SET007 Axioms: SET007+110.ax
%------------------------------------------------------------------------------
% File : SET007+110 : TPTP v9.0.0. Released v3.4.0.
% Domain : Set Theory
% Axioms : The Limit of a Real Function at Infinity
% 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 : limfunc1 [Urb08]
% Status : Satisfiable
% Syntax : Number of formulae : 175 ( 17 unt; 0 def)
% Number of atoms : 1490 ( 97 equ)
% Maximal formula atoms : 24 ( 8 avg)
% Number of connectives : 1466 ( 151 ~; 73 |; 674 &)
% ( 21 <=>; 547 =>; 0 <=; 0 <~>)
% Maximal formula depth : 17 ( 9 avg)
% Maximal term depth : 4 ( 1 avg)
% Number of predicates : 39 ( 37 usr; 1 prp; 0-3 aty)
% Number of functors : 53 ( 53 usr; 5 con; 0-4 aty)
% Number of variables : 482 ( 400 !; 82 ?)
% SPC :
% Comments : The individual reference can be found in [Mat90] by looking for
% the name provided by [Urb08].
% : Translated by MPTP from the Mizar Mathematical Library 4.48.930.
% : These set theory axioms are used in encodings of problems in
% various domains, including ALG, CAT, GRP, LAT, SET, and TOP.
%------------------------------------------------------------------------------
fof(t1_limfunc1,axiom,
$true ).
fof(t2_limfunc1,axiom,
$true ).
fof(t3_limfunc1,axiom,
$true ).
fof(t4_limfunc1,axiom,
$true ).
fof(t5_limfunc1,axiom,
$true ).
fof(t6_limfunc1,axiom,
$true ).
fof(t7_limfunc1,axiom,
$true ).
fof(t8_limfunc1,axiom,
! [A] :
( v1_xreal_0(A)
=> ! [B] :
( v1_xreal_0(B)
=> ( r1_xreal_0(A,B)
=> r1_tarski(k4_limfunc1(B),k4_limfunc1(A)) ) ) ) ).
fof(t9_limfunc1,axiom,
! [A] :
( v1_xreal_0(A)
=> ! [B] :
( v1_xreal_0(B)
=> ( r1_xreal_0(A,B)
=> r1_tarski(k3_limfunc1(B),k3_limfunc1(A)) ) ) ) ).
fof(t10_limfunc1,axiom,
! [A] :
( v1_xreal_0(A)
=> r1_tarski(k4_limfunc1(A),k3_limfunc1(A)) ) ).
fof(t11_limfunc1,axiom,
! [A] :
( v1_xreal_0(A)
=> ! [B] :
( v1_xreal_0(B)
=> r1_tarski(k2_rcomp_1(A,B),k4_limfunc1(A)) ) ) ).
fof(t12_limfunc1,axiom,
! [A] :
( v1_xreal_0(A)
=> ! [B] :
( v1_xreal_0(B)
=> r1_tarski(k1_rcomp_1(A,B),k3_limfunc1(A)) ) ) ).
fof(t13_limfunc1,axiom,
! [A] :
( v1_xreal_0(A)
=> ! [B] :
( v1_xreal_0(B)
=> ( r1_xreal_0(A,B)
=> r1_tarski(k12_prob_1(A),k12_prob_1(B)) ) ) ) ).
fof(t14_limfunc1,axiom,
! [A] :
( v1_xreal_0(A)
=> ! [B] :
( v1_xreal_0(B)
=> ( r1_xreal_0(A,B)
=> r1_tarski(k2_limfunc1(A),k2_limfunc1(B)) ) ) ) ).
fof(t15_limfunc1,axiom,
! [A] :
( v1_xreal_0(A)
=> r1_tarski(k12_prob_1(A),k2_limfunc1(A)) ) ).
fof(t16_limfunc1,axiom,
! [A] :
( v1_xreal_0(A)
=> ! [B] :
( v1_xreal_0(B)
=> r1_tarski(k2_rcomp_1(A,B),k12_prob_1(B)) ) ) ).
fof(t17_limfunc1,axiom,
! [A] :
( v1_xreal_0(A)
=> ! [B] :
( v1_xreal_0(B)
=> r1_tarski(k1_rcomp_1(A,B),k2_limfunc1(B)) ) ) ).
fof(t18_limfunc1,axiom,
! [A] :
( v1_xreal_0(A)
=> ! [B] :
( v1_xreal_0(B)
=> k5_subset_1(k1_numbers,k12_prob_1(A),k4_limfunc1(B)) = k2_rcomp_1(B,A) ) ) ).
fof(t19_limfunc1,axiom,
! [A] :
( v1_xreal_0(A)
=> ! [B] :
( v1_xreal_0(B)
=> k5_subset_1(k1_numbers,k2_limfunc1(A),k3_limfunc1(B)) = k1_rcomp_1(B,A) ) ) ).
fof(t20_limfunc1,axiom,
! [A] :
( v1_xreal_0(A)
=> ! [B] :
( v1_xreal_0(B)
=> ! [C] :
( v1_xreal_0(C)
=> ( r1_xreal_0(A,B)
=> ( r1_tarski(k2_rcomp_1(B,C),k4_limfunc1(A))
& r1_tarski(k1_rcomp_1(B,C),k3_limfunc1(A)) ) ) ) ) ) ).
fof(t21_limfunc1,axiom,
! [A] :
( v1_xreal_0(A)
=> ! [B] :
( v1_xreal_0(B)
=> ! [C] :
( v1_xreal_0(C)
=> ( ~ r1_xreal_0(B,A)
=> r1_tarski(k1_rcomp_1(B,C),k4_limfunc1(A)) ) ) ) ) ).
fof(t22_limfunc1,axiom,
! [A] :
( v1_xreal_0(A)
=> ! [B] :
( v1_xreal_0(B)
=> ! [C] :
( v1_xreal_0(C)
=> ( r1_xreal_0(A,B)
=> ( r1_tarski(k2_rcomp_1(C,A),k12_prob_1(B))
& r1_tarski(k1_rcomp_1(C,A),k2_limfunc1(B)) ) ) ) ) ) ).
fof(t23_limfunc1,axiom,
! [A] :
( v1_xreal_0(A)
=> ! [B] :
( v1_xreal_0(B)
=> ! [C] :
( v1_xreal_0(C)
=> ( ~ r1_xreal_0(B,A)
=> r1_tarski(k1_rcomp_1(C,A),k12_prob_1(B)) ) ) ) ) ).
fof(t24_limfunc1,axiom,
! [A] :
( v1_xreal_0(A)
=> ( k4_xboole_0(k1_numbers,k4_limfunc1(A)) = k2_limfunc1(A)
& k4_xboole_0(k1_numbers,k3_limfunc1(A)) = k12_prob_1(A)
& k4_xboole_0(k1_numbers,k12_prob_1(A)) = k3_limfunc1(A)
& k4_xboole_0(k1_numbers,k2_limfunc1(A)) = k4_limfunc1(A) ) ) ).
fof(t25_limfunc1,axiom,
! [A] :
( v1_xreal_0(A)
=> ! [B] :
( v1_xreal_0(B)
=> ( k4_xboole_0(k1_numbers,k2_rcomp_1(A,B)) = k4_subset_1(k1_numbers,k2_limfunc1(A),k3_limfunc1(B))
& k4_xboole_0(k1_numbers,k1_rcomp_1(A,B)) = k4_subset_1(k1_numbers,k12_prob_1(A),k4_limfunc1(B)) ) ) ) ).
fof(t26_limfunc1,axiom,
! [A] :
( ( v1_funct_1(A)
& v1_funct_2(A,k5_numbers,k1_numbers)
& m2_relset_1(A,k5_numbers,k1_numbers) )
=> ( ( v3_seqm_3(A)
=> v2_seq_2(A) )
& ( v4_seqm_3(A)
=> v1_seq_2(A) ) ) ) ).
fof(t27_limfunc1,axiom,
! [A] :
( ( v1_funct_1(A)
& v1_funct_2(A,k5_numbers,k1_numbers)
& m2_relset_1(A,k5_numbers,k1_numbers) )
=> ( ( v2_relat_1(A)
& v4_seq_2(A)
& k2_seq_2(A) = np__0
& v3_seqm_3(A) )
=> ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> ~ r1_xreal_0(np__0,k2_seq_1(k5_numbers,k1_numbers,A,B)) ) ) ) ).
fof(t28_limfunc1,axiom,
! [A] :
( ( v1_funct_1(A)
& v1_funct_2(A,k5_numbers,k1_numbers)
& m2_relset_1(A,k5_numbers,k1_numbers) )
=> ( ( v2_relat_1(A)
& v4_seq_2(A)
& k2_seq_2(A) = np__0
& v4_seqm_3(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(t29_limfunc1,axiom,
! [A] :
( ( v1_funct_1(A)
& v1_funct_2(A,k5_numbers,k1_numbers)
& m2_relset_1(A,k5_numbers,k1_numbers) )
=> ~ ( v4_seq_2(A)
& ~ r1_xreal_0(k2_seq_2(A),np__0)
& ! [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(t30_limfunc1,axiom,
! [A] :
( ( v1_funct_1(A)
& v1_funct_2(A,k5_numbers,k1_numbers)
& m2_relset_1(A,k5_numbers,k1_numbers) )
=> ~ ( v4_seq_2(A)
& ~ r1_xreal_0(k2_seq_2(A),np__0)
& ! [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),k6_real_1(k2_seq_2(A),np__2)) ) ) ) ) ).
fof(d4_limfunc1,axiom,
! [A] :
( ( v1_funct_1(A)
& v1_funct_2(A,k5_numbers,k1_numbers)
& m2_relset_1(A,k5_numbers,k1_numbers) )
=> ( v1_limfunc1(A)
<=> ! [B] :
( m1_subset_1(B,k1_numbers)
=> ? [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
& ! [D] :
( m2_subset_1(D,k1_numbers,k5_numbers)
=> ~ ( r1_xreal_0(C,D)
& r1_xreal_0(k2_seq_1(k5_numbers,k1_numbers,A,D),B) ) ) ) ) ) ) ).
fof(d5_limfunc1,axiom,
! [A] :
( ( v1_funct_1(A)
& v1_funct_2(A,k5_numbers,k1_numbers)
& m2_relset_1(A,k5_numbers,k1_numbers) )
=> ( v2_limfunc1(A)
<=> ! [B] :
( m1_subset_1(B,k1_numbers)
=> ? [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
& ! [D] :
( m2_subset_1(D,k1_numbers,k5_numbers)
=> ~ ( r1_xreal_0(C,D)
& r1_xreal_0(B,k2_seq_1(k5_numbers,k1_numbers,A,D)) ) ) ) ) ) ) ).
fof(t31_limfunc1,axiom,
$true ).
fof(t32_limfunc1,axiom,
$true ).
fof(t33_limfunc1,axiom,
! [A] :
( ( v1_funct_1(A)
& v1_funct_2(A,k5_numbers,k1_numbers)
& m2_relset_1(A,k5_numbers,k1_numbers) )
=> ~ ( ( v1_limfunc1(A)
| v2_limfunc1(A) )
& ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> ? [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
& r1_xreal_0(B,C)
& ~ v2_relat_1(k1_seqm_3(A,C)) ) ) ) ) ).
fof(t34_limfunc1,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) )
=> ( ( v1_limfunc1(k1_seqm_3(B,A))
=> v1_limfunc1(B) )
& ( v2_limfunc1(k1_seqm_3(B,A))
=> v2_limfunc1(B) ) ) ) ) ).
fof(t35_limfunc1,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) )
=> ( ( v1_limfunc1(A)
& v1_limfunc1(B) )
=> v1_limfunc1(k9_seq_1(A,B)) ) ) ) ).
fof(t36_limfunc1,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) )
=> ( ( v1_limfunc1(A)
& v2_seq_2(B) )
=> v1_limfunc1(k9_seq_1(A,B)) ) ) ) ).
fof(t37_limfunc1,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) )
=> ( ( v1_limfunc1(A)
& v1_limfunc1(B) )
=> v1_limfunc1(k11_seq_1(A,B)) ) ) ) ).
fof(t38_limfunc1,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) )
=> ( ( v2_limfunc1(A)
& v2_limfunc1(B) )
=> v2_limfunc1(k9_seq_1(A,B)) ) ) ) ).
fof(t39_limfunc1,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) )
=> ( ( v2_limfunc1(A)
& v1_seq_2(B) )
=> v2_limfunc1(k9_seq_1(A,B)) ) ) ) ).
fof(t40_limfunc1,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)
=> ( ( v1_limfunc1(A)
=> ( r1_xreal_0(B,np__0)
| v1_limfunc1(k14_seq_1(A,B)) ) )
& ( v1_limfunc1(A)
=> ( r1_xreal_0(np__0,B)
| v2_limfunc1(k14_seq_1(A,B)) ) )
& ( ( v1_limfunc1(A)
& B = np__0 )
=> ( k1_rfunct_2(k14_seq_1(A,B)) = k1_tarski(np__0)
& v5_seqm_3(k14_seq_1(A,B)) ) ) ) ) ) ).
