SET007 Axioms: SET007+175.ax
%------------------------------------------------------------------------------
% File : SET007+175 : TPTP v8.2.0. Released v3.4.0.
% Domain : Set Theory
% Axioms : Some Properties of the Intervals
% 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 : measure6 [Urb08]
% Status : Satisfiable
% Syntax : Number of formulae : 84 ( 10 unt; 0 def)
% Number of atoms : 520 ( 75 equ)
% Maximal formula atoms : 18 ( 6 avg)
% Number of connectives : 532 ( 96 ~; 16 |; 183 &)
% ( 12 <=>; 225 =>; 0 <=; 0 <~>)
% Maximal formula depth : 20 ( 8 avg)
% Maximal term depth : 3 ( 1 avg)
% Number of predicates : 23 ( 21 usr; 1 prp; 0-3 aty)
% Number of functors : 35 ( 35 usr; 11 con; 0-3 aty)
% Number of variables : 183 ( 179 !; 4 ?)
% 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(fc1_measure6,axiom,
! [A,B] :
( ( v5_measure5(A)
& m1_subset_1(A,k1_zfmisc_1(k6_supinf_1))
& v1_xreal_0(B) )
=> ( v5_measure5(k8_measure6(A,B))
& v1_membered(k8_measure6(A,B))
& v2_membered(k8_measure6(A,B)) ) ) ).
fof(t1_measure6,axiom,
? [A] :
( v1_funct_1(A)
& v1_funct_2(A,k5_numbers,k2_zfmisc_1(k5_numbers,k5_numbers))
& m2_relset_1(A,k5_numbers,k2_zfmisc_1(k5_numbers,k5_numbers))
& v2_funct_1(A)
& k1_relat_1(A) = k5_numbers
& k2_relat_1(A) = k2_zfmisc_1(k5_numbers,k5_numbers) ) ).
fof(t2_measure6,axiom,
! [A] :
( ( v1_funct_1(A)
& v1_funct_2(A,k5_numbers,k3_supinf_1)
& m2_relset_1(A,k5_numbers,k3_supinf_1) )
=> ( v6_supinf_2(A,k5_numbers)
=> r1_supinf_1(k1_supinf_2,k19_supinf_2(A)) ) ) ).
fof(t3_measure6,axiom,
! [A] :
( ( v1_funct_1(A)
& v1_funct_2(A,k5_numbers,k3_supinf_1)
& m2_relset_1(A,k5_numbers,k3_supinf_1) )
=> ! [B] :
( m1_subset_1(B,k3_supinf_1)
=> ( v6_supinf_2(A,k5_numbers)
=> ( ! [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
=> ~ r1_supinf_1(B,k13_supinf_2(A,C)) )
| r1_supinf_1(B,k19_supinf_2(A)) ) ) ) ) ).
fof(t4_measure6,axiom,
$true ).
fof(t5_measure6,axiom,
$true ).
fof(t6_measure6,axiom,
$true ).
fof(t7_measure6,axiom,
$true ).
fof(t8_measure6,axiom,
! [A] :
( m1_subset_1(A,k3_supinf_1)
=> ! [B] :
( m1_subset_1(B,k3_supinf_1)
=> ( m1_subset_1(A,k1_numbers)
=> ( k2_supinf_2(k4_supinf_2(B,A),A) = B
& k4_supinf_2(k2_supinf_2(B,A),A) = B ) ) ) ) ).
fof(t9_measure6,axiom,
$true ).
fof(t10_measure6,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)
=> ( r2_hidden(C,k6_supinf_1)
=> ( r1_supinf_1(A,B)
| k4_supinf_2(k2_supinf_2(C,A),k2_supinf_2(C,B)) = k4_supinf_2(A,B) ) ) ) ) ) ).
fof(t11_measure6,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)
=> ( ( r2_hidden(C,k6_supinf_1)
& r1_supinf_1(A,B) )
=> ( r1_supinf_1(k2_supinf_2(C,A),k2_supinf_2(C,B))
& r1_supinf_1(k2_supinf_2(A,C),k2_supinf_2(B,C))
& r1_supinf_1(k4_supinf_2(A,C),k4_supinf_2(B,C)) ) ) ) ) ) ).
