SET007 Axioms: SET007+912.ax
%------------------------------------------------------------------------------
% File : SET007+912 : TPTP v8.2.0. Released v3.4.0.
% Domain : Set Theory
% Axioms : Set Sequences and Monotone Class
% 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_3 [Urb08]
% Status : Satisfiable
% Syntax : Number of formulae : 117 ( 1 unt; 0 def)
% Number of atoms : 639 ( 87 equ)
% Maximal formula atoms : 15 ( 5 avg)
% Number of connectives : 574 ( 52 ~; 0 |; 180 &)
% ( 25 <=>; 317 =>; 0 <=; 0 <~>)
% Maximal formula depth : 17 ( 8 avg)
% Maximal term depth : 4 ( 1 avg)
% Number of predicates : 35 ( 34 usr; 0 prp; 1-3 aty)
% Number of functors : 50 ( 50 usr; 5 con; 0-4 aty)
% Number of variables : 399 ( 386 !; 13 ?)
% 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_3,axiom,
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A)
& v1_finseq_1(A) )
=> ~ r2_hidden(np__0,k5_finsop_1(A)) ) ).
fof(t2_prob_3,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( ( v1_relat_1(B)
& v1_funct_1(B)
& v1_finseq_1(B) )
=> ( r2_hidden(A,k5_finsop_1(B))
<=> ( A != np__0
& r1_xreal_0(A,k3_finseq_1(B)) ) ) ) ) ).
fof(t3_prob_3,axiom,
! [A] :
( v1_xreal_0(A)
=> ! [B] :
( ( v1_funct_1(B)
& v1_funct_2(B,k5_numbers,k1_numbers)
& m2_relset_1(B,k5_numbers,k1_numbers) )
=> ( ? [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
& ! [D] :
( m2_subset_1(D,k1_numbers,k5_numbers)
=> ( r1_xreal_0(C,D)
=> k8_funct_2(k5_numbers,k1_numbers,B,D) = A ) ) )
=> ( v4_seq_2(B)
& k2_seq_2(B) = A ) ) ) ) ).
fof(t4_prob_3,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( ~ v1_xboole_0(B)
=> ! [C] :
( m1_prob_1(C,B)
=> ! [D] :
( m2_prob_1(D,B,C)
=> ! [E] :
( m4_prob_1(E,B,C)
=> r1_xreal_0(np__0,k8_funct_2(k5_numbers,k1_numbers,k9_prob_1(B,C,D,E),A)) ) ) ) ) ) ).
fof(t5_prob_3,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( ~ v1_xboole_0(B)
=> ! [C] :
( m1_prob_1(C,B)
=> ! [D] :
( m2_prob_1(D,B,C)
=> ! [E] :
( m2_prob_1(E,B,C)
=> ! [F] :
( m4_prob_1(F,B,C)
=> ( r1_tarski(k7_kurato_2(B,D,A),k7_kurato_2(B,E,A))
=> r1_xreal_0(k8_funct_2(k5_numbers,k1_numbers,k9_prob_1(B,C,D,F),A),k8_funct_2(k5_numbers,k1_numbers,k9_prob_1(B,C,E,F),A)) ) ) ) ) ) ) ) ).
fof(t6_prob_3,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)
=> ( v3_prob_1(C,A)
=> v3_seqm_3(k9_prob_1(A,B,C,D)) ) ) ) ) ) ).
fof(t7_prob_3,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)
=> ( v2_prob_1(C,A)
=> v4_seqm_3(k9_prob_1(A,B,C,D)) ) ) ) ) ) ).
fof(t8_prob_3,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] :
( v1_funct_1(C)
& v1_funct_2(C,k5_numbers,k1_zfmisc_1(A))
& m2_relset_1(C,k5_numbers,k1_zfmisc_1(A))
& k1_prob_1(A,C,np__0) = k1_prob_1(A,B,np__0)
& ! [D] :
( m2_subset_1(D,k1_numbers,k5_numbers)
=> k1_prob_1(A,C,k1_nat_1(D,np__1)) = k5_subset_1(A,k1_prob_1(A,C,D),k1_prob_1(A,B,k1_nat_1(D,np__1))) ) ) ) ).
fof(t9_prob_3,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] :
( v1_funct_1(C)
& v1_funct_2(C,k5_numbers,k1_zfmisc_1(A))
& m2_relset_1(C,k5_numbers,k1_zfmisc_1(A))
& k1_prob_1(A,C,np__0) = k1_prob_1(A,B,np__0)
& ! [D] :
( m2_subset_1(D,k1_numbers,k5_numbers)
=> k1_prob_1(A,C,k1_nat_1(D,np__1)) = k4_subset_1(A,k1_prob_1(A,C,D),k1_prob_1(A,B,k1_nat_1(D,np__1))) ) ) ) ).
fof(d1_prob_3,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] :
( ( v1_funct_1(C)
& v1_funct_2(C,k5_numbers,k1_zfmisc_1(A))
& m2_relset_1(C,k5_numbers,k1_zfmisc_1(A)) )
=> ( C = k1_prob_3(A,B)
<=> ( k1_prob_1(A,C,np__0) = k1_prob_1(A,B,np__0)
& ! [D] :
( m2_subset_1(D,k1_numbers,k5_numbers)
=> k1_prob_1(A,C,k1_nat_1(D,np__1)) = k5_subset_1(A,k1_prob_1(A,C,D),k1_prob_1(A,B,k1_nat_1(D,np__1))) ) ) ) ) ) ).
