begin
theorem
for
seq1,
seq2,
seq3 being ( (
Function-like V18(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) ,
COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set ) ) ) (
Relation-like NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) )
-defined COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set )
-valued Function-like non
zero V14(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) )
V18(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) ,
COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set ) )
complex-valued )
Complex_Sequence) holds
(seq1 : ( ( Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) -valued Function-like non zero V14( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) complex-valued ) Complex_Sequence) + seq2 : ( ( Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) -valued Function-like non zero V14( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) complex-valued ) Complex_Sequence) ) : ( (
Function-like ) (
Relation-like NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) )
-defined COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set )
-valued Function-like non
zero V14(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) )
V18(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) ,
COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set ) )
complex-valued )
Element of
K19(
K20(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) ,
COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set ) ) : ( ( ) (
complex-valued )
set ) ) : ( ( ) ( )
set ) )
+ seq3 : ( (
Function-like V18(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) ,
COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set ) ) ) (
Relation-like NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) )
-defined COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set )
-valued Function-like non
zero V14(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) )
V18(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) ,
COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set ) )
complex-valued )
Complex_Sequence) : ( (
Function-like ) (
Relation-like NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) )
-defined COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set )
-valued Function-like non
zero V14(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) )
V18(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) ,
COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set ) )
complex-valued )
Element of
K19(
K20(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) ,
COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set ) ) : ( ( ) (
complex-valued )
set ) ) : ( ( ) ( )
set ) )
= seq1 : ( (
Function-like V18(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) ,
COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set ) ) ) (
Relation-like NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) )
-defined COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set )
-valued Function-like non
zero V14(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) )
V18(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) ,
COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set ) )
complex-valued )
Complex_Sequence)
+ (seq2 : ( ( Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) -valued Function-like non zero V14( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) complex-valued ) Complex_Sequence) + seq3 : ( ( Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) -valued Function-like non zero V14( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) complex-valued ) Complex_Sequence) ) : ( (
Function-like ) (
Relation-like NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) )
-defined COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set )
-valued Function-like non
zero V14(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) )
V18(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) ,
COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set ) )
complex-valued )
Element of
K19(
K20(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) ,
COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set ) ) : ( ( ) (
complex-valued )
set ) ) : ( ( ) ( )
set ) ) : ( (
Function-like ) (
Relation-like NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) )
-defined COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set )
-valued Function-like non
zero V14(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) )
V18(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) ,
COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set ) )
complex-valued )
Element of
K19(
K20(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) ,
COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set ) ) : ( ( ) (
complex-valued )
set ) ) : ( ( ) ( )
set ) ) ;
theorem
for
seq1,
seq2,
seq3 being ( (
Function-like V18(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) ,
COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set ) ) ) (
Relation-like NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) )
-defined COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set )
-valued Function-like non
zero V14(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) )
V18(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) ,
COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set ) )
complex-valued )
Complex_Sequence) holds
(seq1 : ( ( Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) -valued Function-like non zero V14( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) complex-valued ) Complex_Sequence) (#) seq2 : ( ( Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) -valued Function-like non zero V14( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) complex-valued ) Complex_Sequence) ) : ( (
Function-like ) (
Relation-like NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) )
-defined COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set )
-valued Function-like non
zero V14(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) )
V18(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) ,
COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set ) )
complex-valued )
Element of
K19(
K20(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) ,
COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set ) ) : ( ( ) (
complex-valued )
set ) ) : ( ( ) ( )
set ) )
(#) seq3 : ( (
Function-like V18(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) ,
COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set ) ) ) (
Relation-like NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) )
-defined COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set )
-valued Function-like non
zero V14(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) )
V18(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) ,
COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set ) )
complex-valued )
Complex_Sequence) : ( (
Function-like ) (
Relation-like NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) )
-defined COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set )
-valued Function-like non
zero V14(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) )
V18(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) ,
COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set ) )
complex-valued )
Element of
K19(
K20(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) ,
COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set ) ) : ( ( ) (
complex-valued )
set ) ) : ( ( ) ( )
set ) )
= seq1 : ( (
Function-like V18(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) ,
COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set ) ) ) (
Relation-like NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) )
-defined COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set )
-valued Function-like non
zero V14(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) )
V18(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) ,
COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set ) )
complex-valued )
Complex_Sequence)
(#) (seq2 : ( ( Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) -valued Function-like non zero V14( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) complex-valued ) Complex_Sequence) (#) seq3 : ( ( Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) -valued Function-like non zero V14( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) complex-valued ) Complex_Sequence) ) : ( (
Function-like ) (
Relation-like NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) )
-defined COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set )
-valued Function-like non
zero V14(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) )
V18(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) ,
COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set ) )
complex-valued )
Element of
K19(
K20(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) ,
COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set ) ) : ( ( ) (
complex-valued )
set ) ) : ( ( ) ( )
set ) ) : ( (
Function-like ) (
Relation-like NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) )
-defined COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set )
-valued Function-like non
zero V14(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) )
V18(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) ,
COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set ) )
complex-valued )
Element of
K19(
K20(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) ,
COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set ) ) : ( ( ) (
complex-valued )
set ) ) : ( ( ) ( )
set ) ) ;
theorem
for
seq1,
seq2,
seq3 being ( (
Function-like V18(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) ,
COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set ) ) ) (
Relation-like NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) )
-defined COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set )
-valued Function-like non
zero V14(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) )
V18(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) ,
COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set ) )
complex-valued )
Complex_Sequence) holds
(seq1 : ( ( Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) -valued Function-like non zero V14( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) complex-valued ) Complex_Sequence) + seq2 : ( ( Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) -valued Function-like non zero V14( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) complex-valued ) Complex_Sequence) ) : ( (
Function-like ) (
Relation-like NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) )
-defined COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set )
-valued Function-like non
zero V14(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) )
V18(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) ,
COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set ) )
complex-valued )
Element of
K19(
K20(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) ,
COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set ) ) : ( ( ) (
complex-valued )
set ) ) : ( ( ) ( )
set ) )
(#) seq3 : ( (
Function-like V18(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) ,
COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set ) ) ) (
Relation-like NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) )
-defined COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set )
-valued Function-like non
zero V14(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) )
V18(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) ,
COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set ) )
complex-valued )
Complex_Sequence) : ( (
Function-like ) (
Relation-like NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) )
-defined COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set )
-valued Function-like non
zero V14(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) )
V18(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) ,
COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set ) )
complex-valued )
Element of
K19(
K20(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) ,
COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set ) ) : ( ( ) (
complex-valued )
set ) ) : ( ( ) ( )
set ) )
= (seq1 : ( ( Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) -valued Function-like non zero V14( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) complex-valued ) Complex_Sequence) (#) seq3 : ( ( Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) -valued Function-like non zero V14( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) complex-valued ) Complex_Sequence) ) : ( (
Function-like ) (
Relation-like NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) )
-defined COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set )
-valued Function-like non
zero V14(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) )
V18(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) ,
COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set ) )
complex-valued )
Element of
K19(
K20(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) ,
COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set ) ) : ( ( ) (
complex-valued )
set ) ) : ( ( ) ( )
set ) )
+ (seq2 : ( ( Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) -valued Function-like non zero V14( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) complex-valued ) Complex_Sequence) (#) seq3 : ( ( Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) -valued Function-like non zero V14( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) complex-valued ) Complex_Sequence) ) : ( (
Function-like ) (
Relation-like NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) )
-defined COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set )
-valued Function-like non
zero V14(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) )
V18(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) ,
COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set ) )
complex-valued )
Element of
K19(
K20(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) ,
COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set ) ) : ( ( ) (
complex-valued )
set ) ) : ( ( ) ( )
set ) ) : ( (
Function-like ) (
Relation-like NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) )
-defined COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set )
-valued Function-like non
zero V14(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) )
V18(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) ,
COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set ) )
complex-valued )
Element of
K19(
K20(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) ,
COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set ) ) : ( ( ) (
complex-valued )
set ) ) : ( ( ) ( )
set ) ) ;
theorem
for
seq3,
seq1,
seq2 being ( (
Function-like V18(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) ,
COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set ) ) ) (
Relation-like NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) )
-defined COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set )
-valued Function-like non
zero V14(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) )
V18(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) ,
COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set ) )
complex-valued )
Complex_Sequence) holds
seq3 : ( (
Function-like V18(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) ,
COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set ) ) ) (
Relation-like NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) )
-defined COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set )
-valued Function-like non
zero V14(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) )
V18(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) ,
COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set ) )
complex-valued )
Complex_Sequence)
(#) (seq1 : ( ( Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) -valued Function-like non zero V14( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) complex-valued ) Complex_Sequence) + seq2 : ( ( Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) -valued Function-like non zero V14( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) complex-valued ) Complex_Sequence) ) : ( (
Function-like ) (
Relation-like NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) )
-defined COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set )
-valued Function-like non
zero V14(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) )
V18(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) ,
COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set ) )
complex-valued )
Element of
K19(
K20(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) ,
COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set ) ) : ( ( ) (
complex-valued )
set ) ) : ( ( ) ( )
set ) ) : ( (
Function-like ) (
Relation-like NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) )
-defined COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set )
-valued Function-like non
zero V14(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) )
V18(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) ,
COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set ) )
complex-valued )
Element of
K19(
K20(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) ,
COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set ) ) : ( ( ) (
complex-valued )
set ) ) : ( ( ) ( )
set ) )
= (seq3 : ( ( Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) -valued Function-like non zero V14( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) complex-valued ) Complex_Sequence) (#) seq1 : ( ( Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) -valued Function-like non zero V14( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) complex-valued ) Complex_Sequence) ) : ( (
Function-like ) (
Relation-like NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) )
-defined COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set )
-valued Function-like non
zero V14(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) )
V18(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) ,
COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set ) )
complex-valued )
Element of
K19(
K20(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) ,
COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set ) ) : ( ( ) (
complex-valued )
set ) ) : ( ( ) ( )
set ) )
+ (seq3 : ( ( Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) -valued Function-like non zero V14( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) complex-valued ) Complex_Sequence) (#) seq2 : ( ( Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) -valued Function-like non zero V14( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) complex-valued ) Complex_Sequence) ) : ( (
Function-like ) (
Relation-like NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) )
-defined COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set )
-valued Function-like non
zero V14(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) )
V18(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) ,
COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set ) )
complex-valued )
Element of
K19(
K20(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) ,
COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set ) ) : ( ( ) (
complex-valued )
set ) ) : ( ( ) ( )
set ) ) : ( (
Function-like ) (
Relation-like NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) )
-defined COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set )
-valued Function-like non
zero V14(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) )
V18(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) ,
COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set ) )
complex-valued )
Element of
K19(
K20(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) ,
COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set ) ) : ( ( ) (
complex-valued )
set ) ) : ( ( ) ( )
set ) ) ;
theorem
for
seq being ( (
Function-like V18(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) ,
COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set ) ) ) (
Relation-like NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) )
-defined COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set )
-valued Function-like non
zero V14(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) )
V18(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) ,
COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set ) )
complex-valued )
Complex_Sequence) holds
- seq : ( (
Function-like V18(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) ,
COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set ) ) ) (
Relation-like NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) )
-defined COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set )
-valued Function-like non
zero V14(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) )
V18(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) ,
COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set ) )
complex-valued )
Complex_Sequence) : ( (
Function-like ) (
Relation-like NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) )
-defined COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set )
-valued Function-like non
zero V14(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) )
V18(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) ,
COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set ) )
complex-valued )
Element of
K19(
K20(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) ,
COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set ) ) : ( ( ) (
complex-valued )
set ) ) : ( ( ) ( )
set ) )
= (- 1r : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) ) : ( ( ) (
complex )
Element of
COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set ) )
(#) seq : ( (
Function-like V18(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) ,
COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set ) ) ) (
Relation-like NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) )
-defined COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set )
-valued Function-like non
zero V14(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) )
V18(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) ,
COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set ) )
complex-valued )
Complex_Sequence) : ( (
Function-like ) (
Relation-like NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) )
-defined COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set )
-valued Function-like non
zero V14(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) )
V18(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) ,
COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set ) )
complex-valued )
Element of
K19(
K20(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) ,
COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set ) ) : ( ( ) (
complex-valued )
set ) ) : ( ( ) ( )
set ) ) ;
theorem
for
r being ( ( ) (
complex )
Element of
COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set ) )
for
seq1,
seq2 being ( (
Function-like V18(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) ,
COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set ) ) ) (
Relation-like NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) )
-defined COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set )
-valued Function-like non
zero V14(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) )
V18(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) ,
COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set ) )
complex-valued )
Complex_Sequence) holds
r : ( ( ) (
complex )
Element of
COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set ) )
(#) (seq1 : ( ( Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) -valued Function-like non zero V14( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) complex-valued ) Complex_Sequence) (#) seq2 : ( ( Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) -valued Function-like non zero V14( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) complex-valued ) Complex_Sequence) ) : ( (
Function-like ) (
Relation-like NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) )
-defined COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set )
-valued Function-like non
zero V14(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) )
V18(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) ,
COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set ) )
complex-valued )
Element of
K19(
K20(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) ,
COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set ) ) : ( ( ) (
complex-valued )
set ) ) : ( ( ) ( )
set ) ) : ( (
Function-like ) (
Relation-like NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) )
-defined COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set )
-valued Function-like non
zero V14(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) )
V18(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) ,
COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set ) )
complex-valued )
Element of
K19(
K20(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) ,
COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set ) ) : ( ( ) (
complex-valued )
set ) ) : ( ( ) ( )
set ) )
= (r : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) (#) seq1 : ( ( Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) -valued Function-like non zero V14( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) complex-valued ) Complex_Sequence) ) : ( (
Function-like ) (
Relation-like NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) )
-defined COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set )
-valued Function-like non
zero V14(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) )
V18(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) ,
COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set ) )
complex-valued )
Element of
K19(
K20(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) ,
COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set ) ) : ( ( ) (
complex-valued )
set ) ) : ( ( ) ( )
set ) )
(#) seq2 : ( (
Function-like V18(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) ,
COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set ) ) ) (
Relation-like NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) )
-defined COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set )
-valued Function-like non
zero V14(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) )
V18(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) ,
COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set ) )
complex-valued )
Complex_Sequence) : ( (
Function-like ) (
Relation-like NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) )
-defined COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set )
-valued Function-like non
zero V14(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) )
V18(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) ,
COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set ) )
complex-valued )
Element of
K19(
K20(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) ,
COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set ) ) : ( ( ) (
complex-valued )
set ) ) : ( ( ) ( )
set ) ) ;
theorem
for
r being ( ( ) (
complex )
Element of
COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set ) )
for
seq1,
seq2 being ( (
Function-like V18(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) ,
COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set ) ) ) (
Relation-like NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) )
-defined COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set )
-valued Function-like non
zero V14(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) )
V18(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) ,
COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set ) )
complex-valued )
Complex_Sequence) holds
r : ( ( ) (
complex )
Element of
COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set ) )
(#) (seq1 : ( ( Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) -valued Function-like non zero V14( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) complex-valued ) Complex_Sequence) (#) seq2 : ( ( Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) -valued Function-like non zero V14( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) complex-valued ) Complex_Sequence) ) : ( (
Function-like ) (
Relation-like NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) )
-defined COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set )
-valued Function-like non
zero V14(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) )
V18(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) ,
COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set ) )
complex-valued )
Element of
K19(
K20(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) ,
COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set ) ) : ( ( ) (
complex-valued )
set ) ) : ( ( ) ( )
set ) ) : ( (
Function-like ) (
Relation-like NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) )
-defined COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set )
-valued Function-like non
zero V14(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) )
V18(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) ,
COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set ) )
complex-valued )
Element of
K19(
K20(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) ,
COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set ) ) : ( ( ) (
complex-valued )
set ) ) : ( ( ) ( )
set ) )
= seq1 : ( (
Function-like V18(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) ,
COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set ) ) ) (
Relation-like NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) )
-defined COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set )
-valued Function-like non
zero V14(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) )
V18(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) ,
COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set ) )
complex-valued )
Complex_Sequence)
(#) (r : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) (#) seq2 : ( ( Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) -valued Function-like non zero V14( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) complex-valued ) Complex_Sequence) ) : ( (
Function-like ) (
Relation-like NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) )
-defined COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set )
-valued Function-like non
zero V14(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) )
V18(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) ,
COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set ) )
complex-valued )
Element of
K19(
K20(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) ,
COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set ) ) : ( ( ) (
complex-valued )
set ) ) : ( ( ) ( )
set ) ) : ( (
Function-like ) (
Relation-like NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) )
-defined COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set )
-valued Function-like non
zero V14(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) )
V18(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) ,
COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set ) )
complex-valued )
Element of
K19(
K20(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) ,
COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set ) ) : ( ( ) (
complex-valued )
set ) ) : ( ( ) ( )
set ) ) ;
theorem
for
seq1,
seq2,
seq3 being ( (
Function-like V18(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) ,
COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set ) ) ) (
Relation-like NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) )
-defined COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set )
-valued Function-like non
zero V14(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) )
V18(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) ,
COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set ) )
complex-valued )
Complex_Sequence) holds
(seq1 : ( ( Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) -valued Function-like non zero V14( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) complex-valued ) Complex_Sequence) - seq2 : ( ( Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) -valued Function-like non zero V14( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) complex-valued ) Complex_Sequence) ) : ( (
Function-like ) (
Relation-like NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) )
-defined COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set )
-valued Function-like non
zero V14(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) )
V18(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) ,
COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set ) )
complex-valued )
Element of
K19(
K20(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) ,
COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set ) ) : ( ( ) (
complex-valued )
set ) ) : ( ( ) ( )
set ) )
(#) seq3 : ( (
Function-like V18(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) ,
COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set ) ) ) (
Relation-like NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) )
-defined COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set )
-valued Function-like non
zero V14(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) )
V18(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) ,
COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set ) )
complex-valued )
Complex_Sequence) : ( (
Function-like ) (
Relation-like NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) )
-defined COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set )
-valued Function-like non
zero V14(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) )
V18(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) ,
COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set ) )
complex-valued )
Element of
K19(
K20(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) ,
COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set ) ) : ( ( ) (
complex-valued )
set ) ) : ( ( ) ( )
set ) )
= (seq1 : ( ( Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) -valued Function-like non zero V14( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) complex-valued ) Complex_Sequence) (#) seq3 : ( ( Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) -valued Function-like non zero V14( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) complex-valued ) Complex_Sequence) ) : ( (
Function-like ) (
Relation-like NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) )
-defined COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set )
-valued Function-like non
zero V14(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) )
V18(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) ,
COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set ) )
complex-valued )
Element of
K19(
K20(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) ,
COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set ) ) : ( ( ) (
complex-valued )
set ) ) : ( ( ) ( )
set ) )
- (seq2 : ( ( Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) -valued Function-like non zero V14( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) complex-valued ) Complex_Sequence) (#) seq3 : ( ( Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) -valued Function-like non zero V14( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) complex-valued ) Complex_Sequence) ) : ( (
Function-like ) (
Relation-like NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) )
-defined COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set )
-valued Function-like non
zero V14(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) )
V18(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) ,
COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set ) )
complex-valued )
Element of
K19(
K20(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) ,
COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set ) ) : ( ( ) (
complex-valued )
set ) ) : ( ( ) ( )
set ) ) : ( (
Function-like ) (
Relation-like NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) )
-defined COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set )
-valued Function-like non
zero V14(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) )
V18(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) ,
COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set ) )
complex-valued )
Element of
K19(
K20(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) ,
COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set ) ) : ( ( ) (
complex-valued )
set ) ) : ( ( ) ( )
set ) ) ;
theorem
for
seq3,
seq1,
seq2 being ( (
Function-like V18(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) ,
COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set ) ) ) (
Relation-like NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) )
-defined COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set )
-valued Function-like non
zero V14(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) )
V18(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) ,
COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set ) )
complex-valued )
Complex_Sequence) holds
(seq3 : ( ( Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) -valued Function-like non zero V14( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) complex-valued ) Complex_Sequence) (#) seq1 : ( ( Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) -valued Function-like non zero V14( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) complex-valued ) Complex_Sequence) ) : ( (
Function-like ) (
Relation-like NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) )
-defined COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set )
-valued Function-like non
zero V14(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) )
V18(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) ,
COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set ) )
complex-valued )
Element of
K19(
K20(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) ,
COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set ) ) : ( ( ) (
complex-valued )
set ) ) : ( ( ) ( )
set ) )
- (seq3 : ( ( Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) -valued Function-like non zero V14( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) complex-valued ) Complex_Sequence) (#) seq2 : ( ( Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) -valued Function-like non zero V14( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) complex-valued ) Complex_Sequence) ) : ( (
Function-like ) (
Relation-like NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) )
-defined COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set )
-valued Function-like non
zero V14(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) )
V18(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) ,
COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set ) )
complex-valued )
Element of
K19(
K20(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) ,
COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set ) ) : ( ( ) (
complex-valued )
set ) ) : ( ( ) ( )
set ) ) : ( (
Function-like ) (
Relation-like NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) )
-defined COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set )
-valued Function-like non
zero V14(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) )
V18(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) ,
COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set ) )
complex-valued )
Element of
K19(
K20(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) ,
COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set ) ) : ( ( ) (
complex-valued )
set ) ) : ( ( ) ( )
set ) )
= seq3 : ( (
Function-like V18(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) ,
COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set ) ) ) (
Relation-like NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) )
-defined COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set )
-valued Function-like non
zero V14(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) )
V18(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) ,
COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set ) )
complex-valued )
Complex_Sequence)
(#) (seq1 : ( ( Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) -valued Function-like non zero V14( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) complex-valued ) Complex_Sequence) - seq2 : ( ( Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) -valued Function-like non zero V14( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) complex-valued ) Complex_Sequence) ) : ( (
Function-like ) (
Relation-like NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) )
-defined COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set )
-valued Function-like non
zero V14(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) )
V18(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) ,
COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set ) )
complex-valued )
Element of
K19(
K20(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) ,
COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set ) ) : ( ( ) (
complex-valued )
set ) ) : ( ( ) ( )
set ) ) : ( (
Function-like ) (
Relation-like NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) )
-defined COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set )
-valued Function-like non
zero V14(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) )
V18(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) ,
COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set ) )
complex-valued )
Element of
K19(
K20(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) ,
COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set ) ) : ( ( ) (
complex-valued )
set ) ) : ( ( ) ( )
set ) ) ;
theorem
for
r being ( ( ) (
complex )
Element of
COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set ) )
for
seq1,
seq2 being ( (
Function-like V18(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) ,
COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set ) ) ) (
Relation-like NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) )
-defined COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set )
-valued Function-like non
zero V14(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) )
V18(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) ,
COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set ) )
complex-valued )
Complex_Sequence) holds
r : ( ( ) (
complex )
Element of
COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set ) )
(#) (seq1 : ( ( Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) -valued Function-like non zero V14( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) complex-valued ) Complex_Sequence) + seq2 : ( ( Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) -valued Function-like non zero V14( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) complex-valued ) Complex_Sequence) ) : ( (
Function-like ) (
Relation-like NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) )
-defined COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set )
-valued Function-like non
zero V14(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) )
V18(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) ,
COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set ) )
complex-valued )
Element of
K19(
K20(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) ,
COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set ) ) : ( ( ) (
complex-valued )
set ) ) : ( ( ) ( )
set ) ) : ( (
Function-like ) (
Relation-like NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) )
-defined COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set )
-valued Function-like non
zero V14(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) )
V18(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) ,
COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set ) )
complex-valued )
Element of
K19(
K20(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) ,
COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set ) ) : ( ( ) (
complex-valued )
set ) ) : ( ( ) ( )
set ) )
= (r : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) (#) seq1 : ( ( Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) -valued Function-like non zero V14( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) complex-valued ) Complex_Sequence) ) : ( (
Function-like ) (
Relation-like NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) )
-defined COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set )
-valued Function-like non
zero V14(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) )
V18(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) ,
COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set ) )
complex-valued )
Element of
K19(
K20(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) ,
COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set ) ) : ( ( ) (
complex-valued )
set ) ) : ( ( ) ( )
set ) )
+ (r : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) (#) seq2 : ( ( Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) -valued Function-like non zero V14( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) complex-valued ) Complex_Sequence) ) : ( (
Function-like ) (
Relation-like NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) )
-defined COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set )
-valued Function-like non
zero V14(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) )
V18(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) ,
COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set ) )
complex-valued )
Element of
K19(
K20(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) ,
COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set ) ) : ( ( ) (
complex-valued )
set ) ) : ( ( ) ( )
set ) ) : ( (
Function-like ) (
Relation-like NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) )
-defined COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set )
-valued Function-like non
zero V14(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) )
V18(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) ,
COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set ) )
