SET007 Axioms: SET007+111.ax
%------------------------------------------------------------------------------
% File : SET007+111 : TPTP v8.2.0. Released v3.4.0.
% Domain : Set Theory
% Axioms : The One-Side Limits of a Real Function at a Point
% 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 : limfunc2 [Urb08]
% Status : Satisfiable
% Syntax : Number of formulae : 96 ( 8 unt; 0 def)
% Number of atoms : 1294 ( 74 equ)
% Maximal formula atoms : 28 ( 13 avg)
% Number of connectives : 1618 ( 420 ~; 60 |; 588 &)
% ( 16 <=>; 534 =>; 0 <=; 0 <~>)
% Maximal formula depth : 23 ( 15 avg)
% Maximal term depth : 4 ( 1 avg)
% Number of predicates : 27 ( 25 usr; 1 prp; 0-3 aty)
% Number of functors : 34 ( 34 usr; 5 con; 0-4 aty)
% Number of variables : 438 ( 428 !; 10 ?)
% 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_limfunc2,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) )
=> ~ ( v4_seq_2(B)
& ~ r1_xreal_0(k2_seq_2(B),A)
& ! [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,B,D),A) ) ) ) ) ) ).
fof(t2_limfunc2,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) )
=> ~ ( v4_seq_2(B)
& ~ r1_xreal_0(A,k2_seq_2(B))
& ! [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
=> ? [D] :
( m2_subset_1(D,k1_numbers,k5_numbers)
& r1_xreal_0(C,D)
& r1_xreal_0(A,k2_seq_1(k5_numbers,k1_numbers,B,D)) ) ) ) ) ) ).
fof(t3_limfunc2,axiom,
! [A] :
( m1_subset_1(A,k1_numbers)
=> ! [B] :
( m1_subset_1(B,k1_numbers)
=> ! [C] :
( ( v1_funct_1(C)
& m2_relset_1(C,k1_numbers,k1_numbers) )
=> ( r1_tarski(k2_rcomp_1(k5_real_1(B,A),B),k4_relset_1(k1_numbers,k1_numbers,C))
=> ( r1_xreal_0(A,np__0)
| ! [D] :
( m1_subset_1(D,k1_numbers)
=> ~ ( ~ r1_xreal_0(B,D)
& ! [E] :
( m1_subset_1(E,k1_numbers)
=> ~ ( ~ r1_xreal_0(E,D)
& ~ r1_xreal_0(B,E)
& r2_hidden(E,k4_relset_1(k1_numbers,k1_numbers,C)) ) ) ) ) ) ) ) ) ) ).
fof(t4_limfunc2,axiom,
! [A] :
( m1_subset_1(A,k1_numbers)
=> ! [B] :
( m1_subset_1(B,k1_numbers)
=> ! [C] :
( ( v1_funct_1(C)
& m2_relset_1(C,k1_numbers,k1_numbers) )
=> ( r1_tarski(k2_rcomp_1(B,k3_real_1(B,A)),k4_relset_1(k1_numbers,k1_numbers,C))
=> ( r1_xreal_0(A,np__0)
| ! [D] :
( m1_subset_1(D,k1_numbers)
=> ~ ( ~ r1_xreal_0(D,B)
& ! [E] :
( m1_subset_1(E,k1_numbers)
=> ~ ( ~ r1_xreal_0(D,E)
& ~ r1_xreal_0(E,B)
& r2_hidden(E,k4_relset_1(k1_numbers,k1_numbers,C)) ) ) ) ) ) ) ) ) ) ).
fof(t5_limfunc2,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) )
=> ! [C] :
( ( v1_funct_1(C)
& m2_relset_1(C,k1_numbers,k1_numbers) )
=> ( ! [D] :
( m2_subset_1(D,k1_numbers,k5_numbers)
=> ( ~ r1_xreal_0(k2_seq_1(k5_numbers,k1_numbers,B,D),k5_real_1(A,k6_real_1(np__1,k1_nat_1(D,np__1))))
& ~ r1_xreal_0(A,k2_seq_1(k5_numbers,k1_numbers,B,D))
& r2_hidden(k2_seq_1(k5_numbers,k1_numbers,B,D),k4_relset_1(k1_numbers,k1_numbers,C)) ) )
=> ( v4_seq_2(B)
& k2_seq_2(B) = A
& r1_tarski(k1_rfunct_2(B),k4_relset_1(k1_numbers,k1_numbers,C))
& r1_tarski(k1_rfunct_2(B),k5_subset_1(k1_numbers,k4_relset_1(k1_numbers,k1_numbers,C),k12_prob_1(A))) ) ) ) ) ) ).
fof(t6_limfunc2,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) )
=> ! [C] :
( ( v1_funct_1(C)
& m2_relset_1(C,k1_numbers,k1_numbers) )
=> ( ! [D] :
( m2_subset_1(D,k1_numbers,k5_numbers)
=> ( ~ r1_xreal_0(k2_seq_1(k5_numbers,k1_numbers,B,D),A)
& ~ r1_xreal_0(k3_real_1(A,k6_real_1(np__1,k1_nat_1(D,np__1))),k2_seq_1(k5_numbers,k1_numbers,B,D))
& r2_hidden(k2_seq_1(k5_numbers,k1_numbers,B,D),k4_relset_1(k1_numbers,k1_numbers,C)) ) )
=> ( v4_seq_2(B)
& k2_seq_2(B) = A
& r1_tarski(k1_rfunct_2(B),k4_relset_1(k1_numbers,k1_numbers,C))
& r1_tarski(k1_rfunct_2(B),k5_subset_1(k1_numbers,k4_relset_1(k1_numbers,k1_numbers,C),k4_limfunc1(A))) ) ) ) ) ) ).
fof(d1_limfunc2,axiom,
! [A] :
( ( v1_funct_1(A)
& m2_relset_1(A,k1_numbers,k1_numbers) )
=> ! [B] :
( m1_subset_1(B,k1_numbers)
=> ( r1_limfunc2(A,B)
<=> ( ! [C] :
( m1_subset_1(C,k1_numbers)
=> ~ ( ~ r1_xreal_0(B,C)
& ! [D] :
( m1_subset_1(D,k1_numbers)
=> ~ ( ~ r1_xreal_0(D,C)
& ~ r1_xreal_0(B,D)
& r2_hidden(D,k4_relset_1(k1_numbers,k1_numbers,A)) ) ) ) )
& ? [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) )
=> ( ( v4_seq_2(D)
& k2_seq_2(D) = B
& r1_tarski(k1_rfunct_2(D),k5_subset_1(k1_numbers,k4_relset_1(k1_numbers,k1_numbers,A),k12_prob_1(B))) )
=> ( v4_seq_2(k2_rfunct_2(A,D))
& k2_seq_2(k2_rfunct_2(A,D)) = C ) ) ) ) ) ) ) ) ).
fof(d2_limfunc2,axiom,
! [A] :
( ( v1_funct_1(A)
& m2_relset_1(A,k1_numbers,k1_numbers) )
=> ! [B] :
( m1_subset_1(B,k1_numbers)
=> ( r2_limfunc2(A,B)
<=> ( ! [C] :
( m1_subset_1(C,k1_numbers)
=> ~ ( ~ r1_xreal_0(B,C)
& ! [D] :
( m1_subset_1(D,k1_numbers)
=> ~ ( ~ r1_xreal_0(D,C)
& ~ r1_xreal_0(B,D)
& r2_hidden(D,k4_relset_1(k1_numbers,k1_numbers,A)) ) ) ) )
& ! [C] :
( ( v1_funct_1(C)
& v1_funct_2(C,k5_numbers,k1_numbers)
& m2_relset_1(C,k5_numbers,k1_numbers) )
=> ( ( v4_seq_2(C)
& k2_seq_2(C) = B
& r1_tarski(k1_rfunct_2(C),k5_subset_1(k1_numbers,k4_relset_1(k1_numbers,k1_numbers,A),k12_prob_1(B))) )
=> v1_limfunc1(k2_rfunct_2(A,C)) ) ) ) ) ) ) ).
fof(d3_limfunc2,axiom,
! [A] :
( ( v1_funct_1(A)
& m2_relset_1(A,k1_numbers,k1_numbers) )
=> ! [B] :
( m1_subset_1(B,k1_numbers)
=> ( r3_limfunc2(A,B)
<=> ( ! [C] :
( m1_subset_1(C,k1_numbers)
=> ~ ( ~ r1_xreal_0(B,C)
& ! [D] :
( m1_subset_1(D,k1_numbers)
=> ~ ( ~ r1_xreal_0(D,C)
& ~ r1_xreal_0(B,D)
& r2_hidden(D,k4_relset_1(k1_numbers,k1_numbers,A)) ) ) ) )
& ! [C] :
( ( v1_funct_1(C)
& v1_funct_2(C,k5_numbers,k1_numbers)
& m2_relset_1(C,k5_numbers,k1_numbers) )
=> ( ( v4_seq_2(C)
& k2_seq_2(C) = B
& r1_tarski(k1_rfunct_2(C),k5_subset_1(k1_numbers,k4_relset_1(k1_numbers,k1_numbers,A),k12_prob_1(B))) )
=> v2_limfunc1(k2_rfunct_2(A,C)) ) ) ) ) ) ) ).
fof(d4_limfunc2,axiom,
! [A] :
( ( v1_funct_1(A)
& m2_relset_1(A,k1_numbers,k1_numbers) )
=> ! [B] :
( m1_subset_1(B,k1_numbers)
=> ( r4_limfunc2(A,B)
<=> ( ! [C] :
( m1_subset_1(C,k1_numbers)
=> ~ ( ~ r1_xreal_0(C,B)
& ! [D] :
( m1_subset_1(D,k1_numbers)
=> ~ ( ~ r1_xreal_0(C,D)
& ~ r1_xreal_0(D,B)
& r2_hidden(D,k4_relset_1(k1_numbers,k1_numbers,A)) ) ) ) )
& ? [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) )
=> ( ( v4_seq_2(D)
& k2_seq_2(D) = B
& r1_tarski(k1_rfunct_2(D),k5_subset_1(k1_numbers,k4_relset_1(k1_numbers,k1_numbers,A),k4_limfunc1(B))) )
=> ( v4_seq_2(k2_rfunct_2(A,D))
& k2_seq_2(k2_rfunct_2(A,D)) = C ) ) ) ) ) ) ) ) ).
fof(d5_limfunc2,axiom,
! [A] :
( ( v1_funct_1(A)
& m2_relset_1(A,k1_numbers,k1_numbers) )
=> ! [B] :
( m1_subset_1(B,k1_numbers)
=> ( r5_limfunc2(A,B)
<=> ( ! [C] :
( m1_subset_1(C,k1_numbers)
=> ~ ( ~ r1_xreal_0(C,B)
& ! [D] :
( m1_subset_1(D,k1_numbers)
=> ~ ( ~ r1_xreal_0(C,D)
& ~ r1_xreal_0(D,B)
& r2_hidden(D,k4_relset_1(k1_numbers,k1_numbers,A)) ) ) ) )
& ! [C] :
( ( v1_funct_1(C)
& v1_funct_2(C,k5_numbers,k1_numbers)
& m2_relset_1(C,k5_numbers,k1_numbers) )
=> ( ( v4_seq_2(C)
& k2_seq_2(C) = B
& r1_tarski(k1_rfunct_2(C),k5_subset_1(k1_numbers,k4_relset_1(k1_numbers,k1_numbers,A),k4_limfunc1(B))) )
=> v1_limfunc1(k2_rfunct_2(A,C)) ) ) ) ) ) ) ).
fof(d6_limfunc2,axiom,
! [A] :
( ( v1_funct_1(A)
& m2_relset_1(A,k1_numbers,k1_numbers) )
=> ! [B] :
( m1_subset_1(B,k1_numbers)
=> ( r6_limfunc2(A,B)
<=> ( ! [C] :
( m1_subset_1(C,k1_numbers)
=> ~ ( ~ r1_xreal_0(C,B)
& ! [D] :
( m1_subset_1(D,k1_numbers)
=> ~ ( ~ r1_xreal_0(C,D)
& ~ r1_xreal_0(D,B)
& r2_hidden(D,k4_relset_1(k1_numbers,k1_numbers,A)) ) ) ) )
& ! [C] :
( ( v1_funct_1(C)
& v1_funct_2(C,k5_numbers,k1_numbers)
& m2_relset_1(C,k5_numbers,k1_numbers) )
=> ( ( v4_seq_2(C)
& k2_seq_2(C) = B
& r1_tarski(k1_rfunct_2(C),k5_subset_1(k1_numbers,k4_relset_1(k1_numbers,k1_numbers,A),k4_limfunc1(B))) )
=> v2_limfunc1(k2_rfunct_2(A,C)) ) ) ) ) ) ) ).
