SET007 Axioms: SET007+628.ax
%------------------------------------------------------------------------------
% File : SET007+628 : TPTP v8.2.0. Released v3.4.0.
% Domain : Set Theory
% Axioms : Asymptotic Notation. Part I: Theory
% 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 : asympt_0 [Urb08]
% Status : Satisfiable
% Syntax : Number of formulae : 113 ( 2 unt; 0 def)
% Number of atoms : 976 ( 53 equ)
% Maximal formula atoms : 18 ( 8 avg)
% Number of connectives : 910 ( 47 ~; 4 |; 605 &)
% ( 29 <=>; 225 =>; 0 <=; 0 <~>)
% Maximal formula depth : 21 ( 9 avg)
% Maximal term depth : 4 ( 1 avg)
% Number of predicates : 32 ( 30 usr; 1 prp; 0-4 aty)
% Number of functors : 59 ( 59 usr; 15 con; 0-4 aty)
% Number of variables : 279 ( 228 !; 51 ?)
% 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_asympt_0,axiom,
? [A] :
( m1_subset_1(A,k1_numbers)
& ~ v1_xboole_0(A)
& v1_xcmplx_0(A)
& v1_xreal_0(A)
& v2_xreal_0(A)
& ~ v3_xreal_0(A) ) ).
fof(rc2_asympt_0,axiom,
? [A] :
( m1_subset_1(A,k1_numbers)
& ~ v1_xboole_0(A)
& v1_xcmplx_0(A)
& v1_xreal_0(A)
& ~ v2_xreal_0(A)
& v3_xreal_0(A) ) ).
fof(rc3_asympt_0,axiom,
? [A] :
( m1_subset_1(A,k1_numbers)
& v1_xcmplx_0(A)
& v1_xreal_0(A)
& v1_asympt_0(A) ) ).
fof(rc4_asympt_0,axiom,
? [A] :
( m1_subset_1(A,k1_numbers)
& v1_xcmplx_0(A)
& v1_xreal_0(A)
& ~ v3_xreal_0(A) ) ).
fof(rc5_asympt_0,axiom,
? [A] :
( m1_subset_1(A,k1_numbers)
& v1_xcmplx_0(A)
& v1_xreal_0(A)
& ~ v2_xreal_0(A) ) ).
fof(rc6_asympt_0,axiom,
? [A] :
( m1_subset_1(A,k1_numbers)
& v1_xcmplx_0(A)
& v1_xreal_0(A)
& ~ v1_asympt_0(A) ) ).
fof(rc7_asympt_0,axiom,
? [A] :
( m1_relset_1(A,k5_numbers,k1_numbers)
& v1_relat_1(A)
& v1_funct_1(A)
& v1_funct_2(A,k5_numbers,k1_numbers)
& v1_seq_1(A)
& v2_asympt_0(A)
& v3_asympt_0(A)
& v4_asympt_0(A)
& v5_asympt_0(A)
& v6_asympt_0(A) ) ).
fof(cc1_asympt_0,axiom,
! [A] :
( m1_relset_1(A,k5_numbers,k1_numbers)
=> ( ( v1_funct_1(A)
& v1_funct_2(A,k5_numbers,k1_numbers)
& v3_asympt_0(A) )
=> ( v1_funct_1(A)
& v1_funct_2(A,k5_numbers,k1_numbers)
& v1_seq_1(A)
& v4_asympt_0(A) ) ) ) ).
fof(cc2_asympt_0,axiom,
! [A] :
( m1_relset_1(A,k5_numbers,k1_numbers)
=> ( ( v1_funct_1(A)
& v1_funct_2(A,k5_numbers,k1_numbers)
& v4_asympt_0(A) )
=> ( v1_funct_1(A)
& v1_funct_2(A,k5_numbers,k1_numbers)
& v1_seq_1(A)
& v2_asympt_0(A)
& v5_asympt_0(A) ) ) ) ).
fof(cc3_asympt_0,axiom,
! [A] :
( m1_relset_1(A,k5_numbers,k1_numbers)
=> ( ( v1_funct_1(A)
& v1_funct_2(A,k5_numbers,k1_numbers)
& v2_asympt_0(A)
& v5_asympt_0(A) )
=> ( v1_funct_1(A)
& v1_funct_2(A,k5_numbers,k1_numbers)
& v1_seq_1(A)
& v4_asympt_0(A) ) ) ) ).
fof(fc1_asympt_0,axiom,
! [A,B] :
( ( v1_funct_1(A)
& v1_funct_2(A,k5_numbers,k1_numbers)
& v2_asympt_0(A)
& m1_relset_1(A,k5_numbers,k1_numbers)
& ~ v3_xreal_0(B)
& m1_subset_1(B,k1_numbers) )
=> ( v1_relat_1(k2_asympt_0(A,B))
& v1_funct_1(k2_asympt_0(A,B))
& v1_funct_2(k2_asympt_0(A,B),k5_numbers,k1_numbers)
& v1_seq_1(k2_asympt_0(A,B))
& v2_asympt_0(k2_asympt_0(A,B)) ) ) ).
fof(fc2_asympt_0,axiom,
! [A,B] :
( ( v1_funct_1(A)
& v1_funct_2(A,k5_numbers,k1_numbers)
& v2_asympt_0(A)
& m1_relset_1(A,k5_numbers,k1_numbers)
& v2_xreal_0(B)
& m1_subset_1(B,k1_numbers) )
=> ( v1_relat_1(k2_asympt_0(A,B))
& v1_funct_1(k2_asympt_0(A,B))
& v1_funct_2(k2_asympt_0(A,B),k5_numbers,k1_numbers)
& v1_seq_1(k2_asympt_0(A,B))
& v2_asympt_0(k2_asympt_0(A,B))
& v4_asympt_0(k2_asympt_0(A,B))
& v5_asympt_0(k2_asympt_0(A,B)) ) ) ).
fof(fc3_asympt_0,axiom,
! [A,B] :
( ( v1_funct_1(A)
& v1_funct_2(A,k5_numbers,k1_numbers)
& m1_relset_1(A,k5_numbers,k1_numbers)
& v1_funct_1(B)
& v1_funct_2(B,k5_numbers,k1_numbers)
& v2_asympt_0(B)
& m1_relset_1(B,k5_numbers,k1_numbers) )
=> ( v1_relat_1(k4_asympt_0(A,B))
& v1_funct_1(k4_asympt_0(A,B))
& v1_funct_2(k4_asympt_0(A,B),k5_numbers,k1_numbers)
& v1_seq_1(k4_asympt_0(A,B))
& v2_asympt_0(k4_asympt_0(A,B)) ) ) ).
fof(fc4_asympt_0,axiom,
! [A,B] :
( ( v1_funct_1(A)
& v1_funct_2(A,k5_numbers,k1_numbers)
& m1_relset_1(A,k5_numbers,k1_numbers)
& v1_funct_1(B)
& v1_funct_2(B,k5_numbers,k1_numbers)
& v4_asympt_0(B)
& m1_relset_1(B,k5_numbers,k1_numbers) )
=> ( v1_relat_1(k4_asympt_0(A,B))
& v1_funct_1(k4_asympt_0(A,B))
& v1_funct_2(k4_asympt_0(A,B),k5_numbers,k1_numbers)
& v1_seq_1(k4_asympt_0(A,B))
& v2_asympt_0(k4_asympt_0(A,B))
& v4_asympt_0(k4_asympt_0(A,B))
& v5_asympt_0(k4_asympt_0(A,B)) ) ) ).
fof(d1_asympt_0,axiom,
$true ).
fof(d2_asympt_0,axiom,
$true ).
fof(d3_asympt_0,axiom,
! [A] :
( v1_xreal_0(A)
=> ( v1_asympt_0(A)
<=> ( ~ r1_xreal_0(A,np__0)
& A != np__1 ) ) ) ).
fof(d4_asympt_0,axiom,
! [A] :
( ( v1_funct_1(A)
& v1_funct_2(A,k5_numbers,k1_numbers)
& m2_relset_1(A,k5_numbers,k1_numbers) )
=> ( v2_asympt_0(A)
<=> ? [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
& ! [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
=> ( r1_xreal_0(B,C)
=> r1_xreal_0(np__0,k2_seq_1(k5_numbers,k1_numbers,A,C)) ) ) ) ) ) ).
