## SET007 Axioms: SET007+46.ax

```%------------------------------------------------------------------------------
% File     : SET007+46 : TPTP v7.5.0. Released v3.4.0.
% Domain   : Set Theory
% Axioms   : Some Properties of Real Numbers
% Version  : [Urb08] axioms.
% English  : Some Properties of Real Numbers. Operations: min, max, square, and
%            square root.

% 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    : square_1 [Urb08]

% Status   : Satisfiable
% Syntax   : Number of formulae    :  138 (  60 unit)
%            Number of atoms       :  344 (  68 equality)
%            Maximal formula depth :   10 (   4 average)
%            Number of connectives :  233 (  27 ~  ;   5  |;  48  &)
%                                         (   3 <=>; 150 =>;   0 <=)
%                                         (   0 <~>;   0 ~|;   0 ~&)
%            Number of predicates  :    8 (   1 propositional; 0-2 arity)
%            Number of functors    :   20 (   6 constant; 0-2 arity)
%            Number of variables   :  138 (   0 singleton; 137 !;   1 ?)
%            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(cc1_square_1,axiom,(
! [A] :
( m1_subset_1(A,k2_numbers)
=> v1_xcmplx_0(A) ) )).

fof(fc1_square_1,axiom,(
! [A] :
( v1_xcmplx_0(A)
=> v1_xcmplx_0(k5_square_1(A)) ) )).

fof(fc2_square_1,axiom,(
! [A] :
( v1_xreal_0(A)
=> ( v1_xcmplx_0(k5_square_1(A))
& v1_xreal_0(k5_square_1(A)) ) ) )).

fof(t1_square_1,axiom,(
\$true )).

fof(t2_square_1,axiom,(
! [A] :
( v1_xreal_0(A)
=> ~ ( ~ r1_xreal_0(A,np__1)
& r1_xreal_0(np__1,k7_xcmplx_0(np__1,A)) ) ) )).

fof(t3_square_1,axiom,(
\$true )).

fof(t4_square_1,axiom,(
\$true )).

fof(t5_square_1,axiom,(
\$true )).

fof(t6_square_1,axiom,(
\$true )).

fof(t7_square_1,axiom,(
\$true )).

fof(t8_square_1,axiom,(
\$true )).

fof(t9_square_1,axiom,(
\$true )).

fof(t10_square_1,axiom,(
\$true )).

fof(t11_square_1,axiom,(
\$true )).

fof(t12_square_1,axiom,(
\$true )).

fof(t13_square_1,axiom,(
\$true )).

fof(t14_square_1,axiom,(
\$true )).

fof(t15_square_1,axiom,(
\$true )).

fof(t16_square_1,axiom,(
\$true )).

fof(t17_square_1,axiom,(
\$true )).

fof(t18_square_1,axiom,(
\$true )).

fof(t19_square_1,axiom,(
\$true )).

fof(t20_square_1,axiom,(
\$true )).

fof(t21_square_1,axiom,(
\$true )).

fof(t22_square_1,axiom,(
\$true )).

fof(t23_square_1,axiom,(
\$true )).

fof(t24_square_1,axiom,(
\$true )).

fof(t25_square_1,axiom,(
! [A] :
( v1_xreal_0(A)
=> ! [B] :
( v1_xreal_0(B)
=> ~ ( r1_xreal_0(np__0,k3_xcmplx_0(A,B))
& ~ ( r1_xreal_0(np__0,A)
& r1_xreal_0(np__0,B) )
& ~ ( r1_xreal_0(A,np__0)
& r1_xreal_0(B,np__0) ) ) ) ) )).

fof(d1_square_1,axiom,(
! [A] :
( v1_xreal_0(A)
=> ! [B] :
( v1_xreal_0(B)
=> ( ( r1_xreal_0(A,B)
=> k1_square_1(A,B) = A )
& ( ~ r1_xreal_0(A,B)
=> k1_square_1(A,B) = B ) ) ) ) )).