fof(t41_limfunc1,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)
=> ( ( v2_limfunc1(A)
=> ( r1_xreal_0(B,np__0)
| v2_limfunc1(k14_seq_1(A,B)) ) )
& ( v2_limfunc1(A)
=> ( r1_xreal_0(np__0,B)
| v1_limfunc1(k14_seq_1(A,B)) ) )
& ( ( v2_limfunc1(A)
& B = np__0 )
=> ( k1_rfunct_2(k14_seq_1(A,B)) = k1_tarski(np__0)
& v5_seqm_3(k14_seq_1(A,B)) ) ) ) ) ) ).
fof(t42_limfunc1,axiom,
! [A] :
( ( v1_funct_1(A)
& v1_funct_2(A,k5_numbers,k1_numbers)
& m2_relset_1(A,k5_numbers,k1_numbers) )
=> ( ( v1_limfunc1(A)
=> v2_limfunc1(k17_seq_1(A)) )
& ( v2_limfunc1(A)
=> v1_limfunc1(k17_seq_1(A)) ) ) ) ).
fof(t43_limfunc1,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) )
=> ( ( v2_seq_2(A)
& v2_limfunc1(B) )
=> v1_limfunc1(k10_seq_1(A,B)) ) ) ) ).
fof(t44_limfunc1,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) )
=> ( ( v1_seq_2(A)
& v1_limfunc1(B) )
=> v2_limfunc1(k10_seq_1(A,B)) ) ) ) ).
fof(t45_limfunc1,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) )
=> ( ( v1_limfunc1(A)
& v4_seq_2(B) )
=> v1_limfunc1(k9_seq_1(A,B)) ) ) ) ).
fof(t46_limfunc1,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) )
=> ( ( v2_limfunc1(A)
& v4_seq_2(B) )
=> v2_limfunc1(k9_seq_1(A,B)) ) ) ) ).
fof(t47_limfunc1,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) = B )
=> v1_limfunc1(A) ) ) ).
fof(t48_limfunc1,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) = k1_real_1(B) )
=> v2_limfunc1(A) ) ) ).
fof(t49_limfunc1,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) )
=> ( v1_limfunc1(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(C,k2_seq_1(k5_numbers,k1_numbers,B,D)) ) ) )
| v1_limfunc1(k11_seq_1(A,B)) ) ) ) ) ).
fof(t50_limfunc1,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) )
=> ( v2_limfunc1(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(C,k2_seq_1(k5_numbers,k1_numbers,B,D)) ) ) )
| v2_limfunc1(k11_seq_1(A,B)) ) ) ) ) ).
fof(t51_limfunc1,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) )
=> ( ( v2_limfunc1(A)
& v2_limfunc1(B) )
=> v1_limfunc1(k11_seq_1(A,B)) ) ) ) ).
fof(t52_limfunc1,axiom,
! [A] :
( ( v1_funct_1(A)
& v1_funct_2(A,k5_numbers,k1_numbers)
& m2_relset_1(A,k5_numbers,k1_numbers) )
=> ( ( v1_limfunc1(A)
| v2_limfunc1(A) )
=> v1_limfunc1(k22_seq_1(A)) ) ) ).
fof(t53_limfunc1,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) )
=> ( ( v1_limfunc1(A)
& m1_seqm_3(B,A) )
=> v1_limfunc1(B) ) ) ) ).
fof(t54_limfunc1,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) )
=> ( ( v2_limfunc1(A)
& m1_seqm_3(B,A) )
=> v2_limfunc1(B) ) ) ) ).
fof(t55_limfunc1,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) )
=> ( ( v1_limfunc1(A)
& v4_seq_2(B) )
=> ( r1_xreal_0(k2_seq_2(B),np__0)
| v1_limfunc1(k11_seq_1(A,B)) ) ) ) ) ).
fof(t56_limfunc1,axiom,
! [A] :
( ( v1_funct_1(A)
& v1_funct_2(A,k5_numbers,k1_numbers)
& m2_relset_1(A,k5_numbers,k1_numbers) )
=> ( v3_seqm_3(A)
=> ( v1_seq_2(A)
| v1_limfunc1(A) ) ) ) ).
fof(t57_limfunc1,axiom,
! [A] :
( ( v1_funct_1(A)
& v1_funct_2(A,k5_numbers,k1_numbers)
& m2_relset_1(A,k5_numbers,k1_numbers) )
=> ( v4_seqm_3(A)
=> ( v2_seq_2(A)
| v2_limfunc1(A) ) ) ) ).
fof(t58_limfunc1,axiom,
! [A] :
( ( v1_funct_1(A)
& v1_funct_2(A,k5_numbers,k1_numbers)
& m2_relset_1(A,k5_numbers,k1_numbers) )
=> ( v1_seqm_3(A)
=> ( v1_seq_2(A)
| v1_limfunc1(A) ) ) ) ).
fof(t59_limfunc1,axiom,
! [A] :
( ( v1_funct_1(A)
& v1_funct_2(A,k5_numbers,k1_numbers)
& m2_relset_1(A,k5_numbers,k1_numbers) )
=> ( v2_seqm_3(A)
=> ( v2_seq_2(A)
| v2_limfunc1(A) ) ) ) ).
fof(t60_limfunc1,axiom,
! [A] :
( ( v1_funct_1(A)
& v1_funct_2(A,k5_numbers,k1_numbers)
& m2_relset_1(A,k5_numbers,k1_numbers) )
=> ~ ( v6_seqm_3(A)
& ~ v4_seq_2(A)
& ~ v1_limfunc1(A)
& ~ v2_limfunc1(A) ) ) ).
fof(t61_limfunc1,axiom,
! [A] :
( ( v1_funct_1(A)
& v1_funct_2(A,k5_numbers,k1_numbers)
& m2_relset_1(A,k5_numbers,k1_numbers) )
=> ( ( v1_limfunc1(A)
| v2_limfunc1(A) )
=> ( v4_seq_2(k18_seq_1(A))
& k2_seq_2(k18_seq_1(A)) = np__0 ) ) ) ).
fof(t62_limfunc1,axiom,
! [A] :
( ( v1_funct_1(A)
& v1_funct_2(A,k5_numbers,k1_numbers)
& m2_relset_1(A,k5_numbers,k1_numbers) )
=> ( ( v2_relat_1(A)
& v4_seq_2(A)
& k2_seq_2(A) = np__0 )
=> ( ! [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) ) )
| v1_limfunc1(k18_seq_1(A)) ) ) ) ).
fof(t63_limfunc1,axiom,
! [A] :
( ( v1_funct_1(A)
& v1_funct_2(A,k5_numbers,k1_numbers)
& m2_relset_1(A,k5_numbers,k1_numbers) )
=> ( ( v2_relat_1(A)
& v4_seq_2(A)
& k2_seq_2(A) = np__0 )
=> ( ! [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)) ) )
| v2_limfunc1(k18_seq_1(A)) ) ) ) ).
fof(t64_limfunc1,axiom,
! [A] :
( ( v1_funct_1(A)
& v1_funct_2(A,k5_numbers,k1_numbers)
& m2_relset_1(A,k5_numbers,k1_numbers) )
=> ( ( v2_relat_1(A)
& v4_seq_2(A)
& k2_seq_2(A) = np__0
& v3_seqm_3(A) )
=> v2_limfunc1(k18_seq_1(A)) ) ) ).
fof(t65_limfunc1,axiom,
! [A] :
( ( v1_funct_1(A)
& v1_funct_2(A,k5_numbers,k1_numbers)
& m2_relset_1(A,k5_numbers,k1_numbers) )
=> ( ( v2_relat_1(A)
& v4_seq_2(A)
& k2_seq_2(A) = np__0
& v4_seqm_3(A) )
=> v1_limfunc1(k18_seq_1(A)) ) ) ).
fof(t66_limfunc1,axiom,
! [A] :
( ( v1_funct_1(A)
& v1_funct_2(A,k5_numbers,k1_numbers)
& m2_relset_1(A,k5_numbers,k1_numbers) )
=> ( ( v2_relat_1(A)
& v4_seq_2(A)
& k2_seq_2(A) = np__0
& v1_seqm_3(A) )
=> v2_limfunc1(k18_seq_1(A)) ) ) ).
fof(t67_limfunc1,axiom,
! [A] :
( ( v1_funct_1(A)
& v1_funct_2(A,k5_numbers,k1_numbers)
& m2_relset_1(A,k5_numbers,k1_numbers) )
=> ( ( v2_relat_1(A)
& v4_seq_2(A)
& k2_seq_2(A) = np__0
& v2_seqm_3(A) )
=> v1_limfunc1(k18_seq_1(A)) ) ) ).
fof(t68_limfunc1,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) )
=> ( ( v3_seq_2(A)
& v2_relat_1(B) )
=> ( ( ~ v1_limfunc1(B)
& ~ v2_limfunc1(B) )
| ( v4_seq_2(k19_seq_1(A,B))
& k2_seq_2(k19_seq_1(A,B)) = np__0 ) ) ) ) ) ).
fof(t69_limfunc1,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) )
=> ( ( v1_limfunc1(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,B,C)) ) )
=> v1_limfunc1(B) ) ) ) ).
fof(t70_limfunc1,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) )
=> ( ( v2_limfunc1(A)
& ! [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
=> r1_xreal_0(k2_seq_1(k5_numbers,k1_numbers,B,C),k2_seq_1(k5_numbers,k1_numbers,A,C)) ) )
=> v2_limfunc1(B) ) ) ) ).
fof(d6_limfunc1,axiom,
! [A] :
( ( v1_funct_1(A)
& m2_relset_1(A,k1_numbers,k1_numbers) )
=> ( v3_limfunc1(A)
<=> ( ! [B] :
( m1_subset_1(B,k1_numbers)
=> ? [C] :
( m1_subset_1(C,k1_numbers)
& ~ r1_xreal_0(C,B)
& r2_hidden(C,k4_relset_1(k1_numbers,k1_numbers,A)) ) )
& ? [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) )
=> ( ( v1_limfunc1(C)
& r1_tarski(k1_rfunct_2(C),k4_relset_1(k1_numbers,k1_numbers,A)) )
=> ( v4_seq_2(k2_rfunct_2(A,C))
& k2_seq_2(k2_rfunct_2(A,C)) = B ) ) ) ) ) ) ) ).
fof(d7_limfunc1,axiom,
! [A] :
( ( v1_funct_1(A)
& m2_relset_1(A,k1_numbers,k1_numbers) )
=> ( v4_limfunc1(A)
<=> ( ! [B] :
( m1_subset_1(B,k1_numbers)
=> ? [C] :
( m1_subset_1(C,k1_numbers)
& ~ r1_xreal_0(C,B)
& r2_hidden(C,k4_relset_1(k1_numbers,k1_numbers,A)) ) )
& ! [B] :
( ( v1_funct_1(B)
& v1_funct_2(B,k5_numbers,k1_numbers)
& m2_relset_1(B,k5_numbers,k1_numbers) )
=> ( ( v1_limfunc1(B)
& r1_tarski(k1_rfunct_2(B),k4_relset_1(k1_numbers,k1_numbers,A)) )
=> v1_limfunc1(k2_rfunct_2(A,B)) ) ) ) ) ) ).
fof(d8_limfunc1,axiom,
! [A] :
( ( v1_funct_1(A)
& m2_relset_1(A,k1_numbers,k1_numbers) )
=> ( v5_limfunc1(A)
<=> ( ! [B] :
( m1_subset_1(B,k1_numbers)
=> ? [C] :
( m1_subset_1(C,k1_numbers)
& ~ r1_xreal_0(C,B)
& r2_hidden(C,k4_relset_1(k1_numbers,k1_numbers,A)) ) )
& ! [B] :
( ( v1_funct_1(B)
& v1_funct_2(B,k5_numbers,k1_numbers)
& m2_relset_1(B,k5_numbers,k1_numbers) )
=> ( ( v1_limfunc1(B)
& r1_tarski(k1_rfunct_2(B),k4_relset_1(k1_numbers,k1_numbers,A)) )
=> v2_limfunc1(k2_rfunct_2(A,B)) ) ) ) ) ) ).
fof(d9_limfunc1,axiom,
! [A] :
( ( v1_funct_1(A)
& m2_relset_1(A,k1_numbers,k1_numbers) )
=> ( v6_limfunc1(A)
<=> ( ! [B] :
( m1_subset_1(B,k1_numbers)
=> ? [C] :
( m1_subset_1(C,k1_numbers)
& ~ r1_xreal_0(B,C)
& r2_hidden(C,k4_relset_1(k1_numbers,k1_numbers,A)) ) )
& ? [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) )
=> ( ( v2_limfunc1(C)
& r1_tarski(k1_rfunct_2(C),k4_relset_1(k1_numbers,k1_numbers,A)) )
=> ( v4_seq_2(k2_rfunct_2(A,C))
& k2_seq_2(k2_rfunct_2(A,C)) = B ) ) ) ) ) ) ) ).
