begin
scheme
UnPartFuncD9{
F1()
-> ( ( ) ( )
set ) ,
F2()
-> ( ( ) ( )
set ) ,
F3()
-> ( ( ) ( )
set ) ,
F4( ( ( ) ( )
set ) )
-> ( ( ) ( )
set ) } :
for
f,
g being ( (
Function-like ) (
Relation-like F1( ( ( ) ( )
set ) ) : ( ( ) ( )
set )
-defined F2( ( ( ) ( )
set ) ) : ( ( ) ( )
set )
-valued Function-like )
PartFunc of ,) st
dom f : ( (
Function-like ) (
Relation-like F1( ( ( ) ( )
set ) ) : ( ( ) ( )
set )
-defined F2( ( ( ) ( )
set ) ) : ( ( ) ( )
set )
-valued Function-like )
PartFunc of ,) : ( ( ) ( )
Element of
K19(
F1( ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) )
= F3( ( ( ) ( )
set ) ) : ( ( ) ( )
set ) & ( for
c being ( ( ) ( )
Element of
F1( ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) st
c : ( ( ) ( )
Element of
F1( ( ( ) ( )
set ) ) : ( ( ) ( )
set ) )
in dom f : ( (
Function-like ) (
Relation-like F1( ( ( ) ( )
set ) ) : ( ( ) ( )
set )
-defined F2( ( ( ) ( )
set ) ) : ( ( ) ( )
set )
-valued Function-like )
PartFunc of ,) : ( ( ) ( )
Element of
K19(
F1( ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) holds
f : ( (
Function-like ) (
Relation-like F1( ( ( ) ( )
set ) ) : ( ( ) ( )
set )
-defined F2( ( ( ) ( )
set ) ) : ( ( ) ( )
set )
-valued Function-like )
PartFunc of ,)
. c : ( ( ) ( )
Element of
F1( ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set )
= F4( ( ( ) ( )
set ) ,
c : ( ( ) ( )
Element of
F1( ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( )
set ) ) &
dom g : ( (
Function-like ) (
Relation-like F1( ( ( ) ( )
set ) ) : ( ( ) ( )
set )
-defined F2( ( ( ) ( )
set ) ) : ( ( ) ( )
set )
-valued Function-like )
PartFunc of ,) : ( ( ) ( )
Element of
K19(
F1( ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) )
= F3( ( ( ) ( )
set ) ) : ( ( ) ( )
set ) & ( for
c being ( ( ) ( )
Element of
F1( ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) st
c : ( ( ) ( )
Element of
F1( ( ( ) ( )
set ) ) : ( ( ) ( )
set ) )
in dom g : ( (
Function-like ) (
Relation-like F1( ( ( ) ( )
set ) ) : ( ( ) ( )
set )
-defined F2( ( ( ) ( )
set ) ) : ( ( ) ( )
set )
-valued Function-like )
PartFunc of ,) : ( ( ) ( )
Element of
K19(
F1( ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) holds
g : ( (
Function-like ) (
Relation-like F1( ( ( ) ( )
set ) ) : ( ( ) ( )
set )
-defined F2( ( ( ) ( )
set ) ) : ( ( ) ( )
set )
-valued Function-like )
PartFunc of ,)
. c : ( ( ) ( )
Element of
F1( ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set )
= F4( ( ( ) ( )
set ) ,
c : ( ( ) ( )
Element of
F1( ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( )
set ) ) holds
f : ( (
Function-like ) (
Relation-like F1( ( ( ) ( )
set ) ) : ( ( ) ( )
set )
-defined F2( ( ( ) ( )
set ) ) : ( ( ) ( )
set )
-valued Function-like )
PartFunc of ,)
= g : ( (
Function-like ) (
Relation-like F1( ( ( ) ( )
set ) ) : ( ( ) ( )
set )
-defined F2( ( ( ) ( )
set ) ) : ( ( ) ( )
set )
-valued Function-like )
PartFunc of ,)
theorem
for
seq,
seq1,
seq2 being ( (
Function-like V18(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) ) (
Relation-like NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) )
-defined REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set )
-valued Function-like non
empty total V18(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) )
complex-valued ext-real-valued real-valued )
Real_Sequence) holds
(
seq : ( (
Function-like V18(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) ) (
Relation-like NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) )
-defined REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set )
-valued Function-like non
empty total V18(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) )
complex-valued ext-real-valued real-valued )
Real_Sequence)
= seq1 : ( (
Function-like V18(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) ) (
Relation-like NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) )
-defined REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set )
-valued Function-like non
empty total V18(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) )
complex-valued ext-real-valued real-valued )
Real_Sequence)
+ seq2 : ( (
Function-like V18(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) ) (
Relation-like NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) )
-defined REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set )
-valued Function-like non
empty total V18(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) )
complex-valued ext-real-valued real-valued )
Real_Sequence) : ( (
Function-like ) (
Relation-like NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) )
-defined REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set )
-valued Function-like non
empty total V18(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) )
complex-valued ext-real-valued real-valued )
Element of
K19(
K20(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) (
complex-valued ext-real-valued real-valued )
set ) ) : ( ( ) ( )
set ) ) iff for
n being ( ( ) (
V21()
epsilon-transitive epsilon-connected ordinal natural V31()
real V33()
V34()
V45()
V46()
V47()
V48()
V49()
V50() )
Element of
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) ) ) holds
seq : ( (
Function-like V18(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) ) (
Relation-like NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) )
-defined REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set )
-valued Function-like non
empty total V18(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) )
complex-valued ext-real-valued real-valued )
Real_Sequence)
. n : ( ( ) (
V21()
epsilon-transitive epsilon-connected ordinal natural V31()
real V33()
V34()
V45()
V46()
V47()
V48()
V49()
V50() )
Element of
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) (
V21()
V31()
real )
Real)
= (seq1 : ( ( Function-like V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K19(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K19(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) -valued Function-like non empty total V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K19(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) . n : ( ( ) ( V21() epsilon-transitive epsilon-connected ordinal natural V31() real V33() V34() V45() V46() V47() V48() V49() V50() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K19(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) (
V21()
V31()
real )
Real)
+ (seq2 : ( ( Function-like V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K19(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K19(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) -valued Function-like non empty total V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K19(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) . n : ( ( ) ( V21() epsilon-transitive epsilon-connected ordinal natural V31() real V33() V34() V45() V46() V47() V48() V49() V50() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K19(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) (
V21()
V31()
real )
Real) : ( ( ) (
V21()
V31()
real )
Element of
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) ) ;
theorem
for
seq,
seq1,
seq2 being ( (
Function-like V18(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) ) (
Relation-like NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) )
-defined REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set )
-valued Function-like non
empty total V18(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) )
complex-valued ext-real-valued real-valued )
Real_Sequence) holds
(
seq : ( (
Function-like V18(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) ) (
Relation-like NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) )
-defined REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set )
-valued Function-like non
empty total V18(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) )
complex-valued ext-real-valued real-valued )
Real_Sequence)
= seq1 : ( (
Function-like V18(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) ) (
Relation-like NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) )
-defined REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set )
-valued Function-like non
empty total V18(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) )
complex-valued ext-real-valued real-valued )
Real_Sequence)
(#) seq2 : ( (
Function-like V18(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) ) (
Relation-like NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) )
-defined REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set )
-valued Function-like non
empty total V18(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) )
complex-valued ext-real-valued real-valued )
Real_Sequence) : ( (
Function-like ) (
Relation-like NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) )
-defined REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set )
-valued Function-like non
empty total V18(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) )
complex-valued ext-real-valued real-valued )
Element of
K19(
K20(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) (
complex-valued ext-real-valued real-valued )
set ) ) : ( ( ) ( )
set ) ) iff for
n being ( ( ) (
V21()
epsilon-transitive epsilon-connected ordinal natural V31()
real V33()
V34()
V45()
V46()
V47()
V48()
V49()
V50() )
Element of
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) ) ) holds
seq : ( (
Function-like V18(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) ) (
Relation-like NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) )
-defined REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set )
-valued Function-like non
empty total V18(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) )
complex-valued ext-real-valued real-valued )
Real_Sequence)
. n : ( ( ) (
V21()
epsilon-transitive epsilon-connected ordinal natural V31()
real V33()
V34()
V45()
V46()
V47()
V48()
V49()
V50() )
Element of
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) (
V21()
V31()
real )
Real)
= (seq1 : ( ( Function-like V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K19(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K19(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) -valued Function-like non empty total V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K19(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) . n : ( ( ) ( V21() epsilon-transitive epsilon-connected ordinal natural V31() real V33() V34() V45() V46() V47() V48() V49() V50() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K19(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) (
V21()
V31()
real )
Real)
* (seq2 : ( ( Function-like V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K19(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K19(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) -valued Function-like non empty total V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K19(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) . n : ( ( ) ( V21() epsilon-transitive epsilon-connected ordinal natural V31() real V33() V34() V45() V46() V47() V48() V49() V50() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K19(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) (
V21()
V31()
real )
Real) : ( ( ) (
V21()
V31()
real )
Element of
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) ) ;
theorem
for
r being ( (
real ) (
V21()
V31()
real )
number )
for
seq1,
seq2 being ( (
Function-like V18(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) ) (
Relation-like NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) )
-defined REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set )
-valued Function-like non
empty total V18(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) )
complex-valued ext-real-valued real-valued )
Real_Sequence) holds
(
seq1 : ( (
Function-like V18(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) ) (
Relation-like NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) )
-defined REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set )
-valued Function-like non
empty total V18(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) )
complex-valued ext-real-valued real-valued )
Real_Sequence)
= r : ( (
real ) (
V21()
V31()
real )
number )
(#) seq2 : ( (
Function-like V18(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) ) (
Relation-like NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) )
-defined REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set )
-valued Function-like non
empty total V18(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) )
complex-valued ext-real-valued real-valued )
Real_Sequence) : ( (
Function-like ) (
Relation-like NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) )
-defined REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set )
-valued Function-like non
empty total V18(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) )
complex-valued ext-real-valued real-valued )
Element of
K19(
K20(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) (
complex-valued ext-real-valued real-valued )
set ) ) : ( ( ) ( )
set ) ) iff for
n being ( ( ) (
V21()
epsilon-transitive epsilon-connected ordinal natural V31()
real V33()
V34()
V45()
V46()
V47()
V48()
V49()
V50() )
Element of
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) ) ) holds
seq1 : ( (
Function-like V18(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) ) (
Relation-like NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) )
-defined REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set )
-valued Function-like non
empty total V18(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) )
complex-valued ext-real-valued real-valued )
Real_Sequence)
. n : ( ( ) (
V21()
epsilon-transitive epsilon-connected ordinal natural V31()
real V33()
V34()
V45()
V46()
V47()
V48()
V49()
V50() )
Element of
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) (
V21()
V31()
real )
Real)
= r : ( (
real ) (
V21()
V31()
real )
number )
* (seq2 : ( ( Function-like V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K19(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K19(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) -valued Function-like non empty total V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K19(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) . n : ( ( ) ( V21() epsilon-transitive epsilon-connected ordinal natural V31() real V33() V34() V45() V46() V47() V48() V49() V50() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K19(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) (
V21()
V31()
real )
Real) : ( ( ) (
V21()
V31()
real )
Element of
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) ) ;
theorem
for
seq1,
seq2 being ( (
Function-like V18(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) ) (
Relation-like NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) )
-defined REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set )
-valued Function-like non
empty total V18(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) )
complex-valued ext-real-valued real-valued )
Real_Sequence) holds
(
seq1 : ( (
Function-like V18(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) ) (
Relation-like NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) )
-defined REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set )
-valued Function-like non
empty total V18(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) )
complex-valued ext-real-valued real-valued )
Real_Sequence)
= - seq2 : ( (
Function-like V18(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) ) (
Relation-like NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) )
-defined REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set )
-valued Function-like non
empty total V18(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) )
complex-valued ext-real-valued real-valued )
Real_Sequence) : ( (
Function-like ) (
Relation-like NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) )
-defined REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set )
-valued Function-like non
empty total V18(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) )
complex-valued ext-real-valued real-valued )
Element of
K19(
K20(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) (
complex-valued ext-real-valued real-valued )
set ) ) : ( ( ) ( )
set ) ) iff for
n being ( ( ) (
V21()
epsilon-transitive epsilon-connected ordinal natural V31()
real V33()
V34()
V45()
V46()
V47()
V48()
V49()
V50() )
Element of
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) ) ) holds
seq1 : ( (
Function-like V18(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) ) (
Relation-like NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) )
-defined REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set )
-valued Function-like non
empty total V18(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) )
complex-valued ext-real-valued real-valued )
Real_Sequence)
. n : ( ( ) (
V21()
epsilon-transitive epsilon-connected ordinal natural V31()
real V33()
V34()
V45()
V46()
V47()
V48()
V49()
V50() )
Element of
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) (
V21()
V31()
real )
Real)
= - (seq2 : ( ( Function-like V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K19(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K19(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) -valued Function-like non empty total V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K19(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) . n : ( ( ) ( V21() epsilon-transitive epsilon-connected ordinal natural V31() real V33() V34() V45() V46() V47() V48() V49() V50() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K19(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) (
V21()
V31()
real )
Real) : ( ( ) (
V21()
V31()
real )
Element of
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) ) ;
theorem
for
seq1,
seq2 being ( (
Function-like V18(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) ) (
Relation-like NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) )
-defined REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set )
-valued Function-like non
empty total V18(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) )
complex-valued ext-real-valued real-valued )
Real_Sequence) holds
seq1 : ( (
Function-like V18(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) ) (
Relation-like NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) )
-defined REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set )
-valued Function-like non
empty total V18(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) )
complex-valued ext-real-valued real-valued )
Real_Sequence)
- seq2 : ( (
Function-like V18(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) ) (
Relation-like NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) )
-defined REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set )
-valued Function-like non
empty total V18(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) )
complex-valued ext-real-valued real-valued )
Real_Sequence) : ( (
Function-like ) (
Relation-like NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) )
-defined REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set )
-valued Function-like non
empty total V18(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) )
complex-valued ext-real-valued real-valued )
Element of
K19(
K20(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) (
complex-valued ext-real-valued real-valued )
set ) ) : ( ( ) ( )
set ) )
= seq1 : ( (
Function-like V18(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) ) (
Relation-like NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) )
-defined REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set )
-valued Function-like non
empty total V18(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) )
complex-valued ext-real-valued real-valued )
Real_Sequence)
+ (- seq2 : ( ( Function-like V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K19(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K19(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) -valued Function-like non empty total V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K19(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) ) : ( (
Function-like ) (
Relation-like NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) )
-defined REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set )
-valued Function-like non
empty total V18(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) )
complex-valued ext-real-valued real-valued )
Element of
K19(
K20(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) (
complex-valued ext-real-valued real-valued )
set ) ) : ( ( ) ( )
set ) ) : ( (
Function-like ) (
Relation-like NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) )
-defined REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set )
-valued Function-like non
empty total V18(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) )
complex-valued ext-real-valued real-valued )
Element of
K19(
K20(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) (
complex-valued ext-real-valued real-valued )
set ) ) : ( ( ) ( )
set ) ) ;
theorem
for
seq1,
seq being ( (
Function-like V18(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) ) (
Relation-like NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) )
-defined REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set )
-valued Function-like non
empty total V18(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) )
complex-valued ext-real-valued real-valued )
Real_Sequence) holds
(
seq1 : ( (
Function-like V18(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) ) (
Relation-like NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) )
-defined REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set )
-valued Function-like non
empty total V18(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) )
complex-valued ext-real-valued real-valued )
Real_Sequence)
= abs seq : ( (
Function-like V18(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) ) (
Relation-like NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) )
-defined REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set )
-valued Function-like non
empty total V18(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) )
complex-valued ext-real-valued real-valued )
Real_Sequence) : ( (
Function-like ) (
Relation-like NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) )
-defined REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set )
-valued Function-like non
empty total V18(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) )
complex-valued ext-real-valued real-valued )
Element of
K19(
K20(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) (
complex-valued ext-real-valued real-valued )
set ) ) : ( ( ) ( )
set ) ) iff for
n being ( ( ) (
V21()
epsilon-transitive epsilon-connected ordinal natural V31()
real V33()
V34()
V45()
V46()
V47()
V48()
V49()
V50() )
Element of
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) ) ) holds
seq1 : ( (
Function-like V18(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) ) (
Relation-like NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) )
-defined REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set )
-valued Function-like non
empty total V18(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) )
complex-valued ext-real-valued real-valued )
Real_Sequence)
. n : ( ( ) (
V21()
epsilon-transitive epsilon-connected ordinal natural V31()
real V33()
V34()
V45()
V46()
V47()
V48()
V49()
V50() )
Element of
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) (
V21()
V31()
real )
Real)
= abs (seq : ( ( Function-like V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K19(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K19(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) -valued Function-like non empty total V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K19(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) . n : ( ( ) ( V21() epsilon-transitive epsilon-connected ordinal natural V31() real V33() V34() V45() V46() V47() V48() V49() V50() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K19(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) (
V21()
V31()
real )
Real) : ( ( ) (
V21()
V31()
real )
Element of
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) ) ;
theorem
for
seq1,
seq2,
seq3 being ( (
Function-like V18(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) ) (
Relation-like NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) )
-defined REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set )
-valued Function-like non
empty total V18(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) )
