SET007 Axioms: SET007+448.ax


%------------------------------------------------------------------------------
% File     : SET007+448 : TPTP v7.5.0. Released v3.4.0.
% Domain   : Set Theory
% Axioms   : Examples of Category Structures
% 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_2 [Urb08]

% Status   : Satisfiable
% Syntax   : Number of formulae    :   92 (   2 unit)
%            Number of atoms       :  546 (  73 equality)
%            Maximal formula depth :   34 (   8 average)
%            Number of connectives :  533 (  79 ~  ;   1  |; 230  &)
%                                         (  19 <=>; 204 =>;   0 <=)
%                                         (   0 <~>;   0 ~|;   0 ~&)
%            Number of predicates  :   39 (   1 propositional; 0-4 arity)
%            Number of functors    :   60 (  12 constant; 0-6 arity)
%            Number of variables   :  244 (   0 singleton; 221 !;  23 ?)
%            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(fc1_altcat_2,axiom,(
    ! [A] :
      ( v1_relat_1(k1_pboole(A))
      & v3_relat_1(k1_pboole(A))
      & v1_funct_1(k1_pboole(A))
      & v1_funcop_1(k1_pboole(A)) ) )).

fof(fc2_altcat_2,axiom,(
    ! [A] :
      ( ( v1_relat_1(A)
        & v1_funct_1(A)
        & v1_funcop_1(A) )
     => ( v1_relat_1(k2_funct_4(A))
        & v1_funct_1(k2_funct_4(A))
        & v1_funcop_1(k2_funct_4(A)) ) ) )).

fof(fc3_altcat_2,axiom,(
    ! [A] :
      ( ( v2_cat_1(A)
        & l1_cat_1(A) )
     => ( ~ v3_struct_0(k6_altcat_2(A))
        & v2_altcat_1(k6_altcat_2(A))
        & v6_altcat_1(k6_altcat_2(A))
        & v11_altcat_1(k6_altcat_2(A))
        & v12_altcat_1(k6_altcat_2(A)) ) ) )).

fof(rc1_altcat_2,axiom,(
    ? [A] :
      ( l1_altcat_1(A)
      & ~ v3_struct_0(A)
      & v1_altcat_1(A) ) )).

fof(cc1_altcat_2,axiom,(
    ! [A] :
      ( l2_altcat_1(A)
     => ( ( ~ v3_struct_0(A)
          & v12_altcat_1(A) )
       => ( ~ v3_struct_0(A)
          & v1_altcat_2(A) ) ) ) )).

fof(rc2_altcat_2,axiom,(
    ? [A] :
      ( l1_altcat_1(A)
      & ~ v3_struct_0(A)
      & v1_altcat_2(A) ) )).

fof(rc3_altcat_2,axiom,(
    ? [A] :
      ( l2_altcat_1(A)
      & ~ v3_struct_0(A)
      & v1_altcat_2(A) ) )).

fof(fc4_altcat_2,axiom,
    ( v3_struct_0(k7_altcat_2)
    & v6_altcat_1(k7_altcat_2) )).

fof(rc4_altcat_2,axiom,(
    ? [A] :
      ( l2_altcat_1(A)
      & v3_struct_0(A)
      & v6_altcat_1(A) ) )).

fof(rc5_altcat_2,axiom,(
    ! [A] :
      ( l2_altcat_1(A)
     => ? [B] :
          ( m1_altcat_2(B,A)
          & v6_altcat_1(B) ) ) )).

fof(fc5_altcat_2,axiom,(
    ! [A,B] :
      ( ( ~ v3_struct_0(A)
        & l2_altcat_1(A)
        & m1_subset_1(B,u1_struct_0(A)) )
     => ( ~ v3_struct_0(k8_altcat_2(A,B))
        & v2_altcat_1(k8_altcat_2(A,B))
        & v6_altcat_1(k8_altcat_2(A,B)) ) ) )).

fof(rc6_altcat_2,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & l2_altcat_1(A) )
     => ? [B] :
          ( m1_altcat_2(B,A)
          & ~ v3_struct_0(B)
          & v2_altcat_1(B)
          & v6_altcat_1(B) ) ) )).

fof(rc7_altcat_2,axiom,(
    ! [A] :
      ( l2_altcat_1(A)
     => ? [B] :
          ( m1_altcat_2(B,A)
          & v6_altcat_1(B)
          & v2_altcat_2(B,A) ) ) )).

fof(rc8_altcat_2,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & l2_altcat_1(A) )
     => ? [B] :
          ( m1_altcat_2(B,A)
          & ~ v3_struct_0(B)
          & v6_altcat_1(B)
          & v2_altcat_2(B,A) ) ) )).

fof(fc6_altcat_2,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)) )
     => ( ~ v3_struct_0(k8_altcat_2(A,B))
        & v2_altcat_1(k8_altcat_2(A,B))
        & v6_altcat_1(k8_altcat_2(A,B))
        & v2_altcat_2(k8_altcat_2(A,B),A)
        & v3_altcat_2(k8_altcat_2(A,B),A) ) ) )).