fof(d5_asympt_0,axiom,
! [A] :
( ( v1_funct_1(A)
& v1_funct_2(A,k5_numbers,k1_numbers)
& m2_relset_1(A,k5_numbers,k1_numbers) )
=> ( v3_asympt_0(A)
<=> ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> ~ r1_xreal_0(k2_seq_1(k5_numbers,k1_numbers,A,B),np__0) ) ) ) ).
fof(d6_asympt_0,axiom,
! [A] :
( ( v1_funct_1(A)
& v1_funct_2(A,k5_numbers,k1_numbers)
& m2_relset_1(A,k5_numbers,k1_numbers) )
=> ( v4_asympt_0(A)
<=> ? [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
& ! [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
=> ~ ( r1_xreal_0(B,C)
& r1_xreal_0(k2_seq_1(k5_numbers,k1_numbers,A,C),np__0) ) ) ) ) ) ).
fof(d7_asympt_0,axiom,
! [A] :
( ( v1_funct_1(A)
& v1_funct_2(A,k5_numbers,k1_numbers)
& m2_relset_1(A,k5_numbers,k1_numbers) )
=> ( v5_asympt_0(A)
<=> ? [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
& ! [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
=> ~ ( r1_xreal_0(B,C)
& k2_seq_1(k5_numbers,k1_numbers,A,C) = np__0 ) ) ) ) ) ).
fof(d8_asympt_0,axiom,
! [A] :
( ( v1_funct_1(A)
& v1_funct_2(A,k5_numbers,k1_numbers)
& m2_relset_1(A,k5_numbers,k1_numbers) )
=> ( v6_asympt_0(A)
<=> ? [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
& ! [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
=> ( r1_xreal_0(B,C)
=> r1_xreal_0(k2_seq_1(k5_numbers,k1_numbers,A,C),k2_seq_1(k5_numbers,k1_numbers,A,k1_nat_1(C,np__1))) ) ) ) ) ) ).
fof(d9_asympt_0,axiom,
! [A] :
( ( v1_funct_1(A)
& v1_funct_2(A,k5_numbers,k1_numbers)
& m2_relset_1(A,k5_numbers,k1_numbers) )
=> ! [B] :
( v1_xreal_0(B)
=> ! [C] :
( ( v1_funct_1(C)
& v1_funct_2(C,k5_numbers,k1_numbers)
& m2_relset_1(C,k5_numbers,k1_numbers) )
=> ( C = k2_asympt_0(A,B)
<=> ! [D] :
( m2_subset_1(D,k1_numbers,k5_numbers)
=> k2_seq_1(k5_numbers,k1_numbers,C,D) = k2_xcmplx_0(B,k2_seq_1(k5_numbers,k1_numbers,A,D)) ) ) ) ) ) ).
fof(d10_asympt_0,axiom,
! [A] :
( ( v1_funct_1(A)
& v1_funct_2(A,k5_numbers,k1_numbers)
& m2_relset_1(A,k5_numbers,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) )
=> ( C = k4_asympt_0(A,B)
<=> ! [D] :
( m2_subset_1(D,k1_numbers,k5_numbers)
=> k2_seq_1(k5_numbers,k1_numbers,C,D) = k4_square_1(k2_seq_1(k5_numbers,k1_numbers,A,D),k2_seq_1(k5_numbers,k1_numbers,B,D)) ) ) ) ) ) ).
fof(d11_asympt_0,axiom,
! [A] :
( ( v1_funct_1(A)
& v1_funct_2(A,k5_numbers,k1_numbers)
& m2_relset_1(A,k5_numbers,k1_numbers) )
=> ! [B] :
( ( v1_funct_1(B)
& v1_funct_2(B,k5_numbers,k1_numbers)
& m2_relset_1(B,k5_numbers,k1_numbers) )
=> ( r1_asympt_0(A,B)
<=> ? [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
& ! [D] :
( m2_subset_1(D,k1_numbers,k5_numbers)
=> ( r1_xreal_0(C,D)
=> r1_xreal_0(k2_seq_1(k5_numbers,k1_numbers,A,D),k2_seq_1(k5_numbers,k1_numbers,B,D)) ) ) ) ) ) ) ).
fof(t1_asympt_0,axiom,
! [A] :
( ( v1_funct_1(A)
& v1_funct_2(A,k5_numbers,k1_numbers)
& m2_relset_1(A,k5_numbers,k1_numbers) )
=> ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> ( ! [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
=> ( r1_xreal_0(B,C)
=> r1_xreal_0(k2_seq_1(k5_numbers,k1_numbers,A,C),k2_seq_1(k5_numbers,k1_numbers,A,k1_nat_1(C,np__1))) ) )
=> ! [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
=> ! [D] :
( m2_subset_1(D,k1_numbers,k5_numbers)
=> ( ( r1_xreal_0(B,C)
& r1_xreal_0(C,D) )
=> r1_xreal_0(k2_seq_1(k5_numbers,k1_numbers,A,C),k2_seq_1(k5_numbers,k1_numbers,A,D)) ) ) ) ) ) ) ).
fof(t2_asympt_0,axiom,
! [A] :
( ( v1_funct_1(A)
& v1_funct_2(A,k5_numbers,k1_numbers)
& v4_asympt_0(A)
& m2_relset_1(A,k5_numbers,k1_numbers) )
=> ! [B] :
( ( v1_funct_1(B)
& v1_funct_2(B,k5_numbers,k1_numbers)
& v4_asympt_0(B)
& m2_relset_1(B,k5_numbers,k1_numbers) )
=> ( v4_seq_2(k19_seq_1(A,B))
=> ( k2_seq_2(k19_seq_1(A,B)) = np__0
| ( v4_seq_2(k19_seq_1(B,A))
& k2_seq_2(k19_seq_1(B,A)) = k2_real_1(k2_seq_2(k19_seq_1(A,B))) ) ) ) ) ) ).
fof(t3_asympt_0,axiom,
! [A] :
( ( v1_funct_1(A)
& v1_funct_2(A,k5_numbers,k1_numbers)
& v2_asympt_0(A)
& m2_relset_1(A,k5_numbers,k1_numbers) )
=> ( v4_seq_2(A)
=> r1_xreal_0(np__0,k2_seq_2(A)) ) ) ).
fof(t4_asympt_0,axiom,
! [A] :
( ( v1_funct_1(A)
& v1_funct_2(A,k5_numbers,k1_numbers)
& m2_relset_1(A,k5_numbers,k1_numbers) )
=> ! [B] :
( ( v1_funct_1(B)
& v1_funct_2(B,k5_numbers,k1_numbers)
& m2_relset_1(B,k5_numbers,k1_numbers) )
=> ( ( v4_seq_2(A)
& v4_seq_2(B)
& r1_asympt_0(A,B) )
=> r1_xreal_0(k2_seq_2(A),k2_seq_2(B)) ) ) ) ).
fof(t5_asympt_0,axiom,
! [A] :
( ( v1_funct_1(A)
& v1_funct_2(A,k5_numbers,k1_numbers)
& m2_relset_1(A,k5_numbers,k1_numbers) )
=> ! [B] :
( ( v1_funct_1(B)
& v1_funct_2(B,k5_numbers,k1_numbers)
& v5_asympt_0(B)
& m2_relset_1(B,k5_numbers,k1_numbers) )
=> ( v1_limfunc1(k19_seq_1(A,B))
=> ( v4_seq_2(k19_seq_1(B,A))
& k2_seq_2(k19_seq_1(B,A)) = np__0 ) ) ) ) ).
fof(t6_asympt_0,axiom,
! [A,B] :
( ( v1_funct_1(B)
& v1_funct_2(B,k5_numbers,k1_numbers)
& v2_asympt_0(B)
& m2_relset_1(B,k5_numbers,k1_numbers) )
=> ( r2_hidden(A,k5_asympt_0(B))
=> ( v1_funct_1(A)
& v1_funct_2(A,k5_numbers,k1_numbers)
& v2_asympt_0(A)
& m2_relset_1(A,k5_numbers,k1_numbers) ) ) ) ).
fof(t7_asympt_0,axiom,
! [A] :
( ( v1_funct_1(A)
& v1_funct_2(A,k5_numbers,k1_numbers)
& v3_asympt_0(A)
& m2_relset_1(A,k5_numbers,k1_numbers) )
=> ! [B] :
( ( v1_funct_1(B)
& v1_funct_2(B,k5_numbers,k1_numbers)
& v2_asympt_0(B)
& m2_relset_1(B,k5_numbers,k1_numbers) )
=> ( r2_hidden(B,k5_asympt_0(A))
<=> ? [C] :
( m1_subset_1(C,k1_numbers)
& ~ r1_xreal_0(C,np__0)
& ! [D] :
( m2_subset_1(D,k1_numbers,k5_numbers)
=> r1_xreal_0(k2_seq_1(k5_numbers,k1_numbers,B,D),k4_real_1(C,k2_seq_1(k5_numbers,k1_numbers,A,D))) ) ) ) ) ) ).
