SET007 Axioms: SET007+62.ax


%------------------------------------------------------------------------------
% File     : SET007+62 : TPTP v7.5.0. Released v3.4.0.
% Domain   : Set Theory
% Axioms   : Cardinal Arithmetics
% 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    : card_2 [Urb08]

% Status   : Satisfiable
% Syntax   : Number of formulae    :   89 (  20 unit)
%            Number of atoms       :  246 (  85 equality)
%            Maximal formula depth :   13 (   5 average)
%            Number of connectives :  188 (  31 ~  ;   4  |;  54  &)
%                                         (   4 <=>;  95 =>;   0 <=)
%                                         (   0 <~>;   0 ~|;   0 ~&)
%            Number of predicates  :   15 (   1 propositional; 0-3 arity)
%            Number of functors    :   43 (  13 constant; 0-8 arity)
%            Number of variables   :  192 (   0 singleton; 189 !;   3 ?)
%            Maximal term depth    :    5 (   2 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_card_2,axiom,(
    $true )).

fof(t2_card_2,axiom,(
    ! [A,B] :
      ( r1_tarski(k1_card_1(A),k1_card_1(B))
    <=> ? [C] :
          ( v1_relat_1(C)
          & v1_funct_1(C)
          & r1_tarski(A,k9_relat_1(C,B)) ) ) )).

fof(t3_card_2,axiom,(
    ! [A,B] :
      ( ( v1_relat_1(B)
        & v1_funct_1(B) )
     => r1_tarski(k1_card_1(k9_relat_1(B,A)),k1_card_1(A)) ) )).

fof(t4_card_2,axiom,(
    ! [A,B] :
      ~ ( r2_hidden(k1_card_1(A),k1_card_1(B))
        & k4_xboole_0(B,A) = k1_xboole_0 ) )).

fof(t5_card_2,axiom,(
    ! [A,B,C] :
      ( ( r2_hidden(A,B)
        & r2_wellord2(B,C) )
     => ( C != k1_xboole_0
        & ? [D] : r2_hidden(D,C) ) ) )).

fof(t6_card_2,axiom,(
    ! [A] :
      ( r2_wellord2(k1_zfmisc_1(A),k1_zfmisc_1(k1_card_1(A)))
      & k1_card_1(k1_zfmisc_1(A)) = k1_card_1(k1_zfmisc_1(k1_card_1(A))) ) )).

fof(t7_card_2,axiom,(
    ! [A,B,C] :
      ( r2_hidden(A,k1_funct_2(B,C))
     => ( r2_wellord2(A,B)
        & k1_card_1(A) = k1_card_1(B) ) ) )).

fof(d1_card_2,axiom,(
    ! [A] :
      ( v1_card_1(A)
     => ! [B] :
          ( v1_card_1(B)
         => k1_card_2(A,B) = k1_card_1(k14_ordinal2(A,B)) ) ) )).

fof(d2_card_2,axiom,(
    ! [A] :
      ( v1_card_1(A)
     => ! [B] :
          ( v1_card_1(B)
         => k2_card_2(A,B) = k1_card_1(k2_zfmisc_1(A,B)) ) ) )).

fof(d3_card_2,axiom,(
    ! [A] :
      ( v1_card_1(A)
     => ! [B] :
          ( v1_card_1(B)
         => k3_card_2(A,B) = k1_card_1(k1_funct_2(B,A)) ) ) )).

fof(t8_card_2,axiom,(
    $true )).

fof(t9_card_2,axiom,(
    $true )).

fof(t10_card_2,axiom,(
    $true )).

fof(t11_card_2,axiom,(
    ! [A,B] :
      ( r2_wellord2(k2_zfmisc_1(A,B),k2_zfmisc_1(B,A))
      & k1_card_1(k2_zfmisc_1(A,B)) = k1_card_1(k2_zfmisc_1(B,A)) ) )).

fof(t12_card_2,axiom,(
    ! [A,B,C] :
      ( r2_wellord2(k2_zfmisc_1(k2_zfmisc_1(A,B),C),k2_zfmisc_1(A,k2_zfmisc_1(B,C)))
      & k1_card_1(k2_zfmisc_1(k2_zfmisc_1(A,B),C)) = k1_card_1(k2_zfmisc_1(A,k2_zfmisc_1(B,C))) ) )).

