## SET007 Axioms: SET007+89.ax

```%------------------------------------------------------------------------------
% File     : SET007+89 : TPTP v7.5.0. Released v3.4.0.
% Domain   : Set Theory
% Axioms   : Independence of Events and Conditional Probability
% 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    : prob_2 [Urb08]

% Status   : Satisfiable
% Syntax   : Number of formulae    :   65 (  10 unit)
%            Number of atoms       :  434 (  55 equality)
%            Maximal formula depth :   22 (  11 average)
%            Number of connectives :  446 (  77 ~  ;  14  |;  70  &)
%                                         (  14 <=>; 271 =>;   0 <=)
%                                         (   0 <~>;   0 ~|;   0 ~&)
%            Number of predicates  :   23 (   1 propositional; 0-6 arity)
%            Number of functors    :   31 (   5 constant; 0-4 arity)
%            Number of variables   :  264 (   0 singleton; 263 !;   1 ?)
%            Maximal term depth    :    7 (   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(t1_prob_2,axiom,(
\$true )).

fof(t2_prob_2,axiom,(
\$true )).

fof(t3_prob_2,axiom,(
\$true )).

fof(t4_prob_2,axiom,(
! [A] :
( m1_subset_1(A,k1_numbers)
=> ! [B] :
( m1_subset_1(B,k1_numbers)
=> ! [C] :
( m1_subset_1(C,k1_numbers)
=> ! [D] :
( m1_subset_1(D,k1_numbers)
=> ~ ( A != np__0
& B != np__0
& ~ ( k6_real_1(D,B) = k6_real_1(C,A)
<=> k4_real_1(D,A) = k4_real_1(C,B) ) ) ) ) ) ) )).

fof(t5_prob_2,axiom,(
! [A] :
( m1_subset_1(A,k1_numbers)
=> ! [B] :
( ( v1_funct_1(B)
& v1_funct_2(B,k5_numbers,k1_numbers)
& m2_relset_1(B,k5_numbers,k1_numbers) )
=> ! [C] :
( ( v1_funct_1(C)
& v1_funct_2(C,k5_numbers,k1_numbers)
& m2_relset_1(C,k5_numbers,k1_numbers) )
=> ( ( v4_seq_2(B)
& ! [D] :
( m2_subset_1(D,k1_numbers,k5_numbers)
=> k2_seq_1(k5_numbers,k1_numbers,C,D) = k5_real_1(A,k2_seq_1(k5_numbers,k1_numbers,B,D)) ) )
=> ( v4_seq_2(C)
& k2_seq_2(C) = k5_real_1(A,k2_seq_2(B)) ) ) ) ) ) )).

fof(t6_prob_2,axiom,(
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m1_prob_1(B,A)
=> ! [C] :
( m3_prob_1(C,A,B)
=> ( k3_xboole_0(C,A) = C
& k6_prob_1(A,B,C,k5_prob_1(A,B)) = C ) ) ) ) )).

fof(d1_prob_2,axiom,(
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m1_prob_1(B,A)
=> ! [C] :
( m2_prob_1(C,A,B)
=> k2_prob_2(A,B,C) = k4_prob_1(A,C) ) ) ) )).

fof(t7_prob_2,axiom,(
\$true )).

fof(t8_prob_2,axiom,(
\$true )).

fof(t9_prob_2,axiom,(
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m1_prob_1(B,A)
=> ! [C] :
( m2_prob_1(C,A,B)
=> ! [D] :
( m3_prob_1(D,A,B)
=> ? [E] :
( m2_prob_1(E,A,B)
& ! [F] :
( m2_subset_1(F,k1_numbers,k5_numbers)
=> k1_prob_2(A,B,E,F) = k6_prob_1(A,B,k1_prob_2(A,B,C,F),D) ) ) ) ) ) ) )).

fof(t10_prob_2,axiom,(
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m1_prob_1(B,A)
=> ! [C] :
( m2_prob_1(C,A,B)
=> ! [D] :
( m2_prob_1(D,A,B)
=> ! [E] :
( m3_prob_1(E,A,B)
=> ( ( v2_prob_1(C,A)
& ! [F] :
( m2_subset_1(F,k1_numbers,k5_numbers)
=> k1_prob_2(A,B,D,F) = k6_prob_1(A,B,k1_prob_2(A,B,C,F),E) ) )
=> v2_prob_1(D,A) ) ) ) ) ) ) )).

