## SET007 Axioms: SET007+929.ax

```%------------------------------------------------------------------------------
% File     : SET007+929 : TPTP v7.5.0. Released v3.4.0.
% Domain   : Set Theory
% Axioms   : The Relevance of Measure and Probability
% Version  : [Urb08] axioms.
% English  : The Relevance of Measure and Probability, and Definition of
%            Completeness of Probability

% 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    : prob_4 [Urb08]

% Status   : Satisfiable
% Syntax   : Number of formulae    :   72 (   0 unit)
%            Number of atoms       :  465 (  61 equality)
%            Maximal formula depth :   18 (  10 average)
%            Number of connectives :  459 (  66 ~  ;   0  |; 150  &)
%                                         (  17 <=>; 226 =>;   0 <=)
%                                         (   0 <~>;   0 ~|;   0 ~&)
%            Number of predicates  :   32 (   0 propositional; 1-4 arity)
%            Number of functors    :   54 (   9 constant; 0-4 arity)
%            Number of variables   :  281 (   0 singleton; 264 !;  17 ?)
%            Maximal term depth    :    5 (   1 average)
% 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_prob_4,axiom,(
! [A] :
? [B] :
( m1_relset_1(B,k5_numbers,k1_zfmisc_1(A))
& ~ v1_xboole_0(B)
& v1_relat_1(B)
& v1_funct_1(B)
& v1_partfun1(B,k5_numbers,k1_zfmisc_1(A))
& v1_funct_2(B,k5_numbers,k1_zfmisc_1(A))
& v1_prob_2(B) ) )).

fof(rc2_prob_4,axiom,(
! [A,B] :
( m1_prob_1(B,A)
=> ? [C] :
( m2_prob_1(C,A,B)
& ~ v1_xboole_0(C)
& v1_relat_1(C)
& v1_funct_1(C)
& v1_partfun1(C,k5_numbers,k1_zfmisc_1(A))
& v1_funct_2(C,k5_numbers,k1_zfmisc_1(A))
& v1_prob_2(C) ) ) )).

fof(t1_prob_4,axiom,(
! [A,B] :
( m1_prob_1(B,A)
=> ! [C] :
( m2_prob_1(C,A,B)
=> r1_tarski(k1_pscomp_1(k5_numbers,k1_zfmisc_1(A),C),B) ) ) )).

fof(t2_prob_4,axiom,(
! [A,B] :
( m1_prob_1(B,A)
=> ! [C] :
( ( v1_relat_1(C)
& v1_funct_1(C) )
=> ( m2_prob_1(C,A,B)
<=> ( v1_funct_1(C)
& v1_funct_2(C,k5_numbers,B)
& m2_relset_1(C,k5_numbers,B) ) ) ) ) )).

fof(t3_prob_4,axiom,(
! [A,B] :
( m1_subset_1(B,k1_zfmisc_1(A))
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(A))
=> ? [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))
& k1_prob_1(A,D,np__0) = B
& k1_prob_1(A,D,np__1) = C
& ! [E] :
( m2_subset_1(E,k1_numbers,k5_numbers)
=> ( ~ r1_xreal_0(E,np__1)
=> k1_prob_1(A,D,E) = k1_xboole_0 ) ) ) ) ) )).

fof(t4_prob_4,axiom,(
! [A,B] :
( ( v1_funct_1(B)
& v1_funct_2(B,k5_numbers,k1_zfmisc_1(A))
& m2_relset_1(B,k5_numbers,k1_zfmisc_1(A)) )
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(A))
=> ! [D] :
( m1_subset_1(D,k1_zfmisc_1(A))
=> ( ( r1_xboole_0(C,D)
& k1_prob_1(A,B,np__0) = C
& k1_prob_1(A,B,np__1) = D
& ! [E] :
( m2_subset_1(E,k1_numbers,k5_numbers)
=> ( ~ r1_xreal_0(E,np__1)
=> k1_prob_1(A,B,E) = k1_xboole_0 ) ) )
=> ( v1_prob_2(B)
& k2_prob_1(A,B) = k4_subset_1(A,C,D) ) ) ) ) ) )).