fof(t13_card_2,axiom,(
    ! [A,B] :
      ( r2_wellord2(A,k2_zfmisc_1(A,k1_tarski(B)))
      & k1_card_1(A) = k1_card_1(k2_zfmisc_1(A,k1_tarski(B))) ) )).

fof(t14_card_2,axiom,(
    ! [A,B] :
      ( r2_wellord2(k2_zfmisc_1(A,B),k2_zfmisc_1(k1_card_1(A),B))
      & r2_wellord2(k2_zfmisc_1(A,B),k2_zfmisc_1(A,k1_card_1(B)))
      & r2_wellord2(k2_zfmisc_1(A,B),k2_zfmisc_1(k1_card_1(A),k1_card_1(B)))
      & k1_card_1(k2_zfmisc_1(A,B)) = k1_card_1(k2_zfmisc_1(k1_card_1(A),B))
      & k1_card_1(k2_zfmisc_1(A,B)) = k1_card_1(k2_zfmisc_1(A,k1_card_1(B)))
      & k1_card_1(k2_zfmisc_1(A,B)) = k1_card_1(k2_zfmisc_1(k1_card_1(A),k1_card_1(B))) ) )).

fof(t15_card_2,axiom,(
    ! [A,B,C,D] :
      ( ( r2_wellord2(A,B)
        & r2_wellord2(C,D) )
     => ( r2_wellord2(k2_zfmisc_1(A,C),k2_zfmisc_1(B,D))
        & k1_card_1(k2_zfmisc_1(A,C)) = k1_card_1(k2_zfmisc_1(B,D)) ) ) )).

fof(t16_card_2,axiom,(
    ! [A] :
      ( v3_ordinal1(A)
     => ! [B] :
          ( v3_ordinal1(B)
         => ! [C,D] :
              ( C != D
             => ( r2_wellord2(k14_ordinal2(A,B),k2_xboole_0(k2_zfmisc_1(A,k1_tarski(C)),k2_zfmisc_1(B,k1_tarski(D))))
                & k1_card_1(k14_ordinal2(A,B)) = k1_card_1(k2_xboole_0(k2_zfmisc_1(A,k1_tarski(C)),k2_zfmisc_1(B,k1_tarski(D)))) ) ) ) ) )).

fof(t17_card_2,axiom,(
    ! [A] :
      ( v1_card_1(A)
     => ! [B] :
          ( v1_card_1(B)
         => ! [C,D] :
              ( C != D
             => ( r2_wellord2(k1_card_2(A,B),k2_xboole_0(k2_zfmisc_1(A,k1_tarski(C)),k2_zfmisc_1(B,k1_tarski(D))))
                & k1_card_2(A,B) = k1_card_1(k2_xboole_0(k2_zfmisc_1(A,k1_tarski(C)),k2_zfmisc_1(B,k1_tarski(D)))) ) ) ) ) )).

fof(t18_card_2,axiom,(
    ! [A] :
      ( v3_ordinal1(A)
     => ! [B] :
          ( v3_ordinal1(B)
         => ( r2_wellord2(k15_ordinal2(A,B),k2_zfmisc_1(A,B))
            & k1_card_1(k15_ordinal2(A,B)) = k1_card_1(k2_zfmisc_1(A,B)) ) ) ) )).

fof(t19_card_2,axiom,
    ( np__0 = k1_card_1(k1_xboole_0)
    & np__1 = k1_card_1(k4_ordinal2)
    & np__2 = k1_card_1(k1_ordinal1(k4_ordinal2)) )).

fof(t20_card_2,axiom,(
    np__1 = k4_ordinal2 )).

fof(t21_card_2,axiom,(
    $true )).

fof(t22_card_2,axiom,
    ( np__2 = k2_tarski(k1_xboole_0,k4_ordinal2)
    & np__2 = k1_ordinal1(k4_ordinal2) )).

