SET007 Axioms: SET007+157.ax
%------------------------------------------------------------------------------
% File : SET007+157 : TPTP v8.2.0. Released v3.4.0.
% Domain : Set Theory
% Axioms : Completeness of the sigma-Additive Measure
% 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 : measure3 [Urb08]
% Status : Satisfiable
% Syntax : Number of formulae : 37 ( 1 unt; 0 def)
% Number of atoms : 412 ( 43 equ)
% Maximal formula atoms : 19 ( 11 avg)
% Number of connectives : 424 ( 49 ~; 7 |; 226 &)
% ( 8 <=>; 134 =>; 0 <=; 0 <~>)
% Maximal formula depth : 16 ( 12 avg)
% Maximal term depth : 5 ( 1 avg)
% Number of predicates : 22 ( 20 usr; 1 prp; 0-4 aty)
% Number of functors : 31 ( 31 usr; 9 con; 0-5 aty)
% Number of variables : 171 ( 163 !; 8 ?)
% 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_measure3,axiom,
! [A,B,C] :
( ( ~ v1_xboole_0(B)
& v1_prob_1(B,A)
& v1_measure1(B,A)
& m1_subset_1(B,k1_zfmisc_1(k1_zfmisc_1(A)))
& m3_measure1(C,A,B) )
=> ( ~ v1_xboole_0(k2_measure3(A,B,C))
& v1_finsub_1(k2_measure3(A,B,C))
& v2_finsub_1(k2_measure3(A,B,C))
& v1_prob_1(k2_measure3(A,B,C),A)
& v1_measure1(k2_measure3(A,B,C),A) ) ) ).
fof(t1_measure3,axiom,
! [A] :
( m1_subset_1(A,k3_supinf_1)
=> ~ ( ~ r1_supinf_1(A,k5_supinf_1)
& ~ r1_supinf_1(k4_supinf_1,A)
& ~ m1_subset_1(A,k1_numbers) ) ) ).
fof(t2_measure3,axiom,
! [A] :
( m1_subset_1(A,k3_supinf_1)
=> ~ ( A != k5_supinf_1
& A != k4_supinf_1
& ~ m1_subset_1(A,k1_numbers) ) ) ).
fof(t3_measure3,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] :
( ( 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(k1_measure1(k5_numbers,k18_supinf_2(A),C),k1_measure1(k5_numbers,k18_supinf_2(B),C)) )
=> r1_supinf_1(k19_supinf_2(A),k19_supinf_2(B)) ) ) ) ).
fof(t4_measure3,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] :
( ( 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)
=> k1_measure1(k5_numbers,k18_supinf_2(A),C) = k1_measure1(k5_numbers,k18_supinf_2(B),C) )
=> k19_supinf_2(A) = k19_supinf_2(B) ) ) ) ).
fof(t5_measure3,axiom,
! [A,B] :
( ( ~ v1_xboole_0(B)
& v1_prob_1(B,A)
& v1_measure1(B,A)
& m1_subset_1(B,k1_zfmisc_1(k1_zfmisc_1(A))) )
=> ! [C] :
( m3_measure1(C,A,B)
=> ! [D] :
( ( v1_funct_1(D)
& v1_funct_2(D,k5_numbers,B)
& m2_relset_1(D,k5_numbers,B) )
=> ! [E] :
( m2_subset_1(E,k1_zfmisc_1(A),B)
=> ( ( r1_tarski(k1_measure2(A,B,k1_measure3(A,B,D)),E)
& ! [F] :
( m2_subset_1(F,k1_numbers,k5_numbers)
=> r1_tarski(E,k8_funct_2(k5_numbers,B,D,F)) ) )
=> k1_measure1(B,C,E) = k1_measure1(B,C,k1_measure2(A,B,k1_measure3(A,B,D))) ) ) ) ) ) ).
