SET007 Axioms: SET007+112.ax
%------------------------------------------------------------------------------
% File : SET007+112 : TPTP v8.2.0. Released v3.4.0.
% Domain : Set Theory
% Axioms : The Limit 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 : limfunc3 [Urb08]
% Status : Satisfiable
% Syntax : Number of formulae : 57 ( 4 unt; 0 def)
% Number of atoms : 924 ( 51 equ)
% Maximal formula atoms : 34 ( 16 avg)
% Number of connectives : 1195 ( 328 ~; 31 |; 465 &)
% ( 11 <=>; 360 =>; 0 <=; 0 <~>)
% Maximal formula depth : 31 ( 18 avg)
% Maximal term depth : 5 ( 1 avg)
% Number of predicates : 30 ( 28 usr; 1 prp; 0-3 aty)
% Number of functors : 37 ( 37 usr; 5 con; 0-4 aty)
% Number of variables : 307 ( 301 !; 6 ?)
% 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_limfunc3,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) )
=> ( ( r1_tarski(k1_rfunct_2(B),k3_xboole_0(k1_relat_1(C),k12_prob_1(A)))
| r1_tarski(k1_rfunct_2(B),k3_xboole_0(k1_relat_1(C),k4_limfunc1(A))) )
=> r1_tarski(k1_rfunct_2(B),k4_xboole_0(k1_relat_1(C),k1_tarski(A))) ) ) ) ) ).
fof(t2_limfunc3,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(k18_complex1(k5_real_1(A,k2_seq_1(k5_numbers,k1_numbers,B,D))),np__0)
& ~ r1_xreal_0(k6_real_1(np__1,k1_nat_1(D,np__1)),k18_complex1(k5_real_1(A,k2_seq_1(k5_numbers,k1_numbers,B,D))))
& r2_hidden(k2_seq_1(k5_numbers,k1_numbers,B,D),k1_relat_1(C)) ) )
=> ( v4_seq_2(B)
& k2_seq_2(B) = A
& r1_tarski(k1_rfunct_2(B),k1_relat_1(C))
& r1_tarski(k1_rfunct_2(B),k4_xboole_0(k1_relat_1(C),k1_tarski(A))) ) ) ) ) ) ).
fof(t3_limfunc3,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) )
=> ( ( v4_seq_2(B)
& k2_seq_2(B) = A
& r1_tarski(k1_rfunct_2(B),k4_xboole_0(k1_relat_1(C),k1_tarski(A))) )
=> ! [D] :
( m1_subset_1(D,k1_numbers)
=> ~ ( ~ r1_xreal_0(D,np__0)
& ! [E] :
( m2_subset_1(E,k1_numbers,k5_numbers)
=> ? [F] :
( m2_subset_1(F,k1_numbers,k5_numbers)
& r1_xreal_0(E,F)
& ~ ( ~ r1_xreal_0(k18_complex1(k5_real_1(A,k2_seq_1(k5_numbers,k1_numbers,B,F))),np__0)
& ~ r1_xreal_0(D,k18_complex1(k5_real_1(A,k2_seq_1(k5_numbers,k1_numbers,B,F))))
& r2_hidden(k2_seq_1(k5_numbers,k1_numbers,B,F),k1_relat_1(C)) ) ) ) ) ) ) ) ) ) ).
fof(t4_limfunc3,axiom,
! [A] :
( m1_subset_1(A,k1_numbers)
=> ! [B] :
( m1_subset_1(B,k1_numbers)
=> ( ~ r1_xreal_0(A,np__0)
=> k4_xboole_0(k2_rcomp_1(k5_real_1(B,A),k3_real_1(B,A)),k1_tarski(B)) = k4_subset_1(k1_numbers,k2_rcomp_1(k5_real_1(B,A),B),k2_rcomp_1(B,k3_real_1(B,A))) ) ) ) ).
fof(t5_limfunc3,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(k4_subset_1(k1_numbers,k2_rcomp_1(k5_real_1(B,A),B),k2_rcomp_1(B,k3_real_1(B,A))),k1_relat_1(C))
=> ( r1_xreal_0(A,np__0)
| ! [D] :
( m1_subset_1(D,k1_numbers)
=> ! [E] :
( m1_subset_1(E,k1_numbers)
=> ~ ( ~ r1_xreal_0(B,D)
& ~ r1_xreal_0(E,B)
& ! [F] :
( m1_subset_1(F,k1_numbers)
=> ! [G] :
( m1_subset_1(G,k1_numbers)
=> ~ ( ~ r1_xreal_0(F,D)
& ~ r1_xreal_0(B,F)
& r2_hidden(F,k1_relat_1(C))
& ~ r1_xreal_0(E,G)
& ~ r1_xreal_0(G,B)
& r2_hidden(G,k1_relat_1(C)) ) ) ) ) ) ) ) ) ) ) ) ).
fof(t6_limfunc3,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),k1_relat_1(C)) ) )
=> ( v4_seq_2(B)
& k2_seq_2(B) = A
& r1_tarski(k1_rfunct_2(B),k4_xboole_0(k1_relat_1(C),k1_tarski(A))) ) ) ) ) ) ).
fof(t7_limfunc3,axiom,
! [A] :
( m1_subset_1(A,k1_numbers)
=> ! [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) )
=> ~ ( v4_seq_2(C)
& k2_seq_2(C) = A
& ~ r1_xreal_0(B,np__0)
& ! [D] :
( m2_subset_1(D,k1_numbers,k5_numbers)
=> ? [E] :
( m2_subset_1(E,k1_numbers,k5_numbers)
& r1_xreal_0(D,E)
& ~ ( ~ r1_xreal_0(k2_seq_1(k5_numbers,k1_numbers,C,E),k5_real_1(A,B))
& ~ r1_xreal_0(k3_real_1(A,B),k2_seq_1(k5_numbers,k1_numbers,C,E)) ) ) ) ) ) ) ) ).
fof(t8_limfunc3,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)
=> ! [D] :
( m1_subset_1(D,k1_numbers)
=> ~ ( ~ r1_xreal_0(A,C)
& ~ r1_xreal_0(D,A)
& ! [E] :
( m1_subset_1(E,k1_numbers)
=> ! [F] :
( m1_subset_1(F,k1_numbers)
=> ~ ( ~ r1_xreal_0(E,C)
& ~ r1_xreal_0(A,E)
& r2_hidden(E,k1_relat_1(B))
& ~ r1_xreal_0(D,F)
& ~ r1_xreal_0(F,A)
& r2_hidden(F,k1_relat_1(B)) ) ) ) ) ) )
<=> ( ! [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,k1_relat_1(B)) ) ) ) )
& ! [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,k1_relat_1(B)) ) ) ) ) ) ) ) ) ).
