SET007 Axioms: SET007+851.ax
%------------------------------------------------------------------------------
% File : SET007+851 : TPTP v8.2.0. Released v3.4.0.
% Domain : Set Theory
% Axioms : Logical Correctness of Vector Calculation Programs
% 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 : prgcor_2 [Urb08]
% Status : Satisfiable
% Syntax : Number of formulae : 34 ( 1 unt; 0 def)
% Number of atoms : 348 ( 82 equ)
% Maximal formula atoms : 21 ( 10 avg)
% Number of connectives : 365 ( 51 ~; 9 |; 167 &)
% ( 11 <=>; 127 =>; 0 <=; 0 <~>)
% Maximal formula depth : 24 ( 12 avg)
% Maximal term depth : 5 ( 1 avg)
% Number of predicates : 21 ( 19 usr; 1 prp; 0-4 aty)
% Number of functors : 29 ( 29 usr; 5 con; 0-5 aty)
% Number of variables : 120 ( 108 !; 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_prgcor_2,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ? [B] :
( m1_ordinal1(B,A)
& ~ v1_xboole_0(B)
& v1_relat_1(B)
& v1_funct_1(B)
& v5_ordinal1(B)
& v1_finset_1(B) ) ) ).
fof(t1_prgcor_2,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> ( r2_hidden(A,B)
<=> ~ r1_xreal_0(B,A) ) ) ) ).
fof(t2_prgcor_2,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( ( ~ v1_xboole_0(B)
& v1_finset_1(B)
& m1_ordinal1(B,A) )
=> ~ r1_xreal_0(k1_afinsq_1(B),np__0) ) ) ).
fof(d1_prgcor_2,axiom,
! [A,B] :
( m2_finseq_1(B,A)
=> ! [C] :
( ( v1_finset_1(C)
& m1_ordinal1(C,A) )
=> ( C = k1_prgcor_2(A,B)
<=> ( k1_afinsq_1(C) = k3_finseq_1(B)
& ! [D] :
( m2_subset_1(D,k1_numbers,k5_numbers)
=> ( ~ r1_xreal_0(k3_finseq_1(B),D)
=> k1_funct_1(B,k1_nat_1(D,np__1)) = k1_funct_1(C,D) ) ) ) ) ) ) ).
fof(d2_prgcor_2,axiom,
! [A,B] :
( ( v1_finset_1(B)
& m1_ordinal1(B,A) )
=> ! [C] :
( m2_finseq_1(C,A)
=> ( C = k2_prgcor_2(A,B)
<=> ( k3_finseq_1(C) = k1_afinsq_1(B)
& ! [D] :
( m2_subset_1(D,k1_numbers,k5_numbers)
=> ( ( r1_xreal_0(np__1,D)
& r1_xreal_0(D,k1_afinsq_1(B)) )
=> k1_funct_1(B,k5_binarith(D,np__1)) = k1_funct_1(C,D) ) ) ) ) ) ) ).
fof(t3_prgcor_2,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( v1_relat_1(k2_funcop_1(A,B))
& v1_funct_1(k2_funcop_1(A,B))
& v5_ordinal1(k2_funcop_1(A,B))
& v1_finset_1(k2_funcop_1(A,B)) ) ) ).
fof(t4_prgcor_2,axiom,
! [A,B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> ! [C] :
( r2_hidden(C,A)
=> ( v1_finset_1(k2_funcop_1(B,C))
& m1_ordinal1(k2_funcop_1(B,C),A)
& ! [D] :
( ( v1_relat_1(D)
& v1_funct_1(D)
& v5_ordinal1(D)
& v1_finset_1(D) )
=> ( D = k2_funcop_1(B,C)
=> k1_afinsq_1(D) = B ) ) ) ) ) ).
