SET007 Axioms: SET007+64.ax
%------------------------------------------------------------------------------
% File : SET007+64 : TPTP v8.2.0. Released v3.4.0.
% Domain : Set Theory
% Axioms : Finite Sequences and Tuples of Elements of a Non-empty Sets
% 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 : finseq_2 [Urb08]
% Status : Satisfiable
% Syntax : Number of formulae : 173 ( 24 unt; 0 def)
% Number of atoms : 1028 ( 170 equ)
% Maximal formula atoms : 17 ( 5 avg)
% Number of connectives : 1012 ( 157 ~; 2 |; 314 &)
% ( 8 <=>; 531 =>; 0 <=; 0 <~>)
% Maximal formula depth : 27 ( 8 avg)
% Maximal term depth : 4 ( 1 avg)
% Number of predicates : 19 ( 17 usr; 1 prp; 0-3 aty)
% Number of functors : 59 ( 59 usr; 10 con; 0-6 aty)
% Number of variables : 562 ( 538 !; 24 ?)
% 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_finseq_2,axiom,
! [A] :
? [B] :
( m1_finseq_2(B,A)
& ~ v1_xboole_0(B) ) ).
fof(fc1_finseq_2,axiom,
! [A,B] :
( ( v4_ordinal2(A)
& ~ v1_xboole_0(B) )
=> ~ v1_xboole_0(k4_finseq_2(A,B)) ) ).
fof(t1_finseq_2,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> ( m2_subset_1(k3_square_1(A,B),k1_numbers,k5_numbers)
& m2_subset_1(k4_square_1(A,B),k1_numbers,k5_numbers) ) ) ) ).
fof(t2_finseq_2,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> ! [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
=> ( A = k3_square_1(B,C)
=> k5_subset_1(k5_numbers,k2_finseq_1(B),k2_finseq_1(C)) = k2_finseq_1(A) ) ) ) ) ).
fof(t3_finseq_2,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> ( r1_xreal_0(A,B)
=> k2_square_1(np__0,k6_xcmplx_0(A,B)) = np__0 ) ) ) ).
fof(t4_finseq_2,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> ( r1_xreal_0(A,B)
=> k2_square_1(np__0,k6_xcmplx_0(B,A)) = k6_xcmplx_0(B,A) ) ) ) ).
fof(t5_finseq_2,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> m2_subset_1(k2_square_1(np__0,k6_xcmplx_0(A,B)),k1_numbers,k5_numbers) ) ) ).
fof(t6_finseq_2,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ( k3_square_1(np__0,A) = np__0
& k3_square_1(A,np__0) = np__0
& k4_square_1(np__0,A) = A
& k4_square_1(A,np__0) = A ) ) ).
fof(t7_finseq_2,axiom,
$true ).
fof(t8_finseq_2,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> ~ ( r2_hidden(A,k2_finseq_1(k1_nat_1(B,np__1)))
& ~ r2_hidden(A,k2_finseq_1(B))
& A != k1_nat_1(B,np__1) ) ) ) ).
fof(t9_finseq_2,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> ! [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
=> ( r2_hidden(A,k2_finseq_1(B))
=> r2_hidden(A,k2_finseq_1(k1_nat_1(B,C))) ) ) ) ) ).
fof(t10_finseq_2,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( ( v1_relat_1(B)
& v1_funct_1(B)
& v1_finseq_1(B) )
=> ! [C] :
( ( v1_relat_1(C)
& v1_funct_1(C)
& v1_finseq_1(C) )
=> ( ( k3_finseq_1(B) = A
& k3_finseq_1(C) = A
& ! [D] :
( m2_subset_1(D,k1_numbers,k5_numbers)
=> ( r2_hidden(D,k2_finseq_1(A))
=> k1_funct_1(B,D) = k1_funct_1(C,D) ) ) )
=> B = C ) ) ) ) ).
fof(t11_finseq_2,axiom,
! [A,B] :
( ( v1_relat_1(B)
& v1_funct_1(B)
& v1_finseq_1(B) )
=> ~ ( r2_hidden(A,k2_relat_1(B))
& ! [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
=> ~ ( r2_hidden(C,k4_finseq_1(B))
& k1_funct_1(B,C) = A ) ) ) ) ).
fof(t12_finseq_2,axiom,
$true ).
fof(t13_finseq_2,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B,C] :
( m2_finseq_1(C,B)
=> ( r2_hidden(A,k4_finseq_1(C))
=> r2_hidden(k1_funct_1(C,A),B) ) ) ) ).
fof(t14_finseq_2,axiom,
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A)
& v1_finseq_1(A) )
=> ! [B] :
( ! [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
=> ( r2_hidden(C,k4_finseq_1(A))
=> r2_hidden(k1_funct_1(A,C),B) ) )
=> m2_finseq_1(A,B) ) ) ).
fof(t15_finseq_2,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m1_subset_1(B,A)
=> ! [C] :
( m1_subset_1(C,A)
=> m2_finseq_1(k10_finseq_1(B,C),A) ) ) ) ).
fof(t16_finseq_2,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m1_subset_1(B,A)
=> ! [C] :
( m1_subset_1(C,A)
=> ! [D] :
( m1_subset_1(D,A)
=> m2_finseq_1(k11_finseq_1(B,C,D),A) ) ) ) ) ).
fof(t17_finseq_2,axiom,
$true ).
fof(t18_finseq_2,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( ( v1_relat_1(B)
& v1_funct_1(B)
& v1_finseq_1(B) )
=> ! [C] :
( ( v1_relat_1(C)
& v1_funct_1(C)
& v1_finseq_1(C) )
=> ( r2_hidden(A,k4_finseq_1(B))
=> r2_hidden(A,k4_finseq_1(k7_finseq_1(B,C))) ) ) ) ) ).
fof(t19_finseq_2,axiom,
! [A,B] :
( ( v1_relat_1(B)
& v1_funct_1(B)
& v1_finseq_1(B) )
=> k3_finseq_1(k7_finseq_1(B,k9_finseq_1(A))) = k1_nat_1(k3_finseq_1(B),np__1) ) ).
fof(t20_finseq_2,axiom,
! [A,B,C] :
( ( v1_relat_1(C)
& v1_funct_1(C)
& v1_finseq_1(C) )
=> ! [D] :
( ( v1_relat_1(D)
& v1_funct_1(D)
& v1_finseq_1(D) )
=> ( k7_finseq_1(C,k9_finseq_1(A)) = k7_finseq_1(D,k9_finseq_1(B))
=> ( C = D
& A = B ) ) ) ) ).
fof(t21_finseq_2,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( ( v1_relat_1(B)
& v1_funct_1(B)
& v1_finseq_1(B) )
=> ~ ( k3_finseq_1(B) = k1_nat_1(A,np__1)
& ! [C] :
( ( v1_relat_1(C)
& v1_funct_1(C)
& v1_finseq_1(C) )
=> ! [D] : B != k7_finseq_1(C,k9_finseq_1(D)) ) ) ) ) ).
fof(t22_finseq_2,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m2_finseq_1(B,A)
=> ~ ( k3_finseq_1(B) != np__0
& ! [C] :
( m2_finseq_1(C,A)
=> ! [D] :
( m1_subset_1(D,A)
=> B != k8_finseq_1(A,C,k12_finseq_1(A,D)) ) ) ) ) ) ).
fof(t23_finseq_2,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( ( v1_relat_1(B)
& v1_funct_1(B)
& v1_finseq_1(B) )
=> ! [C] :
( ( v1_relat_1(C)
& v1_funct_1(C)
& v1_finseq_1(C) )
=> ( ( B = k7_relat_1(C,k2_finseq_1(A))
& r1_xreal_0(k3_finseq_1(C),A) )
=> C = B ) ) ) ) ).
fof(t24_finseq_2,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( ( v1_relat_1(B)
& v1_funct_1(B)
& v1_finseq_1(B) )
=> ! [C] :
( ( v1_relat_1(C)
& v1_funct_1(C)
& v1_finseq_1(C) )
=> ( B = k7_relat_1(C,k2_finseq_1(A))
=> k3_finseq_1(B) = k3_square_1(A,k3_finseq_1(C)) ) ) ) ) ).