complex-valued ext-real-valued real-valued )
Real_Sequence) holds
(seq1 : ( ( Function-like V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K19(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K19(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) -valued Function-like non empty total V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K19(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) + seq2 : ( ( Function-like V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K19(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K19(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) -valued Function-like non empty total V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K19(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) ) : ( (
Function-like ) (
Relation-like NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) )
-defined REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set )
-valued Function-like non
empty total V18(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) )
complex-valued ext-real-valued real-valued )
Element of
K19(
K20(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) (
complex-valued ext-real-valued real-valued )
set ) ) : ( ( ) ( )
set ) )
+ seq3 : ( (
Function-like V18(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) ) (
Relation-like NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) )
-defined REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set )
-valued Function-like non
empty total V18(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) )
complex-valued ext-real-valued real-valued )
Real_Sequence) : ( (
Function-like ) (
Relation-like NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) )
-defined REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set )
-valued Function-like non
empty total V18(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) )
complex-valued ext-real-valued real-valued )
Element of
K19(
K20(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) (
complex-valued ext-real-valued real-valued )
set ) ) : ( ( ) ( )
set ) )
= seq1 : ( (
Function-like V18(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) ) (
Relation-like NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) )
-defined REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set )
-valued Function-like non
empty total V18(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) )
complex-valued ext-real-valued real-valued )
Real_Sequence)
+ (seq2 : ( ( Function-like V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K19(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K19(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) -valued Function-like non empty total V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K19(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) + seq3 : ( ( Function-like V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K19(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K19(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) -valued Function-like non empty total V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K19(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) ) : ( (
Function-like ) (
Relation-like NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) )
-defined REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set )
-valued Function-like non
empty total V18(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) )
complex-valued ext-real-valued real-valued )
Element of
K19(
K20(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) (
complex-valued ext-real-valued real-valued )
set ) ) : ( ( ) ( )
set ) ) : ( (
Function-like ) (
Relation-like NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) )
-defined REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set )
-valued Function-like non
empty total V18(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) )
complex-valued ext-real-valued real-valued )
Element of
K19(
K20(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) (
complex-valued ext-real-valued real-valued )
set ) ) : ( ( ) ( )
set ) ) ;
theorem
for
seq1,
seq2,
seq3 being ( (
Function-like V18(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) ) (
Relation-like NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) )
-defined REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set )
-valued Function-like non
empty total V18(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) )
complex-valued ext-real-valued real-valued )
Real_Sequence) holds
(seq1 : ( ( Function-like V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K19(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K19(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) -valued Function-like non empty total V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K19(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) (#) seq2 : ( ( Function-like V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K19(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K19(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) -valued Function-like non empty total V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K19(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) ) : ( (
Function-like ) (
Relation-like NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) )
-defined REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set )
-valued Function-like non
empty total V18(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) )
complex-valued ext-real-valued real-valued )
Element of
K19(
K20(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) (
complex-valued ext-real-valued real-valued )
set ) ) : ( ( ) ( )
set ) )
(#) seq3 : ( (
Function-like V18(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) ) (
Relation-like NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) )
-defined REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set )
-valued Function-like non
empty total V18(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) )
complex-valued ext-real-valued real-valued )
Real_Sequence) : ( (
Function-like ) (
Relation-like NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) )
-defined REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set )
-valued Function-like non
empty total V18(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) )
complex-valued ext-real-valued real-valued )
Element of
K19(
K20(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) (
complex-valued ext-real-valued real-valued )
set ) ) : ( ( ) ( )
set ) )
= seq1 : ( (
Function-like V18(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) ) (
Relation-like NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) )
-defined REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set )
-valued Function-like non
empty total V18(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) )
complex-valued ext-real-valued real-valued )
Real_Sequence)
(#) (seq2 : ( ( Function-like V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K19(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K19(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) -valued Function-like non empty total V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K19(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) (#) seq3 : ( ( Function-like V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K19(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K19(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) -valued Function-like non empty total V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K19(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) ) : ( (
Function-like ) (
Relation-like NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) )
-defined REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set )
-valued Function-like non
empty total V18(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) )
complex-valued ext-real-valued real-valued )
Element of
K19(
K20(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) (
complex-valued ext-real-valued real-valued )
set ) ) : ( ( ) ( )
set ) ) : ( (
Function-like ) (
Relation-like NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) )
-defined REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set )
-valued Function-like non
empty total V18(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) )
complex-valued ext-real-valued real-valued )
Element of
K19(
K20(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) (
complex-valued ext-real-valued real-valued )
set ) ) : ( ( ) ( )
set ) ) ;
theorem
for
seq1,
seq2,
seq3 being ( (
Function-like V18(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) ) (
Relation-like NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) )
-defined REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set )
-valued Function-like non
empty total V18(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) )
complex-valued ext-real-valued real-valued )
Real_Sequence) holds
(seq1 : ( ( Function-like V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K19(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K19(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) -valued Function-like non empty total V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K19(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) + seq2 : ( ( Function-like V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K19(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K19(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) -valued Function-like non empty total V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K19(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) ) : ( (
Function-like ) (
Relation-like NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) )
-defined REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set )
-valued Function-like non
empty total V18(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) )
complex-valued ext-real-valued real-valued )
Element of
K19(
K20(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) (
complex-valued ext-real-valued real-valued )
set ) ) : ( ( ) ( )
set ) )
(#) seq3 : ( (
Function-like V18(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) ) (
Relation-like NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) )
-defined REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set )
-valued Function-like non
empty total V18(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) )
complex-valued ext-real-valued real-valued )
Real_Sequence) : ( (
Function-like ) (
Relation-like NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) )
-defined REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set )
-valued Function-like non
empty total V18(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) )
complex-valued ext-real-valued real-valued )
Element of
K19(
K20(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) (
complex-valued ext-real-valued real-valued )
set ) ) : ( ( ) ( )
set ) )
= (seq1 : ( ( Function-like V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K19(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K19(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) -valued Function-like non empty total V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K19(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) (#) seq3 : ( ( Function-like V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K19(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K19(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) -valued Function-like non empty total V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K19(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) ) : ( (
Function-like ) (
Relation-like NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) )
-defined REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set )
-valued Function-like non
empty total V18(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) )
complex-valued ext-real-valued real-valued )
Element of
K19(
K20(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) (
complex-valued ext-real-valued real-valued )
set ) ) : ( ( ) ( )
set ) )
+ (seq2 : ( ( Function-like V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K19(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K19(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) -valued Function-like non empty total V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K19(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) (#) seq3 : ( ( Function-like V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K19(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K19(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) -valued Function-like non empty total V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K19(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) ) : ( (
Function-like ) (
Relation-like NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) )
-defined REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set )
-valued Function-like non
empty total V18(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) )
complex-valued ext-real-valued real-valued )
Element of
K19(
K20(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) (
complex-valued ext-real-valued real-valued )
set ) ) : ( ( ) ( )
set ) ) : ( (
Function-like ) (
Relation-like NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) )
-defined REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set )
-valued Function-like non
empty total V18(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) )
complex-valued ext-real-valued real-valued )
Element of
K19(
K20(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) (
complex-valued ext-real-valued real-valued )
set ) ) : ( ( ) ( )
set ) ) ;
theorem
for
seq3,
seq1,
seq2 being ( (
Function-like V18(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) ) (
Relation-like NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) )
-defined REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set )
-valued Function-like non
empty total V18(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) )
complex-valued ext-real-valued real-valued )
Real_Sequence) holds
seq3 : ( (
Function-like V18(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) ) (
Relation-like NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) )
-defined REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set )
-valued Function-like non
empty total V18(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) )
complex-valued ext-real-valued real-valued )
Real_Sequence)
(#) (seq1 : ( ( Function-like V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K19(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K19(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) -valued Function-like non empty total V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K19(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) + seq2 : ( ( Function-like V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K19(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K19(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) -valued Function-like non empty total V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K19(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) ) : ( (
Function-like ) (
Relation-like NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) )
-defined REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set )
-valued Function-like non
empty total V18(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) )
complex-valued ext-real-valued real-valued )
Element of
K19(
K20(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) (
complex-valued ext-real-valued real-valued )
set ) ) : ( ( ) ( )
set ) ) : ( (
Function-like ) (
Relation-like NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) )
-defined REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set )
-valued Function-like non
empty total V18(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) )
complex-valued ext-real-valued real-valued )
Element of
K19(
K20(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) (
complex-valued ext-real-valued real-valued )
set ) ) : ( ( ) ( )
set ) )
= (seq3 : ( ( Function-like V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K19(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K19(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) -valued Function-like non empty total V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K19(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) (#) seq1 : ( ( Function-like V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K19(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K19(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) -valued Function-like non empty total V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K19(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) ) : ( (
Function-like ) (
Relation-like NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) )
-defined REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set )
-valued Function-like non
empty total V18(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) )
complex-valued ext-real-valued real-valued )
Element of
K19(
K20(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) (
complex-valued ext-real-valued real-valued )
set ) ) : ( ( ) ( )
set ) )
+ (seq3 : ( ( Function-like V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K19(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K19(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) -valued Function-like non empty total V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K19(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) (#) seq2 : ( ( Function-like V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K19(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K19(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) -valued Function-like non empty total V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K19(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) ) : ( (
Function-like ) (
Relation-like NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) )
-defined REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set )
-valued Function-like non
empty total V18(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) )
complex-valued ext-real-valued real-valued )
Element of
K19(
K20(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) (
complex-valued ext-real-valued real-valued )
set ) ) : ( ( ) ( )
set ) ) : ( (
Function-like ) (
Relation-like NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) )
-defined REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set )
-valued Function-like non
empty total V18(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) )
complex-valued ext-real-valued real-valued )
Element of
K19(
K20(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) (
complex-valued ext-real-valued real-valued )
set ) ) : ( ( ) ( )
set ) ) ;
theorem
for
seq being ( (
Function-like V18(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) ) (
Relation-like NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) )
-defined REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set )
-valued Function-like non
empty total V18(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) )
complex-valued ext-real-valued real-valued )
Real_Sequence) holds
- seq : ( (
Function-like V18(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) ) (
Relation-like NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) )
-defined REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set )
-valued Function-like non
empty total V18(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) )
complex-valued ext-real-valued real-valued )
Real_Sequence) : ( (
Function-like ) (
Relation-like NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) )
-defined REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set )
-valued Function-like non
empty total V18(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) )
complex-valued ext-real-valued real-valued )
Element of
K19(
K20(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) (
complex-valued ext-real-valued real-valued )
set ) ) : ( ( ) ( )
set ) )
= (- 1 : ( ( ) ( non empty V21() epsilon-transitive epsilon-connected ordinal natural V31() real V33() V34() V45() V46() V47() V48() V49() V50() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K19(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) (
V21()
V31()
real )
Element of
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) )
(#) seq : ( (
Function-like V18(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) ) (
Relation-like NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) )
-defined REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set )
-valued Function-like non
empty total V18(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) )
complex-valued ext-real-valued real-valued )
Real_Sequence) : ( (
Function-like ) (
Relation-like NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) )
-defined REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set )
-valued Function-like non
empty total V18(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) )
complex-valued ext-real-valued real-valued )
Element of
K19(
K20(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) (
complex-valued ext-real-valued real-valued )
set ) ) : ( ( ) ( )
set ) ) ;
theorem
for
r being ( (
real ) (
V21()
V31()
real )
number )
for
seq1,
seq2 being ( (
Function-like V18(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) ) (
Relation-like NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) )
-defined REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set )
-valued Function-like non
empty total V18(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) )
complex-valued ext-real-valued real-valued )
Real_Sequence) holds
r : ( (
real ) (
V21()
V31()
real )
number )
(#) (seq1 : ( ( Function-like V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K19(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K19(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) -valued Function-like non empty total V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K19(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) (#) seq2 : ( ( Function-like V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K19(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K19(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) -valued Function-like non empty total V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K19(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) ) : ( (
Function-like ) (
Relation-like NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) )
-defined REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set )
-valued Function-like non
empty total V18(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) )
complex-valued ext-real-valued real-valued )
Element of
K19(
K20(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) (
complex-valued ext-real-valued real-valued )
set ) ) : ( ( ) ( )
set ) ) : ( (
Function-like ) (
Relation-like NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) )
-defined REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set )
-valued Function-like non
empty total V18(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) )
complex-valued ext-real-valued real-valued )
Element of
K19(
K20(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) (
complex-valued ext-real-valued real-valued )
set ) ) : ( ( ) ( )
set ) )
= (r : ( ( real ) ( V21() V31() real ) number ) (#) seq1 : ( ( Function-like V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K19(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K19(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) -valued Function-like non empty total V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K19(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) ) : ( (
Function-like ) (
Relation-like NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) )
-defined REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set )
-valued Function-like non
empty total V18(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) )
complex-valued ext-real-valued real-valued )
Element of
K19(
K20(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) (
complex-valued ext-real-valued real-valued )
set ) ) : ( ( ) ( )
set ) )
(#) seq2 : ( (
Function-like V18(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) ) (
Relation-like NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) )
-defined REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set )
-valued Function-like non
empty total V18(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) )
complex-valued ext-real-valued real-valued )
