SET007 Axioms: SET007+4.ax


%------------------------------------------------------------------------------
% File     : SET007+4 : TPTP v7.5.0. Released v3.4.0.
% Domain   : Set Theory
% Axioms   : Boolean Properties of Sets - Theorems
% 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    : xboole_1 [Urb08]

% Status   : Satisfiable
% Syntax   : Number of formulae    :  114 (  50 unit)
%            Number of atoms       :  225 (  63 equality)
%            Maximal formula depth :   10 (   5 average)
%            Number of connectives :  140 (  29 ~  ;   0  |;  54  &)
%                                         (   4 <=>;  53 =>;   0 <=)
%                                         (   0 <~>;   0 ~|;   0 ~&)
%            Number of predicates  :    6 (   0 propositional; 1-2 arity)
%            Number of functors    :    5 (   1 constant; 0-2 arity)
%            Number of variables   :  297 (   0 singleton; 297 !;   0 ?)
%            Maximal term depth    :    4 (   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_xboole_1,axiom,(
    ! [A,B,C] :
      ( ( r1_tarski(A,B)
        & r1_tarski(B,C) )
     => r1_tarski(A,C) ) )).

fof(t2_xboole_1,axiom,(
    ! [A] : r1_tarski(k1_xboole_0,A) )).

fof(t3_xboole_1,axiom,(
    ! [A] :
      ( r1_tarski(A,k1_xboole_0)
     => A = k1_xboole_0 ) )).

fof(t4_xboole_1,axiom,(
    ! [A,B,C] : k2_xboole_0(k2_xboole_0(A,B),C) = k2_xboole_0(A,k2_xboole_0(B,C)) )).

fof(t5_xboole_1,axiom,(
    ! [A,B,C] : k2_xboole_0(k2_xboole_0(A,B),C) = k2_xboole_0(k2_xboole_0(A,C),k2_xboole_0(B,C)) )).

fof(t6_xboole_1,axiom,(
    ! [A,B] : k2_xboole_0(A,k2_xboole_0(A,B)) = k2_xboole_0(A,B) )).

fof(t7_xboole_1,axiom,(
    ! [A,B] : r1_tarski(A,k2_xboole_0(A,B)) )).

fof(t8_xboole_1,axiom,(
    ! [A,B,C] :
      ( ( r1_tarski(A,B)
        & r1_tarski(C,B) )
     => r1_tarski(k2_xboole_0(A,C),B) ) )).

fof(t9_xboole_1,axiom,(
    ! [A,B,C] :
      ( r1_tarski(A,B)
     => r1_tarski(k2_xboole_0(A,C),k2_xboole_0(B,C)) ) )).

fof(t10_xboole_1,axiom,(
    ! [A,B,C] :
      ( r1_tarski(A,B)
     => r1_tarski(A,k2_xboole_0(C,B)) ) )).

fof(t11_xboole_1,axiom,(
    ! [A,B,C] :
      ( r1_tarski(k2_xboole_0(A,B),C)
     => r1_tarski(A,C) ) )).

fof(t12_xboole_1,axiom,(
    ! [A,B] :
      ( r1_tarski(A,B)
     => k2_xboole_0(A,B) = B ) )).

fof(t13_xboole_1,axiom,(
    ! [A,B,C,D] :
      ( ( r1_tarski(A,B)
        & r1_tarski(C,D) )
     => r1_tarski(k2_xboole_0(A,C),k2_xboole_0(B,D)) ) )).

fof(t14_xboole_1,axiom,(
    ! [A,B,C] :
      ( ( r1_tarski(A,B)
        & r1_tarski(C,B)
        & ! [D] :
            ( ( r1_tarski(A,D)
              & r1_tarski(C,D) )
           => r1_tarski(B,D) ) )
     => B = k2_xboole_0(A,C) ) )).

fof(t15_xboole_1,axiom,(
    ! [A,B] :
      ( k2_xboole_0(A,B) = k1_xboole_0
     => A = k1_xboole_0 ) )).

fof(t16_xboole_1,axiom,(
    ! [A,B,C] : k3_xboole_0(k3_xboole_0(A,B),C) = k3_xboole_0(A,k3_xboole_0(B,C)) )).

