## SET007 Axioms: SET007+652.ax

```%------------------------------------------------------------------------------
% File     : SET007+652 : TPTP v7.5.0. Released v3.4.0.
% Domain   : Set Theory
% Axioms   : Basic Properties of Fuzzy Set Operation and Membership Function
% 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    : fuzzy_2 [Urb08]

% Status   : Satisfiable
% Syntax   : Number of formulae    :   64 (   0 unit)
%            Number of atoms       :  303 (  39 equality)
%            Maximal formula depth :   13 (   8 average)
%            Number of connectives :  301 (  62 ~  ;   0  |;  19  &)
%                                         (   2 <=>; 218 =>;   0 <=)
%                                         (   0 <~>;   0 ~|;   0 ~&)
%            Number of predicates  :    6 (   0 propositional; 1-2 arity)
%            Number of functors    :   18 (   3 constant; 0-4 arity)
%            Number of variables   :  219 (   0 singleton; 219 !;   0 ?)
%            Maximal term depth    :    5 (   1 average)
% SPC      :

% Comments : The individual reference can be found in [Mat90] by looking for
%            the name provided by [Urb08].
%          : Translated by MPTP from the Mizar Mathematical Library 4.48.930.
%          : These set theory axioms are used in encodings of problems in
%            various domains, including ALG, CAT, GRP, LAT, SET, and TOP.
%------------------------------------------------------------------------------
fof(t1_fuzzy_2,axiom,(
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m1_subset_1(B,A)
=> ! [C] :
( m1_fuzzy_1(C,A)
=> ( r1_xreal_0(np__0,k2_seq_1(A,k1_numbers,C,B))
& r1_xreal_0(k2_seq_1(A,k1_numbers,C,B),np__1) ) ) ) ) )).

fof(t2_fuzzy_2,axiom,(
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m1_fuzzy_1(B,A)
=> ! [C] :
( m1_fuzzy_1(C,A)
=> ! [D] :
( m1_fuzzy_1(D,A)
=> ( r1_fuzzy_1(B,C)
=> k2_fuzzy_1(A,B,k1_fuzzy_1(A,D,C)) = k1_fuzzy_1(A,k2_fuzzy_1(A,B,D),C) ) ) ) ) ) )).

fof(d1_fuzzy_2,axiom,(
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m1_fuzzy_1(B,A)
=> ! [C] :
( m1_fuzzy_1(C,A)
=> k1_fuzzy_2(A,B,C) = k1_fuzzy_1(A,B,k3_fuzzy_1(A,C)) ) ) ) )).

fof(t3_fuzzy_2,axiom,(
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m1_fuzzy_1(B,A)
=> ! [C] :
( m1_fuzzy_1(C,A)
=> k3_fuzzy_1(A,k1_fuzzy_2(A,B,C)) = k2_fuzzy_1(A,k3_fuzzy_1(A,B),C) ) ) ) )).

fof(t4_fuzzy_2,axiom,(
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m1_fuzzy_1(B,A)
=> ! [C] :
( m1_fuzzy_1(C,A)
=> ! [D] :
( m1_fuzzy_1(D,A)
=> ( r1_fuzzy_1(B,C)
=> r1_fuzzy_1(k1_fuzzy_2(A,B,D),k1_fuzzy_2(A,C,D)) ) ) ) ) ) )).

fof(t5_fuzzy_2,axiom,(
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m1_fuzzy_1(B,A)
=> ! [C] :
( m1_fuzzy_1(C,A)
=> ! [D] :
( m1_fuzzy_1(D,A)
=> ( r1_fuzzy_1(B,C)
=> r1_fuzzy_1(k1_fuzzy_2(A,D,C),k1_fuzzy_2(A,D,B)) ) ) ) ) ) )).