fof(t25_finseq_2,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> ! [C] :
( ( v1_relat_1(C)
& v1_funct_1(C)
& v1_finseq_1(C) )
=> ~ ( k3_finseq_1(C) = k1_nat_1(A,B)
& ! [D] :
( ( v1_relat_1(D)
& v1_funct_1(D)
& v1_finseq_1(D) )
=> ! [E] :
( ( v1_relat_1(E)
& v1_funct_1(E)
& v1_finseq_1(E) )
=> ~ ( k3_finseq_1(D) = A
& k3_finseq_1(E) = B
& C = k7_finseq_1(D,E) ) ) ) ) ) ) ) ).
fof(t26_finseq_2,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> ! [C] :
( ~ v1_xboole_0(C)
=> ! [D] :
( m2_finseq_1(D,C)
=> ~ ( k3_finseq_1(D) = k1_nat_1(A,B)
& ! [E] :
( m2_finseq_1(E,C)
=> ! [F] :
( m2_finseq_1(F,C)
=> ~ ( k3_finseq_1(E) = A
& k3_finseq_1(F) = B
& D = k8_finseq_1(C,E,F) ) ) ) ) ) ) ) ) ).
fof(t27_finseq_2,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( ( ~ v1_xboole_0(B)
& m1_subset_1(B,k1_zfmisc_1(A)) )
=> ! [C] :
( m2_finseq_1(C,B)
=> m2_finseq_1(C,A) ) ) ) ).
fof(t28_finseq_2,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( ~ v1_xboole_0(B)
=> ! [C] :
( ( v1_funct_1(C)
& v1_funct_2(C,k2_finseq_1(A),B)
& m2_relset_1(C,k2_finseq_1(A),B) )
=> m2_finseq_1(C,B) ) ) ) ).
fof(t29_finseq_2,axiom,
$true ).
fof(t30_finseq_2,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m2_finseq_1(B,A)
=> ( v1_funct_1(B)
& v1_funct_2(B,k4_finseq_1(B),A)
& m2_relset_1(B,k4_finseq_1(B),A) ) ) ) ).
fof(t31_finseq_2,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( ~ v1_xboole_0(B)
=> ! [C] :
( ( v1_funct_1(C)
& v1_funct_2(C,k5_numbers,B)
& m2_relset_1(C,k5_numbers,B) )
=> m2_finseq_1(k2_partfun1(k5_numbers,B,C,k2_finseq_1(A)),B) ) ) ) ).
fof(t32_finseq_2,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( ~ v1_xboole_0(B)
=> ! [C] :
( ( v1_relat_1(C)
& v1_funct_1(C)
& v1_finseq_1(C) )
=> ! [D] :
( ( v1_funct_1(D)
& v1_funct_2(D,k5_numbers,B)
& m2_relset_1(D,k5_numbers,B) )
=> ( C = k2_partfun1(k5_numbers,B,D,k2_finseq_1(A))
=> k3_finseq_1(C) = A ) ) ) ) ) ).
fof(t33_finseq_2,axiom,
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A)
& v1_finseq_1(A) )
=> ! [B] :
( ( v1_relat_1(B)
& v1_funct_1(B)
& v1_finseq_1(B) )
=> ! [C] :
( ( v1_relat_1(C)
& v1_funct_1(C) )
=> ( ( r1_tarski(k2_relat_1(A),k1_relat_1(C))
& B = k5_relat_1(A,C) )
=> k3_finseq_1(B) = k3_finseq_1(A) ) ) ) ) ).
fof(t34_finseq_2,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( ~ v1_xboole_0(B)
=> ( B = k2_finseq_1(A)
=> ! [C] :
( ( v1_relat_1(C)
& v1_funct_1(C)
& v1_finseq_1(C) )
=> ! [D] :
( m2_finseq_1(D,B)
=> ( r1_xreal_0(A,k3_finseq_1(C))
=> ( v1_relat_1(k5_relat_1(D,C))
& v1_funct_1(k5_relat_1(D,C))
& v1_finseq_1(k5_relat_1(D,C)) ) ) ) ) ) ) ) ).
fof(t35_finseq_2,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( ~ v1_xboole_0(B)
=> ! [C] :
( ~ v1_xboole_0(C)
=> ( B = k2_finseq_1(A)
=> ! [D] :
( m2_finseq_1(D,C)
=> ! [E] :
( m2_finseq_1(E,B)
=> ( r1_xreal_0(A,k3_finseq_1(D))
=> m2_finseq_1(k1_partfun1(k5_numbers,B,k5_numbers,C,E,D),C) ) ) ) ) ) ) ) ).
fof(t36_finseq_2,axiom,
! [A,B,C] :
( m2_finseq_1(C,A)
=> ! [D] :
( ( v1_funct_1(D)
& v1_funct_2(D,A,B)
& m2_relset_1(D,A,B) )
=> m2_finseq_1(k1_partfun1(k5_numbers,A,A,B,C,D),B) ) ) ).
fof(t37_finseq_2,axiom,
! [A,B] :
( ~ v1_xboole_0(B)
=> ! [C] :
( ( v1_relat_1(C)
& v1_funct_1(C)
& v1_finseq_1(C) )
=> ! [D] :
( m2_finseq_1(D,A)
=> ! [E] :
( ( v1_funct_1(E)
& v1_funct_2(E,A,B)
& m2_relset_1(E,A,B) )
=> ( C = k1_partfun1(k5_numbers,A,A,B,D,E)
=> k3_finseq_1(C) = k3_finseq_1(D) ) ) ) ) ) ).
fof(t38_finseq_2,axiom,
! [A,B] :
( ~ v1_xboole_0(B)
=> ! [C] :
( ( v1_funct_1(C)
& v1_funct_2(C,A,B)
& m2_relset_1(C,A,B) )
=> k1_partfun1(k5_numbers,A,A,B,k6_finseq_1(A),C) = k6_finseq_1(B) ) ) ).
fof(t39_finseq_2,axiom,
! [A,B] :
( ~ v1_xboole_0(B)
=> ! [C] :
( ~ v1_xboole_0(C)
=> ! [D] :
( m2_finseq_1(D,B)
=> ! [E] :
( ( v1_funct_1(E)
& v1_funct_2(E,B,C)
& m2_relset_1(E,B,C) )
=> ( D = k9_finseq_1(A)
=> k1_partfun1(k5_numbers,B,B,C,D,E) = k9_finseq_1(k1_funct_1(E,A)) ) ) ) ) ) ).
fof(t40_finseq_2,axiom,
! [A,B,C] :
( ~ v1_xboole_0(C)
=> ! [D] :
( ~ v1_xboole_0(D)
=> ! [E] :
( m2_finseq_1(E,C)
=> ! [F] :
( ( v1_funct_1(F)
& v1_funct_2(F,C,D)
& m2_relset_1(F,C,D) )
=> ( E = k10_finseq_1(A,B)
=> k1_partfun1(k5_numbers,C,C,D,E,F) = k10_finseq_1(k1_funct_1(F,A),k1_funct_1(F,B)) ) ) ) ) ) ).
fof(t41_finseq_2,axiom,
! [A,B,C,D] :
( ~ v1_xboole_0(D)
=> ! [E] :
( ~ v1_xboole_0(E)
=> ! [F] :
( m2_finseq_1(F,D)
=> ! [G] :
( ( v1_funct_1(G)
& v1_funct_2(G,D,E)
& m2_relset_1(G,D,E) )
=> ( F = k11_finseq_1(A,B,C)
=> k1_partfun1(k5_numbers,D,D,E,F,G) = k11_finseq_1(k1_funct_1(G,A),k1_funct_1(G,B),k1_funct_1(G,C)) ) ) ) ) ) ).
fof(t42_finseq_2,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> ! [C] :
( ( v1_relat_1(C)
& v1_funct_1(C)
& v1_finseq_1(C) )
=> ! [D] :
( ( v1_funct_1(D)
& v1_funct_2(D,k2_finseq_1(A),k2_finseq_1(B))
& m2_relset_1(D,k2_finseq_1(A),k2_finseq_1(B)) )
=> ( r1_xreal_0(B,k3_finseq_1(C))
=> ( ( B = np__0
& A != np__0 )
| ( v1_relat_1(k5_relat_1(D,C))
& v1_funct_1(k5_relat_1(D,C))
& v1_finseq_1(k5_relat_1(D,C)) ) ) ) ) ) ) ) ).