fof(t5_prob_4,axiom,(
! [A,B] :
( ~ v1_xboole_0(B)
=> ( m1_prob_1(B,A)
<=> ( r1_tarski(B,k1_zfmisc_1(A))
& ! [C] :
( ( v1_funct_1(C)
& v1_funct_2(C,k5_numbers,k1_zfmisc_1(A))
& m2_relset_1(C,k5_numbers,k1_zfmisc_1(A)) )
=> ( ! [D] :
( m2_subset_1(D,k1_numbers,k5_numbers)
=> r2_hidden(k1_prob_1(A,C,D),B) )
=> r2_hidden(k2_prob_1(A,C),B) ) )
& ! [C] :
( m1_subset_1(C,k1_zfmisc_1(A))
=> ( r2_hidden(C,B)
=> r2_hidden(k3_subset_1(A,C),B) ) ) ) ) ) )).

fof(t6_prob_4,axiom,(
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m1_prob_1(B,A)
=> ! [C] :
( m4_prob_1(C,A,B)
=> ! [D] :
( m3_prob_1(D,A,B)
=> ! [E] :
( m3_prob_1(E,A,B)
=> k10_prob_1(A,B,C,k8_prob_1(A,B,D,E)) = k6_xcmplx_0(k10_prob_1(A,B,C,k7_prob_1(A,B,D,E)),k10_prob_1(A,B,C,E)) ) ) ) ) ) )).

fof(t7_prob_4,axiom,(
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m1_prob_1(B,A)
=> ! [C] :
( m4_prob_1(C,A,B)
=> ! [D] :
( m3_prob_1(D,A,B)
=> ! [E] :
( m3_prob_1(E,A,B)
=> ( ( r1_tarski(D,E)
& k10_prob_1(A,B,C,E) = np__0 )
=> k10_prob_1(A,B,C,D) = np__0 ) ) ) ) ) ) )).

fof(t8_prob_4,axiom,(
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m1_prob_1(B,A)
=> ! [C] :
( m2_prob_1(C,A,B)
=> ! [D] :
( m4_prob_1(D,A,B)
=> ( ! [E] :
( m2_subset_1(E,k1_numbers,k5_numbers)
=> k8_funct_2(B,k1_numbers,D,k1_prob_4(A,B,C,E)) = np__0 )
<=> k1_funct_1(D,k2_prob_1(A,C)) = np__0 ) ) ) ) ) )).

fof(t9_prob_4,axiom,(
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m1_prob_1(B,A)
=> ! [C] :
( m2_prob_1(C,A,B)
=> ! [D] :
( m4_prob_1(D,A,B)
=> ( ! [E] :
( r2_hidden(E,k1_pscomp_1(k5_numbers,k1_zfmisc_1(A),C))
=> k1_funct_1(D,E) = np__0 )
<=> k1_funct_1(D,k5_setfam_1(A,k1_pscomp_1(k5_numbers,k1_zfmisc_1(A),C))) = np__0 ) ) ) ) ) )).

fof(d1_prob_4,axiom,(
! [A,B] :
( m1_prob_1(B,A)
=> k2_prob_4(A,B) = B ) )).

fof(d2_prob_4,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))) )
=> k3_prob_4(A,B) = B ) )).

fof(t10_prob_4,axiom,(
! [A] :
( ( v1_funct_1(A)
& v1_funct_2(A,k5_numbers,k6_supinf_1)
& m2_relset_1(A,k5_numbers,k6_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) )
=> ( A = B
=> k1_series_1(A) = k18_supinf_2(B) ) ) ) )).

fof(t11_prob_4,axiom,(
! [A] :
( ( v1_funct_1(A)
& v1_funct_2(A,k5_numbers,k6_supinf_1)
& m2_relset_1(A,k5_numbers,k6_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) )
=> ( ( A = B
& v1_seq_2(A) )
=> k1_rinfsup1(A) = k9_supinf_1(k1_pscomp_1(k5_numbers,k3_supinf_1,B)) ) ) ) )).

fof(t12_prob_4,axiom,(
! [A] :
( ( v1_funct_1(A)
& v1_funct_2(A,k5_numbers,k6_supinf_1)
& m2_relset_1(A,k5_numbers,k6_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) )
=> ( ( A = B
& v2_seq_2(A) )
=> k2_rinfsup1(A) = k10_supinf_1(k1_pscomp_1(k5_numbers,k3_supinf_1,B)) ) ) ) )).