fof(rc9_altcat_2,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v2_altcat_1(A)
        & v11_altcat_1(A)
        & v12_altcat_1(A)
        & l2_altcat_1(A) )
     => ? [B] :
          ( m1_altcat_2(B,A)
          & ~ v3_struct_0(B)
          & v6_altcat_1(B)
          & v2_altcat_2(B,A)
          & v3_altcat_2(B,A) ) ) )).

fof(cc2_altcat_2,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v2_altcat_1(A)
        & l2_altcat_1(A) )
     => ! [B] :
          ( m1_altcat_2(B,A)
         => ( ( ~ v3_struct_0(B)
              & v2_altcat_2(B,A) )
           => v2_altcat_1(B) ) ) ) )).

fof(cc3_altcat_2,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v2_altcat_1(A)
        & v11_altcat_1(A)
        & l2_altcat_1(A) )
     => ! [B] :
          ( m1_altcat_2(B,A)
         => ( ( ~ v3_struct_0(B)
              & v2_altcat_1(B) )
           => ( ~ v3_struct_0(B)
              & v11_altcat_1(B) ) ) ) ) )).

fof(cc4_altcat_2,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v2_altcat_1(A)
        & v12_altcat_1(A)
        & l2_altcat_1(A) )
     => ! [B] :
          ( m1_altcat_2(B,A)
         => ( ( ~ v3_struct_0(B)
              & v2_altcat_1(B)
              & v3_altcat_2(B,A) )
           => ( ~ v3_struct_0(B)
              & v12_altcat_1(B)
              & v1_altcat_2(B) ) ) ) ) )).

fof(rc10_altcat_2,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v2_altcat_1(A)
        & v11_altcat_1(A)
        & v12_altcat_1(A)
        & l2_altcat_1(A) )
     => ? [B] :
          ( m1_altcat_2(B,A)
          & ~ v3_struct_0(B)
          & v2_altcat_1(B)
          & v11_altcat_1(B)
          & v12_altcat_1(B)
          & v1_altcat_2(B)
          & v3_altcat_2(B,A) ) ) )).

fof(t1_altcat_2,axiom,(
    ! [A,B,C,D] : k15_funct_3(k2_funcop_1(A,C),k2_funcop_1(B,D)) = k2_funcop_1(k2_zfmisc_1(A,B),k4_tarski(C,D)) )).

fof(t2_altcat_2,axiom,(
    ! [A] :
      ( ( v1_relat_1(A)
        & v1_funct_1(A) )
     => ! [B] :
          ( ( v1_relat_1(B)
            & v1_funct_1(B) )
         => k2_funct_4(k5_relat_1(A,B)) = k5_relat_1(k2_funct_4(A),B) ) ) )).

fof(t3_altcat_2,axiom,(
    ! [A] :
      ( ( v1_relat_1(A)
        & v1_funct_1(A) )
     => ! [B] :
          ( ( v1_relat_1(B)
            & v1_funct_1(B) )
         => ! [C] :
              ( ( v1_relat_1(C)
                & v1_funct_1(C) )
             => k2_funct_4(k5_relat_1(k15_funct_3(B,C),A)) = k5_relat_1(k15_funct_3(C,B),k2_funct_4(A)) ) ) ) )).

fof(t4_altcat_2,axiom,(
    ! [A,B] :
      ( m1_pboole(B,A)
     => ! [C] :
          ( m1_pboole(C,A)
         => ! [D] :
              ( m1_pboole(D,A)
             => ( r1_pzfmisc1(A,B,C)
               => ! [E] :
                    ( m3_pboole(E,A,B,C)
                   => ! [F] :
                        ( m3_pboole(F,A,C,D)
                       => m3_pboole(k13_pboole(E,F),A,B,D) ) ) ) ) ) ) )).

fof(t5_altcat_2,axiom,(
    ! [A,B] :
      ( ~ v1_xboole_0(B)
     => ! [C] :
          ( ( v1_funct_1(C)
            & v1_funct_2(C,A,B)
            & m2_relset_1(C,A,B) )
         => ! [D] :
              ( m1_pboole(D,B)
             => ! [E] :
                  ( m1_pboole(E,B)
                 => ! [F] :
                      ( m3_pboole(F,B,D,E)
                     => m3_pboole(k7_pboole(A,B,C,F),A,k7_pboole(A,B,C,D),k7_pboole(A,B,C,E)) ) ) ) ) ) )).

fof(t6_altcat_2,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( ~ v1_xboole_0(B)
         => ! [C] :
              ( m1_pboole(C,k2_zfmisc_1(A,B))
             => ! [D] :
                  ( m1_subset_1(D,A)
                 => ! [E] :
                      ( m1_subset_1(E,B)
                     => k1_binop_1(k18_pralg_1(A,B,C),E,D) = k1_binop_1(C,D,E) ) ) ) ) ) )).