fof(t7_limfunc2,axiom,
$true ).
fof(t8_limfunc2,axiom,
$true ).
fof(t9_limfunc2,axiom,
$true ).
fof(t10_limfunc2,axiom,
$true ).
fof(t11_limfunc2,axiom,
$true ).
fof(t12_limfunc2,axiom,
$true ).
fof(t13_limfunc2,axiom,
! [A] :
( m1_subset_1(A,k1_numbers)
=> ! [B] :
( ( v1_funct_1(B)
& m2_relset_1(B,k1_numbers,k1_numbers) )
=> ( r1_limfunc2(B,A)
<=> ( ! [C] :
( m1_subset_1(C,k1_numbers)
=> ~ ( ~ r1_xreal_0(A,C)
& ! [D] :
( m1_subset_1(D,k1_numbers)
=> ~ ( ~ r1_xreal_0(D,C)
& ~ r1_xreal_0(A,D)
& r2_hidden(D,k4_relset_1(k1_numbers,k1_numbers,B)) ) ) ) )
& ? [C] :
( m1_subset_1(C,k1_numbers)
& ! [D] :
( m1_subset_1(D,k1_numbers)
=> ~ ( ~ r1_xreal_0(D,np__0)
& ! [E] :
( m1_subset_1(E,k1_numbers)
=> ~ ( ~ r1_xreal_0(A,E)
& ! [F] :
( m1_subset_1(F,k1_numbers)
=> ~ ( ~ r1_xreal_0(F,E)
& ~ r1_xreal_0(A,F)
& r2_hidden(F,k4_relset_1(k1_numbers,k1_numbers,B))
& r1_xreal_0(D,k18_complex1(k5_real_1(k2_seq_1(k1_numbers,k1_numbers,B,F),C))) ) ) ) ) ) ) ) ) ) ) ) ).
fof(t14_limfunc2,axiom,
! [A] :
( m1_subset_1(A,k1_numbers)
=> ! [B] :
( ( v1_funct_1(B)
& m2_relset_1(B,k1_numbers,k1_numbers) )
=> ( r2_limfunc2(B,A)
<=> ( ! [C] :
( m1_subset_1(C,k1_numbers)
=> ~ ( ~ r1_xreal_0(A,C)
& ! [D] :
( m1_subset_1(D,k1_numbers)
=> ~ ( ~ r1_xreal_0(D,C)
& ~ r1_xreal_0(A,D)
& r2_hidden(D,k4_relset_1(k1_numbers,k1_numbers,B)) ) ) ) )
& ! [C] :
( m1_subset_1(C,k1_numbers)
=> ? [D] :
( m1_subset_1(D,k1_numbers)
& ~ r1_xreal_0(A,D)
& ! [E] :
( m1_subset_1(E,k1_numbers)
=> ~ ( ~ r1_xreal_0(E,D)
& ~ r1_xreal_0(A,E)
& r2_hidden(E,k4_relset_1(k1_numbers,k1_numbers,B))
& r1_xreal_0(k2_seq_1(k1_numbers,k1_numbers,B,E),C) ) ) ) ) ) ) ) ) ).
fof(t15_limfunc2,axiom,
! [A] :
( m1_subset_1(A,k1_numbers)
=> ! [B] :
( ( v1_funct_1(B)
& m2_relset_1(B,k1_numbers,k1_numbers) )
=> ( r3_limfunc2(B,A)
<=> ( ! [C] :
( m1_subset_1(C,k1_numbers)
=> ~ ( ~ r1_xreal_0(A,C)
& ! [D] :
( m1_subset_1(D,k1_numbers)
=> ~ ( ~ r1_xreal_0(D,C)
& ~ r1_xreal_0(A,D)
& r2_hidden(D,k4_relset_1(k1_numbers,k1_numbers,B)) ) ) ) )
& ! [C] :
( m1_subset_1(C,k1_numbers)
=> ? [D] :
( m1_subset_1(D,k1_numbers)
& ~ r1_xreal_0(A,D)
& ! [E] :
( m1_subset_1(E,k1_numbers)
=> ~ ( ~ r1_xreal_0(E,D)
& ~ r1_xreal_0(A,E)
& r2_hidden(E,k4_relset_1(k1_numbers,k1_numbers,B))
& r1_xreal_0(C,k2_seq_1(k1_numbers,k1_numbers,B,E)) ) ) ) ) ) ) ) ) ).
fof(t16_limfunc2,axiom,
! [A] :
( m1_subset_1(A,k1_numbers)
=> ! [B] :
( ( v1_funct_1(B)
& m2_relset_1(B,k1_numbers,k1_numbers) )
=> ( r4_limfunc2(B,A)
<=> ( ! [C] :
( m1_subset_1(C,k1_numbers)
=> ~ ( ~ r1_xreal_0(C,A)
& ! [D] :
( m1_subset_1(D,k1_numbers)
=> ~ ( ~ r1_xreal_0(C,D)
& ~ r1_xreal_0(D,A)
& r2_hidden(D,k4_relset_1(k1_numbers,k1_numbers,B)) ) ) ) )
& ? [C] :
( m1_subset_1(C,k1_numbers)
& ! [D] :
( m1_subset_1(D,k1_numbers)
=> ~ ( ~ r1_xreal_0(D,np__0)
& ! [E] :
( m1_subset_1(E,k1_numbers)
=> ~ ( ~ r1_xreal_0(E,A)
& ! [F] :
( m1_subset_1(F,k1_numbers)
=> ~ ( ~ r1_xreal_0(E,F)
& ~ r1_xreal_0(F,A)
& r2_hidden(F,k4_relset_1(k1_numbers,k1_numbers,B))
& r1_xreal_0(D,k18_complex1(k5_real_1(k2_seq_1(k1_numbers,k1_numbers,B,F),C))) ) ) ) ) ) ) ) ) ) ) ) ).
fof(t17_limfunc2,axiom,
! [A] :
( m1_subset_1(A,k1_numbers)
=> ! [B] :
( ( v1_funct_1(B)
& m2_relset_1(B,k1_numbers,k1_numbers) )
=> ( r5_limfunc2(B,A)
<=> ( ! [C] :
( m1_subset_1(C,k1_numbers)
=> ~ ( ~ r1_xreal_0(C,A)
& ! [D] :
( m1_subset_1(D,k1_numbers)
=> ~ ( ~ r1_xreal_0(C,D)
& ~ r1_xreal_0(D,A)
& r2_hidden(D,k4_relset_1(k1_numbers,k1_numbers,B)) ) ) ) )
& ! [C] :
( m1_subset_1(C,k1_numbers)
=> ? [D] :
( m1_subset_1(D,k1_numbers)
& ~ r1_xreal_0(D,A)
& ! [E] :
( m1_subset_1(E,k1_numbers)
=> ~ ( ~ r1_xreal_0(D,E)
& ~ r1_xreal_0(E,A)
& r2_hidden(E,k4_relset_1(k1_numbers,k1_numbers,B))
& r1_xreal_0(k2_seq_1(k1_numbers,k1_numbers,B,E),C) ) ) ) ) ) ) ) ) ).
fof(t18_limfunc2,axiom,
! [A] :
( m1_subset_1(A,k1_numbers)
=> ! [B] :
( ( v1_funct_1(B)
& m2_relset_1(B,k1_numbers,k1_numbers) )
=> ( r6_limfunc2(B,A)
<=> ( ! [C] :
( m1_subset_1(C,k1_numbers)
=> ~ ( ~ r1_xreal_0(C,A)
& ! [D] :
( m1_subset_1(D,k1_numbers)
=> ~ ( ~ r1_xreal_0(C,D)
& ~ r1_xreal_0(D,A)
& r2_hidden(D,k4_relset_1(k1_numbers,k1_numbers,B)) ) ) ) )
& ! [C] :
( m1_subset_1(C,k1_numbers)
=> ? [D] :
( m1_subset_1(D,k1_numbers)
& ~ r1_xreal_0(D,A)
& ! [E] :
( m1_subset_1(E,k1_numbers)
=> ~ ( ~ r1_xreal_0(D,E)
& ~ r1_xreal_0(E,A)
& r2_hidden(E,k4_relset_1(k1_numbers,k1_numbers,B))
& r1_xreal_0(C,k2_seq_1(k1_numbers,k1_numbers,B,E)) ) ) ) ) ) ) ) ) ).
fof(t19_limfunc2,axiom,
! [A] :
( m1_subset_1(A,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) )
=> ( ( r2_limfunc2(B,A)
& r2_limfunc2(C,A)
& ! [D] :
( m1_subset_1(D,k1_numbers)
=> ~ ( ~ r1_xreal_0(A,D)
& ! [E] :
( m1_subset_1(E,k1_numbers)
=> ~ ( ~ r1_xreal_0(E,D)
& ~ r1_xreal_0(A,E)
& r2_hidden(E,k5_subset_1(k1_numbers,k4_relset_1(k1_numbers,k1_numbers,B),k4_relset_1(k1_numbers,k1_numbers,C))) ) ) ) ) )
=> ( r2_limfunc2(k6_seq_1(k1_numbers,k1_numbers,B,C),A)
& r2_limfunc2(k8_seq_1(k1_numbers,k1_numbers,B,C),A) ) ) ) ) ) ).
fof(t20_limfunc2,axiom,
! [A] :
( m1_subset_1(A,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) )
=> ( ( r3_limfunc2(B,A)
& r3_limfunc2(C,A)
& ! [D] :
( m1_subset_1(D,k1_numbers)
=> ~ ( ~ r1_xreal_0(A,D)
& ! [E] :
( m1_subset_1(E,k1_numbers)
=> ~ ( ~ r1_xreal_0(E,D)
& ~ r1_xreal_0(A,E)
& r2_hidden(E,k5_subset_1(k1_numbers,k4_relset_1(k1_numbers,k1_numbers,B),k4_relset_1(k1_numbers,k1_numbers,C))) ) ) ) ) )
=> ( r3_limfunc2(k6_seq_1(k1_numbers,k1_numbers,B,C),A)
& r2_limfunc2(k8_seq_1(k1_numbers,k1_numbers,B,C),A) ) ) ) ) ) ).
fof(t21_limfunc2,axiom,
! [A] :
( m1_subset_1(A,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) )
=> ( ( r5_limfunc2(B,A)
& r5_limfunc2(C,A)
& ! [D] :
( m1_subset_1(D,k1_numbers)
=> ~ ( ~ r1_xreal_0(D,A)
& ! [E] :
( m1_subset_1(E,k1_numbers)
=> ~ ( ~ r1_xreal_0(D,E)
& ~ r1_xreal_0(E,A)
& r2_hidden(E,k5_subset_1(k1_numbers,k4_relset_1(k1_numbers,k1_numbers,B),k4_relset_1(k1_numbers,k1_numbers,C))) ) ) ) ) )
=> ( r5_limfunc2(k6_seq_1(k1_numbers,k1_numbers,B,C),A)
& r5_limfunc2(k8_seq_1(k1_numbers,k1_numbers,B,C),A) ) ) ) ) ) ).
fof(t22_limfunc2,axiom,
! [A] :
( m1_subset_1(A,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) )
=> ( ( r6_limfunc2(B,A)
& r6_limfunc2(C,A)
& ! [D] :
( m1_subset_1(D,k1_numbers)
=> ~ ( ~ r1_xreal_0(D,A)
& ! [E] :
( m1_subset_1(E,k1_numbers)
=> ~ ( ~ r1_xreal_0(D,E)
& ~ r1_xreal_0(E,A)
& r2_hidden(E,k5_subset_1(k1_numbers,k4_relset_1(k1_numbers,k1_numbers,B),k4_relset_1(k1_numbers,k1_numbers,C))) ) ) ) ) )
=> ( r6_limfunc2(k6_seq_1(k1_numbers,k1_numbers,B,C),A)
& r5_limfunc2(k8_seq_1(k1_numbers,k1_numbers,B,C),A) ) ) ) ) ) ).