fof(d2_square_1,axiom,(
! [A] :
( v1_xreal_0(A)
=> ! [B] :
( v1_xreal_0(B)
=> ( ( r1_xreal_0(B,A)
=> k2_square_1(A,B) = A )
& ( ~ r1_xreal_0(B,A)
=> k2_square_1(A,B) = B ) ) ) ) )).

fof(t26_square_1,axiom,(
\$true )).

fof(t27_square_1,axiom,(
\$true )).

fof(t28_square_1,axiom,(
\$true )).

fof(t29_square_1,axiom,(
\$true )).

fof(t30_square_1,axiom,(
\$true )).

fof(t31_square_1,axiom,(
\$true )).

fof(t32_square_1,axiom,(
\$true )).

fof(t33_square_1,axiom,(
\$true )).

fof(t34_square_1,axiom,(
\$true )).

fof(t35_square_1,axiom,(
! [A] :
( v1_xreal_0(A)
=> ! [B] :
( v1_xreal_0(B)
=> r1_xreal_0(k1_square_1(A,B),A) ) ) )).

fof(t36_square_1,axiom,(
\$true )).

fof(t37_square_1,axiom,(
\$true )).

fof(t38_square_1,axiom,(
! [A] :
( v1_xreal_0(A)
=> ! [B] :
( v1_xreal_0(B)
=> ( k1_square_1(A,B) = A
| k1_square_1(A,B) = B ) ) ) )).

fof(t39_square_1,axiom,(
! [A] :
( v1_xreal_0(A)
=> ! [B] :
( v1_xreal_0(B)
=> ! [C] :
( v1_xreal_0(C)
=> ( ( r1_xreal_0(A,B)
& r1_xreal_0(A,C) )
<=> r1_xreal_0(A,k1_square_1(B,C)) ) ) ) ) )).

fof(t40_square_1,axiom,(
! [A] :
( v1_xreal_0(A)
=> ! [B] :
( v1_xreal_0(B)
=> ! [C] :
( v1_xreal_0(C)
=> k1_square_1(A,k1_square_1(B,C)) = k1_square_1(k1_square_1(A,B),C) ) ) ) )).

fof(t41_square_1,axiom,(
\$true )).

fof(t42_square_1,axiom,(
\$true )).

fof(t43_square_1,axiom,(
\$true )).

fof(t44_square_1,axiom,(
\$true )).

fof(t45_square_1,axiom,(
\$true )).

fof(t46_square_1,axiom,(
! [A] :
( v1_xreal_0(A)
=> ! [B] :
( v1_xreal_0(B)
=> r1_xreal_0(A,k2_square_1(A,B)) ) ) )).

fof(t47_square_1,axiom,(
\$true )).

fof(t48_square_1,axiom,(
\$true )).

fof(t49_square_1,axiom,(
! [A] :
( v1_xreal_0(A)
=> ! [B] :
( v1_xreal_0(B)
=> ( k2_square_1(A,B) = A
| k2_square_1(A,B) = B ) ) ) )).

fof(t50_square_1,axiom,(
! [A] :
( v1_xreal_0(A)
=> ! [B] :
( v1_xreal_0(B)
=> ! [C] :
( v1_xreal_0(C)
=> ( ( r1_xreal_0(A,B)
& r1_xreal_0(C,B) )
<=> r1_xreal_0(k2_square_1(A,C),B) ) ) ) ) )).

fof(t51_square_1,axiom,(
! [A] :
( v1_xreal_0(A)
=> ! [B] :
( v1_xreal_0(B)
=> ! [C] :
( v1_xreal_0(C)
=> k2_square_1(A,k2_square_1(B,C)) = k2_square_1(k2_square_1(A,B),C) ) ) ) )).

fof(t52_square_1,axiom,(
\$true )).

fof(t53_square_1,axiom,(
! [A] :
( v1_xreal_0(A)
=> ! [B] :
( v1_xreal_0(B)
=> k2_xcmplx_0(k1_square_1(A,B),k2_square_1(A,B)) = k2_xcmplx_0(A,B) ) ) )).