fof(t6_measure3,axiom,
! [A,B] :
( ( ~ v1_xboole_0(B)
& v1_prob_1(B,A)
& v1_measure1(B,A)
& m1_subset_1(B,k1_zfmisc_1(k1_zfmisc_1(A))) )
=> ! [C] :
( ( v1_funct_1(C)
& v1_funct_2(C,k5_numbers,B)
& m2_relset_1(C,k5_numbers,B) )
=> ! [D] :
( ( v1_funct_1(D)
& v1_funct_2(D,k5_numbers,B)
& m2_relset_1(D,k5_numbers,B) )
=> ( ( k8_funct_2(k5_numbers,B,C,np__0) = k1_xboole_0
& ! [E] :
( m2_subset_1(E,k1_numbers,k5_numbers)
=> ( k8_funct_2(k5_numbers,B,C,k1_nat_1(E,np__1)) = k4_measure1(A,B,k8_funct_2(k5_numbers,B,D,np__0),k8_funct_2(k5_numbers,B,D,E))
& r1_tarski(k8_funct_2(k5_numbers,B,D,k1_nat_1(E,np__1)),k8_funct_2(k5_numbers,B,D,E)) ) ) )
=> k2_measure2(A,B,k1_measure3(A,B,C)) = k4_measure1(A,B,k8_funct_2(k5_numbers,B,D,np__0),k1_measure2(A,B,k1_measure3(A,B,D))) ) ) ) ) ).
fof(t7_measure3,axiom,
! [A,B] :
( ( ~ v1_xboole_0(B)
& v1_prob_1(B,A)
& v1_measure1(B,A)
& m1_subset_1(B,k1_zfmisc_1(k1_zfmisc_1(A))) )
=> ! [C] :
( ( v1_funct_1(C)
& v1_funct_2(C,k5_numbers,B)
& m2_relset_1(C,k5_numbers,B) )
=> ! [D] :
( ( v1_funct_1(D)
& v1_funct_2(D,k5_numbers,B)
& m2_relset_1(D,k5_numbers,B) )
=> ( ( k8_funct_2(k5_numbers,B,C,np__0) = k1_xboole_0
& ! [E] :
( m2_subset_1(E,k1_numbers,k5_numbers)
=> ( k8_funct_2(k5_numbers,B,C,k1_nat_1(E,np__1)) = k4_measure1(A,B,k8_funct_2(k5_numbers,B,D,np__0),k8_funct_2(k5_numbers,B,D,E))
& r1_tarski(k8_funct_2(k5_numbers,B,D,k1_nat_1(E,np__1)),k8_funct_2(k5_numbers,B,D,E)) ) ) )
=> k1_measure2(A,B,k1_measure3(A,B,D)) = k4_measure1(A,B,k8_funct_2(k5_numbers,B,D,np__0),k2_measure2(A,B,k1_measure3(A,B,C))) ) ) ) ) ).
fof(t8_measure3,axiom,
! [A,B] :
( ( ~ v1_xboole_0(B)
& v1_prob_1(B,A)
& v1_measure1(B,A)
& m1_subset_1(B,k1_zfmisc_1(k1_zfmisc_1(A))) )
=> ! [C] :
( m3_measure1(C,A,B)
=> ! [D] :
( ( v1_funct_1(D)
& v1_funct_2(D,k5_numbers,B)
& m2_relset_1(D,k5_numbers,B) )
=> ! [E] :
( ( v1_funct_1(E)
& v1_funct_2(E,k5_numbers,B)
& m2_relset_1(E,k5_numbers,B) )
=> ( ( k8_funct_2(k5_numbers,B,D,np__0) = k1_xboole_0
& ! [F] :
( m2_subset_1(F,k1_numbers,k5_numbers)
=> ( k8_funct_2(k5_numbers,B,D,k1_nat_1(F,np__1)) = k4_measure1(A,B,k8_funct_2(k5_numbers,B,E,np__0),k8_funct_2(k5_numbers,B,E,F))
& r1_tarski(k8_funct_2(k5_numbers,B,E,k1_nat_1(F,np__1)),k8_funct_2(k5_numbers,B,E,F)) ) ) )
=> ( r1_supinf_1(k4_supinf_1,k1_measure1(B,C,k8_funct_2(k5_numbers,B,E,np__0)))
| k1_measure1(B,C,k1_measure2(A,B,k1_measure3(A,B,E))) = k4_supinf_2(k1_measure1(B,C,k8_funct_2(k5_numbers,B,E,np__0)),k1_measure1(B,C,k2_measure2(A,B,k1_measure3(A,B,D)))) ) ) ) ) ) ) ).
