SET007 Axioms: SET007+176.ax
%------------------------------------------------------------------------------
% File : SET007+176 : TPTP v9.0.0. Released v3.4.0.
% Domain : Set Theory
% Axioms : Basic Properties of Extended Real Numbers
% 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 : extreal1 [Urb08]
% Status : Satisfiable
% Syntax : Number of formulae : 61 ( 7 unt; 0 def)
% Number of atoms : 400 ( 155 equ)
% Maximal formula atoms : 30 ( 6 avg)
% Number of connectives : 511 ( 172 ~; 26 |; 165 &)
% ( 3 <=>; 145 =>; 0 <=; 0 <~>)
% Maximal formula depth : 21 ( 9 avg)
% Maximal term depth : 3 ( 1 avg)
% Number of predicates : 6 ( 4 usr; 1 prp; 0-2 aty)
% Number of functors : 18 ( 18 usr; 8 con; 0-2 aty)
% Number of variables : 127 ( 125 !; 2 ?)
% 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_extreal1,axiom,
! [A] :
( m1_subset_1(A,k3_supinf_1)
=> ~ ( A != k5_measure6
& A != k4_measure6
& ~ m1_subset_1(A,k1_numbers) ) ) ).
fof(t2_extreal1,axiom,
~ r1_supinf_1(k5_measure6,k4_measure6) ).
fof(t3_extreal1,axiom,
! [A] :
( m1_subset_1(A,k3_supinf_1)
=> ! [B] :
( m1_subset_1(B,k3_supinf_1)
=> ( ~ r1_supinf_1(B,A)
=> ( A != k5_measure6
& B != k4_measure6 ) ) ) ) ).
fof(t4_extreal1,axiom,
! [A] :
( m1_subset_1(A,k3_supinf_1)
=> ( ( A = k5_measure6
=> k3_supinf_2(A) = k4_measure6 )
& ( k3_supinf_2(A) = k4_measure6
=> A = k5_measure6 )
& ( A = k4_measure6
=> k3_supinf_2(A) = k5_measure6 )
& ( k3_supinf_2(A) = k5_measure6
=> A = k4_measure6 ) ) ) ).
fof(t5_extreal1,axiom,
! [A] :
( m1_subset_1(A,k3_supinf_1)
=> ! [B] :
( m1_subset_1(B,k3_supinf_1)
=> k4_supinf_2(A,k3_supinf_2(B)) = k2_supinf_2(A,B) ) ) ).
fof(t6_extreal1,axiom,
$true ).
fof(t7_extreal1,axiom,
! [A] :
( m1_subset_1(A,k3_supinf_1)
=> ! [B] :
( m1_subset_1(B,k3_supinf_1)
=> ( r1_supinf_1(A,B)
=> ( A = k4_measure6
| B = k5_measure6
| ( A != k5_measure6
& B != k4_measure6 ) ) ) ) ) ).
fof(t8_extreal1,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 = k4_measure6 )
& ~ ( B = k4_measure6
& C = k5_measure6 )
& ~ ( A = k5_measure6
& C = k4_measure6 )
& ~ ( A = k4_measure6
& C = k5_measure6 )
& k2_supinf_2(k2_supinf_2(A,B),C) != k2_supinf_2(A,k2_supinf_2(B,C)) ) ) ) ) ).
fof(t9_extreal1,axiom,
! [A] :
( m1_subset_1(A,k3_supinf_1)
=> k2_supinf_2(A,k3_supinf_2(A)) = k1_supinf_2 ) ).
fof(t10_extreal1,axiom,
$true ).
fof(t11_extreal1,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 )
& ~ ( A = k5_measure6
& C = k5_measure6 )
& ~ ( A = k4_measure6
& C = k4_measure6 )
& k4_supinf_2(k2_supinf_2(A,B),C) != k2_supinf_2(A,k4_supinf_2(B,C)) ) ) ) ) ).