fof(t6_fuzzy_2,axiom,(
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m1_fuzzy_1(B,A)
=> ! [C] :
( m1_fuzzy_1(C,A)
=> ! [D] :
( m1_fuzzy_1(D,A)
=> ! [E] :
( m1_fuzzy_1(E,A)
=> ( ( r1_fuzzy_1(B,C)
& r1_fuzzy_1(D,E) )
=> r1_fuzzy_1(k1_fuzzy_2(A,B,E),k1_fuzzy_2(A,C,D)) ) ) ) ) ) ) )).

fof(t7_fuzzy_2,axiom,(
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m1_fuzzy_1(B,A)
=> ! [C] :
( m1_fuzzy_1(C,A)
=> r1_fuzzy_1(k1_fuzzy_2(A,B,C),B) ) ) ) )).

fof(t8_fuzzy_2,axiom,(
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m1_fuzzy_1(B,A)
=> ! [C] :
( m1_fuzzy_1(C,A)
=> r1_fuzzy_1(k1_fuzzy_2(A,B,C),k6_fuzzy_1(A,B,C)) ) ) ) )).

fof(t9_fuzzy_2,axiom,(
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m1_fuzzy_1(B,A)
=> k1_fuzzy_2(A,B,k4_fuzzy_1(A)) = B ) ) )).

fof(t10_fuzzy_2,axiom,(
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m1_fuzzy_1(B,A)
=> k1_fuzzy_2(A,k4_fuzzy_1(A),B) = k4_fuzzy_1(A) ) ) )).

fof(t11_fuzzy_2,axiom,(
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m1_fuzzy_1(B,A)
=> ! [C] :
( m1_fuzzy_1(C,A)
=> r1_fuzzy_1(k1_fuzzy_2(A,B,C),k1_fuzzy_2(A,B,k1_fuzzy_1(A,B,C))) ) ) ) )).

fof(t12_fuzzy_2,axiom,(
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m1_fuzzy_1(B,A)
=> ! [C] :
( m1_fuzzy_1(C,A)
=> r1_fuzzy_1(k2_fuzzy_1(A,k1_fuzzy_1(A,B,C),k1_fuzzy_2(A,B,C)),B) ) ) ) )).

fof(t13_fuzzy_2,axiom,(
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m1_fuzzy_1(B,A)
=> ! [C] :
( m1_fuzzy_1(C,A)
=> r1_fuzzy_1(k2_fuzzy_1(A,B,k1_fuzzy_2(A,C,B)),k2_fuzzy_1(A,B,C)) ) ) ) )).

fof(t14_fuzzy_2,axiom,(
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m1_fuzzy_1(B,A)
=> ! [C] :
( m1_fuzzy_1(C,A)
=> ! [D] :
( m1_fuzzy_1(D,A)
=> k1_fuzzy_2(A,B,k1_fuzzy_2(A,C,D)) = k2_fuzzy_1(A,k1_fuzzy_2(A,B,C),k1_fuzzy_1(A,B,D)) ) ) ) ) )).

fof(t15_fuzzy_2,axiom,(
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m1_fuzzy_1(B,A)
=> ! [C] :
( m1_fuzzy_1(C,A)
=> r1_fuzzy_1(k1_fuzzy_1(A,B,C),k1_fuzzy_2(A,B,k1_fuzzy_2(A,B,C))) ) ) ) )).

fof(t16_fuzzy_2,axiom,(
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m1_fuzzy_1(B,A)
=> ! [C] :
( m1_fuzzy_1(C,A)
=> r1_fuzzy_1(k1_fuzzy_2(A,B,C),k1_fuzzy_2(A,k2_fuzzy_1(A,B,C),C)) ) ) ) )).

fof(t17_fuzzy_2,axiom,(
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m1_fuzzy_1(B,A)
=> ! [C] :
( m1_fuzzy_1(C,A)
=> ! [D] :
( m1_fuzzy_1(D,A)
=> k1_fuzzy_2(A,B,k2_fuzzy_1(A,C,D)) = k1_fuzzy_1(A,k1_fuzzy_2(A,B,C),k1_fuzzy_2(A,B,D)) ) ) ) ) )).