fof(t8_asympt_0,axiom,
! [A] :
( ( v1_funct_1(A)
& v1_funct_2(A,k5_numbers,k1_numbers)
& v4_asympt_0(A)
& m2_relset_1(A,k5_numbers,k1_numbers) )
=> ! [B] :
( ( v1_funct_1(B)
& v1_funct_2(B,k5_numbers,k1_numbers)
& v2_asympt_0(B)
& m2_relset_1(B,k5_numbers,k1_numbers) )
=> ! [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
=> ~ ( r2_hidden(B,k5_asympt_0(A))
& ! [D] :
( m2_subset_1(D,k1_numbers,k5_numbers)
=> ~ ( r1_xreal_0(C,D)
& r1_xreal_0(k2_seq_1(k5_numbers,k1_numbers,A,D),np__0) ) )
& ! [D] :
( m1_subset_1(D,k1_numbers)
=> ~ ( ~ r1_xreal_0(D,np__0)
& ! [E] :
( m2_subset_1(E,k1_numbers,k5_numbers)
=> ( r1_xreal_0(C,E)
=> r1_xreal_0(k2_seq_1(k5_numbers,k1_numbers,B,E),k4_real_1(D,k2_seq_1(k5_numbers,k1_numbers,A,E))) ) ) ) ) ) ) ) ) ).
fof(t9_asympt_0,axiom,
! [A] :
( ( v1_funct_1(A)
& v1_funct_2(A,k5_numbers,k1_numbers)
& v2_asympt_0(A)
& m2_relset_1(A,k5_numbers,k1_numbers) )
=> ! [B] :
( ( v1_funct_1(B)
& v1_funct_2(B,k5_numbers,k1_numbers)
& v2_asympt_0(B)
& m2_relset_1(B,k5_numbers,k1_numbers) )
=> k5_asympt_0(k1_asympt_0(A,B)) = k5_asympt_0(k4_asympt_0(A,B)) ) ) ).
fof(t10_asympt_0,axiom,
! [A] :
( ( v1_funct_1(A)
& v1_funct_2(A,k5_numbers,k1_numbers)
& v2_asympt_0(A)
& m2_relset_1(A,k5_numbers,k1_numbers) )
=> r2_hidden(A,k5_asympt_0(A)) ) ).
fof(t11_asympt_0,axiom,
! [A] :
( ( v1_funct_1(A)
& v1_funct_2(A,k5_numbers,k1_numbers)
& v2_asympt_0(A)
& m2_relset_1(A,k5_numbers,k1_numbers) )
=> ! [B] :
( ( v1_funct_1(B)
& v1_funct_2(B,k5_numbers,k1_numbers)
& v2_asympt_0(B)
& m2_relset_1(B,k5_numbers,k1_numbers) )
=> ( r2_hidden(A,k5_asympt_0(B))
=> r1_tarski(k5_asympt_0(A),k5_asympt_0(B)) ) ) ) ).
fof(t12_asympt_0,axiom,
! [A] :
( ( v1_funct_1(A)
& v1_funct_2(A,k5_numbers,k1_numbers)
& v2_asympt_0(A)
& m2_relset_1(A,k5_numbers,k1_numbers) )
=> ! [B] :
( ( v1_funct_1(B)
& v1_funct_2(B,k5_numbers,k1_numbers)
& v2_asympt_0(B)
& m2_relset_1(B,k5_numbers,k1_numbers) )
=> ! [C] :
( ( v1_funct_1(C)
& v1_funct_2(C,k5_numbers,k1_numbers)
& v2_asympt_0(C)
& m2_relset_1(C,k5_numbers,k1_numbers) )
=> ( ( r2_hidden(A,k5_asympt_0(B))
& r2_hidden(B,k5_asympt_0(C)) )
=> r2_hidden(A,k5_asympt_0(C)) ) ) ) ) ).
fof(t13_asympt_0,axiom,
! [A] :
( ( v1_funct_1(A)
& v1_funct_2(A,k5_numbers,k1_numbers)
& v2_asympt_0(A)
& m2_relset_1(A,k5_numbers,k1_numbers) )
=> ! [B] :
( ( v2_xreal_0(B)
& m1_subset_1(B,k1_numbers) )
=> k5_asympt_0(A) = k5_asympt_0(k3_asympt_0(A,B)) ) ) ).
fof(t14_asympt_0,axiom,
! [A] :
( ( ~ v3_xreal_0(A)
& m1_subset_1(A,k1_numbers) )
=> ! [B] :
( ( v1_funct_1(B)
& v1_funct_2(B,k5_numbers,k1_numbers)
& v2_asympt_0(B)
& m2_relset_1(B,k5_numbers,k1_numbers) )
=> ! [C] :
( ( v1_funct_1(C)
& v1_funct_2(C,k5_numbers,k1_numbers)
& v2_asympt_0(C)
& m2_relset_1(C,k5_numbers,k1_numbers) )
=> ( r2_hidden(B,k5_asympt_0(C))
=> r2_hidden(B,k5_asympt_0(k2_asympt_0(C,A))) ) ) ) ) ).
fof(t15_asympt_0,axiom,
! [A] :
( ( v1_funct_1(A)
& v1_funct_2(A,k5_numbers,k1_numbers)
& v4_asympt_0(A)
& m2_relset_1(A,k5_numbers,k1_numbers) )
=> ! [B] :
( ( v1_funct_1(B)
& v1_funct_2(B,k5_numbers,k1_numbers)
& v4_asympt_0(B)
& m2_relset_1(B,k5_numbers,k1_numbers) )
=> ( v4_seq_2(k19_seq_1(A,B))
=> ( r1_xreal_0(k2_seq_2(k19_seq_1(A,B)),np__0)
| k5_asympt_0(A) = k5_asympt_0(B) ) ) ) ) ).
fof(t16_asympt_0,axiom,
! [A] :
( ( v1_funct_1(A)
& v1_funct_2(A,k5_numbers,k1_numbers)
& v4_asympt_0(A)
& m2_relset_1(A,k5_numbers,k1_numbers) )
=> ! [B] :
( ( v1_funct_1(B)
& v1_funct_2(B,k5_numbers,k1_numbers)
& v4_asympt_0(B)
& m2_relset_1(B,k5_numbers,k1_numbers) )
=> ( ( v4_seq_2(k19_seq_1(A,B))
& k2_seq_2(k19_seq_1(A,B)) = np__0 )
=> ( r2_hidden(A,k5_asympt_0(B))
& ~ r2_hidden(B,k5_asympt_0(A)) ) ) ) ) ).
fof(t17_asympt_0,axiom,
! [A] :
( ( v1_funct_1(A)
& v1_funct_2(A,k5_numbers,k1_numbers)
& v4_asympt_0(A)
& m2_relset_1(A,k5_numbers,k1_numbers) )
=> ! [B] :
( ( v1_funct_1(B)
& v1_funct_2(B,k5_numbers,k1_numbers)
& v4_asympt_0(B)
& m2_relset_1(B,k5_numbers,k1_numbers) )
=> ( v1_limfunc1(k19_seq_1(A,B))
=> ( ~ r2_hidden(A,k5_asympt_0(B))
& r2_hidden(B,k5_asympt_0(A)) ) ) ) ) ).
fof(t18_asympt_0,axiom,
! [A,B] :
( ( v1_funct_1(B)
& v1_funct_2(B,k5_numbers,k1_numbers)
& v2_asympt_0(B)
& m2_relset_1(B,k5_numbers,k1_numbers) )
=> ( r2_hidden(A,k6_asympt_0(B))
=> ( v1_funct_1(A)
& v1_funct_2(A,k5_numbers,k1_numbers)
& v2_asympt_0(A)
& m2_relset_1(A,k5_numbers,k1_numbers) ) ) ) ).
fof(t19_asympt_0,axiom,
! [A] :
( ( v1_funct_1(A)
& v1_funct_2(A,k5_numbers,k1_numbers)
& v2_asympt_0(A)
& m2_relset_1(A,k5_numbers,k1_numbers) )
=> ! [B] :
( ( v1_funct_1(B)
& v1_funct_2(B,k5_numbers,k1_numbers)
& v2_asympt_0(B)
& m2_relset_1(B,k5_numbers,k1_numbers) )
=> ( r2_hidden(A,k6_asympt_0(B))
<=> r2_hidden(B,k5_asympt_0(A)) ) ) ) ).