fof(d1_limfunc3,axiom,
! [A] :
( ( v1_funct_1(A)
& m2_relset_1(A,k1_numbers,k1_numbers) )
=> ! [B] :
( m1_subset_1(B,k1_numbers)
=> ( r1_limfunc3(A,B)
<=> ( ! [C] :
( m1_subset_1(C,k1_numbers)
=> ! [D] :
( m1_subset_1(D,k1_numbers)
=> ~ ( ~ r1_xreal_0(B,C)
& ~ r1_xreal_0(D,B)
& ! [E] :
( m1_subset_1(E,k1_numbers)
=> ! [F] :
( m1_subset_1(F,k1_numbers)
=> ~ ( ~ r1_xreal_0(E,C)
& ~ r1_xreal_0(B,E)
& r2_hidden(E,k1_relat_1(A))
& ~ r1_xreal_0(D,F)
& ~ r1_xreal_0(F,B)
& r2_hidden(F,k1_relat_1(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),k4_xboole_0(k1_relat_1(A),k1_tarski(B))) )
=> ( v4_seq_2(k2_rfunct_2(A,D))
& k2_seq_2(k2_rfunct_2(A,D)) = C ) ) ) ) ) ) ) ) ).
fof(d2_limfunc3,axiom,
! [A] :
( ( v1_funct_1(A)
& m2_relset_1(A,k1_numbers,k1_numbers) )
=> ! [B] :
( m1_subset_1(B,k1_numbers)
=> ( r2_limfunc3(A,B)
<=> ( ! [C] :
( m1_subset_1(C,k1_numbers)
=> ! [D] :
( m1_subset_1(D,k1_numbers)
=> ~ ( ~ r1_xreal_0(B,C)
& ~ r1_xreal_0(D,B)
& ! [E] :
( m1_subset_1(E,k1_numbers)
=> ! [F] :
( m1_subset_1(F,k1_numbers)
=> ~ ( ~ r1_xreal_0(E,C)
& ~ r1_xreal_0(B,E)
& r2_hidden(E,k1_relat_1(A))
& ~ r1_xreal_0(D,F)
& ~ r1_xreal_0(F,B)
& r2_hidden(F,k1_relat_1(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),k4_xboole_0(k1_relat_1(A),k1_tarski(B))) )
=> v1_limfunc1(k2_rfunct_2(A,C)) ) ) ) ) ) ) ).
fof(d3_limfunc3,axiom,
! [A] :
( ( v1_funct_1(A)
& m2_relset_1(A,k1_numbers,k1_numbers) )
=> ! [B] :
( m1_subset_1(B,k1_numbers)
=> ( r3_limfunc3(A,B)
<=> ( ! [C] :
( m1_subset_1(C,k1_numbers)
=> ! [D] :
( m1_subset_1(D,k1_numbers)
=> ~ ( ~ r1_xreal_0(B,C)
& ~ r1_xreal_0(D,B)
& ! [E] :
( m1_subset_1(E,k1_numbers)
=> ! [F] :
( m1_subset_1(F,k1_numbers)
=> ~ ( ~ r1_xreal_0(E,C)
& ~ r1_xreal_0(B,E)
& r2_hidden(E,k1_relat_1(A))
& ~ r1_xreal_0(D,F)
& ~ r1_xreal_0(F,B)
& r2_hidden(F,k1_relat_1(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),k4_xboole_0(k1_relat_1(A),k1_tarski(B))) )
=> v2_limfunc1(k2_rfunct_2(A,C)) ) ) ) ) ) ) ).
fof(t9_limfunc3,axiom,
$true ).
fof(t10_limfunc3,axiom,
$true ).
fof(t11_limfunc3,axiom,
$true ).
fof(t12_limfunc3,axiom,
! [A] :
( m1_subset_1(A,k1_numbers)
=> ! [B] :
( ( v1_funct_1(B)
& m2_relset_1(B,k1_numbers,k1_numbers) )
=> ( r1_limfunc3(B,A)
<=> ( ! [C] :
( m1_subset_1(C,k1_numbers)
=> ! [D] :
( m1_subset_1(D,k1_numbers)
=> ~ ( ~ r1_xreal_0(A,C)
& ~ r1_xreal_0(D,A)
& ! [E] :
( m1_subset_1(E,k1_numbers)
=> ! [F] :
( m1_subset_1(F,k1_numbers)
=> ~ ( ~ r1_xreal_0(E,C)
& ~ r1_xreal_0(A,E)
& r2_hidden(E,k1_relat_1(B))
& ~ r1_xreal_0(D,F)
& ~ r1_xreal_0(F,A)
& r2_hidden(F,k1_relat_1(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,np__0)
& ! [F] :
( m1_subset_1(F,k1_numbers)
=> ~ ( ~ r1_xreal_0(k18_complex1(k5_real_1(A,F)),np__0)
& ~ r1_xreal_0(E,k18_complex1(k5_real_1(A,F)))
& r2_hidden(F,k1_relat_1(B))
& r1_xreal_0(D,k18_complex1(k5_real_1(k2_seq_1(k1_numbers,k1_numbers,B,F),C))) ) ) ) ) ) ) ) ) ) ) ) ).
fof(t13_limfunc3,axiom,
! [A] :
( m1_subset_1(A,k1_numbers)
=> ! [B] :
( ( v1_funct_1(B)
& m2_relset_1(B,k1_numbers,k1_numbers) )
=> ( r2_limfunc3(B,A)
<=> ( ! [C] :
( m1_subset_1(C,k1_numbers)
=> ! [D] :
( m1_subset_1(D,k1_numbers)
=> ~ ( ~ r1_xreal_0(A,C)
& ~ r1_xreal_0(D,A)
& ! [E] :
( m1_subset_1(E,k1_numbers)
=> ! [F] :
( m1_subset_1(F,k1_numbers)
=> ~ ( ~ r1_xreal_0(E,C)
& ~ r1_xreal_0(A,E)
& r2_hidden(E,k1_relat_1(B))
& ~ r1_xreal_0(D,F)
& ~ r1_xreal_0(F,A)
& r2_hidden(F,k1_relat_1(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(k18_complex1(k5_real_1(A,E)),np__0)
& ~ r1_xreal_0(D,k18_complex1(k5_real_1(A,E)))
& r2_hidden(E,k1_relat_1(B))
& r1_xreal_0(k2_seq_1(k1_numbers,k1_numbers,B,E),C) ) ) ) ) ) ) ) ) ).
