SET007 Axioms: SET007+181.ax
%------------------------------------------------------------------------------
% File : SET007+181 : TPTP v8.2.0. Released v3.4.0.
% Domain : Set Theory
% Axioms : The Measurability of Extended Real Valued Functions
% 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 : mesfunc2 [Urb08]
% Status : Satisfiable
% Syntax : Number of formulae : 47 ( 4 unt; 0 def)
% Number of atoms : 371 ( 43 equ)
% Maximal formula atoms : 20 ( 7 avg)
% Number of connectives : 388 ( 64 ~; 5 |; 151 &)
% ( 4 <=>; 164 =>; 0 <=; 0 <~>)
% Maximal formula depth : 22 ( 9 avg)
% Maximal term depth : 6 ( 1 avg)
% Number of predicates : 26 ( 24 usr; 1 prp; 0-4 aty)
% Number of functors : 41 ( 41 usr; 10 con; 0-4 aty)
% Number of variables : 150 ( 147 !; 3 ?)
% 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_mesfunc2,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_finseq_1(C,B)
& v1_relat_1(C)
& v1_funct_1(C)
& v1_finset_1(C)
& v1_finseq_1(C)
& v1_prob_2(C) ) ) ).
fof(d1_mesfunc2,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( ( v1_funct_1(B)
& m2_relset_1(B,A,k3_supinf_1) )
=> ( r1_mesfunc2(A,B)
<=> ! [C] :
( m1_subset_1(C,A)
=> ~ ( r2_hidden(C,k1_relat_1(B))
& r1_supinf_1(k5_measure6,k4_extreal1(k4_mesfunc1(A,B,C))) ) ) ) ) ) ).
fof(t1_mesfunc2,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( ( v1_funct_1(B)
& m2_relset_1(B,A,k3_supinf_1) )
=> B = k8_mesfunc1(A,B,np__1) ) ) ).
fof(t2_mesfunc2,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( ( v1_funct_1(B)
& m2_relset_1(B,A,k3_supinf_1) )
=> ! [C] :
( ( v1_funct_1(C)
& m2_relset_1(C,A,k3_supinf_1) )
=> ( ( r1_mesfunc2(A,B)
| r1_mesfunc2(A,C) )
=> ( k1_relat_1(k13_mesfunc1(A,B,C)) = k3_xboole_0(k1_relat_1(B),k1_relat_1(C))
& k1_relat_1(k6_mesfunc1(A,B,C)) = k3_xboole_0(k1_relat_1(B),k1_relat_1(C)) ) ) ) ) ) ).
fof(t3_mesfunc2,axiom,
! [A] :
( ~ v1_xboole_0(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)
& m2_relset_1(C,A,k3_supinf_1) )
=> ! [D] :
( ( v1_funct_1(D)
& m2_relset_1(D,A,k3_supinf_1) )
=> ! [E] :
( ( v1_funct_1(E)
& v1_funct_2(E,k3_numbers,B)
& m2_relset_1(E,k3_numbers,B) )
=> ! [F] :
( m1_subset_1(F,k1_numbers)
=> ! [G] :
( m1_subset_1(G,B)
=> ( ( r1_mesfunc2(A,C)
& r1_mesfunc2(A,D)
& ! [H] :
( v1_rat_1(H)
=> k1_funct_1(E,H) = k3_xboole_0(k3_xboole_0(G,k15_mesfunc1(A,C,k1_measure6(H))),k3_xboole_0(G,k15_mesfunc1(A,D,k1_measure6(k6_xcmplx_0(F,H))))) ) )
=> k3_xboole_0(G,k15_mesfunc1(A,k13_mesfunc1(A,C,D),k1_measure6(F))) = k3_tarski(k2_relat_1(E)) ) ) ) ) ) ) ) ) ).
