SET007 Axioms: SET007+156.ax
%------------------------------------------------------------------------------
% File : SET007+156 : TPTP v9.0.0. Released v3.4.0.
% Domain : Set Theory
% Axioms : Several Properties 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 : measure2 [Urb08]
% Status : Satisfiable
% Syntax : Number of formulae : 38 ( 4 unt; 0 def)
% Number of atoms : 390 ( 42 equ)
% Maximal formula atoms : 23 ( 10 avg)
% Number of connectives : 390 ( 38 ~; 0 |; 214 &)
% ( 4 <=>; 134 =>; 0 <=; 0 <~>)
% Maximal formula depth : 17 ( 10 avg)
% Maximal term depth : 4 ( 1 avg)
% Number of predicates : 24 ( 22 usr; 1 prp; 0-4 aty)
% Number of functors : 26 ( 26 usr; 6 con; 0-5 aty)
% Number of variables : 161 ( 149 !; 12 ?)
% 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(rc1_measure2,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_measure2(C,A,B)
& ~ v1_xboole_0(C)
& v1_card_4(C)
& v1_measure2(C,A,B) ) ) ).
fof(rc2_measure2,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_measure2(C,A,B)
& ~ v1_xboole_0(C)
& v1_card_4(C)
& v2_measure2(C,A,B) ) ) ).
fof(t1_measure2,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) )
=> v6_supinf_2(k7_funct_2(k5_numbers,B,k3_supinf_1,D,C),k5_numbers) ) ) ) ).
fof(d1_measure2,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_xboole_0(C)
& v1_card_4(C)
& m1_subset_1(C,k1_zfmisc_1(k1_zfmisc_1(A))) )
=> ( m1_measure2(C,A,B)
<=> r1_tarski(C,B) ) ) ) ).
fof(t2_measure2,axiom,
$true ).
fof(t3_measure2,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_measure2(C,A,B)
=> ( r2_hidden(k6_setfam_1(A,C),B)
& r2_hidden(k5_setfam_1(A,C),B) ) ) ) ).
fof(t4_measure2,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,D,np__0) = k8_funct_2(k5_numbers,B,C,np__0)
& ! [E] :
( m2_subset_1(E,k1_numbers,k5_numbers)
=> k8_funct_2(k5_numbers,B,D,k1_nat_1(E,np__1)) = k4_measure1(A,B,k8_funct_2(k5_numbers,B,C,k1_nat_1(E,np__1)),k8_funct_2(k5_numbers,B,C,E)) ) ) ) ) ).
fof(t5_measure2,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,D,np__0) = k8_funct_2(k5_numbers,B,C,np__0)
& ! [E] :
( m2_subset_1(E,k1_numbers,k5_numbers)
=> k8_funct_2(k5_numbers,B,D,k1_nat_1(E,np__1)) = k2_measure1(A,B,k8_funct_2(k5_numbers,B,C,k1_nat_1(E,np__1)),k8_funct_2(k5_numbers,B,D,E)) ) ) ) ) ).
fof(t6_measure2,axiom,
! [A,B] :
( ( ~ v1_xboole_0(B)
& 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,D,np__0) = k8_funct_2(k5_numbers,B,C,np__0)
& ! [E] :
( m2_subset_1(E,k1_numbers,k5_numbers)
=> k8_funct_2(k5_numbers,B,D,k1_nat_1(E,np__1)) = k2_xboole_0(k8_funct_2(k5_numbers,B,C,k1_nat_1(E,np__1)),k8_funct_2(k5_numbers,B,D,E)) ) )
=> ! [E,F] :
( m2_subset_1(F,k1_numbers,k5_numbers)
=> ( r2_hidden(E,k8_funct_2(k5_numbers,B,D,F))
<=> ? [G] :
( m2_subset_1(G,k1_numbers,k5_numbers)
& r1_xreal_0(G,F)
& r2_hidden(E,k8_funct_2(k5_numbers,B,C,G)) ) ) ) ) ) ) ) ).
fof(t7_measure2,axiom,
! [A,B] :
( ( ~ v1_xboole_0(B)
& 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,D,np__0) = k8_funct_2(k5_numbers,B,C,np__0)
& ! [E] :
( m2_subset_1(E,k1_numbers,k5_numbers)
=> k8_funct_2(k5_numbers,B,D,k1_nat_1(E,np__1)) = k2_xboole_0(k8_funct_2(k5_numbers,B,C,k1_nat_1(E,np__1)),k8_funct_2(k5_numbers,B,D,E)) ) )
=> ! [E] :
( m2_subset_1(E,k1_numbers,k5_numbers)
=> ! [F] :
( m2_subset_1(F,k1_numbers,k5_numbers)
=> ( ~ r1_xreal_0(F,E)
=> r1_tarski(k8_funct_2(k5_numbers,B,D,E),k8_funct_2(k5_numbers,B,D,F)) ) ) ) ) ) ) ) ).