complex-valued )
Element of
K19(
K20(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) ,
COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set ) ) : ( ( ) (
complex-valued )
set ) ) : ( ( ) ( )
set ) ) ;
theorem
for
r,
p being ( ( ) (
complex )
Element of
COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set ) )
for
seq being ( (
Function-like V18(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) ,
COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set ) ) ) (
Relation-like NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) )
-defined COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set )
-valued Function-like non
zero V14(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) )
V18(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) ,
COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set ) )
complex-valued )
Complex_Sequence) holds
(r : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) * p : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) ) : ( ( ) (
complex )
Element of
COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set ) )
(#) seq : ( (
Function-like V18(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) ,
COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set ) ) ) (
Relation-like NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) )
-defined COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set )
-valued Function-like non
zero V14(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) )
V18(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) ,
COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set ) )
complex-valued )
Complex_Sequence) : ( (
Function-like ) (
Relation-like NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) )
-defined COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set )
-valued Function-like non
zero V14(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) )
V18(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) ,
COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set ) )
complex-valued )
Element of
K19(
K20(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) ,
COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set ) ) : ( ( ) (
complex-valued )
set ) ) : ( ( ) ( )
set ) )
= r : ( ( ) (
complex )
Element of
COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set ) )
(#) (p : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) (#) seq : ( ( Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) -valued Function-like non zero V14( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) complex-valued ) Complex_Sequence) ) : ( (
Function-like ) (
Relation-like NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) )
-defined COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set )
-valued Function-like non
zero V14(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) )
V18(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) ,
COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set ) )
complex-valued )
Element of
K19(
K20(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) ,
COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set ) ) : ( ( ) (
complex-valued )
set ) ) : ( ( ) ( )
set ) ) : ( (
Function-like ) (
Relation-like NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) )
-defined COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set )
-valued Function-like non
zero V14(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) )
V18(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) ,
COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set ) )
complex-valued )
Element of
K19(
K20(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) ,
COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set ) ) : ( ( ) (
complex-valued )
set ) ) : ( ( ) ( )
set ) ) ;
theorem
for
r being ( ( ) (
complex )
Element of
COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set ) )
for
seq1,
seq2 being ( (
Function-like V18(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) ,
COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set ) ) ) (
Relation-like NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) )
-defined COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set )
-valued Function-like non
zero V14(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) )
V18(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) ,
COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set ) )
complex-valued )
Complex_Sequence) holds
r : ( ( ) (
complex )
Element of
COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set ) )
(#) (seq1 : ( ( Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) -valued Function-like non zero V14( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) complex-valued ) Complex_Sequence) - seq2 : ( ( Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) -valued Function-like non zero V14( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) complex-valued ) Complex_Sequence) ) : ( (
Function-like ) (
Relation-like NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) )
-defined COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set )
-valued Function-like non
zero V14(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) )
V18(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) ,
COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set ) )
complex-valued )
Element of
K19(
K20(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) ,
COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set ) ) : ( ( ) (
complex-valued )
set ) ) : ( ( ) ( )
set ) ) : ( (
Function-like ) (
Relation-like NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) )
-defined COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set )
-valued Function-like non
zero V14(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) )
V18(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) ,
COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set ) )
complex-valued )
Element of
K19(
K20(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) ,
COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set ) ) : ( ( ) (
complex-valued )
set ) ) : ( ( ) ( )
set ) )
= (r : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) (#) seq1 : ( ( Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) -valued Function-like non zero V14( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) complex-valued ) Complex_Sequence) ) : ( (
Function-like ) (
Relation-like NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) )
-defined COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set )
-valued Function-like non
zero V14(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) )
V18(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) ,
COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set ) )
complex-valued )
Element of
K19(
K20(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) ,
COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set ) ) : ( ( ) (
complex-valued )
set ) ) : ( ( ) ( )
set ) )
- (r : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) (#) seq2 : ( ( Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) -valued Function-like non zero V14( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) complex-valued ) Complex_Sequence) ) : ( (
Function-like ) (
Relation-like NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) )
-defined COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set )
-valued Function-like non
zero V14(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) )
V18(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) ,
COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set ) )
complex-valued )
Element of
K19(
K20(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) ,
COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set ) ) : ( ( ) (
complex-valued )
set ) ) : ( ( ) ( )
set ) ) : ( (
Function-like ) (
Relation-like NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) )
-defined COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set )
-valued Function-like non
zero V14(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) )
V18(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) ,
COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set ) )
complex-valued )
Element of
K19(
K20(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) ,
COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set ) ) : ( ( ) (
complex-valued )
set ) ) : ( ( ) ( )
set ) ) ;
theorem
for
r being ( ( ) (
complex )
Element of
COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set ) )
for
seq1,
seq being ( (
Function-like V18(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) ,
COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set ) ) ) (
Relation-like NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) )
-defined COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set )
-valued Function-like non
zero V14(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) )
V18(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) ,
COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set ) )
complex-valued )
Complex_Sequence) holds
r : ( ( ) (
complex )
Element of
COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set ) )
(#) (seq1 : ( ( Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) -valued Function-like non zero V14( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) complex-valued ) Complex_Sequence) /" seq : ( ( Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) -valued Function-like non zero V14( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) complex-valued ) Complex_Sequence) ) : ( (
Function-like ) (
Relation-like NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) )
-defined COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set )
-valued Function-like non
zero V14(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) )
V18(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) ,
COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set ) )
complex-valued )
Element of
K19(
K20(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) ,
COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set ) ) : ( ( ) (
complex-valued )
set ) ) : ( ( ) ( )
set ) ) : ( (
Function-like ) (
Relation-like NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) )
-defined COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set )
-valued Function-like non
zero V14(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) )
V18(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) ,
COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set ) )
complex-valued )
Element of
K19(
K20(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) ,
COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set ) ) : ( ( ) (
complex-valued )
set ) ) : ( ( ) ( )
set ) )
= (r : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) (#) seq1 : ( ( Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) -valued Function-like non zero V14( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) complex-valued ) Complex_Sequence) ) : ( (
Function-like ) (
Relation-like NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) )
-defined COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set )
-valued Function-like non
zero V14(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) )
V18(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) ,
COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set ) )
complex-valued )
Element of
K19(
K20(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) ,
COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set ) ) : ( ( ) (
complex-valued )
set ) ) : ( ( ) ( )
set ) )
/" seq : ( (
Function-like V18(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) ,
COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set ) ) ) (
Relation-like NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) )
-defined COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set )
-valued Function-like non
zero V14(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) )
V18(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) ,
COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set ) )
complex-valued )
Complex_Sequence) : ( (
Function-like ) (
Relation-like NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) )
-defined COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set )
-valued Function-like non
zero V14(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) )
V18(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) ,
COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set ) )
complex-valued )
Element of
K19(
K20(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) ,
COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set ) ) : ( ( ) (
complex-valued )
set ) ) : ( ( ) ( )
set ) ) ;
theorem
for
seq1,
seq2,
seq3 being ( (
Function-like V18(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) ,
COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set ) ) ) (
Relation-like NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) )
-defined COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set )
-valued Function-like non
zero V14(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) )
V18(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) ,
COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set ) )
complex-valued )
Complex_Sequence) holds
seq1 : ( (
Function-like V18(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) ,
COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set ) ) ) (
Relation-like NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) )
-defined COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set )
-valued Function-like non
zero V14(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) )
V18(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) ,
COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set ) )
complex-valued )
Complex_Sequence)
- (seq2 : ( ( Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) -valued Function-like non zero V14( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) complex-valued ) Complex_Sequence) + seq3 : ( ( Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) -valued Function-like non zero V14( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) complex-valued ) Complex_Sequence) ) : ( (
Function-like ) (
Relation-like NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) )
-defined COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set )
-valued Function-like non
zero V14(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) )
V18(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) ,
COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set ) )
complex-valued )
Element of
K19(
K20(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) ,
COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set ) ) : ( ( ) (
complex-valued )
set ) ) : ( ( ) ( )
set ) ) : ( (
Function-like ) (
Relation-like NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) )
-defined COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set )
-valued Function-like non
zero V14(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) )
V18(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) ,
COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set ) )
complex-valued )
Element of
K19(
K20(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) ,
COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set ) ) : ( ( ) (
complex-valued )
set ) ) : ( ( ) ( )
set ) )
= (seq1 : ( ( Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) -valued Function-like non zero V14( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) complex-valued ) Complex_Sequence) - seq2 : ( ( Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) -valued Function-like non zero V14( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) complex-valued ) Complex_Sequence) ) : ( (
Function-like ) (
Relation-like NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) )
-defined COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set )
-valued Function-like non
zero V14(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) )
V18(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) ,
COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set ) )
complex-valued )
Element of
K19(
K20(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) ,
COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set ) ) : ( ( ) (
complex-valued )
set ) ) : ( ( ) ( )
set ) )
- seq3 : ( (
Function-like V18(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) ,
COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set ) ) ) (
Relation-like NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) )
-defined COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set )
-valued Function-like non
zero V14(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) )
V18(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) ,
COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set ) )
complex-valued )
Complex_Sequence) : ( (
Function-like ) (
Relation-like NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) )
-defined COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set )
-valued Function-like non
zero V14(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) )
V18(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) ,
COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set ) )
complex-valued )
Element of
K19(
K20(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) ,
COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set ) ) : ( ( ) (
complex-valued )
set ) ) : ( ( ) ( )
set ) ) ;
theorem
for
seq being ( (
Function-like V18(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) ,
COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set ) ) ) (
Relation-like NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) )
-defined COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set )
-valued Function-like non
zero V14(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) )
V18(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) ,
COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set ) )
complex-valued )
Complex_Sequence) holds
- (- seq : ( ( Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) -valued Function-like non zero V14( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) complex-valued ) Complex_Sequence) ) : ( (
Function-like ) (
Relation-like NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) )
-defined COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set )
-valued Function-like non
zero V14(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) )
V18(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) ,
COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set ) )
complex-valued )
Element of
K19(
K20(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) ,
COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set ) ) : ( ( ) (
complex-valued )
set ) ) : ( ( ) ( )
set ) ) : ( (
Function-like ) (
Relation-like NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) )
-defined COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set )
-valued Function-like non
zero V14(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) )
V18(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) ,
COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set ) )
complex-valued )
Element of
K19(
K20(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) ,
COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set ) ) : ( ( ) (
complex-valued )
set ) ) : ( ( ) ( )
set ) )
= seq : ( (
Function-like V18(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) ,
COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set ) ) ) (
Relation-like NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) )
-defined COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set )
-valued Function-like non
zero V14(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) )
V18(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) ,
COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set ) )
complex-valued )
Complex_Sequence) ;
theorem
for
seq1,
seq2 being ( (
Function-like V18(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) ,
COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set ) ) ) (
Relation-like NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) )
-defined COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set )
-valued Function-like non
zero V14(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) )
V18(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) ,
COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set ) )
complex-valued )
Complex_Sequence) holds
seq1 : ( (
Function-like V18(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) ,
COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set ) ) ) (
Relation-like NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) )
-defined COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set )
-valued Function-like non
zero V14(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) )
V18(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) ,
COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set ) )
complex-valued )
Complex_Sequence)
- (- seq2 : ( ( Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) -valued Function-like non zero V14( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) complex-valued ) Complex_Sequence) ) : ( (
Function-like ) (
Relation-like NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) )
-defined COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set )
-valued Function-like non
zero V14(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) )
V18(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) ,
COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set ) )
complex-valued )
Element of
K19(
K20(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) ,
COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set ) ) : ( ( ) (
complex-valued )
set ) ) : ( ( ) ( )
set ) ) : ( (
Function-like ) (
Relation-like NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) )
-defined COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set )
-valued Function-like non
zero V14(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) )
V18(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) ,
COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set ) )
complex-valued )
Element of
K19(
K20(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) ,
COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set ) ) : ( ( ) (
complex-valued )
set ) ) : ( ( ) ( )
set ) )
= seq1 : ( (
Function-like V18(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) ,
COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set ) ) ) (
Relation-like NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) )
-defined COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set )
-valued Function-like non
zero V14(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) )
V18(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) ,
COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set ) )
complex-valued )
Complex_Sequence)
+ seq2 : ( (
Function-like V18(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) ,
COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set ) ) ) (
Relation-like NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) )