fof(t23_limfunc2,axiom,
! [A] :
( m1_subset_1(A,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) )
=> ( ( r2_limfunc2(B,A)
& ! [D] :
( m1_subset_1(D,k1_numbers)
=> ~ ( ~ r1_xreal_0(A,D)
& ! [E] :
( m1_subset_1(E,k1_numbers)
=> ~ ( ~ r1_xreal_0(E,D)
& ~ r1_xreal_0(A,E)
& r2_hidden(E,k4_relset_1(k1_numbers,k1_numbers,k6_seq_1(k1_numbers,k1_numbers,B,C))) ) ) ) ) )
=> ( ! [D] :
( m1_subset_1(D,k1_numbers)
=> ~ ( ~ r1_xreal_0(D,np__0)
& r2_rfunct_1(C,k2_rcomp_1(k5_real_1(A,D),A)) ) )
| r2_limfunc2(k6_seq_1(k1_numbers,k1_numbers,B,C),A) ) ) ) ) ) ).
fof(t24_limfunc2,axiom,
! [A] :
( m1_subset_1(A,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) )
=> ( ( r2_limfunc2(B,A)
& ! [D] :
( m1_subset_1(D,k1_numbers)
=> ~ ( ~ r1_xreal_0(A,D)
& ! [E] :
( m1_subset_1(E,k1_numbers)
=> ~ ( ~ r1_xreal_0(E,D)
& ~ r1_xreal_0(A,E)
& r2_hidden(E,k4_relset_1(k1_numbers,k1_numbers,k8_seq_1(k1_numbers,k1_numbers,B,C))) ) ) ) ) )
=> ( ! [D] :
( m1_subset_1(D,k1_numbers)
=> ! [E] :
( m1_subset_1(E,k1_numbers)
=> ~ ( ~ r1_xreal_0(D,np__0)
& ~ r1_xreal_0(E,np__0)
& ! [F] :
( m1_subset_1(F,k1_numbers)
=> ( r2_hidden(F,k5_subset_1(k1_numbers,k4_relset_1(k1_numbers,k1_numbers,C),k2_rcomp_1(k5_real_1(A,D),A)))
=> r1_xreal_0(E,k2_seq_1(k1_numbers,k1_numbers,C,F)) ) ) ) ) )
| r2_limfunc2(k8_seq_1(k1_numbers,k1_numbers,B,C),A) ) ) ) ) ) ).
fof(t25_limfunc2,axiom,
! [A] :
( m1_subset_1(A,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) )
=> ( ( r5_limfunc2(B,A)
& ! [D] :
( m1_subset_1(D,k1_numbers)
=> ~ ( ~ r1_xreal_0(D,A)
& ! [E] :
( m1_subset_1(E,k1_numbers)
=> ~ ( ~ r1_xreal_0(D,E)
& ~ r1_xreal_0(E,A)
& r2_hidden(E,k4_relset_1(k1_numbers,k1_numbers,k6_seq_1(k1_numbers,k1_numbers,B,C))) ) ) ) ) )
=> ( ! [D] :
( m1_subset_1(D,k1_numbers)
=> ~ ( ~ r1_xreal_0(D,np__0)
& r2_rfunct_1(C,k2_rcomp_1(A,k3_real_1(A,D))) ) )
| r5_limfunc2(k6_seq_1(k1_numbers,k1_numbers,B,C),A) ) ) ) ) ) ).
fof(t26_limfunc2,axiom,
! [A] :
( m1_subset_1(A,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) )
=> ( ( r5_limfunc2(B,A)
& ! [D] :
( m1_subset_1(D,k1_numbers)
=> ~ ( ~ r1_xreal_0(D,A)
& ! [E] :
( m1_subset_1(E,k1_numbers)
=> ~ ( ~ r1_xreal_0(D,E)
& ~ r1_xreal_0(E,A)
& r2_hidden(E,k4_relset_1(k1_numbers,k1_numbers,k8_seq_1(k1_numbers,k1_numbers,B,C))) ) ) ) ) )
=> ( ! [D] :
( m1_subset_1(D,k1_numbers)
=> ! [E] :
( m1_subset_1(E,k1_numbers)
=> ~ ( ~ r1_xreal_0(D,np__0)
& ~ r1_xreal_0(E,np__0)
& ! [F] :
( m1_subset_1(F,k1_numbers)
=> ( r2_hidden(F,k5_subset_1(k1_numbers,k4_relset_1(k1_numbers,k1_numbers,C),k2_rcomp_1(A,k3_real_1(A,D))))
=> r1_xreal_0(E,k2_seq_1(k1_numbers,k1_numbers,C,F)) ) ) ) ) )
| r5_limfunc2(k8_seq_1(k1_numbers,k1_numbers,B,C),A) ) ) ) ) ) ).
fof(t27_limfunc2,axiom,
! [A] :
( m1_subset_1(A,k1_numbers)
=> ! [B] :
( m1_subset_1(B,k1_numbers)
=> ! [C] :
( ( v1_funct_1(C)
& m2_relset_1(C,k1_numbers,k1_numbers) )
=> ( ( r2_limfunc2(C,A)
=> ( r1_xreal_0(B,np__0)
| r2_limfunc2(k13_seq_1(k1_numbers,k1_numbers,C,B),A) ) )
& ( r2_limfunc2(C,A)
=> ( r1_xreal_0(np__0,B)
| r3_limfunc2(k13_seq_1(k1_numbers,k1_numbers,C,B),A) ) )
& ( r3_limfunc2(C,A)
=> ( r1_xreal_0(B,np__0)
| r3_limfunc2(k13_seq_1(k1_numbers,k1_numbers,C,B),A) ) )
& ( r3_limfunc2(C,A)
=> ( r1_xreal_0(np__0,B)
| r2_limfunc2(k13_seq_1(k1_numbers,k1_numbers,C,B),A) ) ) ) ) ) ) ).
fof(t28_limfunc2,axiom,
! [A] :
( m1_subset_1(A,k1_numbers)
=> ! [B] :
( m1_subset_1(B,k1_numbers)
=> ! [C] :
( ( v1_funct_1(C)
& m2_relset_1(C,k1_numbers,k1_numbers) )
=> ( ( r5_limfunc2(C,A)
=> ( r1_xreal_0(B,np__0)
| r5_limfunc2(k13_seq_1(k1_numbers,k1_numbers,C,B),A) ) )
& ( r5_limfunc2(C,A)
=> ( r1_xreal_0(np__0,B)
| r6_limfunc2(k13_seq_1(k1_numbers,k1_numbers,C,B),A) ) )
& ( r6_limfunc2(C,A)
=> ( r1_xreal_0(B,np__0)
| r6_limfunc2(k13_seq_1(k1_numbers,k1_numbers,C,B),A) ) )
& ( r6_limfunc2(C,A)
=> ( r1_xreal_0(np__0,B)
| r5_limfunc2(k13_seq_1(k1_numbers,k1_numbers,C,B),A) ) ) ) ) ) ) ).
fof(t29_limfunc2,axiom,
! [A] :
( m1_subset_1(A,k1_numbers)
=> ! [B] :
( ( v1_funct_1(B)
& m2_relset_1(B,k1_numbers,k1_numbers) )
=> ( ( r2_limfunc2(B,A)
| r3_limfunc2(B,A) )
=> r2_limfunc2(k21_seq_1(k1_numbers,k1_numbers,B),A) ) ) ) ).
fof(t30_limfunc2,axiom,
! [A] :
( m1_subset_1(A,k1_numbers)
=> ! [B] :
( ( v1_funct_1(B)
& m2_relset_1(B,k1_numbers,k1_numbers) )
=> ( ( r5_limfunc2(B,A)
| r6_limfunc2(B,A) )
=> r5_limfunc2(k21_seq_1(k1_numbers,k1_numbers,B),A) ) ) ) ).
fof(t31_limfunc2,axiom,
! [A] :
( m1_subset_1(A,k1_numbers)
=> ! [B] :
( ( v1_funct_1(B)
& m2_relset_1(B,k1_numbers,k1_numbers) )
=> ( ! [C] :
( m1_subset_1(C,k1_numbers)
=> ~ ( ~ r1_xreal_0(A,C)
& ! [D] :
( m1_subset_1(D,k1_numbers)
=> ~ ( ~ r1_xreal_0(D,C)
& ~ r1_xreal_0(A,D)
& r2_hidden(D,k4_relset_1(k1_numbers,k1_numbers,B)) ) ) ) )
=> ( ! [C] :
( m1_subset_1(C,k1_numbers)
=> ~ ( ~ r1_xreal_0(C,np__0)
& r3_rfunct_2(B,k2_rcomp_1(k5_real_1(A,C),A))
& ~ r1_rfunct_1(B,k2_rcomp_1(k5_real_1(A,C),A)) ) )
| r2_limfunc2(B,A) ) ) ) ) ).
fof(t32_limfunc2,axiom,
! [A] :
( m1_subset_1(A,k1_numbers)
=> ! [B] :
( ( v1_funct_1(B)
& m2_relset_1(B,k1_numbers,k1_numbers) )
=> ( ! [C] :
( m1_subset_1(C,k1_numbers)
=> ~ ( ~ r1_xreal_0(A,C)
& ! [D] :
( m1_subset_1(D,k1_numbers)
=> ~ ( ~ r1_xreal_0(D,C)
& ~ r1_xreal_0(A,D)
& r2_hidden(D,k4_relset_1(k1_numbers,k1_numbers,B)) ) ) ) )
=> ( ! [C] :
( m1_subset_1(C,k1_numbers)
=> ~ ( ~ r1_xreal_0(C,np__0)
& r1_rfunct_2(B,k2_rcomp_1(k5_real_1(A,C),A))
& ~ r1_rfunct_1(B,k2_rcomp_1(k5_real_1(A,C),A)) ) )
| r2_limfunc2(B,A) ) ) ) ) ).
fof(t33_limfunc2,axiom,
! [A] :
( m1_subset_1(A,k1_numbers)
=> ! [B] :
( ( v1_funct_1(B)
& m2_relset_1(B,k1_numbers,k1_numbers) )
=> ( ! [C] :
( m1_subset_1(C,k1_numbers)
=> ~ ( ~ r1_xreal_0(A,C)
& ! [D] :
( m1_subset_1(D,k1_numbers)
=> ~ ( ~ r1_xreal_0(D,C)
& ~ r1_xreal_0(A,D)
& r2_hidden(D,k4_relset_1(k1_numbers,k1_numbers,B)) ) ) ) )
=> ( ! [C] :
( m1_subset_1(C,k1_numbers)
=> ~ ( ~ r1_xreal_0(C,np__0)
& r4_rfunct_2(B,k2_rcomp_1(k5_real_1(A,C),A))
& ~ r2_rfunct_1(B,k2_rcomp_1(k5_real_1(A,C),A)) ) )
| r3_limfunc2(B,A) ) ) ) ) ).
fof(t34_limfunc2,axiom,
! [A] :
( m1_subset_1(A,k1_numbers)
=> ! [B] :
( ( v1_funct_1(B)
& m2_relset_1(B,k1_numbers,k1_numbers) )
=> ( ! [C] :
( m1_subset_1(C,k1_numbers)
=> ~ ( ~ r1_xreal_0(A,C)
& ! [D] :
( m1_subset_1(D,k1_numbers)
=> ~ ( ~ r1_xreal_0(D,C)
& ~ r1_xreal_0(A,D)
& r2_hidden(D,k4_relset_1(k1_numbers,k1_numbers,B)) ) ) ) )
=> ( ! [C] :
( m1_subset_1(C,k1_numbers)
=> ~ ( ~ r1_xreal_0(C,np__0)
& r2_rfunct_2(B,k2_rcomp_1(k5_real_1(A,C),A))
& ~ r2_rfunct_1(B,k2_rcomp_1(k5_real_1(A,C),A)) ) )
| r3_limfunc2(B,A) ) ) ) ) ).