fof(t9_measure3,axiom,
! [A,B] :
( ( ~ v1_xboole_0(B)
& v1_prob_1(B,A)
& v1_measure1(B,A)
& m1_subset_1(B,k1_zfmisc_1(k1_zfmisc_1(A))) )
=> ! [C] :
( m3_measure1(C,A,B)
=> ! [D] :
( ( v1_funct_1(D)
& v1_funct_2(D,k5_numbers,B)
& m2_relset_1(D,k5_numbers,B) )
=> ! [E] :
( ( v1_funct_1(E)
& v1_funct_2(E,k5_numbers,B)
& m2_relset_1(E,k5_numbers,B) )
=> ( ( k8_funct_2(k5_numbers,B,D,np__0) = k1_xboole_0
& ! [F] :
( m2_subset_1(F,k1_numbers,k5_numbers)
=> ( k8_funct_2(k5_numbers,B,D,k1_nat_1(F,np__1)) = k4_measure1(A,B,k8_funct_2(k5_numbers,B,E,np__0),k8_funct_2(k5_numbers,B,E,F))
& r1_tarski(k8_funct_2(k5_numbers,B,E,k1_nat_1(F,np__1)),k8_funct_2(k5_numbers,B,E,F)) ) ) )
=> ( r1_supinf_1(k4_supinf_1,k1_measure1(B,C,k8_funct_2(k5_numbers,B,E,np__0)))
| k1_measure1(B,C,k2_measure2(A,B,k1_measure3(A,B,D))) = k4_supinf_2(k1_measure1(B,C,k8_funct_2(k5_numbers,B,E,np__0)),k1_measure1(B,C,k1_measure2(A,B,k1_measure3(A,B,E)))) ) ) ) ) ) ) ).
fof(t10_measure3,axiom,
! [A,B] :
( ( ~ v1_xboole_0(B)
& v1_prob_1(B,A)
& v1_measure1(B,A)
& m1_subset_1(B,k1_zfmisc_1(k1_zfmisc_1(A))) )
=> ! [C] :
( m3_measure1(C,A,B)
=> ! [D] :
( ( v1_funct_1(D)
& v1_funct_2(D,k5_numbers,B)
& m2_relset_1(D,k5_numbers,B) )
=> ! [E] :
( ( v1_funct_1(E)
& v1_funct_2(E,k5_numbers,B)
& m2_relset_1(E,k5_numbers,B) )
=> ( ( k8_funct_2(k5_numbers,B,D,np__0) = k1_xboole_0
& ! [F] :
( m2_subset_1(F,k1_numbers,k5_numbers)
=> ( k8_funct_2(k5_numbers,B,D,k1_nat_1(F,np__1)) = k4_measure1(A,B,k8_funct_2(k5_numbers,B,E,np__0),k8_funct_2(k5_numbers,B,E,F))
& r1_tarski(k8_funct_2(k5_numbers,B,E,k1_nat_1(F,np__1)),k8_funct_2(k5_numbers,B,E,F)) ) ) )
=> ( r1_supinf_1(k4_supinf_1,k1_measure1(B,C,k8_funct_2(k5_numbers,B,E,np__0)))
| k1_measure1(B,C,k1_measure2(A,B,k1_measure3(A,B,E))) = k4_supinf_2(k1_measure1(B,C,k8_funct_2(k5_numbers,B,E,np__0)),k9_supinf_1(k17_supinf_2(k7_funct_2(k5_numbers,B,k3_supinf_1,D,C)))) ) ) ) ) ) ) ).