Real_Sequence) : ( (
Function-like ) (
Relation-like NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) )
-defined REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set )
-valued Function-like non
empty total V18(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) )
complex-valued ext-real-valued real-valued )
Element of
K19(
K20(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) (
complex-valued ext-real-valued real-valued )
set ) ) : ( ( ) ( )
set ) ) ;
theorem
for
r being ( (
real ) (
V21()
V31()
real )
number )
for
seq1,
seq2 being ( (
Function-like V18(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) ) (
Relation-like NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) )
-defined REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set )
-valued Function-like non
empty total V18(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) )
complex-valued ext-real-valued real-valued )
Real_Sequence) holds
r : ( (
real ) (
V21()
V31()
real )
number )
(#) (seq1 : ( ( Function-like V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K19(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K19(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) -valued Function-like non empty total V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K19(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) (#) seq2 : ( ( Function-like V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K19(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K19(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) -valued Function-like non empty total V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K19(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) ) : ( (
Function-like ) (
Relation-like NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) )
-defined REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set )
-valued Function-like non
empty total V18(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) )
complex-valued ext-real-valued real-valued )
Element of
K19(
K20(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) (
complex-valued ext-real-valued real-valued )
set ) ) : ( ( ) ( )
set ) ) : ( (
Function-like ) (
Relation-like NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) )
-defined REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set )
-valued Function-like non
empty total V18(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) )
complex-valued ext-real-valued real-valued )
Element of
K19(
K20(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) (
complex-valued ext-real-valued real-valued )
set ) ) : ( ( ) ( )
set ) )
= seq1 : ( (
Function-like V18(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) ) (
Relation-like NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) )
-defined REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set )
-valued Function-like non
empty total V18(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) )
complex-valued ext-real-valued real-valued )
Real_Sequence)
(#) (r : ( ( real ) ( V21() V31() real ) number ) (#) seq2 : ( ( Function-like V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K19(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K19(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) -valued Function-like non empty total V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K19(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) ) : ( (
Function-like ) (
Relation-like NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) )
-defined REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set )
-valued Function-like non
empty total V18(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) )
complex-valued ext-real-valued real-valued )
Element of
K19(
K20(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) (
complex-valued ext-real-valued real-valued )
set ) ) : ( ( ) ( )
set ) ) : ( (
Function-like ) (
Relation-like NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) )
-defined REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set )
-valued Function-like non
empty total V18(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) )
complex-valued ext-real-valued real-valued )
Element of
K19(
K20(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) (
complex-valued ext-real-valued real-valued )
set ) ) : ( ( ) ( )
set ) ) ;
theorem
for
seq1,
seq2,
seq3 being ( (
Function-like V18(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) ) (
Relation-like NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) )
-defined REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set )
-valued Function-like non
empty total V18(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) )
complex-valued ext-real-valued real-valued )
Real_Sequence) holds
(seq1 : ( ( Function-like V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K19(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K19(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) -valued Function-like non empty total V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K19(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) - seq2 : ( ( Function-like V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K19(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K19(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) -valued Function-like non empty total V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K19(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) ) : ( (
Function-like ) (
Relation-like NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) )
-defined REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set )
-valued Function-like non
empty total V18(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) )
complex-valued ext-real-valued real-valued )
Element of
K19(
K20(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) (
complex-valued ext-real-valued real-valued )
set ) ) : ( ( ) ( )
set ) )
(#) seq3 : ( (
Function-like V18(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) ) (
Relation-like NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) )
-defined REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set )
-valued Function-like non
empty total V18(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) )
complex-valued ext-real-valued real-valued )
Real_Sequence) : ( (
Function-like ) (
Relation-like NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) )
-defined REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set )
-valued Function-like non
empty total V18(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) )
complex-valued ext-real-valued real-valued )
Element of
K19(
K20(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) (
complex-valued ext-real-valued real-valued )
set ) ) : ( ( ) ( )
set ) )
= (seq1 : ( ( Function-like V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K19(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K19(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) -valued Function-like non empty total V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K19(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) (#) seq3 : ( ( Function-like V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K19(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K19(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) -valued Function-like non empty total V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K19(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) ) : ( (
Function-like ) (
Relation-like NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) )
-defined REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set )
-valued Function-like non
empty total V18(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) )
complex-valued ext-real-valued real-valued )
Element of
K19(
K20(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) (
complex-valued ext-real-valued real-valued )
set ) ) : ( ( ) ( )
set ) )
- (seq2 : ( ( Function-like V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K19(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K19(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) -valued Function-like non empty total V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K19(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) (#) seq3 : ( ( Function-like V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K19(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K19(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) -valued Function-like non empty total V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K19(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) ) : ( (
Function-like ) (
Relation-like NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) )
-defined REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set )
-valued Function-like non
empty total V18(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) )
complex-valued ext-real-valued real-valued )
Element of
K19(
K20(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) (
complex-valued ext-real-valued real-valued )
set ) ) : ( ( ) ( )
set ) ) : ( (
Function-like ) (
Relation-like NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) )
-defined REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set )
-valued Function-like non
empty total V18(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) )
complex-valued ext-real-valued real-valued )
Element of
K19(
K20(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) (
complex-valued ext-real-valued real-valued )
set ) ) : ( ( ) ( )
set ) ) ;
theorem
for
seq3,
seq1,
seq2 being ( (
Function-like V18(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) ) (
Relation-like NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) )
-defined REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set )
-valued Function-like non
empty total V18(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) )
complex-valued ext-real-valued real-valued )
Real_Sequence) holds
(seq3 : ( ( Function-like V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K19(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K19(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) -valued Function-like non empty total V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K19(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) (#) seq1 : ( ( Function-like V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K19(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K19(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) -valued Function-like non empty total V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K19(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) ) : ( (
Function-like ) (
Relation-like NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) )
-defined REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set )
-valued Function-like non
empty total V18(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) )
complex-valued ext-real-valued real-valued )
Element of
K19(
K20(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) (
complex-valued ext-real-valued real-valued )
set ) ) : ( ( ) ( )
set ) )
- (seq3 : ( ( Function-like V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K19(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K19(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) -valued Function-like non empty total V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K19(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) (#) seq2 : ( ( Function-like V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K19(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K19(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) -valued Function-like non empty total V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K19(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) ) : ( (
Function-like ) (
Relation-like NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) )
-defined REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set )
-valued Function-like non
empty total V18(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) )
complex-valued ext-real-valued real-valued )
Element of
K19(
K20(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) (
complex-valued ext-real-valued real-valued )
set ) ) : ( ( ) ( )
set ) ) : ( (
Function-like ) (
Relation-like NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) )
-defined REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set )
-valued Function-like non
empty total V18(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) )
complex-valued ext-real-valued real-valued )
Element of
K19(
K20(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) (
complex-valued ext-real-valued real-valued )
set ) ) : ( ( ) ( )
set ) )
= seq3 : ( (
Function-like V18(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) ) (
Relation-like NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) )
-defined REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set )
-valued Function-like non
empty total V18(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) )
complex-valued ext-real-valued real-valued )
Real_Sequence)
(#) (seq1 : ( ( Function-like V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K19(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K19(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) -valued Function-like non empty total V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K19(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) - seq2 : ( ( Function-like V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K19(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K19(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) -valued Function-like non empty total V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K19(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) ) : ( (
Function-like ) (
Relation-like NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) )
-defined REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set )
-valued Function-like non
empty total V18(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) )
complex-valued ext-real-valued real-valued )
Element of
K19(
K20(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) (
complex-valued ext-real-valued real-valued )
set ) ) : ( ( ) ( )
set ) ) : ( (
Function-like ) (
Relation-like NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) )
-defined REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set )
-valued Function-like non
empty total V18(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) )
complex-valued ext-real-valued real-valued )
Element of
K19(
K20(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) (
complex-valued ext-real-valued real-valued )
set ) ) : ( ( ) ( )
set ) ) ;
theorem
for
r being ( (
real ) (
V21()
V31()
real )
number )
for
seq1,
seq2 being ( (
Function-like V18(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) ) (
Relation-like NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) )
-defined REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set )
-valued Function-like non
empty total V18(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) )
complex-valued ext-real-valued real-valued )
Real_Sequence) holds
r : ( (
real ) (
V21()
V31()
real )
number )
(#) (seq1 : ( ( Function-like V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K19(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K19(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) -valued Function-like non empty total V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K19(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) + seq2 : ( ( Function-like V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K19(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K19(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) -valued Function-like non empty total V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K19(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) ) : ( (
Function-like ) (
Relation-like NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) )
-defined REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set )
-valued Function-like non
empty total V18(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) )
complex-valued ext-real-valued real-valued )
Element of
K19(
K20(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) (
complex-valued ext-real-valued real-valued )
set ) ) : ( ( ) ( )
set ) ) : ( (
Function-like ) (
Relation-like NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) )
-defined REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set )
-valued Function-like non
empty total V18(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) )
complex-valued ext-real-valued real-valued )
Element of
K19(
K20(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) (
complex-valued ext-real-valued real-valued )
set ) ) : ( ( ) ( )
set ) )
= (r : ( ( real ) ( V21() V31() real ) number ) (#) seq1 : ( ( Function-like V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K19(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K19(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) -valued Function-like non empty total V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K19(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) ) : ( (
Function-like ) (
Relation-like NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) )
-defined REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set )
-valued Function-like non
empty total V18(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) )
complex-valued ext-real-valued real-valued )
Element of
K19(
K20(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) (
complex-valued ext-real-valued real-valued )
set ) ) : ( ( ) ( )
set ) )
+ (r : ( ( real ) ( V21() V31() real ) number ) (#) seq2 : ( ( Function-like V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K19(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K19(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) -valued Function-like non empty total V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K19(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) ) : ( (
Function-like ) (
Relation-like NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) )
-defined REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set )
-valued Function-like non
empty total V18(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) )
complex-valued ext-real-valued real-valued )
Element of
K19(
K20(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) (
complex-valued ext-real-valued real-valued )
set ) ) : ( ( ) ( )
set ) ) : ( (
Function-like ) (
Relation-like NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) )
-defined REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set )
-valued Function-like non
empty total V18(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) )
complex-valued ext-real-valued real-valued )
Element of
K19(
K20(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) (
complex-valued ext-real-valued real-valued )
set ) ) : ( ( ) ( )
set ) ) ;
theorem
for
r,
p being ( (
real ) (
V21()
V31()
real )
number )
for
seq being ( (
Function-like V18(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) ) (
Relation-like NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) )
-defined REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set )
-valued Function-like non
empty total V18(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) )
complex-valued ext-real-valued real-valued )
Real_Sequence) holds
(r : ( ( real ) ( V21() V31() real ) number ) * p : ( ( real ) ( V21() V31() real ) number ) ) : ( ( ) (
V21()
V31()
real )
set )
(#) seq : ( (
Function-like V18(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) ) (
Relation-like NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) )
-defined REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set )
-valued Function-like non
empty total V18(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) )
complex-valued ext-real-valued real-valued )
Real_Sequence) : ( (
Function-like ) (
Relation-like NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) )
-defined REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set )
-valued Function-like non
empty total V18(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) )
complex-valued ext-real-valued real-valued )
Element of
K19(
K20(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) (
complex-valued ext-real-valued real-valued )
set ) ) : ( ( ) ( )
set ) )
= r : ( (
real ) (
V21()
V31()
real )
number )
(#) (p : ( ( real ) ( V21() V31() real ) number ) (#) seq : ( ( Function-like V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K19(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K19(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) -valued Function-like non empty total V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K19(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) ) : ( (
Function-like ) (
Relation-like NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) )
-defined REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set )
-valued Function-like non
empty total V18(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) )
complex-valued ext-real-valued real-valued )
Element of
K19(
K20(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) (
complex-valued ext-real-valued real-valued )
set ) ) : ( ( ) ( )
set ) ) : ( (
Function-like ) (
Relation-like NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) )
-defined REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set )
-valued Function-like non
empty total V18(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) )
complex-valued ext-real-valued real-valued )
Element of
K19(
K20(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) (
complex-valued ext-real-valued real-valued )
set ) ) : ( ( ) ( )
set ) ) ;
theorem
for
r being ( (
real ) (
V21()
V31()
real )
number )
for
seq1,
seq2 being ( (
Function-like V18(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) ) (
Relation-like NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) )
-defined REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set )
-valued Function-like non
empty total V18(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) )
complex-valued ext-real-valued real-valued )
Real_Sequence) holds
r : ( (
real ) (
V21()
V31()
real )
number )
(#) (seq1 : ( ( Function-like V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K19(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K19(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) -valued Function-like non empty total V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K19(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) - seq2 : ( ( Function-like V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K19(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K19(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) -valued Function-like non empty total V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K19(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) ) : ( (
Function-like ) (
Relation-like NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) )
-defined REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set )
-valued Function-like non
empty total V18(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) )
complex-valued ext-real-valued real-valued )
Element of
K19(
K20(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) (
complex-valued ext-real-valued real-valued )
set ) ) : ( ( ) ( )
set ) ) : ( (
Function-like ) (
Relation-like NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) )
-defined REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set )
-valued Function-like non
empty total V18(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) )
complex-valued ext-real-valued real-valued )
Element of
K19(
K20(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) (
complex-valued ext-real-valued real-valued )
set ) ) : ( ( ) ( )
set ) )
= (r : ( ( real ) ( V21() V31() real ) number ) (#) seq1 : ( ( Function-like V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K19(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K19(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) -valued Function-like non empty total V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K19(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) ) : ( (
Function-like ) (
Relation-like NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) )
-defined REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set )
-valued Function-like non
empty total V18(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) )
complex-valued ext-real-valued real-valued )
Element of
K19(
K20(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) (