-defined COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set )
-valued Function-like non
zero V14(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) )
V18(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) ,
COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set ) )
complex-valued )
Complex_Sequence) : ( (
Function-like ) (
Relation-like NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) )
-defined COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set )
-valued Function-like non
zero V14(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) )
V18(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) ,
COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set ) )
complex-valued )
Element of
K19(
K20(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) ,
COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set ) ) : ( ( ) (
complex-valued )
set ) ) : ( ( ) ( )
set ) ) ;
theorem
for
seq1,
seq2,
seq3 being ( (
Function-like V18(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) ,
COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set ) ) ) (
Relation-like NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) )
-defined COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set )
-valued Function-like non
zero V14(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) )
V18(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) ,
COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set ) )
complex-valued )
Complex_Sequence) holds
seq1 : ( (
Function-like V18(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) ,
COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set ) ) ) (
Relation-like NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) )
-defined COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set )
-valued Function-like non
zero V14(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) )
V18(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) ,
COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set ) )
complex-valued )
Complex_Sequence)
- (seq2 : ( ( Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) -valued Function-like non zero V14( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) complex-valued ) Complex_Sequence) - seq3 : ( ( Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) -valued Function-like non zero V14( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) complex-valued ) Complex_Sequence) ) : ( (
Function-like ) (
Relation-like NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) )
-defined COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set )
-valued Function-like non
zero V14(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) )
V18(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) ,
COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set ) )
complex-valued )
Element of
K19(
K20(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) ,
COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set ) ) : ( ( ) (
complex-valued )
set ) ) : ( ( ) ( )
set ) ) : ( (
Function-like ) (
Relation-like NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) )
-defined COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set )
-valued Function-like non
zero V14(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) )
V18(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) ,
COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set ) )
complex-valued )
Element of
K19(
K20(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) ,
COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set ) ) : ( ( ) (
complex-valued )
set ) ) : ( ( ) ( )
set ) )
= (seq1 : ( ( Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) -valued Function-like non zero V14( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) complex-valued ) Complex_Sequence) - seq2 : ( ( Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) -valued Function-like non zero V14( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) complex-valued ) Complex_Sequence) ) : ( (
Function-like ) (
Relation-like NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) )
-defined COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set )
-valued Function-like non
zero V14(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) )
V18(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) ,
COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set ) )
complex-valued )
Element of
K19(
K20(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) ,
COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set ) ) : ( ( ) (
complex-valued )
set ) ) : ( ( ) ( )
set ) )
+ seq3 : ( (
Function-like V18(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) ,
COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set ) ) ) (
Relation-like NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) )
-defined COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set )
-valued Function-like non
zero V14(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) )
V18(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) ,
COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set ) )
complex-valued )
Complex_Sequence) : ( (
Function-like ) (
Relation-like NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) )
-defined COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set )
-valued Function-like non
zero V14(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) )
V18(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) ,
COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set ) )
complex-valued )
Element of
K19(
K20(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) ,
COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set ) ) : ( ( ) (
complex-valued )
set ) ) : ( ( ) ( )
set ) ) ;
theorem
for
seq1,
seq2,
seq3 being ( (
Function-like V18(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) ,
COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set ) ) ) (
Relation-like NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) )
-defined COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set )
-valued Function-like non
zero V14(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) )
V18(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) ,
COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set ) )
complex-valued )
Complex_Sequence) holds
seq1 : ( (
Function-like V18(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) ,
COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set ) ) ) (
Relation-like NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) )
-defined COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set )
-valued Function-like non
zero V14(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) )
V18(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) ,
COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set ) )
complex-valued )
Complex_Sequence)
+ (seq2 : ( ( Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) -valued Function-like non zero V14( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) complex-valued ) Complex_Sequence) - seq3 : ( ( Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) -valued Function-like non zero V14( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) complex-valued ) Complex_Sequence) ) : ( (
Function-like ) (
Relation-like NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) )
-defined COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set )
-valued Function-like non
zero V14(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) )
V18(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) ,
COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set ) )
complex-valued )
Element of
K19(
K20(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) ,
COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set ) ) : ( ( ) (
complex-valued )
set ) ) : ( ( ) ( )
set ) ) : ( (
Function-like ) (
Relation-like NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) )
-defined COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set )
-valued Function-like non
zero V14(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) )
V18(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) ,
COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set ) )
complex-valued )
Element of
K19(
K20(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) ,
COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set ) ) : ( ( ) (
complex-valued )
set ) ) : ( ( ) ( )
set ) )
= (seq1 : ( ( Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) -valued Function-like non zero V14( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) complex-valued ) Complex_Sequence) + seq2 : ( ( Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) -valued Function-like non zero V14( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) complex-valued ) Complex_Sequence) ) : ( (
Function-like ) (
Relation-like NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) )
-defined COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set )
-valued Function-like non
zero V14(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) )
V18(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) ,
COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set ) )
complex-valued )
Element of
K19(
K20(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) ,
COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set ) ) : ( ( ) (
complex-valued )
set ) ) : ( ( ) ( )
set ) )
- seq3 : ( (
Function-like V18(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) ,
COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set ) ) ) (
Relation-like NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) )
-defined COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set )
-valued Function-like non
zero V14(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) )
V18(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) ,
COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set ) )
complex-valued )
Complex_Sequence) : ( (
Function-like ) (
Relation-like NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) )
-defined COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set )
-valued Function-like non
zero V14(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) )
V18(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) ,
COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set ) )
complex-valued )
Element of
K19(
K20(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) ,
COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set ) ) : ( ( ) (
complex-valued )
set ) ) : ( ( ) ( )
set ) ) ;
theorem
for
seq1,
seq2 being ( (
Function-like V18(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) ,
COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set ) ) ) (
Relation-like NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) )
-defined COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set )
-valued Function-like non
zero V14(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) )
V18(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) ,
COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set ) )
complex-valued )
Complex_Sequence) holds
(
(- seq1 : ( ( Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) -valued Function-like non zero V14( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) complex-valued ) Complex_Sequence) ) : ( (
Function-like ) (
Relation-like NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) )
-defined COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set )
-valued Function-like non
zero V14(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) )
V18(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) ,
COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set ) )
complex-valued )
Element of
K19(
K20(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) ,
COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set ) ) : ( ( ) (
complex-valued )
set ) ) : ( ( ) ( )
set ) )
(#) seq2 : ( (
Function-like V18(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) ,
COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set ) ) ) (
Relation-like NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) )
-defined COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set )
-valued Function-like non
zero V14(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) )
V18(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) ,
COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set ) )
complex-valued )
Complex_Sequence) : ( (
Function-like ) (
Relation-like NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) )
-defined COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set )
-valued Function-like non
zero V14(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) )
V18(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) ,
COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set ) )
complex-valued )
Element of
K19(
K20(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) ,
COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set ) ) : ( ( ) (
complex-valued )
set ) ) : ( ( ) ( )
set ) )
= - (seq1 : ( ( Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) -valued Function-like non zero V14( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) complex-valued ) Complex_Sequence) (#) seq2 : ( ( Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) -valued Function-like non zero V14( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) complex-valued ) Complex_Sequence) ) : ( (
Function-like ) (
Relation-like NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) )
-defined COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set )
-valued Function-like non
zero V14(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) )
V18(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) ,
COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set ) )
complex-valued )
Element of
K19(
K20(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) ,
COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set ) ) : ( ( ) (
complex-valued )
set ) ) : ( ( ) ( )
set ) ) : ( (
Function-like ) (
Relation-like NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) )
-defined COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set )
-valued Function-like non
zero V14(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) )
V18(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) ,
COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set ) )
complex-valued )
Element of
K19(
K20(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) ,
COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set ) ) : ( ( ) (
complex-valued )
set ) ) : ( ( ) ( )
set ) ) &
seq1 : ( (
Function-like V18(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) ,
COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set ) ) ) (
Relation-like NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) )
-defined COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set )
-valued Function-like non
zero V14(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) )
V18(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) ,
COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set ) )
complex-valued )
Complex_Sequence)
(#) (- seq2 : ( ( Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) -valued Function-like non zero V14( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) complex-valued ) Complex_Sequence) ) : ( (
Function-like ) (
Relation-like NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) )
-defined COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set )
-valued Function-like non
zero V14(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) )
V18(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) ,
COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set ) )
complex-valued )
Element of
K19(
K20(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) ,
COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set ) ) : ( ( ) (
complex-valued )
set ) ) : ( ( ) ( )
set ) ) : ( (
Function-like ) (
Relation-like NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) )
-defined COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set )
-valued Function-like non
zero V14(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) )
V18(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) ,
COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set ) )
complex-valued )
Element of
K19(
K20(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) ,
COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set ) ) : ( ( ) (
complex-valued )
set ) ) : ( ( ) ( )
set ) )
= - (seq1 : ( ( Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) -valued Function-like non zero V14( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) complex-valued ) Complex_Sequence) (#) seq2 : ( ( Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) -valued Function-like non zero V14( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) complex-valued ) Complex_Sequence) ) : ( (
Function-like ) (
Relation-like NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) )
-defined COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set )
-valued Function-like non
zero V14(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) )
V18(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) ,
COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set ) )
complex-valued )
Element of
K19(
K20(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) ,
COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set ) ) : ( ( ) (
complex-valued )
set ) ) : ( ( ) ( )
set ) ) : ( (
Function-like ) (
Relation-like NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) )
-defined COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set )
-valued Function-like non
zero V14(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) )
V18(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) ,
COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set ) )
complex-valued )
Element of
K19(
K20(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) ,
COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set ) ) : ( ( ) (
complex-valued )
set ) ) : ( ( ) ( )
set ) ) ) ;
theorem
for
seq,
seq1 being ( (
Function-like V18(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) ,
COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set ) ) ) (
Relation-like NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) )
-defined COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set )
-valued Function-like non
zero V14(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) )
V18(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) ,
COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set ) )
complex-valued )
Complex_Sequence) holds
( (
seq : ( (
Function-like V18(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) ,
COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set ) ) ) (
Relation-like NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) )
-defined COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set )
-valued Function-like non
zero V14(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) )
V18(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) ,
COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set ) )
complex-valued )
Complex_Sequence) is
non-zero &
seq1 : ( (
Function-like V18(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) ,
COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set ) ) ) (
Relation-like NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) )
-defined COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set )
-valued Function-like non
zero V14(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) )
V18(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) ,
COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set ) )
complex-valued )
Complex_Sequence) is
non-zero ) iff
seq : ( (
Function-like V18(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) ,
COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set ) ) ) (
Relation-like NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) )
-defined COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set )
-valued Function-like non
zero V14(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) )
V18(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) ,
COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set ) )
complex-valued )
Complex_Sequence)
(#) seq1 : ( (
Function-like V18(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) ,
COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set ) ) ) (
Relation-like NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) )
-defined COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set )
-valued Function-like non
zero V14(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) )
V18(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) ,
COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set ) )
complex-valued )
Complex_Sequence) : ( (
Function-like ) (
Relation-like NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) )
-defined COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set )
-valued Function-like non
zero V14(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) )
V18(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) ,
COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set ) )
complex-valued )
Element of
K19(
K20(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) ,
COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set ) ) : ( ( ) (
complex-valued )
set ) ) : ( ( ) ( )
set ) ) is
non-zero ) ;
theorem
for
seq,
seq1 being ( (
Function-like V18(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) ,
COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set ) ) ) (
Relation-like NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) )
-defined COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set )
-valued Function-like non
zero V14(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) )
V18(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) ,
COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set ) )
complex-valued )
Complex_Sequence) holds
(seq : ( ( Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) -valued Function-like non zero V14( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) complex-valued ) Complex_Sequence) ") : ( (
Function-like ) (
Relation-like NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) )
-defined COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set )
-valued Function-like non
zero V14(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) )
V18(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) ,
COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set ) )
complex-valued )
Element of
K19(
K20(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) ,
COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set ) ) : ( ( ) (
complex-valued )
set ) ) : ( ( ) ( )
set ) )
(#) (seq1 : ( ( Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) -valued Function-like non zero V14( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) complex-valued ) Complex_Sequence) ") : ( (
Function-like ) (
Relation-like NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) )
-defined COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set )
-valued Function-like non
zero V14(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) )
V18(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) ,
COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set ) )
complex-valued )
Element of
K19(
K20(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) ,
COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set ) ) : ( ( ) (
complex-valued )
set ) ) : ( ( ) ( )
set ) ) : ( (
Function-like ) (
Relation-like NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) )
-defined COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set )
-valued Function-like non
zero V14(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) )
V18(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) ,
COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set ) )
complex-valued )
Element of
K19(
K20(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) ,
COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set ) ) : ( ( ) (
complex-valued )
set ) ) : ( ( ) ( )
set ) )
= (seq : ( ( Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) -valued Function-like non zero V14( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) complex-valued ) Complex_Sequence) (#) seq1 : ( ( Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) -valued Function-like non zero V14( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) complex-valued ) Complex_Sequence) ) : ( (
Function-like ) (
Relation-like NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) )
-defined COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set )
-valued Function-like non
zero V14(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) )
V18(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) ,
COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set ) )
complex-valued )
Element of
K19(
K20(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) ,
COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set ) ) : ( ( ) (
complex-valued )
set ) ) : ( ( ) ( )
set ) )
" : ( (
Function-like ) (
Relation-like NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) )
-defined COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set )
-valued Function-like non
zero V14(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) )
V18(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) ,
COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set ) )
complex-valued )
Element of
K19(
K20(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) ,
COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set ) ) : ( ( ) (
complex-valued )
set ) ) : ( ( ) ( )
set ) ) ;
theorem
for
seq,
seq1 being ( (
Function-like V18(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) ,
COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set ) ) ) (
Relation-like NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) )
-defined COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set )
-valued Function-like non
zero V14(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) )
V18(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) ,
COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set ) )
complex-valued )
Complex_Sequence) st
seq : ( (
Function-like V18(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) ,
COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set ) ) ) (
Relation-like NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) )
-defined COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set )
-valued Function-like non
zero V14(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) )
V18(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) ,
COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set ) )
complex-valued )
Complex_Sequence) is
non-zero holds
(seq1 : ( ( Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) -valued Function-like non zero V14( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) complex-valued ) Complex_Sequence) /" seq : ( ( Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) -valued Function-like non zero V14( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) complex-valued ) Complex_Sequence) ) : ( (
Function-like ) (
Relation-like NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) )
-defined COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set )
-valued Function-like non
zero V14(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) )
V18(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) ,
COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set ) )
complex-valued )
Element of
K19(
K20(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) ,
COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set ) ) : ( ( ) (
complex-valued )
set ) ) : ( ( ) ( )
set ) )
(#) seq : ( (
Function-like V18(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) ,
COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set ) ) ) (
Relation-like NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) )
-defined COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set )
-valued Function-like non
zero V14(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) )
V18(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) ,
COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set ) )
complex-valued )
Complex_Sequence) : ( (
Function-like ) (
Relation-like NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) )
-defined COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set )
-valued Function-like non
zero V14(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) )
V18(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) ,
COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set ) )
complex-valued )
Element of
K19(
K20(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) ,
COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set ) ) : ( ( ) (
complex-valued )
set ) ) : ( ( ) ( )
set ) )
= seq1 : ( (
Function-like V18(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) ,
COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set ) ) ) (
Relation-like NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) )
-defined COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set )
-valued Function-like non
zero V14(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) )
V18(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) ,
COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set ) )
complex-valued )
Complex_Sequence) ;
theorem
for
seq9,
seq,
seq19,
seq1 being ( (
Function-like V18(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) ,
COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set ) ) ) (
Relation-like NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) )
-defined COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set )
-valued Function-like non
zero V14(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) )
V18(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) ,
COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set ) )
complex-valued )
Complex_Sequence) holds
(seq9 : ( ( Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) -valued Function-like non zero V14( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) complex-valued ) Complex_Sequence) /" seq : ( ( Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) -valued Function-like non zero V14( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) complex-valued ) Complex_Sequence) ) : ( (
Function-like ) (
Relation-like NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) )
-defined COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set )
-valued Function-like non
zero V14(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) )
V18(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) ,
COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set ) )
complex-valued )
Element of
K19(
K20(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) ,
COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set ) ) : ( ( ) (
complex-valued )
set ) ) : ( ( ) ( )
set ) )
(#) (seq19 : ( ( Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) -valued Function-like non zero V14( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) complex-valued ) Complex_Sequence) /" seq1 : ( ( Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) -valued Function-like non zero V14( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) complex-valued ) Complex_Sequence) ) : ( (
Function-like ) (
Relation-like NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) )
-defined COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set )
-valued Function-like non
zero V14(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) )
V18(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) ,
COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set ) )
complex-valued )
Element of
K19(
K20(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) ,
COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set ) ) : ( ( ) (
complex-valued )
set ) ) : ( ( ) ( )
set ) ) : ( (
Function-like ) (
Relation-like NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) )
-defined COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set )
-valued Function-like non
zero V14(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) )
V18(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) ,
COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set ) )
complex-valued )
Element of
K19(
K20(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) ,
COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set ) ) : ( ( ) (
complex-valued )
set ) ) : ( ( ) ( )
set ) )
= (seq9 : ( ( Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) -valued Function-like non zero V14( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) complex-valued ) Complex_Sequence) (#) seq19 : ( ( Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) -valued Function-like non zero V14( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) complex-valued ) Complex_Sequence) ) : ( (
Function-like ) (
Relation-like NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) )
-defined COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set )
-valued Function-like non
zero V14(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) )
V18(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) ,
COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set ) )
complex-valued )
Element of
K19(
K20(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) ,
COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set ) ) : ( ( ) (
complex-valued )
set ) ) : ( ( ) ( )
set ) )
/" (seq : ( ( Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) -valued Function-like non zero V14( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) complex-valued ) Complex_Sequence) (#) seq1 : ( ( Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) -valued Function-like non zero V14( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) complex-valued ) Complex_Sequence) ) : ( (
Function-like ) (
Relation-like NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) )
-defined COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set )
-valued Function-like non
zero V14(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) )
V18(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) ,
COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set ) )
complex-valued )
Element of
K19(
K20(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) ,
COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set ) ) : ( ( ) (
complex-valued )
set ) ) : ( ( ) ( )
set ) ) : ( (
Function-like ) (
Relation-like NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) )
-defined COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set )
-valued Function-like non
zero V14(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) )
V18(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) ,
COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set ) )
complex-valued )
Element of
K19(
K20(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) ,
COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set ) ) : ( ( ) (
complex-valued )
set ) ) : ( ( ) ( )
set ) ) ;
theorem
for
seq,
seq1 being ( (
Function-like V18(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) ,
COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set ) ) ) (
Relation-like NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) )
-defined COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set )
-valued Function-like non
zero V14(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) )
V18(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) ,
COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set ) )
complex-valued )
Complex_Sequence) st
seq : ( (
Function-like V18(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) ,
COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set ) ) ) (
Relation-like NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) )
-defined COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set )
-valued Function-like non
zero V14(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) )
V18(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) ,
COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set ) )
complex-valued )
Complex_Sequence) is
non-zero &
seq1 : ( (
Function-like V18(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) ,
COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set ) ) ) (
Relation-like NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) )
-defined COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set )
-valued Function-like non
zero V14(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) )
V18(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) ,
COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set ) )
complex-valued )
Complex_Sequence) is
non-zero holds
seq : ( (
Function-like V18(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) ,
COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set ) ) ) (
Relation-like NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) )
-defined COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set )
-valued Function-like non
zero V14(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) )
V18(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) ,
COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set ) )
complex-valued )
Complex_Sequence)
/" seq1 : ( (
Function-like V18(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) ,
COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set ) ) ) (
Relation-like NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) )
-defined COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set )
-valued Function-like non
zero V14(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) )
V18(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) ,
COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set ) )
complex-valued )
Complex_Sequence) : ( (
Function-like ) (
Relation-like NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) )
-defined COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set )
-valued Function-like non
zero V14(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) )
V18(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) ,
COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set ) )
complex-valued )
Element of
K19(
K20(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) ,
COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set ) ) : ( ( ) (
complex-valued )
set ) ) : ( ( ) ( )
set ) ) is
non-zero ;
theorem
for
seq,
seq1 being ( (
Function-like V18(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) ,
COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set ) ) ) (
Relation-like NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) )
-defined COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set )
-valued Function-like non
zero V14(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) )
V18(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) ,
COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set ) )
complex-valued )
Complex_Sequence) holds
(seq : ( ( Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) -valued Function-like non zero V14( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) complex-valued ) Complex_Sequence) /" seq1 : ( ( Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) -valued Function-like non zero V14( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) complex-valued ) Complex_Sequence) ) : ( (
Function-like ) (
Relation-like NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) )
-defined COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set )
-valued Function-like non
zero V14(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) )
V18(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) ,
COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set ) )
complex-valued )
Element of
K19(
K20(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) ,
COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set ) ) : ( ( ) (
complex-valued )
set ) ) : ( ( ) ( )
set ) )
" : ( (
Function-like ) (
Relation-like NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) )
-defined COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set )
-valued Function-like non
zero V14(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) )
V18(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) ,
COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set ) )
complex-valued )
Element of
K19(
K20(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) ,
COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set ) ) : ( ( ) (
complex-valued )
set ) ) : ( ( ) ( )
set ) )
= seq1 : ( (
Function-like V18(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) ,
COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set ) ) ) (
Relation-like NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) )
-defined COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set )
-valued Function-like non
zero V14(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) )
V18(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) ,
COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set ) )
complex-valued )
Complex_Sequence)
/" seq : ( (
Function-like V18(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) ,
COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set ) ) ) (
Relation-like NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) )
-defined COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set )
-valued Function-like non
zero V14(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) )
V18(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) ,
COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set ) )
complex-valued )
Complex_Sequence) : ( (
Function-like ) (
Relation-like NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) )
-defined COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set )
-valued Function-like non
zero V14(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) )
V18(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) ,
COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set ) )
complex-valued )
Element of
K19(
K20(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) ,
COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set ) ) : ( ( ) (
complex-valued )
set ) ) : ( ( ) ( )
set ) ) ;
theorem
for
seq2,
seq1,
seq being ( (
Function-like V18(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) ,
COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set ) ) ) (
Relation-like NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) )
-defined COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set )
-valued Function-like non
zero V14(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) )
V18(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) ,
COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set ) )
complex-valued )
Complex_Sequence) holds
seq2 : ( (
Function-like V18(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) ,
COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set ) ) ) (
Relation-like NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) )
-defined COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set )
-valued Function-like non
zero V14(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) )
V18(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) ,
COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set ) )
complex-valued )
Complex_Sequence)
(#) (seq1 : ( ( Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) -valued Function-like non zero V14( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) complex-valued ) Complex_Sequence) /" seq : ( ( Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) -valued Function-like non zero V14( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) complex-valued ) Complex_Sequence) ) : ( (
Function-like ) (
Relation-like NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) )
-defined COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set )
-valued Function-like non
zero V14(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) )
V18(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) ,
COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set ) )
complex-valued )
Element of
K19(
K20(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) ,
COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set ) ) : ( ( ) (
complex-valued )
set ) ) : ( ( ) ( )
set ) ) : ( (
Function-like ) (
Relation-like NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) )
-defined COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set )
-valued Function-like non
zero V14(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) )
V18(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) ,
COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set ) )
complex-valued )
Element of
K19(
K20(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) ,
COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set ) ) : ( ( ) (
complex-valued )
set ) ) : ( ( ) ( )
set ) )
= (seq2 : ( ( Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) -valued Function-like non zero V14( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) complex-valued ) Complex_Sequence) (#) seq1 : ( ( Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) -valued Function-like non zero V14( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) complex-valued ) Complex_Sequence) ) : ( (
Function-like ) (
Relation-like NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) )
-defined COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set )
-valued Function-like non
zero V14(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) )
V18(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) ,
COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set ) )
complex-valued )
Element of
K19(
K20(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) ,
COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set ) ) : ( ( ) (
complex-valued )
set ) ) : ( ( ) ( )
set ) )
/" seq : ( (
Function-like V18(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) ,
COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set ) ) ) (
Relation-like NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) )
-defined COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set )
-valued Function-like non
zero V14(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) )
V18(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) ,
COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set ) )
complex-valued )
Complex_Sequence) : ( (