fof(t11_prob_2,axiom,(
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> ! [C] :
( m1_prob_1(C,A)
=> ! [D] :
( m2_prob_1(D,A,C)
=> ! [E] :
( ( v1_funct_1(E)
& v1_funct_2(E,C,k1_numbers)
& m2_relset_1(E,C,k1_numbers) )
=> k2_seq_1(k5_numbers,k1_numbers,k9_prob_1(A,C,D,E),B) = k10_prob_1(A,C,E,k1_prob_2(A,C,D,B)) ) ) ) ) ) )).

fof(t12_prob_2,axiom,(
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m1_prob_1(B,A)
=> ! [C] :
( m2_prob_1(C,A,B)
=> ! [D] :
( m2_prob_1(D,A,B)
=> ! [E] :
( m3_prob_1(E,A,B)
=> ( ! [F] :
( m2_subset_1(F,k1_numbers,k5_numbers)
=> k1_prob_2(A,B,C,F) = k6_prob_1(A,B,k1_prob_2(A,B,D,F),E) )
=> k5_subset_1(A,k4_prob_1(A,D),E) = k4_prob_1(A,C) ) ) ) ) ) ) )).

fof(t13_prob_2,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,B)
=> ( ! [E] :
( m3_prob_1(E,A,B)
=> k10_prob_1(A,B,C,E) = k10_prob_1(A,B,D,E) )
=> C = D ) ) ) ) ) )).

fof(t14_prob_2,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)) )
=> ( v2_prob_1(B,A)
<=> ! [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
=> r1_tarski(k1_prob_1(A,B,k1_nat_1(C,np__1)),k1_prob_1(A,B,C)) ) ) ) )).

fof(t15_prob_2,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_tarski(k1_prob_1(A,B,C),k1_prob_1(A,B,k1_nat_1(C,np__1))) ) ) ) )).

fof(t16_prob_2,axiom,(
! [A] :
( ~ v1_xboole_0(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] :
( ( 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)
=> k1_prob_1(A,B,D) = k1_prob_1(A,C,D) )
=> B = C ) ) ) ) )).

fof(t17_prob_2,axiom,(
! [A] :
( ~ v1_xboole_0(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)) )
=> ( v2_prob_1(B,A)
<=> v3_prob_1(k3_prob_1(A,B),A) ) ) ) )).

fof(d2_prob_2,axiom,(
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m1_prob_1(B,A)
=> ! [C] :
( m2_prob_1(C,A,B)
=> k3_prob_2(A,B,C) = k3_prob_1(A,C) ) ) ) )).

fof(d3_prob_2,axiom,(
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A) )
=> ( v1_prob_2(A)
<=> ! [B,C] :
( B != C
=> r1_xboole_0(k1_funct_1(A,B),k1_funct_1(A,C)) ) ) ) )).

fof(d4_prob_2,axiom,(
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m1_prob_1(B,A)
=> ! [C] :
( m2_prob_1(C,A,B)
=> ( v1_prob_2(C)
<=> ! [D] :
( m2_subset_1(D,k1_numbers,k5_numbers)
=> ! [E] :
( m2_subset_1(E,k1_numbers,k5_numbers)
=> ( D != E
=> r1_xboole_0(k1_prob_2(A,B,C,D),k1_prob_2(A,B,C,E)) ) ) ) ) ) ) ) )).

fof(t18_prob_2,axiom,(
\$true )).

fof(t19_prob_2,axiom,(
\$true )).

fof(t20_prob_2,axiom,(
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m1_prob_1(B,A)
=> ! [C] :
( ( v1_funct_1(C)
& v1_funct_2(C,B,k1_numbers)
& m2_relset_1(C,B,k1_numbers) )
=> ( m4_prob_1(C,A,B)
<=> ( ! [D] :
( m3_prob_1(D,A,B)
=> r1_xreal_0(np__0,k10_prob_1(A,B,C,D)) )
& k2_seq_1(B,k1_numbers,C,A) = np__1
& ! [D] :
( m3_prob_1(D,A,B)
=> ! [E] :
( m3_prob_1(E,A,B)
=> ( r1_xboole_0(D,E)
=> k10_prob_1(A,B,C,k7_prob_1(A,B,D,E)) = k3_real_1(k10_prob_1(A,B,C,D),k10_prob_1(A,B,C,E)) ) ) )
& ! [D] :
( m2_prob_1(D,A,B)
=> ( v3_prob_1(D,A)
=> ( v4_seq_2(k9_prob_1(A,B,D,C))
& k2_seq_2(k9_prob_1(A,B,D,C)) = k2_seq_1(B,k1_numbers,C,k2_prob_1(A,D)) ) ) ) ) ) ) ) ) )).