fof(t8_measure2,axiom,
! [A,B] :
( ( ~ v1_xboole_0(B)
& 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) )
=> ! [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) = k8_funct_2(k5_numbers,B,C,np__0)
& ! [F] :
( m2_subset_1(F,k1_numbers,k5_numbers)
=> k8_funct_2(k5_numbers,B,D,k1_nat_1(F,np__1)) = k2_xboole_0(k8_funct_2(k5_numbers,B,C,k1_nat_1(F,np__1)),k8_funct_2(k5_numbers,B,D,F)) )
& k8_funct_2(k5_numbers,B,E,np__0) = k8_funct_2(k5_numbers,B,C,np__0)
& ! [F] :
( m2_subset_1(F,k1_numbers,k5_numbers)
=> k8_funct_2(k5_numbers,B,E,k1_nat_1(F,np__1)) = k4_xboole_0(k8_funct_2(k5_numbers,B,C,k1_nat_1(F,np__1)),k8_funct_2(k5_numbers,B,D,F)) ) )
=> ! [F] :
( m2_subset_1(F,k1_numbers,k5_numbers)
=> ! [G] :
( m2_subset_1(G,k1_numbers,k5_numbers)
=> ( r1_xreal_0(F,G)
=> r1_tarski(k8_funct_2(k5_numbers,B,E,F),k8_funct_2(k5_numbers,B,D,G)) ) ) ) ) ) ) ) ) ).
fof(t9_measure2,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) )
=> ? [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,E,np__0) = k8_funct_2(k5_numbers,B,C,np__0)
& ! [F] :
( m2_subset_1(F,k1_numbers,k5_numbers)
=> k8_funct_2(k5_numbers,B,E,k1_nat_1(F,np__1)) = k4_measure1(A,B,k8_funct_2(k5_numbers,B,C,k1_nat_1(F,np__1)),k8_funct_2(k5_numbers,B,D,F)) ) ) ) ) ) ).
fof(t10_measure2,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,D,np__0) = k1_xboole_0
& ! [E] :
( m2_subset_1(E,k1_numbers,k5_numbers)
=> k8_funct_2(k5_numbers,B,D,k1_nat_1(E,np__1)) = k4_measure1(A,B,k8_funct_2(k5_numbers,B,C,np__0),k8_funct_2(k5_numbers,B,C,E)) ) ) ) ) ).
fof(t11_measure2,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) )
=> ! [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) = k8_funct_2(k5_numbers,B,C,np__0)
& ! [F] :
( m2_subset_1(F,k1_numbers,k5_numbers)
=> k8_funct_2(k5_numbers,B,D,k1_nat_1(F,np__1)) = k2_measure1(A,B,k8_funct_2(k5_numbers,B,C,k1_nat_1(F,np__1)),k8_funct_2(k5_numbers,B,D,F)) )
& k8_funct_2(k5_numbers,B,E,np__0) = k8_funct_2(k5_numbers,B,C,np__0)
& ! [F] :
( m2_subset_1(F,k1_numbers,k5_numbers)
=> k8_funct_2(k5_numbers,B,E,k1_nat_1(F,np__1)) = k4_measure1(A,B,k8_funct_2(k5_numbers,B,C,k1_nat_1(F,np__1)),k8_funct_2(k5_numbers,B,D,F)) ) )
=> ! [F] :
( m2_subset_1(F,k1_numbers,k5_numbers)
=> ! [G] :
( m2_subset_1(G,k1_numbers,k5_numbers)
=> ( ~ r1_xreal_0(G,F)
=> r1_xboole_0(k8_funct_2(k5_numbers,B,E,F),k8_funct_2(k5_numbers,B,E,G)) ) ) ) ) ) ) ) ) ).
fof(t12_measure2,axiom,
$true ).
fof(t13_measure2,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_measure2(D,A,B)
=> ! [E] :
( ( v1_funct_1(E)
& v1_funct_2(E,k5_numbers,B)
& m2_relset_1(E,k5_numbers,B) )
=> ( D = k5_measure1(A,B,E)
=> r1_supinf_1(k1_measure1(B,C,k2_measure2(A,B,D)),k19_supinf_2(k7_funct_2(k5_numbers,B,k3_supinf_1,E,C))) ) ) ) ) ) ).
