begin
definition
let X be ( ( non
empty ) ( non
empty )
addLoopStr ) ;
let seq be ( (
V21()
V30(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) , the
carrier of
X : ( ( non
empty ) ( non
empty )
addLoopStr ) : ( ( ) (
V1() )
set ) ) ) (
V1()
V16()
V19(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) )
V20( the
carrier of
X : ( ( non
empty ) ( non
empty )
addLoopStr ) : ( ( ) (
V1() )
set ) )
V21()
V26(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) )
V30(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) , the
carrier of
X : ( ( non
empty ) ( non
empty )
addLoopStr ) : ( ( ) (
V1() )
set ) ) )
sequence of ( ( ) (
V1() )
set ) ) ;
func Partial_Sums seq -> ( (
V21()
V30(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) , the
carrier of
X : ( ( ) ( )
UNITSTR ) : ( ( ) ( )
set ) ) ) (
V1()
V16()
V19(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) )
V20( the
carrier of
X : ( ( ) ( )
UNITSTR ) : ( ( ) ( )
set ) )
V21()
V26(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) )
V30(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) , the
carrier of
X : ( ( ) ( )
UNITSTR ) : ( ( ) ( )
set ) ) )
sequence of ( ( ) ( )
set ) )
means
(
it : ( (
V21()
V30(
K7(
X : ( ( ) ( )
UNITSTR ) ,
X : ( ( ) ( )
UNITSTR ) ) : ( ( ) ( )
set ) ,
X : ( ( ) ( )
UNITSTR ) ) ) (
V16()
V19(
K7(
X : ( ( ) ( )
UNITSTR ) ,
X : ( ( ) ( )
UNITSTR ) ) : ( ( ) ( )
set ) )
V20(
X : ( ( ) ( )
UNITSTR ) )
V21()
V30(
K7(
X : ( ( ) ( )
UNITSTR ) ,
X : ( ( ) ( )
UNITSTR ) ) : ( ( ) ( )
set ) ,
X : ( ( ) ( )
UNITSTR ) ) )
Element of
K6(
K7(
K7(
X : ( ( ) ( )
UNITSTR ) ,
X : ( ( ) ( )
UNITSTR ) ) : ( ( ) ( )
set ) ,
X : ( ( ) ( )
UNITSTR ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) )
. 0 : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural V11()
real ext-real non
positive non
negative integer V58()
V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( )
Element of the
carrier of
X : ( ( ) ( )
UNITSTR ) : ( ( ) ( )
set ) )
= seq : ( ( ) ( )
Element of
X : ( ( ) ( )
UNITSTR ) )
. 0 : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural V11()
real ext-real non
positive non
negative integer V58()
V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( )
Element of the
carrier of
X : ( ( ) ( )
UNITSTR ) : ( ( ) ( )
set ) ) & ( for
n being ( ( ) (
epsilon-transitive epsilon-connected ordinal natural V11()
real ext-real integer V58()
V59()
V60()
V61()
V62()
V63()
V64() )
Element of
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) ) holds
it : ( (
V21()
V30(
K7(
X : ( ( ) ( )
UNITSTR ) ,
X : ( ( ) ( )
UNITSTR ) ) : ( ( ) ( )
set ) ,
X : ( ( ) ( )
UNITSTR ) ) ) (
V16()
V19(
K7(
X : ( ( ) ( )
UNITSTR ) ,
X : ( ( ) ( )
UNITSTR ) ) : ( ( ) ( )
set ) )
V20(
X : ( ( ) ( )
UNITSTR ) )
V21()
V30(
K7(
X : ( ( ) ( )
UNITSTR ) ,
X : ( ( ) ( )
UNITSTR ) ) : ( ( ) ( )
set ) ,
X : ( ( ) ( )
UNITSTR ) ) )
Element of
K6(
K7(
K7(
X : ( ( ) ( )
UNITSTR ) ,
X : ( ( ) ( )
UNITSTR ) ) : ( ( ) ( )
set ) ,
X : ( ( ) ( )
UNITSTR ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) )
. (n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real integer V58() V59() V60() V61() V62() V63() V64() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) ) + 1 : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative integer V58() V59() V60() V61() V62() V63() V64() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) (
epsilon-transitive epsilon-connected ordinal natural V11()
real ext-real integer V58()
V59()
V60()
V61()
V62()
V63()
V64() )
Element of
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( )
Element of the
carrier of
X : ( ( ) ( )
UNITSTR ) : ( ( ) ( )
set ) )
= (it : ( ( V21() V30(K7(X : ( ( ) ( ) UNITSTR ) ,X : ( ( ) ( ) UNITSTR ) ) : ( ( ) ( ) set ) ,X : ( ( ) ( ) UNITSTR ) ) ) ( V16() V19(K7(X : ( ( ) ( ) UNITSTR ) ,X : ( ( ) ( ) UNITSTR ) ) : ( ( ) ( ) set ) ) V20(X : ( ( ) ( ) UNITSTR ) ) V21() V30(K7(X : ( ( ) ( ) UNITSTR ) ,X : ( ( ) ( ) UNITSTR ) ) : ( ( ) ( ) set ) ,X : ( ( ) ( ) UNITSTR ) ) ) Element of K6(K7(K7(X : ( ( ) ( ) UNITSTR ) ,X : ( ( ) ( ) UNITSTR ) ) : ( ( ) ( ) set ) ,X : ( ( ) ( ) UNITSTR ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) . n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real integer V58() V59() V60() V61() V62() V63() V64() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( )
Element of the
carrier of
X : ( ( ) ( )
UNITSTR ) : ( ( ) ( )
set ) )
+ (seq : ( ( ) ( ) Element of X : ( ( ) ( ) UNITSTR ) ) . (n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real integer V58() V59() V60() V61() V62() V63() V64() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) ) + 1 : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative integer V58() V59() V60() V61() V62() V63() V64() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real integer V58() V59() V60() V61() V62() V63() V64() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( )
Element of the
carrier of
X : ( ( ) ( )
UNITSTR ) : ( ( ) ( )
set ) ) : ( ( ) ( )
Element of the
carrier of
X : ( ( ) ( )
UNITSTR ) : ( ( ) ( )
set ) ) ) );
end;
theorem
for
X being ( ( non
empty Abelian add-associative ) ( non
empty Abelian add-associative )
addLoopStr )
for
seq1,
seq2 being ( (
V21()
V30(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) , the
carrier of
b1 : ( ( non
empty Abelian add-associative ) ( non
empty Abelian add-associative )
addLoopStr ) : ( ( ) (
V1() )
set ) ) ) (
V1()
V16()
V19(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) )
V20( the
carrier of
b1 : ( ( non
empty Abelian add-associative ) ( non
empty Abelian add-associative )
addLoopStr ) : ( ( ) (
V1() )
set ) )
V21()
V26(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) )
V30(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) , the
carrier of
b1 : ( ( non
empty Abelian add-associative ) ( non
empty Abelian add-associative )
addLoopStr ) : ( ( ) (
V1() )
set ) ) )
sequence of ( ( ) (
V1() )
set ) ) holds
(Partial_Sums seq1 : ( ( V21() V30( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of b1 : ( ( non empty Abelian add-associative ) ( non empty Abelian add-associative ) addLoopStr ) : ( ( ) ( V1() ) set ) ) ) ( V1() V16() V19( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) ) V20( the carrier of b1 : ( ( non empty Abelian add-associative ) ( non empty Abelian add-associative ) addLoopStr ) : ( ( ) ( V1() ) set ) ) V21() V26( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of b1 : ( ( non empty Abelian add-associative ) ( non empty Abelian add-associative ) addLoopStr ) : ( ( ) ( V1() ) set ) ) ) sequence of ( ( ) ( V1() ) set ) ) ) : ( (
V21()
V30(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) , the
carrier of
b1 : ( ( non
empty Abelian add-associative ) ( non
empty Abelian add-associative )
addLoopStr ) : ( ( ) (
V1() )
set ) ) ) (
V1()
V16()
V19(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) )
V20( the
carrier of
b1 : ( ( non
empty Abelian add-associative ) ( non
empty Abelian add-associative )
addLoopStr ) : ( ( ) (
V1() )
set ) )
V21()
V26(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) )
V30(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) , the
carrier of
b1 : ( ( non
empty Abelian add-associative ) ( non
empty Abelian add-associative )
addLoopStr ) : ( ( ) (
V1() )
set ) ) )
sequence of ( ( ) (
V1() )
set ) )
+ (Partial_Sums seq2 : ( ( V21() V30( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of b1 : ( ( non empty Abelian add-associative ) ( non empty Abelian add-associative ) addLoopStr ) : ( ( ) ( V1() ) set ) ) ) ( V1() V16() V19( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) ) V20( the carrier of b1 : ( ( non empty Abelian add-associative ) ( non empty Abelian add-associative ) addLoopStr ) : ( ( ) ( V1() ) set ) ) V21() V26( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of b1 : ( ( non empty Abelian add-associative ) ( non empty Abelian add-associative ) addLoopStr ) : ( ( ) ( V1() ) set ) ) ) sequence of ( ( ) ( V1() ) set ) ) ) : ( (
V21()
V30(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) , the
carrier of
b1 : ( ( non
empty Abelian add-associative ) ( non
empty Abelian add-associative )
addLoopStr ) : ( ( ) (
V1() )
set ) ) ) (
V1()
V16()
V19(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) )
V20( the
carrier of
b1 : ( ( non
empty Abelian add-associative ) ( non
empty Abelian add-associative )
addLoopStr ) : ( ( ) (
V1() )
set ) )
V21()
V26(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) )
V30(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) , the
carrier of
b1 : ( ( non
empty Abelian add-associative ) ( non
empty Abelian add-associative )
addLoopStr ) : ( ( ) (
V1() )
set ) ) )
sequence of ( ( ) (
V1() )
set ) ) : ( (
V21()
V30(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) , the
carrier of
b1 : ( ( non
empty Abelian add-associative ) ( non
empty Abelian add-associative )
addLoopStr ) : ( ( ) (
V1() )
set ) ) ) (
V1()
V16()
V19(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) )
V20( the
carrier of
b1 : ( ( non
empty Abelian add-associative ) ( non
empty Abelian add-associative )
addLoopStr ) : ( ( ) (
V1() )
set ) )
V21()
V26(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) )
V30(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) , the
carrier of
b1 : ( ( non
empty Abelian add-associative ) ( non
empty Abelian add-associative )
addLoopStr ) : ( ( ) (
V1() )
set ) ) )
Element of
K6(
K7(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) , the
carrier of
b1 : ( ( non
empty Abelian add-associative ) ( non
empty Abelian add-associative )
addLoopStr ) : ( ( ) (
V1() )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) )
= Partial_Sums (seq1 : ( ( V21() V30( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of b1 : ( ( non empty Abelian add-associative ) ( non empty Abelian add-associative ) addLoopStr ) : ( ( ) ( V1() ) set ) ) ) ( V1() V16() V19( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) ) V20( the carrier of b1 : ( ( non empty Abelian add-associative ) ( non empty Abelian add-associative ) addLoopStr ) : ( ( ) ( V1() ) set ) ) V21() V26( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of b1 : ( ( non empty Abelian add-associative ) ( non empty Abelian add-associative ) addLoopStr ) : ( ( ) ( V1() ) set ) ) ) sequence of ( ( ) ( V1() ) set ) ) + seq2 : ( ( V21() V30( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of b1 : ( ( non empty Abelian add-associative ) ( non empty Abelian add-associative ) addLoopStr ) : ( ( ) ( V1() ) set ) ) ) ( V1() V16() V19( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) ) V20( the carrier of b1 : ( ( non empty Abelian add-associative ) ( non empty Abelian add-associative ) addLoopStr ) : ( ( ) ( V1() ) set ) ) V21() V26( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of b1 : ( ( non empty Abelian add-associative ) ( non empty Abelian add-associative ) addLoopStr ) : ( ( ) ( V1() ) set ) ) ) sequence of ( ( ) ( V1() ) set ) ) ) : ( (
V21()
V30(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) , the
carrier of
b1 : ( ( non
empty Abelian add-associative ) ( non
empty Abelian add-associative )
addLoopStr ) : ( ( ) (
V1() )
set ) ) ) (
V1()
V16()
V19(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) )
V20( the
carrier of
b1 : ( ( non
empty Abelian add-associative ) ( non
empty Abelian add-associative )
addLoopStr ) : ( ( ) (
V1() )
set ) )
V21()
V26(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) )
V30(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) , the
carrier of
b1 : ( ( non
empty Abelian add-associative ) ( non
empty Abelian add-associative )
addLoopStr ) : ( ( ) (
V1() )
set ) ) )
Element of
K6(
K7(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) , the
carrier of
b1 : ( ( non
empty Abelian add-associative ) ( non
empty Abelian add-associative )
addLoopStr ) : ( ( ) (
V1() )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) : ( (
V21()
V30(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) , the
carrier of
b1 : ( ( non
empty Abelian add-associative ) ( non
empty Abelian add-associative )
addLoopStr ) : ( ( ) (
V1() )
set ) ) ) (
V1()
V16()
V19(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) )
V20( the
carrier of
b1 : ( ( non
empty Abelian add-associative ) ( non
empty Abelian add-associative )
addLoopStr ) : ( ( ) (
V1() )
set ) )
V21()
V26(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) )
V30(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) , the
carrier of
b1 : ( ( non
empty Abelian add-associative ) ( non
empty Abelian add-associative )
addLoopStr ) : ( ( ) (
V1() )
set ) ) )
sequence of ( ( ) (
V1() )
set ) ) ;
theorem
for
X being ( ( non
empty right_complementable Abelian add-associative right_zeroed ) ( non
empty left_complementable right_complementable Abelian add-associative right_zeroed zeroed )
addLoopStr )
for
seq1,
seq2 being ( (
V21()
V30(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) , the
carrier of
b1 : ( ( non
empty right_complementable Abelian add-associative right_zeroed ) ( non
empty left_complementable right_complementable Abelian add-associative right_zeroed zeroed )
addLoopStr ) : ( ( ) (
V1() )
set ) ) ) (
V1()
V16()
V19(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) )
V20( the
carrier of
b1 : ( ( non
empty right_complementable Abelian add-associative right_zeroed ) ( non
empty left_complementable right_complementable Abelian add-associative right_zeroed zeroed )
addLoopStr ) : ( ( ) (
V1() )
set ) )
V21()
V26(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) )
V30(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) , the
carrier of
b1 : ( ( non
empty right_complementable Abelian add-associative right_zeroed ) ( non
empty left_complementable right_complementable Abelian add-associative right_zeroed zeroed )
addLoopStr ) : ( ( ) (
V1() )
set ) ) )
sequence of ( ( ) (
V1() )
set ) ) holds
(Partial_Sums seq1 : ( ( V21() V30( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed zeroed ) addLoopStr ) : ( ( ) ( V1() ) set ) ) ) ( V1() V16() V19( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) ) V20( the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed zeroed ) addLoopStr ) : ( ( ) ( V1() ) set ) ) V21() V26( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed zeroed ) addLoopStr ) : ( ( ) ( V1() ) set ) ) ) sequence of ( ( ) ( V1() ) set ) ) ) : ( (
V21()
V30(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) , the
carrier of
b1 : ( ( non
empty right_complementable Abelian add-associative right_zeroed ) ( non
empty left_complementable right_complementable Abelian add-associative right_zeroed zeroed )
addLoopStr ) : ( ( ) (
V1() )
set ) ) ) (
V1()
V16()
V19(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) )
V20( the
carrier of
b1 : ( ( non
empty right_complementable Abelian add-associative right_zeroed ) ( non
empty left_complementable right_complementable Abelian add-associative right_zeroed zeroed )
addLoopStr ) : ( ( ) (
V1() )
set ) )
V21()
V26(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) )
V30(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) , the
carrier of
b1 : ( ( non
empty right_complementable Abelian add-associative right_zeroed ) ( non
empty left_complementable right_complementable Abelian add-associative right_zeroed zeroed )
addLoopStr ) : ( ( ) (
V1() )
set ) ) )
sequence of ( ( ) (
V1() )
set ) )
- (Partial_Sums seq2 : ( ( V21() V30( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed zeroed ) addLoopStr ) : ( ( ) ( V1() ) set ) ) ) ( V1() V16() V19( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) ) V20( the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed zeroed ) addLoopStr ) : ( ( ) ( V1() ) set ) ) V21() V26( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed zeroed ) addLoopStr ) : ( ( ) ( V1() ) set ) ) ) sequence of ( ( ) ( V1() ) set ) ) ) : ( (
V21()
V30(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) , the
carrier of
b1 : ( ( non
empty right_complementable Abelian add-associative right_zeroed ) ( non
empty left_complementable right_complementable Abelian add-associative right_zeroed zeroed )
addLoopStr ) : ( ( ) (
V1() )
set ) ) ) (
V1()
V16()
V19(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) )
V20( the
carrier of
b1 : ( ( non
empty right_complementable Abelian add-associative right_zeroed ) ( non
empty left_complementable right_complementable Abelian add-associative right_zeroed zeroed )
addLoopStr ) : ( ( ) (
V1() )
set ) )
V21()
V26(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) )
V30(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) , the
carrier of
b1 : ( ( non
empty right_complementable Abelian add-associative right_zeroed ) ( non
empty left_complementable right_complementable Abelian add-associative right_zeroed zeroed )
addLoopStr ) : ( ( ) (
V1() )
set ) ) )
sequence of ( ( ) (
V1() )
set ) ) : ( (
V21()
V30(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) , the
carrier of
b1 : ( ( non
empty right_complementable Abelian add-associative right_zeroed ) ( non
empty left_complementable right_complementable Abelian add-associative right_zeroed zeroed )
addLoopStr ) : ( ( ) (
V1() )
set ) ) ) (
V1()
V16()
V19(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) )
V20( the
carrier of
b1 : ( ( non
empty right_complementable Abelian add-associative right_zeroed ) ( non
empty left_complementable right_complementable Abelian add-associative right_zeroed zeroed )
addLoopStr ) : ( ( ) (
V1() )
set ) )
V21()
V26(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) )
V30(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) , the
carrier of
b1 : ( ( non
empty right_complementable Abelian add-associative right_zeroed ) ( non
empty left_complementable right_complementable Abelian add-associative right_zeroed zeroed )
addLoopStr ) : ( ( ) (
V1() )
set ) ) )
Element of
K6(
K7(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) , the
carrier of
b1 : ( ( non
empty right_complementable Abelian add-associative right_zeroed ) ( non
empty left_complementable right_complementable Abelian add-associative right_zeroed zeroed )
addLoopStr ) : ( ( ) (
V1() )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) )
= Partial_Sums (seq1 : ( ( V21() V30( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed zeroed ) addLoopStr ) : ( ( ) ( V1() ) set ) ) ) ( V1() V16() V19( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) ) V20( the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed zeroed ) addLoopStr ) : ( ( ) ( V1() ) set ) ) V21() V26( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed zeroed ) addLoopStr ) : ( ( ) ( V1() ) set ) ) ) sequence of ( ( ) ( V1() ) set ) ) - seq2 : ( ( V21() V30( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed zeroed ) addLoopStr ) : ( ( ) ( V1() ) set ) ) ) ( V1() V16() V19( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) ) V20( the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed zeroed ) addLoopStr ) : ( ( ) ( V1() ) set ) ) V21() V26( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed zeroed ) addLoopStr ) : ( ( ) ( V1() ) set ) ) ) sequence of ( ( ) ( V1() ) set ) ) ) : ( (
V21()
V30(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) , the
carrier of
b1 : ( ( non
empty right_complementable Abelian add-associative right_zeroed ) ( non
empty left_complementable right_complementable Abelian add-associative right_zeroed zeroed )
addLoopStr ) : ( ( ) (
V1() )
set ) ) ) (
V1()
V16()
V19(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) )
V20( the
carrier of
b1 : ( ( non
empty right_complementable Abelian add-associative right_zeroed ) ( non
empty left_complementable right_complementable Abelian add-associative right_zeroed zeroed )
addLoopStr ) : ( ( ) (
V1() )
set ) )
V21()
V26(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) )
V30(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) , the
carrier of
b1 : ( ( non
empty right_complementable Abelian add-associative right_zeroed ) ( non
empty left_complementable right_complementable Abelian add-associative right_zeroed zeroed )
addLoopStr ) : ( ( ) (
V1() )
set ) ) )
Element of
K6(
K7(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) , the
carrier of
b1 : ( ( non
empty right_complementable Abelian add-associative right_zeroed ) ( non
empty left_complementable right_complementable Abelian add-associative right_zeroed zeroed )
addLoopStr ) : ( ( ) (
V1() )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) : ( (
V21()
V30(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) , the
carrier of
b1 : ( ( non
empty right_complementable Abelian add-associative right_zeroed ) ( non
empty left_complementable right_complementable Abelian add-associative right_zeroed zeroed )
addLoopStr ) : ( ( ) (
V1() )
set ) ) ) (
V1()
V16()
V19(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) )
V20( the
carrier of
b1 : ( ( non
empty right_complementable Abelian add-associative right_zeroed ) ( non
empty left_complementable right_complementable Abelian add-associative right_zeroed zeroed )
addLoopStr ) : ( ( ) (
V1() )
set ) )
V21()
V26(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) )
V30(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) , the
carrier of
b1 : ( ( non
empty right_complementable Abelian add-associative right_zeroed ) ( non
empty left_complementable right_complementable Abelian add-associative right_zeroed zeroed )
addLoopStr ) : ( ( ) (
V1() )
set ) ) )
sequence of ( ( ) (
V1() )
set ) ) ;
theorem
for
a being ( ( ) (
V11()
real ext-real )
Real)
for
X being ( ( non
empty vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non
empty vector-distributive scalar-distributive scalar-associative scalar-unital )
RLSStruct )
for
seq being ( (
V21()
V30(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) , the
carrier of
b2 : ( ( non
empty vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non
empty vector-distributive scalar-distributive scalar-associative scalar-unital )
RLSStruct ) : ( ( ) (
V1() )
set ) ) ) (
V1()
V16()
V19(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) )
V20( the
carrier of
b2 : ( ( non
empty vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non
empty vector-distributive scalar-distributive scalar-associative scalar-unital )
RLSStruct ) : ( ( ) (
V1() )
set ) )
V21()
V26(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) )
V30(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) , the
carrier of
b2 : ( ( non
empty vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non
empty vector-distributive scalar-distributive scalar-associative scalar-unital )
RLSStruct ) : ( ( ) (
V1() )
set ) ) )
sequence of ( ( ) (
V1() )
set ) ) holds
Partial_Sums (a : ( ( ) ( V11() real ext-real ) Real) * seq : ( ( V21() V30( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of b2 : ( ( non empty vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty vector-distributive scalar-distributive scalar-associative scalar-unital ) RLSStruct ) : ( ( ) ( V1() ) set ) ) ) ( V1() V16() V19( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) ) V20( the carrier of b2 : ( ( non empty vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty vector-distributive scalar-distributive scalar-associative scalar-unital ) RLSStruct ) : ( ( ) ( V1() ) set ) ) V21() V26( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of b2 : ( ( non empty vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty vector-distributive scalar-distributive scalar-associative scalar-unital ) RLSStruct ) : ( ( ) ( V1() ) set ) ) ) sequence of ( ( ) ( V1() ) set ) ) ) : ( (
V21()
V30(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) , the
carrier of
b2 : ( ( non
empty vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non
empty vector-distributive scalar-distributive scalar-associative scalar-unital )
RLSStruct ) : ( ( ) (
V1() )
set ) ) ) (
V1()
V16()
V19(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) )
V20( the
carrier of
b2 : ( ( non
empty vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non
empty vector-distributive scalar-distributive scalar-associative scalar-unital )
RLSStruct ) : ( ( ) (
V1() )
set ) )
V21()
V26(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) )
V30(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) , the
carrier of
b2 : ( ( non
empty vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non
empty vector-distributive scalar-distributive scalar-associative scalar-unital )
RLSStruct ) : ( ( ) (
V1() )
set ) ) )
Element of
K6(
K7(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) , the
carrier of
b2 : ( ( non
empty vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non
empty vector-distributive scalar-distributive scalar-associative scalar-unital )
RLSStruct ) : ( ( ) (
V1() )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) : ( (
V21()
V30(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) , the
carrier of
b2 : ( ( non
empty vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non
empty vector-distributive scalar-distributive scalar-associative scalar-unital )
RLSStruct ) : ( ( ) (
V1() )
set ) ) ) (
V1()
V16()
V19(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) )
V20( the
carrier of
b2 : ( ( non
empty vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non
empty vector-distributive scalar-distributive scalar-associative scalar-unital )
RLSStruct ) : ( ( ) (
V1() )
set ) )
V21()
V26(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) )
V30(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) , the
carrier of
b2 : ( ( non
empty vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non
empty vector-distributive scalar-distributive scalar-associative scalar-unital )
RLSStruct ) : ( ( ) (
V1() )
set ) ) )
sequence of ( ( ) (
V1() )
set ) )
= a : ( ( ) (
V11()
real ext-real )
Real)
* (Partial_Sums seq : ( ( V21() V30( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of b2 : ( ( non empty vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty vector-distributive scalar-distributive scalar-associative scalar-unital ) RLSStruct ) : ( ( ) ( V1() ) set ) ) ) ( V1() V16() V19( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) ) V20( the carrier of b2 : ( ( non empty vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty vector-distributive scalar-distributive scalar-associative scalar-unital ) RLSStruct ) : ( ( ) ( V1() ) set ) ) V21() V26( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of b2 : ( ( non empty vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty vector-distributive scalar-distributive scalar-associative scalar-unital ) RLSStruct ) : ( ( ) ( V1() ) set ) ) ) sequence of ( ( ) ( V1() ) set ) ) ) : ( (
V21()
V30(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) , the
carrier of
b2 : ( ( non
empty vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non
empty vector-distributive scalar-distributive scalar-associative scalar-unital )
RLSStruct ) : ( ( ) (
V1() )
set ) ) ) (
V1()
V16()
V19(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) )
V20( the
carrier of
b2 : ( ( non
empty vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non
empty vector-distributive scalar-distributive scalar-associative scalar-unital )
RLSStruct ) : ( ( ) (
V1() )
set ) )
V21()
V26(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) )
V30(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) , the
carrier of
b2 : ( ( non
empty vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non
empty vector-distributive scalar-distributive scalar-associative scalar-unital )
RLSStruct ) : ( ( ) (
V1() )
set ) ) )
sequence of ( ( ) (
V1() )
set ) ) : ( (
V21()
V30(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) , the
carrier of
b2 : ( ( non
empty vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non
empty vector-distributive scalar-distributive scalar-associative scalar-unital )
RLSStruct ) : ( ( ) (
V1() )
set ) ) ) (
V1()
V16()
V19(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) )
V20( the
carrier of
b2 : ( ( non
empty vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non
empty vector-distributive scalar-distributive scalar-associative scalar-unital )
RLSStruct ) : ( ( ) (
V1() )
set ) )
V21()
V26(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) )
V30(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) , the
carrier of
b2 : ( ( non
empty vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non
empty vector-distributive scalar-distributive scalar-associative scalar-unital )
RLSStruct ) : ( ( ) (
V1() )
set ) ) )
Element of
K6(
K7(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) , the
carrier of
b2 : ( ( non
empty vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non
empty vector-distributive scalar-distributive scalar-associative scalar-unital )
RLSStruct ) : ( ( ) (
V1() )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) ;
theorem
for
X being ( ( non
empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non
empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like )
RealUnitarySpace)
for
seq being ( (
V21()
V30(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) , the
carrier of
b1 : ( ( non
empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non
empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like )
RealUnitarySpace) : ( ( ) (
V1() )
set ) ) ) (
V1()
V16()
V19(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) )
V20( the
carrier of
b1 : ( ( non
empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non
empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like )
RealUnitarySpace) : ( ( ) (
V1() )
set ) )
V21()
V26(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) )
V30(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) , the
carrier of
b1 : ( ( non
empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non
empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like )
RealUnitarySpace) : ( ( ) (
V1() )
set ) ) )
sequence of ( ( ) (
V1() )
set ) ) holds
Partial_Sums (- seq : ( ( V21() V30( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V1() ) set ) ) ) ( V1() V16() V19( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) ) V20( the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V1() ) set ) ) V21() V26( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V1() ) set ) ) ) sequence of ( ( ) ( V1() ) set ) ) ) : ( (
V21() ) (
V1()
V16()
V19(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) )
V20( the
carrier of
b1 : ( ( non
empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non
empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like )
RealUnitarySpace) : ( ( ) (
V1() )
set ) )
V21()
V26(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) )
V30(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) , the
carrier of
b1 : ( ( non
empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non
empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like )