fof(t54_square_1,axiom,(
! [A] :
( v1_xreal_0(A)
=> ! [B] :
( v1_xreal_0(B)
=> k2_square_1(A,k1_square_1(A,B)) = A ) ) )).

fof(t55_square_1,axiom,(
! [A] :
( v1_xreal_0(A)
=> ! [B] :
( v1_xreal_0(B)
=> k1_square_1(A,k2_square_1(A,B)) = A ) ) )).

fof(t56_square_1,axiom,(
! [A] :
( v1_xreal_0(A)
=> ! [B] :
( v1_xreal_0(B)
=> ! [C] :
( v1_xreal_0(C)
=> k1_square_1(A,k2_square_1(B,C)) = k2_square_1(k1_square_1(A,B),k1_square_1(A,C)) ) ) ) )).

fof(t57_square_1,axiom,(
! [A] :
( v1_xreal_0(A)
=> ! [B] :
( v1_xreal_0(B)
=> ! [C] :
( v1_xreal_0(C)
=> k2_square_1(A,k1_square_1(B,C)) = k1_square_1(k2_square_1(A,B),k2_square_1(A,C)) ) ) ) )).

fof(d3_square_1,axiom,(
! [A] :
( v1_xcmplx_0(A)
=> k5_square_1(A) = k3_xcmplx_0(A,A) ) )).

fof(t58_square_1,axiom,(
\$true )).

fof(t59_square_1,axiom,(
k7_square_1(np__1) = np__1 )).

fof(t60_square_1,axiom,(
k7_square_1(np__0) = np__0 )).

fof(t61_square_1,axiom,(
! [A] :
( v1_xcmplx_0(A)
=> k5_square_1(A) = k5_square_1(k4_xcmplx_0(A)) ) )).

fof(t62_square_1,axiom,(
\$true )).

fof(t63_square_1,axiom,(
! [A] :
( v1_xcmplx_0(A)
=> ! [B] :
( v1_xcmplx_0(B)
=> k5_square_1(k2_xcmplx_0(A,B)) = k2_xcmplx_0(k2_xcmplx_0(k5_square_1(A),k3_xcmplx_0(k3_xcmplx_0(np__2,A),B)),k5_square_1(B)) ) ) )).

fof(t64_square_1,axiom,(
! [A] :
( v1_xcmplx_0(A)
=> ! [B] :
( v1_xcmplx_0(B)
=> k5_square_1(k6_xcmplx_0(A,B)) = k2_xcmplx_0(k6_xcmplx_0(k5_square_1(A),k3_xcmplx_0(k3_xcmplx_0(np__2,A),B)),k5_square_1(B)) ) ) )).

fof(t65_square_1,axiom,(
! [A] :
( v1_xcmplx_0(A)
=> k5_square_1(k2_xcmplx_0(A,np__1)) = k2_xcmplx_0(k2_xcmplx_0(k5_square_1(A),k3_xcmplx_0(np__2,A)),np__1) ) )).

fof(t66_square_1,axiom,(
! [A] :
( v1_xcmplx_0(A)
=> k5_square_1(k6_xcmplx_0(A,np__1)) = k2_xcmplx_0(k6_xcmplx_0(k5_square_1(A),k3_xcmplx_0(np__2,A)),np__1) ) )).

fof(t67_square_1,axiom,(
! [A] :
( v1_xcmplx_0(A)
=> ! [B] :
( v1_xcmplx_0(B)
=> k3_xcmplx_0(k6_xcmplx_0(A,B),k2_xcmplx_0(A,B)) = k6_xcmplx_0(k5_square_1(A),k5_square_1(B)) ) ) )).

fof(t68_square_1,axiom,(
! [A] :
( v1_xcmplx_0(A)
=> ! [B] :
( v1_xcmplx_0(B)
=> k5_square_1(k3_xcmplx_0(A,B)) = k3_xcmplx_0(k5_square_1(A),k5_square_1(B)) ) ) )).

