SET007 Axioms: SET007+629.ax
%------------------------------------------------------------------------------
% File : SET007+629 : TPTP v9.0.0. Released v3.4.0.
% Domain : Set Theory
% Axioms : Asymptotic Notation. Part II: Examples and Problems
% 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_1 [Urb08]
% Status : Satisfiable
% Syntax : Number of formulae : 127 ( 14 unt; 0 def)
% Number of atoms : 946 ( 168 equ)
% Maximal formula atoms : 31 ( 7 avg)
% Number of connectives : 961 ( 142 ~; 8 |; 487 &)
% ( 13 <=>; 311 =>; 0 <=; 0 <~>)
% Maximal formula depth : 26 ( 8 avg)
% Maximal term depth : 7 ( 1 avg)
% Number of predicates : 27 ( 26 usr; 0 prp; 1-3 aty)
% Number of functors : 81 ( 81 usr; 29 con; 0-4 aty)
% Number of variables : 268 ( 248 !; 20 ?)
% 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(fc1_asympt_1,axiom,
! [A,B,C] :
( ( v2_xreal_0(A)
& m1_subset_1(A,k1_numbers)
& m1_subset_1(B,k1_numbers)
& m1_subset_1(C,k1_numbers) )
=> ( v1_relat_1(k1_asympt_1(A,B,C))
& v1_funct_1(k1_asympt_1(A,B,C))
& v1_funct_2(k1_asympt_1(A,B,C),k5_numbers,k1_numbers)
& v1_seq_1(k1_asympt_1(A,B,C))
& v2_asympt_0(k1_asympt_1(A,B,C))
& v4_asympt_0(k1_asympt_1(A,B,C))
& v5_asympt_0(k1_asympt_1(A,B,C)) ) ) ).
fof(fc2_asympt_1,axiom,
( v1_relat_1(k2_asympt_1)
& v1_funct_1(k2_asympt_1)
& v1_funct_2(k2_asympt_1,k5_numbers,k1_numbers)
& v1_seq_1(k2_asympt_1)
& v2_asympt_0(k2_asympt_1)
& v4_asympt_0(k2_asympt_1)
& v5_asympt_0(k2_asympt_1) ) ).
fof(fc3_asympt_1,axiom,
! [A] :
( m1_subset_1(A,k1_numbers)
=> ( v1_relat_1(k3_asympt_1(A))
& v1_funct_1(k3_asympt_1(A))
& v1_funct_2(k3_asympt_1(A),k5_numbers,k1_numbers)
& v1_seq_1(k3_asympt_1(A))
& v2_asympt_0(k3_asympt_1(A))
& v4_asympt_0(k3_asympt_1(A))
& v5_asympt_0(k3_asympt_1(A)) ) ) ).
fof(fc4_asympt_1,axiom,
( v1_relat_1(k4_asympt_1(np__1))
& v1_funct_1(k4_asympt_1(np__1))
& v1_funct_2(k4_asympt_1(np__1),k5_numbers,k1_numbers)
& v2_asympt_0(k4_asympt_1(np__1)) ) ).
fof(fc5_asympt_1,axiom,
! [A] :
( m1_subset_1(A,k5_numbers)
=> ( v1_relat_1(k5_asympt_1(A))
& v1_funct_1(k5_asympt_1(A))
& v1_funct_2(k5_asympt_1(A),k5_numbers,k1_numbers)
& v1_seq_1(k5_asympt_1(A))
& v2_asympt_0(k5_asympt_1(A))
& v4_asympt_0(k5_asympt_1(A))
& v5_asympt_0(k5_asympt_1(A)) ) ) ).
fof(t1_asympt_1,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) )
=> ~ ( k2_seq_1(k5_numbers,k1_numbers,A,np__0) = np__0
& ! [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
=> ( ~ r1_xreal_0(C,np__0)
=> k2_seq_1(k5_numbers,k1_numbers,A,C) = k3_real_1(k3_real_1(k5_real_1(k4_real_1(k2_nat_1(np__12,k3_series_1(C,np__3)),k6_power(np__2,C)),k4_real_1(np__5,k7_square_1(C))),k7_square_1(k6_power(np__2,C))),np__36) ) )
& k2_seq_1(k5_numbers,k1_numbers,B,np__0) = np__0
& ! [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
=> ( ~ r1_xreal_0(C,np__0)
=> k2_seq_1(k5_numbers,k1_numbers,B,C) = k4_real_1(k3_series_1(C,np__3),k6_power(np__2,C)) ) )
& ! [C] :
( ( v1_funct_1(C)
& v1_funct_2(C,k5_numbers,k1_numbers)
& v4_asympt_0(C)
& m2_relset_1(C,k5_numbers,k1_numbers) )
=> ! [D] :
( ( v1_funct_1(D)
& v1_funct_2(D,k5_numbers,k1_numbers)
& v4_asympt_0(D)
& m2_relset_1(D,k5_numbers,k1_numbers) )
=> ~ ( C = A
& D = B
& r2_hidden(C,k5_asympt_0(D)) ) ) ) ) ) ) ).
fof(t2_asympt_1,axiom,
! [A] :
( ( v1_asympt_0(A)
& m1_subset_1(A,k1_numbers) )
=> ! [B] :
( ( v1_asympt_0(B)
& m1_subset_1(B,k1_numbers) )
=> ! [C] :
( ( v1_funct_1(C)
& v1_funct_2(C,k5_numbers,k1_numbers)
& m2_relset_1(C,k5_numbers,k1_numbers) )
=> ! [D] :
( ( v1_funct_1(D)
& v1_funct_2(D,k5_numbers,k1_numbers)
& m2_relset_1(D,k5_numbers,k1_numbers) )
=> ~ ( ~ r1_xreal_0(A,np__1)
& ~ r1_xreal_0(B,np__1)
& k2_seq_1(k5_numbers,k1_numbers,C,np__0) = np__0
& ! [E] :
( m2_subset_1(E,k1_numbers,k5_numbers)
=> ( ~ r1_xreal_0(E,np__0)
=> k2_seq_1(k5_numbers,k1_numbers,C,E) = k6_power(A,E) ) )
& k2_seq_1(k5_numbers,k1_numbers,D,np__0) = np__0
& ! [E] :
( m2_subset_1(E,k1_numbers,k5_numbers)
=> ( ~ r1_xreal_0(E,np__0)
=> k2_seq_1(k5_numbers,k1_numbers,D,E) = k6_power(B,E) ) )
& ! [E] :
( ( v1_funct_1(E)
& v1_funct_2(E,k5_numbers,k1_numbers)
& v4_asympt_0(E)
& m2_relset_1(E,k5_numbers,k1_numbers) )
=> ! [F] :
( ( v1_funct_1(F)
& v1_funct_2(F,k5_numbers,k1_numbers)
& v4_asympt_0(F)
& m2_relset_1(F,k5_numbers,k1_numbers) )
=> ~ ( E = C
& F = D
& k5_asympt_0(E) = k5_asympt_0(F) ) ) ) ) ) ) ) ) ).