fof(d1_extreal1,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)
=> ( C = k1_extreal1(A,B)
<=> ~ ( ! [D] :
( m1_subset_1(D,k1_numbers)
=> ! [E] :
( m1_subset_1(E,k1_numbers)
=> ~ ( A = D
& B = E
& C = k4_real_1(D,E) ) ) )
& ~ ( ~ ( ~ ( ~ r1_supinf_1(A,k1_supinf_2)
& B = k5_measure6 )
& ~ ( ~ r1_supinf_1(B,k1_supinf_2)
& A = k5_measure6 )
& ~ ( ~ r1_supinf_1(k1_supinf_2,A)
& B = k4_measure6 )
& ~ ( ~ r1_supinf_1(k1_supinf_2,B)
& A = k4_measure6 ) )
& C = k5_measure6 )
& ~ ( ~ ( ~ ( ~ r1_supinf_1(k1_supinf_2,A)
& B = k5_measure6 )
& ~ ( ~ r1_supinf_1(k1_supinf_2,B)
& A = k5_measure6 )
& ~ ( ~ r1_supinf_1(A,k1_supinf_2)
& B = k4_measure6 )
& ~ ( ~ r1_supinf_1(B,k1_supinf_2)
& A = k4_measure6 ) )
& C = k4_measure6 )
& ~ ( ( A = k1_supinf_2
| B = k1_supinf_2 )
& C = k1_supinf_2 ) ) ) ) ) ) ).
fof(t12_extreal1,axiom,
$true ).
fof(t13_extreal1,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_extreal1(A,B) = k4_real_1(C,D) ) ) ) ) ) ).
fof(t14_extreal1,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(B,k1_supinf_2) )
| ( ~ r1_supinf_1(A,k1_supinf_2)
& r1_supinf_1(k1_supinf_2,B) ) )
& r1_supinf_1(k2_supinf_2(A,B),k1_supinf_2) ) ) ) ).
fof(t15_extreal1,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_supinf_2(A,B)) ) ) ) ).
fof(t16_extreal1,axiom,
! [A] :
( m1_subset_1(A,k3_supinf_1)
=> ~ ( r2_hidden(A,k6_supinf_1)
& ~ ( ~ r1_supinf_1(A,k4_measure6)
& ~ r1_supinf_1(k1_supinf_2,A) )
& A != k1_supinf_2
& ~ ( ~ r1_supinf_1(A,k1_supinf_2)
& ~ r1_supinf_1(k5_measure6,A) ) ) ) ).
fof(t17_extreal1,axiom,
! [A] :
( m1_subset_1(A,k3_supinf_1)
=> ! [B] :
( m1_subset_1(B,k3_supinf_1)
=> k1_extreal1(A,B) = k1_extreal1(B,A) ) ) ).
fof(t18_extreal1,axiom,
! [A] :
( m1_subset_1(A,k3_supinf_1)
=> ! [B] :
( m1_subset_1(B,k1_numbers)
=> ( A = B
=> ( ~ ( ~ r1_xreal_0(B,np__0)
& r1_supinf_1(A,k1_supinf_2) )
& ~ ( ~ r1_supinf_1(A,k1_supinf_2)
& r1_xreal_0(B,np__0) ) ) ) ) ) ).
fof(t19_extreal1,axiom,
! [A] :
( m1_subset_1(A,k3_supinf_1)
=> ! [B] :
( m1_subset_1(B,k1_numbers)
=> ( A = B
=> ( ~ ( ~ r1_xreal_0(np__0,B)
& r1_supinf_1(k1_supinf_2,A) )
& ~ ( ~ r1_supinf_1(k1_supinf_2,A)
& r1_xreal_0(np__0,B) ) ) ) ) ) ).
fof(t20_extreal1,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) ) ) ) ).
fof(t21_extreal1,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)) ) ) ) ).
fof(t22_extreal1,axiom,
! [A] :
( m1_subset_1(A,k3_supinf_1)
=> ! [B] :
( m1_subset_1(B,k3_supinf_1)
=> ( k2_extreal1(A,B) = k1_supinf_2
<=> ( A = k1_supinf_2
| B = k1_supinf_2 ) ) ) ) ).