fof(t18_fuzzy_2,axiom,(
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m1_fuzzy_1(B,A)
=> ! [C] :
( m1_fuzzy_1(C,A)
=> ! [D] :
( m1_fuzzy_1(D,A)
=> k1_fuzzy_2(A,B,k1_fuzzy_1(A,C,D)) = k2_fuzzy_1(A,k1_fuzzy_2(A,B,C),k1_fuzzy_2(A,B,D)) ) ) ) ) )).

fof(t19_fuzzy_2,axiom,(
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m1_fuzzy_1(B,A)
=> ! [C] :
( m1_fuzzy_1(C,A)
=> ! [D] :
( m1_fuzzy_1(D,A)
=> k1_fuzzy_2(A,k1_fuzzy_2(A,B,C),D) = k1_fuzzy_2(A,B,k2_fuzzy_1(A,C,D)) ) ) ) ) )).

fof(t20_fuzzy_2,axiom,(
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m1_fuzzy_1(B,A)
=> ! [C] :
( m1_fuzzy_1(C,A)
=> r1_fuzzy_1(k6_fuzzy_1(A,B,C),k1_fuzzy_2(A,k2_fuzzy_1(A,B,C),k1_fuzzy_1(A,B,C))) ) ) ) )).

fof(t21_fuzzy_2,axiom,(
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m1_fuzzy_1(B,A)
=> ! [C] :
( m1_fuzzy_1(C,A)
=> ! [D] :
( m1_fuzzy_1(D,A)
=> k1_fuzzy_2(A,k2_fuzzy_1(A,B,C),D) = k2_fuzzy_1(A,k1_fuzzy_2(A,B,D),k1_fuzzy_2(A,C,D)) ) ) ) ) )).

fof(t22_fuzzy_2,axiom,(
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m1_fuzzy_1(B,A)
=> ! [C] :
( m1_fuzzy_1(C,A)
=> ! [D] :
( m1_fuzzy_1(D,A)
=> ( ( r1_fuzzy_1(k1_fuzzy_2(A,B,C),D)
& r1_fuzzy_1(k1_fuzzy_2(A,C,B),D) )
=> r1_fuzzy_1(k6_fuzzy_1(A,B,C),D) ) ) ) ) ) )).

fof(t23_fuzzy_2,axiom,(
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m1_fuzzy_1(B,A)
=> ! [C] :
( m1_fuzzy_1(C,A)
=> ! [D] :
( m1_fuzzy_1(D,A)
=> k1_fuzzy_1(A,B,k1_fuzzy_2(A,C,D)) = k1_fuzzy_2(A,k1_fuzzy_1(A,B,C),D) ) ) ) ) )).

fof(t24_fuzzy_2,axiom,(
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m1_fuzzy_1(B,A)
=> ! [C] :
( m1_fuzzy_1(C,A)
=> ! [D] :
( m1_fuzzy_1(D,A)
=> r1_fuzzy_1(k1_fuzzy_1(A,B,k1_fuzzy_2(A,C,D)),k1_fuzzy_2(A,k1_fuzzy_1(A,B,C),k1_fuzzy_1(A,B,D))) ) ) ) ) )).

fof(t25_fuzzy_2,axiom,(
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m1_fuzzy_1(B,A)
=> ! [C] :
( m1_fuzzy_1(C,A)
=> r1_fuzzy_1(k6_fuzzy_1(A,B,C),k1_fuzzy_2(A,k2_fuzzy_1(A,B,C),k1_fuzzy_1(A,B,C))) ) ) ) )).

fof(t26_fuzzy_2,axiom,(
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m1_fuzzy_1(B,A)
=> ! [C] :
( m1_fuzzy_1(C,A)
=> r1_fuzzy_1(k2_fuzzy_1(A,k1_fuzzy_1(A,B,C),k3_fuzzy_1(A,k2_fuzzy_1(A,B,C))),k3_fuzzy_1(A,k6_fuzzy_1(A,B,C))) ) ) ) )).