fof(d1_asympt_1,axiom,
! [A] :
( m1_subset_1(A,k1_numbers)
=> ! [B] :
( m1_subset_1(B,k1_numbers)
=> ! [C] :
( m1_subset_1(C,k1_numbers)
=> ! [D] :
( ( v1_funct_1(D)
& v1_funct_2(D,k5_numbers,k1_numbers)
& m2_relset_1(D,k5_numbers,k1_numbers) )
=> ( D = k1_asympt_1(A,B,C)
<=> ! [E] :
( m2_subset_1(E,k1_numbers,k5_numbers)
=> k2_seq_1(k5_numbers,k1_numbers,D,E) = k4_power(A,k3_real_1(k4_real_1(B,E),C)) ) ) ) ) ) ) ).
fof(t3_asympt_1,axiom,
! [A] :
( ( v2_xreal_0(A)
& m1_subset_1(A,k1_numbers) )
=> ! [B] :
( ( v2_xreal_0(B)
& m1_subset_1(B,k1_numbers) )
=> ~ ( ~ r1_xreal_0(B,A)
& r2_hidden(k1_asympt_1(B,np__1,np__0),k5_asympt_0(k1_asympt_1(A,np__1,np__0))) ) ) ) ).
fof(d2_asympt_1,axiom,
! [A] :
( ( v1_funct_1(A)
& v1_funct_2(A,k5_numbers,k1_numbers)
& m2_relset_1(A,k5_numbers,k1_numbers) )
=> ( A = k2_asympt_1
<=> ( k2_seq_1(k5_numbers,k1_numbers,A,np__0) = np__0
& ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> ( ~ r1_xreal_0(B,np__0)
=> k2_seq_1(k5_numbers,k1_numbers,A,B) = k6_power(np__2,B) ) ) ) ) ) ).
fof(d3_asympt_1,axiom,
! [A] :
( m1_subset_1(A,k1_numbers)
=> ! [B] :
( ( v1_funct_1(B)
& v1_funct_2(B,k5_numbers,k1_numbers)
& m2_relset_1(B,k5_numbers,k1_numbers) )
=> ( B = k3_asympt_1(A)
<=> ( k2_seq_1(k5_numbers,k1_numbers,B,np__0) = np__0
& ! [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
=> ( ~ r1_xreal_0(C,np__0)
=> k2_seq_1(k5_numbers,k1_numbers,B,C) = k4_power(C,A) ) ) ) ) ) ) ).
fof(t4_asympt_1,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) )
=> ( ( r1_tarski(k5_asympt_0(A),k5_asympt_0(B))
=> ( k5_asympt_0(A) = k5_asympt_0(B)
| ( r2_hidden(A,k5_asympt_0(B))
& ~ r2_hidden(A,k6_asympt_0(B)) ) ) )
& ( r2_hidden(A,k5_asympt_0(B))
=> ( r2_hidden(A,k6_asympt_0(B))
| ( r1_tarski(k5_asympt_0(A),k5_asympt_0(B))
& k5_asympt_0(A) != k5_asympt_0(B) ) ) ) ) ) ) ).
fof(t5_asympt_1,axiom,
( r1_tarski(k5_asympt_0(k2_asympt_1),k5_asympt_0(k3_asympt_1(k6_real_1(np__1,np__2))))
& k5_asympt_0(k2_asympt_1) != k5_asympt_0(k3_asympt_1(k6_real_1(np__1,np__2))) ) ).
fof(t6_asympt_1,axiom,
( r2_hidden(k3_asympt_1(k6_real_1(np__1,np__2)),k6_asympt_0(k2_asympt_1))
& ~ r2_hidden(k2_asympt_1,k6_asympt_0(k3_asympt_1(k6_real_1(np__1,np__2)))) ) ).
fof(t7_asympt_1,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)
=> k2_seq_1(k5_numbers,k1_numbers,A,C) = k5_bhsp_4(k3_asympt_1(B),C) )
=> r2_hidden(A,k7_asympt_0(k3_asympt_1(k1_nat_1(B,np__1)))) ) ) ) ).
fof(t8_asympt_1,axiom,
! [A] :
( ( v1_funct_1(A)
& v1_funct_2(A,k5_numbers,k1_numbers)
& m2_relset_1(A,k5_numbers,k1_numbers) )
=> ~ ( k2_seq_1(k5_numbers,k1_numbers,A,np__0) = np__0
& ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> ( ~ r1_xreal_0(B,np__0)
=> k2_seq_1(k5_numbers,k1_numbers,A,B) = k4_power(B,k6_power(np__2,B)) ) )
& ! [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) )
=> ~ ( B = A
& ~ v7_asympt_0(B) ) ) ) ) ).
fof(d4_asympt_1,axiom,
! [A] :
( m1_subset_1(A,k1_numbers)
=> k4_asympt_1(A) = k2_pre_circ(k5_numbers,A) ) ).
fof(t9_asympt_1,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] :
( m1_fraenkel(B,k5_numbers,k1_numbers)
& B = k6_domain_1(k1_zfmisc_1(k2_zfmisc_1(k5_numbers,k1_numbers)),k3_asympt_1(np__1))
& ~ ( r2_hidden(A,k14_asympt_0(B,k5_asympt_0(k4_asympt_1(np__1))))
& ! [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
=> ! [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(C,F)
=> ( r1_xreal_0(np__1,k2_seq_1(k5_numbers,k1_numbers,A,F))
& r1_xreal_0(k2_seq_1(k5_numbers,k1_numbers,A,F),k4_real_1(D,k2_seq_1(k5_numbers,k1_numbers,k3_asympt_1(E),F))) ) ) ) ) ) ) ) )
& ( ? [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
& ? [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(C,F)
=> ( r1_xreal_0(np__1,k2_seq_1(k5_numbers,k1_numbers,A,F))
& r1_xreal_0(k2_seq_1(k5_numbers,k1_numbers,A,F),k4_real_1(D,k2_seq_1(k5_numbers,k1_numbers,k3_asympt_1(E),F))) ) ) ) ) ) )
=> r2_hidden(A,k14_asympt_0(B,k5_asympt_0(k4_asympt_1(np__1)))) ) ) ) ).
fof(t10_asympt_1,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)
=> k2_seq_1(k5_numbers,k1_numbers,A,B) = k3_real_1(k5_real_1(k2_nat_1(np__3,k3_series_1(np__10,np__6)),k2_nat_1(k2_nat_1(np__18,k3_series_1(np__10,np__3)),B)),k4_real_1(np__27,k7_square_1(B))) )
=> r2_hidden(A,k5_asympt_0(k3_asympt_1(np__2))) ) ) ).
fof(t11_asympt_1,axiom,
r2_hidden(k3_asympt_1(np__2),k5_asympt_0(k3_asympt_1(np__3))) ).
fof(t12_asympt_1,axiom,
~ r2_hidden(k3_asympt_1(np__2),k6_asympt_0(k3_asympt_1(np__3))) ).
fof(t13_asympt_1,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)
& A = k1_asympt_1(np__2,np__1,np__1)
& r2_hidden(k1_asympt_1(np__2,np__1,np__0),k7_asympt_0(A)) ) ).
fof(d5_asympt_1,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( ( v1_funct_1(B)
& v1_funct_2(B,k5_numbers,k1_numbers)
& m2_relset_1(B,k5_numbers,k1_numbers) )
=> ( B = k5_asympt_1(A)
<=> ! [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
=> k2_seq_1(k5_numbers,k1_numbers,B,C) = k11_newton(k1_nat_1(C,A)) ) ) ) ) ).