fof(d3_prgcor_2,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m2_finseq_1(B,A)
=> ! [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
=> ( r1_tarski(k5_numbers,A)
=> ( r1_xreal_0(C,k3_finseq_1(B))
| ! [D] :
( ( ~ v1_xboole_0(D)
& v1_finset_1(D)
& m1_ordinal1(D,A) )
=> ( D = k3_prgcor_2(A,B,C)
<=> ( k3_finseq_1(B) = k1_funct_1(D,np__0)
& k1_afinsq_1(D) = C
& ! [E] :
( m2_subset_1(E,k1_numbers,k5_numbers)
=> ( ( r1_xreal_0(np__1,E)
& r1_xreal_0(E,k3_finseq_1(B)) )
=> k1_funct_1(D,E) = k1_funct_1(B,E) ) )
& ! [E] :
( m2_subset_1(E,k1_numbers,k5_numbers)
=> ~ ( ~ r1_xreal_0(E,k3_finseq_1(B))
& ~ r1_xreal_0(C,E)
& k1_funct_1(D,E) != np__0 ) ) ) ) ) ) ) ) ) ) ).
fof(d4_prgcor_2,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( ( ~ v1_xboole_0(B)
& v1_finset_1(B)
& m1_ordinal1(B,A) )
=> ( ( r1_tarski(k5_numbers,A)
& m2_subset_1(k1_funct_1(B,np__0),k1_numbers,k5_numbers)
& r2_hidden(k1_funct_1(B,np__0),k1_afinsq_1(B)) )
=> ! [C] :
( m2_finseq_1(C,A)
=> ( C = k4_prgcor_2(A,B)
<=> ! [D] :
( m2_subset_1(D,k1_numbers,k5_numbers)
=> ( D = k1_funct_1(B,np__0)
=> ( k3_finseq_1(C) = D
& ! [E] :
( m2_subset_1(E,k1_numbers,k5_numbers)
=> ( ( r1_xreal_0(np__1,E)
& r1_xreal_0(E,D) )
=> k1_funct_1(C,E) = k1_funct_1(B,E) ) ) ) ) ) ) ) ) ) ) ).
fof(t5_prgcor_2,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( ( ~ v1_xboole_0(B)
& v1_finset_1(B)
& m1_ordinal1(B,A) )
=> ( ( r1_tarski(k5_numbers,A)
& k1_funct_1(B,np__0) = np__0 )
=> ( r1_xreal_0(k1_afinsq_1(B),np__0)
| k4_prgcor_2(A,B) = k1_xboole_0 ) ) ) ) ).
fof(d5_prgcor_2,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( ( v1_finset_1(B)
& m1_ordinal1(B,A) )
=> ! [C] :
( m2_finseq_1(C,A)
=> ( r1_prgcor_2(A,B,C)
<=> ( r1_tarski(k5_numbers,A)
& k1_funct_1(B,np__0) = k3_finseq_1(C)
& ~ r1_xreal_0(k1_afinsq_1(B),k3_finseq_1(C))
& ! [D] :
( m2_subset_1(D,k1_numbers,k5_numbers)
=> ( ( r1_xreal_0(np__1,D)
& r1_xreal_0(D,k3_finseq_1(C)) )
=> k1_funct_1(B,D) = k1_funct_1(C,D) ) ) ) ) ) ) ) ).
fof(t6_prgcor_2,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( ( ~ v1_xboole_0(B)
& v1_finset_1(B)
& m1_ordinal1(B,A) )
=> ( ( r1_tarski(k5_numbers,A)
& m2_subset_1(k1_funct_1(B,np__0),k1_numbers,k5_numbers)
& r2_hidden(k1_funct_1(B,np__0),k1_afinsq_1(B)) )
=> r1_prgcor_2(A,B,k4_prgcor_2(A,B)) ) ) ) ).
fof(d6_prgcor_2,axiom,
! [A,B,C,D,E] :
( ( r2_hidden(A,B)
=> k5_prgcor_2(A,B,C,D,E) = C )
& ( A = B
=> k5_prgcor_2(A,B,C,D,E) = D )
& ~ ( ~ r2_hidden(A,B)
& A != B
& k5_prgcor_2(A,B,C,D,E) != E ) ) ).