fof(t14_measure2,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_measure2(C,A,B)
=> ? [D] :
( v1_funct_1(D)
& v1_funct_2(D,k5_numbers,B)
& m2_relset_1(D,k5_numbers,B)
& C = k5_measure1(A,B,D) ) ) ) ).
fof(t15_measure2,axiom,
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A) )
=> ! [B] :
( ( v1_relat_1(B)
& v1_funct_1(B) )
=> ( ( k1_funct_1(B,np__0) = k1_xboole_0
& ! [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
=> ( k1_funct_1(B,k1_nat_1(C,np__1)) = k4_xboole_0(k1_funct_1(A,np__0),k1_funct_1(A,C))
& r1_tarski(k1_funct_1(A,k1_nat_1(C,np__1)),k1_funct_1(A,C)) ) ) )
=> ! [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
=> r1_tarski(k1_funct_1(B,C),k1_funct_1(B,k1_nat_1(C,np__1))) ) ) ) ) ).
fof(t16_measure2,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_measure2(D,A,B)
=> ( ! [E] :
( r2_hidden(E,D)
=> m4_measure1(E,A,B,C) )
=> m4_measure1(k2_measure2(A,B,D),A,B,C) ) ) ) ) ).
fof(t17_measure2,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_measure2(D,A,B)
=> ( ? [E] :
( r2_hidden(E,D)
& m4_measure1(E,A,B,C) )
=> m4_measure1(k1_measure2(A,B,D),A,B,C) ) ) ) ) ).
fof(t18_measure2,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_measure2(D,A,B)
=> ( ! [E] :
( r2_hidden(E,D)
=> m4_measure1(E,A,B,C) )
=> m4_measure1(k1_measure2(A,B,D),A,B,C) ) ) ) ) ).
fof(d2_measure2,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_measure2(C,A,B)
=> ( v1_measure2(C,A,B)
<=> ? [D] :
( v1_funct_1(D)
& v1_funct_2(D,k5_numbers,B)
& m2_relset_1(D,k5_numbers,B)
& C = k5_measure1(A,B,D)
& ! [E] :
( m2_subset_1(E,k1_numbers,k5_numbers)
=> r1_tarski(k8_funct_2(k5_numbers,B,D,E),k8_funct_2(k5_numbers,B,D,k1_nat_1(E,np__1))) ) ) ) ) ) ).
fof(d3_measure2,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_measure2(C,A,B)
=> ( v2_measure2(C,A,B)
<=> ? [D] :
( v1_funct_1(D)
& v1_funct_2(D,k5_numbers,B)
& m2_relset_1(D,k5_numbers,B)
& C = k5_measure1(A,B,D)
& ! [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)) ) ) ) ) ) ).
fof(t19_measure2,axiom,
$true ).
fof(t20_measure2,axiom,
$true ).
fof(t21_measure2,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,D,np__0) = k1_xboole_0
& ! [E] :
( m2_subset_1(E,k1_numbers,k5_numbers)
=> ( k8_funct_2(k5_numbers,B,D,k1_nat_1(E,np__1)) = k4_measure1(A,B,k8_funct_2(k5_numbers,B,C,np__0),k8_funct_2(k5_numbers,B,C,E))
& r1_tarski(k8_funct_2(k5_numbers,B,C,k1_nat_1(E,np__1)),k8_funct_2(k5_numbers,B,C,E)) ) ) )
=> ( v1_measure2(k5_measure1(A,B,D),A,B)
& m1_measure2(k5_measure1(A,B,D),A,B) ) ) ) ) ) ).
fof(t22_measure2,axiom,
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A) )
=> ( ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> r1_tarski(k1_funct_1(A,B),k1_funct_1(A,k1_nat_1(B,np__1))) )
=> ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> ! [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
=> ( r1_xreal_0(C,B)
=> r1_tarski(k1_funct_1(A,C),k1_funct_1(A,B)) ) ) ) ) ) ).
fof(t23_measure2,axiom,
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A) )
=> ! [B] :
( ( v1_relat_1(B)
& v1_funct_1(B) )
=> ( ( k1_funct_1(B,np__0) = k1_funct_1(A,np__0)
& ! [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
=> ( k1_funct_1(B,k1_nat_1(C,np__1)) = k4_xboole_0(k1_funct_1(A,k1_nat_1(C,np__1)),k1_funct_1(A,C))
& r1_tarski(k1_funct_1(A,C),k1_funct_1(A,k1_nat_1(C,np__1))) ) ) )
=> ! [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
=> ! [D] :
( m2_subset_1(D,k1_numbers,k5_numbers)
=> ( ~ r1_xreal_0(D,C)
=> r1_xboole_0(k1_funct_1(B,C),k1_funct_1(B,D)) ) ) ) ) ) ) ).
