begin
theorem
for
X being ( ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like )
ComplexNormSpace)
for
seq being ( (
Function-like V30(
NAT : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal V52()
V53()
V54()
V55()
V56()
V57()
V58() )
Element of
K32(
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4() )
set ) ) , the
carrier of
b1 : ( ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like )
ComplexNormSpace) : ( ( ) (
V4() )
set ) ) ) (
V4()
Relation-like NAT : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal V52()
V53()
V54()
V55()
V56()
V57()
V58() )
Element of
K32(
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4() )
set ) )
-defined the
carrier of
b1 : ( ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like )
ComplexNormSpace) : ( ( ) (
V4() )
set )
-valued Function-like V29(
NAT : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal V52()
V53()
V54()
V55()
V56()
V57()
V58() )
Element of
K32(
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4() )
set ) ) )
V30(
NAT : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal V52()
V53()
V54()
V55()
V56()
V57()
V58() )
Element of
K32(
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4() )
set ) ) , the
carrier of
b1 : ( ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like )
ComplexNormSpace) : ( ( ) (
V4() )
set ) ) )
sequence of ( ( ) (
V4() )
set ) ) st ( for
n being ( ( ) (
epsilon-transitive epsilon-connected ordinal natural V24()
real ext-real V45()
V46()
V52()
V53()
V54()
V55()
V56()
V57() )
Element of
NAT : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal V52()
V53()
V54()
V55()
V56()
V57()
V58() )
Element of
K32(
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4() )
set ) ) ) holds
seq : ( (
Function-like V30(
NAT : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal V52()
V53()
V54()
V55()
V56()
V57()
V58() )
Element of
K32(
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4() )
set ) ) , the
carrier of
b1 : ( ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like )
ComplexNormSpace) : ( ( ) (
V4() )
set ) ) ) (
V4()
Relation-like NAT : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal V52()
V53()
V54()
V55()
V56()
V57()
V58() )
Element of
K32(
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4() )
set ) )
-defined the
carrier of
b1 : ( ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like )
ComplexNormSpace) : ( ( ) (
V4() )
set )
-valued Function-like V29(
NAT : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal V52()
V53()
V54()
V55()
V56()
V57()
V58() )
Element of
K32(
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4() )
set ) ) )
V30(
NAT : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal V52()
V53()
V54()
V55()
V56()
V57()
V58() )
Element of
K32(
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4() )
set ) ) , the
carrier of
b1 : ( ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like )
ComplexNormSpace) : ( ( ) (
V4() )
set ) ) )
sequence of ( ( ) (
V4() )
set ) )
. n : ( ( ) (
epsilon-transitive epsilon-connected ordinal natural V24()
real ext-real V45()
V46()
V52()
V53()
V54()
V55()
V56()
V57() )
Element of
NAT : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal V52()
V53()
V54()
V55()
V56()
V57()
V58() )
Element of
K32(
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4() )
set ) ) ) : ( ( ) (
right_complementable )
Element of the
carrier of
b1 : ( ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like )
ComplexNormSpace) : ( ( ) (
V4() )
set ) )
= 0. X : ( ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like )
ComplexNormSpace) : ( ( ) (
V79(
b1 : ( ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like )
ComplexNormSpace) )
right_complementable )
Element of the
carrier of
b1 : ( ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like )
ComplexNormSpace) : ( ( ) (
V4() )
set ) ) ) holds
for
m being ( ( ) (
epsilon-transitive epsilon-connected ordinal natural V24()
real ext-real V45()
V46()
V52()
V53()
V54()
V55()
V56()
V57() )
Element of
NAT : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal V52()
V53()
V54()
V55()
V56()
V57()
V58() )
Element of
K32(
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4() )
set ) ) ) holds
(Partial_Sums ||.seq : ( ( Function-like V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed V133() V134() vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed V133() V134() vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( V4() ) set ) ) ) ( V4() Relation-like NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) -defined the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed V133() V134() vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed V133() V134() vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( V4() ) set ) -valued Function-like V29( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) ) V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed V133() V134() vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed V133() V134() vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( V4() ) set ) ) ) sequence of ( ( ) ( V4() ) set ) ) .|| : ( ( Function-like V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) ) ( V4() Relation-like NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) -defined REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) -valued Function-like V29( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) ) V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) complex-valued ext-real-valued real-valued ) Element of K32(K33(NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) ,REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( V4() ) set ) ) ) : ( (
Function-like V30(
NAT : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal V52()
V53()
V54()
V55()
V56()
V57()
V58() )
Element of
K32(
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4() )
set ) ) ,
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) ) (
V4()
Relation-like NAT : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal V52()
V53()
V54()
V55()
V56()
V57()
V58() )
Element of
K32(
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4() )
set ) )
-defined REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set )
-valued Function-like V29(
NAT : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal V52()
V53()
V54()
V55()
V56()
V57()
V58() )
Element of
K32(
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4() )
set ) ) )
V30(
NAT : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal V52()
V53()
V54()
V55()
V56()
V57()
V58() )
Element of
K32(
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4() )
set ) ) ,
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) )
complex-valued ext-real-valued real-valued )
Element of
K32(
K33(
NAT : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal V52()
V53()
V54()
V55()
V56()
V57()
V58() )
Element of
K32(
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4() )
set ) ) ,
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4()
complex-valued ext-real-valued real-valued )
set ) ) : ( ( ) (
V4() )
set ) )
. m : ( ( ) (
epsilon-transitive epsilon-connected ordinal natural V24()
real ext-real V45()
V46()
V52()
V53()
V54()
V55()
V56()
V57() )
Element of
NAT : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal V52()
V53()
V54()
V55()
V56()
V57()
V58() )
Element of
K32(
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4() )
set ) ) ) : ( ( ) (
V24()
real ext-real )
Element of
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) )
= 0 : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural V24()
real ext-real non
positive non
negative V45()
V46()
V52()
V53()
V54()
V55()
V56()
V57()
V58() )
Element of
NAT : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal V52()
V53()
V54()
V55()
V56()
V57()
V58() )
Element of
K32(
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4() )
set ) ) ) ;
theorem
for
X being ( ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like )
ComplexNormSpace)
for
seq,
seq1 being ( (
Function-like V30(
NAT : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal V52()
V53()
V54()
V55()
V56()
V57()
V58() )
Element of
K32(
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4() )
set ) ) , the
carrier of
b1 : ( ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like )
ComplexNormSpace) : ( ( ) (
V4() )
set ) ) ) (
V4()
Relation-like NAT : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal V52()
V53()
V54()
V55()
V56()
V57()
V58() )
Element of
K32(
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4() )
set ) )
-defined the
carrier of
b1 : ( ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like )
ComplexNormSpace) : ( ( ) (
V4() )
set )
-valued Function-like V29(
NAT : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal V52()
V53()
V54()
V55()
V56()
V57()
V58() )
Element of
K32(
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4() )
set ) ) )
V30(
NAT : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal V52()
V53()
V54()
V55()
V56()
V57()
V58() )
Element of
K32(
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4() )
set ) ) , the
carrier of
b1 : ( ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like )
ComplexNormSpace) : ( ( ) (
V4() )
set ) ) )
sequence of ( ( ) (
V4() )
set ) ) st
seq1 : ( (
Function-like V30(
NAT : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal V52()
V53()
V54()
V55()
V56()
V57()
V58() )
Element of
K32(
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4() )
set ) ) , the
carrier of
b1 : ( ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like )
ComplexNormSpace) : ( ( ) (
V4() )
set ) ) ) (
V4()
Relation-like NAT : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal V52()
V53()
V54()
V55()
V56()
V57()
V58() )
Element of
K32(
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4() )
set ) )
-defined the
carrier of
b1 : ( ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like )
ComplexNormSpace) : ( ( ) (
V4() )
set )
-valued Function-like V29(
NAT : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal V52()
V53()
V54()
V55()
V56()
V57()
V58() )
Element of
K32(
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4() )
set ) ) )
V30(
NAT : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal V52()
V53()
V54()
V55()
V56()
V57()
V58() )
Element of
K32(
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4() )
set ) ) , the
carrier of
b1 : ( ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like )
ComplexNormSpace) : ( ( ) (
V4() )
set ) ) )
sequence of ( ( ) (
V4() )
set ) ) is ( ( ) (
V4()
Relation-like NAT : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal V52()
V53()
V54()
V55()
V56()
V57()
V58() )
Element of
K32(
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4() )
set ) )
-defined the
carrier of
b1 : ( ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like )
ComplexNormSpace) : ( ( ) (
V4() )
set )
-valued Function-like V29(
NAT : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal V52()
V53()
V54()
V55()
V56()
V57()
V58() )
Element of
K32(
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4() )
set ) ) )
V30(
NAT : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal V52()
V53()
V54()
V55()
V56()
V57()
V58() )
Element of
K32(
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4() )
set ) ) , the
carrier of
b1 : ( ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like )
ComplexNormSpace) : ( ( ) (
V4() )
set ) ) )
subsequence of
seq : ( (
Function-like V30(
NAT : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal V52()
V53()
V54()
V55()
V56()
V57()
V58() )
Element of
K32(
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4() )
set ) ) , the
carrier of
b1 : ( ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like )
ComplexNormSpace) : ( ( ) (
V4() )
set ) ) ) (
V4()
Relation-like NAT : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal V52()
V53()
V54()
V55()
V56()
V57()
V58() )
Element of
K32(
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4() )
set ) )
-defined the
carrier of
b1 : ( ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like )
ComplexNormSpace) : ( ( ) (
V4() )
set )
-valued Function-like V29(
NAT : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal V52()
V53()
V54()
V55()
V56()
V57()
V58() )
Element of
K32(
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4() )
set ) ) )
V30(
NAT : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal V52()
V53()
V54()
V55()
V56()
V57()
V58() )
Element of
K32(
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4() )
set ) ) , the
carrier of
b1 : ( ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like )
ComplexNormSpace) : ( ( ) (
V4() )
set ) ) )
sequence of ( ( ) (
V4() )
set ) ) ) &
seq : ( (
Function-like V30(
NAT : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal V52()
V53()
V54()
V55()
V56()
V57()
V58() )
Element of
K32(
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4() )
set ) ) , the
carrier of
b1 : ( ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like )
ComplexNormSpace) : ( ( ) (
V4() )
set ) ) ) (
V4()
Relation-like NAT : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal V52()
V53()
V54()
V55()
V56()
V57()
V58() )
Element of
K32(
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4() )
set ) )
-defined the
carrier of
b1 : ( ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like )
ComplexNormSpace) : ( ( ) (
V4() )
set )
-valued Function-like V29(
NAT : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal V52()
V53()
V54()
V55()
V56()
V57()
V58() )
Element of
K32(
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4() )
set ) ) )
V30(
NAT : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal V52()
V53()
V54()
V55()
V56()
V57()
V58() )
Element of
K32(
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4() )
set ) ) , the
carrier of
b1 : ( ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like )
ComplexNormSpace) : ( ( ) (
V4() )
set ) ) )
sequence of ( ( ) (
V4() )
set ) ) is
convergent holds
seq1 : ( (
Function-like V30(
NAT : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal V52()
V53()
V54()
V55()
V56()
V57()
V58() )
Element of
K32(
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4() )
set ) ) , the
carrier of
b1 : ( ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like )
ComplexNormSpace) : ( ( ) (
V4() )
set ) ) ) (
V4()
Relation-like NAT : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal V52()
V53()
V54()
V55()
V56()
V57()
V58() )
Element of
K32(
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4() )
set ) )
-defined the
carrier of
b1 : ( ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like )
ComplexNormSpace) : ( ( ) (
V4() )
set )
-valued Function-like V29(
NAT : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal V52()
V53()
V54()
V55()
V56()
V57()
V58() )
Element of
K32(
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4() )
set ) ) )
V30(
NAT : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal V52()
V53()
V54()
V55()
V56()
V57()
V58() )
Element of
K32(
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4() )
set ) ) , the
carrier of
b1 : ( ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like )
ComplexNormSpace) : ( ( ) (
V4() )
set ) ) )
sequence of ( ( ) (
V4() )
set ) ) is
convergent ;
theorem
for
X being ( ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like )
ComplexNormSpace)
for
seq,
seq1 being ( (
Function-like V30(
NAT : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal V52()
V53()
V54()
V55()
V56()
V57()
V58() )
Element of
K32(
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4() )
set ) ) , the
carrier of
b1 : ( ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like )
ComplexNormSpace) : ( ( ) (
V4() )
set ) ) ) (
V4()
Relation-like NAT : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal V52()
V53()
V54()
V55()
V56()
V57()
V58() )
Element of
K32(
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4() )
set ) )
-defined the
carrier of
b1 : ( ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like )
ComplexNormSpace) : ( ( ) (
V4() )
set )
-valued Function-like V29(
NAT : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal V52()
V53()
V54()
V55()
V56()
V57()
V58() )
Element of
K32(
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4() )
set ) ) )
V30(
NAT : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal V52()
V53()
V54()
V55()
V56()
V57()
V58() )
Element of
K32(
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4() )
set ) ) , the
carrier of
b1 : ( ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like )
ComplexNormSpace) : ( ( ) (
V4() )
set ) ) )
sequence of ( ( ) (
V4() )
set ) ) st
seq1 : ( (
Function-like V30(
NAT : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal V52()
V53()
V54()
V55()
V56()
V57()
V58() )
Element of
K32(
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4() )
set ) ) , the
carrier of
b1 : ( ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like )
ComplexNormSpace) : ( ( ) (
V4() )
set ) ) ) (
V4()
Relation-like NAT : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal V52()
V53()
V54()
V55()
V56()
V57()
V58() )
Element of
K32(
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4() )
set ) )
-defined the
carrier of
b1 : ( ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like )
ComplexNormSpace) : ( ( ) (
V4() )
set )
-valued Function-like V29(
NAT : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal V52()
V53()
V54()
V55()
V56()
V57()
V58() )
Element of
K32(
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4() )
set ) ) )
V30(
NAT : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal V52()
V53()
V54()
V55()
V56()
V57()
V58() )
Element of
K32(
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4() )
set ) ) , the
carrier of
b1 : ( ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like )
ComplexNormSpace) : ( ( ) (
V4() )
set ) ) )
sequence of ( ( ) (
V4() )
set ) ) is ( ( ) (
V4()
Relation-like NAT : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal V52()
V53()
V54()
V55()
V56()
V57()
V58() )
Element of
K32(
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4() )
set ) )
-defined the
carrier of
b1 : ( ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like )
ComplexNormSpace) : ( ( ) (
V4() )
set )
-valued Function-like V29(
NAT : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal V52()
V53()
V54()
V55()
V56()
V57()
V58() )
Element of
K32(
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4() )
set ) ) )
V30(
NAT : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal V52()
V53()
V54()
V55()
V56()
V57()
V58() )
Element of
K32(
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4() )
set ) ) , the
carrier of
b1 : ( ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like )
ComplexNormSpace) : ( ( ) (
V4() )
set ) ) )
subsequence of
seq : ( (
Function-like V30(
NAT : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal V52()
V53()
V54()
V55()
V56()
V57()
V58() )
Element of
K32(
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4() )
set ) ) , the
carrier of
b1 : ( ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like )
ComplexNormSpace) : ( ( ) (
V4() )
set ) ) ) (
V4()
Relation-like NAT : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal V52()
V53()
V54()
V55()
V56()
V57()
V58() )
Element of
K32(
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4() )
set ) )
-defined the
carrier of
b1 : ( ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like )
ComplexNormSpace) : ( ( ) (
V4() )
set )
-valued Function-like V29(
NAT : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal V52()
V53()
V54()
V55()
V56()
V57()
V58() )
Element of
K32(
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4() )
set ) ) )
V30(
NAT : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal V52()
V53()
V54()
V55()
V56()
V57()
V58() )
Element of
K32(
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4() )
set ) ) , the
carrier of
b1 : ( ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like )
ComplexNormSpace) : ( ( ) (
V4() )
set ) ) )
sequence of ( ( ) (
V4() )
set ) ) ) &
seq : ( (
Function-like V30(
NAT : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal V52()
V53()
V54()
V55()
V56()
V57()
V58() )
Element of
K32(
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4() )
set ) ) , the
carrier of
b1 : ( ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like )
ComplexNormSpace) : ( ( ) (
V4() )
set ) ) ) (
V4()
Relation-like NAT : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal V52()
V53()
V54()
V55()
V56()
V57()
V58() )
Element of
K32(
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4() )
set ) )
-defined the
carrier of
b1 : ( ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like )
ComplexNormSpace) : ( ( ) (
V4() )
set )
-valued Function-like V29(
NAT : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal V52()
V53()
V54()
V55()
V56()
V57()
V58() )
Element of
K32(
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4() )
set ) ) )
V30(
NAT : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal V52()
V53()
V54()
V55()
V56()
V57()
V58() )
Element of
K32(
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4() )
set ) ) , the
carrier of
b1 : ( ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like )
ComplexNormSpace) : ( ( ) (
V4() )
set ) ) )
sequence of ( ( ) (
V4() )
set ) ) is
convergent holds
lim seq1 : ( (
Function-like V30(
NAT : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal V52()
V53()
V54()
V55()
V56()
V57()
V58() )
Element of
K32(
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4() )
set ) ) , the
carrier of
b1 : ( ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like )
ComplexNormSpace) : ( ( ) (
V4() )
set ) ) ) (
V4()
Relation-like NAT : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal V52()
V53()
V54()
V55()
V56()
V57()
V58() )
Element of
K32(
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4() )
set ) )
-defined the
carrier of
b1 : ( ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like )
ComplexNormSpace) : ( ( ) (
V4() )
set )
-valued Function-like V29(
NAT : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal V52()
V53()
V54()
V55()
V56()
V57()
V58() )
Element of
K32(
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4() )
set ) ) )
V30(
NAT : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal V52()
V53()
V54()
V55()
V56()
V57()
V58() )
Element of
K32(
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4() )
set ) ) , the
carrier of
b1 : ( ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like )
ComplexNormSpace) : ( ( ) (
V4() )
set ) ) )
sequence of ( ( ) (
V4() )
set ) ) : ( ( ) (
right_complementable )
Element of the
carrier of
b1 : ( ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like )
ComplexNormSpace) : ( ( ) (
V4() )
set ) )
= lim seq : ( (
Function-like V30(
NAT : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal V52()
V53()
V54()
V55()
V56()
V57()
V58() )
Element of
K32(
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4() )
set ) ) , the
carrier of
b1 : ( ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like )
ComplexNormSpace) : ( ( ) (
V4() )
set ) ) ) (
V4()
Relation-like NAT : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal V52()
V53()
V54()
V55()
V56()
V57()
V58() )
Element of
K32(
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4() )
set ) )
-defined the
carrier of
b1 : ( ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like )
ComplexNormSpace) : ( ( ) (
V4() )
set )
-valued Function-like V29(
NAT : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal V52()
V53()
V54()
V55()
V56()
V57()
V58() )
Element of
K32(
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4() )
set ) ) )
V30(
NAT : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal V52()
V53()
V54()
V55()
V56()
V57()
V58() )
Element of
K32(
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4() )
set ) ) , the
carrier of
b1 : ( ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like )
ComplexNormSpace) : ( ( ) (
V4() )
set ) ) )
sequence of ( ( ) (
V4() )
set ) ) : ( ( ) (
right_complementable )
Element of the
carrier of
b1 : ( ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like )
ComplexNormSpace) : ( ( ) (
V4() )
set ) ) ;
theorem
for
X being ( ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like )
ComplexNormSpace)
for
seq,
seq1 being ( (
Function-like V30(
NAT : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal V52()
V53()
V54()
V55()
V56()
V57()
V58() )
Element of
K32(
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4() )
set ) ) , the
carrier of
b1 : ( ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like )
ComplexNormSpace) : ( ( ) (
V4() )
set ) ) ) (
V4()
Relation-like NAT : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal V52()
V53()
V54()
V55()
V56()
V57()
V58() )
Element of
K32(
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4() )
set ) )
-defined the
carrier of
b1 : ( ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like )
ComplexNormSpace) : ( ( ) (
V4() )
set )
-valued Function-like V29(
NAT : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal V52()
V53()
V54()
V55()
V56()
V57()
V58() )
Element of
K32(
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4() )
set ) ) )
V30(
NAT : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal V52()
V53()
V54()
V55()
V56()
V57()
V58() )
Element of
K32(
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4() )
set ) ) , the
carrier of
b1 : ( ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like )
ComplexNormSpace) : ( ( ) (
V4() )
set ) ) )
sequence of ( ( ) (
V4() )
set ) )
for
k being ( ( ) (
epsilon-transitive epsilon-connected ordinal natural V24()
real ext-real V45()
V46()
V52()
V53()
V54()
V55()
V56()
V57() )
Element of
NAT : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal V52()
V53()
V54()
V55()
V56()
V57()
V58() )
Element of
K32(
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4() )
set ) ) ) st
seq : ( (
Function-like V30(
NAT : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal V52()
V53()
V54()
V55()
V56()
V57()
V58() )
Element of
K32(
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4() )
set ) ) , the
carrier of
b1 : ( ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like )
ComplexNormSpace) : ( ( ) (
V4() )
set ) ) ) (
V4()
Relation-like NAT : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal V52()
V53()
V54()
V55()
V56()
V57()
V58() )
Element of
K32(
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4() )
set ) )
-defined the
carrier of
b1 : ( ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like )
ComplexNormSpace) : ( ( ) (
V4() )
set )
-valued Function-like V29(
NAT : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal V52()
V53()
V54()
V55()
V56()
V57()
V58() )
Element of
K32(
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4() )
set ) ) )
V30(
NAT : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal V52()
V53()
V54()
V55()
V56()
V57()
V58() )
Element of
K32(
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4() )
set ) ) , the
carrier of
b1 : ( ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like )
ComplexNormSpace) : ( ( ) (
V4() )
set ) ) )
sequence of ( ( ) (
V4() )
set ) ) is
convergent holds
(
seq : ( (
Function-like V30(
NAT : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal V52()
V53()
V54()
V55()
V56()
V57()
V58() )
Element of
K32(
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4() )
set ) ) , the
carrier of
b1 : ( ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like )
ComplexNormSpace) : ( ( ) (
V4() )
set ) ) ) (
V4()
Relation-like NAT : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal V52()
V53()
V54()
V55()
V56()
V57()
V58() )
Element of
K32(
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4() )
set ) )
-defined the
carrier of
b1 : ( ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like )
ComplexNormSpace) : ( ( ) (
V4() )
set )
-valued Function-like V29(
NAT : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal V52()
V53()
V54()
V55()
V56()
V57()
V58() )
Element of
K32(
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4() )
set ) ) )
V30(
NAT : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal V52()
V53()
V54()
V55()
V56()
V57()
V58() )
Element of
K32(
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4() )
set ) ) , the
carrier of
b1 : ( ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like )
ComplexNormSpace) : ( ( ) (
V4() )
set ) ) )
sequence of ( ( ) (
V4() )
set ) )
^\ k : ( ( ) (
epsilon-transitive epsilon-connected ordinal natural V24()
real ext-real V45()
V46()
V52()
V53()
V54()
V55()
V56()
V57() )
Element of
NAT : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal V52()
V53()
V54()
V55()
V56()
V57()
V58() )
Element of
K32(
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4() )
set ) ) ) : ( ( ) (
V4()
Relation-like NAT : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal V52()
V53()
V54()
V55()
V56()
V57()
V58() )
Element of
K32(
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4() )
set ) )
-defined the
carrier of
b1 : ( ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like )
ComplexNormSpace) : ( ( ) (
V4() )
set )
-valued Function-like V29(
NAT : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal V52()
V53()
V54()
V55()
V56()
V57()
V58() )
Element of
K32(
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4() )
set ) ) )
V30(
NAT : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal V52()
V53()
V54()
V55()
V56()
V57()
V58() )
Element of
K32(
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4() )
set ) ) , the
carrier of
b1 : ( ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like )
ComplexNormSpace) : ( ( ) (
V4() )
set ) ) )
subsequence of
b2 : ( (
Function-like V30(
NAT : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal V52()
V53()
V54()
V55()
V56()
V57()
V58() )
Element of
K32(
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4() )
set ) ) , the
carrier of
b1 : ( ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like )
ComplexNormSpace) : ( ( ) (
V4() )
set ) ) ) (
V4()
Relation-like NAT : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal V52()
V53()
V54()
V55()
V56()
V57()
V58() )
Element of
K32(
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4() )
set ) )
-defined the
carrier of
b1 : ( ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like )
ComplexNormSpace) : ( ( ) (
V4() )
set )
-valued Function-like V29(
NAT : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal V52()
V53()
V54()
V55()
V56()
V57()
V58() )
Element of
K32(
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4() )
set ) ) )
V30(
NAT : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal V52()
V53()
V54()
V55()
V56()
V57()
V58() )
Element of
K32(
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4() )
set ) ) , the
carrier of
b1 : ( ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like )
ComplexNormSpace) : ( ( ) (
V4() )
set ) ) )
sequence of ( ( ) (
V4() )
set ) ) ) is
convergent &
lim (seq : ( ( Function-like V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed V133() V134() vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed V133() V134() vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( V4() ) set ) ) ) ( V4() Relation-like NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) -defined the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed V133() V134() vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed V133() V134() vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( V4() ) set ) -valued Function-like V29( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) ) V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed V133() V134() vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed V133() V134() vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( V4() ) set ) ) ) sequence of ( ( ) ( V4() ) set ) ) ^\ k : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V24() real ext-real V45() V46() V52() V53() V54() V55() V56() V57() ) Element of NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) ) ) : ( ( ) (
V4()
Relation-like NAT : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal V52()
V53()
V54()
V55()
V56()
V57()
V58() )
Element of
K32(
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4() )
set ) )
-defined the
carrier of
b1 : ( ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like )
ComplexNormSpace) : ( ( ) (
V4() )
set )
-valued Function-like V29(
NAT : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal V52()
V53()
V54()
V55()
V56()
V57()
V58() )
Element of
K32(
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4() )
set ) ) )
V30(
NAT : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal V52()
V53()
V54()
V55()
V56()
V57()
V58() )
Element of
K32(
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4() )
set ) ) , the
carrier of
b1 : ( ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like )
ComplexNormSpace) : ( ( ) (
V4() )
set ) ) )
subsequence of
b2 : ( (
Function-like V30(
NAT : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal V52()
V53()
V54()
V55()
V56()
V57()
V58() )
Element of
K32(
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4() )
set ) ) , the
carrier of
b1 : ( ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like )
ComplexNormSpace) : ( ( ) (
V4() )
set ) ) ) (
V4()
Relation-like NAT : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal V52()
V53()
V54()
V55()
V56()
V57()
V58() )
Element of
K32(
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4() )
set ) )
-defined the
carrier of
b1 : ( ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like )
ComplexNormSpace) : ( ( ) (
V4() )
set )
-valued Function-like V29(
NAT : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal V52()
V53()
V54()
V55()
V56()
V57()
V58() )
Element of
K32(
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4() )
set ) ) )
V30(
NAT : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal V52()
V53()
V54()
V55()
V56()
V57()
V58() )
Element of
K32(
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4() )
set ) ) , the
carrier of
b1 : ( ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like )
ComplexNormSpace) : ( ( ) (
V4() )
set ) ) )
sequence of ( ( ) (
V4() )
set ) ) ) : ( ( ) (
right_complementable )
Element of the
carrier of
b1 : ( ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like )
ComplexNormSpace) : ( ( ) (
V4() )
set ) )
= lim seq : ( (
Function-like V30(
NAT : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal V52()
V53()
V54()
V55()
V56()
V57()
V58() )
Element of
K32(
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4() )
set ) ) , the
carrier of
b1 : ( ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like )
ComplexNormSpace) : ( ( ) (
V4() )
set ) ) ) (
V4()
Relation-like NAT : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal V52()
V53()
V54()
V55()
V56()
V57()
V58() )
Element of
K32(
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4() )
set ) )
-defined the
carrier of
b1 : ( ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like )
ComplexNormSpace) : ( ( ) (
V4() )
set )
-valued Function-like V29(
NAT : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal V52()
V53()
V54()
V55()
V56()
V57()
V58() )
Element of
K32(
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4() )
set ) ) )
V30(
NAT : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal V52()
V53()
V54()
V55()
V56()
V57()
V58() )
Element of
K32(
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4() )
set ) ) , the
carrier of
b1 : ( ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like )
ComplexNormSpace) : ( ( ) (
V4() )
set ) ) )
sequence of ( ( ) (
V4() )
set ) ) : ( ( ) (
right_complementable )
Element of the
carrier of
b1 : ( ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like )
ComplexNormSpace) : ( ( ) (
V4() )
set ) ) ) ;
theorem
for
X being ( ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like )
ComplexNormSpace)
for
seq,
seq1 being ( (
Function-like V30(
NAT : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal V52()
V53()
V54()
V55()
V56()
V57()
V58() )
Element of
K32(
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4() )
set ) ) , the
carrier of
b1 : ( ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like )
ComplexNormSpace) : ( ( ) (
V4() )
set ) ) ) (
V4()
Relation-like NAT : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal V52()
V53()
V54()
V55()
V56()
V57()
V58() )
Element of
K32(
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4() )
set ) )
-defined the
carrier of
b1 : ( ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like )
ComplexNormSpace) : ( ( ) (
V4() )
set )
-valued Function-like V29(
NAT : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal V52()
V53()
V54()
V55()
V56()
V57()
V58() )
Element of
K32(
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4() )
set ) ) )
V30(
NAT : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal V52()
V53()
V54()
V55()
V56()
V57()
V58() )
Element of
K32(
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4() )
set ) ) , the
carrier of
b1 : ( ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like )
ComplexNormSpace) : ( ( ) (
V4() )
set ) ) )
sequence of ( ( ) (
V4() )
set ) ) st
seq : ( (
Function-like V30(
