## SET007 Axioms: SET007+178.ax

```%------------------------------------------------------------------------------
% File     : SET007+178 : TPTP v7.5.0. Released v3.4.0.
% Domain   : Set Theory
% Axioms   : Properties of Extended Real Numbers Operations: abs, min and max
% 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    : extreal2 [Urb08]

% Status   : Satisfiable
% Syntax   : Number of formulae    :  112 (  13 unit)
%            Number of atoms       :  643 ( 273 equality)
%            Maximal formula depth :   16 (   8 average)
%            Number of connectives :  784 ( 253 ~  ;  27  |; 264  &)
%                                         (   6 <=>; 234 =>;   0 <=)
%                                         (   0 <~>;   0 ~|;   0 ~&)
%            Number of predicates  :    6 (   1 propositional; 0-2 arity)
%            Number of functors    :   21 (   9 constant; 0-2 arity)
%            Number of variables   :  202 (   0 singleton; 202 !;   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_extreal2,axiom,(
\$true )).

fof(t2_extreal2,axiom,(
! [A] :
( m1_subset_1(A,k3_supinf_1)
=> ~ ( A != k5_measure6
& A != k4_measure6
& ! [B] :
( m1_subset_1(B,k3_supinf_1)
=> k2_supinf_2(A,B) != k1_supinf_2 ) ) ) )).

fof(t3_extreal2,axiom,(
! [A] :
( m1_subset_1(A,k3_supinf_1)
=> ~ ( A != k5_measure6
& A != k4_measure6
& A != k1_supinf_2
& ! [B] :
( m1_subset_1(B,k3_supinf_1)
=> k2_extreal1(A,B) != k10_mesfunc1 ) ) ) )).

fof(t4_extreal2,axiom,(
! [A] :
( m1_subset_1(A,k3_supinf_1)
=> ( k2_extreal1(k10_mesfunc1,A) = A
& k2_extreal1(A,k10_mesfunc1) = A
& k2_extreal1(k1_measure6(np__1),A) = A
& k2_extreal1(A,k1_measure6(np__1)) = A ) ) )).

fof(t5_extreal2,axiom,(
! [A] :
( m1_subset_1(A,k3_supinf_1)
=> k4_supinf_2(k1_supinf_2,A) = k3_supinf_2(A) ) )).

fof(t6_extreal2,axiom,(
\$true )).

fof(t7_extreal2,axiom,(
! [A] :
( m1_subset_1(A,k3_supinf_1)
=> ! [B] :
( m1_subset_1(B,k3_supinf_1)
=> ( ( r1_supinf_1(k1_supinf_2,A)
& r1_supinf_1(k1_supinf_2,B) )
=> r1_supinf_1(k1_supinf_2,k2_supinf_2(A,B)) ) ) ) )).

fof(t8_extreal2,axiom,(
\$true )).

fof(t9_extreal2,axiom,(
! [A] :
( m1_subset_1(A,k3_supinf_1)
=> ! [B] :
( m1_subset_1(B,k3_supinf_1)
=> ( ( r1_supinf_1(A,k1_supinf_2)
& r1_supinf_1(B,k1_supinf_2) )
=> r1_supinf_1(k2_supinf_2(A,B),k1_supinf_2) ) ) ) )).

fof(t10_extreal2,axiom,(
\$true )).

fof(t11_extreal2,axiom,(
! [A] :
( m1_subset_1(A,k3_supinf_1)
=> ! [B] :
( m1_subset_1(B,k3_supinf_1)
=> ! [C] :
( m1_subset_1(C,k3_supinf_1)
=> ( k2_supinf_2(B,A) = C
=> ( A = k5_measure6
| A = k4_measure6
| B = k4_supinf_2(C,A) ) ) ) ) ) )).

fof(t12_extreal2,axiom,(
! [A] :
( m1_subset_1(A,k3_supinf_1)
=> ~ ( A != k5_measure6
& A != k4_measure6
& A != k1_supinf_2
& ~ ( k2_extreal1(A,k3_extreal1(k10_mesfunc1,A)) = k10_mesfunc1
& k2_extreal1(k3_extreal1(k10_mesfunc1,A),A) = k10_mesfunc1 ) ) ) )).

fof(t13_extreal2,axiom,(
! [A] :
( m1_subset_1(A,k3_supinf_1)
=> ~ ( A != k5_measure6
& A != k4_measure6
& k4_supinf_2(A,A) != k1_supinf_2 ) ) )).

fof(t14_extreal2,axiom,(
! [A] :
( m1_subset_1(A,k3_supinf_1)
=> ! [B] :
( m1_subset_1(B,k3_supinf_1)
=> ~ ( ~ ( A = k5_measure6
& B = k4_measure6 )
& ~ ( A = k4_measure6
& B = k5_measure6 )
& ~ ( k3_supinf_2(k2_supinf_2(A,B)) = k2_supinf_2(k3_supinf_2(A),k3_supinf_2(B))
& k3_supinf_2(k2_supinf_2(A,B)) = k4_supinf_2(k3_supinf_2(B),A)
& k3_supinf_2(k2_supinf_2(A,B)) = k4_supinf_2(k3_supinf_2(A),B) ) ) ) ) )).

fof(t15_extreal2,axiom,(
! [A] :
( m1_subset_1(A,k3_supinf_1)
=> ! [B] :
( m1_subset_1(B,k3_supinf_1)
=> ~ ( ~ ( A = k5_measure6
& B = k5_measure6 )
& ~ ( A = k4_measure6
& B = k4_measure6 )
& ~ ( k3_supinf_2(k4_supinf_2(A,B)) = k2_supinf_2(k3_supinf_2(A),B)
& k3_supinf_2(k4_supinf_2(A,B)) = k4_supinf_2(B,A) ) ) ) ) )).