fof(t14_limfunc3,axiom,
! [A] :
( m1_subset_1(A,k1_numbers)
=> ! [B] :
( ( v1_funct_1(B)
& m2_relset_1(B,k1_numbers,k1_numbers) )
=> ( r3_limfunc3(B,A)
<=> ( ! [C] :
( m1_subset_1(C,k1_numbers)
=> ! [D] :
( m1_subset_1(D,k1_numbers)
=> ~ ( ~ r1_xreal_0(A,C)
& ~ r1_xreal_0(D,A)
& ! [E] :
( m1_subset_1(E,k1_numbers)
=> ! [F] :
( m1_subset_1(F,k1_numbers)
=> ~ ( ~ r1_xreal_0(E,C)
& ~ r1_xreal_0(A,E)
& r2_hidden(E,k1_relat_1(B))
& ~ r1_xreal_0(D,F)
& ~ r1_xreal_0(F,A)
& r2_hidden(F,k1_relat_1(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(k18_complex1(k5_real_1(A,E)),np__0)
& ~ r1_xreal_0(D,k18_complex1(k5_real_1(A,E)))
& r2_hidden(E,k1_relat_1(B))
& r1_xreal_0(C,k2_seq_1(k1_numbers,k1_numbers,B,E)) ) ) ) ) ) ) ) ) ).
fof(t15_limfunc3,axiom,
! [A] :
( m1_subset_1(A,k1_numbers)
=> ! [B] :
( ( v1_funct_1(B)
& m2_relset_1(B,k1_numbers,k1_numbers) )
=> ( r2_limfunc3(B,A)
<=> ( r2_limfunc2(B,A)
& r5_limfunc2(B,A) ) ) ) ) ).
fof(t16_limfunc3,axiom,
! [A] :
( m1_subset_1(A,k1_numbers)
=> ! [B] :
( ( v1_funct_1(B)
& m2_relset_1(B,k1_numbers,k1_numbers) )
=> ( r3_limfunc3(B,A)
<=> ( r3_limfunc2(B,A)
& r6_limfunc2(B,A) ) ) ) ) ).
fof(t17_limfunc3,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_limfunc3(B,A)
& r2_limfunc3(C,A)
& ! [D] :
( m1_subset_1(D,k1_numbers)
=> ! [E] :
( m1_subset_1(E,k1_numbers)
=> ~ ( ~ r1_xreal_0(A,D)
& ~ r1_xreal_0(E,A)
& ! [F] :
( m1_subset_1(F,k1_numbers)
=> ! [G] :
( m1_subset_1(G,k1_numbers)
=> ~ ( ~ r1_xreal_0(F,D)
& ~ r1_xreal_0(A,F)
& r2_hidden(F,k3_xboole_0(k1_relat_1(B),k1_relat_1(C)))
& ~ r1_xreal_0(E,G)
& ~ r1_xreal_0(G,A)
& r2_hidden(G,k3_xboole_0(k1_relat_1(B),k1_relat_1(C))) ) ) ) ) ) ) )
=> ( r2_limfunc3(k6_seq_1(k1_numbers,k1_numbers,B,C),A)
& r2_limfunc3(k8_seq_1(k1_numbers,k1_numbers,B,C),A) ) ) ) ) ) ).
fof(t18_limfunc3,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_limfunc3(B,A)
& r3_limfunc3(C,A)
& ! [D] :
( m1_subset_1(D,k1_numbers)
=> ! [E] :
( m1_subset_1(E,k1_numbers)
=> ~ ( ~ r1_xreal_0(A,D)
& ~ r1_xreal_0(E,A)
& ! [F] :
( m1_subset_1(F,k1_numbers)
=> ! [G] :
( m1_subset_1(G,k1_numbers)
=> ~ ( ~ r1_xreal_0(F,D)
& ~ r1_xreal_0(A,F)
& r2_hidden(F,k3_xboole_0(k1_relat_1(B),k1_relat_1(C)))
& ~ r1_xreal_0(E,G)
& ~ r1_xreal_0(G,A)
& r2_hidden(G,k3_xboole_0(k1_relat_1(B),k1_relat_1(C))) ) ) ) ) ) ) )
=> ( r3_limfunc3(k6_seq_1(k1_numbers,k1_numbers,B,C),A)
& r2_limfunc3(k8_seq_1(k1_numbers,k1_numbers,B,C),A) ) ) ) ) ) ).
fof(t19_limfunc3,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_limfunc3(B,A)
& ! [D] :
( m1_subset_1(D,k1_numbers)
=> ! [E] :
( m1_subset_1(E,k1_numbers)
=> ~ ( ~ r1_xreal_0(A,D)
& ~ r1_xreal_0(E,A)
& ! [F] :
( m1_subset_1(F,k1_numbers)
=> ! [G] :
( m1_subset_1(G,k1_numbers)
=> ~ ( ~ r1_xreal_0(F,D)
& ~ r1_xreal_0(A,F)
& r2_hidden(F,k1_relat_1(k6_seq_1(k1_numbers,k1_numbers,B,C)))
& ~ r1_xreal_0(E,G)
& ~ r1_xreal_0(G,A)
& r2_hidden(G,k1_relat_1(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,k4_subset_1(k1_numbers,k2_rcomp_1(k5_real_1(A,D),A),k2_rcomp_1(A,k3_real_1(A,D)))) ) )
| r2_limfunc3(k6_seq_1(k1_numbers,k1_numbers,B,C),A) ) ) ) ) ) ).
fof(t20_limfunc3,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_limfunc3(B,A)
& ! [D] :
( m1_subset_1(D,k1_numbers)
=> ! [E] :
( m1_subset_1(E,k1_numbers)
=> ~ ( ~ r1_xreal_0(A,D)
& ~ r1_xreal_0(E,A)
& ! [F] :
( m1_subset_1(F,k1_numbers)
=> ! [G] :
( m1_subset_1(G,k1_numbers)
=> ~ ( ~ r1_xreal_0(F,D)
& ~ r1_xreal_0(A,F)
& r2_hidden(F,k1_relat_1(k8_seq_1(k1_numbers,k1_numbers,B,C)))
& ~ r1_xreal_0(E,G)
& ~ r1_xreal_0(G,A)
& r2_hidden(G,k1_relat_1(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,k3_xboole_0(k1_relat_1(C),k4_subset_1(k1_numbers,k2_rcomp_1(k5_real_1(A,D),A),k2_rcomp_1(A,k3_real_1(A,D)))))
=> r1_xreal_0(E,k2_seq_1(k1_numbers,k1_numbers,C,F)) ) ) ) ) )
| r2_limfunc3(k8_seq_1(k1_numbers,k1_numbers,B,C),A) ) ) ) ) ) ).
fof(t21_limfunc3,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_limfunc3(C,A)
=> ( r1_xreal_0(B,np__0)
| r2_limfunc3(k13_seq_1(k1_numbers,k1_numbers,C,B),A) ) )
& ( r2_limfunc3(C,A)
=> ( r1_xreal_0(np__0,B)
| r3_limfunc3(k13_seq_1(k1_numbers,k1_numbers,C,B),A) ) )
& ( r3_limfunc3(C,A)
=> ( r1_xreal_0(B,np__0)
| r3_limfunc3(k13_seq_1(k1_numbers,k1_numbers,C,B),A) ) )
& ( r3_limfunc3(C,A)
=> ( r1_xreal_0(np__0,B)
| r2_limfunc3(k13_seq_1(k1_numbers,k1_numbers,C,B),A) ) ) ) ) ) ) ).