fof(t35_limfunc2,axiom,
! [A] :
( m1_subset_1(A,k1_numbers)
=> ! [B] :
( ( v1_funct_1(B)
& m2_relset_1(B,k1_numbers,k1_numbers) )
=> ( ! [C] :
( m1_subset_1(C,k1_numbers)
=> ~ ( ~ r1_xreal_0(C,A)
& ! [D] :
( m1_subset_1(D,k1_numbers)
=> ~ ( ~ r1_xreal_0(C,D)
& ~ r1_xreal_0(D,A)
& r2_hidden(D,k4_relset_1(k1_numbers,k1_numbers,B)) ) ) ) )
=> ( ! [C] :
( m1_subset_1(C,k1_numbers)
=> ~ ( ~ r1_xreal_0(C,np__0)
& r4_rfunct_2(B,k2_rcomp_1(A,k3_real_1(A,C)))
& ~ r1_rfunct_1(B,k2_rcomp_1(A,k3_real_1(A,C))) ) )
| r5_limfunc2(B,A) ) ) ) ) ).
fof(t36_limfunc2,axiom,
! [A] :
( m1_subset_1(A,k1_numbers)
=> ! [B] :
( ( v1_funct_1(B)
& m2_relset_1(B,k1_numbers,k1_numbers) )
=> ( ! [C] :
( m1_subset_1(C,k1_numbers)
=> ~ ( ~ r1_xreal_0(C,A)
& ! [D] :
( m1_subset_1(D,k1_numbers)
=> ~ ( ~ r1_xreal_0(C,D)
& ~ r1_xreal_0(D,A)
& r2_hidden(D,k4_relset_1(k1_numbers,k1_numbers,B)) ) ) ) )
=> ( ! [C] :
( m1_subset_1(C,k1_numbers)
=> ~ ( ~ r1_xreal_0(C,np__0)
& r2_rfunct_2(B,k2_rcomp_1(A,k3_real_1(A,C)))
& ~ r1_rfunct_1(B,k2_rcomp_1(A,k3_real_1(A,C))) ) )
| r5_limfunc2(B,A) ) ) ) ) ).
fof(t37_limfunc2,axiom,
! [A] :
( m1_subset_1(A,k1_numbers)
=> ! [B] :
( ( v1_funct_1(B)
& m2_relset_1(B,k1_numbers,k1_numbers) )
=> ( ! [C] :
( m1_subset_1(C,k1_numbers)
=> ~ ( ~ r1_xreal_0(C,A)
& ! [D] :
( m1_subset_1(D,k1_numbers)
=> ~ ( ~ r1_xreal_0(C,D)
& ~ r1_xreal_0(D,A)
& r2_hidden(D,k4_relset_1(k1_numbers,k1_numbers,B)) ) ) ) )
=> ( ! [C] :
( m1_subset_1(C,k1_numbers)
=> ~ ( ~ r1_xreal_0(C,np__0)
& r3_rfunct_2(B,k2_rcomp_1(A,k3_real_1(A,C)))
& ~ r2_rfunct_1(B,k2_rcomp_1(A,k3_real_1(A,C))) ) )
| r6_limfunc2(B,A) ) ) ) ) ).
fof(t38_limfunc2,axiom,
! [A] :
( m1_subset_1(A,k1_numbers)
=> ! [B] :
( ( v1_funct_1(B)
& m2_relset_1(B,k1_numbers,k1_numbers) )
=> ( ! [C] :
( m1_subset_1(C,k1_numbers)
=> ~ ( ~ r1_xreal_0(C,A)
& ! [D] :
( m1_subset_1(D,k1_numbers)
=> ~ ( ~ r1_xreal_0(C,D)
& ~ r1_xreal_0(D,A)
& r2_hidden(D,k4_relset_1(k1_numbers,k1_numbers,B)) ) ) ) )
=> ( ! [C] :
( m1_subset_1(C,k1_numbers)
=> ~ ( ~ r1_xreal_0(C,np__0)
& r1_rfunct_2(B,k2_rcomp_1(A,k3_real_1(A,C)))
& ~ r2_rfunct_1(B,k2_rcomp_1(A,k3_real_1(A,C))) ) )
| r6_limfunc2(B,A) ) ) ) ) ).
fof(t39_limfunc2,axiom,
! [A] :
( m1_subset_1(A,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) )
=> ( ( r2_limfunc2(B,A)
& ! [D] :
( m1_subset_1(D,k1_numbers)
=> ~ ( ~ r1_xreal_0(A,D)
& ! [E] :
( m1_subset_1(E,k1_numbers)
=> ~ ( ~ r1_xreal_0(E,D)
& ~ r1_xreal_0(A,E)
& r2_hidden(E,k4_relset_1(k1_numbers,k1_numbers,C)) ) ) ) ) )
=> ( ! [D] :
( m1_subset_1(D,k1_numbers)
=> ~ ( ~ r1_xreal_0(D,np__0)
& r1_tarski(k5_subset_1(k1_numbers,k4_relset_1(k1_numbers,k1_numbers,C),k2_rcomp_1(k5_real_1(A,D),A)),k5_subset_1(k1_numbers,k4_relset_1(k1_numbers,k1_numbers,B),k2_rcomp_1(k5_real_1(A,D),A)))
& ! [E] :
( m1_subset_1(E,k1_numbers)
=> ( r2_hidden(E,k5_subset_1(k1_numbers,k4_relset_1(k1_numbers,k1_numbers,C),k2_rcomp_1(k5_real_1(A,D),A)))
=> r1_xreal_0(k2_seq_1(k1_numbers,k1_numbers,B,E),k2_seq_1(k1_numbers,k1_numbers,C,E)) ) ) ) )
| r2_limfunc2(C,A) ) ) ) ) ) ).
fof(t40_limfunc2,axiom,
! [A] :
( m1_subset_1(A,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) )
=> ( ( r3_limfunc2(B,A)
& ! [D] :
( m1_subset_1(D,k1_numbers)
=> ~ ( ~ r1_xreal_0(A,D)
& ! [E] :
( m1_subset_1(E,k1_numbers)
=> ~ ( ~ r1_xreal_0(E,D)
& ~ r1_xreal_0(A,E)
& r2_hidden(E,k4_relset_1(k1_numbers,k1_numbers,C)) ) ) ) ) )
=> ( ! [D] :
( m1_subset_1(D,k1_numbers)
=> ~ ( ~ r1_xreal_0(D,np__0)
& r1_tarski(k5_subset_1(k1_numbers,k4_relset_1(k1_numbers,k1_numbers,C),k2_rcomp_1(k5_real_1(A,D),A)),k5_subset_1(k1_numbers,k4_relset_1(k1_numbers,k1_numbers,B),k2_rcomp_1(k5_real_1(A,D),A)))
& ! [E] :
( m1_subset_1(E,k1_numbers)
=> ( r2_hidden(E,k5_subset_1(k1_numbers,k4_relset_1(k1_numbers,k1_numbers,C),k2_rcomp_1(k5_real_1(A,D),A)))
=> r1_xreal_0(k2_seq_1(k1_numbers,k1_numbers,C,E),k2_seq_1(k1_numbers,k1_numbers,B,E)) ) ) ) )
| r3_limfunc2(C,A) ) ) ) ) ) ).
fof(t41_limfunc2,axiom,
! [A] :
( m1_subset_1(A,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) )
=> ( ( r5_limfunc2(B,A)
& ! [D] :
( m1_subset_1(D,k1_numbers)
=> ~ ( ~ r1_xreal_0(D,A)
& ! [E] :
( m1_subset_1(E,k1_numbers)
=> ~ ( ~ r1_xreal_0(D,E)
& ~ r1_xreal_0(E,A)
& r2_hidden(E,k4_relset_1(k1_numbers,k1_numbers,C)) ) ) ) ) )
=> ( ! [D] :
( m1_subset_1(D,k1_numbers)
=> ~ ( ~ r1_xreal_0(D,np__0)
& r1_tarski(k5_subset_1(k1_numbers,k4_relset_1(k1_numbers,k1_numbers,C),k2_rcomp_1(A,k3_real_1(A,D))),k5_subset_1(k1_numbers,k4_relset_1(k1_numbers,k1_numbers,B),k2_rcomp_1(A,k3_real_1(A,D))))
& ! [E] :
( m1_subset_1(E,k1_numbers)
=> ( r2_hidden(E,k5_subset_1(k1_numbers,k4_relset_1(k1_numbers,k1_numbers,C),k2_rcomp_1(A,k3_real_1(A,D))))
=> r1_xreal_0(k2_seq_1(k1_numbers,k1_numbers,B,E),k2_seq_1(k1_numbers,k1_numbers,C,E)) ) ) ) )
| r5_limfunc2(C,A) ) ) ) ) ) ).
fof(t42_limfunc2,axiom,
! [A] :
( m1_subset_1(A,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) )
=> ( ( r6_limfunc2(B,A)
& ! [D] :
( m1_subset_1(D,k1_numbers)
=> ~ ( ~ r1_xreal_0(D,A)
& ! [E] :
( m1_subset_1(E,k1_numbers)
=> ~ ( ~ r1_xreal_0(D,E)
& ~ r1_xreal_0(E,A)
& r2_hidden(E,k4_relset_1(k1_numbers,k1_numbers,C)) ) ) ) ) )
=> ( ! [D] :
( m1_subset_1(D,k1_numbers)
=> ~ ( ~ r1_xreal_0(D,np__0)
& r1_tarski(k5_subset_1(k1_numbers,k4_relset_1(k1_numbers,k1_numbers,C),k2_rcomp_1(A,k3_real_1(A,D))),k5_subset_1(k1_numbers,k4_relset_1(k1_numbers,k1_numbers,B),k2_rcomp_1(A,k3_real_1(A,D))))
& ! [E] :
( m1_subset_1(E,k1_numbers)
=> ( r2_hidden(E,k5_subset_1(k1_numbers,k4_relset_1(k1_numbers,k1_numbers,C),k2_rcomp_1(A,k3_real_1(A,D))))
=> r1_xreal_0(k2_seq_1(k1_numbers,k1_numbers,C,E),k2_seq_1(k1_numbers,k1_numbers,B,E)) ) ) ) )
| r6_limfunc2(C,A) ) ) ) ) ) ).
fof(t43_limfunc2,axiom,
! [A] :
( m1_subset_1(A,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) )
=> ( r2_limfunc2(B,A)
=> ( ! [D] :
( m1_subset_1(D,k1_numbers)
=> ~ ( ~ r1_xreal_0(D,np__0)
& r1_tarski(k2_rcomp_1(k5_real_1(A,D),A),k5_subset_1(k1_numbers,k4_relset_1(k1_numbers,k1_numbers,C),k4_relset_1(k1_numbers,k1_numbers,B)))
& ! [E] :
( m1_subset_1(E,k1_numbers)
=> ( r2_hidden(E,k2_rcomp_1(k5_real_1(A,D),A))
=> r1_xreal_0(k2_seq_1(k1_numbers,k1_numbers,B,E),k2_seq_1(k1_numbers,k1_numbers,C,E)) ) ) ) )
| r2_limfunc2(C,A) ) ) ) ) ) ).
fof(t44_limfunc2,axiom,
! [A] :
( m1_subset_1(A,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) )
=> ( r3_limfunc2(B,A)
=> ( ! [D] :
( m1_subset_1(D,k1_numbers)
=> ~ ( ~ r1_xreal_0(D,np__0)
& r1_tarski(k2_rcomp_1(k5_real_1(A,D),A),k5_subset_1(k1_numbers,k4_relset_1(k1_numbers,k1_numbers,C),k4_relset_1(k1_numbers,k1_numbers,B)))
& ! [E] :
( m1_subset_1(E,k1_numbers)
=> ( r2_hidden(E,k2_rcomp_1(k5_real_1(A,D),A))
=> r1_xreal_0(k2_seq_1(k1_numbers,k1_numbers,C,E),k2_seq_1(k1_numbers,k1_numbers,B,E)) ) ) ) )
| r3_limfunc2(C,A) ) ) ) ) ) ).