fof(t23_extreal1,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_extreal1(k2_extreal1(A,B),C) = k2_extreal1(A,k2_extreal1(B,C)) ) ) ) ).
fof(t24_extreal1,axiom,
k3_supinf_2(k1_supinf_2) = k1_supinf_2 ).
fof(t25_extreal1,axiom,
! [A] :
( m1_subset_1(A,k3_supinf_1)
=> ( ~ ( ~ r1_supinf_1(A,k1_supinf_2)
& r1_supinf_1(k1_supinf_2,k3_supinf_2(A)) )
& ~ ( ~ r1_supinf_1(k1_supinf_2,k3_supinf_2(A))
& r1_supinf_1(A,k1_supinf_2) )
& ~ ( ~ r1_supinf_1(k1_supinf_2,A)
& r1_supinf_1(k3_supinf_2(A),k1_supinf_2) )
& ~ ( ~ r1_supinf_1(k3_supinf_2(A),k1_supinf_2)
& r1_supinf_1(k1_supinf_2,A) ) ) ) ).
fof(t26_extreal1,axiom,
! [A] :
( m1_subset_1(A,k3_supinf_1)
=> ! [B] :
( m1_subset_1(B,k3_supinf_1)
=> ( k3_supinf_2(k2_extreal1(A,B)) = k2_extreal1(A,k3_supinf_2(B))
& k3_supinf_2(k2_extreal1(A,B)) = k2_extreal1(k3_supinf_2(A),B) ) ) ) ).
fof(t27_extreal1,axiom,
! [A] :
( m1_subset_1(A,k3_supinf_1)
=> ! [B] :
( m1_subset_1(B,k3_supinf_1)
=> ~ ( A != k5_measure6
& A != k4_measure6
& k2_extreal1(A,B) = k5_measure6
& B != k5_measure6
& B != k4_measure6 ) ) ) ).
fof(t28_extreal1,axiom,
! [A] :
( m1_subset_1(A,k3_supinf_1)
=> ! [B] :
( m1_subset_1(B,k3_supinf_1)
=> ~ ( A != k5_measure6
& A != k4_measure6
& k2_extreal1(A,B) = k4_measure6
& B != k5_measure6
& B != k4_measure6 ) ) ) ).
fof(t29_extreal1,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
& A != k4_measure6 )
& ~ ( B = k4_measure6
& C = k5_measure6 )
& ~ ( ~ r1_supinf_1(k1_supinf_2,B)
& ~ r1_supinf_1(k1_supinf_2,C) )
& B != k1_supinf_2
& C != k1_supinf_2
& ~ ( ~ r1_supinf_1(B,k1_supinf_2)
& ~ r1_supinf_1(C,k1_supinf_2) )
& ~ ( B = k5_measure6
& C = k4_measure6 ) )
=> k2_extreal1(A,k2_supinf_2(B,C)) = k2_supinf_2(k2_extreal1(A,B),k2_extreal1(A,C)) ) ) ) ) ).
fof(t30_extreal1,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 )
& C != k5_measure6
& C != k4_measure6
& k2_extreal1(C,k4_supinf_2(A,B)) != k4_supinf_2(k2_extreal1(C,A),k2_extreal1(C,B)) ) ) ) ) ).
fof(d2_extreal1,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
& ~ ! [C] :
( m1_subset_1(C,k3_supinf_1)
=> ( C = k3_extreal1(A,B)
<=> ~ ( ! [D] :
( m1_subset_1(D,k1_numbers)
=> ! [E] :
( m1_subset_1(E,k1_numbers)
=> ~ ( A = D
& B = E
& C = k6_real_1(D,E) ) ) )
& ~ ( ( ( A = k5_measure6
& ~ r1_supinf_1(B,k1_supinf_2) )
| ( A = k4_measure6
& ~ r1_supinf_1(k1_supinf_2,B) ) )
& C = k5_measure6 )
& ~ ( ( ( A = k4_measure6
& ~ r1_supinf_1(B,k1_supinf_2) )
| ( A = k5_measure6
& ~ r1_supinf_1(k1_supinf_2,B) ) )
& C = k4_measure6 )
& ~ ( ( B = k4_measure6
| B = k5_measure6 )
& C = k1_supinf_2 ) ) ) ) ) ) ) ).