fof(t22_limfunc3,axiom,
! [A] :
( m1_subset_1(A,k1_numbers)
=> ! [B] :
( ( v1_funct_1(B)
& m2_relset_1(B,k1_numbers,k1_numbers) )
=> ( ( r2_limfunc3(B,A)
| r3_limfunc3(B,A) )
=> r2_limfunc3(k21_seq_1(k1_numbers,k1_numbers,B),A) ) ) ) ).
fof(t23_limfunc3,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)
=> ! [D] :
( m1_subset_1(D,k1_numbers)
=> ~ ( ~ r1_xreal_0(A,C)
& ~ r1_xreal_0(D,A)
& ! [E] :
( m1_subset_1(E,k1_numbers)
=> ! [F] :
( m1_subset_1(F,k1_numbers)
=> ~ ( ~ r1_xreal_0(E,C)
& ~ r1_xreal_0(A,E)
& r2_hidden(E,k1_relat_1(B))
& ~ r1_xreal_0(D,F)
& ~ r1_xreal_0(F,A)
& r2_hidden(F,k1_relat_1(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))
& r4_rfunct_2(B,k2_rcomp_1(A,k3_real_1(A,C)))
& ~ r1_rfunct_1(B,k2_rcomp_1(k5_real_1(A,C),A))
& ~ r1_rfunct_1(B,k2_rcomp_1(A,k3_real_1(A,C))) ) )
| r2_limfunc3(B,A) ) ) ) ) ).
fof(t24_limfunc3,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)
=> ! [D] :
( m1_subset_1(D,k1_numbers)
=> ~ ( ~ r1_xreal_0(A,C)
& ~ r1_xreal_0(D,A)
& ! [E] :
( m1_subset_1(E,k1_numbers)
=> ! [F] :
( m1_subset_1(F,k1_numbers)
=> ~ ( ~ r1_xreal_0(E,C)
& ~ r1_xreal_0(A,E)
& r2_hidden(E,k1_relat_1(B))
& ~ r1_xreal_0(D,F)
& ~ r1_xreal_0(F,A)
& r2_hidden(F,k1_relat_1(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))
& r2_rfunct_2(B,k2_rcomp_1(A,k3_real_1(A,C)))
& ~ r1_rfunct_1(B,k2_rcomp_1(k5_real_1(A,C),A))
& ~ r1_rfunct_1(B,k2_rcomp_1(A,k3_real_1(A,C))) ) )
| r2_limfunc3(B,A) ) ) ) ) ).
fof(t25_limfunc3,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)
=> ! [D] :
( m1_subset_1(D,k1_numbers)
=> ~ ( ~ r1_xreal_0(A,C)
& ~ r1_xreal_0(D,A)
& ! [E] :
( m1_subset_1(E,k1_numbers)
=> ! [F] :
( m1_subset_1(F,k1_numbers)
=> ~ ( ~ r1_xreal_0(E,C)
& ~ r1_xreal_0(A,E)
& r2_hidden(E,k1_relat_1(B))
& ~ r1_xreal_0(D,F)
& ~ r1_xreal_0(F,A)
& r2_hidden(F,k1_relat_1(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))
& r3_rfunct_2(B,k2_rcomp_1(A,k3_real_1(A,C)))
& ~ r2_rfunct_1(B,k2_rcomp_1(k5_real_1(A,C),A))
& ~ r2_rfunct_1(B,k2_rcomp_1(A,k3_real_1(A,C))) ) )
| r3_limfunc3(B,A) ) ) ) ) ).
fof(t26_limfunc3,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)
=> ! [D] :
( m1_subset_1(D,k1_numbers)
=> ~ ( ~ r1_xreal_0(A,C)
& ~ r1_xreal_0(D,A)
& ! [E] :
( m1_subset_1(E,k1_numbers)
=> ! [F] :
( m1_subset_1(F,k1_numbers)
=> ~ ( ~ r1_xreal_0(E,C)
& ~ r1_xreal_0(A,E)
& r2_hidden(E,k1_relat_1(B))
& ~ r1_xreal_0(D,F)
& ~ r1_xreal_0(F,A)
& r2_hidden(F,k1_relat_1(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))
& r1_rfunct_2(B,k2_rcomp_1(A,k3_real_1(A,C)))
& ~ r2_rfunct_1(B,k2_rcomp_1(k5_real_1(A,C),A))
& ~ r2_rfunct_1(B,k2_rcomp_1(A,k3_real_1(A,C))) ) )
| r3_limfunc3(B,A) ) ) ) ) ).
fof(t27_limfunc3,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_limfunc3(B,A)
& ! [D] :
( m1_subset_1(D,k1_numbers)
=> ! [E] :
( m1_subset_1(E,k1_numbers)
=> ~ ( ~ r1_xreal_0(A,D)
& ~ r1_xreal_0(E,A)
& ! [F] :
( m1_subset_1(F,k1_numbers)
=> ! [G] :
( m1_subset_1(G,k1_numbers)
=> ~ ( ~ r1_xreal_0(F,D)
& ~ r1_xreal_0(A,F)
& r2_hidden(F,k1_relat_1(C))
& ~ r1_xreal_0(E,G)
& ~ r1_xreal_0(G,A)
& r2_hidden(G,k1_relat_1(C)) ) ) ) ) ) ) )
=> ( ! [D] :
( m1_subset_1(D,k1_numbers)
=> ~ ( ~ r1_xreal_0(D,np__0)
& r1_tarski(k3_xboole_0(k1_relat_1(C),k4_subset_1(k1_numbers,k2_rcomp_1(k5_real_1(A,D),A),k2_rcomp_1(A,k3_real_1(A,D)))),k3_xboole_0(k1_relat_1(B),k4_subset_1(k1_numbers,k2_rcomp_1(k5_real_1(A,D),A),k2_rcomp_1(A,k3_real_1(A,D)))))
& ! [E] :
( m1_subset_1(E,k1_numbers)
=> ( r2_hidden(E,k3_xboole_0(k1_relat_1(C),k4_subset_1(k1_numbers,k2_rcomp_1(k5_real_1(A,D),A),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)) ) ) ) )
| r2_limfunc3(C,A) ) ) ) ) ) ).