fof(d1_altcat_2,axiom,(
    ! [A] :
      ( ( v1_relat_1(A)
        & v1_funct_1(A) )
     => ! [B] :
          ( ( v1_relat_1(B)
            & v1_funct_1(B) )
         => ( r1_altcat_2(A,B)
          <=> ( r1_tarski(k1_relat_1(A),k1_relat_1(B))
              & ! [C] :
                  ( r2_hidden(C,k1_relat_1(A))
                 => r1_tarski(k1_funct_1(A,C),k1_funct_1(B,C)) ) ) ) ) ) )).

fof(d2_altcat_2,axiom,(
    ! [A,B,C] :
      ( m1_pboole(C,A)
     => ! [D] :
          ( m1_pboole(D,B)
         => ( r2_altcat_2(A,B,C,D)
          <=> ( r1_tarski(A,B)
              & ! [E] :
                  ( r2_hidden(E,A)
                 => r1_tarski(k1_funct_1(C,E),k1_funct_1(D,E)) ) ) ) ) ) )).

fof(t7_altcat_2,axiom,(
    $true )).

fof(t8_altcat_2,axiom,(
    ! [A,B,C] :
      ( m1_pboole(C,A)
     => ! [D] :
          ( m1_pboole(D,B)
         => ( ( r2_altcat_2(A,B,C,D)
              & r2_altcat_2(B,A,D,C) )
           => C = D ) ) ) )).

fof(t9_altcat_2,axiom,(
    ! [A,B,C,D] :
      ( m1_pboole(D,A)
     => ! [E] :
          ( m1_pboole(E,B)
         => ! [F] :
              ( m1_pboole(F,C)
             => ( ( r2_altcat_2(A,B,D,E)
                  & r2_altcat_2(B,C,E,F) )
               => r2_altcat_2(A,C,D,F) ) ) ) ) )).

fof(t10_altcat_2,axiom,(
    ! [A,B] :
      ( m1_pboole(B,A)
     => ! [C] :
          ( m1_pboole(C,A)
         => ( r2_altcat_2(A,A,B,C)
          <=> r2_pboole(A,B,C) ) ) ) )).

fof(t11_altcat_2,axiom,(
    ! [A] :
      ( ( v2_cat_1(A)
        & l1_cat_1(A) )
     => ! [B] :
          ( m1_subset_1(B,u1_cat_1(A))
         => ! [C] :
              ( m1_subset_1(C,u1_cat_1(A))
             => ! [D] :
                  ( m1_subset_1(D,u1_cat_1(A))
                 => r1_tarski(k12_mcart_1(u2_cat_1(A),u2_cat_1(A),k6_cat_1(A,C,D),k6_cat_1(A,B,C)),k1_relat_1(u5_cat_1(A))) ) ) ) ) )).

fof(t12_altcat_2,axiom,(
    ! [A] :
      ( ( v2_cat_1(A)
        & l1_cat_1(A) )
     => ! [B] :
          ( m1_subset_1(B,u1_cat_1(A))
         => ! [C] :
              ( m1_subset_1(C,u1_cat_1(A))
             => ! [D] :
                  ( m1_subset_1(D,u1_cat_1(A))
                 => r1_tarski(k9_relat_1(u5_cat_1(A),k12_mcart_1(u2_cat_1(A),u2_cat_1(A),k6_cat_1(A,C,D),k6_cat_1(A,B,C))),k6_cat_1(A,B,D)) ) ) ) ) )).

fof(d3_altcat_2,axiom,(
    ! [A] :
      ( l1_cat_1(A)
     => ! [B] :
          ( m1_pboole(B,k2_zfmisc_1(u1_cat_1(A),u1_cat_1(A)))
         => ( B = k4_altcat_2(A)
          <=> ! [C] :
                ( m1_subset_1(C,u1_cat_1(A))
               => ! [D] :
                    ( m1_subset_1(D,u1_cat_1(A))
                   => k1_binop_1(B,C,D) = k6_cat_1(A,C,D) ) ) ) ) ) )).

fof(t13_altcat_2,axiom,(
    ! [A] :
      ( ( v2_cat_1(A)
        & l1_cat_1(A) )
     => ! [B] :
          ( m1_subset_1(B,u1_cat_1(A))
         => r2_hidden(k10_cat_1(A,B),k1_binop_1(k4_altcat_2(A),B,B)) ) ) )).

fof(d4_altcat_2,axiom,(
    ! [A] :
      ( ( v2_cat_1(A)
        & l1_cat_1(A) )
     => ! [B] :
          ( m3_pboole(B,k3_zfmisc_1(u1_cat_1(A),u1_cat_1(A),u1_cat_1(A)),k3_altcat_1(u1_cat_1(A),k4_altcat_2(A),k4_altcat_2(A)),k2_altcat_1(u1_cat_1(A),k4_altcat_2(A)))
         => ( B = k5_altcat_2(A)
          <=> ! [C] :
                ( m1_subset_1(C,u1_cat_1(A))
               => ! [D] :
                    ( m1_subset_1(D,u1_cat_1(A))
                   => ! [E] :
                        ( m1_subset_1(E,u1_cat_1(A))
                       => k4_altcat_1(u1_cat_1(A),k4_altcat_2(A),B,C,D,E) = k7_relat_1(u5_cat_1(A),k2_zfmisc_1(k1_binop_1(k4_altcat_2(A),D,E),k1_binop_1(k4_altcat_2(A),C,D))) ) ) ) ) ) ) )).