complex-valued ext-real-valued real-valued )
set ) ) : ( ( ) ( )
set ) )
- (r : ( ( real ) ( V21() V31() real ) number ) (#) seq2 : ( ( Function-like V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K19(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K19(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) -valued Function-like non empty total V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K19(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) ) : ( (
Function-like ) (
Relation-like NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) )
-defined REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set )
-valued Function-like non
empty total V18(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) )
complex-valued ext-real-valued real-valued )
Element of
K19(
K20(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) (
complex-valued ext-real-valued real-valued )
set ) ) : ( ( ) ( )
set ) ) : ( (
Function-like ) (
Relation-like NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) )
-defined REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set )
-valued Function-like non
empty total V18(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) )
complex-valued ext-real-valued real-valued )
Element of
K19(
K20(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) (
complex-valued ext-real-valued real-valued )
set ) ) : ( ( ) ( )
set ) ) ;
theorem
for
r being ( (
real ) (
V21()
V31()
real )
number )
for
seq1,
seq being ( (
Function-like V18(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) ) (
Relation-like NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) )
-defined REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set )
-valued Function-like non
empty total V18(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) )
complex-valued ext-real-valued real-valued )
Real_Sequence) holds
r : ( (
real ) (
V21()
V31()
real )
number )
(#) (seq1 : ( ( Function-like V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K19(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K19(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) -valued Function-like non empty total V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K19(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) /" seq : ( ( Function-like V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K19(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K19(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) -valued Function-like non empty total V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K19(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) ) : ( (
Function-like ) (
Relation-like NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) )
-defined REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set )
-valued Function-like non
empty total V18(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) )
complex-valued ext-real-valued real-valued )
Element of
K19(
K20(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) (
complex-valued ext-real-valued real-valued )
set ) ) : ( ( ) ( )
set ) ) : ( (
Function-like ) (
Relation-like NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) )
-defined REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set )
-valued Function-like non
empty total V18(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) )
complex-valued ext-real-valued real-valued )
Element of
K19(
K20(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) (
complex-valued ext-real-valued real-valued )
set ) ) : ( ( ) ( )
set ) )
= (r : ( ( real ) ( V21() V31() real ) number ) (#) seq1 : ( ( Function-like V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K19(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K19(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) -valued Function-like non empty total V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K19(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) ) : ( (
Function-like ) (
Relation-like NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) )
-defined REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set )
-valued Function-like non
empty total V18(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) )
complex-valued ext-real-valued real-valued )
Element of
K19(
K20(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) (
complex-valued ext-real-valued real-valued )
set ) ) : ( ( ) ( )
set ) )
/" seq : ( (
Function-like V18(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) ) (
Relation-like NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) )
-defined REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set )
-valued Function-like non
empty total V18(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) )
complex-valued ext-real-valued real-valued )
Real_Sequence) : ( (
Function-like ) (
Relation-like NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) )
-defined REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set )
-valued Function-like non
empty total V18(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) )
complex-valued ext-real-valued real-valued )
Element of
K19(
K20(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) (
complex-valued ext-real-valued real-valued )
set ) ) : ( ( ) ( )
set ) ) ;
theorem
for
seq1,
seq2,
seq3 being ( (
Function-like V18(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) ) (
Relation-like NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) )
-defined REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set )
-valued Function-like non
empty total V18(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) )
complex-valued ext-real-valued real-valued )
Real_Sequence) holds
seq1 : ( (
Function-like V18(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) ) (
Relation-like NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) )
-defined REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set )
-valued Function-like non
empty total V18(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) )
complex-valued ext-real-valued real-valued )
Real_Sequence)
- (seq2 : ( ( Function-like V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K19(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K19(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) -valued Function-like non empty total V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K19(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) + seq3 : ( ( Function-like V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K19(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K19(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) -valued Function-like non empty total V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K19(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) ) : ( (
Function-like ) (
Relation-like NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) )
-defined REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set )
-valued Function-like non
empty total V18(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) )
complex-valued ext-real-valued real-valued )
Element of
K19(
K20(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) (
complex-valued ext-real-valued real-valued )
set ) ) : ( ( ) ( )
set ) ) : ( (
Function-like ) (
Relation-like NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) )
-defined REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set )
-valued Function-like non
empty total V18(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) )
complex-valued ext-real-valued real-valued )
Element of
K19(
K20(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) (
complex-valued ext-real-valued real-valued )
set ) ) : ( ( ) ( )
set ) )
= (seq1 : ( ( Function-like V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K19(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K19(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) -valued Function-like non empty total V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K19(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) - seq2 : ( ( Function-like V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K19(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K19(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) -valued Function-like non empty total V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K19(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) ) : ( (
Function-like ) (
Relation-like NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) )
-defined REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set )
-valued Function-like non
empty total V18(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) )
complex-valued ext-real-valued real-valued )
Element of
K19(
K20(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) (
complex-valued ext-real-valued real-valued )
set ) ) : ( ( ) ( )
set ) )
- seq3 : ( (
Function-like V18(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) ) (
Relation-like NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) )
-defined REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set )
-valued Function-like non
empty total V18(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) )
complex-valued ext-real-valued real-valued )
Real_Sequence) : ( (
Function-like ) (
Relation-like NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) )
-defined REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set )
-valued Function-like non
empty total V18(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) )
complex-valued ext-real-valued real-valued )
Element of
K19(
K20(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) (
complex-valued ext-real-valued real-valued )
set ) ) : ( ( ) ( )
set ) ) ;
theorem
for
seq1,
seq2 being ( (
Function-like V18(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) ) (
Relation-like NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) )
-defined REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set )
-valued Function-like non
empty total V18(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) )
complex-valued ext-real-valued real-valued )
Real_Sequence) holds
seq1 : ( (
Function-like V18(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) ) (
Relation-like NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) )
-defined REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set )
-valued Function-like non
empty total V18(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) )
complex-valued ext-real-valued real-valued )
Real_Sequence)
- (- seq2 : ( ( Function-like V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K19(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K19(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) -valued Function-like non empty total V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K19(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) ) : ( (
Function-like ) (
Relation-like NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) )
-defined REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set )
-valued Function-like non
empty total V18(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) )
complex-valued ext-real-valued real-valued )
Element of
K19(
K20(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) (
complex-valued ext-real-valued real-valued )
set ) ) : ( ( ) ( )
set ) ) : ( (
Function-like ) (
Relation-like NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) )
-defined REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set )
-valued Function-like non
empty total V18(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) )
complex-valued ext-real-valued real-valued )
Element of
K19(
K20(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) (
complex-valued ext-real-valued real-valued )
set ) ) : ( ( ) ( )
set ) )
= seq1 : ( (
Function-like V18(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) ) (
Relation-like NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) )
-defined REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set )
-valued Function-like non
empty total V18(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) )
complex-valued ext-real-valued real-valued )
Real_Sequence)
+ seq2 : ( (
Function-like V18(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) ) (
Relation-like NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) )
-defined REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set )
-valued Function-like non
empty total V18(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) )
complex-valued ext-real-valued real-valued )
Real_Sequence) : ( (
Function-like ) (
Relation-like NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) )
-defined REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set )
-valued Function-like non
empty total V18(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) )
complex-valued ext-real-valued real-valued )
Element of
K19(
K20(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) (
complex-valued ext-real-valued real-valued )
set ) ) : ( ( ) ( )
set ) ) ;
theorem
for
seq1,
seq2,
seq3 being ( (
Function-like V18(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) ) (
Relation-like NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) )
-defined REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set )
-valued Function-like non
empty total V18(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) )
complex-valued ext-real-valued real-valued )
Real_Sequence) holds
seq1 : ( (
Function-like V18(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) ) (
Relation-like NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) )
-defined REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set )
-valued Function-like non
empty total V18(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) )
complex-valued ext-real-valued real-valued )
Real_Sequence)
- (seq2 : ( ( Function-like V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K19(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K19(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) -valued Function-like non empty total V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K19(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) - seq3 : ( ( Function-like V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K19(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K19(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) -valued Function-like non empty total V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K19(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) ) : ( (
Function-like ) (
Relation-like NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) )
-defined REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set )
-valued Function-like non
empty total V18(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) )
complex-valued ext-real-valued real-valued )
Element of
K19(
K20(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) (
complex-valued ext-real-valued real-valued )
set ) ) : ( ( ) ( )
set ) ) : ( (
Function-like ) (
Relation-like NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) )
-defined REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set )
-valued Function-like non
empty total V18(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) )
complex-valued ext-real-valued real-valued )
Element of
K19(
K20(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) (
complex-valued ext-real-valued real-valued )
set ) ) : ( ( ) ( )
set ) )
= (seq1 : ( ( Function-like V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K19(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K19(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) -valued Function-like non empty total V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K19(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) - seq2 : ( ( Function-like V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K19(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K19(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) -valued Function-like non empty total V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K19(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) ) : ( (
Function-like ) (
Relation-like NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) )
-defined REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set )
-valued Function-like non
empty total V18(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) )
complex-valued ext-real-valued real-valued )
Element of
K19(
K20(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) (
complex-valued ext-real-valued real-valued )
set ) ) : ( ( ) ( )
set ) )
+ seq3 : ( (
Function-like V18(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) ) (
Relation-like NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) )
-defined REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set )
-valued Function-like non
empty total V18(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) )
complex-valued ext-real-valued real-valued )
Real_Sequence) : ( (
Function-like ) (
Relation-like NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) )
-defined REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set )
-valued Function-like non
empty total V18(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) )
complex-valued ext-real-valued real-valued )
Element of
K19(
K20(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) (
complex-valued ext-real-valued real-valued )
set ) ) : ( ( ) ( )
set ) ) ;
theorem
for
seq1,
seq2,
seq3 being ( (
Function-like V18(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) ) (
Relation-like NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) )
-defined REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set )
-valued Function-like non
empty total V18(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) )
complex-valued ext-real-valued real-valued )
Real_Sequence) holds
seq1 : ( (
Function-like V18(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) ) (
Relation-like NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) )
-defined REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set )
-valued Function-like non
empty total V18(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) )
complex-valued ext-real-valued real-valued )
Real_Sequence)
+ (seq2 : ( ( Function-like V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K19(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K19(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) -valued Function-like non empty total V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K19(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) - seq3 : ( ( Function-like V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K19(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K19(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) -valued Function-like non empty total V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K19(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) ) : ( (
Function-like ) (
Relation-like NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) )
-defined REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set )
-valued Function-like non
empty total V18(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) )
complex-valued ext-real-valued real-valued )
Element of
K19(
K20(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) (
complex-valued ext-real-valued real-valued )
set ) ) : ( ( ) ( )
set ) ) : ( (
Function-like ) (
Relation-like NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) )
-defined REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set )
-valued Function-like non
empty total V18(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) )
complex-valued ext-real-valued real-valued )
Element of
K19(
K20(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) (
complex-valued ext-real-valued real-valued )
set ) ) : ( ( ) ( )
set ) )
= (seq1 : ( ( Function-like V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K19(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K19(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) -valued Function-like non empty total V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K19(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) + seq2 : ( ( Function-like V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K19(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K19(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) -valued Function-like non empty total V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K19(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) ) : ( (
Function-like ) (
Relation-like NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) )
-defined REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set )
-valued Function-like non
empty total V18(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) )
complex-valued ext-real-valued real-valued )
Element of
K19(
K20(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) (
complex-valued ext-real-valued real-valued )
set ) ) : ( ( ) ( )
set ) )
- seq3 : ( (
Function-like V18(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) ) (
Relation-like NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) )
-defined REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set )
-valued Function-like non
empty total V18(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) )
complex-valued ext-real-valued real-valued )
Real_Sequence) : ( (
Function-like ) (
Relation-like NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) )
-defined REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set )
-valued Function-like non
empty total V18(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) )
complex-valued ext-real-valued real-valued )
Element of
K19(
K20(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) (
complex-valued ext-real-valued real-valued )
set ) ) : ( ( ) ( )
set ) ) ;
theorem
for
seq1,
seq2 being ( (
Function-like V18(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) ) (
Relation-like NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) )
-defined REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set )
-valued Function-like non
empty total V18(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) )
complex-valued ext-real-valued real-valued )
Real_Sequence) holds
(
(- seq1 : ( ( Function-like V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K19(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K19(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) -valued Function-like non empty total V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K19(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) ) : ( (
Function-like ) (
Relation-like NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) )
-defined REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set )
-valued Function-like non
empty total V18(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) )
complex-valued ext-real-valued real-valued )
Element of
K19(
K20(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) (
complex-valued ext-real-valued real-valued )
set ) ) : ( ( ) ( )
set ) )
(#) seq2 : ( (
Function-like V18(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) ) (
Relation-like NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) )
-defined REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set )
-valued Function-like non
empty total V18(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) )
complex-valued ext-real-valued real-valued )
Real_Sequence) : ( (
Function-like ) (
Relation-like NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) )
-defined REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set )
-valued Function-like non
empty total V18(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) )
complex-valued ext-real-valued real-valued )
Element of
K19(
K20(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) (
complex-valued ext-real-valued real-valued )
set ) ) : ( ( ) ( )
set ) )
= - (seq1 : ( ( Function-like V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K19(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K19(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) -valued Function-like non empty total V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K19(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) (#) seq2 : ( ( Function-like V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K19(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K19(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) -valued Function-like non empty total V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K19(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) ) : ( (
Function-like ) (
Relation-like NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) )
-defined REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set )