fof(t14_asympt_1,axiom,
~ r2_hidden(k5_asympt_1(np__0),k7_asympt_0(k5_asympt_1(np__1))) ).
fof(t15_asympt_1,axiom,
! [A] :
( ( v1_funct_1(A)
& v1_funct_2(A,k5_numbers,k1_numbers)
& m2_relset_1(A,k5_numbers,k1_numbers) )
=> ( r2_hidden(A,k5_asympt_0(k3_asympt_1(np__1)))
=> r2_hidden(k11_seq_1(A,A),k5_asympt_0(k3_asympt_1(np__2))) ) ) ).
fof(t16_asympt_1,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)
& A = k1_asympt_1(np__2,np__1,np__0)
& r2_hidden(k3_asympt_0(k3_asympt_1(np__1),np__2),k5_asympt_0(k3_asympt_1(np__1)))
& ~ r2_hidden(k1_asympt_1(np__2,np__2,np__0),k5_asympt_0(A)) ) ).
fof(t17_asympt_1,axiom,
( ~ r1_xreal_0(k6_real_1(np__159,np__100),k6_power(np__2,np__3))
=> ( r2_hidden(k3_asympt_1(k6_power(np__2,np__3)),k5_asympt_0(k3_asympt_1(k6_real_1(np__159,np__100))))
& ~ r2_hidden(k3_asympt_1(k6_power(np__2,np__3)),k6_asympt_0(k3_asympt_1(k6_real_1(np__159,np__100))))
& ~ r2_hidden(k3_asympt_1(k6_power(np__2,np__3)),k7_asympt_0(k3_asympt_1(k6_real_1(np__159,np__100)))) ) ) ).
fof(t18_asympt_1,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] :
( m2_subset_1(C,k1_numbers,k5_numbers)
=> k2_seq_1(k5_numbers,k1_numbers,A,C) = k4_nat_1(C,np__2) )
& ! [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
=> k2_seq_1(k5_numbers,k1_numbers,B,C) = k4_nat_1(k1_nat_1(C,np__1),np__2) )
& ! [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) )
=> ! [D] :
( ( v1_funct_1(D)
& v1_funct_2(D,k5_numbers,k1_numbers)
& v2_asympt_0(D)
& m2_relset_1(D,k5_numbers,k1_numbers) )
=> ~ ( C = A
& D = B
& ~ r2_hidden(C,k5_asympt_0(D))
& ~ r2_hidden(D,k5_asympt_0(C)) ) ) ) ) ) ) ).
fof(t19_asympt_1,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(A) = k5_asympt_0(B)
<=> r2_hidden(A,k7_asympt_0(B)) ) ) ) ).
fof(t20_asympt_1,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))
<=> k7_asympt_0(A) = k7_asympt_0(B) ) ) ) ).
fof(t21_asympt_1,axiom,
! [A] :
( m1_subset_1(A,k1_numbers)
=> ! [B] :
( ( v1_funct_1(B)
& v1_funct_2(B,k5_numbers,k1_numbers)
& m2_relset_1(B,k5_numbers,k1_numbers) )
=> ~ ( ~ r1_xreal_0(A,np__0)
& k2_seq_1(k5_numbers,k1_numbers,B,np__0) = np__0
& ! [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
=> ( ~ r1_xreal_0(C,np__0)
=> k2_seq_1(k5_numbers,k1_numbers,B,C) = k4_real_1(C,k6_power(np__2,C)) ) )
& ! [C] :
( ( v1_funct_1(C)
& v1_funct_2(C,k5_numbers,k1_numbers)
& v4_asympt_0(C)
& m2_relset_1(C,k5_numbers,k1_numbers) )
=> ~ ( C = B
& r1_tarski(k5_asympt_0(C),k5_asympt_0(k3_asympt_1(k3_real_1(np__1,A))))
& k5_asympt_0(C) != k5_asympt_0(k3_asympt_1(k3_real_1(np__1,A))) ) ) ) ) ) ).
fof(t22_asympt_1,axiom,
! [A] :
( m1_subset_1(A,k1_numbers)
=> ! [B] :
( ( v1_funct_1(B)
& v1_funct_2(B,k5_numbers,k1_numbers)
& m2_relset_1(B,k5_numbers,k1_numbers) )
=> ~ ( ~ r1_xreal_0(A,np__0)
& ~ r1_xreal_0(np__1,A)
& k2_seq_1(k5_numbers,k1_numbers,B,np__0) = np__0
& k2_seq_1(k5_numbers,k1_numbers,B,np__1) = np__0
& ! [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
=> ( ~ r1_xreal_0(C,np__1)
=> k2_seq_1(k5_numbers,k1_numbers,B,C) = k6_real_1(k3_series_1(C,np__2),k6_power(np__2,C)) ) )
& ! [C] :
( ( v1_funct_1(C)
& v1_funct_2(C,k5_numbers,k1_numbers)
& v4_asympt_0(C)
& m2_relset_1(C,k5_numbers,k1_numbers) )
=> ~ ( C = B
& r1_tarski(k5_asympt_0(k3_asympt_1(k3_real_1(np__1,A))),k5_asympt_0(C))
& k5_asympt_0(k3_asympt_1(k3_real_1(np__1,A))) != k5_asympt_0(C) ) ) ) ) ) ).
fof(t23_asympt_1,axiom,
! [A] :
( ( v1_funct_1(A)
& v1_funct_2(A,k5_numbers,k1_numbers)
& m2_relset_1(A,k5_numbers,k1_numbers) )
=> ~ ( k2_seq_1(k5_numbers,k1_numbers,A,np__0) = np__0
& k2_seq_1(k5_numbers,k1_numbers,A,np__1) = np__0
& ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> ( ~ r1_xreal_0(B,np__1)
=> k2_seq_1(k5_numbers,k1_numbers,A,B) = k6_real_1(k3_series_1(B,np__2),k6_power(np__2,B)) ) )
& ! [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) )
=> ~ ( B = A
& r1_tarski(k5_asympt_0(B),k5_asympt_0(k3_asympt_1(np__8)))
& k5_asympt_0(B) != k5_asympt_0(k3_asympt_1(np__8)) ) ) ) ) ).
fof(t24_asympt_1,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)
=> k2_seq_1(k5_numbers,k1_numbers,A,B) = k4_power(k3_real_1(k5_real_1(k7_square_1(B),B),np__1),np__4) )
& ! [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) )
=> ~ ( B = A
& k5_asympt_0(k3_asympt_1(np__8)) = k5_asympt_0(B) ) ) ) ) ).
fof(t25_asympt_1,axiom,
! [A] :
( m1_subset_1(A,k1_numbers)
=> ~ ( ~ r1_xreal_0(A,np__0)
& ~ r1_xreal_0(np__1,A)
& ! [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) )
=> ~ ( B = k1_asympt_1(k3_real_1(np__1,A),np__1,np__0)
& r1_tarski(k5_asympt_0(k3_asympt_1(np__8)),k5_asympt_0(B))
& k5_asympt_0(k3_asympt_1(np__8)) != k5_asympt_0(B) ) ) ) ) ).