fof(d10_limfunc1,axiom,
! [A] :
( ( v1_funct_1(A)
& m2_relset_1(A,k1_numbers,k1_numbers) )
=> ( v7_limfunc1(A)
<=> ( ! [B] :
( m1_subset_1(B,k1_numbers)
=> ? [C] :
( m1_subset_1(C,k1_numbers)
& ~ r1_xreal_0(B,C)
& r2_hidden(C,k4_relset_1(k1_numbers,k1_numbers,A)) ) )
& ! [B] :
( ( v1_funct_1(B)
& v1_funct_2(B,k5_numbers,k1_numbers)
& m2_relset_1(B,k5_numbers,k1_numbers) )
=> ( ( v2_limfunc1(B)
& r1_tarski(k1_rfunct_2(B),k4_relset_1(k1_numbers,k1_numbers,A)) )
=> v1_limfunc1(k2_rfunct_2(A,B)) ) ) ) ) ) ).
fof(d11_limfunc1,axiom,
! [A] :
( ( v1_funct_1(A)
& m2_relset_1(A,k1_numbers,k1_numbers) )
=> ( v8_limfunc1(A)
<=> ( ! [B] :
( m1_subset_1(B,k1_numbers)
=> ? [C] :
( m1_subset_1(C,k1_numbers)
& ~ r1_xreal_0(B,C)
& r2_hidden(C,k4_relset_1(k1_numbers,k1_numbers,A)) ) )
& ! [B] :
( ( v1_funct_1(B)
& v1_funct_2(B,k5_numbers,k1_numbers)
& m2_relset_1(B,k5_numbers,k1_numbers) )
=> ( ( v2_limfunc1(B)
& r1_tarski(k1_rfunct_2(B),k4_relset_1(k1_numbers,k1_numbers,A)) )
=> v2_limfunc1(k2_rfunct_2(A,B)) ) ) ) ) ) ).
fof(t71_limfunc1,axiom,
$true ).
fof(t72_limfunc1,axiom,
$true ).
fof(t73_limfunc1,axiom,
$true ).
fof(t74_limfunc1,axiom,
$true ).
fof(t75_limfunc1,axiom,
$true ).
fof(t76_limfunc1,axiom,
$true ).
fof(t77_limfunc1,axiom,
! [A] :
( ( v1_funct_1(A)
& m2_relset_1(A,k1_numbers,k1_numbers) )
=> ( v3_limfunc1(A)
<=> ( ! [B] :
( m1_subset_1(B,k1_numbers)
=> ? [C] :
( m1_subset_1(C,k1_numbers)
& ~ r1_xreal_0(C,B)
& r2_hidden(C,k4_relset_1(k1_numbers,k1_numbers,A)) ) )
& ? [B] :
( m1_subset_1(B,k1_numbers)
& ! [C] :
( m1_subset_1(C,k1_numbers)
=> ~ ( ~ r1_xreal_0(C,np__0)
& ! [D] :
( m1_subset_1(D,k1_numbers)
=> ? [E] :
( m1_subset_1(E,k1_numbers)
& ~ r1_xreal_0(E,D)
& r2_hidden(E,k4_relset_1(k1_numbers,k1_numbers,A))
& r1_xreal_0(C,k18_complex1(k5_real_1(k2_seq_1(k1_numbers,k1_numbers,A,E),B))) ) ) ) ) ) ) ) ) ).
fof(t78_limfunc1,axiom,
! [A] :
( ( v1_funct_1(A)
& m2_relset_1(A,k1_numbers,k1_numbers) )
=> ( v6_limfunc1(A)
<=> ( ! [B] :
( m1_subset_1(B,k1_numbers)
=> ? [C] :
( m1_subset_1(C,k1_numbers)
& ~ r1_xreal_0(B,C)
& r2_hidden(C,k4_relset_1(k1_numbers,k1_numbers,A)) ) )
& ? [B] :
( m1_subset_1(B,k1_numbers)
& ! [C] :
( m1_subset_1(C,k1_numbers)
=> ~ ( ~ r1_xreal_0(C,np__0)
& ! [D] :
( m1_subset_1(D,k1_numbers)
=> ? [E] :
( m1_subset_1(E,k1_numbers)
& ~ r1_xreal_0(D,E)
& r2_hidden(E,k4_relset_1(k1_numbers,k1_numbers,A))
& r1_xreal_0(C,k18_complex1(k5_real_1(k2_seq_1(k1_numbers,k1_numbers,A,E),B))) ) ) ) ) ) ) ) ) ).
fof(t79_limfunc1,axiom,
! [A] :
( ( v1_funct_1(A)
& m2_relset_1(A,k1_numbers,k1_numbers) )
=> ( v4_limfunc1(A)
<=> ( ! [B] :
( m1_subset_1(B,k1_numbers)
=> ? [C] :
( m1_subset_1(C,k1_numbers)
& ~ r1_xreal_0(C,B)
& r2_hidden(C,k4_relset_1(k1_numbers,k1_numbers,A)) ) )
& ! [B] :
( m1_subset_1(B,k1_numbers)
=> ? [C] :
( m1_subset_1(C,k1_numbers)
& ! [D] :
( m1_subset_1(D,k1_numbers)
=> ~ ( ~ r1_xreal_0(D,C)
& r2_hidden(D,k4_relset_1(k1_numbers,k1_numbers,A))
& r1_xreal_0(k2_seq_1(k1_numbers,k1_numbers,A,D),B) ) ) ) ) ) ) ) ).
fof(t80_limfunc1,axiom,
! [A] :
( ( v1_funct_1(A)
& m2_relset_1(A,k1_numbers,k1_numbers) )
=> ( v5_limfunc1(A)
<=> ( ! [B] :
( m1_subset_1(B,k1_numbers)
=> ? [C] :
( m1_subset_1(C,k1_numbers)
& ~ r1_xreal_0(C,B)
& r2_hidden(C,k4_relset_1(k1_numbers,k1_numbers,A)) ) )
& ! [B] :
( m1_subset_1(B,k1_numbers)
=> ? [C] :
( m1_subset_1(C,k1_numbers)
& ! [D] :
( m1_subset_1(D,k1_numbers)
=> ~ ( ~ r1_xreal_0(D,C)
& r2_hidden(D,k4_relset_1(k1_numbers,k1_numbers,A))
& r1_xreal_0(B,k2_seq_1(k1_numbers,k1_numbers,A,D)) ) ) ) ) ) ) ) ).
fof(t81_limfunc1,axiom,
! [A] :
( ( v1_funct_1(A)
& m2_relset_1(A,k1_numbers,k1_numbers) )
=> ( v7_limfunc1(A)
<=> ( ! [B] :
( m1_subset_1(B,k1_numbers)
=> ? [C] :
( m1_subset_1(C,k1_numbers)
& ~ r1_xreal_0(B,C)
& r2_hidden(C,k4_relset_1(k1_numbers,k1_numbers,A)) ) )
& ! [B] :
( m1_subset_1(B,k1_numbers)
=> ? [C] :
( m1_subset_1(C,k1_numbers)
& ! [D] :
( m1_subset_1(D,k1_numbers)
=> ~ ( ~ r1_xreal_0(C,D)
& r2_hidden(D,k4_relset_1(k1_numbers,k1_numbers,A))
& r1_xreal_0(k2_seq_1(k1_numbers,k1_numbers,A,D),B) ) ) ) ) ) ) ) ).
fof(t82_limfunc1,axiom,
! [A] :
( ( v1_funct_1(A)
& m2_relset_1(A,k1_numbers,k1_numbers) )
=> ( v8_limfunc1(A)
<=> ( ! [B] :
( m1_subset_1(B,k1_numbers)
=> ? [C] :
( m1_subset_1(C,k1_numbers)
& ~ r1_xreal_0(B,C)
& r2_hidden(C,k4_relset_1(k1_numbers,k1_numbers,A)) ) )
& ! [B] :
( m1_subset_1(B,k1_numbers)
=> ? [C] :
( m1_subset_1(C,k1_numbers)
& ! [D] :
( m1_subset_1(D,k1_numbers)
=> ~ ( ~ r1_xreal_0(C,D)
& r2_hidden(D,k4_relset_1(k1_numbers,k1_numbers,A))
& r1_xreal_0(B,k2_seq_1(k1_numbers,k1_numbers,A,D)) ) ) ) ) ) ) ) ).
fof(t83_limfunc1,axiom,
! [A] :
( ( v1_funct_1(A)
& m2_relset_1(A,k1_numbers,k1_numbers) )
=> ! [B] :
( ( v1_funct_1(B)
& m2_relset_1(B,k1_numbers,k1_numbers) )
=> ( ( v4_limfunc1(A)
& v4_limfunc1(B)
& ! [C] :
( m1_subset_1(C,k1_numbers)
=> ? [D] :
( m1_subset_1(D,k1_numbers)
& ~ r1_xreal_0(D,C)
& r2_hidden(D,k5_subset_1(k1_numbers,k4_relset_1(k1_numbers,k1_numbers,A),k4_relset_1(k1_numbers,k1_numbers,B))) ) ) )
=> ( v4_limfunc1(k6_seq_1(k1_numbers,k1_numbers,A,B))
& v4_limfunc1(k8_seq_1(k1_numbers,k1_numbers,A,B)) ) ) ) ) ).
fof(t84_limfunc1,axiom,
! [A] :
( ( v1_funct_1(A)
& m2_relset_1(A,k1_numbers,k1_numbers) )
=> ! [B] :
( ( v1_funct_1(B)
& m2_relset_1(B,k1_numbers,k1_numbers) )
=> ( ( v5_limfunc1(A)
& v5_limfunc1(B)
& ! [C] :
( m1_subset_1(C,k1_numbers)
=> ? [D] :
( m1_subset_1(D,k1_numbers)
& ~ r1_xreal_0(D,C)
& r2_hidden(D,k5_subset_1(k1_numbers,k4_relset_1(k1_numbers,k1_numbers,A),k4_relset_1(k1_numbers,k1_numbers,B))) ) ) )
=> ( v5_limfunc1(k6_seq_1(k1_numbers,k1_numbers,A,B))
& v4_limfunc1(k8_seq_1(k1_numbers,k1_numbers,A,B)) ) ) ) ) ).
fof(t85_limfunc1,axiom,
! [A] :
( ( v1_funct_1(A)
& m2_relset_1(A,k1_numbers,k1_numbers) )
=> ! [B] :
( ( v1_funct_1(B)
& m2_relset_1(B,k1_numbers,k1_numbers) )
=> ( ( v7_limfunc1(A)
& v7_limfunc1(B)
& ! [C] :
( m1_subset_1(C,k1_numbers)
=> ? [D] :
( m1_subset_1(D,k1_numbers)
& ~ r1_xreal_0(C,D)
& r2_hidden(D,k5_subset_1(k1_numbers,k4_relset_1(k1_numbers,k1_numbers,A),k4_relset_1(k1_numbers,k1_numbers,B))) ) ) )
=> ( v7_limfunc1(k6_seq_1(k1_numbers,k1_numbers,A,B))
& v7_limfunc1(k8_seq_1(k1_numbers,k1_numbers,A,B)) ) ) ) ) ).
fof(t86_limfunc1,axiom,
! [A] :
( ( v1_funct_1(A)
& m2_relset_1(A,k1_numbers,k1_numbers) )
=> ! [B] :
( ( v1_funct_1(B)
& m2_relset_1(B,k1_numbers,k1_numbers) )
=> ( ( v8_limfunc1(A)
& v8_limfunc1(B)
& ! [C] :
( m1_subset_1(C,k1_numbers)
=> ? [D] :
( m1_subset_1(D,k1_numbers)
& ~ r1_xreal_0(C,D)
& r2_hidden(D,k5_subset_1(k1_numbers,k4_relset_1(k1_numbers,k1_numbers,A),k4_relset_1(k1_numbers,k1_numbers,B))) ) ) )
=> ( v8_limfunc1(k6_seq_1(k1_numbers,k1_numbers,A,B))
& v7_limfunc1(k8_seq_1(k1_numbers,k1_numbers,A,B)) ) ) ) ) ).
fof(t87_limfunc1,axiom,
! [A] :
( ( v1_funct_1(A)
& m2_relset_1(A,k1_numbers,k1_numbers) )
=> ! [B] :
( ( v1_funct_1(B)
& m2_relset_1(B,k1_numbers,k1_numbers) )
=> ( ( v4_limfunc1(A)
& ! [C] :
( m1_subset_1(C,k1_numbers)
=> ? [D] :
( m1_subset_1(D,k1_numbers)
& ~ r1_xreal_0(D,C)
& r2_hidden(D,k4_relset_1(k1_numbers,k1_numbers,k6_seq_1(k1_numbers,k1_numbers,A,B))) ) ) )
=> ( ! [C] :
( m1_subset_1(C,k1_numbers)
=> ~ r2_rfunct_1(B,k4_limfunc1(C)) )
| v4_limfunc1(k6_seq_1(k1_numbers,k1_numbers,A,B)) ) ) ) ) ).
