )).

fof(t26_altcat_3,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v2_altcat_1(A)
        & v11_altcat_1(A)
        & v12_altcat_1(A)
        & l2_altcat_1(A) )
     => ! [B] :
          ( m1_subset_1(B,u1_struct_0(A))
         => ! [C] :
              ( m1_subset_1(C,u1_struct_0(A))
             => ( ( v6_altcat_3(B,A)
                  & v6_altcat_3(C,A) )
               => r2_altcat_3(A,B,C) ) ) ) ) )).

fof(d10_altcat_3,axiom,(
    ! [A] :
      ( l1_altcat_1(A)
     => ! [B] :
          ( m1_subset_1(B,u1_struct_0(A))
         => ( v7_altcat_3(B,A)
          <=> ! [C] :
                ( m1_subset_1(C,u1_struct_0(A))
               => ? [D] :
                    ( m1_subset_1(D,k1_altcat_1(A,C,B))
                    & r2_hidden(D,k1_altcat_1(A,C,B))
                    & v1_realset1(k1_altcat_1(A,C,B)) ) ) ) ) ) )).

fof(t27_altcat_3,axiom,(
    ! [A] :
      ( l1_altcat_1(A)
     => ! [B] :
          ( m1_subset_1(B,u1_struct_0(A))
         => ( v7_altcat_3(B,A)
          <=> ! [C] :
                ( m1_subset_1(C,u1_struct_0(A))
               => ? [D] :
                    ( m1_subset_1(D,k1_altcat_1(A,C,B))
                    & r2_hidden(D,k1_altcat_1(A,C,B))
                    & ! [E] :
                        ( m1_subset_1(E,k1_altcat_1(A,C,B))
                       => ( r2_hidden(E,k1_altcat_1(A,C,B))
                         => D = E ) ) ) ) ) ) ) )).

fof(t28_altcat_3,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v2_altcat_1(A)
        & v11_altcat_1(A)
        & v12_altcat_1(A)
        & l2_altcat_1(A) )
     => ! [B] :
          ( m1_subset_1(B,u1_struct_0(A))
         => ! [C] :
              ( m1_subset_1(C,u1_struct_0(A))
             => ( ( v7_altcat_3(B,A)
                  & v7_altcat_3(C,A) )
               => r2_altcat_3(A,B,C) ) ) ) ) )).

fof(d11_altcat_3,axiom,(
    ! [A] :
      ( l1_altcat_1(A)
     => ! [B] :
          ( m1_subset_1(B,u1_struct_0(A))
         => ( v8_altcat_3(B,A)
          <=> ( v6_altcat_3(B,A)
              & v7_altcat_3(B,A) ) ) ) ) )).

fof(t29_altcat_3,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v2_altcat_1(A)
        & v11_altcat_1(A)
        & v12_altcat_1(A)
        & l2_altcat_1(A) )
     => ! [B] :
          ( m1_subset_1(B,u1_struct_0(A))
         => ! [C] :
              ( m1_subset_1(C,u1_struct_0(A))
             => ( ( v8_altcat_3(B,A)
                  & v8_altcat_3(C,A) )
               => r2_altcat_3(A,B,C) ) ) ) ) )).

fof(d12_altcat_3,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & l2_altcat_1(A) )
     => ! [B] :
          ( m1_subset_1(B,u1_struct_0(A))
         => ! [C] :
              ( m1_subset_1(C,u1_struct_0(A))
             => ! [D] :
                  ( m1_subset_1(D,k1_altcat_1(A,B,C))
                 => ( v9_altcat_3(D,A,B,C)
                  <=> ! [E] :
                        ( m1_subset_1(E,u1_struct_0(A))
                       => ( v8_altcat_3(E,A)
                         => ! [F] :
                              ( m1_subset_1(F,k1_altcat_1(A,B,E))
                             => ! [G] :
                                  ( m1_subset_1(G,k1_altcat_1(A,E,C))
                                 => D = k5_altcat_1(A,B,E,C,F,G) ) ) ) ) ) ) ) ) ) )).

fof(t30_altcat_3,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v2_altcat_1(A)
        & v11_altcat_1(A)
        & v12_altcat_1(A)
        & l2_altcat_1(A) )
     => ! [B] :
          ( m1_subset_1(B,u1_struct_0(A))
         => ! [C] :
              ( m1_subset_1(C,u1_struct_0(A))
             => ! [D] :
                  ( m1_subset_1(D,u1_struct_0(A))
                 => ! [E] :
                      ( m1_subset_1(E,k1_altcat_1(A,B,C))
                     => ! [F] :
                          ( m1_subset_1(F,k1_altcat_1(A,C,D))
                         => ( ( v9_altcat_3(E,A,B,C)
                              & v9_altcat_3(F,A,C,D) )
                           => v9_altcat_3(k5_altcat_1(A,B,C,D,E,F),A,B,D) ) ) ) ) ) ) ) )).

fof(symmetry_r2_altcat_3,axiom,(
    ! [A,B,C] :
      ( ( ~ v3_struct_0(A)
        & v2_altcat_1(A)
        & v11_altcat_1(A)
        & v12_altcat_1(A)
        & l2_altcat_1(A)
        & m1_subset_1(B,u1_struct_0(A))
        & m1_subset_1(C,u1_struct_0(A)) )
     => ( r2_altcat_3(A,B,C)
       => r2_altcat_3(A,C,B) ) ) )).

fof(reflexivity_r2_altcat_3,axiom,(
    ! [A,B,C] :
      ( ( ~ v3_struct_0(A)
        & v2_altcat_1(A)
        & v11_altcat_1(A)
        & v12_altcat_1(A)
        & l2_altcat_1(A)
        & m1_subset_1(B,u1_struct_0(A))
        & m1_subset_1(C,u1_struct_0(A)) )
     => r2_altcat_3(A,B,B) ) )).

fof(dt_k1_altcat_3,axiom,(
    ! [A,B,C,D] :
      ( ( ~ v3_struct_0(A)
        & v2_altcat_1(A)
        & v11_altcat_1(A)
        & v12_altcat_1(A)
        & l2_altcat_1(A)
        & m1_subset_1(B,u1_struct_0(A))
        & m1_subset_1(C,u1_struct_0(A))
        & m1_subset_1(D,k1_altcat_1(A,B,C)) )
     => m1_subset_1(k1_altcat_3(A,B,C,D),k1_altcat_1(A,C,B)) ) )).
%------------------------------------------------------------------------------