fof(t26_asympt_1,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) )
=> ~ ( k2_seq_1(k5_numbers,k1_numbers,A,np__0) = np__0
& ! [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
=> ( ~ r1_xreal_0(C,np__0)
=> k2_seq_1(k5_numbers,k1_numbers,A,C) = k4_power(C,k6_power(np__2,C)) ) )
& k2_seq_1(k5_numbers,k1_numbers,B,np__0) = np__0
& ! [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
=> ( ~ r1_xreal_0(C,np__0)
=> k2_seq_1(k5_numbers,k1_numbers,B,C) = k4_power(C,k9_square_1(C)) ) )
& ! [C] :
( ( v1_funct_1(C)
& v1_funct_2(C,k5_numbers,k1_numbers)
& v4_asympt_0(C)
& m2_relset_1(C,k5_numbers,k1_numbers) )
=> ! [D] :
( ( v1_funct_1(D)
& v1_funct_2(D,k5_numbers,k1_numbers)
& v4_asympt_0(D)
& m2_relset_1(D,k5_numbers,k1_numbers) )
=> ~ ( C = A
& D = B
& r1_tarski(k5_asympt_0(C),k5_asympt_0(D))
& k5_asympt_0(C) != k5_asympt_0(D) ) ) ) ) ) ) ).
fof(t27_asympt_1,axiom,
! [A] :
( ( v1_funct_1(A)
& v1_funct_2(A,k5_numbers,k1_numbers)
& m2_relset_1(A,k5_numbers,k1_numbers) )
=> ~ ( k2_seq_1(k5_numbers,k1_numbers,A,np__0) = np__0
& ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> ( ~ r1_xreal_0(B,np__0)
=> k2_seq_1(k5_numbers,k1_numbers,A,B) = k4_power(B,k9_square_1(B)) ) )
& ! [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) )
=> ! [C] :
( ( v1_funct_1(C)
& v1_funct_2(C,k5_numbers,k1_numbers)
& v4_asympt_0(C)
& m2_relset_1(C,k5_numbers,k1_numbers) )
=> ~ ( B = A
& C = k1_asympt_1(np__2,np__1,np__0)
& r1_tarski(k5_asympt_0(B),k5_asympt_0(C))
& k5_asympt_0(B) != k5_asympt_0(C) ) ) ) ) ) ).
fof(t28_asympt_1,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)
& A = k1_asympt_1(np__2,np__1,np__0)
& B = k1_asympt_1(np__2,np__1,np__1)
& k5_asympt_0(A) = k5_asympt_0(B) ) ) ).
fof(t29_asympt_1,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)
& A = k1_asympt_1(np__2,np__1,np__0)
& B = k1_asympt_1(np__2,np__2,np__0)
& r1_tarski(k5_asympt_0(A),k5_asympt_0(B))
& k5_asympt_0(A) != k5_asympt_0(B) ) ) ).
fof(t30_asympt_1,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)
& A = k1_asympt_1(np__2,np__2,np__0)
& r1_tarski(k5_asympt_0(A),k5_asympt_0(k5_asympt_1(np__0)))
& k5_asympt_0(A) != k5_asympt_0(k5_asympt_1(np__0)) ) ).
fof(t31_asympt_1,axiom,
( r1_tarski(k5_asympt_0(k5_asympt_1(np__0)),k5_asympt_0(k5_asympt_1(np__1)))
& k5_asympt_0(k5_asympt_1(np__0)) != k5_asympt_0(k5_asympt_1(np__1)) ) ).
fof(t32_asympt_1,axiom,
! [A] :
( ( v1_funct_1(A)
& v1_funct_2(A,k5_numbers,k1_numbers)
& m2_relset_1(A,k5_numbers,k1_numbers) )
=> ~ ( k2_seq_1(k5_numbers,k1_numbers,A,np__0) = np__0
& ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> ( ~ r1_xreal_0(B,np__0)
=> k2_seq_1(k5_numbers,k1_numbers,A,B) = k3_series_1(B,B) ) )
& ! [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) )
=> ~ ( B = A
& r1_tarski(k5_asympt_0(k5_asympt_1(np__1)),k5_asympt_0(B))
& k5_asympt_0(k5_asympt_1(np__1)) != k5_asympt_0(B) ) ) ) ) ).
fof(t33_asympt_1,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ( r1_xreal_0(np__1,A)
=> ! [B] :
( ( v1_funct_1(B)
& v1_funct_2(B,k5_numbers,k1_numbers)
& m2_relset_1(B,k5_numbers,k1_numbers) )
=> ! [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
=> ( ! [D] :
( m2_subset_1(D,k1_numbers,k5_numbers)
=> k2_seq_1(k5_numbers,k1_numbers,B,D) = k5_bhsp_4(k3_asympt_1(C),D) )
=> r1_xreal_0(k6_real_1(k3_series_1(A,k1_nat_1(C,np__1)),k1_nat_1(C,np__1)),k2_seq_1(k5_numbers,k1_numbers,B,A)) ) ) ) ) ) ).
fof(t34_asympt_1,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) )
=> ~ ( k2_seq_1(k5_numbers,k1_numbers,B,np__0) = np__0
& ! [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
=> ( ~ r1_xreal_0(C,np__0)
=> k2_seq_1(k5_numbers,k1_numbers,B,C) = k4_real_1(C,k6_power(np__2,C)) ) )
& ! [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
=> k2_seq_1(k5_numbers,k1_numbers,A,C) = k6_power(np__2,k11_newton(C)) )
& ! [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) )
=> ~ ( C = B
& r2_hidden(A,k7_asympt_0(C)) ) ) ) ) ) ).
fof(t35_asympt_1,axiom,
! [A] :
( ( v1_funct_1(A)
& v1_funct_2(A,k5_numbers,k1_numbers)
& v2_asympt_0(A)
& v6_asympt_0(A)
& 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] :
( m2_subset_1(C,k1_numbers,k5_numbers)
=> ( ( k4_nat_1(C,np__2) = np__0
=> k2_seq_1(k5_numbers,k1_numbers,B,C) = np__1 )
& ( k4_nat_1(C,np__2) = np__1
=> k2_seq_1(k5_numbers,k1_numbers,B,C) = C ) ) )
& r2_hidden(B,k7_asympt_0(A)) ) ) ) ).
fof(d6_asympt_1,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( ( v2_xreal_0(B)
& m1_subset_1(B,k1_numbers) )
=> ! [C] :
( ( v2_xreal_0(C)
& m1_subset_1(C,k1_numbers) )
=> ! [D] :
( m1_subset_1(D,k1_numbers)
=> ( ( A = np__0
=> ( D = k7_asympt_1(A,B,C)
<=> D = np__0 ) )
& ( A != np__0
=> ( D = k7_asympt_1(A,B,C)
<=> ? [E] :
( m2_subset_1(E,k1_numbers,k5_numbers)
& ? [F] :
( v1_funct_1(F)
& v1_funct_2(F,k5_numbers,k3_finseq_2(k1_numbers))
& m2_relset_1(F,k5_numbers,k3_finseq_2(k1_numbers))
& k1_nat_1(E,np__1) = A
& D = k4_finseq_4(k5_numbers,k1_numbers,k6_asympt_1(F,E),A)
& k6_asympt_1(F,np__0) = k12_finseq_1(k1_numbers,B)
& ! [G] :
( m2_subset_1(G,k1_numbers,k5_numbers)
=> ? [H] :
( m2_subset_1(H,k1_numbers,k5_numbers)
& H = k2_int_1(k6_real_1(k1_nat_1(k1_nat_1(G,np__1),np__1),np__2))
& k6_asympt_1(F,k1_nat_1(G,np__1)) = k8_finseq_1(k1_numbers,k6_asympt_1(F,G),k12_finseq_1(k1_numbers,k3_real_1(k4_real_1(np__4,k4_finseq_4(k5_numbers,k1_numbers,k6_asympt_1(F,G),H)),k4_real_1(C,k1_nat_1(k1_nat_1(G,np__1),np__1))))) ) ) ) ) ) ) ) ) ) ) ) ).