fof(t17_xboole_1,axiom,(
    ! [A,B] : r1_tarski(k3_xboole_0(A,B),A) )).

fof(t18_xboole_1,axiom,(
    ! [A,B,C] :
      ( r1_tarski(A,k3_xboole_0(B,C))
     => r1_tarski(A,B) ) )).

fof(t19_xboole_1,axiom,(
    ! [A,B,C] :
      ( ( r1_tarski(A,B)
        & r1_tarski(A,C) )
     => r1_tarski(A,k3_xboole_0(B,C)) ) )).

fof(t20_xboole_1,axiom,(
    ! [A,B,C] :
      ( ( r1_tarski(A,B)
        & r1_tarski(A,C)
        & ! [D] :
            ( ( r1_tarski(D,B)
              & r1_tarski(D,C) )
           => r1_tarski(D,A) ) )
     => A = k3_xboole_0(B,C) ) )).

fof(t21_xboole_1,axiom,(
    ! [A,B] : k3_xboole_0(A,k2_xboole_0(A,B)) = A )).

fof(t22_xboole_1,axiom,(
    ! [A,B] : k2_xboole_0(A,k3_xboole_0(A,B)) = A )).

fof(t23_xboole_1,axiom,(
    ! [A,B,C] : k3_xboole_0(A,k2_xboole_0(B,C)) = k2_xboole_0(k3_xboole_0(A,B),k3_xboole_0(A,C)) )).

fof(t24_xboole_1,axiom,(
    ! [A,B,C] : k2_xboole_0(A,k3_xboole_0(B,C)) = k3_xboole_0(k2_xboole_0(A,B),k2_xboole_0(A,C)) )).

fof(t25_xboole_1,axiom,(
    ! [A,B,C] : k2_xboole_0(k2_xboole_0(k3_xboole_0(A,B),k3_xboole_0(B,C)),k3_xboole_0(C,A)) = k3_xboole_0(k3_xboole_0(k2_xboole_0(A,B),k2_xboole_0(B,C)),k2_xboole_0(C,A)) )).

fof(t26_xboole_1,axiom,(
    ! [A,B,C] :
      ( r1_tarski(A,B)
     => r1_tarski(k3_xboole_0(A,C),k3_xboole_0(B,C)) ) )).

fof(t27_xboole_1,axiom,(
    ! [A,B,C,D] :
      ( ( r1_tarski(A,B)
        & r1_tarski(C,D) )
     => r1_tarski(k3_xboole_0(A,C),k3_xboole_0(B,D)) ) )).

fof(t28_xboole_1,axiom,(
    ! [A,B] :
      ( r1_tarski(A,B)
     => k3_xboole_0(A,B) = A ) )).

fof(t29_xboole_1,axiom,(
    ! [A,B,C] : r1_tarski(k3_xboole_0(A,B),k2_xboole_0(A,C)) )).

fof(t30_xboole_1,axiom,(
    ! [A,B,C] :
      ( r1_tarski(A,B)
     => k2_xboole_0(A,k3_xboole_0(C,B)) = k3_xboole_0(k2_xboole_0(A,C),B) ) )).

fof(t31_xboole_1,axiom,(
    ! [A,B,C] : r1_tarski(k2_xboole_0(k3_xboole_0(A,B),k3_xboole_0(A,C)),k2_xboole_0(B,C)) )).

fof(t32_xboole_1,axiom,(
    ! [A,B] :
      ( k4_xboole_0(A,B) = k4_xboole_0(B,A)
     => A = B ) )).

fof(t33_xboole_1,axiom,(
    ! [A,B,C] :
      ( r1_tarski(A,B)
     => r1_tarski(k4_xboole_0(A,C),k4_xboole_0(B,C)) ) )).

fof(t34_xboole_1,axiom,(
    ! [A,B,C] :
      ( r1_tarski(A,B)
     => r1_tarski(k4_xboole_0(C,B),k4_xboole_0(C,A)) ) )).

fof(t35_xboole_1,axiom,(
    ! [A,B,C,D] :
      ( ( r1_tarski(A,B)
        & r1_tarski(C,D) )
     => r1_tarski(k4_xboole_0(A,D),k4_xboole_0(B,C)) ) )).