fof(t20_asympt_0,axiom,
! [A] :
( ( v1_funct_1(A)
& v1_funct_2(A,k5_numbers,k1_numbers)
& v2_asympt_0(A)
& m2_relset_1(A,k5_numbers,k1_numbers) )
=> r2_hidden(A,k6_asympt_0(A)) ) ).
fof(t21_asympt_0,axiom,
! [A] :
( ( v1_funct_1(A)
& v1_funct_2(A,k5_numbers,k1_numbers)
& v2_asympt_0(A)
& m2_relset_1(A,k5_numbers,k1_numbers) )
=> ! [B] :
( ( v1_funct_1(B)
& v1_funct_2(B,k5_numbers,k1_numbers)
& v2_asympt_0(B)
& m2_relset_1(B,k5_numbers,k1_numbers) )
=> ! [C] :
( ( v1_funct_1(C)
& v1_funct_2(C,k5_numbers,k1_numbers)
& v2_asympt_0(C)
& m2_relset_1(C,k5_numbers,k1_numbers) )
=> ( ( r2_hidden(A,k6_asympt_0(B))
& r2_hidden(B,k6_asympt_0(C)) )
=> r2_hidden(A,k6_asympt_0(C)) ) ) ) ) ).
fof(t22_asympt_0,axiom,
! [A] :
( ( v1_funct_1(A)
& v1_funct_2(A,k5_numbers,k1_numbers)
& v4_asympt_0(A)
& m2_relset_1(A,k5_numbers,k1_numbers) )
=> ! [B] :
( ( v1_funct_1(B)
& v1_funct_2(B,k5_numbers,k1_numbers)
& v4_asympt_0(B)
& m2_relset_1(B,k5_numbers,k1_numbers) )
=> ( v4_seq_2(k19_seq_1(A,B))
=> ( r1_xreal_0(k2_seq_2(k19_seq_1(A,B)),np__0)
| k6_asympt_0(A) = k6_asympt_0(B) ) ) ) ) ).
fof(t23_asympt_0,axiom,
! [A] :
( ( v1_funct_1(A)
& v1_funct_2(A,k5_numbers,k1_numbers)
& v4_asympt_0(A)
& m2_relset_1(A,k5_numbers,k1_numbers) )
=> ! [B] :
( ( v1_funct_1(B)
& v1_funct_2(B,k5_numbers,k1_numbers)
& v4_asympt_0(B)
& m2_relset_1(B,k5_numbers,k1_numbers) )
=> ( ( v4_seq_2(k19_seq_1(A,B))
& k2_seq_2(k19_seq_1(A,B)) = np__0 )
=> ( r2_hidden(B,k6_asympt_0(A))
& ~ r2_hidden(A,k6_asympt_0(B)) ) ) ) ) ).
fof(t24_asympt_0,axiom,
! [A] :
( ( v1_funct_1(A)
& v1_funct_2(A,k5_numbers,k1_numbers)
& v4_asympt_0(A)
& m2_relset_1(A,k5_numbers,k1_numbers) )
=> ! [B] :
( ( v1_funct_1(B)
& v1_funct_2(B,k5_numbers,k1_numbers)
& v4_asympt_0(B)
& m2_relset_1(B,k5_numbers,k1_numbers) )
=> ( v1_limfunc1(k19_seq_1(A,B))
=> ( ~ r2_hidden(B,k6_asympt_0(A))
& r2_hidden(A,k6_asympt_0(B)) ) ) ) ) ).
fof(t25_asympt_0,axiom,
! [A] :
( ( v1_funct_1(A)
& v1_funct_2(A,k5_numbers,k1_numbers)
& v3_asympt_0(A)
& m2_relset_1(A,k5_numbers,k1_numbers) )
=> ! [B] :
( ( v1_funct_1(B)
& v1_funct_2(B,k5_numbers,k1_numbers)
& v3_asympt_0(B)
& m2_relset_1(B,k5_numbers,k1_numbers) )
=> ( r2_hidden(B,k6_asympt_0(A))
<=> ? [C] :
( m1_subset_1(C,k1_numbers)
& ~ r1_xreal_0(C,np__0)
& ! [D] :
( m2_subset_1(D,k1_numbers,k5_numbers)
=> r1_xreal_0(k4_real_1(C,k2_seq_1(k5_numbers,k1_numbers,A,D)),k2_seq_1(k5_numbers,k1_numbers,B,D)) ) ) ) ) ) ).
fof(t26_asympt_0,axiom,
! [A] :
( ( v1_funct_1(A)
& v1_funct_2(A,k5_numbers,k1_numbers)
& v2_asympt_0(A)
& m2_relset_1(A,k5_numbers,k1_numbers) )
=> ! [B] :
( ( v1_funct_1(B)
& v1_funct_2(B,k5_numbers,k1_numbers)
& v2_asympt_0(B)
& m2_relset_1(B,k5_numbers,k1_numbers) )
=> k6_asympt_0(k1_asympt_0(A,B)) = k6_asympt_0(k4_asympt_0(A,B)) ) ) ).
fof(d14_asympt_0,axiom,
! [A] :
( ( v1_funct_1(A)
& v1_funct_2(A,k5_numbers,k1_numbers)
& v2_asympt_0(A)
& m2_relset_1(A,k5_numbers,k1_numbers) )
=> k7_asympt_0(A) = k3_xboole_0(k5_asympt_0(A),k6_asympt_0(A)) ) ).
fof(t28_asympt_0,axiom,
! [A] :
( ( v1_funct_1(A)
& v1_funct_2(A,k5_numbers,k1_numbers)
& v2_asympt_0(A)
& m2_relset_1(A,k5_numbers,k1_numbers) )
=> r2_hidden(A,k7_asympt_0(A)) ) ).
fof(t29_asympt_0,axiom,
! [A] :
( ( v1_funct_1(A)
& v1_funct_2(A,k5_numbers,k1_numbers)
& v2_asympt_0(A)
& m2_relset_1(A,k5_numbers,k1_numbers) )
=> ! [B] :
( ( v1_funct_1(B)
& v1_funct_2(B,k5_numbers,k1_numbers)
& v2_asympt_0(B)
& m2_relset_1(B,k5_numbers,k1_numbers) )
=> ( r2_hidden(A,k7_asympt_0(B))
=> r2_hidden(B,k7_asympt_0(A)) ) ) ) ).
fof(t30_asympt_0,axiom,
! [A] :
( ( v1_funct_1(A)
& v1_funct_2(A,k5_numbers,k1_numbers)
& v2_asympt_0(A)
& m2_relset_1(A,k5_numbers,k1_numbers) )
=> ! [B] :
( ( v1_funct_1(B)
& v1_funct_2(B,k5_numbers,k1_numbers)
& v2_asympt_0(B)
& m2_relset_1(B,k5_numbers,k1_numbers) )
=> ! [C] :
( ( v1_funct_1(C)
& v1_funct_2(C,k5_numbers,k1_numbers)
& v2_asympt_0(C)
& m2_relset_1(C,k5_numbers,k1_numbers) )
=> ( ( r2_hidden(A,k7_asympt_0(B))
& r2_hidden(B,k7_asympt_0(C)) )
=> r2_hidden(A,k7_asympt_0(C)) ) ) ) ) ).
fof(t31_asympt_0,axiom,
! [A] :
( ( v1_funct_1(A)
& v1_funct_2(A,k5_numbers,k1_numbers)
& v3_asympt_0(A)
& m2_relset_1(A,k5_numbers,k1_numbers) )
=> ! [B] :
( ( v1_funct_1(B)
& v1_funct_2(B,k5_numbers,k1_numbers)
& v3_asympt_0(B)
& m2_relset_1(B,k5_numbers,k1_numbers) )
=> ( r2_hidden(B,k7_asympt_0(A))
<=> ? [C] :
( m1_subset_1(C,k1_numbers)
& ? [D] :
( m1_subset_1(D,k1_numbers)
& ~ r1_xreal_0(C,np__0)
& ~ r1_xreal_0(D,np__0)
& ! [E] :
( m2_subset_1(E,k1_numbers,k5_numbers)
=> ( r1_xreal_0(k4_real_1(D,k2_seq_1(k5_numbers,k1_numbers,A,E)),k2_seq_1(k5_numbers,k1_numbers,B,E))
& r1_xreal_0(k2_seq_1(k5_numbers,k1_numbers,B,E),k4_real_1(C,k2_seq_1(k5_numbers,k1_numbers,A,E))) ) ) ) ) ) ) ) ).