NAT : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal V52()
V53()
V54()
V55()
V56()
V57()
V58() )
Element of
K32(
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4() )
set ) ) , the
carrier of
b1 : ( ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like )
ComplexNormSpace) : ( ( ) (
V4() )
set ) ) ) (
V4()
Relation-like NAT : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal V52()
V53()
V54()
V55()
V56()
V57()
V58() )
Element of
K32(
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4() )
set ) )
-defined the
carrier of
b1 : ( ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like )
ComplexNormSpace) : ( ( ) (
V4() )
set )
-valued Function-like V29(
NAT : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal V52()
V53()
V54()
V55()
V56()
V57()
V58() )
Element of
K32(
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4() )
set ) ) )
V30(
NAT : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal V52()
V53()
V54()
V55()
V56()
V57()
V58() )
Element of
K32(
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4() )
set ) ) , the
carrier of
b1 : ( ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like )
ComplexNormSpace) : ( ( ) (
V4() )
set ) ) )
sequence of ( ( ) (
V4() )
set ) ) is
convergent & ex
k being ( ( ) (
epsilon-transitive epsilon-connected ordinal natural V24()
real ext-real V45()
V46()
V52()
V53()
V54()
V55()
V56()
V57() )
Element of
NAT : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal V52()
V53()
V54()
V55()
V56()
V57()
V58() )
Element of
K32(
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4() )
set ) ) ) st
seq : ( (
Function-like V30(
NAT : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal V52()
V53()
V54()
V55()
V56()
V57()
V58() )
Element of
K32(
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4() )
set ) ) , the
carrier of
b1 : ( ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like )
ComplexNormSpace) : ( ( ) (
V4() )
set ) ) ) (
V4()
Relation-like NAT : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal V52()
V53()
V54()
V55()
V56()
V57()
V58() )
Element of
K32(
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4() )
set ) )
-defined the
carrier of
b1 : ( ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like )
ComplexNormSpace) : ( ( ) (
V4() )
set )
-valued Function-like V29(
NAT : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal V52()
V53()
V54()
V55()
V56()
V57()
V58() )
Element of
K32(
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4() )
set ) ) )
V30(
NAT : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal V52()
V53()
V54()
V55()
V56()
V57()
V58() )
Element of
K32(
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4() )
set ) ) , the
carrier of
b1 : ( ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like )
ComplexNormSpace) : ( ( ) (
V4() )
set ) ) )
sequence of ( ( ) (
V4() )
set ) )
= seq1 : ( (
Function-like V30(
NAT : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal V52()
V53()
V54()
V55()
V56()
V57()
V58() )
Element of
K32(
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4() )
set ) ) , the
carrier of
b1 : ( ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like )
ComplexNormSpace) : ( ( ) (
V4() )
set ) ) ) (
V4()
Relation-like NAT : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal V52()
V53()
V54()
V55()
V56()
V57()
V58() )
Element of
K32(
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4() )
set ) )
-defined the
carrier of
b1 : ( ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like )
ComplexNormSpace) : ( ( ) (
V4() )
set )
-valued Function-like V29(
NAT : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal V52()
V53()
V54()
V55()
V56()
V57()
V58() )
Element of
K32(
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4() )
set ) ) )
V30(
NAT : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal V52()
V53()
V54()
V55()
V56()
V57()
V58() )
Element of
K32(
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4() )
set ) ) , the
carrier of
b1 : ( ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like )
ComplexNormSpace) : ( ( ) (
V4() )
set ) ) )
sequence of ( ( ) (
V4() )
set ) )
^\ k : ( ( ) (
epsilon-transitive epsilon-connected ordinal natural V24()
real ext-real V45()
V46()
V52()
V53()
V54()
V55()
V56()
V57() )
Element of
NAT : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal V52()
V53()
V54()
V55()
V56()
V57()
V58() )
Element of
K32(
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4() )
set ) ) ) : ( ( ) (
V4()
Relation-like NAT : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal V52()
V53()
V54()
V55()
V56()
V57()
V58() )
Element of
K32(
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4() )
set ) )
-defined the
carrier of
b1 : ( ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like )
ComplexNormSpace) : ( ( ) (
V4() )
set )
-valued Function-like V29(
NAT : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal V52()
V53()
V54()
V55()
V56()
V57()
V58() )
Element of
K32(
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4() )
set ) ) )
V30(
NAT : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal V52()
V53()
V54()
V55()
V56()
V57()
V58() )
Element of
K32(
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4() )
set ) ) , the
carrier of
b1 : ( ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like )
ComplexNormSpace) : ( ( ) (
V4() )
set ) ) )
subsequence of
b3 : ( (
Function-like V30(
NAT : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal V52()
V53()
V54()
V55()
V56()
V57()
V58() )
Element of
K32(
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4() )
set ) ) , the
carrier of
b1 : ( ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like )
ComplexNormSpace) : ( ( ) (
V4() )
set ) ) ) (
V4()
Relation-like NAT : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal V52()
V53()
V54()
V55()
V56()
V57()
V58() )
Element of
K32(
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4() )
set ) )
-defined the
carrier of
b1 : ( ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like )
ComplexNormSpace) : ( ( ) (
V4() )
set )
-valued Function-like V29(
NAT : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal V52()
V53()
V54()
V55()
V56()
V57()
V58() )
Element of
K32(
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4() )
set ) ) )
V30(
NAT : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal V52()
V53()
V54()
V55()
V56()
V57()
V58() )
Element of
K32(
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4() )
set ) ) , the
carrier of
b1 : ( ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like )
ComplexNormSpace) : ( ( ) (
V4() )
set ) ) )
sequence of ( ( ) (
V4() )
set ) ) ) holds
seq1 : ( (
Function-like V30(
NAT : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal V52()
V53()
V54()
V55()
V56()
V57()
V58() )
Element of
K32(
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4() )
set ) ) , the
carrier of
b1 : ( ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like )
ComplexNormSpace) : ( ( ) (
V4() )
set ) ) ) (
V4()
Relation-like NAT : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal V52()
V53()
V54()
V55()
V56()
V57()
V58() )
Element of
K32(
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4() )
set ) )
-defined the
carrier of
b1 : ( ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like )
ComplexNormSpace) : ( ( ) (
V4() )
set )
-valued Function-like V29(
NAT : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal V52()
V53()
V54()
V55()
V56()
V57()
V58() )
Element of
K32(
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4() )
set ) ) )
V30(
NAT : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal V52()
V53()
V54()
V55()
V56()
V57()
V58() )
Element of
K32(
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4() )
set ) ) , the
carrier of
b1 : ( ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like )
ComplexNormSpace) : ( ( ) (
V4() )
set ) ) )
sequence of ( ( ) (
V4() )
set ) ) is
convergent ;
theorem
for
X being ( ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like )
ComplexNormSpace)
for
seq,
seq1 being ( (
Function-like V30(
NAT : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal V52()
V53()
V54()
V55()
V56()
V57()
V58() )
Element of
K32(
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4() )
set ) ) , the
carrier of
b1 : ( ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like )
ComplexNormSpace) : ( ( ) (
V4() )
set ) ) ) (
V4()
Relation-like NAT : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal V52()
V53()
V54()
V55()
V56()
V57()
V58() )
Element of
K32(
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4() )
set ) )
-defined the
carrier of
b1 : ( ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like )
ComplexNormSpace) : ( ( ) (
V4() )
set )
-valued Function-like V29(
NAT : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal V52()
V53()
V54()
V55()
V56()
V57()
V58() )
Element of
K32(
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4() )
set ) ) )
V30(
NAT : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal V52()
V53()
V54()
V55()
V56()
V57()
V58() )
Element of
K32(
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4() )
set ) ) , the
carrier of
b1 : ( ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like )
ComplexNormSpace) : ( ( ) (
V4() )
set ) ) )
sequence of ( ( ) (
V4() )
set ) ) st
seq : ( (
Function-like V30(
NAT : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal V52()
V53()
V54()
V55()
V56()
V57()
V58() )
Element of
K32(
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4() )
set ) ) , the
carrier of
b1 : ( ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like )
ComplexNormSpace) : ( ( ) (
V4() )
set ) ) ) (
V4()
Relation-like NAT : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal V52()
V53()
V54()
V55()
V56()
V57()
V58() )
Element of
K32(
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4() )
set ) )
-defined the
carrier of
b1 : ( ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like )
ComplexNormSpace) : ( ( ) (
V4() )
set )
-valued Function-like V29(
NAT : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal V52()
V53()
V54()
V55()
V56()
V57()
V58() )
Element of
K32(
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4() )
set ) ) )
V30(
NAT : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal V52()
V53()
V54()
V55()
V56()
V57()
V58() )
Element of
K32(
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4() )
set ) ) , the
carrier of
b1 : ( ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like )
ComplexNormSpace) : ( ( ) (
V4() )
set ) ) )
sequence of ( ( ) (
V4() )
set ) ) is
convergent & ex
k being ( ( ) (
epsilon-transitive epsilon-connected ordinal natural V24()
real ext-real V45()
V46()
V52()
V53()
V54()
V55()
V56()
V57() )
Element of
NAT : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal V52()
V53()
V54()
V55()
V56()
V57()
V58() )
Element of
K32(
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4() )
set ) ) ) st
seq : ( (
Function-like V30(
NAT : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal V52()
V53()
V54()
V55()
V56()
V57()
V58() )
Element of
K32(
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4() )
set ) ) , the
carrier of
b1 : ( ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like )
ComplexNormSpace) : ( ( ) (
V4() )
set ) ) ) (
V4()
Relation-like NAT : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal V52()
V53()
V54()
V55()
V56()
V57()
V58() )
Element of
K32(
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4() )
set ) )
-defined the
carrier of
b1 : ( ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like )
ComplexNormSpace) : ( ( ) (
V4() )
set )
-valued Function-like V29(
NAT : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal V52()
V53()
V54()
V55()
V56()
V57()
V58() )
Element of
K32(
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4() )
set ) ) )
V30(
NAT : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal V52()
V53()
V54()
V55()
V56()
V57()
V58() )
Element of
K32(
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4() )
set ) ) , the
carrier of
b1 : ( ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like )
ComplexNormSpace) : ( ( ) (
V4() )
set ) ) )
sequence of ( ( ) (
V4() )
set ) )
= seq1 : ( (
Function-like V30(
NAT : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal V52()
V53()
V54()
V55()
V56()
V57()
V58() )
Element of
K32(
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4() )
set ) ) , the
carrier of
b1 : ( ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like )
ComplexNormSpace) : ( ( ) (
V4() )
set ) ) ) (
V4()
Relation-like NAT : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal V52()
V53()
V54()
V55()
V56()
V57()
V58() )
Element of
K32(
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4() )
set ) )
-defined the
carrier of
b1 : ( ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like )
ComplexNormSpace) : ( ( ) (
V4() )
set )
-valued Function-like V29(
NAT : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal V52()
V53()
V54()
V55()
V56()
V57()
V58() )
Element of
K32(
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4() )
set ) ) )
V30(
NAT : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal V52()
V53()
V54()
V55()
V56()
V57()
V58() )
Element of
K32(
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4() )
set ) ) , the
carrier of
b1 : ( ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like )
ComplexNormSpace) : ( ( ) (
V4() )
set ) ) )
sequence of ( ( ) (
V4() )
set ) )
^\ k : ( ( ) (
epsilon-transitive epsilon-connected ordinal natural V24()
real ext-real V45()
V46()
V52()
V53()
V54()
V55()
V56()
V57() )
Element of
NAT : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal V52()
V53()
V54()
V55()
V56()
V57()
V58() )
Element of
K32(
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4() )
set ) ) ) : ( ( ) (
V4()
Relation-like NAT : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal V52()
V53()
V54()
V55()
V56()
V57()
V58() )
Element of
K32(
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4() )
set ) )
-defined the
carrier of
b1 : ( ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like )
ComplexNormSpace) : ( ( ) (
V4() )
set )
-valued Function-like V29(
NAT : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal V52()
V53()
V54()
V55()
V56()
V57()
V58() )
Element of
K32(
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4() )
set ) ) )
V30(
NAT : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal V52()
V53()
V54()
V55()
V56()
V57()
V58() )
Element of
K32(
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4() )
set ) ) , the
carrier of
b1 : ( ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like )
ComplexNormSpace) : ( ( ) (
V4() )
set ) ) )
subsequence of
b3 : ( (
Function-like V30(
NAT : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal V52()
V53()
V54()
V55()
V56()
V57()
V58() )
Element of
K32(
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4() )
set ) ) , the
carrier of
b1 : ( ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like )
ComplexNormSpace) : ( ( ) (
V4() )
set ) ) ) (
V4()
Relation-like NAT : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal V52()
V53()
V54()
V55()
V56()
V57()
V58() )
Element of
K32(
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4() )
set ) )
-defined the
carrier of
b1 : ( ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like )
ComplexNormSpace) : ( ( ) (
V4() )
set )
-valued Function-like V29(
NAT : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal V52()
V53()
V54()
V55()
V56()
V57()
V58() )
Element of
K32(
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4() )
set ) ) )
V30(
NAT : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal V52()
V53()
V54()
V55()
V56()
V57()
V58() )
Element of
K32(
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4() )
set ) ) , the
carrier of
b1 : ( ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like )
ComplexNormSpace) : ( ( ) (
V4() )
set ) ) )
sequence of ( ( ) (
V4() )
set ) ) ) holds
lim seq1 : ( (
Function-like V30(
NAT : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal V52()
V53()
V54()
V55()
V56()
V57()
V58() )
Element of
K32(
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4() )
set ) ) , the
carrier of
b1 : ( ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like )
ComplexNormSpace) : ( ( ) (
V4() )
set ) ) ) (
V4()
Relation-like NAT : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal V52()
V53()
V54()
V55()
V56()
V57()
V58() )
Element of
K32(
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4() )
set ) )
-defined the
carrier of
b1 : ( ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like )
ComplexNormSpace) : ( ( ) (
V4() )
set )
-valued Function-like V29(
NAT : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal V52()
V53()
V54()
V55()
V56()
V57()
V58() )
Element of
K32(
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4() )
set ) ) )
V30(
NAT : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal V52()
V53()
V54()
V55()
V56()
V57()
V58() )
Element of
K32(
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4() )
set ) ) , the
carrier of
b1 : ( ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like )
ComplexNormSpace) : ( ( ) (
V4() )
set ) ) )
sequence of ( ( ) (
V4() )
set ) ) : ( ( ) (
right_complementable )
Element of the
carrier of
b1 : ( ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like )
ComplexNormSpace) : ( ( ) (
V4() )
set ) )
= lim seq : ( (
Function-like V30(
NAT : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal V52()
V53()
V54()
V55()
V56()
V57()
V58() )
Element of
K32(
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4() )
set ) ) , the
carrier of
b1 : ( ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like )
ComplexNormSpace) : ( ( ) (
V4() )
set ) ) ) (
V4()
Relation-like NAT : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal V52()
V53()
V54()
V55()
V56()
V57()
V58() )
Element of
K32(
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4() )
set ) )
-defined the
carrier of
b1 : ( ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like )
ComplexNormSpace) : ( ( ) (
V4() )
set )
-valued Function-like V29(
NAT : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal V52()
V53()
V54()
V55()
V56()
V57()
V58() )
Element of
K32(
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4() )
set ) ) )
V30(
NAT : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal V52()
V53()
V54()
V55()
V56()
V57()
V58() )
Element of
K32(
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4() )
set ) ) , the
carrier of
b1 : ( ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like )
ComplexNormSpace) : ( ( ) (
V4() )
set ) ) )
sequence of ( ( ) (
V4() )
set ) ) : ( ( ) (
right_complementable )
Element of the
carrier of
b1 : ( ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like )
ComplexNormSpace) : ( ( ) (
V4() )
set ) ) ;
theorem
for
X being ( ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like )
ComplexNormSpace)
for
s1,
s2 being ( (
Function-like V30(
NAT : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal V52()
V53()
V54()
V55()
V56()
V57()
V58() )
Element of
K32(
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4() )
set ) ) , the
carrier of
b1 : ( ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like )
ComplexNormSpace) : ( ( ) (
V4() )
set ) ) ) (
V4()
Relation-like NAT : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal V52()
V53()
V54()
V55()
V56()
V57()
V58() )
Element of
K32(
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4() )
set ) )
-defined the
carrier of
b1 : ( ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like )
ComplexNormSpace) : ( ( ) (
V4() )
set )
-valued Function-like V29(
NAT : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal V52()
V53()
V54()
V55()
V56()
V57()
V58() )
Element of
K32(
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4() )
set ) ) )
V30(
NAT : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal V52()
V53()
V54()
V55()
V56()
V57()
V58() )
Element of
K32(
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4() )
set ) ) , the
carrier of
b1 : ( ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like )
ComplexNormSpace) : ( ( ) (
V4() )
set ) ) )
sequence of ( ( ) (
V4() )
set ) ) holds
(Partial_Sums s1 : ( ( Function-like V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed V133() V134() vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed V133() V134() vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( V4() ) set ) ) ) ( V4() Relation-like NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) -defined the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed V133() V134() vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed V133() V134() vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( V4() ) set ) -valued Function-like V29( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) ) V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed V133() V134() vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed V133() V134() vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( V4() ) set ) ) ) sequence of ( ( ) ( V4() ) set ) ) ) : ( (
Function-like V30(
NAT : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal V52()
V53()
V54()
V55()
V56()
V57()
V58() )
Element of
K32(
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4() )
set ) ) , the
carrier of
b1 : ( ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like )
ComplexNormSpace) : ( ( ) (
V4() )
set ) ) ) (
V4()
Relation-like NAT : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal V52()
V53()
V54()
V55()
V56()
V57()
V58() )
Element of
K32(
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4() )
set ) )
-defined the
carrier of
b1 : ( ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like )
ComplexNormSpace) : ( ( ) (
V4() )
set )
-valued Function-like V29(
NAT : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal V52()
V53()
V54()
V55()
V56()
V57()
V58() )
Element of
K32(
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4() )
set ) ) )
V30(
NAT : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal V52()
V53()
V54()
V55()
V56()
V57()
V58() )
Element of
K32(
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4() )
set ) ) , the
carrier of
b1 : ( ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like )
ComplexNormSpace) : ( ( ) (
V4() )
set ) ) )
Element of
K32(
K33(
NAT : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal V52()
V53()
V54()
V55()
V56()
V57()
V58() )
Element of
K32(
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4() )
set ) ) , the
carrier of
b1 : ( ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like )
ComplexNormSpace) : ( ( ) (
V4() )
set ) ) : ( ( ) (
V4() )
set ) ) : ( ( ) (
V4() )
set ) )
+ (Partial_Sums s2 : ( ( Function-like V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed V133() V134() vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed V133() V134() vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( V4() ) set ) ) ) ( V4() Relation-like NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) -defined the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed V133() V134() vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed V133() V134() vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( V4() ) set ) -valued Function-like V29( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) ) V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed V133() V134() vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed V133() V134() vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( V4() ) set ) ) ) sequence of ( ( ) ( V4() ) set ) ) ) : ( (
Function-like V30(
NAT : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal V52()
V53()
V54()
V55()
V56()
V57()
V58() )
Element of
K32(
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4() )
set ) ) , the
carrier of
b1 : ( ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like )
ComplexNormSpace) : ( ( ) (
V4() )
set ) ) ) (
V4()
Relation-like NAT : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal V52()
V53()
V54()
V55()
V56()
V57()
V58() )
Element of
K32(
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4() )
set ) )
-defined the
carrier of
b1 : ( ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like )
ComplexNormSpace) : ( ( ) (
V4() )
set )
-valued Function-like V29(
NAT : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal V52()
V53()
V54()
V55()
V56()
V57()
V58() )
Element of
K32(
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4() )
set ) ) )
V30(
NAT : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal V52()
V53()
V54()
V55()
V56()
V57()
V58() )
Element of
K32(
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4() )
set ) ) , the
carrier of
b1 : ( ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like )
ComplexNormSpace) : ( ( ) (
V4() )
set ) ) )
Element of
K32(
K33(
NAT : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal V52()
V53()
V54()
V55()
V56()
V57()
V58() )
Element of
K32(
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4() )
set ) ) , the
carrier of
b1 : ( ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like )
ComplexNormSpace) : ( ( ) (
V4() )
set ) ) : ( ( ) (
V4() )
set ) ) : ( ( ) (
V4() )
set ) ) : ( (
Function-like V30(
NAT : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal V52()
V53()
V54()
V55()
V56()
V57()
V58() )
Element of
K32(
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4() )
set ) ) , the
carrier of
b1 : ( ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like )
ComplexNormSpace) : ( ( ) (
V4() )
set ) ) ) (
V4()
Relation-like NAT : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal V52()
V53()
V54()
V55()
V56()
V57()
V58() )
Element of
K32(
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4() )
set ) )
-defined the
carrier of
b1 : ( ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like )
ComplexNormSpace) : ( ( ) (
V4() )
set )
-valued Function-like V29(
NAT : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal V52()
V53()
V54()
V55()
V56()
V57()
V58() )
Element of
K32(
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4() )
set ) ) )
V30(
NAT : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal V52()
V53()
V54()
V55()
V56()
V57()
V58() )
Element of
K32(
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4() )
set ) ) , the
carrier of
b1 : ( ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like )
ComplexNormSpace) : ( ( ) (
V4() )
set ) ) )
Element of
K32(
K33(
NAT : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal V52()
V53()
V54()
V55()
V56()
V57()
V58() )
Element of
K32(
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4() )
set ) ) , the
carrier of
b1 : ( ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like )
ComplexNormSpace) : ( ( ) (
V4() )
set ) ) : ( ( ) (
V4() )
set ) ) : ( ( ) (
V4() )
set ) )
= Partial_Sums (s1 : ( ( Function-like V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed V133() V134() vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed V133() V134() vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( V4() ) set ) ) ) ( V4() Relation-like NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) -defined the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed V133() V134() vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed V133() V134() vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( V4() ) set ) -valued Function-like V29( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) ) V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed V133() V134() vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed V133() V134() vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( V4() ) set ) ) ) sequence of ( ( ) ( V4() ) set ) ) + s2 : ( ( Function-like V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed V133() V134() vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed V133() V134() vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( V4() ) set ) ) ) ( V4() Relation-like NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) -defined the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed V133() V134() vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed V133() V134() vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( V4() ) set ) -valued Function-like V29( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) ) V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed V133() V134() vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed V133() V134() vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( V4() ) set ) ) ) sequence of ( ( ) ( V4() ) set ) ) ) : ( (
Function-like V30(
NAT : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal V52()
V53()
V54()
V55()
V56()
V57()
V58() )
Element of
K32(
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4() )
set ) ) , the
carrier of
b1 : ( ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like )
ComplexNormSpace) : ( ( ) (
V4() )
set ) ) ) (
V4()
Relation-like NAT : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal V52()
V53()
V54()
V55()
V56()
V57()
V58() )
Element of
K32(
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4() )
set ) )
-defined the
carrier of
b1 : ( ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like )
ComplexNormSpace) : ( ( ) (
V4() )
set )
-valued Function-like V29(
NAT : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal V52()
V53()
V54()
V55()
V56()
V57()
V58() )
Element of
K32(
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4() )
set ) ) )
V30(
NAT : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal V52()
V53()
V54()
V55()
V56()
V57()
V58() )
Element of
K32(
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4() )
set ) ) , the
carrier of
b1 : ( ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like )
ComplexNormSpace) : ( ( ) (
V4() )
set ) ) )
Element of
K32(
K33(
NAT : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal V52()
V53()
V54()
V55()
V56()
V57()
V58() )
Element of
K32(
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4() )
set ) ) , the
carrier of
b1 : ( ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like )
ComplexNormSpace) : ( ( ) (
V4() )
set ) ) : ( ( ) (
V4() )
set ) ) : ( ( ) (
V4() )
set ) ) : ( (
Function-like V30(
NAT : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal V52()
V53()
V54()
V55()
V56()
V57()
V58() )
Element of
K32(
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4() )
set ) ) , the
carrier of
b1 : ( ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like )
ComplexNormSpace) : ( ( ) (
V4() )
set ) ) ) (
V4()
Relation-like NAT : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal V52()
V53()
V54()
V55()
V56()
V57()
V58() )
Element of
K32(
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4() )
set ) )
-defined the
carrier of
b1 : ( ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like )
ComplexNormSpace) : ( ( ) (
V4() )
set )
-valued Function-like V29(
NAT : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal V52()
V53()
V54()
V55()
V56()
V57()
V58() )
Element of
K32(
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4() )
set ) ) )
V30(
NAT : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal V52()
V53()
V54()
V55()
V56()
V57()
V58() )
Element of
K32(
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4() )
set ) ) , the
carrier of
b1 : ( ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like )
ComplexNormSpace) : ( ( ) (
V4() )
set ) ) )
Element of
K32(
K33(
NAT : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal V52()
V53()
V54()
V55()
V56()
V57()
V58() )
Element of
K32(
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4() )
set ) ) , the
carrier of
b1 : ( ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like )
ComplexNormSpace) : ( ( ) (
V4() )
set ) ) : ( ( ) (
V4() )
set ) ) : ( ( ) (
V4() )
set ) ) ;
theorem
for
X being ( ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like )
ComplexNormSpace)
for
s1,
s2 being ( (
Function-like V30(
NAT : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal V52()
V53()
V54()
V55()
V56()
V57()
V58() )
Element of
K32(
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4() )
set ) ) , the
carrier of
b1 : ( ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like )
ComplexNormSpace) : ( ( ) (
V4() )
set ) ) ) (
V4()
Relation-like NAT : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal V52()
V53()
V54()
V55()
V56()
V57()
V58() )
Element of
K32(
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4() )
set ) )
-defined the
carrier of
b1 : ( ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like )
ComplexNormSpace) : ( ( ) (
V4() )
set )
-valued Function-like V29(
NAT : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal V52()
V53()
V54()
V55()
V56()
V57()
V58() )
Element of
K32(
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4() )
set ) ) )
V30(
NAT : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal V52()
V53()
V54()
V55()
V56()
V57()
V58() )
Element of
K32(
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4() )
set ) ) , the
carrier of
b1 : ( ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like )
ComplexNormSpace) : ( ( ) (
V4() )
set ) ) )
sequence of ( ( ) (
V4() )
set ) ) holds
(Partial_Sums s1 : ( ( Function-like V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed V133() V134() vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed V133() V134() vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( V4() ) set ) ) ) ( V4() Relation-like NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) -defined the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed V133() V134() vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed V133() V134() vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( V4() ) set ) -valued Function-like V29( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) ) V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed V133() V134() vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed V133() V134() vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( V4() ) set ) ) ) sequence of ( ( ) ( V4() ) set ) ) ) : ( (
Function-like V30(
NAT : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal V52()
V53()
V54()
V55()
V56()
V57()
V58() )
Element of
K32(
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4() )
set ) ) , the
carrier of
b1 : ( ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like )
ComplexNormSpace) : ( ( ) (
V4() )
set ) ) ) (
V4()
Relation-like NAT : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal V52()
V53()
V54()
V55()
V56()
V57()
V58() )
Element of
K32(
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4() )
set ) )
-defined the
carrier of
b1 : ( ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like )
ComplexNormSpace) : ( ( ) (
V4() )
set )
-valued Function-like V29(
NAT : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal V52()
V53()
V54()
V55()
V56()
V57()
V58() )
Element of
K32(
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4() )
set ) ) )
V30(
NAT : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal V52()
V53()
V54()
V55()
V56()
V57()
V58() )
Element of
K32(
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4() )
set ) ) , the
carrier of
b1 : ( ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like )
ComplexNormSpace) : ( ( ) (
V4() )
set ) ) )
Element of
K32(
K33(
NAT : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal V52()
V53()
V54()
V55()
V56()
V57()
V58() )
Element of
K32(
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4() )
set ) ) , the
carrier of
b1 : ( ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like )
ComplexNormSpace) : ( ( ) (
V4() )
set ) ) : ( ( ) (
V4() )
set ) ) : ( ( ) (
V4() )
set ) )
- (Partial_Sums s2 : ( ( Function-like V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed V133() V134() vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed V133() V134() vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( V4() ) set ) ) ) ( V4() Relation-like NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) -defined the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed V133() V134() vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed V133() V134() vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( V4() ) set ) -valued Function-like V29( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) ) V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed V133() V134() vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed V133() V134() vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( V4() ) set ) ) ) sequence of ( ( ) ( V4() ) set ) ) ) : ( (
Function-like V30(
NAT : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal V52()
V53()
V54()
V55()
V56()
V57()
V58() )
Element of
K32(
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4() )
set ) ) , the
carrier of
b1 : ( ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like )
ComplexNormSpace) : ( ( ) (
V4() )
set ) ) ) (
V4()
Relation-like NAT : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal V52()
V53()
V54()
V55()
V56()
V57()
V58() )
Element of
K32(
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4() )
set ) )
-defined the
carrier of
b1 : ( ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like )
ComplexNormSpace) : ( ( ) (
V4() )
set )
-valued Function-like V29(