fof(t27_fuzzy_2,axiom,(
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m1_fuzzy_1(B,A)
=> ! [C] :
( m1_fuzzy_1(C,A)
=> ! [D] :
( m1_fuzzy_1(D,A)
=> k1_fuzzy_2(A,k6_fuzzy_1(A,B,C),D) = k2_fuzzy_1(A,k1_fuzzy_2(A,B,k2_fuzzy_1(A,C,D)),k1_fuzzy_2(A,C,k2_fuzzy_1(A,B,D))) ) ) ) ) )).

fof(t28_fuzzy_2,axiom,(
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m1_fuzzy_1(B,A)
=> ! [C] :
( m1_fuzzy_1(C,A)
=> ! [D] :
( m1_fuzzy_1(D,A)
=> r1_fuzzy_1(k2_fuzzy_1(A,k1_fuzzy_2(A,B,k2_fuzzy_1(A,C,D)),k1_fuzzy_1(A,k1_fuzzy_1(A,B,C),D)),k1_fuzzy_2(A,B,k6_fuzzy_1(A,C,D))) ) ) ) ) )).

fof(t29_fuzzy_2,axiom,(
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m1_fuzzy_1(B,A)
=> ! [C] :
( m1_fuzzy_1(C,A)
=> ( r1_fuzzy_1(B,C)
=> r1_fuzzy_1(k2_fuzzy_1(A,B,k1_fuzzy_2(A,C,B)),C) ) ) ) ) )).

fof(t30_fuzzy_2,axiom,(
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m1_fuzzy_1(B,A)
=> ! [C] :
( m1_fuzzy_1(C,A)
=> r1_fuzzy_1(k2_fuzzy_1(A,k6_fuzzy_1(A,B,C),k1_fuzzy_1(A,B,C)),k2_fuzzy_1(A,B,C)) ) ) ) )).

fof(t31_fuzzy_2,axiom,(
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m1_fuzzy_1(B,A)
=> ! [C] :
( m1_fuzzy_1(C,A)
=> ( k1_fuzzy_2(A,B,C) = k4_fuzzy_1(A)
=> r1_fuzzy_1(B,C) ) ) ) ) )).

fof(t32_fuzzy_2,axiom,(
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m1_fuzzy_1(B,A)
=> ! [C] :
( m1_fuzzy_1(C,A)
=> ( k1_fuzzy_1(A,B,C) = k4_fuzzy_1(A)
=> k1_fuzzy_2(A,B,C) = B ) ) ) ) )).

fof(d2_fuzzy_2,axiom,(
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m1_fuzzy_1(B,A)
=> ! [C] :
( m1_fuzzy_1(C,A)
=> ! [D] :
( m1_fuzzy_1(D,A)
=> ( D = k2_fuzzy_2(A,B,C)
<=> ! [E] :
( m1_subset_1(E,A)
=> k2_seq_1(A,k1_numbers,D,E) = k4_real_1(k2_seq_1(A,k1_numbers,B,E),k2_seq_1(A,k1_numbers,C,E)) ) ) ) ) ) ) )).

fof(d3_fuzzy_2,axiom,(
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m1_fuzzy_1(B,A)
=> ! [C] :
( m1_fuzzy_1(C,A)
=> ! [D] :
( m1_fuzzy_1(D,A)
=> ( D = k3_fuzzy_2(A,B,C)
<=> ! [E] :
( m1_subset_1(E,A)
=> k2_seq_1(A,k1_numbers,D,E) = k5_real_1(k3_real_1(k2_seq_1(A,k1_numbers,B,E),k2_seq_1(A,k1_numbers,C,E)),k4_real_1(k2_seq_1(A,k1_numbers,B,E),k2_seq_1(A,k1_numbers,C,E))) ) ) ) ) ) ) )).

fof(t33_fuzzy_2,axiom,(
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m1_fuzzy_1(B,A)
=> ( r1_fuzzy_1(k2_fuzzy_2(A,B,B),B)
& r1_fuzzy_1(B,k3_fuzzy_2(A,B,B)) ) ) ) )).