fof(t32_asympt_0,axiom,
! [A] :
( ( v1_funct_1(A)
& v1_funct_2(A,k5_numbers,k1_numbers)
& v2_asympt_0(A)
& m2_relset_1(A,k5_numbers,k1_numbers) )
=> ! [B] :
( ( v1_funct_1(B)
& v1_funct_2(B,k5_numbers,k1_numbers)
& v2_asympt_0(B)
& m2_relset_1(B,k5_numbers,k1_numbers) )
=> k7_asympt_0(k1_asympt_0(A,B)) = k7_asympt_0(k4_asympt_0(A,B)) ) ) ).
fof(t33_asympt_0,axiom,
! [A] :
( ( v1_funct_1(A)
& v1_funct_2(A,k5_numbers,k1_numbers)
& v4_asympt_0(A)
& m2_relset_1(A,k5_numbers,k1_numbers) )
=> ! [B] :
( ( v1_funct_1(B)
& v1_funct_2(B,k5_numbers,k1_numbers)
& v4_asympt_0(B)
& m2_relset_1(B,k5_numbers,k1_numbers) )
=> ( v4_seq_2(k19_seq_1(A,B))
=> ( r1_xreal_0(k2_seq_2(k19_seq_1(A,B)),np__0)
| r2_hidden(A,k7_asympt_0(B)) ) ) ) ) ).
fof(t34_asympt_0,axiom,
! [A] :
( ( v1_funct_1(A)
& v1_funct_2(A,k5_numbers,k1_numbers)
& v4_asympt_0(A)
& m2_relset_1(A,k5_numbers,k1_numbers) )
=> ! [B] :
( ( v1_funct_1(B)
& v1_funct_2(B,k5_numbers,k1_numbers)
& v4_asympt_0(B)
& m2_relset_1(B,k5_numbers,k1_numbers) )
=> ( ( v4_seq_2(k19_seq_1(A,B))
& k2_seq_2(k19_seq_1(A,B)) = np__0 )
=> ( r2_hidden(A,k5_asympt_0(B))
& ~ r2_hidden(A,k7_asympt_0(B)) ) ) ) ) ).
fof(t35_asympt_0,axiom,
! [A] :
( ( v1_funct_1(A)
& v1_funct_2(A,k5_numbers,k1_numbers)
& v4_asympt_0(A)
& m2_relset_1(A,k5_numbers,k1_numbers) )
=> ! [B] :
( ( v1_funct_1(B)
& v1_funct_2(B,k5_numbers,k1_numbers)
& v4_asympt_0(B)
& m2_relset_1(B,k5_numbers,k1_numbers) )
=> ( v1_limfunc1(k19_seq_1(A,B))
=> ( r2_hidden(A,k6_asympt_0(B))
& ~ r2_hidden(A,k7_asympt_0(B)) ) ) ) ) ).
fof(t36_asympt_0,axiom,
! [A] :
( ( v1_funct_1(A)
& v1_funct_2(A,k5_numbers,k1_numbers)
& v2_asympt_0(A)
& m2_relset_1(A,k5_numbers,k1_numbers) )
=> ! [B] : k10_asympt_0(A,B) = k3_xboole_0(k8_asympt_0(A,B),k9_asympt_0(A,B)) ) ).
fof(d18_asympt_0,axiom,
! [A] :
( ( v1_funct_1(A)
& v1_funct_2(A,k5_numbers,k1_numbers)
& m2_relset_1(A,k5_numbers,k1_numbers) )
=> ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> ! [C] :
( ( v1_funct_1(C)
& v1_funct_2(C,k5_numbers,k1_numbers)
& m2_relset_1(C,k5_numbers,k1_numbers) )
=> ( C = k11_asympt_0(A,B)
<=> ! [D] :
( m2_subset_1(D,k1_numbers,k5_numbers)
=> k2_seq_1(k5_numbers,k1_numbers,C,D) = k2_seq_1(k5_numbers,k1_numbers,A,k2_nat_1(B,D)) ) ) ) ) ) ).
fof(d19_asympt_0,axiom,
! [A] :
( ( v1_funct_1(A)
& v1_funct_2(A,k5_numbers,k1_numbers)
& v2_asympt_0(A)
& m2_relset_1(A,k5_numbers,k1_numbers) )
=> ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> ( r2_asympt_0(A,B)
<=> ( v6_asympt_0(A)
& r2_hidden(k11_asympt_0(A,B),k5_asympt_0(A)) ) ) ) ) ).
fof(d20_asympt_0,axiom,
! [A] :
( ( v1_funct_1(A)
& v1_funct_2(A,k5_numbers,k1_numbers)
& v2_asympt_0(A)
& m2_relset_1(A,k5_numbers,k1_numbers) )
=> ( v7_asympt_0(A)
<=> ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> ( r1_xreal_0(np__2,B)
=> r2_asympt_0(A,B) ) ) ) ) ).
fof(t37_asympt_0,axiom,
! [A] :
( ( v1_funct_1(A)
& v1_funct_2(A,k5_numbers,k1_numbers)
& v2_asympt_0(A)
& m2_relset_1(A,k5_numbers,k1_numbers) )
=> ( ? [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
& r1_xreal_0(np__2,B)
& r2_asympt_0(A,B) )
=> v7_asympt_0(A) ) ) ).
fof(t41_asympt_0,axiom,
! [A] :
( ( v1_funct_1(A)
& v1_funct_2(A,k5_numbers,k1_numbers)
& v2_asympt_0(A)
& m2_relset_1(A,k5_numbers,k1_numbers) )
=> ! [B] :
( ( v1_funct_1(B)
& v1_funct_2(B,k5_numbers,k1_numbers)
& v2_asympt_0(B)
& m2_relset_1(B,k5_numbers,k1_numbers) )
=> k12_asympt_0(k5_numbers,k5_asympt_0(A),k5_asympt_0(B)) = k5_asympt_0(k1_asympt_0(A,B)) ) ) ).
fof(t42_asympt_0,axiom,
! [A] :
( ( v1_funct_1(A)
& v1_funct_2(A,k5_numbers,k1_numbers)
& v2_asympt_0(A)
& m2_relset_1(A,k5_numbers,k1_numbers) )
=> ! [B] :
( ( v1_funct_1(B)
& v1_funct_2(B,k5_numbers,k1_numbers)
& v2_asympt_0(B)
& m2_relset_1(B,k5_numbers,k1_numbers) )
=> k13_asympt_0(k5_numbers,k5_asympt_0(A),k5_asympt_0(B)) = k5_asympt_0(k4_asympt_0(A,B)) ) ) ).
fof(dt_k1_asympt_0,axiom,
! [A,B] :
( ( v1_funct_1(A)
& v1_funct_2(A,k5_numbers,k1_numbers)
& v2_asympt_0(A)
& m1_relset_1(A,k5_numbers,k1_numbers)
& v1_funct_1(B)
& v1_funct_2(B,k5_numbers,k1_numbers)
& v2_asympt_0(B)
& m1_relset_1(B,k5_numbers,k1_numbers) )
=> ( v1_funct_1(k1_asympt_0(A,B))
& v1_funct_2(k1_asympt_0(A,B),k5_numbers,k1_numbers)
& v2_asympt_0(k1_asympt_0(A,B))
& m2_relset_1(k1_asympt_0(A,B),k5_numbers,k1_numbers) ) ) ).
fof(commutativity_k1_asympt_0,axiom,
! [A,B] :
( ( v1_funct_1(A)
& v1_funct_2(A,k5_numbers,k1_numbers)
& v2_asympt_0(A)
& m1_relset_1(A,k5_numbers,k1_numbers)
& v1_funct_1(B)
& v1_funct_2(B,k5_numbers,k1_numbers)
& v2_asympt_0(B)
& m1_relset_1(B,k5_numbers,k1_numbers) )
=> k1_asympt_0(A,B) = k1_asympt_0(B,A) ) ).