fof(t16_extreal2,axiom,(
! [A] :
( m1_subset_1(A,k3_supinf_1)
=> ! [B] :
( m1_subset_1(B,k3_supinf_1)
=> ~ ( ~ ( A = k5_measure6
& B = k5_measure6 )
& ~ ( A = k4_measure6
& B = k4_measure6 )
& ~ ( k3_supinf_2(k2_supinf_2(k3_supinf_2(A),B)) = k4_supinf_2(A,B)
& k3_supinf_2(k2_supinf_2(k3_supinf_2(A),B)) = k2_supinf_2(A,k3_supinf_2(B)) ) ) ) ) )).

fof(t17_extreal2,axiom,(
! [A] :
( m1_subset_1(A,k3_supinf_1)
=> ! [B] :
( m1_subset_1(B,k3_supinf_1)
=> ( ( ( A = k5_measure6
& ~ r1_supinf_1(B,k1_supinf_2)
& ~ r1_supinf_1(k5_measure6,B) )
| ( A = k4_measure6
& ~ r1_supinf_1(k1_supinf_2,B)
& ~ r1_supinf_1(B,k4_measure6) ) )
=> k3_extreal1(A,B) = k5_measure6 ) ) ) )).

fof(t18_extreal2,axiom,(
! [A] :
( m1_subset_1(A,k3_supinf_1)
=> ! [B] :
( m1_subset_1(B,k3_supinf_1)
=> ( ( ( A = k5_measure6
& ~ r1_supinf_1(k1_supinf_2,B)
& ~ r1_supinf_1(B,k4_measure6) )
| ( A = k4_measure6
& ~ r1_supinf_1(B,k1_supinf_2)
& ~ r1_supinf_1(k5_measure6,B) ) )
=> k3_extreal1(A,B) = k4_measure6 ) ) ) )).

fof(t19_extreal2,axiom,(
! [A] :
( m1_subset_1(A,k3_supinf_1)
=> ! [B] :
( m1_subset_1(B,k3_supinf_1)
=> ~ ( ~ r1_supinf_1(A,k4_measure6)
& ~ r1_supinf_1(k5_measure6,A)
& A != k1_supinf_2
& ~ ( k3_extreal1(k2_extreal1(B,A),A) = B
& k2_extreal1(B,k3_extreal1(A,A)) = B ) ) ) ) )).

fof(t20_extreal2,axiom,
( ~ r1_supinf_1(k5_measure6,k10_mesfunc1)
& ~ r1_supinf_1(k10_mesfunc1,k4_measure6) )).

fof(t21_extreal2,axiom,(
! [A] :
( m1_subset_1(A,k3_supinf_1)
=> ( ( A = k5_measure6
| A = k4_measure6 )
=> ! [B] :
( m1_subset_1(B,k3_supinf_1)
=> ~ ( r2_hidden(B,k6_supinf_1)
& k2_supinf_2(A,B) = k1_supinf_2 ) ) ) ) )).

fof(t22_extreal2,axiom,(
! [A] :
( m1_subset_1(A,k3_supinf_1)
=> ( ( A = k5_measure6
| A = k4_measure6 )
=> ! [B] :
( m1_subset_1(B,k3_supinf_1)
=> k2_extreal1(A,B) != k10_mesfunc1 ) ) ) )).

fof(t23_extreal2,axiom,(
! [A] :
( m1_subset_1(A,k3_supinf_1)
=> ! [B] :
( m1_subset_1(B,k3_supinf_1)
=> ~ ( ~ ( A = k5_measure6
& B = k4_measure6 )
& ~ ( A = k4_measure6
& B = k5_measure6 )
& ~ r1_supinf_1(k5_measure6,k2_supinf_2(A,B))
& ~ ( A != k5_measure6
& B != k5_measure6 ) ) ) ) )).

fof(t24_extreal2,axiom,(
! [A] :
( m1_subset_1(A,k3_supinf_1)
=> ! [B] :
( m1_subset_1(B,k3_supinf_1)
=> ~ ( ~ ( A = k5_measure6
& B = k4_measure6 )
& ~ ( A = k4_measure6
& B = k5_measure6 )
& ~ r1_supinf_1(k2_supinf_2(A,B),k4_measure6)
& ~ ( A != k4_measure6
& B != k4_measure6 ) ) ) ) )).

fof(t25_extreal2,axiom,(
! [A] :
( m1_subset_1(A,k3_supinf_1)
=> ! [B] :
( m1_subset_1(B,k3_supinf_1)
=> ~ ( ~ ( A = k5_measure6
& B = k5_measure6 )
& ~ ( A = k4_measure6
& B = k4_measure6 )
& ~ r1_supinf_1(k5_measure6,k4_supinf_2(A,B))
& ~ ( A != k5_measure6
& B != k4_measure6 ) ) ) ) )).

fof(t26_extreal2,axiom,(
! [A] :
( m1_subset_1(A,k3_supinf_1)
=> ! [B] :
( m1_subset_1(B,k3_supinf_1)
=> ~ ( ~ ( A = k5_measure6
& B = k5_measure6 )
& ~ ( A = k4_measure6
& B = k4_measure6 )
& ~ r1_supinf_1(k4_supinf_2(A,B),k4_measure6)
& ~ ( A != k4_measure6
& B != k5_measure6 ) ) ) ) )).

fof(t27_extreal2,axiom,(
! [A] :
( m1_subset_1(A,k3_supinf_1)
=> ! [B] :
( m1_subset_1(B,k3_supinf_1)
=> ! [C] :
( m1_subset_1(C,k3_supinf_1)
=> ~ ( ~ ( A = k5_measure6
& B = k4_measure6 )
& ~ ( A = k4_measure6
& B = k5_measure6 )
& ~ r1_supinf_1(C,k2_supinf_2(A,B))
& ~ ( A != k5_measure6
& B != k5_measure6
& C != k4_measure6
& ~ r1_supinf_1(k4_supinf_2(C,B),A) ) ) ) ) ) )).