-valued Function-like non
empty total V18(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) )
complex-valued ext-real-valued real-valued )
Element of
K19(
K20(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) (
complex-valued ext-real-valued real-valued )
set ) ) : ( ( ) ( )
set ) ) : ( (
Function-like ) (
Relation-like NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) )
-defined REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set )
-valued Function-like non
empty total V18(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) )
complex-valued ext-real-valued real-valued )
Element of
K19(
K20(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) (
complex-valued ext-real-valued real-valued )
set ) ) : ( ( ) ( )
set ) ) &
seq1 : ( (
Function-like V18(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) ) (
Relation-like NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) )
-defined REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set )
-valued Function-like non
empty total V18(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) )
complex-valued ext-real-valued real-valued )
Real_Sequence)
(#) (- seq2 : ( ( Function-like V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K19(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K19(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) -valued Function-like non empty total V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K19(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) ) : ( (
Function-like ) (
Relation-like NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) )
-defined REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set )
-valued Function-like non
empty total V18(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) )
complex-valued ext-real-valued real-valued )
Element of
K19(
K20(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) (
complex-valued ext-real-valued real-valued )
set ) ) : ( ( ) ( )
set ) ) : ( (
Function-like ) (
Relation-like NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) )
-defined REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set )
-valued Function-like non
empty total V18(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) )
complex-valued ext-real-valued real-valued )
Element of
K19(
K20(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) (
complex-valued ext-real-valued real-valued )
set ) ) : ( ( ) ( )
set ) )
= - (seq1 : ( ( Function-like V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K19(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K19(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) -valued Function-like non empty total V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K19(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) (#) seq2 : ( ( Function-like V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K19(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K19(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) -valued Function-like non empty total V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K19(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) ) : ( (
Function-like ) (
Relation-like NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) )
-defined REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set )
-valued Function-like non
empty total V18(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) )
complex-valued ext-real-valued real-valued )
Element of
K19(
K20(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) (
complex-valued ext-real-valued real-valued )
set ) ) : ( ( ) ( )
set ) ) : ( (
Function-like ) (
Relation-like NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) )
-defined REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set )
-valued Function-like non
empty total V18(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) )
complex-valued ext-real-valued real-valued )
Element of
K19(
K20(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) (
complex-valued ext-real-valued real-valued )
set ) ) : ( ( ) ( )
set ) ) ) ;
theorem
for
seq,
seq1 being ( (
Function-like V18(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) ) (
Relation-like NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) )
-defined REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set )
-valued Function-like non
empty total V18(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) )
complex-valued ext-real-valued real-valued )
Real_Sequence) holds
( (
seq : ( (
Function-like V18(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) ) (
Relation-like NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) )
-defined REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set )
-valued Function-like non
empty total V18(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) )
complex-valued ext-real-valued real-valued )
Real_Sequence) is
non-zero &
seq1 : ( (
Function-like V18(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) ) (
Relation-like NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) )
-defined REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set )
-valued Function-like non
empty total V18(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) )
complex-valued ext-real-valued real-valued )
Real_Sequence) is
non-zero ) iff
seq : ( (
Function-like V18(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) ) (
Relation-like NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) )
-defined REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set )
-valued Function-like non
empty total V18(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) )
complex-valued ext-real-valued real-valued )
Real_Sequence)
(#) seq1 : ( (
Function-like V18(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) ) (
Relation-like NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) )
-defined REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set )
-valued Function-like non
empty total V18(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) )
complex-valued ext-real-valued real-valued )
Real_Sequence) : ( (
Function-like ) (
Relation-like NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) )
-defined REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set )
-valued Function-like non
empty total V18(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) )
complex-valued ext-real-valued real-valued )
Element of
K19(
K20(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) (
complex-valued ext-real-valued real-valued )
set ) ) : ( ( ) ( )
set ) ) is
non-zero ) ;
theorem
for
seq,
seq1 being ( (
Function-like V18(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) ) (
Relation-like NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) )
-defined REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set )
-valued Function-like non
empty total V18(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) )
complex-valued ext-real-valued real-valued )
Real_Sequence) holds
(seq : ( ( Function-like V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K19(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K19(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) -valued Function-like non empty total V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K19(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) ") : ( (
Function-like ) (
Relation-like NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) )
-defined REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set )
-valued Function-like non
empty total V18(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) )
complex-valued ext-real-valued real-valued )
Element of
K19(
K20(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) (
complex-valued ext-real-valued real-valued )
set ) ) : ( ( ) ( )
set ) )
(#) (seq1 : ( ( Function-like V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K19(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K19(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) -valued Function-like non empty total V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K19(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) ") : ( (
Function-like ) (
Relation-like NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) )
-defined REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set )
-valued Function-like non
empty total V18(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) )
complex-valued ext-real-valued real-valued )
Element of
K19(
K20(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) (
complex-valued ext-real-valued real-valued )
set ) ) : ( ( ) ( )
set ) ) : ( (
Function-like ) (
Relation-like NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) )
-defined REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set )
-valued Function-like non
empty total V18(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) )
complex-valued ext-real-valued real-valued )
Element of
K19(
K20(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) (
complex-valued ext-real-valued real-valued )
set ) ) : ( ( ) ( )
set ) )
= (seq : ( ( Function-like V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K19(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K19(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) -valued Function-like non empty total V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K19(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) (#) seq1 : ( ( Function-like V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K19(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K19(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) -valued Function-like non empty total V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K19(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) ) : ( (
Function-like ) (
Relation-like NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) )
-defined REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set )
-valued Function-like non
empty total V18(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) )
complex-valued ext-real-valued real-valued )
Element of
K19(
K20(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) (
complex-valued ext-real-valued real-valued )
set ) ) : ( ( ) ( )
set ) )
" : ( (
Function-like ) (
Relation-like NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) )
-defined REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set )
-valued Function-like non
empty total V18(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) )
complex-valued ext-real-valued real-valued )
Element of
K19(
K20(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) (
complex-valued ext-real-valued real-valued )
set ) ) : ( ( ) ( )
set ) ) ;
theorem
for
seq,
seq1 being ( (
Function-like V18(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) ) (
Relation-like NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) )
-defined REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set )
-valued Function-like non
empty total V18(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) )
complex-valued ext-real-valued real-valued )
Real_Sequence) st
seq : ( (
Function-like V18(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) ) (
Relation-like NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) )
-defined REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set )
-valued Function-like non
empty total V18(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) )
complex-valued ext-real-valued real-valued )
Real_Sequence) is
non-zero holds
(seq1 : ( ( Function-like V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K19(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K19(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) -valued Function-like non empty total V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K19(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) /" seq : ( ( Function-like V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K19(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K19(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) -valued Function-like non empty total V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K19(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) ) : ( (
Function-like ) (
Relation-like NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) )
-defined REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set )
-valued Function-like non
empty total V18(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) )
complex-valued ext-real-valued real-valued )
Element of
K19(
K20(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) (
complex-valued ext-real-valued real-valued )
set ) ) : ( ( ) ( )
set ) )
(#) seq : ( (
Function-like V18(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) ) (
Relation-like NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) )
-defined REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set )
-valued Function-like non
empty total V18(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) )
complex-valued ext-real-valued real-valued )
Real_Sequence) : ( (
Function-like ) (
Relation-like NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) )
-defined REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set )
-valued Function-like non
empty total V18(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) )
complex-valued ext-real-valued real-valued )
Element of
K19(
K20(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) (
complex-valued ext-real-valued real-valued )
set ) ) : ( ( ) ( )
set ) )
= seq1 : ( (
Function-like V18(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) ) (
Relation-like NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) )
-defined REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set )
-valued Function-like non
empty total V18(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) )
complex-valued ext-real-valued real-valued )
Real_Sequence) ;
theorem
for
seq9,
seq,
seq19,
seq1 being ( (
Function-like V18(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) ) (
Relation-like NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) )
-defined REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set )
-valued Function-like non
empty total V18(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) )
complex-valued ext-real-valued real-valued )
Real_Sequence) holds
(seq9 : ( ( Function-like V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K19(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K19(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) -valued Function-like non empty total V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K19(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) /" seq : ( ( Function-like V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K19(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K19(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) -valued Function-like non empty total V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K19(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) ) : ( (
Function-like ) (
Relation-like NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) )
-defined REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set )
-valued Function-like non
empty total V18(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) )
complex-valued ext-real-valued real-valued )
Element of
K19(
K20(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) (
complex-valued ext-real-valued real-valued )
set ) ) : ( ( ) ( )
set ) )
(#) (seq19 : ( ( Function-like V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K19(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K19(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) -valued Function-like non empty total V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K19(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) /" seq1 : ( ( Function-like V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K19(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K19(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) -valued Function-like non empty total V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K19(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) ) : ( (
Function-like ) (
Relation-like NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) )
-defined REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set )
-valued Function-like non
empty total V18(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) )
complex-valued ext-real-valued real-valued )
Element of
K19(
K20(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) (
complex-valued ext-real-valued real-valued )
set ) ) : ( ( ) ( )
set ) ) : ( (
Function-like ) (
Relation-like NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) )
-defined REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set )
-valued Function-like non
empty total V18(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) )
complex-valued ext-real-valued real-valued )
Element of
K19(
K20(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) (
complex-valued ext-real-valued real-valued )
set ) ) : ( ( ) ( )
set ) )
= (seq9 : ( ( Function-like V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K19(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K19(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) -valued Function-like non empty total V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K19(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) (#) seq19 : ( ( Function-like V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K19(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K19(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) -valued Function-like non empty total V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K19(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) ) : ( (
Function-like ) (
Relation-like NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) )
-defined REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set )
-valued Function-like non
empty total V18(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) )
complex-valued ext-real-valued real-valued )
Element of
K19(
K20(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) (
complex-valued ext-real-valued real-valued )
set ) ) : ( ( ) ( )
set ) )
/" (seq : ( ( Function-like V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K19(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K19(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) -valued Function-like non empty total V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K19(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) (#) seq1 : ( ( Function-like V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K19(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K19(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) -valued Function-like non empty total V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K19(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) ) : ( (
Function-like ) (
Relation-like NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) )
-defined REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set )
-valued Function-like non
empty total V18(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) )
complex-valued ext-real-valued real-valued )
Element of
K19(
K20(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) (
complex-valued ext-real-valued real-valued )
set ) ) : ( ( ) ( )
set ) ) : ( (
Function-like ) (
Relation-like NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) )
-defined REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set )
-valued Function-like non
empty total V18(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) )
complex-valued ext-real-valued real-valued )
Element of
K19(
K20(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) (
complex-valued ext-real-valued real-valued )
set ) ) : ( ( ) ( )
set ) ) ;
theorem
for
seq,
seq1 being ( (
Function-like V18(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) ) (
Relation-like NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) )
-defined REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set )
-valued Function-like non
empty total V18(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) )
complex-valued ext-real-valued real-valued )
Real_Sequence) st
seq : ( (
Function-like V18(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) ) (
Relation-like NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) )
-defined REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set )
-valued Function-like non
empty total V18(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) )
complex-valued ext-real-valued real-valued )
Real_Sequence) is
non-zero &
seq1 : ( (
Function-like V18(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) ) (
Relation-like NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) )
-defined REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set )
-valued Function-like non
empty total V18(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) )
complex-valued ext-real-valued real-valued )
Real_Sequence) is
non-zero holds
seq : ( (
Function-like V18(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) ) (
Relation-like NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) )
-defined REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set )
-valued Function-like non
empty total V18(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) )
complex-valued ext-real-valued real-valued )
Real_Sequence)
/" seq1 : ( (
Function-like V18(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) ) (
Relation-like NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) )
-defined REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set )
-valued Function-like non
empty total V18(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) )
complex-valued ext-real-valued real-valued )
Real_Sequence) : ( (
Function-like ) (
Relation-like NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) )
-defined REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set )
-valued Function-like non
empty total V18(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) )
complex-valued ext-real-valued real-valued )
Element of
K19(
K20(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) (
complex-valued ext-real-valued real-valued )
set ) ) : ( ( ) ( )
set ) ) is
non-zero ;
theorem
for
seq,
seq1 being ( (
Function-like V18(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) ) (
Relation-like NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) )
-defined REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set )
-valued Function-like non
empty total V18(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) )
complex-valued ext-real-valued real-valued )
Real_Sequence) holds
(seq : ( ( Function-like V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K19(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K19(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) -valued Function-like non empty total V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K19(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) /" seq1 : ( ( Function-like V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K19(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K19(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) -valued Function-like non empty total V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K19(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) ) : ( (
Function-like ) (
Relation-like NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) )
-defined REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set )
-valued Function-like non
empty total V18(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) )
complex-valued ext-real-valued real-valued )
Element of
K19(
K20(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) (
complex-valued ext-real-valued real-valued )
set ) ) : ( ( ) ( )
set ) )
" : ( (
Function-like ) (
Relation-like NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) )
-defined REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set )
-valued Function-like non
empty total V18(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) )
complex-valued ext-real-valued real-valued )
Element of
K19(
K20(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) (
complex-valued ext-real-valued real-valued )
set ) ) : ( ( ) ( )
set ) )
= seq1 : ( (
Function-like V18(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) ) (
Relation-like NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) )
-defined REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set )
-valued Function-like non
empty total V18(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) )
complex-valued ext-real-valued real-valued )
Real_Sequence)
/" seq : ( (