fof(redefinition_k1_asympt_0,axiom,
! [A,B] :
( ( v1_funct_1(A)
& v1_funct_2(A,k5_numbers,k1_numbers)
& v2_asympt_0(A)
& m1_relset_1(A,k5_numbers,k1_numbers)
& v1_funct_1(B)
& v1_funct_2(B,k5_numbers,k1_numbers)
& v2_asympt_0(B)
& m1_relset_1(B,k5_numbers,k1_numbers) )
=> k1_asympt_0(A,B) = k3_seq_1(A,B) ) ).
fof(dt_k2_asympt_0,axiom,
! [A,B] :
( ( v1_funct_1(A)
& v1_funct_2(A,k5_numbers,k1_numbers)
& m1_relset_1(A,k5_numbers,k1_numbers)
& v1_xreal_0(B) )
=> ( v1_funct_1(k2_asympt_0(A,B))
& v1_funct_2(k2_asympt_0(A,B),k5_numbers,k1_numbers)
& m2_relset_1(k2_asympt_0(A,B),k5_numbers,k1_numbers) ) ) ).
fof(dt_k3_asympt_0,axiom,
! [A,B] :
( ( v1_funct_1(A)
& v1_funct_2(A,k5_numbers,k1_numbers)
& v2_asympt_0(A)
& m1_relset_1(A,k5_numbers,k1_numbers)
& v2_xreal_0(B)
& m1_subset_1(B,k1_numbers) )
=> ( v1_funct_1(k3_asympt_0(A,B))
& v1_funct_2(k3_asympt_0(A,B),k5_numbers,k1_numbers)
& v2_asympt_0(k3_asympt_0(A,B))
& m2_relset_1(k3_asympt_0(A,B),k5_numbers,k1_numbers) ) ) ).
fof(redefinition_k3_asympt_0,axiom,
! [A,B] :
( ( v1_funct_1(A)
& v1_funct_2(A,k5_numbers,k1_numbers)
& v2_asympt_0(A)
& m1_relset_1(A,k5_numbers,k1_numbers)
& v2_xreal_0(B)
& m1_subset_1(B,k1_numbers) )
=> k3_asympt_0(A,B) = k12_seq_1(A,B) ) ).
fof(dt_k4_asympt_0,axiom,
! [A,B] :
( ( v1_funct_1(A)
& v1_funct_2(A,k5_numbers,k1_numbers)
& m1_relset_1(A,k5_numbers,k1_numbers)
& v1_funct_1(B)
& v1_funct_2(B,k5_numbers,k1_numbers)
& m1_relset_1(B,k5_numbers,k1_numbers) )
=> ( v1_funct_1(k4_asympt_0(A,B))
& v1_funct_2(k4_asympt_0(A,B),k5_numbers,k1_numbers)
& m2_relset_1(k4_asympt_0(A,B),k5_numbers,k1_numbers) ) ) ).
fof(commutativity_k4_asympt_0,axiom,
! [A,B] :
( ( v1_funct_1(A)
& v1_funct_2(A,k5_numbers,k1_numbers)
& m1_relset_1(A,k5_numbers,k1_numbers)
& v1_funct_1(B)
& v1_funct_2(B,k5_numbers,k1_numbers)
& m1_relset_1(B,k5_numbers,k1_numbers) )
=> k4_asympt_0(A,B) = k4_asympt_0(B,A) ) ).
fof(dt_k5_asympt_0,axiom,
! [A] :
( ( v1_funct_1(A)
& v1_funct_2(A,k5_numbers,k1_numbers)
& v2_asympt_0(A)
& m1_relset_1(A,k5_numbers,k1_numbers) )
=> m1_fraenkel(k5_asympt_0(A),k5_numbers,k1_numbers) ) ).
fof(dt_k6_asympt_0,axiom,
! [A] :
( ( v1_funct_1(A)
& v1_funct_2(A,k5_numbers,k1_numbers)
& v2_asympt_0(A)
& m1_relset_1(A,k5_numbers,k1_numbers) )
=> m1_fraenkel(k6_asympt_0(A),k5_numbers,k1_numbers) ) ).
fof(dt_k7_asympt_0,axiom,
! [A] :
( ( v1_funct_1(A)
& v1_funct_2(A,k5_numbers,k1_numbers)
& v2_asympt_0(A)
& m1_relset_1(A,k5_numbers,k1_numbers) )
=> m1_fraenkel(k7_asympt_0(A),k5_numbers,k1_numbers) ) ).
fof(dt_k8_asympt_0,axiom,
! [A,B] :
( ( v1_funct_1(A)
& v1_funct_2(A,k5_numbers,k1_numbers)
& v2_asympt_0(A)
& m1_relset_1(A,k5_numbers,k1_numbers) )
=> m1_fraenkel(k8_asympt_0(A,B),k5_numbers,k1_numbers) ) ).
fof(dt_k9_asympt_0,axiom,
! [A,B] :
( ( v1_funct_1(A)
& v1_funct_2(A,k5_numbers,k1_numbers)
& v2_asympt_0(A)
& m1_relset_1(A,k5_numbers,k1_numbers) )
=> m1_fraenkel(k9_asympt_0(A,B),k5_numbers,k1_numbers) ) ).
fof(dt_k10_asympt_0,axiom,
! [A,B] :
( ( v1_funct_1(A)
& v1_funct_2(A,k5_numbers,k1_numbers)
& v2_asympt_0(A)
& m1_relset_1(A,k5_numbers,k1_numbers) )
=> m1_fraenkel(k10_asympt_0(A,B),k5_numbers,k1_numbers) ) ).
fof(dt_k11_asympt_0,axiom,
! [A,B] :
( ( v1_funct_1(A)
& v1_funct_2(A,k5_numbers,k1_numbers)
& m1_relset_1(A,k5_numbers,k1_numbers)
& m1_subset_1(B,k5_numbers) )
=> ( v1_funct_1(k11_asympt_0(A,B))
& v1_funct_2(k11_asympt_0(A,B),k5_numbers,k1_numbers)
& m2_relset_1(k11_asympt_0(A,B),k5_numbers,k1_numbers) ) ) ).
fof(dt_k12_asympt_0,axiom,
! [A,B,C] :
( ( ~ v1_xboole_0(A)
& m1_fraenkel(B,A,k1_numbers)
& m1_fraenkel(C,A,k1_numbers) )
=> m1_fraenkel(k12_asympt_0(A,B,C),A,k1_numbers) ) ).
fof(dt_k13_asympt_0,axiom,
! [A,B,C] :
( ( ~ v1_xboole_0(A)
& m1_fraenkel(B,A,k1_numbers)
& m1_fraenkel(C,A,k1_numbers) )
=> m1_fraenkel(k13_asympt_0(A,B,C),A,k1_numbers) ) ).
fof(dt_k14_asympt_0,axiom,
! [A,B] :
( ( m1_fraenkel(A,k5_numbers,k1_numbers)
& m1_fraenkel(B,k5_numbers,k1_numbers) )
=> m1_fraenkel(k14_asympt_0(A,B),k5_numbers,k1_numbers) ) ).
fof(d12_asympt_0,axiom,
! [A] :
( ( v1_funct_1(A)
& v1_funct_2(A,k5_numbers,k1_numbers)
& v2_asympt_0(A)
& m2_relset_1(A,k5_numbers,k1_numbers) )
=> k5_asympt_0(A) = a_1_0_asympt_0(A) ) ).
fof(d13_asympt_0,axiom,
! [A] :
( ( v1_funct_1(A)
& v1_funct_2(A,k5_numbers,k1_numbers)
& v2_asympt_0(A)
& m2_relset_1(A,k5_numbers,k1_numbers) )
=> k6_asympt_0(A) = a_1_1_asympt_0(A) ) ).
fof(t27_asympt_0,axiom,
! [A] :
( ( v1_funct_1(A)
& v1_funct_2(A,k5_numbers,k1_numbers)
& v2_asympt_0(A)
& m2_relset_1(A,k5_numbers,k1_numbers) )
=> k7_asympt_0(A) = a_1_2_asympt_0(A) ) ).
fof(d15_asympt_0,axiom,
! [A] :
( ( v1_funct_1(A)
& v1_funct_2(A,k5_numbers,k1_numbers)
& v2_asympt_0(A)
& m2_relset_1(A,k5_numbers,k1_numbers) )
=> ! [B] : k8_asympt_0(A,B) = a_2_0_asympt_0(A,B) ) ).
fof(d16_asympt_0,axiom,
! [A] :
( ( v1_funct_1(A)
& v1_funct_2(A,k5_numbers,k1_numbers)
& v2_asympt_0(A)
& m2_relset_1(A,k5_numbers,k1_numbers) )
=> ! [B] : k9_asympt_0(A,B) = a_2_1_asympt_0(A,B) ) ).