fof(t31_extreal1,axiom,
$true ).
fof(t32_extreal1,axiom,
! [A] :
( m1_subset_1(A,k3_supinf_1)
=> ! [B] :
( m1_subset_1(B,k3_supinf_1)
=> ( B != k1_supinf_2
=> ! [C] :
( m1_subset_1(C,k1_numbers)
=> ! [D] :
( m1_subset_1(D,k1_numbers)
=> ( ( A = C
& B = D )
=> k3_extreal1(A,B) = k6_real_1(C,D) ) ) ) ) ) ) ).
fof(t33_extreal1,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 )
& k3_extreal1(A,B) != k1_supinf_2 ) ) ) ).
fof(t34_extreal1,axiom,
! [A] :
( m1_subset_1(A,k3_supinf_1)
=> ~ ( A != k4_measure6
& A != k5_measure6
& A != k1_supinf_2
& k3_extreal1(A,A) != np__1 ) ) ).
fof(d3_extreal1,axiom,
! [A] :
( m1_subset_1(A,k3_supinf_1)
=> ( ( r1_supinf_1(k1_supinf_2,A)
=> k4_extreal1(A) = A )
& ( ~ r1_supinf_1(k1_supinf_2,A)
=> k4_extreal1(A) = k3_supinf_2(A) ) ) ) ).
fof(t35_extreal1,axiom,
$true ).
fof(t36_extreal1,axiom,
! [A] :
( m1_subset_1(A,k3_supinf_1)
=> ( ~ r1_supinf_1(A,k1_supinf_2)
=> k4_extreal1(A) = A ) ) ).
fof(t37_extreal1,axiom,
! [A] :
( m1_subset_1(A,k3_supinf_1)
=> ( ~ r1_supinf_1(k1_supinf_2,A)
=> k4_extreal1(A) = k3_supinf_2(A) ) ) ).
fof(t38_extreal1,axiom,
! [A] :
( m1_subset_1(A,k1_numbers)
=> ! [B] :
( m1_subset_1(B,k1_numbers)
=> k1_measure6(k4_real_1(A,B)) = k2_extreal1(k1_measure6(A),k1_measure6(B)) ) ) ).
fof(t39_extreal1,axiom,
! [A] :
( m1_subset_1(A,k1_numbers)
=> ! [B] :
( m1_subset_1(B,k1_numbers)
=> ( B != np__0
=> k1_measure6(k6_real_1(A,B)) = k3_extreal1(k1_measure6(A),k1_measure6(B)) ) ) ) ).
fof(t40_extreal1,axiom,
! [A] :
( m1_subset_1(A,k3_supinf_1)
=> ! [B] :
( m1_subset_1(B,k3_supinf_1)
=> ( r1_supinf_1(A,B)
=> ( r1_supinf_1(k5_measure6,A)
| r1_supinf_1(B,k4_measure6)
| r1_supinf_1(k1_supinf_2,k4_supinf_2(B,A)) ) ) ) ) ).
fof(t41_extreal1,axiom,
! [A] :
( m1_subset_1(A,k3_supinf_1)
=> ! [B] :
( m1_subset_1(B,k3_supinf_1)
=> ~ ( ~ r1_supinf_1(B,A)
& ~ r1_supinf_1(k5_measure6,A)
& ~ r1_supinf_1(B,k4_measure6)
& r1_supinf_1(k4_supinf_2(B,A),k1_supinf_2) ) ) ) ).
fof(t42_extreal1,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(k1_supinf_2,C) )
=> r1_supinf_1(k2_extreal1(A,C),k2_extreal1(B,C)) ) ) ) ) ).
fof(t43_extreal1,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,k1_supinf_2) )
=> r1_supinf_1(k2_extreal1(B,C),k2_extreal1(A,C)) ) ) ) ) ).