fof(t34_fuzzy_2,axiom,(
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m1_fuzzy_1(B,A)
=> ! [C] :
( m1_fuzzy_1(C,A)
=> ! [D] :
( m1_fuzzy_1(D,A)
=> k2_fuzzy_2(A,k2_fuzzy_2(A,B,C),D) = k2_fuzzy_2(A,B,k2_fuzzy_2(A,C,D)) ) ) ) ) )).

fof(t35_fuzzy_2,axiom,(
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m1_fuzzy_1(B,A)
=> ! [C] :
( m1_fuzzy_1(C,A)
=> ! [D] :
( m1_fuzzy_1(D,A)
=> k3_fuzzy_2(A,k3_fuzzy_2(A,B,C),D) = k3_fuzzy_2(A,B,k3_fuzzy_2(A,C,D)) ) ) ) ) )).

fof(t36_fuzzy_2,axiom,(
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m1_fuzzy_1(B,A)
=> ! [C] :
( m1_fuzzy_1(C,A)
=> ( r1_fuzzy_1(k2_fuzzy_2(A,B,k3_fuzzy_2(A,B,C)),B)
& r1_fuzzy_1(B,k3_fuzzy_2(A,B,k2_fuzzy_2(A,B,C))) ) ) ) ) )).

fof(t37_fuzzy_2,axiom,(
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m1_fuzzy_1(B,A)
=> ! [C] :
( m1_fuzzy_1(C,A)
=> ! [D] :
( m1_fuzzy_1(D,A)
=> r1_fuzzy_1(k2_fuzzy_2(A,B,k3_fuzzy_2(A,C,D)),k3_fuzzy_2(A,k2_fuzzy_2(A,B,C),k2_fuzzy_2(A,B,D))) ) ) ) ) )).

fof(t38_fuzzy_2,axiom,(
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m1_fuzzy_1(B,A)
=> ! [C] :
( m1_fuzzy_1(C,A)
=> ! [D] :
( m1_fuzzy_1(D,A)
=> r1_fuzzy_1(k2_fuzzy_2(A,k3_fuzzy_2(A,B,C),k3_fuzzy_2(A,B,D)),k3_fuzzy_2(A,B,k2_fuzzy_2(A,C,D))) ) ) ) ) )).

fof(t39_fuzzy_2,axiom,(
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m1_fuzzy_1(B,A)
=> ! [C] :
( m1_fuzzy_1(C,A)
=> k3_fuzzy_1(A,k2_fuzzy_2(A,B,C)) = k3_fuzzy_2(A,k3_fuzzy_1(A,B),k3_fuzzy_1(A,C)) ) ) ) )).

fof(t40_fuzzy_2,axiom,(
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m1_fuzzy_1(B,A)
=> ! [C] :
( m1_fuzzy_1(C,A)
=> k3_fuzzy_1(A,k3_fuzzy_2(A,B,C)) = k2_fuzzy_2(A,k3_fuzzy_1(A,B),k3_fuzzy_1(A,C)) ) ) ) )).

fof(t41_fuzzy_2,axiom,(
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m1_fuzzy_1(B,A)
=> ! [C] :
( m1_fuzzy_1(C,A)
=> k3_fuzzy_2(A,B,C) = k3_fuzzy_1(A,k2_fuzzy_2(A,k3_fuzzy_1(A,B),k3_fuzzy_1(A,C))) ) ) ) )).

fof(t42_fuzzy_2,axiom,(
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m1_fuzzy_1(B,A)
=> ( k2_fuzzy_2(A,B,k4_fuzzy_1(A)) = k4_fuzzy_1(A)
& k2_fuzzy_2(A,B,k5_fuzzy_1(A)) = B ) ) ) )).

fof(t43_fuzzy_2,axiom,(
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m1_fuzzy_1(B,A)
=> ( k3_fuzzy_2(A,B,k4_fuzzy_1(A)) = B
& k3_fuzzy_2(A,B,k5_fuzzy_1(A)) = k5_fuzzy_1(A) ) ) ) )).