fof(d17_asympt_0,axiom,
! [A] :
( ( v1_funct_1(A)
& v1_funct_2(A,k5_numbers,k1_numbers)
& v2_asympt_0(A)
& m2_relset_1(A,k5_numbers,k1_numbers) )
=> ! [B] : k10_asympt_0(A,B) = a_2_2_asympt_0(A,B) ) ).
fof(t38_asympt_0,axiom,
! [A] :
( ( v1_funct_1(A)
& v1_funct_2(A,k5_numbers,k1_numbers)
& v2_asympt_0(A)
& m2_relset_1(A,k5_numbers,k1_numbers) )
=> ! [B] :
( ( v1_funct_1(B)
& v1_funct_2(B,k5_numbers,k1_numbers)
& v2_asympt_0(B)
& v6_asympt_0(B)
& m2_relset_1(B,k5_numbers,k1_numbers) )
=> ! [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
=> ( ( v7_asympt_0(A)
& r1_xreal_0(np__2,C)
& r2_hidden(B,k8_asympt_0(A,a_1_3_asympt_0(C))) )
=> r2_hidden(B,k5_asympt_0(A)) ) ) ) ) ).
fof(t39_asympt_0,axiom,
! [A] :
( ( v1_funct_1(A)
& v1_funct_2(A,k5_numbers,k1_numbers)
& v2_asympt_0(A)
& m2_relset_1(A,k5_numbers,k1_numbers) )
=> ! [B] :
( ( v1_funct_1(B)
& v1_funct_2(B,k5_numbers,k1_numbers)
& v2_asympt_0(B)
& v6_asympt_0(B)
& m2_relset_1(B,k5_numbers,k1_numbers) )
=> ! [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
=> ( ( v7_asympt_0(A)
& r1_xreal_0(np__2,C)
& r2_hidden(B,k9_asympt_0(A,a_1_3_asympt_0(C))) )
=> r2_hidden(B,k6_asympt_0(A)) ) ) ) ) ).
fof(t40_asympt_0,axiom,
! [A] :
( ( v1_funct_1(A)
& v1_funct_2(A,k5_numbers,k1_numbers)
& v2_asympt_0(A)
& m2_relset_1(A,k5_numbers,k1_numbers) )
=> ! [B] :
( ( v1_funct_1(B)
& v1_funct_2(B,k5_numbers,k1_numbers)
& v2_asympt_0(B)
& v6_asympt_0(B)
& m2_relset_1(B,k5_numbers,k1_numbers) )
=> ! [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
=> ( ( v7_asympt_0(A)
& r1_xreal_0(np__2,C)
& r2_hidden(B,k10_asympt_0(A,a_1_3_asympt_0(C))) )
=> r2_hidden(B,k7_asympt_0(A)) ) ) ) ) ).
fof(d21_asympt_0,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m1_fraenkel(B,A,k1_numbers)
=> ! [C] :
( m1_fraenkel(C,A,k1_numbers)
=> k12_asympt_0(A,B,C) = a_3_0_asympt_0(A,B,C) ) ) ) ).
fof(d22_asympt_0,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m1_fraenkel(B,A,k1_numbers)
=> ! [C] :
( m1_fraenkel(C,A,k1_numbers)
=> k13_asympt_0(A,B,C) = a_3_1_asympt_0(A,B,C) ) ) ) ).
fof(d23_asympt_0,axiom,
! [A] :
( m1_fraenkel(A,k5_numbers,k1_numbers)
=> ! [B] :
( m1_fraenkel(B,k5_numbers,k1_numbers)
=> k14_asympt_0(A,B) = a_2_3_asympt_0(A,B) ) ) ).
fof(s1_asympt_0,axiom,
( r1_xreal_0(f1_s1_asympt_0,f2_s1_asympt_0)
=> ( ~ v1_xboole_0(a_0_0_asympt_0)
& v1_finset_1(a_0_0_asympt_0)
& m1_subset_1(a_0_0_asympt_0,k1_zfmisc_1(f3_s1_asympt_0)) ) ) ).
fof(s2_asympt_0,axiom,
( ~ v1_xboole_0(a_0_1_asympt_0)
& v1_finset_1(a_0_1_asympt_0)
& m1_subset_1(a_0_1_asympt_0,k1_zfmisc_1(f2_s2_asympt_0)) ) ).
fof(s3_asympt_0,axiom,
( ~ r1_xreal_0(f1_s3_asympt_0,np__0)
=> ( ~ v1_xboole_0(a_0_2_asympt_0)
& v1_finset_1(a_0_2_asympt_0)
& m1_subset_1(a_0_2_asympt_0,k1_zfmisc_1(f2_s3_asympt_0)) ) ) ).
fof(fraenkel_a_1_0_asympt_0,axiom,
! [A,B] :
( ( v1_funct_1(B)
& v1_funct_2(B,k5_numbers,k1_numbers)
& v2_asympt_0(B)
& m2_relset_1(B,k5_numbers,k1_numbers) )
=> ( r2_hidden(A,a_1_0_asympt_0(B))
<=> ? [C] :
( m2_fraenkel(C,k5_numbers,k1_numbers,k1_fraenkel(k5_numbers,k1_numbers))
& A = C
& ? [D] :
( m1_subset_1(D,k1_numbers)
& ? [E] :
( m2_subset_1(E,k1_numbers,k5_numbers)
& ~ r1_xreal_0(D,np__0)
& ! [F] :
( m2_subset_1(F,k1_numbers,k5_numbers)
=> ( r1_xreal_0(E,F)
=> ( r1_xreal_0(k2_seq_1(k5_numbers,k1_numbers,C,F),k4_real_1(D,k2_seq_1(k5_numbers,k1_numbers,B,F)))
& r1_xreal_0(np__0,k2_seq_1(k5_numbers,k1_numbers,C,F)) ) ) ) ) ) ) ) ) ).
fof(fraenkel_a_1_1_asympt_0,axiom,
! [A,B] :
( ( v1_funct_1(B)
& v1_funct_2(B,k5_numbers,k1_numbers)
& v2_asympt_0(B)
& m2_relset_1(B,k5_numbers,k1_numbers) )
=> ( r2_hidden(A,a_1_1_asympt_0(B))
<=> ? [C] :
( m2_fraenkel(C,k5_numbers,k1_numbers,k1_fraenkel(k5_numbers,k1_numbers))
& A = C
& ? [D] :
( m1_subset_1(D,k1_numbers)
& ? [E] :
( m2_subset_1(E,k1_numbers,k5_numbers)
& ~ r1_xreal_0(D,np__0)
& ! [F] :
( m2_subset_1(F,k1_numbers,k5_numbers)
=> ( r1_xreal_0(E,F)
=> ( r1_xreal_0(k4_real_1(D,k2_seq_1(k5_numbers,k1_numbers,B,F)),k2_seq_1(k5_numbers,k1_numbers,C,F))
& r1_xreal_0(np__0,k2_seq_1(k5_numbers,k1_numbers,C,F)) ) ) ) ) ) ) ) ) ).
fof(fraenkel_a_1_2_asympt_0,axiom,
! [A,B] :
( ( v1_funct_1(B)
& v1_funct_2(B,k5_numbers,k1_numbers)
& v2_asympt_0(B)
& m2_relset_1(B,k5_numbers,k1_numbers) )
=> ( r2_hidden(A,a_1_2_asympt_0(B))
<=> ? [C] :
( m2_fraenkel(C,k5_numbers,k1_numbers,k1_fraenkel(k5_numbers,k1_numbers))
& A = C
& ? [D] :
( m1_subset_1(D,k1_numbers)
& ? [E] :
( m1_subset_1(E,k1_numbers)
& ? [F] :
( m2_subset_1(F,k1_numbers,k5_numbers)
& ~ r1_xreal_0(D,np__0)
& ~ r1_xreal_0(E,np__0)
& ! [G] :
( m2_subset_1(G,k1_numbers,k5_numbers)
=> ( r1_xreal_0(F,G)
=> ( r1_xreal_0(k4_real_1(E,k2_seq_1(k5_numbers,k1_numbers,B,G)),k2_seq_1(k5_numbers,k1_numbers,C,G))
& r1_xreal_0(k2_seq_1(k5_numbers,k1_numbers,C,G),k4_real_1(D,k2_seq_1(k5_numbers,k1_numbers,B,G))) ) ) ) ) ) ) ) ) ) ).