RealUnitarySpace) : ( ( ) (
V1() )
set ) ) )
Element of
K6(
K7(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) , the
carrier of
b1 : ( ( non
empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non
empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like )
RealUnitarySpace) : ( ( ) (
V1() )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) : ( (
V21()
V30(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) , the
carrier of
b1 : ( ( non
empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non
empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like )
RealUnitarySpace) : ( ( ) (
V1() )
set ) ) ) (
V1()
V16()
V19(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) )
V20( the
carrier of
b1 : ( ( non
empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non
empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like )
RealUnitarySpace) : ( ( ) (
V1() )
set ) )
V21()
V26(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) )
V30(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) , the
carrier of
b1 : ( ( non
empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non
empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like )
RealUnitarySpace) : ( ( ) (
V1() )
set ) ) )
sequence of ( ( ) (
V1() )
set ) )
= - (Partial_Sums seq : ( ( V21() V30( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V1() ) set ) ) ) ( V1() V16() V19( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) ) V20( the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V1() ) set ) ) V21() V26( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V1() ) set ) ) ) sequence of ( ( ) ( V1() ) set ) ) ) : ( (
V21()
V30(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) , the
carrier of
b1 : ( ( non
empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non
empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like )
RealUnitarySpace) : ( ( ) (
V1() )
set ) ) ) (
V1()
V16()
V19(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) )
V20( the
carrier of
b1 : ( ( non
empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non
empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like )
RealUnitarySpace) : ( ( ) (
V1() )
set ) )
V21()
V26(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) )
V30(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) , the
carrier of
b1 : ( ( non
empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non
empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like )
RealUnitarySpace) : ( ( ) (
V1() )
set ) ) )
sequence of ( ( ) (
V1() )
set ) ) : ( (
V21() ) (
V1()
V16()
V19(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) )
V20( the
carrier of
b1 : ( ( non
empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non
empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like )
RealUnitarySpace) : ( ( ) (
V1() )
set ) )
V21()
V26(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) )
V30(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) , the
carrier of
b1 : ( ( non
empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non
empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like )
RealUnitarySpace) : ( ( ) (
V1() )
set ) ) )
Element of
K6(
K7(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) , the
carrier of
b1 : ( ( non
empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non
empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like )
RealUnitarySpace) : ( ( ) (
V1() )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) ;
theorem
for
a,
b being ( ( ) (
V11()
real ext-real )
Real)
for
X being ( ( non
empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non
empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like )
RealUnitarySpace)
for
seq1,
seq2 being ( (
V21()
V30(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) , the
carrier of
b3 : ( ( non
empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non
empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like )
RealUnitarySpace) : ( ( ) (
V1() )
set ) ) ) (
V1()
V16()
V19(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) )
V20( the
carrier of
b3 : ( ( non
empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non
empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like )
RealUnitarySpace) : ( ( ) (
V1() )
set ) )
V21()
V26(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) )
V30(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) , the
carrier of
b3 : ( ( non
empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non
empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like )
RealUnitarySpace) : ( ( ) (
V1() )
set ) ) )
sequence of ( ( ) (
V1() )
set ) ) holds
(a : ( ( ) ( V11() real ext-real ) Real) * (Partial_Sums seq1 : ( ( V21() V30( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of b3 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V1() ) set ) ) ) ( V1() V16() V19( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) ) V20( the carrier of b3 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V1() ) set ) ) V21() V26( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of b3 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V1() ) set ) ) ) sequence of ( ( ) ( V1() ) set ) ) ) : ( ( V21() V30( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of b3 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V1() ) set ) ) ) ( V1() V16() V19( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) ) V20( the carrier of b3 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V1() ) set ) ) V21() V26( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of b3 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V1() ) set ) ) ) sequence of ( ( ) ( V1() ) set ) ) ) : ( (
V21()
V30(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) , the
carrier of
b3 : ( ( non
empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non
empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like )
RealUnitarySpace) : ( ( ) (
V1() )
set ) ) ) (
V1()
V16()
V19(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) )
V20( the
carrier of
b3 : ( ( non
empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non
empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like )
RealUnitarySpace) : ( ( ) (
V1() )
set ) )
V21()
V26(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) )
V30(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) , the
carrier of
b3 : ( ( non
empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non
empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like )
RealUnitarySpace) : ( ( ) (
V1() )
set ) ) )
Element of
K6(
K7(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) , the
carrier of
b3 : ( ( non
empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non
empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like )
RealUnitarySpace) : ( ( ) (
V1() )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) )
+ (b : ( ( ) ( V11() real ext-real ) Real) * (Partial_Sums seq2 : ( ( V21() V30( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of b3 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V1() ) set ) ) ) ( V1() V16() V19( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) ) V20( the carrier of b3 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V1() ) set ) ) V21() V26( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of b3 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V1() ) set ) ) ) sequence of ( ( ) ( V1() ) set ) ) ) : ( ( V21() V30( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of b3 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V1() ) set ) ) ) ( V1() V16() V19( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) ) V20( the carrier of b3 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V1() ) set ) ) V21() V26( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of b3 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V1() ) set ) ) ) sequence of ( ( ) ( V1() ) set ) ) ) : ( (
V21()
V30(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) , the
carrier of
b3 : ( ( non
empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non
empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like )
RealUnitarySpace) : ( ( ) (
V1() )
set ) ) ) (
V1()
V16()
V19(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) )
V20( the
carrier of
b3 : ( ( non
empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non
empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like )
RealUnitarySpace) : ( ( ) (
V1() )
set ) )
V21()
V26(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) )
V30(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) , the
carrier of
b3 : ( ( non
empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non
empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like )
RealUnitarySpace) : ( ( ) (
V1() )
set ) ) )
Element of
K6(
K7(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) , the
carrier of
b3 : ( ( non
empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non
empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like )
RealUnitarySpace) : ( ( ) (
V1() )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) : ( (
V21()
V30(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) , the
carrier of
b3 : ( ( non
empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non
empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like )
RealUnitarySpace) : ( ( ) (
V1() )
set ) ) ) (
V1()
V16()
V19(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) )
V20( the
carrier of
b3 : ( ( non
empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non
empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like )
RealUnitarySpace) : ( ( ) (
V1() )
set ) )
V21()
V26(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) )
V30(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) , the
carrier of
b3 : ( ( non
empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non
empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like )
RealUnitarySpace) : ( ( ) (
V1() )
set ) ) )
Element of
K6(
K7(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) , the
carrier of
b3 : ( ( non
empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non
empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like )
RealUnitarySpace) : ( ( ) (
V1() )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) )
= Partial_Sums ((a : ( ( ) ( V11() real ext-real ) Real) * seq1 : ( ( V21() V30( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of b3 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V1() ) set ) ) ) ( V1() V16() V19( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) ) V20( the carrier of b3 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V1() ) set ) ) V21() V26( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of b3 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V1() ) set ) ) ) sequence of ( ( ) ( V1() ) set ) ) ) : ( ( V21() V30( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of b3 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V1() ) set ) ) ) ( V1() V16() V19( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) ) V20( the carrier of b3 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V1() ) set ) ) V21() V26( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of b3 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V1() ) set ) ) ) Element of K6(K7(NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of b3 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V1() ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) + (b : ( ( ) ( V11() real ext-real ) Real) * seq2 : ( ( V21() V30( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of b3 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V1() ) set ) ) ) ( V1() V16() V19( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) ) V20( the carrier of b3 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V1() ) set ) ) V21() V26( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of b3 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V1() ) set ) ) ) sequence of ( ( ) ( V1() ) set ) ) ) : ( ( V21() V30( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of b3 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V1() ) set ) ) ) ( V1() V16() V19( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) ) V20( the carrier of b3 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V1() ) set ) ) V21() V26( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of b3 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V1() ) set ) ) ) Element of K6(K7(NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of b3 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V1() ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( (
V21()
V30(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) , the
carrier of
b3 : ( ( non
empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non
empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like )
RealUnitarySpace) : ( ( ) (
V1() )
set ) ) ) (
V1()
V16()
V19(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) )
V20( the
carrier of
b3 : ( ( non
empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non
empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like )
RealUnitarySpace) : ( ( ) (
V1() )
set ) )
V21()
V26(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) )
V30(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) , the
carrier of
b3 : ( ( non
empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non
empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like )
RealUnitarySpace) : ( ( ) (
V1() )
set ) ) )
Element of
K6(
K7(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) , the
carrier of
b3 : ( ( non
empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non
empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like )
RealUnitarySpace) : ( ( ) (
V1() )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) : ( (
V21()
V30(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) , the
carrier of
b3 : ( ( non
empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non
empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like )
RealUnitarySpace) : ( ( ) (
V1() )
set ) ) ) (
V1()
V16()
V19(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) )
V20( the
carrier of
b3 : ( ( non
empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non
empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like )
RealUnitarySpace) : ( ( ) (
V1() )
set ) )
V21()
V26(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) )
V30(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) , the
carrier of
b3 : ( ( non
empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non
empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like )
RealUnitarySpace) : ( ( ) (
V1() )
set ) ) )
sequence of ( ( ) (
V1() )
set ) ) ;
theorem
for
X being ( ( non
empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non
empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like )
RealUnitarySpace)
for
seq1,
seq2 being ( (
V21()
V30(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) , the
carrier of
b1 : ( ( non
empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non
empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like )
RealUnitarySpace) : ( ( ) (
V1() )
set ) ) ) (
V1()
V16()
V19(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) )
V20( the
carrier of
b1 : ( ( non
empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non
empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like )
RealUnitarySpace) : ( ( ) (
V1() )
set ) )
V21()
V26(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) )
V30(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) , the
carrier of
b1 : ( ( non
empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non
empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like )
RealUnitarySpace) : ( ( ) (
V1() )
set ) ) )
sequence of ( ( ) (
V1() )
set ) ) st
seq1 : ( (
V21()
V30(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) , the
carrier of
b1 : ( ( non
empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non
empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like )
RealUnitarySpace) : ( ( ) (
V1() )
set ) ) ) (
V1()
V16()
V19(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) )
V20( the
carrier of
b1 : ( ( non
empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non
empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like )
RealUnitarySpace) : ( ( ) (
V1() )
set ) )
V21()
V26(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) )
V30(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) , the
carrier of
b1 : ( ( non
empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non
empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like )
RealUnitarySpace) : ( ( ) (
V1() )
set ) ) )
sequence of ( ( ) (
V1() )
set ) ) is
summable &
seq2 : ( (
V21()
V30(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) , the
carrier of
b1 : ( ( non
empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non
empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like )
RealUnitarySpace) : ( ( ) (
V1() )
set ) ) ) (
V1()
V16()
V19(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) )
V20( the
carrier of
b1 : ( ( non
empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non
empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like )
RealUnitarySpace) : ( ( ) (
V1() )
set ) )
V21()
V26(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) )
V30(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) , the
carrier of
b1 : ( ( non
empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non
empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like )
RealUnitarySpace) : ( ( ) (
V1() )
set ) ) )
sequence of ( ( ) (
V1() )
set ) ) is
summable holds
(
seq1 : ( (
V21()
V30(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) , the
carrier of
b1 : ( ( non
empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non
empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like )
RealUnitarySpace) : ( ( ) (
V1() )
set ) ) ) (
V1()
V16()
V19(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) )
V20( the
carrier of
b1 : ( ( non
empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non
empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like )
RealUnitarySpace) : ( ( ) (
V1() )
set ) )
V21()
V26(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) )
V30(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) , the
carrier of
b1 : ( ( non
empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non
empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like )
RealUnitarySpace) : ( ( ) (
V1() )
set ) ) )
sequence of ( ( ) (
V1() )
set ) )
+ seq2 : ( (
V21()
V30(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) , the
carrier of
b1 : ( ( non
empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non
empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like )
RealUnitarySpace) : ( ( ) (
V1() )
set ) ) ) (
V1()
V16()
V19(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) )
V20( the
carrier of
b1 : ( ( non
empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non
empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like )
RealUnitarySpace) : ( ( ) (
V1() )
set ) )
V21()
V26(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) )
V30(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) , the
carrier of
b1 : ( ( non
empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non
empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like )
RealUnitarySpace) : ( ( ) (
V1() )
set ) ) )
sequence of ( ( ) (
V1() )
set ) ) : ( (
V21()
V30(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) , the
carrier of
b1 : ( ( non
empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non
empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like )
RealUnitarySpace) : ( ( ) (
V1() )
set ) ) ) (
V1()
V16()
V19(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) )
V20( the
carrier of
b1 : ( ( non
empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non
empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like )
RealUnitarySpace) : ( ( ) (
V1() )
set ) )
V21()
V26(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) )
V30(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) , the
carrier of
b1 : ( ( non
empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non
empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like )
RealUnitarySpace) : ( ( ) (
V1() )
set ) ) )
Element of
K6(
K7(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) , the
carrier of
b1 : ( ( non
empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non
empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like )
RealUnitarySpace) : ( ( ) (
V1() )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) is
summable &
Sum (seq1 : ( ( V21() V30( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V1() ) set ) ) ) ( V1() V16() V19( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) ) V20( the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V1() ) set ) ) V21() V26( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V1() ) set ) ) ) sequence of ( ( ) ( V1() ) set ) ) + seq2 : ( ( V21() V30( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V1() ) set ) ) ) ( V1() V16() V19( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) ) V20( the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V1() ) set ) ) V21() V26( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V1() ) set ) ) ) sequence of ( ( ) ( V1() ) set ) ) ) : ( (
V21()
V30(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) , the
carrier of
b1 : ( ( non
empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non
empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like )
RealUnitarySpace) : ( ( ) (
V1() )
set ) ) ) (
V1()
V16()
V19(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) )
V20( the
carrier of
b1 : ( ( non
empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non
empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like )
RealUnitarySpace) : ( ( ) (
V1() )
set ) )
V21()
V26(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) )
V30(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) , the
carrier of
b1 : ( ( non
empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non
empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like )
RealUnitarySpace) : ( ( ) (
V1() )
set ) ) )
Element of
K6(
K7(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) , the
carrier of
b1 : ( ( non
empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non
empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like )
RealUnitarySpace) : ( ( ) (
V1() )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
Point of ( ( ) (
V1() )
set ) )
= (Sum seq1 : ( ( V21() V30( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V1() ) set ) ) ) ( V1() V16() V19( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) ) V20( the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V1() ) set ) ) V21() V26( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V1() ) set ) ) ) sequence of ( ( ) ( V1() ) set ) ) ) : ( ( ) ( )
Point of ( ( ) (
V1() )
set ) )
+ (Sum seq2 : ( ( V21() V30( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V1() ) set ) ) ) ( V1() V16() V19( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) ) V20( the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V1() ) set ) ) V21() V26( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V1() ) set ) ) ) sequence of ( ( ) ( V1() ) set ) ) ) : ( ( ) ( )
Point of ( ( ) (
V1() )
set ) ) : ( ( ) ( )
Element of the
carrier of
b1 : ( ( non
empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non
empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like )
RealUnitarySpace) : ( ( ) (
V1() )
set ) ) ) ;
theorem
for
X being ( ( non
empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non
empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like )
RealUnitarySpace)
for
seq1,
seq2 being ( (
V21()
V30(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) , the
carrier of
b1 : ( ( non
empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non
empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like )
RealUnitarySpace) : ( ( ) (
V1() )
set ) ) ) (
V1()
V16()
V19(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) )
V20( the
carrier of
b1 : ( ( non
empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non
empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like )
RealUnitarySpace) : ( ( ) (
V1() )
set ) )
V21()
V26(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) )
V30(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) , the
carrier of
b1 : ( ( non
empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non
empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like )
RealUnitarySpace) : ( ( ) (
V1() )
set ) ) )
sequence of ( ( ) (
V1() )
set ) ) st
seq1 : ( (
V21()
V30(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) , the
carrier of
b1 : ( ( non
empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non
empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like )
RealUnitarySpace) : ( ( ) (
V1() )
set ) ) ) (
V1()
V16()
V19(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) )
V20( the
carrier of
b1 : ( ( non
empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non
empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like )
RealUnitarySpace) : ( ( ) (
V1() )
set ) )
V21()
V26(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) )
V30(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) , the
carrier of
b1 : ( ( non
empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non
empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like )
RealUnitarySpace) : ( ( ) (
V1() )
set ) ) )
sequence of ( ( ) (
V1() )
set ) ) is
summable &
seq2 : ( (
V21()
V30(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) , the
carrier of
b1 : ( ( non
empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non
empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like )
RealUnitarySpace) : ( ( ) (
V1() )
set ) ) ) (
V1()
V16()
V19(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) )
V20( the
carrier of
b1 : ( ( non
empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non
empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like )
RealUnitarySpace) : ( ( ) (
V1() )
set ) )
V21()
V26(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) )
V30(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) , the
carrier of
b1 : ( ( non
empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non
empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like )
RealUnitarySpace) : ( ( ) (
V1() )
set ) ) )
sequence of ( ( ) (
V1() )
set ) ) is
summable holds
(
seq1 : ( (
V21()
V30(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) , the
carrier of
b1 : ( ( non
empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non
empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like )
RealUnitarySpace) : ( ( ) (
V1() )
set ) ) ) (
V1()
V16()
V19(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) )
V20( the
carrier of
b1 : ( ( non
empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non
empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like )
RealUnitarySpace) : ( ( ) (
V1() )
set ) )
V21()
V26(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) )
V30(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) , the
carrier of
b1 : ( ( non
empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non
empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like )
RealUnitarySpace) : ( ( ) (
V1() )
set ) ) )
sequence of ( ( ) (
V1() )
set ) )
- seq2 : ( (
V21()
V30(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) , the
carrier of
b1 : ( ( non
empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non
empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like )
RealUnitarySpace) : ( ( ) (
V1() )
set ) ) ) (
V1()
V16()
V19(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) )
V20( the
carrier of
b1 : ( ( non
empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non
empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like )
RealUnitarySpace) : ( ( ) (
V1() )
set ) )
V21()
V26(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) )
V30(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) , the
carrier of
b1 : ( ( non
empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non
empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like )
RealUnitarySpace) : ( ( ) (
V1() )
set ) ) )
sequence of ( ( ) (
V1() )
set ) ) : ( (
V21()
V30(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) , the
carrier of
b1 : ( ( non
empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non
empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like )
RealUnitarySpace) : ( ( ) (
V1() )
set ) ) ) (
V1()
V16()
V19(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) )
V20( the
carrier of
b1 : ( ( non
empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non
empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like )
RealUnitarySpace) : ( ( ) (
V1() )
set ) )
V21()
V26(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) )
V30(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) , the
carrier of
b1 : ( ( non
empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non
empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like )
RealUnitarySpace) : ( ( ) (
V1() )
set ) ) )
Element of
K6(
K7(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) , the
carrier of
b1 : ( ( non
empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non
empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like )
RealUnitarySpace) : ( ( ) (
V1() )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) is
summable &
Sum (seq1 : ( ( V21() V30( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V1() ) set ) ) ) ( V1() V16() V19( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) ) V20( the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V1() ) set ) ) V21() V26( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V1() ) set ) ) ) sequence of ( ( ) ( V1() ) set ) ) - seq2 : ( ( V21() V30( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V1() ) set ) ) ) ( V1() V16() V19( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) ) V20( the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V1() ) set ) ) V21() V26( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V1() ) set ) ) ) sequence of ( ( ) ( V1() ) set ) ) ) : ( (
V21()
V30(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) , the
carrier of
b1 : ( ( non
empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non
empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like )
RealUnitarySpace) : ( ( ) (
V1() )
set ) ) ) (
V1()
V16()
V19(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) )
V20( the
carrier of
b1 : ( ( non
empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non
empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like )
RealUnitarySpace) : ( ( ) (
V1() )
set ) )
V21()
V26(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) )