fof(t11_measure3,axiom,
! [A,B] :
( ( ~ v1_xboole_0(B)
& v1_prob_1(B,A)
& v1_measure1(B,A)
& m1_subset_1(B,k1_zfmisc_1(k1_zfmisc_1(A))) )
=> ! [C] :
( m3_measure1(C,A,B)
=> ! [D] :
( ( v1_funct_1(D)
& v1_funct_2(D,k5_numbers,B)
& m2_relset_1(D,k5_numbers,B) )
=> ! [E] :
( ( v1_funct_1(E)
& v1_funct_2(E,k5_numbers,B)
& m2_relset_1(E,k5_numbers,B) )
=> ( ( k8_funct_2(k5_numbers,B,D,np__0) = k1_xboole_0
& ! [F] :
( m2_subset_1(F,k1_numbers,k5_numbers)
=> ( k8_funct_2(k5_numbers,B,D,k1_nat_1(F,np__1)) = k4_measure1(A,B,k8_funct_2(k5_numbers,B,E,np__0),k8_funct_2(k5_numbers,B,E,F))
& r1_tarski(k8_funct_2(k5_numbers,B,E,k1_nat_1(F,np__1)),k8_funct_2(k5_numbers,B,E,F)) ) ) )
=> ( r1_supinf_1(k4_supinf_1,k1_measure1(B,C,k8_funct_2(k5_numbers,B,E,np__0)))
| ( m1_subset_1(k9_supinf_1(k17_supinf_2(k7_funct_2(k5_numbers,B,k3_supinf_1,D,C))),k1_numbers)
& m1_subset_1(k1_measure1(B,C,k8_funct_2(k5_numbers,B,E,np__0)),k1_numbers)
& m1_subset_1(k10_supinf_1(k17_supinf_2(k7_funct_2(k5_numbers,B,k3_supinf_1,E,C))),k1_numbers) ) ) ) ) ) ) ) ).
fof(t12_measure3,axiom,
! [A,B] :
( ( ~ v1_xboole_0(B)
& v1_prob_1(B,A)
& v1_measure1(B,A)
& m1_subset_1(B,k1_zfmisc_1(k1_zfmisc_1(A))) )
=> ! [C] :
( m3_measure1(C,A,B)
=> ! [D] :
( ( v1_funct_1(D)
& v1_funct_2(D,k5_numbers,B)
& m2_relset_1(D,k5_numbers,B) )
=> ! [E] :
( ( v1_funct_1(E)
& v1_funct_2(E,k5_numbers,B)
& m2_relset_1(E,k5_numbers,B) )
=> ( ( k8_funct_2(k5_numbers,B,D,np__0) = k1_xboole_0
& ! [F] :
( m2_subset_1(F,k1_numbers,k5_numbers)
=> ( k8_funct_2(k5_numbers,B,D,k1_nat_1(F,np__1)) = k4_measure1(A,B,k8_funct_2(k5_numbers,B,E,np__0),k8_funct_2(k5_numbers,B,E,F))
& r1_tarski(k8_funct_2(k5_numbers,B,E,k1_nat_1(F,np__1)),k8_funct_2(k5_numbers,B,E,F)) ) ) )
=> ( r1_supinf_1(k4_supinf_1,k1_measure1(B,C,k8_funct_2(k5_numbers,B,E,np__0)))
| k9_supinf_1(k17_supinf_2(k7_funct_2(k5_numbers,B,k3_supinf_1,D,C))) = k4_supinf_2(k1_measure1(B,C,k8_funct_2(k5_numbers,B,E,np__0)),k10_supinf_1(k17_supinf_2(k7_funct_2(k5_numbers,B,k3_supinf_1,E,C)))) ) ) ) ) ) ) ).
fof(t13_measure3,axiom,
! [A,B] :
( ( ~ v1_xboole_0(B)
& v1_prob_1(B,A)
& v1_measure1(B,A)
& m1_subset_1(B,k1_zfmisc_1(k1_zfmisc_1(A))) )
=> ! [C] :
( m3_measure1(C,A,B)
=> ! [D] :
( ( v1_funct_1(D)
& v1_funct_2(D,k5_numbers,B)
& m2_relset_1(D,k5_numbers,B) )
=> ! [E] :
( ( v1_funct_1(E)
& v1_funct_2(E,k5_numbers,B)
& m2_relset_1(E,k5_numbers,B) )
=> ( ( k8_funct_2(k5_numbers,B,D,np__0) = k1_xboole_0
& ! [F] :
( m2_subset_1(F,k1_numbers,k5_numbers)
=> ( k8_funct_2(k5_numbers,B,D,k1_nat_1(F,np__1)) = k4_measure1(A,B,k8_funct_2(k5_numbers,B,E,np__0),k8_funct_2(k5_numbers,B,E,F))
& r1_tarski(k8_funct_2(k5_numbers,B,E,k1_nat_1(F,np__1)),k8_funct_2(k5_numbers,B,E,F)) ) ) )
=> ( r1_supinf_1(k4_supinf_1,k1_measure1(B,C,k8_funct_2(k5_numbers,B,E,np__0)))
| k10_supinf_1(k17_supinf_2(k7_funct_2(k5_numbers,B,k3_supinf_1,E,C))) = k4_supinf_2(k1_measure1(B,C,k8_funct_2(k5_numbers,B,E,np__0)),k9_supinf_1(k17_supinf_2(k7_funct_2(k5_numbers,B,k3_supinf_1,D,C)))) ) ) ) ) ) ) ).