fof(t7_prgcor_2,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m2_finseq_1(B,A)
=> ! [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
=> ~ ( r1_tarski(k5_numbers,A)
& ~ r1_xreal_0(C,k3_finseq_1(B))
& ! [D] :
( ( v1_finset_1(D)
& m1_ordinal1(D,A) )
=> ~ ( k1_afinsq_1(D) = C
& r1_prgcor_2(A,D,B) ) ) ) ) ) ) ).
fof(d7_prgcor_2,axiom,
! [A] :
( ( v1_finset_1(A)
& m1_ordinal1(A,k1_numbers) )
=> ! [B] :
( ( v1_finset_1(B)
& m1_ordinal1(B,k1_numbers) )
=> ( ( m2_subset_1(k6_prgcor_2(B,np__0),k1_numbers,k5_numbers)
& r1_xreal_0(np__0,k6_prgcor_2(B,np__0)) )
=> ( r1_xreal_0(k1_afinsq_1(A),k6_prgcor_2(B,np__0))
| ! [C] :
( m1_subset_1(C,k1_numbers)
=> ( C = k7_prgcor_2(A,B)
<=> ? [D] :
( v1_finset_1(D)
& m1_ordinal1(D,k1_numbers)
& ? [E] :
( v1_int_1(E)
& k1_afinsq_1(D) = k1_afinsq_1(A)
& k6_prgcor_2(D,np__0) = np__0
& E = k6_prgcor_2(B,np__0)
& ( E != np__0
=> ! [F] :
( m2_subset_1(F,k1_numbers,k5_numbers)
=> ( ~ r1_xreal_0(E,F)
=> k6_prgcor_2(D,k1_nat_1(F,np__1)) = k3_real_1(k6_prgcor_2(D,F),k4_real_1(k6_prgcor_2(A,k1_nat_1(F,np__1)),k6_prgcor_2(B,k1_nat_1(F,np__1)))) ) ) )
& C = k1_funct_1(D,E) ) ) ) ) ) ) ) ) ).
fof(t8_prgcor_2,axiom,
! [A] :
( m2_finseq_1(A,k1_numbers)
=> ! [B] :
( ( v1_finset_1(B)
& m1_ordinal1(B,k1_numbers) )
=> ( ( k6_prgcor_2(B,np__0) = np__0
& ! [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
=> ( ~ r1_xreal_0(k3_finseq_1(A),C)
=> k6_prgcor_2(B,k1_nat_1(C,np__1)) = k3_real_1(k6_prgcor_2(B,C),k2_seq_1(k5_numbers,k1_numbers,A,k1_nat_1(C,np__1))) ) ) )
=> ( r1_xreal_0(k1_afinsq_1(B),k3_finseq_1(A))
| k15_rvsum_1(A) = k6_prgcor_2(B,k3_finseq_1(A)) ) ) ) ) ).
fof(t9_prgcor_2,axiom,
! [A] :
( m2_finseq_1(A,k1_numbers)
=> ? [B] :
( v1_finset_1(B)
& m1_ordinal1(B,k1_numbers)
& k1_afinsq_1(B) = k1_nat_1(k3_finseq_1(A),np__1)
& k6_prgcor_2(B,np__0) = np__0
& ! [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
=> ( ~ r1_xreal_0(k3_finseq_1(A),C)
=> k6_prgcor_2(B,k1_nat_1(C,np__1)) = k3_real_1(k6_prgcor_2(B,C),k2_seq_1(k5_numbers,k1_numbers,A,k1_nat_1(C,np__1))) ) )
& k15_rvsum_1(A) = k6_prgcor_2(B,k3_finseq_1(A)) ) ) ).