fof(t4_mesfunc2,axiom,
? [A] :
( v1_funct_1(A)
& v1_funct_2(A,k5_numbers,k3_numbers)
& m2_relset_1(A,k5_numbers,k3_numbers)
& v2_funct_1(A)
& k1_relat_1(A) = k5_numbers
& k2_relat_1(A) = k3_numbers ) ).
fof(t5_mesfunc2,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( ~ v1_xboole_0(B)
=> ! [C] :
( ~ v1_xboole_0(C)
=> ! [D] :
( ( v1_funct_1(D)
& v1_funct_2(D,A,C)
& m2_relset_1(D,A,C) )
=> ~ ( r2_wellord2(A,B)
& ! [E] :
( ( v1_funct_1(E)
& v1_funct_2(E,B,C)
& m2_relset_1(E,B,C) )
=> k2_relat_1(D) != k2_relat_1(E) ) ) ) ) ) ) ).
fof(t6_mesfunc2,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m1_subset_1(B,k1_numbers)
=> ! [C] :
( ( ~ v1_xboole_0(C)
& v1_prob_1(C,A)
& v1_measure1(C,A)
& m1_subset_1(C,k1_zfmisc_1(k1_zfmisc_1(A))) )
=> ! [D] :
( ( v1_funct_1(D)
& m2_relset_1(D,A,k3_supinf_1) )
=> ! [E] :
( ( v1_funct_1(E)
& m2_relset_1(E,A,k3_supinf_1) )
=> ! [F] :
( m1_subset_1(F,C)
=> ~ ( r2_mesfunc1(A,C,D,F)
& r2_mesfunc1(A,C,E,F)
& ! [G] :
( ( v1_funct_1(G)
& v1_funct_2(G,k3_numbers,C)
& m2_relset_1(G,k3_numbers,C) )
=> ~ ! [H] :
( v1_rat_1(H)
=> k1_funct_1(G,H) = k3_xboole_0(k3_xboole_0(F,k15_mesfunc1(A,D,k1_measure6(H))),k3_xboole_0(F,k15_mesfunc1(A,E,k1_measure6(k6_xcmplx_0(B,H))))) ) ) ) ) ) ) ) ) ) ).
fof(t7_mesfunc2,axiom,
! [A] :
( ~ v1_xboole_0(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)
& m2_relset_1(C,A,k3_supinf_1) )
=> ! [D] :
( ( v1_funct_1(D)
& m2_relset_1(D,A,k3_supinf_1) )
=> ! [E] :
( m1_subset_1(E,B)
=> ( ( r1_mesfunc2(A,C)
& r1_mesfunc2(A,D)
& r2_mesfunc1(A,B,C,E)
& r2_mesfunc1(A,B,D,E) )
=> r2_mesfunc1(A,B,k13_mesfunc1(A,C,D),E) ) ) ) ) ) ) ).
fof(t8_mesfunc2,axiom,
$true ).
fof(t9_mesfunc2,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( ( v1_funct_1(B)
& m2_relset_1(B,A,k3_supinf_1) )
=> ! [C] :
( ( v1_funct_1(C)
& m2_relset_1(C,A,k3_supinf_1) )
=> k6_mesfunc1(A,B,C) = k13_mesfunc1(A,B,k9_mesfunc1(A,C)) ) ) ) ).
fof(t10_mesfunc2,axiom,
! [A] :
( m1_subset_1(A,k1_numbers)
=> k1_measure6(k1_real_1(A)) = k3_supinf_2(k1_measure6(A)) ) ).
fof(t11_mesfunc2,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( ( v1_funct_1(B)
& m2_relset_1(B,A,k3_supinf_1) )
=> k9_mesfunc1(A,B) = k8_mesfunc1(A,B,k1_real_1(np__1)) ) ) ).
fof(t12_mesfunc2,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( ( v1_funct_1(B)
& m2_relset_1(B,A,k3_supinf_1) )
=> ! [C] :
( m1_subset_1(C,k1_numbers)
=> ( r1_mesfunc2(A,B)
=> r1_mesfunc2(A,k8_mesfunc1(A,B,C)) ) ) ) ) ).