fof(t14_measure3,axiom,
! [A,B] :
( ( ~ v1_xboole_0(B)
& v1_prob_1(B,A)
& v1_measure1(B,A)
& m1_subset_1(B,k1_zfmisc_1(k1_zfmisc_1(A))) )
=> ! [C] :
( m3_measure1(C,A,B)
=> ! [D] :
( ( v1_funct_1(D)
& v1_funct_2(D,k5_numbers,B)
& m2_relset_1(D,k5_numbers,B) )
=> ( ! [E] :
( m2_subset_1(E,k1_numbers,k5_numbers)
=> r1_tarski(k8_funct_2(k5_numbers,B,D,k1_nat_1(E,np__1)),k8_funct_2(k5_numbers,B,D,E)) )
=> ( r1_supinf_1(k4_supinf_1,k1_measure1(B,C,k8_funct_2(k5_numbers,B,D,np__0)))
| k1_measure1(B,C,k1_measure2(A,B,k1_measure3(A,B,D))) = k10_supinf_1(k17_supinf_2(k7_funct_2(k5_numbers,B,k3_supinf_1,D,C))) ) ) ) ) ) ).
fof(t15_measure3,axiom,
! [A,B] :
( ( ~ v1_xboole_0(B)
& v1_prob_1(B,A)
& v1_measure1(B,A)
& m1_subset_1(B,k1_zfmisc_1(k1_zfmisc_1(A))) )
=> ! [C] :
( m1_measure1(C,A,B)
=> ! [D] :
( m1_measure2(D,A,B)
=> ! [E] :
( ( v1_funct_1(E)
& v1_funct_2(E,k5_numbers,B)
& v1_prob_2(E)
& m2_relset_1(E,k5_numbers,B) )
=> ( D = k1_measure3(A,B,E)
=> r1_supinf_1(k19_supinf_2(k7_funct_2(k5_numbers,B,k3_supinf_1,E,C)),k1_measure1(B,C,k2_measure2(A,B,D))) ) ) ) ) ) ).
fof(t16_measure3,axiom,
! [A,B] :
( ( ~ v1_xboole_0(B)
& v1_prob_1(B,A)
& v1_measure1(B,A)
& m1_subset_1(B,k1_zfmisc_1(k1_zfmisc_1(A))) )
=> ! [C] :
( m1_measure1(C,A,B)
=> ! [D] :
( ( v1_funct_1(D)
& v1_funct_2(D,k5_numbers,B)
& v1_prob_2(D)
& m2_relset_1(D,k5_numbers,B) )
=> r1_supinf_1(k19_supinf_2(k7_funct_2(k5_numbers,B,k3_supinf_1,D,C)),k1_measure1(B,C,k2_measure2(A,B,k1_measure3(A,B,D)))) ) ) ) ).
fof(t17_measure3,axiom,
! [A,B] :
( ( ~ v1_xboole_0(B)
& v1_prob_1(B,A)
& v1_measure1(B,A)
& m1_subset_1(B,k1_zfmisc_1(k1_zfmisc_1(A))) )
=> ! [C] :
( m1_measure1(C,A,B)
=> ( ! [D] :
( ( v1_funct_1(D)
& v1_funct_2(D,k5_numbers,B)
& v1_prob_2(D)
& m2_relset_1(D,k5_numbers,B) )
=> r1_supinf_1(k1_measure1(B,C,k2_measure2(A,B,k1_measure3(A,B,D))),k19_supinf_2(k7_funct_2(k5_numbers,B,k3_supinf_1,D,C))) )
=> m3_measure1(C,A,B) ) ) ) ).
fof(d1_measure3,axiom,
$true ).