fof(t44_extreal1,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(B,A)
& ~ r1_supinf_1(C,k1_supinf_2)
& C != k5_measure6
& r1_supinf_1(k2_extreal1(B,C),k2_extreal1(A,C)) ) ) ) ) ).
fof(t45_extreal1,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(B,A)
& ~ r1_supinf_1(k1_supinf_2,C)
& C != k4_measure6
& r1_supinf_1(k2_extreal1(A,C),k2_extreal1(B,C)) ) ) ) ) ).
fof(t46_extreal1,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(B,A)
& ! [C] :
( m1_subset_1(C,k1_numbers)
=> ! [D] :
( m1_subset_1(D,k1_numbers)
=> ~ ( C = A
& D = B
& ~ r1_xreal_0(D,C) ) ) ) )
& ~ ( ? [C] :
( m1_subset_1(C,k1_numbers)
& ? [D] :
( m1_subset_1(D,k1_numbers)
& C = A
& D = B
& ~ r1_xreal_0(D,C) ) )
& r1_supinf_1(B,A) ) ) ) ) ) ).
fof(t47_extreal1,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)
=> ( A = k4_measure6
| B = k5_measure6
| r1_supinf_1(C,k1_supinf_2)
| r1_supinf_1(k3_extreal1(A,C),k3_extreal1(B,C)) ) ) ) ) ) ).
fof(t48_extreal1,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,k1_supinf_2)
| C = k5_measure6
| r1_supinf_1(k3_extreal1(A,C),k3_extreal1(B,C)) ) ) ) ) ) ).
fof(t49_extreal1,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)
=> ( A = k4_measure6
| B = k5_measure6
| r1_supinf_1(k1_supinf_2,C)
| r1_supinf_1(k3_extreal1(B,C),k3_extreal1(A,C)) ) ) ) ) ) ).
fof(t50_extreal1,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(k1_supinf_2,C)
| C = k4_measure6
| r1_supinf_1(k3_extreal1(B,C),k3_extreal1(A,C)) ) ) ) ) ) ).
fof(t51_extreal1,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(B,A)
& ~ r1_supinf_1(C,k1_supinf_2)
& C != k5_measure6
& r1_supinf_1(k3_extreal1(B,C),k3_extreal1(A,C)) ) ) ) ) ).
fof(t52_extreal1,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(B,A)
& ~ r1_supinf_1(k1_supinf_2,C)
& C != k4_measure6
& r1_supinf_1(k3_extreal1(A,C),k3_extreal1(B,C)) ) ) ) ) ).
fof(dt_k1_extreal1,axiom,
! [A,B] :
( ( m1_subset_1(A,k3_supinf_1)
& m1_subset_1(B,k3_supinf_1) )
=> m1_subset_1(k1_extreal1(A,B),k3_supinf_1) ) ).
fof(dt_k2_extreal1,axiom,
! [A,B] :
( ( m1_subset_1(A,k3_supinf_1)
& m1_subset_1(B,k3_supinf_1) )
=> m1_subset_1(k2_extreal1(A,B),k3_supinf_1) ) ).
fof(commutativity_k2_extreal1,axiom,
! [A,B] :
( ( m1_subset_1(A,k3_supinf_1)
& m1_subset_1(B,k3_supinf_1) )
=> k2_extreal1(A,B) = k2_extreal1(B,A) ) ).
fof(redefinition_k2_extreal1,axiom,
! [A,B] :
( ( m1_subset_1(A,k3_supinf_1)
& m1_subset_1(B,k3_supinf_1) )
=> k2_extreal1(A,B) = k1_extreal1(A,B) ) ).
fof(dt_k3_extreal1,axiom,
! [A,B] :
( ( m1_subset_1(A,k3_supinf_1)
& m1_subset_1(B,k3_supinf_1) )
=> m1_subset_1(k3_extreal1(A,B),k3_supinf_1) ) ).
fof(dt_k4_extreal1,axiom,
! [A] :
( m1_subset_1(A,k3_supinf_1)
=> m1_subset_1(k4_extreal1(A),k3_supinf_1) ) ).
%------------------------------------------------------------------------------