fof(t13_mesfunc2,axiom,
! [A] :
( ~ v1_xboole_0(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)
& m2_relset_1(C,A,k3_supinf_1) )
=> ! [D] :
( ( v1_funct_1(D)
& m2_relset_1(D,A,k3_supinf_1) )
=> ! [E] :
( m1_subset_1(E,B)
=> ( ( r1_mesfunc2(A,C)
& r1_mesfunc2(A,D)
& r2_mesfunc1(A,B,C,E)
& r2_mesfunc1(A,B,D,E)
& r1_tarski(E,k1_relat_1(D)) )
=> r2_mesfunc1(A,B,k6_mesfunc1(A,C,D),E) ) ) ) ) ) ) ).
fof(d2_mesfunc2,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( ( v1_funct_1(B)
& m2_relset_1(B,A,k3_supinf_1) )
=> ! [C] :
( ( v1_funct_1(C)
& m2_relset_1(C,A,k3_supinf_1) )
=> ( C = k1_mesfunc2(A,B)
<=> ( k1_relat_1(C) = k1_relat_1(B)
& ! [D] :
( m1_subset_1(D,A)
=> ( r2_hidden(D,k1_relat_1(C))
=> k4_mesfunc1(A,C,D) = k2_extreal2(k4_mesfunc1(A,B,D),k1_supinf_2) ) ) ) ) ) ) ) ).
fof(d3_mesfunc2,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( ( v1_funct_1(B)
& m2_relset_1(B,A,k3_supinf_1) )
=> ! [C] :
( ( v1_funct_1(C)
& m2_relset_1(C,A,k3_supinf_1) )
=> ( C = k2_mesfunc2(A,B)
<=> ( k1_relat_1(C) = k1_relat_1(B)
& ! [D] :
( m1_subset_1(D,A)
=> ( r2_hidden(D,k1_relat_1(C))
=> k4_mesfunc1(A,C,D) = k2_extreal2(k3_supinf_2(k4_mesfunc1(A,B,D)),k1_supinf_2) ) ) ) ) ) ) ) ).
fof(t14_mesfunc2,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( ( v1_funct_1(B)
& m2_relset_1(B,A,k3_supinf_1) )
=> ! [C] :
( m1_subset_1(C,A)
=> ( r2_hidden(C,k1_relat_1(B))
=> r1_supinf_1(k1_supinf_2,k4_mesfunc1(A,k1_mesfunc2(A,B),C)) ) ) ) ) ).
fof(t15_mesfunc2,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( ( v1_funct_1(B)
& m2_relset_1(B,A,k3_supinf_1) )
=> ! [C] :
( m1_subset_1(C,A)
=> ( r2_hidden(C,k1_relat_1(B))
=> r1_supinf_1(k1_supinf_2,k4_mesfunc1(A,k2_mesfunc2(A,B),C)) ) ) ) ) ).
fof(t16_mesfunc2,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( ( v1_funct_1(B)
& m2_relset_1(B,A,k3_supinf_1) )
=> k2_mesfunc2(A,B) = k1_mesfunc2(A,k9_mesfunc1(A,B)) ) ) ).
fof(t17_mesfunc2,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( ( v1_funct_1(B)
& m2_relset_1(B,A,k3_supinf_1) )
=> ! [C] :
( m1_subset_1(C,A)
=> ( r2_hidden(C,k1_relat_1(B))
=> ( r1_supinf_1(k4_mesfunc1(A,k1_mesfunc2(A,B),C),k1_supinf_2)
| k4_mesfunc1(A,k2_mesfunc2(A,B),C) = k1_supinf_2 ) ) ) ) ) ).