fof(d7_asympt_1,axiom,
! [A] :
( ( v2_xreal_0(A)
& m1_subset_1(A,k1_numbers) )
=> ! [B] :
( ( v2_xreal_0(B)
& m1_subset_1(B,k1_numbers) )
=> ! [C] :
( ( v1_funct_1(C)
& v1_funct_2(C,k5_numbers,k1_numbers)
& m2_relset_1(C,k5_numbers,k1_numbers) )
=> ( C = k8_asympt_1(A,B)
<=> ! [D] :
( m2_subset_1(D,k1_numbers,k5_numbers)
=> k2_seq_1(k5_numbers,k1_numbers,C,D) = k7_asympt_1(D,A,B) ) ) ) ) ) ).
fof(t36_asympt_1,axiom,
! [A] :
( ( v2_xreal_0(A)
& m1_subset_1(A,k1_numbers) )
=> ! [B] :
( ( v2_xreal_0(B)
& m1_subset_1(B,k1_numbers) )
=> v6_asympt_0(k8_asympt_1(A,B)) ) ) ).
fof(t37_asympt_1,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)
=> ( ( r2_hidden(B,k9_asympt_1)
=> k2_seq_1(k5_numbers,k1_numbers,A,B) = B )
& ( ~ r2_hidden(B,k9_asympt_1)
=> k2_seq_1(k5_numbers,k1_numbers,A,B) = k3_series_1(np__2,B) ) ) )
=> ( r2_hidden(A,k10_asympt_0(k3_asympt_1(np__1),k9_asympt_1))
& ~ r2_hidden(A,k7_asympt_0(k3_asympt_1(np__1)))
& v7_asympt_0(k3_asympt_1(np__1))
& ~ v6_asympt_0(A) ) ) ) ).
fof(t38_asympt_1,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) )
=> ~ ( k2_seq_1(k5_numbers,k1_numbers,A,np__0) = np__0
& ! [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
=> ( ~ r1_xreal_0(C,np__0)
=> k2_seq_1(k5_numbers,k1_numbers,A,C) = k3_power(C,k3_power(np__2,k1_int_1(k6_power(np__2,C)))) ) )
& k2_seq_1(k5_numbers,k1_numbers,B,np__0) = np__0
& ! [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
=> ( ~ r1_xreal_0(C,np__0)
=> k2_seq_1(k5_numbers,k1_numbers,B,C) = k3_series_1(C,C) ) )
& ! [C] :
( ( v1_funct_1(C)
& v1_funct_2(C,k5_numbers,k1_numbers)
& v4_asympt_0(C)
& m2_relset_1(C,k5_numbers,k1_numbers) )
=> ~ ( C = B
& r2_hidden(A,k10_asympt_0(C,k9_asympt_1))
& ~ r2_hidden(A,k7_asympt_0(C))
& v6_asympt_0(A)
& v6_asympt_0(C)
& ~ r2_asympt_0(C,np__2) ) ) ) ) ) ).
fof(t39_asympt_1,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)
=> ( ( r2_hidden(B,k9_asympt_1)
=> k2_seq_1(k5_numbers,k1_numbers,A,B) = B )
& ( ~ r2_hidden(B,k9_asympt_1)
=> k2_seq_1(k5_numbers,k1_numbers,A,B) = k3_series_1(B,np__2) ) ) )
& ! [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) )
=> ~ ( B = A
& r2_hidden(k3_asympt_1(np__1),k10_asympt_0(B,k9_asympt_1))
& ~ r2_hidden(k3_asympt_1(np__1),k7_asympt_0(B))
& r2_hidden(k11_asympt_0(B,np__2),k5_asympt_0(B))
& v6_asympt_0(k3_asympt_1(np__1))
& ~ v6_asympt_0(B) ) ) ) ) ).
fof(d9_asympt_1,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> ( ( A != np__0
=> ( B = k10_asympt_1(A)
<=> ? [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
& r1_xreal_0(k11_newton(C),A)
& ~ r1_xreal_0(k11_newton(k1_nat_1(C,np__1)),A)
& B = k11_newton(C) ) ) )
& ( A = np__0
=> ( B = k10_asympt_1(A)
<=> B = np__0 ) ) ) ) ) ).
fof(t40_asympt_1,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)
=> k2_seq_1(k5_numbers,k1_numbers,A,B) = k10_asympt_1(B) )
& ! [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) )
=> ~ ( B = A
& v6_asympt_0(A)
& ! [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
=> r1_xreal_0(k2_seq_1(k5_numbers,k1_numbers,A,C),k2_seq_1(k5_numbers,k1_numbers,k3_asympt_1(np__1),C)) )
& ~ v7_asympt_0(B) ) ) ) ) ).
fof(t41_asympt_1,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) )
=> ( A = k10_seq_1(k3_asympt_1(np__1),k4_asympt_1(np__1))
=> k12_asympt_0(k5_numbers,k7_asympt_0(A),k7_asympt_0(k3_asympt_1(np__1))) = k7_asympt_0(k3_asympt_1(np__1)) ) ) ).
fof(t42_asympt_1,axiom,
? [A] :
( m1_fraenkel(A,k5_numbers,k1_numbers)
& A = k6_domain_1(k1_zfmisc_1(k2_zfmisc_1(k5_numbers,k1_numbers)),k3_asympt_1(np__1))
& ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> r1_xreal_0(k2_seq_1(k5_numbers,k1_numbers,k3_asympt_1(k1_real_1(np__1)),B),k2_seq_1(k5_numbers,k1_numbers,k3_asympt_1(np__1),B)) )
& ~ r2_hidden(k3_asympt_1(k1_real_1(np__1)),k14_asympt_0(A,k5_asympt_0(k4_asympt_1(np__1)))) ) ).
fof(t43_asympt_1,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(k2_asympt_0(C,A)))
=> ( ! [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(D,k2_seq_1(k5_numbers,k1_numbers,C,F)) ) ) ) ) )
| r2_hidden(B,k5_asympt_0(C)) ) ) ) ) ) ).
fof(t44_asympt_1,axiom,
k3_series_1(np__2,np__2) = np__4 ).
fof(t45_asympt_1,axiom,
k3_series_1(np__2,np__3) = np__8 ).
fof(t46_asympt_1,axiom,
k3_series_1(np__2,np__4) = np__16 ).