fof(t69_square_1,axiom,(
! [A] :
( v1_xcmplx_0(A)
=> ! [B] :
( v1_xcmplx_0(B)
=> k5_square_1(k7_xcmplx_0(A,B)) = k7_xcmplx_0(k5_square_1(A),k5_square_1(B)) ) ) )).

fof(t70_square_1,axiom,(
! [A] :
( v1_xcmplx_0(A)
=> ! [B] :
( v1_xcmplx_0(B)
=> ( k6_xcmplx_0(k5_square_1(A),k5_square_1(B)) != np__0
=> k7_xcmplx_0(np__1,k2_xcmplx_0(A,B)) = k7_xcmplx_0(k6_xcmplx_0(A,B),k6_xcmplx_0(k5_square_1(A),k5_square_1(B))) ) ) ) )).

fof(t71_square_1,axiom,(
! [A] :
( v1_xcmplx_0(A)
=> ! [B] :
( v1_xcmplx_0(B)
=> ( k6_xcmplx_0(k5_square_1(A),k5_square_1(B)) != np__0
=> k7_xcmplx_0(np__1,k6_xcmplx_0(A,B)) = k7_xcmplx_0(k2_xcmplx_0(A,B),k6_xcmplx_0(k5_square_1(A),k5_square_1(B))) ) ) ) )).

fof(t72_square_1,axiom,(
! [A] :
( v1_xreal_0(A)
=> r1_xreal_0(np__0,k5_square_1(A)) ) )).

fof(t73_square_1,axiom,(
! [A] :
( v1_xcmplx_0(A)
=> ( k5_square_1(A) = np__0
=> A = np__0 ) ) )).

fof(t74_square_1,axiom,(
! [A] :
( v1_xreal_0(A)
=> ~ ( np__0 != A
& r1_xreal_0(k5_square_1(A),np__0) ) ) )).

fof(t75_square_1,axiom,(
! [A] :
( v1_xreal_0(A)
=> ~ ( ~ r1_xreal_0(A,np__0)
& ~ r1_xreal_0(np__1,A)
& r1_xreal_0(A,k5_square_1(A)) ) ) )).

fof(t76_square_1,axiom,(
! [A] :
( v1_xreal_0(A)
=> ~ ( ~ r1_xreal_0(A,np__1)
& r1_xreal_0(k5_square_1(A),A) ) ) )).

fof(t77_square_1,axiom,(
! [A] :
( v1_xreal_0(A)
=> ! [B] :
( v1_xreal_0(B)
=> ( ( r1_xreal_0(np__0,A)
& r1_xreal_0(A,B) )
=> r1_xreal_0(k5_square_1(A),k5_square_1(B)) ) ) ) )).

fof(t78_square_1,axiom,(
! [A] :
( v1_xreal_0(A)
=> ! [B] :
( v1_xreal_0(B)
=> ~ ( r1_xreal_0(np__0,A)
& ~ r1_xreal_0(B,A)
& r1_xreal_0(k5_square_1(B),k5_square_1(A)) ) ) ) )).

fof(d4_square_1,axiom,(
! [A] :
( v1_xreal_0(A)
=> ( r1_xreal_0(np__0,A)
=> ! [B] :
( v1_xreal_0(B)
=> ( B = k8_square_1(A)
<=> ( r1_xreal_0(np__0,B)
& k5_square_1(B) = A ) ) ) ) ) )).

fof(t79_square_1,axiom,(
\$true )).

fof(t80_square_1,axiom,(
\$true )).

fof(t81_square_1,axiom,(
\$true )).

fof(t82_square_1,axiom,(
k9_square_1(np__0) = np__0 )).

fof(t83_square_1,axiom,(
k9_square_1(np__1) = np__1 )).

fof(t84_square_1,axiom,(
~ r1_xreal_0(k9_square_1(np__2),np__1) )).