fof(t43_finseq_2,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( ( v1_relat_1(B)
& v1_funct_1(B)
& v1_finseq_1(B) )
=> ! [C] :
( ( v1_funct_1(C)
& v1_funct_2(C,k2_finseq_1(A),k2_finseq_1(A))
& m2_relset_1(C,k2_finseq_1(A),k2_finseq_1(A)) )
=> ( r1_xreal_0(A,k3_finseq_1(B))
=> ( v1_relat_1(k5_relat_1(C,B))
& v1_funct_1(k5_relat_1(C,B))
& v1_finseq_1(k5_relat_1(C,B)) ) ) ) ) ) ).
fof(t44_finseq_2,axiom,
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A)
& v1_finseq_1(A) )
=> ! [B] :
( ( v1_funct_1(B)
& v1_funct_2(B,k4_finseq_1(A),k4_finseq_1(A))
& m2_relset_1(B,k4_finseq_1(A),k4_finseq_1(A)) )
=> ( v1_relat_1(k5_relat_1(B,A))
& v1_funct_1(k5_relat_1(B,A))
& v1_finseq_1(k5_relat_1(B,A)) ) ) ) ).
fof(t45_finseq_2,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( ( v1_relat_1(B)
& v1_funct_1(B)
& v1_finseq_1(B) )
=> ! [C] :
( ( v1_relat_1(C)
& v1_funct_1(C)
& v1_finseq_1(C) )
=> ! [D] :
( ( v1_funct_1(D)
& v1_funct_2(D,k2_finseq_1(A),k2_finseq_1(A))
& m2_relset_1(D,k2_finseq_1(A),k2_finseq_1(A)) )
=> ( ( k2_relat_1(D) = k2_finseq_1(A)
& r1_xreal_0(A,k3_finseq_1(B))
& C = k5_relat_1(D,B) )
=> k3_finseq_1(C) = A ) ) ) ) ) ).
fof(t46_finseq_2,axiom,
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A)
& v1_finseq_1(A) )
=> ! [B] :
( ( v1_relat_1(B)
& v1_funct_1(B)
& v1_finseq_1(B) )
=> ! [C] :
( ( v1_funct_1(C)
& v1_funct_2(C,k4_finseq_1(A),k4_finseq_1(A))
& m2_relset_1(C,k4_finseq_1(A),k4_finseq_1(A)) )
=> ( ( k2_relat_1(C) = k4_finseq_1(A)
& B = k5_relat_1(C,A) )
=> k3_finseq_1(B) = k3_finseq_1(A) ) ) ) ) ).
fof(t47_finseq_2,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( ( v1_relat_1(B)
& v1_funct_1(B)
& v1_finseq_1(B) )
=> ! [C] :
( ( v1_relat_1(C)
& v1_funct_1(C)
& v1_finseq_1(C) )
=> ! [D] :
( ( v1_funct_1(D)
& v1_funct_2(D,k2_finseq_1(A),k2_finseq_1(A))
& v3_funct_2(D,k2_finseq_1(A),k2_finseq_1(A))
& m2_relset_1(D,k2_finseq_1(A),k2_finseq_1(A)) )
=> ( ( r1_xreal_0(A,k3_finseq_1(B))
& C = k5_relat_1(D,B) )
=> k3_finseq_1(C) = A ) ) ) ) ) ).
fof(t48_finseq_2,axiom,
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A)
& v1_finseq_1(A) )
=> ! [B] :
( ( v1_relat_1(B)
& v1_funct_1(B)
& v1_finseq_1(B) )
=> ! [C] :
( ( v1_funct_1(C)
& v1_funct_2(C,k4_finseq_1(A),k4_finseq_1(A))
& v3_funct_2(C,k4_finseq_1(A),k4_finseq_1(A))
& m2_relset_1(C,k4_finseq_1(A),k4_finseq_1(A)) )
=> ( B = k5_relat_1(C,A)
=> k3_finseq_1(B) = k3_finseq_1(A) ) ) ) ) ).
fof(t49_finseq_2,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> ! [C] :
( ~ v1_xboole_0(C)
=> ! [D] :
( m2_finseq_1(D,C)
=> ! [E] :
( ( v1_funct_1(E)
& v1_funct_2(E,k2_finseq_1(A),k2_finseq_1(B))
& m2_relset_1(E,k2_finseq_1(A),k2_finseq_1(B)) )
=> ( r1_xreal_0(B,k3_finseq_1(D))
=> ( ( B = np__0
& A != np__0 )
| m2_finseq_1(k1_partfun1(k2_finseq_1(A),k2_finseq_1(B),k5_numbers,C,E,D),C) ) ) ) ) ) ) ) ).
fof(t50_finseq_2,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( ~ v1_xboole_0(B)
=> ! [C] :
( m2_finseq_1(C,B)
=> ! [D] :
( ( v1_funct_1(D)
& v1_funct_2(D,k2_finseq_1(A),k2_finseq_1(A))
& m2_relset_1(D,k2_finseq_1(A),k2_finseq_1(A)) )
=> ( r1_xreal_0(A,k3_finseq_1(C))
=> m2_finseq_1(k1_partfun1(k2_finseq_1(A),k2_finseq_1(A),k5_numbers,B,D,C),B) ) ) ) ) ) ).
fof(t51_finseq_2,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m2_finseq_1(B,A)
=> ! [C] :
( ( v1_funct_1(C)
& v1_funct_2(C,k4_finseq_1(B),k4_finseq_1(B))
& m2_relset_1(C,k4_finseq_1(B),k4_finseq_1(B)) )
=> m2_finseq_1(k1_partfun1(k4_finseq_1(B),k4_finseq_1(B),k5_numbers,A,C,B),A) ) ) ) ).
fof(t52_finseq_2,axiom,
! [A] :
( v4_ordinal2(A)
=> m2_finseq_1(k6_partfun1(k2_finseq_1(A)),k5_numbers) ) ).
fof(d1_finseq_2,axiom,
! [A] :
( v4_ordinal2(A)
=> k1_finseq_2(A) = k6_partfun1(k2_finseq_1(A)) ) ).
fof(t53_finseq_2,axiom,
$true ).
fof(t54_finseq_2,axiom,
! [A] :
( v4_ordinal2(A)
=> k4_finseq_1(k1_finseq_2(A)) = k2_finseq_1(A) ) ).
fof(t55_finseq_2,axiom,
! [A] :
( v4_ordinal2(A)
=> k3_finseq_1(k1_finseq_2(A)) = A ) ).
fof(t56_finseq_2,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> ( r2_hidden(A,k2_finseq_1(B))
=> k1_funct_1(k1_finseq_2(B),A) = A ) ) ) ).
fof(t57_finseq_2,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ( A != np__0
=> ! [B] :
( m1_subset_1(B,k2_finseq_1(A))
=> k1_funct_1(k1_finseq_2(A),B) = B ) ) ) ).
fof(t58_finseq_2,axiom,
k1_finseq_2(np__0) = k1_xboole_0 ).
fof(t59_finseq_2,axiom,
k1_finseq_2(np__1) = k12_finseq_1(k5_numbers,np__1) ).
fof(t60_finseq_2,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> k1_finseq_2(k1_nat_1(A,np__1)) = k7_finseq_1(k1_finseq_2(A),k12_finseq_1(k5_numbers,k1_nat_1(A,np__1))) ) ).
fof(t61_finseq_2,axiom,
k1_finseq_2(np__2) = k10_finseq_1(np__1,np__2) ).
fof(t62_finseq_2,axiom,
k1_finseq_2(np__3) = k11_finseq_1(np__1,np__2,np__3) ).
fof(t63_finseq_2,axiom,
! [A] :
( v4_ordinal2(A)
=> ! [B] :
( ( v1_relat_1(B)
& v1_funct_1(B)
& v1_finseq_1(B) )
=> k5_relat_1(k1_finseq_2(A),B) = k7_relat_1(B,k2_finseq_1(A)) ) ) ).
fof(t64_finseq_2,axiom,
! [A] :
( v4_ordinal2(A)
=> ! [B] :
( ( v1_relat_1(B)
& v1_funct_1(B)
& v1_finseq_1(B) )
=> ( r1_xreal_0(k3_finseq_1(B),A)
=> k5_relat_1(k1_finseq_2(A),B) = B ) ) ) ).