fof(t23_card_2,axiom,(
    ! [A,B,C,D,E,F,G,H] :
      ( ( r2_wellord2(A,B)
        & r2_wellord2(C,D) )
     => ( E = F
        | G = H
        | ( r2_wellord2(k2_xboole_0(k2_zfmisc_1(A,k1_tarski(E)),k2_zfmisc_1(C,k1_tarski(F))),k2_xboole_0(k2_zfmisc_1(B,k1_tarski(G)),k2_zfmisc_1(D,k1_tarski(H))))
          & k1_card_1(k2_xboole_0(k2_zfmisc_1(A,k1_tarski(E)),k2_zfmisc_1(C,k1_tarski(F)))) = k1_card_1(k2_xboole_0(k2_zfmisc_1(B,k1_tarski(G)),k2_zfmisc_1(D,k1_tarski(H)))) ) ) ) )).

fof(t24_card_2,axiom,(
    ! [A] :
      ( v3_ordinal1(A)
     => ! [B] :
          ( v3_ordinal1(B)
         => k1_card_1(k14_ordinal2(A,B)) = k1_card_2(k1_card_1(A),k1_card_1(B)) ) ) )).

fof(t25_card_2,axiom,(
    ! [A] :
      ( v3_ordinal1(A)
     => ! [B] :
          ( v3_ordinal1(B)
         => k1_card_1(k15_ordinal2(A,B)) = k2_card_2(k1_card_1(A),k1_card_1(B)) ) ) )).

fof(t26_card_2,axiom,(
    ! [A,B] :
      ( r2_wellord2(k2_xboole_0(k2_zfmisc_1(A,k1_tarski(np__0)),k2_zfmisc_1(B,k1_tarski(np__1))),k2_xboole_0(k2_zfmisc_1(B,k1_tarski(np__0)),k2_zfmisc_1(A,k1_tarski(np__1))))
      & k1_card_1(k2_xboole_0(k2_zfmisc_1(A,k1_tarski(np__0)),k2_zfmisc_1(B,k1_tarski(np__1)))) = k1_card_1(k2_xboole_0(k2_zfmisc_1(B,k1_tarski(np__0)),k2_zfmisc_1(A,k1_tarski(np__1)))) ) )).

fof(t27_card_2,axiom,(
    ! [A,B,C,D] :
      ( r2_wellord2(k2_xboole_0(k2_zfmisc_1(A,B),k2_zfmisc_1(C,D)),k2_xboole_0(k2_zfmisc_1(B,A),k2_zfmisc_1(D,C)))
      & k1_card_1(k2_xboole_0(k2_zfmisc_1(A,B),k2_zfmisc_1(C,D))) = k1_card_1(k2_xboole_0(k2_zfmisc_1(B,A),k2_zfmisc_1(D,C))) ) )).

fof(t28_card_2,axiom,(
    ! [A,B,C,D] :
      ( A != B
     => k1_card_2(k1_card_1(C),k1_card_1(D)) = k1_card_1(k2_xboole_0(k2_zfmisc_1(C,k1_tarski(A)),k2_zfmisc_1(D,k1_tarski(B)))) ) )).

fof(t29_card_2,axiom,(
    ! [A] :
      ( v1_card_1(A)
     => k1_card_2(A,np__0) = A ) )).

fof(t30_card_2,axiom,(
    $true )).

fof(t31_card_2,axiom,(
    ! [A] :
      ( v1_card_1(A)
     => ! [B] :
          ( v1_card_1(B)
         => ! [C] :
              ( v1_card_1(C)
             => k1_card_2(k1_card_2(A,B),C) = k1_card_2(A,k1_card_2(B,C)) ) ) ) )).

fof(t32_card_2,axiom,(
    ! [A] :
      ( v1_card_1(A)
     => k2_card_2(A,np__0) = np__0 ) )).

fof(t33_card_2,axiom,(
    ! [A] :
      ( v1_card_1(A)
     => k2_card_2(A,np__1) = A ) )).

fof(t34_card_2,axiom,(
    $true )).

fof(t35_card_2,axiom,(
    ! [A] :
      ( v1_card_1(A)
     => ! [B] :
          ( v1_card_1(B)
         => ! [C] :
              ( v1_card_1(C)
             => k2_card_2(k2_card_2(A,B),C) = k2_card_2(A,k2_card_2(B,C)) ) ) ) )).