fof(t88_limfunc1,axiom,
! [A] :
( ( v1_funct_1(A)
& m2_relset_1(A,k1_numbers,k1_numbers) )
=> ! [B] :
( ( v1_funct_1(B)
& m2_relset_1(B,k1_numbers,k1_numbers) )
=> ( ( v4_limfunc1(A)
& ! [C] :
( m1_subset_1(C,k1_numbers)
=> ? [D] :
( m1_subset_1(D,k1_numbers)
& ~ r1_xreal_0(D,C)
& r2_hidden(D,k4_relset_1(k1_numbers,k1_numbers,k8_seq_1(k1_numbers,k1_numbers,A,B))) ) ) )
=> ( ! [C] :
( m1_subset_1(C,k1_numbers)
=> ! [D] :
( m1_subset_1(D,k1_numbers)
=> ~ ( ~ r1_xreal_0(C,np__0)
& ! [E] :
( m1_subset_1(E,k1_numbers)
=> ( r2_hidden(E,k5_subset_1(k1_numbers,k4_relset_1(k1_numbers,k1_numbers,B),k4_limfunc1(D)))
=> r1_xreal_0(C,k2_seq_1(k1_numbers,k1_numbers,B,E)) ) ) ) ) )
| v4_limfunc1(k8_seq_1(k1_numbers,k1_numbers,A,B)) ) ) ) ) ).
fof(t89_limfunc1,axiom,
! [A] :
( ( v1_funct_1(A)
& m2_relset_1(A,k1_numbers,k1_numbers) )
=> ! [B] :
( ( v1_funct_1(B)
& m2_relset_1(B,k1_numbers,k1_numbers) )
=> ( ( v7_limfunc1(A)
& ! [C] :
( m1_subset_1(C,k1_numbers)
=> ? [D] :
( m1_subset_1(D,k1_numbers)
& ~ r1_xreal_0(C,D)
& r2_hidden(D,k4_relset_1(k1_numbers,k1_numbers,k6_seq_1(k1_numbers,k1_numbers,A,B))) ) ) )
=> ( ! [C] :
( m1_subset_1(C,k1_numbers)
=> ~ r2_rfunct_1(B,k12_prob_1(C)) )
| v7_limfunc1(k6_seq_1(k1_numbers,k1_numbers,A,B)) ) ) ) ) ).
fof(t90_limfunc1,axiom,
! [A] :
( ( v1_funct_1(A)
& m2_relset_1(A,k1_numbers,k1_numbers) )
=> ! [B] :
( ( v1_funct_1(B)
& m2_relset_1(B,k1_numbers,k1_numbers) )
=> ( ( v7_limfunc1(A)
& ! [C] :
( m1_subset_1(C,k1_numbers)
=> ? [D] :
( m1_subset_1(D,k1_numbers)
& ~ r1_xreal_0(C,D)
& r2_hidden(D,k4_relset_1(k1_numbers,k1_numbers,k8_seq_1(k1_numbers,k1_numbers,A,B))) ) ) )
=> ( ! [C] :
( m1_subset_1(C,k1_numbers)
=> ! [D] :
( m1_subset_1(D,k1_numbers)
=> ~ ( ~ r1_xreal_0(C,np__0)
& ! [E] :
( m1_subset_1(E,k1_numbers)
=> ( r2_hidden(E,k5_subset_1(k1_numbers,k4_relset_1(k1_numbers,k1_numbers,B),k12_prob_1(D)))
=> r1_xreal_0(C,k2_seq_1(k1_numbers,k1_numbers,B,E)) ) ) ) ) )
| v7_limfunc1(k8_seq_1(k1_numbers,k1_numbers,A,B)) ) ) ) ) ).
fof(t91_limfunc1,axiom,
! [A] :
( ( v1_funct_1(A)
& m2_relset_1(A,k1_numbers,k1_numbers) )
=> ! [B] :
( m1_subset_1(B,k1_numbers)
=> ( ( v4_limfunc1(A)
=> ( r1_xreal_0(B,np__0)
| v4_limfunc1(k13_seq_1(k1_numbers,k1_numbers,A,B)) ) )
& ( v4_limfunc1(A)
=> ( r1_xreal_0(np__0,B)
| v5_limfunc1(k13_seq_1(k1_numbers,k1_numbers,A,B)) ) )
& ( v5_limfunc1(A)
=> ( r1_xreal_0(B,np__0)
| v5_limfunc1(k13_seq_1(k1_numbers,k1_numbers,A,B)) ) )
& ( v5_limfunc1(A)
=> ( r1_xreal_0(np__0,B)
| v4_limfunc1(k13_seq_1(k1_numbers,k1_numbers,A,B)) ) ) ) ) ) ).
fof(t92_limfunc1,axiom,
! [A] :
( ( v1_funct_1(A)
& m2_relset_1(A,k1_numbers,k1_numbers) )
=> ! [B] :
( m1_subset_1(B,k1_numbers)
=> ( ( v7_limfunc1(A)
=> ( r1_xreal_0(B,np__0)
| v7_limfunc1(k13_seq_1(k1_numbers,k1_numbers,A,B)) ) )
& ( v7_limfunc1(A)
=> ( r1_xreal_0(np__0,B)
| v8_limfunc1(k13_seq_1(k1_numbers,k1_numbers,A,B)) ) )
& ( v8_limfunc1(A)
=> ( r1_xreal_0(B,np__0)
| v8_limfunc1(k13_seq_1(k1_numbers,k1_numbers,A,B)) ) )
& ( v8_limfunc1(A)
=> ( r1_xreal_0(np__0,B)
| v7_limfunc1(k13_seq_1(k1_numbers,k1_numbers,A,B)) ) ) ) ) ) ).
fof(t93_limfunc1,axiom,
! [A] :
( ( v1_funct_1(A)
& m2_relset_1(A,k1_numbers,k1_numbers) )
=> ( ( v4_limfunc1(A)
| v5_limfunc1(A) )
=> v4_limfunc1(k21_seq_1(k1_numbers,k1_numbers,A)) ) ) ).
fof(t94_limfunc1,axiom,
! [A] :
( ( v1_funct_1(A)
& m2_relset_1(A,k1_numbers,k1_numbers) )
=> ( ( v7_limfunc1(A)
| v8_limfunc1(A) )
=> v7_limfunc1(k21_seq_1(k1_numbers,k1_numbers,A)) ) ) ).
fof(t95_limfunc1,axiom,
! [A] :
( ( v1_funct_1(A)
& m2_relset_1(A,k1_numbers,k1_numbers) )
=> ( ! [B] :
( m1_subset_1(B,k1_numbers)
=> ? [C] :
( m1_subset_1(C,k1_numbers)
& ~ r1_xreal_0(C,B)
& r2_hidden(C,k4_relset_1(k1_numbers,k1_numbers,A)) ) )
=> ( ! [B] :
( m1_subset_1(B,k1_numbers)
=> ~ ( r3_rfunct_2(A,k4_limfunc1(B))
& ~ r1_rfunct_1(A,k4_limfunc1(B)) ) )
| v4_limfunc1(A) ) ) ) ).
fof(t96_limfunc1,axiom,
! [A] :
( ( v1_funct_1(A)
& m2_relset_1(A,k1_numbers,k1_numbers) )
=> ( ! [B] :
( m1_subset_1(B,k1_numbers)
=> ? [C] :
( m1_subset_1(C,k1_numbers)
& ~ r1_xreal_0(C,B)
& r2_hidden(C,k4_relset_1(k1_numbers,k1_numbers,A)) ) )
=> ( ! [B] :
( m1_subset_1(B,k1_numbers)
=> ~ ( r1_rfunct_2(A,k4_limfunc1(B))
& ~ r1_rfunct_1(A,k4_limfunc1(B)) ) )
| v4_limfunc1(A) ) ) ) ).
fof(t97_limfunc1,axiom,
! [A] :
( ( v1_funct_1(A)
& m2_relset_1(A,k1_numbers,k1_numbers) )
=> ( ! [B] :
( m1_subset_1(B,k1_numbers)
=> ? [C] :
( m1_subset_1(C,k1_numbers)
& ~ r1_xreal_0(C,B)
& r2_hidden(C,k4_relset_1(k1_numbers,k1_numbers,A)) ) )
=> ( ! [B] :
( m1_subset_1(B,k1_numbers)
=> ~ ( r4_rfunct_2(A,k4_limfunc1(B))
& ~ r2_rfunct_1(A,k4_limfunc1(B)) ) )
| v5_limfunc1(A) ) ) ) ).
fof(t98_limfunc1,axiom,
! [A] :
( ( v1_funct_1(A)
& m2_relset_1(A,k1_numbers,k1_numbers) )
=> ( ! [B] :
( m1_subset_1(B,k1_numbers)
=> ? [C] :
( m1_subset_1(C,k1_numbers)
& ~ r1_xreal_0(C,B)
& r2_hidden(C,k4_relset_1(k1_numbers,k1_numbers,A)) ) )
=> ( ! [B] :
( m1_subset_1(B,k1_numbers)
=> ~ ( r2_rfunct_2(A,k4_limfunc1(B))
& ~ r2_rfunct_1(A,k4_limfunc1(B)) ) )
| v5_limfunc1(A) ) ) ) ).
fof(t99_limfunc1,axiom,
! [A] :
( ( v1_funct_1(A)
& m2_relset_1(A,k1_numbers,k1_numbers) )
=> ( ! [B] :
( m1_subset_1(B,k1_numbers)
=> ? [C] :
( m1_subset_1(C,k1_numbers)
& ~ r1_xreal_0(B,C)
& r2_hidden(C,k4_relset_1(k1_numbers,k1_numbers,A)) ) )
=> ( ! [B] :
( m1_subset_1(B,k1_numbers)
=> ~ ( r4_rfunct_2(A,k12_prob_1(B))
& ~ r1_rfunct_1(A,k12_prob_1(B)) ) )
| v7_limfunc1(A) ) ) ) ).
fof(t100_limfunc1,axiom,
! [A] :
( ( v1_funct_1(A)
& m2_relset_1(A,k1_numbers,k1_numbers) )
=> ( ! [B] :
( m1_subset_1(B,k1_numbers)
=> ? [C] :
( m1_subset_1(C,k1_numbers)
& ~ r1_xreal_0(B,C)
& r2_hidden(C,k4_relset_1(k1_numbers,k1_numbers,A)) ) )
=> ( ! [B] :
( m1_subset_1(B,k1_numbers)
=> ~ ( r2_rfunct_2(A,k12_prob_1(B))
& ~ r1_rfunct_1(A,k12_prob_1(B)) ) )
| v7_limfunc1(A) ) ) ) ).
fof(t101_limfunc1,axiom,
! [A] :
( ( v1_funct_1(A)
& m2_relset_1(A,k1_numbers,k1_numbers) )
=> ( ! [B] :
( m1_subset_1(B,k1_numbers)
=> ? [C] :
( m1_subset_1(C,k1_numbers)
& ~ r1_xreal_0(B,C)
& r2_hidden(C,k4_relset_1(k1_numbers,k1_numbers,A)) ) )
=> ( ! [B] :
( m1_subset_1(B,k1_numbers)
=> ~ ( r3_rfunct_2(A,k12_prob_1(B))
& ~ r2_rfunct_1(A,k12_prob_1(B)) ) )
| v8_limfunc1(A) ) ) ) ).
fof(t102_limfunc1,axiom,
! [A] :
( ( v1_funct_1(A)
& m2_relset_1(A,k1_numbers,k1_numbers) )
=> ( ! [B] :
( m1_subset_1(B,k1_numbers)
=> ? [C] :
( m1_subset_1(C,k1_numbers)
& ~ r1_xreal_0(B,C)
& r2_hidden(C,k4_relset_1(k1_numbers,k1_numbers,A)) ) )
=> ( ! [B] :
( m1_subset_1(B,k1_numbers)
=> ~ ( r1_rfunct_2(A,k12_prob_1(B))
& ~ r2_rfunct_1(A,k12_prob_1(B)) ) )
| v8_limfunc1(A) ) ) ) ).
fof(t103_limfunc1,axiom,
! [A] :
( ( v1_funct_1(A)
& m2_relset_1(A,k1_numbers,k1_numbers) )
=> ! [B] :
( ( v1_funct_1(B)
& m2_relset_1(B,k1_numbers,k1_numbers) )
=> ( ( v4_limfunc1(A)
& ! [C] :
( m1_subset_1(C,k1_numbers)
=> ? [D] :
( m1_subset_1(D,k1_numbers)
& ~ r1_xreal_0(D,C)
& r2_hidden(D,k4_relset_1(k1_numbers,k1_numbers,B)) ) ) )
=> ( ! [C] :
( m1_subset_1(C,k1_numbers)
=> ~ ( r1_tarski(k5_subset_1(k1_numbers,k4_relset_1(k1_numbers,k1_numbers,B),k4_limfunc1(C)),k5_subset_1(k1_numbers,k4_relset_1(k1_numbers,k1_numbers,A),k4_limfunc1(C)))
& ! [D] :
( m1_subset_1(D,k1_numbers)
=> ( r2_hidden(D,k5_subset_1(k1_numbers,k4_relset_1(k1_numbers,k1_numbers,B),k4_limfunc1(C)))
=> r1_xreal_0(k2_seq_1(k1_numbers,k1_numbers,A,D),k2_seq_1(k1_numbers,k1_numbers,B,D)) ) ) ) )
| v4_limfunc1(B) ) ) ) ) ).