fof(t13_prob_4,axiom,(
! [A] :
( ( v1_funct_1(A)
& v1_funct_2(A,k5_numbers,k6_supinf_1)
& m2_relset_1(A,k5_numbers,k6_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) )
=> ( ( A = B
& v4_partfun3(A)
& v1_series_1(A) )
=> k2_series_1(A) = k19_supinf_2(B) ) ) ) )).

fof(t14_prob_4,axiom,(
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m1_prob_1(B,A)
=> ! [C] :
( m4_prob_1(C,A,B)
=> m3_measure1(C,A,k2_prob_4(A,B)) ) ) ) )).

fof(d3_prob_4,axiom,(
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m1_prob_1(B,A)
=> ! [C] :
( m4_prob_1(C,A,B)
=> k4_prob_4(A,B,C) = C ) ) ) )).

fof(t15_prob_4,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] :
( m3_measure1(C,A,B)
=> ( k1_funct_1(C,A) = k1_measure6(np__1)
=> m4_prob_1(C,A,k3_prob_4(A,B)) ) ) ) ) )).

fof(d4_prob_4,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] :
( m3_measure1(C,A,B)
=> ( k1_funct_1(C,A) = k1_measure6(np__1)
=> k5_prob_4(A,B,C) = C ) ) ) ) )).

fof(t16_prob_4,axiom,(
! [A,B] :
( ( v1_funct_1(B)
& v1_funct_2(B,k5_numbers,k1_zfmisc_1(A))
& m2_relset_1(B,k5_numbers,k1_zfmisc_1(A)) )
=> ( v3_prob_1(B,A)
=> k2_prob_3(A,B) = B ) ) )).

fof(t17_prob_4,axiom,(
! [A,B] :
( ( v1_funct_1(B)
& v1_funct_2(B,k5_numbers,k1_zfmisc_1(A))
& m2_relset_1(B,k5_numbers,k1_zfmisc_1(A)) )
=> ( v3_prob_1(B,A)
=> ( k1_prob_1(A,k3_prob_3(A,B),np__0) = k1_prob_1(A,B,np__0)
& ! [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
=> k1_prob_1(A,k3_prob_3(A,B),k1_nat_1(C,np__1)) = k6_subset_1(A,k1_prob_1(A,B,k1_nat_1(C,np__1)),k1_prob_1(A,B,C)) ) ) ) ) )).

fof(t18_prob_4,axiom,(
! [A,B] :
( ( v1_funct_1(B)
& v1_funct_2(B,k5_numbers,k1_zfmisc_1(A))
& m2_relset_1(B,k5_numbers,k1_zfmisc_1(A)) )
=> ( v3_prob_1(B,A)
=> ! [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
=> k1_prob_1(A,B,k1_nat_1(C,np__1)) = k4_subset_1(A,k1_prob_1(A,k3_prob_3(A,B),k1_nat_1(C,np__1)),k1_prob_1(A,B,C)) ) ) ) )).

fof(t19_prob_4,axiom,(
! [A,B] :
( ( v1_funct_1(B)
& v1_funct_2(B,k5_numbers,k1_zfmisc_1(A))
& m2_relset_1(B,k5_numbers,k1_zfmisc_1(A)) )
=> ( v3_prob_1(B,A)
=> ! [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
=> r1_xboole_0(k1_prob_1(A,k3_prob_3(A,B),k1_nat_1(C,np__1)),k1_prob_1(A,B,C)) ) ) ) )).

fof(t20_prob_4,axiom,(
! [A,B] :
( m1_prob_1(B,A)
=> ! [C] :
( m2_prob_1(C,A,B)
=> ( v3_prob_1(C,A)
=> k5_prob_3(A,B,C) = C ) ) ) )).

fof(t21_prob_4,axiom,(
! [A,B] :
( m1_prob_1(B,A)
=> ! [C] :
( m2_prob_1(C,A,B)
=> ( v3_prob_1(C,A)
=> ( k1_prob_4(A,B,k6_prob_3(A,B,C),np__0) = k1_prob_4(A,B,C,np__0)
& ! [D] :
( m2_subset_1(D,k1_numbers,k5_numbers)
=> k1_prob_4(A,B,k6_prob_3(A,B,C),k1_nat_1(D,np__1)) = k4_measure1(A,B,k1_prob_4(A,B,C,k1_nat_1(D,np__1)),k1_prob_4(A,B,C,D)) ) ) ) ) ) )).