fof(t10_prgcor_2,axiom,
! [A] :
( m2_finseq_1(A,k1_numbers)
=> ! [B] :
( m2_finseq_1(B,k1_numbers)
=> ! [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
=> ( k3_finseq_1(A) = k3_finseq_1(B)
=> ( r1_xreal_0(C,k3_finseq_1(A))
| k1_euclid_2(A,B) = k7_prgcor_2(k3_prgcor_2(k1_numbers,A,C),k3_prgcor_2(k1_numbers,B,C)) ) ) ) ) ) ).
fof(d8_prgcor_2,axiom,
! [A] :
( ( v1_finset_1(A)
& m1_ordinal1(A,k1_numbers) )
=> ! [B] :
( ( v1_finset_1(B)
& m1_ordinal1(B,k1_numbers) )
=> ! [C] :
( m1_subset_1(C,k1_numbers)
=> ! [D] :
( v1_int_1(D)
=> ( r2_prgcor_2(A,B,C,D)
<=> ( k1_afinsq_1(B) = D
& k1_afinsq_1(A) = D
& ? [E] :
( v1_int_1(E)
& k6_prgcor_2(B,np__0) = k6_prgcor_2(A,np__0)
& E = k6_prgcor_2(A,np__0)
& ( E != np__0
=> ! [F] :
( m2_subset_1(F,k1_numbers,k5_numbers)
=> ( ( r1_xreal_0(np__1,F)
& r1_xreal_0(F,E) )
=> k6_prgcor_2(B,F) = k4_real_1(C,k6_prgcor_2(A,F)) ) ) ) ) ) ) ) ) ) ) ).
fof(t11_prgcor_2,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& v1_finset_1(A)
& m1_ordinal1(A,k1_numbers) )
=> ! [B] :
( m1_subset_1(B,k1_numbers)
=> ! [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
=> ( ( m2_subset_1(k6_prgcor_2(A,np__0),k1_numbers,k5_numbers)
& k1_afinsq_1(A) = C
& r1_xreal_0(np__0,k6_prgcor_2(A,np__0)) )
=> ( r1_xreal_0(C,k6_prgcor_2(A,np__0))
| ( ? [D] :
( v1_finset_1(D)
& m1_ordinal1(D,k1_numbers)
& r2_prgcor_2(A,D,B,C) )
& ! [D] :
( ( ~ v1_xboole_0(D)
& v1_finset_1(D)
& m1_ordinal1(D,k1_numbers) )
=> ( r2_prgcor_2(A,D,B,C)
=> k4_prgcor_2(k1_numbers,D) = k9_rvsum_1(B,k4_prgcor_2(k1_numbers,A)) ) ) ) ) ) ) ) ) ).
fof(d9_prgcor_2,axiom,
! [A] :
( ( v1_finset_1(A)
& m1_ordinal1(A,k1_numbers) )
=> ! [B] :
( ( v1_finset_1(B)
& m1_ordinal1(B,k1_numbers) )
=> ! [C] :
( v1_int_1(C)
=> ( r3_prgcor_2(A,B,C)
<=> ( k1_afinsq_1(B) = C
& k1_afinsq_1(A) = C
& ? [D] :
( v1_int_1(D)
& k6_prgcor_2(B,np__0) = k6_prgcor_2(A,np__0)
& D = k6_prgcor_2(A,np__0)
& ( D != np__0
=> ! [E] :
( m2_subset_1(E,k1_numbers,k5_numbers)
=> ( ( r1_xreal_0(np__1,E)
& r1_xreal_0(E,D) )
=> k6_prgcor_2(B,E) = k1_real_1(k6_prgcor_2(A,E)) ) ) ) ) ) ) ) ) ) ).