Function-like V18(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) ) (
Relation-like NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) )
-defined REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set )
-valued Function-like non
empty total V18(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) )
complex-valued ext-real-valued real-valued )
Real_Sequence) : ( (
Function-like ) (
Relation-like NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) )
-defined REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set )
-valued Function-like non
empty total V18(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) )
complex-valued ext-real-valued real-valued )
Element of
K19(
K20(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) (
complex-valued ext-real-valued real-valued )
set ) ) : ( ( ) ( )
set ) ) ;
theorem
for
seq2,
seq1,
seq being ( (
Function-like V18(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) ) (
Relation-like NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) )
-defined REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set )
-valued Function-like non
empty total V18(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) )
complex-valued ext-real-valued real-valued )
Real_Sequence) holds
seq2 : ( (
Function-like V18(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) ) (
Relation-like NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) )
-defined REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set )
-valued Function-like non
empty total V18(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) )
complex-valued ext-real-valued real-valued )
Real_Sequence)
(#) (seq1 : ( ( Function-like V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K19(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K19(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) -valued Function-like non empty total V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K19(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) /" seq : ( ( Function-like V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K19(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K19(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) -valued Function-like non empty total V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K19(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) ) : ( (
Function-like ) (
Relation-like NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) )
-defined REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set )
-valued Function-like non
empty total V18(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) )
complex-valued ext-real-valued real-valued )
Element of
K19(
K20(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) (
complex-valued ext-real-valued real-valued )
set ) ) : ( ( ) ( )
set ) ) : ( (
Function-like ) (
Relation-like NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) )
-defined REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set )
-valued Function-like non
empty total V18(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) )
complex-valued ext-real-valued real-valued )
Element of
K19(
K20(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) (
complex-valued ext-real-valued real-valued )
set ) ) : ( ( ) ( )
set ) )
= (seq2 : ( ( Function-like V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K19(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K19(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) -valued Function-like non empty total V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K19(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) (#) seq1 : ( ( Function-like V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K19(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K19(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) -valued Function-like non empty total V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K19(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) ) : ( (
Function-like ) (
Relation-like NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) )
-defined REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set )
-valued Function-like non
empty total V18(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) )
complex-valued ext-real-valued real-valued )
Element of
K19(
K20(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) (
complex-valued ext-real-valued real-valued )
set ) ) : ( ( ) ( )
set ) )
/" seq : ( (
Function-like V18(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) ) (
Relation-like NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) )
-defined REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set )
-valued Function-like non
empty total V18(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) )
complex-valued ext-real-valued real-valued )
Real_Sequence) : ( (
Function-like ) (
Relation-like NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) )
-defined REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set )
-valued Function-like non
empty total V18(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) )
complex-valued ext-real-valued real-valued )
Element of
K19(
K20(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) (
complex-valued ext-real-valued real-valued )
set ) ) : ( ( ) ( )
set ) ) ;
theorem
for
seq2,
seq,
seq1 being ( (
Function-like V18(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) ) (
Relation-like NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) )
-defined REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set )
-valued Function-like non
empty total V18(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) )
complex-valued ext-real-valued real-valued )
Real_Sequence) holds
seq2 : ( (
Function-like V18(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) ) (
Relation-like NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) )
-defined REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set )
-valued Function-like non
empty total V18(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) )
complex-valued ext-real-valued real-valued )
Real_Sequence)
/" (seq : ( ( Function-like V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K19(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K19(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) -valued Function-like non empty total V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K19(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) /" seq1 : ( ( Function-like V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K19(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K19(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) -valued Function-like non empty total V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K19(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) ) : ( (
Function-like ) (
Relation-like NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) )
-defined REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set )
-valued Function-like non
empty total V18(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) )
complex-valued ext-real-valued real-valued )
Element of
K19(
K20(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) (
complex-valued ext-real-valued real-valued )
set ) ) : ( ( ) ( )
set ) ) : ( (
Function-like ) (
Relation-like NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) )
-defined REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set )
-valued Function-like non
empty total V18(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) )
complex-valued ext-real-valued real-valued )
Element of
K19(
K20(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) (
complex-valued ext-real-valued real-valued )
set ) ) : ( ( ) ( )
set ) )
= (seq2 : ( ( Function-like V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K19(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K19(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) -valued Function-like non empty total V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K19(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) (#) seq1 : ( ( Function-like V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K19(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K19(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) -valued Function-like non empty total V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K19(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) ) : ( (
Function-like ) (
Relation-like NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) )
-defined REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set )
-valued Function-like non
empty total V18(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) )
complex-valued ext-real-valued real-valued )
Element of
K19(
K20(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) (
complex-valued ext-real-valued real-valued )
set ) ) : ( ( ) ( )
set ) )
/" seq : ( (
Function-like V18(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) ) (
Relation-like NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) )
-defined REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set )
-valued Function-like non
empty total V18(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) )
complex-valued ext-real-valued real-valued )
Real_Sequence) : ( (
Function-like ) (
Relation-like NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) )
-defined REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set )
-valued Function-like non
empty total V18(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) )
complex-valued ext-real-valued real-valued )
Element of
K19(
K20(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) (
complex-valued ext-real-valued real-valued )
set ) ) : ( ( ) ( )
set ) ) ;
theorem
for
seq1,
seq2,
seq being ( (
Function-like V18(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) ) (
Relation-like NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) )
-defined REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set )
-valued Function-like non
empty total V18(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) )
complex-valued ext-real-valued real-valued )
Real_Sequence) st
seq1 : ( (
Function-like V18(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) ) (
Relation-like NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) )
-defined REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set )
-valued Function-like non
empty total V18(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) )
complex-valued ext-real-valued real-valued )
Real_Sequence) is
non-zero holds
seq2 : ( (
Function-like V18(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) ) (
Relation-like NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) )
-defined REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set )
-valued Function-like non
empty total V18(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) )
complex-valued ext-real-valued real-valued )
Real_Sequence)
/" seq : ( (
Function-like V18(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) ) (
Relation-like NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) )
-defined REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set )
-valued Function-like non
empty total V18(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) )
complex-valued ext-real-valued real-valued )
Real_Sequence) : ( (
Function-like ) (
Relation-like NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) )
-defined REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set )
-valued Function-like non
empty total V18(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) )
complex-valued ext-real-valued real-valued )
Element of
K19(
K20(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) (
complex-valued ext-real-valued real-valued )
set ) ) : ( ( ) ( )
set ) )
= (seq2 : ( ( Function-like V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K19(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K19(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) -valued Function-like non empty total V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K19(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) (#) seq1 : ( ( Function-like V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K19(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K19(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) -valued Function-like non empty total V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K19(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) ) : ( (
Function-like ) (
Relation-like NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) )
-defined REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set )
-valued Function-like non
empty total V18(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) )
complex-valued ext-real-valued real-valued )
Element of
K19(
K20(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) (
complex-valued ext-real-valued real-valued )
set ) ) : ( ( ) ( )
set ) )
/" (seq : ( ( Function-like V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K19(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K19(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) -valued Function-like non empty total V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K19(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) (#) seq1 : ( ( Function-like V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K19(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K19(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) -valued Function-like non empty total V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K19(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) ) : ( (
Function-like ) (
Relation-like NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) )
-defined REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set )
-valued Function-like non
empty total V18(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) )
complex-valued ext-real-valued real-valued )
Element of
K19(
K20(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) (
complex-valued ext-real-valued real-valued )
set ) ) : ( ( ) ( )
set ) ) : ( (
Function-like ) (
Relation-like NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) )
-defined REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set )
-valued Function-like non
empty total V18(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) )
complex-valued ext-real-valued real-valued )
Element of
K19(
K20(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) (
complex-valued ext-real-valued real-valued )
set ) ) : ( ( ) ( )
set ) ) ;
theorem
for
r being ( (
real ) (
V21()
V31()
real )
number )
for
seq being ( (
Function-like V18(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) ) (
Relation-like NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) )
-defined REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set )
-valued Function-like non
empty total V18(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) )
complex-valued ext-real-valued real-valued )
Real_Sequence) holds
(r : ( ( real ) ( V21() V31() real ) number ) (#) seq : ( ( Function-like V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K19(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K19(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) -valued Function-like non empty total V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K19(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) ) : ( (
Function-like ) (
Relation-like NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) )
-defined REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set )
-valued Function-like non
empty total V18(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) )
complex-valued ext-real-valued real-valued )
Element of
K19(
K20(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) (
complex-valued ext-real-valued real-valued )
set ) ) : ( ( ) ( )
set ) )
" : ( (
Function-like ) (
Relation-like NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) )
-defined REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set )
-valued Function-like non
empty total V18(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) )
complex-valued ext-real-valued real-valued )
Element of
K19(
K20(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) (
complex-valued ext-real-valued real-valued )
set ) ) : ( ( ) ( )
set ) )
= (r : ( ( real ) ( V21() V31() real ) number ) ") : ( (
V31() ) (
V21()
V31()
real )
set )
(#) (seq : ( ( Function-like V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K19(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K19(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) -valued Function-like non empty total V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K19(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) ") : ( (
Function-like ) (
Relation-like NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) )
-defined REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set )
-valued Function-like non
empty total V18(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) )
complex-valued ext-real-valued real-valued )
Element of
K19(
K20(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) (
complex-valued ext-real-valued real-valued )
set ) ) : ( ( ) ( )
set ) ) : ( (
Function-like ) (
Relation-like NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) )
-defined REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set )
-valued Function-like non
empty total V18(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) )
complex-valued ext-real-valued real-valued )
Element of
K19(
K20(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) (
complex-valued ext-real-valued real-valued )
set ) ) : ( ( ) ( )
set ) ) ;
theorem
for
seq being ( (
Function-like V18(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) ) (
Relation-like NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) )
-defined REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set )
-valued Function-like non
empty total V18(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) )
complex-valued ext-real-valued real-valued )
Real_Sequence) holds
(- seq : ( ( Function-like V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K19(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K19(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) -valued Function-like non empty total V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K19(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) ) : ( (
Function-like ) (
Relation-like NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) )
-defined REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set )
-valued Function-like non
empty total V18(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) )
complex-valued ext-real-valued real-valued )
Element of
K19(
K20(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) (
complex-valued ext-real-valued real-valued )
set ) ) : ( ( ) ( )
set ) )
" : ( (
Function-like ) (
Relation-like NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) )
-defined REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set )
-valued Function-like non
empty total V18(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) )
complex-valued ext-real-valued real-valued )
Element of
K19(
K20(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) (
complex-valued ext-real-valued real-valued )
set ) ) : ( ( ) ( )
set ) )
= (- 1 : ( ( ) ( non empty V21() epsilon-transitive epsilon-connected ordinal natural V31() real V33() V34() V45() V46() V47() V48() V49() V50() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K19(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) (
V21()
V31()
real )
Element of
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) )
(#) (seq : ( ( Function-like V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K19(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K19(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) -valued Function-like non empty total V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K19(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) ") : ( (
Function-like ) (
Relation-like NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) )
-defined REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set )
-valued Function-like non
empty total V18(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) )
complex-valued ext-real-valued real-valued )
Element of
K19(
K20(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) (
complex-valued ext-real-valued real-valued )
set ) ) : ( ( ) ( )
set ) ) : ( (
Function-like ) (
Relation-like NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) )
-defined REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set )
-valued Function-like non
empty total V18(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) )
complex-valued ext-real-valued real-valued )
Element of
K19(
K20(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) (
complex-valued ext-real-valued real-valued )
set ) ) : ( ( ) ( )
set ) ) ;
theorem
for
seq1,
seq being ( (
Function-like V18(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) ) (
Relation-like NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) )
-defined REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set )
-valued Function-like non
empty total V18(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) )
complex-valued ext-real-valued real-valued )
Real_Sequence) holds
(
- (seq1 : ( ( Function-like V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K19(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K19(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) -valued Function-like non empty total V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K19(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) /" seq : ( ( Function-like V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K19(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K19(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) -valued Function-like non empty total V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K19(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) ) : ( (
Function-like ) (
Relation-like NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) )
-defined REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set )
-valued Function-like non
empty total V18(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) )
complex-valued ext-real-valued real-valued )
Element of
K19(
K20(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) (
complex-valued ext-real-valued real-valued )
set ) ) : ( ( ) ( )
set ) ) : ( (
Function-like ) (
Relation-like NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) )
-defined REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set )
-valued Function-like non
empty total V18(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) )
complex-valued ext-real-valued real-valued )
Element of
K19(
K20(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) (
complex-valued ext-real-valued real-valued )
set ) ) : ( ( ) ( )
set ) )
= (- seq1 : ( ( Function-like V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K19(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K19(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) -valued Function-like non empty total V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K19(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) ) : ( (
Function-like ) (
Relation-like NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) )
-defined REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set )
-valued Function-like non
empty total V18(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) )
complex-valued ext-real-valued real-valued )
Element of
K19(
K20(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) (
complex-valued ext-real-valued real-valued )
set ) ) : ( ( ) ( )
set ) )
/" seq : ( (
Function-like V18(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) ) (
Relation-like NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) )
-defined REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set )
-valued Function-like non
empty total V18(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) )
complex-valued ext-real-valued real-valued )
Real_Sequence) : ( (
Function-like ) (
Relation-like NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) )
-defined REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set )
-valued Function-like non
empty total V18(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) )
complex-valued ext-real-valued real-valued )
Element of
K19(
K20(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) (
complex-valued ext-real-valued real-valued )
set ) ) : ( ( ) ( )
set ) ) &
seq1 : ( (
Function-like V18(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) ) (