V30(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) , the
carrier of
b1 : ( ( non
empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non
empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like )
RealUnitarySpace) : ( ( ) (
V1() )
set ) ) )
Element of
K6(
K7(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) , the
carrier of
b1 : ( ( non
empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non
empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like )
RealUnitarySpace) : ( ( ) (
V1() )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
Point of ( ( ) (
V1() )
set ) )
= (Sum seq1 : ( ( V21() V30( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V1() ) set ) ) ) ( V1() V16() V19( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) ) V20( the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V1() ) set ) ) V21() V26( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V1() ) set ) ) ) sequence of ( ( ) ( V1() ) set ) ) ) : ( ( ) ( )
Point of ( ( ) (
V1() )
set ) )
- (Sum seq2 : ( ( V21() V30( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V1() ) set ) ) ) ( V1() V16() V19( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) ) V20( the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V1() ) set ) ) V21() V26( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V1() ) set ) ) ) sequence of ( ( ) ( V1() ) set ) ) ) : ( ( ) ( )
Point of ( ( ) (
V1() )
set ) ) : ( ( ) ( )
Element of the
carrier of
b1 : ( ( non
empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non
empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like )
RealUnitarySpace) : ( ( ) (
V1() )
set ) ) ) ;
theorem
for
a being ( ( ) (
V11()
real ext-real )
Real)
for
X being ( ( non
empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non
empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like )
RealUnitarySpace)
for
seq being ( (
V21()
V30(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) , the
carrier of
b2 : ( ( non
empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non
empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like )
RealUnitarySpace) : ( ( ) (
V1() )
set ) ) ) (
V1()
V16()
V19(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) )
V20( the
carrier of
b2 : ( ( non
empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non
empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like )
RealUnitarySpace) : ( ( ) (
V1() )
set ) )
V21()
V26(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) )
V30(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) , the
carrier of
b2 : ( ( non
empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non
empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like )
RealUnitarySpace) : ( ( ) (
V1() )
set ) ) )
sequence of ( ( ) (
V1() )
set ) ) st
seq : ( (
V21()
V30(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) , the
carrier of
b2 : ( ( non
empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non
empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like )
RealUnitarySpace) : ( ( ) (
V1() )
set ) ) ) (
V1()
V16()
V19(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) )
V20( the
carrier of
b2 : ( ( non
empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non
empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like )
RealUnitarySpace) : ( ( ) (
V1() )
set ) )
V21()
V26(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) )
V30(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) , the
carrier of
b2 : ( ( non
empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non
empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like )
RealUnitarySpace) : ( ( ) (
V1() )
set ) ) )
sequence of ( ( ) (
V1() )
set ) ) is
summable holds
(
a : ( ( ) (
V11()
real ext-real )
Real)
* seq : ( (
V21()
V30(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) , the
carrier of
b2 : ( ( non
empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non
empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like )
RealUnitarySpace) : ( ( ) (
V1() )
set ) ) ) (
V1()
V16()
V19(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) )
V20( the
carrier of
b2 : ( ( non
empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non
empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like )
RealUnitarySpace) : ( ( ) (
V1() )
set ) )
V21()
V26(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) )
V30(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) , the
carrier of
b2 : ( ( non
empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non
empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like )
RealUnitarySpace) : ( ( ) (
V1() )
set ) ) )
sequence of ( ( ) (
V1() )
set ) ) : ( (
V21()
V30(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) , the
carrier of
b2 : ( ( non
empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non
empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like )
RealUnitarySpace) : ( ( ) (
V1() )
set ) ) ) (
V1()
V16()
V19(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) )
V20( the
carrier of
b2 : ( ( non
empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non
empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like )
RealUnitarySpace) : ( ( ) (
V1() )
set ) )
V21()
V26(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) )
V30(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) , the
carrier of
b2 : ( ( non
empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non
empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like )
RealUnitarySpace) : ( ( ) (
V1() )
set ) ) )
Element of
K6(
K7(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) , the
carrier of
b2 : ( ( non
empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non
empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like )
RealUnitarySpace) : ( ( ) (
V1() )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) is
summable &
Sum (a : ( ( ) ( V11() real ext-real ) Real) * seq : ( ( V21() V30( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V1() ) set ) ) ) ( V1() V16() V19( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) ) V20( the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V1() ) set ) ) V21() V26( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V1() ) set ) ) ) sequence of ( ( ) ( V1() ) set ) ) ) : ( (
V21()
V30(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) , the
carrier of
b2 : ( ( non
empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non
empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like )
RealUnitarySpace) : ( ( ) (
V1() )
set ) ) ) (
V1()
V16()
V19(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) )
V20( the
carrier of
b2 : ( ( non
empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non
empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like )
RealUnitarySpace) : ( ( ) (
V1() )
set ) )
V21()
V26(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) )
V30(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) , the
carrier of
b2 : ( ( non
empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non
empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like )
RealUnitarySpace) : ( ( ) (
V1() )
set ) ) )
Element of
K6(
K7(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) , the
carrier of
b2 : ( ( non
empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non
empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like )
RealUnitarySpace) : ( ( ) (
V1() )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
Point of ( ( ) (
V1() )
set ) )
= a : ( ( ) (
V11()
real ext-real )
Real)
* (Sum seq : ( ( V21() V30( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V1() ) set ) ) ) ( V1() V16() V19( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) ) V20( the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V1() ) set ) ) V21() V26( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V1() ) set ) ) ) sequence of ( ( ) ( V1() ) set ) ) ) : ( ( ) ( )
Point of ( ( ) (
V1() )
set ) ) : ( ( ) ( )
Element of the
carrier of
b2 : ( ( non
empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non
empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like )
RealUnitarySpace) : ( ( ) (
V1() )
set ) ) ) ;
theorem
for
X being ( ( non
empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like V144() ) ( non
empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like V144() )
RealHilbertSpace)
for
seq being ( (
V21()
V30(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) , the
carrier of
b1 : ( ( non
empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like V144() ) ( non
empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like V144() )
RealHilbertSpace) : ( ( ) (
V1() )
set ) ) ) (
V1()
V16()
V19(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) )
V20( the
carrier of
b1 : ( ( non
empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like V144() ) ( non
empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like V144() )
RealHilbertSpace) : ( ( ) (
V1() )
set ) )
V21()
V26(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) )
V30(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) , the
carrier of
b1 : ( ( non
empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like V144() ) ( non
empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like V144() )
RealHilbertSpace) : ( ( ) (
V1() )
set ) ) )
sequence of ( ( ) (
V1() )
set ) ) holds
(
seq : ( (
V21()
V30(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) , the
carrier of
b1 : ( ( non
empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like V144() ) ( non
empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like V144() )
RealHilbertSpace) : ( ( ) (
V1() )
set ) ) ) (
V1()
V16()
V19(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) )
V20( the
carrier of
b1 : ( ( non
empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like V144() ) ( non
empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like V144() )
RealHilbertSpace) : ( ( ) (
V1() )
set ) )
V21()
V26(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) )
V30(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) , the
carrier of
b1 : ( ( non
empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like V144() ) ( non
empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like V144() )
RealHilbertSpace) : ( ( ) (
V1() )
set ) ) )
sequence of ( ( ) (
V1() )
set ) ) is
summable iff for
r being ( ( ) (
V11()
real ext-real )
Real) st
r : ( ( ) (
V11()
real ext-real )
Real)
> 0 : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural V11()
real ext-real non
positive non
negative integer V58()
V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) ) holds
ex
k being ( ( ) (
epsilon-transitive epsilon-connected ordinal natural V11()
real ext-real integer V58()
V59()
V60()
V61()
V62()
V63()
V64() )
Element of
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) ) st
for
n,
m being ( ( ) (
epsilon-transitive epsilon-connected ordinal natural V11()
real ext-real integer V58()
V59()
V60()
V61()
V62()
V63()
V64() )
Element of
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) ) st
n : ( ( ) (
epsilon-transitive epsilon-connected ordinal natural V11()
real ext-real integer V58()
V59()
V60()
V61()
V62()
V63()
V64() )
Element of
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) )
>= k : ( ( ) (
epsilon-transitive epsilon-connected ordinal natural V11()
real ext-real integer V58()
V59()
V60()
V61()
V62()
V63()
V64() )
Element of
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) ) &
m : ( ( ) (
epsilon-transitive epsilon-connected ordinal natural V11()
real ext-real integer V58()
V59()
V60()
V61()
V62()
V63()
V64() )
Element of
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) )
>= k : ( ( ) (
epsilon-transitive epsilon-connected ordinal natural V11()
real ext-real integer V58()
V59()
V60()
V61()
V62()
V63()
V64() )
Element of
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) ) holds
||.(((Partial_Sums seq : ( ( V21() V30( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like V144() ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like V144() ) RealHilbertSpace) : ( ( ) ( V1() ) set ) ) ) ( V1() V16() V19( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) ) V20( the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like V144() ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like V144() ) RealHilbertSpace) : ( ( ) ( V1() ) set ) ) V21() V26( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like V144() ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like V144() ) RealHilbertSpace) : ( ( ) ( V1() ) set ) ) ) sequence of ( ( ) ( V1() ) set ) ) ) : ( ( V21() V30( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like V144() ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like V144() ) RealHilbertSpace) : ( ( ) ( V1() ) set ) ) ) ( V1() V16() V19( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) ) V20( the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like V144() ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like V144() ) RealHilbertSpace) : ( ( ) ( V1() ) set ) ) V21() V26( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like V144() ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like V144() ) RealHilbertSpace) : ( ( ) ( V1() ) set ) ) ) sequence of ( ( ) ( V1() ) set ) ) . n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real integer V58() V59() V60() V61() V62() V63() V64() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ) Element of the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like V144() ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like V144() ) RealHilbertSpace) : ( ( ) ( V1() ) set ) ) - ((Partial_Sums seq : ( ( V21() V30( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like V144() ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like V144() ) RealHilbertSpace) : ( ( ) ( V1() ) set ) ) ) ( V1() V16() V19( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) ) V20( the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like V144() ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like V144() ) RealHilbertSpace) : ( ( ) ( V1() ) set ) ) V21() V26( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like V144() ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like V144() ) RealHilbertSpace) : ( ( ) ( V1() ) set ) ) ) sequence of ( ( ) ( V1() ) set ) ) ) : ( ( V21() V30( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like V144() ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like V144() ) RealHilbertSpace) : ( ( ) ( V1() ) set ) ) ) ( V1() V16() V19( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) ) V20( the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like V144() ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like V144() ) RealHilbertSpace) : ( ( ) ( V1() ) set ) ) V21() V26( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like V144() ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like V144() ) RealHilbertSpace) : ( ( ) ( V1() ) set ) ) ) sequence of ( ( ) ( V1() ) set ) ) . m : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real integer V58() V59() V60() V61() V62() V63() V64() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ) Element of the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like V144() ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like V144() ) RealHilbertSpace) : ( ( ) ( V1() ) set ) ) ) : ( ( ) ( ) Element of the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like V144() ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like V144() ) RealHilbertSpace) : ( ( ) ( V1() ) set ) ) .|| : ( ( ) (
V11()
real ext-real )
Element of
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) )
< r : ( ( ) (
V11()
real ext-real )
Real) ) ;
theorem
for
X being ( ( non
empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non
empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like )
RealUnitarySpace)
for
seq,
seq1 being ( (
V21()
V30(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) , the
carrier of
b1 : ( ( non
empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non
empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like )
RealUnitarySpace) : ( ( ) (
V1() )
set ) ) ) (
V1()
V16()
V19(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) )
V20( the
carrier of
b1 : ( ( non
empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non
empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like )
RealUnitarySpace) : ( ( ) (
V1() )
set ) )
V21()
V26(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) )
V30(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) , the
carrier of
b1 : ( ( non
empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non
empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like )
RealUnitarySpace) : ( ( ) (
V1() )
set ) ) )
sequence of ( ( ) (
V1() )
set ) ) st ( for
n being ( ( ) (
epsilon-transitive epsilon-connected ordinal natural V11()
real ext-real integer V58()
V59()
V60()
V61()
V62()
V63()
V64() )
Element of
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) ) holds
seq1 : ( (
V21()
V30(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) , the
carrier of
b1 : ( ( non
empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non
empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like )
RealUnitarySpace) : ( ( ) (
V1() )
set ) ) ) (
V1()
V16()
V19(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) )
V20( the
carrier of
b1 : ( ( non
empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non
empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like )
RealUnitarySpace) : ( ( ) (
V1() )
set ) )
V21()
V26(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) )
V30(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) , the
carrier of
b1 : ( ( non
empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non
empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like )
RealUnitarySpace) : ( ( ) (
V1() )
set ) ) )
sequence of ( ( ) (
V1() )
set ) )
. n : ( ( ) (
epsilon-transitive epsilon-connected ordinal natural V11()
real ext-real integer V58()
V59()
V60()
V61()
V62()
V63()
V64() )
Element of
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( )
Element of the
carrier of
b1 : ( ( non
empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non
empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like )
RealUnitarySpace) : ( ( ) (
V1() )
set ) )
= seq : ( (
V21()
V30(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) , the
carrier of
b1 : ( ( non
empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non
empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like )
RealUnitarySpace) : ( ( ) (
V1() )
set ) ) ) (
V1()
V16()
V19(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) )
V20( the
carrier of
b1 : ( ( non
empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non
empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like )
RealUnitarySpace) : ( ( ) (
V1() )
set ) )
V21()
V26(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) )
V30(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) , the
carrier of
b1 : ( ( non
empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non
empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like )
RealUnitarySpace) : ( ( ) (
V1() )
set ) ) )
sequence of ( ( ) (
V1() )
set ) )
. 0 : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural V11()
real ext-real non
positive non
negative integer V58()
V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( )
Element of the
carrier of
b1 : ( ( non
empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non
empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like )
RealUnitarySpace) : ( ( ) (
V1() )
set ) ) ) holds
Partial_Sums (seq : ( ( V21() V30( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V1() ) set ) ) ) ( V1() V16() V19( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) ) V20( the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V1() ) set ) ) V21() V26( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V1() ) set ) ) ) sequence of ( ( ) ( V1() ) set ) ) ^\ 1 : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative integer V58() V59() V60() V61() V62() V63() V64() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) (
V1()
V16()
V19(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) )
V20( the
carrier of
b1 : ( ( non
empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non
empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like )
RealUnitarySpace) : ( ( ) (
V1() )
set ) )
V21()
V26(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) )
V30(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) , the
carrier of
b1 : ( ( non
empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non
empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like )
RealUnitarySpace) : ( ( ) (
V1() )
set ) ) )
subsequence of
b2 : ( (
V21()
V30(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) , the
carrier of
b1 : ( ( non
empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non
empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like )
RealUnitarySpace) : ( ( ) (
V1() )
set ) ) ) (
V1()
V16()
V19(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) )
V20( the
carrier of
b1 : ( ( non
empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non
empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like )
RealUnitarySpace) : ( ( ) (
V1() )
set ) )
V21()
V26(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) )
V30(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) , the
carrier of
b1 : ( ( non
empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non
empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like )
RealUnitarySpace) : ( ( ) (
V1() )
set ) ) )
sequence of ( ( ) (
V1() )
set ) ) ) : ( (
V21()
V30(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) , the
carrier of
b1 : ( ( non
empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non
empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like )
RealUnitarySpace) : ( ( ) (
V1() )
set ) ) ) (
V1()
V16()
V19(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) )
V20( the
carrier of
b1 : ( ( non
empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non
empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like )
RealUnitarySpace) : ( ( ) (
V1() )
set ) )
V21()
V26(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) )
V30(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) , the
carrier of
b1 : ( ( non
empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non
empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like )
RealUnitarySpace) : ( ( ) (
V1() )
set ) ) )
sequence of ( ( ) (
V1() )
set ) )
= ((Partial_Sums seq : ( ( V21() V30( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V1() ) set ) ) ) ( V1() V16() V19( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) ) V20( the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V1() ) set ) ) V21() V26( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V1() ) set ) ) ) sequence of ( ( ) ( V1() ) set ) ) ) : ( ( V21() V30( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V1() ) set ) ) ) ( V1() V16() V19( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) ) V20( the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V1() ) set ) ) V21() V26( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V1() ) set ) ) ) sequence of ( ( ) ( V1() ) set ) ) ^\ 1 : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative integer V58() V59() V60() V61() V62() V63() V64() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) (
V1()
V16()
V19(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) )
V20( the
carrier of
b1 : ( ( non
empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non
empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like )
RealUnitarySpace) : ( ( ) (
V1() )
set ) )
V21()
V26(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) )
V30(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) , the
carrier of
b1 : ( ( non
empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non
empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like )
RealUnitarySpace) : ( ( ) (
V1() )
set ) ) )
subsequence of
Partial_Sums b2 : ( (
V21()
V30(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) , the
carrier of
b1 : ( ( non
empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non
empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like )
RealUnitarySpace) : ( ( ) (
V1() )
set ) ) ) (
V1()
V16()
V19(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) )
V20( the
carrier of
b1 : ( ( non
empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non
empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like )
RealUnitarySpace) : ( ( ) (
V1() )
set ) )
V21()
V26(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) )
V30(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) , the
carrier of
b1 : ( ( non
empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non
empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like )
RealUnitarySpace) : ( ( ) (
V1() )
set ) ) )
sequence of ( ( ) (
V1() )
set ) ) : ( (
V21()
V30(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) , the
carrier of
b1 : ( ( non
empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non
empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like )
RealUnitarySpace) : ( ( ) (
V1() )
set ) ) ) (
V1()
V16()
V19(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) )
V20( the
carrier of
b1 : ( ( non
empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non
empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like )
RealUnitarySpace) : ( ( ) (
V1() )
set ) )
V21()
V26(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) )
V30(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) , the
carrier of
b1 : ( ( non
empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non
empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like )
RealUnitarySpace) : ( ( ) (
V1() )
set ) ) )
sequence of ( ( ) (
V1() )
set ) ) )
- seq1 : ( (
V21()
V30(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) , the
carrier of
b1 : ( ( non
empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non
empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like )
RealUnitarySpace) : ( ( ) (
V1() )
set ) ) ) (
V1()
V16()
V19(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) )
V20( the
carrier of
b1 : ( ( non
empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non
empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like )
RealUnitarySpace) : ( ( ) (
V1() )
set ) )
V21()
V26(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) )
V30(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) , the
carrier of
b1 : ( ( non
empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non
empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like )
RealUnitarySpace) : ( ( ) (
V1() )
set ) ) )
sequence of ( ( ) (
V1() )
set ) ) : ( (
V21()
V30(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) , the
carrier of
b1 : ( ( non
empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non
empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like )
RealUnitarySpace) : ( ( ) (
V1() )
set ) ) ) (
V1()
V16()
V19(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) )
V20( the
carrier of
b1 : ( ( non
empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non
empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like )
RealUnitarySpace) : ( ( ) (
V1() )
set ) )
V21()
V26(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) )
V30(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) , the
carrier of
b1 : ( ( non
empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non
empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like )
RealUnitarySpace) : ( ( ) (
V1() )
set ) ) )
Element of
K6(
K7(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) , the
carrier of
b1 : ( ( non
empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non
empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like )
RealUnitarySpace) : ( ( ) (
V1() )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) ;
definition
let X be ( ( non
empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non
empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like )
RealUnitarySpace) ;
let seq be ( (
V21()
V30(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) , the
carrier of
X : ( ( non
empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non
empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like )
RealUnitarySpace) : ( ( ) (
V1() )
set ) ) ) (
V1()
V16()
V19(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) )
V20( the
carrier of
X : ( ( non
empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non
empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like )
RealUnitarySpace) : ( ( ) (
V1() )
set ) )
V21()
V26(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) )
V30(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) , the
carrier of
X : ( ( non
empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non
empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like )
RealUnitarySpace) : ( ( ) (
V1() )
set ) ) )
sequence of ( ( ) (
V1() )
set ) ) ;
let n be ( ( ) (
epsilon-transitive epsilon-connected ordinal natural V11()
real ext-real integer V58()
V59()
V60()
V61()
V62()
V63()
V64() )
Element of
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) ) ;
func Sum (
seq,
n)
-> ( ( ) ( )
Point of ( ( ) ( )
set ) )
equals
(Partial_Sums seq : ( ( ) ( ) Element of X : ( ( ) ( ) UNITSTR ) ) ) : ( (
V21()
V30(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) , the
carrier of
X : ( ( ) ( )
UNITSTR ) : ( ( ) ( )
set ) ) ) (
V1()
V16()
V19(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) )
V20( the
carrier of
X : ( ( ) ( )
UNITSTR ) : ( ( ) ( )
set ) )
V21()
V26(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) )
V30(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) , the
carrier of
X : ( ( ) ( )
UNITSTR ) : ( ( ) ( )
set ) ) )
sequence of ( ( ) ( )
set ) )
. n : ( (
V21()
V30(
K7(
X : ( ( ) ( )
UNITSTR ) ,
X : ( ( ) ( )
UNITSTR ) ) : ( ( ) ( )
set ) ,
X : ( ( ) ( )
UNITSTR ) ) ) (
V16()
V19(
K7(
X : ( ( ) ( )
UNITSTR ) ,
X : ( ( ) ( )
UNITSTR ) ) : ( ( ) ( )
set ) )
V20(
X : ( ( ) ( )
UNITSTR ) )
V21()
V30(
K7(
X : ( ( ) ( )
UNITSTR ) ,
X : ( ( ) ( )
UNITSTR ) ) : ( ( ) ( )
set ) ,
X : ( ( ) ( )
UNITSTR ) ) )
Element of
K6(
K7(
K7(
X : ( ( ) ( )
UNITSTR ) ,
X : ( ( ) ( )
UNITSTR ) ) : ( ( ) ( )
set ) ,
X : ( ( ) ( )
UNITSTR ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
Element of the
carrier of
X : ( ( ) ( )
UNITSTR ) : ( ( ) ( )
set ) ) ;
end;
theorem
for
X being ( ( non
empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non
empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like )
RealUnitarySpace)
for
seq being ( (
V21()
V30(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) , the
carrier of
b1 : ( ( non
empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non
empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like )
RealUnitarySpace) : ( ( ) (
V1() )
set ) ) ) (
V1()
V16()
V19(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) )
V20( the
carrier of
b1 : ( ( non
empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non
empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like )
RealUnitarySpace) : ( ( ) (
V1() )
set ) )
V21()
V26(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) )
V30(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) , the
carrier of
b1 : ( ( non
empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non
empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like )
RealUnitarySpace) : ( ( ) (
V1() )
set ) ) )
sequence of ( ( ) (
V1() )
set ) ) holds
Sum (
seq : ( (
V21()
V30(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) , the
carrier of
b1 : ( ( non
empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non
empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like )
RealUnitarySpace) : ( ( ) (
V1() )
set ) ) ) (
V1()
V16()
V19(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) )
V20( the
carrier of
b1 : ( ( non
empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non
empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like )
RealUnitarySpace) : ( ( ) (
V1() )
set ) )
V21()
V26(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) )
V30(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) , the
carrier of
b1 : ( ( non
empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non
empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like )
RealUnitarySpace) : ( ( ) (
V1() )
set ) ) )
sequence of ( ( ) (
V1() )
set ) ) ,1 : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal natural V11()
real ext-real positive non
negative integer V58()
V59()
V60()
V61()
V62()
V63()
V64() )
Element of
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) ) ) : ( ( ) ( )
Point of ( ( ) (
V1() )
set ) )
= (Sum (seq : ( ( V21() V30( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V1() ) set ) ) ) ( V1() V16() V19( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) ) V20( the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V1() ) set ) ) V21() V26( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V1() ) set ) ) ) sequence of ( ( ) ( V1() ) set ) ) ,0 : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural V11() real ext-real non positive non negative integer V58() V59() V60() V61() V62() V63() V64() V65() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( ) ( )
Point of ( ( ) (
V1() )
set ) )
+ (seq : ( ( V21() V30( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V1() ) set ) ) ) ( V1() V16() V19( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) ) V20( the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V1() ) set ) ) V21() V26( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V1() ) set ) ) ) sequence of ( ( ) ( V1() ) set ) ) . 1 : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative integer V58() V59() V60() V61() V62() V63() V64() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( )
Element of the
carrier of
b1 : ( ( non
empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non
empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like )
RealUnitarySpace) : ( ( ) (
V1() )
set ) ) : ( ( ) ( )
Element of the
carrier of
b1 : ( ( non
empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non
empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like )
RealUnitarySpace) : ( ( ) (
V1() )
set ) ) ;
theorem
for
n being ( ( ) (
epsilon-transitive epsilon-connected ordinal natural V11()
real ext-real integer V58()
V59()
V60()
V61()
V62()
V63()
V64() )
Element of
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) )
for
X being ( ( non
empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non
empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like )
RealUnitarySpace)
for
seq being ( (
V21()
V30(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) , the
carrier of
b2 : ( ( non
empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non
empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like )
RealUnitarySpace) : ( ( ) (
V1() )
set ) ) ) (
V1()
V16()
V19(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) )
V20( the
carrier of
b2 : ( ( non
empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non
empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like )
RealUnitarySpace) : ( ( ) (
V1() )
set ) )
V21()
V26(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) )
V30(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) , the
carrier of
b2 : ( ( non
empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non
empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like )
RealUnitarySpace) : ( ( ) (
V1() )
set ) ) )
sequence of ( ( ) (
V1() )
set ) ) holds
Sum (
seq : ( (
V21()
V30(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) , the
carrier of
b2 : ( ( non
empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non
empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like )
RealUnitarySpace) : ( ( ) (
V1() )
set ) ) ) (
V1()
V16()
V19(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) )
V20( the
carrier of
b2 : ( ( non
empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non
empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like )
RealUnitarySpace) : ( ( ) (
V1() )
set ) )
V21()
V26(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) )
V30(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) , the
carrier of
b2 : ( ( non
empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non
empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like )
RealUnitarySpace) : ( ( ) (
V1() )
set ) ) )
sequence of ( ( ) (
V1() )
set ) ) ,
(n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real integer V58() V59() V60() V61() V62() V63() V64() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) ) + 1 : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative integer V58() V59() V60() V61() V62() V63() V64() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) (
epsilon-transitive epsilon-connected ordinal natural V11()
real ext-real integer V58()
V59()
V60()
V61()
V62()
V63()
V64() )
Element of
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) ) ) : ( ( ) ( )
Point of ( ( ) (
V1() )
set ) )
= (Sum (seq : ( ( V21() V30( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V1() ) set ) ) ) ( V1() V16() V19( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) ) V20( the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V1() ) set ) ) V21() V26( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V1() ) set ) ) ) sequence of ( ( ) ( V1() ) set ) ) ,n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real integer V58() V59() V60() V61() V62() V63() V64() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( ) ( )
Point of ( ( ) (
V1() )
set ) )
+ (seq : ( ( V21() V30( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V1() ) set ) ) ) ( V1() V16() V19( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) ) V20( the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V1() ) set ) ) V21() V26( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V1() ) set ) ) ) sequence of ( ( ) ( V1() ) set ) ) . (n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real integer V58() V59() V60() V61() V62() V63() V64() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) ) + 1 : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative integer V58() V59() V60() V61() V62() V63() V64() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real integer V58() V59() V60() V61() V62() V63() V64() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( )
Element of the
carrier of
b2 : ( ( non
empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non
empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like )
RealUnitarySpace) : ( ( ) (
V1() )
set ) ) : ( ( ) ( )
Element of the
carrier of
b2 : ( ( non
empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non
empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like )
RealUnitarySpace) : ( ( ) (
V1() )
set ) ) ;
theorem
for
n being ( ( ) (
epsilon-transitive epsilon-connected ordinal natural V11()
real ext-real integer V58()
V59()
V60()
V61()
V62()
V63()
V64() )
Element of
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) )
for
X being ( ( non
empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non
empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like )
RealUnitarySpace)
for
seq being ( (
V21()
V30(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) , the
carrier of
b2 : ( ( non
empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non
empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like )
RealUnitarySpace) : ( ( ) (
V1() )
set ) ) ) (
V1()
V16()
V19(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) )
V20( the
carrier of
b2 : ( ( non
empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non
empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like )
RealUnitarySpace) : ( ( ) (
V1() )
set ) )
V21()
V26(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) )
V30(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) , the
carrier of
b2 : ( ( non
empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non
empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like )
RealUnitarySpace) : ( ( ) (
V1() )
set ) ) )
sequence of ( ( ) (
V1() )
set ) ) holds
seq : ( (
V21()
V30(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) , the
carrier of
b2 : ( ( non
empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non
empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like )
RealUnitarySpace) : ( ( ) (
V1() )
set ) ) ) (
V1()
V16()
V19(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) )
V20( the
carrier of
b2 : ( ( non
empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non
empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like )
RealUnitarySpace) : ( ( ) (
V1() )
set ) )
V21()
V26(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) )
V30(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) , the
carrier of
b2 : ( ( non
empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non
empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like )
RealUnitarySpace) : ( ( ) (
V1() )
set ) ) )
sequence of ( ( ) (
V1() )
set ) )
. (n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real integer V58() V59() V60() V61() V62() V63() V64() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) ) + 1 : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative integer V58() V59() V60() V61() V62() V63() V64() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) (
epsilon-transitive epsilon-connected ordinal natural V11()
real ext-real integer V58()
V59()
V60()
V61()
V62()
V63()
V64() )
Element of
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( )
Element of the
carrier of
b2 : ( ( non
empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non
empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like )
RealUnitarySpace) : ( ( ) (
V1() )
set ) )
= (Sum (seq : ( ( V21() V30( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V1() ) set ) ) ) ( V1() V16() V19( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) ) V20( the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V1() ) set ) ) V21() V26( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V1() ) set ) ) ) sequence of ( ( ) ( V1() ) set ) ) ,(n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real integer V58() V59() V60() V61() V62() V63() V64() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) ) + 1 : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative integer V58() V59() V60() V61() V62() V63() V64() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real integer V58() V59() V60() V61() V62() V63() V64() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( ) ( )
Point of ( ( ) (
V1() )
set ) )
- (Sum (seq : ( ( V21() V30( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V1() ) set ) ) ) ( V1() V16() V19( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) ) V20( the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V1() ) set ) ) V21() V26( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V1() ) set ) ) ) sequence of ( ( ) ( V1() ) set ) ) ,n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real integer V58() V59() V60() V61() V62() V63() V64() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( ) ( )
Point of ( ( ) (
V1() )
set ) ) : ( ( ) ( )
Element of the
carrier of
b2 : ( ( non
empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non
empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like )
RealUnitarySpace) : ( ( ) (
V1() )
set ) ) ;
theorem
for
X being ( ( non
empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non
empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like )
RealUnitarySpace)
for
seq being ( (
V21()
V30(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) , the
carrier of
b1 : ( ( non
empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non
empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like )
RealUnitarySpace) : ( ( ) (
V1() )
set ) ) ) (
V1()
V16()
V19(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) )
V20( the
carrier of
b1 : ( ( non
empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non
empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like )
RealUnitarySpace) : ( ( ) (
V1() )
set ) )
V21()
V26(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) )
V30(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) , the
carrier of
b1 : ( ( non
empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non
empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like )
RealUnitarySpace) : ( ( ) (
V1() )
set ) ) )
sequence of ( ( ) (
V1() )
set ) ) holds
seq : ( (
V21()
V30(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) , the
carrier of
b1 : ( ( non
empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non
empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like )
RealUnitarySpace) : ( ( ) (
V1() )
set ) ) ) (
V1()
V16()
V19(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) )
V20( the
carrier of
b1 : ( ( non
empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non
empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like )
RealUnitarySpace) : ( ( ) (
V1() )
set ) )
V21()
V26(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) )
V30(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) , the
carrier of
b1 : ( ( non
empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non
empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like )
RealUnitarySpace) : ( ( ) (
V1() )
set ) ) )
sequence of ( ( ) (
V1() )
set ) )
. 1 : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal natural V11()
real ext-real positive non
negative integer V58()
V59()
V60()
V61()
V62()
V63()
V64() )
Element of
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( )
Element of the
carrier of
b1 : ( ( non
empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non
empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like )
RealUnitarySpace) : ( ( ) (
V1() )
set ) )
= (Sum (seq : ( ( V21() V30( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V1() ) set ) ) ) ( V1() V16() V19( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) ) V20( the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V1() ) set ) ) V21() V26( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V1() ) set ) ) ) sequence of ( ( ) ( V1() ) set ) ) ,1 : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative integer V58() V59() V60() V61() V62() V63() V64() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( ) ( )
Point of ( ( ) (
V1() )
set ) )
- (Sum (seq : ( ( V21() V30( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V1() ) set ) ) ) ( V1() V16() V19( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) ) V20( the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V1() ) set ) ) V21() V26( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V1() ) set ) ) ) sequence of ( ( ) ( V1() ) set ) ) ,0 : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural V11() real ext-real non positive non negative integer V58() V59() V60() V61() V62() V63() V64() V65() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( ) ( )
Point of ( ( ) (
V1() )
set ) ) : ( ( ) ( )
Element of the
carrier of
b1 : ( ( non
empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non
empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like )
RealUnitarySpace) : ( ( ) (
V1() )
set ) ) ;
definition
let X be ( ( non
empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non
empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like )
RealUnitarySpace) ;
let seq be ( (
V21()
V30(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) , the
carrier of
X : ( ( non
empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non
empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like )
RealUnitarySpace) : ( ( ) (
V1() )
set ) ) ) (
V1()
V16()
V19(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) )
V20( the
carrier of
X : ( ( non
empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non
empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like )
RealUnitarySpace) : ( ( ) (
V1() )
set ) )
V21()
V26(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) )
V30(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) , the
carrier of
X : ( ( non
empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non
empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like )
RealUnitarySpace) : ( ( ) (
V1() )
set ) ) )
sequence of ( ( ) (
V1() )
set ) ) ;
let n,
m be ( ( ) (
epsilon-transitive epsilon-connected ordinal natural V11()
real ext-real integer V58()
V59()
V60()
V61()
V62()
V63()
V64() )
Element of
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) ) ;
func Sum (
seq,
n,
m)
-> ( ( ) ( )
Point of ( ( ) ( )
set ) )
equals
(Sum (seq : ( ( ) ( ) Element of X : ( ( ) ( ) UNITSTR ) ) ,n : ( ( V21() V30(K7(X : ( ( ) ( ) UNITSTR ) ,X : ( ( ) ( ) UNITSTR ) ) : ( ( ) ( ) set ) ,X : ( ( ) ( ) UNITSTR ) ) ) ( V16() V19(K7(X : ( ( ) ( ) UNITSTR ) ,X : ( ( ) ( ) UNITSTR ) ) : ( ( ) ( ) set ) ) V20(X : ( ( ) ( ) UNITSTR ) ) V21() V30(K7(X : ( ( ) ( ) UNITSTR ) ,X : ( ( ) ( ) UNITSTR ) ) : ( ( ) ( ) set ) ,X : ( ( ) ( ) UNITSTR ) ) ) Element of K6(K7(K7(X : ( ( ) ( ) UNITSTR ) ,X : ( ( ) ( ) UNITSTR ) ) : ( ( ) ( ) set ) ,X : ( ( ) ( ) UNITSTR ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) )) : ( ( ) ( )
Point of ( ( ) ( )
set ) )
- (Sum (seq : ( ( ) ( ) Element of X : ( ( ) ( ) UNITSTR ) ) ,m : ( ( V21() V30(K7(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ,X : ( ( ) ( ) UNITSTR ) ) : ( ( ) ( ) set ) ,X : ( ( ) ( ) UNITSTR ) ) ) ( V16() V19(K7(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ,X : ( ( ) ( ) UNITSTR ) ) : ( ( ) ( ) set ) ) V20(X : ( ( ) ( ) UNITSTR ) ) V21() V30(K7(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ,X : ( ( ) ( ) UNITSTR ) ) : ( ( ) ( ) set ) ,X : ( ( ) ( ) UNITSTR ) ) ) Element of K6(K7(K7(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ,X : ( ( ) ( ) UNITSTR ) ) : ( ( ) ( ) set ) ,X : ( ( ) ( ) UNITSTR ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) )) : ( ( ) ( )
Point of ( ( ) ( )
set ) ) : ( ( ) ( )
Element of the
carrier of
X : ( ( ) ( )
UNITSTR ) : ( ( ) ( )
set ) ) ;
end;
theorem
for
X being ( ( non
empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like V144() ) ( non
empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like V144() )
RealHilbertSpace)
for
seq being ( (
V21()
V30(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) , the
carrier of
b1 : ( ( non
empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like V144() ) ( non
empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like V144() )
RealHilbertSpace) : ( ( ) (
V1() )
set ) ) ) (
V1()
V16()
V19(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) )
V20( the
carrier of
b1 : ( ( non
empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like V144() ) ( non
empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like V144() )
RealHilbertSpace) : ( ( ) (
V1() )
set ) )
V21()
V26(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) )
V30(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) , the
carrier of
b1 : ( ( non
empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like V144() ) ( non
empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like V144() )
RealHilbertSpace) : ( ( ) (
V1() )
set ) ) )
sequence of ( ( ) (
V1() )
set ) ) holds
(
seq : ( (
V21()
V30(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) , the
carrier of
b1 : ( ( non
empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like V144() ) ( non
empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like V144() )
RealHilbertSpace) : ( ( ) (
V1() )
set ) ) ) (
V1()
V16()
V19(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) )
V20( the
carrier of
b1 : ( ( non
empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like V144() ) ( non
empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like V144() )
RealHilbertSpace) : ( ( ) (
V1() )
set ) )
V21()
V26(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) )
V30(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) , the
carrier of
b1 : ( ( non
empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like V144() ) ( non
empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like V144() )
RealHilbertSpace) : ( ( ) (
V1() )
set ) ) )
sequence of ( ( ) (
V1() )
set ) ) is
summable iff for
r being ( ( ) (
V11()
real ext-real )
Real) st
r : ( ( ) (
V11()
real ext-real )
Real)
> 0 : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural V11()
real ext-real non
positive non
negative integer V58()
V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) ) holds
ex
k being ( ( ) (
epsilon-transitive epsilon-connected ordinal natural V11()
real ext-real integer V58()
V59()
V60()
V61()
V62()
V63()
V64() )
Element of
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) ) st
for
n,
m being ( ( ) (
epsilon-transitive epsilon-connected ordinal natural V11()
real ext-real integer V58()
V59()
V60()
V61()
V62()
V63()
V64() )
Element of
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) ) st
n : ( ( ) (
epsilon-transitive epsilon-connected ordinal natural V11()
real ext-real integer V58()
V59()
V60()
V61()
V62()
V63()
V64() )
Element of
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) )
>= k : ( ( ) (
epsilon-transitive epsilon-connected ordinal natural V11()
real ext-real integer V58()
V59()
V60()
V61()
V62()
V63()
V64() )
Element of
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) ) &
m : ( ( ) (
epsilon-transitive epsilon-connected ordinal natural V11()
real ext-real integer V58()
V59()
V60()
V61()
V62()
V63()
V64() )
Element of
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) )
>= k : ( ( ) (
epsilon-transitive epsilon-connected ordinal natural V11()
real ext-real integer V58()
V59()
V60()
V61()
V62()
V63()
V64() )
Element of
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) ) holds
||.((Sum (seq : ( ( V21() V30( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like V144() ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like V144() ) RealHilbertSpace) : ( ( ) ( V1() ) set ) ) ) ( V1() V16() V19( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) ) V20( the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like V144() ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like V144() ) RealHilbertSpace) : ( ( ) ( V1() ) set ) ) V21() V26( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like V144() ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like V144() ) RealHilbertSpace) : ( ( ) ( V1() ) set ) ) ) sequence of ( ( ) ( V1() ) set ) ) ,n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real integer V58() V59() V60() V61() V62() V63() V64() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( ) ( ) Point of ( ( ) ( V1() ) set ) ) - (Sum (seq : ( ( V21() V30( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like V144() ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like V144() ) RealHilbertSpace) : ( ( ) ( V1() ) set ) ) ) ( V1() V16() V19( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) ) V20( the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like V144() ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like V144() ) RealHilbertSpace) : ( ( ) ( V1() ) set ) ) V21() V26( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like V144() ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like V144() ) RealHilbertSpace) : ( ( ) ( V1() ) set ) ) ) sequence of ( ( ) ( V1() ) set ) ) ,m : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real integer V58() V59() V60() V61() V62() V63() V64() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( ) ( ) Point of ( ( ) ( V1() ) set ) ) ) : ( ( ) ( ) Element of the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like V144() ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like V144() ) RealHilbertSpace) : ( ( ) ( V1() ) set ) ) .|| : ( ( ) (
V11()
real ext-real )
Element of
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) )
< r : ( ( ) (
V11()
real ext-real )
Real) ) ;
theorem
for
X being ( ( non
empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non
empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like )
RealUnitarySpace)
for
seq1,
seq2 being ( (
V21()
V30(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) , the
carrier of
b1 : ( ( non
empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non
empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like )
RealUnitarySpace) : ( ( ) (
V1() )
set ) ) ) (
V1()
V16()
V19(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) )
V20( the
carrier of
b1 : ( ( non
empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non
empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like )
RealUnitarySpace) : ( ( ) (
V1() )
set ) )
V21()
V26(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) )
V30(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) , the
carrier of
b1 : ( ( non
empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non
empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like )
RealUnitarySpace) : ( ( ) (
V1() )
set ) ) )
sequence of ( ( ) (
V1() )
set ) ) st
seq1 : ( (
V21()
V30(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) , the
carrier of
b1 : ( ( non
empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non
empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like )
RealUnitarySpace) : ( ( ) (
V1() )
set ) ) ) (
V1()
V16()
V19(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) )
V20( the
carrier of
b1 : ( ( non
empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non
empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like )
RealUnitarySpace) : ( ( ) (
V1() )
set ) )
V21()
V26(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) )
V30(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) , the
carrier of
b1 : ( ( non
empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non
empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like )
RealUnitarySpace) : ( ( ) (
V1() )
set ) ) )
sequence of ( ( ) (
V1() )
set ) ) is
absolutely_summable &
seq2 : ( (
V21()
V30(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) , the
carrier of
b1 : ( ( non
empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non
empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like )
RealUnitarySpace) : ( ( ) (
V1() )
set ) ) ) (
V1()
V16()
V19(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) )
V20( the
carrier of
b1 : ( ( non
empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non
empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like )
RealUnitarySpace) : ( ( ) (
V1() )
set ) )
V21()
V26(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) )
V30(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) , the
carrier of
b1 : ( ( non
empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non
empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like )
RealUnitarySpace) : ( ( ) (
V1() )
set ) ) )
sequence of ( ( ) (
V1() )
set ) ) is
absolutely_summable holds
seq1 : ( (
V21()
V30(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) , the
carrier of
b1 : ( ( non
empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non
empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like )
RealUnitarySpace) : ( ( ) (
V1() )
set ) ) ) (
V1()
V16()
V19(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) )
V20( the
carrier of
b1 : ( ( non
empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non
empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like )
RealUnitarySpace) : ( ( ) (
V1() )
set ) )
V21()
V26(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) )
V30(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) , the
carrier of
b1 : ( ( non
empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non
empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like )
RealUnitarySpace) : ( ( ) (
V1() )
set ) ) )
sequence of ( ( ) (
V1() )
set ) )
+ seq2 : ( (
V21()
V30(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) , the
carrier of
b1 : ( ( non
empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non
empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like )
RealUnitarySpace) : ( ( ) (
V1() )
set ) ) ) (
V1()
V16()
V19(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) )
V20( the
carrier of
b1 : ( ( non
empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non
empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like )
RealUnitarySpace) : ( ( ) (
V1() )
set ) )
V21()
V26(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) )
V30(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) , the
carrier of
b1 : ( ( non
empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non
empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like )
RealUnitarySpace) : ( ( ) (
V1() )
set ) ) )
sequence of ( ( ) (
V1() )
set ) ) : ( (
V21()
V30(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) , the
carrier of
b1 : ( ( non
empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non
empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like )
RealUnitarySpace) : ( ( ) (
V1() )
set ) ) ) (
V1()
V16()
V19(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) )
V20( the
carrier of
b1 : ( ( non
empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non
empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like )
RealUnitarySpace) : ( ( ) (
V1() )
set ) )
V21()
V26(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) )
V30(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) , the
carrier of
b1 : ( ( non
empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non
empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like )
RealUnitarySpace) : ( ( ) (
V1() )
set ) ) )
Element of
K6(
K7(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) , the