fof(t104_limfunc1,axiom,
! [A] :
( ( v1_funct_1(A)
& m2_relset_1(A,k1_numbers,k1_numbers) )
=> ! [B] :
( ( v1_funct_1(B)
& m2_relset_1(B,k1_numbers,k1_numbers) )
=> ( ( v5_limfunc1(A)
& ! [C] :
( m1_subset_1(C,k1_numbers)
=> ? [D] :
( m1_subset_1(D,k1_numbers)
& ~ r1_xreal_0(D,C)
& r2_hidden(D,k4_relset_1(k1_numbers,k1_numbers,B)) ) ) )
=> ( ! [C] :
( m1_subset_1(C,k1_numbers)
=> ~ ( r1_tarski(k5_subset_1(k1_numbers,k4_relset_1(k1_numbers,k1_numbers,B),k4_limfunc1(C)),k5_subset_1(k1_numbers,k4_relset_1(k1_numbers,k1_numbers,A),k4_limfunc1(C)))
& ! [D] :
( m1_subset_1(D,k1_numbers)
=> ( r2_hidden(D,k5_subset_1(k1_numbers,k4_relset_1(k1_numbers,k1_numbers,B),k4_limfunc1(C)))
=> r1_xreal_0(k2_seq_1(k1_numbers,k1_numbers,B,D),k2_seq_1(k1_numbers,k1_numbers,A,D)) ) ) ) )
| v5_limfunc1(B) ) ) ) ) ).
fof(t105_limfunc1,axiom,
! [A] :
( ( v1_funct_1(A)
& m2_relset_1(A,k1_numbers,k1_numbers) )
=> ! [B] :
( ( v1_funct_1(B)
& m2_relset_1(B,k1_numbers,k1_numbers) )
=> ( ( v7_limfunc1(A)
& ! [C] :
( m1_subset_1(C,k1_numbers)
=> ? [D] :
( m1_subset_1(D,k1_numbers)
& ~ r1_xreal_0(C,D)
& r2_hidden(D,k4_relset_1(k1_numbers,k1_numbers,B)) ) ) )
=> ( ! [C] :
( m1_subset_1(C,k1_numbers)
=> ~ ( r1_tarski(k5_subset_1(k1_numbers,k4_relset_1(k1_numbers,k1_numbers,B),k12_prob_1(C)),k5_subset_1(k1_numbers,k4_relset_1(k1_numbers,k1_numbers,A),k12_prob_1(C)))
& ! [D] :
( m1_subset_1(D,k1_numbers)
=> ( r2_hidden(D,k5_subset_1(k1_numbers,k4_relset_1(k1_numbers,k1_numbers,B),k12_prob_1(C)))
=> r1_xreal_0(k2_seq_1(k1_numbers,k1_numbers,A,D),k2_seq_1(k1_numbers,k1_numbers,B,D)) ) ) ) )
| v7_limfunc1(B) ) ) ) ) ).
fof(t106_limfunc1,axiom,
! [A] :
( ( v1_funct_1(A)
& m2_relset_1(A,k1_numbers,k1_numbers) )
=> ! [B] :
( ( v1_funct_1(B)
& m2_relset_1(B,k1_numbers,k1_numbers) )
=> ( ( v8_limfunc1(A)
& ! [C] :
( m1_subset_1(C,k1_numbers)
=> ? [D] :
( m1_subset_1(D,k1_numbers)
& ~ r1_xreal_0(C,D)
& r2_hidden(D,k4_relset_1(k1_numbers,k1_numbers,B)) ) ) )
=> ( ! [C] :
( m1_subset_1(C,k1_numbers)
=> ~ ( r1_tarski(k5_subset_1(k1_numbers,k4_relset_1(k1_numbers,k1_numbers,B),k12_prob_1(C)),k5_subset_1(k1_numbers,k4_relset_1(k1_numbers,k1_numbers,A),k12_prob_1(C)))
& ! [D] :
( m1_subset_1(D,k1_numbers)
=> ( r2_hidden(D,k5_subset_1(k1_numbers,k4_relset_1(k1_numbers,k1_numbers,B),k12_prob_1(C)))
=> r1_xreal_0(k2_seq_1(k1_numbers,k1_numbers,B,D),k2_seq_1(k1_numbers,k1_numbers,A,D)) ) ) ) )
| v8_limfunc1(B) ) ) ) ) ).
fof(t107_limfunc1,axiom,
! [A] :
( ( v1_funct_1(A)
& m2_relset_1(A,k1_numbers,k1_numbers) )
=> ! [B] :
( ( v1_funct_1(B)
& m2_relset_1(B,k1_numbers,k1_numbers) )
=> ( v4_limfunc1(A)
=> ( ! [C] :
( m1_subset_1(C,k1_numbers)
=> ~ ( r1_tarski(k4_limfunc1(C),k5_subset_1(k1_numbers,k4_relset_1(k1_numbers,k1_numbers,B),k4_relset_1(k1_numbers,k1_numbers,A)))
& ! [D] :
( m1_subset_1(D,k1_numbers)
=> ( r2_hidden(D,k4_limfunc1(C))
=> r1_xreal_0(k2_seq_1(k1_numbers,k1_numbers,A,D),k2_seq_1(k1_numbers,k1_numbers,B,D)) ) ) ) )
| v4_limfunc1(B) ) ) ) ) ).
fof(t108_limfunc1,axiom,
! [A] :
( ( v1_funct_1(A)
& m2_relset_1(A,k1_numbers,k1_numbers) )
=> ! [B] :
( ( v1_funct_1(B)
& m2_relset_1(B,k1_numbers,k1_numbers) )
=> ( v5_limfunc1(A)
=> ( ! [C] :
( m1_subset_1(C,k1_numbers)
=> ~ ( r1_tarski(k4_limfunc1(C),k5_subset_1(k1_numbers,k4_relset_1(k1_numbers,k1_numbers,B),k4_relset_1(k1_numbers,k1_numbers,A)))
& ! [D] :
( m1_subset_1(D,k1_numbers)
=> ( r2_hidden(D,k4_limfunc1(C))
=> r1_xreal_0(k2_seq_1(k1_numbers,k1_numbers,B,D),k2_seq_1(k1_numbers,k1_numbers,A,D)) ) ) ) )
| v5_limfunc1(B) ) ) ) ) ).
fof(t109_limfunc1,axiom,
! [A] :
( ( v1_funct_1(A)
& m2_relset_1(A,k1_numbers,k1_numbers) )
=> ! [B] :
( ( v1_funct_1(B)
& m2_relset_1(B,k1_numbers,k1_numbers) )
=> ( v7_limfunc1(A)
=> ( ! [C] :
( m1_subset_1(C,k1_numbers)
=> ~ ( r1_tarski(k12_prob_1(C),k5_subset_1(k1_numbers,k4_relset_1(k1_numbers,k1_numbers,B),k4_relset_1(k1_numbers,k1_numbers,A)))
& ! [D] :
( m1_subset_1(D,k1_numbers)
=> ( r2_hidden(D,k12_prob_1(C))
=> r1_xreal_0(k2_seq_1(k1_numbers,k1_numbers,A,D),k2_seq_1(k1_numbers,k1_numbers,B,D)) ) ) ) )
| v7_limfunc1(B) ) ) ) ) ).
fof(t110_limfunc1,axiom,
! [A] :
( ( v1_funct_1(A)
& m2_relset_1(A,k1_numbers,k1_numbers) )
=> ! [B] :
( ( v1_funct_1(B)
& m2_relset_1(B,k1_numbers,k1_numbers) )
=> ( v8_limfunc1(A)
=> ( ! [C] :
( m1_subset_1(C,k1_numbers)
=> ~ ( r1_tarski(k12_prob_1(C),k5_subset_1(k1_numbers,k4_relset_1(k1_numbers,k1_numbers,B),k4_relset_1(k1_numbers,k1_numbers,A)))
& ! [D] :
( m1_subset_1(D,k1_numbers)
=> ( r2_hidden(D,k12_prob_1(C))
=> r1_xreal_0(k2_seq_1(k1_numbers,k1_numbers,B,D),k2_seq_1(k1_numbers,k1_numbers,A,D)) ) ) ) )
| v8_limfunc1(B) ) ) ) ) ).
fof(d12_limfunc1,axiom,
! [A] :
( ( v1_funct_1(A)
& m2_relset_1(A,k1_numbers,k1_numbers) )
=> ( v3_limfunc1(A)
=> ! [B] :
( m1_subset_1(B,k1_numbers)
=> ( B = k5_limfunc1(A)
<=> ! [C] :
( ( v1_funct_1(C)
& v1_funct_2(C,k5_numbers,k1_numbers)
& m2_relset_1(C,k5_numbers,k1_numbers) )
=> ( ( v1_limfunc1(C)
& r1_tarski(k1_rfunct_2(C),k4_relset_1(k1_numbers,k1_numbers,A)) )
=> ( v4_seq_2(k2_rfunct_2(A,C))
& k2_seq_2(k2_rfunct_2(A,C)) = B ) ) ) ) ) ) ) ).
fof(d13_limfunc1,axiom,
! [A] :
( ( v1_funct_1(A)
& m2_relset_1(A,k1_numbers,k1_numbers) )
=> ( v6_limfunc1(A)
=> ! [B] :
( m1_subset_1(B,k1_numbers)
=> ( B = k6_limfunc1(A)
<=> ! [C] :
( ( v1_funct_1(C)
& v1_funct_2(C,k5_numbers,k1_numbers)
& m2_relset_1(C,k5_numbers,k1_numbers) )
=> ( ( v2_limfunc1(C)
& r1_tarski(k1_rfunct_2(C),k4_relset_1(k1_numbers,k1_numbers,A)) )
=> ( v4_seq_2(k2_rfunct_2(A,C))
& k2_seq_2(k2_rfunct_2(A,C)) = B ) ) ) ) ) ) ) ).
fof(t111_limfunc1,axiom,
$true ).
fof(t112_limfunc1,axiom,
$true ).
fof(t113_limfunc1,axiom,
! [A] :
( ( v1_funct_1(A)
& m2_relset_1(A,k1_numbers,k1_numbers) )
=> ! [B] :
( m1_subset_1(B,k1_numbers)
=> ( v6_limfunc1(A)
=> ( k6_limfunc1(A) = B
<=> ! [C] :
( m1_subset_1(C,k1_numbers)
=> ~ ( ~ r1_xreal_0(C,np__0)
& ! [D] :
( m1_subset_1(D,k1_numbers)
=> ? [E] :
( m1_subset_1(E,k1_numbers)
& ~ r1_xreal_0(D,E)
& r2_hidden(E,k4_relset_1(k1_numbers,k1_numbers,A))
& r1_xreal_0(C,k18_complex1(k5_real_1(k2_seq_1(k1_numbers,k1_numbers,A,E),B))) ) ) ) ) ) ) ) ) ).
fof(t114_limfunc1,axiom,
! [A] :
( ( v1_funct_1(A)
& m2_relset_1(A,k1_numbers,k1_numbers) )
=> ! [B] :
( m1_subset_1(B,k1_numbers)
=> ( v3_limfunc1(A)
=> ( k5_limfunc1(A) = B
<=> ! [C] :
( m1_subset_1(C,k1_numbers)
=> ~ ( ~ r1_xreal_0(C,np__0)
& ! [D] :
( m1_subset_1(D,k1_numbers)
=> ? [E] :
( m1_subset_1(E,k1_numbers)
& ~ r1_xreal_0(E,D)
& r2_hidden(E,k4_relset_1(k1_numbers,k1_numbers,A))
& r1_xreal_0(C,k18_complex1(k5_real_1(k2_seq_1(k1_numbers,k1_numbers,A,E),B))) ) ) ) ) ) ) ) ) ).
fof(t115_limfunc1,axiom,
! [A] :
( ( v1_funct_1(A)
& m2_relset_1(A,k1_numbers,k1_numbers) )
=> ! [B] :
( m1_subset_1(B,k1_numbers)
=> ( v3_limfunc1(A)
=> ( v3_limfunc1(k13_seq_1(k1_numbers,k1_numbers,A,B))
& k5_limfunc1(k13_seq_1(k1_numbers,k1_numbers,A,B)) = k4_real_1(B,k5_limfunc1(A)) ) ) ) ) ).
fof(t116_limfunc1,axiom,
! [A] :
( ( v1_funct_1(A)
& m2_relset_1(A,k1_numbers,k1_numbers) )
=> ( v3_limfunc1(A)
=> ( v3_limfunc1(k16_seq_1(k1_numbers,k1_numbers,A))
& k5_limfunc1(k16_seq_1(k1_numbers,k1_numbers,A)) = k1_real_1(k5_limfunc1(A)) ) ) ) ).
fof(t117_limfunc1,axiom,
! [A] :
( ( v1_funct_1(A)
& m2_relset_1(A,k1_numbers,k1_numbers) )
=> ! [B] :
( ( v1_funct_1(B)
& m2_relset_1(B,k1_numbers,k1_numbers) )
=> ( ( v3_limfunc1(A)
& v3_limfunc1(B)
& ! [C] :
( m1_subset_1(C,k1_numbers)
=> ? [D] :
( m1_subset_1(D,k1_numbers)
& ~ r1_xreal_0(D,C)
& r2_hidden(D,k4_relset_1(k1_numbers,k1_numbers,k6_seq_1(k1_numbers,k1_numbers,A,B))) ) ) )
=> ( v3_limfunc1(k6_seq_1(k1_numbers,k1_numbers,A,B))
& k5_limfunc1(k6_seq_1(k1_numbers,k1_numbers,A,B)) = k3_real_1(k5_limfunc1(A),k5_limfunc1(B)) ) ) ) ) ).