fof(t36_card_2,axiom,(
    ! [A] :
      ( v1_card_1(A)
     => k2_card_2(np__2,A) = k1_card_2(A,A) ) )).

fof(t37_card_2,axiom,(
    ! [A] :
      ( v1_card_1(A)
     => ! [B] :
          ( v1_card_1(B)
         => ! [C] :
              ( v1_card_1(C)
             => k2_card_2(A,k1_card_2(B,C)) = k1_card_2(k2_card_2(A,B),k2_card_2(A,C)) ) ) ) )).

fof(t38_card_2,axiom,(
    ! [A] :
      ( v1_card_1(A)
     => k3_card_2(A,np__0) = np__1 ) )).

fof(t39_card_2,axiom,(
    ! [A] :
      ( v1_card_1(A)
     => ( A != np__0
       => k3_card_2(np__0,A) = np__0 ) ) )).

fof(t40_card_2,axiom,(
    ! [A] :
      ( v1_card_1(A)
     => ( k3_card_2(A,np__1) = A
        & k3_card_2(np__1,A) = np__1 ) ) )).

fof(t41_card_2,axiom,(
    ! [A] :
      ( v1_card_1(A)
     => ! [B] :
          ( v1_card_1(B)
         => ! [C] :
              ( v1_card_1(C)
             => k3_card_2(A,k1_card_2(B,C)) = k2_card_2(k3_card_2(A,B),k3_card_2(A,C)) ) ) ) )).

fof(t42_card_2,axiom,(
    ! [A] :
      ( v1_card_1(A)
     => ! [B] :
          ( v1_card_1(B)
         => ! [C] :
              ( v1_card_1(C)
             => k3_card_2(k2_card_2(A,B),C) = k2_card_2(k3_card_2(A,C),k3_card_2(B,C)) ) ) ) )).

fof(t43_card_2,axiom,(
    ! [A] :
      ( v1_card_1(A)
     => ! [B] :
          ( v1_card_1(B)
         => ! [C] :
              ( v1_card_1(C)
             => k3_card_2(A,k2_card_2(B,C)) = k3_card_2(k3_card_2(A,B),C) ) ) ) )).

fof(t44_card_2,axiom,(
    ! [A] : k3_card_2(np__2,k1_card_1(A)) = k1_card_1(k1_zfmisc_1(A)) )).

fof(t45_card_2,axiom,(
    ! [A] :
      ( v1_card_1(A)
     => k3_card_2(A,np__2) = k2_card_2(A,A) ) )).

fof(t46_card_2,axiom,(
    ! [A] :
      ( v1_card_1(A)
     => ! [B] :
          ( v1_card_1(B)
         => k3_card_2(k1_card_2(A,B),np__2) = k1_card_2(k1_card_2(k2_card_2(A,A),k2_card_2(k2_card_2(np__2,A),B)),k2_card_2(B,B)) ) ) )).

fof(t47_card_2,axiom,(
    ! [A,B] : r1_tarski(k1_card_1(k2_xboole_0(A,B)),k1_card_2(k1_card_1(A),k1_card_1(B))) )).

fof(t48_card_2,axiom,(
    ! [A,B] :
      ( r1_xboole_0(A,B)
     => k1_card_1(k2_xboole_0(A,B)) = k1_card_2(k1_card_1(A),k1_card_1(B)) ) )).

fof(t49_card_2,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( m2_subset_1(B,k1_numbers,k5_numbers)
         => k1_nat_1(A,B) = k14_ordinal2(A,B) ) ) )).

fof(t50_card_2,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( m2_subset_1(B,k1_numbers,k5_numbers)
         => k2_nat_1(A,B) = k15_ordinal2(A,B) ) ) )).

fof(t51_card_2,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( m2_subset_1(B,k1_numbers,k5_numbers)
         => k1_card_1(k1_nat_1(A,B)) = k1_card_2(k1_card_1(A),k1_card_1(B)) ) ) )).