fof(t12_prgcor_2,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& v1_finset_1(A)
& m1_ordinal1(A,k1_numbers) )
=> ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> ( ( m2_subset_1(k6_prgcor_2(A,np__0),k1_numbers,k5_numbers)
& k1_afinsq_1(A) = B
& r1_xreal_0(np__0,k6_prgcor_2(A,np__0)) )
=> ( r1_xreal_0(B,k6_prgcor_2(A,np__0))
| ( ? [C] :
( v1_finset_1(C)
& m1_ordinal1(C,k1_numbers)
& r3_prgcor_2(A,C,B) )
& ! [C] :
( ( ~ v1_xboole_0(C)
& v1_finset_1(C)
& m1_ordinal1(C,k1_numbers) )
=> ( r3_prgcor_2(A,C,B)
=> k4_prgcor_2(k1_numbers,C) = k5_rvsum_1(k4_prgcor_2(k1_numbers,A)) ) ) ) ) ) ) ) ).
fof(d10_prgcor_2,axiom,
! [A] :
( ( v1_finset_1(A)
& m1_ordinal1(A,k1_numbers) )
=> ! [B] :
( ( v1_finset_1(B)
& m1_ordinal1(B,k1_numbers) )
=> ! [C] :
( ( v1_finset_1(C)
& m1_ordinal1(C,k1_numbers) )
=> ! [D] :
( v1_int_1(D)
=> ( r4_prgcor_2(A,B,C,D)
<=> ( k1_afinsq_1(C) = D
& k1_afinsq_1(A) = D
& k1_afinsq_1(B) = D
& ? [E] :
( v1_int_1(E)
& k6_prgcor_2(C,np__0) = k6_prgcor_2(B,np__0)
& E = k6_prgcor_2(B,np__0)
& ( E != np__0
=> ! [F] :
( m2_subset_1(F,k1_numbers,k5_numbers)
=> ( ( r1_xreal_0(np__1,F)
& r1_xreal_0(F,E) )
=> k6_prgcor_2(C,F) = k3_real_1(k6_prgcor_2(A,F),k6_prgcor_2(B,F)) ) ) ) ) ) ) ) ) ) ) ).
fof(t13_prgcor_2,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& v1_finset_1(A)
& m1_ordinal1(A,k1_numbers) )
=> ! [B] :
( ( ~ v1_xboole_0(B)
& v1_finset_1(B)
& m1_ordinal1(B,k1_numbers) )
=> ! [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
=> ( ( m2_subset_1(k6_prgcor_2(B,np__0),k1_numbers,k5_numbers)
& k1_afinsq_1(A) = C
& k1_afinsq_1(B) = C
& k6_prgcor_2(A,np__0) = k6_prgcor_2(B,np__0)
& r1_xreal_0(np__0,k6_prgcor_2(B,np__0)) )
=> ( r1_xreal_0(C,k6_prgcor_2(B,np__0))
| ( ? [D] :
( v1_finset_1(D)
& m1_ordinal1(D,k1_numbers)
& r4_prgcor_2(A,B,D,C) )
& ! [D] :
( ( ~ v1_xboole_0(D)
& v1_finset_1(D)
& m1_ordinal1(D,k1_numbers) )
=> ( r4_prgcor_2(A,B,D,C)
=> k4_prgcor_2(k1_numbers,D) = k3_rvsum_1(k4_prgcor_2(k1_numbers,A),k4_prgcor_2(k1_numbers,B)) ) ) ) ) ) ) ) ) ).
fof(d11_prgcor_2,axiom,
! [A] :
( ( v1_finset_1(A)
& m1_ordinal1(A,k1_numbers) )
=> ! [B] :
( ( v1_finset_1(B)
& m1_ordinal1(B,k1_numbers) )
=> ! [C] :
( ( v1_finset_1(C)
& m1_ordinal1(C,k1_numbers) )
=> ! [D] :
( v1_int_1(D)
=> ( r5_prgcor_2(A,B,C,D)
<=> ( k1_afinsq_1(C) = D
& k1_afinsq_1(A) = D
& k1_afinsq_1(B) = D
& ? [E] :
( v1_int_1(E)
& k6_prgcor_2(C,np__0) = k6_prgcor_2(B,np__0)
& E = k6_prgcor_2(B,np__0)
& ( E != np__0
=> ! [F] :
( m2_subset_1(F,k1_numbers,k5_numbers)
=> ( ( r1_xreal_0(np__1,F)
& r1_xreal_0(F,E) )
=> k6_prgcor_2(C,F) = k5_real_1(k6_prgcor_2(A,F),k6_prgcor_2(B,F)) ) ) ) ) ) ) ) ) ) ) ).