fof(t36_xboole_1,axiom,(
    ! [A,B] : r1_tarski(k4_xboole_0(A,B),A) )).

fof(t37_xboole_1,axiom,(
    ! [A,B] :
      ( k4_xboole_0(A,B) = k1_xboole_0
    <=> r1_tarski(A,B) ) )).

fof(t38_xboole_1,axiom,(
    ! [A,B] :
      ( r1_tarski(A,k4_xboole_0(B,A))
     => A = k1_xboole_0 ) )).

fof(t39_xboole_1,axiom,(
    ! [A,B] : k2_xboole_0(A,k4_xboole_0(B,A)) = k2_xboole_0(A,B) )).

fof(t40_xboole_1,axiom,(
    ! [A,B] : k4_xboole_0(k2_xboole_0(A,B),B) = k4_xboole_0(A,B) )).

fof(t41_xboole_1,axiom,(
    ! [A,B,C] : k4_xboole_0(k4_xboole_0(A,B),C) = k4_xboole_0(A,k2_xboole_0(B,C)) )).

fof(t42_xboole_1,axiom,(
    ! [A,B,C] : k4_xboole_0(k2_xboole_0(A,B),C) = k2_xboole_0(k4_xboole_0(A,C),k4_xboole_0(B,C)) )).

fof(t43_xboole_1,axiom,(
    ! [A,B,C] :
      ( r1_tarski(A,k2_xboole_0(B,C))
     => r1_tarski(k4_xboole_0(A,B),C) ) )).

fof(t44_xboole_1,axiom,(
    ! [A,B,C] :
      ( r1_tarski(k4_xboole_0(A,B),C)
     => r1_tarski(A,k2_xboole_0(B,C)) ) )).

fof(t45_xboole_1,axiom,(
    ! [A,B] :
      ( r1_tarski(A,B)
     => B = k2_xboole_0(A,k4_xboole_0(B,A)) ) )).

fof(t46_xboole_1,axiom,(
    ! [A,B] : k4_xboole_0(A,k2_xboole_0(A,B)) = k1_xboole_0 )).

fof(t47_xboole_1,axiom,(
    ! [A,B] : k4_xboole_0(A,k3_xboole_0(A,B)) = k4_xboole_0(A,B) )).

fof(t48_xboole_1,axiom,(
    ! [A,B] : k4_xboole_0(A,k4_xboole_0(A,B)) = k3_xboole_0(A,B) )).

fof(t49_xboole_1,axiom,(
    ! [A,B,C] : k3_xboole_0(A,k4_xboole_0(B,C)) = k4_xboole_0(k3_xboole_0(A,B),C) )).

fof(t50_xboole_1,axiom,(
    ! [A,B,C] : k3_xboole_0(A,k4_xboole_0(B,C)) = k4_xboole_0(k3_xboole_0(A,B),k3_xboole_0(A,C)) )).

fof(t51_xboole_1,axiom,(
    ! [A,B] : k2_xboole_0(k3_xboole_0(A,B),k4_xboole_0(A,B)) = A )).

fof(t52_xboole_1,axiom,(
    ! [A,B,C] : k4_xboole_0(A,k4_xboole_0(B,C)) = k2_xboole_0(k4_xboole_0(A,B),k3_xboole_0(A,C)) )).

fof(t53_xboole_1,axiom,(
    ! [A,B,C] : k4_xboole_0(A,k2_xboole_0(B,C)) = k3_xboole_0(k4_xboole_0(A,B),k4_xboole_0(A,C)) )).

fof(t54_xboole_1,axiom,(
    ! [A,B,C] : k4_xboole_0(A,k3_xboole_0(B,C)) = k2_xboole_0(k4_xboole_0(A,B),k4_xboole_0(A,C)) )).

fof(t55_xboole_1,axiom,(
    ! [A,B] : k4_xboole_0(k2_xboole_0(A,B),k3_xboole_0(A,B)) = k2_xboole_0(k4_xboole_0(A,B),k4_xboole_0(B,A)) )).

fof(t56_xboole_1,axiom,(
    ! [A,B,C] :
      ( ( r2_xboole_0(A,B)
        & r2_xboole_0(B,C) )
     => r2_xboole_0(A,C) ) )).