NAT : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal V52()
V53()
V54()
V55()
V56()
V57()
V58() )
Element of
K32(
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4() )
set ) ) )
V30(
NAT : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal V52()
V53()
V54()
V55()
V56()
V57()
V58() )
Element of
K32(
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4() )
set ) ) , the
carrier of
b1 : ( ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like )
ComplexNormSpace) : ( ( ) (
V4() )
set ) ) )
Element of
K32(
K33(
NAT : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal V52()
V53()
V54()
V55()
V56()
V57()
V58() )
Element of
K32(
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4() )
set ) ) , the
carrier of
b1 : ( ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like )
ComplexNormSpace) : ( ( ) (
V4() )
set ) ) : ( ( ) (
V4() )
set ) ) : ( ( ) (
V4() )
set ) ) : ( (
Function-like V30(
NAT : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal V52()
V53()
V54()
V55()
V56()
V57()
V58() )
Element of
K32(
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4() )
set ) ) , the
carrier of
b1 : ( ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like )
ComplexNormSpace) : ( ( ) (
V4() )
set ) ) ) (
V4()
Relation-like NAT : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal V52()
V53()
V54()
V55()
V56()
V57()
V58() )
Element of
K32(
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4() )
set ) )
-defined the
carrier of
b1 : ( ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like )
ComplexNormSpace) : ( ( ) (
V4() )
set )
-valued Function-like V29(
NAT : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal V52()
V53()
V54()
V55()
V56()
V57()
V58() )
Element of
K32(
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4() )
set ) ) )
V30(
NAT : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal V52()
V53()
V54()
V55()
V56()
V57()
V58() )
Element of
K32(
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4() )
set ) ) , the
carrier of
b1 : ( ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like )
ComplexNormSpace) : ( ( ) (
V4() )
set ) ) )
Element of
K32(
K33(
NAT : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal V52()
V53()
V54()
V55()
V56()
V57()
V58() )
Element of
K32(
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4() )
set ) ) , the
carrier of
b1 : ( ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like )
ComplexNormSpace) : ( ( ) (
V4() )
set ) ) : ( ( ) (
V4() )
set ) ) : ( ( ) (
V4() )
set ) )
= Partial_Sums (s1 : ( ( Function-like V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed V133() V134() vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed V133() V134() vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( V4() ) set ) ) ) ( V4() Relation-like NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) -defined the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed V133() V134() vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed V133() V134() vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( V4() ) set ) -valued Function-like V29( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) ) V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed V133() V134() vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed V133() V134() vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( V4() ) set ) ) ) sequence of ( ( ) ( V4() ) set ) ) - s2 : ( ( Function-like V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed V133() V134() vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed V133() V134() vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( V4() ) set ) ) ) ( V4() Relation-like NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) -defined the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed V133() V134() vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed V133() V134() vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( V4() ) set ) -valued Function-like V29( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) ) V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed V133() V134() vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed V133() V134() vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( V4() ) set ) ) ) sequence of ( ( ) ( V4() ) set ) ) ) : ( (
Function-like V30(
NAT : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal V52()
V53()
V54()
V55()
V56()
V57()
V58() )
Element of
K32(
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4() )
set ) ) , the
carrier of
b1 : ( ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like )
ComplexNormSpace) : ( ( ) (
V4() )
set ) ) ) (
V4()
Relation-like NAT : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal V52()
V53()
V54()
V55()
V56()
V57()
V58() )
Element of
K32(
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4() )
set ) )
-defined the
carrier of
b1 : ( ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like )
ComplexNormSpace) : ( ( ) (
V4() )
set )
-valued Function-like V29(
NAT : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal V52()
V53()
V54()
V55()
V56()
V57()
V58() )
Element of
K32(
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4() )
set ) ) )
V30(
NAT : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal V52()
V53()
V54()
V55()
V56()
V57()
V58() )
Element of
K32(
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4() )
set ) ) , the
carrier of
b1 : ( ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like )
ComplexNormSpace) : ( ( ) (
V4() )
set ) ) )
Element of
K32(
K33(
NAT : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal V52()
V53()
V54()
V55()
V56()
V57()
V58() )
Element of
K32(
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4() )
set ) ) , the
carrier of
b1 : ( ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like )
ComplexNormSpace) : ( ( ) (
V4() )
set ) ) : ( ( ) (
V4() )
set ) ) : ( ( ) (
V4() )
set ) ) : ( (
Function-like V30(
NAT : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal V52()
V53()
V54()
V55()
V56()
V57()
V58() )
Element of
K32(
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4() )
set ) ) , the
carrier of
b1 : ( ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like )
ComplexNormSpace) : ( ( ) (
V4() )
set ) ) ) (
V4()
Relation-like NAT : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal V52()
V53()
V54()
V55()
V56()
V57()
V58() )
Element of
K32(
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4() )
set ) )
-defined the
carrier of
b1 : ( ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like )
ComplexNormSpace) : ( ( ) (
V4() )
set )
-valued Function-like V29(
NAT : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal V52()
V53()
V54()
V55()
V56()
V57()
V58() )
Element of
K32(
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4() )
set ) ) )
V30(
NAT : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal V52()
V53()
V54()
V55()
V56()
V57()
V58() )
Element of
K32(
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4() )
set ) ) , the
carrier of
b1 : ( ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like )
ComplexNormSpace) : ( ( ) (
V4() )
set ) ) )
Element of
K32(
K33(
NAT : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal V52()
V53()
V54()
V55()
V56()
V57()
V58() )
Element of
K32(
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4() )
set ) ) , the
carrier of
b1 : ( ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like )
ComplexNormSpace) : ( ( ) (
V4() )
set ) ) : ( ( ) (
V4() )
set ) ) : ( ( ) (
V4() )
set ) ) ;
theorem
for
X being ( ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like )
ComplexNormSpace)
for
seq1,
seq2 being ( (
Function-like V30(
NAT : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal V52()
V53()
V54()
V55()
V56()
V57()
V58() )
Element of
K32(
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4() )
set ) ) , the
carrier of
b1 : ( ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like )
ComplexNormSpace) : ( ( ) (
V4() )
set ) ) ) (
V4()
Relation-like NAT : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal V52()
V53()
V54()
V55()
V56()
V57()
V58() )
Element of
K32(
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4() )
set ) )
-defined the
carrier of
b1 : ( ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like )
ComplexNormSpace) : ( ( ) (
V4() )
set )
-valued Function-like V29(
NAT : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal V52()
V53()
V54()
V55()
V56()
V57()
V58() )
Element of
K32(
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4() )
set ) ) )
V30(
NAT : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal V52()
V53()
V54()
V55()
V56()
V57()
V58() )
Element of
K32(
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4() )
set ) ) , the
carrier of
b1 : ( ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like )
ComplexNormSpace) : ( ( ) (
V4() )
set ) ) )
sequence of ( ( ) (
V4() )
set ) ) st
seq1 : ( (
Function-like V30(
NAT : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal V52()
V53()
V54()
V55()
V56()
V57()
V58() )
Element of
K32(
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4() )
set ) ) , the
carrier of
b1 : ( ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like )
ComplexNormSpace) : ( ( ) (
V4() )
set ) ) ) (
V4()
Relation-like NAT : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal V52()
V53()
V54()
V55()
V56()
V57()
V58() )
Element of
K32(
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4() )
set ) )
-defined the
carrier of
b1 : ( ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like )
ComplexNormSpace) : ( ( ) (
V4() )
set )
-valued Function-like V29(
NAT : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal V52()
V53()
V54()
V55()
V56()
V57()
V58() )
Element of
K32(
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4() )
set ) ) )
V30(
NAT : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal V52()
V53()
V54()
V55()
V56()
V57()
V58() )
Element of
K32(
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4() )
set ) ) , the
carrier of
b1 : ( ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like )
ComplexNormSpace) : ( ( ) (
V4() )
set ) ) )
sequence of ( ( ) (
V4() )
set ) ) is
summable &
seq2 : ( (
Function-like V30(
NAT : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal V52()
V53()
V54()
V55()
V56()
V57()
V58() )
Element of
K32(
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4() )
set ) ) , the
carrier of
b1 : ( ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like )
ComplexNormSpace) : ( ( ) (
V4() )
set ) ) ) (
V4()
Relation-like NAT : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal V52()
V53()
V54()
V55()
V56()
V57()
V58() )
Element of
K32(
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4() )
set ) )
-defined the
carrier of
b1 : ( ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like )
ComplexNormSpace) : ( ( ) (
V4() )
set )
-valued Function-like V29(
NAT : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal V52()
V53()
V54()
V55()
V56()
V57()
V58() )
Element of
K32(
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4() )
set ) ) )
V30(
NAT : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal V52()
V53()
V54()
V55()
V56()
V57()
V58() )
Element of
K32(
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4() )
set ) ) , the
carrier of
b1 : ( ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like )
ComplexNormSpace) : ( ( ) (
V4() )
set ) ) )
sequence of ( ( ) (
V4() )
set ) ) is
summable holds
(
seq1 : ( (
Function-like V30(
NAT : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal V52()
V53()
V54()
V55()
V56()
V57()
V58() )
Element of
K32(
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4() )
set ) ) , the
carrier of
b1 : ( ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like )
ComplexNormSpace) : ( ( ) (
V4() )
set ) ) ) (
V4()
Relation-like NAT : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal V52()
V53()
V54()
V55()
V56()
V57()
V58() )
Element of
K32(
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4() )
set ) )
-defined the
carrier of
b1 : ( ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like )
ComplexNormSpace) : ( ( ) (
V4() )
set )
-valued Function-like V29(
NAT : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal V52()
V53()
V54()
V55()
V56()
V57()
V58() )
Element of
K32(
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4() )
set ) ) )
V30(
NAT : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal V52()
V53()
V54()
V55()
V56()
V57()
V58() )
Element of
K32(
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4() )
set ) ) , the
carrier of
b1 : ( ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like )
ComplexNormSpace) : ( ( ) (
V4() )
set ) ) )
sequence of ( ( ) (
V4() )
set ) )
+ seq2 : ( (
Function-like V30(
NAT : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal V52()
V53()
V54()
V55()
V56()
V57()
V58() )
Element of
K32(
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4() )
set ) ) , the
carrier of
b1 : ( ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like )
ComplexNormSpace) : ( ( ) (
V4() )
set ) ) ) (
V4()
Relation-like NAT : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal V52()
V53()
V54()
V55()
V56()
V57()
V58() )
Element of
K32(
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4() )
set ) )
-defined the
carrier of
b1 : ( ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like )
ComplexNormSpace) : ( ( ) (
V4() )
set )
-valued Function-like V29(
NAT : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal V52()
V53()
V54()
V55()
V56()
V57()
V58() )
Element of
K32(
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4() )
set ) ) )
V30(
NAT : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal V52()
V53()
V54()
V55()
V56()
V57()
V58() )
Element of
K32(
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4() )
set ) ) , the
carrier of
b1 : ( ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like )
ComplexNormSpace) : ( ( ) (
V4() )
set ) ) )
sequence of ( ( ) (
V4() )
set ) ) : ( (
Function-like V30(
NAT : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal V52()
V53()
V54()
V55()
V56()
V57()
V58() )
Element of
K32(
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4() )
set ) ) , the
carrier of
b1 : ( ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like )
ComplexNormSpace) : ( ( ) (
V4() )
set ) ) ) (
V4()
Relation-like NAT : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal V52()
V53()
V54()
V55()
V56()
V57()
V58() )
Element of
K32(
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4() )
set ) )
-defined the
carrier of
b1 : ( ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like )
ComplexNormSpace) : ( ( ) (
V4() )
set )
-valued Function-like V29(
NAT : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal V52()
V53()
V54()
V55()
V56()
V57()
V58() )
Element of
K32(
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4() )
set ) ) )
V30(
NAT : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal V52()
V53()
V54()
V55()
V56()
V57()
V58() )
Element of
K32(
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4() )
set ) ) , the
carrier of
b1 : ( ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like )
ComplexNormSpace) : ( ( ) (
V4() )
set ) ) )
Element of
K32(
K33(
NAT : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal V52()
V53()
V54()
V55()
V56()
V57()
V58() )
Element of
K32(
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4() )
set ) ) , the
carrier of
b1 : ( ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like )
ComplexNormSpace) : ( ( ) (
V4() )
set ) ) : ( ( ) (
V4() )
set ) ) : ( ( ) (
V4() )
set ) ) is
summable &
Sum (seq1 : ( ( Function-like V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed V133() V134() vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed V133() V134() vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( V4() ) set ) ) ) ( V4() Relation-like NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) -defined the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed V133() V134() vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed V133() V134() vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( V4() ) set ) -valued Function-like V29( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) ) V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed V133() V134() vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed V133() V134() vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( V4() ) set ) ) ) sequence of ( ( ) ( V4() ) set ) ) + seq2 : ( ( Function-like V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed V133() V134() vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed V133() V134() vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( V4() ) set ) ) ) ( V4() Relation-like NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) -defined the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed V133() V134() vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed V133() V134() vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( V4() ) set ) -valued Function-like V29( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) ) V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed V133() V134() vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed V133() V134() vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( V4() ) set ) ) ) sequence of ( ( ) ( V4() ) set ) ) ) : ( (
Function-like V30(
NAT : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal V52()
V53()
V54()
V55()
V56()
V57()
V58() )
Element of
K32(
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4() )
set ) ) , the
carrier of
b1 : ( ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like )
ComplexNormSpace) : ( ( ) (
V4() )
set ) ) ) (
V4()
Relation-like NAT : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal V52()
V53()
V54()
V55()
V56()
V57()
V58() )
Element of
K32(
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4() )
set ) )
-defined the
carrier of
b1 : ( ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like )
ComplexNormSpace) : ( ( ) (
V4() )
set )
-valued Function-like V29(
NAT : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal V52()
V53()
V54()
V55()
V56()
V57()
V58() )
Element of
K32(
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4() )
set ) ) )
V30(
NAT : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal V52()
V53()
V54()
V55()
V56()
V57()
V58() )
Element of
K32(
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4() )
set ) ) , the
carrier of
b1 : ( ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like )
ComplexNormSpace) : ( ( ) (
V4() )
set ) ) )
Element of
K32(
K33(
NAT : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal V52()
V53()
V54()
V55()
V56()
V57()
V58() )
Element of
K32(
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4() )
set ) ) , the
carrier of
b1 : ( ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like )
ComplexNormSpace) : ( ( ) (
V4() )
set ) ) : ( ( ) (
V4() )
set ) ) : ( ( ) (
V4() )
set ) ) : ( ( ) (
right_complementable )
Element of ( ( ) (
V4() )
set ) )
= (Sum seq1 : ( ( Function-like V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed V133() V134() vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed V133() V134() vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( V4() ) set ) ) ) ( V4() Relation-like NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) -defined the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed V133() V134() vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed V133() V134() vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( V4() ) set ) -valued Function-like V29( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) ) V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed V133() V134() vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed V133() V134() vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( V4() ) set ) ) ) sequence of ( ( ) ( V4() ) set ) ) ) : ( ( ) (
right_complementable )
Element of ( ( ) (
V4() )
set ) )
+ (Sum seq2 : ( ( Function-like V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed V133() V134() vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed V133() V134() vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( V4() ) set ) ) ) ( V4() Relation-like NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) -defined the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed V133() V134() vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed V133() V134() vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( V4() ) set ) -valued Function-like V29( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) ) V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed V133() V134() vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed V133() V134() vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( V4() ) set ) ) ) sequence of ( ( ) ( V4() ) set ) ) ) : ( ( ) (
right_complementable )
Element of ( ( ) (
V4() )
set ) ) : ( ( ) (
right_complementable )
Element of the
carrier of
b1 : ( ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like )
ComplexNormSpace) : ( ( ) (
V4() )
set ) ) ) ;
theorem
for
X being ( ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like )
ComplexNormSpace)
for
seq1,
seq2 being ( (
Function-like V30(
NAT : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal V52()
V53()
V54()
V55()
V56()
V57()
V58() )
Element of
K32(
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4() )
set ) ) , the
carrier of
b1 : ( ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like )
ComplexNormSpace) : ( ( ) (
V4() )
set ) ) ) (
V4()
Relation-like NAT : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal V52()
V53()
V54()
V55()
V56()
V57()
V58() )
Element of
K32(
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4() )
set ) )
-defined the
carrier of
b1 : ( ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like )
ComplexNormSpace) : ( ( ) (
V4() )
set )
-valued Function-like V29(
NAT : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal V52()
V53()
V54()
V55()
V56()
V57()
V58() )
Element of
K32(
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4() )
set ) ) )
V30(
NAT : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal V52()
V53()
V54()
V55()
V56()
V57()
V58() )
Element of
K32(
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4() )
set ) ) , the
carrier of
b1 : ( ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like )
ComplexNormSpace) : ( ( ) (
V4() )
set ) ) )
sequence of ( ( ) (
V4() )
set ) ) st
seq1 : ( (
Function-like V30(
NAT : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal V52()
V53()
V54()
V55()
V56()
V57()
V58() )
Element of
K32(
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4() )
set ) ) , the
carrier of
b1 : ( ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like )
ComplexNormSpace) : ( ( ) (
V4() )
set ) ) ) (
V4()
Relation-like NAT : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal V52()
V53()
V54()
V55()
V56()
V57()
V58() )
Element of
K32(
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4() )
set ) )
-defined the
carrier of
b1 : ( ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like )
ComplexNormSpace) : ( ( ) (
V4() )
set )
-valued Function-like V29(
NAT : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal V52()
V53()
V54()
V55()
V56()
V57()
V58() )
Element of
K32(
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4() )
set ) ) )
V30(
NAT : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal V52()
V53()
V54()
V55()
V56()
V57()
V58() )
Element of
K32(
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4() )
set ) ) , the
carrier of
b1 : ( ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like )
ComplexNormSpace) : ( ( ) (
V4() )
set ) ) )
sequence of ( ( ) (
V4() )
set ) ) is
summable &
seq2 : ( (
Function-like V30(
NAT : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal V52()
V53()
V54()
V55()
V56()
V57()
V58() )
Element of
K32(
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4() )
set ) ) , the
carrier of
b1 : ( ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like )
ComplexNormSpace) : ( ( ) (
V4() )
set ) ) ) (
V4()
Relation-like NAT : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal V52()
V53()
V54()
V55()
V56()
V57()
V58() )
Element of
K32(
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4() )
set ) )
-defined the
carrier of
b1 : ( ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like )
ComplexNormSpace) : ( ( ) (
V4() )
set )
-valued Function-like V29(
NAT : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal V52()
V53()
V54()
V55()
V56()
V57()
V58() )
Element of
K32(
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4() )
set ) ) )
V30(
NAT : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal V52()
V53()
V54()
V55()
V56()
V57()
V58() )
Element of
K32(
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4() )
set ) ) , the
carrier of
b1 : ( ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like )
ComplexNormSpace) : ( ( ) (
V4() )
set ) ) )
sequence of ( ( ) (
V4() )
set ) ) is
summable holds
(
seq1 : ( (
Function-like V30(
NAT : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal V52()
V53()
V54()
V55()
V56()
V57()
V58() )
Element of
K32(
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4() )
set ) ) , the
carrier of
b1 : ( ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like )
ComplexNormSpace) : ( ( ) (
V4() )
set ) ) ) (
V4()
Relation-like NAT : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal V52()
V53()
V54()
V55()
V56()
V57()
V58() )
Element of
K32(
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4() )
set ) )
-defined the
carrier of
b1 : ( ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like )
ComplexNormSpace) : ( ( ) (
V4() )
set )
-valued Function-like V29(
NAT : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal V52()
V53()
V54()
V55()
V56()
V57()
V58() )
Element of
K32(
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4() )
set ) ) )
V30(
NAT : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal V52()
V53()
V54()
V55()
V56()
V57()
V58() )
Element of
K32(
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4() )
set ) ) , the
carrier of
b1 : ( ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like )
ComplexNormSpace) : ( ( ) (
V4() )
set ) ) )
sequence of ( ( ) (
V4() )
set ) )
- seq2 : ( (
Function-like V30(
NAT : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal V52()
V53()
V54()
V55()
V56()
V57()
V58() )
Element of
K32(
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4() )
set ) ) , the
carrier of
b1 : ( ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like )
ComplexNormSpace) : ( ( ) (
V4() )
set ) ) ) (
V4()
Relation-like NAT : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal V52()
V53()
V54()
V55()
V56()
V57()
V58() )
Element of
K32(
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4() )
set ) )
-defined the
carrier of
b1 : ( ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like )
ComplexNormSpace) : ( ( ) (
V4() )
set )
-valued Function-like V29(
NAT : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal V52()
V53()
V54()
V55()
V56()
V57()
V58() )
Element of
K32(
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4() )
set ) ) )
V30(
NAT : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal V52()
V53()
V54()
V55()
V56()
V57()
V58() )
Element of
K32(
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4() )
set ) ) , the
carrier of
b1 : ( ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like )
ComplexNormSpace) : ( ( ) (
V4() )
set ) ) )
sequence of ( ( ) (
V4() )
set ) ) : ( (
Function-like V30(
NAT : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal V52()
V53()
V54()
V55()
V56()
V57()
V58() )
Element of
K32(
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4() )
set ) ) , the
carrier of
b1 : ( ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like )
ComplexNormSpace) : ( ( ) (
V4() )
set ) ) ) (
V4()
Relation-like NAT : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal V52()
V53()
V54()
V55()
V56()
V57()
V58() )
Element of
K32(
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4() )
set ) )
-defined the
carrier of
b1 : ( ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like )
ComplexNormSpace) : ( ( ) (
V4() )
set )
-valued Function-like V29(
NAT : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal V52()
V53()
V54()
V55()
V56()
V57()
V58() )
Element of
K32(
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4() )
set ) ) )
V30(
NAT : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal V52()
V53()
V54()
V55()
V56()
V57()
V58() )
Element of
K32(
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4() )
set ) ) , the
carrier of
b1 : ( ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like )
ComplexNormSpace) : ( ( ) (
V4() )
set ) ) )
Element of
K32(
K33(
NAT : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal V52()
V53()
V54()
V55()
V56()
V57()
V58() )
Element of
K32(
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4() )
set ) ) , the
carrier of
b1 : ( ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like )
ComplexNormSpace) : ( ( ) (
V4() )
set ) ) : ( ( ) (
V4() )
set ) ) : ( ( ) (
V4() )
set ) ) is
summable &
Sum (seq1 : ( ( Function-like V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed V133() V134() vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed V133() V134() vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( V4() ) set ) ) ) ( V4() Relation-like NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) -defined the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed V133() V134() vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed V133() V134() vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( V4() ) set ) -valued Function-like V29( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) ) V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed V133() V134() vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed V133() V134() vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( V4() ) set ) ) ) sequence of ( ( ) ( V4() ) set ) ) - seq2 : ( ( Function-like V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed V133() V134() vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed V133() V134() vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( V4() ) set ) ) ) ( V4() Relation-like NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) -defined the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed V133() V134() vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed V133() V134() vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( V4() ) set ) -valued Function-like V29( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) ) V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed V133() V134() vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed V133() V134() vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( V4() ) set ) ) ) sequence of ( ( ) ( V4() ) set ) ) ) : ( (
Function-like V30(
NAT : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal V52()
V53()
V54()
V55()
V56()
V57()
V58() )
Element of
K32(
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4() )
set ) ) , the
carrier of
b1 : ( ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like )
ComplexNormSpace) : ( ( ) (
V4() )
set ) ) ) (
V4()
Relation-like NAT : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal V52()
V53()
V54()
V55()
V56()
V57()
V58() )
Element of
K32(
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4() )
set ) )
-defined the
carrier of
b1 : ( ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like )
ComplexNormSpace) : ( ( ) (
V4() )
set )
-valued Function-like V29(
NAT : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal V52()
V53()
V54()
V55()
V56()
V57()
V58() )
Element of
K32(
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4() )
set ) ) )
V30(
NAT : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal V52()
V53()
V54()
V55()
V56()
V57()
V58() )
Element of
K32(
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4() )
set ) ) , the
carrier of
b1 : ( ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like )
ComplexNormSpace) : ( ( ) (
V4() )
set ) ) )
Element of
K32(
K33(
NAT : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal V52()
V53()
V54()
V55()
V56()
V57()
V58() )
Element of
K32(
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4() )
set ) ) , the
carrier of
b1 : ( ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like )
ComplexNormSpace) : ( ( ) (
V4() )
set ) ) : ( ( ) (
V4() )
set ) ) : ( ( ) (
V4() )
set ) ) : ( ( ) (
right_complementable )
Element of ( ( ) (
V4() )
set ) )
= (Sum seq1 : ( ( Function-like V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed V133() V134() vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed V133() V134() vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( V4() ) set ) ) ) ( V4() Relation-like NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) -defined the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed V133() V134() vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed V133() V134() vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( V4() ) set ) -valued Function-like V29( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) ) V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed V133() V134() vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed V133() V134() vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( V4() ) set ) ) ) sequence of ( ( ) ( V4() ) set ) ) ) : ( ( ) (
right_complementable )
Element of ( ( ) (
V4() )
set ) )
- (Sum seq2 : ( ( Function-like V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed V133() V134() vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed V133() V134() vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( V4() ) set ) ) ) ( V4() Relation-like NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) -defined the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed V133() V134() vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed V133() V134() vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( V4() ) set ) -valued Function-like V29( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) ) V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed V133() V134() vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed V133() V134() vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( V4() ) set ) ) ) sequence of ( ( ) ( V4() ) set ) ) ) : ( ( ) (
right_complementable )
Element of ( ( ) (
V4() )
set ) ) : ( ( ) (
right_complementable )
Element of the
carrier of
b1 : ( ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like )
ComplexNormSpace) : ( ( ) (
V4() )
set ) ) ) ;
registration
let X be ( ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like )
ComplexNormSpace) ;
let seq1,
seq2 be ( (
Function-like V30(
NAT : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal V52()
V53()
V54()
V55()
V56()
V57()
V58() )
Element of
K32(
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4() )
set ) ) , the
carrier of
X : ( ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like )
ComplexNormSpace) : ( ( ) (
V4() )
set ) )
summable ) (
V4()
Relation-like NAT : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal V52()
V53()
V54()
V55()
V56()
V57()
V58() )
Element of
K32(
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4() )
set ) )
-defined the
carrier of
X : ( ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like )
ComplexNormSpace) : ( ( ) (
V4() )
set )
-valued Function-like V29(
NAT : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal V52()
V53()
V54()
V55()
V56()
V57()
V58() )
Element of
K32(
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4() )
set ) ) )
V30(
NAT : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal V52()
V53()
V54()
V55()
V56()
V57()
V58() )
Element of
K32(
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4() )
set ) ) , the
carrier of
X : ( ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like )
ComplexNormSpace) : ( ( ) (
V4() )
set ) )
convergent summable )
sequence of ( ( ) (
V4() )
set ) ) ;
cluster seq1 : ( (
Function-like V30(
NAT : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal V52()
V53()
V54()
V55()
V56()
V57()
V58() )
Element of
K32(
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4() )
set ) ) , the
carrier of
X : ( ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like )
CNORMSTR ) : ( ( ) (
V4() )
set ) )
summable ) (
V4()
Relation-like NAT : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal V52()
V53()
V54()
V55()
V56()
V57()
V58() )
Element of
K32(
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4() )
set ) )
-defined the
carrier of
X : ( ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like )
CNORMSTR ) : ( ( ) (
V4() )
set )
-valued Function-like V29(
NAT : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal V52()
V53()
V54()
V55()
V56()
V57()
V58() )
Element of
K32(
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4() )
set ) ) )
V30(
NAT : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal V52()
V53()
V54()
V55()
V56()
V57()
V58() )
Element of
K32(
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4() )
set ) ) , the
carrier of
X : ( ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like )
CNORMSTR ) : ( ( ) (
V4() )