fof(t47_asympt_1,axiom,
k3_series_1(np__2,np__5) = np__32 ).
fof(t48_asympt_1,axiom,
k3_series_1(np__2,np__6) = np__64 ).
fof(t49_asympt_1,axiom,
k3_series_1(np__2,np__12) = np__4096 ).
fof(t50_asympt_1,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ~ ( r1_xreal_0(np__3,A)
& r1_xreal_0(k7_square_1(A),k1_nat_1(k2_nat_1(np__2,A),np__1)) ) ) ).
fof(t51_asympt_1,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ~ ( r1_xreal_0(np__10,A)
& r1_xreal_0(k4_power(np__2,k5_real_1(A,np__1)),k7_square_1(k2_nat_1(np__2,A))) ) ) ).
fof(t52_asympt_1,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ~ ( r1_xreal_0(np__9,A)
& r1_xreal_0(k2_nat_1(np__2,k3_series_1(A,np__6)),k3_series_1(k1_nat_1(A,np__1),np__6)) ) ) ).
fof(t53_asympt_1,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ~ ( r1_xreal_0(np__30,A)
& r1_xreal_0(k3_series_1(np__2,A),k3_series_1(A,np__6)) ) ) ).
fof(t54_asympt_1,axiom,
! [A] :
( m1_subset_1(A,k1_numbers)
=> ~ ( ~ r1_xreal_0(A,np__9)
& r1_xreal_0(k4_power(np__2,A),k7_square_1(k4_real_1(np__2,A))) ) ) ).
fof(t55_asympt_1,axiom,
? [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
& ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> ~ ( r1_xreal_0(A,B)
& r1_xreal_0(k5_real_1(k9_square_1(B),k6_power(np__2,B)),np__1) ) ) ) ).
fof(t56_asympt_1,axiom,
! [A] :
( m1_subset_1(A,k1_numbers)
=> ! [B] :
( m1_subset_1(B,k1_numbers)
=> ! [C] :
( m1_subset_1(C,k1_numbers)
=> ~ ( ~ r1_xreal_0(A,np__0)
& ~ r1_xreal_0(C,np__0)
& C != np__1
& k4_power(A,B) != k4_power(C,k4_real_1(B,k6_power(C,A))) ) ) ) ) ).
fof(t57_asympt_1,axiom,
k11_newton(k1_nat_1(np__4,np__1)) = np__120 ).
fof(t58_asympt_1,axiom,
k3_series_1(np__5,np__5) = np__3125 ).
fof(t59_asympt_1,axiom,
k3_series_1(np__4,np__4) = np__256 ).
fof(t60_asympt_1,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ~ r1_xreal_0(k3_real_1(k5_real_1(k7_square_1(A),A),np__1),np__0) ) ).
fof(t61_asympt_1,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ~ ( r1_xreal_0(np__2,A)
& r1_xreal_0(k11_newton(A),np__1) ) ) ).
fof(t62_asympt_1,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> ( r1_xreal_0(B,A)
=> r1_xreal_0(k11_newton(B),k11_newton(A)) ) ) ) ).
fof(t63_asympt_1,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ~ ( r1_xreal_0(np__1,A)
& ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> ~ ( r1_xreal_0(k11_newton(B),A)
& ~ r1_xreal_0(k11_newton(k1_nat_1(B,np__1)),A)
& ! [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
=> ( r1_xreal_0(k11_newton(C),A)
=> ( r1_xreal_0(k11_newton(k1_nat_1(C,np__1)),A)
| C = B ) ) ) ) ) ) ) ).
fof(t64_asympt_1,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ~ ( r1_xreal_0(np__2,A)
& r1_xreal_0(A,k2_int_1(k6_real_1(A,np__2))) ) ) ).
fof(t65_asympt_1,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ~ ( r1_xreal_0(np__3,A)
& r1_xreal_0(k11_newton(A),A) ) ) ).
fof(t66_asympt_1,axiom,
v4_asympt_0(k10_seq_1(k3_asympt_1(np__1),k4_asympt_1(np__1))) ).
fof(t67_asympt_1,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ~ ( r1_xreal_0(np__2,A)
& r1_xreal_0(k3_series_1(np__2,A),k1_nat_1(A,np__1)) ) ) ).
fof(t68_asympt_1,axiom,
! [A] :
( ( v1_asympt_0(A)
& m1_subset_1(A,k1_numbers) )
=> ! [B] :
( ( v1_funct_1(B)
& v1_funct_2(B,k5_numbers,k1_numbers)
& m2_relset_1(B,k5_numbers,k1_numbers) )
=> ( ( k2_seq_1(k5_numbers,k1_numbers,B,np__0) = np__0
& ! [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
=> ( ~ r1_xreal_0(C,np__0)
=> k2_seq_1(k5_numbers,k1_numbers,B,C) = k6_power(A,C) ) ) )
=> ( r1_xreal_0(A,np__1)
| v4_asympt_0(B) ) ) ) ) ).
fof(t69_asympt_1,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))
& r2_hidden(B,k5_asympt_0(A)) )
<=> k5_asympt_0(A) = k5_asympt_0(B) ) ) ) ).
fof(t70_asympt_1,axiom,
! [A] :
( m1_subset_1(A,k1_numbers)
=> ! [B] :
( m1_subset_1(B,k1_numbers)
=> ! [C] :
( m1_subset_1(C,k1_numbers)
=> ( ( r1_xreal_0(A,B)
& r1_xreal_0(np__0,C) )
=> ( r1_xreal_0(A,np__0)
| r1_xreal_0(k4_power(A,C),k4_power(B,C)) ) ) ) ) ) ).
fof(t71_asympt_1,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ~ ( r1_xreal_0(np__4,A)
& r1_xreal_0(k3_series_1(np__2,A),k1_nat_1(k2_nat_1(np__2,A),np__3)) ) ) ).
fof(t72_asympt_1,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ~ ( r1_xreal_0(np__6,A)
& r1_xreal_0(k3_series_1(np__2,A),k7_square_1(k1_nat_1(A,np__1))) ) ) ).
fof(t73_asympt_1,axiom,
! [A] :
( m1_subset_1(A,k1_numbers)
=> ~ ( ~ r1_xreal_0(A,np__6)
& r1_xreal_0(k4_power(np__2,A),k7_square_1(A)) ) ) ).
fof(t74_asympt_1,axiom,
! [A] :
( ( v2_xreal_0(A)
& m1_subset_1(A,k1_numbers) )
=> ! [B] :
( ( v1_funct_1(B)
& v1_funct_2(B,k5_numbers,k1_numbers)
& m2_relset_1(B,k5_numbers,k1_numbers) )
=> ( ( k2_seq_1(k5_numbers,k1_numbers,B,np__0) = np__0
& ! [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
=> ( ~ r1_xreal_0(C,np__0)
=> k2_seq_1(k5_numbers,k1_numbers,B,C) = k6_power(np__2,k4_power(C,A)) ) ) )
=> ( v4_seq_2(k19_seq_1(B,k3_asympt_1(A)))
& k2_seq_2(k19_seq_1(B,k3_asympt_1(A))) = np__0 ) ) ) ) ).