fof(t45_limfunc2,axiom,
! [A] :
( m1_subset_1(A,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) )
=> ( r5_limfunc2(B,A)
=> ( ! [D] :
( m1_subset_1(D,k1_numbers)
=> ~ ( ~ r1_xreal_0(D,np__0)
& r1_tarski(k2_rcomp_1(A,k3_real_1(A,D)),k5_subset_1(k1_numbers,k4_relset_1(k1_numbers,k1_numbers,C),k4_relset_1(k1_numbers,k1_numbers,B)))
& ! [E] :
( m1_subset_1(E,k1_numbers)
=> ( r2_hidden(E,k2_rcomp_1(A,k3_real_1(A,D)))
=> r1_xreal_0(k2_seq_1(k1_numbers,k1_numbers,B,E),k2_seq_1(k1_numbers,k1_numbers,C,E)) ) ) ) )
| r5_limfunc2(C,A) ) ) ) ) ) ).
fof(t46_limfunc2,axiom,
! [A] :
( m1_subset_1(A,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) )
=> ( r6_limfunc2(B,A)
=> ( ! [D] :
( m1_subset_1(D,k1_numbers)
=> ~ ( ~ r1_xreal_0(D,np__0)
& r1_tarski(k2_rcomp_1(A,k3_real_1(A,D)),k5_subset_1(k1_numbers,k4_relset_1(k1_numbers,k1_numbers,C),k4_relset_1(k1_numbers,k1_numbers,B)))
& ! [E] :
( m1_subset_1(E,k1_numbers)
=> ( r2_hidden(E,k2_rcomp_1(A,k3_real_1(A,D)))
=> r1_xreal_0(k2_seq_1(k1_numbers,k1_numbers,C,E),k2_seq_1(k1_numbers,k1_numbers,B,E)) ) ) ) )
| r6_limfunc2(C,A) ) ) ) ) ) ).
fof(d7_limfunc2,axiom,
! [A] :
( ( v1_funct_1(A)
& m2_relset_1(A,k1_numbers,k1_numbers) )
=> ! [B] :
( m1_subset_1(B,k1_numbers)
=> ( r1_limfunc2(A,B)
=> ! [C] :
( m1_subset_1(C,k1_numbers)
=> ( C = k1_limfunc2(A,B)
<=> ! [D] :
( ( v1_funct_1(D)
& v1_funct_2(D,k5_numbers,k1_numbers)
& m2_relset_1(D,k5_numbers,k1_numbers) )
=> ( ( v4_seq_2(D)
& k2_seq_2(D) = B
& r1_tarski(k1_rfunct_2(D),k5_subset_1(k1_numbers,k4_relset_1(k1_numbers,k1_numbers,A),k12_prob_1(B))) )
=> ( v4_seq_2(k2_rfunct_2(A,D))
& k2_seq_2(k2_rfunct_2(A,D)) = C ) ) ) ) ) ) ) ) ).
fof(d8_limfunc2,axiom,
! [A] :
( ( v1_funct_1(A)
& m2_relset_1(A,k1_numbers,k1_numbers) )
=> ! [B] :
( m1_subset_1(B,k1_numbers)
=> ( r4_limfunc2(A,B)
=> ! [C] :
( m1_subset_1(C,k1_numbers)
=> ( C = k2_limfunc2(A,B)
<=> ! [D] :
( ( v1_funct_1(D)
& v1_funct_2(D,k5_numbers,k1_numbers)
& m2_relset_1(D,k5_numbers,k1_numbers) )
=> ( ( v4_seq_2(D)
& k2_seq_2(D) = B
& r1_tarski(k1_rfunct_2(D),k5_subset_1(k1_numbers,k4_relset_1(k1_numbers,k1_numbers,A),k4_limfunc1(B))) )
=> ( v4_seq_2(k2_rfunct_2(A,D))
& k2_seq_2(k2_rfunct_2(A,D)) = C ) ) ) ) ) ) ) ) ).
fof(t47_limfunc2,axiom,
$true ).
fof(t48_limfunc2,axiom,
$true ).
fof(t49_limfunc2,axiom,
! [A] :
( m1_subset_1(A,k1_numbers)
=> ! [B] :
( m1_subset_1(B,k1_numbers)
=> ! [C] :
( ( v1_funct_1(C)
& m2_relset_1(C,k1_numbers,k1_numbers) )
=> ( r1_limfunc2(C,A)
=> ( k1_limfunc2(C,A) = B
<=> ! [D] :
( m1_subset_1(D,k1_numbers)
=> ~ ( ~ r1_xreal_0(D,np__0)
& ! [E] :
( m1_subset_1(E,k1_numbers)
=> ~ ( ~ r1_xreal_0(A,E)
& ! [F] :
( m1_subset_1(F,k1_numbers)
=> ~ ( ~ r1_xreal_0(F,E)
& ~ r1_xreal_0(A,F)
& r2_hidden(F,k4_relset_1(k1_numbers,k1_numbers,C))
& r1_xreal_0(D,k18_complex1(k5_real_1(k2_seq_1(k1_numbers,k1_numbers,C,F),B))) ) ) ) ) ) ) ) ) ) ) ) ).
fof(t50_limfunc2,axiom,
! [A] :
( m1_subset_1(A,k1_numbers)
=> ! [B] :
( m1_subset_1(B,k1_numbers)
=> ! [C] :
( ( v1_funct_1(C)
& m2_relset_1(C,k1_numbers,k1_numbers) )
=> ( r4_limfunc2(C,A)
=> ( k2_limfunc2(C,A) = B
<=> ! [D] :
( m1_subset_1(D,k1_numbers)
=> ~ ( ~ r1_xreal_0(D,np__0)
& ! [E] :
( m1_subset_1(E,k1_numbers)
=> ~ ( ~ r1_xreal_0(E,A)
& ! [F] :
( m1_subset_1(F,k1_numbers)
=> ~ ( ~ r1_xreal_0(E,F)
& ~ r1_xreal_0(F,A)
& r2_hidden(F,k4_relset_1(k1_numbers,k1_numbers,C))
& r1_xreal_0(D,k18_complex1(k5_real_1(k2_seq_1(k1_numbers,k1_numbers,C,F),B))) ) ) ) ) ) ) ) ) ) ) ) ).
fof(t51_limfunc2,axiom,
! [A] :
( m1_subset_1(A,k1_numbers)
=> ! [B] :
( m1_subset_1(B,k1_numbers)
=> ! [C] :
( ( v1_funct_1(C)
& m2_relset_1(C,k1_numbers,k1_numbers) )
=> ( r1_limfunc2(C,A)
=> ( r1_limfunc2(k13_seq_1(k1_numbers,k1_numbers,C,B),A)
& k1_limfunc2(k13_seq_1(k1_numbers,k1_numbers,C,B),A) = k4_real_1(B,k1_limfunc2(C,A)) ) ) ) ) ) ).
fof(t52_limfunc2,axiom,
! [A] :
( m1_subset_1(A,k1_numbers)
=> ! [B] :
( ( v1_funct_1(B)
& m2_relset_1(B,k1_numbers,k1_numbers) )
=> ( r1_limfunc2(B,A)
=> ( r1_limfunc2(k16_seq_1(k1_numbers,k1_numbers,B),A)
& k1_limfunc2(k16_seq_1(k1_numbers,k1_numbers,B),A) = k1_real_1(k1_limfunc2(B,A)) ) ) ) ) ).
fof(t53_limfunc2,axiom,
! [A] :
( m1_subset_1(A,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) )
=> ( ( r1_limfunc2(B,A)
& r1_limfunc2(C,A)
& ! [D] :
( m1_subset_1(D,k1_numbers)
=> ~ ( ~ r1_xreal_0(A,D)
& ! [E] :
( m1_subset_1(E,k1_numbers)
=> ~ ( ~ r1_xreal_0(E,D)
& ~ r1_xreal_0(A,E)
& r2_hidden(E,k4_relset_1(k1_numbers,k1_numbers,k6_seq_1(k1_numbers,k1_numbers,B,C))) ) ) ) ) )
=> ( r1_limfunc2(k6_seq_1(k1_numbers,k1_numbers,B,C),A)
& k1_limfunc2(k6_seq_1(k1_numbers,k1_numbers,B,C),A) = k3_real_1(k1_limfunc2(B,A),k1_limfunc2(C,A)) ) ) ) ) ) ).
fof(t54_limfunc2,axiom,
! [A] :
( m1_subset_1(A,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) )
=> ( ( r1_limfunc2(B,A)
& r1_limfunc2(C,A)
& ! [D] :
( m1_subset_1(D,k1_numbers)
=> ~ ( ~ r1_xreal_0(A,D)
& ! [E] :
( m1_subset_1(E,k1_numbers)
=> ~ ( ~ r1_xreal_0(E,D)
& ~ r1_xreal_0(A,E)
& r2_hidden(E,k4_relset_1(k1_numbers,k1_numbers,k7_seq_1(k1_numbers,k1_numbers,B,C))) ) ) ) ) )
=> ( r1_limfunc2(k7_seq_1(k1_numbers,k1_numbers,B,C),A)
& k1_limfunc2(k7_seq_1(k1_numbers,k1_numbers,B,C),A) = k5_real_1(k1_limfunc2(B,A),k1_limfunc2(C,A)) ) ) ) ) ) ).
fof(t55_limfunc2,axiom,
! [A] :
( m1_subset_1(A,k1_numbers)
=> ! [B] :
( ( v1_funct_1(B)
& m2_relset_1(B,k1_numbers,k1_numbers) )
=> ( ( r1_limfunc2(B,A)
& k11_relset_1(k1_numbers,k1_numbers,B,k1_tarski(np__0)) = k1_xboole_0 )
=> ( k1_limfunc2(B,A) = np__0
| ( r1_limfunc2(k4_rfunct_1(k1_numbers,k1_numbers,B),A)
& k1_limfunc2(k4_rfunct_1(k1_numbers,k1_numbers,B),A) = k2_real_1(k1_limfunc2(B,A)) ) ) ) ) ) ).
fof(t56_limfunc2,axiom,
! [A] :
( m1_subset_1(A,k1_numbers)
=> ! [B] :
( ( v1_funct_1(B)
& m2_relset_1(B,k1_numbers,k1_numbers) )
=> ( r1_limfunc2(B,A)
=> ( r1_limfunc2(k21_seq_1(k1_numbers,k1_numbers,B),A)
& k1_limfunc2(k21_seq_1(k1_numbers,k1_numbers,B),A) = k18_complex1(k1_limfunc2(B,A)) ) ) ) ) ).
fof(t57_limfunc2,axiom,
! [A] :
( m1_subset_1(A,k1_numbers)
=> ! [B] :
( ( v1_funct_1(B)
& m2_relset_1(B,k1_numbers,k1_numbers) )
=> ( ( r1_limfunc2(B,A)
& ! [C] :
( m1_subset_1(C,k1_numbers)
=> ~ ( ~ r1_xreal_0(A,C)
& ! [D] :
( m1_subset_1(D,k1_numbers)
=> ~ ( ~ r1_xreal_0(D,C)
& ~ r1_xreal_0(A,D)
& r2_hidden(D,k4_relset_1(k1_numbers,k1_numbers,B))
& k2_seq_1(k1_numbers,k1_numbers,B,D) != np__0 ) ) ) ) )
=> ( k1_limfunc2(B,A) = np__0
| ( r1_limfunc2(k4_rfunct_1(k1_numbers,k1_numbers,B),A)
& k1_limfunc2(k4_rfunct_1(k1_numbers,k1_numbers,B),A) = k2_real_1(k1_limfunc2(B,A)) ) ) ) ) ) ).
fof(t58_limfunc2,axiom,
! [A] :
( m1_subset_1(A,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) )
=> ( ( r1_limfunc2(B,A)
& r1_limfunc2(C,A)
& ! [D] :
( m1_subset_1(D,k1_numbers)
=> ~ ( ~ r1_xreal_0(A,D)
& ! [E] :
( m1_subset_1(E,k1_numbers)
=> ~ ( ~ r1_xreal_0(E,D)
& ~ r1_xreal_0(A,E)
& r2_hidden(E,k4_relset_1(k1_numbers,k1_numbers,k8_seq_1(k1_numbers,k1_numbers,B,C))) ) ) ) ) )
=> ( r1_limfunc2(k8_seq_1(k1_numbers,k1_numbers,B,C),A)
& k1_limfunc2(k8_seq_1(k1_numbers,k1_numbers,B,C),A) = k4_real_1(k1_limfunc2(B,A),k1_limfunc2(C,A)) ) ) ) ) ) ).