fof(t21_prob_2,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)
=> ! [F] :
( m3_prob_1(F,A,B)
=> k10_prob_1(A,B,C,k7_prob_1(A,B,k7_prob_1(A,B,D,E),F)) = k3_real_1(k5_real_1(k3_real_1(k3_real_1(k10_prob_1(A,B,C,D),k10_prob_1(A,B,C,E)),k10_prob_1(A,B,C,F)),k3_real_1(k3_real_1(k10_prob_1(A,B,C,k6_prob_1(A,B,D,E)),k10_prob_1(A,B,C,k6_prob_1(A,B,E,F))),k10_prob_1(A,B,C,k6_prob_1(A,B,D,F)))),k10_prob_1(A,B,C,k6_prob_1(A,B,k6_prob_1(A,B,D,E),F))) ) ) ) ) ) ) )).

fof(t22_prob_2,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,k6_prob_1(A,B,D,E))) = k5_real_1(k10_prob_1(A,B,C,D),k10_prob_1(A,B,C,k6_prob_1(A,B,D,E))) ) ) ) ) ) )).

fof(t23_prob_2,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_xreal_0(k10_prob_1(A,B,C,k6_prob_1(A,B,D,E)),k10_prob_1(A,B,C,E))
& r1_xreal_0(k10_prob_1(A,B,C,k6_prob_1(A,B,D,E)),k10_prob_1(A,B,C,D)) ) ) ) ) ) ) )).

fof(t24_prob_2,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)
=> ! [F] :
( m3_prob_1(F,A,B)
=> ( D = k3_subset_1(A,E)
=> k10_prob_1(A,B,C,F) = k3_real_1(k10_prob_1(A,B,C,k6_prob_1(A,B,F,E)),k10_prob_1(A,B,C,k6_prob_1(A,B,F,D))) ) ) ) ) ) ) ) )).

fof(t25_prob_2,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_xreal_0(k5_real_1(k3_real_1(k10_prob_1(A,B,C,D),k10_prob_1(A,B,C,E)),np__1),k10_prob_1(A,B,C,k6_prob_1(A,B,D,E))) ) ) ) ) ) )).

fof(t26_prob_2,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) = k5_real_1(np__1,k10_prob_1(A,B,C,k8_prob_1(A,B,k5_prob_1(A,B),D))) ) ) ) ) )).

fof(t27_prob_2,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)
=> ( ~ ( ~ r1_xreal_0(np__1,k10_prob_1(A,B,C,D))
& r1_xreal_0(k10_prob_1(A,B,C,k8_prob_1(A,B,k5_prob_1(A,B),D)),np__0) )
& ~ ( ~ r1_xreal_0(k10_prob_1(A,B,C,k8_prob_1(A,B,k5_prob_1(A,B),D)),np__0)
& r1_xreal_0(np__1,k10_prob_1(A,B,C,D)) ) ) ) ) ) ) )).

fof(t28_prob_2,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)
=> ( ~ ( ~ r1_xreal_0(np__1,k10_prob_1(A,B,C,k8_prob_1(A,B,k5_prob_1(A,B),D)))
& r1_xreal_0(k10_prob_1(A,B,C,D),np__0) )
& ~ ( ~ r1_xreal_0(k10_prob_1(A,B,C,D),np__0)
& r1_xreal_0(np__1,k10_prob_1(A,B,C,k8_prob_1(A,B,k5_prob_1(A,B),D))) ) ) ) ) ) ) )).

fof(d5_prob_2,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_prob_2(A,B,C,D,E)
<=> k10_prob_1(A,B,C,k6_prob_1(A,B,D,E)) = k4_real_1(k10_prob_1(A,B,C,D),k10_prob_1(A,B,C,E)) ) ) ) ) ) ) )).