fof(t12_measure6,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)
=> ( r2_hidden(C,k6_supinf_1)
=> ( r1_supinf_1(B,A)
| ( ~ r1_supinf_1(k2_supinf_2(C,B),k2_supinf_2(C,A))
& ~ r1_supinf_1(k2_supinf_2(B,C),k2_supinf_2(A,C))
& ~ r1_supinf_1(k4_supinf_2(B,C),k4_supinf_2(A,C)) ) ) ) ) ) ) ).
fof(d1_measure6,axiom,
! [A] :
( v1_xreal_0(A)
=> k1_measure6(A) = A ) ).
fof(t13_measure6,axiom,
! [A] :
( v1_xreal_0(A)
=> ! [B] :
( v1_xreal_0(B)
=> ( r1_xreal_0(A,B)
<=> r1_supinf_1(k1_measure6(A),k1_measure6(B)) ) ) ) ).
fof(t14_measure6,axiom,
! [A] :
( v1_xreal_0(A)
=> ! [B] :
( v1_xreal_0(B)
=> ( ~ ( ~ r1_xreal_0(B,A)
& r1_supinf_1(k1_measure6(B),k1_measure6(A)) )
& ~ ( ~ r1_supinf_1(k1_measure6(B),k1_measure6(A))
& r1_xreal_0(B,A) ) ) ) ) ).
fof(t15_measure6,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,B)
& ~ m1_subset_1(B,k1_numbers) ) ) ) ) ).
fof(t16_measure6,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)
=> ( ( m1_subset_1(A,k1_numbers)
& m1_subset_1(C,k1_numbers)
& r1_supinf_1(A,B)
& r1_supinf_1(B,C) )
=> m1_subset_1(B,k1_numbers) ) ) ) ) ).
fof(t17_measure6,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)
=> ( ( m1_subset_1(A,k1_numbers)
& r1_supinf_1(A,B) )
=> ( r1_supinf_1(C,B)
| m1_subset_1(B,k1_numbers) ) ) ) ) ) ).
fof(t18_measure6,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,C)
& m1_subset_1(C,k1_numbers) )
=> ( r1_supinf_1(B,A)
| m1_subset_1(B,k1_numbers) ) ) ) ) ) ).
fof(t19_measure6,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,A)
& r1_supinf_1(k4_supinf_2(B,A),k1_supinf_2) ) ) ) ).
fof(t20_measure6,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(k1_supinf_2,A)
& r1_supinf_1(k1_supinf_2,C)
& ~ r1_supinf_1(B,k2_supinf_2(C,A))
& r1_supinf_1(k4_supinf_2(B,A),C) ) ) ) ) ).
fof(t21_measure6,axiom,
! [A] :
( m1_subset_1(A,k3_supinf_1)
=> k4_supinf_2(A,k1_supinf_2) = A ) ).
fof(t22_measure6,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(k1_supinf_2,A)
& r1_supinf_1(k1_supinf_2,C) )
=> ( r1_supinf_1(B,k2_supinf_2(C,A))
| r1_supinf_1(C,B) ) ) ) ) ) ).
fof(t23_measure6,axiom,
! [A] :
( m1_subset_1(A,k3_supinf_1)
=> ~ ( ~ r1_supinf_1(A,k1_supinf_2)
& ! [B] :
( m1_subset_1(B,k3_supinf_1)
=> ~ ( ~ r1_supinf_1(B,k1_supinf_2)
& ~ r1_supinf_1(A,B) ) ) ) ) ).
fof(t24_measure6,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,A)
& ! [C] :
( m1_subset_1(C,k3_supinf_1)
=> ~ ( ~ r1_supinf_1(C,k1_supinf_2)
& ~ r1_supinf_1(B,k2_supinf_2(A,C))
& r2_hidden(C,k6_supinf_1) ) ) ) ) ) ).
fof(t25_measure6,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,A)
& ! [C] :
( m1_subset_1(C,k3_supinf_1)
=> ~ ( ~ r1_supinf_1(C,k1_supinf_2)
& ~ r1_supinf_1(B,k2_supinf_2(A,C))
& r2_hidden(C,k6_supinf_1) ) ) ) ) ) ).
fof(t26_measure6,axiom,
! [A] :
( m1_subset_1(A,k3_supinf_1)
=> ~ ( ~ r1_supinf_1(A,k1_supinf_2)
& ! [B] :
( m1_subset_1(B,k3_supinf_1)
=> ~ ( ~ r1_supinf_1(B,k1_supinf_2)
& ~ r1_supinf_1(A,k2_supinf_2(B,B)) ) ) ) ) ).