Relation-like NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) )
-defined REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set )
-valued Function-like non
empty total V18(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) )
complex-valued ext-real-valued real-valued )
Real_Sequence)
/" (- seq : ( ( Function-like V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K19(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K19(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) -valued Function-like non empty total V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K19(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) ) : ( (
Function-like ) (
Relation-like NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) )
-defined REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set )
-valued Function-like non
empty total V18(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) )
complex-valued ext-real-valued real-valued )
Element of
K19(
K20(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) (
complex-valued ext-real-valued real-valued )
set ) ) : ( ( ) ( )
set ) ) : ( (
Function-like ) (
Relation-like NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) )
-defined REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set )
-valued Function-like non
empty total V18(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) )
complex-valued ext-real-valued real-valued )
Element of
K19(
K20(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) (
complex-valued ext-real-valued real-valued )
set ) ) : ( ( ) ( )
set ) )
= - (seq1 : ( ( Function-like V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K19(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K19(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) -valued Function-like non empty total V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K19(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) /" seq : ( ( Function-like V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K19(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K19(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) -valued Function-like non empty total V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K19(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) ) : ( (
Function-like ) (
Relation-like NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) )
-defined REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set )
-valued Function-like non
empty total V18(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) )
complex-valued ext-real-valued real-valued )
Element of
K19(
K20(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) (
complex-valued ext-real-valued real-valued )
set ) ) : ( ( ) ( )
set ) ) : ( (
Function-like ) (
Relation-like NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) )
-defined REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set )
-valued Function-like non
empty total V18(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) )
complex-valued ext-real-valued real-valued )
Element of
K19(
K20(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) (
complex-valued ext-real-valued real-valued )
set ) ) : ( ( ) ( )
set ) ) ) ;
theorem
for
seq1,
seq,
seq19 being ( (
Function-like V18(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) ) (
Relation-like NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) )
-defined REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set )
-valued Function-like non
empty total V18(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) )
complex-valued ext-real-valued real-valued )
Real_Sequence) holds
(
(seq1 : ( ( Function-like V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K19(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K19(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) -valued Function-like non empty total V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K19(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) /" seq : ( ( Function-like V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K19(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K19(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) -valued Function-like non empty total V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K19(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) ) : ( (
Function-like ) (
Relation-like NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) )
-defined REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set )
-valued Function-like non
empty total V18(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) )
complex-valued ext-real-valued real-valued )
Element of
K19(
K20(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) (
complex-valued ext-real-valued real-valued )
set ) ) : ( ( ) ( )
set ) )
+ (seq19 : ( ( Function-like V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K19(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K19(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) -valued Function-like non empty total V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K19(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) /" seq : ( ( Function-like V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K19(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K19(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) -valued Function-like non empty total V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K19(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) ) : ( (
Function-like ) (
Relation-like NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) )
-defined REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set )
-valued Function-like non
empty total V18(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) )
complex-valued ext-real-valued real-valued )
Element of
K19(
K20(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) (
complex-valued ext-real-valued real-valued )
set ) ) : ( ( ) ( )
set ) ) : ( (
Function-like ) (
Relation-like NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) )
-defined REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set )
-valued Function-like non
empty total V18(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) )
complex-valued ext-real-valued real-valued )
Element of
K19(
K20(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) (
complex-valued ext-real-valued real-valued )
set ) ) : ( ( ) ( )
set ) )
= (seq1 : ( ( Function-like V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K19(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K19(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) -valued Function-like non empty total V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K19(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) + seq19 : ( ( Function-like V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K19(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K19(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) -valued Function-like non empty total V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K19(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) ) : ( (
Function-like ) (
Relation-like NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) )
-defined REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set )
-valued Function-like non
empty total V18(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) )
complex-valued ext-real-valued real-valued )
Element of
K19(
K20(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) (
complex-valued ext-real-valued real-valued )
set ) ) : ( ( ) ( )
set ) )
/" seq : ( (
Function-like V18(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) ) (
Relation-like NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) )
-defined REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set )
-valued Function-like non
empty total V18(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) )
complex-valued ext-real-valued real-valued )
Real_Sequence) : ( (
Function-like ) (
Relation-like NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) )
-defined REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set )
-valued Function-like non
empty total V18(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) )
complex-valued ext-real-valued real-valued )
Element of
K19(
K20(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) (
complex-valued ext-real-valued real-valued )
set ) ) : ( ( ) ( )
set ) ) &
(seq1 : ( ( Function-like V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K19(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K19(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) -valued Function-like non empty total V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K19(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) /" seq : ( ( Function-like V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K19(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K19(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) -valued Function-like non empty total V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K19(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) ) : ( (
Function-like ) (
Relation-like NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) )
-defined REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set )
-valued Function-like non
empty total V18(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) )
complex-valued ext-real-valued real-valued )
Element of
K19(
K20(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) (
complex-valued ext-real-valued real-valued )
set ) ) : ( ( ) ( )
set ) )
- (seq19 : ( ( Function-like V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K19(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K19(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) -valued Function-like non empty total V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K19(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) /" seq : ( ( Function-like V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K19(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K19(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) -valued Function-like non empty total V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K19(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) ) : ( (
Function-like ) (
Relation-like NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) )
-defined REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set )
-valued Function-like non
empty total V18(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) )
complex-valued ext-real-valued real-valued )
Element of
K19(
K20(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) (
complex-valued ext-real-valued real-valued )
set ) ) : ( ( ) ( )
set ) ) : ( (
Function-like ) (
Relation-like NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) )
-defined REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set )
-valued Function-like non
empty total V18(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) )
complex-valued ext-real-valued real-valued )
Element of
K19(
K20(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) (
complex-valued ext-real-valued real-valued )
set ) ) : ( ( ) ( )
set ) )
= (seq1 : ( ( Function-like V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K19(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K19(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) -valued Function-like non empty total V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K19(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) - seq19 : ( ( Function-like V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K19(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K19(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) -valued Function-like non empty total V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K19(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) ) : ( (
Function-like ) (
Relation-like NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) )
-defined REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set )
-valued Function-like non
empty total V18(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) )
complex-valued ext-real-valued real-valued )
Element of
K19(
K20(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) (
complex-valued ext-real-valued real-valued )
set ) ) : ( ( ) ( )
set ) )
/" seq : ( (
Function-like V18(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) ) (
Relation-like NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) )
-defined REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set )
-valued Function-like non
empty total V18(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) )
complex-valued ext-real-valued real-valued )
Real_Sequence) : ( (
Function-like ) (
Relation-like NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) )
-defined REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set )
-valued Function-like non
empty total V18(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) )
complex-valued ext-real-valued real-valued )
Element of
K19(
K20(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) (
complex-valued ext-real-valued real-valued )
set ) ) : ( ( ) ( )
set ) ) ) ;
theorem
for
seq,
seq9,
seq1,
seq19 being ( (
Function-like V18(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) ) (
Relation-like NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) )
-defined REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set )
-valued Function-like non
empty total V18(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) )
complex-valued ext-real-valued real-valued )
Real_Sequence) st
seq : ( (
Function-like V18(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) ) (
Relation-like NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) )
-defined REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set )
-valued Function-like non
empty total V18(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) )
complex-valued ext-real-valued real-valued )
Real_Sequence) is
non-zero &
seq9 : ( (
Function-like V18(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) ) (
Relation-like NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) )
-defined REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set )
-valued Function-like non
empty total V18(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) )
complex-valued ext-real-valued real-valued )
Real_Sequence) is
non-zero holds
(
(seq1 : ( ( Function-like V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K19(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K19(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) -valued Function-like non empty total V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K19(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) /" seq : ( ( Function-like V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K19(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K19(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) -valued Function-like non empty total V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K19(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) ) : ( (
Function-like ) (
Relation-like NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) )
-defined REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set )
-valued Function-like non
empty total V18(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) )
complex-valued ext-real-valued real-valued )
Element of
K19(
K20(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) (
complex-valued ext-real-valued real-valued )
set ) ) : ( ( ) ( )
set ) )
+ (seq19 : ( ( Function-like V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K19(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K19(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) -valued Function-like non empty total V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K19(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) /" seq9 : ( ( Function-like V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K19(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K19(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) -valued Function-like non empty total V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K19(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) ) : ( (
Function-like ) (
Relation-like NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) )
-defined REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set )
-valued Function-like non
empty total V18(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) )
complex-valued ext-real-valued real-valued )
Element of
K19(
K20(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) (
complex-valued ext-real-valued real-valued )
set ) ) : ( ( ) ( )
set ) ) : ( (
Function-like ) (
Relation-like NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) )
-defined REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set )
-valued Function-like non
empty total V18(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) )
complex-valued ext-real-valued real-valued )
Element of
K19(
K20(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) (
complex-valued ext-real-valued real-valued )
set ) ) : ( ( ) ( )
set ) )
= ((seq1 : ( ( Function-like V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K19(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K19(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) -valued Function-like non empty total V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K19(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) (#) seq9 : ( ( Function-like V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K19(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K19(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) -valued Function-like non empty total V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K19(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) ) : ( ( Function-like ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K19(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) -valued Function-like non empty total V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K19(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) complex-valued ext-real-valued real-valued ) Element of K19(K20(NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K19(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) ,REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) + (seq19 : ( ( Function-like V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K19(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K19(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) -valued Function-like non empty total V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K19(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) (#) seq : ( ( Function-like V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K19(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K19(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) -valued Function-like non empty total V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K19(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) ) : ( ( Function-like ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K19(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) -valued Function-like non empty total V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K19(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) complex-valued ext-real-valued real-valued ) Element of K19(K20(NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K19(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) ,REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) ) : ( (
Function-like ) (
Relation-like NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) )
-defined REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set )
-valued Function-like non
empty total V18(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) )
complex-valued ext-real-valued real-valued )
Element of
K19(
K20(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) (
complex-valued ext-real-valued real-valued )
set ) ) : ( ( ) ( )
set ) )
/" (seq : ( ( Function-like V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K19(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K19(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) -valued Function-like non empty total V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K19(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) (#) seq9 : ( ( Function-like V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K19(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K19(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) -valued Function-like non empty total V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K19(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) ) : ( (
Function-like ) (
Relation-like NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) )
-defined REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set )
-valued Function-like non
empty total V18(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) )
complex-valued ext-real-valued real-valued )
Element of
K19(
K20(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) (
complex-valued ext-real-valued real-valued )
set ) ) : ( ( ) ( )
set ) ) : ( (
Function-like ) (
Relation-like NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) )
-defined REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set )
-valued Function-like non
empty total V18(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) )
complex-valued ext-real-valued real-valued )
Element of
K19(
K20(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) (
complex-valued ext-real-valued real-valued )
set ) ) : ( ( ) ( )
set ) ) &
(seq1 : ( ( Function-like V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K19(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K19(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) -valued Function-like non empty total V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K19(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) /" seq : ( ( Function-like V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K19(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K19(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) -valued Function-like non empty total V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K19(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) ) : ( (
Function-like ) (
Relation-like NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) )
-defined REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set )
-valued Function-like non
empty total V18(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) )
complex-valued ext-real-valued real-valued )
Element of
K19(
K20(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) (
complex-valued ext-real-valued real-valued )
set ) ) : ( ( ) ( )
set ) )
- (seq19 : ( ( Function-like V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K19(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K19(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) -valued Function-like non empty total V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K19(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) /" seq9 : ( ( Function-like V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K19(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K19(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) -valued Function-like non empty total V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K19(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) ) : ( (
Function-like ) (
Relation-like NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) )
-defined REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set )
-valued Function-like non
empty total V18(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) )
complex-valued ext-real-valued real-valued )
Element of
K19(
K20(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) (
complex-valued ext-real-valued real-valued )
set ) ) : ( ( ) ( )
set ) ) : ( (
Function-like ) (
Relation-like NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) )
-defined REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set )
-valued Function-like non
empty total V18(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) )
complex-valued ext-real-valued real-valued )
Element of
K19(
K20(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) (
complex-valued ext-real-valued real-valued )
set ) ) : ( ( ) ( )
set ) )