set ) )
convergent summable )
Element of
K32(
K33(
NAT : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal V52()
V53()
V54()
V55()
V56()
V57()
V58() )
Element of
K32(
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4() )
set ) ) , the
carrier of
X : ( ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like )
CNORMSTR ) : ( ( ) (
V4() )
set ) ) : ( ( ) (
V4() )
set ) ) : ( ( ) (
V4() )
set ) )
+ seq2 : ( (
Function-like V30(
NAT : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal V52()
V53()
V54()
V55()
V56()
V57()
V58() )
Element of
K32(
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4() )
set ) ) , the
carrier of
X : ( ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like )
CNORMSTR ) : ( ( ) (
V4() )
set ) )
summable ) (
V4()
Relation-like NAT : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal V52()
V53()
V54()
V55()
V56()
V57()
V58() )
Element of
K32(
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4() )
set ) )
-defined the
carrier of
X : ( ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like )
CNORMSTR ) : ( ( ) (
V4() )
set )
-valued Function-like V29(
NAT : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal V52()
V53()
V54()
V55()
V56()
V57()
V58() )
Element of
K32(
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4() )
set ) ) )
V30(
NAT : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal V52()
V53()
V54()
V55()
V56()
V57()
V58() )
Element of
K32(
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4() )
set ) ) , the
carrier of
X : ( ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like )
CNORMSTR ) : ( ( ) (
V4() )
set ) )
convergent summable )
Element of
K32(
K33(
NAT : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal V52()
V53()
V54()
V55()
V56()
V57()
V58() )
Element of
K32(
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4() )
set ) ) , the
carrier of
X : ( ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like )
CNORMSTR ) : ( ( ) (
V4() )
set ) ) : ( ( ) (
V4() )
set ) ) : ( ( ) (
V4() )
set ) ) : ( (
Function-like V30(
NAT : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal V52()
V53()
V54()
V55()
V56()
V57()
V58() )
Element of
K32(
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4() )
set ) ) , the
carrier of
X : ( ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like )
CNORMSTR ) : ( ( ) (
V4() )
set ) ) ) (
V4()
Relation-like NAT : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal V52()
V53()
V54()
V55()
V56()
V57()
V58() )
Element of
K32(
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4() )
set ) )
-defined the
carrier of
X : ( ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like )
CNORMSTR ) : ( ( ) (
V4() )
set )
-valued Function-like V29(
NAT : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal V52()
V53()
V54()
V55()
V56()
V57()
V58() )
Element of
K32(
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4() )
set ) ) )
V30(
NAT : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal V52()
V53()
V54()
V55()
V56()
V57()
V58() )
Element of
K32(
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4() )
set ) ) , the
carrier of
X : ( ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like )
CNORMSTR ) : ( ( ) (
V4() )
set ) ) )
Element of
K32(
K33(
NAT : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal V52()
V53()
V54()
V55()
V56()
V57()
V58() )
Element of
K32(
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4() )
set ) ) , the
carrier of
X : ( ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like )
CNORMSTR ) : ( ( ) (
V4() )
set ) ) : ( ( ) (
V4() )
set ) ) : ( ( ) (
V4() )
set ) )
-> Function-like V30(
NAT : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal V52()
V53()
V54()
V55()
V56()
V57()
V58() )
Element of
K32(
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4() )
set ) ) , the
carrier of
X : ( ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like )
CNORMSTR ) : ( ( ) (
V4() )
set ) )
summable ;
cluster seq1 : ( (
Function-like V30(
NAT : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal V52()
V53()
V54()
V55()
V56()
V57()
V58() )
Element of
K32(
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4() )
set ) ) , the
carrier of
X : ( ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like )
CNORMSTR ) : ( ( ) (
V4() )
set ) )
summable ) (
V4()
Relation-like NAT : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal V52()
V53()
V54()
V55()
V56()
V57()
V58() )
Element of
K32(
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4() )
set ) )
-defined the
carrier of
X : ( ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like )
CNORMSTR ) : ( ( ) (
V4() )
set )
-valued Function-like V29(
NAT : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal V52()
V53()
V54()
V55()
V56()
V57()
V58() )
Element of
K32(
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4() )
set ) ) )
V30(
NAT : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal V52()
V53()
V54()
V55()
V56()
V57()
V58() )
Element of
K32(
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4() )
set ) ) , the
carrier of
X : ( ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like )
CNORMSTR ) : ( ( ) (
V4() )
set ) )
convergent summable )
Element of
K32(
K33(
NAT : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal V52()
V53()
V54()
V55()
V56()
V57()
V58() )
Element of
K32(
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4() )
set ) ) , the
carrier of
X : ( ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like )
CNORMSTR ) : ( ( ) (
V4() )
set ) ) : ( ( ) (
V4() )
set ) ) : ( ( ) (
V4() )
set ) )
- seq2 : ( (
Function-like V30(
NAT : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal V52()
V53()
V54()
V55()
V56()
V57()
V58() )
Element of
K32(
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4() )
set ) ) , the
carrier of
X : ( ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like )
CNORMSTR ) : ( ( ) (
V4() )
set ) )
summable ) (
V4()
Relation-like NAT : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal V52()
V53()
V54()
V55()
V56()
V57()
V58() )
Element of
K32(
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4() )
set ) )
-defined the
carrier of
X : ( ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like )
CNORMSTR ) : ( ( ) (
V4() )
set )
-valued Function-like V29(
NAT : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal V52()
V53()
V54()
V55()
V56()
V57()
V58() )
Element of
K32(
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4() )
set ) ) )
V30(
NAT : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal V52()
V53()
V54()
V55()
V56()
V57()
V58() )
Element of
K32(
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4() )
set ) ) , the
carrier of
X : ( ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like )
CNORMSTR ) : ( ( ) (
V4() )
set ) )
convergent summable )
Element of
K32(
K33(
NAT : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal V52()
V53()
V54()
V55()
V56()
V57()
V58() )
Element of
K32(
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4() )
set ) ) , the
carrier of
X : ( ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like )
CNORMSTR ) : ( ( ) (
V4() )
set ) ) : ( ( ) (
V4() )
set ) ) : ( ( ) (
V4() )
set ) ) : ( (
Function-like V30(
NAT : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal V52()
V53()
V54()
V55()
V56()
V57()
V58() )
Element of
K32(
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4() )
set ) ) , the
carrier of
X : ( ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like )
CNORMSTR ) : ( ( ) (
V4() )
set ) ) ) (
V4()
Relation-like NAT : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal V52()
V53()
V54()
V55()
V56()
V57()
V58() )
Element of
K32(
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4() )
set ) )
-defined the
carrier of
X : ( ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like )
CNORMSTR ) : ( ( ) (
V4() )
set )
-valued Function-like V29(
NAT : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal V52()
V53()
V54()
V55()
V56()
V57()
V58() )
Element of
K32(
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4() )
set ) ) )
V30(
NAT : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal V52()
V53()
V54()
V55()
V56()
V57()
V58() )
Element of
K32(
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4() )
set ) ) , the
carrier of
X : ( ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like )
CNORMSTR ) : ( ( ) (
V4() )
set ) ) )
Element of
K32(
K33(
NAT : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal V52()
V53()
V54()
V55()
V56()
V57()
V58() )
Element of
K32(
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4() )
set ) ) , the
carrier of
X : ( ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like )
CNORMSTR ) : ( ( ) (
V4() )
set ) ) : ( ( ) (
V4() )
set ) ) : ( ( ) (
V4() )
set ) )
-> Function-like V30(
NAT : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal V52()
V53()
V54()
V55()
V56()
V57()
V58() )
Element of
K32(
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4() )
set ) ) , the
carrier of
X : ( ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like )
CNORMSTR ) : ( ( ) (
V4() )
set ) )
summable ;
end;
theorem
for
X being ( ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like )
ComplexNormSpace)
for
seq being ( (
Function-like V30(
NAT : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal V52()
V53()
V54()
V55()
V56()
V57()
V58() )
Element of
K32(
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4() )
set ) ) , the
carrier of
b1 : ( ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like )
ComplexNormSpace) : ( ( ) (
V4() )
set ) ) ) (
V4()
Relation-like NAT : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal V52()
V53()
V54()
V55()
V56()
V57()
V58() )
Element of
K32(
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4() )
set ) )
-defined the
carrier of
b1 : ( ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like )
ComplexNormSpace) : ( ( ) (
V4() )
set )
-valued Function-like V29(
NAT : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal V52()
V53()
V54()
V55()
V56()
V57()
V58() )
Element of
K32(
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4() )
set ) ) )
V30(
NAT : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal V52()
V53()
V54()
V55()
V56()
V57()
V58() )
Element of
K32(
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4() )
set ) ) , the
carrier of
b1 : ( ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like )
ComplexNormSpace) : ( ( ) (
V4() )
set ) ) )
sequence of ( ( ) (
V4() )
set ) )
for
z being ( (
V24() ) (
V24() )
Complex) holds
Partial_Sums (z : ( ( V24() ) ( V24() ) Complex) * seq : ( ( Function-like V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed V133() V134() vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed V133() V134() vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( V4() ) set ) ) ) ( V4() Relation-like NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) -defined the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed V133() V134() vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed V133() V134() vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( V4() ) set ) -valued Function-like V29( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) ) V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed V133() V134() vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed V133() V134() vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( V4() ) set ) ) ) sequence of ( ( ) ( V4() ) set ) ) ) : ( (
Function-like V30(
NAT : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal V52()
V53()
V54()
V55()
V56()
V57()
V58() )
Element of
K32(
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4() )
set ) ) , the
carrier of
b1 : ( ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like )
ComplexNormSpace) : ( ( ) (
V4() )
set ) ) ) (
V4()
Relation-like NAT : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal V52()
V53()
V54()
V55()
V56()
V57()
V58() )
Element of
K32(
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4() )
set ) )
-defined the
carrier of
b1 : ( ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like )
ComplexNormSpace) : ( ( ) (
V4() )
set )
-valued Function-like V29(
NAT : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal V52()
V53()
V54()
V55()
V56()
V57()
V58() )
Element of
K32(
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4() )
set ) ) )
V30(
NAT : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal V52()
V53()
V54()
V55()
V56()
V57()
V58() )
Element of
K32(
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4() )
set ) ) , the
carrier of
b1 : ( ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like )
ComplexNormSpace) : ( ( ) (
V4() )
set ) ) )
Element of
K32(
K33(
NAT : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal V52()
V53()
V54()
V55()
V56()
V57()
V58() )
Element of
K32(
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4() )
set ) ) , the
carrier of
b1 : ( ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like )
ComplexNormSpace) : ( ( ) (
V4() )
set ) ) : ( ( ) (
V4() )
set ) ) : ( ( ) (
V4() )
set ) ) : ( (
Function-like V30(
NAT : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal V52()
V53()
V54()
V55()
V56()
V57()
V58() )
Element of
K32(
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4() )
set ) ) , the
carrier of
b1 : ( ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like )
ComplexNormSpace) : ( ( ) (
V4() )
set ) ) ) (
V4()
Relation-like NAT : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal V52()
V53()
V54()
V55()
V56()
V57()
V58() )
Element of
K32(
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4() )
set ) )
-defined the
carrier of
b1 : ( ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like )
ComplexNormSpace) : ( ( ) (
V4() )
set )
-valued Function-like V29(
NAT : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal V52()
V53()
V54()
V55()
V56()
V57()
V58() )
Element of
K32(
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4() )
set ) ) )
V30(
NAT : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal V52()
V53()
V54()
V55()
V56()
V57()
V58() )
Element of
K32(
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4() )
set ) ) , the
carrier of
b1 : ( ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like )
ComplexNormSpace) : ( ( ) (
V4() )
set ) ) )
Element of
K32(
K33(
NAT : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal V52()
V53()
V54()
V55()
V56()
V57()
V58() )
Element of
K32(
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4() )
set ) ) , the
carrier of
b1 : ( ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like )
ComplexNormSpace) : ( ( ) (
V4() )
set ) ) : ( ( ) (
V4() )
set ) ) : ( ( ) (
V4() )
set ) )
= z : ( (
V24() ) (
V24() )
Complex)
* (Partial_Sums seq : ( ( Function-like V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed V133() V134() vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed V133() V134() vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( V4() ) set ) ) ) ( V4() Relation-like NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) -defined the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed V133() V134() vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed V133() V134() vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( V4() ) set ) -valued Function-like V29( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) ) V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed V133() V134() vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed V133() V134() vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( V4() ) set ) ) ) sequence of ( ( ) ( V4() ) set ) ) ) : ( (
Function-like V30(
NAT : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal V52()
V53()
V54()
V55()
V56()
V57()
V58() )
Element of
K32(
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4() )
set ) ) , the
carrier of
b1 : ( ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like )
ComplexNormSpace) : ( ( ) (
V4() )
set ) ) ) (
V4()
Relation-like NAT : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal V52()
V53()
V54()
V55()
V56()
V57()
V58() )
Element of
K32(
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4() )
set ) )
-defined the
carrier of
b1 : ( ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like )
ComplexNormSpace) : ( ( ) (
V4() )
set )
-valued Function-like V29(
NAT : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal V52()
V53()
V54()
V55()
V56()
V57()
V58() )
Element of
K32(
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4() )
set ) ) )
V30(
NAT : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal V52()
V53()
V54()
V55()
V56()
V57()
V58() )
Element of
K32(
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4() )
set ) ) , the
carrier of
b1 : ( ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like )
ComplexNormSpace) : ( ( ) (
V4() )
set ) ) )
Element of
K32(
K33(
NAT : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal V52()
V53()
V54()
V55()
V56()
V57()
V58() )
Element of
K32(
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4() )
set ) ) , the
carrier of
b1 : ( ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like )
ComplexNormSpace) : ( ( ) (
V4() )
set ) ) : ( ( ) (
V4() )
set ) ) : ( ( ) (
V4() )
set ) ) : ( (
Function-like V30(
NAT : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal V52()
V53()
V54()
V55()
V56()
V57()
V58() )
Element of
K32(
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4() )
set ) ) , the
carrier of
b1 : ( ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like )
ComplexNormSpace) : ( ( ) (
V4() )
set ) ) ) (
V4()
Relation-like NAT : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal V52()
V53()
V54()
V55()
V56()
V57()
V58() )
Element of
K32(
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4() )
set ) )
-defined the
carrier of
b1 : ( ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like )
ComplexNormSpace) : ( ( ) (
V4() )
set )
-valued Function-like V29(
NAT : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal V52()
V53()
V54()
V55()
V56()
V57()
V58() )
Element of
K32(
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4() )
set ) ) )
V30(
NAT : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal V52()
V53()
V54()
V55()
V56()
V57()
V58() )
Element of
K32(
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4() )
set ) ) , the
carrier of
b1 : ( ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like )
ComplexNormSpace) : ( ( ) (
V4() )
set ) ) )
Element of
K32(
K33(
NAT : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal V52()
V53()
V54()
V55()
V56()
V57()
V58() )
Element of
K32(
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4() )
set ) ) , the
carrier of
b1 : ( ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like )
ComplexNormSpace) : ( ( ) (
V4() )
set ) ) : ( ( ) (
V4() )
set ) ) : ( ( ) (
V4() )
set ) ) ;
theorem
for
X being ( ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like )
ComplexNormSpace)
for
seq being ( (
Function-like V30(
NAT : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal V52()
V53()
V54()
V55()
V56()
V57()
V58() )
Element of
K32(
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4() )
set ) ) , the
carrier of
b1 : ( ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like )
ComplexNormSpace) : ( ( ) (
V4() )
set ) )
summable ) (
V4()
Relation-like NAT : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal V52()
V53()
V54()
V55()
V56()
V57()
V58() )
Element of
K32(
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4() )
set ) )
-defined the
carrier of
b1 : ( ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like )
ComplexNormSpace) : ( ( ) (
V4() )
set )
-valued Function-like V29(
NAT : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal V52()
V53()
V54()
V55()
V56()
V57()
V58() )
Element of
K32(
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4() )
set ) ) )
V30(
NAT : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal V52()
V53()
V54()
V55()
V56()
V57()
V58() )
Element of
K32(
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4() )
set ) ) , the
carrier of
b1 : ( ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like )
ComplexNormSpace) : ( ( ) (
V4() )
set ) )
convergent summable )
sequence of ( ( ) (
V4() )
set ) )
for
z being ( (
V24() ) (
V24() )
Complex) holds
(
z : ( (
V24() ) (
V24() )
Complex)
* seq : ( (
Function-like V30(
NAT : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal V52()
V53()
V54()
V55()
V56()
V57()
V58() )
Element of
K32(
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4() )
set ) ) , the
carrier of
b1 : ( ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like )
ComplexNormSpace) : ( ( ) (
V4() )
set ) )
summable ) (
V4()
Relation-like NAT : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal V52()
V53()
V54()
V55()
V56()
V57()
V58() )
Element of
K32(
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4() )
set ) )
-defined the
carrier of
b1 : ( ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like )
ComplexNormSpace) : ( ( ) (
V4() )
set )
-valued Function-like V29(
NAT : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal V52()
V53()
V54()
V55()
V56()
V57()
V58() )
Element of
K32(
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4() )
set ) ) )
V30(
NAT : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal V52()
V53()
V54()
V55()
V56()
V57()
V58() )
Element of
K32(
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4() )
set ) ) , the
carrier of
b1 : ( ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like )
ComplexNormSpace) : ( ( ) (
V4() )
set ) )
convergent summable )
sequence of ( ( ) (
V4() )
set ) ) : ( (
Function-like V30(
NAT : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal V52()
V53()
V54()
V55()
V56()
V57()
V58() )
Element of
K32(
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4() )
set ) ) , the
carrier of
b1 : ( ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like )
ComplexNormSpace) : ( ( ) (
V4() )
set ) ) ) (
V4()
Relation-like NAT : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal V52()
V53()
V54()
V55()
V56()
V57()
V58() )
Element of
K32(
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4() )
set ) )
-defined the
carrier of
b1 : ( ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like )
ComplexNormSpace) : ( ( ) (
V4() )
set )
-valued Function-like V29(
NAT : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal V52()
V53()
V54()
V55()
V56()
V57()
V58() )
Element of
K32(
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4() )
set ) ) )
V30(
NAT : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal V52()
V53()
V54()
V55()
V56()
V57()
V58() )
Element of
K32(
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4() )
set ) ) , the
carrier of
b1 : ( ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like )
ComplexNormSpace) : ( ( ) (
V4() )
set ) ) )
Element of
K32(
K33(
NAT : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal V52()
V53()
V54()
V55()
V56()
V57()
V58() )
Element of
K32(
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4() )
set ) ) , the
carrier of
b1 : ( ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like )
ComplexNormSpace) : ( ( ) (
V4() )
set ) ) : ( ( ) (
V4() )
set ) ) : ( ( ) (
V4() )
set ) ) is
summable &
Sum (z : ( ( V24() ) ( V24() ) Complex) * seq : ( ( Function-like V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed V133() V134() vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed V133() V134() vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( V4() ) set ) ) summable ) ( V4() Relation-like NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) -defined the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed V133() V134() vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed V133() V134() vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( V4() ) set ) -valued Function-like V29( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) ) V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed V133() V134() vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed V133() V134() vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( V4() ) set ) ) convergent summable ) sequence of ( ( ) ( V4() ) set ) ) ) : ( (
Function-like V30(
NAT : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal V52()
V53()
V54()
V55()
V56()
V57()
V58() )
Element of
K32(
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4() )
set ) ) , the
carrier of
b1 : ( ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like )
ComplexNormSpace) : ( ( ) (
V4() )
set ) ) ) (
V4()
Relation-like NAT : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal V52()
V53()
V54()
V55()
V56()
V57()
V58() )
Element of
K32(
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4() )
set ) )
-defined the
carrier of
b1 : ( ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like )
ComplexNormSpace) : ( ( ) (
V4() )
set )
-valued Function-like V29(
NAT : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal V52()
V53()
V54()
V55()
V56()
V57()
V58() )
Element of
K32(
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4() )
set ) ) )
V30(
NAT : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal V52()
V53()
V54()
V55()
V56()
V57()
V58() )
Element of
K32(
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4() )
set ) ) , the
carrier of
b1 : ( ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like )
ComplexNormSpace) : ( ( ) (
V4() )
set ) ) )
Element of
K32(
K33(
NAT : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal V52()
V53()
V54()
V55()
V56()
V57()
V58() )
Element of
K32(
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4() )
set ) ) , the
carrier of
b1 : ( ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like )
ComplexNormSpace) : ( ( ) (
V4() )
set ) ) : ( ( ) (
V4() )
set ) ) : ( ( ) (
V4() )
set ) ) : ( ( ) (
right_complementable )
Element of ( ( ) (
V4() )
set ) )
= z : ( (
V24() ) (
V24() )
Complex)
* (Sum seq : ( ( Function-like V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed V133() V134() vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed V133() V134() vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( V4() ) set ) ) summable ) ( V4() Relation-like NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) -defined the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed V133() V134() vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed V133() V134() vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( V4() ) set ) -valued Function-like V29( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) ) V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed V133() V134() vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed V133() V134() vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( V4() ) set ) ) convergent summable ) sequence of ( ( ) ( V4() ) set ) ) ) : ( ( ) (
right_complementable )
Element of ( ( ) (
V4() )
set ) ) : ( ( ) (
right_complementable )
Element of the
carrier of
b1 : ( ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like )
ComplexNormSpace) : ( ( ) (
V4() )
set ) ) ) ;
theorem
for
X being ( ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like )
ComplexNormSpace)
for
s,
s1 being ( (
Function-like V30(
NAT : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal V52()
V53()
V54()
V55()
V56()
V57()
V58() )
Element of
K32(
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4() )
set ) ) , the
carrier of
b1 : ( ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like )
ComplexNormSpace) : ( ( ) (
V4() )
set ) ) ) (
V4()
Relation-like NAT : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal V52()
V53()
V54()
V55()
V56()
V57()
V58() )
Element of
K32(
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4() )
set ) )
-defined the
carrier of
b1 : ( ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like )
ComplexNormSpace) : ( ( ) (
V4() )
set )
-valued Function-like V29(
NAT : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal V52()
V53()
V54()
V55()
V56()
V57()
V58() )
Element of
K32(
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4() )
set ) ) )
V30(
NAT : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal V52()
V53()
V54()
V55()
V56()
V57()
V58() )
Element of
K32(
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4() )
set ) ) , the
carrier of
b1 : ( ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like )
ComplexNormSpace) : ( ( ) (
V4() )
set ) ) )
sequence of ( ( ) (
V4() )
set ) ) st ( for
n being ( ( ) (
epsilon-transitive epsilon-connected ordinal natural V24()
real ext-real V45()
V46()
V52()
V53()
V54()
V55()
V56()
V57() )
Element of
NAT : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal V52()
V53()
V54()
V55()
V56()
V57()
V58() )
Element of
K32(
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4() )
set ) ) ) holds
s1 : ( (
Function-like V30(
NAT : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal V52()
V53()
V54()
V55()
V56()
V57()
V58() )
Element of
K32(
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4() )
set ) ) , the
carrier of
b1 : ( ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like )
ComplexNormSpace) : ( ( ) (
V4() )
set ) ) ) (
V4()
Relation-like NAT : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal V52()
V53()
V54()
V55()
V56()
V57()
V58() )
Element of
K32(
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4() )
set ) )
-defined the
carrier of
b1 : ( ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like )
ComplexNormSpace) : ( ( ) (
V4() )
set )
-valued Function-like V29(
NAT : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal V52()
V53()
V54()
V55()
V56()
V57()
V58() )
Element of
K32(
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4() )
set ) ) )
V30(
NAT : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal V52()
V53()
V54()
V55()
V56()
V57()
V58() )
Element of
K32(
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4() )
set ) ) , the
carrier of
b1 : ( ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like )
ComplexNormSpace) : ( ( ) (
V4() )
set ) ) )
sequence of ( ( ) (
V4() )
set ) )
. n : ( ( ) (
epsilon-transitive epsilon-connected ordinal natural V24()
real ext-real V45()
V46()
V52()
V53()
V54()
V55()
V56()
V57() )
Element of
NAT : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal V52()
V53()
V54()
V55()
V56()
V57()
V58() )
Element of
K32(
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4() )
set ) ) ) : ( ( ) (
right_complementable )
Element of the
carrier of
b1 : ( ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like )
ComplexNormSpace) : ( ( ) (
V4() )
set ) )
= s : ( (
Function-like V30(
NAT : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal V52()
V53()
V54()
V55()
V56()
V57()
V58() )
Element of
K32(
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4() )
set ) ) , the
carrier of
b1 : ( ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like )
ComplexNormSpace) : ( ( ) (
V4() )
set ) ) ) (
V4()
Relation-like NAT : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal V52()
V53()
V54()
V55()
V56()
V57()
V58() )
Element of
K32(
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4() )
set ) )
-defined the
carrier of
b1 : ( ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like )
ComplexNormSpace) : ( ( ) (
V4() )
set )
-valued Function-like V29(
NAT : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal V52()
V53()
V54()
V55()
V56()
V57()
V58() )
Element of
K32(
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4() )
set ) ) )
V30(
NAT : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal V52()
V53()
V54()
V55()
V56()
V57()
V58() )
Element of
K32(
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4() )
set ) ) , the
carrier of
b1 : ( ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like )
ComplexNormSpace) : ( ( ) (
V4() )
set ) ) )
sequence of ( ( ) (
V4() )
set ) )
. 0 : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural V24()
real ext-real non
positive non
negative V45()
V46()
V52()
V53()
V54()
V55()
V56()
V57()
V58() )
Element of
NAT : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal V52()
V53()
V54()
V55()
V56()
V57()
V58() )
Element of
K32(
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4() )
set ) ) ) : ( ( ) (
right_complementable )
Element of the
carrier of
b1 : ( ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like )
ComplexNormSpace) : ( ( ) (
V4() )
set ) ) ) holds
Partial_Sums (s : ( ( Function-like V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed V133() V134() vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed V133() V134() vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( V4() ) set ) ) ) ( V4() Relation-like NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) -defined the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed V133() V134() vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed V133() V134() vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( V4() ) set ) -valued Function-like V29( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) ) V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed V133() V134() vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed V133() V134() vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( V4() ) set ) ) ) sequence of ( ( ) ( V4() ) set ) ) ^\ 1 : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal natural V24() real ext-real positive non negative V45() V46() V52() V53() V54() V55() V56() V57() ) Element of NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) ) ) : ( ( ) (
V4()
Relation-like NAT : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal V52()
V53()
V54()
V55()
V56()
V57()
V58() )
Element of
K32(
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4() )
set ) )
-defined the
carrier of
b1 : ( ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like )
ComplexNormSpace) : ( ( ) (
V4() )
set )
-valued Function-like V29(
NAT : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal V52()
V53()
V54()
V55()
V56()
V57()
V58() )
Element of
K32(
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4() )
set ) ) )
V30(
NAT : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal V52()
V53()
V54()
V55()
V56()
V57()
V58() )
Element of
K32(
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4() )
set ) ) , the
carrier of
b1 : ( ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like )
ComplexNormSpace) : ( ( ) (
V4() )
set ) ) )
subsequence of
b2 : ( (
Function-like V30(
NAT : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal V52()
V53()
V54()
V55()
V56()
V57()
V58() )
Element of
K32(
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4() )
set ) ) , the
carrier of
b1 : ( ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like )
ComplexNormSpace) : ( ( ) (
V4() )
set ) ) ) (
V4()
Relation-like NAT : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal V52()
V53()
V54()
V55()
V56()
V57()
V58() )
Element of
K32(
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4() )
set ) )
-defined the
carrier of
b1 : ( ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like )
ComplexNormSpace) : ( ( ) (
V4() )
set )
-valued Function-like V29(
NAT : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal V52()
V53()
V54()
V55()
V56()
V57()
V58() )
Element of
K32(
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4() )
set ) ) )
V30(
NAT : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal V52()
V53()
V54()
V55()
V56()
V57()
V58() )
Element of
K32(
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4() )
set ) ) , the
carrier of
b1 : ( ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like )
ComplexNormSpace) : ( ( ) (
V4() )
set ) ) )
sequence of ( ( ) (
V4() )
set ) ) ) : ( (
Function-like V30(
NAT : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal V52()
V53()
V54()
V55()
V56()
V57()
V58() )
Element of
K32(
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4() )
set ) ) , the
carrier of
b1 : ( ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like )
ComplexNormSpace) : ( ( ) (
V4() )
set ) ) ) (
V4()
Relation-like NAT : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal V52()
V53()
V54()
V55()
V56()
V57()
V58() )
Element of
K32(
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4() )
set ) )
-defined the
carrier of
b1 : ( ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like )
ComplexNormSpace) : ( ( ) (
V4() )
set )
-valued Function-like V29(
NAT : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal V52()
V53()
V54()
V55()
V56()
V57()
V58() )
Element of
K32(
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4() )
set ) ) )
V30(
NAT : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal V52()
V53()
V54()
V55()
V56()
V57()
V58() )
Element of
K32(
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4() )
set ) ) , the
carrier of
b1 : ( ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like )
ComplexNormSpace) : ( ( ) (
V4() )
set ) ) )
Element of
K32(
K33(
NAT : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal V52()
V53()
V54()
V55()
V56()
V57()
V58() )
Element of
K32(
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4() )
set ) ) , the
carrier of
b1 : ( ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like )
ComplexNormSpace) : ( ( ) (
V4() )
set ) ) : ( ( ) (
V4() )
set ) ) : ( ( ) (
V4() )
set ) )