fof(t14_altcat_2,axiom,(
    ! [A] :
      ( ( v2_cat_1(A)
        & l1_cat_1(A) )
     => ! [B] :
          ( m1_subset_1(B,u1_cat_1(A))
         => ! [C] :
              ( m1_subset_1(C,u1_cat_1(A))
             => ! [D] :
                  ( m1_subset_1(D,u1_cat_1(A))
                 => ~ ( k6_cat_1(A,B,C) != k1_xboole_0
                      & k6_cat_1(A,C,D) != k1_xboole_0
                      & ~ ! [E] :
                            ( m1_cat_1(E,A,B,C)
                           => ! [F] :
                                ( m1_cat_1(F,A,C,D)
                               => k1_binop_1(k4_altcat_1(u1_cat_1(A),k4_altcat_2(A),k5_altcat_2(A),B,C,D),F,E) = k9_cat_1(A,B,C,D,E,F) ) ) ) ) ) ) ) )).

fof(t15_altcat_2,axiom,(
    ! [A] :
      ( ( v2_cat_1(A)
        & l1_cat_1(A) )
     => v3_altcat_1(k5_altcat_2(A),u1_cat_1(A),k4_altcat_2(A)) ) )).

fof(t16_altcat_2,axiom,(
    ! [A] :
      ( ( v2_cat_1(A)
        & l1_cat_1(A) )
     => ( v5_altcat_1(k5_altcat_2(A),u1_cat_1(A),k4_altcat_2(A))
        & v4_altcat_1(k5_altcat_2(A),u1_cat_1(A),k4_altcat_2(A)) ) ) )).

fof(d5_altcat_2,axiom,(
    ! [A] :
      ( ( v2_cat_1(A)
        & l1_cat_1(A) )
     => k6_altcat_2(A) = g2_altcat_1(u1_cat_1(A),k4_altcat_2(A),k5_altcat_2(A)) ) )).

fof(t17_altcat_2,axiom,(
    ! [A] :
      ( ( v2_cat_1(A)
        & l1_cat_1(A) )
     => v11_altcat_1(k6_altcat_2(A)) ) )).

fof(t18_altcat_2,axiom,(
    ! [A] :
      ( ( v2_cat_1(A)
        & l1_cat_1(A) )
     => v12_altcat_1(k6_altcat_2(A)) ) )).

fof(t19_altcat_2,axiom,(
    ! [A] :
      ( ( v2_cat_1(A)
        & l1_cat_1(A) )
     => v2_altcat_1(k6_altcat_2(A)) ) )).

fof(d6_altcat_2,axiom,(
    ! [A] :
      ( l1_altcat_1(A)
     => ( v1_altcat_2(A)
      <=> ! [B] :
            ~ ( r2_hidden(B,u1_struct_0(A))
              & k1_binop_1(u1_altcat_1(A),B,B) = k1_xboole_0 ) ) ) )).

fof(d7_altcat_2,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & l1_altcat_1(A) )
     => ( v1_altcat_2(A)
      <=> ! [B] :
            ( m1_subset_1(B,u1_struct_0(A))
           => k1_altcat_1(A,B,B) != k1_xboole_0 ) ) ) )).

fof(d8_altcat_2,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v2_altcat_1(A)
        & l2_altcat_1(A) )
     => ( v11_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,u1_struct_0(A))
                       => ! [F] :
                            ( m1_subset_1(F,k1_altcat_1(A,B,C))
                           => ! [G] :
                                ( m1_subset_1(G,k1_altcat_1(A,C,D))
                               => ! [H] :
                                    ( m1_subset_1(H,k1_altcat_1(A,D,E))
                                   => ~ ( k1_altcat_1(A,B,C) != k1_xboole_0
                                        & k1_altcat_1(A,C,D) != k1_xboole_0
                                        & k1_altcat_1(A,D,E) != k1_xboole_0
                                        & k5_altcat_1(A,B,C,E,F,k5_altcat_1(A,C,D,E,G,H)) != k5_altcat_1(A,B,D,E,k5_altcat_1(A,B,C,D,F,G),H) ) ) ) ) ) ) ) ) ) ) )).

fof(d9_altcat_2,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & l2_altcat_1(A) )
     => ( v12_altcat_1(A)
      <=> ! [B] :
            ( m1_subset_1(B,u1_struct_0(A))
           => ( k1_altcat_1(A,B,B) != k1_xboole_0
              & ? [C] :
                  ( m1_subset_1(C,k1_altcat_1(A,B,B))
                  & ! [D] :
                      ( m1_subset_1(D,u1_struct_0(A))
                     => ! [E] :
                          ( m1_subset_1(E,k1_altcat_1(A,D,B))
                         => ! [F] :
                              ( m1_subset_1(F,k1_altcat_1(A,B,D))
                             => ( ( k1_altcat_1(A,D,B) != k1_xboole_0
                                 => k5_altcat_1(A,D,B,B,E,C) = E )
                                & ( k1_altcat_1(A,B,D) != k1_xboole_0
                                 => k5_altcat_1(A,B,B,D,C,F) = F ) ) ) ) ) ) ) ) ) ) )).