fof(d2_measure3,axiom,
! [A,B] :
( ( ~ v1_xboole_0(B)
& v1_prob_1(B,A)
& v1_measure1(B,A)
& m1_subset_1(B,k1_zfmisc_1(k1_zfmisc_1(A))) )
=> ! [C] :
( m3_measure1(C,A,B)
=> ( r1_measure3(A,B,C)
<=> ! [D] :
( m1_subset_1(D,k1_zfmisc_1(A))
=> ! [E] :
( ( r2_hidden(E,B)
& r1_tarski(D,E)
& k1_funct_1(C,E) = k1_supinf_2 )
=> r2_hidden(D,B) ) ) ) ) ) ).
fof(d3_measure3,axiom,
! [A,B] :
( ( ~ v1_xboole_0(B)
& v1_prob_1(B,A)
& v1_measure1(B,A)
& m1_subset_1(B,k1_zfmisc_1(k1_zfmisc_1(A))) )
=> ! [C] :
( m3_measure1(C,A,B)
=> ! [D] :
( m1_subset_1(D,k1_zfmisc_1(A))
=> ( m1_measure3(D,A,B,C)
<=> ? [E] :
( r2_hidden(E,B)
& r1_tarski(D,E)
& k1_funct_1(C,E) = k1_supinf_2 ) ) ) ) ) ).
fof(d4_measure3,axiom,
! [A,B] :
( ( ~ v1_xboole_0(B)
& v1_prob_1(B,A)
& v1_measure1(B,A)
& m1_subset_1(B,k1_zfmisc_1(k1_zfmisc_1(A))) )
=> ! [C] :
( m3_measure1(C,A,B)
=> ! [D] :
( ( ~ v1_xboole_0(D)
& m1_subset_1(D,k1_zfmisc_1(k1_zfmisc_1(A))) )
=> ( D = k2_measure3(A,B,C)
<=> ! [E] :
( r2_hidden(E,D)
<=> ? [F] :
( r2_hidden(F,B)
& ? [G] :
( m1_measure3(G,A,B,C)
& E = k2_xboole_0(F,G) ) ) ) ) ) ) ) ).
fof(d5_measure3,axiom,
! [A,B] :
( ( ~ v1_xboole_0(B)
& v1_prob_1(B,A)
& v1_measure1(B,A)
& m1_subset_1(B,k1_zfmisc_1(k1_zfmisc_1(A))) )
=> ! [C] :
( m3_measure1(C,A,B)
=> ! [D] :
( m2_subset_1(D,k1_zfmisc_1(A),k2_measure3(A,B,C))
=> ! [E] :
( ( ~ v1_xboole_0(E)
& m1_subset_1(E,k1_zfmisc_1(k1_zfmisc_1(A))) )
=> ( E = k3_measure3(A,B,C,D)
<=> ! [F] :
( r2_hidden(F,E)
<=> ( r2_hidden(F,B)
& r1_tarski(F,D)
& m1_measure3(k4_xboole_0(D,F),A,B,C) ) ) ) ) ) ) ) ).
fof(t18_measure3,axiom,
! [A,B] :
( ( ~ v1_xboole_0(B)
& v1_prob_1(B,A)
& v1_measure1(B,A)
& m1_subset_1(B,k1_zfmisc_1(k1_zfmisc_1(A))) )
=> ! [C] :
( m3_measure1(C,A,B)
=> ! [D] :
( ( v1_funct_1(D)
& v1_funct_2(D,k5_numbers,k2_measure3(A,B,C))
& m2_relset_1(D,k5_numbers,k2_measure3(A,B,C)) )
=> ? [E] :
( v1_funct_1(E)
& v1_funct_2(E,k5_numbers,B)
& m2_relset_1(E,k5_numbers,B)
& ! [F] :
( m2_subset_1(F,k1_numbers,k5_numbers)
=> r2_hidden(k8_funct_2(k5_numbers,B,E,F),k3_measure3(A,B,C,k8_funct_2(k5_numbers,k2_measure3(A,B,C),D,F))) ) ) ) ) ) ).