fof(t22_prob_4,axiom,(
! [A,B] :
( m1_prob_1(B,A)
=> ! [C] :
( m2_prob_1(C,A,B)
=> ( v3_prob_1(C,A)
=> ! [D] :
( m2_subset_1(D,k1_numbers,k5_numbers)
=> r1_xboole_0(k1_prob_4(A,B,k6_prob_3(A,B,C),k1_nat_1(D,np__1)),k1_prob_4(A,B,C,D)) ) ) ) ) )).

fof(d5_prob_4,axiom,(
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m1_prob_1(B,A)
=> ! [C] :
( m4_prob_1(C,A,B)
=> ( r1_prob_4(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) = np__0 )
=> r2_hidden(D,B) ) ) ) ) ) ) )).

fof(t23_prob_4,axiom,(
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m1_prob_1(B,A)
=> ! [C] :
( m4_prob_1(C,A,B)
=> ( r1_prob_4(A,B,C)
<=> r1_measure3(A,k2_prob_4(A,B),k4_prob_4(A,B,C)) ) ) ) ) )).

fof(d6_prob_4,axiom,(
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m1_prob_1(B,A)
=> ! [C] :
( m4_prob_1(C,A,B)
=> ! [D] :
( m1_subset_1(D,k1_zfmisc_1(A))
=> ( m1_prob_4(D,A,B,C)
<=> ? [E] :
( r2_hidden(E,B)
& r1_tarski(D,E)
& k1_funct_1(C,E) = np__0 ) ) ) ) ) ) )).

fof(t24_prob_4,axiom,(
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m1_prob_1(B,A)
=> ! [C] :
( m4_prob_1(C,A,B)
=> ! [D] :
( m1_subset_1(D,k1_zfmisc_1(A))
=> ( m1_prob_4(D,A,B,C)
<=> m1_measure3(D,A,k2_prob_4(A,B),k4_prob_4(A,B,C)) ) ) ) ) ) )).

fof(t25_prob_4,axiom,(
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m1_prob_1(B,A)
=> ! [C] :
( m4_prob_1(C,A,B)
=> m1_prob_4(k1_xboole_0,A,B,C) ) ) ) )).

fof(t26_prob_4,axiom,(
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m1_prob_1(B,A)
=> ! [C] :
( m4_prob_1(C,A,B)
=> ! [D,E] :
( ( r2_hidden(D,B)
& r2_hidden(E,B) )
=> ! [F] :
( m1_prob_4(F,A,B,C)
=> ! [G] :
( m1_prob_4(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(d7_prob_4,axiom,(
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m1_prob_1(B,A)
=> ! [C] :
( m4_prob_1(C,A,B)
=> ! [D] :
( ( ~ v1_xboole_0(D)
& m1_subset_1(D,k1_zfmisc_1(k1_zfmisc_1(A))) )
=> ( D = k6_prob_4(A,B,C)
<=> ! [E] :
( r2_hidden(E,D)
<=> ? [F] :
( r2_hidden(F,B)
& ? [G] :
( m1_prob_4(G,A,B,C)
& E = k2_xboole_0(F,G) ) ) ) ) ) ) ) ) )).

fof(t27_prob_4,axiom,(
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m1_prob_1(B,A)
=> ! [C] :
( m4_prob_1(C,A,B)
=> ! [D] :
( m1_prob_4(D,A,B,C)
=> r2_hidden(D,k6_prob_4(A,B,C)) ) ) ) ) )).

fof(t28_prob_4,axiom,(
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m1_prob_1(B,A)
=> ! [C] :
( m4_prob_1(C,A,B)
=> k6_prob_4(A,B,C) = k2_measure3(A,k2_prob_4(A,B),k4_prob_4(A,B,C)) ) ) ) )).

fof(d8_prob_4,axiom,(
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m1_prob_1(B,A)
=> ! [C] :
( m4_prob_1(C,A,B)
=> ! [D] :
( m2_subset_1(D,k1_zfmisc_1(A),k6_prob_4(A,B,C))
=> k7_prob_4(A,B,C,D) = D ) ) ) ) )).

fof(t29_prob_4,axiom,(
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m1_prob_1(B,A)
=> ! [C] :
( m4_prob_1(C,A,B)
=> r1_tarski(B,k6_prob_4(A,B,C)) ) ) ) )).