fof(t65_finseq_2,axiom,
! [A] :
( v4_ordinal2(A)
=> ( v1_funct_1(k1_finseq_2(A))
& v1_funct_2(k1_finseq_2(A),k2_finseq_1(A),k2_finseq_1(A))
& v3_funct_2(k1_finseq_2(A),k2_finseq_1(A),k2_finseq_1(A))
& m2_relset_1(k1_finseq_2(A),k2_finseq_1(A),k2_finseq_1(A)) ) ) ).
fof(t66_finseq_2,axiom,
! [A] :
( v4_ordinal2(A)
=> ! [B] :
( v1_relat_1(k2_funcop_1(k2_finseq_1(A),B))
& v1_funct_1(k2_funcop_1(k2_finseq_1(A),B))
& v1_finseq_1(k2_funcop_1(k2_finseq_1(A),B)) ) ) ).
fof(d2_finseq_2,axiom,
! [A] :
( v4_ordinal2(A)
=> ! [B] : k2_finseq_2(A,B) = k2_funcop_1(k2_finseq_1(A),B) ) ).
fof(t67_finseq_2,axiom,
$true ).
fof(t68_finseq_2,axiom,
! [A] :
( v4_ordinal2(A)
=> ! [B] : k4_finseq_1(k2_finseq_2(A,B)) = k2_finseq_1(A) ) ).
fof(t69_finseq_2,axiom,
! [A] :
( v4_ordinal2(A)
=> ! [B] : k3_finseq_1(k2_finseq_2(A,B)) = A ) ).
fof(t70_finseq_2,axiom,
! [A] :
( v4_ordinal2(A)
=> ! [B,C] :
( r2_hidden(B,k2_finseq_1(A))
=> k1_funct_1(k2_finseq_2(A,C),B) = C ) ) ).
fof(t71_finseq_2,axiom,
! [A] :
( v4_ordinal2(A)
=> ! [B] :
( ~ v1_xboole_0(B)
=> ! [C] :
( m1_subset_1(C,B)
=> ( A != np__0
=> ! [D] :
( m1_subset_1(D,k2_finseq_1(A))
=> k1_funct_1(k2_finseq_2(A,C),D) = C ) ) ) ) ) ).
fof(t72_finseq_2,axiom,
! [A] : k2_finseq_2(np__0,A) = k1_xboole_0 ).
fof(t73_finseq_2,axiom,
! [A] : k2_finseq_2(np__1,A) = k9_finseq_1(A) ).
fof(t74_finseq_2,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] : k2_finseq_2(k1_nat_1(A,np__1),B) = k7_finseq_1(k2_finseq_2(A,B),k9_finseq_1(B)) ) ).
fof(t75_finseq_2,axiom,
! [A] : k2_finseq_2(np__2,A) = k10_finseq_1(A,A) ).
fof(t76_finseq_2,axiom,
! [A] : k2_finseq_2(np__3,A) = k11_finseq_1(A,A,A) ).
fof(t77_finseq_2,axiom,
! [A] :
( v4_ordinal2(A)
=> ! [B] :
( ~ v1_xboole_0(B)
=> ! [C] :
( m1_subset_1(C,B)
=> m2_finseq_1(k2_finseq_2(A,C),B) ) ) ) ).
fof(t78_finseq_2,axiom,
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A)
& v1_finseq_1(A) )
=> ! [B] :
( ( v1_relat_1(B)
& v1_funct_1(B)
& v1_finseq_1(B) )
=> ! [C] :
( ( v1_relat_1(C)
& v1_funct_1(C) )
=> ( r1_tarski(k2_zfmisc_1(k2_relat_1(A),k2_relat_1(B)),k1_relat_1(C))
=> ( v1_relat_1(k3_funcop_1(C,A,B))
& v1_funct_1(k3_funcop_1(C,A,B))
& v1_finseq_1(k3_funcop_1(C,A,B)) ) ) ) ) ) ).
fof(t79_finseq_2,axiom,
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A)
& v1_finseq_1(A) )
=> ! [B] :
( ( v1_relat_1(B)
& v1_funct_1(B)
& v1_finseq_1(B) )
=> ! [C] :
( ( v1_relat_1(C)
& v1_funct_1(C)
& v1_finseq_1(C) )
=> ! [D] :
( ( v1_relat_1(D)
& v1_funct_1(D) )
=> ( ( r1_tarski(k2_zfmisc_1(k2_relat_1(A),k2_relat_1(B)),k1_relat_1(D))
& C = k3_funcop_1(D,A,B) )
=> k3_finseq_1(C) = k3_square_1(k3_finseq_1(A),k3_finseq_1(B)) ) ) ) ) ) ).
fof(t80_finseq_2,axiom,
! [A,B] :
( ( v1_relat_1(B)
& v1_funct_1(B)
& v1_finseq_1(B) )
=> ! [C] :
( ( v1_relat_1(C)
& v1_funct_1(C) )
=> ( r1_tarski(k2_zfmisc_1(k1_tarski(A),k2_relat_1(B)),k1_relat_1(C))
=> ( v1_relat_1(k5_funcop_1(C,A,B))
& v1_funct_1(k5_funcop_1(C,A,B))
& v1_finseq_1(k5_funcop_1(C,A,B)) ) ) ) ) ).
fof(t81_finseq_2,axiom,
! [A,B] :
( ( v1_relat_1(B)
& v1_funct_1(B)
& v1_finseq_1(B) )
=> ! [C] :
( ( v1_relat_1(C)
& v1_funct_1(C)
& v1_finseq_1(C) )
=> ! [D] :
( ( v1_relat_1(D)
& v1_funct_1(D) )
=> ( ( r1_tarski(k2_zfmisc_1(k1_tarski(A),k2_relat_1(B)),k1_relat_1(D))
& C = k5_funcop_1(D,A,B) )
=> k3_finseq_1(C) = k3_finseq_1(B) ) ) ) ) ).
fof(t82_finseq_2,axiom,
! [A,B] :
( ( v1_relat_1(B)
& v1_funct_1(B)
& v1_finseq_1(B) )
=> ! [C] :
( ( v1_relat_1(C)
& v1_funct_1(C) )
=> ( r1_tarski(k2_zfmisc_1(k2_relat_1(B),k1_tarski(A)),k1_relat_1(C))
=> ( v1_relat_1(k4_funcop_1(C,B,A))
& v1_funct_1(k4_funcop_1(C,B,A))
& v1_finseq_1(k4_funcop_1(C,B,A)) ) ) ) ) ).
fof(t83_finseq_2,axiom,
! [A,B] :
( ( v1_relat_1(B)
& v1_funct_1(B)
& v1_finseq_1(B) )
=> ! [C] :
( ( v1_relat_1(C)
& v1_funct_1(C)
& v1_finseq_1(C) )
=> ! [D] :
( ( v1_relat_1(D)
& v1_funct_1(D) )
=> ( ( r1_tarski(k2_zfmisc_1(k2_relat_1(B),k1_tarski(A)),k1_relat_1(D))
& C = k4_funcop_1(D,B,A) )
=> k3_finseq_1(C) = k3_finseq_1(B) ) ) ) ) ).
fof(t84_finseq_2,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,k2_zfmisc_1(A,B),C)
& m2_relset_1(D,k2_zfmisc_1(A,B),C) )
=> ! [E] :
( m2_finseq_1(E,A)
=> ! [F] :
( m2_finseq_1(F,B)
=> m2_finseq_1(k3_funcop_1(D,E,F),C) ) ) ) ) ) ) ).
fof(t85_finseq_2,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( ~ v1_xboole_0(B)
=> ! [C] :
( ~ v1_xboole_0(C)
=> ! [D] :
( ( v1_relat_1(D)
& v1_funct_1(D)
& v1_finseq_1(D) )
=> ! [E] :
( ( v1_funct_1(E)
& v1_funct_2(E,k2_zfmisc_1(A,B),C)
& m2_relset_1(E,k2_zfmisc_1(A,B),C) )
=> ! [F] :
( m2_finseq_1(F,A)
=> ! [G] :
( m2_finseq_1(G,B)
=> ( D = k3_funcop_1(E,F,G)
=> k3_finseq_1(D) = k3_square_1(k3_finseq_1(F),k3_finseq_1(G)) ) ) ) ) ) ) ) ) ).