fof(d6_prob_2,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)
=> ! [F] :
( m3_prob_1(F,A,B)
=> ( r2_prob_2(A,B,C,D,E,F)
<=> ( k10_prob_1(A,B,C,k6_prob_1(A,B,k6_prob_1(A,B,D,E),F)) = k4_real_1(k4_real_1(k10_prob_1(A,B,C,D),k10_prob_1(A,B,C,E)),k10_prob_1(A,B,C,F))
& k10_prob_1(A,B,C,k6_prob_1(A,B,D,E)) = k4_real_1(k10_prob_1(A,B,C,D),k10_prob_1(A,B,C,E))
& k10_prob_1(A,B,C,k6_prob_1(A,B,D,F)) = k4_real_1(k10_prob_1(A,B,C,D),k10_prob_1(A,B,C,F))
& k10_prob_1(A,B,C,k6_prob_1(A,B,E,F)) = k4_real_1(k10_prob_1(A,B,C,E),k10_prob_1(A,B,C,F)) ) ) ) ) ) ) ) ) )).

fof(t29_prob_2,axiom,(
\$true )).

fof(t30_prob_2,axiom,(
\$true )).

fof(t31_prob_2,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_prob_2(A,B,C,D,E)
<=> r1_prob_2(A,B,C,E,D) ) ) ) ) ) ) )).

fof(t32_prob_2,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)
=> ! [F] :
( m3_prob_1(F,A,B)
=> ( r2_prob_2(A,B,C,D,E,F)
<=> ( k10_prob_1(A,B,C,k6_prob_1(A,B,k6_prob_1(A,B,D,E),F)) = k4_real_1(k4_real_1(k10_prob_1(A,B,C,D),k10_prob_1(A,B,C,E)),k10_prob_1(A,B,C,F))
& r1_prob_2(A,B,C,D,E)
& r1_prob_2(A,B,C,E,F)
& r1_prob_2(A,B,C,D,F) ) ) ) ) ) ) ) ) )).

fof(t33_prob_2,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)
=> ! [F] :
( m3_prob_1(F,A,B)
=> ( r2_prob_2(A,B,C,D,E,F)
=> r2_prob_2(A,B,C,E,D,F) ) ) ) ) ) ) ) )).

fof(t34_prob_2,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)
=> ! [F] :
( m3_prob_1(F,A,B)
=> ( r2_prob_2(A,B,C,D,E,F)
=> r2_prob_2(A,B,C,D,F,E) ) ) ) ) ) ) ) )).

fof(t35_prob_2,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)
=> ( E = k1_xboole_0
=> r1_prob_2(A,B,C,D,E) ) ) ) ) ) ) )).

fof(t36_prob_2,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)
=> r1_prob_2(A,B,C,D,k5_prob_1(A,B)) ) ) ) ) )).

fof(t37_prob_2,axiom,(
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m1_prob_1(B,A)
=> ! [C] :
( m3_prob_1(C,A,B)
=> ! [D] :
( m3_prob_1(D,A,B)
=> ! [E] :
( m4_prob_1(E,A,B)
=> ( r1_prob_2(A,B,E,C,D)
=> r1_prob_2(A,B,E,C,k8_prob_1(A,B,k5_prob_1(A,B),D)) ) ) ) ) ) ) )).

fof(t38_prob_2,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_prob_2(A,B,C,D,E)
=> r1_prob_2(A,B,C,k8_prob_1(A,B,k5_prob_1(A,B),D),k8_prob_1(A,B,k5_prob_1(A,B),E)) ) ) ) ) ) ) )).

fof(t39_prob_2,axiom,(
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m1_prob_1(B,A)
=> ! [C] :
( m3_prob_1(C,A,B)
=> ! [D] :
( m3_prob_1(D,A,B)
=> ! [E] :
( m3_prob_1(E,A,B)
=> ! [F] :
( m4_prob_1(F,A,B)
=> ( ( r1_prob_2(A,B,F,C,D)
& r1_prob_2(A,B,F,C,E)
& r1_xboole_0(D,E) )
=> r1_prob_2(A,B,F,C,k7_prob_1(A,B,D,E)) ) ) ) ) ) ) ) )).