carrier of
b1 : ( ( non
empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non
empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like )
RealUnitarySpace) : ( ( ) (
V1() )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) is
absolutely_summable ;
theorem
for
Rseq being ( (
V21()
V30(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) ) (
V1()
V16()
V19(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) )
V20(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) )
V21()
V26(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) )
V30(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) )
complex-valued ext-real-valued real-valued )
Real_Sequence)
for
X being ( ( non
empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non
empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like )
RealUnitarySpace)
for
seq being ( (
V21()
V30(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) , the
carrier of
b2 : ( ( non
empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non
empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like )
RealUnitarySpace) : ( ( ) (
V1() )
set ) ) ) (
V1()
V16()
V19(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) )
V20( the
carrier of
b2 : ( ( non
empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non
empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like )
RealUnitarySpace) : ( ( ) (
V1() )
set ) )
V21()
V26(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) )
V30(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) , the
carrier of
b2 : ( ( non
empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non
empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like )
RealUnitarySpace) : ( ( ) (
V1() )
set ) ) )
sequence of ( ( ) (
V1() )
set ) ) st ( for
n being ( ( ) (
epsilon-transitive epsilon-connected ordinal natural V11()
real ext-real integer V58()
V59()
V60()
V61()
V62()
V63()
V64() )
Element of
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) ) holds
||.seq : ( ( V21() V30( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V1() ) set ) ) ) ( V1() V16() V19( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) ) V20( the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V1() ) set ) ) V21() V26( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V1() ) set ) ) ) sequence of ( ( ) ( V1() ) set ) ) .|| : ( (
V21()
V30(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) ) (
V1()
V16()
V19(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) )
V20(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) )
V21()
V26(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) )
V30(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) )
complex-valued ext-real-valued real-valued )
Element of
K6(
K7(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) (
complex-valued ext-real-valued real-valued )
set ) ) : ( ( ) ( )
set ) )
. n : ( ( ) (
epsilon-transitive epsilon-connected ordinal natural V11()
real ext-real integer V58()
V59()
V60()
V61()
V62()
V63()
V64() )
Element of
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) (
V11()
real ext-real )
Element of
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) )
<= Rseq : ( (
V21()
V30(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) ) (
V1()
V16()
V19(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) )
V20(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) )
V21()
V26(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) )
V30(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) )
complex-valued ext-real-valued real-valued )
Real_Sequence)
. n : ( ( ) (
epsilon-transitive epsilon-connected ordinal natural V11()
real ext-real integer V58()
V59()
V60()
V61()
V62()
V63()
V64() )
Element of
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) (
V11()
real ext-real )
Element of
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) ) &
Rseq : ( (
V21()
V30(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) ) (
V1()
V16()
V19(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) )
V20(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) )
V21()
V26(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) )
V30(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) )
complex-valued ext-real-valued real-valued )
Real_Sequence) is
summable holds
seq : ( (
V21()
V30(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) , the
carrier of
b2 : ( ( non
empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non
empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like )
RealUnitarySpace) : ( ( ) (
V1() )
set ) ) ) (
V1()
V16()
V19(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) )
V20( the
carrier of
b2 : ( ( non
empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non
empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like )
RealUnitarySpace) : ( ( ) (
V1() )
set ) )
V21()
V26(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) )
V30(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) , the
carrier of
b2 : ( ( non
empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non
empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like )
RealUnitarySpace) : ( ( ) (
V1() )
set ) ) )
sequence of ( ( ) (
V1() )
set ) ) is
absolutely_summable ;
theorem
for
Rseq being ( (
V21()
V30(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) ) (
V1()
V16()
V19(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) )
V20(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) )
V21()
V26(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) )
V30(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) )
complex-valued ext-real-valued real-valued )
Real_Sequence)
for
X being ( ( non
empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non
empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like )
RealUnitarySpace)
for
seq being ( (
V21()
V30(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) , the
carrier of
b2 : ( ( non
empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non
empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like )
RealUnitarySpace) : ( ( ) (
V1() )
set ) ) ) (
V1()
V16()
V19(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) )
V20( the
carrier of
b2 : ( ( non
empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non
empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like )
RealUnitarySpace) : ( ( ) (
V1() )
set ) )
V21()
V26(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) )
V30(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) , the
carrier of
b2 : ( ( non
empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non
empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like )
RealUnitarySpace) : ( ( ) (
V1() )
set ) ) )
sequence of ( ( ) (
V1() )
set ) ) st ( for
n being ( ( ) (
epsilon-transitive epsilon-connected ordinal natural V11()
real ext-real integer V58()
V59()
V60()
V61()
V62()
V63()
V64() )
Element of
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) ) holds
(
seq : ( (
V21()
V30(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) , the
carrier of
b2 : ( ( non
empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non
empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like )
RealUnitarySpace) : ( ( ) (
V1() )
set ) ) ) (
V1()
V16()
V19(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) )
V20( the
carrier of
b2 : ( ( non
empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non
empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like )
RealUnitarySpace) : ( ( ) (
V1() )
set ) )
V21()
V26(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) )
V30(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) , the
carrier of
b2 : ( ( non
empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non
empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like )
RealUnitarySpace) : ( ( ) (
V1() )
set ) ) )
sequence of ( ( ) (
V1() )
set ) )
. n : ( ( ) (
epsilon-transitive epsilon-connected ordinal natural V11()
real ext-real integer V58()
V59()
V60()
V61()
V62()
V63()
V64() )
Element of
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( )
Element of the
carrier of
b2 : ( ( non
empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non
empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like )
RealUnitarySpace) : ( ( ) (
V1() )
set ) )
<> 0. X : ( ( non
empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non
empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like )
RealUnitarySpace) : ( ( ) (
V80(
b2 : ( ( non
empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non
empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like )
RealUnitarySpace) ) )
Element of the
carrier of
b2 : ( ( non
empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non
empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like )
RealUnitarySpace) : ( ( ) (
V1() )
set ) ) &
Rseq : ( (
V21()
V30(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) ) (
V1()
V16()
V19(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) )
V20(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) )
V21()
V26(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) )
V30(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) )
complex-valued ext-real-valued real-valued )
Real_Sequence)
. n : ( ( ) (
epsilon-transitive epsilon-connected ordinal natural V11()
real ext-real integer V58()
V59()
V60()
V61()
V62()
V63()
V64() )
Element of
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) (
V11()
real ext-real )
Element of
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) )
= ||.(seq : ( ( V21() V30( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V1() ) set ) ) ) ( V1() V16() V19( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) ) V20( the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V1() ) set ) ) V21() V26( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V1() ) set ) ) ) sequence of ( ( ) ( V1() ) set ) ) . (n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real integer V58() V59() V60() V61() V62() V63() V64() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) ) + 1 : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative integer V58() V59() V60() V61() V62() V63() V64() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real integer V58() V59() V60() V61() V62() V63() V64() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ) Element of the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V1() ) set ) ) .|| : ( ( ) (
V11()
real ext-real )
Element of
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) )
/ ||.(seq : ( ( V21() V30( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V1() ) set ) ) ) ( V1() V16() V19( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) ) V20( the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V1() ) set ) ) V21() V26( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V1() ) set ) ) ) sequence of ( ( ) ( V1() ) set ) ) . n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real integer V58() V59() V60() V61() V62() V63() V64() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ) Element of the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V1() ) set ) ) .|| : ( ( ) (
V11()
real ext-real )
Element of
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) (
V11()
real ext-real )
Element of
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) ) ) &
Rseq : ( (
V21()
V30(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) ) (
V1()
V16()
V19(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) )
V20(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) )
V21()
V26(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) )
V30(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) )
complex-valued ext-real-valued real-valued )
Real_Sequence) is
convergent &
lim Rseq : ( (
V21()
V30(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) ) (
V1()
V16()
V19(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) )
V20(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) )
V21()
V26(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) )
V30(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) )
complex-valued ext-real-valued real-valued )
Real_Sequence) : ( ( ) (
V11()
real ext-real )
Element of
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) )
< 1 : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal natural V11()
real ext-real positive non
negative integer V58()
V59()
V60()
V61()
V62()
V63()
V64() )
Element of
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) ) holds
seq : ( (
V21()
V30(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) , the
carrier of
b2 : ( ( non
empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non
empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like )
RealUnitarySpace) : ( ( ) (
V1() )
set ) ) ) (
V1()
V16()
V19(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) )
V20( the
carrier of
b2 : ( ( non
empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non
empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like )
RealUnitarySpace) : ( ( ) (
V1() )
set ) )
V21()
V26(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) )
V30(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) , the
carrier of
b2 : ( ( non
empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non
empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like )
RealUnitarySpace) : ( ( ) (
V1() )
set ) ) )
sequence of ( ( ) (
V1() )
set ) ) is
absolutely_summable ;
theorem
for
X being ( ( non
empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non
empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like )
RealUnitarySpace)
for
seq being ( (
V21()
V30(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) , the
carrier of
b1 : ( ( non
empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non
empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like )
RealUnitarySpace) : ( ( ) (
V1() )
set ) ) ) (
V1()
V16()
V19(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) )
V20( the
carrier of
b1 : ( ( non
empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non
empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like )
RealUnitarySpace) : ( ( ) (
V1() )
set ) )
V21()
V26(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) )
V30(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) , the
carrier of
b1 : ( ( non
empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non
empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like )
RealUnitarySpace) : ( ( ) (
V1() )
set ) ) )
sequence of ( ( ) (
V1() )
set ) ) st ( for
n being ( ( ) (
epsilon-transitive epsilon-connected ordinal natural V11()
real ext-real integer V58()
V59()
V60()
V61()
V62()
V63()
V64() )
Element of
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) ) holds
seq : ( (
V21()
V30(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) , the
carrier of
b1 : ( ( non
empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non
empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like )
RealUnitarySpace) : ( ( ) (
V1() )
set ) ) ) (
V1()
V16()
V19(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) )
V20( the
carrier of
b1 : ( ( non
empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non
empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like )
RealUnitarySpace) : ( ( ) (
V1() )
set ) )
V21()
V26(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) )
V30(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) , the
carrier of
b1 : ( ( non
empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non
empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like )
RealUnitarySpace) : ( ( ) (
V1() )
set ) ) )
sequence of ( ( ) (
V1() )
set ) )
. n : ( ( ) (
epsilon-transitive epsilon-connected ordinal natural V11()
real ext-real integer V58()
V59()
V60()
V61()
V62()
V63()
V64() )
Element of
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( )
Element of the
carrier of
b1 : ( ( non
empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non
empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like )
RealUnitarySpace) : ( ( ) (
V1() )
set ) )
<> 0. X : ( ( non
empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non
empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like )
RealUnitarySpace) : ( ( ) (
V80(
b1 : ( ( non
empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non
empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like )
RealUnitarySpace) ) )
Element of the
carrier of
b1 : ( ( non
empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non
empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like )
RealUnitarySpace) : ( ( ) (
V1() )
set ) ) ) & ex
m being ( ( ) (
epsilon-transitive epsilon-connected ordinal natural V11()
real ext-real integer V58()
V59()
V60()
V61()
V62()
V63()
V64() )
Element of
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) ) st
for
n being ( ( ) (
epsilon-transitive epsilon-connected ordinal natural V11()
real ext-real integer V58()
V59()
V60()
V61()
V62()
V63()
V64() )
Element of
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) ) st
n : ( ( ) (
epsilon-transitive epsilon-connected ordinal natural V11()
real ext-real integer V58()
V59()
V60()
V61()
V62()
V63()
V64() )
Element of
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) )
>= m : ( ( ) (
epsilon-transitive epsilon-connected ordinal natural V11()
real ext-real integer V58()
V59()
V60()
V61()
V62()
V63()
V64() )
Element of
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) ) holds
||.(seq : ( ( V21() V30( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V1() ) set ) ) ) ( V1() V16() V19( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) ) V20( the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V1() ) set ) ) V21() V26( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V1() ) set ) ) ) sequence of ( ( ) ( V1() ) set ) ) . (n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real integer V58() V59() V60() V61() V62() V63() V64() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) ) + 1 : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative integer V58() V59() V60() V61() V62() V63() V64() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real integer V58() V59() V60() V61() V62() V63() V64() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ) Element of the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V1() ) set ) ) .|| : ( ( ) (
V11()
real ext-real )
Element of
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) )
/ ||.(seq : ( ( V21() V30( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V1() ) set ) ) ) ( V1() V16() V19( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) ) V20( the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V1() ) set ) ) V21() V26( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V1() ) set ) ) ) sequence of ( ( ) ( V1() ) set ) ) . n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real integer V58() V59() V60() V61() V62() V63() V64() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ) Element of the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V1() ) set ) ) .|| : ( ( ) (
V11()
real ext-real )
Element of
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) (
V11()
real ext-real )
Element of
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) )
>= 1 : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal natural V11()
real ext-real positive non
negative integer V58()
V59()
V60()
V61()
V62()
V63()
V64() )
Element of
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) ) holds
not
seq : ( (
V21()
V30(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) , the
carrier of
b1 : ( ( non
empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non
empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like )
RealUnitarySpace) : ( ( ) (
V1() )
set ) ) ) (
V1()
V16()
V19(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) )
V20( the
carrier of
b1 : ( ( non
empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non
empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like )
RealUnitarySpace) : ( ( ) (
V1() )
set ) )
V21()
V26(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) )
V30(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) , the
carrier of
b1 : ( ( non
empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non
empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like )
RealUnitarySpace) : ( ( ) (
V1() )
set ) ) )
sequence of ( ( ) (
V1() )
set ) ) is
summable ;
theorem
for
Rseq being ( (
V21()
V30(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) ) (
V1()
V16()
V19(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) )
V20(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) )
V21()
V26(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) )
V30(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) )
complex-valued ext-real-valued real-valued )
Real_Sequence)
for
X being ( ( non
empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non
empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like )
RealUnitarySpace)
for
seq being ( (
V21()
V30(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) , the
carrier of
b2 : ( ( non
empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non
empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like )
RealUnitarySpace) : ( ( ) (
V1() )
set ) ) ) (
V1()
V16()
V19(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) )
V20( the
carrier of
b2 : ( ( non
empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non
empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like )
RealUnitarySpace) : ( ( ) (
V1() )
set ) )
V21()
V26(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) )
V30(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) , the
carrier of
b2 : ( ( non
empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non
empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like )
RealUnitarySpace) : ( ( ) (
V1() )
set ) ) )
sequence of ( ( ) (
V1() )
set ) ) st ( for
n being ( ( ) (
epsilon-transitive epsilon-connected ordinal natural V11()
real ext-real integer V58()
V59()
V60()
V61()
V62()
V63()
V64() )
Element of
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) ) holds
seq : ( (
V21()
V30(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) , the
carrier of
b2 : ( ( non
empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non
empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like )
RealUnitarySpace) : ( ( ) (
V1() )
set ) ) ) (
V1()
V16()
V19(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) )
V20( the
carrier of
b2 : ( ( non
empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non
empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like )
RealUnitarySpace) : ( ( ) (
V1() )
set ) )
V21()
V26(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) )
V30(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) , the
carrier of
b2 : ( ( non
empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non
empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like )
RealUnitarySpace) : ( ( ) (
V1() )
set ) ) )
sequence of ( ( ) (
V1() )
set ) )
. n : ( ( ) (
epsilon-transitive epsilon-connected ordinal natural V11()
real ext-real integer V58()
V59()
V60()
V61()
V62()
V63()
V64() )
Element of
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( )
Element of the
carrier of
b2 : ( ( non
empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non
empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like )
RealUnitarySpace) : ( ( ) (
V1() )
set ) )
<> 0. X : ( ( non
empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non
empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like )
RealUnitarySpace) : ( ( ) (
V80(
b2 : ( ( non
empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non
empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like )
RealUnitarySpace) ) )
Element of the
carrier of
b2 : ( ( non
empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non
empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like )
RealUnitarySpace) : ( ( ) (
V1() )
set ) ) ) & ( for
n being ( ( ) (
epsilon-transitive epsilon-connected ordinal natural V11()
real ext-real integer V58()
V59()
V60()
V61()
V62()
V63()
V64() )
Element of
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) ) holds
Rseq : ( (
V21()
V30(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) ) (
V1()
V16()
V19(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) )
V20(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) )
V21()
V26(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) )
V30(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) )
complex-valued ext-real-valued real-valued )
Real_Sequence)
. n : ( ( ) (
epsilon-transitive epsilon-connected ordinal natural V11()
real ext-real integer V58()
V59()
V60()
V61()
V62()
V63()
V64() )
Element of
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) (
V11()
real ext-real )
Element of
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) )
= ||.(seq : ( ( V21() V30( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V1() ) set ) ) ) ( V1() V16() V19( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) ) V20( the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V1() ) set ) ) V21() V26( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V1() ) set ) ) ) sequence of ( ( ) ( V1() ) set ) ) . (n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real integer V58() V59() V60() V61() V62() V63() V64() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) ) + 1 : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative integer V58() V59() V60() V61() V62() V63() V64() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real integer V58() V59() V60() V61() V62() V63() V64() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ) Element of the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V1() ) set ) ) .|| : ( ( ) (
V11()
real ext-real )
Element of
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) )
/ ||.(seq : ( ( V21() V30( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V1() ) set ) ) ) ( V1() V16() V19( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) ) V20( the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V1() ) set ) ) V21() V26( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V1() ) set ) ) ) sequence of ( ( ) ( V1() ) set ) ) . n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real integer V58() V59() V60() V61() V62() V63() V64() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ) Element of the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V1() ) set ) ) .|| : ( ( ) (
V11()
real ext-real )
Element of
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) (
V11()
real ext-real )
Element of
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) ) &
Rseq : ( (
V21()
V30(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) ) (
V1()
V16()
V19(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) )
V20(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) )
V21()
V26(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) )
V30(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) )
complex-valued ext-real-valued real-valued )
Real_Sequence) is
convergent &
lim Rseq : ( (
V21()
V30(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) ) (
V1()
V16()
V19(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) )
V20(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) )
V21()
V26(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) )
V30(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) )
complex-valued ext-real-valued real-valued )
Real_Sequence) : ( ( ) (
V11()
real ext-real )
Element of
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) )
> 1 : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal natural V11()
real ext-real positive non
negative integer V58()
V59()
V60()
V61()
V62()
V63()
V64() )
Element of
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) ) holds
not
seq : ( (
V21()
V30(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) , the
carrier of
b2 : ( ( non
empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non
empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like )
RealUnitarySpace) : ( ( ) (
V1() )
set ) ) ) (
V1()
V16()
V19(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) )
V20( the
carrier of
b2 : ( ( non
empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non
empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like )
RealUnitarySpace) : ( ( ) (
V1() )
set ) )
V21()
V26(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) )
V30(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) , the
carrier of
b2 : ( ( non
empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non
empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like )
RealUnitarySpace) : ( ( ) (
V1() )
set ) ) )
sequence of ( ( ) (
V1() )
set ) ) is
summable ;
theorem
for
Rseq being ( (
V21()
V30(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) ) (
V1()
V16()
V19(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) )
V20(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) )
V21()
V26(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) )
V30(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) )
complex-valued ext-real-valued real-valued )
Real_Sequence)
for
X being ( ( non
empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non
empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like )
RealUnitarySpace)
for
seq being ( (
V21()
V30(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) , the
carrier of
b2 : ( ( non
empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non
empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like )
RealUnitarySpace) : ( ( ) (
V1() )
set ) ) ) (
V1()
V16()
V19(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) )
V20( the
carrier of
b2 : ( ( non
empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non
empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like )
RealUnitarySpace) : ( ( ) (
V1() )
set ) )
V21()
V26(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) )
V30(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) , the
carrier of
b2 : ( ( non
empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non
empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like )
RealUnitarySpace) : ( ( ) (
V1() )
set ) ) )
sequence of ( ( ) (
V1() )
set ) ) st ( for
n being ( ( ) (
epsilon-transitive epsilon-connected ordinal natural V11()
real ext-real integer V58()
V59()
V60()
V61()
V62()
V63()
V64() )
Element of
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) ) holds
Rseq : ( (
V21()
V30(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) ) (
V1()
V16()
V19(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) )
V20(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) )
V21()
V26(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) )
V30(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) )
complex-valued ext-real-valued real-valued )
Real_Sequence)
. n : ( ( ) (