fof(t52_card_2,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( m2_subset_1(B,k1_numbers,k5_numbers)
         => k1_card_1(k2_nat_1(A,B)) = k2_card_2(k1_card_1(A),k1_card_1(B)) ) ) )).

fof(t53_card_2,axiom,(
    ! [A] :
      ( v1_finset_1(A)
     => ! [B] :
          ( v1_finset_1(B)
         => ( r1_xboole_0(A,B)
           => k4_card_1(k2_xboole_0(A,B)) = k1_nat_1(k4_card_1(A),k4_card_1(B)) ) ) ) )).

fof(t54_card_2,axiom,(
    ! [A,B] :
      ( v1_finset_1(B)
     => ( ~ r2_hidden(A,B)
       => k4_card_1(k2_xboole_0(B,k1_tarski(A))) = k1_nat_1(k4_card_1(B),np__1) ) ) )).

fof(t55_card_2,axiom,(
    $true )).

fof(t56_card_2,axiom,(
    $true )).

fof(t57_card_2,axiom,(
    ! [A] :
      ( v1_finset_1(A)
     => ! [B] :
          ( v1_finset_1(B)
         => ( r1_tarski(k1_card_1(A),k1_card_1(B))
          <=> r1_xreal_0(k4_card_1(A),k4_card_1(B)) ) ) ) )).

fof(t58_card_2,axiom,(
    ! [A] :
      ( v1_finset_1(A)
     => ! [B] :
          ( v1_finset_1(B)
         => ( r2_hidden(k1_card_1(A),k1_card_1(B))
          <=> ~ r1_xreal_0(k4_card_1(B),k4_card_1(A)) ) ) ) )).

fof(t59_card_2,axiom,(
    ! [A] :
      ( k1_card_1(A) = np__0
     => A = k1_xboole_0 ) )).

fof(t60_card_2,axiom,(
    ! [A] :
      ( k1_card_1(A) = np__1
    <=> ? [B] : A = k1_tarski(B) ) )).

fof(t61_card_2,axiom,(
    ! [A] :
      ( v1_finset_1(A)
     => ( r2_wellord2(A,k4_card_1(A))
        & r2_wellord2(A,k1_card_1(k4_card_1(A)))
        & r2_wellord2(A,k2_finseq_1(k4_card_1(A))) ) ) )).

fof(t62_card_2,axiom,(
    ! [A] :
      ( v1_finset_1(A)
     => ! [B] :
          ( v1_finset_1(B)
         => r1_xreal_0(k4_card_1(k2_xboole_0(A,B)),k1_nat_1(k4_card_1(A),k4_card_1(B))) ) ) )).

fof(t63_card_2,axiom,(
    ! [A] :
      ( v1_finset_1(A)
     => ! [B] :
          ( v1_finset_1(B)
         => ( r1_tarski(B,A)
           => k4_card_1(k4_xboole_0(A,B)) = k5_real_1(k4_card_1(A),k4_card_1(B)) ) ) ) )).

fof(t64_card_2,axiom,(
    ! [A] :
      ( v1_finset_1(A)
     => ! [B] :
          ( v1_finset_1(B)
         => k4_card_1(k2_xboole_0(A,B)) = k5_real_1(k1_nat_1(k4_card_1(A),k4_card_1(B)),k4_card_1(k3_xboole_0(A,B))) ) ) )).

fof(t65_card_2,axiom,(
    ! [A] :
      ( v1_finset_1(A)
     => ! [B] :
          ( v1_finset_1(B)
         => k4_card_1(k2_zfmisc_1(A,B)) = k2_nat_1(k4_card_1(A),k4_card_1(B)) ) ) )).

fof(t66_card_2,axiom,(
    $true )).

fof(t67_card_2,axiom,(
    ! [A] :
      ( v1_finset_1(A)
     => ! [B] :
          ( v1_finset_1(B)
         => ( r2_xboole_0(A,B)
           => ( ~ r1_xreal_0(k4_card_1(B),k4_card_1(A))
              & r2_hidden(k1_card_1(A),k1_card_1(B)) ) ) ) ) )).