fof(t118_limfunc1,axiom,
! [A] :
( ( v1_funct_1(A)
& m2_relset_1(A,k1_numbers,k1_numbers) )
=> ! [B] :
( ( v1_funct_1(B)
& m2_relset_1(B,k1_numbers,k1_numbers) )
=> ( ( v3_limfunc1(A)
& v3_limfunc1(B)
& ! [C] :
( m1_subset_1(C,k1_numbers)
=> ? [D] :
( m1_subset_1(D,k1_numbers)
& ~ r1_xreal_0(D,C)
& r2_hidden(D,k4_relset_1(k1_numbers,k1_numbers,k7_seq_1(k1_numbers,k1_numbers,A,B))) ) ) )
=> ( v3_limfunc1(k7_seq_1(k1_numbers,k1_numbers,A,B))
& k5_limfunc1(k7_seq_1(k1_numbers,k1_numbers,A,B)) = k5_real_1(k5_limfunc1(A),k5_limfunc1(B)) ) ) ) ) ).
fof(t119_limfunc1,axiom,
! [A] :
( ( v1_funct_1(A)
& m2_relset_1(A,k1_numbers,k1_numbers) )
=> ( ( v3_limfunc1(A)
& k11_relset_1(k1_numbers,k1_numbers,A,k1_tarski(np__0)) = k1_xboole_0 )
=> ( k5_limfunc1(A) = np__0
| ( v3_limfunc1(k4_rfunct_1(k1_numbers,k1_numbers,A))
& k5_limfunc1(k4_rfunct_1(k1_numbers,k1_numbers,A)) = k2_real_1(k5_limfunc1(A)) ) ) ) ) ).
fof(t120_limfunc1,axiom,
! [A] :
( ( v1_funct_1(A)
& m2_relset_1(A,k1_numbers,k1_numbers) )
=> ( v3_limfunc1(A)
=> ( v3_limfunc1(k21_seq_1(k1_numbers,k1_numbers,A))
& k5_limfunc1(k21_seq_1(k1_numbers,k1_numbers,A)) = k18_complex1(k5_limfunc1(A)) ) ) ) ).
fof(t121_limfunc1,axiom,
! [A] :
( ( v1_funct_1(A)
& m2_relset_1(A,k1_numbers,k1_numbers) )
=> ( ( v3_limfunc1(A)
& ! [B] :
( m1_subset_1(B,k1_numbers)
=> ? [C] :
( m1_subset_1(C,k1_numbers)
& ~ r1_xreal_0(C,B)
& r2_hidden(C,k4_relset_1(k1_numbers,k1_numbers,A))
& k2_seq_1(k1_numbers,k1_numbers,A,C) != np__0 ) ) )
=> ( k5_limfunc1(A) = np__0
| ( v3_limfunc1(k4_rfunct_1(k1_numbers,k1_numbers,A))
& k5_limfunc1(k4_rfunct_1(k1_numbers,k1_numbers,A)) = k2_real_1(k5_limfunc1(A)) ) ) ) ) ).
fof(t122_limfunc1,axiom,
! [A] :
( ( v1_funct_1(A)
& m2_relset_1(A,k1_numbers,k1_numbers) )
=> ! [B] :
( ( v1_funct_1(B)
& m2_relset_1(B,k1_numbers,k1_numbers) )
=> ( ( v3_limfunc1(A)
& v3_limfunc1(B)
& ! [C] :
( m1_subset_1(C,k1_numbers)
=> ? [D] :
( m1_subset_1(D,k1_numbers)
& ~ r1_xreal_0(D,C)
& r2_hidden(D,k4_relset_1(k1_numbers,k1_numbers,k8_seq_1(k1_numbers,k1_numbers,A,B))) ) ) )
=> ( v3_limfunc1(k8_seq_1(k1_numbers,k1_numbers,A,B))
& k5_limfunc1(k8_seq_1(k1_numbers,k1_numbers,A,B)) = k4_real_1(k5_limfunc1(A),k5_limfunc1(B)) ) ) ) ) ).
fof(t123_limfunc1,axiom,
! [A] :
( ( v1_funct_1(A)
& m2_relset_1(A,k1_numbers,k1_numbers) )
=> ! [B] :
( ( v1_funct_1(B)
& m2_relset_1(B,k1_numbers,k1_numbers) )
=> ( ( v3_limfunc1(A)
& v3_limfunc1(B)
& ! [C] :
( m1_subset_1(C,k1_numbers)
=> ? [D] :
( m1_subset_1(D,k1_numbers)
& ~ r1_xreal_0(D,C)
& r2_hidden(D,k4_relset_1(k1_numbers,k1_numbers,k2_rfunct_1(k1_numbers,k1_numbers,A,B))) ) ) )
=> ( k5_limfunc1(B) = np__0
| ( v3_limfunc1(k2_rfunct_1(k1_numbers,k1_numbers,A,B))
& k5_limfunc1(k2_rfunct_1(k1_numbers,k1_numbers,A,B)) = k6_real_1(k5_limfunc1(A),k5_limfunc1(B)) ) ) ) ) ) ).
fof(t124_limfunc1,axiom,
! [A] :
( ( v1_funct_1(A)
& m2_relset_1(A,k1_numbers,k1_numbers) )
=> ! [B] :
( m1_subset_1(B,k1_numbers)
=> ( v6_limfunc1(A)
=> ( v6_limfunc1(k13_seq_1(k1_numbers,k1_numbers,A,B))
& k6_limfunc1(k13_seq_1(k1_numbers,k1_numbers,A,B)) = k4_real_1(B,k6_limfunc1(A)) ) ) ) ) ).
fof(t125_limfunc1,axiom,
! [A] :
( ( v1_funct_1(A)
& m2_relset_1(A,k1_numbers,k1_numbers) )
=> ( v6_limfunc1(A)
=> ( v6_limfunc1(k16_seq_1(k1_numbers,k1_numbers,A))
& k6_limfunc1(k16_seq_1(k1_numbers,k1_numbers,A)) = k1_real_1(k6_limfunc1(A)) ) ) ) ).
fof(t126_limfunc1,axiom,
! [A] :
( ( v1_funct_1(A)
& m2_relset_1(A,k1_numbers,k1_numbers) )
=> ! [B] :
( ( v1_funct_1(B)
& m2_relset_1(B,k1_numbers,k1_numbers) )
=> ( ( v6_limfunc1(A)
& v6_limfunc1(B)
& ! [C] :
( m1_subset_1(C,k1_numbers)
=> ? [D] :
( m1_subset_1(D,k1_numbers)
& ~ r1_xreal_0(C,D)
& r2_hidden(D,k4_relset_1(k1_numbers,k1_numbers,k6_seq_1(k1_numbers,k1_numbers,A,B))) ) ) )
=> ( v6_limfunc1(k6_seq_1(k1_numbers,k1_numbers,A,B))
& k6_limfunc1(k6_seq_1(k1_numbers,k1_numbers,A,B)) = k3_real_1(k6_limfunc1(A),k6_limfunc1(B)) ) ) ) ) ).
fof(t127_limfunc1,axiom,
! [A] :
( ( v1_funct_1(A)
& m2_relset_1(A,k1_numbers,k1_numbers) )
=> ! [B] :
( ( v1_funct_1(B)
& m2_relset_1(B,k1_numbers,k1_numbers) )
=> ( ( v6_limfunc1(A)
& v6_limfunc1(B)
& ! [C] :
( m1_subset_1(C,k1_numbers)
=> ? [D] :
( m1_subset_1(D,k1_numbers)
& ~ r1_xreal_0(C,D)
& r2_hidden(D,k4_relset_1(k1_numbers,k1_numbers,k7_seq_1(k1_numbers,k1_numbers,A,B))) ) ) )
=> ( v6_limfunc1(k7_seq_1(k1_numbers,k1_numbers,A,B))
& k6_limfunc1(k7_seq_1(k1_numbers,k1_numbers,A,B)) = k5_real_1(k6_limfunc1(A),k6_limfunc1(B)) ) ) ) ) ).
fof(t128_limfunc1,axiom,
! [A] :
( ( v1_funct_1(A)
& m2_relset_1(A,k1_numbers,k1_numbers) )
=> ( ( v6_limfunc1(A)
& k11_relset_1(k1_numbers,k1_numbers,A,k1_tarski(np__0)) = k1_xboole_0 )
=> ( k6_limfunc1(A) = np__0
| ( v6_limfunc1(k4_rfunct_1(k1_numbers,k1_numbers,A))
& k6_limfunc1(k4_rfunct_1(k1_numbers,k1_numbers,A)) = k2_real_1(k6_limfunc1(A)) ) ) ) ) ).
fof(t129_limfunc1,axiom,
! [A] :
( ( v1_funct_1(A)
& m2_relset_1(A,k1_numbers,k1_numbers) )
=> ( v6_limfunc1(A)
=> ( v6_limfunc1(k21_seq_1(k1_numbers,k1_numbers,A))
& k6_limfunc1(k21_seq_1(k1_numbers,k1_numbers,A)) = k18_complex1(k6_limfunc1(A)) ) ) ) ).
fof(t130_limfunc1,axiom,
! [A] :
( ( v1_funct_1(A)
& m2_relset_1(A,k1_numbers,k1_numbers) )
=> ( ( v6_limfunc1(A)
& ! [B] :
( m1_subset_1(B,k1_numbers)
=> ? [C] :
( m1_subset_1(C,k1_numbers)
& ~ r1_xreal_0(B,C)
& r2_hidden(C,k4_relset_1(k1_numbers,k1_numbers,A))
& k2_seq_1(k1_numbers,k1_numbers,A,C) != np__0 ) ) )
=> ( k6_limfunc1(A) = np__0
| ( v6_limfunc1(k4_rfunct_1(k1_numbers,k1_numbers,A))
& k6_limfunc1(k4_rfunct_1(k1_numbers,k1_numbers,A)) = k2_real_1(k6_limfunc1(A)) ) ) ) ) ).
fof(t131_limfunc1,axiom,
! [A] :
( ( v1_funct_1(A)
& m2_relset_1(A,k1_numbers,k1_numbers) )
=> ! [B] :
( ( v1_funct_1(B)
& m2_relset_1(B,k1_numbers,k1_numbers) )
=> ( ( v6_limfunc1(A)
& v6_limfunc1(B)
& ! [C] :
( m1_subset_1(C,k1_numbers)
=> ? [D] :
( m1_subset_1(D,k1_numbers)
& ~ r1_xreal_0(C,D)
& r2_hidden(D,k4_relset_1(k1_numbers,k1_numbers,k8_seq_1(k1_numbers,k1_numbers,A,B))) ) ) )
=> ( v6_limfunc1(k8_seq_1(k1_numbers,k1_numbers,A,B))
& k6_limfunc1(k8_seq_1(k1_numbers,k1_numbers,A,B)) = k4_real_1(k6_limfunc1(A),k6_limfunc1(B)) ) ) ) ) ).
fof(t132_limfunc1,axiom,
! [A] :
( ( v1_funct_1(A)
& m2_relset_1(A,k1_numbers,k1_numbers) )
=> ! [B] :
( ( v1_funct_1(B)
& m2_relset_1(B,k1_numbers,k1_numbers) )
=> ( ( v6_limfunc1(A)
& v6_limfunc1(B)
& ! [C] :
( m1_subset_1(C,k1_numbers)
=> ? [D] :
( m1_subset_1(D,k1_numbers)
& ~ r1_xreal_0(C,D)
& r2_hidden(D,k4_relset_1(k1_numbers,k1_numbers,k2_rfunct_1(k1_numbers,k1_numbers,A,B))) ) ) )
=> ( k6_limfunc1(B) = np__0
| ( v6_limfunc1(k2_rfunct_1(k1_numbers,k1_numbers,A,B))
& k6_limfunc1(k2_rfunct_1(k1_numbers,k1_numbers,A,B)) = k6_real_1(k6_limfunc1(A),k6_limfunc1(B)) ) ) ) ) ) ).
fof(t133_limfunc1,axiom,
! [A] :
( ( v1_funct_1(A)
& m2_relset_1(A,k1_numbers,k1_numbers) )
=> ! [B] :
( ( v1_funct_1(B)
& m2_relset_1(B,k1_numbers,k1_numbers) )
=> ( ( v3_limfunc1(A)
& k5_limfunc1(A) = np__0
& ! [C] :
( m1_subset_1(C,k1_numbers)
=> ? [D] :
( m1_subset_1(D,k1_numbers)
& ~ r1_xreal_0(D,C)
& r2_hidden(D,k4_relset_1(k1_numbers,k1_numbers,k8_seq_1(k1_numbers,k1_numbers,A,B))) ) ) )
=> ( ! [C] :
( m1_subset_1(C,k1_numbers)
=> ~ r3_rfunct_1(B,k4_limfunc1(C)) )
| ( v3_limfunc1(k8_seq_1(k1_numbers,k1_numbers,A,B))
& k5_limfunc1(k8_seq_1(k1_numbers,k1_numbers,A,B)) = np__0 ) ) ) ) ) ).