fof(t19_measure3,axiom,
! [A,B] :
( ( ~ v1_xboole_0(B)
& v1_prob_1(B,A)
& v1_measure1(B,A)
& m1_subset_1(B,k1_zfmisc_1(k1_zfmisc_1(A))) )
=> ! [C] :
( m3_measure1(C,A,B)
=> ! [D] :
( ( v1_funct_1(D)
& v1_funct_2(D,k5_numbers,k2_measure3(A,B,C))
& m2_relset_1(D,k5_numbers,k2_measure3(A,B,C)) )
=> ! [E] :
( ( v1_funct_1(E)
& v1_funct_2(E,k5_numbers,B)
& m2_relset_1(E,k5_numbers,B) )
=> ? [F] :
( v1_funct_1(F)
& v1_funct_2(F,k5_numbers,k1_zfmisc_1(A))
& m2_relset_1(F,k5_numbers,k1_zfmisc_1(A))
& ! [G] :
( m2_subset_1(G,k1_numbers,k5_numbers)
=> k8_funct_2(k5_numbers,k1_zfmisc_1(A),F,G) = k4_xboole_0(k8_funct_2(k5_numbers,k2_measure3(A,B,C),D,G),k8_funct_2(k5_numbers,B,E,G)) ) ) ) ) ) ) ).
fof(t20_measure3,axiom,
! [A,B] :
( ( ~ v1_xboole_0(B)
& v1_prob_1(B,A)
& v1_measure1(B,A)
& m1_subset_1(B,k1_zfmisc_1(k1_zfmisc_1(A))) )
=> ! [C] :
( m3_measure1(C,A,B)
=> ! [D] :
( ( v1_funct_1(D)
& v1_funct_2(D,k5_numbers,k1_zfmisc_1(A))
& m2_relset_1(D,k5_numbers,k1_zfmisc_1(A)) )
=> ~ ( ! [E] :
( m2_subset_1(E,k1_numbers,k5_numbers)
=> m1_measure3(k8_funct_2(k5_numbers,k1_zfmisc_1(A),D,E),A,B,C) )
& ! [E] :
( ( v1_funct_1(E)
& v1_funct_2(E,k5_numbers,B)
& m2_relset_1(E,k5_numbers,B) )
=> ~ ! [F] :
( m2_subset_1(F,k1_numbers,k5_numbers)
=> ( r1_tarski(k8_funct_2(k5_numbers,k1_zfmisc_1(A),D,F),k8_funct_2(k5_numbers,B,E,F))
& k1_measure1(B,C,k8_funct_2(k5_numbers,B,E,F)) = k1_supinf_2 ) ) ) ) ) ) ) ).
fof(t21_measure3,axiom,
! [A,B] :
( ( ~ v1_xboole_0(B)
& v1_prob_1(B,A)
& v1_measure1(B,A)
& m1_subset_1(B,k1_zfmisc_1(k1_zfmisc_1(A))) )
=> ! [C] :
( m3_measure1(C,A,B)
=> ! [D] :
( ( ~ v1_xboole_0(D)
& m1_subset_1(D,k1_zfmisc_1(k1_zfmisc_1(A))) )
=> ( ! [E] :
( r2_hidden(E,D)
<=> ? [F] :
( r2_hidden(F,B)
& ? [G] :
( m1_measure3(G,A,B,C)
& E = k2_xboole_0(F,G) ) ) )
=> ( ~ v1_xboole_0(D)
& v1_prob_1(D,A)
& v1_measure1(D,A)
& m1_subset_1(D,k1_zfmisc_1(k1_zfmisc_1(A))) ) ) ) ) ) ).
fof(t22_measure3,axiom,
! [A,B] :
( ( ~ v1_xboole_0(B)
& v1_prob_1(B,A)
& v1_measure1(B,A)
& m1_subset_1(B,k1_zfmisc_1(k1_zfmisc_1(A))) )
=> ! [C] :
( m3_measure1(C,A,B)
=> ! [D,E] :
( ( r2_hidden(D,B)
& r2_hidden(E,B) )
=> ! [F] :
( m1_measure3(F,A,B,C)
=> ! [G] :
( m1_measure3(G,A,B,C)
=> ( k2_xboole_0(D,F) = k2_xboole_0(E,G)
=> k1_funct_1(C,D) = k1_funct_1(C,E) ) ) ) ) ) ) ).