fof(d2_measure6,axiom,
! [A] :
( m1_subset_1(A,k3_supinf_1)
=> ( ~ r1_supinf_1(A,k1_supinf_2)
=> ! [B] :
( ( ~ v1_xboole_0(B)
& m1_subset_1(B,k1_zfmisc_1(k3_supinf_1)) )
=> ( B = k2_measure6(A)
<=> ! [C] :
( m1_subset_1(C,k3_supinf_1)
=> ( r2_hidden(C,B)
<=> ( ~ r1_supinf_1(C,k1_supinf_2)
& ~ r1_supinf_1(A,k2_supinf_2(C,C)) ) ) ) ) ) ) ) ).
fof(d3_measure6,axiom,
! [A] :
( m1_subset_1(A,k3_supinf_1)
=> k3_measure6(A) = k9_supinf_1(k2_measure6(A)) ) ).
fof(t27_measure6,axiom,
! [A] :
( m1_subset_1(A,k3_supinf_1)
=> ~ ( ~ r1_supinf_1(A,k1_supinf_2)
& r1_supinf_1(k3_measure6(A),k1_supinf_2) ) ) ).
fof(t28_measure6,axiom,
! [A] :
( m1_subset_1(A,k3_supinf_1)
=> ( ~ r1_supinf_1(A,k1_supinf_2)
=> r1_supinf_1(k3_measure6(A),A) ) ) ).
fof(t29_measure6,axiom,
! [A] :
( m1_subset_1(A,k3_supinf_1)
=> ~ ( ~ r1_supinf_1(A,k1_supinf_2)
& ~ r1_supinf_1(k4_supinf_1,A)
& ~ m1_subset_1(k3_measure6(A),k1_numbers) ) ) ).
fof(t30_measure6,axiom,
! [A] :
( m1_subset_1(A,k3_supinf_1)
=> ( ~ r1_supinf_1(A,k1_supinf_2)
=> r1_supinf_1(k2_supinf_2(k3_measure6(A),k3_measure6(A)),A) ) ) ).
fof(t31_measure6,axiom,
! [A] :
( m1_subset_1(A,k3_supinf_1)
=> ~ ( ~ r1_supinf_1(A,k1_supinf_2)
& ! [B] :
( ( v1_funct_1(B)
& v1_funct_2(B,k5_numbers,k3_supinf_1)
& m2_relset_1(B,k5_numbers,k3_supinf_1) )
=> ~ ( ! [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
=> ~ r1_supinf_1(k13_supinf_2(B,C),k1_supinf_2) )
& ~ r1_supinf_1(A,k19_supinf_2(B)) ) ) ) ) ).
fof(t32_measure6,axiom,
! [A] :
( m1_subset_1(A,k3_supinf_1)
=> ! [B] :
( ( ~ v1_xboole_0(B)
& m1_subset_1(B,k1_zfmisc_1(k3_supinf_1)) )
=> ~ ( ~ r1_supinf_1(A,k1_supinf_2)
& m1_subset_1(k10_supinf_1(B),k1_numbers)
& ! [C] :
( m1_subset_1(C,k3_supinf_1)
=> ~ ( r2_hidden(C,B)
& ~ r1_supinf_1(k2_supinf_2(k10_supinf_1(B),A),C) ) ) ) ) ) ).
fof(t33_measure6,axiom,
! [A] :
( m1_subset_1(A,k3_supinf_1)
=> ! [B] :
( ( ~ v1_xboole_0(B)
& m1_subset_1(B,k1_zfmisc_1(k3_supinf_1)) )
=> ~ ( ~ r1_supinf_1(A,k1_supinf_2)
& m1_subset_1(k9_supinf_1(B),k1_numbers)
& ! [C] :
( m1_subset_1(C,k3_supinf_1)
=> ~ ( r2_hidden(C,B)
& ~ r1_supinf_1(C,k4_supinf_2(k9_supinf_1(B),A)) ) ) ) ) ) ).
fof(t34_measure6,axiom,
! [A] :
( ( v1_funct_1(A)
& v1_funct_2(A,k5_numbers,k3_supinf_1)
& m2_relset_1(A,k5_numbers,k3_supinf_1) )
=> ( v6_supinf_2(A,k5_numbers)
=> ( r1_supinf_1(k4_supinf_1,k19_supinf_2(A))
| ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> r2_hidden(k13_supinf_2(A,B),k6_supinf_1) ) ) ) ) ).