= ((Partial_Sums s : ( ( Function-like V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed V133() V134() vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed V133() V134() vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( V4() ) set ) ) ) ( V4() Relation-like NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) -defined the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed V133() V134() vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed V133() V134() vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( V4() ) set ) -valued Function-like V29( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) ) V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed V133() V134() vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed V133() V134() vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( V4() ) set ) ) ) sequence of ( ( ) ( V4() ) set ) ) ) : ( ( Function-like V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed V133() V134() vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed V133() V134() vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( V4() ) set ) ) ) ( V4() Relation-like NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) -defined the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed V133() V134() vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed V133() V134() vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( V4() ) set ) -valued Function-like V29( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) ) V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed V133() V134() vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed V133() V134() vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( V4() ) set ) ) ) Element of K32(K33(NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed V133() V134() vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed V133() V134() vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( V4() ) set ) ) : ( ( ) ( V4() ) set ) ) : ( ( ) ( V4() ) set ) ) ^\ 1 : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal natural V24() real ext-real positive non negative V45() V46() V52() V53() V54() V55() V56() V57() ) Element of NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) ) ) : ( ( ) (
V4()
Relation-like NAT : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal V52()
V53()
V54()
V55()
V56()
V57()
V58() )
Element of
K32(
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4() )
set ) )
-defined the
carrier of
b1 : ( ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like )
ComplexNormSpace) : ( ( ) (
V4() )
set )
-valued Function-like V29(
NAT : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal V52()
V53()
V54()
V55()
V56()
V57()
V58() )
Element of
K32(
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4() )
set ) ) )
V30(
NAT : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal V52()
V53()
V54()
V55()
V56()
V57()
V58() )
Element of
K32(
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4() )
set ) ) , the
carrier of
b1 : ( ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like )
ComplexNormSpace) : ( ( ) (
V4() )
set ) ) )
subsequence of
Partial_Sums b2 : ( (
Function-like V30(
NAT : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal V52()
V53()
V54()
V55()
V56()
V57()
V58() )
Element of
K32(
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4() )
set ) ) , the
carrier of
b1 : ( ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like )
ComplexNormSpace) : ( ( ) (
V4() )
set ) ) ) (
V4()
Relation-like NAT : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal V52()
V53()
V54()
V55()
V56()
V57()
V58() )
Element of
K32(
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4() )
set ) )
-defined the
carrier of
b1 : ( ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like )
ComplexNormSpace) : ( ( ) (
V4() )
set )
-valued Function-like V29(
NAT : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal V52()
V53()
V54()
V55()
V56()
V57()
V58() )
Element of
K32(
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4() )
set ) ) )
V30(
NAT : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal V52()
V53()
V54()
V55()
V56()
V57()
V58() )
Element of
K32(
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4() )
set ) ) , the
carrier of
b1 : ( ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like )
ComplexNormSpace) : ( ( ) (
V4() )
set ) ) )
sequence of ( ( ) (
V4() )
set ) ) : ( (
Function-like V30(
NAT : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal V52()
V53()
V54()
V55()
V56()
V57()
V58() )
Element of
K32(
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4() )
set ) ) , the
carrier of
b1 : ( ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like )
ComplexNormSpace) : ( ( ) (
V4() )
set ) ) ) (
V4()
Relation-like NAT : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal V52()
V53()
V54()
V55()
V56()
V57()
V58() )
Element of
K32(
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4() )
set ) )
-defined the
carrier of
b1 : ( ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like )
ComplexNormSpace) : ( ( ) (
V4() )
set )
-valued Function-like V29(
NAT : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal V52()
V53()
V54()
V55()
V56()
V57()
V58() )
Element of
K32(
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4() )
set ) ) )
V30(
NAT : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal V52()
V53()
V54()
V55()
V56()
V57()
V58() )
Element of
K32(
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4() )
set ) ) , the
carrier of
b1 : ( ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like )
ComplexNormSpace) : ( ( ) (
V4() )
set ) ) )
Element of
K32(
K33(
NAT : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal V52()
V53()
V54()
V55()
V56()
V57()
V58() )
Element of
K32(
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4() )
set ) ) , the
carrier of
b1 : ( ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like )
ComplexNormSpace) : ( ( ) (
V4() )
set ) ) : ( ( ) (
V4() )
set ) ) : ( ( ) (
V4() )
set ) ) )
- s1 : ( (
Function-like V30(
NAT : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal V52()
V53()
V54()
V55()
V56()
V57()
V58() )
Element of
K32(
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4() )
set ) ) , the
carrier of
b1 : ( ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like )
ComplexNormSpace) : ( ( ) (
V4() )
set ) ) ) (
V4()
Relation-like NAT : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal V52()
V53()
V54()
V55()
V56()
V57()
V58() )
Element of
K32(
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4() )
set ) )
-defined the
carrier of
b1 : ( ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like )
ComplexNormSpace) : ( ( ) (
V4() )
set )
-valued Function-like V29(
NAT : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal V52()
V53()
V54()
V55()
V56()
V57()
V58() )
Element of
K32(
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4() )
set ) ) )
V30(
NAT : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal V52()
V53()
V54()
V55()
V56()
V57()
V58() )
Element of
K32(
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4() )
set ) ) , the
carrier of
b1 : ( ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like )
ComplexNormSpace) : ( ( ) (
V4() )
set ) ) )
sequence of ( ( ) (
V4() )
set ) ) : ( (
Function-like V30(
NAT : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal V52()
V53()
V54()
V55()
V56()
V57()
V58() )
Element of
K32(
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4() )
set ) ) , the
carrier of
b1 : ( ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like )
ComplexNormSpace) : ( ( ) (
V4() )
set ) ) ) (
V4()
Relation-like NAT : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal V52()
V53()
V54()
V55()
V56()
V57()
V58() )
Element of
K32(
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4() )
set ) )
-defined the
carrier of
b1 : ( ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like )
ComplexNormSpace) : ( ( ) (
V4() )
set )
-valued Function-like V29(
NAT : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal V52()
V53()
V54()
V55()
V56()
V57()
V58() )
Element of
K32(
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4() )
set ) ) )
V30(
NAT : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal V52()
V53()
V54()
V55()
V56()
V57()
V58() )
Element of
K32(
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4() )
set ) ) , the
carrier of
b1 : ( ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like )
ComplexNormSpace) : ( ( ) (
V4() )
set ) ) )
Element of
K32(
K33(
NAT : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal V52()
V53()
V54()
V55()
V56()
V57()
V58() )
Element of
K32(
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4() )
set ) ) , the
carrier of
b1 : ( ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like )
ComplexNormSpace) : ( ( ) (
V4() )
set ) ) : ( ( ) (
V4() )
set ) ) : ( ( ) (
V4() )
set ) ) ;
theorem
for
X being ( ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like )
ComplexNormSpace)
for
seq being ( (
Function-like V30(
NAT : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal V52()
V53()
V54()
V55()
V56()
V57()
V58() )
Element of
K32(
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4() )
set ) ) , the
carrier of
b1 : ( ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like )
ComplexNormSpace) : ( ( ) (
V4() )
set ) ) ) (
V4()
Relation-like NAT : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal V52()
V53()
V54()
V55()
V56()
V57()
V58() )
Element of
K32(
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4() )
set ) )
-defined the
carrier of
b1 : ( ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like )
ComplexNormSpace) : ( ( ) (
V4() )
set )
-valued Function-like V29(
NAT : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal V52()
V53()
V54()
V55()
V56()
V57()
V58() )
Element of
K32(
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4() )
set ) ) )
V30(
NAT : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal V52()
V53()
V54()
V55()
V56()
V57()
V58() )
Element of
K32(
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4() )
set ) ) , the
carrier of
b1 : ( ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like )
ComplexNormSpace) : ( ( ) (
V4() )
set ) ) )
sequence of ( ( ) (
V4() )
set ) ) holds
(
Partial_Sums ||.seq : ( ( Function-like V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed V133() V134() vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed V133() V134() vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( V4() ) set ) ) ) ( V4() Relation-like NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) -defined the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed V133() V134() vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed V133() V134() vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( V4() ) set ) -valued Function-like V29( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) ) V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed V133() V134() vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed V133() V134() vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( V4() ) set ) ) ) sequence of ( ( ) ( V4() ) set ) ) .|| : ( (
Function-like V30(
NAT : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal V52()
V53()
V54()
V55()
V56()
V57()
V58() )
Element of
K32(
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4() )
set ) ) ,
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) ) (
V4()
Relation-like NAT : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal V52()
V53()
V54()
V55()
V56()
V57()
V58() )
Element of
K32(
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4() )
set ) )
-defined REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set )
-valued Function-like V29(
NAT : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal V52()
V53()
V54()
V55()
V56()
V57()
V58() )
Element of
K32(
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4() )
set ) ) )
V30(
NAT : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal V52()
V53()
V54()
V55()
V56()
V57()
V58() )
Element of
K32(
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4() )
set ) ) ,
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) )
complex-valued ext-real-valued real-valued )
Element of
K32(
K33(
NAT : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal V52()
V53()
V54()
V55()
V56()
V57()
V58() )
Element of
K32(
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4() )
set ) ) ,
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4()
complex-valued ext-real-valued real-valued )
set ) ) : ( ( ) (
V4() )
set ) ) : ( (
Function-like V30(
NAT : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal V52()
V53()
V54()
V55()
V56()
V57()
V58() )
Element of
K32(
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4() )
set ) ) ,
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) ) (
V4()
Relation-like NAT : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal V52()
V53()
V54()
V55()
V56()
V57()
V58() )
Element of
K32(
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4() )
set ) )
-defined REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set )
-valued Function-like V29(
NAT : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal V52()
V53()
V54()
V55()
V56()
V57()
V58() )
Element of
K32(
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4() )
set ) ) )
V30(
NAT : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal V52()
V53()
V54()
V55()
V56()
V57()
V58() )
Element of
K32(
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4() )
set ) ) ,
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) )
complex-valued ext-real-valued real-valued )
Element of
K32(
K33(
NAT : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal V52()
V53()
V54()
V55()
V56()
V57()
V58() )
Element of
K32(
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4() )
set ) ) ,
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4()
complex-valued ext-real-valued real-valued )
set ) ) : ( ( ) (
V4() )
set ) ) is
bounded_above iff
seq : ( (
Function-like V30(
NAT : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal V52()
V53()
V54()
V55()
V56()
V57()
V58() )
Element of
K32(
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4() )
set ) ) , the
carrier of
b1 : ( ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like )
ComplexNormSpace) : ( ( ) (
V4() )
set ) ) ) (
V4()
Relation-like NAT : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal V52()
V53()
V54()
V55()
V56()
V57()
V58() )
Element of
K32(
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4() )
set ) )
-defined the
carrier of
b1 : ( ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like )
ComplexNormSpace) : ( ( ) (
V4() )
set )
-valued Function-like V29(
NAT : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal V52()
V53()
V54()
V55()
V56()
V57()
V58() )
Element of
K32(
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4() )
set ) ) )
V30(
NAT : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal V52()
V53()
V54()
V55()
V56()
V57()
V58() )
Element of
K32(
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4() )
set ) ) , the
carrier of
b1 : ( ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like )
ComplexNormSpace) : ( ( ) (
V4() )
set ) ) )
sequence of ( ( ) (
V4() )
set ) ) is
norm_summable ) ;
theorem
for
X being ( ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like complete ) ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like complete )
ComplexBanachSpace)
for
seq being ( (
Function-like V30(
NAT : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal V52()
V53()
V54()
V55()
V56()
V57()
V58() )
Element of
K32(
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4() )
set ) ) , the
carrier of
b1 : ( ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like complete ) ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like complete )
ComplexBanachSpace) : ( ( ) (
V4() )
set ) ) ) (
V4()
Relation-like NAT : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal V52()
V53()
V54()
V55()
V56()
V57()
V58() )
Element of
K32(
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4() )
set ) )
-defined the
carrier of
b1 : ( ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like complete ) ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like complete )
ComplexBanachSpace) : ( ( ) (
V4() )
set )
-valued Function-like V29(
NAT : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal V52()
V53()
V54()
V55()
V56()
V57()
V58() )
Element of
K32(
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4() )
set ) ) )
V30(
NAT : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal V52()
V53()
V54()
V55()
V56()
V57()
V58() )
Element of
K32(
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4() )
set ) ) , the
carrier of
b1 : ( ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like complete ) ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like complete )
ComplexBanachSpace) : ( ( ) (
V4() )
set ) ) )
sequence of ( ( ) (
V4() )
set ) ) holds
(
seq : ( (
Function-like V30(
NAT : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal V52()
V53()
V54()
V55()
V56()
V57()
V58() )
Element of
K32(
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4() )
set ) ) , the
carrier of
b1 : ( ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like complete ) ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like complete )
ComplexBanachSpace) : ( ( ) (
V4() )
set ) ) ) (
V4()
Relation-like NAT : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal V52()
V53()
V54()
V55()
V56()
V57()
V58() )
Element of
K32(
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4() )
set ) )
-defined the
carrier of
b1 : ( ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like complete ) ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like complete )
ComplexBanachSpace) : ( ( ) (
V4() )
set )
-valued Function-like V29(
NAT : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal V52()
V53()
V54()
V55()
V56()
V57()
V58() )
Element of
K32(
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4() )
set ) ) )
V30(
NAT : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal V52()
V53()
V54()
V55()
V56()
V57()
V58() )
Element of
K32(
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4() )
set ) ) , the
carrier of
b1 : ( ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like complete ) ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like complete )
ComplexBanachSpace) : ( ( ) (
V4() )
set ) ) )
sequence of ( ( ) (
V4() )
set ) ) is
summable iff for
p being ( ( ) (
V24()
real ext-real )
Real) st
0 : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural V24()
real ext-real non
positive non
negative V45()
V46()
V52()
V53()
V54()
V55()
V56()
V57()
V58() )
Element of
NAT : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal V52()
V53()
V54()
V55()
V56()
V57()
V58() )
Element of
K32(
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4() )
set ) ) )
< p : ( ( ) (
V24()
real ext-real )
Real) holds
ex
n being ( ( ) (
epsilon-transitive epsilon-connected ordinal natural V24()
real ext-real V45()
V46()
V52()
V53()
V54()
V55()
V56()
V57() )
Element of
NAT : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal V52()
V53()
V54()
V55()
V56()
V57()
V58() )
Element of
K32(
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4() )
set ) ) ) st
for
m being ( ( ) (
epsilon-transitive epsilon-connected ordinal natural V24()
real ext-real V45()
V46()
V52()
V53()
V54()
V55()
V56()
V57() )
Element of
NAT : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal V52()
V53()
V54()
V55()
V56()
V57()
V58() )
Element of
K32(
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4() )
set ) ) ) st
n : ( ( ) (
epsilon-transitive epsilon-connected ordinal natural V24()
real ext-real V45()
V46()
V52()
V53()
V54()
V55()
V56()
V57() )
Element of
NAT : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal V52()
V53()
V54()
V55()
V56()
V57()
V58() )
Element of
K32(
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4() )
set ) ) )
<= m : ( ( ) (
epsilon-transitive epsilon-connected ordinal natural V24()
real ext-real V45()
V46()
V52()
V53()
V54()
V55()
V56()
V57() )
Element of
NAT : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal V52()
V53()
V54()
V55()
V56()
V57()
V58() )
Element of
K32(
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4() )
set ) ) ) holds
||.(((Partial_Sums seq : ( ( Function-like V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed V133() V134() vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like complete ) ( non empty right_complementable Abelian add-associative right_zeroed V133() V134() vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like complete ) ComplexBanachSpace) : ( ( ) ( V4() ) set ) ) ) ( V4() Relation-like NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) -defined the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed V133() V134() vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like complete ) ( non empty right_complementable Abelian add-associative right_zeroed V133() V134() vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like complete ) ComplexBanachSpace) : ( ( ) ( V4() ) set ) -valued Function-like V29( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) ) V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed V133() V134() vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like complete ) ( non empty right_complementable Abelian add-associative right_zeroed V133() V134() vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like complete ) ComplexBanachSpace) : ( ( ) ( V4() ) set ) ) ) sequence of ( ( ) ( V4() ) set ) ) ) : ( ( Function-like V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed V133() V134() vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like complete ) ( non empty right_complementable Abelian add-associative right_zeroed V133() V134() vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like complete ) ComplexBanachSpace) : ( ( ) ( V4() ) set ) ) ) ( V4() Relation-like NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) -defined the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed V133() V134() vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like complete ) ( non empty right_complementable Abelian add-associative right_zeroed V133() V134() vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like complete ) ComplexBanachSpace) : ( ( ) ( V4() ) set ) -valued Function-like V29( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) ) V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed V133() V134() vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like complete ) ( non empty right_complementable Abelian add-associative right_zeroed V133() V134() vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like complete ) ComplexBanachSpace) : ( ( ) ( V4() ) set ) ) ) Element of K32(K33(NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed V133() V134() vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like complete ) ( non empty right_complementable Abelian add-associative right_zeroed V133() V134() vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like complete ) ComplexBanachSpace) : ( ( ) ( V4() ) set ) ) : ( ( ) ( V4() ) set ) ) : ( ( ) ( V4() ) set ) ) . m : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V24() real ext-real V45() V46() V52() V53() V54() V55() V56() V57() ) Element of NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) ) ) : ( ( ) ( right_complementable ) Element of the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed V133() V134() vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like complete ) ( non empty right_complementable Abelian add-associative right_zeroed V133() V134() vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like complete ) ComplexBanachSpace) : ( ( ) ( V4() ) set ) ) - ((Partial_Sums seq : ( ( Function-like V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed V133() V134() vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like complete ) ( non empty right_complementable Abelian add-associative right_zeroed V133() V134() vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like complete ) ComplexBanachSpace) : ( ( ) ( V4() ) set ) ) ) ( V4() Relation-like NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) -defined the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed V133() V134() vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like complete ) ( non empty right_complementable Abelian add-associative right_zeroed V133() V134() vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like complete ) ComplexBanachSpace) : ( ( ) ( V4() ) set ) -valued Function-like V29( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) ) V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed V133() V134() vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like complete ) ( non empty right_complementable Abelian add-associative right_zeroed V133() V134() vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like complete ) ComplexBanachSpace) : ( ( ) ( V4() ) set ) ) ) sequence of ( ( ) ( V4() ) set ) ) ) : ( ( Function-like V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed V133() V134() vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like complete ) ( non empty right_complementable Abelian add-associative right_zeroed V133() V134() vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like complete ) ComplexBanachSpace) : ( ( ) ( V4() ) set ) ) ) ( V4() Relation-like NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) -defined the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed V133() V134() vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like complete ) ( non empty right_complementable Abelian add-associative right_zeroed V133() V134() vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like complete ) ComplexBanachSpace) : ( ( ) ( V4() ) set ) -valued Function-like V29( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) ) V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed V133() V134() vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like complete ) ( non empty right_complementable Abelian add-associative right_zeroed V133() V134() vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like complete ) ComplexBanachSpace) : ( ( ) ( V4() ) set ) ) ) Element of K32(K33(NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed V133() V134() vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like complete ) ( non empty right_complementable Abelian add-associative right_zeroed V133() V134() vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like complete ) ComplexBanachSpace) : ( ( ) ( V4() ) set ) ) : ( ( ) ( V4() ) set ) ) : ( ( ) ( V4() ) set ) ) . n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V24() real ext-real V45() V46() V52() V53() V54() V55() V56() V57() ) Element of NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) ) ) : ( ( ) ( right_complementable ) Element of the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed V133() V134() vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like complete ) ( non empty right_complementable Abelian add-associative right_zeroed V133() V134() vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like complete ) ComplexBanachSpace) : ( ( ) ( V4() ) set ) ) ) : ( ( ) ( right_complementable ) Element of the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed V133() V134() vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like complete ) ( non empty right_complementable Abelian add-associative right_zeroed V133() V134() vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like complete ) ComplexBanachSpace) : ( ( ) ( V4() ) set ) ) .|| : ( ( ) (
V24()
real ext-real )
Element of
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) )
< p : ( ( ) (
V24()
real ext-real )
Real) ) ;
theorem
for
X being ( ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like )
ComplexNormSpace)
for
s being ( (
Function-like V30(
NAT : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal V52()
V53()
V54()
V55()
V56()
V57()
V58() )
Element of
K32(
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4() )
set ) ) , the
carrier of
b1 : ( ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like )
ComplexNormSpace) : ( ( ) (
V4() )
set ) ) ) (
V4()
Relation-like NAT : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal V52()
V53()
V54()
V55()
V56()
V57()
V58() )
Element of
K32(
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4() )
set ) )
-defined the
carrier of
b1 : ( ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like )
ComplexNormSpace) : ( ( ) (
V4() )
set )
-valued Function-like V29(
NAT : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal V52()
V53()
V54()
V55()
V56()
V57()
V58() )
Element of
K32(
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4() )
set ) ) )
V30(
NAT : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal V52()
V53()
V54()
V55()
V56()
V57()
V58() )
Element of
K32(
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4() )
set ) ) , the
carrier of
b1 : ( ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like )
ComplexNormSpace) : ( ( ) (
V4() )
set ) ) )
sequence of ( ( ) (
V4() )
set ) )
for
n,
m being ( ( ) (
epsilon-transitive epsilon-connected ordinal natural V24()
real ext-real V45()
V46()
V52()
V53()
V54()
V55()
V56()
V57() )
Element of
NAT : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal V52()
V53()
V54()
V55()
V56()
V57()
V58() )
Element of
K32(
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4() )
set ) ) ) st
n : ( ( ) (
epsilon-transitive epsilon-connected ordinal natural V24()
real ext-real V45()
V46()
V52()
V53()
V54()
V55()
V56()
V57() )
Element of
NAT : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal V52()
V53()
V54()
V55()
V56()
V57()
V58() )
Element of
K32(
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4() )
set ) ) )
<= m : ( ( ) (
epsilon-transitive epsilon-connected ordinal natural V24()
real ext-real V45()
V46()
V52()
V53()
V54()
V55()
V56()
V57() )
Element of
NAT : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal V52()
V53()
V54()
V55()
V56()
V57()
V58() )
Element of
K32(
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4() )
set ) ) ) holds
||.(((Partial_Sums s : ( ( Function-like V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed V133() V134() vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed V133() V134() vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( V4() ) set ) ) ) ( V4() Relation-like NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) -defined the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed V133() V134() vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed V133() V134() vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( V4() ) set ) -valued Function-like V29( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) ) V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed V133() V134() vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed V133() V134() vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( V4() ) set ) ) ) sequence of ( ( ) ( V4() ) set ) ) ) : ( ( Function-like V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed V133() V134() vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed V133() V134() vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( V4() ) set ) ) ) ( V4() Relation-like NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) -defined the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed V133() V134() vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed V133() V134() vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( V4() ) set ) -valued Function-like V29( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) ) V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed V133() V134() vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed V133() V134() vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( V4() ) set ) ) ) Element of K32(K33(NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed V133() V134() vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed V133() V134() vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( V4() ) set ) ) : ( ( ) ( V4() ) set ) ) : ( ( ) ( V4() ) set ) ) . m : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V24() real ext-real V45() V46() V52() V53() V54() V55() V56() V57() ) Element of NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) ) ) : ( ( ) ( right_complementable ) Element of the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed V133() V134() vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed V133() V134() vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( V4() ) set ) ) - ((Partial_Sums s : ( ( Function-like V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed V133() V134() vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed V133() V134() vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( V4() ) set ) ) ) ( V4() Relation-like NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) -defined the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed V133() V134() vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed V133() V134() vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( V4() ) set ) -valued Function-like V29( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) ) V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed V133() V134() vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed V133() V134() vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( V4() ) set ) ) ) sequence of ( ( ) ( V4() ) set ) ) ) : ( ( Function-like V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed V133() V134() vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed V133() V134() vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( V4() ) set ) ) ) ( V4() Relation-like NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) -defined the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed V133() V134() vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed V133() V134() vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( V4() ) set ) -valued Function-like V29( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) ) V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed V133() V134() vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed V133() V134() vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( V4() ) set ) ) ) Element of K32(K33(NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed V133() V134() vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed V133() V134() vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( V4() ) set ) ) : ( ( ) ( V4() ) set ) ) : ( ( ) ( V4() ) set ) ) . n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V24() real ext-real V45() V46() V52() V53() V54() V55() V56() V57() ) Element of NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) ) ) : ( ( ) ( right_complementable ) Element of the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed V133() V134() vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed V133() V134() vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( V4() ) set ) ) ) : ( ( ) ( right_complementable ) Element of the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed V133() V134() vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed V133() V134() vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( V4() ) set ) ) .|| : ( ( ) (
V24()
real ext-real )
Element of
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) )