fof(d6_measure3,axiom,
! [A,B] :
( ( ~ v1_xboole_0(B)
& v1_prob_1(B,A)
& v1_measure1(B,A)
& m1_subset_1(B,k1_zfmisc_1(k1_zfmisc_1(A))) )
=> ! [C] :
( m3_measure1(C,A,B)
=> ! [D] :
( m3_measure1(D,A,k2_measure3(A,B,C))
=> ( D = k4_measure3(A,B,C)
<=> ! [E] :
( r2_hidden(E,B)
=> ! [F] :
( m1_measure3(F,A,B,C)
=> k1_funct_1(D,k2_xboole_0(E,F)) = k1_funct_1(C,E) ) ) ) ) ) ) ).
fof(t23_measure3,axiom,
! [A,B] :
( ( ~ v1_xboole_0(B)
& v1_prob_1(B,A)
& v1_measure1(B,A)
& m1_subset_1(B,k1_zfmisc_1(k1_zfmisc_1(A))) )
=> ! [C] :
( m3_measure1(C,A,B)
=> r1_measure3(A,k2_measure3(A,B,C),k4_measure3(A,B,C)) ) ) ).
fof(dt_m1_measure3,axiom,
! [A,B,C] :
( ( ~ v1_xboole_0(B)
& v1_prob_1(B,A)
& v1_measure1(B,A)
& m1_subset_1(B,k1_zfmisc_1(k1_zfmisc_1(A)))
& m3_measure1(C,A,B) )
=> ! [D] :
( m1_measure3(D,A,B,C)
=> m1_subset_1(D,k1_zfmisc_1(A)) ) ) ).
fof(existence_m1_measure3,axiom,
! [A,B,C] :
( ( ~ v1_xboole_0(B)
& v1_prob_1(B,A)
& v1_measure1(B,A)
& m1_subset_1(B,k1_zfmisc_1(k1_zfmisc_1(A)))
& m3_measure1(C,A,B) )
=> ? [D] : m1_measure3(D,A,B,C) ) ).
fof(dt_k1_measure3,axiom,
! [A,B,C] :
( ( ~ v1_xboole_0(B)
& v1_prob_1(B,A)
& v1_measure1(B,A)
& m1_subset_1(B,k1_zfmisc_1(k1_zfmisc_1(A)))
& v1_funct_1(C)
& v1_funct_2(C,k5_numbers,B)
& m1_relset_1(C,k5_numbers,B) )
=> m1_measure2(k1_measure3(A,B,C),A,B) ) ).
fof(redefinition_k1_measure3,axiom,
! [A,B,C] :
( ( ~ v1_xboole_0(B)
& v1_prob_1(B,A)
& v1_measure1(B,A)
& m1_subset_1(B,k1_zfmisc_1(k1_zfmisc_1(A)))
& v1_funct_1(C)
& v1_funct_2(C,k5_numbers,B)
& m1_relset_1(C,k5_numbers,B) )
=> k1_measure3(A,B,C) = k2_relat_1(C) ) ).
fof(dt_k2_measure3,axiom,
! [A,B,C] :
( ( ~ v1_xboole_0(B)
& v1_prob_1(B,A)
& v1_measure1(B,A)
& m1_subset_1(B,k1_zfmisc_1(k1_zfmisc_1(A)))
& m3_measure1(C,A,B) )
=> ( ~ v1_xboole_0(k2_measure3(A,B,C))
& m1_subset_1(k2_measure3(A,B,C),k1_zfmisc_1(k1_zfmisc_1(A))) ) ) ).
fof(dt_k3_measure3,axiom,
! [A,B,C,D] :
( ( ~ v1_xboole_0(B)
& v1_prob_1(B,A)
& v1_measure1(B,A)
& m1_subset_1(B,k1_zfmisc_1(k1_zfmisc_1(A)))
& m3_measure1(C,A,B)
& m1_subset_1(D,k2_measure3(A,B,C)) )
=> ( ~ v1_xboole_0(k3_measure3(A,B,C,D))
& m1_subset_1(k3_measure3(A,B,C,D),k1_zfmisc_1(k1_zfmisc_1(A))) ) ) ).
fof(dt_k4_measure3,axiom,
! [A,B,C] :
( ( ~ v1_xboole_0(B)
& v1_prob_1(B,A)
& v1_measure1(B,A)
& m1_subset_1(B,k1_zfmisc_1(k1_zfmisc_1(A)))
& m3_measure1(C,A,B) )
=> m3_measure1(k4_measure3(A,B,C),A,k2_measure3(A,B,C)) ) ).
%------------------------------------------------------------------------------