fof(t18_mesfunc2,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( ( v1_funct_1(B)
& m2_relset_1(B,A,k3_supinf_1) )
=> ! [C] :
( m1_subset_1(C,A)
=> ( r2_hidden(C,k1_relat_1(B))
=> ( r1_supinf_1(k4_mesfunc1(A,k2_mesfunc2(A,B),C),k1_supinf_2)
| k4_mesfunc1(A,k1_mesfunc2(A,B),C) = k1_supinf_2 ) ) ) ) ) ).
fof(t19_mesfunc2,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( ( v1_funct_1(B)
& m2_relset_1(B,A,k3_supinf_1) )
=> ( k1_relat_1(B) = k1_relat_1(k6_mesfunc1(A,k1_mesfunc2(A,B),k2_mesfunc2(A,B)))
& k1_relat_1(B) = k1_relat_1(k13_mesfunc1(A,k1_mesfunc2(A,B),k2_mesfunc2(A,B))) ) ) ) ).
fof(t20_mesfunc2,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( ( v1_funct_1(B)
& m2_relset_1(B,A,k3_supinf_1) )
=> ! [C] :
( m1_subset_1(C,A)
=> ( r2_hidden(C,k1_relat_1(B))
=> ( ( k4_mesfunc1(A,k1_mesfunc2(A,B),C) = k4_mesfunc1(A,B,C)
| k4_mesfunc1(A,k1_mesfunc2(A,B),C) = k1_supinf_2 )
& ( k4_mesfunc1(A,k2_mesfunc2(A,B),C) = k3_supinf_2(k4_mesfunc1(A,B,C))
| k4_mesfunc1(A,k2_mesfunc2(A,B),C) = k1_supinf_2 ) ) ) ) ) ) ).
fof(t21_mesfunc2,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( ( v1_funct_1(B)
& m2_relset_1(B,A,k3_supinf_1) )
=> ! [C] :
( m1_subset_1(C,A)
=> ( ( r2_hidden(C,k1_relat_1(B))
& k4_mesfunc1(A,k1_mesfunc2(A,B),C) = k4_mesfunc1(A,B,C) )
=> k4_mesfunc1(A,k2_mesfunc2(A,B),C) = k1_supinf_2 ) ) ) ) ).
fof(t22_mesfunc2,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( ( v1_funct_1(B)
& m2_relset_1(B,A,k3_supinf_1) )
=> ! [C] :
( m1_subset_1(C,A)
=> ( ( r2_hidden(C,k1_relat_1(B))
& k4_mesfunc1(A,k1_mesfunc2(A,B),C) = k1_supinf_2 )
=> k4_mesfunc1(A,k2_mesfunc2(A,B),C) = k3_supinf_2(k4_mesfunc1(A,B,C)) ) ) ) ) ).
fof(t23_mesfunc2,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( ( v1_funct_1(B)
& m2_relset_1(B,A,k3_supinf_1) )
=> ! [C] :
( m1_subset_1(C,A)
=> ( ( r2_hidden(C,k1_relat_1(B))
& k4_mesfunc1(A,k2_mesfunc2(A,B),C) = k3_supinf_2(k4_mesfunc1(A,B,C)) )
=> k4_mesfunc1(A,k1_mesfunc2(A,B),C) = k1_supinf_2 ) ) ) ) ).
fof(t24_mesfunc2,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( ( v1_funct_1(B)
& m2_relset_1(B,A,k3_supinf_1) )
=> ! [C] :
( m1_subset_1(C,A)
=> ( ( r2_hidden(C,k1_relat_1(B))
& k4_mesfunc1(A,k2_mesfunc2(A,B),C) = k1_supinf_2 )
=> k4_mesfunc1(A,k1_mesfunc2(A,B),C) = k4_mesfunc1(A,B,C) ) ) ) ) ).
fof(t25_mesfunc2,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( ( v1_funct_1(B)
& m2_relset_1(B,A,k3_supinf_1) )
=> B = k6_mesfunc1(A,k1_mesfunc2(A,B),k2_mesfunc2(A,B)) ) ) ).