fof(t40_prob_2,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_prob_2(A,B,C,D,E)
& ~ r1_xreal_0(np__1,k10_prob_1(A,B,C,D))
& ~ r1_xreal_0(np__1,k10_prob_1(A,B,C,E))
& r1_xreal_0(np__1,k10_prob_1(A,B,C,k7_prob_1(A,B,D,E))) ) ) ) ) ) ) )).

fof(d7_prob_2,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)
=> ( ~ r1_xreal_0(k10_prob_1(A,B,C,D),np__0)
=> ! [E] :
( m4_prob_1(E,A,B)
=> ( E = k4_prob_2(A,B,C,D)
<=> ! [F] :
( m3_prob_1(F,A,B)
=> k10_prob_1(A,B,E,F) = k6_real_1(k10_prob_1(A,B,C,k6_prob_1(A,B,F,D)),k10_prob_1(A,B,C,D)) ) ) ) ) ) ) ) ) )).

fof(t41_prob_2,axiom,(
\$true )).

fof(t42_prob_2,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_xreal_0(k10_prob_1(A,B,C,D),np__0)
=> k10_prob_1(A,B,C,k6_prob_1(A,B,E,D)) = k4_real_1(k10_prob_1(A,B,k4_prob_2(A,B,C,D),E),k10_prob_1(A,B,C,D)) ) ) ) ) ) ) )).

fof(t43_prob_2,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)
=> ! [F] :
( m3_prob_1(F,A,B)
=> ( ~ r1_xreal_0(k10_prob_1(A,B,C,k6_prob_1(A,B,D,E)),np__0)
=> k10_prob_1(A,B,C,k6_prob_1(A,B,k6_prob_1(A,B,D,E),F)) = k4_real_1(k4_real_1(k10_prob_1(A,B,C,D),k10_prob_1(A,B,k4_prob_2(A,B,C,D),E)),k10_prob_1(A,B,k4_prob_2(A,B,C,k6_prob_1(A,B,D,E)),F)) ) ) ) ) ) ) ) )).

fof(t44_prob_2,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)
=> ! [F] :
( m3_prob_1(F,A,B)
=> ( F = k3_subset_1(A,E)
=> ( r1_xreal_0(k10_prob_1(A,B,C,E),np__0)
| r1_xreal_0(k10_prob_1(A,B,C,F),np__0)
| k10_prob_1(A,B,C,D) = k3_real_1(k4_real_1(k10_prob_1(A,B,k4_prob_2(A,B,C,E),D),k10_prob_1(A,B,C,E)),k4_real_1(k10_prob_1(A,B,k4_prob_2(A,B,C,F),D),k10_prob_1(A,B,C,F))) ) ) ) ) ) ) ) ) )).

fof(t45_prob_2,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)
=> ! [F] :
( m3_prob_1(F,A,B)
=> ! [G] :
( m3_prob_1(G,A,B)
=> ( ( r1_xboole_0(E,F)
& G = k3_subset_1(A,k7_prob_1(A,B,E,F)) )
=> ( r1_xreal_0(k10_prob_1(A,B,C,E),np__0)
| r1_xreal_0(k10_prob_1(A,B,C,F),np__0)
| r1_xreal_0(k10_prob_1(A,B,C,G),np__0)
| k10_prob_1(A,B,C,D) = k3_real_1(k3_real_1(k4_real_1(k10_prob_1(A,B,k4_prob_2(A,B,C,E),D),k10_prob_1(A,B,C,E)),k4_real_1(k10_prob_1(A,B,k4_prob_2(A,B,C,F),D),k10_prob_1(A,B,C,F))),k4_real_1(k10_prob_1(A,B,k4_prob_2(A,B,C,G),D),k10_prob_1(A,B,C,G))) ) ) ) ) ) ) ) ) ) )).

fof(t46_prob_2,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_xreal_0(k10_prob_1(A,B,C,E),np__0)
=> ( k10_prob_1(A,B,k4_prob_2(A,B,C,E),D) = k10_prob_1(A,B,C,D)
<=> r1_prob_2(A,B,C,D,E) ) ) ) ) ) ) ) )).

fof(t47_prob_2,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,k4_prob_2(A,B,C,E),D) = k10_prob_1(A,B,k4_prob_2(A,B,C,k8_prob_1(A,B,k5_prob_1(A,B),E)),D)
=> ( r1_xreal_0(k10_prob_1(A,B,C,E),np__0)
| r1_xreal_0(np__1,k10_prob_1(A,B,C,E))
| r1_prob_2(A,B,C,D,E) ) ) ) ) ) ) ) )).