fof(t57_xboole_1,axiom,(
    ! [A,B] :
      ~ ( r2_xboole_0(A,B)
        & r2_xboole_0(B,A) ) )).

fof(t58_xboole_1,axiom,(
    ! [A,B,C] :
      ( ( r2_xboole_0(A,B)
        & r1_tarski(B,C) )
     => r2_xboole_0(A,C) ) )).

fof(t59_xboole_1,axiom,(
    ! [A,B,C] :
      ( ( r1_tarski(A,B)
        & r2_xboole_0(B,C) )
     => r2_xboole_0(A,C) ) )).

fof(t60_xboole_1,axiom,(
    ! [A,B] :
      ~ ( r1_tarski(A,B)
        & r2_xboole_0(B,A) ) )).

fof(t61_xboole_1,axiom,(
    ! [A] :
      ( A != k1_xboole_0
     => r2_xboole_0(k1_xboole_0,A) ) )).

fof(t62_xboole_1,axiom,(
    ! [A] : ~ r2_xboole_0(A,k1_xboole_0) )).

fof(t63_xboole_1,axiom,(
    ! [A,B,C] :
      ( ( r1_tarski(A,B)
        & r1_xboole_0(B,C) )
     => r1_xboole_0(A,C) ) )).

fof(t64_xboole_1,axiom,(
    ! [A,B,C,D] :
      ( ( r1_tarski(A,B)
        & r1_tarski(C,D)
        & r1_xboole_0(B,D) )
     => r1_xboole_0(A,C) ) )).

fof(t65_xboole_1,axiom,(
    ! [A] : r1_xboole_0(A,k1_xboole_0) )).

fof(t66_xboole_1,axiom,(
    ! [A] :
      ( ~ ( ~ r1_xboole_0(A,A)
          & A = k1_xboole_0 )
      & ~ ( A != k1_xboole_0
          & r1_xboole_0(A,A) ) ) )).

fof(t67_xboole_1,axiom,(
    ! [A,B,C] :
      ( ( r1_tarski(A,B)
        & r1_tarski(A,C)
        & r1_xboole_0(B,C) )
     => A = k1_xboole_0 ) )).

fof(t68_xboole_1,axiom,(
    ! [A,B,C] :
      ( ~ v1_xboole_0(C)
     => ~ ( r1_tarski(C,A)
          & r1_tarski(C,B)
          & r1_xboole_0(A,B) ) ) )).

fof(t69_xboole_1,axiom,(
    ! [A,B] :
      ( ~ v1_xboole_0(B)
     => ~ ( r1_tarski(B,A)
          & r1_xboole_0(B,A) ) ) )).

fof(t70_xboole_1,axiom,(
    ! [A,B,C] :
      ( ~ ( ~ r1_xboole_0(A,k2_xboole_0(B,C))
          & r1_xboole_0(A,B)
          & r1_xboole_0(A,C) )
      & ~ ( ~ ( r1_xboole_0(A,B)
              & r1_xboole_0(A,C) )
          & r1_xboole_0(A,k2_xboole_0(B,C)) ) ) )).

fof(t71_xboole_1,axiom,(
    ! [A,B,C] :
      ( ( k2_xboole_0(A,B) = k2_xboole_0(C,B)
        & r1_xboole_0(A,B)
        & r1_xboole_0(C,B) )
     => A = C ) )).

fof(t72_xboole_1,axiom,(
    ! [A,B,C,D] :
      ( ( k2_xboole_0(A,B) = k2_xboole_0(C,D)
        & r1_xboole_0(C,A)
        & r1_xboole_0(D,B) )
     => C = B ) )).

fof(t73_xboole_1,axiom,(
    ! [A,B,C] :
      ( ( r1_tarski(A,k2_xboole_0(B,C))
        & r1_xboole_0(A,C) )
     => r1_tarski(A,B) ) )).

fof(t74_xboole_1,axiom,(
    ! [A,B,C] :
      ~ ( ~ r1_xboole_0(A,k3_xboole_0(B,C))
        & r1_xboole_0(A,B) ) )).

fof(t75_xboole_1,axiom,(
    ! [A,B] :
      ~ ( ~ r1_xboole_0(A,B)
        & r1_xboole_0(k3_xboole_0(A,B),B) ) )).