epsilon-transitive epsilon-connected ordinal natural V11()
real ext-real integer V58()
V59()
V60()
V61()
V62()
V63()
V64() )
Element of
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) (
V11()
real ext-real )
Element of
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) )
= n : ( ( ) (
epsilon-transitive epsilon-connected ordinal natural V11()
real ext-real integer V58()
V59()
V60()
V61()
V62()
V63()
V64() )
Element of
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) )
-root ||.(seq : ( ( V21() V30( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V1() ) set ) ) ) ( V1() V16() V19( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) ) V20( the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V1() ) set ) ) V21() V26( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V1() ) set ) ) ) sequence of ( ( ) ( V1() ) set ) ) . n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real integer V58() V59() V60() V61() V62() V63() V64() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ) Element of the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V1() ) set ) ) .|| : ( ( ) (
V11()
real ext-real )
Element of
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) (
V11()
real ext-real )
Element of
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) ) &
Rseq : ( (
V21()
V30(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) ) (
V1()
V16()
V19(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) )
V20(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) )
V21()
V26(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) )
V30(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) )
complex-valued ext-real-valued real-valued )
Real_Sequence) is
convergent &
lim Rseq : ( (
V21()
V30(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) ) (
V1()
V16()
V19(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) )
V20(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) )
V21()
V26(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) )
V30(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) )
complex-valued ext-real-valued real-valued )
Real_Sequence) : ( ( ) (
V11()
real ext-real )
Element of
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) )
< 1 : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal natural V11()
real ext-real positive non
negative integer V58()
V59()
V60()
V61()
V62()
V63()
V64() )
Element of
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) ) holds
seq : ( (
V21()
V30(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) , the
carrier of
b2 : ( ( non
empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non
empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like )
RealUnitarySpace) : ( ( ) (
V1() )
set ) ) ) (
V1()
V16()
V19(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) )
V20( the
carrier of
b2 : ( ( non
empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non
empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like )
RealUnitarySpace) : ( ( ) (
V1() )
set ) )
V21()
V26(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) )
V30(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) , the
carrier of
b2 : ( ( non
empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non
empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like )
RealUnitarySpace) : ( ( ) (
V1() )
set ) ) )
sequence of ( ( ) (
V1() )
set ) ) is
absolutely_summable ;
theorem
for
Rseq being ( (
V21()
V30(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) ) (
V1()
V16()
V19(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) )
V20(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) )
V21()
V26(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) )
V30(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) )
complex-valued ext-real-valued real-valued )
Real_Sequence)
for
X being ( ( non
empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non
empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like )
RealUnitarySpace)
for
seq being ( (
V21()
V30(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) , the
carrier of
b2 : ( ( non
empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non
empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like )
RealUnitarySpace) : ( ( ) (
V1() )
set ) ) ) (
V1()
V16()
V19(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) )
V20( the
carrier of
b2 : ( ( non
empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non
empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like )
RealUnitarySpace) : ( ( ) (
V1() )
set ) )
V21()
V26(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) )
V30(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) , the
carrier of
b2 : ( ( non
empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non
empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like )
RealUnitarySpace) : ( ( ) (
V1() )
set ) ) )
sequence of ( ( ) (
V1() )
set ) ) st ( for
n being ( ( ) (
epsilon-transitive epsilon-connected ordinal natural V11()
real ext-real integer V58()
V59()
V60()
V61()
V62()
V63()
V64() )
Element of
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) ) holds
Rseq : ( (
V21()
V30(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) ) (
V1()
V16()
V19(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) )
V20(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) )
V21()
V26(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) )
V30(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) )
complex-valued ext-real-valued real-valued )
Real_Sequence)
. n : ( ( ) (
epsilon-transitive epsilon-connected ordinal natural V11()
real ext-real integer V58()
V59()
V60()
V61()
V62()
V63()
V64() )
Element of
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) (
V11()
real ext-real )
Element of
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) )
= n : ( ( ) (
epsilon-transitive epsilon-connected ordinal natural V11()
real ext-real integer V58()
V59()
V60()
V61()
V62()
V63()
V64() )
Element of
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) )
-root (||.seq : ( ( V21() V30( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V1() ) set ) ) ) ( V1() V16() V19( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) ) V20( the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V1() ) set ) ) V21() V26( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V1() ) set ) ) ) sequence of ( ( ) ( V1() ) set ) ) .|| : ( ( V21() V30( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) ) ( V1() V16() V19( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) V21() V26( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) complex-valued ext-real-valued real-valued ) Element of K6(K7(NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) ,REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) . n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real integer V58() V59() V60() V61() V62() V63() V64() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) (
V11()
real ext-real )
Element of
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) (
V11()
real ext-real )
Element of
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) ) & ex
m being ( ( ) (
epsilon-transitive epsilon-connected ordinal natural V11()
real ext-real integer V58()
V59()
V60()
V61()
V62()
V63()
V64() )
Element of
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) ) st
for
n being ( ( ) (
epsilon-transitive epsilon-connected ordinal natural V11()
real ext-real integer V58()
V59()
V60()
V61()
V62()
V63()
V64() )
Element of
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) ) st
n : ( ( ) (
epsilon-transitive epsilon-connected ordinal natural V11()
real ext-real integer V58()
V59()
V60()
V61()
V62()
V63()
V64() )
Element of
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) )
>= m : ( ( ) (
epsilon-transitive epsilon-connected ordinal natural V11()
real ext-real integer V58()
V59()
V60()
V61()
V62()
V63()
V64() )
Element of
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) ) holds
Rseq : ( (
V21()
V30(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) ) (
V1()
V16()
V19(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) )
V20(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) )
V21()
V26(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) )
V30(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) )
complex-valued ext-real-valued real-valued )
Real_Sequence)
. n : ( ( ) (
epsilon-transitive epsilon-connected ordinal natural V11()
real ext-real integer V58()
V59()
V60()
V61()
V62()
V63()
V64() )
Element of
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) (
V11()
real ext-real )
Element of
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) )
>= 1 : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal natural V11()
real ext-real positive non
negative integer V58()
V59()
V60()
V61()
V62()
V63()
V64() )
Element of
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) ) holds
not
seq : ( (
V21()
V30(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) , the
carrier of
b2 : ( ( non
empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non
empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like )
RealUnitarySpace) : ( ( ) (
V1() )
set ) ) ) (
V1()
V16()
V19(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) )
V20( the
carrier of
b2 : ( ( non
empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non
empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like )
RealUnitarySpace) : ( ( ) (
V1() )
set ) )
V21()
V26(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) )
V30(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) , the
carrier of
b2 : ( ( non
empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non
empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like )
RealUnitarySpace) : ( ( ) (
V1() )
set ) ) )
sequence of ( ( ) (
V1() )
set ) ) is
summable ;
theorem
for
Rseq being ( (
V21()
V30(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) ) (
V1()
V16()
V19(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) )
V20(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) )
V21()
V26(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) )
V30(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) )
complex-valued ext-real-valued real-valued )
Real_Sequence)
for
X being ( ( non
empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non
empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like )
RealUnitarySpace)
for
seq being ( (
V21()
V30(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) , the
carrier of
b2 : ( ( non
empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non
empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like )
RealUnitarySpace) : ( ( ) (
V1() )
set ) ) ) (
V1()
V16()
V19(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) )
V20( the
carrier of
b2 : ( ( non
empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non
empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like )
RealUnitarySpace) : ( ( ) (
V1() )
set ) )
V21()
V26(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) )
V30(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) , the
carrier of
b2 : ( ( non
empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non
empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like )
RealUnitarySpace) : ( ( ) (
V1() )
set ) ) )
sequence of ( ( ) (
V1() )
set ) ) st ( for
n being ( ( ) (
epsilon-transitive epsilon-connected ordinal natural V11()
real ext-real integer V58()
V59()
V60()
V61()
V62()
V63()
V64() )
Element of
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) ) holds
Rseq : ( (
V21()
V30(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) ) (
V1()
V16()
V19(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) )
V20(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) )
V21()
V26(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) )
V30(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) )
complex-valued ext-real-valued real-valued )
Real_Sequence)
. n : ( ( ) (
epsilon-transitive epsilon-connected ordinal natural V11()
real ext-real integer V58()
V59()
V60()
V61()
V62()
V63()
V64() )
Element of
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) (
V11()
real ext-real )
Element of
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) )
= n : ( ( ) (
epsilon-transitive epsilon-connected ordinal natural V11()
real ext-real integer V58()
V59()
V60()
V61()
V62()
V63()
V64() )
Element of
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) )
-root (||.seq : ( ( V21() V30( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V1() ) set ) ) ) ( V1() V16() V19( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) ) V20( the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V1() ) set ) ) V21() V26( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V1() ) set ) ) ) sequence of ( ( ) ( V1() ) set ) ) .|| : ( ( V21() V30( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) ) ( V1() V16() V19( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) V21() V26( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) complex-valued ext-real-valued real-valued ) Element of K6(K7(NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) ,REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) . n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real integer V58() V59() V60() V61() V62() V63() V64() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) (
V11()
real ext-real )
Element of
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) (
V11()
real ext-real )
Element of
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) ) &
Rseq : ( (
V21()
V30(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) ) (
V1()
V16()
V19(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) )
V20(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) )
V21()
V26(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) )
V30(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) )
complex-valued ext-real-valued real-valued )
Real_Sequence) is
convergent &
lim Rseq : ( (
V21()
V30(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) ) (
V1()
V16()
V19(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) )
V20(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) )
V21()
V26(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) )
V30(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) )
complex-valued ext-real-valued real-valued )
Real_Sequence) : ( ( ) (
V11()
real ext-real )
Element of
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) )
> 1 : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal natural V11()
real ext-real positive non
negative integer V58()
V59()
V60()
V61()
V62()
V63()
V64() )
Element of
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) ) holds
not
seq : ( (
V21()
V30(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) , the
carrier of
b2 : ( ( non
empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non
empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like )
RealUnitarySpace) : ( ( ) (
V1() )
set ) ) ) (
V1()
V16()
V19(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) )
V20( the
carrier of
b2 : ( ( non
empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non
empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like )
RealUnitarySpace) : ( ( ) (
V1() )
set ) )
V21()
V26(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) )
V30(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) , the
carrier of
b2 : ( ( non
empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non
empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like )
RealUnitarySpace) : ( ( ) (
V1() )
set ) ) )
sequence of ( ( ) (
V1() )
set ) ) is
summable ;
theorem
for
X being ( ( non
empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non
empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like )
RealUnitarySpace)
for
seq being ( (
V21()
V30(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) , the
carrier of
b1 : ( ( non
empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non
empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like )
RealUnitarySpace) : ( ( ) (
V1() )
set ) ) ) (
V1()
V16()
V19(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) )
V20( the
carrier of
b1 : ( ( non
empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non
empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like )
RealUnitarySpace) : ( ( ) (
V1() )
set ) )
V21()
V26(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) )
V30(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) , the
carrier of
b1 : ( ( non
empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non
empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like )
RealUnitarySpace) : ( ( ) (
V1() )
set ) ) )
sequence of ( ( ) (
V1() )
set ) ) holds
Partial_Sums ||.seq : ( ( V21() V30( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V1() ) set ) ) ) ( V1() V16() V19( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) ) V20( the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V1() ) set ) ) V21() V26( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V1() ) set ) ) ) sequence of ( ( ) ( V1() ) set ) ) .|| : ( (
V21()
V30(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) ) (
V1()
V16()
V19(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) )
V20(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) )
V21()
V26(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) )
V30(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) )
complex-valued ext-real-valued real-valued )
Element of
K6(
K7(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) (
complex-valued ext-real-valued real-valued )
set ) ) : ( ( ) ( )
set ) ) : ( (
V21()
V30(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) ) (
V1()
V16()
V19(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) )
V20(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) )
V21()
V26(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) )
V30(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) )
complex-valued ext-real-valued real-valued )
Element of
K6(
K7(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) (
complex-valued ext-real-valued real-valued )
set ) ) : ( ( ) ( )
set ) ) is
non-decreasing ;
theorem
for
X being ( ( non
empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non
empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like )
RealUnitarySpace)
for
seq being ( (
V21()
V30(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) , the
carrier of
b1 : ( ( non
empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non
empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like )
RealUnitarySpace) : ( ( ) (
V1() )
set ) ) ) (
V1()
V16()
V19(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) )
V20( the
carrier of
b1 : ( ( non
empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non
empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like )
RealUnitarySpace) : ( ( ) (
V1() )
set ) )
V21()
V26(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) )
V30(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) , the
carrier of
b1 : ( ( non
empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non
empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like )
RealUnitarySpace) : ( ( ) (
V1() )
set ) ) )
sequence of ( ( ) (
V1() )
set ) )
for
n being ( ( ) (
epsilon-transitive epsilon-connected ordinal natural V11()
real ext-real integer V58()
V59()
V60()
V61()
V62()
V63()
V64() )
Element of
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) ) holds
(Partial_Sums ||.seq : ( ( V21() V30( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V1() ) set ) ) ) ( V1() V16() V19( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) ) V20( the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V1() ) set ) ) V21() V26( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V1() ) set ) ) ) sequence of ( ( ) ( V1() ) set ) ) .|| : ( ( V21() V30( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) ) ( V1() V16() V19( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) V21() V26( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) complex-valued ext-real-valued real-valued ) Element of K6(K7(NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) ,REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) ) : ( (
V21()
V30(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) ) (
V1()
V16()
V19(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) )
V20(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) )
V21()
V26(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) )
V30(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) )
complex-valued ext-real-valued real-valued )
Element of
K6(
K7(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) (
complex-valued ext-real-valued real-valued )
set ) ) : ( ( ) ( )
set ) )
. n : ( ( ) (
epsilon-transitive epsilon-connected ordinal natural V11()
real ext-real integer V58()
V59()
V60()
V61()
V62()
V63()
V64() )
Element of
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) (
V11()
real ext-real )
Element of
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) )
>= 0 : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural V11()
real ext-real non
positive non
negative integer V58()
V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) ) ;
theorem
for
X being ( ( non
empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non
empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like )
RealUnitarySpace)
for
seq being ( (
V21()
V30(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) , the
carrier of
b1 : ( ( non
empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non
empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like )
RealUnitarySpace) : ( ( ) (
V1() )
set ) ) ) (
V1()
V16()
V19(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) )
V20( the
carrier of
b1 : ( ( non
empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non
empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like )
RealUnitarySpace) : ( ( ) (
V1() )
set ) )
V21()
V26(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) )
V30(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) , the
carrier of
b1 : ( ( non
empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non
empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like )
RealUnitarySpace) : ( ( ) (
V1() )
set ) ) )
sequence of ( ( ) (
V1() )
set ) )
for
n being ( ( ) (
epsilon-transitive epsilon-connected ordinal natural V11()
real ext-real integer V58()
V59()
V60()
V61()
V62()
V63()
V64() )
Element of
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) ) holds
||.((Partial_Sums seq : ( ( V21() V30( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V1() ) set ) ) ) ( V1() V16() V19( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) ) V20( the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V1() ) set ) ) V21() V26( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V1() ) set ) ) ) sequence of ( ( ) ( V1() ) set ) ) ) : ( ( V21() V30( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V1() ) set ) ) ) ( V1() V16() V19( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) ) V20( the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V1() ) set ) ) V21() V26( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V1() ) set ) ) ) sequence of ( ( ) ( V1() ) set ) ) . n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real integer V58() V59() V60() V61() V62() V63() V64() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ) Element of the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V1() ) set ) ) .|| : ( ( ) (
V11()
real ext-real )
Element of
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) )
<= (Partial_Sums ||.seq : ( ( V21() V30( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V1() ) set ) ) ) ( V1() V16() V19( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) ) V20( the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V1() ) set ) ) V21() V26( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V1() ) set ) ) ) sequence of ( ( ) ( V1() ) set ) ) .|| : ( ( V21() V30( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) ) ( V1() V16() V19( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) V21() V26( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) complex-valued ext-real-valued real-valued ) Element of K6(K7(NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) ,REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) ) : ( (
V21()
V30(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) ) (
V1()
V16()
V19(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) )
V20(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) )
V21()
V26(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) )
V30(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) )
complex-valued ext-real-valued real-valued )
Element of
K6(
K7(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) (
complex-valued ext-real-valued real-valued )
set ) ) : ( ( ) ( )
set ) )
. n : ( ( ) (
epsilon-transitive epsilon-connected ordinal natural V11()
real ext-real integer V58()
V59()
V60()
V61()
V62()
V63()
V64() )
Element of
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) (
V11()
real ext-real )
Element of
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) ;
theorem
for
X being ( ( non
empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non
empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like )
RealUnitarySpace)
for
seq being ( (
V21()
V30(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) , the
carrier of
b1 : ( ( non
empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non
empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like )
RealUnitarySpace) : ( ( ) (
V1() )
set ) ) ) (
V1()
V16()
V19(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) )
V20( the
carrier of
b1 : ( ( non
empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non
empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like )
RealUnitarySpace) : ( ( ) (
V1() )
set ) )
V21()
V26(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) )
V30(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) , the
carrier of
b1 : ( ( non
empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non
empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like )
RealUnitarySpace) : ( ( ) (
V1() )
set ) ) )
sequence of ( ( ) (
V1() )
set ) )
for
n being ( ( ) (
epsilon-transitive epsilon-connected ordinal natural V11()
real ext-real integer V58()
V59()
V60()
V61()
V62()
V63()
V64() )
Element of
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) ) holds
||.(Sum (seq : ( ( V21() V30( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V1() ) set ) ) ) ( V1() V16() V19( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) ) V20( the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V1() ) set ) ) V21() V26( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V1() ) set ) ) ) sequence of ( ( ) ( V1() ) set ) ) ,n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real integer V58() V59() V60() V61() V62() V63() V64() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( ) ( ) Point of ( ( ) ( V1() ) set ) ) .|| : ( ( ) (
V11()
real ext-real )
Element of
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) )
<= Sum (
||.seq : ( ( V21() V30( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V1() ) set ) ) ) ( V1() V16() V19( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) ) V20( the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V1() ) set ) ) V21() V26( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V1() ) set ) ) ) sequence of ( ( ) ( V1() ) set ) ) .|| : ( (
V21()
V30(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) ) (
V1()
V16()
V19(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) )
V20(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) )
V21()
V26(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) )
V30(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) )
complex-valued ext-real-valued real-valued )
Element of
K6(
K7(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) (
complex-valued ext-real-valued real-valued )
set ) ) : ( ( ) ( )
set ) ) ,
n : ( ( ) (
epsilon-transitive epsilon-connected ordinal natural V11()
real ext-real integer V58()
V59()
V60()
V61()
V62()
V63()
V64() )
Element of
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) ) ) : ( ( ) (
V11()
real ext-real )
Element of
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) ;
theorem
for
X being ( ( non
empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non
empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like )
RealUnitarySpace)
for
seq being ( (
V21()
V30(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) , the
carrier of
b1 : ( ( non
empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non
empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like )
RealUnitarySpace) : ( ( ) (
V1() )
set ) ) ) (
V1()
V16()
V19(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) )
V20( the
carrier of
b1 : ( ( non
empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non
empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like )
RealUnitarySpace) : ( ( ) (
V1() )
set ) )
V21()
V26(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) )
V30(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) , the
carrier of
b1 : ( ( non
empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non
empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like )
RealUnitarySpace) : ( ( ) (
V1() )
set ) ) )
sequence of ( ( ) (
V1() )
set ) )
for
n,
m being ( ( ) (
epsilon-transitive epsilon-connected ordinal natural V11()
real ext-real integer V58()
V59()
V60()
V61()
V62()
V63()
V64() )
Element of
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) ) holds
||.(((Partial_Sums seq : ( ( V21() V30( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V1() ) set ) ) ) ( V1() V16() V19( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) ) V20( the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V1() ) set ) ) V21() V26( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V1() ) set ) ) ) sequence of ( ( ) ( V1() ) set ) ) ) : ( ( V21() V30( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V1() ) set ) ) ) ( V1() V16() V19( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) ) V20( the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V1() ) set ) ) V21() V26( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V1() ) set ) ) ) sequence of ( ( ) ( V1() ) set ) ) . m : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real integer V58() V59() V60() V61() V62() V63() V64() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ) Element of the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V1() ) set ) ) - ((Partial_Sums seq : ( ( V21() V30( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V1() ) set ) ) ) ( V1() V16() V19( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) ) V20( the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V1() ) set ) ) V21() V26( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V1() ) set ) ) ) sequence of ( ( ) ( V1() ) set ) ) ) : ( ( V21() V30( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V1() ) set ) ) ) ( V1() V16() V19( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) ) V20( the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V1() ) set ) ) V21() V26( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V1() ) set ) ) ) sequence of ( ( ) ( V1() ) set ) ) . n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real integer V58() V59() V60() V61() V62() V63() V64() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ) Element of the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V1() ) set ) ) ) : ( ( ) ( ) Element of the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V1() ) set ) ) .|| : ( ( ) (