fof(d9_prob_4,axiom,(
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m1_prob_1(B,A)
=> ! [C] :
( m4_prob_1(C,A,B)
=> ! [D] :
( m2_subset_1(D,k1_zfmisc_1(A),k6_prob_4(A,B,C))
=> ! [E] :
( ( ~ v1_xboole_0(E)
& m1_subset_1(E,k1_zfmisc_1(k1_zfmisc_1(A))) )
=> ( E = k8_prob_4(A,B,C,D)
<=> ! [F] :
( r2_hidden(F,E)
<=> ( r2_hidden(F,B)
& r1_tarski(F,D)
& m1_prob_4(k4_xboole_0(D,F),A,B,C) ) ) ) ) ) ) ) ) )).

fof(t30_prob_4,axiom,(
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m1_prob_1(B,A)
=> ! [C] :
( m4_prob_1(C,A,B)
=> ! [D] :
( m2_subset_1(D,k1_zfmisc_1(A),k6_prob_4(A,B,C))
=> k8_prob_4(A,B,C,D) = k3_measure3(A,k2_prob_4(A,B),k4_prob_4(A,B,C),k7_prob_4(A,B,C,D)) ) ) ) ) )).

fof(t31_prob_4,axiom,(
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m1_prob_1(B,A)
=> ! [C] :
( m4_prob_1(C,A,B)
=> ! [D] :
( m2_subset_1(D,k1_zfmisc_1(A),k6_prob_4(A,B,C))
=> ! [E,F] :
( ( r2_hidden(E,k8_prob_4(A,B,C,D))
& r2_hidden(F,k8_prob_4(A,B,C,D)) )
=> k1_funct_1(C,E) = k1_funct_1(C,F) ) ) ) ) ) )).

fof(t32_prob_4,axiom,(
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m1_prob_1(B,A)
=> ! [C] :
( m4_prob_1(C,A,B)
=> ! [D] :
( ( v1_funct_1(D)
& v1_funct_2(D,k5_numbers,k6_prob_4(A,B,C))
& m2_relset_1(D,k5_numbers,k6_prob_4(A,B,C)) )
=> ? [E] :
( m2_prob_1(E,A,B)
& ! [F] :
( m2_subset_1(F,k1_numbers,k5_numbers)
=> r2_hidden(k1_prob_4(A,B,E,F),k8_prob_4(A,B,C,k8_funct_2(k5_numbers,k6_prob_4(A,B,C),D,F))) ) ) ) ) ) ) )).

fof(t33_prob_4,axiom,(
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m1_prob_1(B,A)
=> ! [C] :
( m4_prob_1(C,A,B)
=> ! [D] :
( ( v1_funct_1(D)
& v1_funct_2(D,k5_numbers,k6_prob_4(A,B,C))
& m2_relset_1(D,k5_numbers,k6_prob_4(A,B,C)) )
=> ! [E] :
( m2_prob_1(E,A,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)
=> k1_prob_1(A,F,G) = k4_xboole_0(k8_funct_2(k5_numbers,k6_prob_4(A,B,C),D,G),k1_prob_4(A,B,E,G)) ) ) ) ) ) ) ) )).

fof(t34_prob_4,axiom,(
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m1_prob_1(B,A)
=> ! [C] :
( m4_prob_1(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_prob_4(k1_prob_1(A,D,E),A,B,C) )
& ! [E] :
( m2_prob_1(E,A,B)
=> ~ ! [F] :
( m2_subset_1(F,k1_numbers,k5_numbers)
=> ( r1_tarski(k1_prob_1(A,D,F),k1_prob_4(A,B,E,F))
& k8_funct_2(B,k1_numbers,C,k1_prob_4(A,B,E,F)) = np__0 ) ) ) ) ) ) ) ) )).

fof(t35_prob_4,axiom,(
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m1_prob_1(B,A)
=> ! [C] :
( m4_prob_1(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_prob_4(G,A,B,C)
& E = k2_xboole_0(F,G) ) ) )
=> m1_prob_1(D,A) ) ) ) ) ) )).

fof(t36_prob_4,axiom,(
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m1_prob_1(B,A)
=> ! [C] :
( m4_prob_1(C,A,B)
=> ! [D] :
( r2_hidden(D,k9_prob_4(A,B,C))
<=> ? [E,F] :
( r2_hidden(E,B)
& r2_hidden(F,B)
& r1_tarski(E,D)
& r1_tarski(D,F)
& k1_funct_1(C,k4_xboole_0(F,E)) = np__0 ) ) ) ) ) )).