fof(t35_measure6,axiom,
( v5_measure5(k6_supinf_1)
& m1_subset_1(k6_supinf_1,k1_zfmisc_1(k6_supinf_1))
& k6_supinf_1 = k2_measure5(k4_measure6,k5_measure6)
& k6_supinf_1 = k1_measure5(k4_measure6,k5_measure6)
& k6_supinf_1 = k4_measure5(k4_measure6,k5_measure6)
& k6_supinf_1 = k3_measure5(k4_measure6,k5_measure6) ) ).
fof(t36_measure6,axiom,
! [A] :
( m1_subset_1(A,k3_supinf_1)
=> ! [B] :
( m1_subset_1(B,k3_supinf_1)
=> ( B = k4_measure6
=> ( k2_measure5(A,B) = k6_measure5
& k1_measure5(A,B) = k6_measure5
& k4_measure5(A,B) = k6_measure5
& k3_measure5(A,B) = k6_measure5 ) ) ) ) ).
fof(t37_measure6,axiom,
! [A] :
( m1_subset_1(A,k3_supinf_1)
=> ! [B] :
( m1_subset_1(B,k3_supinf_1)
=> ( A = k5_measure6
=> ( k2_measure5(A,B) = k6_measure5
& k1_measure5(A,B) = k6_measure5
& k4_measure5(A,B) = k6_measure5
& k3_measure5(A,B) = k6_measure5 ) ) ) ) ).
fof(t38_measure6,axiom,
! [A] :
( ( v5_measure5(A)
& m1_subset_1(A,k1_zfmisc_1(k6_supinf_1)) )
=> ! [B] :
( m1_subset_1(B,k3_supinf_1)
=> ! [C] :
( m1_subset_1(C,k3_supinf_1)
=> ! [D] :
( m1_subset_1(D,k1_numbers)
=> ! [E] :
( m1_subset_1(E,k1_numbers)
=> ! [F] :
( m1_subset_1(F,k1_numbers)
=> ( ( A = k2_measure5(B,C)
& r2_hidden(D,A)
& r2_hidden(E,A)
& r1_xreal_0(D,F)
& r1_xreal_0(F,E) )
=> r2_hidden(F,A) ) ) ) ) ) ) ) ).
fof(t39_measure6,axiom,
! [A] :
( ( v5_measure5(A)
& m1_subset_1(A,k1_zfmisc_1(k6_supinf_1)) )
=> ! [B] :
( m1_subset_1(B,k3_supinf_1)
=> ! [C] :
( m1_subset_1(C,k3_supinf_1)
=> ! [D] :
( m1_subset_1(D,k1_numbers)
=> ! [E] :
( m1_subset_1(E,k1_numbers)
=> ! [F] :
( m1_subset_1(F,k1_numbers)
=> ( ( A = k1_measure5(B,C)
& r2_hidden(D,A)
& r2_hidden(E,A)
& r1_xreal_0(D,F)
& r1_xreal_0(F,E) )
=> r2_hidden(F,A) ) ) ) ) ) ) ) ).
fof(t40_measure6,axiom,
! [A] :
( ( v5_measure5(A)
& m1_subset_1(A,k1_zfmisc_1(k6_supinf_1)) )
=> ! [B] :
( m1_subset_1(B,k3_supinf_1)
=> ! [C] :
( m1_subset_1(C,k3_supinf_1)
=> ! [D] :
( m1_subset_1(D,k1_numbers)
=> ! [E] :
( m1_subset_1(E,k1_numbers)
=> ! [F] :
( m1_subset_1(F,k1_numbers)
=> ( ( A = k3_measure5(B,C)
& r2_hidden(D,A)
& r2_hidden(E,A)
& r1_xreal_0(D,F)
& r1_xreal_0(F,E) )
=> r2_hidden(F,A) ) ) ) ) ) ) ) ).