fof(t26_mesfunc2,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( ( v1_funct_1(B)
& m2_relset_1(B,A,k3_supinf_1) )
=> k12_mesfunc1(A,B) = k13_mesfunc1(A,k1_mesfunc2(A,B),k2_mesfunc2(A,B)) ) ) ).
fof(t27_mesfunc2,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( ( v1_funct_1(B)
& m2_relset_1(B,A,k3_supinf_1) )
=> ! [C] :
( ( ~ v1_xboole_0(C)
& v1_prob_1(C,A)
& v1_measure1(C,A)
& m1_subset_1(C,k1_zfmisc_1(k1_zfmisc_1(A))) )
=> ! [D] :
( m1_subset_1(D,C)
=> ( r2_mesfunc1(A,C,B,D)
=> r2_mesfunc1(A,C,k1_mesfunc2(A,B),D) ) ) ) ) ) ).
fof(t28_mesfunc2,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( ( v1_funct_1(B)
& m2_relset_1(B,A,k3_supinf_1) )
=> ! [C] :
( ( ~ v1_xboole_0(C)
& v1_prob_1(C,A)
& v1_measure1(C,A)
& m1_subset_1(C,k1_zfmisc_1(k1_zfmisc_1(A))) )
=> ! [D] :
( m1_subset_1(D,C)
=> ( ( r2_mesfunc1(A,C,B,D)
& r1_tarski(D,k1_relat_1(B)) )
=> r2_mesfunc1(A,C,k2_mesfunc2(A,B),D) ) ) ) ) ) ).
fof(t29_mesfunc2,axiom,
! [A] :
( ~ v1_xboole_0(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)
& m2_relset_1(C,A,k3_supinf_1) )
=> ! [D] :
( m1_subset_1(D,B)
=> ( ( r2_mesfunc1(A,B,C,D)
& r1_tarski(D,k1_relat_1(C)) )
=> r2_mesfunc1(A,B,k12_mesfunc1(A,C),D) ) ) ) ) ) ).
fof(t30_mesfunc2,axiom,
! [A,B] : r1_tarski(k2_relat_1(k5_funct_3(A,B)),k12_supinf_1(k1_supinf_2,k10_mesfunc1)) ).
fof(t31_mesfunc2,axiom,
! [A] :
( ~ v1_xboole_0(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_subset_1(C,B)
=> r1_mesfunc2(A,k3_mesfunc2(C,A)) ) ) ) ).
fof(t32_mesfunc2,axiom,
! [A] :
( ~ v1_xboole_0(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_subset_1(C,B)
=> ! [D] :
( m1_subset_1(D,B)
=> r2_mesfunc1(A,B,k3_mesfunc2(C,A),D) ) ) ) ) ).
fof(t33_mesfunc2,axiom,
! [A] :
( ~ v1_xboole_0(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_relat_1(C)
& v1_funct_1(C) )
=> ~ ( v1_prob_2(C)
& m2_finseq_1(C,B)
& ! [D] :
( ( v1_funct_1(D)
& v1_funct_2(D,k5_numbers,B)
& v1_prob_2(D)
& m2_relset_1(D,k5_numbers,B) )
=> ~ ( k3_tarski(k2_relat_1(C)) = k2_measure2(A,B,k1_measure3(A,B,D))
& ! [E] :
( m2_subset_1(E,k1_numbers,k5_numbers)
=> ( r2_hidden(E,k1_relat_1(C))
=> k1_funct_1(C,E) = k8_funct_2(k5_numbers,B,D,E) ) )
& ! [E] :
( m2_subset_1(E,k1_numbers,k5_numbers)
=> ( ~ r2_hidden(E,k1_relat_1(C))
=> k8_funct_2(k5_numbers,B,D,E) = k1_xboole_0 ) ) ) ) ) ) ) ) ).