fof(t37_prob_4,axiom,(
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m1_prob_1(B,A)
=> ! [C] :
( m4_prob_1(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,G] :
( r2_hidden(F,B)
& r2_hidden(G,B)
& r1_tarski(F,E)
& r1_tarski(E,G)
& k1_funct_1(C,k4_xboole_0(G,F)) = np__0 ) )
=> D = k9_prob_4(A,B,C) ) ) ) ) ) )).

fof(d10_prob_4,axiom,(
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m1_prob_1(B,A)
=> ! [C] :
( m4_prob_1(C,A,B)
=> ! [D] :
( m4_prob_1(D,A,k9_prob_4(A,B,C))
=> ( D = k10_prob_4(A,B,C)
<=> ! [E] :
( r2_hidden(E,B)
=> ! [F] :
( m2_prob_4(F,A,B,C)
=> k1_funct_1(D,k2_xboole_0(E,F)) = k1_funct_1(C,E) ) ) ) ) ) ) ) )).

fof(t38_prob_4,axiom,(
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m1_prob_1(B,A)
=> ! [C] :
( m4_prob_1(C,A,B)
=> k10_prob_4(A,B,C) = k4_measure3(A,k2_prob_4(A,B),k4_prob_4(A,B,C)) ) ) ) )).

fof(t39_prob_4,axiom,(
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m1_prob_1(B,A)
=> ! [C] :
( m4_prob_1(C,A,B)
=> r1_prob_4(A,k9_prob_4(A,B,C),k10_prob_4(A,B,C)) ) ) ) )).

fof(t40_prob_4,axiom,(
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m1_prob_1(B,A)
=> ! [C] :
( m4_prob_1(C,A,B)
=> ! [D] :
( m3_prob_1(D,A,B)
=> k10_prob_1(A,B,C,D) = k1_funct_1(k10_prob_4(A,B,C),D) ) ) ) ) )).

fof(t41_prob_4,axiom,(
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m1_prob_1(B,A)
=> ! [C] :
( m4_prob_1(C,A,B)
=> ! [D] :
( m2_prob_4(D,A,B,C)
=> k10_prob_1(A,k9_prob_4(A,B,C),k10_prob_4(A,B,C),D) = np__0 ) ) ) ) )).

fof(t42_prob_4,axiom,(
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m1_prob_1(B,A)
=> ! [C] :
( m4_prob_1(C,A,B)
=> ! [D] :
( m2_subset_1(D,k1_zfmisc_1(A),k9_prob_4(A,B,C))
=> ! [E] :
( r2_hidden(E,k8_prob_4(A,B,C,D))
=> k1_funct_1(C,E) = k8_funct_2(k9_prob_4(A,B,C),k1_numbers,k10_prob_4(A,B,C),D) ) ) ) ) ) )).

fof(s1_prob_4,axiom,(
? [A] :
( m2_prob_1(A,f1_s1_prob_4,f2_s1_prob_4)
& ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> k1_prob_4(f1_s1_prob_4,f2_s1_prob_4,A,B) = f3_s1_prob_4(B) ) ) )).

fof(dt_m1_prob_4,axiom,(
! [A,B,C] :
( ( ~ v1_xboole_0(A)
& m1_prob_1(B,A)
& m4_prob_1(C,A,B) )
=> ! [D] :
( m1_prob_4(D,A,B,C)
=> m1_subset_1(D,k1_zfmisc_1(A)) ) ) )).

fof(existence_m1_prob_4,axiom,(
! [A,B,C] :
( ( ~ v1_xboole_0(A)
& m1_prob_1(B,A)
& m4_prob_1(C,A,B) )
=> ? [D] : m1_prob_4(D,A,B,C) ) )).

fof(dt_m2_prob_4,axiom,(
! [A,B,C] :
( ( ~ v1_xboole_0(A)
& m1_prob_1(B,A)
& m4_prob_1(C,A,B) )
=> ! [D] :
( m2_prob_4(D,A,B,C)
=> m3_prob_1(D,A,k9_prob_4(A,B,C)) ) ) )).

fof(existence_m2_prob_4,axiom,(
! [A,B,C] :
( ( ~ v1_xboole_0(A)
& m1_prob_1(B,A)
& m4_prob_1(C,A,B) )
=> ? [D] : m2_prob_4(D,A,B,C) ) )).