fof(d2_prob_3,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] :
( ( v1_funct_1(C)
& v1_funct_2(C,k5_numbers,k1_zfmisc_1(A))
& m2_relset_1(C,k5_numbers,k1_zfmisc_1(A)) )
=> ( C = k2_prob_3(A,B)
<=> ( k1_prob_1(A,C,np__0) = k1_prob_1(A,B,np__0)
& ! [D] :
( m2_subset_1(D,k1_numbers,k5_numbers)
=> k1_prob_1(A,C,k1_nat_1(D,np__1)) = k4_subset_1(A,k1_prob_1(A,C,D),k1_prob_1(A,B,k1_nat_1(D,np__1))) ) ) ) ) ) ).
fof(t10_prob_3,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B,C] :
( ( v1_funct_1(C)
& v1_funct_2(C,k5_numbers,k1_zfmisc_1(B))
& m2_relset_1(C,k5_numbers,k1_zfmisc_1(B)) )
=> r1_tarski(k1_prob_1(B,k1_prob_3(B,C),A),k1_prob_1(B,C,A)) ) ) ).
fof(t11_prob_3,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B,C] :
( ( v1_funct_1(C)
& v1_funct_2(C,k5_numbers,k1_zfmisc_1(B))
& m2_relset_1(C,k5_numbers,k1_zfmisc_1(B)) )
=> r1_tarski(k1_prob_1(B,C,A),k1_prob_1(B,k2_prob_3(B,C),A)) ) ) ).
fof(t12_prob_3,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(k1_prob_3(A,B),A) ) ).
fof(t13_prob_3,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(k2_prob_3(A,B),A) ) ).
fof(t14_prob_3,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B,C,D] :
( ( v1_funct_1(D)
& v1_funct_2(D,k5_numbers,k1_zfmisc_1(B))
& m2_relset_1(D,k5_numbers,k1_zfmisc_1(B)) )
=> ( r2_hidden(C,k1_prob_1(B,k1_prob_3(B,D),A))
<=> ! [E] :
( m2_subset_1(E,k1_numbers,k5_numbers)
=> ( r1_xreal_0(E,A)
=> r2_hidden(C,k1_prob_1(B,D,E)) ) ) ) ) ) ).
fof(t15_prob_3,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B,C,D] :
( ( v1_funct_1(D)
& v1_funct_2(D,k5_numbers,k1_zfmisc_1(B))
& m2_relset_1(D,k5_numbers,k1_zfmisc_1(B)) )
=> ( r2_hidden(C,k1_prob_1(B,k2_prob_3(B,D),A))
<=> ? [E] :
( m2_subset_1(E,k1_numbers,k5_numbers)
& r1_xreal_0(E,A)
& r2_hidden(C,k1_prob_1(B,D,E)) ) ) ) ) ).
fof(t16_prob_3,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)) )
=> k4_prob_1(A,k1_prob_3(A,B)) = k4_prob_1(A,B) ) ).
fof(t17_prob_3,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)) )
=> k1_kurato_2(A,k2_prob_3(A,B)) = k1_kurato_2(A,B) ) ).
fof(t18_prob_3,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] :
( v1_funct_1(C)
& v1_funct_2(C,k5_numbers,k1_zfmisc_1(A))
& m2_relset_1(C,k5_numbers,k1_zfmisc_1(A))
& k1_prob_1(A,C,np__0) = k1_prob_1(A,B,np__0)
& ! [D] :
( m2_subset_1(D,k1_numbers,k5_numbers)
=> k1_prob_1(A,C,k1_nat_1(D,np__1)) = k6_subset_1(A,k1_prob_1(A,B,k1_nat_1(D,np__1)),k1_prob_1(A,k2_prob_3(A,B),D)) ) ) ) ).
fof(d3_prob_3,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] :
( ( v1_funct_1(C)
& v1_funct_2(C,k5_numbers,k1_zfmisc_1(A))
& m2_relset_1(C,k5_numbers,k1_zfmisc_1(A)) )
=> ( C = k3_prob_3(A,B)
<=> ( k1_prob_1(A,C,np__0) = k1_prob_1(A,B,np__0)
& ! [D] :
( m2_subset_1(D,k1_numbers,k5_numbers)
=> k1_prob_1(A,C,k1_nat_1(D,np__1)) = k6_subset_1(A,k1_prob_1(A,B,k1_nat_1(D,np__1)),k1_prob_1(A,k2_prob_3(A,B),D)) ) ) ) ) ) ).
fof(t19_prob_3,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B,C,D] :
( ( v1_funct_1(D)
& v1_funct_2(D,k5_numbers,k1_zfmisc_1(B))
& m2_relset_1(D,k5_numbers,k1_zfmisc_1(B)) )
=> ( r2_hidden(C,k1_prob_1(B,k3_prob_3(B,D),A))
<=> ( r2_hidden(C,k1_prob_1(B,D,A))
& ! [E] :
( m2_subset_1(E,k1_numbers,k5_numbers)
=> ~ ( ~ r1_xreal_0(A,E)
& r2_hidden(C,k1_prob_1(B,D,E)) ) ) ) ) ) ) ).
fof(t20_prob_3,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B,C] :
( ( v1_funct_1(C)
& v1_funct_2(C,k5_numbers,k1_zfmisc_1(B))
& m2_relset_1(C,k5_numbers,k1_zfmisc_1(B)) )
=> r1_tarski(k1_prob_1(B,k3_prob_3(B,C),A),k1_prob_1(B,C,A)) ) ) ).
fof(t21_prob_3,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B,C] :
( ( v1_funct_1(C)
& v1_funct_2(C,k5_numbers,k1_zfmisc_1(B))
& m2_relset_1(C,k5_numbers,k1_zfmisc_1(B)) )
=> r1_tarski(k1_prob_1(B,k3_prob_3(B,C),A),k1_prob_1(B,k2_prob_3(B,C),A)) ) ) ).
fof(t22_prob_3,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)) )
=> k2_prob_3(A,k3_prob_3(A,B)) = k2_prob_3(A,B) ) ).