fof(d10_altcat_2,axiom,(
    ! [A] :
      ( ( v6_altcat_1(A)
        & l2_altcat_1(A) )
     => ( A = k7_altcat_2
      <=> v1_xboole_0(u1_struct_0(A)) ) ) )).

fof(t20_altcat_2,axiom,(
    ! [A] :
      ( ( v3_struct_0(A)
        & v6_altcat_1(A)
        & l2_altcat_1(A) )
     => A = k7_altcat_2 ) )).

fof(d11_altcat_2,axiom,(
    ! [A] :
      ( l2_altcat_1(A)
     => ! [B] :
          ( l2_altcat_1(B)
         => ( m1_altcat_2(B,A)
          <=> ( r1_tarski(u1_struct_0(B),u1_struct_0(A))
              & r2_altcat_2(k2_zfmisc_1(u1_struct_0(B),u1_struct_0(B)),k2_zfmisc_1(u1_struct_0(A),u1_struct_0(A)),u1_altcat_1(B),u1_altcat_1(A))
              & r2_altcat_2(k3_zfmisc_1(u1_struct_0(B),u1_struct_0(B),u1_struct_0(B)),k3_zfmisc_1(u1_struct_0(A),u1_struct_0(A),u1_struct_0(A)),u2_altcat_1(B),u2_altcat_1(A)) ) ) ) ) )).

fof(t21_altcat_2,axiom,(
    ! [A] :
      ( l2_altcat_1(A)
     => m1_altcat_2(A,A) ) )).

fof(t22_altcat_2,axiom,(
    ! [A] :
      ( l2_altcat_1(A)
     => ! [B] :
          ( l2_altcat_1(B)
         => ! [C] :
              ( l2_altcat_1(C)
             => ( ( m1_altcat_2(A,B)
                  & m1_altcat_2(B,C) )
               => m1_altcat_2(A,C) ) ) ) ) )).

fof(t23_altcat_2,axiom,(
    ! [A] :
      ( l2_altcat_1(A)
     => ! [B] :
          ( l2_altcat_1(B)
         => ( ( m1_altcat_2(A,B)
              & m1_altcat_2(B,A) )
           => g2_altcat_1(u1_struct_0(A),u1_altcat_1(A),u2_altcat_1(A)) = g2_altcat_1(u1_struct_0(B),u1_altcat_1(B),u2_altcat_1(B)) ) ) ) )).

fof(d12_altcat_2,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & l2_altcat_1(A) )
     => ! [B] :
          ( m1_subset_1(B,u1_struct_0(A))
         => ! [C] :
              ( ( v6_altcat_1(C)
                & m1_altcat_2(C,A) )
             => ( C = k8_altcat_2(A,B)
              <=> ( u1_struct_0(C) = k1_struct_0(A,B)
                  & u1_altcat_1(C) = k9_funct_2(B,B,k1_altcat_1(A,B,B))
                  & u2_altcat_1(C) = k7_cat_1(k3_mcart_1(B,B,B),k4_altcat_1(u1_struct_0(A),u1_altcat_1(A),u2_altcat_1(A),B,B,B)) ) ) ) ) ) )).

fof(t24_altcat_2,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(k8_altcat_2(A,B)))
             => C = B ) ) ) )).

fof(t25_altcat_2,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v2_altcat_1(A)
        & l2_altcat_1(A) )
     => ! [B] :
          ( ( ~ v3_struct_0(B)
            & v2_altcat_1(B)
            & m1_altcat_2(B,A) )
         => ! [C] :
              ( ( ~ v3_struct_0(C)
                & v2_altcat_1(C)
                & m1_altcat_2(C,A) )
             => ( ( r1_tarski(u1_struct_0(B),u1_struct_0(C))
                  & r2_altcat_2(k2_zfmisc_1(u1_struct_0(B),u1_struct_0(B)),k2_zfmisc_1(u1_struct_0(C),u1_struct_0(C)),u1_altcat_1(B),u1_altcat_1(C)) )
               => m1_altcat_2(B,C) ) ) ) ) )).

fof(d13_altcat_2,axiom,(
    ! [A] :
      ( l2_altcat_1(A)
     => ! [B] :
          ( m1_altcat_2(B,A)
         => ( v2_altcat_2(B,A)
          <=> u1_altcat_1(B) = k1_realset1(u1_altcat_1(A),u1_struct_0(B)) ) ) ) )).