fof(t48_prob_2,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_xreal_0(k10_prob_1(A,B,C,E),np__0)
=> r1_xreal_0(k6_real_1(k5_real_1(k3_real_1(k10_prob_1(A,B,C,D),k10_prob_1(A,B,C,E)),np__1),k10_prob_1(A,B,C,E)),k10_prob_1(A,B,k4_prob_2(A,B,C,E),D)) ) ) ) ) ) ) )).

fof(t49_prob_2,axiom,(
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m1_prob_1(B,A)
=> ! [C] :
( m3_prob_1(C,A,B)
=> ! [D] :
( m3_prob_1(D,A,B)
=> ! [E] :
( m4_prob_1(E,A,B)
=> ~ ( ~ r1_xreal_0(k10_prob_1(A,B,E,C),np__0)
& ~ r1_xreal_0(k10_prob_1(A,B,E,D),np__0)
& k10_prob_1(A,B,k4_prob_2(A,B,E,D),C) != k6_real_1(k4_real_1(k10_prob_1(A,B,k4_prob_2(A,B,E,C),D),k10_prob_1(A,B,E,C)),k10_prob_1(A,B,E,D)) ) ) ) ) ) ) )).

fof(t50_prob_2,axiom,(
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m1_prob_1(B,A)
=> ! [C] :
( m3_prob_1(C,A,B)
=> ! [D] :
( m3_prob_1(D,A,B)
=> ! [E] :
( m3_prob_1(E,A,B)
=> ! [F] :
( m4_prob_1(F,A,B)
=> ( E = k3_subset_1(A,D)
=> ( r1_xreal_0(k10_prob_1(A,B,F,C),np__0)
| r1_xreal_0(k10_prob_1(A,B,F,D),np__0)
| r1_xreal_0(k10_prob_1(A,B,F,E),np__0)
| ( k10_prob_1(A,B,k4_prob_2(A,B,F,C),D) = k6_real_1(k4_real_1(k10_prob_1(A,B,k4_prob_2(A,B,F,D),C),k10_prob_1(A,B,F,D)),k3_real_1(k4_real_1(k10_prob_1(A,B,k4_prob_2(A,B,F,D),C),k10_prob_1(A,B,F,D)),k4_real_1(k10_prob_1(A,B,k4_prob_2(A,B,F,E),C),k10_prob_1(A,B,F,E))))
& k10_prob_1(A,B,k4_prob_2(A,B,F,C),E) = k6_real_1(k4_real_1(k10_prob_1(A,B,k4_prob_2(A,B,F,E),C),k10_prob_1(A,B,F,E)),k3_real_1(k4_real_1(k10_prob_1(A,B,k4_prob_2(A,B,F,D),C),k10_prob_1(A,B,F,D)),k4_real_1(k10_prob_1(A,B,k4_prob_2(A,B,F,E),C),k10_prob_1(A,B,F,E)))) ) ) ) ) ) ) ) ) ) )).