fof(t28_limfunc3,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_limfunc3(B,A)
& ! [D] :
( m1_subset_1(D,k1_numbers)
=> ! [E] :
( m1_subset_1(E,k1_numbers)
=> ~ ( ~ r1_xreal_0(A,D)
& ~ r1_xreal_0(E,A)
& ! [F] :
( m1_subset_1(F,k1_numbers)
=> ! [G] :
( m1_subset_1(G,k1_numbers)
=> ~ ( ~ r1_xreal_0(F,D)
& ~ r1_xreal_0(A,F)
& r2_hidden(F,k1_relat_1(C))
& ~ r1_xreal_0(E,G)
& ~ r1_xreal_0(G,A)
& r2_hidden(G,k1_relat_1(C)) ) ) ) ) ) ) )
=> ( ! [D] :
( m1_subset_1(D,k1_numbers)
=> ~ ( ~ r1_xreal_0(D,np__0)
& r1_tarski(k3_xboole_0(k1_relat_1(C),k4_subset_1(k1_numbers,k2_rcomp_1(k5_real_1(A,D),A),k2_rcomp_1(A,k3_real_1(A,D)))),k3_xboole_0(k1_relat_1(B),k4_subset_1(k1_numbers,k2_rcomp_1(k5_real_1(A,D),A),k2_rcomp_1(A,k3_real_1(A,D)))))
& ! [E] :
( m1_subset_1(E,k1_numbers)
=> ( r2_hidden(E,k3_xboole_0(k1_relat_1(C),k4_subset_1(k1_numbers,k2_rcomp_1(k5_real_1(A,D),A),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)) ) ) ) )
| r3_limfunc3(C,A) ) ) ) ) ) ).
fof(t29_limfunc3,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_limfunc3(B,A)
=> ( ! [D] :
( m1_subset_1(D,k1_numbers)
=> ~ ( ~ r1_xreal_0(D,np__0)
& r1_tarski(k4_subset_1(k1_numbers,k2_rcomp_1(k5_real_1(A,D),A),k2_rcomp_1(A,k3_real_1(A,D))),k3_xboole_0(k1_relat_1(C),k1_relat_1(B)))
& ! [E] :
( m1_subset_1(E,k1_numbers)
=> ( r2_hidden(E,k4_subset_1(k1_numbers,k2_rcomp_1(k5_real_1(A,D),A),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)) ) ) ) )
| r2_limfunc3(C,A) ) ) ) ) ) ).
fof(t30_limfunc3,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_limfunc3(B,A)
=> ( ! [D] :
( m1_subset_1(D,k1_numbers)
=> ~ ( ~ r1_xreal_0(D,np__0)
& r1_tarski(k4_subset_1(k1_numbers,k2_rcomp_1(k5_real_1(A,D),A),k2_rcomp_1(A,k3_real_1(A,D))),k3_xboole_0(k1_relat_1(C),k1_relat_1(B)))
& ! [E] :
( m1_subset_1(E,k1_numbers)
=> ( r2_hidden(E,k4_subset_1(k1_numbers,k2_rcomp_1(k5_real_1(A,D),A),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)) ) ) ) )
| r3_limfunc3(C,A) ) ) ) ) ) ).
fof(d4_limfunc3,axiom,
! [A] :
( ( v1_funct_1(A)
& m2_relset_1(A,k1_numbers,k1_numbers) )
=> ! [B] :
( m1_subset_1(B,k1_numbers)
=> ( r1_limfunc3(A,B)
=> ! [C] :
( m1_subset_1(C,k1_numbers)
=> ( C = k1_limfunc3(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),k4_xboole_0(k1_relat_1(A),k1_tarski(B))) )
=> ( v4_seq_2(k2_rfunct_2(A,D))
& k2_seq_2(k2_rfunct_2(A,D)) = C ) ) ) ) ) ) ) ) ).
fof(t31_limfunc3,axiom,
$true ).
fof(t32_limfunc3,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_limfunc3(C,A)
=> ( k1_limfunc3(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,np__0)
& ! [F] :
( m1_subset_1(F,k1_numbers)
=> ~ ( ~ r1_xreal_0(k18_complex1(k5_real_1(A,F)),np__0)
& ~ r1_xreal_0(E,k18_complex1(k5_real_1(A,F)))
& r2_hidden(F,k1_relat_1(C))
& r1_xreal_0(D,k18_complex1(k5_real_1(k2_seq_1(k1_numbers,k1_numbers,C,F),B))) ) ) ) ) ) ) ) ) ) ) ) ).
fof(t33_limfunc3,axiom,
! [A] :
( m1_subset_1(A,k1_numbers)
=> ! [B] :
( ( v1_funct_1(B)
& m2_relset_1(B,k1_numbers,k1_numbers) )
=> ( r1_limfunc3(B,A)
=> ( r1_limfunc2(B,A)
& r4_limfunc2(B,A)
& k1_limfunc2(B,A) = k2_limfunc2(B,A)
& k1_limfunc3(B,A) = k1_limfunc2(B,A)
& k1_limfunc3(B,A) = k2_limfunc2(B,A) ) ) ) ) ).
fof(t34_limfunc3,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)
& r4_limfunc2(B,A)
& k1_limfunc2(B,A) = k2_limfunc2(B,A) )
=> ( r1_limfunc3(B,A)
& k1_limfunc3(B,A) = k1_limfunc2(B,A)
& k1_limfunc3(B,A) = k2_limfunc2(B,A) ) ) ) ) ).
fof(t35_limfunc3,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_limfunc3(C,A)
=> ( r1_limfunc3(k13_seq_1(k1_numbers,k1_numbers,C,B),A)
& k1_limfunc3(k13_seq_1(k1_numbers,k1_numbers,C,B),A) = k4_real_1(B,k1_limfunc3(C,A)) ) ) ) ) ) ).
fof(t36_limfunc3,axiom,
! [A] :
( m1_subset_1(A,k1_numbers)
=> ! [B] :
( ( v1_funct_1(B)
& m2_relset_1(B,k1_numbers,k1_numbers) )
=> ( r1_limfunc3(B,A)
=> ( r1_limfunc3(k16_seq_1(k1_numbers,k1_numbers,B),A)
& k1_limfunc3(k16_seq_1(k1_numbers,k1_numbers,B),A) = k1_real_1(k1_limfunc3(B,A)) ) ) ) ) ).
fof(t37_limfunc3,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_limfunc3(B,A)
& r1_limfunc3(C,A)
& ! [D] :
( m1_subset_1(D,k1_numbers)
=> ! [E] :
( m1_subset_1(E,k1_numbers)
=> ~ ( ~ r1_xreal_0(A,D)
& ~ r1_xreal_0(E,A)
& ! [F] :
( m1_subset_1(F,k1_numbers)
=> ! [G] :
( m1_subset_1(G,k1_numbers)
=> ~ ( ~ r1_xreal_0(F,D)
& ~ r1_xreal_0(A,F)
& r2_hidden(F,k1_relat_1(k6_seq_1(k1_numbers,k1_numbers,B,C)))
& ~ r1_xreal_0(E,G)
& ~ r1_xreal_0(G,A)
& r2_hidden(G,k1_relat_1(k6_seq_1(k1_numbers,k1_numbers,B,C))) ) ) ) ) ) ) )
=> ( r1_limfunc3(k6_seq_1(k1_numbers,k1_numbers,B,C),A)
& k1_limfunc3(k6_seq_1(k1_numbers,k1_numbers,B,C),A) = k3_real_1(k1_limfunc3(B,A),k1_limfunc3(C,A)) ) ) ) ) ) ).