fof(t75_asympt_1,axiom,
! [A] :
( m1_subset_1(A,k1_numbers)
=> ( ~ r1_xreal_0(A,np__0)
=> ( v4_seq_2(k19_seq_1(k2_asympt_1,k3_asympt_1(A)))
& k2_seq_2(k19_seq_1(k2_asympt_1,k3_asympt_1(A))) = np__0 ) ) ) ).
fof(t76_asympt_1,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(C,B)
=> r1_xreal_0(np__0,k2_seq_1(k5_numbers,k1_numbers,A,C)) ) )
=> r1_xreal_0(np__0,k5_bhsp_4(A,B)) ) ) ) ).
fof(t77_asympt_1,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] :
( m2_subset_1(C,k1_numbers,k5_numbers)
=> ( ! [D] :
( m2_subset_1(D,k1_numbers,k5_numbers)
=> ( r1_xreal_0(D,C)
=> r1_xreal_0(k2_seq_1(k5_numbers,k1_numbers,A,D),k2_seq_1(k5_numbers,k1_numbers,B,D)) ) )
=> r1_xreal_0(k5_bhsp_4(A,C),k5_bhsp_4(B,C)) ) ) ) ) ).
fof(t78_asympt_1,axiom,
! [A] :
( ( v1_funct_1(A)
& v1_funct_2(A,k5_numbers,k1_numbers)
& m2_relset_1(A,k5_numbers,k1_numbers) )
=> ! [B] :
( m1_subset_1(B,k1_numbers)
=> ( ( k2_seq_1(k5_numbers,k1_numbers,A,np__0) = np__0
& ! [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
=> ( ~ r1_xreal_0(C,np__0)
=> k2_seq_1(k5_numbers,k1_numbers,A,C) = B ) ) )
=> ! [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
=> k5_bhsp_4(A,C) = k4_real_1(B,C) ) ) ) ) ).
fof(t79_asympt_1,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)
=> k3_real_1(k6_bhsp_4(A,B,C),k2_seq_1(k5_numbers,k1_numbers,A,k1_nat_1(B,np__1))) = k6_bhsp_4(A,k1_nat_1(B,np__1),C) ) ) ) ).
fof(t80_asympt_1,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] :
( m2_subset_1(C,k1_numbers,k5_numbers)
=> ! [D] :
( m2_subset_1(D,k1_numbers,k5_numbers)
=> ( ( r1_xreal_0(k1_nat_1(C,np__1),D)
& ! [E] :
( m2_subset_1(E,k1_numbers,k5_numbers)
=> ( ( r1_xreal_0(k1_nat_1(C,np__1),E)
& r1_xreal_0(E,D) )
=> r1_xreal_0(k2_seq_1(k5_numbers,k1_numbers,A,E),k2_seq_1(k5_numbers,k1_numbers,B,E)) ) ) )
=> r1_xreal_0(k6_bhsp_4(A,D,C),k6_bhsp_4(B,D,C)) ) ) ) ) ) ).
fof(t81_asympt_1,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> r1_xreal_0(k2_int_1(k6_real_1(A,np__2)),A) ) ).
fof(t82_asympt_1,axiom,
! [A] :
( ( v1_funct_1(A)
& v1_funct_2(A,k5_numbers,k1_numbers)
& m2_relset_1(A,k5_numbers,k1_numbers) )
=> ! [B] :
( m1_subset_1(B,k1_numbers)
=> ! [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
=> ( ( k2_seq_1(k5_numbers,k1_numbers,A,np__0) = np__0
& ! [D] :
( m2_subset_1(D,k1_numbers,k5_numbers)
=> ( ~ r1_xreal_0(D,np__0)
=> k2_seq_1(k5_numbers,k1_numbers,A,D) = B ) ) )
=> ! [D] :
( m2_subset_1(D,k1_numbers,k5_numbers)
=> k6_bhsp_4(A,C,D) = k4_real_1(B,k5_real_1(C,D)) ) ) ) ) ) ).
fof(t83_asympt_1,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] :
( m2_subset_1(C,k1_numbers,k5_numbers)
=> ! [D] :
( m1_subset_1(D,k1_numbers)
=> ( ( v4_seq_2(A)
& k2_seq_2(A) = D
& ! [E] :
( m2_subset_1(E,k1_numbers,k5_numbers)
=> ( r1_xreal_0(C,E)
=> k2_seq_1(k5_numbers,k1_numbers,A,E) = k2_seq_1(k5_numbers,k1_numbers,B,E) ) ) )
=> ( v4_seq_2(B)
& k2_seq_2(B) = D ) ) ) ) ) ) ).
fof(t84_asympt_1,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ( r1_xreal_0(np__1,A)
=> r1_xreal_0(k3_real_1(k5_real_1(k7_square_1(A),A),np__1),k7_square_1(A)) ) ) ).
fof(t85_asympt_1,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ( r1_xreal_0(np__1,A)
=> r1_xreal_0(k7_square_1(A),k4_real_1(np__2,k3_real_1(k5_real_1(k7_square_1(A),A),np__1))) ) ) ).
fof(t86_asympt_1,axiom,
! [A] :
( m1_subset_1(A,k1_numbers)
=> ~ ( ~ r1_xreal_0(A,np__0)
& ~ r1_xreal_0(np__1,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(k5_real_1(k4_real_1(C,k6_power(np__2,k3_real_1(np__1,A))),k4_real_1(np__8,k6_power(np__2,C))),k4_real_1(np__8,k6_power(np__2,C))) ) ) ) ) ).
fof(t87_asympt_1,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ~ ( r1_xreal_0(np__10,A)
& r1_xreal_0(k6_real_1(np__1,k4_power(np__2,k5_real_1(A,np__9))),k6_real_1(k3_series_1(np__2,k2_nat_1(np__2,A)),k11_newton(A))) ) ) ).
fof(t88_asympt_1,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ( r1_xreal_0(np__3,A)
=> r1_xreal_0(k5_real_1(A,np__1),k4_real_1(np__2,k5_real_1(A,np__2))) ) ) ).
fof(t89_asympt_1,axiom,
! [A] :
( v1_xreal_0(A)
=> ( r1_xreal_0(np__0,A)
=> k3_power(A,k6_real_1(np__1,np__2)) = k8_square_1(A) ) ) ).
fof(t90_asympt_1,axiom,
? [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
& ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> ~ ( r1_xreal_0(A,B)
& r1_xreal_0(k5_real_1(B,k4_real_1(k9_square_1(B),k6_power(np__2,B))),k6_real_1(B,np__2)) ) ) ) ).
fof(t91_asympt_1,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)
=> k2_seq_1(k5_numbers,k1_numbers,A,B) = k4_power(k3_real_1(np__1,k6_real_1(np__1,k1_nat_1(B,np__1))),k1_nat_1(B,np__1)) )
=> v3_seqm_3(A) ) ) ).
fof(t92_asympt_1,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ( r1_xreal_0(np__1,A)
=> r1_xreal_0(k4_power(k6_real_1(k1_nat_1(A,np__1),A),A),k4_power(k6_real_1(k1_nat_1(A,np__2),k1_nat_1(A,np__1)),k1_nat_1(A,np__1))) ) ) ).
fof(t93_asympt_1,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> ( r1_xreal_0(A,B)
=> r1_xreal_0(k6_real_1(k8_newton(A,k1_nat_1(B,np__1)),k1_nat_1(B,np__1)),k8_newton(A,B)) ) ) ) ).