<= abs (((Partial_Sums ||.s : ( ( Function-like V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed V133() V134() vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed V133() V134() vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( V4() ) set ) ) ) ( V4() Relation-like NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) -defined the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed V133() V134() vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed V133() V134() vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( V4() ) set ) -valued Function-like V29( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) ) V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed V133() V134() vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed V133() V134() vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( V4() ) set ) ) ) sequence of ( ( ) ( V4() ) set ) ) .|| : ( ( Function-like V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) ) ( V4() Relation-like NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) -defined REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) -valued Function-like V29( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) ) V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) complex-valued ext-real-valued real-valued ) Element of K32(K33(NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) ,REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( V4() ) set ) ) ) : ( ( Function-like V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) ) ( V4() Relation-like NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) -defined REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) -valued Function-like V29( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) ) V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) complex-valued ext-real-valued real-valued ) Element of K32(K33(NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) ,REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( V4() ) set ) ) . m : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V24() real ext-real V45() V46() V52() V53() V54() V55() V56() V57() ) Element of NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) ) ) : ( ( ) ( V24() real ext-real ) Element of REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) - ((Partial_Sums ||.s : ( ( Function-like V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed V133() V134() vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed V133() V134() vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( V4() ) set ) ) ) ( V4() Relation-like NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) -defined the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed V133() V134() vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed V133() V134() vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( V4() ) set ) -valued Function-like V29( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) ) V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed V133() V134() vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed V133() V134() vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( V4() ) set ) ) ) sequence of ( ( ) ( V4() ) set ) ) .|| : ( ( Function-like V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) ) ( V4() Relation-like NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) -defined REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) -valued Function-like V29( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) ) V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) complex-valued ext-real-valued real-valued ) Element of K32(K33(NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) ,REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( V4() ) set ) ) ) : ( ( Function-like V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) ) ( V4() Relation-like NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) -defined REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) -valued Function-like V29( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) ) V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) complex-valued ext-real-valued real-valued ) Element of K32(K33(NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) ,REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( V4() ) set ) ) . n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V24() real ext-real V45() V46() V52() V53() V54() V55() V56() V57() ) Element of NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) ) ) : ( ( ) ( V24() real ext-real ) Element of REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) ) : ( ( ) (
V24()
real ext-real )
Element of
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V24()
real ext-real )
Element of
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) ;
theorem
for
X being ( ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like )
ComplexNormSpace)
for
rseq1 being ( (
Function-like V30(
NAT : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal V52()
V53()
V54()
V55()
V56()
V57()
V58() )
Element of
K32(
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4() )
set ) ) ,
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) ) (
V4()
Relation-like NAT : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal V52()
V53()
V54()
V55()
V56()
V57()
V58() )
Element of
K32(
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4() )
set ) )
-defined REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set )
-valued Function-like V29(
NAT : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal V52()
V53()
V54()
V55()
V56()
V57()
V58() )
Element of
K32(
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4() )
set ) ) )
V30(
NAT : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal V52()
V53()
V54()
V55()
V56()
V57()
V58() )
Element of
K32(
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4() )
set ) ) ,
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) )
complex-valued ext-real-valued real-valued )
Real_Sequence)
for
seq2 being ( (
Function-like V30(
NAT : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal V52()
V53()
V54()
V55()
V56()
V57()
V58() )
Element of
K32(
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4() )
set ) ) , the
carrier of
b1 : ( ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like )
ComplexNormSpace) : ( ( ) (
V4() )
set ) ) ) (
V4()
Relation-like NAT : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal V52()
V53()
V54()
V55()
V56()
V57()
V58() )
Element of
K32(
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4() )
set ) )
-defined the
carrier of
b1 : ( ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like )
ComplexNormSpace) : ( ( ) (
V4() )
set )
-valued Function-like V29(
NAT : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal V52()
V53()
V54()
V55()
V56()
V57()
V58() )
Element of
K32(
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4() )
set ) ) )
V30(
NAT : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal V52()
V53()
V54()
V55()
V56()
V57()
V58() )
Element of
K32(
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4() )
set ) ) , the
carrier of
b1 : ( ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like )
ComplexNormSpace) : ( ( ) (
V4() )
set ) ) )
sequence of ( ( ) (
V4() )
set ) ) st
rseq1 : ( (
Function-like V30(
NAT : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal V52()
V53()
V54()
V55()
V56()
V57()
V58() )
Element of
K32(
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4() )
set ) ) ,
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) ) (
V4()
Relation-like NAT : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal V52()
V53()
V54()
V55()
V56()
V57()
V58() )
Element of
K32(
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4() )
set ) )
-defined REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set )
-valued Function-like V29(
NAT : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal V52()
V53()
V54()
V55()
V56()
V57()
V58() )
Element of
K32(
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4() )
set ) ) )
V30(
NAT : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal V52()
V53()
V54()
V55()
V56()
V57()
V58() )
Element of
K32(
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4() )
set ) ) ,
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) )
complex-valued ext-real-valued real-valued )
Real_Sequence) is
summable & ex
m being ( ( ) (
epsilon-transitive epsilon-connected ordinal natural V24()
real ext-real V45()
V46()
V52()
V53()
V54()
V55()
V56()
V57() )
Element of
NAT : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal V52()
V53()
V54()
V55()
V56()
V57()
V58() )
Element of
K32(
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4() )
set ) ) ) st
for
n being ( ( ) (
epsilon-transitive epsilon-connected ordinal natural V24()
real ext-real V45()
V46()
V52()
V53()
V54()
V55()
V56()
V57() )
Element of
NAT : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal V52()
V53()
V54()
V55()
V56()
V57()
V58() )
Element of
K32(
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4() )
set ) ) ) st
m : ( ( ) (
epsilon-transitive epsilon-connected ordinal natural V24()
real ext-real V45()
V46()
V52()
V53()
V54()
V55()
V56()
V57() )
Element of
NAT : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal V52()
V53()
V54()
V55()
V56()
V57()
V58() )
Element of
K32(
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4() )
set ) ) )
<= n : ( ( ) (
epsilon-transitive epsilon-connected ordinal natural V24()
real ext-real V45()
V46()
V52()
V53()
V54()
V55()
V56()
V57() )
Element of
NAT : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal V52()
V53()
V54()
V55()
V56()
V57()
V58() )
Element of
K32(
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4() )
set ) ) ) holds
||.(seq2 : ( ( Function-like V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed V133() V134() vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed V133() V134() vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( V4() ) set ) ) ) ( V4() Relation-like NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) -defined the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed V133() V134() vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed V133() V134() vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( V4() ) set ) -valued Function-like V29( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) ) V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed V133() V134() vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed V133() V134() vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( V4() ) set ) ) ) sequence of ( ( ) ( V4() ) set ) ) . n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V24() real ext-real V45() V46() V52() V53() V54() V55() V56() V57() ) Element of NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) ) ) : ( ( ) ( right_complementable ) Element of the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed V133() V134() vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed V133() V134() vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( V4() ) set ) ) .|| : ( ( ) (
V24()
real ext-real )
Element of
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) )
<= rseq1 : ( (
Function-like V30(
NAT : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal V52()
V53()
V54()
V55()
V56()
V57()
V58() )
Element of
K32(
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4() )
set ) ) ,
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) ) (
V4()
Relation-like NAT : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal V52()
V53()
V54()
V55()
V56()
V57()
V58() )
Element of
K32(
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4() )
set ) )
-defined REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set )
-valued Function-like V29(
NAT : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal V52()
V53()
V54()
V55()
V56()
V57()
V58() )
Element of
K32(
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4() )
set ) ) )
V30(
NAT : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal V52()
V53()
V54()
V55()
V56()
V57()
V58() )
Element of
K32(
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4() )
set ) ) ,
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) )
complex-valued ext-real-valued real-valued )
Real_Sequence)
. n : ( ( ) (
epsilon-transitive epsilon-connected ordinal natural V24()
real ext-real V45()
V46()
V52()
V53()
V54()
V55()
V56()
V57() )
Element of
NAT : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal V52()
V53()
V54()
V55()
V56()
V57()
V58() )
Element of
K32(
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4() )
set ) ) ) : ( ( ) (
V24()
real ext-real )
Element of
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) holds
seq2 : ( (
Function-like V30(
NAT : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal V52()
V53()
V54()
V55()
V56()
V57()
V58() )
Element of
K32(
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4() )
set ) ) , the
carrier of
b1 : ( ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like )
ComplexNormSpace) : ( ( ) (
V4() )
set ) ) ) (
V4()
Relation-like NAT : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal V52()
V53()
V54()
V55()
V56()
V57()
V58() )
Element of
K32(
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4() )
set ) )
-defined the
carrier of
b1 : ( ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like )
ComplexNormSpace) : ( ( ) (
V4() )
set )
-valued Function-like V29(
NAT : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal V52()
V53()
V54()
V55()
V56()
V57()
V58() )
Element of
K32(
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4() )
set ) ) )
V30(
NAT : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal V52()
V53()
V54()
V55()
V56()
V57()
V58() )
Element of
K32(
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4() )
set ) ) , the
carrier of
b1 : ( ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like )
ComplexNormSpace) : ( ( ) (
V4() )
set ) ) )
sequence of ( ( ) (
V4() )
set ) ) is
norm_summable ;
theorem
for
X being ( ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like )
ComplexNormSpace)
for
seq1,
seq2 being ( (
Function-like V30(
NAT : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal V52()
V53()
V54()
V55()
V56()
V57()
V58() )
Element of
K32(
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4() )
set ) ) , the
carrier of
b1 : ( ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like )
ComplexNormSpace) : ( ( ) (
V4() )
set ) ) ) (
V4()
Relation-like NAT : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal V52()
V53()
V54()
V55()
V56()
V57()
V58() )
Element of
K32(
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4() )
set ) )
-defined the
carrier of
b1 : ( ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like )
ComplexNormSpace) : ( ( ) (
V4() )
set )
-valued Function-like V29(
NAT : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal V52()
V53()
V54()
V55()
V56()
V57()
V58() )
Element of
K32(
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4() )
set ) ) )
V30(
NAT : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal V52()
V53()
V54()
V55()
V56()
V57()
V58() )
Element of
K32(
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4() )
set ) ) , the
carrier of
b1 : ( ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like )
ComplexNormSpace) : ( ( ) (
V4() )
set ) ) )
sequence of ( ( ) (
V4() )
set ) ) st ( for
n being ( ( ) (
epsilon-transitive epsilon-connected ordinal natural V24()
real ext-real V45()
V46()
V52()
V53()
V54()
V55()
V56()
V57() )
Element of
NAT : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal V52()
V53()
V54()
V55()
V56()
V57()
V58() )
Element of
K32(
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4() )
set ) ) ) holds
(
0 : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural V24()
real ext-real non
positive non
negative V45()
V46()
V52()
V53()
V54()
V55()
V56()
V57()
V58() )
Element of
NAT : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal V52()
V53()
V54()
V55()
V56()
V57()
V58() )
Element of
K32(
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4() )
set ) ) )
<= ||.seq1 : ( ( Function-like V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed V133() V134() vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed V133() V134() vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( V4() ) set ) ) ) ( V4() Relation-like NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) -defined the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed V133() V134() vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed V133() V134() vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( V4() ) set ) -valued Function-like V29( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) ) V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed V133() V134() vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed V133() V134() vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( V4() ) set ) ) ) sequence of ( ( ) ( V4() ) set ) ) .|| : ( (
Function-like V30(
NAT : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal V52()
V53()
V54()
V55()
V56()
V57()
V58() )
Element of
K32(
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4() )
set ) ) ,
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) ) (
V4()
Relation-like NAT : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal V52()
V53()
V54()
V55()
V56()
V57()
V58() )
Element of
K32(
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4() )
set ) )
-defined REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set )
-valued Function-like V29(
NAT : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal V52()
V53()
V54()
V55()
V56()
V57()
V58() )
Element of
K32(
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4() )
set ) ) )
V30(
NAT : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal V52()
V53()
V54()
V55()
V56()
V57()
V58() )
Element of
K32(
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4() )
set ) ) ,
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) )
complex-valued ext-real-valued real-valued )
Element of
K32(
K33(
NAT : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal V52()
V53()
V54()
V55()
V56()
V57()
V58() )
Element of
K32(
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4() )
set ) ) ,
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4()
complex-valued ext-real-valued real-valued )
set ) ) : ( ( ) (
V4() )
set ) )
. n : ( ( ) (
epsilon-transitive epsilon-connected ordinal natural V24()
real ext-real V45()
V46()
V52()
V53()
V54()
V55()
V56()
V57() )
Element of
NAT : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal V52()
V53()
V54()
V55()
V56()
V57()
V58() )
Element of
K32(
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4() )
set ) ) ) : ( ( ) (
V24()
real ext-real )
Element of
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) &
||.seq1 : ( ( Function-like V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed V133() V134() vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed V133() V134() vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( V4() ) set ) ) ) ( V4() Relation-like NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) -defined the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed V133() V134() vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed V133() V134() vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( V4() ) set ) -valued Function-like V29( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) ) V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed V133() V134() vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed V133() V134() vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( V4() ) set ) ) ) sequence of ( ( ) ( V4() ) set ) ) .|| : ( (
Function-like V30(
NAT : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal V52()
V53()
V54()
V55()
V56()
V57()
V58() )
Element of
K32(
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4() )
set ) ) ,
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) ) (
V4()
Relation-like NAT : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal V52()
V53()
V54()
V55()
V56()
V57()
V58() )
Element of
K32(
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4() )
set ) )
-defined REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set )
-valued Function-like V29(
NAT : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal V52()
V53()
V54()
V55()
V56()
V57()
V58() )
Element of
K32(
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4() )
set ) ) )
V30(
NAT : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal V52()
V53()
V54()
V55()
V56()
V57()
V58() )
Element of
K32(
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4() )
set ) ) ,
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) )
complex-valued ext-real-valued real-valued )
Element of
K32(
K33(
NAT : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal V52()
V53()
V54()
V55()
V56()
V57()
V58() )
Element of
K32(
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4() )
set ) ) ,
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4()
complex-valued ext-real-valued real-valued )
set ) ) : ( ( ) (
V4() )
set ) )
. n : ( ( ) (
epsilon-transitive epsilon-connected ordinal natural V24()
real ext-real V45()
V46()
V52()
V53()
V54()
V55()
V56()
V57() )
Element of
NAT : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal V52()
V53()
V54()
V55()
V56()
V57()
V58() )
Element of
K32(
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4() )
set ) ) ) : ( ( ) (
V24()
real ext-real )
Element of
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) )
<= ||.seq2 : ( ( Function-like V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed V133() V134() vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed V133() V134() vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( V4() ) set ) ) ) ( V4() Relation-like NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) -defined the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed V133() V134() vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed V133() V134() vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( V4() ) set ) -valued Function-like V29( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) ) V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed V133() V134() vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed V133() V134() vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( V4() ) set ) ) ) sequence of ( ( ) ( V4() ) set ) ) .|| : ( (
Function-like V30(
NAT : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal V52()
V53()
V54()
V55()
V56()
V57()
V58() )
Element of
K32(
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4() )
set ) ) ,
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) ) (
V4()
Relation-like NAT : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal V52()
V53()
V54()
V55()
V56()
V57()
V58() )
Element of
K32(
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4() )
set ) )
-defined REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set )
-valued Function-like V29(
NAT : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal V52()
V53()
V54()
V55()
V56()
V57()
V58() )
Element of
K32(
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4() )
set ) ) )
V30(
NAT : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal V52()
V53()
V54()
V55()
V56()
V57()
V58() )
Element of
K32(
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4() )
set ) ) ,
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) )
complex-valued ext-real-valued real-valued )
Element of
K32(
K33(
NAT : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal V52()
V53()
V54()
V55()
V56()
V57()
V58() )
Element of
K32(
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4() )
set ) ) ,
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4()
complex-valued ext-real-valued real-valued )
set ) ) : ( ( ) (
V4() )
set ) )
. n : ( ( ) (
epsilon-transitive epsilon-connected ordinal natural V24()
real ext-real V45()
V46()
V52()
V53()
V54()
V55()
V56()
V57() )
Element of
NAT : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal V52()
V53()
V54()
V55()
V56()
V57()
V58() )
Element of
K32(
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4() )
set ) ) ) : ( ( ) (
V24()
real ext-real )
Element of
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) ) ) &
seq2 : ( (
Function-like V30(
NAT : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal V52()
V53()
V54()
V55()
V56()
V57()
V58() )
Element of
K32(
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4() )
set ) ) , the
carrier of
b1 : ( ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like )
ComplexNormSpace) : ( ( ) (
V4() )
set ) ) ) (
V4()
Relation-like NAT : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal V52()
V53()
V54()
V55()
V56()
V57()
V58() )
Element of
K32(
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4() )
set ) )
-defined the
carrier of
b1 : ( ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like )
ComplexNormSpace) : ( ( ) (
V4() )
set )
-valued Function-like V29(
NAT : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal V52()
V53()
V54()
V55()
V56()
V57()
V58() )
Element of
K32(
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4() )
set ) ) )
V30(
NAT : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal V52()
V53()
V54()
V55()
V56()
V57()
V58() )
Element of
K32(
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4() )
set ) ) , the
carrier of
b1 : ( ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like )
ComplexNormSpace) : ( ( ) (
V4() )
set ) ) )
sequence of ( ( ) (
V4() )
set ) ) is
norm_summable holds
(
seq1 : ( (
Function-like V30(
NAT : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal V52()
V53()
V54()
V55()
V56()
V57()
V58() )
Element of
K32(
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4() )
set ) ) , the
carrier of
b1 : ( ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like )
ComplexNormSpace) : ( ( ) (
V4() )
set ) ) ) (
V4()
Relation-like NAT : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal V52()
V53()
V54()
V55()
V56()
V57()
V58() )
Element of
K32(
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4() )
set ) )
-defined the
carrier of
b1 : ( ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like )
ComplexNormSpace) : ( ( ) (
V4() )
set )
-valued Function-like V29(
NAT : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal V52()
V53()
V54()
V55()
V56()
V57()
V58() )
Element of
K32(
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4() )
set ) ) )
V30(
NAT : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal V52()
V53()
V54()
V55()
V56()
V57()
V58() )
Element of
K32(
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4() )
set ) ) , the
carrier of
b1 : ( ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like )
ComplexNormSpace) : ( ( ) (
V4() )
set ) ) )
sequence of ( ( ) (
V4() )
set ) ) is
norm_summable &
Sum ||.seq1 : ( ( Function-like V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed V133() V134() vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed V133() V134() vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( V4() ) set ) ) ) ( V4() Relation-like NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) -defined the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed V133() V134() vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed V133() V134() vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( V4() ) set ) -valued Function-like V29( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) ) V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed V133() V134() vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed V133() V134() vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( V4() ) set ) ) ) sequence of ( ( ) ( V4() ) set ) ) .|| : ( (
Function-like V30(
NAT : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal V52()
V53()
V54()
V55()
V56()
V57()
V58() )
Element of
K32(
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4() )
set ) ) ,
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) ) (
V4()
Relation-like NAT : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal V52()
V53()
V54()
V55()
V56()
V57()
V58() )
Element of
K32(
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4() )
set ) )
-defined REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set )
-valued Function-like V29(
NAT : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal V52()
V53()
V54()
V55()
V56()
V57()
V58() )
Element of
K32(
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4() )
set ) ) )
V30(
NAT : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal V52()
V53()
V54()
V55()
V56()
V57()
V58() )
Element of
K32(
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4() )
set ) ) ,
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) )
complex-valued ext-real-valued real-valued )
Element of
K32(
K33(
NAT : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal V52()
V53()
V54()
V55()
V56()
V57()
V58() )
Element of
K32(
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4() )
set ) ) ,
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4()
complex-valued ext-real-valued real-valued )
set ) ) : ( ( ) (
V4() )
set ) ) : ( ( ) (
V24()
real ext-real )
Element of
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) )
<= Sum ||.seq2 : ( ( Function-like V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed V133() V134() vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed V133() V134() vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( V4() ) set ) ) ) ( V4() Relation-like NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) -defined the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed V133() V134() vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed V133() V134() vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( V4() ) set ) -valued Function-like V29( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) ) V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed V133() V134() vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed V133() V134() vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( V4() ) set ) ) ) sequence of ( ( ) ( V4() ) set ) ) .|| : ( (
Function-like V30(
NAT : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal V52()
V53()
V54()
V55()
V56()
V57()
V58() )
Element of
K32(
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4() )
set ) ) ,
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) ) (
V4()
Relation-like NAT : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal V52()
V53()
V54()
V55()
V56()
V57()
V58() )
Element of
K32(
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4() )
set ) )
-defined REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set )
-valued Function-like V29(
NAT : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal V52()
V53()
V54()
V55()
V56()
V57()
V58() )
Element of
K32(
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4() )
set ) ) )
V30(
NAT : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal V52()
V53()
V54()
V55()
V56()
V57()
V58() )
Element of
K32(
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4() )
set ) ) ,
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) )
complex-valued ext-real-valued real-valued )
Element of
K32(
K33(
NAT : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal V52()
V53()
V54()
V55()
V56()
V57()
V58() )
Element of
K32(
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4() )
set ) ) ,
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4()
complex-valued ext-real-valued real-valued )
set ) ) : ( ( ) (
V4() )
set ) ) : ( ( ) (
V24()
real ext-real )
Element of
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) ) ;
theorem
for
X being ( ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like )
ComplexNormSpace)
for
seq being ( (
Function-like V30(
NAT : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal V52()
V53()
V54()
V55()
V56()
V57()
V58() )
Element of
K32(
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4() )
set ) ) , the
carrier of
b1 : ( ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like )
ComplexNormSpace) : ( ( ) (
V4() )
set ) ) ) (
V4()
Relation-like NAT : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal V52()
V53()
V54()
V55()
V56()
V57()
V58() )
Element of
K32(
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4() )
set ) )
-defined the
carrier of
b1 : ( ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like )
ComplexNormSpace) : ( ( ) (
V4() )
set )
-valued Function-like V29(
NAT : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal V52()
V53()
V54()
V55()
V56()
V57()
V58() )
Element of
K32(
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4() )
set ) ) )
V30(
NAT : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal V52()
V53()
V54()
V55()
V56()
V57()
V58() )
Element of
K32(
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4() )
set ) ) , the
carrier of
b1 : ( ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like )
ComplexNormSpace) : ( ( ) (
V4() )
set ) ) )
sequence of ( ( ) (
V4() )
set ) ) st ( for
n being ( ( ) (
epsilon-transitive epsilon-connected ordinal natural V24()
real ext-real V45()
V46()
V52()
V53()
V54()
V55()
V56()
V57() )
Element of
NAT : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal V52()
V53()
V54()
V55()
V56()
V57()
V58() )
Element of
K32(
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4() )
set ) ) ) holds
||.seq : ( ( Function-like V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed V133() V134() vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed V133() V134() vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( V4() ) set ) ) ) ( V4() Relation-like NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) -defined the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed V133() V134() vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed V133() V134() vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( V4() ) set ) -valued Function-like V29( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) ) V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed V133() V134() vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed V133() V134() vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( V4() ) set ) ) ) sequence of ( ( ) ( V4() ) set ) ) .|| : ( (
Function-like V30(
NAT : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal V52()
V53()
V54()
V55()
V56()
V57()
V58() )
Element of
K32(
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4() )
set ) ) ,
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) ) (
V4()
Relation-like NAT : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal V52()
V53()
V54()
V55()
V56()
V57()
V58() )
Element of
K32(
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4() )
set ) )
-defined REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set )
-valued Function-like V29(
NAT : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal V52()
V53()
V54()
V55()
V56()
V57()
V58() )
Element of
K32(
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4() )
set ) ) )
V30(
NAT : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal V52()
V53()
V54()
V55()
V56()
V57()
V58() )
Element of
K32(
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4() )
set ) ) ,
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) )
complex-valued ext-real-valued real-valued )
Element of
K32(
K33(
NAT : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal V52()
V53()
V54()
V55()
V56()
V57()
V58() )
Element of
K32(
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4() )
set ) ) ,
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4()
complex-valued ext-real-valued real-valued )
set ) ) : ( ( ) (
V4() )
set ) )
. n : ( ( ) (
epsilon-transitive epsilon-connected ordinal natural V24()
real ext-real V45()
V46()
V52()
V53()
V54()
V55()
V56()
V57() )
Element of
NAT : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal V52()
V53()
V54()
V55()
V56()
V57()
V58() )
Element of
K32(
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4() )
set ) ) ) : ( ( ) (
V24()
real ext-real )
Element of
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) )
> 0 : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural V24()
real ext-real non
positive non
negative V45()
V46()
V52()
V53()
V54()
V55()
V56()
V57()
V58() )
Element of
NAT : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal V52()
V53()
V54()
V55()
V56()
V57()
V58() )
Element of
K32(
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4() )
set ) ) ) ) & ex
m being ( ( ) (
epsilon-transitive epsilon-connected ordinal natural V24()
real ext-real V45()
V46()
V52()
V53()
V54()
V55()
V56()
V57() )
Element of
NAT : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal V52()
V53()
V54()
V55()
V56()
V57()
V58() )
Element of
K32(
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4() )
set ) ) ) st
for
n being ( ( ) (
epsilon-transitive epsilon-connected ordinal natural V24()
real ext-real V45()
V46()
V52()
V53()
V54()
V55()
V56()
V57() )
Element of
NAT : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal V52()
V53()
V54()
V55()
V56()
V57()
V58() )
Element of
K32(
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4() )
set ) ) ) st
n : ( ( ) (
epsilon-transitive epsilon-connected ordinal natural V24()
real ext-real V45()
V46()
V52()
V53()
V54()
V55()
V56()
V57() )
Element of
NAT : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal V52()
V53()
V54()
V55()
V56()
V57()
V58() )
Element of
K32(
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4() )
set ) ) )
>= m : ( ( ) (
epsilon-transitive epsilon-connected ordinal natural V24()
real ext-real V45()
V46()
V52()
V53()
V54()
V55()
V56()
V57() )
Element of
NAT : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal V52()
V53()
V54()
V55()
V56()
V57()
V58() )
Element of
K32(
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4() )
set ) ) ) holds
(||.seq : ( ( Function-like V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed V133() V134() vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed V133() V134() vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( V4() ) set ) ) ) ( V4() Relation-like NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) -defined the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed V133() V134() vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed V133() V134() vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( V4() ) set ) -valued Function-like V29( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) ) V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed V133() V134() vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed V133() V134() vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( V4() ) set ) ) ) sequence of ( ( ) ( V4() ) set ) ) .|| : ( ( Function-like V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) ) ( V4() Relation-like NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) -defined REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) -valued Function-like V29( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) ) V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) complex-valued ext-real-valued real-valued ) Element of K32(K33(NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) ,REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( V4() ) set ) ) . (n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V24() real ext-real V45() V46() V52() V53() V54() V55() V56() V57() ) Element of NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) ) + 1 : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal natural V24() real ext-real positive non negative V45() V46() V52() V53() V54() V55() V56() V57() ) Element of NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) ) ) : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V24() real ext-real V45() V46() V52() V53() V54() V55() V56() V57() ) Element of NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) ) ) : ( ( ) (