fof(t38_limfunc3,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_limfunc3(B,A)
& r1_limfunc3(C,A)
& ! [D] :
( m1_subset_1(D,k1_numbers)
=> ! [E] :
( m1_subset_1(E,k1_numbers)
=> ~ ( ~ r1_xreal_0(A,D)
& ~ r1_xreal_0(E,A)
& ! [F] :
( m1_subset_1(F,k1_numbers)
=> ! [G] :
( m1_subset_1(G,k1_numbers)
=> ~ ( ~ r1_xreal_0(F,D)
& ~ r1_xreal_0(A,F)
& r2_hidden(F,k1_relat_1(k7_seq_1(k1_numbers,k1_numbers,B,C)))
& ~ r1_xreal_0(E,G)
& ~ r1_xreal_0(G,A)
& r2_hidden(G,k1_relat_1(k7_seq_1(k1_numbers,k1_numbers,B,C))) ) ) ) ) ) ) )
=> ( r1_limfunc3(k7_seq_1(k1_numbers,k1_numbers,B,C),A)
& k1_limfunc3(k7_seq_1(k1_numbers,k1_numbers,B,C),A) = k5_real_1(k1_limfunc3(B,A),k1_limfunc3(C,A)) ) ) ) ) ) ).
fof(t39_limfunc3,axiom,
! [A] :
( m1_subset_1(A,k1_numbers)
=> ! [B] :
( ( v1_funct_1(B)
& m2_relset_1(B,k1_numbers,k1_numbers) )
=> ( ( r1_limfunc3(B,A)
& k3_funct_2(k1_numbers,k1_numbers,B,k1_tarski(np__0)) = k1_xboole_0 )
=> ( k1_limfunc3(B,A) = np__0
| ( r1_limfunc3(k4_rfunct_1(k1_numbers,k1_numbers,B),A)
& k1_limfunc3(k4_rfunct_1(k1_numbers,k1_numbers,B),A) = k2_real_1(k1_limfunc3(B,A)) ) ) ) ) ) ).
fof(t40_limfunc3,axiom,
! [A] :
( m1_subset_1(A,k1_numbers)
=> ! [B] :
( ( v1_funct_1(B)
& m2_relset_1(B,k1_numbers,k1_numbers) )
=> ( r1_limfunc3(B,A)
=> ( r1_limfunc3(k21_seq_1(k1_numbers,k1_numbers,B),A)
& k1_limfunc3(k21_seq_1(k1_numbers,k1_numbers,B),A) = k18_complex1(k1_limfunc3(B,A)) ) ) ) ) ).
fof(t41_limfunc3,axiom,
! [A] :
( m1_subset_1(A,k1_numbers)
=> ! [B] :
( ( v1_funct_1(B)
& m2_relset_1(B,k1_numbers,k1_numbers) )
=> ( ( r1_limfunc3(B,A)
& ! [C] :
( m1_subset_1(C,k1_numbers)
=> ! [D] :
( m1_subset_1(D,k1_numbers)
=> ~ ( ~ r1_xreal_0(A,C)
& ~ r1_xreal_0(D,A)
& ! [E] :
( m1_subset_1(E,k1_numbers)
=> ! [F] :
( m1_subset_1(F,k1_numbers)
=> ~ ( ~ r1_xreal_0(E,C)
& ~ r1_xreal_0(A,E)
& r2_hidden(E,k1_relat_1(B))
& ~ r1_xreal_0(D,F)
& ~ r1_xreal_0(F,A)
& r2_hidden(F,k1_relat_1(B))
& k2_seq_1(k1_numbers,k1_numbers,B,E) != np__0
& k2_seq_1(k1_numbers,k1_numbers,B,F) != np__0 ) ) ) ) ) ) )
=> ( k1_limfunc3(B,A) = np__0
| ( r1_limfunc3(k4_rfunct_1(k1_numbers,k1_numbers,B),A)
& k1_limfunc3(k4_rfunct_1(k1_numbers,k1_numbers,B),A) = k2_real_1(k1_limfunc3(B,A)) ) ) ) ) ) ).
fof(t42_limfunc3,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_limfunc3(B,A)
& r1_limfunc3(C,A)
& ! [D] :
( m1_subset_1(D,k1_numbers)
=> ! [E] :
( m1_subset_1(E,k1_numbers)
=> ~ ( ~ r1_xreal_0(A,D)
& ~ r1_xreal_0(E,A)
& ! [F] :
( m1_subset_1(F,k1_numbers)
=> ! [G] :
( m1_subset_1(G,k1_numbers)
=> ~ ( ~ r1_xreal_0(F,D)
& ~ r1_xreal_0(A,F)
& r2_hidden(F,k1_relat_1(k8_seq_1(k1_numbers,k1_numbers,B,C)))
& ~ r1_xreal_0(E,G)
& ~ r1_xreal_0(G,A)
& r2_hidden(G,k1_relat_1(k8_seq_1(k1_numbers,k1_numbers,B,C))) ) ) ) ) ) ) )
=> ( r1_limfunc3(k8_seq_1(k1_numbers,k1_numbers,B,C),A)
& k1_limfunc3(k8_seq_1(k1_numbers,k1_numbers,B,C),A) = k4_real_1(k1_limfunc3(B,A),k1_limfunc3(C,A)) ) ) ) ) ) ).
fof(t43_limfunc3,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_limfunc3(B,A)
& r1_limfunc3(C,A)
& ! [D] :
( m1_subset_1(D,k1_numbers)
=> ! [E] :
( m1_subset_1(E,k1_numbers)
=> ~ ( ~ r1_xreal_0(A,D)
& ~ r1_xreal_0(E,A)
& ! [F] :
( m1_subset_1(F,k1_numbers)
=> ! [G] :
( m1_subset_1(G,k1_numbers)
=> ~ ( ~ r1_xreal_0(F,D)
& ~ r1_xreal_0(A,F)
& r2_hidden(F,k1_relat_1(k2_rfunct_1(k1_numbers,k1_numbers,B,C)))
& ~ r1_xreal_0(E,G)
& ~ r1_xreal_0(G,A)
& r2_hidden(G,k1_relat_1(k2_rfunct_1(k1_numbers,k1_numbers,B,C))) ) ) ) ) ) ) )
=> ( k1_limfunc3(C,A) = np__0
| ( r1_limfunc3(k2_rfunct_1(k1_numbers,k1_numbers,B,C),A)
& k1_limfunc3(k2_rfunct_1(k1_numbers,k1_numbers,B,C),A) = k6_real_1(k1_limfunc3(B,A),k1_limfunc3(C,A)) ) ) ) ) ) ) ).