fof(fraenkel_a_2_0_asympt_0,axiom,
! [A,B,C] :
( ( v1_funct_1(B)
& v1_funct_2(B,k5_numbers,k1_numbers)
& v2_asympt_0(B)
& m2_relset_1(B,k5_numbers,k1_numbers) )
=> ( r2_hidden(A,a_2_0_asympt_0(B,C))
<=> ? [D] :
( m2_fraenkel(D,k5_numbers,k1_numbers,k1_fraenkel(k5_numbers,k1_numbers))
& A = D
& ? [E] :
( m1_subset_1(E,k1_numbers)
& ? [F] :
( m2_subset_1(F,k1_numbers,k5_numbers)
& ~ r1_xreal_0(E,np__0)
& ! [G] :
( m2_subset_1(G,k1_numbers,k5_numbers)
=> ( ( r1_xreal_0(F,G)
& r2_hidden(G,C) )
=> ( r1_xreal_0(k2_seq_1(k5_numbers,k1_numbers,D,G),k4_real_1(E,k2_seq_1(k5_numbers,k1_numbers,B,G)))
& r1_xreal_0(np__0,k2_seq_1(k5_numbers,k1_numbers,D,G)) ) ) ) ) ) ) ) ) ).
fof(fraenkel_a_2_1_asympt_0,axiom,
! [A,B,C] :
( ( v1_funct_1(B)
& v1_funct_2(B,k5_numbers,k1_numbers)
& v2_asympt_0(B)
& m2_relset_1(B,k5_numbers,k1_numbers) )
=> ( r2_hidden(A,a_2_1_asympt_0(B,C))
<=> ? [D] :
( m2_fraenkel(D,k5_numbers,k1_numbers,k1_fraenkel(k5_numbers,k1_numbers))
& A = D
& ? [E] :
( m1_subset_1(E,k1_numbers)
& ? [F] :
( m2_subset_1(F,k1_numbers,k5_numbers)
& ~ r1_xreal_0(E,np__0)
& ! [G] :
( m2_subset_1(G,k1_numbers,k5_numbers)
=> ( ( r1_xreal_0(F,G)
& r2_hidden(G,C) )
=> ( r1_xreal_0(k4_real_1(E,k2_seq_1(k5_numbers,k1_numbers,B,G)),k2_seq_1(k5_numbers,k1_numbers,D,G))
& r1_xreal_0(np__0,k2_seq_1(k5_numbers,k1_numbers,D,G)) ) ) ) ) ) ) ) ) ).
fof(fraenkel_a_2_2_asympt_0,axiom,
! [A,B,C] :
( ( v1_funct_1(B)
& v1_funct_2(B,k5_numbers,k1_numbers)
& v2_asympt_0(B)
& m2_relset_1(B,k5_numbers,k1_numbers) )
=> ( r2_hidden(A,a_2_2_asympt_0(B,C))
<=> ? [D] :
( m2_fraenkel(D,k5_numbers,k1_numbers,k1_fraenkel(k5_numbers,k1_numbers))
& A = D
& ? [E] :
( m1_subset_1(E,k1_numbers)
& ? [F] :
( m1_subset_1(F,k1_numbers)
& ? [G] :
( m2_subset_1(G,k1_numbers,k5_numbers)
& ~ r1_xreal_0(E,np__0)
& ~ r1_xreal_0(F,np__0)
& ! [H] :
( m2_subset_1(H,k1_numbers,k5_numbers)
=> ( ( r1_xreal_0(G,H)
& r2_hidden(H,C) )
=> ( r1_xreal_0(k4_real_1(F,k2_seq_1(k5_numbers,k1_numbers,B,H)),k2_seq_1(k5_numbers,k1_numbers,D,H))
& r1_xreal_0(k2_seq_1(k5_numbers,k1_numbers,D,H),k4_real_1(E,k2_seq_1(k5_numbers,k1_numbers,B,H))) ) ) ) ) ) ) ) ) ) ).
fof(fraenkel_a_1_3_asympt_0,axiom,
! [A,B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> ( r2_hidden(A,a_1_3_asympt_0(B))
<=> ? [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
& A = k3_newton(B,C) ) ) ) ).
fof(fraenkel_a_3_0_asympt_0,axiom,
! [A,B,C,D] :
( ( ~ v1_xboole_0(B)
& m1_fraenkel(C,B,k1_numbers)
& m1_fraenkel(D,B,k1_numbers) )
=> ( r2_hidden(A,a_3_0_asympt_0(B,C,D))
<=> ? [E] :
( m2_fraenkel(E,B,k1_numbers,k1_fraenkel(B,k1_numbers))
& A = E
& ? [F] :
( m2_fraenkel(F,B,k1_numbers,k1_fraenkel(B,k1_numbers))
& ? [G] :
( m2_fraenkel(G,B,k1_numbers,k1_fraenkel(B,k1_numbers))
& r2_hidden(F,C)
& r2_hidden(G,D)
& ! [H] :
( m1_subset_1(H,B)
=> k2_seq_1(B,k1_numbers,E,H) = k3_real_1(k2_seq_1(B,k1_numbers,F,H),k2_seq_1(B,k1_numbers,G,H)) ) ) ) ) ) ) ).
fof(fraenkel_a_3_1_asympt_0,axiom,
! [A,B,C,D] :
( ( ~ v1_xboole_0(B)
& m1_fraenkel(C,B,k1_numbers)
& m1_fraenkel(D,B,k1_numbers) )
=> ( r2_hidden(A,a_3_1_asympt_0(B,C,D))
<=> ? [E] :
( m2_fraenkel(E,B,k1_numbers,k1_fraenkel(B,k1_numbers))
& A = E
& ? [F] :
( m2_fraenkel(F,B,k1_numbers,k1_fraenkel(B,k1_numbers))
& ? [G] :
( m2_fraenkel(G,B,k1_numbers,k1_fraenkel(B,k1_numbers))
& r2_hidden(F,C)
& r2_hidden(G,D)
& ! [H] :
( m1_subset_1(H,B)
=> k2_seq_1(B,k1_numbers,E,H) = k4_square_1(k2_seq_1(B,k1_numbers,F,H),k2_seq_1(B,k1_numbers,G,H)) ) ) ) ) ) ) ).
fof(fraenkel_a_2_3_asympt_0,axiom,
! [A,B,C] :
( ( m1_fraenkel(B,k5_numbers,k1_numbers)
& m1_fraenkel(C,k5_numbers,k1_numbers) )
=> ( r2_hidden(A,a_2_3_asympt_0(B,C))
<=> ? [D] :
( m2_fraenkel(D,k5_numbers,k1_numbers,k1_fraenkel(k5_numbers,k1_numbers))
& A = D
& ? [E] :
( m2_fraenkel(E,k5_numbers,k1_numbers,k1_fraenkel(k5_numbers,k1_numbers))
& ? [F] :
( m2_fraenkel(F,k5_numbers,k1_numbers,k1_fraenkel(k5_numbers,k1_numbers))
& ? [G] :
( m2_subset_1(G,k1_numbers,k5_numbers)
& r2_hidden(E,B)
& r2_hidden(F,C)
& ! [H] :
( m2_subset_1(H,k1_numbers,k5_numbers)
=> ( r1_xreal_0(G,H)
=> k2_seq_1(k5_numbers,k1_numbers,D,H) = k4_power(k2_seq_1(k5_numbers,k1_numbers,E,H),k2_seq_1(k5_numbers,k1_numbers,F,H)) ) ) ) ) ) ) ) ) ).
fof(fraenkel_a_0_0_asympt_0,axiom,
! [A] :
( r2_hidden(A,a_0_0_asympt_0)
<=> ? [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
& A = f4_s1_asympt_0(B)
& r1_xreal_0(f1_s1_asympt_0,B)
& r1_xreal_0(B,f2_s1_asympt_0) ) ) ).
fof(fraenkel_a_0_1_asympt_0,axiom,
! [A] :
( r2_hidden(A,a_0_1_asympt_0)
<=> ? [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
& A = f3_s2_asympt_0(B)
& r1_xreal_0(B,f1_s2_asympt_0) ) ) ).
fof(fraenkel_a_0_2_asympt_0,axiom,
! [A] :
( r2_hidden(A,a_0_2_asympt_0)
<=> ? [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
& A = f3_s3_asympt_0(B)
& ~ r1_xreal_0(f1_s3_asympt_0,B) ) ) ).
%------------------------------------------------------------------------------