V24()
real ext-real )
Element of
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) )
/ (||.seq : ( ( Function-like V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed V133() V134() vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed V133() V134() vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( V4() ) set ) ) ) ( V4() Relation-like NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) -defined the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed V133() V134() vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed V133() V134() vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( V4() ) set ) -valued Function-like V29( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) ) V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed V133() V134() vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed V133() V134() vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( V4() ) set ) ) ) sequence of ( ( ) ( V4() ) set ) ) .|| : ( ( Function-like V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) ) ( V4() Relation-like NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) -defined REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) -valued Function-like V29( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) ) V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) complex-valued ext-real-valued real-valued ) Element of K32(K33(NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) ,REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( V4() ) set ) ) . n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V24() real ext-real V45() V46() V52() V53() V54() V55() V56() V57() ) Element of NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) ) ) : ( ( ) (
V24()
real ext-real )
Element of
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V24()
real ext-real )
Element of
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) )
>= 1 : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal natural V24()
real ext-real positive non
negative V45()
V46()
V52()
V53()
V54()
V55()
V56()
V57() )
Element of
NAT : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal V52()
V53()
V54()
V55()
V56()
V57()
V58() )
Element of
K32(
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4() )
set ) ) ) holds
not
seq : ( (
Function-like V30(
NAT : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal V52()
V53()
V54()
V55()
V56()
V57()
V58() )
Element of
K32(
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4() )
set ) ) , the
carrier of
b1 : ( ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like )
ComplexNormSpace) : ( ( ) (
V4() )
set ) ) ) (
V4()
Relation-like NAT : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal V52()
V53()
V54()
V55()
V56()
V57()
V58() )
Element of
K32(
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4() )
set ) )
-defined the
carrier of
b1 : ( ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like )
ComplexNormSpace) : ( ( ) (
V4() )
set )
-valued Function-like V29(
NAT : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal V52()
V53()
V54()
V55()
V56()
V57()
V58() )
Element of
K32(
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4() )
set ) ) )
V30(
NAT : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal V52()
V53()
V54()
V55()
V56()
V57()
V58() )
Element of
K32(
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4() )
set ) ) , the
carrier of
b1 : ( ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like )
ComplexNormSpace) : ( ( ) (
V4() )
set ) ) )
sequence of ( ( ) (
V4() )
set ) ) is
norm_summable ;
theorem
for
X being ( ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like )
ComplexNormSpace)
for
seq being ( (
Function-like V30(
NAT : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal V52()
V53()
V54()
V55()
V56()
V57()
V58() )
Element of
K32(
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4() )
set ) ) , the
carrier of
b1 : ( ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like )
ComplexNormSpace) : ( ( ) (
V4() )
set ) ) ) (
V4()
Relation-like NAT : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal V52()
V53()
V54()
V55()
V56()
V57()
V58() )
Element of
K32(
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4() )
set ) )
-defined the
carrier of
b1 : ( ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like )
ComplexNormSpace) : ( ( ) (
V4() )
set )
-valued Function-like V29(
NAT : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal V52()
V53()
V54()
V55()
V56()
V57()
V58() )
Element of
K32(
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4() )
set ) ) )
V30(
NAT : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal V52()
V53()
V54()
V55()
V56()
V57()
V58() )
Element of
K32(
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4() )
set ) ) , the
carrier of
b1 : ( ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like )
ComplexNormSpace) : ( ( ) (
V4() )
set ) ) )
sequence of ( ( ) (
V4() )
set ) )
for
rseq1 being ( (
Function-like V30(
NAT : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal V52()
V53()
V54()
V55()
V56()
V57()
V58() )
Element of
K32(
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4() )
set ) ) ,
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) ) (
V4()
Relation-like NAT : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal V52()
V53()
V54()
V55()
V56()
V57()
V58() )
Element of
K32(
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4() )
set ) )
-defined REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set )
-valued Function-like V29(
NAT : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal V52()
V53()
V54()
V55()
V56()
V57()
V58() )
Element of
K32(
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4() )
set ) ) )
V30(
NAT : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal V52()
V53()
V54()
V55()
V56()
V57()
V58() )
Element of
K32(
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4() )
set ) ) ,
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) )
complex-valued ext-real-valued real-valued )
Real_Sequence) st ( for
n being ( ( ) (
epsilon-transitive epsilon-connected ordinal natural V24()
real ext-real V45()
V46()
V52()
V53()
V54()
V55()
V56()
V57() )
Element of
NAT : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal V52()
V53()
V54()
V55()
V56()
V57()
V58() )
Element of
K32(
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4() )
set ) ) ) holds
rseq1 : ( (
Function-like V30(
NAT : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal V52()
V53()
V54()
V55()
V56()
V57()
V58() )
Element of
K32(
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4() )
set ) ) ,
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) ) (
V4()
Relation-like NAT : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal V52()
V53()
V54()
V55()
V56()
V57()
V58() )
Element of
K32(
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4() )
set ) )
-defined REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set )
-valued Function-like V29(
NAT : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal V52()
V53()
V54()
V55()
V56()
V57()
V58() )
Element of
K32(
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4() )
set ) ) )
V30(
NAT : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal V52()
V53()
V54()
V55()
V56()
V57()
V58() )
Element of
K32(
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4() )
set ) ) ,
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) )
complex-valued ext-real-valued real-valued )
Real_Sequence)
. n : ( ( ) (
epsilon-transitive epsilon-connected ordinal natural V24()
real ext-real V45()
V46()
V52()
V53()
V54()
V55()
V56()
V57() )
Element of
NAT : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal V52()
V53()
V54()
V55()
V56()
V57()
V58() )
Element of
K32(
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4() )
set ) ) ) : ( ( ) (
V24()
real ext-real )
Element of
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) )
= n : ( ( ) (
epsilon-transitive epsilon-connected ordinal natural V24()
real ext-real V45()
V46()
V52()
V53()
V54()
V55()
V56()
V57() )
Element of
NAT : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal V52()
V53()
V54()
V55()
V56()
V57()
V58() )
Element of
K32(
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4() )
set ) ) )
-root (||.seq : ( ( Function-like V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed V133() V134() vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed V133() V134() vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( V4() ) set ) ) ) ( V4() Relation-like NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) -defined the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed V133() V134() vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed V133() V134() vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( V4() ) set ) -valued Function-like V29( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) ) V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed V133() V134() vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed V133() V134() vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( V4() ) set ) ) ) sequence of ( ( ) ( V4() ) set ) ) .|| : ( ( Function-like V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) ) ( V4() Relation-like NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) -defined REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) -valued Function-like V29( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) ) V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) complex-valued ext-real-valued real-valued ) Element of K32(K33(NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) ,REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( V4() ) set ) ) . n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V24() real ext-real V45() V46() V52() V53() V54() V55() V56() V57() ) Element of NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) ) ) : ( ( ) (
V24()
real ext-real )
Element of
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V24()
real ext-real )
Element of
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) ) &
rseq1 : ( (
Function-like V30(
NAT : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal V52()
V53()
V54()
V55()
V56()
V57()
V58() )
Element of
K32(
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4() )
set ) ) ,
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) ) (
V4()
Relation-like NAT : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal V52()
V53()
V54()
V55()
V56()
V57()
V58() )
Element of
K32(
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4() )
set ) )
-defined REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set )
-valued Function-like V29(
NAT : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal V52()
V53()
V54()
V55()
V56()
V57()
V58() )
Element of
K32(
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4() )
set ) ) )
V30(
NAT : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal V52()
V53()
V54()
V55()
V56()
V57()
V58() )
Element of
K32(
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4() )
set ) ) ,
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) )
complex-valued ext-real-valued real-valued )
Real_Sequence) is
convergent &
lim rseq1 : ( (
Function-like V30(
NAT : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal V52()
V53()
V54()
V55()
V56()
V57()
V58() )
Element of
K32(
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4() )
set ) ) ,
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) ) (
V4()
Relation-like NAT : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal V52()
V53()
V54()
V55()
V56()
V57()
V58() )
Element of
K32(
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4() )
set ) )
-defined REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set )
-valued Function-like V29(
NAT : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal V52()
V53()
V54()
V55()
V56()
V57()
V58() )
Element of
K32(
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4() )
set ) ) )
V30(
NAT : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal V52()
V53()
V54()
V55()
V56()
V57()
V58() )
Element of
K32(
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4() )
set ) ) ,
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) )
complex-valued ext-real-valued real-valued )
Real_Sequence) : ( ( ) (
V24()
real ext-real )
Element of
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) )
< 1 : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal natural V24()
real ext-real positive non
negative V45()
V46()
V52()
V53()
V54()
V55()
V56()
V57() )
Element of
NAT : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal V52()
V53()
V54()
V55()
V56()
V57()
V58() )
Element of
K32(
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4() )
set ) ) ) holds
seq : ( (
Function-like V30(
NAT : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal V52()
V53()
V54()
V55()
V56()
V57()
V58() )
Element of
K32(
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4() )
set ) ) , the
carrier of
b1 : ( ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like )
ComplexNormSpace) : ( ( ) (
V4() )
set ) ) ) (
V4()
Relation-like NAT : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal V52()
V53()
V54()
V55()
V56()
V57()
V58() )
Element of
K32(
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4() )
set ) )
-defined the
carrier of
b1 : ( ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like )
ComplexNormSpace) : ( ( ) (
V4() )
set )
-valued Function-like V29(
NAT : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal V52()
V53()
V54()
V55()
V56()
V57()
V58() )
Element of
K32(
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4() )
set ) ) )
V30(
NAT : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal V52()
V53()
V54()
V55()
V56()
V57()
V58() )
Element of
K32(
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4() )
set ) ) , the
carrier of
b1 : ( ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like )
ComplexNormSpace) : ( ( ) (
V4() )
set ) ) )
sequence of ( ( ) (
V4() )
set ) ) is
norm_summable ;
theorem
for
X being ( ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like )
ComplexNormSpace)
for
seq being ( (
Function-like V30(
NAT : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal V52()
V53()
V54()
V55()
V56()
V57()
V58() )
Element of
K32(
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4() )
set ) ) , the
carrier of
b1 : ( ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like )
ComplexNormSpace) : ( ( ) (
V4() )
set ) ) ) (
V4()
Relation-like NAT : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal V52()
V53()
V54()
V55()
V56()
V57()
V58() )
Element of
K32(
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4() )
set ) )
-defined the
carrier of
b1 : ( ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like )
ComplexNormSpace) : ( ( ) (
V4() )
set )
-valued Function-like V29(
NAT : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal V52()
V53()
V54()
V55()
V56()
V57()
V58() )
Element of
K32(
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4() )
set ) ) )
V30(
NAT : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal V52()
V53()
V54()
V55()
V56()
V57()
V58() )
Element of
K32(
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4() )
set ) ) , the
carrier of
b1 : ( ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like )
ComplexNormSpace) : ( ( ) (
V4() )
set ) ) )
sequence of ( ( ) (
V4() )
set ) )
for
rseq1 being ( (
Function-like V30(
NAT : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal V52()
V53()
V54()
V55()
V56()
V57()
V58() )
Element of
K32(
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4() )
set ) ) ,
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) ) (
V4()
Relation-like NAT : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal V52()
V53()
V54()
V55()
V56()
V57()
V58() )
Element of
K32(
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4() )
set ) )
-defined REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set )
-valued Function-like V29(
NAT : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal V52()
V53()
V54()
V55()
V56()
V57()
V58() )
Element of
K32(
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4() )
set ) ) )
V30(
NAT : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal V52()
V53()
V54()
V55()
V56()
V57()
V58() )
Element of
K32(
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4() )
set ) ) ,
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) )
complex-valued ext-real-valued real-valued )
Real_Sequence) st ( for
n being ( ( ) (
epsilon-transitive epsilon-connected ordinal natural V24()
real ext-real V45()
V46()
V52()
V53()
V54()
V55()
V56()
V57() )
Element of
NAT : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal V52()
V53()
V54()
V55()
V56()
V57()
V58() )
Element of
K32(
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4() )
set ) ) ) holds
rseq1 : ( (
Function-like V30(
NAT : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal V52()
V53()
V54()
V55()
V56()
V57()
V58() )
Element of
K32(
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4() )
set ) ) ,
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) ) (
V4()
Relation-like NAT : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal V52()
V53()
V54()
V55()
V56()
V57()
V58() )
Element of
K32(
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4() )
set ) )
-defined REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set )
-valued Function-like V29(
NAT : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal V52()
V53()
V54()
V55()
V56()
V57()
V58() )
Element of
K32(
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4() )
set ) ) )
V30(
NAT : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal V52()
V53()
V54()
V55()
V56()
V57()
V58() )
Element of
K32(
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4() )
set ) ) ,
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) )
complex-valued ext-real-valued real-valued )
Real_Sequence)
. n : ( ( ) (
epsilon-transitive epsilon-connected ordinal natural V24()
real ext-real V45()
V46()
V52()
V53()
V54()
V55()
V56()
V57() )
Element of
NAT : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal V52()
V53()
V54()
V55()
V56()
V57()
V58() )
Element of
K32(
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4() )
set ) ) ) : ( ( ) (
V24()
real ext-real )
Element of
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) )
= n : ( ( ) (
epsilon-transitive epsilon-connected ordinal natural V24()
real ext-real V45()
V46()
V52()
V53()
V54()
V55()
V56()
V57() )
Element of
NAT : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal V52()
V53()
V54()
V55()
V56()
V57()
V58() )
Element of
K32(
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4() )
set ) ) )
-root (||.seq : ( ( Function-like V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed V133() V134() vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed V133() V134() vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( V4() ) set ) ) ) ( V4() Relation-like NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) -defined the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed V133() V134() vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed V133() V134() vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( V4() ) set ) -valued Function-like V29( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) ) V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed V133() V134() vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed V133() V134() vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( V4() ) set ) ) ) sequence of ( ( ) ( V4() ) set ) ) .|| : ( ( Function-like V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) ) ( V4() Relation-like NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) -defined REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) -valued Function-like V29( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) ) V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) complex-valued ext-real-valued real-valued ) Element of K32(K33(NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) ,REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( V4() ) set ) ) . n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V24() real ext-real V45() V46() V52() V53() V54() V55() V56() V57() ) Element of NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) ) ) : ( ( ) (
V24()
real ext-real )
Element of
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V24()
real ext-real )
Element of
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) ) & ex
m being ( ( ) (
epsilon-transitive epsilon-connected ordinal natural V24()
real ext-real V45()
V46()
V52()
V53()
V54()
V55()
V56()
V57() )
Element of
NAT : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal V52()
V53()
V54()
V55()
V56()
V57()
V58() )
Element of
K32(
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4() )
set ) ) ) st
for
n being ( ( ) (
epsilon-transitive epsilon-connected ordinal natural V24()
real ext-real V45()
V46()
V52()
V53()
V54()
V55()
V56()
V57() )
Element of
NAT : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal V52()
V53()
V54()
V55()
V56()
V57()
V58() )
Element of
K32(
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4() )
set ) ) ) st
m : ( ( ) (
epsilon-transitive epsilon-connected ordinal natural V24()
real ext-real V45()
V46()
V52()
V53()
V54()
V55()
V56()
V57() )
Element of
NAT : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal V52()
V53()
V54()
V55()
V56()
V57()
V58() )
Element of
K32(
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4() )
set ) ) )
<= n : ( ( ) (
epsilon-transitive epsilon-connected ordinal natural V24()
real ext-real V45()
V46()
V52()
V53()
V54()
V55()
V56()
V57() )
Element of
NAT : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal V52()
V53()
V54()
V55()
V56()
V57()
V58() )
Element of
K32(
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4() )
set ) ) ) holds
rseq1 : ( (
Function-like V30(
NAT : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal V52()
V53()
V54()
V55()
V56()
V57()
V58() )
Element of
K32(
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4() )
set ) ) ,
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) ) (
V4()
Relation-like NAT : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal V52()
V53()
V54()
V55()
V56()
V57()
V58() )
Element of
K32(
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4() )
set ) )
-defined REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set )
-valued Function-like V29(
NAT : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal V52()
V53()
V54()
V55()
V56()
V57()
V58() )
Element of
K32(
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4() )
set ) ) )
V30(
NAT : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal V52()
V53()
V54()
V55()
V56()
V57()
V58() )
Element of
K32(
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4() )
set ) ) ,
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) )
complex-valued ext-real-valued real-valued )
Real_Sequence)
. n : ( ( ) (
epsilon-transitive epsilon-connected ordinal natural V24()
real ext-real V45()
V46()
V52()
V53()
V54()
V55()
V56()
V57() )
Element of
NAT : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal V52()
V53()
V54()
V55()
V56()
V57()
V58() )
Element of
K32(
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4() )
set ) ) ) : ( ( ) (
V24()
real ext-real )
Element of
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) )
>= 1 : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal natural V24()
real ext-real positive non
negative V45()
V46()
V52()
V53()
V54()
V55()
V56()
V57() )
Element of
NAT : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal V52()
V53()
V54()
V55()
V56()
V57()
V58() )
Element of
K32(
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4() )
set ) ) ) holds
not
||.seq : ( ( Function-like V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed V133() V134() vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed V133() V134() vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( V4() ) set ) ) ) ( V4() Relation-like NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) -defined the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed V133() V134() vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed V133() V134() vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( V4() ) set ) -valued Function-like V29( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) ) V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed V133() V134() vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed V133() V134() vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( V4() ) set ) ) ) sequence of ( ( ) ( V4() ) set ) ) .|| : ( (
Function-like V30(
NAT : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal V52()
V53()
V54()
V55()
V56()
V57()
V58() )
Element of
K32(
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4() )
set ) ) ,
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) ) (
V4()
Relation-like NAT : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal V52()
V53()
V54()
V55()
V56()
V57()
V58() )
Element of
K32(
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4() )
set ) )
-defined REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set )
-valued Function-like V29(
NAT : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal V52()
V53()
V54()
V55()
V56()
V57()
V58() )
Element of
K32(
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4() )
set ) ) )
V30(
NAT : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal V52()
V53()
V54()
V55()
V56()
V57()
V58() )
Element of
K32(
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4() )
set ) ) ,
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) )
complex-valued ext-real-valued real-valued )
Element of
K32(
K33(
NAT : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal V52()
V53()
V54()
V55()
V56()
V57()
V58() )
Element of
K32(
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4() )
set ) ) ,
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4()
complex-valued ext-real-valued real-valued )
set ) ) : ( ( ) (
V4() )
set ) ) is
summable ;
theorem
for
X being ( ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like )
ComplexNormSpace)
for
seq being ( (
Function-like V30(
NAT : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal V52()
V53()
V54()
V55()
V56()
V57()
V58() )
Element of
K32(
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4() )
set ) ) , the
carrier of
b1 : ( ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like )
ComplexNormSpace) : ( ( ) (
V4() )
set ) ) ) (
V4()
Relation-like NAT : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal V52()
V53()
V54()
V55()
V56()
V57()
V58() )
Element of
K32(
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4() )
set ) )
-defined the
carrier of
b1 : ( ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like )
ComplexNormSpace) : ( ( ) (
V4() )
set )
-valued Function-like V29(
NAT : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal V52()
V53()
V54()
V55()
V56()
V57()
V58() )
Element of
K32(
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4() )
set ) ) )
V30(
NAT : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal V52()
V53()
V54()
V55()
V56()
V57()
V58() )
Element of
K32(
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4() )
set ) ) , the
carrier of
b1 : ( ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like )
ComplexNormSpace) : ( ( ) (
V4() )
set ) ) )
sequence of ( ( ) (
V4() )
set ) )
for
rseq1 being ( (
Function-like V30(
NAT : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal V52()
V53()
V54()
V55()
V56()
V57()
V58() )
Element of
K32(
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4() )
set ) ) ,
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) ) (
V4()
Relation-like NAT : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal V52()
V53()
V54()
V55()
V56()
V57()
V58() )
Element of
K32(
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4() )
set ) )
-defined REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set )
-valued Function-like V29(
NAT : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal V52()
V53()
V54()
V55()
V56()
V57()
V58() )
Element of
K32(
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4() )
set ) ) )
V30(
NAT : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal V52()
V53()
V54()
V55()
V56()
V57()
V58() )
Element of
K32(
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4() )
set ) ) ,
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) )
complex-valued ext-real-valued real-valued )
Real_Sequence) st ( for
n being ( ( ) (
epsilon-transitive epsilon-connected ordinal natural V24()
real ext-real V45()
V46()
V52()
V53()
V54()
V55()
V56()
V57() )
Element of
NAT : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal V52()
V53()
V54()
V55()
V56()
V57()
V58() )
Element of
K32(
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4() )
set ) ) ) holds
rseq1 : ( (
Function-like V30(
NAT : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal V52()
V53()
V54()
V55()
V56()
V57()
V58() )
Element of
K32(
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4() )
set ) ) ,
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) ) (
V4()
Relation-like NAT : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal V52()
V53()
V54()
V55()
V56()
V57()
V58() )
Element of
K32(
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4() )
set ) )
-defined REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set )
-valued Function-like V29(
NAT : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal V52()
V53()
V54()
V55()
V56()
V57()
V58() )
Element of
K32(
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4() )
set ) ) )
V30(
NAT : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal V52()
V53()
V54()
V55()
V56()
V57()
V58() )
Element of
K32(
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4() )
set ) ) ,
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) )
complex-valued ext-real-valued real-valued )
Real_Sequence)
. n : ( ( ) (
epsilon-transitive epsilon-connected ordinal natural V24()
real ext-real V45()
V46()
V52()
V53()
V54()
V55()
V56()
V57() )
Element of
NAT : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal V52()
V53()
V54()
V55()
V56()
V57()
V58() )
Element of
K32(
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4() )
set ) ) ) : ( ( ) (
V24()
real ext-real )
Element of
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) )
= n : ( ( ) (
epsilon-transitive epsilon-connected ordinal natural V24()
real ext-real V45()
V46()
V52()
V53()
V54()
V55()
V56()
V57() )
Element of
NAT : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal V52()
V53()
V54()
V55()
V56()
V57()
V58() )
Element of
K32(
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4() )
set ) ) )
-root (||.seq : ( ( Function-like V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed V133() V134() vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed V133() V134() vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( V4() ) set ) ) ) ( V4() Relation-like NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) -defined the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed V133() V134() vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed V133() V134() vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( V4() ) set ) -valued Function-like V29( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) ) V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed V133() V134() vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed V133() V134() vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( V4() ) set ) ) ) sequence of ( ( ) ( V4() ) set ) ) .|| : ( ( Function-like V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) ) ( V4() Relation-like NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) -defined REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) -valued Function-like V29( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) ) V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) complex-valued ext-real-valued real-valued ) Element of K32(K33(NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) ,REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( V4() ) set ) ) . n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V24() real ext-real V45() V46() V52() V53() V54() V55() V56() V57() ) Element of NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) ) ) : ( ( ) (
V24()
real ext-real )
Element of
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V24()
real ext-real )
Element of
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) ) &
rseq1 : ( (
Function-like V30(
NAT : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal V52()
V53()
V54()
V55()
V56()
V57()
V58() )
Element of
K32(
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4() )
set ) ) ,
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) ) (
V4()
Relation-like NAT : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal V52()
V53()
V54()
V55()
V56()
V57()
V58() )
Element of
K32(
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4() )
set ) )
-defined REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set )
-valued Function-like V29(
NAT : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal V52()
V53()
V54()
V55()
V56()
V57()
V58() )
Element of
K32(
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4() )
set ) ) )
V30(
NAT : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal V52()
V53()
V54()
V55()
V56()
V57()
V58() )
Element of
K32(
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4() )
set ) ) ,
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) )
complex-valued ext-real-valued real-valued )
Real_Sequence) is
convergent &
lim rseq1 : ( (
Function-like V30(
NAT : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal V52()
V53()
V54()
V55()
V56()
V57()
V58() )
Element of
K32(
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4() )
set ) ) ,
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) ) (
V4()
Relation-like NAT : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal V52()
V53()
V54()
V55()
V56()
V57()
V58() )
Element of
K32(
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4() )
set ) )
-defined REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set )
-valued Function-like V29(
NAT : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal V52()
V53()
V54()
V55()
V56()
V57()
V58() )
Element of
K32(
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4() )
set ) ) )
V30(
NAT : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal V52()
V53()
V54()
V55()
V56()
V57()
V58() )
Element of
K32(
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4() )
set ) ) ,
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) )
complex-valued ext-real-valued real-valued )
Real_Sequence) : ( ( ) (
V24()
real ext-real )
Element of
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) )
> 1 : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal natural V24()
real ext-real positive non
negative V45()
V46()
V52()
V53()
V54()
V55()
V56()
V57() )
Element of
NAT : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal V52()
V53()
V54()
V55()
V56()
V57()
V58() )
Element of
K32(
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4() )
set ) ) ) holds
not
seq : ( (
Function-like V30(
NAT : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal V52()
V53()
V54()
V55()
V56()
V57()
V58() )
Element of
K32(
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4() )
set ) ) , the
carrier of
b1 : ( ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like )
ComplexNormSpace) : ( ( ) (
V4() )
set ) ) ) (
V4()
Relation-like NAT : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal V52()
V53()
V54()
V55()
V56()
V57()
V58() )
Element of
K32(
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4() )
set ) )
-defined the
carrier of
b1 : ( ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like )
ComplexNormSpace) : ( ( ) (
V4() )
set )
-valued Function-like V29(
NAT : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal V52()
V53()
V54()
V55()
V56()
V57()
V58() )
Element of
K32(
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4() )
set ) ) )
V30(
NAT : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal V52()
V53()
V54()
V55()
V56()
V57()
V58() )
Element of
K32(
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4() )
set ) ) , the
carrier of
b1 : ( ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like )