fof(t44_limfunc3,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_limfunc3(B,A)
& k1_limfunc3(B,A) = np__0
& ! [D] :
( m1_subset_1(D,k1_numbers)
=> ! [E] :
( m1_subset_1(E,k1_numbers)
=> ~ ( ~ r1_xreal_0(A,D)
& ~ r1_xreal_0(E,A)
& ! [F] :
( m1_subset_1(F,k1_numbers)
=> ! [G] :
( m1_subset_1(G,k1_numbers)
=> ~ ( ~ r1_xreal_0(F,D)
& ~ r1_xreal_0(A,F)
& r2_hidden(F,k1_relat_1(k8_seq_1(k1_numbers,k1_numbers,B,C)))
& ~ r1_xreal_0(E,G)
& ~ r1_xreal_0(G,A)
& r2_hidden(G,k1_relat_1(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,k4_subset_1(k1_numbers,k2_rcomp_1(k5_real_1(A,D),A),k2_rcomp_1(A,k3_real_1(A,D)))) ) )
| ( r1_limfunc3(k8_seq_1(k1_numbers,k1_numbers,B,C),A)
& k1_limfunc3(k8_seq_1(k1_numbers,k1_numbers,B,C),A) = np__0 ) ) ) ) ) ) ).
fof(t45_limfunc3,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_limfunc3(B,A)
& r1_limfunc3(C,A)
& k1_limfunc3(B,A) = k1_limfunc3(C,A)
& ! [E] :
( m1_subset_1(E,k1_numbers)
=> ! [F] :
( m1_subset_1(F,k1_numbers)
=> ~ ( ~ r1_xreal_0(A,E)
& ~ r1_xreal_0(F,A)
& ! [G] :
( m1_subset_1(G,k1_numbers)
=> ! [H] :
( m1_subset_1(H,k1_numbers)
=> ~ ( ~ r1_xreal_0(G,E)
& ~ r1_xreal_0(A,G)
& r2_hidden(G,k1_relat_1(D))
& ~ r1_xreal_0(F,H)
& ~ r1_xreal_0(H,A)
& r2_hidden(H,k1_relat_1(D)) ) ) ) ) ) ) )
=> ( ! [E] :
( m1_subset_1(E,k1_numbers)
=> ~ ( ~ r1_xreal_0(E,np__0)
& ! [F] :
( m1_subset_1(F,k1_numbers)
=> ( r2_hidden(F,k3_xboole_0(k1_relat_1(D),k4_subset_1(k1_numbers,k2_rcomp_1(k5_real_1(A,E),A),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(k3_xboole_0(k1_relat_1(B),k4_subset_1(k1_numbers,k2_rcomp_1(k5_real_1(A,E),A),k2_rcomp_1(A,k3_real_1(A,E)))),k3_xboole_0(k1_relat_1(C),k4_subset_1(k1_numbers,k2_rcomp_1(k5_real_1(A,E),A),k2_rcomp_1(A,k3_real_1(A,E)))))
& r1_tarski(k3_xboole_0(k1_relat_1(D),k4_subset_1(k1_numbers,k2_rcomp_1(k5_real_1(A,E),A),k2_rcomp_1(A,k3_real_1(A,E)))),k3_xboole_0(k1_relat_1(B),k4_subset_1(k1_numbers,k2_rcomp_1(k5_real_1(A,E),A),k2_rcomp_1(A,k3_real_1(A,E))))) )
| ( r1_tarski(k3_xboole_0(k1_relat_1(C),k4_subset_1(k1_numbers,k2_rcomp_1(k5_real_1(A,E),A),k2_rcomp_1(A,k3_real_1(A,E)))),k3_xboole_0(k1_relat_1(B),k4_subset_1(k1_numbers,k2_rcomp_1(k5_real_1(A,E),A),k2_rcomp_1(A,k3_real_1(A,E)))))
& r1_tarski(k3_xboole_0(k1_relat_1(D),k4_subset_1(k1_numbers,k2_rcomp_1(k5_real_1(A,E),A),k2_rcomp_1(A,k3_real_1(A,E)))),k3_xboole_0(k1_relat_1(C),k4_subset_1(k1_numbers,k2_rcomp_1(k5_real_1(A,E),A),k2_rcomp_1(A,k3_real_1(A,E))))) ) ) ) )
| ( r1_limfunc3(D,A)
& k1_limfunc3(D,A) = k1_limfunc3(B,A) ) ) ) ) ) ) ) ).
fof(t46_limfunc3,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_limfunc3(B,A)
& r1_limfunc3(C,A)
& k1_limfunc3(B,A) = k1_limfunc3(C,A) )
=> ( ! [E] :
( m1_subset_1(E,k1_numbers)
=> ~ ( ~ r1_xreal_0(E,np__0)
& r1_tarski(k4_subset_1(k1_numbers,k2_rcomp_1(k5_real_1(A,E),A),k2_rcomp_1(A,k3_real_1(A,E))),k3_xboole_0(k3_xboole_0(k1_relat_1(B),k1_relat_1(C)),k1_relat_1(D)))
& ! [F] :
( m1_subset_1(F,k1_numbers)
=> ( r2_hidden(F,k4_subset_1(k1_numbers,k2_rcomp_1(k5_real_1(A,E),A),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_limfunc3(D,A)
& k1_limfunc3(D,A) = k1_limfunc3(B,A) ) ) ) ) ) ) ) ).
fof(t47_limfunc3,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_limfunc3(B,A)
& r1_limfunc3(C,A) )
=> ( ! [D] :
( m1_subset_1(D,k1_numbers)
=> ~ ( ~ r1_xreal_0(D,np__0)
& ( ( r1_tarski(k3_xboole_0(k1_relat_1(B),k4_subset_1(k1_numbers,k2_rcomp_1(k5_real_1(A,D),A),k2_rcomp_1(A,k3_real_1(A,D)))),k3_xboole_0(k1_relat_1(C),k4_subset_1(k1_numbers,k2_rcomp_1(k5_real_1(A,D),A),k2_rcomp_1(A,k3_real_1(A,D)))))
& ! [E] :
( m1_subset_1(E,k1_numbers)
=> ( r2_hidden(E,k3_xboole_0(k1_relat_1(B),k4_subset_1(k1_numbers,k2_rcomp_1(k5_real_1(A,D),A),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(k3_xboole_0(k1_relat_1(C),k4_subset_1(k1_numbers,k2_rcomp_1(k5_real_1(A,D),A),k2_rcomp_1(A,k3_real_1(A,D)))),k3_xboole_0(k1_relat_1(B),k4_subset_1(k1_numbers,k2_rcomp_1(k5_real_1(A,D),A),k2_rcomp_1(A,k3_real_1(A,D)))))
& ! [E] :
( m1_subset_1(E,k1_numbers)
=> ( r2_hidden(E,k3_xboole_0(k1_relat_1(C),k4_subset_1(k1_numbers,k2_rcomp_1(k5_real_1(A,D),A),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(k1_limfunc3(B,A),k1_limfunc3(C,A)) ) ) ) ) ) ).