fof(t23_prob_3,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)) )
=> k1_kurato_2(A,k3_prob_3(A,B)) = k1_kurato_2(A,B) ) ).
fof(d4_prob_3,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)) )
=> ( v1_prob_2(B)
<=> ! [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
=> ! [D] :
( m2_subset_1(D,k1_numbers,k5_numbers)
=> ( C != D
=> r1_xboole_0(k1_prob_1(A,B,C),k1_prob_1(A,B,D)) ) ) ) ) ) ).
fof(t24_prob_3,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)) )
=> v1_prob_2(k3_prob_3(A,B)) ) ).
fof(d5_prob_3,axiom,
! [A,B] :
( m1_prob_1(B,A)
=> ! [C] :
( m2_prob_1(C,A,B)
=> k4_prob_3(A,B,C) = k1_prob_3(A,C) ) ) ).
fof(d6_prob_3,axiom,
! [A,B] :
( m1_prob_1(B,A)
=> ! [C] :
( m2_prob_1(C,A,B)
=> k5_prob_3(A,B,C) = k2_prob_3(A,C) ) ) ).
fof(d7_prob_3,axiom,
! [A,B] :
( m1_prob_1(B,A)
=> ! [C] :
( m2_prob_1(C,A,B)
=> k6_prob_3(A,B,C) = k3_prob_3(A,C) ) ) ).
fof(t25_prob_3,axiom,
! [A,B] :
( m1_prob_1(B,A)
=> ! [C] :
( m2_prob_1(C,A,B)
=> ! [D] :
( m2_prob_1(D,A,B)
=> ( C = k4_prob_3(A,B,D)
=> ( k1_prob_1(A,C,np__0) = k1_prob_1(A,D,np__0)
& ! [E] :
( m2_subset_1(E,k1_numbers,k5_numbers)
=> k1_prob_1(A,C,k1_nat_1(E,np__1)) = k5_subset_1(A,k1_prob_1(A,C,E),k1_prob_1(A,D,k1_nat_1(E,np__1))) ) ) ) ) ) ) ).
fof(t26_prob_3,axiom,
! [A,B] :
( m1_prob_1(B,A)
=> ! [C] :
( m2_prob_1(C,A,B)
=> ! [D] :
( m2_prob_1(D,A,B)
=> ( C = k5_prob_3(A,B,D)
=> ( k1_prob_1(A,C,np__0) = k1_prob_1(A,D,np__0)
& ! [E] :
( m2_subset_1(E,k1_numbers,k5_numbers)
=> k1_prob_1(A,C,k1_nat_1(E,np__1)) = k4_subset_1(A,k1_prob_1(A,C,E),k1_prob_1(A,D,k1_nat_1(E,np__1))) ) ) ) ) ) ) ).
fof(t27_prob_3,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B,C] :
( m1_prob_1(C,B)
=> ! [D] :
( m2_prob_1(D,B,C)
=> r1_tarski(k1_prob_1(B,k4_prob_3(B,C,D),A),k1_prob_1(B,D,A)) ) ) ) ).
fof(t28_prob_3,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B,C] :
( m1_prob_1(C,B)
=> ! [D] :
( m2_prob_1(D,B,C)
=> r1_tarski(k1_prob_1(B,D,A),k1_prob_1(B,k5_prob_3(B,C,D),A)) ) ) ) ).
fof(t29_prob_3,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B,C,D] :
( m1_prob_1(D,B)
=> ! [E] :
( m2_prob_1(E,B,D)
=> ( r2_hidden(C,k1_prob_1(B,k4_prob_3(B,D,E),A))
<=> ! [F] :
( m2_subset_1(F,k1_numbers,k5_numbers)
=> ( r1_xreal_0(F,A)
=> r2_hidden(C,k1_prob_1(B,E,F)) ) ) ) ) ) ) ).
fof(t30_prob_3,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B,C,D] :
( m1_prob_1(D,B)
=> ! [E] :
( m2_prob_1(E,B,D)
=> ( r2_hidden(C,k1_prob_1(B,k5_prob_3(B,D,E),A))
<=> ? [F] :
( m2_subset_1(F,k1_numbers,k5_numbers)
& r1_xreal_0(F,A)
& r2_hidden(C,k1_prob_1(B,E,F)) ) ) ) ) ) ).
fof(t31_prob_3,axiom,
! [A,B] :
( m1_prob_1(B,A)
=> ! [C] :
( m2_prob_1(C,A,B)
=> v2_prob_1(k4_prob_3(A,B,C),A) ) ) ).
fof(t32_prob_3,axiom,
! [A,B] :
( m1_prob_1(B,A)
=> ! [C] :
( m2_prob_1(C,A,B)
=> v3_prob_1(k5_prob_3(A,B,C),A) ) ) ).
fof(t33_prob_3,axiom,
! [A,B] :
( m1_prob_1(B,A)
=> ! [C] :
( m2_prob_1(C,A,B)
=> k4_prob_1(A,k4_prob_3(A,B,C)) = k4_prob_1(A,C) ) ) ).
fof(t34_prob_3,axiom,
! [A,B] :
( m1_prob_1(B,A)
=> ! [C] :
( m2_prob_1(C,A,B)
=> k1_kurato_2(A,k5_prob_3(A,B,C)) = k1_kurato_2(A,C) ) ) ).
fof(t35_prob_3,axiom,
! [A,B] :
( m1_prob_1(B,A)
=> ! [C] :
( m2_prob_1(C,A,B)
=> ! [D] :
( m2_prob_1(D,A,B)
=> ( C = k6_prob_3(A,B,D)
=> ( k1_prob_1(A,C,np__0) = k1_prob_1(A,D,np__0)
& ! [E] :
( m2_subset_1(E,k1_numbers,k5_numbers)
=> k1_prob_1(A,C,k1_nat_1(E,np__1)) = k6_subset_1(A,k1_prob_1(A,D,k1_nat_1(E,np__1)),k1_prob_1(A,k5_prob_3(A,B,D),E)) ) ) ) ) ) ) ).