ComplexNormSpace) : ( ( ) (
V4() )
set ) ) )
sequence of ( ( ) (
V4() )
set ) ) is
norm_summable ;
theorem
for
X being ( ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like )
ComplexNormSpace)
for
seq being ( (
Function-like V30(
NAT : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal V52()
V53()
V54()
V55()
V56()
V57()
V58() )
Element of
K32(
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4() )
set ) ) , the
carrier of
b1 : ( ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like )
ComplexNormSpace) : ( ( ) (
V4() )
set ) ) ) (
V4()
Relation-like NAT : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal V52()
V53()
V54()
V55()
V56()
V57()
V58() )
Element of
K32(
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4() )
set ) )
-defined the
carrier of
b1 : ( ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like )
ComplexNormSpace) : ( ( ) (
V4() )
set )
-valued Function-like V29(
NAT : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal V52()
V53()
V54()
V55()
V56()
V57()
V58() )
Element of
K32(
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4() )
set ) ) )
V30(
NAT : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal V52()
V53()
V54()
V55()
V56()
V57()
V58() )
Element of
K32(
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4() )
set ) ) , the
carrier of
b1 : ( ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like )
ComplexNormSpace) : ( ( ) (
V4() )
set ) ) )
sequence of ( ( ) (
V4() )
set ) )
for
rseq1 being ( (
Function-like V30(
NAT : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal V52()
V53()
V54()
V55()
V56()
V57()
V58() )
Element of
K32(
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4() )
set ) ) ,
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) ) (
V4()
Relation-like NAT : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal V52()
V53()
V54()
V55()
V56()
V57()
V58() )
Element of
K32(
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4() )
set ) )
-defined REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set )
-valued Function-like V29(
NAT : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal V52()
V53()
V54()
V55()
V56()
V57()
V58() )
Element of
K32(
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4() )
set ) ) )
V30(
NAT : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal V52()
V53()
V54()
V55()
V56()
V57()
V58() )
Element of
K32(
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4() )
set ) ) ,
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) )
complex-valued ext-real-valued real-valued )
Real_Sequence) st
||.seq : ( ( Function-like V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed V133() V134() vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed V133() V134() vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( V4() ) set ) ) ) ( V4() Relation-like NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) -defined the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed V133() V134() vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed V133() V134() vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( V4() ) set ) -valued Function-like V29( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) ) V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed V133() V134() vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed V133() V134() vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( V4() ) set ) ) ) sequence of ( ( ) ( V4() ) set ) ) .|| : ( (
Function-like V30(
NAT : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal V52()
V53()
V54()
V55()
V56()
V57()
V58() )
Element of
K32(
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4() )
set ) ) ,
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) ) (
V4()
Relation-like NAT : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal V52()
V53()
V54()
V55()
V56()
V57()
V58() )
Element of
K32(
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4() )
set ) )
-defined REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set )
-valued Function-like V29(
NAT : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal V52()
V53()
V54()
V55()
V56()
V57()
V58() )
Element of
K32(
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4() )
set ) ) )
V30(
NAT : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal V52()
V53()
V54()
V55()
V56()
V57()
V58() )
Element of
K32(
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4() )
set ) ) ,
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) )
complex-valued ext-real-valued real-valued )
Element of
K32(
K33(
NAT : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal V52()
V53()
V54()
V55()
V56()
V57()
V58() )
Element of
K32(
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4() )
set ) ) ,
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4()
complex-valued ext-real-valued real-valued )
set ) ) : ( ( ) (
V4() )
set ) ) is
non-increasing & ( for
n being ( ( ) (
epsilon-transitive epsilon-connected ordinal natural V24()
real ext-real V45()
V46()
V52()
V53()
V54()
V55()
V56()
V57() )
Element of
NAT : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal V52()
V53()
V54()
V55()
V56()
V57()
V58() )
Element of
K32(
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4() )
set ) ) ) holds
rseq1 : ( (
Function-like V30(
NAT : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal V52()
V53()
V54()
V55()
V56()
V57()
V58() )
Element of
K32(
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4() )
set ) ) ,
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) ) (
V4()
Relation-like NAT : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal V52()
V53()
V54()
V55()
V56()
V57()
V58() )
Element of
K32(
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4() )
set ) )
-defined REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set )
-valued Function-like V29(
NAT : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal V52()
V53()
V54()
V55()
V56()
V57()
V58() )
Element of
K32(
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4() )
set ) ) )
V30(
NAT : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal V52()
V53()
V54()
V55()
V56()
V57()
V58() )
Element of
K32(
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4() )
set ) ) ,
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) )
complex-valued ext-real-valued real-valued )
Real_Sequence)
. n : ( ( ) (
epsilon-transitive epsilon-connected ordinal natural V24()
real ext-real V45()
V46()
V52()
V53()
V54()
V55()
V56()
V57() )
Element of
NAT : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal V52()
V53()
V54()
V55()
V56()
V57()
V58() )
Element of
K32(
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4() )
set ) ) ) : ( ( ) (
V24()
real ext-real )
Element of
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) )
= (2 : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal natural V24() real ext-real positive non negative V45() V46() V52() V53() V54() V55() V56() V57() ) Element of NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) ) to_power n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V24() real ext-real V45() V46() V52() V53() V54() V55() V56() V57() ) Element of NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) ) ) : ( ( ) (
V24()
real ext-real )
Element of
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) )
* (||.seq : ( ( Function-like V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed V133() V134() vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed V133() V134() vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( V4() ) set ) ) ) ( V4() Relation-like NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) -defined the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed V133() V134() vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed V133() V134() vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( V4() ) set ) -valued Function-like V29( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) ) V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed V133() V134() vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed V133() V134() vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( V4() ) set ) ) ) sequence of ( ( ) ( V4() ) set ) ) .|| : ( ( Function-like V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) ) ( V4() Relation-like NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) -defined REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) -valued Function-like V29( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) ) V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) complex-valued ext-real-valued real-valued ) Element of K32(K33(NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) ,REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( V4() ) set ) ) . (2 : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal natural V24() real ext-real positive non negative V45() V46() V52() V53() V54() V55() V56() V57() ) Element of NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) ) to_power n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V24() real ext-real V45() V46() V52() V53() V54() V55() V56() V57() ) Element of NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) ) ) : ( ( ) ( V24() real ext-real ) Element of REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) ) : ( ( ) (
V24()
real ext-real )
Element of
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V24()
real ext-real )
Element of
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) ) holds
(
seq : ( (
Function-like V30(
NAT : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal V52()
V53()
V54()
V55()
V56()
V57()
V58() )
Element of
K32(
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4() )
set ) ) , the
carrier of
b1 : ( ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like )
ComplexNormSpace) : ( ( ) (
V4() )
set ) ) ) (
V4()
Relation-like NAT : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal V52()
V53()
V54()
V55()
V56()
V57()
V58() )
Element of
K32(
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4() )
set ) )
-defined the
carrier of
b1 : ( ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like )
ComplexNormSpace) : ( ( ) (
V4() )
set )
-valued Function-like V29(
NAT : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal V52()
V53()
V54()
V55()
V56()
V57()
V58() )
Element of
K32(
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4() )
set ) ) )
V30(
NAT : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal V52()
V53()
V54()
V55()
V56()
V57()
V58() )
Element of
K32(
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4() )
set ) ) , the
carrier of
b1 : ( ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like )
ComplexNormSpace) : ( ( ) (
V4() )
set ) ) )
sequence of ( ( ) (
V4() )
set ) ) is
norm_summable iff
rseq1 : ( (
Function-like V30(
NAT : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal V52()
V53()
V54()
V55()
V56()
V57()
V58() )
Element of
K32(
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4() )
set ) ) ,
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) ) (
V4()
Relation-like NAT : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal V52()
V53()
V54()
V55()
V56()
V57()
V58() )
Element of
K32(
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4() )
set ) )
-defined REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set )
-valued Function-like V29(
NAT : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal V52()
V53()
V54()
V55()
V56()
V57()
V58() )
Element of
K32(
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4() )
set ) ) )
V30(
NAT : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal V52()
V53()
V54()
V55()
V56()
V57()
V58() )
Element of
K32(
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4() )
set ) ) ,
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) )
complex-valued ext-real-valued real-valued )
Real_Sequence) is
summable ) ;
theorem
for
X being ( ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like )
ComplexNormSpace)
for
seq being ( (
Function-like V30(
NAT : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal V52()
V53()
V54()
V55()
V56()
V57()
V58() )
Element of
K32(
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4() )
set ) ) , the
carrier of
b1 : ( ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like )
ComplexNormSpace) : ( ( ) (
V4() )
set ) ) ) (
V4()
Relation-like NAT : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal V52()
V53()
V54()
V55()
V56()
V57()
V58() )
Element of
K32(
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4() )
set ) )
-defined the
carrier of
b1 : ( ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like )
ComplexNormSpace) : ( ( ) (
V4() )
set )
-valued Function-like V29(
NAT : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal V52()
V53()
V54()
V55()
V56()
V57()
V58() )
Element of
K32(
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4() )
set ) ) )
V30(
NAT : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal V52()
V53()
V54()
V55()
V56()
V57()
V58() )
Element of
K32(
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4() )
set ) ) , the
carrier of
b1 : ( ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like )
ComplexNormSpace) : ( ( ) (
V4() )
set ) ) )
sequence of ( ( ) (
V4() )
set ) )
for
rseq1 being ( (
Function-like V30(
NAT : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal V52()
V53()
V54()
V55()
V56()
V57()
V58() )
Element of
K32(
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4() )
set ) ) ,
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) ) (
V4()
Relation-like NAT : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal V52()
V53()
V54()
V55()
V56()
V57()
V58() )
Element of
K32(
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4() )
set ) )
-defined REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set )
-valued Function-like V29(
NAT : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal V52()
V53()
V54()
V55()
V56()
V57()
V58() )
Element of
K32(
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4() )
set ) ) )
V30(
NAT : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal V52()
V53()
V54()
V55()
V56()
V57()
V58() )
Element of
K32(
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4() )
set ) ) ,
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) )
complex-valued ext-real-valued real-valued )
Real_Sequence) st ( for
n being ( ( ) (
epsilon-transitive epsilon-connected ordinal natural V24()
real ext-real V45()
V46()
V52()
V53()
V54()
V55()
V56()
V57() )
Element of
NAT : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal V52()
V53()
V54()
V55()
V56()
V57()
V58() )
Element of
K32(
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4() )
set ) ) ) holds
(
seq : ( (
Function-like V30(
NAT : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal V52()
V53()
V54()
V55()
V56()
V57()
V58() )
Element of
K32(
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4() )
set ) ) , the
carrier of
b1 : ( ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like )
ComplexNormSpace) : ( ( ) (
V4() )
set ) ) ) (
V4()
Relation-like NAT : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal V52()
V53()
V54()
V55()
V56()
V57()
V58() )
Element of
K32(
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4() )
set ) )
-defined the
carrier of
b1 : ( ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like )
ComplexNormSpace) : ( ( ) (
V4() )
set )
-valued Function-like V29(
NAT : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal V52()
V53()
V54()
V55()
V56()
V57()
V58() )
Element of
K32(
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4() )
set ) ) )
V30(
NAT : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal V52()
V53()
V54()
V55()
V56()
V57()
V58() )
Element of
K32(
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4() )
set ) ) , the
carrier of
b1 : ( ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like )
ComplexNormSpace) : ( ( ) (
V4() )
set ) ) )
sequence of ( ( ) (
V4() )
set ) )
. n : ( ( ) (
epsilon-transitive epsilon-connected ordinal natural V24()
real ext-real V45()
V46()
V52()
V53()
V54()
V55()
V56()
V57() )
Element of
NAT : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal V52()
V53()
V54()
V55()
V56()
V57()
V58() )
Element of
K32(
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4() )
set ) ) ) : ( ( ) (
right_complementable )
Element of the
carrier of
b1 : ( ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like )
ComplexNormSpace) : ( ( ) (
V4() )
set ) )
<> 0. X : ( ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like )
ComplexNormSpace) : ( ( ) (
V79(
b1 : ( ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like )
ComplexNormSpace) )
right_complementable )
Element of the
carrier of
b1 : ( ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like )
ComplexNormSpace) : ( ( ) (
V4() )
set ) ) &
rseq1 : ( (
Function-like V30(
NAT : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal V52()
V53()
V54()
V55()
V56()
V57()
V58() )
Element of
K32(
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4() )
set ) ) ,
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) ) (
V4()
Relation-like NAT : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal V52()
V53()
V54()
V55()
V56()
V57()
V58() )
Element of
K32(
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4() )
set ) )
-defined REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set )
-valued Function-like V29(
NAT : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal V52()
V53()
V54()
V55()
V56()
V57()
V58() )
Element of
K32(
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4() )
set ) ) )
V30(
NAT : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal V52()
V53()
V54()
V55()
V56()
V57()
V58() )
Element of
K32(
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4() )
set ) ) ,
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) )
complex-valued ext-real-valued real-valued )
Real_Sequence)
. n : ( ( ) (
epsilon-transitive epsilon-connected ordinal natural V24()
real ext-real V45()
V46()
V52()
V53()
V54()
V55()
V56()
V57() )
Element of
NAT : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal V52()
V53()
V54()
V55()
V56()
V57()
V58() )
Element of
K32(
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4() )
set ) ) ) : ( ( ) (
V24()
real ext-real )
Element of
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) )
= (||.seq : ( ( Function-like V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed V133() V134() vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed V133() V134() vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( V4() ) set ) ) ) ( V4() Relation-like NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) -defined the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed V133() V134() vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed V133() V134() vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( V4() ) set ) -valued Function-like V29( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) ) V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed V133() V134() vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed V133() V134() vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( V4() ) set ) ) ) sequence of ( ( ) ( V4() ) set ) ) .|| : ( ( Function-like V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) ) ( V4() Relation-like NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) -defined REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) -valued Function-like V29( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) ) V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) complex-valued ext-real-valued real-valued ) Element of K32(K33(NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) ,REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( V4() ) set ) ) . (n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V24() real ext-real V45() V46() V52() V53() V54() V55() V56() V57() ) Element of NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) ) + 1 : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal natural V24() real ext-real positive non negative V45() V46() V52() V53() V54() V55() V56() V57() ) Element of NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) ) ) : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V24() real ext-real V45() V46() V52() V53() V54() V55() V56() V57() ) Element of NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) ) ) : ( ( ) (
V24()
real ext-real )
Element of
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) )
/ (||.seq : ( ( Function-like V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed V133() V134() vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed V133() V134() vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( V4() ) set ) ) ) ( V4() Relation-like NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) -defined the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed V133() V134() vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed V133() V134() vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( V4() ) set ) -valued Function-like V29( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) ) V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed V133() V134() vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed V133() V134() vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( V4() ) set ) ) ) sequence of ( ( ) ( V4() ) set ) ) .|| : ( ( Function-like V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) ) ( V4() Relation-like NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) -defined REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) -valued Function-like V29( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) ) V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) complex-valued ext-real-valued real-valued ) Element of K32(K33(NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) ,REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( V4() ) set ) ) . n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V24() real ext-real V45() V46() V52() V53() V54() V55() V56() V57() ) Element of NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) ) ) : ( ( ) (
V24()
real ext-real )
Element of
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V24()
real ext-real )
Element of
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) ) ) &
rseq1 : ( (
Function-like V30(
NAT : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal V52()
V53()
V54()
V55()
V56()
V57()
V58() )
Element of
K32(
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4() )
set ) ) ,
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) ) (
V4()
Relation-like NAT : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal V52()
V53()
V54()
V55()
V56()
V57()
V58() )
Element of
K32(
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4() )
set ) )
-defined REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set )
-valued Function-like V29(
NAT : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal V52()
V53()
V54()
V55()
V56()
V57()
V58() )
Element of
K32(
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4() )
set ) ) )
V30(
NAT : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal V52()
V53()
V54()
V55()
V56()
V57()
V58() )
Element of
K32(
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4() )
set ) ) ,
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) )
complex-valued ext-real-valued real-valued )
Real_Sequence) is
convergent &
lim rseq1 : ( (
Function-like V30(
NAT : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal V52()
V53()
V54()
V55()
V56()
V57()
V58() )
Element of
K32(
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4() )
set ) ) ,
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) ) (
V4()
Relation-like NAT : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal V52()
V53()
V54()
V55()
V56()
V57()
V58() )
Element of
K32(
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4() )
set ) )
-defined REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set )
-valued Function-like V29(
NAT : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal V52()
V53()
V54()
V55()
V56()
V57()
V58() )
Element of
K32(
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4() )
set ) ) )
V30(
NAT : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal V52()
V53()
V54()
V55()
V56()
V57()
V58() )
Element of
K32(
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4() )
set ) ) ,
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) )
complex-valued ext-real-valued real-valued )
Real_Sequence) : ( ( ) (
V24()
real ext-real )
Element of
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) )
< 1 : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal natural V24()
real ext-real positive non
negative V45()
V46()
V52()
V53()
V54()
V55()
V56()
V57() )
Element of
NAT : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal V52()
V53()
V54()
V55()
V56()
V57()
V58() )
Element of
K32(
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4() )
set ) ) ) holds
seq : ( (
Function-like V30(
NAT : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal V52()
V53()
V54()
V55()
V56()
V57()
V58() )
Element of
K32(
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4() )
set ) ) , the
carrier of
b1 : ( ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like )
ComplexNormSpace) : ( ( ) (
V4() )
set ) ) ) (
V4()
Relation-like NAT : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal V52()
V53()
V54()
V55()
V56()
V57()
V58() )
Element of
K32(
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4() )
set ) )
-defined the
carrier of
b1 : ( ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like )
ComplexNormSpace) : ( ( ) (
V4() )
set )
-valued Function-like V29(
NAT : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal V52()
V53()
V54()
V55()
V56()
V57()
V58() )
Element of
K32(
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4() )
set ) ) )
V30(
NAT : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal V52()
V53()
V54()
V55()
V56()
V57()
V58() )
Element of
K32(
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4() )
set ) ) , the
carrier of
b1 : ( ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like )
ComplexNormSpace) : ( ( ) (
V4() )
set ) ) )
sequence of ( ( ) (
V4() )
set ) ) is
norm_summable ;
theorem
for
X being ( ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like )
ComplexNormSpace)
for
seq being ( (
Function-like V30(
NAT : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal V52()
V53()
V54()
V55()
V56()
V57()
V58() )
Element of
K32(
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4() )
set ) ) , the
carrier of
b1 : ( ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like )
ComplexNormSpace) : ( ( ) (
V4() )
set ) ) ) (
V4()
Relation-like NAT : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal V52()
V53()
V54()
V55()
V56()
V57()
V58() )
Element of
K32(
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4() )
set ) )
-defined the
carrier of
b1 : ( ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like )
ComplexNormSpace) : ( ( ) (
V4() )
set )
-valued Function-like V29(
NAT : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal V52()
V53()
V54()
V55()
V56()
V57()
V58() )
Element of
K32(
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4() )
set ) ) )
V30(
NAT : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal V52()
V53()
V54()
V55()
V56()
V57()
V58() )
Element of
K32(
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4() )
set ) ) , the
carrier of
b1 : ( ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like )
ComplexNormSpace) : ( ( ) (
V4() )
set ) ) )
sequence of ( ( ) (
V4() )
set ) ) st ( for
n being ( ( ) (
epsilon-transitive epsilon-connected ordinal natural V24()
real ext-real V45()
V46()
V52()
V53()
V54()
V55()
V56()
V57() )
Element of
NAT : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal V52()
V53()
V54()
V55()
V56()
V57()
V58() )
Element of
K32(
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4() )
set ) ) ) holds
seq : ( (
Function-like V30(
NAT : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal V52()
V53()
V54()
V55()
V56()
V57()
V58() )
Element of
K32(
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4() )
set ) ) , the
carrier of
b1 : ( ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like )
ComplexNormSpace) : ( ( ) (
V4() )
set ) ) ) (
V4()
Relation-like NAT : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal V52()
V53()
V54()
V55()
V56()
V57()
V58() )
Element of
K32(
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4() )
set ) )
-defined the
carrier of
b1 : ( ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like )
ComplexNormSpace) : ( ( ) (
V4() )
set )
-valued Function-like V29(
NAT : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal V52()
V53()
V54()
V55()
V56()
V57()
V58() )
Element of
K32(
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4() )
set ) ) )
V30(
NAT : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal V52()
V53()
V54()
V55()
V56()
V57()
V58() )
Element of
K32(
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4() )
set ) ) , the
carrier of
b1 : ( ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like )
ComplexNormSpace) : ( ( ) (
V4() )
set ) ) )
sequence of ( ( ) (
V4() )
set ) )
. n : ( ( ) (
epsilon-transitive epsilon-connected ordinal natural V24()
real ext-real V45()
V46()
V52()
V53()
V54()
V55()
V56()
V57() )
Element of
NAT : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal V52()
V53()
V54()
V55()
V56()
V57()
V58() )
Element of
K32(
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4() )
set ) ) ) : ( ( ) (
right_complementable )
Element of the
carrier of
b1 : ( ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like )
ComplexNormSpace) : ( ( ) (
V4() )
set ) )
<> 0. X : ( ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like )
ComplexNormSpace) : ( ( ) (
V79(
b1 : ( ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like )
ComplexNormSpace) )
right_complementable )
Element of the
carrier of
b1 : ( ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like )
ComplexNormSpace) : ( ( ) (
V4() )
set ) ) ) & ex
m being ( ( ) (
epsilon-transitive epsilon-connected ordinal natural V24()
real ext-real V45()
V46()
V52()
V53()
V54()
V55()
V56()
V57() )
Element of
NAT : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal V52()
V53()
V54()
V55()
V56()
V57()
V58() )
Element of
K32(
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4() )
set ) ) ) st
for
n being ( ( ) (
epsilon-transitive epsilon-connected ordinal natural V24()
real ext-real V45()
V46()
V52()
V53()
V54()
V55()
V56()
V57() )
Element of
NAT : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal V52()
V53()
V54()
V55()
V56()
V57()
V58() )
Element of
K32(
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4() )
set ) ) ) st
n : ( ( ) (
epsilon-transitive epsilon-connected ordinal natural V24()
real ext-real V45()
V46()
V52()
V53()
V54()
V55()
V56()
V57() )
Element of
NAT : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal V52()
V53()
V54()
V55()
V56()
V57()
V58() )
Element of
K32(
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4() )
set ) ) )
>= m : ( ( ) (
epsilon-transitive epsilon-connected ordinal natural V24()
real ext-real V45()
V46()
V52()
V53()
V54()
V55()
V56()
V57() )
Element of
NAT : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal V52()
V53()
V54()
V55()
V56()
V57()
V58() )
Element of
K32(
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4() )
set ) ) ) holds
(||.seq : ( ( Function-like V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed V133() V134() vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed V133() V134() vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( V4() ) set ) ) ) ( V4() Relation-like NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) -defined the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed V133() V134() vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed V133() V134() vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( V4() ) set ) -valued Function-like V29( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) ) V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed V133() V134() vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed V133() V134() vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( V4() ) set ) ) ) sequence of ( ( ) ( V4() ) set ) ) .|| : ( ( Function-like V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) ) ( V4() Relation-like NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) -defined REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) -valued Function-like V29( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) ) V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) complex-valued ext-real-valued real-valued ) Element of K32(K33(NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) ,REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( V4() ) set ) ) . (n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V24() real ext-real V45() V46() V52() V53() V54() V55() V56() V57() ) Element of NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) ) + 1 : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal natural V24() real ext-real positive non negative V45() V46() V52() V53() V54() V55() V56() V57() ) Element of NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) ) ) : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V24() real ext-real V45() V46() V52() V53() V54() V55() V56() V57() ) Element of NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) ) ) : ( ( ) (
V24()
real ext-real )
Element of
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) )
/ (||.seq : ( ( Function-like V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed V133() V134() vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed V133() V134() vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( V4() ) set ) ) ) ( V4() Relation-like NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) -defined the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed V133() V134() vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed V133() V134() vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( V4() ) set ) -valued Function-like V29( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) ) V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed V133() V134() vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed V133() V134() vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( V4() ) set ) ) ) sequence of ( ( ) ( V4() ) set ) ) .|| : ( ( Function-like V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) ) ( V4() Relation-like NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) -defined REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) -valued Function-like V29( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) ) V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) complex-valued ext-real-valued real-valued ) Element of K32(K33(NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) ,REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( V4() ) set ) ) . n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V24() real ext-real V45() V46() V52() V53() V54() V55() V56() V57() ) Element of NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) ) ) : ( ( ) (
V24()
real ext-real )
Element of
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V24()
real ext-real )
Element of
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) )
>= 1 : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal natural V24()
real ext-real positive non
negative V45()
V46()
V52()
V53()
V54()
V55()
V56()
V57() )
Element of
NAT : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal V52()
V53()
V54()
V55()
V56()
V57()
V58() )
Element of
K32(
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4() )
set ) ) ) holds
not
seq : ( (
Function-like V30(
NAT : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal V52()
V53()
V54()
V55()
V56()
V57()
V58() )
Element of
K32(
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4() )
set ) ) , the
carrier of
b1 : ( ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like )
ComplexNormSpace) : ( ( ) (
V4() )
set ) ) ) (
V4()
Relation-like NAT : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal V52()
V53()
V54()
V55()
V56()
V57()
V58() )
Element of
K32(
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4() )
set ) )
-defined the
carrier of
b1 : ( ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like )
ComplexNormSpace) : ( ( ) (
V4() )
set )
-valued Function-like V29(
NAT : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal V52()
V53()
V54()
V55()
V56()
V57()
V58() )
Element of
K32(
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4() )
set ) ) )
V30(
NAT : ( ( ) (
V4()
epsilon-transitive epsilon-connected ordinal V52()
V53()
V54()
V55()
V56()
V57()
V58() )
Element of
K32(
REAL : ( ( ) (
V4()
V47()
V52()
V53()
V54()
V58() )
set ) ) : ( ( ) (
V4() )
set ) ) , the
carrier of
b1 : ( ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non
empty right_complementable Abelian add-associative right_zeroed V133()
V134()
vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like )
ComplexNormSpace) : ( ( ) (
V4() )
set ) ) )
sequence of ( ( ) (
V4() )
set ) ) is
norm_summable ;
begin