fof(t48_limfunc3,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)
=> ! [D] :
( m1_subset_1(D,k1_numbers)
=> ~ ( ~ r1_xreal_0(A,C)
& ~ r1_xreal_0(D,A)
& ! [E] :
( m1_subset_1(E,k1_numbers)
=> ! [F] :
( m1_subset_1(F,k1_numbers)
=> ~ ( ~ r1_xreal_0(E,C)
& ~ r1_xreal_0(A,E)
& r2_hidden(E,k1_relat_1(B))
& ~ r1_xreal_0(D,F)
& ~ r1_xreal_0(F,A)
& r2_hidden(F,k1_relat_1(B))
& k2_seq_1(k1_numbers,k1_numbers,B,E) != np__0
& k2_seq_1(k1_numbers,k1_numbers,B,F) != np__0 ) ) ) ) ) )
=> ( ( ~ r2_limfunc3(B,A)
& ~ r3_limfunc3(B,A) )
| ( r1_limfunc3(k4_rfunct_1(k1_numbers,k1_numbers,B),A)
& k1_limfunc3(k4_rfunct_1(k1_numbers,k1_numbers,B),A) = np__0 ) ) ) ) ) ).
fof(t49_limfunc3,axiom,
! [A] :
( m1_subset_1(A,k1_numbers)
=> ! [B] :
( ( v1_funct_1(B)
& m2_relset_1(B,k1_numbers,k1_numbers) )
=> ( ( r1_limfunc3(B,A)
& k1_limfunc3(B,A) = np__0
& ! [C] :
( m1_subset_1(C,k1_numbers)
=> ! [D] :
( m1_subset_1(D,k1_numbers)
=> ~ ( ~ r1_xreal_0(A,C)
& ~ r1_xreal_0(D,A)
& ! [E] :
( m1_subset_1(E,k1_numbers)
=> ! [F] :
( m1_subset_1(F,k1_numbers)
=> ~ ( ~ r1_xreal_0(E,C)
& ~ r1_xreal_0(A,E)
& r2_hidden(E,k1_relat_1(B))
& ~ r1_xreal_0(D,F)
& ~ r1_xreal_0(F,A)
& r2_hidden(F,k1_relat_1(B))
& k2_seq_1(k1_numbers,k1_numbers,B,E) != np__0
& k2_seq_1(k1_numbers,k1_numbers,B,F) != 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,k3_xboole_0(k1_relat_1(B),k4_subset_1(k1_numbers,k2_rcomp_1(k5_real_1(A,C),A),k2_rcomp_1(A,k3_real_1(A,C)))))
=> r1_xreal_0(np__0,k2_seq_1(k1_numbers,k1_numbers,B,D)) ) ) ) )
| r2_limfunc3(k4_rfunct_1(k1_numbers,k1_numbers,B),A) ) ) ) ) ).
fof(t50_limfunc3,axiom,
! [A] :
( m1_subset_1(A,k1_numbers)
=> ! [B] :
( ( v1_funct_1(B)
& m2_relset_1(B,k1_numbers,k1_numbers) )
=> ( ( r1_limfunc3(B,A)
& k1_limfunc3(B,A) = np__0
& ! [C] :
( m1_subset_1(C,k1_numbers)
=> ! [D] :
( m1_subset_1(D,k1_numbers)
=> ~ ( ~ r1_xreal_0(A,C)
& ~ r1_xreal_0(D,A)
& ! [E] :
( m1_subset_1(E,k1_numbers)
=> ! [F] :
( m1_subset_1(F,k1_numbers)
=> ~ ( ~ r1_xreal_0(E,C)
& ~ r1_xreal_0(A,E)
& r2_hidden(E,k1_relat_1(B))
& ~ r1_xreal_0(D,F)
& ~ r1_xreal_0(F,A)
& r2_hidden(F,k1_relat_1(B))
& k2_seq_1(k1_numbers,k1_numbers,B,E) != np__0
& k2_seq_1(k1_numbers,k1_numbers,B,F) != 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,k3_xboole_0(k1_relat_1(B),k4_subset_1(k1_numbers,k2_rcomp_1(k5_real_1(A,C),A),k2_rcomp_1(A,k3_real_1(A,C)))))
=> r1_xreal_0(k2_seq_1(k1_numbers,k1_numbers,B,D),np__0) ) ) ) )
| r3_limfunc3(k4_rfunct_1(k1_numbers,k1_numbers,B),A) ) ) ) ) ).
fof(t51_limfunc3,axiom,
! [A] :
( m1_subset_1(A,k1_numbers)
=> ! [B] :
( ( v1_funct_1(B)
& m2_relset_1(B,k1_numbers,k1_numbers) )
=> ( ( r1_limfunc3(B,A)
& k1_limfunc3(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,k3_xboole_0(k1_relat_1(B),k4_subset_1(k1_numbers,k2_rcomp_1(k5_real_1(A,C),A),k2_rcomp_1(A,k3_real_1(A,C)))))
& r1_xreal_0(k2_seq_1(k1_numbers,k1_numbers,B,D),np__0) ) ) ) )
| r2_limfunc3(k4_rfunct_1(k1_numbers,k1_numbers,B),A) ) ) ) ) ).
fof(t52_limfunc3,axiom,
! [A] :
( m1_subset_1(A,k1_numbers)
=> ! [B] :
( ( v1_funct_1(B)
& m2_relset_1(B,k1_numbers,k1_numbers) )
=> ( ( r1_limfunc3(B,A)
& k1_limfunc3(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,k3_xboole_0(k1_relat_1(B),k4_subset_1(k1_numbers,k2_rcomp_1(k5_real_1(A,C),A),k2_rcomp_1(A,k3_real_1(A,C)))))
& r1_xreal_0(np__0,k2_seq_1(k1_numbers,k1_numbers,B,D)) ) ) ) )
| r3_limfunc3(k4_rfunct_1(k1_numbers,k1_numbers,B),A) ) ) ) ) ).
fof(dt_k1_limfunc3,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_limfunc3(A,B),k1_numbers) ) ).
%------------------------------------------------------------------------------