fof(t41_measure6,axiom,
! [A] :
( ( v5_measure5(A)
& m1_subset_1(A,k1_zfmisc_1(k6_supinf_1)) )
=> ! [B] :
( m1_subset_1(B,k3_supinf_1)
=> ! [C] :
( m1_subset_1(C,k3_supinf_1)
=> ! [D] :
( m1_subset_1(D,k1_numbers)
=> ! [E] :
( m1_subset_1(E,k1_numbers)
=> ! [F] :
( m1_subset_1(F,k1_numbers)
=> ( ( A = k4_measure5(B,C)
& r2_hidden(D,A)
& r2_hidden(E,A)
& r1_xreal_0(D,F)
& r1_xreal_0(F,E) )
=> r2_hidden(F,A) ) ) ) ) ) ) ) ).
fof(t42_measure6,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& m1_subset_1(A,k1_zfmisc_1(k3_supinf_1)) )
=> ! [B] :
( m1_subset_1(B,k3_supinf_1)
=> ! [C] :
( m1_subset_1(C,k3_supinf_1)
=> ( ( B = k10_supinf_1(A)
& C = k9_supinf_1(A)
& ! [D] :
( m1_subset_1(D,k1_numbers)
=> ! [E] :
( m1_subset_1(E,k1_numbers)
=> ( ( r2_hidden(D,A)
& r2_hidden(E,A) )
=> ! [F] :
( m1_subset_1(F,k1_numbers)
=> ( ( r1_xreal_0(D,F)
& r1_xreal_0(F,E) )
=> r2_hidden(F,A) ) ) ) ) ) )
=> ( r2_hidden(B,A)
| r2_hidden(C,A)
| A = k2_measure5(B,C) ) ) ) ) ) ).
fof(t43_measure6,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& m1_subset_1(A,k1_zfmisc_1(k3_supinf_1)) )
=> ! [B] :
( m1_subset_1(B,k3_supinf_1)
=> ! [C] :
( m1_subset_1(C,k3_supinf_1)
=> ( ( B = k10_supinf_1(A)
& C = k9_supinf_1(A)
& ! [D] :
( m1_subset_1(D,k1_numbers)
=> ! [E] :
( m1_subset_1(E,k1_numbers)
=> ( ( r2_hidden(D,A)
& r2_hidden(E,A) )
=> ! [F] :
( m1_subset_1(F,k1_numbers)
=> ( ( r1_xreal_0(D,F)
& r1_xreal_0(F,E) )
=> r2_hidden(F,A) ) ) ) ) )
& r2_hidden(B,A)
& r2_hidden(C,A)
& r1_tarski(A,k6_supinf_1) )
=> A = k1_measure5(B,C) ) ) ) ) ).
fof(t44_measure6,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& m1_subset_1(A,k1_zfmisc_1(k3_supinf_1)) )
=> ! [B] :
( m1_subset_1(B,k3_supinf_1)
=> ! [C] :
( m1_subset_1(C,k3_supinf_1)
=> ( ( B = k10_supinf_1(A)
& C = k9_supinf_1(A)
& ! [D] :
( m1_subset_1(D,k1_numbers)
=> ! [E] :
( m1_subset_1(E,k1_numbers)
=> ( ( r2_hidden(D,A)
& r2_hidden(E,A) )
=> ! [F] :
( m1_subset_1(F,k1_numbers)
=> ( ( r1_xreal_0(D,F)
& r1_xreal_0(F,E) )
=> r2_hidden(F,A) ) ) ) ) )
& r2_hidden(B,A)
& r1_tarski(A,k6_supinf_1) )
=> ( r2_hidden(C,A)
| A = k4_measure5(B,C) ) ) ) ) ) ).
fof(t45_measure6,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& m1_subset_1(A,k1_zfmisc_1(k3_supinf_1)) )
=> ! [B] :
( m1_subset_1(B,k3_supinf_1)
=> ! [C] :
( m1_subset_1(C,k3_supinf_1)
=> ( ( B = k10_supinf_1(A)
& C = k9_supinf_1(A)
& ! [D] :
( m1_subset_1(D,k1_numbers)
=> ! [E] :
( m1_subset_1(E,k1_numbers)
=> ( ( r2_hidden(D,A)
& r2_hidden(E,A) )
=> ! [F] :
( m1_subset_1(F,k1_numbers)
=> ( ( r1_xreal_0(D,F)
& r1_xreal_0(F,E) )
=> r2_hidden(F,A) ) ) ) ) )
& r2_hidden(C,A)
& r1_tarski(A,k6_supinf_1) )
=> ( r2_hidden(B,A)
| A = k3_measure5(B,C) ) ) ) ) ) ).