V11()
real ext-real )
Element of
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) )
<= abs (((Partial_Sums ||.seq : ( ( V21() V30( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V1() ) set ) ) ) ( V1() V16() V19( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) ) V20( the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V1() ) set ) ) V21() V26( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V1() ) set ) ) ) sequence of ( ( ) ( V1() ) set ) ) .|| : ( ( V21() V30( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) ) ( V1() V16() V19( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) V21() V26( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) complex-valued ext-real-valued real-valued ) Element of K6(K7(NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) ,REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V21() V30( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) ) ( V1() V16() V19( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) V21() V26( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) complex-valued ext-real-valued real-valued ) Element of K6(K7(NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) ,REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) . m : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real integer V58() V59() V60() V61() V62() V63() V64() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) - ((Partial_Sums ||.seq : ( ( V21() V30( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V1() ) set ) ) ) ( V1() V16() V19( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) ) V20( the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V1() ) set ) ) V21() V26( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V1() ) set ) ) ) sequence of ( ( ) ( V1() ) set ) ) .|| : ( ( V21() V30( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) ) ( V1() V16() V19( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) V21() V26( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) complex-valued ext-real-valued real-valued ) Element of K6(K7(NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) ,REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V21() V30( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) ) ( V1() V16() V19( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) V21() V26( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) complex-valued ext-real-valued real-valued ) Element of K6(K7(NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) ,REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) . n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real integer V58() V59() V60() V61() V62() V63() V64() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) ) : ( ( ) (
V11()
real ext-real )
Element of
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) (
V11()
real ext-real )
Element of
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) ;
theorem
for
X being ( ( non
empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non
empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like )
RealUnitarySpace)
for
seq being ( (
V21()
V30(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) , the
carrier of
b1 : ( ( non
empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non
empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like )
RealUnitarySpace) : ( ( ) (
V1() )
set ) ) ) (
V1()
V16()
V19(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) )
V20( the
carrier of
b1 : ( ( non
empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non
empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like )
RealUnitarySpace) : ( ( ) (
V1() )
set ) )
V21()
V26(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) )
V30(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) , the
carrier of
b1 : ( ( non
empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non
empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like )
RealUnitarySpace) : ( ( ) (
V1() )
set ) ) )
sequence of ( ( ) (
V1() )
set ) )
for
n,
m being ( ( ) (
epsilon-transitive epsilon-connected ordinal natural V11()
real ext-real integer V58()
V59()
V60()
V61()
V62()
V63()
V64() )
Element of
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) ) holds
||.((Sum (seq : ( ( V21() V30( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V1() ) set ) ) ) ( V1() V16() V19( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) ) V20( the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V1() ) set ) ) V21() V26( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V1() ) set ) ) ) sequence of ( ( ) ( V1() ) set ) ) ,m : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real integer V58() V59() V60() V61() V62() V63() V64() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( ) ( ) Point of ( ( ) ( V1() ) set ) ) - (Sum (seq : ( ( V21() V30( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V1() ) set ) ) ) ( V1() V16() V19( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) ) V20( the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V1() ) set ) ) V21() V26( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V1() ) set ) ) ) sequence of ( ( ) ( V1() ) set ) ) ,n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real integer V58() V59() V60() V61() V62() V63() V64() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( ) ( ) Point of ( ( ) ( V1() ) set ) ) ) : ( ( ) ( ) Element of the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V1() ) set ) ) .|| : ( ( ) (
V11()
real ext-real )
Element of
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) )
<= abs ((Sum (||.seq : ( ( V21() V30( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V1() ) set ) ) ) ( V1() V16() V19( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) ) V20( the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V1() ) set ) ) V21() V26( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V1() ) set ) ) ) sequence of ( ( ) ( V1() ) set ) ) .|| : ( ( V21() V30( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) ) ( V1() V16() V19( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) V21() V26( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) complex-valued ext-real-valued real-valued ) Element of K6(K7(NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) ,REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) ,m : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real integer V58() V59() V60() V61() V62() V63() V64() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) - (Sum (||.seq : ( ( V21() V30( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V1() ) set ) ) ) ( V1() V16() V19( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) ) V20( the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V1() ) set ) ) V21() V26( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V1() ) set ) ) ) sequence of ( ( ) ( V1() ) set ) ) .|| : ( ( V21() V30( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) ) ( V1() V16() V19( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) V21() V26( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) complex-valued ext-real-valued real-valued ) Element of K6(K7(NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) ,REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) ,n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real integer V58() V59() V60() V61() V62() V63() V64() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) ) : ( ( ) (
V11()
real ext-real )
Element of
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) (
V11()
real ext-real )
Element of
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) ;
theorem
for
X being ( ( non
empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non
empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like )
RealUnitarySpace)
for
seq being ( (
V21()
V30(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) , the
carrier of
b1 : ( ( non
empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non
empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like )
RealUnitarySpace) : ( ( ) (
V1() )
set ) ) ) (
V1()
V16()
V19(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) )
V20( the
carrier of
b1 : ( ( non
empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non
empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like )
RealUnitarySpace) : ( ( ) (
V1() )
set ) )
V21()
V26(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) )
V30(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) , the
carrier of
b1 : ( ( non
empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non
empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like )
RealUnitarySpace) : ( ( ) (
V1() )
set ) ) )
sequence of ( ( ) (
V1() )
set ) )
for
n,
m being ( ( ) (
epsilon-transitive epsilon-connected ordinal natural V11()
real ext-real integer V58()
V59()
V60()
V61()
V62()
V63()
V64() )
Element of
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) ) holds
||.(Sum (seq : ( ( V21() V30( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V1() ) set ) ) ) ( V1() V16() V19( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) ) V20( the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V1() ) set ) ) V21() V26( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V1() ) set ) ) ) sequence of ( ( ) ( V1() ) set ) ) ,m : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real integer V58() V59() V60() V61() V62() V63() V64() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) ) ,n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real integer V58() V59() V60() V61() V62() V63() V64() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( ) ( ) Point of ( ( ) ( V1() ) set ) ) .|| : ( ( ) (
V11()
real ext-real )
Element of
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) )
<= abs (Sum (||.seq : ( ( V21() V30( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V1() ) set ) ) ) ( V1() V16() V19( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) ) V20( the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V1() ) set ) ) V21() V26( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V1() ) set ) ) ) sequence of ( ( ) ( V1() ) set ) ) .|| : ( ( V21() V30( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) ) ( V1() V16() V19( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) V21() V26( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) complex-valued ext-real-valued real-valued ) Element of K6(K7(NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) ,REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) ,m : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real integer V58() V59() V60() V61() V62() V63() V64() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) ) ,n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real integer V58() V59() V60() V61() V62() V63() V64() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( ) (
V11()
real ext-real )
Element of
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) (
V11()
real ext-real )
Element of
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) ;
definition
let X be ( ( non
empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non
empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like )
RealUnitarySpace) ;
let seq be ( (
V21()
V30(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) , the
carrier of
X : ( ( non
empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non
empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like )
RealUnitarySpace) : ( ( ) (
V1() )
set ) ) ) (
V1()
V16()
V19(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) )
V20( the
carrier of
X : ( ( non
empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non
empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like )
RealUnitarySpace) : ( ( ) (
V1() )
set ) )
V21()
V26(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) )
V30(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) , the
carrier of
X : ( ( non
empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non
empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like )
RealUnitarySpace) : ( ( ) (
V1() )
set ) ) )
sequence of ( ( ) (
V1() )
set ) ) ;
let Rseq be ( (
V21()
V30(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) ) (
V1()
V16()
V19(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) )
V20(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) )
V21()
V26(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) )
V30(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) )
complex-valued ext-real-valued real-valued )
Real_Sequence) ;
func Rseq * seq -> ( (
V21()
V30(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) , the
carrier of
X : ( ( ) ( )
UNITSTR ) : ( ( ) ( )
set ) ) ) (
V1()
V16()
V19(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) )
V20( the
carrier of
X : ( ( ) ( )
UNITSTR ) : ( ( ) ( )
set ) )
V21()
V26(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) )
V30(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) , the
carrier of
X : ( ( ) ( )
UNITSTR ) : ( ( ) ( )
set ) ) )
sequence of ( ( ) ( )
set ) )
means
for
n being ( ( ) (
epsilon-transitive epsilon-connected ordinal natural V11()
real ext-real integer V58()
V59()
V60()
V61()
V62()
V63()
V64() )
Element of
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) ) holds
it : ( (
V21()
V30(
K7(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ,
X : ( ( ) ( )
UNITSTR ) ) : ( ( ) ( )
set ) ,
X : ( ( ) ( )
UNITSTR ) ) ) (
V16()
V19(
K7(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ,
X : ( ( ) ( )
UNITSTR ) ) : ( ( ) ( )
set ) )
V20(
X : ( ( ) ( )
UNITSTR ) )
V21()
V30(
K7(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ,
X : ( ( ) ( )
UNITSTR ) ) : ( ( ) ( )
set ) ,
X : ( ( ) ( )
UNITSTR ) ) )
Element of
K6(
K7(
K7(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ,
X : ( ( ) ( )
UNITSTR ) ) : ( ( ) ( )
set ) ,
X : ( ( ) ( )
UNITSTR ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) )
. n : ( ( ) (
epsilon-transitive epsilon-connected ordinal natural V11()
real ext-real integer V58()
V59()
V60()
V61()
V62()
V63()
V64() )
Element of
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( )
Element of the
carrier of
X : ( ( ) ( )
UNITSTR ) : ( ( ) ( )
set ) )
= (Rseq : ( ( V21() V30(K7(X : ( ( ) ( ) UNITSTR ) ,X : ( ( ) ( ) UNITSTR ) ) : ( ( ) ( ) set ) ,X : ( ( ) ( ) UNITSTR ) ) ) ( V16() V19(K7(X : ( ( ) ( ) UNITSTR ) ,X : ( ( ) ( ) UNITSTR ) ) : ( ( ) ( ) set ) ) V20(X : ( ( ) ( ) UNITSTR ) ) V21() V30(K7(X : ( ( ) ( ) UNITSTR ) ,X : ( ( ) ( ) UNITSTR ) ) : ( ( ) ( ) set ) ,X : ( ( ) ( ) UNITSTR ) ) ) Element of K6(K7(K7(X : ( ( ) ( ) UNITSTR ) ,X : ( ( ) ( ) UNITSTR ) ) : ( ( ) ( ) set ) ,X : ( ( ) ( ) UNITSTR ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) . n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real integer V58() V59() V60() V61() V62() V63() V64() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) (
V11()
real ext-real )
Element of
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) )
* (seq : ( ( ) ( ) Element of X : ( ( ) ( ) UNITSTR ) ) . n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real integer V58() V59() V60() V61() V62() V63() V64() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( )
Element of the
carrier of
X : ( ( ) ( )
UNITSTR ) : ( ( ) ( )
set ) ) : ( ( ) ( )
Element of the
carrier of
X : ( ( ) ( )
UNITSTR ) : ( ( ) ( )
set ) ) ;
end;
theorem
for
Rseq being ( (
V21()
V30(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) ) (
V1()
V16()
V19(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) )
V20(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) )
V21()
V26(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) )
V30(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) )
complex-valued ext-real-valued real-valued )
Real_Sequence)
for
X being ( ( non
empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non
empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like )
RealUnitarySpace)
for
seq1,
seq2 being ( (
V21()
V30(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) , the
carrier of
b2 : ( ( non
empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non
empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like )
RealUnitarySpace) : ( ( ) (
V1() )
set ) ) ) (
V1()
V16()
V19(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) )
V20( the
carrier of
b2 : ( ( non
empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non
empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like )
RealUnitarySpace) : ( ( ) (
V1() )
set ) )
V21()
V26(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) )
V30(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) , the
carrier of
b2 : ( ( non
empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non
empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like )
RealUnitarySpace) : ( ( ) (
V1() )
set ) ) )
sequence of ( ( ) (
V1() )
set ) ) holds
Rseq : ( (
V21()
V30(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) ) (
V1()
V16()
V19(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) )
V20(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) )
V21()
V26(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) )
V30(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) )
complex-valued ext-real-valued real-valued )
Real_Sequence)
* (seq1 : ( ( V21() V30( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V1() ) set ) ) ) ( V1() V16() V19( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) ) V20( the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V1() ) set ) ) V21() V26( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V1() ) set ) ) ) sequence of ( ( ) ( V1() ) set ) ) + seq2 : ( ( V21() V30( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V1() ) set ) ) ) ( V1() V16() V19( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) ) V20( the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V1() ) set ) ) V21() V26( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V1() ) set ) ) ) sequence of ( ( ) ( V1() ) set ) ) ) : ( (
V21()
V30(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) , the
carrier of
b2 : ( ( non
empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non
empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like )
RealUnitarySpace) : ( ( ) (
V1() )
set ) ) ) (
V1()
V16()
V19(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) )
V20( the
carrier of
b2 : ( ( non
empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non
empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like )
RealUnitarySpace) : ( ( ) (
V1() )
set ) )
V21()
V26(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) )
V30(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) , the
carrier of
b2 : ( ( non
empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non
empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like )
RealUnitarySpace) : ( ( ) (
V1() )
set ) ) )
Element of
K6(
K7(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) , the
carrier of
b2 : ( ( non
empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non
empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like )
RealUnitarySpace) : ( ( ) (
V1() )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) : ( (
V21()
V30(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) , the
carrier of
b2 : ( ( non
empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non
empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like )
RealUnitarySpace) : ( ( ) (
V1() )
set ) ) ) (
V1()
V16()
V19(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) )
V20( the
carrier of
b2 : ( ( non
empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non
empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like )
RealUnitarySpace) : ( ( ) (
V1() )
set ) )
V21()
V26(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) )
V30(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) , the
carrier of
b2 : ( ( non
empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non
empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like )
RealUnitarySpace) : ( ( ) (
V1() )
set ) ) )
sequence of ( ( ) (
V1() )
set ) )
= (Rseq : ( ( V21() V30( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) ) ( V1() V16() V19( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) V21() V26( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) * seq1 : ( ( V21() V30( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V1() ) set ) ) ) ( V1() V16() V19( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) ) V20( the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V1() ) set ) ) V21() V26( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V1() ) set ) ) ) sequence of ( ( ) ( V1() ) set ) ) ) : ( (
V21()
V30(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) , the
carrier of
b2 : ( ( non
empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non
empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like )
RealUnitarySpace) : ( ( ) (
V1() )
set ) ) ) (
V1()
V16()
V19(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) )
V20( the
carrier of
b2 : ( ( non
empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non
empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like )
RealUnitarySpace) : ( ( ) (
V1() )
set ) )
V21()
V26(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) )
V30(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) , the
carrier of
b2 : ( ( non
empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non
empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like )
RealUnitarySpace) : ( ( ) (
V1() )
set ) ) )
sequence of ( ( ) (
V1() )
set ) )
+ (Rseq : ( ( V21() V30( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) ) ( V1() V16() V19( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) V21() V26( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) * seq2 : ( ( V21() V30( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V1() ) set ) ) ) ( V1() V16() V19( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) ) V20( the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V1() ) set ) ) V21() V26( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V1() ) set ) ) ) sequence of ( ( ) ( V1() ) set ) ) ) : ( (
V21()
V30(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) , the
carrier of
b2 : ( ( non
empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non
empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like )
RealUnitarySpace) : ( ( ) (
V1() )
set ) ) ) (
V1()
V16()
V19(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) )
V20( the
carrier of
b2 : ( ( non
empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non
empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like )
RealUnitarySpace) : ( ( ) (
V1() )
set ) )
V21()
V26(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) )
V30(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) , the
carrier of
b2 : ( ( non
empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non
empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like )
RealUnitarySpace) : ( ( ) (
V1() )
set ) ) )
sequence of ( ( ) (
V1() )
set ) ) : ( (
V21()
V30(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) , the
carrier of
b2 : ( ( non
empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non
empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like )
RealUnitarySpace) : ( ( ) (
V1() )
set ) ) ) (
V1()
V16()
V19(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) )
V20( the
carrier of
b2 : ( ( non
empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non
empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like )
RealUnitarySpace) : ( ( ) (
V1() )
set ) )
V21()
V26(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) )
V30(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) , the
carrier of
b2 : ( ( non
empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non
empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like )
RealUnitarySpace) : ( ( ) (
V1() )
set ) ) )
Element of
K6(
K7(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) , the
carrier of
b2 : ( ( non
empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non
empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like )
RealUnitarySpace) : ( ( ) (
V1() )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) ;
theorem
for
Rseq1,
Rseq2 being ( (
V21()
V30(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) ) (
V1()
V16()
V19(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) )
V20(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) )
V21()
V26(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) )
V30(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) )
complex-valued ext-real-valued real-valued )
Real_Sequence)
for
X being ( ( non
empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non
empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like )
RealUnitarySpace)
for
seq being ( (
V21()
V30(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) , the
carrier of
b3 : ( ( non
empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non
empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like )
RealUnitarySpace) : ( ( ) (
V1() )
set ) ) ) (
V1()
V16()
V19(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) )
V20( the
carrier of
b3 : ( ( non
empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non
empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like )
RealUnitarySpace) : ( ( ) (
V1() )
set ) )
V21()
V26(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) )
V30(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) , the
carrier of
b3 : ( ( non
empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non
empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like )
RealUnitarySpace) : ( ( ) (
V1() )
set ) ) )
sequence of ( ( ) (
V1() )
set ) ) holds
(Rseq1 : ( ( V21() V30( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) ) ( V1() V16() V19( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) V21() V26( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) + Rseq2 : ( ( V21() V30( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) ) ( V1() V16() V19( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) V21() V26( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) ) : ( (
V21()
V30(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) ) (
V1()
V16()
V19(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) )
V20(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) )
V21()
V26(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) )
V30(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) )
complex-valued ext-real-valued real-valued )
Element of
K6(
K7(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) (
complex-valued ext-real-valued real-valued )
set ) ) : ( ( ) ( )
set ) )
* seq : ( (
V21()
V30(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) , the
carrier of
b3 : ( ( non
empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non
empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like )
RealUnitarySpace) : ( ( ) (
V1() )
set ) ) ) (
V1()
V16()
V19(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) )
V20( the
carrier of
b3 : ( ( non
empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non
empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like )
RealUnitarySpace) : ( ( ) (
V1() )
set ) )
V21()
V26(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) )
V30(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) , the
carrier of
b3 : ( ( non
empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non
empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like )
RealUnitarySpace) : ( ( ) (
V1() )
set ) ) )
sequence of ( ( ) (
V1() )
set ) ) : ( (
V21()
V30(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) , the
carrier of
b3 : ( ( non
empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non
empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like )
RealUnitarySpace) : ( ( ) (
V1() )
set ) ) ) (
V1()
V16()
V19(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) )
V20( the
carrier of
b3 : ( ( non
empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non
empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like )
RealUnitarySpace) : ( ( ) (
V1() )
set ) )
V21()
V26(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) )
V30(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) , the
carrier of
b3 : ( ( non
empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non
empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like )
RealUnitarySpace) : ( ( ) (
V1() )
set ) ) )
sequence of ( ( ) (
V1() )
set ) )
= (Rseq1 : ( ( V21() V30( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) ) ( V1() V16() V19( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) V21() V26( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) * seq : ( ( V21() V30( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of b3 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V1() ) set ) ) ) ( V1() V16() V19( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) ) V20( the carrier of b3 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V1() ) set ) ) V21() V26( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of b3 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V1() ) set ) ) ) sequence of ( ( ) ( V1() ) set ) ) ) : ( (
V21()
V30(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) , the
carrier of
b3 : ( ( non
empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non
empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like )
RealUnitarySpace) : ( ( ) (
V1() )
set ) ) ) (
V1()
V16()
V19(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) )
V20( the
carrier of
b3 : ( ( non
empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non
empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like )
RealUnitarySpace) : ( ( ) (
V1() )
set ) )
V21()
V26(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) )
V30(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) , the
carrier of
b3 : ( ( non
empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non
empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like )