fof(t44_fuzzy_2,axiom,(
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m1_fuzzy_1(B,A)
=> ! [C] :
( m1_fuzzy_1(C,A)
=> r1_fuzzy_1(k2_fuzzy_2(A,B,C),k1_fuzzy_1(A,B,C)) ) ) ) )).

fof(t45_fuzzy_2,axiom,(
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m1_fuzzy_1(B,A)
=> ! [C] :
( m1_fuzzy_1(C,A)
=> r1_fuzzy_1(k2_fuzzy_1(A,B,C),k3_fuzzy_2(A,B,C)) ) ) ) )).

fof(t46_fuzzy_2,axiom,(
! [A] :
( m1_subset_1(A,k1_numbers)
=> ! [B] :
( m1_subset_1(B,k1_numbers)
=> ! [C] :
( m1_subset_1(C,k1_numbers)
=> ( r1_xreal_0(np__0,C)
=> ( k4_real_1(C,k4_square_1(A,B)) = k4_square_1(k4_real_1(C,A),k4_real_1(C,B))
& k4_real_1(C,k3_square_1(A,B)) = k3_square_1(k4_real_1(C,A),k4_real_1(C,B)) ) ) ) ) ) )).

fof(t47_fuzzy_2,axiom,(
! [A] :
( m1_subset_1(A,k1_numbers)
=> ! [B] :
( m1_subset_1(B,k1_numbers)
=> ! [C] :
( m1_subset_1(C,k1_numbers)
=> ( k3_real_1(C,k4_square_1(A,B)) = k4_square_1(k3_real_1(C,A),k3_real_1(C,B))
& k3_real_1(C,k3_square_1(A,B)) = k3_square_1(k3_real_1(C,A),k3_real_1(C,B)) ) ) ) ) )).

fof(t48_fuzzy_2,axiom,(
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m1_fuzzy_1(B,A)
=> ! [C] :
( m1_fuzzy_1(C,A)
=> ! [D] :
( m1_fuzzy_1(D,A)
=> k2_fuzzy_2(A,B,k2_fuzzy_1(A,C,D)) = k2_fuzzy_1(A,k2_fuzzy_2(A,B,C),k2_fuzzy_2(A,B,D)) ) ) ) ) )).

fof(t49_fuzzy_2,axiom,(
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m1_fuzzy_1(B,A)
=> ! [C] :
( m1_fuzzy_1(C,A)
=> ! [D] :
( m1_fuzzy_1(D,A)
=> k2_fuzzy_2(A,B,k1_fuzzy_1(A,C,D)) = k1_fuzzy_1(A,k2_fuzzy_2(A,B,C),k2_fuzzy_2(A,B,D)) ) ) ) ) )).

fof(t50_fuzzy_2,axiom,(
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m1_fuzzy_1(B,A)
=> ! [C] :
( m1_fuzzy_1(C,A)
=> ! [D] :
( m1_fuzzy_1(D,A)
=> k3_fuzzy_2(A,B,k2_fuzzy_1(A,C,D)) = k2_fuzzy_1(A,k3_fuzzy_2(A,B,C),k3_fuzzy_2(A,B,D)) ) ) ) ) )).

fof(t51_fuzzy_2,axiom,(
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m1_fuzzy_1(B,A)
=> ! [C] :
( m1_fuzzy_1(C,A)
=> ! [D] :
( m1_fuzzy_1(D,A)
=> k3_fuzzy_2(A,B,k1_fuzzy_1(A,C,D)) = k1_fuzzy_1(A,k3_fuzzy_2(A,B,C),k3_fuzzy_2(A,B,D)) ) ) ) ) )).