fof(t46_measure6,axiom,
! [A] :
( m1_subset_1(A,k1_zfmisc_1(k6_supinf_1))
=> ( ( v5_measure5(A)
& m1_subset_1(A,k1_zfmisc_1(k6_supinf_1)) )
<=> ! [B] :
( m1_subset_1(B,k1_numbers)
=> ! [C] :
( m1_subset_1(C,k1_numbers)
=> ( ( r2_hidden(B,A)
& r2_hidden(C,A) )
=> ! [D] :
( m1_subset_1(D,k1_numbers)
=> ( ( r1_xreal_0(B,D)
& r1_xreal_0(D,C) )
=> r2_hidden(D,A) ) ) ) ) ) ) ) ).
fof(t47_measure6,axiom,
! [A] :
( ( v5_measure5(A)
& m1_subset_1(A,k1_zfmisc_1(k6_supinf_1)) )
=> ! [B] :
( ( v5_measure5(B)
& m1_subset_1(B,k1_zfmisc_1(k6_supinf_1)) )
=> ( ~ r1_xboole_0(A,B)
=> ( v5_measure5(k4_subset_1(k6_supinf_1,A,B))
& m1_subset_1(k4_subset_1(k6_supinf_1,A,B),k1_zfmisc_1(k6_supinf_1)) ) ) ) ) ).
fof(d4_measure6,axiom,
! [A] :
( ( v5_measure5(A)
& m1_subset_1(A,k1_zfmisc_1(k6_supinf_1)) )
=> ( A != k6_measure5
=> ! [B] :
( m1_subset_1(B,k3_supinf_1)
=> ( B = k6_measure6(A)
<=> ? [C] :
( m1_subset_1(C,k3_supinf_1)
& r1_supinf_1(B,C)
& ~ ( A != k2_measure5(B,C)
& A != k3_measure5(B,C)
& A != k1_measure5(B,C)
& A != k4_measure5(B,C) ) ) ) ) ) ) ).
fof(d5_measure6,axiom,
! [A] :
( ( v5_measure5(A)
& m1_subset_1(A,k1_zfmisc_1(k6_supinf_1)) )
=> ( A != k6_measure5
=> ! [B] :
( m1_subset_1(B,k3_supinf_1)
=> ( B = k7_measure6(A)
<=> ? [C] :
( m1_subset_1(C,k3_supinf_1)
& r1_supinf_1(C,B)
& ~ ( A != k2_measure5(C,B)
& A != k3_measure5(C,B)
& A != k1_measure5(C,B)
& A != k4_measure5(C,B) ) ) ) ) ) ) ).
fof(t48_measure6,axiom,
! [A] :
( ( v5_measure5(A)
& m1_subset_1(A,k1_zfmisc_1(k6_supinf_1)) )
=> ( v1_measure5(A)
=> ( A = k6_measure5
| ( r1_supinf_1(k6_measure6(A),k7_measure6(A))
& A = k2_measure5(k6_measure6(A),k7_measure6(A)) ) ) ) ) ).
fof(t49_measure6,axiom,
! [A] :
( ( v5_measure5(A)
& m1_subset_1(A,k1_zfmisc_1(k6_supinf_1)) )
=> ( v2_measure5(A)
=> ( A = k6_measure5
| ( r1_supinf_1(k6_measure6(A),k7_measure6(A))
& A = k1_measure5(k6_measure6(A),k7_measure6(A)) ) ) ) ) ).
fof(t50_measure6,axiom,
! [A] :
( ( v5_measure5(A)
& m1_subset_1(A,k1_zfmisc_1(k6_supinf_1)) )
=> ( v3_measure5(A)
=> ( A = k6_measure5
| ( r1_supinf_1(k6_measure6(A),k7_measure6(A))
& A = k4_measure5(k6_measure6(A),k7_measure6(A)) ) ) ) ) ).
fof(t51_measure6,axiom,
! [A] :
( ( v5_measure5(A)
& m1_subset_1(A,k1_zfmisc_1(k6_supinf_1)) )
=> ( v4_measure5(A)
=> ( A = k6_measure5
| ( r1_supinf_1(k6_measure6(A),k7_measure6(A))
& A = k3_measure5(k6_measure6(A),k7_measure6(A)) ) ) ) ) ).