= ((seq1 : ( ( Function-like V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K19(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K19(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) -valued Function-like non empty total V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K19(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) (#) seq9 : ( ( Function-like V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K19(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K19(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) -valued Function-like non empty total V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K19(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) ) : ( ( Function-like ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K19(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) -valued Function-like non empty total V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K19(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) complex-valued ext-real-valued real-valued ) Element of K19(K20(NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K19(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) ,REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) - (seq19 : ( ( Function-like V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K19(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K19(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) -valued Function-like non empty total V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K19(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) (#) seq : ( ( Function-like V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K19(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K19(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) -valued Function-like non empty total V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K19(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) ) : ( ( Function-like ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K19(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) -valued Function-like non empty total V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K19(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) complex-valued ext-real-valued real-valued ) Element of K19(K20(NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K19(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) ,REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) ) : ( (
Function-like ) (
Relation-like NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) )
-defined REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set )
-valued Function-like non
empty total V18(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) )
complex-valued ext-real-valued real-valued )
Element of
K19(
K20(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) (
complex-valued ext-real-valued real-valued )
set ) ) : ( ( ) ( )
set ) )
/" (seq : ( ( Function-like V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K19(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K19(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) -valued Function-like non empty total V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K19(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) (#) seq9 : ( ( Function-like V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K19(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K19(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) -valued Function-like non empty total V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K19(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) ) : ( (
Function-like ) (
Relation-like NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) )
-defined REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set )
-valued Function-like non
empty total V18(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) )
complex-valued ext-real-valued real-valued )
Element of
K19(
K20(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) (
complex-valued ext-real-valued real-valued )
set ) ) : ( ( ) ( )
set ) ) : ( (
Function-like ) (
Relation-like NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) )
-defined REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set )
-valued Function-like non
empty total V18(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) )
complex-valued ext-real-valued real-valued )
Element of
K19(
K20(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) (
complex-valued ext-real-valued real-valued )
set ) ) : ( ( ) ( )
set ) ) ) ;
theorem
for
seq19,
seq,
seq9,
seq1 being ( (
Function-like V18(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) ) (
Relation-like NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) )
-defined REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set )
-valued Function-like non
empty total V18(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) )
complex-valued ext-real-valued real-valued )
Real_Sequence) holds
(seq19 : ( ( Function-like V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K19(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K19(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) -valued Function-like non empty total V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K19(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) /" seq : ( ( Function-like V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K19(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K19(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) -valued Function-like non empty total V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K19(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) ) : ( (
Function-like ) (
Relation-like NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) )
-defined REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set )
-valued Function-like non
empty total V18(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) )
complex-valued ext-real-valued real-valued )
Element of
K19(
K20(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) (
complex-valued ext-real-valued real-valued )
set ) ) : ( ( ) ( )
set ) )
/" (seq9 : ( ( Function-like V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K19(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K19(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) -valued Function-like non empty total V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K19(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) /" seq1 : ( ( Function-like V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K19(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K19(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) -valued Function-like non empty total V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K19(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) ) : ( (
Function-like ) (
Relation-like NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) )
-defined REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set )
-valued Function-like non
empty total V18(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) )
complex-valued ext-real-valued real-valued )
Element of
K19(
K20(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) (
complex-valued ext-real-valued real-valued )
set ) ) : ( ( ) ( )
set ) ) : ( (
Function-like ) (
Relation-like NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) )
-defined REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set )
-valued Function-like non
empty total V18(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) )
complex-valued ext-real-valued real-valued )
Element of
K19(
K20(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) (
complex-valued ext-real-valued real-valued )
set ) ) : ( ( ) ( )
set ) )
= (seq19 : ( ( Function-like V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K19(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K19(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) -valued Function-like non empty total V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K19(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) (#) seq1 : ( ( Function-like V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K19(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K19(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) -valued Function-like non empty total V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K19(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) ) : ( (
Function-like ) (
Relation-like NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) )
-defined REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set )
-valued Function-like non
empty total V18(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) )
complex-valued ext-real-valued real-valued )
Element of
K19(
K20(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) (
complex-valued ext-real-valued real-valued )
set ) ) : ( ( ) ( )
set ) )
/" (seq : ( ( Function-like V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K19(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K19(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) -valued Function-like non empty total V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K19(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) (#) seq9 : ( ( Function-like V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K19(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K19(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) -valued Function-like non empty total V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K19(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) ) : ( (
Function-like ) (
Relation-like NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) )
-defined REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set )
-valued Function-like non
empty total V18(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) )
complex-valued ext-real-valued real-valued )
Element of
K19(
K20(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) (
complex-valued ext-real-valued real-valued )
set ) ) : ( ( ) ( )
set ) ) : ( (
Function-like ) (
Relation-like NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) )
-defined REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set )
-valued Function-like non
empty total V18(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) )
complex-valued ext-real-valued real-valued )
Element of
K19(
K20(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) (
complex-valued ext-real-valued real-valued )
set ) ) : ( ( ) ( )
set ) ) ;
theorem
for
seq,
seq9 being ( (
Function-like V18(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) ) (
Relation-like NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) )
-defined REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set )
-valued Function-like non
empty total V18(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) )
complex-valued ext-real-valued real-valued )
Real_Sequence) holds
abs (seq : ( ( Function-like V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K19(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K19(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) -valued Function-like non empty total V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K19(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) (#) seq9 : ( ( Function-like V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K19(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K19(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) -valued Function-like non empty total V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K19(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) ) : ( (
Function-like ) (
Relation-like NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) )
-defined REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set )
-valued Function-like non
empty total V18(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) )
complex-valued ext-real-valued real-valued )
Element of
K19(
K20(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) (
complex-valued ext-real-valued real-valued )
set ) ) : ( ( ) ( )
set ) ) : ( (
Function-like ) (
Relation-like NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) )
-defined REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set )
-valued Function-like non
empty total V18(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) )
complex-valued ext-real-valued real-valued )
Element of
K19(
K20(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) (
complex-valued ext-real-valued real-valued )
set ) ) : ( ( ) ( )
set ) )
= (abs seq : ( ( Function-like V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K19(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K19(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) -valued Function-like non empty total V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K19(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) ) : ( (
Function-like ) (
Relation-like NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) )
-defined REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set )
-valued Function-like non
empty total V18(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) )
complex-valued ext-real-valued real-valued )
Element of
K19(
K20(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) (
complex-valued ext-real-valued real-valued )
set ) ) : ( ( ) ( )
set ) )
(#) (abs seq9 : ( ( Function-like V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K19(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K19(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) -valued Function-like non empty total V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K19(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) ) : ( (
Function-like ) (
Relation-like NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) )
-defined REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set )
-valued Function-like non
empty total V18(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) )
complex-valued ext-real-valued real-valued )
Element of
K19(
K20(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) (
complex-valued ext-real-valued real-valued )
set ) ) : ( ( ) ( )
set ) ) : ( (
Function-like ) (
Relation-like NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) )
-defined REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set )
-valued Function-like non
empty total V18(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) )
complex-valued ext-real-valued real-valued )
Element of
K19(
K20(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) (
complex-valued ext-real-valued real-valued )
set ) ) : ( ( ) ( )
set ) ) ;
theorem
for
seq being ( (
Function-like V18(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) ) (
Relation-like NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) )
-defined REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set )
-valued Function-like non
empty total V18(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) )
complex-valued ext-real-valued real-valued )
Real_Sequence) holds
(abs seq : ( ( Function-like V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K19(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K19(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) -valued Function-like non empty total V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K19(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) ) : ( (
Function-like ) (
Relation-like NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) )
-defined REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set )
-valued Function-like non
empty total V18(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) )
complex-valued ext-real-valued real-valued )
Element of
K19(
K20(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) (
complex-valued ext-real-valued real-valued )
set ) ) : ( ( ) ( )
set ) )
" : ( (
Function-like ) (
Relation-like NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) )
-defined REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set )
-valued Function-like non
empty total V18(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) )
complex-valued ext-real-valued real-valued )
Element of
K19(
K20(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) (
complex-valued ext-real-valued real-valued )
set ) ) : ( ( ) ( )
set ) )
= abs (seq : ( ( Function-like V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K19(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K19(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) -valued Function-like non empty total V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K19(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) ") : ( (
Function-like ) (
Relation-like NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) )
-defined REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set )
-valued Function-like non
empty total V18(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) )
complex-valued ext-real-valued real-valued )
Element of
K19(
K20(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) (
complex-valued ext-real-valued real-valued )
set ) ) : ( ( ) ( )
set ) ) : ( (
Function-like ) (
Relation-like NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) )
-defined REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set )
-valued Function-like non
empty total V18(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) )
complex-valued ext-real-valued real-valued )
Element of
K19(
K20(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) (
complex-valued ext-real-valued real-valued )
set ) ) : ( ( ) ( )
set ) ) ;
theorem
for
seq9,
seq being ( (
Function-like V18(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) ) (
Relation-like NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) )
-defined REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set )
-valued Function-like non
empty total V18(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) )
complex-valued ext-real-valued real-valued )
Real_Sequence) holds
abs (seq9 : ( ( Function-like V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K19(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K19(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) -valued Function-like non empty total V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K19(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) /" seq : ( ( Function-like V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K19(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K19(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) -valued Function-like non empty total V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K19(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) ) : ( (
Function-like ) (
Relation-like NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) )
-defined REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set )
-valued Function-like non
empty total V18(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) )
complex-valued ext-real-valued real-valued )
Element of
K19(
K20(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) (
complex-valued ext-real-valued real-valued )
set ) ) : ( ( ) ( )
set ) ) : ( (
Function-like ) (
Relation-like NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) )
-defined REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set )
-valued Function-like non
empty total V18(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) )
complex-valued ext-real-valued real-valued )
Element of
K19(
K20(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) (
complex-valued ext-real-valued real-valued )
set ) ) : ( ( ) ( )
set ) )
= (abs seq9 : ( ( Function-like V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K19(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K19(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) -valued Function-like non empty total V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K19(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) ) : ( (
Function-like ) (
Relation-like NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) )
-defined REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set )
-valued Function-like non
empty total V18(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) )
complex-valued ext-real-valued real-valued )
Element of
K19(
K20(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) (
complex-valued ext-real-valued real-valued )
set ) ) : ( ( ) ( )
set ) )
/" (abs seq : ( ( Function-like V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K19(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K19(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) -valued Function-like non empty total V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K19(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) ) : ( (
Function-like ) (
Relation-like NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) )
-defined REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set )
-valued Function-like non
empty total V18(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) )
complex-valued ext-real-valued real-valued )
Element of
K19(
K20(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) (
complex-valued ext-real-valued real-valued )
set ) ) : ( ( ) ( )
set ) ) : ( (
Function-like ) (
Relation-like NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) )
-defined REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set )
-valued Function-like non
empty total V18(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) )
complex-valued ext-real-valued real-valued )
Element of
K19(
K20(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) (
complex-valued ext-real-valued real-valued )
set ) ) : ( ( ) ( )
set ) ) ;
theorem
for
r being ( (
real ) (
V21()
V31()
real )
number )
for
seq being ( (
Function-like V18(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) ) (
Relation-like NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) )
-defined REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set )
-valued Function-like non
empty total V18(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) )
complex-valued ext-real-valued real-valued )
Real_Sequence) holds
abs (r : ( ( real ) ( V21() V31() real ) number ) (#) seq : ( ( Function-like V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K19(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K19(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) -valued Function-like non empty total V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K19(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) ) : ( (
Function-like ) (
Relation-like NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) )
-defined REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set )
-valued Function-like non
empty total V18(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) )
complex-valued ext-real-valued real-valued )
Element of
K19(
K20(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) (
complex-valued ext-real-valued real-valued )
set ) ) : ( ( ) ( )
set ) ) : ( (
Function-like ) (
Relation-like NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) )
-defined REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set )
-valued Function-like non
empty total V18(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) )
complex-valued ext-real-valued real-valued )
Element of
K19(
K20(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) (
complex-valued ext-real-valued real-valued )
set ) ) : ( ( ) ( )
set ) )
= (abs r : ( ( real ) ( V21() V31() real ) number ) ) : ( ( ) (
V21()
V31()
real )
Element of
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) )
(#) (abs seq : ( ( Function-like V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K19(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K19(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) -valued Function-like non empty total V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K19(REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V45() V46() V47() V51() V52() ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) ) : ( (
Function-like ) (
Relation-like NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) )
-defined REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set )
-valued Function-like non
empty total V18(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) )
complex-valued ext-real-valued real-valued )
Element of
K19(
K20(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) (
complex-valued ext-real-valued real-valued )
set ) ) : ( ( ) ( )
set ) ) : ( (
Function-like ) (
Relation-like NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) )
-defined REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set )
-valued Function-like non
empty total V18(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) )
complex-valued ext-real-valued real-valued )
Element of
K19(
K20(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V45()
V46()
V47()
V48()
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V45()
V46()
V47()
V51()
V52() )
set ) ) : ( ( ) (
complex-valued ext-real-valued real-valued )
set ) ) : ( ( ) ( )
set ) ) ;