fof(t36_prob_3,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B,C,D] :
( m1_prob_1(D,B)
=> ! [E] :
( m2_prob_1(E,B,D)
=> ( r2_hidden(C,k1_prob_1(B,k6_prob_3(B,D,E),A))
<=> ( r2_hidden(C,k1_prob_1(B,E,A))
& ! [F] :
( m2_subset_1(F,k1_numbers,k5_numbers)
=> ~ ( ~ r1_xreal_0(A,F)
& r2_hidden(C,k1_prob_1(B,E,F)) ) ) ) ) ) ) ) ).
fof(t37_prob_3,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B,C] :
( m1_prob_1(C,B)
=> ! [D] :
( m2_prob_1(D,B,C)
=> r1_tarski(k1_prob_1(B,k6_prob_3(B,C,D),A),k1_prob_1(B,D,A)) ) ) ) ).
fof(t38_prob_3,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B,C] :
( m1_prob_1(C,B)
=> ! [D] :
( m2_prob_1(D,B,C)
=> r1_tarski(k1_prob_1(B,k6_prob_3(B,C,D),A),k1_prob_1(B,k5_prob_3(B,C,D),A)) ) ) ) ).
fof(t39_prob_3,axiom,
! [A,B] :
( m1_prob_1(B,A)
=> ! [C] :
( m2_prob_1(C,A,B)
=> k5_prob_3(A,B,k6_prob_3(A,B,C)) = k5_prob_3(A,B,C) ) ) ).
fof(t40_prob_3,axiom,
! [A,B] :
( m1_prob_1(B,A)
=> ! [C] :
( m2_prob_1(C,A,B)
=> k1_kurato_2(A,k6_prob_3(A,B,C)) = k1_kurato_2(A,C) ) ) ).
fof(t41_prob_3,axiom,
! [A,B] :
( m1_prob_1(B,A)
=> ! [C] :
( m2_prob_1(C,A,B)
=> v1_prob_2(k6_prob_3(A,B,C)) ) ) ).
fof(t42_prob_3,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)
=> v3_seqm_3(k9_prob_1(A,B,k5_prob_3(A,B,C),D)) ) ) ) ) ).
fof(t43_prob_3,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)
=> v4_seqm_3(k9_prob_1(A,B,k4_prob_3(A,B,C),D)) ) ) ) ) ).
fof(t44_prob_3,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)
=> v3_seqm_3(k1_series_1(k9_prob_1(A,B,C,D))) ) ) ) ) ).
fof(t45_prob_3,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)
=> k8_funct_2(k5_numbers,k1_numbers,k9_prob_1(A,B,k5_prob_3(A,B,C),D),np__0) = k8_funct_2(k5_numbers,k1_numbers,k1_series_1(k9_prob_1(A,B,C,D)),np__0) ) ) ) ) ).
fof(t46_prob_3,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)
=> ( v4_seq_2(k9_prob_1(A,B,k5_prob_3(A,B,C),D))
& k2_seq_2(k9_prob_1(A,B,k5_prob_3(A,B,C),D)) = k1_rinfsup1(k9_prob_1(A,B,k5_prob_3(A,B,C),D))
& k2_seq_2(k9_prob_1(A,B,k5_prob_3(A,B,C),D)) = k1_funct_1(D,k1_kurato_2(A,C)) ) ) ) ) ) ).
fof(t47_prob_3,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)
=> ( ~ r1_xreal_0(E,D)
=> r1_xboole_0(k7_kurato_2(A,k5_prob_3(A,B,C),D),k7_kurato_2(A,C,E)) ) ) ) ) ) ) ) ).
fof(t48_prob_3,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( ~ v1_xboole_0(B)
=> ! [C] :
( m1_prob_1(C,B)
=> ! [D] :
( m2_prob_1(D,B,C)
=> ! [E] :
( m4_prob_1(E,B,C)
=> ( v1_prob_2(D)
=> k8_funct_2(k5_numbers,k1_numbers,k9_prob_1(B,C,k5_prob_3(B,C,D),E),A) = k8_funct_2(k5_numbers,k1_numbers,k1_series_1(k9_prob_1(B,C,D,E)),A) ) ) ) ) ) ) ).
fof(t49_prob_3,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)
=> ( v1_prob_2(C)
=> k9_prob_1(A,B,k5_prob_3(A,B,C),D) = k1_series_1(k9_prob_1(A,B,C,D)) ) ) ) ) ) ).
fof(t50_prob_3,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)
=> ( v1_prob_2(C)
=> ( v4_seq_2(k1_series_1(k9_prob_1(A,B,C,D)))
& k2_seq_2(k1_series_1(k9_prob_1(A,B,C,D))) = k1_rinfsup1(k1_series_1(k9_prob_1(A,B,C,D)))
& k2_seq_2(k1_series_1(k9_prob_1(A,B,C,D))) = k1_funct_1(D,k1_kurato_2(A,C)) ) ) ) ) ) ) ).
fof(t51_prob_3,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)
=> ( v1_prob_2(C)
=> k1_funct_1(D,k1_kurato_2(A,C)) = k2_series_1(k9_prob_1(A,B,C,D)) ) ) ) ) ) ).
fof(t52_prob_3,axiom,
! [A] :
? [B] :
( m2_finseq_1(B,k1_pcomps_1(A))
& ! [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
=> ( r2_hidden(C,k4_relset_1(k5_numbers,k1_pcomps_1(A),B))
=> k7_prob_3(A,B,C) = A ) ) ) ).