fof(t28_extreal2,axiom,(
! [A] :
( m1_subset_1(A,k3_supinf_1)
=> ! [B] :
( m1_subset_1(B,k3_supinf_1)
=> ! [C] :
( m1_subset_1(C,k3_supinf_1)
=> ~ ( ~ ( A = k5_measure6
& B = k5_measure6 )
& ~ ( A = k4_measure6
& B = k4_measure6 )
& ~ r1_supinf_1(k4_supinf_2(A,B),C)
& ~ ( C != k5_measure6
& B != k5_measure6
& A != k4_measure6
& ~ r1_supinf_1(A,k2_supinf_2(C,B)) ) ) ) ) ) )).

fof(t29_extreal2,axiom,(
! [A] :
( m1_subset_1(A,k3_supinf_1)
=> ! [B] :
( m1_subset_1(B,k3_supinf_1)
=> ! [C] :
( m1_subset_1(C,k3_supinf_1)
=> ~ ( ~ ( A = k5_measure6
& B = k5_measure6 )
& ~ ( A = k4_measure6
& B = k4_measure6 )
& ~ r1_supinf_1(C,k4_supinf_2(A,B))
& ~ ( A != k5_measure6
& B != k4_measure6
& C != k4_measure6
& ~ r1_supinf_1(k2_supinf_2(C,B),A) ) ) ) ) ) )).

fof(t30_extreal2,axiom,(
! [A] :
( m1_subset_1(A,k3_supinf_1)
=> ! [B] :
( m1_subset_1(B,k3_supinf_1)
=> ! [C] :
( m1_subset_1(C,k3_supinf_1)
=> ~ ( ~ ( A = k5_measure6
& B = k4_measure6 )
& ~ ( A = k4_measure6
& B = k5_measure6 )
& ~ r1_supinf_1(k2_supinf_2(A,B),C)
& ~ ( C != k5_measure6
& B != k4_measure6
& A != k4_measure6
& ~ r1_supinf_1(A,k4_supinf_2(C,B)) ) ) ) ) ) )).

fof(t31_extreal2,axiom,(
! [A] :
( m1_subset_1(A,k3_supinf_1)
=> ! [B] :
( m1_subset_1(B,k3_supinf_1)
=> ! [C] :
( m1_subset_1(C,k3_supinf_1)
=> ~ ( ~ ( A = k5_measure6
& B = k4_measure6 )
& ~ ( A = k4_measure6
& B = k5_measure6 )
& ~ ( B = k5_measure6
& C = k5_measure6 )
& ~ ( B = k4_measure6
& C = k4_measure6 )
& r1_supinf_1(k2_supinf_2(A,B),C)
& ~ ( B != k5_measure6
& r1_supinf_1(A,k4_supinf_2(C,B)) ) ) ) ) ) )).

fof(t32_extreal2,axiom,(
! [A] :
( m1_subset_1(A,k3_supinf_1)
=> ! [B] :
( m1_subset_1(B,k3_supinf_1)
=> ! [C] :
( m1_subset_1(C,k3_supinf_1)
=> ( r1_supinf_1(A,k4_supinf_2(C,B))
=> ( ( A = k5_measure6
& B = k4_measure6 )
| ( A = k4_measure6
& B = k5_measure6 )
| ( B = k5_measure6
& C = k5_measure6 )
| ( B != k5_measure6
& r1_supinf_1(k2_supinf_2(A,B),C) ) ) ) ) ) ) )).

fof(t33_extreal2,axiom,(
! [A] :
( m1_subset_1(A,k3_supinf_1)
=> ! [B] :
( m1_subset_1(B,k3_supinf_1)
=> ! [C] :
( m1_subset_1(C,k3_supinf_1)
=> ~ ( ~ ( A = k5_measure6
& B = k5_measure6 )
& ~ ( A = k4_measure6
& B = k4_measure6 )
& ~ ( B = k5_measure6
& C = k4_measure6 )
& ~ ( B = k4_measure6
& C = k5_measure6 )
& r1_supinf_1(k4_supinf_2(A,B),C)
& ~ ( B != k4_measure6
& r1_supinf_1(A,k2_supinf_2(C,B)) ) ) ) ) ) )).

fof(t34_extreal2,axiom,(
! [A] :
( m1_subset_1(A,k3_supinf_1)
=> ! [B] :
( m1_subset_1(B,k3_supinf_1)
=> ! [C] :
( m1_subset_1(C,k3_supinf_1)
=> ( r1_supinf_1(A,k2_supinf_2(C,B))
=> ( ( A = k5_measure6
& B = k5_measure6 )
| ( A = k4_measure6
& B = k4_measure6 )
| ( B = k4_measure6
& C = k5_measure6 )
| ( B != k4_measure6
& r1_supinf_1(k4_supinf_2(A,B),C) ) ) ) ) ) ) )).

fof(t35_extreal2,axiom,(
\$true )).

fof(t36_extreal2,axiom,(
\$true )).

fof(t37_extreal2,axiom,(
\$true )).

fof(t38_extreal2,axiom,(
\$true )).

fof(t39_extreal2,axiom,(
\$true )).

fof(t40_extreal2,axiom,(
! [A] :
( m1_subset_1(A,k3_supinf_1)
=> ! [B] :
( m1_subset_1(B,k3_supinf_1)
=> ! [C] :
( m1_subset_1(C,k3_supinf_1)
=> ~ ( ~ ( A = k5_measure6
& B = k5_measure6 )
& ~ ( A = k4_measure6
& B = k4_measure6 )
& ~ ( B = k5_measure6
& C = k4_measure6 )
& ~ ( B = k4_measure6
& C = k5_measure6 )
& ~ ( A = k5_measure6
& C = k5_measure6 )
& ~ ( A = k4_measure6
& C = k4_measure6 )
& k4_supinf_2(k4_supinf_2(A,B),C) != k4_supinf_2(A,k2_supinf_2(B,C)) ) ) ) ) )).