fof(t52_measure6,axiom,
! [A] :
( ( v5_measure5(A)
& m1_subset_1(A,k1_zfmisc_1(k6_supinf_1)) )
=> ( A != k6_measure5
=> ( r1_supinf_1(k6_measure6(A),k7_measure6(A))
& ~ ( A != k2_measure5(k6_measure6(A),k7_measure6(A))
& A != k3_measure5(k6_measure6(A),k7_measure6(A))
& A != k1_measure5(k6_measure6(A),k7_measure6(A))
& A != k4_measure5(k6_measure6(A),k7_measure6(A)) ) ) ) ) ).
fof(t53_measure6,axiom,
$true ).
fof(t54_measure6,axiom,
! [A] :
( ( v5_measure5(A)
& m1_subset_1(A,k1_zfmisc_1(k6_supinf_1)) )
=> ! [B] :
( v1_xreal_0(B)
=> ( r2_hidden(B,A)
=> ( r1_supinf_1(k6_measure6(A),k1_measure6(B))
& r1_supinf_1(k1_measure6(B),k7_measure6(A)) ) ) ) ) ).
fof(t55_measure6,axiom,
! [A] :
( ( v5_measure5(A)
& m1_subset_1(A,k1_zfmisc_1(k6_supinf_1)) )
=> ! [B] :
( ( v5_measure5(B)
& m1_subset_1(B,k1_zfmisc_1(k6_supinf_1)) )
=> ! [C] :
( v1_xreal_0(C)
=> ! [D] :
( v1_xreal_0(D)
=> ( ( r2_hidden(C,A)
& r2_hidden(D,B)
& r1_supinf_1(k7_measure6(A),k6_measure6(B)) )
=> r1_xreal_0(C,D) ) ) ) ) ) ).
fof(t56_measure6,axiom,
! [A] :
( ( v5_measure5(A)
& m1_subset_1(A,k1_zfmisc_1(k6_supinf_1)) )
=> ! [B] :
( m1_subset_1(B,k3_supinf_1)
=> ( r2_hidden(B,A)
=> ( r1_supinf_1(k6_measure6(A),B)
& r1_supinf_1(B,k7_measure6(A)) ) ) ) ) ).
fof(t57_measure6,axiom,
! [A] :
( ( v5_measure5(A)
& m1_subset_1(A,k1_zfmisc_1(k6_supinf_1)) )
=> ( A != k6_measure5
=> ! [B] :
( m1_subset_1(B,k3_supinf_1)
=> ~ ( ~ r1_supinf_1(B,k6_measure6(A))
& ~ r1_supinf_1(k7_measure6(A),B)
& ~ r2_hidden(B,A) ) ) ) ) ).
fof(t58_measure6,axiom,
! [A] :
( ( v5_measure5(A)
& m1_subset_1(A,k1_zfmisc_1(k6_supinf_1)) )
=> ! [B] :
( ( v5_measure5(B)
& m1_subset_1(B,k1_zfmisc_1(k6_supinf_1)) )
=> ( k7_measure6(A) = k6_measure6(B)
=> ( ( ~ r2_hidden(k7_measure6(A),A)
& ~ r2_hidden(k6_measure6(B),B) )
| ( v5_measure5(k4_subset_1(k6_supinf_1,A,B))
& m1_subset_1(k4_subset_1(k6_supinf_1,A,B),k1_zfmisc_1(k6_supinf_1)) ) ) ) ) ) ).
fof(d6_measure6,axiom,
! [A] :
( m1_subset_1(A,k1_zfmisc_1(k6_supinf_1))
=> ! [B] :
( v1_xreal_0(B)
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(k6_supinf_1))
=> ( C = k8_measure6(A,B)
<=> ! [D] :
( m1_subset_1(D,k1_numbers)
=> ( r2_hidden(D,C)
<=> ? [E] :
( m1_subset_1(E,k1_numbers)
& r2_hidden(E,A)
& D = k2_xcmplx_0(B,E) ) ) ) ) ) ) ) ).
fof(t59_measure6,axiom,
! [A] :
( m1_subset_1(A,k1_zfmisc_1(k6_supinf_1))
=> ! [B] :
( v1_xreal_0(B)
=> k8_measure6(k8_measure6(A,B),k4_xcmplx_0(B)) = A ) ) ).