fof(t53_prob_3,axiom,
! [A,B] :
( m2_finseq_1(B,k1_pcomps_1(A))
=> m1_subset_1(k3_tarski(k2_relat_1(B)),k1_zfmisc_1(A)) ) ).
fof(t54_prob_3,axiom,
! [A,B,C] :
( m2_finseq_1(C,k1_pcomps_1(A))
=> ( r2_hidden(B,k8_prob_3(A,C))
<=> ? [D] :
( m2_subset_1(D,k1_numbers,k5_numbers)
& r2_hidden(D,k4_relset_1(k5_numbers,k1_pcomps_1(A),C))
& r2_hidden(B,k7_prob_3(A,C,D)) ) ) ) ).
fof(d8_prob_3,axiom,
! [A,B] :
( m2_finseq_1(B,k1_pcomps_1(A))
=> ! [C] :
( m2_finseq_1(C,k1_pcomps_1(A))
=> ( C = k9_prob_3(A,B)
<=> ( k3_finseq_1(C) = k3_finseq_1(B)
& ! [D] :
( m2_subset_1(D,k1_numbers,k5_numbers)
=> ( r2_hidden(D,k4_relset_1(k5_numbers,k1_pcomps_1(A),C))
=> k7_prob_3(A,C,D) = k3_subset_1(A,k7_prob_3(A,B,D)) ) ) ) ) ) ) ).
fof(d9_prob_3,axiom,
! [A,B] :
( m2_finseq_1(B,k1_pcomps_1(A))
=> ( ( B != k1_xboole_0
=> k10_prob_3(A,B) = k3_subset_1(A,k8_prob_3(A,k9_prob_3(A,B))) )
& ( B = k1_xboole_0
=> k10_prob_3(A,B) = k1_xboole_0 ) ) ) ).
fof(t55_prob_3,axiom,
! [A,B] :
( m2_finseq_1(B,k1_pcomps_1(A))
=> k4_relset_1(k5_numbers,k1_pcomps_1(A),k9_prob_3(A,B)) = k4_relset_1(k5_numbers,k1_pcomps_1(A),B) ) ).
fof(t56_prob_3,axiom,
! [A,B,C] :
( m2_finseq_1(C,k1_pcomps_1(A))
=> ( C != k1_xboole_0
=> ( r2_hidden(B,k10_prob_3(A,C))
<=> ! [D] :
( m2_subset_1(D,k1_numbers,k5_numbers)
=> ( r2_hidden(D,k4_relset_1(k5_numbers,k1_pcomps_1(A),C))
=> r2_hidden(B,k7_prob_3(A,C,D)) ) ) ) ) ) ).
fof(t57_prob_3,axiom,
! [A,B,C] :
( m2_finseq_1(C,k1_pcomps_1(A))
=> ( C != k1_xboole_0
=> ( r2_hidden(B,k1_setfam_1(k2_relat_1(C)))
<=> ! [D] :
( m2_subset_1(D,k1_numbers,k5_numbers)
=> ( r2_hidden(D,k4_relset_1(k5_numbers,k1_pcomps_1(A),C))
=> r2_hidden(B,k7_prob_3(A,C,D)) ) ) ) ) ) ).
fof(t58_prob_3,axiom,
! [A,B] :
( m2_finseq_1(B,k1_pcomps_1(A))
=> k10_prob_3(A,B) = k1_setfam_1(k2_relat_1(B)) ) ).
fof(t59_prob_3,axiom,
! [A,B] :
( m2_finseq_1(B,k1_pcomps_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(D,k4_relset_1(k5_numbers,k1_pcomps_1(A),B))
=> k1_prob_1(A,C,D) = k7_prob_3(A,B,D) ) )
& ! [D] :
( m2_subset_1(D,k1_numbers,k5_numbers)
=> ( ~ r2_hidden(D,k4_relset_1(k5_numbers,k1_pcomps_1(A),B))
=> k1_prob_1(A,C,D) = k1_xboole_0 ) ) ) ) ).
fof(t60_prob_3,axiom,
! [A,B] :
( m2_finseq_1(B,k1_pcomps_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(D,k4_relset_1(k5_numbers,k1_pcomps_1(A),B))
=> k1_prob_1(A,C,D) = k7_prob_3(A,B,D) ) )
& ! [D] :
( m2_subset_1(D,k1_numbers,k5_numbers)
=> ( ~ r2_hidden(D,k4_relset_1(k5_numbers,k1_pcomps_1(A),B))
=> k1_prob_1(A,C,D) = k1_xboole_0 ) ) )
=> ( k1_prob_1(A,C,np__0) = k1_xboole_0
& k1_kurato_2(A,C) = k8_prob_3(A,B) ) ) ) ) ).
fof(d10_prob_3,axiom,
! [A,B] :
( m1_prob_1(B,A)
=> ! [C] :
( m2_finseq_1(C,k1_pcomps_1(A))
=> ( m1_prob_3(C,A,B)
<=> ! [D] :
( m2_subset_1(D,k1_numbers,k5_numbers)
=> ( r2_hidden(D,k4_relset_1(k5_numbers,k1_pcomps_1(A),C))
=> r2_hidden(k7_prob_3(A,C,D),B) ) ) ) ) ) ).
fof(t61_prob_3,axiom,
! [A,B] :
( m1_prob_1(B,A)
=> ! [C] :
( m1_prob_3(C,A,B)
=> ? [D] :
( m2_prob_1(D,A,B)
& ! [E] :
( m2_subset_1(E,k1_numbers,k5_numbers)
=> ( r2_hidden(E,k4_relset_1(k5_numbers,k1_pcomps_1(A),C))
=> k1_prob_1(A,D,E) = k11_prob_3(A,B,C,E) ) )
& ! [E] :
( m2_subset_1(E,k1_numbers,k5_numbers)
=> ( ~ r2_hidden(E,k4_relset_1(k5_numbers,k1_pcomps_1(A),C))
=> k1_prob_1(A,D,E) = k1_xboole_0 ) ) ) ) ) ).