fof(t59_limfunc2,axiom,
! [A] :
( m1_subset_1(A,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) )
=> ( ( r1_limfunc2(B,A)
& r1_limfunc2(C,A)
& ! [D] :
( m1_subset_1(D,k1_numbers)
=> ~ ( ~ r1_xreal_0(A,D)
& ! [E] :
( m1_subset_1(E,k1_numbers)
=> ~ ( ~ r1_xreal_0(E,D)
& ~ r1_xreal_0(A,E)
& r2_hidden(E,k4_relset_1(k1_numbers,k1_numbers,k2_rfunct_1(k1_numbers,k1_numbers,B,C))) ) ) ) ) )
=> ( k1_limfunc2(C,A) = np__0
| ( r1_limfunc2(k2_rfunct_1(k1_numbers,k1_numbers,B,C),A)
& k1_limfunc2(k2_rfunct_1(k1_numbers,k1_numbers,B,C),A) = k6_real_1(k1_limfunc2(B,A),k1_limfunc2(C,A)) ) ) ) ) ) ) ).
fof(t60_limfunc2,axiom,
! [A] :
( m1_subset_1(A,k1_numbers)
=> ! [B] :
( m1_subset_1(B,k1_numbers)
=> ! [C] :
( ( v1_funct_1(C)
& m2_relset_1(C,k1_numbers,k1_numbers) )
=> ( r4_limfunc2(C,A)
=> ( r4_limfunc2(k13_seq_1(k1_numbers,k1_numbers,C,B),A)
& k2_limfunc2(k13_seq_1(k1_numbers,k1_numbers,C,B),A) = k4_real_1(B,k2_limfunc2(C,A)) ) ) ) ) ) ).
fof(t61_limfunc2,axiom,
! [A] :
( m1_subset_1(A,k1_numbers)
=> ! [B] :
( ( v1_funct_1(B)
& m2_relset_1(B,k1_numbers,k1_numbers) )
=> ( r4_limfunc2(B,A)
=> ( r4_limfunc2(k16_seq_1(k1_numbers,k1_numbers,B),A)
& k2_limfunc2(k16_seq_1(k1_numbers,k1_numbers,B),A) = k1_real_1(k2_limfunc2(B,A)) ) ) ) ) ).
fof(t62_limfunc2,axiom,
! [A] :
( m1_subset_1(A,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) )
=> ( ( r4_limfunc2(B,A)
& r4_limfunc2(C,A)
& ! [D] :
( m1_subset_1(D,k1_numbers)
=> ~ ( ~ r1_xreal_0(D,A)
& ! [E] :
( m1_subset_1(E,k1_numbers)
=> ~ ( ~ r1_xreal_0(D,E)
& ~ r1_xreal_0(E,A)
& r2_hidden(E,k4_relset_1(k1_numbers,k1_numbers,k6_seq_1(k1_numbers,k1_numbers,B,C))) ) ) ) ) )
=> ( r4_limfunc2(k6_seq_1(k1_numbers,k1_numbers,B,C),A)
& k2_limfunc2(k6_seq_1(k1_numbers,k1_numbers,B,C),A) = k3_real_1(k2_limfunc2(B,A),k2_limfunc2(C,A)) ) ) ) ) ) ).
fof(t63_limfunc2,axiom,
! [A] :
( m1_subset_1(A,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) )
=> ( ( r4_limfunc2(B,A)
& r4_limfunc2(C,A)
& ! [D] :
( m1_subset_1(D,k1_numbers)
=> ~ ( ~ r1_xreal_0(D,A)
& ! [E] :
( m1_subset_1(E,k1_numbers)
=> ~ ( ~ r1_xreal_0(D,E)
& ~ r1_xreal_0(E,A)
& r2_hidden(E,k4_relset_1(k1_numbers,k1_numbers,k7_seq_1(k1_numbers,k1_numbers,B,C))) ) ) ) ) )
=> ( r4_limfunc2(k7_seq_1(k1_numbers,k1_numbers,B,C),A)
& k2_limfunc2(k7_seq_1(k1_numbers,k1_numbers,B,C),A) = k5_real_1(k2_limfunc2(B,A),k2_limfunc2(C,A)) ) ) ) ) ) ).
fof(t64_limfunc2,axiom,
! [A] :
( m1_subset_1(A,k1_numbers)
=> ! [B] :
( ( v1_funct_1(B)
& m2_relset_1(B,k1_numbers,k1_numbers) )
=> ( ( r4_limfunc2(B,A)
& k11_relset_1(k1_numbers,k1_numbers,B,k1_tarski(np__0)) = k1_xboole_0 )
=> ( k2_limfunc2(B,A) = np__0
| ( r4_limfunc2(k4_rfunct_1(k1_numbers,k1_numbers,B),A)
& k2_limfunc2(k4_rfunct_1(k1_numbers,k1_numbers,B),A) = k2_real_1(k2_limfunc2(B,A)) ) ) ) ) ) ).
fof(t65_limfunc2,axiom,
! [A] :
( m1_subset_1(A,k1_numbers)
=> ! [B] :
( ( v1_funct_1(B)
& m2_relset_1(B,k1_numbers,k1_numbers) )
=> ( r4_limfunc2(B,A)
=> ( r4_limfunc2(k21_seq_1(k1_numbers,k1_numbers,B),A)
& k2_limfunc2(k21_seq_1(k1_numbers,k1_numbers,B),A) = k18_complex1(k2_limfunc2(B,A)) ) ) ) ) ).
fof(t66_limfunc2,axiom,
! [A] :
( m1_subset_1(A,k1_numbers)
=> ! [B] :
( ( v1_funct_1(B)
& m2_relset_1(B,k1_numbers,k1_numbers) )
=> ( ( r4_limfunc2(B,A)
& ! [C] :
( m1_subset_1(C,k1_numbers)
=> ~ ( ~ r1_xreal_0(C,A)
& ! [D] :
( m1_subset_1(D,k1_numbers)
=> ~ ( ~ r1_xreal_0(C,D)
& ~ r1_xreal_0(D,A)
& r2_hidden(D,k4_relset_1(k1_numbers,k1_numbers,B))
& k2_seq_1(k1_numbers,k1_numbers,B,D) != np__0 ) ) ) ) )
=> ( k2_limfunc2(B,A) = np__0
| ( r4_limfunc2(k4_rfunct_1(k1_numbers,k1_numbers,B),A)
& k2_limfunc2(k4_rfunct_1(k1_numbers,k1_numbers,B),A) = k2_real_1(k2_limfunc2(B,A)) ) ) ) ) ) ).
fof(t67_limfunc2,axiom,
! [A] :
( m1_subset_1(A,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) )
=> ( ( r4_limfunc2(B,A)
& r4_limfunc2(C,A)
& ! [D] :
( m1_subset_1(D,k1_numbers)
=> ~ ( ~ r1_xreal_0(D,A)
& ! [E] :
( m1_subset_1(E,k1_numbers)
=> ~ ( ~ r1_xreal_0(D,E)
& ~ r1_xreal_0(E,A)
& r2_hidden(E,k4_relset_1(k1_numbers,k1_numbers,k8_seq_1(k1_numbers,k1_numbers,B,C))) ) ) ) ) )
=> ( r4_limfunc2(k8_seq_1(k1_numbers,k1_numbers,B,C),A)
& k2_limfunc2(k8_seq_1(k1_numbers,k1_numbers,B,C),A) = k4_real_1(k2_limfunc2(B,A),k2_limfunc2(C,A)) ) ) ) ) ) ).
fof(t68_limfunc2,axiom,
! [A] :
( m1_subset_1(A,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) )
=> ( ( r4_limfunc2(B,A)
& r4_limfunc2(C,A)
& ! [D] :
( m1_subset_1(D,k1_numbers)
=> ~ ( ~ r1_xreal_0(D,A)
& ! [E] :
( m1_subset_1(E,k1_numbers)
=> ~ ( ~ r1_xreal_0(D,E)
& ~ r1_xreal_0(E,A)
& r2_hidden(E,k4_relset_1(k1_numbers,k1_numbers,k2_rfunct_1(k1_numbers,k1_numbers,B,C))) ) ) ) ) )
=> ( k2_limfunc2(C,A) = np__0
| ( r4_limfunc2(k2_rfunct_1(k1_numbers,k1_numbers,B,C),A)
& k2_limfunc2(k2_rfunct_1(k1_numbers,k1_numbers,B,C),A) = k6_real_1(k2_limfunc2(B,A),k2_limfunc2(C,A)) ) ) ) ) ) ) ).
fof(t69_limfunc2,axiom,
! [A] :
( m1_subset_1(A,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) )
=> ( ( r1_limfunc2(B,A)
& k1_limfunc2(B,A) = np__0
& ! [D] :
( m1_subset_1(D,k1_numbers)
=> ~ ( ~ r1_xreal_0(A,D)
& ! [E] :
( m1_subset_1(E,k1_numbers)
=> ~ ( ~ r1_xreal_0(E,D)
& ~ r1_xreal_0(A,E)
& r2_hidden(E,k4_relset_1(k1_numbers,k1_numbers,k8_seq_1(k1_numbers,k1_numbers,B,C))) ) ) ) ) )
=> ( ! [D] :
( m1_subset_1(D,k1_numbers)
=> ~ ( ~ r1_xreal_0(D,np__0)
& r3_rfunct_1(C,k2_rcomp_1(k5_real_1(A,D),A)) ) )
| ( r1_limfunc2(k8_seq_1(k1_numbers,k1_numbers,B,C),A)
& k1_limfunc2(k8_seq_1(k1_numbers,k1_numbers,B,C),A) = np__0 ) ) ) ) ) ) ).
fof(t70_limfunc2,axiom,
! [A] :
( m1_subset_1(A,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) )
=> ( ( r4_limfunc2(B,A)
& k2_limfunc2(B,A) = np__0
& ! [D] :
( m1_subset_1(D,k1_numbers)
=> ~ ( ~ r1_xreal_0(D,A)
& ! [E] :
( m1_subset_1(E,k1_numbers)
=> ~ ( ~ r1_xreal_0(D,E)
& ~ r1_xreal_0(E,A)
& r2_hidden(E,k4_relset_1(k1_numbers,k1_numbers,k8_seq_1(k1_numbers,k1_numbers,B,C))) ) ) ) ) )
=> ( ! [D] :
( m1_subset_1(D,k1_numbers)
=> ~ ( ~ r1_xreal_0(D,np__0)
& r3_rfunct_1(C,k2_rcomp_1(A,k3_real_1(A,D))) ) )
| ( r4_limfunc2(k8_seq_1(k1_numbers,k1_numbers,B,C),A)
& k2_limfunc2(k8_seq_1(k1_numbers,k1_numbers,B,C),A) = np__0 ) ) ) ) ) ) ).