fof(t34_mesfunc2,axiom,
! [A] :
( ~ v1_xboole_0(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_relat_1(C)
& v1_funct_1(C) )
=> ( ( v1_prob_2(C)
& m2_finseq_1(C,B) )
=> r2_hidden(k3_tarski(k2_relat_1(C)),B) ) ) ) ) ).
fof(d4_mesfunc2,axiom,
$true ).
fof(d5_mesfunc2,axiom,
! [A] :
( ~ v1_xboole_0(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)
& m2_relset_1(C,A,k3_supinf_1) )
=> ( r2_mesfunc2(A,B,C)
<=> ( r1_mesfunc2(A,C)
& ? [D] :
( v1_prob_2(D)
& m2_finseq_1(D,B)
& k1_relat_1(C) = k3_tarski(k2_relat_1(D))
& ! [E] :
( m2_subset_1(E,k1_numbers,k5_numbers)
=> ! [F] :
( m1_subset_1(F,A)
=> ! [G] :
( m1_subset_1(G,A)
=> ( ( r2_hidden(E,k4_finseq_1(D))
& r2_hidden(F,k1_funct_1(D,E))
& r2_hidden(G,k1_funct_1(D,E)) )
=> k4_mesfunc1(A,C,F) = k4_mesfunc1(A,C,G) ) ) ) ) ) ) ) ) ) ) ).
fof(t35_mesfunc2,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( ( v1_funct_1(B)
& m2_relset_1(B,A,k3_supinf_1) )
=> ( r1_mesfunc2(A,B)
=> m1_subset_1(k2_relat_1(B),k1_zfmisc_1(k6_supinf_1)) ) ) ) ).
fof(t36_mesfunc2,axiom,
! [A] :
( ~ v1_xboole_0(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] :
( m2_subset_1(C,k1_numbers,k5_numbers)
=> ! [D] :
( v1_relat_1(D)
=> ( ( v1_prob_2(D)
& m2_finseq_1(D,B) )
=> ( v1_prob_2(k7_relat_1(D,k2_finseq_1(C)))
& m2_finseq_1(k7_relat_1(D,k2_finseq_1(C)),B) ) ) ) ) ) ) ).
fof(t37_mesfunc2,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( ( v1_funct_1(B)
& m2_relset_1(B,A,k3_supinf_1) )
=> ! [C] :
( ( ~ v1_xboole_0(C)
& v1_prob_1(C,A)
& v1_measure1(C,A)
& m1_subset_1(C,k1_zfmisc_1(k1_zfmisc_1(A))) )
=> ! [D] :
( m1_subset_1(D,C)
=> ( r2_mesfunc2(A,C,B)
=> r2_mesfunc1(A,C,B,D) ) ) ) ) ) ).
fof(dt_k1_mesfunc2,axiom,
! [A,B] :
( ( ~ v1_xboole_0(A)
& v1_funct_1(B)
& m1_relset_1(B,A,k3_supinf_1) )
=> ( v1_funct_1(k1_mesfunc2(A,B))
& m2_relset_1(k1_mesfunc2(A,B),A,k3_supinf_1) ) ) ).
fof(dt_k2_mesfunc2,axiom,
! [A,B] :
( ( ~ v1_xboole_0(A)
& v1_funct_1(B)
& m1_relset_1(B,A,k3_supinf_1) )
=> ( v1_funct_1(k2_mesfunc2(A,B))
& m2_relset_1(k2_mesfunc2(A,B),A,k3_supinf_1) ) ) ).
fof(dt_k3_mesfunc2,axiom,
! [A,B] :
( v1_funct_1(k3_mesfunc2(A,B))
& m2_relset_1(k3_mesfunc2(A,B),B,k3_supinf_1) ) ).
fof(redefinition_k3_mesfunc2,axiom,
! [A,B] : k3_mesfunc2(A,B) = k4_funct_3(A,B) ).
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