fof(t94_asympt_1,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)
=> k2_seq_1(k5_numbers,k1_numbers,A,B) = k6_power(np__2,k11_newton(B)) )
=> ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> k2_seq_1(k5_numbers,k1_numbers,A,B) = k5_bhsp_4(k2_asympt_1,B) ) ) ) ).
fof(t95_asympt_1,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ( r1_xreal_0(np__4,A)
=> r1_xreal_0(k2_nat_1(np__2,A),k4_real_1(A,k6_power(np__2,A))) ) ) ).
fof(t96_asympt_1,axiom,
! [A] :
( ( v2_xreal_0(A)
& m1_subset_1(A,k1_numbers) )
=> ! [B] :
( ( v2_xreal_0(B)
& m1_subset_1(B,k1_numbers) )
=> ( k7_asympt_1(np__0,A,B) = np__0
& k7_asympt_1(np__1,A,B) = A
& ! [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
=> ~ ( r1_xreal_0(np__2,C)
& ! [D] :
( m2_subset_1(D,k1_numbers,k5_numbers)
=> ~ ( D = k2_int_1(k6_real_1(C,np__2))
& k7_asympt_1(C,A,B) = k3_real_1(k4_real_1(np__4,k7_asympt_1(D,A,B)),k4_real_1(B,C)) ) ) ) ) ) ) ) ).
fof(t97_asympt_1,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ~ ( r1_xreal_0(np__2,A)
& r1_xreal_0(k7_square_1(A),k1_nat_1(A,np__1)) ) ) ).
fof(t98_asympt_1,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ~ ( r1_xreal_0(np__1,A)
& r1_xreal_0(k5_real_1(k3_series_1(np__2,k1_nat_1(A,np__1)),k3_series_1(np__2,A)),np__1) ) ) ).
fof(t99_asympt_1,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ~ ( r1_xreal_0(np__2,A)
& r2_hidden(k5_real_1(k3_series_1(np__2,A),np__1),k9_asympt_1) ) ) ).
fof(t100_asympt_1,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> ( ( r1_xreal_0(np__1,B)
& r1_xreal_0(k11_newton(A),B) )
=> ( r1_xreal_0(k11_newton(k1_nat_1(A,np__1)),B)
| k10_asympt_1(B) = k11_newton(A) ) ) ) ) ).
fof(t101_asympt_1,axiom,
! [A] :
( m1_subset_1(A,k1_numbers)
=> ! [B] :
( m1_subset_1(B,k1_numbers)
=> ! [C] :
( m1_subset_1(C,k1_numbers)
=> ( ( r1_xreal_0(A,B)
& r1_xreal_0(np__1,C) )
=> ( r1_xreal_0(A,np__1)
| r1_xreal_0(k6_power(B,C),k6_power(A,C)) ) ) ) ) ) ).
fof(dt_k1_asympt_1,axiom,
! [A,B,C] :
( ( m1_subset_1(A,k1_numbers)
& m1_subset_1(B,k1_numbers)
& m1_subset_1(C,k1_numbers) )
=> ( v1_funct_1(k1_asympt_1(A,B,C))
& v1_funct_2(k1_asympt_1(A,B,C),k5_numbers,k1_numbers)
& m2_relset_1(k1_asympt_1(A,B,C),k5_numbers,k1_numbers) ) ) ).
fof(dt_k2_asympt_1,axiom,
( v1_funct_1(k2_asympt_1)
& v1_funct_2(k2_asympt_1,k5_numbers,k1_numbers)
& m2_relset_1(k2_asympt_1,k5_numbers,k1_numbers) ) ).
fof(dt_k3_asympt_1,axiom,
! [A] :
( m1_subset_1(A,k1_numbers)
=> ( v1_funct_1(k3_asympt_1(A))
& v1_funct_2(k3_asympt_1(A),k5_numbers,k1_numbers)
& m2_relset_1(k3_asympt_1(A),k5_numbers,k1_numbers) ) ) ).
fof(dt_k4_asympt_1,axiom,
! [A] :
( m1_subset_1(A,k1_numbers)
=> ( v1_funct_1(k4_asympt_1(A))
& v1_funct_2(k4_asympt_1(A),k5_numbers,k1_numbers)
& m2_relset_1(k4_asympt_1(A),k5_numbers,k1_numbers) ) ) ).
fof(dt_k5_asympt_1,axiom,
! [A] :
( m1_subset_1(A,k5_numbers)
=> ( v1_funct_1(k5_asympt_1(A))
& v1_funct_2(k5_asympt_1(A),k5_numbers,k1_numbers)
& m2_relset_1(k5_asympt_1(A),k5_numbers,k1_numbers) ) ) ).
fof(dt_k6_asympt_1,axiom,
! [A,B] :
( ( v1_funct_1(A)
& v1_funct_2(A,k5_numbers,k3_finseq_2(k1_numbers))
& m1_relset_1(A,k5_numbers,k3_finseq_2(k1_numbers))
& m1_subset_1(B,k5_numbers) )
=> m2_finseq_1(k6_asympt_1(A,B),k1_numbers) ) ).
fof(redefinition_k6_asympt_1,axiom,
! [A,B] :
( ( v1_funct_1(A)
& v1_funct_2(A,k5_numbers,k3_finseq_2(k1_numbers))
& m1_relset_1(A,k5_numbers,k3_finseq_2(k1_numbers))
& m1_subset_1(B,k5_numbers) )
=> k6_asympt_1(A,B) = k1_funct_1(A,B) ) ).
fof(dt_k7_asympt_1,axiom,
! [A,B,C] :
( ( m1_subset_1(A,k5_numbers)
& v2_xreal_0(B)
& m1_subset_1(B,k1_numbers)
& v2_xreal_0(C)
& m1_subset_1(C,k1_numbers) )
=> m1_subset_1(k7_asympt_1(A,B,C),k1_numbers) ) ).
fof(dt_k8_asympt_1,axiom,
! [A,B] :
( ( v2_xreal_0(A)
& m1_subset_1(A,k1_numbers)
& v2_xreal_0(B)
& m1_subset_1(B,k1_numbers) )
=> ( v1_funct_1(k8_asympt_1(A,B))
& v1_funct_2(k8_asympt_1(A,B),k5_numbers,k1_numbers)
& m2_relset_1(k8_asympt_1(A,B),k5_numbers,k1_numbers) ) ) ).
fof(dt_k9_asympt_1,axiom,
( ~ v1_xboole_0(k9_asympt_1)
& m1_subset_1(k9_asympt_1,k1_zfmisc_1(k5_numbers)) ) ).
fof(dt_k10_asympt_1,axiom,
! [A] :
( m1_subset_1(A,k5_numbers)
=> m2_subset_1(k10_asympt_1(A),k1_numbers,k5_numbers) ) ).
fof(d8_asympt_1,axiom,
k9_asympt_1 = a_0_0_asympt_1 ).
fof(fraenkel_a_0_0_asympt_1,axiom,
! [A] :
( r2_hidden(A,a_0_0_asympt_1)
<=> ? [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
& A = k3_series_1(np__2,B) ) ) ).
%------------------------------------------------------------------------------