fof(t86_finseq_2,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( ~ v1_xboole_0(B)
=> ! [C] :
( ~ v1_xboole_0(C)
=> ! [D] :
( ( v1_relat_1(D)
& v1_funct_1(D)
& v1_finseq_1(D) )
=> ! [E] :
( ( v1_funct_1(E)
& v1_funct_2(E,k2_zfmisc_1(A,B),C)
& m2_relset_1(E,k2_zfmisc_1(A,B),C) )
=> ! [F] :
( m2_finseq_1(F,A)
=> ! [G] :
( m2_finseq_1(G,B)
=> ( ( k3_finseq_1(F) = k3_finseq_1(G)
& D = k3_funcop_1(E,F,G) )
=> ( k3_finseq_1(D) = k3_finseq_1(F)
& k3_finseq_1(D) = k3_finseq_1(G) ) ) ) ) ) ) ) ) ) ).
fof(t87_finseq_2,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,k2_zfmisc_1(A,B),C)
& m2_relset_1(D,k2_zfmisc_1(A,B),C) )
=> ! [E] :
( m2_finseq_1(E,A)
=> ! [F] :
( m2_finseq_1(F,B)
=> ( k3_funcop_1(D,k6_finseq_1(A),F) = k6_finseq_1(C)
& k3_funcop_1(D,E,k6_finseq_1(B)) = k6_finseq_1(C) ) ) ) ) ) ) ) ).
fof(t88_finseq_2,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( ~ v1_xboole_0(B)
=> ! [C] :
( ~ v1_xboole_0(C)
=> ! [D] :
( m1_subset_1(D,A)
=> ! [E] :
( m1_subset_1(E,B)
=> ! [F] :
( ( v1_funct_1(F)
& v1_funct_2(F,k2_zfmisc_1(A,B),C)
& m2_relset_1(F,k2_zfmisc_1(A,B),C) )
=> ! [G] :
( m2_finseq_1(G,A)
=> ! [H] :
( m2_finseq_1(H,B)
=> ( ( G = k12_finseq_1(A,D)
& H = k12_finseq_1(B,E) )
=> k3_funcop_1(F,G,H) = k12_finseq_1(C,k2_binop_1(A,B,C,F,D,E)) ) ) ) ) ) ) ) ) ) ).
fof(t89_finseq_2,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( ~ v1_xboole_0(B)
=> ! [C] :
( ~ v1_xboole_0(C)
=> ! [D] :
( m1_subset_1(D,A)
=> ! [E] :
( m1_subset_1(E,A)
=> ! [F] :
( m1_subset_1(F,B)
=> ! [G] :
( m1_subset_1(G,B)
=> ! [H] :
( ( v1_funct_1(H)
& v1_funct_2(H,k2_zfmisc_1(A,B),C)
& m2_relset_1(H,k2_zfmisc_1(A,B),C) )
=> ! [I] :
( m2_finseq_1(I,A)
=> ! [J] :
( m2_finseq_1(J,B)
=> ( ( I = k10_finseq_1(D,E)
& J = k10_finseq_1(F,G) )
=> k3_funcop_1(H,I,J) = k10_finseq_1(k2_binop_1(A,B,C,H,D,F),k2_binop_1(A,B,C,H,E,G)) ) ) ) ) ) ) ) ) ) ) ) ).
fof(t90_finseq_2,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( ~ v1_xboole_0(B)
=> ! [C] :
( ~ v1_xboole_0(C)
=> ! [D] :
( m1_subset_1(D,A)
=> ! [E] :
( m1_subset_1(E,A)
=> ! [F] :
( m1_subset_1(F,A)
=> ! [G] :
( m1_subset_1(G,B)
=> ! [H] :
( m1_subset_1(H,B)
=> ! [I] :
( m1_subset_1(I,B)
=> ! [J] :
( ( v1_funct_1(J)
& v1_funct_2(J,k2_zfmisc_1(A,B),C)
& m2_relset_1(J,k2_zfmisc_1(A,B),C) )
=> ! [K] :
( m2_finseq_1(K,A)
=> ! [L] :
( m2_finseq_1(L,B)
=> ( ( K = k11_finseq_1(D,E,F)
& L = k11_finseq_1(G,H,I) )
=> k3_funcop_1(J,K,L) = k11_finseq_1(k2_binop_1(A,B,C,J,D,G),k2_binop_1(A,B,C,J,E,H),k2_binop_1(A,B,C,J,F,I)) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
fof(t91_finseq_2,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( ~ v1_xboole_0(B)
=> ! [C] :
( ~ v1_xboole_0(C)
=> ! [D] :
( m1_subset_1(D,A)
=> ! [E] :
( ( v1_funct_1(E)
& v1_funct_2(E,k2_zfmisc_1(A,B),C)
& m2_relset_1(E,k2_zfmisc_1(A,B),C) )
=> ! [F] :
( m2_finseq_1(F,B)
=> m2_finseq_1(k5_funcop_1(E,D,F),C) ) ) ) ) ) ) ).
fof(t92_finseq_2,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( ~ v1_xboole_0(B)
=> ! [C] :
( ~ v1_xboole_0(C)
=> ! [D] :
( m1_subset_1(D,A)
=> ! [E] :
( ( v1_relat_1(E)
& v1_funct_1(E)
& v1_finseq_1(E) )
=> ! [F] :
( ( v1_funct_1(F)
& v1_funct_2(F,k2_zfmisc_1(A,B),C)
& m2_relset_1(F,k2_zfmisc_1(A,B),C) )
=> ! [G] :
( m2_finseq_1(G,B)
=> ( E = k5_funcop_1(F,D,G)
=> k3_finseq_1(E) = k3_finseq_1(G) ) ) ) ) ) ) ) ) ).
fof(t93_finseq_2,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( ~ v1_xboole_0(B)
=> ! [C] :
( ~ v1_xboole_0(C)
=> ! [D] :
( m1_subset_1(D,A)
=> ! [E] :
( ( v1_funct_1(E)
& v1_funct_2(E,k2_zfmisc_1(A,B),C)
& m2_relset_1(E,k2_zfmisc_1(A,B),C) )
=> k5_funcop_1(E,D,k6_finseq_1(B)) = k6_finseq_1(C) ) ) ) ) ) ).
fof(t94_finseq_2,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( ~ v1_xboole_0(B)
=> ! [C] :
( ~ v1_xboole_0(C)
=> ! [D] :
( m1_subset_1(D,A)
=> ! [E] :
( m1_subset_1(E,B)
=> ! [F] :
( ( v1_funct_1(F)
& v1_funct_2(F,k2_zfmisc_1(A,B),C)
& m2_relset_1(F,k2_zfmisc_1(A,B),C) )
=> ! [G] :
( m2_finseq_1(G,B)
=> ( G = k12_finseq_1(B,E)
=> k5_funcop_1(F,D,G) = k12_finseq_1(C,k2_binop_1(A,B,C,F,D,E)) ) ) ) ) ) ) ) ) ).
fof(t95_finseq_2,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( ~ v1_xboole_0(B)
=> ! [C] :
( ~ v1_xboole_0(C)
=> ! [D] :
( m1_subset_1(D,A)
=> ! [E] :
( m1_subset_1(E,B)
=> ! [F] :
( m1_subset_1(F,B)
=> ! [G] :
( ( v1_funct_1(G)
& v1_funct_2(G,k2_zfmisc_1(A,B),C)
& m2_relset_1(G,k2_zfmisc_1(A,B),C) )
=> ! [H] :
( m2_finseq_1(H,B)
=> ( H = k10_finseq_1(E,F)
=> k5_funcop_1(G,D,H) = k10_finseq_1(k2_binop_1(A,B,C,G,D,E),k2_binop_1(A,B,C,G,D,F)) ) ) ) ) ) ) ) ) ) ).
fof(t96_finseq_2,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( ~ v1_xboole_0(B)
=> ! [C] :
( ~ v1_xboole_0(C)
=> ! [D] :
( m1_subset_1(D,A)
=> ! [E] :
( m1_subset_1(E,B)
=> ! [F] :
( m1_subset_1(F,B)
=> ! [G] :
( m1_subset_1(G,B)
=> ! [H] :
( ( v1_funct_1(H)
& v1_funct_2(H,k2_zfmisc_1(A,B),C)
& m2_relset_1(H,k2_zfmisc_1(A,B),C) )
=> ! [I] :
( m2_finseq_1(I,B)
=> ( I = k11_finseq_1(E,F,G)
=> k5_funcop_1(H,D,I) = k11_finseq_1(k2_binop_1(A,B,C,H,D,E),k2_binop_1(A,B,C,H,D,F),k2_binop_1(A,B,C,H,D,G)) ) ) ) ) ) ) ) ) ) ) ).