fof(t60_measure6,axiom,
! [A] :
( v1_xreal_0(A)
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(k6_supinf_1))
=> ( B = k6_supinf_1
=> k8_measure6(B,A) = B ) ) ) ).
fof(t61_measure6,axiom,
! [A] :
( v1_xreal_0(A)
=> k8_measure6(k6_measure5,A) = k6_measure5 ) ).
fof(t62_measure6,axiom,
! [A] :
( ( v5_measure5(A)
& m1_subset_1(A,k1_zfmisc_1(k6_supinf_1)) )
=> ! [B] :
( v1_xreal_0(B)
=> ( v1_measure5(A)
<=> v1_measure5(k8_measure6(A,B)) ) ) ) ).
fof(t63_measure6,axiom,
! [A] :
( ( v5_measure5(A)
& m1_subset_1(A,k1_zfmisc_1(k6_supinf_1)) )
=> ! [B] :
( v1_xreal_0(B)
=> ( v2_measure5(A)
<=> v2_measure5(k8_measure6(A,B)) ) ) ) ).
fof(t64_measure6,axiom,
! [A] :
( ( v5_measure5(A)
& m1_subset_1(A,k1_zfmisc_1(k6_supinf_1)) )
=> ! [B] :
( v1_xreal_0(B)
=> ( v3_measure5(A)
<=> v3_measure5(k8_measure6(A,B)) ) ) ) ).
fof(t65_measure6,axiom,
! [A] :
( ( v5_measure5(A)
& m1_subset_1(A,k1_zfmisc_1(k6_supinf_1)) )
=> ! [B] :
( v1_xreal_0(B)
=> ( v4_measure5(A)
<=> v4_measure5(k8_measure6(A,B)) ) ) ) ).
fof(t66_measure6,axiom,
! [A] :
( ( v5_measure5(A)
& m1_subset_1(A,k1_zfmisc_1(k6_supinf_1)) )
=> ! [B] :
( v1_xreal_0(B)
=> ( v5_measure5(k8_measure6(A,B))
& m1_subset_1(k8_measure6(A,B),k1_zfmisc_1(k6_supinf_1)) ) ) ) ).
fof(t67_measure6,axiom,
! [A] :
( ( v5_measure5(A)
& m1_subset_1(A,k1_zfmisc_1(k6_supinf_1)) )
=> ! [B] :
( v1_xreal_0(B)
=> k5_measure5(A) = k5_measure5(k8_measure6(A,B)) ) ) ).
fof(dt_k1_measure6,axiom,
! [A] :
( v1_xreal_0(A)
=> m1_subset_1(k1_measure6(A),k3_supinf_1) ) ).
fof(dt_k2_measure6,axiom,
! [A] :
( m1_subset_1(A,k3_supinf_1)
=> ( ~ v1_xboole_0(k2_measure6(A))
& m1_subset_1(k2_measure6(A),k1_zfmisc_1(k3_supinf_1)) ) ) ).
fof(dt_k3_measure6,axiom,
! [A] :
( m1_subset_1(A,k3_supinf_1)
=> m1_subset_1(k3_measure6(A),k3_supinf_1) ) ).
fof(dt_k4_measure6,axiom,
m1_subset_1(k4_measure6,k3_supinf_1) ).
fof(redefinition_k4_measure6,axiom,
k4_measure6 = k2_supinf_1 ).
fof(dt_k5_measure6,axiom,
m1_subset_1(k5_measure6,k3_supinf_1) ).
fof(redefinition_k5_measure6,axiom,
k5_measure6 = k1_supinf_1 ).
fof(dt_k6_measure6,axiom,
! [A] :
( ( v5_measure5(A)
& m1_subset_1(A,k1_zfmisc_1(k6_supinf_1)) )
=> m1_subset_1(k6_measure6(A),k3_supinf_1) ) ).
fof(dt_k7_measure6,axiom,
! [A] :
( ( v5_measure5(A)
& m1_subset_1(A,k1_zfmisc_1(k6_supinf_1)) )
=> m1_subset_1(k7_measure6(A),k3_supinf_1) ) ).
fof(dt_k8_measure6,axiom,
! [A,B] :
( ( m1_subset_1(A,k1_zfmisc_1(k6_supinf_1))
& v1_xreal_0(B) )
=> m1_subset_1(k8_measure6(A,B),k1_zfmisc_1(k6_supinf_1)) ) ).
%------------------------------------------------------------------------------