fof(t62_prob_3,axiom,
! [A,B] :
( m1_prob_1(B,A)
=> ! [C] :
( m1_prob_3(C,A,B)
=> r2_hidden(k8_prob_3(A,C),B) ) ) ).
fof(d11_prob_3,axiom,
! [A,B] :
( m1_prob_1(B,A)
=> ! [C] :
( m1_prob_3(C,A,B)
=> k12_prob_3(A,B,C) = k9_prob_3(A,C) ) ) ).
fof(t63_prob_3,axiom,
! [A,B] :
( m1_prob_1(B,A)
=> ! [C] :
( m1_prob_3(C,A,B)
=> r2_hidden(k10_prob_3(A,C),B) ) ) ).
fof(t64_prob_3,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m1_prob_1(B,A)
=> ! [C] :
( m4_prob_1(C,A,B)
=> ! [D] :
( m1_prob_3(D,A,B)
=> k1_relat_1(k5_relat_1(D,C)) = k4_relset_1(k5_numbers,k1_pcomps_1(A),D) ) ) ) ) ).
fof(t65_prob_3,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m1_prob_1(B,A)
=> ! [C] :
( m4_prob_1(C,A,B)
=> ! [D] :
( m1_prob_3(D,A,B)
=> m2_finseq_1(k5_relat_1(D,C),k1_numbers) ) ) ) ) ).
fof(t66_prob_3,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m1_prob_1(B,A)
=> ! [C] :
( m4_prob_1(C,A,B)
=> ! [D] :
( m1_prob_3(D,A,B)
=> k3_finseq_1(k13_prob_3(A,B,D,C)) = k3_finseq_1(D) ) ) ) ) ).
fof(t67_prob_3,axiom,
! [A] :
( m2_finseq_1(A,k1_numbers)
=> ( k3_finseq_1(A) = np__0
=> k15_rvsum_1(A) = np__0 ) ) ).
fof(t68_prob_3,axiom,
! [A] :
( m2_finseq_1(A,k1_numbers)
=> ~ ( r1_xreal_0(np__1,k3_finseq_1(A))
& ! [B] :
( ( v1_funct_1(B)
& v1_funct_2(B,k5_numbers,k1_numbers)
& m2_relset_1(B,k5_numbers,k1_numbers) )
=> ~ ( k8_funct_2(k5_numbers,k1_numbers,B,np__1) = k1_funct_1(A,np__1)
& ! [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
=> ~ ( np__0 != C
& ~ r1_xreal_0(k3_finseq_1(A),C)
& k8_funct_2(k5_numbers,k1_numbers,B,k1_nat_1(C,np__1)) != k2_xcmplx_0(k8_funct_2(k5_numbers,k1_numbers,B,C),k1_funct_1(A,k1_nat_1(C,np__1))) ) )
& k15_rvsum_1(A) = k8_funct_2(k5_numbers,k1_numbers,B,k3_finseq_1(A)) ) ) ) ) ).
fof(t69_prob_3,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m1_prob_1(B,A)
=> ! [C] :
( m4_prob_1(C,A,B)
=> ! [D] :
( m1_prob_3(D,A,B)
=> ! [E] :
( m2_prob_1(E,A,B)
=> ( ( ! [F] :
( m2_subset_1(F,k1_numbers,k5_numbers)
=> ( r2_hidden(F,k4_relset_1(k5_numbers,k1_pcomps_1(A),D))
=> k7_kurato_2(A,E,F) = k11_prob_3(A,B,D,F) ) )
& ! [F] :
( m2_subset_1(F,k1_numbers,k5_numbers)
=> ( ~ r2_hidden(F,k4_relset_1(k5_numbers,k1_pcomps_1(A),D))
=> k7_kurato_2(A,E,F) = k1_xboole_0 ) ) )
=> ( v4_seq_2(k1_series_1(k9_prob_1(A,B,E,C)))
& k2_series_1(k9_prob_1(A,B,E,C)) = k8_funct_2(k5_numbers,k1_numbers,k1_series_1(k9_prob_1(A,B,E,C)),k3_finseq_1(D))
& r1_xreal_0(k1_funct_1(C,k1_kurato_2(A,E)),k2_series_1(k9_prob_1(A,B,E,C)))
& k15_rvsum_1(k13_prob_3(A,B,D,C)) = k2_series_1(k9_prob_1(A,B,E,C)) ) ) ) ) ) ) ) ).
fof(t70_prob_3,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m1_prob_1(B,A)
=> ! [C] :
( m4_prob_1(C,A,B)
=> ! [D] :
( m1_prob_3(D,A,B)
=> ( r1_xreal_0(k1_funct_1(C,k8_prob_3(A,D)),k15_rvsum_1(k13_prob_3(A,B,D,C)))
& ( v1_prob_2(D)
=> k1_funct_1(C,k8_prob_3(A,D)) = k15_rvsum_1(k13_prob_3(A,B,D,C)) ) ) ) ) ) ) ).
fof(d12_prob_3,axiom,
! [A,B] :
( m1_subset_1(B,k1_zfmisc_1(k1_zfmisc_1(A)))
=> ( v2_prob_3(B,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)) )
=> ( ( v3_prob_1(C,A)
& ! [D] :
( m2_subset_1(D,k1_numbers,k5_numbers)
=> r2_hidden(k1_prob_1(A,C,D),B) ) )
=> r2_hidden(k1_kurato_2(A,C),B) ) ) ) ) ).