fof(t97_finseq_2,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( ~ v1_xboole_0(B)
=> ! [C] :
( ~ v1_xboole_0(C)
=> ! [D] :
( m1_subset_1(D,B)
=> ! [E] :
( ( v1_funct_1(E)
& v1_funct_2(E,k2_zfmisc_1(A,B),C)
& m2_relset_1(E,k2_zfmisc_1(A,B),C) )
=> ! [F] :
( m2_finseq_1(F,A)
=> m2_finseq_1(k4_funcop_1(E,F,D),C) ) ) ) ) ) ) ).
fof(t98_finseq_2,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( ~ v1_xboole_0(B)
=> ! [C] :
( ~ v1_xboole_0(C)
=> ! [D] :
( m1_subset_1(D,B)
=> ! [E] :
( ( v1_relat_1(E)
& v1_funct_1(E)
& v1_finseq_1(E) )
=> ! [F] :
( ( v1_funct_1(F)
& v1_funct_2(F,k2_zfmisc_1(A,B),C)
& m2_relset_1(F,k2_zfmisc_1(A,B),C) )
=> ! [G] :
( m2_finseq_1(G,A)
=> ( E = k4_funcop_1(F,G,D)
=> k3_finseq_1(E) = k3_finseq_1(G) ) ) ) ) ) ) ) ) ).
fof(t99_finseq_2,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( ~ v1_xboole_0(B)
=> ! [C] :
( ~ v1_xboole_0(C)
=> ! [D] :
( m1_subset_1(D,B)
=> ! [E] :
( ( v1_funct_1(E)
& v1_funct_2(E,k2_zfmisc_1(A,B),C)
& m2_relset_1(E,k2_zfmisc_1(A,B),C) )
=> k4_funcop_1(E,k6_finseq_1(A),D) = k6_finseq_1(C) ) ) ) ) ) ).
fof(t100_finseq_2,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( ~ v1_xboole_0(B)
=> ! [C] :
( ~ v1_xboole_0(C)
=> ! [D] :
( m1_subset_1(D,A)
=> ! [E] :
( m1_subset_1(E,B)
=> ! [F] :
( ( v1_funct_1(F)
& v1_funct_2(F,k2_zfmisc_1(A,B),C)
& m2_relset_1(F,k2_zfmisc_1(A,B),C) )
=> ! [G] :
( m2_finseq_1(G,A)
=> ( G = k12_finseq_1(A,D)
=> k4_funcop_1(F,G,E) = k12_finseq_1(C,k2_binop_1(A,B,C,F,D,E)) ) ) ) ) ) ) ) ) ).
fof(t101_finseq_2,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( ~ v1_xboole_0(B)
=> ! [C] :
( ~ v1_xboole_0(C)
=> ! [D] :
( m1_subset_1(D,A)
=> ! [E] :
( m1_subset_1(E,A)
=> ! [F] :
( m1_subset_1(F,B)
=> ! [G] :
( ( v1_funct_1(G)
& v1_funct_2(G,k2_zfmisc_1(A,B),C)
& m2_relset_1(G,k2_zfmisc_1(A,B),C) )
=> ! [H] :
( m2_finseq_1(H,A)
=> ( H = k10_finseq_1(D,E)
=> k4_funcop_1(G,H,F) = k10_finseq_1(k2_binop_1(A,B,C,G,D,F),k2_binop_1(A,B,C,G,E,F)) ) ) ) ) ) ) ) ) ) ).
fof(t102_finseq_2,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( ~ v1_xboole_0(B)
=> ! [C] :
( ~ v1_xboole_0(C)
=> ! [D] :
( m1_subset_1(D,A)
=> ! [E] :
( m1_subset_1(E,A)
=> ! [F] :
( m1_subset_1(F,A)
=> ! [G] :
( m1_subset_1(G,B)
=> ! [H] :
( ( v1_funct_1(H)
& v1_funct_2(H,k2_zfmisc_1(A,B),C)
& m2_relset_1(H,k2_zfmisc_1(A,B),C) )
=> ! [I] :
( m2_finseq_1(I,A)
=> ( I = k11_finseq_1(D,E,F)
=> k4_funcop_1(H,I,G) = k11_finseq_1(k2_binop_1(A,B,C,H,D,G),k2_binop_1(A,B,C,H,E,G),k2_binop_1(A,B,C,H,F,G)) ) ) ) ) ) ) ) ) ) ) ).
fof(d3_finseq_2,axiom,
! [A,B] :
( m1_finseq_2(B,A)
<=> ! [C] :
( r2_hidden(C,B)
=> m2_finseq_1(C,A) ) ) ).
fof(t103_finseq_2,axiom,
$true ).
fof(t104_finseq_2,axiom,
! [A] :
( ~ v1_xboole_0(k13_finseq_1(A))
& m1_finseq_2(k13_finseq_1(A),A) ) ).
fof(t105_finseq_2,axiom,
! [A,B] :
( ( ~ v1_xboole_0(B)
& m1_finseq_2(B,A) )
=> r1_tarski(B,k3_finseq_2(A)) ) ).
fof(t106_finseq_2,axiom,
$true ).
fof(t107_finseq_2,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( ( ~ v1_xboole_0(B)
& m1_subset_1(B,k1_zfmisc_1(A)) )
=> ! [C] :
( ( ~ v1_xboole_0(C)
& m1_finseq_2(C,B) )
=> ( ~ v1_xboole_0(C)
& m1_finseq_2(C,A) ) ) ) ) ).
fof(t108_finseq_2,axiom,
$true ).
fof(t109_finseq_2,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( ~ v1_xboole_0(B)
=> ! [C] :
( m2_finseq_2(C,B,k4_finseq_2(A,B))
=> k3_finseq_1(C) = A ) ) ) ).
fof(t110_finseq_2,axiom,
! [A,B] :
( m2_finseq_1(B,A)
=> m1_subset_1(B,k4_finseq_2(k3_finseq_1(B),A)) ) ).
fof(t111_finseq_2,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( ~ v1_xboole_0(B)
=> k4_finseq_2(A,B) = k1_funct_2(k2_finseq_1(A),B) ) ) ).
fof(t112_finseq_2,axiom,
! [A] : k4_finseq_2(np__0,A) = k6_domain_1(k1_zfmisc_1(k2_zfmisc_1(k5_numbers,A)),k6_finseq_1(A)) ).
fof(t113_finseq_2,axiom,
! [A,B] :
( m1_subset_1(B,k4_finseq_2(np__0,A))
=> B = k6_finseq_1(A) ) ).
fof(t114_finseq_2,axiom,
! [A] : m1_subset_1(k6_finseq_1(A),k4_finseq_2(np__0,A)) ).
fof(t115_finseq_2,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( ~ v1_xboole_0(B)
=> ! [C] :
( m2_finseq_2(C,B,k4_finseq_2(np__0,B))
=> ! [D] :
( m2_finseq_2(D,B,k4_finseq_2(A,B))
=> ( k8_finseq_1(B,C,D) = D
& k8_finseq_1(B,D,C) = D ) ) ) ) ) ).
fof(t117_finseq_2,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m2_finseq_2(B,A,k4_finseq_2(np__1,A))
=> ? [C] :
( m1_subset_1(C,A)
& B = k12_finseq_1(A,C) ) ) ) ).
fof(t118_finseq_2,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m1_subset_1(B,A)
=> m2_finseq_2(k12_finseq_1(A,B),A,k4_finseq_2(np__1,A)) ) ) ).
fof(t120_finseq_2,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m2_finseq_2(B,A,k4_finseq_2(np__2,A))
=> ? [C] :
( m1_subset_1(C,A)
& ? [D] :
( m1_subset_1(D,A)
& B = k10_finseq_1(C,D) ) ) ) ) ).
fof(t121_finseq_2,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m1_subset_1(B,A)
=> ! [C] :
( m1_subset_1(C,A)
=> m2_finseq_2(k10_finseq_1(B,C),A,k4_finseq_2(np__2,A)) ) ) ) ).