fof(t24_measure2,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,D,np__0) = k8_funct_2(k5_numbers,B,C,np__0)
& ! [E] :
( m2_subset_1(E,k1_numbers,k5_numbers)
=> ( k8_funct_2(k5_numbers,B,D,k1_nat_1(E,np__1)) = k4_measure1(A,B,k8_funct_2(k5_numbers,B,C,k1_nat_1(E,np__1)),k8_funct_2(k5_numbers,B,C,E))
& r1_tarski(k8_funct_2(k5_numbers,B,C,E),k8_funct_2(k5_numbers,B,C,k1_nat_1(E,np__1))) ) ) )
=> k5_setfam_1(A,k5_measure1(A,B,D)) = k5_setfam_1(A,k5_measure1(A,B,C)) ) ) ) ) ).
fof(t25_measure2,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,D,np__0) = k8_funct_2(k5_numbers,B,C,np__0)
& ! [E] :
( m2_subset_1(E,k1_numbers,k5_numbers)
=> ( k8_funct_2(k5_numbers,B,D,k1_nat_1(E,np__1)) = k4_measure1(A,B,k8_funct_2(k5_numbers,B,C,k1_nat_1(E,np__1)),k8_funct_2(k5_numbers,B,C,E))
& r1_tarski(k8_funct_2(k5_numbers,B,C,E),k8_funct_2(k5_numbers,B,C,k1_nat_1(E,np__1))) ) ) )
=> ( v1_funct_1(D)
& v1_funct_2(D,k5_numbers,B)
& v1_prob_2(D)
& m2_relset_1(D,k5_numbers,B) ) ) ) ) ) ).
fof(t26_measure2,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,D,np__0) = k8_funct_2(k5_numbers,B,C,np__0)
& ! [E] :
( m2_subset_1(E,k1_numbers,k5_numbers)
=> ( k8_funct_2(k5_numbers,B,D,k1_nat_1(E,np__1)) = k4_measure1(A,B,k8_funct_2(k5_numbers,B,C,k1_nat_1(E,np__1)),k8_funct_2(k5_numbers,B,C,E))
& r1_tarski(k8_funct_2(k5_numbers,B,C,E),k8_funct_2(k5_numbers,B,C,k1_nat_1(E,np__1))) ) ) )
=> ( k8_funct_2(k5_numbers,B,C,np__0) = k8_funct_2(k5_numbers,B,D,np__0)
& ! [E] :
( m2_subset_1(E,k1_numbers,k5_numbers)
=> k8_funct_2(k5_numbers,B,C,k1_nat_1(E,np__1)) = k2_measure1(A,B,k8_funct_2(k5_numbers,B,D,k1_nat_1(E,np__1)),k8_funct_2(k5_numbers,B,C,E)) ) ) ) ) ) ) ).
fof(t27_measure2,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,E),k8_funct_2(k5_numbers,B,D,k1_nat_1(E,np__1))) )
=> k1_funct_1(C,k5_setfam_1(A,k5_measure1(A,B,D))) = k9_supinf_1(k17_supinf_2(k7_funct_2(k5_numbers,B,k3_supinf_1,D,C))) ) ) ) ) ).
fof(dt_m1_measure2,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_measure2(C,A,B)
=> ( ~ v1_xboole_0(C)
& v1_card_4(C)
& m1_subset_1(C,k1_zfmisc_1(k1_zfmisc_1(A))) ) ) ) ).
fof(existence_m1_measure2,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_measure2(C,A,B) ) ).
fof(dt_k1_measure2,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)))
& m1_measure2(C,A,B) )
=> m2_subset_1(k1_measure2(A,B,C),k1_zfmisc_1(A),B) ) ).
fof(redefinition_k1_measure2,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)))
& m1_measure2(C,A,B) )
=> k1_measure2(A,B,C) = k1_setfam_1(C) ) ).
fof(dt_k2_measure2,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)))
& m1_measure2(C,A,B) )
=> m2_subset_1(k2_measure2(A,B,C),k1_zfmisc_1(A),B) ) ).
fof(redefinition_k2_measure2,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)))
& m1_measure2(C,A,B) )
=> k2_measure2(A,B,C) = k3_tarski(C) ) ).
%------------------------------------------------------------------------------