fof(t76_xboole_1,axiom,(
    ! [A,B,C] :
      ( r1_xboole_0(A,B)
     => r1_xboole_0(k3_xboole_0(C,A),k3_xboole_0(C,B)) ) )).

fof(t77_xboole_1,axiom,(
    ! [A,B,C] :
      ~ ( ~ r1_xboole_0(A,B)
        & r1_tarski(A,C)
        & r1_xboole_0(A,k3_xboole_0(B,C)) ) )).

fof(t78_xboole_1,axiom,(
    ! [A,B,C] :
      ( r1_xboole_0(A,B)
     => k3_xboole_0(A,k2_xboole_0(B,C)) = k3_xboole_0(A,C) ) )).

fof(t79_xboole_1,axiom,(
    ! [A,B] : r1_xboole_0(k4_xboole_0(A,B),B) )).

fof(t80_xboole_1,axiom,(
    ! [A,B,C] :
      ( r1_xboole_0(A,B)
     => r1_xboole_0(A,k4_xboole_0(B,C)) ) )).

fof(t81_xboole_1,axiom,(
    ! [A,B,C] :
      ( r1_xboole_0(A,k4_xboole_0(B,C))
     => r1_xboole_0(B,k4_xboole_0(A,C)) ) )).

fof(t82_xboole_1,axiom,(
    ! [A,B] : r1_xboole_0(k4_xboole_0(A,B),k4_xboole_0(B,A)) )).

fof(t83_xboole_1,axiom,(
    ! [A,B] :
      ( r1_xboole_0(A,B)
    <=> k4_xboole_0(A,B) = A ) )).

fof(t84_xboole_1,axiom,(
    ! [A,B,C] :
      ~ ( ~ r1_xboole_0(A,B)
        & r1_xboole_0(A,C)
        & r1_xboole_0(A,k4_xboole_0(B,C)) ) )).

fof(t85_xboole_1,axiom,(
    ! [A,B,C] :
      ( r1_tarski(A,B)
     => r1_xboole_0(A,k4_xboole_0(C,B)) ) )).

fof(t86_xboole_1,axiom,(
    ! [A,B,C] :
      ( ( r1_tarski(A,B)
        & r1_xboole_0(A,C) )
     => r1_tarski(A,k4_xboole_0(B,C)) ) )).

fof(t87_xboole_1,axiom,(
    ! [A,B,C] :
      ( r1_xboole_0(A,B)
     => k2_xboole_0(k4_xboole_0(C,A),B) = k4_xboole_0(k2_xboole_0(C,B),A) ) )).

fof(t88_xboole_1,axiom,(
    ! [A,B] :
      ( r1_xboole_0(A,B)
     => k4_xboole_0(k2_xboole_0(A,B),B) = A ) )).

fof(t89_xboole_1,axiom,(
    ! [A,B] : r1_xboole_0(k3_xboole_0(A,B),k4_xboole_0(A,B)) )).

fof(t90_xboole_1,axiom,(
    ! [A,B] : r1_xboole_0(k4_xboole_0(A,k3_xboole_0(A,B)),B) )).

fof(t91_xboole_1,axiom,(
    ! [A,B,C] : k5_xboole_0(k5_xboole_0(A,B),C) = k5_xboole_0(A,k5_xboole_0(B,C)) )).

fof(t92_xboole_1,axiom,(
    ! [A] : k5_xboole_0(A,A) = k1_xboole_0 )).

fof(t93_xboole_1,axiom,(
    ! [A,B] : k2_xboole_0(A,B) = k2_xboole_0(k5_xboole_0(A,B),k3_xboole_0(A,B)) )).

fof(t94_xboole_1,axiom,(
    ! [A,B] : k2_xboole_0(A,B) = k5_xboole_0(k5_xboole_0(A,B),k3_xboole_0(A,B)) )).

fof(t95_xboole_1,axiom,(
    ! [A,B] : k3_xboole_0(A,B) = k5_xboole_0(k5_xboole_0(A,B),k2_xboole_0(A,B)) )).

fof(t96_xboole_1,axiom,(
    ! [A,B] : r1_tarski(k4_xboole_0(A,B),k5_xboole_0(A,B)) )).