fof(t14_prgcor_2,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& v1_finset_1(A)
& m1_ordinal1(A,k1_numbers) )
=> ! [B] :
( ( ~ v1_xboole_0(B)
& v1_finset_1(B)
& m1_ordinal1(B,k1_numbers) )
=> ! [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
=> ( ( m2_subset_1(k6_prgcor_2(B,np__0),k1_numbers,k5_numbers)
& k1_afinsq_1(A) = C
& k1_afinsq_1(B) = C
& k6_prgcor_2(A,np__0) = k6_prgcor_2(B,np__0)
& r1_xreal_0(np__0,k6_prgcor_2(B,np__0)) )
=> ( r1_xreal_0(C,k6_prgcor_2(B,np__0))
| ( ? [D] :
( v1_finset_1(D)
& m1_ordinal1(D,k1_numbers)
& r5_prgcor_2(A,B,D,C) )
& ! [D] :
( ( ~ v1_xboole_0(D)
& v1_finset_1(D)
& m1_ordinal1(D,k1_numbers) )
=> ( r5_prgcor_2(A,B,D,C)
=> k4_prgcor_2(k1_numbers,D) = k7_rvsum_1(k4_prgcor_2(k1_numbers,A),k4_prgcor_2(k1_numbers,B)) ) ) ) ) ) ) ) ) ).
fof(dt_k1_prgcor_2,axiom,
! [A,B] :
( m1_finseq_1(B,A)
=> ( v1_finset_1(k1_prgcor_2(A,B))
& m1_ordinal1(k1_prgcor_2(A,B),A) ) ) ).
fof(dt_k2_prgcor_2,axiom,
! [A,B] :
( ( v1_finset_1(B)
& m1_ordinal1(B,A) )
=> m2_finseq_1(k2_prgcor_2(A,B),A) ) ).
fof(dt_k3_prgcor_2,axiom,
! [A,B,C] :
( ( ~ v1_xboole_0(A)
& m1_finseq_1(B,A)
& m1_subset_1(C,k5_numbers) )
=> ( ~ v1_xboole_0(k3_prgcor_2(A,B,C))
& v1_finset_1(k3_prgcor_2(A,B,C))
& m1_ordinal1(k3_prgcor_2(A,B,C),A) ) ) ).
fof(dt_k4_prgcor_2,axiom,
! [A,B] :
( ( ~ v1_xboole_0(A)
& ~ v1_xboole_0(B)
& v1_finset_1(B)
& m1_ordinal1(B,A) )
=> m2_finseq_1(k4_prgcor_2(A,B),A) ) ).
fof(dt_k5_prgcor_2,axiom,
$true ).
fof(dt_k6_prgcor_2,axiom,
! [A,B] :
( ( v1_finset_1(A)
& m1_ordinal1(A,k1_numbers)
& m1_subset_1(B,k5_numbers) )
=> m1_subset_1(k6_prgcor_2(A,B),k1_numbers) ) ).
fof(redefinition_k6_prgcor_2,axiom,
! [A,B] :
( ( v1_finset_1(A)
& m1_ordinal1(A,k1_numbers)
& m1_subset_1(B,k5_numbers) )
=> k6_prgcor_2(A,B) = k1_funct_1(A,B) ) ).
fof(dt_k7_prgcor_2,axiom,
! [A,B] :
( ( v1_finset_1(A)
& m1_ordinal1(A,k1_numbers)
& v1_finset_1(B)
& m1_ordinal1(B,k1_numbers) )
=> m1_subset_1(k7_prgcor_2(A,B),k1_numbers) ) ).
%------------------------------------------------------------------------------