fof(t71_limfunc2,axiom,
! [A] :
( m1_subset_1(A,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) )
=> ! [D] :
( ( v1_funct_1(D)
& m2_relset_1(D,k1_numbers,k1_numbers) )
=> ( ( r1_limfunc2(B,A)
& r1_limfunc2(C,A)
& k1_limfunc2(B,A) = k1_limfunc2(C,A)
& ! [E] :
( m1_subset_1(E,k1_numbers)
=> ~ ( ~ r1_xreal_0(A,E)
& ! [F] :
( m1_subset_1(F,k1_numbers)
=> ~ ( ~ r1_xreal_0(F,E)
& ~ r1_xreal_0(A,F)
& r2_hidden(F,k4_relset_1(k1_numbers,k1_numbers,D)) ) ) ) ) )
=> ( ! [E] :
( m1_subset_1(E,k1_numbers)
=> ~ ( ~ r1_xreal_0(E,np__0)
& ! [F] :
( m1_subset_1(F,k1_numbers)
=> ( r2_hidden(F,k5_subset_1(k1_numbers,k4_relset_1(k1_numbers,k1_numbers,D),k2_rcomp_1(k5_real_1(A,E),A)))
=> ( r1_xreal_0(k2_seq_1(k1_numbers,k1_numbers,B,F),k2_seq_1(k1_numbers,k1_numbers,D,F))
& r1_xreal_0(k2_seq_1(k1_numbers,k1_numbers,D,F),k2_seq_1(k1_numbers,k1_numbers,C,F)) ) ) )
& ( ( r1_tarski(k5_subset_1(k1_numbers,k4_relset_1(k1_numbers,k1_numbers,B),k2_rcomp_1(k5_real_1(A,E),A)),k5_subset_1(k1_numbers,k4_relset_1(k1_numbers,k1_numbers,C),k2_rcomp_1(k5_real_1(A,E),A)))
& r1_tarski(k5_subset_1(k1_numbers,k4_relset_1(k1_numbers,k1_numbers,D),k2_rcomp_1(k5_real_1(A,E),A)),k5_subset_1(k1_numbers,k4_relset_1(k1_numbers,k1_numbers,B),k2_rcomp_1(k5_real_1(A,E),A))) )
| ( r1_tarski(k5_subset_1(k1_numbers,k4_relset_1(k1_numbers,k1_numbers,C),k2_rcomp_1(k5_real_1(A,E),A)),k5_subset_1(k1_numbers,k4_relset_1(k1_numbers,k1_numbers,B),k2_rcomp_1(k5_real_1(A,E),A)))
& r1_tarski(k5_subset_1(k1_numbers,k4_relset_1(k1_numbers,k1_numbers,D),k2_rcomp_1(k5_real_1(A,E),A)),k5_subset_1(k1_numbers,k4_relset_1(k1_numbers,k1_numbers,C),k2_rcomp_1(k5_real_1(A,E),A))) ) ) ) )
| ( r1_limfunc2(D,A)
& k1_limfunc2(D,A) = k1_limfunc2(B,A) ) ) ) ) ) ) ) ).
fof(t72_limfunc2,axiom,
! [A] :
( m1_subset_1(A,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) )
=> ! [D] :
( ( v1_funct_1(D)
& m2_relset_1(D,k1_numbers,k1_numbers) )
=> ( ( r1_limfunc2(B,A)
& r1_limfunc2(C,A)
& k1_limfunc2(B,A) = k1_limfunc2(C,A) )
=> ( ! [E] :
( m1_subset_1(E,k1_numbers)
=> ~ ( ~ r1_xreal_0(E,np__0)
& r1_tarski(k2_rcomp_1(k5_real_1(A,E),A),k5_subset_1(k1_numbers,k5_subset_1(k1_numbers,k4_relset_1(k1_numbers,k1_numbers,B),k4_relset_1(k1_numbers,k1_numbers,C)),k4_relset_1(k1_numbers,k1_numbers,D)))
& ! [F] :
( m1_subset_1(F,k1_numbers)
=> ( r2_hidden(F,k2_rcomp_1(k5_real_1(A,E),A))
=> ( r1_xreal_0(k2_seq_1(k1_numbers,k1_numbers,B,F),k2_seq_1(k1_numbers,k1_numbers,D,F))
& r1_xreal_0(k2_seq_1(k1_numbers,k1_numbers,D,F),k2_seq_1(k1_numbers,k1_numbers,C,F)) ) ) ) ) )
| ( r1_limfunc2(D,A)
& k1_limfunc2(D,A) = k1_limfunc2(B,A) ) ) ) ) ) ) ) ).
fof(t73_limfunc2,axiom,
! [A] :
( m1_subset_1(A,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) )
=> ! [D] :
( ( v1_funct_1(D)
& m2_relset_1(D,k1_numbers,k1_numbers) )
=> ( ( r4_limfunc2(B,A)
& r4_limfunc2(C,A)
& k2_limfunc2(B,A) = k2_limfunc2(C,A)
& ! [E] :
( m1_subset_1(E,k1_numbers)
=> ~ ( ~ r1_xreal_0(E,A)
& ! [F] :
( m1_subset_1(F,k1_numbers)
=> ~ ( ~ r1_xreal_0(E,F)
& ~ r1_xreal_0(F,A)
& r2_hidden(F,k4_relset_1(k1_numbers,k1_numbers,D)) ) ) ) ) )
=> ( ! [E] :
( m1_subset_1(E,k1_numbers)
=> ~ ( ~ r1_xreal_0(E,np__0)
& ! [F] :
( m1_subset_1(F,k1_numbers)
=> ( r2_hidden(F,k5_subset_1(k1_numbers,k4_relset_1(k1_numbers,k1_numbers,D),k2_rcomp_1(A,k3_real_1(A,E))))
=> ( r1_xreal_0(k2_seq_1(k1_numbers,k1_numbers,B,F),k2_seq_1(k1_numbers,k1_numbers,D,F))
& r1_xreal_0(k2_seq_1(k1_numbers,k1_numbers,D,F),k2_seq_1(k1_numbers,k1_numbers,C,F)) ) ) )
& ( ( r1_tarski(k5_subset_1(k1_numbers,k4_relset_1(k1_numbers,k1_numbers,B),k2_rcomp_1(A,k3_real_1(A,E))),k5_subset_1(k1_numbers,k4_relset_1(k1_numbers,k1_numbers,C),k2_rcomp_1(A,k3_real_1(A,E))))
& r1_tarski(k5_subset_1(k1_numbers,k4_relset_1(k1_numbers,k1_numbers,D),k2_rcomp_1(A,k3_real_1(A,E))),k5_subset_1(k1_numbers,k4_relset_1(k1_numbers,k1_numbers,B),k2_rcomp_1(A,k3_real_1(A,E)))) )
| ( r1_tarski(k5_subset_1(k1_numbers,k4_relset_1(k1_numbers,k1_numbers,C),k2_rcomp_1(A,k3_real_1(A,E))),k5_subset_1(k1_numbers,k4_relset_1(k1_numbers,k1_numbers,B),k2_rcomp_1(A,k3_real_1(A,E))))
& r1_tarski(k5_subset_1(k1_numbers,k4_relset_1(k1_numbers,k1_numbers,D),k2_rcomp_1(A,k3_real_1(A,E))),k5_subset_1(k1_numbers,k4_relset_1(k1_numbers,k1_numbers,C),k2_rcomp_1(A,k3_real_1(A,E)))) ) ) ) )
| ( r4_limfunc2(D,A)
& k2_limfunc2(D,A) = k2_limfunc2(B,A) ) ) ) ) ) ) ) ).
fof(t74_limfunc2,axiom,
! [A] :
( m1_subset_1(A,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) )
=> ! [D] :
( ( v1_funct_1(D)
& m2_relset_1(D,k1_numbers,k1_numbers) )
=> ( ( r4_limfunc2(B,A)
& r4_limfunc2(C,A)
& k2_limfunc2(B,A) = k2_limfunc2(C,A) )
=> ( ! [E] :
( m1_subset_1(E,k1_numbers)
=> ~ ( ~ r1_xreal_0(E,np__0)
& r1_tarski(k2_rcomp_1(A,k3_real_1(A,E)),k5_subset_1(k1_numbers,k5_subset_1(k1_numbers,k4_relset_1(k1_numbers,k1_numbers,B),k4_relset_1(k1_numbers,k1_numbers,C)),k4_relset_1(k1_numbers,k1_numbers,D)))
& ! [F] :
( m1_subset_1(F,k1_numbers)
=> ( r2_hidden(F,k2_rcomp_1(A,k3_real_1(A,E)))
=> ( r1_xreal_0(k2_seq_1(k1_numbers,k1_numbers,B,F),k2_seq_1(k1_numbers,k1_numbers,D,F))
& r1_xreal_0(k2_seq_1(k1_numbers,k1_numbers,D,F),k2_seq_1(k1_numbers,k1_numbers,C,F)) ) ) ) ) )
| ( r4_limfunc2(D,A)
& k2_limfunc2(D,A) = k2_limfunc2(B,A) ) ) ) ) ) ) ) ).
fof(t75_limfunc2,axiom,
! [A] :
( m1_subset_1(A,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) )
=> ( ( r1_limfunc2(B,A)
& r1_limfunc2(C,A) )
=> ( ! [D] :
( m1_subset_1(D,k1_numbers)
=> ~ ( ~ r1_xreal_0(D,np__0)
& ( ( r1_tarski(k5_subset_1(k1_numbers,k4_relset_1(k1_numbers,k1_numbers,B),k2_rcomp_1(k5_real_1(A,D),A)),k5_subset_1(k1_numbers,k4_relset_1(k1_numbers,k1_numbers,C),k2_rcomp_1(k5_real_1(A,D),A)))
& ! [E] :
( m1_subset_1(E,k1_numbers)
=> ( r2_hidden(E,k5_subset_1(k1_numbers,k4_relset_1(k1_numbers,k1_numbers,B),k2_rcomp_1(k5_real_1(A,D),A)))
=> r1_xreal_0(k2_seq_1(k1_numbers,k1_numbers,B,E),k2_seq_1(k1_numbers,k1_numbers,C,E)) ) ) )
| ( r1_tarski(k5_subset_1(k1_numbers,k4_relset_1(k1_numbers,k1_numbers,C),k2_rcomp_1(k5_real_1(A,D),A)),k5_subset_1(k1_numbers,k4_relset_1(k1_numbers,k1_numbers,B),k2_rcomp_1(k5_real_1(A,D),A)))
& ! [E] :
( m1_subset_1(E,k1_numbers)
=> ( r2_hidden(E,k5_subset_1(k1_numbers,k4_relset_1(k1_numbers,k1_numbers,C),k2_rcomp_1(k5_real_1(A,D),A)))
=> r1_xreal_0(k2_seq_1(k1_numbers,k1_numbers,B,E),k2_seq_1(k1_numbers,k1_numbers,C,E)) ) ) ) ) ) )
| r1_xreal_0(k1_limfunc2(B,A),k1_limfunc2(C,A)) ) ) ) ) ) ).
fof(t76_limfunc2,axiom,
! [A] :
( m1_subset_1(A,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) )
=> ( ( r4_limfunc2(B,A)
& r4_limfunc2(C,A) )
=> ( ! [D] :
( m1_subset_1(D,k1_numbers)
=> ~ ( ~ r1_xreal_0(D,np__0)
& ( ( r1_tarski(k5_subset_1(k1_numbers,k4_relset_1(k1_numbers,k1_numbers,B),k2_rcomp_1(A,k3_real_1(A,D))),k5_subset_1(k1_numbers,k4_relset_1(k1_numbers,k1_numbers,C),k2_rcomp_1(A,k3_real_1(A,D))))
& ! [E] :
( m1_subset_1(E,k1_numbers)
=> ( r2_hidden(E,k5_subset_1(k1_numbers,k4_relset_1(k1_numbers,k1_numbers,B),k2_rcomp_1(A,k3_real_1(A,D))))
=> r1_xreal_0(k2_seq_1(k1_numbers,k1_numbers,B,E),k2_seq_1(k1_numbers,k1_numbers,C,E)) ) ) )
| ( r1_tarski(k5_subset_1(k1_numbers,k4_relset_1(k1_numbers,k1_numbers,C),k2_rcomp_1(A,k3_real_1(A,D))),k5_subset_1(k1_numbers,k4_relset_1(k1_numbers,k1_numbers,B),k2_rcomp_1(A,k3_real_1(A,D))))
& ! [E] :
( m1_subset_1(E,k1_numbers)
=> ( r2_hidden(E,k5_subset_1(k1_numbers,k4_relset_1(k1_numbers,k1_numbers,C),k2_rcomp_1(A,k3_real_1(A,D))))
=> r1_xreal_0(k2_seq_1(k1_numbers,k1_numbers,B,E),k2_seq_1(k1_numbers,k1_numbers,C,E)) ) ) ) ) ) )
| r1_xreal_0(k2_limfunc2(B,A),k2_limfunc2(C,A)) ) ) ) ) ) ).