fof(t52_fuzzy_2,axiom,(
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m1_fuzzy_1(B,A)
=> ! [C] :
( m1_fuzzy_1(C,A)
=> ! [D] :
( m1_fuzzy_1(D,A)
=> r1_fuzzy_1(k2_fuzzy_2(A,k2_fuzzy_1(A,B,C),k2_fuzzy_1(A,B,D)),k2_fuzzy_1(A,B,k2_fuzzy_2(A,C,D))) ) ) ) ) )).

fof(t53_fuzzy_2,axiom,(
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m1_fuzzy_1(B,A)
=> ! [C] :
( m1_fuzzy_1(C,A)
=> ! [D] :
( m1_fuzzy_1(D,A)
=> r1_fuzzy_1(k2_fuzzy_2(A,k1_fuzzy_1(A,B,C),k1_fuzzy_1(A,B,D)),k1_fuzzy_1(A,B,k2_fuzzy_2(A,C,D))) ) ) ) ) )).

fof(t54_fuzzy_2,axiom,(
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m1_subset_1(B,A)
=> ! [C] :
( m1_fuzzy_1(C,A)
=> ! [D] :
( m1_fuzzy_1(D,A)
=> k2_seq_1(A,k1_numbers,k3_fuzzy_2(A,C,D),B) = k5_real_1(np__1,k4_real_1(k5_real_1(np__1,k2_seq_1(A,k1_numbers,C,B)),k5_real_1(np__1,k2_seq_1(A,k1_numbers,D,B)))) ) ) ) ) )).

fof(t55_fuzzy_2,axiom,(
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m1_fuzzy_1(B,A)
=> ! [C] :
( m1_fuzzy_1(C,A)
=> ! [D] :
( m1_fuzzy_1(D,A)
=> r1_fuzzy_1(k2_fuzzy_1(A,B,k3_fuzzy_2(A,C,D)),k3_fuzzy_2(A,k2_fuzzy_1(A,B,C),k2_fuzzy_1(A,B,D))) ) ) ) ) )).

fof(t56_fuzzy_2,axiom,(
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m1_fuzzy_1(B,A)
=> ! [C] :
( m1_fuzzy_1(C,A)
=> ! [D] :
( m1_fuzzy_1(D,A)
=> r1_fuzzy_1(k1_fuzzy_1(A,B,k3_fuzzy_2(A,C,D)),k3_fuzzy_2(A,k1_fuzzy_1(A,B,C),k1_fuzzy_1(A,B,D))) ) ) ) ) )).

fof(dt_k1_fuzzy_2,axiom,(
! [A,B,C] :
( ( ~ v1_xboole_0(A)
& m1_fuzzy_1(B,A)
& m1_fuzzy_1(C,A) )
=> m1_fuzzy_1(k1_fuzzy_2(A,B,C),A) ) )).

fof(dt_k2_fuzzy_2,axiom,(
! [A,B,C] :
( ( ~ v1_xboole_0(A)
& m1_fuzzy_1(B,A)
& m1_fuzzy_1(C,A) )
=> m1_fuzzy_1(k2_fuzzy_2(A,B,C),A) ) )).

fof(commutativity_k2_fuzzy_2,axiom,(
! [A,B,C] :
( ( ~ v1_xboole_0(A)
& m1_fuzzy_1(B,A)
& m1_fuzzy_1(C,A) )
=> k2_fuzzy_2(A,B,C) = k2_fuzzy_2(A,C,B) ) )).

fof(dt_k3_fuzzy_2,axiom,(
! [A,B,C] :
( ( ~ v1_xboole_0(A)
& m1_fuzzy_1(B,A)
& m1_fuzzy_1(C,A) )
=> m1_fuzzy_1(k3_fuzzy_2(A,B,C),A) ) )).

fof(commutativity_k3_fuzzy_2,axiom,(
! [A,B,C] :
( ( ~ v1_xboole_0(A)
& m1_fuzzy_1(B,A)
& m1_fuzzy_1(C,A) )
=> k3_fuzzy_2(A,B,C) = k3_fuzzy_2(A,C,B) ) )).
%------------------------------------------------------------------------------
```