fof(t41_extreal2,axiom,(
! [A] :
( m1_subset_1(A,k3_supinf_1)
=> ! [B] :
( m1_subset_1(B,k3_supinf_1)
=> ! [C] :
( m1_subset_1(C,k3_supinf_1)
=> ~ ( ~ ( A = k5_measure6
& B = k5_measure6 )
& ~ ( A = k4_measure6
& B = k4_measure6 )
& ~ ( B = k5_measure6
& C = k5_measure6 )
& ~ ( B = k4_measure6
& C = k4_measure6 )
& ~ ( A = k5_measure6
& C = k4_measure6 )
& ~ ( A = k4_measure6
& C = k5_measure6 )
& k2_supinf_2(k4_supinf_2(A,B),C) != k4_supinf_2(A,k4_supinf_2(B,C)) ) ) ) ) )).

fof(t42_extreal2,axiom,(
! [A] :
( m1_subset_1(A,k3_supinf_1)
=> ! [B] :
( m1_subset_1(B,k3_supinf_1)
=> ~ ( k2_extreal1(A,B) != k5_measure6
& k2_extreal1(A,B) != k4_measure6
& ~ m1_subset_1(A,k1_numbers)
& ~ m1_subset_1(B,k1_numbers) ) ) ) )).

fof(t43_extreal2,axiom,(
! [A] :
( m1_subset_1(A,k3_supinf_1)
=> ! [B] :
( m1_subset_1(B,k3_supinf_1)
=> ( ~ ( ( ( ~ r1_supinf_1(A,k1_supinf_2)
& ~ r1_supinf_1(B,k1_supinf_2) )
| ( ~ r1_supinf_1(k1_supinf_2,A)
& ~ r1_supinf_1(k1_supinf_2,B) ) )
& r1_supinf_1(k2_extreal1(A,B),k1_supinf_2) )
& ~ ( ~ r1_supinf_1(k2_extreal1(A,B),k1_supinf_2)
& ~ ( ~ r1_supinf_1(A,k1_supinf_2)
& ~ r1_supinf_1(B,k1_supinf_2) )
& ~ ( ~ r1_supinf_1(k1_supinf_2,A)
& ~ r1_supinf_1(k1_supinf_2,B) ) ) ) ) ) )).

fof(t44_extreal2,axiom,(
! [A] :
( m1_subset_1(A,k3_supinf_1)
=> ! [B] :
( m1_subset_1(B,k3_supinf_1)
=> ( ~ ( ( ( ~ r1_supinf_1(A,k1_supinf_2)
& ~ r1_supinf_1(k1_supinf_2,B) )
| ( ~ r1_supinf_1(k1_supinf_2,A)
& ~ r1_supinf_1(B,k1_supinf_2) ) )
& r1_supinf_1(k1_supinf_2,k2_extreal1(A,B)) )
& ~ ( ~ r1_supinf_1(k1_supinf_2,k2_extreal1(A,B))
& ~ ( ~ r1_supinf_1(A,k1_supinf_2)
& ~ r1_supinf_1(k1_supinf_2,B) )
& ~ ( ~ r1_supinf_1(k1_supinf_2,A)
& ~ r1_supinf_1(B,k1_supinf_2) ) ) ) ) ) )).

fof(t45_extreal2,axiom,(
! [A] :
( m1_subset_1(A,k3_supinf_1)
=> ! [B] :
( m1_subset_1(B,k3_supinf_1)
=> ( ( ( ~ ( ~ r1_supinf_1(k1_supinf_2,A)
& r1_supinf_1(A,k1_supinf_2) )
& ~ ( ~ r1_supinf_1(k1_supinf_2,B)
& r1_supinf_1(B,k1_supinf_2) ) )
| ( ~ ( ~ r1_supinf_1(A,k1_supinf_2)
& r1_supinf_1(k1_supinf_2,A) )
& ~ ( ~ r1_supinf_1(B,k1_supinf_2)
& r1_supinf_1(k1_supinf_2,B) ) ) )
<=> r1_supinf_1(k1_supinf_2,k2_extreal1(A,B)) ) ) ) )).

fof(t46_extreal2,axiom,(
! [A] :
( m1_subset_1(A,k3_supinf_1)
=> ! [B] :
( m1_subset_1(B,k3_supinf_1)
=> ( ( ( ~ ( ~ r1_supinf_1(A,k1_supinf_2)
& r1_supinf_1(k1_supinf_2,A) )
& ~ ( ~ r1_supinf_1(k1_supinf_2,B)
& r1_supinf_1(B,k1_supinf_2) ) )
| ( ~ ( ~ r1_supinf_1(k1_supinf_2,A)
& r1_supinf_1(A,k1_supinf_2) )
& ~ ( ~ r1_supinf_1(B,k1_supinf_2)
& r1_supinf_1(k1_supinf_2,B) ) ) )
<=> r1_supinf_1(k2_extreal1(A,B),k1_supinf_2) ) ) ) )).

fof(t47_extreal2,axiom,(
! [A] :
( m1_subset_1(A,k3_supinf_1)
=> ! [B] :
( m1_subset_1(B,k3_supinf_1)
=> ( ( r1_supinf_1(A,k3_supinf_2(B))
=> r1_supinf_1(B,k3_supinf_2(A)) )
& ( r1_supinf_1(B,k3_supinf_2(A))
=> r1_supinf_1(A,k3_supinf_2(B)) )
& ( r1_supinf_1(k3_supinf_2(A),B)
=> r1_supinf_1(k3_supinf_2(B),A) )
& ( r1_supinf_1(k3_supinf_2(B),A)
=> r1_supinf_1(k3_supinf_2(A),B) ) ) ) ) )).