fof(redefinition_m2_prob_4,axiom,(
! [A,B,C] :
( ( ~ v1_xboole_0(A)
& m1_prob_1(B,A)
& m4_prob_1(C,A,B) )
=> ! [D] :
( m2_prob_4(D,A,B,C)
<=> m1_prob_4(D,A,B,C) ) ) )).

fof(dt_k1_prob_4,axiom,(
! [A,B,C,D] :
( ( m1_prob_1(B,A)
& m2_prob_1(C,A,B)
& m1_subset_1(D,k5_numbers) )
=> m2_subset_1(k1_prob_4(A,B,C,D),k1_zfmisc_1(A),B) ) )).

fof(redefinition_k1_prob_4,axiom,(
! [A,B,C,D] :
( ( m1_prob_1(B,A)
& m2_prob_1(C,A,B)
& m1_subset_1(D,k5_numbers) )
=> k1_prob_4(A,B,C,D) = k1_funct_1(C,D) ) )).

fof(dt_k2_prob_4,axiom,(
! [A,B] :
( m1_prob_1(B,A)
=> ( ~ v1_xboole_0(k2_prob_4(A,B))
& v1_prob_1(k2_prob_4(A,B),A)
& v1_measure1(k2_prob_4(A,B),A)
& m1_subset_1(k2_prob_4(A,B),k1_zfmisc_1(k1_zfmisc_1(A))) ) ) )).

fof(dt_k3_prob_4,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))) )
=> m1_prob_1(k3_prob_4(A,B),A) ) )).

fof(dt_k4_prob_4,axiom,(
! [A,B,C] :
( ( ~ v1_xboole_0(A)
& m1_prob_1(B,A)
& m4_prob_1(C,A,B) )
=> m3_measure1(k4_prob_4(A,B,C),A,k2_prob_4(A,B)) ) )).

fof(dt_k5_prob_4,axiom,(
! [A,B,C] :
( ( ~ v1_xboole_0(A)
& ~ 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) )
=> m4_prob_1(k5_prob_4(A,B,C),A,k3_prob_4(A,B)) ) )).

fof(dt_k6_prob_4,axiom,(
! [A,B,C] :
( ( ~ v1_xboole_0(A)
& m1_prob_1(B,A)
& m4_prob_1(C,A,B) )
=> ( ~ v1_xboole_0(k6_prob_4(A,B,C))
& m1_subset_1(k6_prob_4(A,B,C),k1_zfmisc_1(k1_zfmisc_1(A))) ) ) )).

fof(dt_k7_prob_4,axiom,(
! [A,B,C,D] :
( ( ~ v1_xboole_0(A)
& m1_prob_1(B,A)
& m4_prob_1(C,A,B)
& m1_subset_1(D,k6_prob_4(A,B,C)) )
=> m2_subset_1(k7_prob_4(A,B,C,D),k1_zfmisc_1(A),k2_measure3(A,k2_prob_4(A,B),k4_prob_4(A,B,C))) ) )).

fof(dt_k8_prob_4,axiom,(
! [A,B,C,D] :
( ( ~ v1_xboole_0(A)
& m1_prob_1(B,A)
& m4_prob_1(C,A,B)
& m1_subset_1(D,k6_prob_4(A,B,C)) )
=> ( ~ v1_xboole_0(k8_prob_4(A,B,C,D))
& m1_subset_1(k8_prob_4(A,B,C,D),k1_zfmisc_1(k1_zfmisc_1(A))) ) ) )).

fof(dt_k9_prob_4,axiom,(
! [A,B,C] :
( ( ~ v1_xboole_0(A)
& m1_prob_1(B,A)
& m4_prob_1(C,A,B) )
=> m1_prob_1(k9_prob_4(A,B,C),A) ) )).

fof(redefinition_k9_prob_4,axiom,(
! [A,B,C] :
( ( ~ v1_xboole_0(A)
& m1_prob_1(B,A)
& m4_prob_1(C,A,B) )
=> k9_prob_4(A,B,C) = k6_prob_4(A,B,C) ) )).

fof(dt_k10_prob_4,axiom,(
! [A,B,C] :
( ( ~ v1_xboole_0(A)
& m1_prob_1(B,A)
& m4_prob_1(C,A,B) )
=> m4_prob_1(k10_prob_4(A,B,C),A,k9_prob_4(A,B,C)) ) )).
%------------------------------------------------------------------------------
```