Function-like ) (
Relation-like NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) )
-defined COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set )
-valued Function-like non
zero V14(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) )
V18(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) ,
COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set ) )
complex-valued )
Element of
K19(
K20(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) ,
COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set ) ) : ( ( ) (
complex-valued )
set ) ) : ( ( ) ( )
set ) ) ;
theorem
for
seq2,
seq,
seq1 being ( (
Function-like V18(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) ,
COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set ) ) ) (
Relation-like NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) )
-defined COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set )
-valued Function-like non
zero V14(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) )
V18(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) ,
COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set ) )
complex-valued )
Complex_Sequence) holds
seq2 : ( (
Function-like V18(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) ,
COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set ) ) ) (
Relation-like NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) )
-defined COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set )
-valued Function-like non
zero V14(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) )
V18(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) ,
COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set ) )
complex-valued )
Complex_Sequence)
/" (seq : ( ( Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) -valued Function-like non zero V14( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) complex-valued ) Complex_Sequence) /" seq1 : ( ( Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) -valued Function-like non zero V14( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) complex-valued ) Complex_Sequence) ) : ( (
Function-like ) (
Relation-like NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) )
-defined COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set )
-valued Function-like non
zero V14(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) )
V18(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) ,
COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set ) )
complex-valued )
Element of
K19(
K20(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) ,
COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set ) ) : ( ( ) (
complex-valued )
set ) ) : ( ( ) ( )
set ) ) : ( (
Function-like ) (
Relation-like NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) )
-defined COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set )
-valued Function-like non
zero V14(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) )
V18(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) ,
COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set ) )
complex-valued )
Element of
K19(
K20(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) ,
COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set ) ) : ( ( ) (
complex-valued )
set ) ) : ( ( ) ( )
set ) )
= (seq2 : ( ( Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) -valued Function-like non zero V14( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) complex-valued ) Complex_Sequence) (#) seq1 : ( ( Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) -valued Function-like non zero V14( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) complex-valued ) Complex_Sequence) ) : ( (
Function-like ) (
Relation-like NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) )
-defined COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set )
-valued Function-like non
zero V14(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) )
V18(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) ,
COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set ) )
complex-valued )
Element of
K19(
K20(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) ,
COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set ) ) : ( ( ) (
complex-valued )
set ) ) : ( ( ) ( )
set ) )
/" seq : ( (
Function-like V18(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) ,
COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set ) ) ) (
Relation-like NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) )
-defined COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set )
-valued Function-like non
zero V14(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) )
V18(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) ,
COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set ) )
complex-valued )
Complex_Sequence) : ( (
Function-like ) (
Relation-like NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) )
-defined COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set )
-valued Function-like non
zero V14(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) )
V18(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) ,
COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set ) )
complex-valued )
Element of
K19(
K20(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) ,
COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set ) ) : ( ( ) (
complex-valued )
set ) ) : ( ( ) ( )
set ) ) ;
theorem
for
seq1,
seq2,
seq being ( (
Function-like V18(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) ,
COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set ) ) ) (
Relation-like NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) )
-defined COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set )
-valued Function-like non
zero V14(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) )
V18(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) ,
COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set ) )
complex-valued )
Complex_Sequence) st
seq1 : ( (
Function-like V18(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) ,
COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set ) ) ) (
Relation-like NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) )
-defined COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set )
-valued Function-like non
zero V14(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) )
V18(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) ,
COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set ) )
complex-valued )
Complex_Sequence) is
non-zero holds
seq2 : ( (
Function-like V18(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) ,
COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set ) ) ) (
Relation-like NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) )
-defined COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set )
-valued Function-like non
zero V14(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) )
V18(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) ,
COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set ) )
complex-valued )
Complex_Sequence)
/" seq : ( (
Function-like V18(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) ,
COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set ) ) ) (
Relation-like NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) )
-defined COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set )
-valued Function-like non
zero V14(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) )
V18(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) ,
COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set ) )
complex-valued )
Complex_Sequence) : ( (
Function-like ) (
Relation-like NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) )
-defined COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set )
-valued Function-like non
zero V14(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) )
V18(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) ,
COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set ) )
complex-valued )
Element of
K19(
K20(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) ,
COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set ) ) : ( ( ) (
complex-valued )
set ) ) : ( ( ) ( )
set ) )
= (seq2 : ( ( Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) -valued Function-like non zero V14( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) complex-valued ) Complex_Sequence) (#) seq1 : ( ( Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) -valued Function-like non zero V14( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) complex-valued ) Complex_Sequence) ) : ( (
Function-like ) (
Relation-like NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) )
-defined COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set )
-valued Function-like non
zero V14(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) )
V18(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) ,
COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set ) )
complex-valued )
Element of
K19(
K20(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) ,
COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set ) ) : ( ( ) (
complex-valued )
set ) ) : ( ( ) ( )
set ) )
/" (seq : ( ( Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) -valued Function-like non zero V14( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) complex-valued ) Complex_Sequence) (#) seq1 : ( ( Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) -valued Function-like non zero V14( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) complex-valued ) Complex_Sequence) ) : ( (
Function-like ) (
Relation-like NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) )
-defined COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set )
-valued Function-like non
zero V14(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) )
V18(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) ,
COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set ) )
complex-valued )
Element of
K19(
K20(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) ,
COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set ) ) : ( ( ) (
complex-valued )
set ) ) : ( ( ) ( )
set ) ) : ( (
Function-like ) (
Relation-like NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) )
-defined COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set )
-valued Function-like non
zero V14(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) )
V18(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) ,
COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set ) )
complex-valued )
Element of
K19(
K20(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) ,
COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set ) ) : ( ( ) (
complex-valued )
set ) ) : ( ( ) ( )
set ) ) ;
theorem
for
r being ( ( ) (
complex )
Element of
COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set ) )
for
seq being ( (
Function-like V18(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) ,
COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set ) ) ) (
Relation-like NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) )
-defined COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set )
-valued Function-like non
zero V14(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) )
V18(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) ,
COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set ) )
complex-valued )
Complex_Sequence) holds
(r : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) (#) seq : ( ( Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) -valued Function-like non zero V14( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) complex-valued ) Complex_Sequence) ) : ( (
Function-like ) (
Relation-like NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) )
-defined COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set )
-valued Function-like non
zero V14(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) )
V18(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) ,
COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set ) )
complex-valued )
Element of
K19(
K20(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) ,
COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set ) ) : ( ( ) (
complex-valued )
set ) ) : ( ( ) ( )
set ) )
" : ( (
Function-like ) (
Relation-like NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) )
-defined COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set )
-valued Function-like non
zero V14(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) )
V18(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) ,
COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set ) )
complex-valued )
Element of
K19(
K20(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) ,
COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set ) ) : ( ( ) (
complex-valued )
set ) ) : ( ( ) ( )
set ) )
= (r : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) ") : ( ( ) (
complex )
Element of
COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set ) )
(#) (seq : ( ( Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) -valued Function-like non zero V14( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) complex-valued ) Complex_Sequence) ") : ( (
Function-like ) (
Relation-like NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) )
-defined COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set )
-valued Function-like non
zero V14(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) )
V18(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) ,
COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set ) )
complex-valued )
Element of
K19(
K20(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) ,
COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set ) ) : ( ( ) (
complex-valued )
set ) ) : ( ( ) ( )
set ) ) : ( (
Function-like ) (
Relation-like NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) )
-defined COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set )
-valued Function-like non
zero V14(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) )
V18(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) ,
COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set ) )
complex-valued )
Element of
K19(
K20(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) ,
COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set ) ) : ( ( ) (
complex-valued )
set ) ) : ( ( ) ( )
set ) ) ;
theorem
for
seq being ( (
Function-like V18(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) ,
COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set ) ) ) (
Relation-like NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) )
-defined COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set )
-valued Function-like non
zero V14(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) )
V18(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) ,
COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set ) )
complex-valued )
Complex_Sequence) st
seq : ( (
Function-like V18(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) ,
COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set ) ) ) (
Relation-like NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) )
-defined COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set )
-valued Function-like non
zero V14(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) )
V18(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) ,
COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set ) )
complex-valued )
Complex_Sequence) is
non-zero holds
(- seq : ( ( Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) -valued Function-like non zero V14( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) complex-valued ) Complex_Sequence) ) : ( (
Function-like ) (
Relation-like NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) )
-defined COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set )
-valued Function-like non
zero V14(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) )
V18(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) ,
COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set ) )
complex-valued )
Element of
K19(
K20(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) ,
COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set ) ) : ( ( ) (
complex-valued )
set ) ) : ( ( ) ( )
set ) )
" : ( (
Function-like ) (
Relation-like NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) )
-defined COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set )
-valued Function-like non
zero V14(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) )
V18(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) ,
COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set ) )
complex-valued )
Element of
K19(
K20(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) ,
COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set ) ) : ( ( ) (
complex-valued )
set ) ) : ( ( ) ( )
set ) )
= (- 1r : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) ) : ( ( ) (
complex )
Element of
COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set ) )
(#) (seq : ( ( Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) -valued Function-like non zero V14( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) complex-valued ) Complex_Sequence) ") : ( (
Function-like ) (
Relation-like NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) )
-defined COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set )
-valued Function-like non
zero V14(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) )
V18(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) ,
COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set ) )
complex-valued )
Element of
K19(
K20(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) ,
COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set ) ) : ( ( ) (
complex-valued )
set ) ) : ( ( ) ( )
set ) ) : ( (
Function-like ) (
Relation-like NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) )
-defined COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set )
-valued Function-like non
zero V14(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) )
V18(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) ,
COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set ) )
complex-valued )
Element of
K19(
K20(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) ,
COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set ) ) : ( ( ) (
complex-valued )
set ) ) : ( ( ) ( )
set ) ) ;
theorem
for
seq,
seq1 being ( (
Function-like V18(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) ,
COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set ) ) ) (
Relation-like NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) )
-defined COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set )
-valued Function-like non
zero V14(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) )
V18(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) ,
COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set ) )
complex-valued )
Complex_Sequence) st
seq : ( (
Function-like V18(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) ,
COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set ) ) ) (
Relation-like NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) )
-defined COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set )
-valued Function-like non
zero V14(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) )
V18(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) ,
COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set ) )
complex-valued )
Complex_Sequence) is
non-zero holds
(
- (seq1 : ( ( Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) -valued Function-like non zero V14( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) complex-valued ) Complex_Sequence) /" seq : ( ( Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) -valued Function-like non zero V14( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) complex-valued ) Complex_Sequence) ) : ( (
Function-like ) (
Relation-like NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) )
-defined COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set )
-valued Function-like non
zero V14(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) )
V18(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) ,
COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set ) )
complex-valued )
Element of
K19(
K20(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) ,
COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set ) ) : ( ( ) (
complex-valued )
set ) ) : ( ( ) ( )
set ) ) : ( (
Function-like ) (
Relation-like NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) )
-defined COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set )
-valued Function-like non
zero V14(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) )
V18(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) ,
COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set ) )
complex-valued )
Element of
K19(
K20(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) ,
COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set ) ) : ( ( ) (
complex-valued )
set ) ) : ( ( ) ( )
set ) )
= (- seq1 : ( ( Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) -valued Function-like non zero V14( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) complex-valued ) Complex_Sequence) ) : ( (
Function-like ) (
Relation-like NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) )
-defined COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set )
-valued Function-like non
zero V14(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) )
V18(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) ,
COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set ) )
complex-valued )
Element of
K19(
K20(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) ,
COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set ) ) : ( ( ) (
complex-valued )
set ) ) : ( ( ) ( )
set ) )
/" seq : ( (
Function-like V18(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) ,
COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set ) ) ) (
Relation-like NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) )
-defined COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set )
-valued Function-like non
zero V14(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) )
V18(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) ,
COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set ) )
complex-valued )
Complex_Sequence) : ( (
Function-like ) (
Relation-like NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) )
-defined COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set )
-valued Function-like non
zero V14(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) )
V18(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) ,
COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set ) )
complex-valued )
Element of
K19(
K20(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) ,
COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set ) ) : ( ( ) (
complex-valued )
set ) ) : ( ( ) ( )
set ) ) &
seq1 : ( (
Function-like V18(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) ,
COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set ) ) ) (
Relation-like NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) )
-defined COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set )