fof(t48_extreal2,axiom,(
! [A] :
( m1_subset_1(A,k3_supinf_1)
=> ! [B] :
( m1_subset_1(B,k3_supinf_1)
=> ( ~ ( ~ r1_supinf_1(k3_supinf_2(B),A)
& r1_supinf_1(k3_supinf_2(A),B) )
& ~ ( ~ r1_supinf_1(k3_supinf_2(A),B)
& r1_supinf_1(k3_supinf_2(B),A) )
& ~ ( ~ r1_supinf_1(B,k3_supinf_2(A))
& r1_supinf_1(A,k3_supinf_2(B)) )
& ~ ( ~ r1_supinf_1(A,k3_supinf_2(B))
& r1_supinf_1(B,k3_supinf_2(A)) ) ) ) ) )).

fof(t49_extreal2,axiom,(
! [A] :
( m1_subset_1(A,k3_supinf_1)
=> ! [B] :
( m1_subset_1(B,k1_numbers)
=> ( A = B
=> k4_extreal1(A) = k18_complex1(B) ) ) ) )).

fof(t50_extreal2,axiom,(
! [A] :
( m1_subset_1(A,k3_supinf_1)
=> ( k4_extreal1(A) = A
| k4_extreal1(A) = k3_supinf_2(A) ) ) )).

fof(t51_extreal2,axiom,(
! [A] :
( m1_subset_1(A,k3_supinf_1)
=> r1_supinf_1(k1_supinf_2,k4_extreal1(A)) ) )).

fof(t52_extreal2,axiom,(
! [A] :
( m1_subset_1(A,k3_supinf_1)
=> ~ ( A != k1_supinf_2
& r1_supinf_1(k4_extreal1(A),k1_supinf_2) ) ) )).

fof(t53_extreal2,axiom,(
! [A] :
( m1_subset_1(A,k3_supinf_1)
=> ( A = k1_supinf_2
<=> k4_extreal1(A) = k1_supinf_2 ) ) )).

fof(t54_extreal2,axiom,(
! [A] :
( m1_subset_1(A,k3_supinf_1)
=> ~ ( k4_extreal1(A) = k3_supinf_2(A)
& A != k1_supinf_2
& r1_supinf_1(k1_supinf_2,A) ) ) )).

fof(t55_extreal2,axiom,(
! [A] :
( m1_subset_1(A,k3_supinf_1)
=> ( r1_supinf_1(A,k1_supinf_2)
=> k4_extreal1(A) = k3_supinf_2(A) ) ) )).

fof(t56_extreal2,axiom,(
! [A] :
( m1_subset_1(A,k3_supinf_1)
=> ! [B] :
( m1_subset_1(B,k3_supinf_1)
=> k4_extreal1(k2_extreal1(A,B)) = k2_extreal1(k4_extreal1(A),k4_extreal1(B)) ) ) )).

fof(t57_extreal2,axiom,(
! [A] :
( m1_subset_1(A,k3_supinf_1)
=> ( r1_supinf_1(k3_supinf_2(k4_extreal1(A)),A)
& r1_supinf_1(A,k4_extreal1(A)) ) ) )).

fof(t58_extreal2,axiom,(
! [A] :
( m1_subset_1(A,k3_supinf_1)
=> ! [B] :
( m1_subset_1(B,k3_supinf_1)
=> ( ~ r1_supinf_1(B,k4_extreal1(A))
=> ( ~ r1_supinf_1(A,k3_supinf_2(B))
& ~ r1_supinf_1(B,A) ) ) ) ) )).

fof(t59_extreal2,axiom,(
! [A] :
( m1_subset_1(A,k3_supinf_1)
=> ! [B] :
( m1_subset_1(B,k3_supinf_1)
=> ~ ( ~ r1_supinf_1(B,k3_supinf_2(A))
& ~ r1_supinf_1(A,B)
& ~ ( ~ r1_supinf_1(A,k1_supinf_2)
& ~ r1_supinf_1(A,k4_extreal1(B)) ) ) ) ) )).

fof(t60_extreal2,axiom,(
! [A] :
( m1_subset_1(A,k3_supinf_1)
=> ! [B] :
( m1_subset_1(B,k3_supinf_1)
=> ( ( r1_supinf_1(k3_supinf_2(A),B)
& r1_supinf_1(B,A) )
<=> r1_supinf_1(k4_extreal1(B),A) ) ) ) )).

fof(t61_extreal2,axiom,(
! [A] :
( m1_subset_1(A,k3_supinf_1)
=> ! [B] :
( m1_subset_1(B,k3_supinf_1)
=> ~ ( ~ ( A = k5_measure6
& B = k4_measure6 )
& ~ ( A = k4_measure6
& B = k5_measure6 )
& ~ r1_supinf_1(k4_extreal1(k2_supinf_2(A,B)),k2_supinf_2(k4_extreal1(A),k4_extreal1(B))) ) ) ) )).

fof(t62_extreal2,axiom,(
! [A] :
( m1_subset_1(A,k3_supinf_1)
=> ~ ( ~ r1_supinf_1(A,k4_measure6)
& ~ r1_supinf_1(k5_measure6,A)
& A != k1_supinf_2
& k2_extreal1(k4_extreal1(A),k4_extreal1(k3_extreal1(k10_mesfunc1,A))) != k10_mesfunc1 ) ) )).

fof(t63_extreal2,axiom,(
! [A] :
( m1_subset_1(A,k3_supinf_1)
=> ( ( A = k5_measure6
| A = k4_measure6 )
=> k2_extreal1(k4_extreal1(A),k4_extreal1(k3_extreal1(k10_mesfunc1,A))) = k1_supinf_2 ) ) )).

fof(t64_extreal2,axiom,(
! [A] :
( m1_subset_1(A,k3_supinf_1)
=> ( A != k1_supinf_2
=> k4_extreal1(k3_extreal1(k10_mesfunc1,A)) = k3_extreal1(k10_mesfunc1,k4_extreal1(A)) ) ) )).