fof(t85_square_1,axiom,(
k9_square_1(np__4) = np__2 )).

fof(t86_square_1,axiom,(
~ r1_xreal_0(np__2,k9_square_1(np__2)) )).

fof(t87_square_1,axiom,(
\$true )).

fof(t88_square_1,axiom,(
\$true )).

fof(t89_square_1,axiom,(
! [A] :
( v1_xreal_0(A)
=> ( r1_xreal_0(np__0,A)
=> k8_square_1(k5_square_1(A)) = A ) ) )).

fof(t90_square_1,axiom,(
! [A] :
( v1_xreal_0(A)
=> ( r1_xreal_0(A,np__0)
=> k8_square_1(k5_square_1(A)) = k4_xcmplx_0(A) ) ) )).

fof(t91_square_1,axiom,(
\$true )).

fof(t92_square_1,axiom,(
! [A] :
( v1_xreal_0(A)
=> ( ( r1_xreal_0(np__0,A)
& k8_square_1(A) = np__0 )
=> A = np__0 ) ) )).

fof(t93_square_1,axiom,(
! [A] :
( v1_xreal_0(A)
=> ~ ( ~ r1_xreal_0(A,np__0)
& r1_xreal_0(k8_square_1(A),np__0) ) ) )).

fof(t94_square_1,axiom,(
! [A] :
( v1_xreal_0(A)
=> ! [B] :
( v1_xreal_0(B)
=> ( ( r1_xreal_0(np__0,A)
& r1_xreal_0(A,B) )
=> r1_xreal_0(k8_square_1(A),k8_square_1(B)) ) ) ) )).

fof(t95_square_1,axiom,(
! [A] :
( v1_xreal_0(A)
=> ! [B] :
( v1_xreal_0(B)
=> ~ ( r1_xreal_0(np__0,A)
& ~ r1_xreal_0(B,A)
& r1_xreal_0(k8_square_1(B),k8_square_1(A)) ) ) ) )).

fof(t96_square_1,axiom,(
! [A] :
( v1_xreal_0(A)
=> ! [B] :
( v1_xreal_0(B)
=> ( ( r1_xreal_0(np__0,A)
& r1_xreal_0(np__0,B)
& k8_square_1(A) = k8_square_1(B) )
=> A = B ) ) ) )).

fof(t97_square_1,axiom,(
! [A] :
( v1_xreal_0(A)
=> ! [B] :
( v1_xreal_0(B)
=> ( ( r1_xreal_0(np__0,A)
& r1_xreal_0(np__0,B) )
=> k8_square_1(k3_xcmplx_0(A,B)) = k3_xcmplx_0(k8_square_1(A),k8_square_1(B)) ) ) ) )).

fof(t98_square_1,axiom,(
\$true )).

fof(t99_square_1,axiom,(
! [A] :
( v1_xreal_0(A)
=> ! [B] :
( v1_xreal_0(B)
=> ( ( r1_xreal_0(np__0,A)
& r1_xreal_0(np__0,B) )
=> k8_square_1(k7_xcmplx_0(A,B)) = k7_xcmplx_0(k8_square_1(A),k8_square_1(B)) ) ) ) )).

fof(t100_square_1,axiom,(
\$true )).

fof(t101_square_1,axiom,(
! [A] :
( v1_xreal_0(A)
=> ( ~ r1_xreal_0(A,np__0)
=> k8_square_1(k7_xcmplx_0(np__1,A)) = k7_xcmplx_0(np__1,k8_square_1(A)) ) ) )).

fof(t102_square_1,axiom,(
! [A] :
( v1_xreal_0(A)
=> ( ~ r1_xreal_0(A,np__0)
=> k7_xcmplx_0(k8_square_1(A),A) = k7_xcmplx_0(np__1,k8_square_1(A)) ) ) )).

fof(t103_square_1,axiom,(
! [A] :
( v1_xreal_0(A)
=> ( ~ r1_xreal_0(A,np__0)
=> k7_xcmplx_0(A,k8_square_1(A)) = k8_square_1(A) ) ) )).

