SET007 Axioms: SET007+178.ax
%------------------------------------------------------------------------------
% File : SET007+178 : TPTP v8.2.0. 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 unt; 0 def)
% Number of atoms : 643 ( 273 equ)
% Maximal formula atoms : 16 ( 5 avg)
% Number of connectives : 784 ( 253 ~; 27 |; 264 &)
% ( 6 <=>; 234 =>; 0 <=; 0 <~>)
% Maximal formula depth : 16 ( 8 avg)
% Maximal term depth : 5 ( 1 avg)
% Number of predicates : 6 ( 4 usr; 1 prp; 0-2 aty)
% Number of functors : 21 ( 21 usr; 9 con; 0-2 aty)
% Number of variables : 202 ( 202 !; 0 ?)
% 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 ) ).
%------------------------------------------------------------------------------