fof(t65_extreal2,axiom,(
! [A] :
( m1_subset_1(A,k3_supinf_1)
=> ! [B] :
( m1_subset_1(B,k3_supinf_1)
=> ~ ( ~ ( ( A = k4_measure6
| A = k5_measure6 )
& ( B = k4_measure6
| B = k5_measure6 ) )
& B != k1_supinf_2
& k4_extreal1(k3_extreal1(A,B)) != k3_extreal1(k4_extreal1(A),k4_extreal1(B)) ) ) ) )).

fof(t66_extreal2,axiom,(
! [A] :
( m1_subset_1(A,k3_supinf_1)
=> k4_extreal1(A) = k4_extreal1(k3_supinf_2(A)) ) )).

fof(t67_extreal2,axiom,(
! [A] :
( m1_subset_1(A,k3_supinf_1)
=> ( ( A = k5_measure6
| A = k4_measure6 )
=> k4_extreal1(A) = k5_measure6 ) ) )).

fof(t68_extreal2,axiom,(
! [A] :
( m1_subset_1(A,k3_supinf_1)
=> ! [B] :
( m1_subset_1(B,k3_supinf_1)
=> ( ( m1_subset_1(A,k1_numbers)
| m1_subset_1(B,k1_numbers) )
=> r1_supinf_1(k4_supinf_2(k4_extreal1(A),k4_extreal1(B)),k4_extreal1(k4_supinf_2(A,B))) ) ) ) )).

fof(t69_extreal2,axiom,(
! [A] :
( m1_subset_1(A,k3_supinf_1)
=> ! [B] :
( m1_subset_1(B,k3_supinf_1)
=> ~ ( ~ ( A = k5_measure6
& B = k5_measure6 )
& ~ ( A = k4_measure6
& B = k4_measure6 )
& ~ r1_supinf_1(k4_extreal1(k4_supinf_2(A,B)),k2_supinf_2(k4_extreal1(A),k4_extreal1(B))) ) ) ) )).

fof(t70_extreal2,axiom,(
! [A] :
( m1_subset_1(A,k3_supinf_1)
=> k4_extreal1(k4_extreal1(A)) = k4_extreal1(A) ) )).

fof(t71_extreal2,axiom,(
! [A] :
( m1_subset_1(A,k3_supinf_1)
=> ! [B] :
( m1_subset_1(B,k3_supinf_1)
=> ! [C] :
( m1_subset_1(C,k3_supinf_1)
=> ! [D] :
( m1_subset_1(D,k3_supinf_1)
=> ~ ( ~ ( A = k5_measure6
& B = k4_measure6 )
& ~ ( A = k4_measure6
& B = k5_measure6 )
& r1_supinf_1(k4_extreal1(A),C)
& r1_supinf_1(k4_extreal1(B),D)
& ~ r1_supinf_1(k4_extreal1(k2_supinf_2(A,B)),k2_supinf_2(C,D)) ) ) ) ) ) )).

fof(t72_extreal2,axiom,(
! [A] :
( m1_subset_1(A,k3_supinf_1)
=> ! [B] :
( m1_subset_1(B,k3_supinf_1)
=> ( ( m1_subset_1(A,k1_numbers)
| m1_subset_1(B,k1_numbers) )
=> r1_supinf_1(k4_extreal1(k4_supinf_2(k4_extreal1(A),k4_extreal1(B))),k4_extreal1(k4_supinf_2(A,B))) ) ) ) )).

fof(t73_extreal2,axiom,(
! [A] :
( m1_subset_1(A,k3_supinf_1)
=> ! [B] :
( m1_subset_1(B,k3_supinf_1)
=> ( r1_supinf_1(k1_supinf_2,k2_extreal1(A,B))
=> k4_extreal1(k2_supinf_2(A,B)) = k2_supinf_2(k4_extreal1(A),k4_extreal1(B)) ) ) ) )).

fof(t74_extreal2,axiom,(
! [A] :
( m1_subset_1(A,k3_supinf_1)
=> ! [B] :
( m1_subset_1(B,k3_supinf_1)
=> ! [C] :
( m1_subset_1(C,k1_numbers)
=> ! [D] :
( m1_subset_1(D,k1_numbers)
=> ( ( A = C
& B = D )
=> ( ~ ( ~ r1_xreal_0(C,D)
& r1_supinf_1(A,B) )
& ~ ( ~ r1_supinf_1(A,B)
& r1_xreal_0(C,D) )
& ( r1_xreal_0(D,C)
=> r1_supinf_1(B,A) )
& ( r1_supinf_1(B,A)
=> r1_xreal_0(D,C) ) ) ) ) ) ) ) )).

fof(d1_extreal2,axiom,(
! [A] :
( m1_subset_1(A,k3_supinf_1)
=> ! [B] :
( m1_subset_1(B,k3_supinf_1)
=> ( ( r1_supinf_1(A,B)
=> k1_extreal2(A,B) = A )
& ( ~ r1_supinf_1(A,B)
=> k1_extreal2(A,B) = B ) ) ) ) )).

fof(d2_extreal2,axiom,(
! [A] :
( m1_subset_1(A,k3_supinf_1)
=> ! [B] :
( m1_subset_1(B,k3_supinf_1)
=> ( ( r1_supinf_1(B,A)
=> k2_extreal2(A,B) = A )
& ( ~ r1_supinf_1(B,A)
=> k2_extreal2(A,B) = B ) ) ) ) )).

fof(t75_extreal2,axiom,(
! [A] :
( m1_subset_1(A,k3_supinf_1)
=> ! [B] :
( m1_subset_1(B,k3_supinf_1)
=> ( ( A = k4_measure6
| B = k4_measure6 )
=> k1_extreal2(A,B) = k4_measure6 ) ) ) )).