fof(t51_prob_2,axiom,(
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m1_prob_1(B,A)
=> ! [C] :
( m3_prob_1(C,A,B)
=> ! [D] :
( m3_prob_1(D,A,B)
=> ! [E] :
( m3_prob_1(E,A,B)
=> ! [F] :
( m3_prob_1(F,A,B)
=> ! [G] :
( m4_prob_1(G,A,B)
=> ( ( r1_xboole_0(D,E)
& F = k3_subset_1(A,k7_prob_1(A,B,D,E)) )
=> ( r1_xreal_0(k10_prob_1(A,B,G,C),np__0)
| r1_xreal_0(k10_prob_1(A,B,G,D),np__0)
| r1_xreal_0(k10_prob_1(A,B,G,E),np__0)
| r1_xreal_0(k10_prob_1(A,B,G,F),np__0)
| ( k10_prob_1(A,B,k4_prob_2(A,B,G,C),D) = k6_real_1(k4_real_1(k10_prob_1(A,B,k4_prob_2(A,B,G,D),C),k10_prob_1(A,B,G,D)),k3_real_1(k3_real_1(k4_real_1(k10_prob_1(A,B,k4_prob_2(A,B,G,D),C),k10_prob_1(A,B,G,D)),k4_real_1(k10_prob_1(A,B,k4_prob_2(A,B,G,E),C),k10_prob_1(A,B,G,E))),k4_real_1(k10_prob_1(A,B,k4_prob_2(A,B,G,F),C),k10_prob_1(A,B,G,F))))
& k10_prob_1(A,B,k4_prob_2(A,B,G,C),E) = k6_real_1(k4_real_1(k10_prob_1(A,B,k4_prob_2(A,B,G,E),C),k10_prob_1(A,B,G,E)),k3_real_1(k3_real_1(k4_real_1(k10_prob_1(A,B,k4_prob_2(A,B,G,D),C),k10_prob_1(A,B,G,D)),k4_real_1(k10_prob_1(A,B,k4_prob_2(A,B,G,E),C),k10_prob_1(A,B,G,E))),k4_real_1(k10_prob_1(A,B,k4_prob_2(A,B,G,F),C),k10_prob_1(A,B,G,F))))
& k10_prob_1(A,B,k4_prob_2(A,B,G,C),F) = k6_real_1(k4_real_1(k10_prob_1(A,B,k4_prob_2(A,B,G,F),C),k10_prob_1(A,B,G,F)),k3_real_1(k3_real_1(k4_real_1(k10_prob_1(A,B,k4_prob_2(A,B,G,D),C),k10_prob_1(A,B,G,D)),k4_real_1(k10_prob_1(A,B,k4_prob_2(A,B,G,E),C),k10_prob_1(A,B,G,E))),k4_real_1(k10_prob_1(A,B,k4_prob_2(A,B,G,F),C),k10_prob_1(A,B,G,F)))) ) ) ) ) ) ) ) ) ) ) )).

fof(t52_prob_2,axiom,(
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m1_prob_1(B,A)
=> ! [C] :
( m3_prob_1(C,A,B)
=> ! [D] :
( m3_prob_1(D,A,B)
=> ! [E] :
( m4_prob_1(E,A,B)
=> ( ~ r1_xreal_0(k10_prob_1(A,B,E,D),np__0)
=> r1_xreal_0(k5_real_1(np__1,k6_real_1(k10_prob_1(A,B,E,k8_prob_1(A,B,k5_prob_1(A,B),C)),k10_prob_1(A,B,E,D))),k10_prob_1(A,B,k4_prob_2(A,B,E,D),C)) ) ) ) ) ) ) )).

fof(redefinition_v2_prob_2,axiom,(
! [A,B,C] :
( ( ~ v1_xboole_0(A)
& m1_prob_1(B,A)
& m2_prob_1(C,A,B) )
=> ( v2_prob_2(C,A,B)
<=> v1_prob_2(C) ) ) )).

fof(dt_k1_prob_2,axiom,(
! [A,B,C,D] :
( ( ~ v1_xboole_0(A)
& m1_prob_1(B,A)
& m2_prob_1(C,A,B)
& m1_subset_1(D,k5_numbers) )
=> m3_prob_1(k1_prob_2(A,B,C,D),A,B) ) )).

fof(redefinition_k1_prob_2,axiom,(
! [A,B,C,D] :
( ( ~ v1_xboole_0(A)
& m1_prob_1(B,A)
& m2_prob_1(C,A,B)
& m1_subset_1(D,k5_numbers) )
=> k1_prob_2(A,B,C,D) = k1_funct_1(C,D) ) )).

fof(dt_k2_prob_2,axiom,(
! [A,B,C] :
( ( ~ v1_xboole_0(A)
& m1_prob_1(B,A)
& m2_prob_1(C,A,B) )
=> m3_prob_1(k2_prob_2(A,B,C),A,B) ) )).

fof(dt_k3_prob_2,axiom,(
! [A,B,C] :
( ( ~ v1_xboole_0(A)
& m1_prob_1(B,A)
& m2_prob_1(C,A,B) )
=> m2_prob_1(k3_prob_2(A,B,C),A,B) ) )).

fof(dt_k4_prob_2,axiom,(
! [A,B,C,D] :
( ( ~ v1_xboole_0(A)
& m1_prob_1(B,A)
& m4_prob_1(C,A,B)
& m3_prob_1(D,A,B) )
=> m4_prob_1(k4_prob_2(A,B,C,D),A,B) ) )).
%------------------------------------------------------------------------------
```