fof(t123_finseq_2,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m2_finseq_2(B,A,k4_finseq_2(np__3,A))
=> ? [C] :
( m1_subset_1(C,A)
& ? [D] :
( m1_subset_1(D,A)
& ? [E] :
( m1_subset_1(E,A)
& B = k11_finseq_1(C,D,E) ) ) ) ) ) ).
fof(t124_finseq_2,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m1_subset_1(B,A)
=> ! [C] :
( m1_subset_1(C,A)
=> ! [D] :
( m1_subset_1(D,A)
=> m2_finseq_2(k11_finseq_1(B,C,D),A,k4_finseq_2(np__3,A)) ) ) ) ) ).
fof(t126_finseq_2,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> ! [C] :
( ~ v1_xboole_0(C)
=> ! [D] :
( m2_finseq_2(D,C,k4_finseq_2(k1_nat_1(A,B),C))
=> ? [E] :
( m2_finseq_2(E,C,k4_finseq_2(A,C))
& ? [F] :
( m2_finseq_2(F,C,k4_finseq_2(B,C))
& D = k8_finseq_1(C,E,F) ) ) ) ) ) ) ).
fof(t127_finseq_2,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> ! [C] :
( ~ v1_xboole_0(C)
=> ! [D] :
( m2_finseq_2(D,C,k4_finseq_2(A,C))
=> ! [E] :
( m2_finseq_2(E,C,k4_finseq_2(B,C))
=> m2_finseq_2(k8_finseq_1(C,D,E),C,k4_finseq_2(k1_nat_1(A,B),C)) ) ) ) ) ) ).
fof(t129_finseq_2,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( ~ v1_xboole_0(B)
=> ! [C] :
( ( ~ v1_xboole_0(C)
& m1_subset_1(C,k1_zfmisc_1(B)) )
=> ! [D] :
( m2_finseq_2(D,C,k4_finseq_2(A,C))
=> m2_finseq_2(D,B,k4_finseq_2(A,B)) ) ) ) ) ).
fof(t130_finseq_2,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> ! [C] :
( ~ v1_xboole_0(C)
=> ( k4_finseq_2(A,C) = k4_finseq_2(B,C)
=> A = B ) ) ) ) ).
fof(t131_finseq_2,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> m2_finseq_2(k1_finseq_2(A),k5_numbers,k4_finseq_2(A,k5_numbers)) ) ).
fof(t132_finseq_2,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( ~ v1_xboole_0(B)
=> ! [C] :
( m1_subset_1(C,B)
=> m2_finseq_2(k2_finseq_2(A,C),B,k4_finseq_2(A,B)) ) ) ) ).
fof(t133_finseq_2,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( ~ v1_xboole_0(B)
=> ! [C] :
( ~ v1_xboole_0(C)
=> ! [D] :
( m2_finseq_2(D,B,k4_finseq_2(A,B))
=> ! [E] :
( ( v1_funct_1(E)
& v1_funct_2(E,B,C)
& m2_relset_1(E,B,C) )
=> m2_finseq_2(k1_partfun1(k5_numbers,B,B,C,D,E),C,k4_finseq_2(A,C)) ) ) ) ) ) ).
fof(t134_finseq_2,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( ~ v1_xboole_0(B)
=> ! [C] :
( m2_finseq_2(C,B,k4_finseq_2(A,B))
=> ! [D] :
( ( v1_funct_1(D)
& v1_funct_2(D,k2_finseq_1(A),k2_finseq_1(A))
& m2_relset_1(D,k2_finseq_1(A),k2_finseq_1(A)) )
=> ( k2_relat_1(D) = k2_finseq_1(A)
=> m2_finseq_2(k1_partfun1(k2_finseq_1(A),k2_finseq_1(A),k5_numbers,B,D,C),B,k4_finseq_2(A,B)) ) ) ) ) ) ).
fof(t135_finseq_2,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( ~ v1_xboole_0(B)
=> ! [C] :
( m2_finseq_2(C,B,k4_finseq_2(A,B))
=> ! [D] :
( ( v1_funct_1(D)
& v1_funct_2(D,k2_finseq_1(A),k2_finseq_1(A))
& v3_funct_2(D,k2_finseq_1(A),k2_finseq_1(A))
& m2_relset_1(D,k2_finseq_1(A),k2_finseq_1(A)) )
=> m2_finseq_2(k1_partfun1(k2_finseq_1(A),k2_finseq_1(A),k5_numbers,B,D,C),B,k4_finseq_2(A,B)) ) ) ) ) ).
fof(t136_finseq_2,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( ~ v1_xboole_0(B)
=> ! [C] :
( m2_finseq_2(C,B,k4_finseq_2(A,B))
=> ! [D] :
( m1_subset_1(D,B)
=> k1_funct_1(k8_finseq_1(B,C,k12_finseq_1(B,D)),k1_nat_1(A,np__1)) = D ) ) ) ) ).
fof(t137_finseq_2,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( ~ v1_xboole_0(B)
=> ! [C] :
( m2_finseq_2(C,B,k4_finseq_2(k1_nat_1(A,np__1),B))
=> ? [D] :
( m2_finseq_2(D,B,k4_finseq_2(A,B))
& ? [E] :
( m1_subset_1(E,B)
& C = k8_finseq_1(B,D,k12_finseq_1(B,E)) ) ) ) ) ) ).
fof(t138_finseq_2,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( ~ v1_xboole_0(B)
=> ! [C] :
( m2_finseq_2(C,B,k4_finseq_2(A,B))
=> k5_relat_1(k1_finseq_2(A),C) = C ) ) ) ).
fof(t139_finseq_2,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( ~ v1_xboole_0(B)
=> ! [C] :
( m2_finseq_2(C,B,k4_finseq_2(A,B))
=> ! [D] :
( m2_finseq_2(D,B,k4_finseq_2(A,B))
=> ( ! [E] :
( m2_subset_1(E,k1_numbers,k5_numbers)
=> ( r2_hidden(E,k2_finseq_1(A))
=> k1_funct_1(C,E) = k1_funct_1(D,E) ) )
=> C = D ) ) ) ) ) ).
fof(t140_finseq_2,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( ~ v1_xboole_0(B)
=> ! [C] :
( ~ v1_xboole_0(C)
=> ! [D] :
( ~ v1_xboole_0(D)
=> ! [E] :
( ( v1_funct_1(E)
& v1_funct_2(E,k2_zfmisc_1(B,C),D)
& m2_relset_1(E,k2_zfmisc_1(B,C),D) )
=> ! [F] :
( m2_finseq_2(F,B,k4_finseq_2(A,B))
=> ! [G] :
( m2_finseq_2(G,C,k4_finseq_2(A,C))
=> m2_finseq_2(k3_funcop_1(E,F,G),D,k4_finseq_2(A,D)) ) ) ) ) ) ) ) ).
fof(t141_finseq_2,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( ~ v1_xboole_0(B)
=> ! [C] :
( ~ v1_xboole_0(C)
=> ! [D] :
( ~ v1_xboole_0(D)
=> ! [E] :
( m1_subset_1(E,B)
=> ! [F] :
( ( v1_funct_1(F)
& v1_funct_2(F,k2_zfmisc_1(B,C),D)
& m2_relset_1(F,k2_zfmisc_1(B,C),D) )
=> ! [G] :
( m2_finseq_2(G,C,k4_finseq_2(A,C))
=> m2_finseq_2(k5_funcop_1(F,E,G),D,k4_finseq_2(A,D)) ) ) ) ) ) ) ) ).
fof(t142_finseq_2,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( ~ v1_xboole_0(B)
=> ! [C] :
( ~ v1_xboole_0(C)
=> ! [D] :
( ~ v1_xboole_0(D)
=> ! [E] :
( m1_subset_1(E,C)
=> ! [F] :
( ( v1_funct_1(F)
& v1_funct_2(F,k2_zfmisc_1(B,C),D)
& m2_relset_1(F,k2_zfmisc_1(B,C),D) )
=> ! [G] :
( m2_finseq_2(G,B,k4_finseq_2(A,B))
=> m2_finseq_2(k4_funcop_1(F,G,E),D,k4_finseq_2(A,D)) ) ) ) ) ) ) ) ).