fof(t97_xboole_1,axiom,(
    ! [A,B,C] :
      ( ( r1_tarski(k4_xboole_0(A,B),C)
        & r1_tarski(k4_xboole_0(B,A),C) )
     => r1_tarski(k5_xboole_0(A,B),C) ) )).

fof(t98_xboole_1,axiom,(
    ! [A,B] : k2_xboole_0(A,B) = k5_xboole_0(A,k4_xboole_0(B,A)) )).

fof(t99_xboole_1,axiom,(
    ! [A,B,C] : k4_xboole_0(k5_xboole_0(A,B),C) = k2_xboole_0(k4_xboole_0(A,k2_xboole_0(B,C)),k4_xboole_0(B,k2_xboole_0(A,C))) )).

fof(t100_xboole_1,axiom,(
    ! [A,B] : k4_xboole_0(A,B) = k5_xboole_0(A,k3_xboole_0(A,B)) )).

fof(t101_xboole_1,axiom,(
    ! [A,B] : k5_xboole_0(A,B) = k4_xboole_0(k2_xboole_0(A,B),k3_xboole_0(A,B)) )).

fof(t102_xboole_1,axiom,(
    ! [A,B,C] : k4_xboole_0(A,k5_xboole_0(B,C)) = k2_xboole_0(k4_xboole_0(A,k2_xboole_0(B,C)),k3_xboole_0(k3_xboole_0(A,B),C)) )).

fof(t103_xboole_1,axiom,(
    ! [A,B] : r1_xboole_0(k3_xboole_0(A,B),k5_xboole_0(A,B)) )).

fof(t104_xboole_1,axiom,(
    ! [A,B] :
      ( ~ ( ~ r2_xboole_0(A,B)
          & A != B
          & ~ r2_xboole_0(B,A) )
    <=> r3_xboole_0(A,B) ) )).

fof(t105_xboole_1,axiom,(
    ! [A,B] :
      ~ ( r2_xboole_0(A,B)
        & k4_xboole_0(B,A) = k1_xboole_0 ) )).

fof(t106_xboole_1,axiom,(
    ! [A,B,C] :
      ( r1_tarski(A,k4_xboole_0(B,C))
     => ( r1_tarski(A,B)
        & r1_xboole_0(A,C) ) ) )).

fof(t107_xboole_1,axiom,(
    ! [A,B,C] :
      ( r1_tarski(A,k5_xboole_0(B,C))
    <=> ( r1_tarski(A,k2_xboole_0(B,C))
        & r1_xboole_0(A,k3_xboole_0(B,C)) ) ) )).

fof(t108_xboole_1,axiom,(
    ! [A,B,C] :
      ( r1_tarski(A,B)
     => r1_tarski(k3_xboole_0(A,C),B) ) )).

fof(t109_xboole_1,axiom,(
    ! [A,B,C] :
      ( r1_tarski(A,B)
     => r1_tarski(k4_xboole_0(A,C),B) ) )).

fof(t110_xboole_1,axiom,(
    ! [A,B,C] :
      ( ( r1_tarski(A,B)
        & r1_tarski(C,B) )
     => r1_tarski(k5_xboole_0(A,C),B) ) )).

fof(t111_xboole_1,axiom,(
    ! [A,B,C] : k4_xboole_0(k3_xboole_0(A,B),k3_xboole_0(C,B)) = k3_xboole_0(k4_xboole_0(A,C),B) )).

fof(t112_xboole_1,axiom,(
    ! [A,B,C] : k5_xboole_0(k3_xboole_0(A,B),k3_xboole_0(C,B)) = k3_xboole_0(k5_xboole_0(A,C),B) )).

fof(t113_xboole_1,axiom,(
    ! [A,B,C,D] : k2_xboole_0(k2_xboole_0(k2_xboole_0(A,B),C),D) = k2_xboole_0(A,k2_xboole_0(k2_xboole_0(B,C),D)) )).

fof(t114_xboole_1,axiom,(
    ! [A,B,C,D] :
      ( ( r1_xboole_0(A,D)
        & r1_xboole_0(B,D)
        & r1_xboole_0(C,D) )
     => r1_xboole_0(k2_xboole_0(k2_xboole_0(A,B),C),D) ) )).
%------------------------------------------------------------------------------