fof(t68_card_2,axiom,(
    ! [A,B] :
      ( v1_finset_1(B)
     => ( ( ~ r1_tarski(k1_card_1(A),k1_card_1(B))
          & ~ r2_hidden(k1_card_1(A),k1_card_1(B)) )
        | v1_finset_1(A) ) ) )).

fof(t69_card_2,axiom,(
    ! [A,B] : r1_xreal_0(k4_card_1(k2_tarski(A,B)),np__2) )).

fof(t70_card_2,axiom,(
    ! [A,B,C] : r1_xreal_0(k4_card_1(k1_enumset1(A,B,C)),np__3) )).

fof(t71_card_2,axiom,(
    ! [A,B,C,D] : r1_xreal_0(k4_card_1(k2_enumset1(A,B,C,D)),np__4) )).

fof(t72_card_2,axiom,(
    ! [A,B,C,D,E] : r1_xreal_0(k4_card_1(k3_enumset1(A,B,C,D,E)),np__5) )).

fof(t73_card_2,axiom,(
    ! [A,B,C,D,E,F] : r1_xreal_0(k4_card_1(k4_enumset1(A,B,C,D,E,F)),np__6) )).

fof(t74_card_2,axiom,(
    ! [A,B,C,D,E,F,G] : r1_xreal_0(k4_card_1(k5_enumset1(A,B,C,D,E,F,G)),np__7) )).

fof(t75_card_2,axiom,(
    ! [A,B,C,D,E,F,G,H] : r1_xreal_0(k4_card_1(k6_enumset1(A,B,C,D,E,F,G,H)),np__8) )).

fof(t76_card_2,axiom,(
    ! [A,B] :
      ( A != B
     => k4_card_1(k2_tarski(A,B)) = np__2 ) )).

fof(t77_card_2,axiom,(
    ! [A,B,C] :
      ~ ( A != B
        & A != C
        & B != C
        & k4_card_1(k1_enumset1(A,B,C)) != np__3 ) )).

fof(t78_card_2,axiom,(
    ! [A,B,C,D] :
      ~ ( A != B
        & A != C
        & A != D
        & B != C
        & B != D
        & C != D
        & k4_card_1(k2_enumset1(A,B,C,D)) != np__4 ) )).

fof(t79_card_2,axiom,(
    ! [A] :
      ( v1_finset_1(A)
     => ~ ( k4_card_1(A) = np__2
          & ! [B,C] :
              ~ ( B != C
                & A = k2_tarski(B,C) ) ) ) )).

fof(t80_card_2,axiom,(
    ! [A] :
      ( ( v1_relat_1(A)
        & v1_funct_1(A) )
     => r1_ordinal1(k1_card_1(k2_relat_1(A)),k1_card_1(k1_relat_1(A))) ) )).

fof(t81_card_2,axiom,(
    ! [A] :
      ( ( v1_finset_1(A)
        & ! [B,C] :
            ~ ( r2_hidden(B,A)
              & r2_hidden(C,A)
              & ~ r1_tarski(B,C)
              & ~ r1_tarski(C,B) ) )
     => ( A = k1_xboole_0
        | r2_hidden(k3_tarski(A),A) ) ) )).

fof(dt_k1_card_2,axiom,(
    ! [A,B] :
      ( ( v1_card_1(A)
        & v1_card_1(B) )
     => v1_card_1(k1_card_2(A,B)) ) )).

fof(commutativity_k1_card_2,axiom,(
    ! [A,B] :
      ( ( v1_card_1(A)
        & v1_card_1(B) )
     => k1_card_2(A,B) = k1_card_2(B,A) ) )).

fof(dt_k2_card_2,axiom,(
    ! [A,B] :
      ( ( v1_card_1(A)
        & v1_card_1(B) )
     => v1_card_1(k2_card_2(A,B)) ) )).

fof(commutativity_k2_card_2,axiom,(
    ! [A,B] :
      ( ( v1_card_1(A)
        & v1_card_1(B) )
     => k2_card_2(A,B) = k2_card_2(B,A) ) )).

fof(dt_k3_card_2,axiom,(
    ! [A,B] :
      ( ( v1_card_1(A)
        & v1_card_1(B) )
     => v1_card_1(k3_card_2(A,B)) ) )).
%------------------------------------------------------------------------------