fof(t77_limfunc2,axiom,
! [A] :
( m1_subset_1(A,k1_numbers)
=> ! [B] :
( ( v1_funct_1(B)
& m2_relset_1(B,k1_numbers,k1_numbers) )
=> ( ! [C] :
( m1_subset_1(C,k1_numbers)
=> ~ ( ~ r1_xreal_0(A,C)
& ! [D] :
( m1_subset_1(D,k1_numbers)
=> ~ ( ~ r1_xreal_0(D,C)
& ~ r1_xreal_0(A,D)
& r2_hidden(D,k4_relset_1(k1_numbers,k1_numbers,B))
& k2_seq_1(k1_numbers,k1_numbers,B,D) != np__0 ) ) ) )
=> ( ( ~ r2_limfunc2(B,A)
& ~ r3_limfunc2(B,A) )
| ( r1_limfunc2(k4_rfunct_1(k1_numbers,k1_numbers,B),A)
& k1_limfunc2(k4_rfunct_1(k1_numbers,k1_numbers,B),A) = np__0 ) ) ) ) ) ).
fof(t78_limfunc2,axiom,
! [A] :
( m1_subset_1(A,k1_numbers)
=> ! [B] :
( ( v1_funct_1(B)
& m2_relset_1(B,k1_numbers,k1_numbers) )
=> ( ! [C] :
( m1_subset_1(C,k1_numbers)
=> ~ ( ~ r1_xreal_0(C,A)
& ! [D] :
( m1_subset_1(D,k1_numbers)
=> ~ ( ~ r1_xreal_0(C,D)
& ~ r1_xreal_0(D,A)
& r2_hidden(D,k4_relset_1(k1_numbers,k1_numbers,B))
& k2_seq_1(k1_numbers,k1_numbers,B,D) != np__0 ) ) ) )
=> ( ( ~ r5_limfunc2(B,A)
& ~ r6_limfunc2(B,A) )
| ( r4_limfunc2(k4_rfunct_1(k1_numbers,k1_numbers,B),A)
& k2_limfunc2(k4_rfunct_1(k1_numbers,k1_numbers,B),A) = np__0 ) ) ) ) ) ).
fof(t79_limfunc2,axiom,
! [A] :
( m1_subset_1(A,k1_numbers)
=> ! [B] :
( ( v1_funct_1(B)
& m2_relset_1(B,k1_numbers,k1_numbers) )
=> ( ( r1_limfunc2(B,A)
& k1_limfunc2(B,A) = np__0 )
=> ( ! [C] :
( m1_subset_1(C,k1_numbers)
=> ~ ( ~ r1_xreal_0(C,np__0)
& ! [D] :
( m1_subset_1(D,k1_numbers)
=> ~ ( r2_hidden(D,k5_subset_1(k1_numbers,k4_relset_1(k1_numbers,k1_numbers,B),k2_rcomp_1(k5_real_1(A,C),A)))
& r1_xreal_0(k2_seq_1(k1_numbers,k1_numbers,B,D),np__0) ) ) ) )
| r2_limfunc2(k4_rfunct_1(k1_numbers,k1_numbers,B),A) ) ) ) ) ).
fof(t80_limfunc2,axiom,
! [A] :
( m1_subset_1(A,k1_numbers)
=> ! [B] :
( ( v1_funct_1(B)
& m2_relset_1(B,k1_numbers,k1_numbers) )
=> ( ( r1_limfunc2(B,A)
& k1_limfunc2(B,A) = np__0 )
=> ( ! [C] :
( m1_subset_1(C,k1_numbers)
=> ~ ( ~ r1_xreal_0(C,np__0)
& ! [D] :
( m1_subset_1(D,k1_numbers)
=> ~ ( r2_hidden(D,k5_subset_1(k1_numbers,k4_relset_1(k1_numbers,k1_numbers,B),k2_rcomp_1(k5_real_1(A,C),A)))
& r1_xreal_0(np__0,k2_seq_1(k1_numbers,k1_numbers,B,D)) ) ) ) )
| r3_limfunc2(k4_rfunct_1(k1_numbers,k1_numbers,B),A) ) ) ) ) ).
fof(t81_limfunc2,axiom,
! [A] :
( m1_subset_1(A,k1_numbers)
=> ! [B] :
( ( v1_funct_1(B)
& m2_relset_1(B,k1_numbers,k1_numbers) )
=> ( ( r4_limfunc2(B,A)
& k2_limfunc2(B,A) = np__0 )
=> ( ! [C] :
( m1_subset_1(C,k1_numbers)
=> ~ ( ~ r1_xreal_0(C,np__0)
& ! [D] :
( m1_subset_1(D,k1_numbers)
=> ~ ( r2_hidden(D,k5_subset_1(k1_numbers,k4_relset_1(k1_numbers,k1_numbers,B),k2_rcomp_1(A,k3_real_1(A,C))))
& r1_xreal_0(k2_seq_1(k1_numbers,k1_numbers,B,D),np__0) ) ) ) )
| r5_limfunc2(k4_rfunct_1(k1_numbers,k1_numbers,B),A) ) ) ) ) ).
fof(t82_limfunc2,axiom,
! [A] :
( m1_subset_1(A,k1_numbers)
=> ! [B] :
( ( v1_funct_1(B)
& m2_relset_1(B,k1_numbers,k1_numbers) )
=> ( ( r4_limfunc2(B,A)
& k2_limfunc2(B,A) = np__0 )
=> ( ! [C] :
( m1_subset_1(C,k1_numbers)
=> ~ ( ~ r1_xreal_0(C,np__0)
& ! [D] :
( m1_subset_1(D,k1_numbers)
=> ~ ( r2_hidden(D,k5_subset_1(k1_numbers,k4_relset_1(k1_numbers,k1_numbers,B),k2_rcomp_1(A,k3_real_1(A,C))))
& r1_xreal_0(np__0,k2_seq_1(k1_numbers,k1_numbers,B,D)) ) ) ) )
| r6_limfunc2(k4_rfunct_1(k1_numbers,k1_numbers,B),A) ) ) ) ) ).
fof(t83_limfunc2,axiom,
! [A] :
( m1_subset_1(A,k1_numbers)
=> ! [B] :
( ( v1_funct_1(B)
& m2_relset_1(B,k1_numbers,k1_numbers) )
=> ( ( r1_limfunc2(B,A)
& k1_limfunc2(B,A) = np__0
& ! [C] :
( m1_subset_1(C,k1_numbers)
=> ~ ( ~ r1_xreal_0(A,C)
& ! [D] :
( m1_subset_1(D,k1_numbers)
=> ~ ( ~ r1_xreal_0(D,C)
& ~ r1_xreal_0(A,D)
& r2_hidden(D,k4_relset_1(k1_numbers,k1_numbers,B))
& k2_seq_1(k1_numbers,k1_numbers,B,D) != np__0 ) ) ) ) )
=> ( ! [C] :
( m1_subset_1(C,k1_numbers)
=> ~ ( ~ r1_xreal_0(C,np__0)
& ! [D] :
( m1_subset_1(D,k1_numbers)
=> ( r2_hidden(D,k5_subset_1(k1_numbers,k4_relset_1(k1_numbers,k1_numbers,B),k2_rcomp_1(k5_real_1(A,C),A)))
=> r1_xreal_0(np__0,k2_seq_1(k1_numbers,k1_numbers,B,D)) ) ) ) )
| r2_limfunc2(k4_rfunct_1(k1_numbers,k1_numbers,B),A) ) ) ) ) ).
fof(t84_limfunc2,axiom,
! [A] :
( m1_subset_1(A,k1_numbers)
=> ! [B] :
( ( v1_funct_1(B)
& m2_relset_1(B,k1_numbers,k1_numbers) )
=> ( ( r1_limfunc2(B,A)
& k1_limfunc2(B,A) = np__0
& ! [C] :
( m1_subset_1(C,k1_numbers)
=> ~ ( ~ r1_xreal_0(A,C)
& ! [D] :
( m1_subset_1(D,k1_numbers)
=> ~ ( ~ r1_xreal_0(D,C)
& ~ r1_xreal_0(A,D)
& r2_hidden(D,k4_relset_1(k1_numbers,k1_numbers,B))
& k2_seq_1(k1_numbers,k1_numbers,B,D) != np__0 ) ) ) ) )
=> ( ! [C] :
( m1_subset_1(C,k1_numbers)
=> ~ ( ~ r1_xreal_0(C,np__0)
& ! [D] :
( m1_subset_1(D,k1_numbers)
=> ( r2_hidden(D,k5_subset_1(k1_numbers,k4_relset_1(k1_numbers,k1_numbers,B),k2_rcomp_1(k5_real_1(A,C),A)))
=> r1_xreal_0(k2_seq_1(k1_numbers,k1_numbers,B,D),np__0) ) ) ) )
| r3_limfunc2(k4_rfunct_1(k1_numbers,k1_numbers,B),A) ) ) ) ) ).
fof(t85_limfunc2,axiom,
! [A] :
( m1_subset_1(A,k1_numbers)
=> ! [B] :
( ( v1_funct_1(B)
& m2_relset_1(B,k1_numbers,k1_numbers) )
=> ( ( r4_limfunc2(B,A)
& k2_limfunc2(B,A) = np__0
& ! [C] :
( m1_subset_1(C,k1_numbers)
=> ~ ( ~ r1_xreal_0(C,A)
& ! [D] :
( m1_subset_1(D,k1_numbers)
=> ~ ( ~ r1_xreal_0(C,D)
& ~ r1_xreal_0(D,A)
& r2_hidden(D,k4_relset_1(k1_numbers,k1_numbers,B))
& k2_seq_1(k1_numbers,k1_numbers,B,D) != np__0 ) ) ) ) )
=> ( ! [C] :
( m1_subset_1(C,k1_numbers)
=> ~ ( ~ r1_xreal_0(C,np__0)
& ! [D] :
( m1_subset_1(D,k1_numbers)
=> ( r2_hidden(D,k5_subset_1(k1_numbers,k4_relset_1(k1_numbers,k1_numbers,B),k2_rcomp_1(A,k3_real_1(A,C))))
=> r1_xreal_0(np__0,k2_seq_1(k1_numbers,k1_numbers,B,D)) ) ) ) )
| r5_limfunc2(k4_rfunct_1(k1_numbers,k1_numbers,B),A) ) ) ) ) ).
fof(t86_limfunc2,axiom,
! [A] :
( m1_subset_1(A,k1_numbers)
=> ! [B] :
( ( v1_funct_1(B)
& m2_relset_1(B,k1_numbers,k1_numbers) )
=> ( ( r4_limfunc2(B,A)
& k2_limfunc2(B,A) = np__0
& ! [C] :
( m1_subset_1(C,k1_numbers)
=> ~ ( ~ r1_xreal_0(C,A)
& ! [D] :
( m1_subset_1(D,k1_numbers)
=> ~ ( ~ r1_xreal_0(C,D)
& ~ r1_xreal_0(D,A)
& r2_hidden(D,k4_relset_1(k1_numbers,k1_numbers,B))
& k2_seq_1(k1_numbers,k1_numbers,B,D) != np__0 ) ) ) ) )
=> ( ! [C] :
( m1_subset_1(C,k1_numbers)
=> ~ ( ~ r1_xreal_0(C,np__0)
& ! [D] :
( m1_subset_1(D,k1_numbers)
=> ( r2_hidden(D,k5_subset_1(k1_numbers,k4_relset_1(k1_numbers,k1_numbers,B),k2_rcomp_1(A,k3_real_1(A,C))))
=> r1_xreal_0(k2_seq_1(k1_numbers,k1_numbers,B,D),np__0) ) ) ) )
| r6_limfunc2(k4_rfunct_1(k1_numbers,k1_numbers,B),A) ) ) ) ) ).
fof(dt_k1_limfunc2,axiom,
! [A,B] :
( ( v1_funct_1(A)
& m1_relset_1(A,k1_numbers,k1_numbers)
& m1_subset_1(B,k1_numbers) )
=> m1_subset_1(k1_limfunc2(A,B),k1_numbers) ) ).
fof(dt_k2_limfunc2,axiom,
! [A,B] :
( ( v1_funct_1(A)
& m1_relset_1(A,k1_numbers,k1_numbers)
& m1_subset_1(B,k1_numbers) )
=> m1_subset_1(k2_limfunc2(A,B),k1_numbers) ) ).
%------------------------------------------------------------------------------