fof(d14_altcat_2,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v12_altcat_1(A)
        & l2_altcat_1(A) )
     => ! [B] :
          ( m1_altcat_2(B,A)
         => ( ( ~ v3_struct_0(B)
             => ( v3_altcat_2(B,A)
              <=> ! [C] :
                    ( m1_subset_1(C,u1_struct_0(B))
                   => ! [D] :
                        ( m1_subset_1(D,u1_struct_0(A))
                       => ( C = D
                         => r2_hidden(k8_altcat_1(A,D),k1_altcat_1(B,C,C)) ) ) ) ) )
            & ( v3_struct_0(B)
             => v3_altcat_2(B,A) ) ) ) ) )).

fof(t26_altcat_2,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v2_altcat_1(A)
        & l2_altcat_1(A) )
     => ! [B] :
          ( m1_altcat_2(B,A)
         => ( ( u1_struct_0(B) = u1_struct_0(A)
              & u1_altcat_1(B) = u1_altcat_1(A) )
           => g2_altcat_1(u1_struct_0(B),u1_altcat_1(B),u2_altcat_1(B)) = g2_altcat_1(u1_struct_0(A),u1_altcat_1(A),u2_altcat_1(A)) ) ) ) )).

fof(t27_altcat_2,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v2_altcat_1(A)
        & l2_altcat_1(A) )
     => ! [B] :
          ( ( ~ v3_struct_0(B)
            & v2_altcat_1(B)
            & m1_altcat_2(B,A) )
         => ! [C] :
              ( ( ~ v3_struct_0(C)
                & v2_altcat_1(C)
                & m1_altcat_2(C,A) )
             => ( ( u1_struct_0(B) = u1_struct_0(C)
                  & u1_altcat_1(B) = u1_altcat_1(C) )
               => g2_altcat_1(u1_struct_0(B),u1_altcat_1(B),u2_altcat_1(B)) = g2_altcat_1(u1_struct_0(C),u1_altcat_1(C),u2_altcat_1(C)) ) ) ) ) )).

fof(t28_altcat_2,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v2_altcat_1(A)
        & l2_altcat_1(A) )
     => ! [B] :
          ( ( v2_altcat_2(B,A)
            & m1_altcat_2(B,A) )
         => ( u1_struct_0(B) = u1_struct_0(A)
           => g2_altcat_1(u1_struct_0(B),u1_altcat_1(B),u2_altcat_1(B)) = g2_altcat_1(u1_struct_0(A),u1_altcat_1(A),u2_altcat_1(A)) ) ) ) )).

fof(t29_altcat_2,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & l2_altcat_1(A) )
     => ! [B] :
          ( ( ~ v3_struct_0(B)
            & v2_altcat_2(B,A)
            & m1_altcat_2(B,A) )
         => ! [C] :
              ( m1_subset_1(C,u1_struct_0(A))
             => ! [D] :
                  ( m1_subset_1(D,u1_struct_0(A))
                 => ! [E] :
                      ( m1_subset_1(E,u1_struct_0(B))
                     => ! [F] :
                          ( m1_subset_1(F,u1_struct_0(B))
                         => ( ( C = E
                              & D = F )
                           => k1_altcat_1(A,C,D) = k1_altcat_1(B,E,F) ) ) ) ) ) ) ) )).

fof(t30_altcat_2,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & l2_altcat_1(A) )
     => ! [B] :
          ( ( ~ v3_struct_0(B)
            & m1_altcat_2(B,A) )
         => ! [C] :
              ( m1_subset_1(C,u1_struct_0(B))
             => m1_subset_1(C,u1_struct_0(A)) ) ) ) )).

fof(t31_altcat_2,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v2_altcat_1(A)
        & l2_altcat_1(A) )
     => ! [B] :
          ( ( ~ v3_struct_0(B)
            & v2_altcat_2(B,A)
            & m1_altcat_2(B,A) )
         => ! [C] :
              ( ( ~ v3_struct_0(C)
                & v2_altcat_2(C,A)
                & m1_altcat_2(C,A) )
             => ( u1_struct_0(B) = u1_struct_0(C)
               => g2_altcat_1(u1_struct_0(B),u1_altcat_1(B),u2_altcat_1(B)) = g2_altcat_1(u1_struct_0(C),u1_altcat_1(C),u2_altcat_1(C)) ) ) ) ) )).

fof(t32_altcat_2,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & l2_altcat_1(A) )
     => ! [B] :
          ( ( ~ v3_struct_0(B)
            & m1_altcat_2(B,A) )
         => ! [C] :
              ( m1_subset_1(C,u1_struct_0(A))
             => ! [D] :
                  ( m1_subset_1(D,u1_struct_0(A))
                 => ! [E] :
                      ( m1_subset_1(E,u1_struct_0(B))
                     => ! [F] :
                          ( m1_subset_1(F,u1_struct_0(B))
                         => ( ( C = E
                              & D = F )
                           => r1_tarski(k1_altcat_1(B,E,F),k1_altcat_1(A,C,D)) ) ) ) ) ) ) ) )).