fof(t76_extreal2,axiom,(
! [A] :
( m1_subset_1(A,k3_supinf_1)
=> ! [B] :
( m1_subset_1(B,k3_supinf_1)
=> ( ( A = k5_measure6
| B = k5_measure6 )
=> k2_extreal2(A,B) = k5_measure6 ) ) ) )).

fof(t77_extreal2,axiom,(
! [A] :
( m1_subset_1(A,k3_supinf_1)
=> ! [B] :
( m1_subset_1(B,k3_supinf_1)
=> ! [C] :
( m1_subset_1(C,k1_numbers)
=> ! [D] :
( m1_subset_1(D,k1_numbers)
=> ( ( A = C
& B = D )
=> ( k1_extreal2(A,B) = k3_square_1(C,D)
& k2_extreal2(A,B) = k4_square_1(C,D) ) ) ) ) ) ) )).

fof(t78_extreal2,axiom,(
! [A] :
( m1_subset_1(A,k3_supinf_1)
=> ! [B] :
( m1_subset_1(B,k3_supinf_1)
=> ( r1_supinf_1(A,B)
=> k1_extreal2(B,A) = A ) ) ) )).

fof(t79_extreal2,axiom,(
! [A] :
( m1_subset_1(A,k3_supinf_1)
=> ! [B] :
( m1_subset_1(B,k3_supinf_1)
=> ( ~ r1_supinf_1(A,B)
=> k1_extreal2(B,A) = B ) ) ) )).

fof(t80_extreal2,axiom,(
! [A] :
( m1_subset_1(A,k3_supinf_1)
=> ! [B] :
( m1_subset_1(B,k3_supinf_1)
=> ~ ( A != k5_measure6
& B != k5_measure6
& ~ ( A = k5_measure6
& B = k5_measure6 )
& ~ ( A = k4_measure6
& B = k4_measure6 )
& k1_extreal2(A,B) != k3_extreal1(k4_supinf_2(k2_supinf_2(A,B),k4_extreal1(k4_supinf_2(A,B))),k1_measure6(np__2)) ) ) ) )).

fof(t81_extreal2,axiom,(
! [A] :
( m1_subset_1(A,k3_supinf_1)
=> ! [B] :
( m1_subset_1(B,k3_supinf_1)
=> ( r1_supinf_1(k1_extreal2(A,B),A)
& r1_supinf_1(k1_extreal2(B,A),A) ) ) ) )).

fof(t82_extreal2,axiom,(
\$true )).

fof(t83_extreal2,axiom,(
! [A] :
( m1_subset_1(A,k3_supinf_1)
=> ! [B] :
( m1_subset_1(B,k3_supinf_1)
=> k1_extreal2(A,B) = k1_extreal2(B,A) ) ) )).

fof(t84_extreal2,axiom,(
! [A] :
( m1_subset_1(A,k3_supinf_1)
=> ! [B] :
( m1_subset_1(B,k3_supinf_1)
=> ( k1_extreal2(A,B) = A
| k1_extreal2(A,B) = B ) ) ) )).

fof(t85_extreal2,axiom,(
! [A] :
( m1_subset_1(A,k3_supinf_1)
=> ! [B] :
( m1_subset_1(B,k3_supinf_1)
=> ! [C] :
( m1_subset_1(C,k3_supinf_1)
=> ( ( r1_supinf_1(A,B)
& r1_supinf_1(A,C) )
<=> r1_supinf_1(A,k1_extreal2(B,C)) ) ) ) ) )).

fof(t86_extreal2,axiom,(
\$true )).

fof(t87_extreal2,axiom,(
! [A] :
( m1_subset_1(A,k3_supinf_1)
=> ! [B] :
( m1_subset_1(B,k3_supinf_1)
=> ( k1_extreal2(A,B) = B
=> r1_supinf_1(B,A) ) ) ) )).

fof(t88_extreal2,axiom,(
! [A] :
( m1_subset_1(A,k3_supinf_1)
=> ! [B] :
( m1_subset_1(B,k3_supinf_1)
=> ! [C] :
( m1_subset_1(C,k3_supinf_1)
=> k1_extreal2(A,k1_extreal2(B,C)) = k1_extreal2(k1_extreal2(A,B),C) ) ) ) )).

fof(t89_extreal2,axiom,(
! [A] :
( m1_subset_1(A,k3_supinf_1)
=> ! [B] :
( m1_subset_1(B,k3_supinf_1)
=> ( r1_supinf_1(A,B)
=> k2_extreal2(A,B) = B ) ) ) )).

fof(t90_extreal2,axiom,(
! [A] :
( m1_subset_1(A,k3_supinf_1)
=> ! [B] :
( m1_subset_1(B,k3_supinf_1)
=> ( ~ r1_supinf_1(A,B)
=> k2_extreal2(A,B) = A ) ) ) )).

fof(t91_extreal2,axiom,(
! [A] :
( m1_subset_1(A,k3_supinf_1)
=> ! [B] :
( m1_subset_1(B,k3_supinf_1)
=> ~ ( A != k4_measure6
& B != k4_measure6
& ~ ( A = k5_measure6
& B = k5_measure6 )
& ~ ( A = k4_measure6
& B = k4_measure6 )
& k2_extreal2(A,B) != k3_extreal1(k2_supinf_2(k2_supinf_2(A,B),k4_extreal1(k4_supinf_2(A,B))),k1_measure6(np__2)) ) ) ) )).

fof(t92_extreal2,axiom,(
! [A] :
( m1_subset_1(A,k3_supinf_1)
=> ! [B] :
( m1_subset_1(B,k3_supinf_1)
=> ( r1_supinf_1(A,k2_extreal2(A,B))
& r1_supinf_1(A,k2_extreal2(B,A)) ) ) ) )).

fof(t93_extreal2,axiom,(
\$true )).