fof(t104_square_1,axiom,(
! [A] :
( v1_xreal_0(A)
=> ! [B] :
( v1_xreal_0(B)
=> ( ( r1_xreal_0(np__0,A)
& r1_xreal_0(np__0,B) )
=> k3_xcmplx_0(k6_xcmplx_0(k8_square_1(A),k8_square_1(B)),k2_xcmplx_0(k8_square_1(A),k8_square_1(B))) = k6_xcmplx_0(A,B) ) ) ) )).

fof(t105_square_1,axiom,(
! [A] :
( v1_xreal_0(A)
=> ! [B] :
( v1_xreal_0(B)
=> ( ( r1_xreal_0(np__0,A)
& r1_xreal_0(np__0,B) )
=> ( A = B
| k7_xcmplx_0(np__1,k2_xcmplx_0(k8_square_1(A),k8_square_1(B))) = k7_xcmplx_0(k6_xcmplx_0(k8_square_1(A),k8_square_1(B)),k6_xcmplx_0(A,B)) ) ) ) ) )).

fof(t106_square_1,axiom,(
! [A] :
( v1_xreal_0(A)
=> ! [B] :
( v1_xreal_0(B)
=> ( r1_xreal_0(np__0,A)
=> ( r1_xreal_0(B,A)
| k7_xcmplx_0(np__1,k2_xcmplx_0(k8_square_1(B),k8_square_1(A))) = k7_xcmplx_0(k6_xcmplx_0(k8_square_1(B),k8_square_1(A)),k6_xcmplx_0(B,A)) ) ) ) ) )).

fof(t107_square_1,axiom,(
! [A] :
( v1_xreal_0(A)
=> ! [B] :
( v1_xreal_0(B)
=> ( ( r1_xreal_0(np__0,A)
& r1_xreal_0(np__0,B) )
=> k7_xcmplx_0(np__1,k6_xcmplx_0(k8_square_1(A),k8_square_1(B))) = k7_xcmplx_0(k2_xcmplx_0(k8_square_1(A),k8_square_1(B)),k6_xcmplx_0(A,B)) ) ) ) )).

fof(t108_square_1,axiom,(
! [A] :
( v1_xreal_0(A)
=> ! [B] :
( v1_xreal_0(B)
=> ( r1_xreal_0(np__0,A)
=> ( r1_xreal_0(B,A)
| k7_xcmplx_0(np__1,k6_xcmplx_0(k8_square_1(B),k8_square_1(A))) = k7_xcmplx_0(k2_xcmplx_0(k8_square_1(B),k8_square_1(A)),k6_xcmplx_0(B,A)) ) ) ) ) )).

fof(s1_square_1,axiom,
( ! [A] :
( v1_xreal_0(A)
=> ! [B] :
( v1_xreal_0(B)
=> ( ( p1_s1_square_1(A)
& p2_s1_square_1(B) )
=> r1_xreal_0(A,B) ) ) )
=> ? [A] :
( v1_xreal_0(A)
& ! [B] :
( v1_xreal_0(B)
=> ! [C] :
( v1_xreal_0(C)
=> ( ( p1_s1_square_1(B)
& p2_s1_square_1(C) )
=> ( r1_xreal_0(B,A)
& r1_xreal_0(A,C) ) ) ) ) ) )).

fof(dt_k1_square_1,axiom,(
! [A,B] :
( ( v1_xreal_0(A)
& v1_xreal_0(B) )
=> v1_xreal_0(k1_square_1(A,B)) ) )).

fof(commutativity_k1_square_1,axiom,(
! [A,B] :
( ( v1_xreal_0(A)
& v1_xreal_0(B) )
=> k1_square_1(A,B) = k1_square_1(B,A) ) )).

fof(idempotence_k1_square_1,axiom,(
! [A,B] :
( ( v1_xreal_0(A)
& v1_xreal_0(B) )
=> k1_square_1(A,A) = A ) )).