fof(t33_altcat_2,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v2_altcat_1(A)
        & l2_altcat_1(A) )
     => ! [B] :
          ( ( ~ v3_struct_0(B)
            & v2_altcat_1(B)
            & m1_altcat_2(B,A) )
         => ! [C] :
              ( m1_subset_1(C,u1_struct_0(B))
             => ! [D] :
                  ( m1_subset_1(D,u1_struct_0(B))
                 => ! [E] :
                      ( m1_subset_1(E,u1_struct_0(B))
                     => ~ ( k1_altcat_1(B,C,D) != k1_xboole_0
                          & k1_altcat_1(B,D,E) != k1_xboole_0
                          & ? [F] :
                              ( m1_subset_1(F,u1_struct_0(A))
                              & ? [G] :
                                  ( m1_subset_1(G,u1_struct_0(A))
                                  & ? [H] :
                                      ( m1_subset_1(H,u1_struct_0(A))
                                      & F = C
                                      & G = D
                                      & H = E
                                      & ? [I] :
                                          ( m1_subset_1(I,k1_altcat_1(A,F,G))
                                          & ? [J] :
                                              ( m1_subset_1(J,k1_altcat_1(A,G,H))
                                              & ? [K] :
                                                  ( m1_subset_1(K,k1_altcat_1(B,C,D))
                                                  & ? [L] :
                                                      ( m1_subset_1(L,k1_altcat_1(B,D,E))
                                                      & I = K
                                                      & J = L
                                                      & k5_altcat_1(A,F,G,H,I,J) != k5_altcat_1(B,C,D,E,K,L) ) ) ) ) ) ) ) ) ) ) ) ) ) )).

fof(t34_altcat_2,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & l2_altcat_1(A) )
     => ! [B] :
          ( ( ~ v3_struct_0(B)
            & m1_altcat_2(B,A) )
         => ! [C] :
              ( m1_subset_1(C,u1_struct_0(A))
             => ! [D] :
                  ( m1_subset_1(D,u1_struct_0(A))
                 => ! [E] :
                      ( m1_subset_1(E,u1_struct_0(B))
                     => ! [F] :
                          ( m1_subset_1(F,u1_struct_0(B))
                         => ( ( C = E
                              & D = F )
                           => ( k1_altcat_1(B,E,F) = k1_xboole_0
                              | ! [G] :
                                  ( m1_subset_1(G,k1_altcat_1(B,E,F))
                                 => m1_subset_1(G,k1_altcat_1(A,C,D)) ) ) ) ) ) ) ) ) ) )).

fof(t35_altcat_2,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v2_altcat_1(A)
        & v11_altcat_1(A)
        & v12_altcat_1(A)
        & l2_altcat_1(A) )
     => ! [B] :
          ( ( ~ v3_struct_0(B)
            & v2_altcat_1(B)
            & v3_altcat_2(B,A)
            & m1_altcat_2(B,A) )
         => ! [C] :
              ( m1_subset_1(C,u1_struct_0(B))
             => ! [D] :
                  ( m1_subset_1(D,u1_struct_0(A))
                 => ( C = D
                   => k8_altcat_1(B,C) = k8_altcat_1(A,D) ) ) ) ) ) )).

fof(dt_m1_altcat_2,axiom,(
    ! [A] :
      ( l2_altcat_1(A)
     => ! [B] :
          ( m1_altcat_2(B,A)
         => l2_altcat_1(B) ) ) )).

fof(existence_m1_altcat_2,axiom,(
    ! [A] :
      ( l2_altcat_1(A)
     => ? [B] : m1_altcat_2(B,A) ) )).

fof(reflexivity_r1_altcat_2,axiom,(
    ! [A,B] :
      ( ( v1_relat_1(A)
        & v1_funct_1(A)
        & v1_relat_1(B)
        & v1_funct_1(B) )
     => r1_altcat_2(A,A) ) )).

fof(reflexivity_r2_altcat_2,axiom,(
    ! [A,B,C,D] :
      ( ( m1_pboole(C,A)
        & m1_pboole(D,B) )
     => r2_altcat_2(A,B,C,C) ) )).

fof(redefinition_r2_altcat_2,axiom,(
    ! [A,B,C,D] :
      ( ( m1_pboole(C,A)
        & m1_pboole(D,B) )
     => ( r2_altcat_2(A,B,C,D)
      <=> r1_altcat_2(C,D) ) ) )).

fof(dt_k1_altcat_2,axiom,(
    ! [A,B] :
      ( m1_pboole(B,k2_zfmisc_1(A,A))
     => m1_pboole(k1_altcat_2(A,B),k2_zfmisc_1(A,A)) ) )).

fof(redefinition_k1_altcat_2,axiom,(
    ! [A,B] :
      ( m1_pboole(B,k2_zfmisc_1(A,A))
     => k1_altcat_2(A,B) = k2_funct_4(B) ) )).

fof(dt_k2_altcat_2,axiom,(
    ! [A,B,C,D] :
      ( ( m1_pboole(B,k2_zfmisc_1(A,A))
        & m1_pboole(C,k2_zfmisc_1(A,A))
        & m3_pboole(D,k2_zfmisc_1(A,A),B,C) )
     => m3_pboole(k2_altcat_2(A,B,C,D),k2_zfmisc_1(A,A),k1_altcat_2(A,B),k1_altcat_2(A,C)) ) )).