fof(t134_limfunc1,axiom,
! [A] :
( ( v1_funct_1(A)
& m2_relset_1(A,k1_numbers,k1_numbers) )
=> ! [B] :
( ( v1_funct_1(B)
& m2_relset_1(B,k1_numbers,k1_numbers) )
=> ( ( v6_limfunc1(A)
& k6_limfunc1(A) = np__0
& ! [C] :
( m1_subset_1(C,k1_numbers)
=> ? [D] :
( m1_subset_1(D,k1_numbers)
& ~ r1_xreal_0(C,D)
& r2_hidden(D,k4_relset_1(k1_numbers,k1_numbers,k8_seq_1(k1_numbers,k1_numbers,A,B))) ) ) )
=> ( ! [C] :
( m1_subset_1(C,k1_numbers)
=> ~ r3_rfunct_1(B,k12_prob_1(C)) )
| ( v6_limfunc1(k8_seq_1(k1_numbers,k1_numbers,A,B))
& k6_limfunc1(k8_seq_1(k1_numbers,k1_numbers,A,B)) = np__0 ) ) ) ) ) ).
fof(t135_limfunc1,axiom,
! [A] :
( ( v1_funct_1(A)
& m2_relset_1(A,k1_numbers,k1_numbers) )
=> ! [B] :
( ( v1_funct_1(B)
& m2_relset_1(B,k1_numbers,k1_numbers) )
=> ! [C] :
( ( v1_funct_1(C)
& m2_relset_1(C,k1_numbers,k1_numbers) )
=> ( ( v3_limfunc1(A)
& v3_limfunc1(B)
& k5_limfunc1(A) = k5_limfunc1(B)
& ! [D] :
( m1_subset_1(D,k1_numbers)
=> ? [E] :
( m1_subset_1(E,k1_numbers)
& ~ r1_xreal_0(E,D)
& r2_hidden(E,k4_relset_1(k1_numbers,k1_numbers,C)) ) ) )
=> ( ! [D] :
( m1_subset_1(D,k1_numbers)
=> ~ ( ( ( r1_tarski(k5_subset_1(k1_numbers,k4_relset_1(k1_numbers,k1_numbers,A),k4_limfunc1(D)),k5_subset_1(k1_numbers,k4_relset_1(k1_numbers,k1_numbers,B),k4_limfunc1(D)))
& r1_tarski(k5_subset_1(k1_numbers,k4_relset_1(k1_numbers,k1_numbers,C),k4_limfunc1(D)),k5_subset_1(k1_numbers,k4_relset_1(k1_numbers,k1_numbers,A),k4_limfunc1(D))) )
| ( r1_tarski(k5_subset_1(k1_numbers,k4_relset_1(k1_numbers,k1_numbers,B),k4_limfunc1(D)),k5_subset_1(k1_numbers,k4_relset_1(k1_numbers,k1_numbers,A),k4_limfunc1(D)))
& r1_tarski(k5_subset_1(k1_numbers,k4_relset_1(k1_numbers,k1_numbers,C),k4_limfunc1(D)),k5_subset_1(k1_numbers,k4_relset_1(k1_numbers,k1_numbers,B),k4_limfunc1(D))) ) )
& ! [E] :
( m1_subset_1(E,k1_numbers)
=> ( r2_hidden(E,k5_subset_1(k1_numbers,k4_relset_1(k1_numbers,k1_numbers,C),k4_limfunc1(D)))
=> ( r1_xreal_0(k2_seq_1(k1_numbers,k1_numbers,A,E),k2_seq_1(k1_numbers,k1_numbers,C,E))
& r1_xreal_0(k2_seq_1(k1_numbers,k1_numbers,C,E),k2_seq_1(k1_numbers,k1_numbers,B,E)) ) ) ) ) )
| ( v3_limfunc1(C)
& k5_limfunc1(C) = k5_limfunc1(A) ) ) ) ) ) ) ).
fof(t136_limfunc1,axiom,
! [A] :
( ( v1_funct_1(A)
& m2_relset_1(A,k1_numbers,k1_numbers) )
=> ! [B] :
( ( v1_funct_1(B)
& m2_relset_1(B,k1_numbers,k1_numbers) )
=> ! [C] :
( ( v1_funct_1(C)
& m2_relset_1(C,k1_numbers,k1_numbers) )
=> ( ( v3_limfunc1(A)
& v3_limfunc1(B)
& k5_limfunc1(A) = k5_limfunc1(B) )
=> ( ! [D] :
( m1_subset_1(D,k1_numbers)
=> ~ ( r1_tarski(k4_limfunc1(D),k5_subset_1(k1_numbers,k5_subset_1(k1_numbers,k4_relset_1(k1_numbers,k1_numbers,A),k4_relset_1(k1_numbers,k1_numbers,B)),k4_relset_1(k1_numbers,k1_numbers,C)))
& ! [E] :
( m1_subset_1(E,k1_numbers)
=> ( r2_hidden(E,k4_limfunc1(D))
=> ( r1_xreal_0(k2_seq_1(k1_numbers,k1_numbers,A,E),k2_seq_1(k1_numbers,k1_numbers,C,E))
& r1_xreal_0(k2_seq_1(k1_numbers,k1_numbers,C,E),k2_seq_1(k1_numbers,k1_numbers,B,E)) ) ) ) ) )
| ( v3_limfunc1(C)
& k5_limfunc1(C) = k5_limfunc1(A) ) ) ) ) ) ) ).
fof(t137_limfunc1,axiom,
! [A] :
( ( v1_funct_1(A)
& m2_relset_1(A,k1_numbers,k1_numbers) )
=> ! [B] :
( ( v1_funct_1(B)
& m2_relset_1(B,k1_numbers,k1_numbers) )
=> ! [C] :
( ( v1_funct_1(C)
& m2_relset_1(C,k1_numbers,k1_numbers) )
=> ( ( v6_limfunc1(A)
& v6_limfunc1(B)
& k6_limfunc1(A) = k6_limfunc1(B)
& ! [D] :
( m1_subset_1(D,k1_numbers)
=> ? [E] :
( m1_subset_1(E,k1_numbers)
& ~ r1_xreal_0(D,E)
& r2_hidden(E,k4_relset_1(k1_numbers,k1_numbers,C)) ) ) )
=> ( ! [D] :
( m1_subset_1(D,k1_numbers)
=> ~ ( ( ( r1_tarski(k5_subset_1(k1_numbers,k4_relset_1(k1_numbers,k1_numbers,A),k12_prob_1(D)),k5_subset_1(k1_numbers,k4_relset_1(k1_numbers,k1_numbers,B),k12_prob_1(D)))
& r1_tarski(k5_subset_1(k1_numbers,k4_relset_1(k1_numbers,k1_numbers,C),k12_prob_1(D)),k5_subset_1(k1_numbers,k4_relset_1(k1_numbers,k1_numbers,A),k12_prob_1(D))) )
| ( r1_tarski(k5_subset_1(k1_numbers,k4_relset_1(k1_numbers,k1_numbers,B),k12_prob_1(D)),k5_subset_1(k1_numbers,k4_relset_1(k1_numbers,k1_numbers,A),k12_prob_1(D)))
& r1_tarski(k5_subset_1(k1_numbers,k4_relset_1(k1_numbers,k1_numbers,C),k12_prob_1(D)),k5_subset_1(k1_numbers,k4_relset_1(k1_numbers,k1_numbers,B),k12_prob_1(D))) ) )
& ! [E] :
( m1_subset_1(E,k1_numbers)
=> ( r2_hidden(E,k5_subset_1(k1_numbers,k4_relset_1(k1_numbers,k1_numbers,C),k12_prob_1(D)))
=> ( r1_xreal_0(k2_seq_1(k1_numbers,k1_numbers,A,E),k2_seq_1(k1_numbers,k1_numbers,C,E))
& r1_xreal_0(k2_seq_1(k1_numbers,k1_numbers,C,E),k2_seq_1(k1_numbers,k1_numbers,B,E)) ) ) ) ) )
| ( v6_limfunc1(C)
& k6_limfunc1(C) = k6_limfunc1(A) ) ) ) ) ) ) ).
fof(t138_limfunc1,axiom,
! [A] :
( ( v1_funct_1(A)
& m2_relset_1(A,k1_numbers,k1_numbers) )
=> ! [B] :
( ( v1_funct_1(B)
& m2_relset_1(B,k1_numbers,k1_numbers) )
=> ! [C] :
( ( v1_funct_1(C)
& m2_relset_1(C,k1_numbers,k1_numbers) )
=> ( ( v6_limfunc1(A)
& v6_limfunc1(B)
& k6_limfunc1(A) = k6_limfunc1(B) )
=> ( ! [D] :
( m1_subset_1(D,k1_numbers)
=> ~ ( r1_tarski(k12_prob_1(D),k5_subset_1(k1_numbers,k5_subset_1(k1_numbers,k4_relset_1(k1_numbers,k1_numbers,A),k4_relset_1(k1_numbers,k1_numbers,B)),k4_relset_1(k1_numbers,k1_numbers,C)))
& ! [E] :
( m1_subset_1(E,k1_numbers)
=> ( r2_hidden(E,k12_prob_1(D))
=> ( r1_xreal_0(k2_seq_1(k1_numbers,k1_numbers,A,E),k2_seq_1(k1_numbers,k1_numbers,C,E))
& r1_xreal_0(k2_seq_1(k1_numbers,k1_numbers,C,E),k2_seq_1(k1_numbers,k1_numbers,B,E)) ) ) ) ) )
| ( v6_limfunc1(C)
& k6_limfunc1(C) = k6_limfunc1(A) ) ) ) ) ) ) ).
fof(t139_limfunc1,axiom,
! [A] :
( ( v1_funct_1(A)
& m2_relset_1(A,k1_numbers,k1_numbers) )
=> ! [B] :
( ( v1_funct_1(B)
& m2_relset_1(B,k1_numbers,k1_numbers) )
=> ( ( v3_limfunc1(A)
& v3_limfunc1(B) )
=> ( ! [C] :
( m1_subset_1(C,k1_numbers)
=> ( ~ ( r1_tarski(k5_subset_1(k1_numbers,k4_relset_1(k1_numbers,k1_numbers,A),k4_limfunc1(C)),k5_subset_1(k1_numbers,k4_relset_1(k1_numbers,k1_numbers,B),k4_limfunc1(C)))
& ! [D] :
( m1_subset_1(D,k1_numbers)
=> ( r2_hidden(D,k5_subset_1(k1_numbers,k4_relset_1(k1_numbers,k1_numbers,A),k4_limfunc1(C)))
=> r1_xreal_0(k2_seq_1(k1_numbers,k1_numbers,A,D),k2_seq_1(k1_numbers,k1_numbers,B,D)) ) ) )
& ~ ( r1_tarski(k5_subset_1(k1_numbers,k4_relset_1(k1_numbers,k1_numbers,B),k4_limfunc1(C)),k5_subset_1(k1_numbers,k4_relset_1(k1_numbers,k1_numbers,A),k4_limfunc1(C)))
& ! [D] :
( m1_subset_1(D,k1_numbers)
=> ( r2_hidden(D,k5_subset_1(k1_numbers,k4_relset_1(k1_numbers,k1_numbers,B),k4_limfunc1(C)))
=> r1_xreal_0(k2_seq_1(k1_numbers,k1_numbers,A,D),k2_seq_1(k1_numbers,k1_numbers,B,D)) ) ) ) ) )
| r1_xreal_0(k5_limfunc1(A),k5_limfunc1(B)) ) ) ) ) ).
fof(t140_limfunc1,axiom,
! [A] :
( ( v1_funct_1(A)
& m2_relset_1(A,k1_numbers,k1_numbers) )
=> ! [B] :
( ( v1_funct_1(B)
& m2_relset_1(B,k1_numbers,k1_numbers) )
=> ( ( v6_limfunc1(A)
& v6_limfunc1(B) )
=> ( ! [C] :
( m1_subset_1(C,k1_numbers)
=> ( ~ ( r1_tarski(k5_subset_1(k1_numbers,k4_relset_1(k1_numbers,k1_numbers,A),k12_prob_1(C)),k5_subset_1(k1_numbers,k4_relset_1(k1_numbers,k1_numbers,B),k12_prob_1(C)))
& ! [D] :
( m1_subset_1(D,k1_numbers)
=> ( r2_hidden(D,k5_subset_1(k1_numbers,k4_relset_1(k1_numbers,k1_numbers,A),k12_prob_1(C)))
=> r1_xreal_0(k2_seq_1(k1_numbers,k1_numbers,A,D),k2_seq_1(k1_numbers,k1_numbers,B,D)) ) ) )
& ~ ( r1_tarski(k5_subset_1(k1_numbers,k4_relset_1(k1_numbers,k1_numbers,B),k12_prob_1(C)),k5_subset_1(k1_numbers,k4_relset_1(k1_numbers,k1_numbers,A),k12_prob_1(C)))
& ! [D] :
( m1_subset_1(D,k1_numbers)
=> ( r2_hidden(D,k5_subset_1(k1_numbers,k4_relset_1(k1_numbers,k1_numbers,B),k12_prob_1(C)))
=> r1_xreal_0(k2_seq_1(k1_numbers,k1_numbers,A,D),k2_seq_1(k1_numbers,k1_numbers,B,D)) ) ) ) ) )
| r1_xreal_0(k6_limfunc1(A),k6_limfunc1(B)) ) ) ) ) ).