fof(t94_extreal2,axiom,(
! [A] :
( m1_subset_1(A,k3_supinf_1)
=> ! [B] :
( m1_subset_1(B,k3_supinf_1)
=> k2_extreal2(A,B) = k2_extreal2(B,A) ) ) )).

fof(t95_extreal2,axiom,(
! [A] :
( m1_subset_1(A,k3_supinf_1)
=> ! [B] :
( m1_subset_1(B,k3_supinf_1)
=> ( k2_extreal2(A,B) = A
| k2_extreal2(A,B) = B ) ) ) )).

fof(t96_extreal2,axiom,(
! [A] :
( m1_subset_1(A,k3_supinf_1)
=> ! [B] :
( m1_subset_1(B,k3_supinf_1)
=> ! [C] :
( m1_subset_1(C,k3_supinf_1)
=> ( ( r1_supinf_1(A,B)
& r1_supinf_1(C,B) )
<=> r1_supinf_1(k2_extreal2(A,C),B) ) ) ) ) )).

fof(t97_extreal2,axiom,(
\$true )).

fof(t98_extreal2,axiom,(
! [A] :
( m1_subset_1(A,k3_supinf_1)
=> ! [B] :
( m1_subset_1(B,k3_supinf_1)
=> ( k2_extreal2(A,B) = B
=> r1_supinf_1(A,B) ) ) ) )).

fof(t99_extreal2,axiom,(
! [A] :
( m1_subset_1(A,k3_supinf_1)
=> ! [B] :
( m1_subset_1(B,k3_supinf_1)
=> ! [C] :
( m1_subset_1(C,k3_supinf_1)
=> k2_extreal2(A,k2_extreal2(B,C)) = k2_extreal2(k2_extreal2(A,B),C) ) ) ) )).

fof(t100_extreal2,axiom,(
! [A] :
( m1_subset_1(A,k3_supinf_1)
=> ! [B] :
( m1_subset_1(B,k3_supinf_1)
=> k2_supinf_2(k1_extreal2(A,B),k2_extreal2(A,B)) = k2_supinf_2(A,B) ) ) )).

fof(t101_extreal2,axiom,(
! [A] :
( m1_subset_1(A,k3_supinf_1)
=> ! [B] :
( m1_subset_1(B,k3_supinf_1)
=> ( k2_extreal2(A,k1_extreal2(A,B)) = A
& k2_extreal2(k1_extreal2(A,B),A) = A
& k2_extreal2(k1_extreal2(B,A),A) = A
& k2_extreal2(A,k1_extreal2(B,A)) = A ) ) ) )).

fof(t102_extreal2,axiom,(
! [A] :
( m1_subset_1(A,k3_supinf_1)
=> ! [B] :
( m1_subset_1(B,k3_supinf_1)
=> ( k1_extreal2(A,k2_extreal2(A,B)) = A
& k1_extreal2(k2_extreal2(A,B),A) = A
& k1_extreal2(k2_extreal2(B,A),A) = A
& k1_extreal2(A,k2_extreal2(B,A)) = A ) ) ) )).

fof(t103_extreal2,axiom,(
! [A] :
( m1_subset_1(A,k3_supinf_1)
=> ! [B] :
( m1_subset_1(B,k3_supinf_1)
=> ! [C] :
( m1_subset_1(C,k3_supinf_1)
=> ( k1_extreal2(A,k2_extreal2(B,C)) = k2_extreal2(k1_extreal2(A,B),k1_extreal2(A,C))
& k1_extreal2(k2_extreal2(B,C),A) = k2_extreal2(k1_extreal2(B,A),k1_extreal2(C,A)) ) ) ) ) )).

fof(t104_extreal2,axiom,(
! [A] :
( m1_subset_1(A,k3_supinf_1)
=> ! [B] :
( m1_subset_1(B,k3_supinf_1)
=> ! [C] :
( m1_subset_1(C,k3_supinf_1)
=> ( k2_extreal2(A,k1_extreal2(B,C)) = k1_extreal2(k2_extreal2(A,B),k2_extreal2(A,C))
& k2_extreal2(k1_extreal2(B,C),A) = k1_extreal2(k2_extreal2(B,A),k2_extreal2(C,A)) ) ) ) ) )).

fof(dt_k1_extreal2,axiom,(
! [A,B] :
( ( m1_subset_1(A,k3_supinf_1)
& m1_subset_1(B,k3_supinf_1) )
=> m1_subset_1(k1_extreal2(A,B),k3_supinf_1) ) )).

fof(commutativity_k1_extreal2,axiom,(
! [A,B] :
( ( m1_subset_1(A,k3_supinf_1)
& m1_subset_1(B,k3_supinf_1) )
=> k1_extreal2(A,B) = k1_extreal2(B,A) ) )).

fof(idempotence_k1_extreal2,axiom,(
! [A,B] :
( ( m1_subset_1(A,k3_supinf_1)
& m1_subset_1(B,k3_supinf_1) )
=> k1_extreal2(A,A) = A ) )).

fof(dt_k2_extreal2,axiom,(
! [A,B] :
( ( m1_subset_1(A,k3_supinf_1)
& m1_subset_1(B,k3_supinf_1) )
=> m1_subset_1(k2_extreal2(A,B),k3_supinf_1) ) )).

fof(commutativity_k2_extreal2,axiom,(
! [A,B] :
( ( m1_subset_1(A,k3_supinf_1)
& m1_subset_1(B,k3_supinf_1) )
=> k2_extreal2(A,B) = k2_extreal2(B,A) ) )).

fof(idempotence_k2_extreal2,axiom,(
! [A,B] :
( ( m1_subset_1(A,k3_supinf_1)
& m1_subset_1(B,k3_supinf_1) )
=> k2_extreal2(A,A) = A ) )).
%------------------------------------------------------------------------------
```