fof(d13_prob_3,axiom,
! [A,B] :
( m1_subset_1(B,k1_zfmisc_1(k1_zfmisc_1(A)))
=> ( v3_prob_3(B,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)) )
=> ( ( v2_prob_1(C,A)
& ! [D] :
( m2_subset_1(D,k1_numbers,k5_numbers)
=> r2_hidden(k1_prob_1(A,C,D),B) ) )
=> r2_hidden(k4_prob_1(A,C),B) ) ) ) ) ).
fof(t71_prob_3,axiom,
! [A,B] :
( m1_subset_1(B,k1_zfmisc_1(k1_zfmisc_1(A)))
=> ( v2_prob_3(B,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)) )
=> ( ( v3_prob_1(C,A)
& ! [D] :
( m2_subset_1(D,k1_numbers,k5_numbers)
=> r2_hidden(k1_prob_1(A,C,D),B) ) )
=> r2_hidden(k3_setlim_1(A,C),B) ) ) ) ) ).
fof(t72_prob_3,axiom,
! [A,B] :
( m1_subset_1(B,k1_zfmisc_1(k1_zfmisc_1(A)))
=> ( v3_prob_3(B,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)) )
=> ( ( v2_prob_1(C,A)
& ! [D] :
( m2_subset_1(D,k1_numbers,k5_numbers)
=> r2_hidden(k1_prob_1(A,C,D),B) ) )
=> r2_hidden(k3_setlim_1(A,C),B) ) ) ) ) ).
fof(t73_prob_3,axiom,
! [A] :
( v2_prob_3(k1_pcomps_1(A),A)
& v3_prob_3(k1_pcomps_1(A),A) ) ).
fof(d14_prob_3,axiom,
! [A,B] :
( m1_subset_1(B,k1_zfmisc_1(k1_zfmisc_1(A)))
=> ( m2_prob_3(B,A)
<=> ( v2_prob_3(B,A)
& v3_prob_3(B,A) ) ) ) ).
fof(t74_prob_3,axiom,
! [A,B] :
( m2_prob_3(A,B)
<=> ( r1_tarski(A,k1_pcomps_1(B))
& ! [C] :
( ( v1_funct_1(C)
& v1_funct_2(C,k5_numbers,k1_zfmisc_1(B))
& m2_relset_1(C,k5_numbers,k1_zfmisc_1(B)) )
=> ( ( v1_setlim_1(C,B)
& ! [D] :
( m2_subset_1(D,k1_numbers,k5_numbers)
=> r2_hidden(k1_prob_1(B,C,D),A) ) )
=> r2_hidden(k3_setlim_1(B,C),A) ) ) ) ) ).
fof(t75_prob_3,axiom,
! [A,B] :
( ( ~ v1_xboole_0(B)
& v1_prob_1(B,A)
& v2_finsub_1(B)
& m1_subset_1(B,k1_zfmisc_1(k1_zfmisc_1(A))) )
=> ( m1_prob_1(B,A)
<=> m2_prob_3(B,A) ) ) ).
fof(t76_prob_3,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> m2_prob_3(k1_pcomps_1(A),A) ) ).
fof(t77_prob_3,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(k1_zfmisc_1(A)))
=> ? [C] :
( m2_prob_3(C,A)
& r1_tarski(B,C)
& ! [D] :
( ( r1_tarski(B,D)
& m2_prob_3(D,A) )
=> r1_tarski(C,D) ) ) ) ) ).
fof(d15_prob_3,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(k1_zfmisc_1(A)))
=> ! [C] :
( m2_prob_3(C,A)
=> ( C = k14_prob_3(A,B)
<=> ( r1_tarski(B,C)
& ! [D] :
( ( r1_tarski(B,D)
& m2_prob_3(D,A) )
=> r1_tarski(C,D) ) ) ) ) ) ) ).
fof(t78_prob_3,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( ( ~ v1_xboole_0(B)
& v1_prob_1(B,A)
& v2_finsub_1(B)
& m1_subset_1(B,k1_zfmisc_1(k1_zfmisc_1(A))) )
=> ( ~ v1_xboole_0(k14_prob_3(A,B))
& v1_prob_1(k14_prob_3(A,B),A)
& v2_finsub_1(k14_prob_3(A,B))
& m1_subset_1(k14_prob_3(A,B),k1_zfmisc_1(k1_zfmisc_1(A))) ) ) ) ).
fof(t79_prob_3,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( ( ~ v1_xboole_0(B)
& v1_prob_1(B,A)
& v2_finsub_1(B)
& m1_subset_1(B,k1_zfmisc_1(k1_zfmisc_1(A))) )
=> k11_prob_1(A,B) = k14_prob_3(A,B) ) ) ).
fof(dt_m1_prob_3,axiom,
! [A,B] :
( m1_prob_1(B,A)
=> ! [C] :
( m1_prob_3(C,A,B)
=> m2_finseq_1(C,k1_pcomps_1(A)) ) ) ).
fof(existence_m1_prob_3,axiom,
! [A,B] :
( m1_prob_1(B,A)
=> ? [C] : m1_prob_3(C,A,B) ) ).
fof(dt_m2_prob_3,axiom,
! [A,B] :
( m2_prob_3(B,A)
=> m1_subset_1(B,k1_zfmisc_1(k1_zfmisc_1(A))) ) ).
fof(existence_m2_prob_3,axiom,
! [A] :
? [B] : m2_prob_3(B,A) ).
fof(redefinition_v1_prob_3,axiom,
! [A,B] :
( ( v1_funct_1(B)
& v1_funct_2(B,k5_numbers,k1_zfmisc_1(A))
& m1_relset_1(B,k5_numbers,k1_zfmisc_1(A)) )
=> ( v1_prob_3(B,A)
<=> v1_prob_2(B) ) ) ).