fof(dt_k2_square_1,axiom,(
! [A,B] :
( ( v1_xreal_0(A)
& v1_xreal_0(B) )
=> v1_xreal_0(k2_square_1(A,B)) ) )).

fof(commutativity_k2_square_1,axiom,(
! [A,B] :
( ( v1_xreal_0(A)
& v1_xreal_0(B) )
=> k2_square_1(A,B) = k2_square_1(B,A) ) )).

fof(idempotence_k2_square_1,axiom,(
! [A,B] :
( ( v1_xreal_0(A)
& v1_xreal_0(B) )
=> k2_square_1(A,A) = A ) )).

fof(dt_k3_square_1,axiom,(
! [A,B] :
( ( m1_subset_1(A,k1_numbers)
& m1_subset_1(B,k1_numbers) )
=> m1_subset_1(k3_square_1(A,B),k1_numbers) ) )).

fof(commutativity_k3_square_1,axiom,(
! [A,B] :
( ( m1_subset_1(A,k1_numbers)
& m1_subset_1(B,k1_numbers) )
=> k3_square_1(A,B) = k3_square_1(B,A) ) )).

fof(idempotence_k3_square_1,axiom,(
! [A,B] :
( ( m1_subset_1(A,k1_numbers)
& m1_subset_1(B,k1_numbers) )
=> k3_square_1(A,A) = A ) )).

fof(redefinition_k3_square_1,axiom,(
! [A,B] :
( ( m1_subset_1(A,k1_numbers)
& m1_subset_1(B,k1_numbers) )
=> k3_square_1(A,B) = k1_square_1(A,B) ) )).

fof(dt_k4_square_1,axiom,(
! [A,B] :
( ( m1_subset_1(A,k1_numbers)
& m1_subset_1(B,k1_numbers) )
=> m1_subset_1(k4_square_1(A,B),k1_numbers) ) )).

fof(commutativity_k4_square_1,axiom,(
! [A,B] :
( ( m1_subset_1(A,k1_numbers)
& m1_subset_1(B,k1_numbers) )
=> k4_square_1(A,B) = k4_square_1(B,A) ) )).

fof(idempotence_k4_square_1,axiom,(
! [A,B] :
( ( m1_subset_1(A,k1_numbers)
& m1_subset_1(B,k1_numbers) )
=> k4_square_1(A,A) = A ) )).

fof(redefinition_k4_square_1,axiom,(
! [A,B] :
( ( m1_subset_1(A,k1_numbers)
& m1_subset_1(B,k1_numbers) )
=> k4_square_1(A,B) = k2_square_1(A,B) ) )).

fof(dt_k5_square_1,axiom,(
\$true )).

fof(dt_k6_square_1,axiom,(
! [A] :
( m1_subset_1(A,k2_numbers)
=> m1_subset_1(k6_square_1(A),k2_numbers) ) )).

fof(redefinition_k6_square_1,axiom,(
! [A] :
( m1_subset_1(A,k2_numbers)
=> k6_square_1(A) = k5_square_1(A) ) )).

fof(dt_k7_square_1,axiom,(
! [A] :
( m1_subset_1(A,k1_numbers)
=> m1_subset_1(k7_square_1(A),k1_numbers) ) )).

fof(redefinition_k7_square_1,axiom,(
! [A] :
( m1_subset_1(A,k1_numbers)
=> k7_square_1(A) = k5_square_1(A) ) )).

fof(dt_k8_square_1,axiom,(
! [A] :
( v1_xreal_0(A)
=> v1_xreal_0(k8_square_1(A)) ) )).

fof(dt_k9_square_1,axiom,(
! [A] :
( m1_subset_1(A,k1_numbers)
=> m1_subset_1(k9_square_1(A),k1_numbers) ) )).

fof(redefinition_k9_square_1,axiom,(
! [A] :
( m1_subset_1(A,k1_numbers)
=> k9_square_1(A) = k8_square_1(A) ) )).
%------------------------------------------------------------------------------
```