fof(redefinition_k2_altcat_2,axiom,(
    ! [A,B,C,D] :
      ( ( m1_pboole(B,k2_zfmisc_1(A,A))
        & m1_pboole(C,k2_zfmisc_1(A,A))
        & m3_pboole(D,k2_zfmisc_1(A,A),B,C) )
     => k2_altcat_2(A,B,C,D) = k2_funct_4(D) ) )).

fof(dt_k3_altcat_2,axiom,(
    ! [A,B,C] :
      ( ( v1_funcop_1(B)
        & m1_pboole(B,A)
        & v1_funcop_1(C)
        & m1_pboole(C,A) )
     => ( v1_funcop_1(k3_altcat_2(A,B,C))
        & m1_pboole(k3_altcat_2(A,B,C),A) ) ) )).

fof(redefinition_k3_altcat_2,axiom,(
    ! [A,B,C] :
      ( ( v1_funcop_1(B)
        & m1_pboole(B,A)
        & v1_funcop_1(C)
        & m1_pboole(C,A) )
     => k3_altcat_2(A,B,C) = k13_pboole(B,C) ) )).

fof(dt_k4_altcat_2,axiom,(
    ! [A] :
      ( l1_cat_1(A)
     => m1_pboole(k4_altcat_2(A),k2_zfmisc_1(u1_cat_1(A),u1_cat_1(A))) ) )).

fof(dt_k5_altcat_2,axiom,(
    ! [A] :
      ( ( v2_cat_1(A)
        & l1_cat_1(A) )
     => m3_pboole(k5_altcat_2(A),k3_zfmisc_1(u1_cat_1(A),u1_cat_1(A),u1_cat_1(A)),k3_altcat_1(u1_cat_1(A),k4_altcat_2(A),k4_altcat_2(A)),k2_altcat_1(u1_cat_1(A),k4_altcat_2(A))) ) )).

fof(dt_k6_altcat_2,axiom,(
    ! [A] :
      ( ( v2_cat_1(A)
        & l1_cat_1(A) )
     => ( ~ v3_struct_0(k6_altcat_2(A))
        & v6_altcat_1(k6_altcat_2(A))
        & l2_altcat_1(k6_altcat_2(A)) ) ) )).

fof(dt_k7_altcat_2,axiom,
    ( v6_altcat_1(k7_altcat_2)
    & l2_altcat_1(k7_altcat_2) )).

fof(dt_k8_altcat_2,axiom,(
    ! [A,B] :
      ( ( ~ v3_struct_0(A)
        & l2_altcat_1(A)
        & m1_subset_1(B,u1_struct_0(A)) )
     => ( v6_altcat_1(k8_altcat_2(A,B))
        & m1_altcat_2(k8_altcat_2(A,B),A) ) ) )).

fof(s1_altcat_2,axiom,
    ( v1_relat_1(a_0_0_altcat_2)
    & v1_funct_1(a_0_0_altcat_2) )).

fof(s2_altcat_2,axiom,
    ( f2_s2_altcat_2 = a_0_1_altcat_2
   => k1_relat_1(f2_s2_altcat_2) = a_0_2_altcat_2 )).

fof(s3_altcat_2,axiom,
    ( ( f2_s3_altcat_2 = a_0_3_altcat_2
      & p1_s3_altcat_2(f3_s3_altcat_2) )
   => k1_funct_1(f2_s3_altcat_2,f3_s3_altcat_2) = f4_s3_altcat_2(f3_s3_altcat_2) )).

fof(fraenkel_a_0_0_altcat_2,axiom,(
    ! [A] :
      ( r2_hidden(A,a_0_0_altcat_2)
    <=> ? [B] :
          ( m1_subset_1(B,f1_s1_altcat_2)
          & A = k4_tarski(B,f2_s1_altcat_2(B))
          & p1_s1_altcat_2(B) ) ) )).

fof(fraenkel_a_0_1_altcat_2,axiom,(
    ! [A] :
      ( r2_hidden(A,a_0_1_altcat_2)
    <=> ? [B] :
          ( m1_subset_1(B,f1_s2_altcat_2)
          & A = k4_tarski(B,f3_s2_altcat_2(B))
          & p1_s2_altcat_2(B) ) ) )).

fof(fraenkel_a_0_2_altcat_2,axiom,(
    ! [A] :
      ( r2_hidden(A,a_0_2_altcat_2)
    <=> ? [B] :
          ( m1_subset_1(B,f1_s2_altcat_2)
          & A = B
          & p1_s2_altcat_2(B) ) ) )).

fof(fraenkel_a_0_3_altcat_2,axiom,(
    ! [A] :
      ( r2_hidden(A,a_0_3_altcat_2)
    <=> ? [B] :
          ( m1_subset_1(B,f1_s3_altcat_2)
          & A = k4_tarski(B,f4_s3_altcat_2(B))
          & p1_s3_altcat_2(B) ) ) )).
%------------------------------------------------------------------------------