-valued Function-like non
zero V14(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) )
V18(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) ,
COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set ) )
complex-valued )
Complex_Sequence)
/" (- seq : ( ( Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) -valued Function-like non zero V14( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) complex-valued ) Complex_Sequence) ) : ( (
Function-like ) (
Relation-like NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) )
-defined COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set )
-valued Function-like non
zero V14(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) )
V18(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) ,
COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set ) )
complex-valued )
Element of
K19(
K20(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) ,
COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set ) ) : ( ( ) (
complex-valued )
set ) ) : ( ( ) ( )
set ) ) : ( (
Function-like ) (
Relation-like NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) )
-defined COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set )
-valued Function-like non
zero V14(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) )
V18(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) ,
COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set ) )
complex-valued )
Element of
K19(
K20(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) ,
COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set ) ) : ( ( ) (
complex-valued )
set ) ) : ( ( ) ( )
set ) )
= - (seq1 : ( ( Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) -valued Function-like non zero V14( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) complex-valued ) Complex_Sequence) /" seq : ( ( Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) -valued Function-like non zero V14( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) complex-valued ) Complex_Sequence) ) : ( (
Function-like ) (
Relation-like NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) )
-defined COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set )
-valued Function-like non
zero V14(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) )
V18(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) ,
COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set ) )
complex-valued )
Element of
K19(
K20(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) ,
COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set ) ) : ( ( ) (
complex-valued )
set ) ) : ( ( ) ( )
set ) ) : ( (
Function-like ) (
Relation-like NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) )
-defined COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set )
-valued Function-like non
zero V14(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) )
V18(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) ,
COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set ) )
complex-valued )
Element of
K19(
K20(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) ,
COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set ) ) : ( ( ) (
complex-valued )
set ) ) : ( ( ) ( )
set ) ) ) ;
theorem
for
seq1,
seq,
seq19 being ( (
Function-like V18(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) ,
COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set ) ) ) (
Relation-like NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) )
-defined COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set )
-valued Function-like non
zero V14(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) )
V18(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) ,
COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set ) )
complex-valued )
Complex_Sequence) holds
(
(seq1 : ( ( Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) -valued Function-like non zero V14( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) complex-valued ) Complex_Sequence) /" seq : ( ( Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) -valued Function-like non zero V14( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) complex-valued ) Complex_Sequence) ) : ( (
Function-like ) (
Relation-like NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) )
-defined COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set )
-valued Function-like non
zero V14(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) )
V18(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) ,
COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set ) )
complex-valued )
Element of
K19(
K20(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) ,
COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set ) ) : ( ( ) (
complex-valued )
set ) ) : ( ( ) ( )
set ) )
+ (seq19 : ( ( Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) -valued Function-like non zero V14( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) complex-valued ) Complex_Sequence) /" seq : ( ( Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) -valued Function-like non zero V14( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) complex-valued ) Complex_Sequence) ) : ( (
Function-like ) (
Relation-like NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) )
-defined COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set )
-valued Function-like non
zero V14(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) )
V18(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) ,
COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set ) )
complex-valued )
Element of
K19(
K20(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) ,
COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set ) ) : ( ( ) (
complex-valued )
set ) ) : ( ( ) ( )
set ) ) : ( (
Function-like ) (
Relation-like NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) )
-defined COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set )
-valued Function-like non
zero V14(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) )
V18(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) ,
COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set ) )
complex-valued )
Element of
K19(
K20(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) ,
COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set ) ) : ( ( ) (
complex-valued )
set ) ) : ( ( ) ( )
set ) )
= (seq1 : ( ( Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) -valued Function-like non zero V14( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) complex-valued ) Complex_Sequence) + seq19 : ( ( Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) -valued Function-like non zero V14( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) complex-valued ) Complex_Sequence) ) : ( (
Function-like ) (
Relation-like NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) )
-defined COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set )
-valued Function-like non
zero V14(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) )
V18(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) ,
COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set ) )
complex-valued )
Element of
K19(
K20(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) ,
COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set ) ) : ( ( ) (
complex-valued )
set ) ) : ( ( ) ( )
set ) )
/" seq : ( (
Function-like V18(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) ,
COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set ) ) ) (
Relation-like NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) )
-defined COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set )
-valued Function-like non
zero V14(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) )
V18(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) ,
COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set ) )
complex-valued )
Complex_Sequence) : ( (
Function-like ) (
Relation-like NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) )
-defined COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set )
-valued Function-like non
zero V14(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) )
V18(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) ,
COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set ) )
complex-valued )
Element of
K19(
K20(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) ,
COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set ) ) : ( ( ) (
complex-valued )
set ) ) : ( ( ) ( )
set ) ) &
(seq1 : ( ( Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) -valued Function-like non zero V14( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) complex-valued ) Complex_Sequence) /" seq : ( ( Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) -valued Function-like non zero V14( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) complex-valued ) Complex_Sequence) ) : ( (
Function-like ) (
Relation-like NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) )
-defined COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set )
-valued Function-like non
zero V14(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) )
V18(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) ,
COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set ) )
complex-valued )
Element of
K19(
K20(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) ,
COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set ) ) : ( ( ) (
complex-valued )
set ) ) : ( ( ) ( )
set ) )
- (seq19 : ( ( Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) -valued Function-like non zero V14( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) complex-valued ) Complex_Sequence) /" seq : ( ( Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) -valued Function-like non zero V14( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) complex-valued ) Complex_Sequence) ) : ( (
Function-like ) (
Relation-like NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) )
-defined COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set )
-valued Function-like non
zero V14(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) )
V18(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) ,
COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set ) )
complex-valued )
Element of
K19(
K20(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) ,
COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set ) ) : ( ( ) (
complex-valued )
set ) ) : ( ( ) ( )
set ) ) : ( (
Function-like ) (
Relation-like NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) )
-defined COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set )
-valued Function-like non
zero V14(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) )
V18(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) ,
COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set ) )
complex-valued )
Element of
K19(
K20(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) ,
COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set ) ) : ( ( ) (
complex-valued )
set ) ) : ( ( ) ( )
set ) )
= (seq1 : ( ( Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) -valued Function-like non zero V14( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) complex-valued ) Complex_Sequence) - seq19 : ( ( Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) -valued Function-like non zero V14( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) complex-valued ) Complex_Sequence) ) : ( (
Function-like ) (
Relation-like NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) )
-defined COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set )
-valued Function-like non
zero V14(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) )
V18(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) ,
COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set ) )
complex-valued )
Element of
K19(
K20(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) ,
COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set ) ) : ( ( ) (
complex-valued )
set ) ) : ( ( ) ( )
set ) )
/" seq : ( (
Function-like V18(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) ,
COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set ) ) ) (
Relation-like NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) )
-defined COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set )
-valued Function-like non
zero V14(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) )
V18(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) ,
COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set ) )
complex-valued )
Complex_Sequence) : ( (
Function-like ) (
Relation-like NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) )
-defined COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set )
-valued Function-like non
zero V14(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) )
V18(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) ,
COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set ) )
complex-valued )
Element of
K19(
K20(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) ,
COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set ) ) : ( ( ) (
complex-valued )
set ) ) : ( ( ) ( )
set ) ) ) ;
theorem
for
seq,
seq9,
seq1,
seq19 being ( (
Function-like V18(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) ,
COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set ) ) ) (
Relation-like NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) )
-defined COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set )
-valued Function-like non
zero V14(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) )
V18(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) ,
COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set ) )
complex-valued )
Complex_Sequence) st
seq : ( (
Function-like V18(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) ,
COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set ) ) ) (
Relation-like NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) )
-defined COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set )
-valued Function-like non
zero V14(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) )
V18(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) ,
COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set ) )
complex-valued )
Complex_Sequence) is
non-zero &
seq9 : ( (
Function-like V18(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) ,
COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set ) ) ) (
Relation-like NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) )
-defined COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set )
-valued Function-like non
zero V14(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) )
V18(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) ,
COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set ) )
complex-valued )
Complex_Sequence) is
non-zero holds
(
(seq1 : ( ( Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) -valued Function-like non zero V14( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) complex-valued ) Complex_Sequence) /" seq : ( ( Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) -valued Function-like non zero V14( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) complex-valued ) Complex_Sequence) ) : ( (
Function-like ) (
Relation-like NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) )
-defined COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set )
-valued Function-like non
zero V14(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) )
V18(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) ,
COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set ) )
complex-valued )
Element of
K19(
K20(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) ,
COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set ) ) : ( ( ) (
complex-valued )
set ) ) : ( ( ) ( )
set ) )
+ (seq19 : ( ( Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) -valued Function-like non zero V14( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) complex-valued ) Complex_Sequence) /" seq9 : ( ( Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) -valued Function-like non zero V14( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) complex-valued ) Complex_Sequence) ) : ( (
Function-like ) (
Relation-like NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) )
-defined COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set )
-valued Function-like non
zero V14(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) )
V18(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) ,
COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set ) )
complex-valued )
Element of
K19(
K20(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) ,
COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set ) ) : ( ( ) (
complex-valued )
set ) ) : ( ( ) ( )
set ) ) : ( (
Function-like ) (
Relation-like NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) )
-defined COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set )
-valued Function-like non
zero V14(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) )
V18(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) ,
COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set ) )
complex-valued )
Element of
K19(
K20(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) ,
COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set ) ) : ( ( ) (
complex-valued )
set ) ) : ( ( ) ( )
set ) )
= ((seq1 : ( ( Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) -valued Function-like non zero V14( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) complex-valued ) Complex_Sequence) (#) seq9 : ( ( Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) -valued Function-like non zero V14( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) complex-valued ) Complex_Sequence) ) : ( ( Function-like ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) -valued Function-like non zero V14( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) complex-valued ) Element of K19(K20(NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ,COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) : ( ( ) ( complex-valued ) set ) ) : ( ( ) ( ) set ) ) + (seq19 : ( ( Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) -valued Function-like non zero V14( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) complex-valued ) Complex_Sequence) (#) seq : ( ( Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) -valued Function-like non zero V14( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) complex-valued ) Complex_Sequence) ) : ( ( Function-like ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) -valued Function-like non zero V14( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) complex-valued ) Element of K19(K20(NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ,COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) : ( ( ) ( complex-valued ) set ) ) : ( ( ) ( ) set ) ) ) : ( (
Function-like ) (
Relation-like NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) )
-defined COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set )
-valued Function-like non
zero V14(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) )
V18(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) ,
COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set ) )
complex-valued )
Element of
K19(
K20(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) ,
COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set ) ) : ( ( ) (
complex-valued )
set ) ) : ( ( ) ( )
set ) )
/" (seq : ( ( Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) -valued Function-like non zero V14( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) complex-valued ) Complex_Sequence) (#) seq9 : ( ( Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) -valued Function-like non zero V14( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) complex-valued ) Complex_Sequence) ) : ( (
Function-like ) (
Relation-like NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) )
-defined COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set )
-valued Function-like non
zero V14(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) )
V18(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) ,
COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set ) )
complex-valued )
Element of
K19(
K20(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) ,
COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set ) ) : ( ( ) (
complex-valued )
set ) ) : ( ( ) ( )
set ) ) : ( (
Function-like ) (
Relation-like NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) )
-defined COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set )
-valued Function-like non
zero V14(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) )
V18(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) ,
COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set ) )
complex-valued )
Element of
K19(
K20(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) ,
COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set ) ) : ( ( ) (
complex-valued )
set ) ) : ( ( ) ( )
set ) ) &
(seq1 : ( ( Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) -valued Function-like non zero V14( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) complex-valued ) Complex_Sequence) /" seq : ( ( Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) -valued Function-like non zero V14( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) complex-valued ) Complex_Sequence) ) : ( (
Function-like ) (
Relation-like NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) )
-defined COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set )
-valued Function-like non
zero V14(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) )
V18(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) ,
COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set ) )
complex-valued )
Element of
K19(
K20(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) ,
COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set ) ) : ( ( ) (
complex-valued )
set ) ) : ( ( ) ( )
set ) )
- (seq19 : ( ( Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) -valued Function-like non zero V14( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) complex-valued ) Complex_Sequence) /" seq9 : ( ( Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) -valued Function-like non zero V14( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) complex-valued ) Complex_Sequence) ) : ( (
Function-like ) (
Relation-like NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) )
-defined COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set )
-valued Function-like non
zero V14(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) )
V18(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) ,
COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set ) )
complex-valued )
Element of
K19(
K20(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) ,
COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set ) ) : ( ( ) (
complex-valued )
set ) ) : ( ( ) ( )
set ) ) : ( (
Function-like ) (
Relation-like NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) )
-defined COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set )
-valued Function-like non
zero V14(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) )
V18(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) ,
COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set ) )
complex-valued )
Element of
K19(
K20(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) ,
COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set ) ) : ( ( ) (
complex-valued )
set ) ) : ( ( ) ( )
set ) )
= ((seq1 : ( ( Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) -valued Function-like non zero V14( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) complex-valued ) Complex_Sequence) (#) seq9 : ( ( Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) -valued Function-like non zero V14( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) complex-valued ) Complex_Sequence) ) : ( ( Function-like ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) -valued Function-like non zero V14( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) complex-valued ) Element of K19(K20(NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ,COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) : ( ( ) ( complex-valued ) set ) ) : ( ( ) ( ) set ) ) - (seq19 : ( ( Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) -valued Function-like non zero V14( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) complex-valued ) Complex_Sequence) (#) seq : ( ( Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) -valued Function-like non zero V14( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) complex-valued ) Complex_Sequence) ) : ( ( Function-like ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) -valued Function-like non zero V14( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) complex-valued ) Element of K19(K20(NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ,COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) : ( ( ) ( complex-valued ) set ) ) : ( ( ) ( ) set ) ) ) : ( (
Function-like ) (
Relation-like NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) )
-defined COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set )
-valued Function-like non
zero V14(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) )
V18(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) ,
COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set ) )
complex-valued )
Element of
K19(
K20(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) ,
COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set ) ) : ( ( ) (
complex-valued )
set ) ) : ( ( ) ( )
set ) )
/" (seq : ( ( Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) -valued Function-like non zero V14( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) complex-valued ) Complex_Sequence) (#) seq9 : ( ( Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) -valued Function-like non zero V14( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) complex-valued ) Complex_Sequence) ) : ( (
Function-like ) (
Relation-like NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) )
-defined COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set )
-valued Function-like non
zero V14(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) )
V18(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) ,
COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set ) )
complex-valued )
Element of
K19(
K20(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) ,
COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set ) ) : ( ( ) (
complex-valued )
set ) ) : ( ( ) ( )
set ) ) : ( (
Function-like ) (
Relation-like NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) )
-defined COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set )
-valued Function-like non
zero V14(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) )
V18(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) ,
COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set ) )
complex-valued )
Element of
K19(
K20(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) ,
COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set ) ) : ( ( ) (
complex-valued )
set ) ) : ( ( ) ( )
set ) ) ) ;
theorem
for
seq19,
seq,
seq9,
seq1 being ( (
Function-like V18(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) ,
COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set ) ) ) (
Relation-like NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) )
-defined COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set )
-valued Function-like non
zero V14(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) )
V18(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) ,
COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set ) )
complex-valued )
Complex_Sequence) holds
(seq19 : ( ( Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) -valued Function-like non zero V14( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) complex-valued ) Complex_Sequence) /" seq : ( ( Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) -valued Function-like non zero V14( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) complex-valued ) Complex_Sequence) ) : ( (
Function-like ) (
Relation-like NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) )
-defined COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set )
-valued Function-like non
zero V14(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) )
V18(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) ,
COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set ) )
complex-valued )
Element of
K19(
K20(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) ,
COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set ) ) : ( ( ) (
complex-valued )
set ) ) : ( ( ) ( )
set ) )
/" (seq9 : ( ( Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) -valued Function-like non zero V14( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) complex-valued ) Complex_Sequence) /" seq1 : ( ( Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) -valued Function-like non zero V14( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) complex-valued ) Complex_Sequence) ) : ( (
Function-like ) (
Relation-like NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) )
-defined COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set )
-valued Function-like non
zero V14(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) )
V18(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) ,
COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set ) )
complex-valued )
Element of
K19(
K20(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) ,
COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set ) ) : ( ( ) (
complex-valued )
set ) ) : ( ( ) ( )
set ) ) : ( (
Function-like ) (
Relation-like NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) )
-defined COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set )
-valued Function-like non
zero V14(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) )
V18(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) ,
COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set ) )
complex-valued )
Element of
K19(
K20(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) ,
COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set ) ) : ( ( ) (
complex-valued )
set ) ) : ( ( ) ( )
set ) )
= (seq19 : ( ( Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) -valued Function-like non zero V14( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) complex-valued ) Complex_Sequence) (#) seq1 : ( ( Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) -valued Function-like non zero V14( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) complex-valued ) Complex_Sequence) ) : ( (
Function-like ) (
Relation-like NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) )
-defined COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set )
-valued Function-like non
zero V14(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) )
V18(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) ,
COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set ) )
complex-valued )
Element of
K19(
K20(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) ,
COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set ) ) : ( ( ) (
complex-valued )
set ) ) : ( ( ) ( )
set ) )
/" (seq : ( ( Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) -valued Function-like non zero V14( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) complex-valued ) Complex_Sequence) (#) seq9 : ( ( Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) -valued Function-like non zero V14( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) complex-valued ) Complex_Sequence) ) : ( (
Function-like ) (
Relation-like NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) )
-defined COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set )
-valued Function-like non
zero V14(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) )
V18(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) ,
COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set ) )
complex-valued )
Element of
K19(
K20(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) ,
COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set ) ) : ( ( ) (
complex-valued )
set ) ) : ( ( ) ( )
set ) ) : ( (
Function-like ) (
Relation-like NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) )
-defined COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set )
-valued Function-like non
zero V14(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) )
V18(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) ,
COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set ) )
complex-valued )
Element of
K19(
K20(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) ,
COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set ) ) : ( ( ) (
complex-valued )
set ) ) : ( ( ) ( )
set ) ) ;
theorem
for
seq,
seq9 being ( (
Function-like V18(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) ,
COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set ) ) ) (
Relation-like NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) )
-defined COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set )
-valued Function-like non
zero V14(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) )
V18(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) ,
COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set ) )
complex-valued )
Complex_Sequence) holds
|.(seq : ( ( Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) -valued Function-like non zero V14( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) complex-valued ) Complex_Sequence) (#) seq9 : ( ( Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) -valued Function-like non zero V14( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) complex-valued ) Complex_Sequence) ) : ( ( Function-like ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) -valued Function-like non zero V14( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) complex-valued ) Element of K19(K20(NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ,COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) : ( ( ) ( complex-valued ) set ) ) : ( ( ) ( ) set ) ) .| : ( (
Function-like ) (
Relation-like NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) )
-defined REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set )
-valued Function-like non
zero V14(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) )
V18(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) )
complex-valued ext-real-valued real-valued )
Element of
K19(
K20(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) (
complex-valued ext-real-valued real-valued )
set ) ) : ( ( ) ( )
set ) )
= |.seq : ( ( Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) -valued Function-like non zero V14( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) complex-valued ) Complex_Sequence) .| : ( (
Function-like ) (
Relation-like NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) )
-defined REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set )
-valued Function-like non
zero V14(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) )
V18(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) )
complex-valued ext-real-valued real-valued )
Element of
K19(
K20(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) (
complex-valued ext-real-valued real-valued )
set ) ) : ( ( ) ( )
set ) )
(#) |.seq9 : ( ( Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) -valued Function-like non zero V14( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) complex-valued ) Complex_Sequence) .| : ( (
Function-like ) (
Relation-like NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) )
-defined REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set )
-valued Function-like non
zero V14(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) )
V18(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) )
complex-valued ext-real-valued real-valued )
Element of
K19(
K20(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) (
complex-valued ext-real-valued real-valued )
set ) ) : ( ( ) ( )
set ) ) : ( (
Function-like ) (
Relation-like NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) )
-defined REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set )
-valued Function-like non
zero V14(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) )
V18(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) )
complex-valued ext-real-valued real-valued )
Element of
K19(
K20(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) (
complex-valued ext-real-valued real-valued )
set ) ) : ( ( ) ( )
set ) ) ;
theorem
for
seq being ( (
Function-like V18(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) ,
COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set ) ) ) (
Relation-like NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) )
-defined COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set )
-valued Function-like non
zero V14(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) )
V18(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) ,
COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set ) )
complex-valued )
Complex_Sequence) holds
|.seq : ( ( Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) -valued Function-like non zero V14( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) complex-valued ) Complex_Sequence) .| : ( (
Function-like ) (
Relation-like NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) )
-defined REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set )
-valued Function-like non
zero V14(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) )
V18(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) )
complex-valued ext-real-valued real-valued )
Element of
K19(
K20(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) (
complex-valued ext-real-valued real-valued )
set ) ) : ( ( ) ( )
set ) )
" : ( (
Function-like ) (
Relation-like NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) )
-defined REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set )
-valued Function-like non
zero V14(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) )
V18(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) )
complex-valued ext-real-valued real-valued )
Element of
K19(
K20(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) (
complex-valued ext-real-valued real-valued )
set ) ) : ( ( ) ( )
set ) )
= |.(seq : ( ( Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) -valued Function-like non zero V14( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) complex-valued ) Complex_Sequence) ") : ( ( Function-like ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) -valued Function-like non zero V14( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) complex-valued ) Element of K19(K20(NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ,COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) : ( ( ) ( complex-valued ) set ) ) : ( ( ) ( ) set ) ) .| : ( (
Function-like ) (
Relation-like NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) )
-defined REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set )
-valued Function-like non
zero V14(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) )
V18(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) )
complex-valued ext-real-valued real-valued )
Element of
K19(
K20(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) (
complex-valued ext-real-valued real-valued )
set ) ) : ( ( ) ( )
set ) ) ;
theorem
for
seq9,
seq being ( (
Function-like V18(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) ,
COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set ) ) ) (
Relation-like NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) )
-defined COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set )
-valued Function-like non
zero V14(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) )
V18(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) ,
COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set ) )
complex-valued )
Complex_Sequence) holds
|.(seq9 : ( ( Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) -valued Function-like non zero V14( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) complex-valued ) Complex_Sequence) /" seq : ( ( Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) -valued Function-like non zero V14( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) complex-valued ) Complex_Sequence) ) : ( ( Function-like ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) -valued Function-like non zero V14( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) complex-valued ) Element of K19(K20(NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ,COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) : ( ( ) ( complex-valued ) set ) ) : ( ( ) ( ) set ) ) .| : ( (
Function-like ) (
Relation-like NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) )
-defined REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set )
-valued Function-like non
zero V14(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) )
V18(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) )
complex-valued ext-real-valued real-valued )
Element of
K19(
K20(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) (
complex-valued ext-real-valued real-valued )
set ) ) : ( ( ) ( )
set ) )
= |.seq9 : ( ( Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) -valued Function-like non zero V14( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) complex-valued ) Complex_Sequence) .| : ( (
Function-like ) (
Relation-like NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) )
-defined REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set )
-valued Function-like non
zero V14(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) )
V18(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) )
complex-valued ext-real-valued real-valued )
Element of
K19(
K20(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) (
complex-valued ext-real-valued real-valued )
set ) ) : ( ( ) ( )
set ) )
/" |.seq : ( ( Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) -valued Function-like non zero V14( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) complex-valued ) Complex_Sequence) .| : ( (
Function-like ) (
Relation-like NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) )
-defined REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set )
-valued Function-like non
zero V14(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) )
V18(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) )
complex-valued ext-real-valued real-valued )
Element of
K19(
K20(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) (
complex-valued ext-real-valued real-valued )
set ) ) : ( ( ) ( )
set ) ) : ( (
Function-like ) (
Relation-like NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) )
-defined REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set )
-valued Function-like non
zero V14(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) )
V18(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) )
complex-valued ext-real-valued real-valued )
Element of
K19(
K20(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) (
complex-valued ext-real-valued real-valued )
set ) ) : ( ( ) ( )
set ) ) ;
theorem
for
r being ( ( ) (
complex )
Element of
COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set ) )
for
seq being ( (
Function-like V18(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) ,
COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set ) ) ) (
Relation-like NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) )
-defined COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set )
-valued Function-like non
zero V14(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) )
V18(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) ,
COMPLEX : ( ( ) ( non
zero V47()
V53()
V54() )
set ) )
complex-valued )
Complex_Sequence) holds
|.(r : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) (#) seq : ( ( Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) -valued Function-like non zero V14( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) complex-valued ) Complex_Sequence) ) : ( ( Function-like ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) -valued Function-like non zero V14( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) complex-valued ) Element of K19(K20(NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ,COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) : ( ( ) ( complex-valued ) set ) ) : ( ( ) ( ) set ) ) .| : ( (
Function-like ) (
Relation-like NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) )
-defined REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set )
-valued Function-like non
zero V14(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) )
V18(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) )
complex-valued ext-real-valued real-valued )
Element of
K19(
K20(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) (
complex-valued ext-real-valued real-valued )
set ) ) : ( ( ) ( )
set ) )
= |.r : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) .| : ( ( ) (
complex V31()
V32() )
Element of
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) )
(#) |.seq : ( ( Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) -valued Function-like non zero V14( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) complex-valued ) Complex_Sequence) .| : ( (
Function-like ) (
Relation-like NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) )
-defined REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set )
-valued Function-like non
zero V14(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) )
V18(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) )
complex-valued ext-real-valued real-valued )
Element of
K19(
K20(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) (
complex-valued ext-real-valued real-valued )
set ) ) : ( ( ) ( )
set ) ) : ( (
Function-like ) (
Relation-like NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) )
-defined REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set )
-valued Function-like non
zero V14(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) )
V18(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) )
complex-valued ext-real-valued real-valued )
Element of
K19(
K20(
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V47()
V48()
V49()
V50()
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
zero V47()
V48()
V49()
V53()
V54() )
set ) ) : ( ( ) (
complex-valued ext-real-valued real-valued )
set ) ) : ( ( ) ( )
set ) ) ;