fof(t141_limfunc1,axiom,
! [A] :
( ( v1_funct_1(A)
& m2_relset_1(A,k1_numbers,k1_numbers) )
=> ( ! [B] :
( m1_subset_1(B,k1_numbers)
=> ? [C] :
( m1_subset_1(C,k1_numbers)
& ~ r1_xreal_0(C,B)
& r2_hidden(C,k4_relset_1(k1_numbers,k1_numbers,A))
& k2_seq_1(k1_numbers,k1_numbers,A,C) != np__0 ) )
=> ( ( ~ v4_limfunc1(A)
& ~ v5_limfunc1(A) )
| ( v3_limfunc1(k4_rfunct_1(k1_numbers,k1_numbers,A))
& k5_limfunc1(k4_rfunct_1(k1_numbers,k1_numbers,A)) = np__0 ) ) ) ) ).
fof(t142_limfunc1,axiom,
! [A] :
( ( v1_funct_1(A)
& m2_relset_1(A,k1_numbers,k1_numbers) )
=> ( ! [B] :
( m1_subset_1(B,k1_numbers)
=> ? [C] :
( m1_subset_1(C,k1_numbers)
& ~ r1_xreal_0(B,C)
& r2_hidden(C,k4_relset_1(k1_numbers,k1_numbers,A))
& k2_seq_1(k1_numbers,k1_numbers,A,C) != np__0 ) )
=> ( ( ~ v7_limfunc1(A)
& ~ v8_limfunc1(A) )
| ( v6_limfunc1(k4_rfunct_1(k1_numbers,k1_numbers,A))
& k6_limfunc1(k4_rfunct_1(k1_numbers,k1_numbers,A)) = np__0 ) ) ) ) ).
fof(t143_limfunc1,axiom,
! [A] :
( ( v1_funct_1(A)
& m2_relset_1(A,k1_numbers,k1_numbers) )
=> ( ( v3_limfunc1(A)
& k5_limfunc1(A) = np__0
& ! [B] :
( m1_subset_1(B,k1_numbers)
=> ? [C] :
( m1_subset_1(C,k1_numbers)
& ~ r1_xreal_0(C,B)
& r2_hidden(C,k4_relset_1(k1_numbers,k1_numbers,A))
& k2_seq_1(k1_numbers,k1_numbers,A,C) != np__0 ) ) )
=> ( ! [B] :
( m1_subset_1(B,k1_numbers)
=> ? [C] :
( m1_subset_1(C,k1_numbers)
& r2_hidden(C,k5_subset_1(k1_numbers,k4_relset_1(k1_numbers,k1_numbers,A),k4_limfunc1(B)))
& ~ r1_xreal_0(np__0,k2_seq_1(k1_numbers,k1_numbers,A,C)) ) )
| v4_limfunc1(k4_rfunct_1(k1_numbers,k1_numbers,A)) ) ) ) ).
fof(t144_limfunc1,axiom,
! [A] :
( ( v1_funct_1(A)
& m2_relset_1(A,k1_numbers,k1_numbers) )
=> ( ( v3_limfunc1(A)
& k5_limfunc1(A) = np__0
& ! [B] :
( m1_subset_1(B,k1_numbers)
=> ? [C] :
( m1_subset_1(C,k1_numbers)
& ~ r1_xreal_0(C,B)
& r2_hidden(C,k4_relset_1(k1_numbers,k1_numbers,A))
& k2_seq_1(k1_numbers,k1_numbers,A,C) != np__0 ) ) )
=> ( ! [B] :
( m1_subset_1(B,k1_numbers)
=> ? [C] :
( m1_subset_1(C,k1_numbers)
& r2_hidden(C,k5_subset_1(k1_numbers,k4_relset_1(k1_numbers,k1_numbers,A),k4_limfunc1(B)))
& ~ r1_xreal_0(k2_seq_1(k1_numbers,k1_numbers,A,C),np__0) ) )
| v5_limfunc1(k4_rfunct_1(k1_numbers,k1_numbers,A)) ) ) ) ).
fof(t145_limfunc1,axiom,
! [A] :
( ( v1_funct_1(A)
& m2_relset_1(A,k1_numbers,k1_numbers) )
=> ( ( v6_limfunc1(A)
& k6_limfunc1(A) = np__0
& ! [B] :
( m1_subset_1(B,k1_numbers)
=> ? [C] :
( m1_subset_1(C,k1_numbers)
& ~ r1_xreal_0(B,C)
& r2_hidden(C,k4_relset_1(k1_numbers,k1_numbers,A))
& k2_seq_1(k1_numbers,k1_numbers,A,C) != np__0 ) ) )
=> ( ! [B] :
( m1_subset_1(B,k1_numbers)
=> ? [C] :
( m1_subset_1(C,k1_numbers)
& r2_hidden(C,k5_subset_1(k1_numbers,k4_relset_1(k1_numbers,k1_numbers,A),k12_prob_1(B)))
& ~ r1_xreal_0(np__0,k2_seq_1(k1_numbers,k1_numbers,A,C)) ) )
| v7_limfunc1(k4_rfunct_1(k1_numbers,k1_numbers,A)) ) ) ) ).
fof(t146_limfunc1,axiom,
! [A] :
( ( v1_funct_1(A)
& m2_relset_1(A,k1_numbers,k1_numbers) )
=> ( ( v6_limfunc1(A)
& k6_limfunc1(A) = np__0
& ! [B] :
( m1_subset_1(B,k1_numbers)
=> ? [C] :
( m1_subset_1(C,k1_numbers)
& ~ r1_xreal_0(B,C)
& r2_hidden(C,k4_relset_1(k1_numbers,k1_numbers,A))
& k2_seq_1(k1_numbers,k1_numbers,A,C) != np__0 ) ) )
=> ( ! [B] :
( m1_subset_1(B,k1_numbers)
=> ? [C] :
( m1_subset_1(C,k1_numbers)
& r2_hidden(C,k5_subset_1(k1_numbers,k4_relset_1(k1_numbers,k1_numbers,A),k12_prob_1(B)))
& ~ r1_xreal_0(k2_seq_1(k1_numbers,k1_numbers,A,C),np__0) ) )
| v8_limfunc1(k4_rfunct_1(k1_numbers,k1_numbers,A)) ) ) ) ).
fof(t147_limfunc1,axiom,
! [A] :
( ( v1_funct_1(A)
& m2_relset_1(A,k1_numbers,k1_numbers) )
=> ( ( v3_limfunc1(A)
& k5_limfunc1(A) = np__0 )
=> ( ! [B] :
( m1_subset_1(B,k1_numbers)
=> ? [C] :
( m1_subset_1(C,k1_numbers)
& r2_hidden(C,k5_subset_1(k1_numbers,k4_relset_1(k1_numbers,k1_numbers,A),k4_limfunc1(B)))
& r1_xreal_0(k2_seq_1(k1_numbers,k1_numbers,A,C),np__0) ) )
| v4_limfunc1(k4_rfunct_1(k1_numbers,k1_numbers,A)) ) ) ) ).
fof(t148_limfunc1,axiom,
! [A] :
( ( v1_funct_1(A)
& m2_relset_1(A,k1_numbers,k1_numbers) )
=> ( ( v3_limfunc1(A)
& k5_limfunc1(A) = np__0 )
=> ( ! [B] :
( m1_subset_1(B,k1_numbers)
=> ? [C] :
( m1_subset_1(C,k1_numbers)
& r2_hidden(C,k5_subset_1(k1_numbers,k4_relset_1(k1_numbers,k1_numbers,A),k4_limfunc1(B)))
& r1_xreal_0(np__0,k2_seq_1(k1_numbers,k1_numbers,A,C)) ) )
| v5_limfunc1(k4_rfunct_1(k1_numbers,k1_numbers,A)) ) ) ) ).
fof(t149_limfunc1,axiom,
! [A] :
( ( v1_funct_1(A)
& m2_relset_1(A,k1_numbers,k1_numbers) )
=> ( ( v6_limfunc1(A)
& k6_limfunc1(A) = np__0 )
=> ( ! [B] :
( m1_subset_1(B,k1_numbers)
=> ? [C] :
( m1_subset_1(C,k1_numbers)
& r2_hidden(C,k5_subset_1(k1_numbers,k4_relset_1(k1_numbers,k1_numbers,A),k12_prob_1(B)))
& r1_xreal_0(k2_seq_1(k1_numbers,k1_numbers,A,C),np__0) ) )
| v7_limfunc1(k4_rfunct_1(k1_numbers,k1_numbers,A)) ) ) ) ).
fof(t150_limfunc1,axiom,
! [A] :
( ( v1_funct_1(A)
& m2_relset_1(A,k1_numbers,k1_numbers) )
=> ( ( v6_limfunc1(A)
& k6_limfunc1(A) = np__0 )
=> ( ! [B] :
( m1_subset_1(B,k1_numbers)
=> ? [C] :
( m1_subset_1(C,k1_numbers)
& r2_hidden(C,k5_subset_1(k1_numbers,k4_relset_1(k1_numbers,k1_numbers,A),k12_prob_1(B)))
& r1_xreal_0(np__0,k2_seq_1(k1_numbers,k1_numbers,A,C)) ) )
| v8_limfunc1(k4_rfunct_1(k1_numbers,k1_numbers,A)) ) ) ) ).
fof(dt_k1_limfunc1,axiom,
! [A,B] :
( ( m1_subset_1(A,k5_numbers)
& m1_subset_1(B,k5_numbers) )
=> m2_subset_1(k1_limfunc1(A,B),k1_numbers,k5_numbers) ) ).
fof(commutativity_k1_limfunc1,axiom,
! [A,B] :
( ( m1_subset_1(A,k5_numbers)
& m1_subset_1(B,k5_numbers) )
=> k1_limfunc1(A,B) = k1_limfunc1(B,A) ) ).
fof(idempotence_k1_limfunc1,axiom,
! [A,B] :
( ( m1_subset_1(A,k5_numbers)
& m1_subset_1(B,k5_numbers) )
=> k1_limfunc1(A,A) = A ) ).
fof(redefinition_k1_limfunc1,axiom,
! [A,B] :
( ( m1_subset_1(A,k5_numbers)
& m1_subset_1(B,k5_numbers) )
=> k1_limfunc1(A,B) = k2_square_1(A,B) ) ).
fof(dt_k2_limfunc1,axiom,
! [A] :
( v1_xreal_0(A)
=> m1_subset_1(k2_limfunc1(A),k1_zfmisc_1(k1_numbers)) ) ).
fof(dt_k3_limfunc1,axiom,
! [A] :
( v1_xreal_0(A)
=> m1_subset_1(k3_limfunc1(A),k1_zfmisc_1(k1_numbers)) ) ).
fof(dt_k4_limfunc1,axiom,
! [A] :
( v1_xreal_0(A)
=> m1_subset_1(k4_limfunc1(A),k1_zfmisc_1(k1_numbers)) ) ).
fof(dt_k5_limfunc1,axiom,
! [A] :
( ( v1_funct_1(A)
& m1_relset_1(A,k1_numbers,k1_numbers) )
=> m1_subset_1(k5_limfunc1(A),k1_numbers) ) ).
fof(dt_k6_limfunc1,axiom,
! [A] :
( ( v1_funct_1(A)
& m1_relset_1(A,k1_numbers,k1_numbers) )
=> m1_subset_1(k6_limfunc1(A),k1_numbers) ) ).
fof(d1_limfunc1,axiom,
! [A] :
( v1_xreal_0(A)
=> k2_limfunc1(A) = a_1_0_limfunc1(A) ) ).
fof(d2_limfunc1,axiom,
! [A] :
( v1_xreal_0(A)
=> k3_limfunc1(A) = a_1_1_limfunc1(A) ) ).
fof(d3_limfunc1,axiom,
! [A] :
( v1_xreal_0(A)
=> k4_limfunc1(A) = a_1_2_limfunc1(A) ) ).
fof(fraenkel_a_1_0_limfunc1,axiom,
! [A,B] :
( v1_xreal_0(B)
=> ( r2_hidden(A,a_1_0_limfunc1(B))
<=> ? [C] :
( m1_subset_1(C,k1_numbers)
& A = C
& r1_xreal_0(C,B) ) ) ) ).
fof(fraenkel_a_1_1_limfunc1,axiom,
! [A,B] :
( v1_xreal_0(B)
=> ( r2_hidden(A,a_1_1_limfunc1(B))
<=> ? [C] :
( m1_subset_1(C,k1_numbers)
& A = C
& r1_xreal_0(B,C) ) ) ) ).
fof(fraenkel_a_1_2_limfunc1,axiom,
! [A,B] :
( v1_xreal_0(B)
=> ( r2_hidden(A,a_1_2_limfunc1(B))
<=> ? [C] :
( m1_subset_1(C,k1_numbers)
& A = C
& ~ r1_xreal_0(C,B) ) ) ) ).
%------------------------------------------------------------------------------