fof(t143_finseq_2,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> ! [C] : k2_finseq_2(k1_nat_1(A,B),C) = k7_finseq_1(k2_finseq_2(A,C),k2_finseq_2(B,C)) ) ) ).
fof(t144_finseq_2,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( ~ v1_xboole_0(B)
=> ! [C] :
( m2_finseq_2(C,B,k4_finseq_2(A,B))
=> k4_finseq_1(C) = k2_finseq_1(A) ) ) ) ).
fof(t145_finseq_2,axiom,
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A) )
=> ! [B,C] :
( ( r2_hidden(B,k1_relat_1(A))
& r2_hidden(C,k1_relat_1(A)) )
=> k5_relat_1(k10_finseq_1(B,C),A) = k10_finseq_1(k1_funct_1(A,B),k1_funct_1(A,C)) ) ) ).
fof(t146_finseq_2,axiom,
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A) )
=> ! [B,C,D] :
( ( r2_hidden(B,k1_relat_1(A))
& r2_hidden(C,k1_relat_1(A))
& r2_hidden(D,k1_relat_1(A)) )
=> k5_relat_1(k11_finseq_1(B,C,D),A) = k11_finseq_1(k1_funct_1(A,B),k1_funct_1(A,C),k1_funct_1(A,D)) ) ) ).
fof(t147_finseq_2,axiom,
! [A,B] : k2_relat_1(k10_finseq_1(A,B)) = k2_tarski(A,B) ).
fof(t148_finseq_2,axiom,
! [A,B,C] : k2_relat_1(k11_finseq_1(A,B,C)) = k1_enumset1(A,B,C) ).
fof(t149_finseq_2,axiom,
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A)
& v1_finseq_1(A) )
=> ! [B] :
( ( v1_relat_1(B)
& v1_funct_1(B)
& v1_finseq_1(B) )
=> ! [C] :
( ( v1_relat_1(C)
& v1_funct_1(C)
& v1_finseq_1(C) )
=> ( ( r1_tarski(A,C)
& r1_tarski(B,C)
& k3_finseq_1(A) = k3_finseq_1(B) )
=> A = B ) ) ) ) ).
fof(s1_finseq_2,axiom,
? [A] :
( m2_finseq_1(A,f2_s1_finseq_2)
& k3_finseq_1(A) = f1_s1_finseq_2
& ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> ( r2_hidden(B,k2_finseq_1(f1_s1_finseq_2))
=> k1_funct_1(A,B) = f3_s1_finseq_2(B) ) ) ) ).
fof(s2_finseq_2,axiom,
( ( p1_s2_finseq_2(k6_finseq_1(f1_s2_finseq_2))
& ! [A] :
( m2_finseq_1(A,f1_s2_finseq_2)
=> ! [B] :
( m1_subset_1(B,f1_s2_finseq_2)
=> ( p1_s2_finseq_2(A)
=> p1_s2_finseq_2(k8_finseq_1(f1_s2_finseq_2,A,k12_finseq_1(f1_s2_finseq_2,B))) ) ) ) )
=> ! [A] :
( m2_finseq_1(A,f1_s2_finseq_2)
=> p1_s2_finseq_2(A) ) ) ).
fof(dt_m1_finseq_2,axiom,
$true ).
fof(existence_m1_finseq_2,axiom,
! [A] :
? [B] : m1_finseq_2(B,A) ).
fof(dt_m2_finseq_2,axiom,
! [A,B] :
( ( ~ v1_xboole_0(B)
& m1_finseq_2(B,A) )
=> ! [C] :
( m2_finseq_2(C,A,B)
=> m2_finseq_1(C,A) ) ) ).
fof(existence_m2_finseq_2,axiom,
! [A,B] :
( ( ~ v1_xboole_0(B)
& m1_finseq_2(B,A) )
=> ? [C] : m2_finseq_2(C,A,B) ) ).
fof(redefinition_m2_finseq_2,axiom,
! [A,B] :
( ( ~ v1_xboole_0(B)
& m1_finseq_2(B,A) )
=> ! [C] :
( m2_finseq_2(C,A,B)
<=> m1_subset_1(C,B) ) ) ).
fof(dt_k1_finseq_2,axiom,
! [A] :
( v4_ordinal2(A)
=> ( v1_relat_1(k1_finseq_2(A))
& v1_funct_1(k1_finseq_2(A))
& v1_finseq_1(k1_finseq_2(A)) ) ) ).
fof(dt_k2_finseq_2,axiom,
! [A,B] :
( v4_ordinal2(A)
=> ( v1_relat_1(k2_finseq_2(A,B))
& v1_funct_1(k2_finseq_2(A,B))
& v1_finseq_1(k2_finseq_2(A,B)) ) ) ).
fof(dt_k3_finseq_2,axiom,
! [A] :
( ~ v1_xboole_0(k3_finseq_2(A))
& m1_finseq_2(k3_finseq_2(A),A) ) ).
fof(redefinition_k3_finseq_2,axiom,
! [A] : k3_finseq_2(A) = k13_finseq_1(A) ).
fof(dt_k4_finseq_2,axiom,
! [A,B] :
( v4_ordinal2(A)
=> m1_finseq_2(k4_finseq_2(A,B),B) ) ).
fof(d4_finseq_2,axiom,
! [A] :
( v4_ordinal2(A)
=> ! [B] : k4_finseq_2(A,B) = a_2_0_finseq_2(A,B) ) ).
fof(t116_finseq_2,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> k4_finseq_2(np__1,A) = a_1_0_finseq_2(A) ) ).
fof(t119_finseq_2,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> k4_finseq_2(np__2,A) = a_1_1_finseq_2(A) ) ).
fof(t122_finseq_2,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> k4_finseq_2(np__3,A) = a_1_2_finseq_2(A) ) ).
fof(t125_finseq_2,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> ! [C] :
( ~ v1_xboole_0(C)
=> k4_finseq_2(k1_nat_1(A,B),C) = a_3_0_finseq_2(A,B,C) ) ) ) ).
fof(t128_finseq_2,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> k3_finseq_2(A) = k3_tarski(a_1_3_finseq_2(A)) ) ).
fof(fraenkel_a_2_0_finseq_2,axiom,
! [A,B,C] :
( v4_ordinal2(B)
=> ( r2_hidden(A,a_2_0_finseq_2(B,C))
<=> ? [D] :
( m2_finseq_2(D,C,k3_finseq_2(C))
& A = D
& k3_finseq_1(D) = B ) ) ) ).
fof(fraenkel_a_1_0_finseq_2,axiom,
! [A,B] :
( ~ v1_xboole_0(B)
=> ( r2_hidden(A,a_1_0_finseq_2(B))
<=> ? [C] :
( m1_subset_1(C,B)
& A = k12_finseq_1(B,C) ) ) ) ).
fof(fraenkel_a_1_1_finseq_2,axiom,
! [A,B] :
( ~ v1_xboole_0(B)
=> ( r2_hidden(A,a_1_1_finseq_2(B))
<=> ? [C,D] :
( m1_subset_1(C,B)
& m1_subset_1(D,B)
& A = k10_finseq_1(C,D) ) ) ) ).
fof(fraenkel_a_1_2_finseq_2,axiom,
! [A,B] :
( ~ v1_xboole_0(B)
=> ( r2_hidden(A,a_1_2_finseq_2(B))
<=> ? [C,D,E] :
( m1_subset_1(C,B)
& m1_subset_1(D,B)
& m1_subset_1(E,B)
& A = k11_finseq_1(C,D,E) ) ) ) ).
fof(fraenkel_a_3_0_finseq_2,axiom,
! [A,B,C,D] :
( ( m2_subset_1(B,k1_numbers,k5_numbers)
& m2_subset_1(C,k1_numbers,k5_numbers)
& ~ v1_xboole_0(D) )
=> ( r2_hidden(A,a_3_0_finseq_2(B,C,D))
<=> ? [E,F] :
( m2_finseq_2(E,D,k4_finseq_2(B,D))
& m2_finseq_2(F,D,k4_finseq_2(C,D))
& A = k8_finseq_1(D,E,F) ) ) ) ).
fof(fraenkel_a_1_3_finseq_2,axiom,
! [A,B] :
( ~ v1_xboole_0(B)
=> ( r2_hidden(A,a_1_3_finseq_2(B))
<=> ? [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
& A = k4_finseq_2(C,B) ) ) ) ).
%------------------------------------------------------------------------------