fof(dt_k1_prob_3,axiom,
! [A,B] :
( ( v1_funct_1(B)
& v1_funct_2(B,k5_numbers,k1_zfmisc_1(A))
& m1_relset_1(B,k5_numbers,k1_zfmisc_1(A)) )
=> ( v1_funct_1(k1_prob_3(A,B))
& v1_funct_2(k1_prob_3(A,B),k5_numbers,k1_zfmisc_1(A))
& m2_relset_1(k1_prob_3(A,B),k5_numbers,k1_zfmisc_1(A)) ) ) ).
fof(dt_k2_prob_3,axiom,
! [A,B] :
( ( v1_funct_1(B)
& v1_funct_2(B,k5_numbers,k1_zfmisc_1(A))
& m1_relset_1(B,k5_numbers,k1_zfmisc_1(A)) )
=> ( v1_funct_1(k2_prob_3(A,B))
& v1_funct_2(k2_prob_3(A,B),k5_numbers,k1_zfmisc_1(A))
& m2_relset_1(k2_prob_3(A,B),k5_numbers,k1_zfmisc_1(A)) ) ) ).
fof(dt_k3_prob_3,axiom,
! [A,B] :
( ( v1_funct_1(B)
& v1_funct_2(B,k5_numbers,k1_zfmisc_1(A))
& m1_relset_1(B,k5_numbers,k1_zfmisc_1(A)) )
=> ( v1_funct_1(k3_prob_3(A,B))
& v1_funct_2(k3_prob_3(A,B),k5_numbers,k1_zfmisc_1(A))
& m2_relset_1(k3_prob_3(A,B),k5_numbers,k1_zfmisc_1(A)) ) ) ).
fof(dt_k4_prob_3,axiom,
! [A,B,C] :
( ( m1_prob_1(B,A)
& m2_prob_1(C,A,B) )
=> m2_prob_1(k4_prob_3(A,B,C),A,B) ) ).
fof(dt_k5_prob_3,axiom,
! [A,B,C] :
( ( m1_prob_1(B,A)
& m2_prob_1(C,A,B) )
=> m2_prob_1(k5_prob_3(A,B,C),A,B) ) ).
fof(dt_k6_prob_3,axiom,
! [A,B,C] :
( ( m1_prob_1(B,A)
& m2_prob_1(C,A,B) )
=> m2_prob_1(k6_prob_3(A,B,C),A,B) ) ).
fof(dt_k7_prob_3,axiom,
! [A,B,C] :
( ( m1_finseq_1(B,k1_pcomps_1(A))
& m1_subset_1(C,k5_numbers) )
=> m1_subset_1(k7_prob_3(A,B,C),k1_zfmisc_1(A)) ) ).
fof(redefinition_k7_prob_3,axiom,
! [A,B,C] :
( ( m1_finseq_1(B,k1_pcomps_1(A))
& m1_subset_1(C,k5_numbers) )
=> k7_prob_3(A,B,C) = k1_funct_1(B,C) ) ).
fof(dt_k8_prob_3,axiom,
! [A,B] :
( m1_finseq_1(B,k1_pcomps_1(A))
=> m1_subset_1(k8_prob_3(A,B),k1_zfmisc_1(A)) ) ).
fof(redefinition_k8_prob_3,axiom,
! [A,B] :
( m1_finseq_1(B,k1_pcomps_1(A))
=> k8_prob_3(A,B) = k3_card_3(B) ) ).
fof(dt_k9_prob_3,axiom,
! [A,B] :
( m1_finseq_1(B,k1_pcomps_1(A))
=> m2_finseq_1(k9_prob_3(A,B),k1_pcomps_1(A)) ) ).
fof(dt_k10_prob_3,axiom,
! [A,B] :
( m1_finseq_1(B,k1_pcomps_1(A))
=> m1_subset_1(k10_prob_3(A,B),k1_zfmisc_1(A)) ) ).
fof(dt_k11_prob_3,axiom,
! [A,B,C,D] :
( ( m1_prob_1(B,A)
& m1_prob_3(C,A,B)
& m1_subset_1(D,k5_numbers) )
=> m3_prob_1(k11_prob_3(A,B,C,D),A,B) ) ).
fof(redefinition_k11_prob_3,axiom,
! [A,B,C,D] :
( ( m1_prob_1(B,A)
& m1_prob_3(C,A,B)
& m1_subset_1(D,k5_numbers) )
=> k11_prob_3(A,B,C,D) = k1_funct_1(C,D) ) ).
fof(dt_k12_prob_3,axiom,
! [A,B,C] :
( ( m1_prob_1(B,A)
& m1_prob_3(C,A,B) )
=> m1_prob_3(k12_prob_3(A,B,C),A,B) ) ).
fof(dt_k13_prob_3,axiom,
! [A,B,C,D] :
( ( ~ v1_xboole_0(A)
& m1_prob_1(B,A)
& m1_prob_3(C,A,B)
& m4_prob_1(D,A,B) )
=> m2_finseq_1(k13_prob_3(A,B,C,D),k1_numbers) ) ).
fof(redefinition_k13_prob_3,axiom,
! [A,B,C,D] :
( ( ~ v1_xboole_0(A)
& m1_prob_1(B,A)
& m1_prob_3(C,A,B)
& m4_prob_1(D,A,B) )
=> k13_prob_3(A,B,C,D) = k5_relat_1(C,D) ) ).
fof(dt_k14_prob_3,axiom,
! [A,B] :
( ( ~ v1_xboole_0(A)
& m1_subset_1(B,k1_zfmisc_1(k1_zfmisc_1(A))) )
=> m2_prob_3(k14_prob_3(A,B),A) ) ).
%------------------------------------------------------------------------------