RealUnitarySpace) : ( ( ) (
V1() )
set ) ) )
sequence of ( ( ) (
V1() )
set ) )
+ (Rseq2 : ( ( V21() V30( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) ) ( V1() V16() V19( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) V21() V26( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) * seq : ( ( V21() V30( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of b3 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V1() ) set ) ) ) ( V1() V16() V19( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) ) V20( the carrier of b3 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V1() ) set ) ) V21() V26( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of b3 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V1() ) set ) ) ) sequence of ( ( ) ( V1() ) set ) ) ) : ( (
V21()
V30(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) , the
carrier of
b3 : ( ( non
empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non
empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like )
RealUnitarySpace) : ( ( ) (
V1() )
set ) ) ) (
V1()
V16()
V19(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) )
V20( the
carrier of
b3 : ( ( non
empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non
empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like )
RealUnitarySpace) : ( ( ) (
V1() )
set ) )
V21()
V26(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) )
V30(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) , the
carrier of
b3 : ( ( non
empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non
empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like )
RealUnitarySpace) : ( ( ) (
V1() )
set ) ) )
sequence of ( ( ) (
V1() )
set ) ) : ( (
V21()
V30(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) , the
carrier of
b3 : ( ( non
empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non
empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like )
RealUnitarySpace) : ( ( ) (
V1() )
set ) ) ) (
V1()
V16()
V19(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) )
V20( the
carrier of
b3 : ( ( non
empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non
empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like )
RealUnitarySpace) : ( ( ) (
V1() )
set ) )
V21()
V26(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) )
V30(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) , the
carrier of
b3 : ( ( non
empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non
empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like )
RealUnitarySpace) : ( ( ) (
V1() )
set ) ) )
Element of
K6(
K7(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) , the
carrier of
b3 : ( ( non
empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non
empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like )
RealUnitarySpace) : ( ( ) (
V1() )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) ;
theorem
for
Rseq1,
Rseq2 being ( (
V21()
V30(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) ) (
V1()
V16()
V19(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) )
V20(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) )
V21()
V26(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) )
V30(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) )
complex-valued ext-real-valued real-valued )
Real_Sequence)
for
X being ( ( non
empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non
empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like )
RealUnitarySpace)
for
seq being ( (
V21()
V30(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) , the
carrier of
b3 : ( ( non
empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non
empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like )
RealUnitarySpace) : ( ( ) (
V1() )
set ) ) ) (
V1()
V16()
V19(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) )
V20( the
carrier of
b3 : ( ( non
empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non
empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like )
RealUnitarySpace) : ( ( ) (
V1() )
set ) )
V21()
V26(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) )
V30(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) , the
carrier of
b3 : ( ( non
empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non
empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like )
RealUnitarySpace) : ( ( ) (
V1() )
set ) ) )
sequence of ( ( ) (
V1() )
set ) ) holds
(Rseq1 : ( ( V21() V30( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) ) ( V1() V16() V19( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) V21() V26( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) (#) Rseq2 : ( ( V21() V30( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) ) ( V1() V16() V19( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) V21() V26( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) ) : ( (
V21() ) (
V1()
V16()
V19(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) )
V20(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) )
V21()
V26(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) )
V30(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) )
complex-valued ext-real-valued real-valued )
Element of
K6(
K7(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) (
complex-valued ext-real-valued real-valued )
set ) ) : ( ( ) ( )
set ) )
* seq : ( (
V21()
V30(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) , the
carrier of
b3 : ( ( non
empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non
empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like )
RealUnitarySpace) : ( ( ) (
V1() )
set ) ) ) (
V1()
V16()
V19(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) )
V20( the
carrier of
b3 : ( ( non
empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non
empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like )
RealUnitarySpace) : ( ( ) (
V1() )
set ) )
V21()
V26(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) )
V30(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) , the
carrier of
b3 : ( ( non
empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non
empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like )
RealUnitarySpace) : ( ( ) (
V1() )
set ) ) )
sequence of ( ( ) (
V1() )
set ) ) : ( (
V21()
V30(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) , the
carrier of
b3 : ( ( non
empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non
empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like )
RealUnitarySpace) : ( ( ) (
V1() )
set ) ) ) (
V1()
V16()
V19(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) )
V20( the
carrier of
b3 : ( ( non
empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non
empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like )
RealUnitarySpace) : ( ( ) (
V1() )
set ) )
V21()
V26(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) )
V30(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) , the
carrier of
b3 : ( ( non
empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non
empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like )
RealUnitarySpace) : ( ( ) (
V1() )
set ) ) )
sequence of ( ( ) (
V1() )
set ) )
= Rseq1 : ( (
V21()
V30(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) ) (
V1()
V16()
V19(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) )
V20(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) )
V21()
V26(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) )
V30(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) )
complex-valued ext-real-valued real-valued )
Real_Sequence)
* (Rseq2 : ( ( V21() V30( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) ) ( V1() V16() V19( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) V21() V26( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) * seq : ( ( V21() V30( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of b3 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V1() ) set ) ) ) ( V1() V16() V19( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) ) V20( the carrier of b3 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V1() ) set ) ) V21() V26( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of b3 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V1() ) set ) ) ) sequence of ( ( ) ( V1() ) set ) ) ) : ( (
V21()
V30(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) , the
carrier of
b3 : ( ( non
empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non
empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like )
RealUnitarySpace) : ( ( ) (
V1() )
set ) ) ) (
V1()
V16()
V19(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) )
V20( the
carrier of
b3 : ( ( non
empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non
empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like )
RealUnitarySpace) : ( ( ) (
V1() )
set ) )
V21()
V26(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) )
V30(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) , the
carrier of
b3 : ( ( non
empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non
empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like )
RealUnitarySpace) : ( ( ) (
V1() )
set ) ) )
sequence of ( ( ) (
V1() )
set ) ) : ( (
V21()
V30(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) , the
carrier of
b3 : ( ( non
empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non
empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like )
RealUnitarySpace) : ( ( ) (
V1() )
set ) ) ) (
V1()
V16()
V19(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) )
V20( the
carrier of
b3 : ( ( non
empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non
empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like )
RealUnitarySpace) : ( ( ) (
V1() )
set ) )
V21()
V26(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) )
V30(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) , the
carrier of
b3 : ( ( non
empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non
empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like )
RealUnitarySpace) : ( ( ) (
V1() )
set ) ) )
sequence of ( ( ) (
V1() )
set ) ) ;
theorem
for
a being ( ( ) (
V11()
real ext-real )
Real)
for
Rseq being ( (
V21()
V30(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) ) (
V1()
V16()
V19(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) )
V20(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) )
V21()
V26(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) )
V30(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) )
complex-valued ext-real-valued real-valued )
Real_Sequence)
for
X being ( ( non
empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non
empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like )
RealUnitarySpace)
for
seq being ( (
V21()
V30(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) , the
carrier of
b3 : ( ( non
empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non
empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like )
RealUnitarySpace) : ( ( ) (
V1() )
set ) ) ) (
V1()
V16()
V19(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) )
V20( the
carrier of
b3 : ( ( non
empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non
empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like )
RealUnitarySpace) : ( ( ) (
V1() )
set ) )
V21()
V26(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) )
V30(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) , the
carrier of
b3 : ( ( non
empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non
empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like )
RealUnitarySpace) : ( ( ) (
V1() )
set ) ) )
sequence of ( ( ) (
V1() )
set ) ) holds
(a : ( ( ) ( V11() real ext-real ) Real) (#) Rseq : ( ( V21() V30( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) ) ( V1() V16() V19( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) V21() V26( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) ) : ( (
V21() ) (
V1()
V16()
V19(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) )
V20(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) )
V21()
V26(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) )
V30(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) )
complex-valued ext-real-valued real-valued )
Element of
K6(
K7(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) (
complex-valued ext-real-valued real-valued )
set ) ) : ( ( ) ( )
set ) )
* seq : ( (
V21()
V30(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) , the
carrier of
b3 : ( ( non
empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non
empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like )
RealUnitarySpace) : ( ( ) (
V1() )
set ) ) ) (
V1()
V16()
V19(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) )
V20( the
carrier of
b3 : ( ( non
empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non
empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like )
RealUnitarySpace) : ( ( ) (
V1() )
set ) )
V21()
V26(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) )
V30(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) , the
carrier of
b3 : ( ( non
empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non
empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like )
RealUnitarySpace) : ( ( ) (
V1() )
set ) ) )
sequence of ( ( ) (
V1() )
set ) ) : ( (
V21()
V30(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) , the
carrier of
b3 : ( ( non
empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non
empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like )
RealUnitarySpace) : ( ( ) (
V1() )
set ) ) ) (
V1()
V16()
V19(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) )
V20( the
carrier of
b3 : ( ( non
empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non
empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like )
RealUnitarySpace) : ( ( ) (
V1() )
set ) )
V21()
V26(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) )
V30(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) , the
carrier of
b3 : ( ( non
empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non
empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like )
RealUnitarySpace) : ( ( ) (
V1() )
set ) ) )
sequence of ( ( ) (
V1() )
set ) )
= a : ( ( ) (
V11()
real ext-real )
Real)
* (Rseq : ( ( V21() V30( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) ) ( V1() V16() V19( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) V21() V26( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) * seq : ( ( V21() V30( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of b3 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V1() ) set ) ) ) ( V1() V16() V19( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) ) V20( the carrier of b3 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V1() ) set ) ) V21() V26( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of b3 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V1() ) set ) ) ) sequence of ( ( ) ( V1() ) set ) ) ) : ( (
V21()
V30(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) , the
carrier of
b3 : ( ( non
empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non
empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like )
RealUnitarySpace) : ( ( ) (
V1() )
set ) ) ) (
V1()
V16()
V19(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) )
V20( the
carrier of
b3 : ( ( non
empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non
empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like )
RealUnitarySpace) : ( ( ) (
V1() )
set ) )
V21()
V26(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) )
V30(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) , the
carrier of
b3 : ( ( non
empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non
empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like )
RealUnitarySpace) : ( ( ) (
V1() )
set ) ) )
sequence of ( ( ) (
V1() )
set ) ) : ( (
V21()
V30(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) , the
carrier of
b3 : ( ( non
empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non
empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like )
RealUnitarySpace) : ( ( ) (
V1() )
set ) ) ) (
V1()
V16()
V19(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) )
V20( the
carrier of
b3 : ( ( non
empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non
empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like )
RealUnitarySpace) : ( ( ) (
V1() )
set ) )
V21()
V26(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) )
V30(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) , the
carrier of
b3 : ( ( non
empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non
empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like )
RealUnitarySpace) : ( ( ) (
V1() )
set ) ) )
Element of
K6(
K7(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) , the
carrier of
b3 : ( ( non
empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non
empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like )
RealUnitarySpace) : ( ( ) (
V1() )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) ;
theorem
for
Rseq being ( (
V21()
V30(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) ) (
V1()
V16()
V19(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) )
V20(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) )
V21()
V26(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) )
V30(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) )
complex-valued ext-real-valued real-valued )
Real_Sequence)
for
X being ( ( non
empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non
empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like )
RealUnitarySpace)
for
seq being ( (
V21()
V30(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) , the
carrier of
b2 : ( ( non
empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non
empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like )
RealUnitarySpace) : ( ( ) (
V1() )
set ) ) ) (
V1()
V16()
V19(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) )
V20( the
carrier of
b2 : ( ( non
empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non
empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like )
RealUnitarySpace) : ( ( ) (
V1() )
set ) )
V21()
V26(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) )
V30(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) , the
carrier of
b2 : ( ( non
empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non
empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like )
RealUnitarySpace) : ( ( ) (
V1() )
set ) ) )
sequence of ( ( ) (
V1() )
set ) ) holds
Rseq : ( (
V21()
V30(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) ) (
V1()
V16()
V19(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) )
V20(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) )
V21()
V26(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) )
V30(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) )
complex-valued ext-real-valued real-valued )
Real_Sequence)
* (- seq : ( ( V21() V30( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V1() ) set ) ) ) ( V1() V16() V19( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) ) V20( the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V1() ) set ) ) V21() V26( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V1() ) set ) ) ) sequence of ( ( ) ( V1() ) set ) ) ) : ( (
V21() ) (
V1()
V16()
V19(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) )
V20( the
carrier of
b2 : ( ( non
empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non
empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like )
RealUnitarySpace) : ( ( ) (
V1() )
set ) )
V21()
V26(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) )
V30(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) , the
carrier of
b2 : ( ( non
empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non
empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like )
RealUnitarySpace) : ( ( ) (
V1() )
set ) ) )
Element of
K6(
K7(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) , the
carrier of
b2 : ( ( non
empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non
empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like )
RealUnitarySpace) : ( ( ) (
V1() )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) : ( (
V21()
V30(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) , the
carrier of
b2 : ( ( non
empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non
empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like )
RealUnitarySpace) : ( ( ) (
V1() )
set ) ) ) (
V1()
V16()
V19(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) )
V20( the
carrier of
b2 : ( ( non
empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non
empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like )
RealUnitarySpace) : ( ( ) (
V1() )
set ) )
V21()
V26(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) )
V30(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) , the
carrier of
b2 : ( ( non
empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non
empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like )
RealUnitarySpace) : ( ( ) (
V1() )
set ) ) )
sequence of ( ( ) (
V1() )
set ) )
= (- Rseq : ( ( V21() V30( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) ) ( V1() V16() V19( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) V21() V26( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) ) : ( (
V21() ) (
V1()
V16()
V19(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) )
V20(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) )
V21()
V26(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) )
V30(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) )
complex-valued ext-real-valued real-valued )
Element of
K6(
K7(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) (
complex-valued ext-real-valued real-valued )
set ) ) : ( ( ) ( )
set ) )
* seq : ( (
V21()
V30(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) , the
carrier of
b2 : ( ( non
empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non
empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like )
RealUnitarySpace) : ( ( ) (
V1() )
set ) ) ) (
V1()
V16()
V19(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) )
V20( the
carrier of
b2 : ( ( non
empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non
empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like )
RealUnitarySpace) : ( ( ) (
V1() )
set ) )
V21()
V26(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) )
V30(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) , the
carrier of
b2 : ( ( non
empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non
empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like )
RealUnitarySpace) : ( ( ) (
V1() )
set ) ) )
sequence of ( ( ) (
V1() )
set ) ) : ( (
V21()
V30(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) , the
carrier of
b2 : ( ( non
empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non
empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like )
RealUnitarySpace) : ( ( ) (
V1() )
set ) ) ) (
V1()
V16()
V19(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) )
V20( the
carrier of
b2 : ( ( non
empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non
empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like )
RealUnitarySpace) : ( ( ) (
V1() )
set ) )
V21()
V26(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) )
V30(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) , the
carrier of
b2 : ( ( non
empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non
empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like )
RealUnitarySpace) : ( ( ) (
V1() )
set ) ) )
sequence of ( ( ) (
V1() )
set ) ) ;
theorem
for
Rseq being ( (
V21()
V30(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) ) (
V1()
V16()
V19(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) )
V20(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) )
V21()
V26(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) )
V30(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) )
complex-valued ext-real-valued real-valued )
Real_Sequence)
for
X being ( ( non
empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non
empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like )
RealUnitarySpace)
for
seq being ( (
V21()
V30(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) , the
carrier of
b2 : ( ( non
empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non
empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like )
RealUnitarySpace) : ( ( ) (
V1() )
set ) ) ) (
V1()
V16()
V19(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) )
V20( the
carrier of
b2 : ( ( non
empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non
empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like )
RealUnitarySpace) : ( ( ) (
V1() )
set ) )
V21()
V26(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) )
V30(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) , the
carrier of
b2 : ( ( non
empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non
empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like )
RealUnitarySpace) : ( ( ) (
V1() )
set ) ) )
sequence of ( ( ) (
V1() )
set ) ) st
Rseq : ( (
V21()
V30(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) ) (
V1()
V16()
V19(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) )
V20(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) )
V21()
V26(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) )
V30(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) )
complex-valued ext-real-valued real-valued )
Real_Sequence) is
convergent &
seq : ( (
V21()
V30(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) , the
carrier of
b2 : ( ( non
empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non
empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like )
RealUnitarySpace) : ( ( ) (
V1() )
set ) ) ) (
V1()
V16()
V19(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) )
V20( the
carrier of
b2 : ( ( non
empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non
empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like )
RealUnitarySpace) : ( ( ) (
V1() )
set ) )
V21()
V26(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) )
V30(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) , the
carrier of
b2 : ( ( non
empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non
empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like )
RealUnitarySpace) : ( ( ) (
V1() )
set ) ) )
sequence of ( ( ) (
V1() )
set ) ) is
convergent holds
Rseq : ( (
V21()
V30(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) ) (
V1()
V16()
V19(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) )
V20(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) )
V21()
V26(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) )
V30(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) )
complex-valued ext-real-valued real-valued )
Real_Sequence)
* seq : ( (
V21()
V30(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) , the
carrier of
b2 : ( ( non
empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non
empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like )
RealUnitarySpace) : ( ( ) (
V1() )
set ) ) ) (
V1()
V16()
V19(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) )
V20( the
carrier of
b2 : ( ( non
empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non
empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like )
RealUnitarySpace) : ( ( ) (
V1() )
set ) )
V21()
V26(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) )
V30(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) , the
carrier of
b2 : ( ( non
empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non
empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like )
RealUnitarySpace) : ( ( ) (
V1() )
set ) ) )
sequence of ( ( ) (
V1() )
set ) ) : ( (
V21()
V30(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) , the
carrier of
b2 : ( ( non
empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non
empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like )
RealUnitarySpace) : ( ( ) (
V1() )
set ) ) ) (
V1()
V16()
V19(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) )
V20( the
carrier of
b2 : ( ( non
empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non
empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like )
RealUnitarySpace) : ( ( ) (
V1() )
set ) )
V21()
V26(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) )
V30(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) , the
carrier of
b2 : ( ( non
empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non
empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like )
RealUnitarySpace) : ( ( ) (
V1() )
set ) ) )
sequence of ( ( ) (
V1() )
set ) ) is
convergent ;
theorem
for
Rseq being ( (
V21()
V30(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) ) (
V1()
V16()
V19(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) )
V20(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) )
V21()
V26(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) )
V30(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) )
complex-valued ext-real-valued real-valued )
Real_Sequence)
for
X being ( ( non
empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non
empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like )
RealUnitarySpace)
for
seq being ( (
V21()
V30(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) , the
carrier of
b2 : ( ( non
empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non
empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like )
RealUnitarySpace) : ( ( ) (
V1() )
set ) ) ) (
V1()
V16()
V19(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) )
V20( the
carrier of
b2 : ( ( non
empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non
empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like )
RealUnitarySpace) : ( ( ) (
V1() )
set ) )
V21()
V26(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) )
V30(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) , the
carrier of
b2 : ( ( non
empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non
empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like )
RealUnitarySpace) : ( ( ) (
V1() )
set ) ) )
sequence of ( ( ) (
V1() )
set ) ) st
Rseq : ( (
V21()
V30(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) ) (
V1()
V16()
V19(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) )
V20(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) )
V21()
V26(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) )
V30(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) )
complex-valued ext-real-valued real-valued )
Real_Sequence) is
bounded &
seq : ( (
V21()
V30(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) , the
carrier of
b2 : ( ( non
empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non
empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like )
RealUnitarySpace) : ( ( ) (
V1() )
set ) ) ) (
V1()
V16()
V19(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) )
V20( the
carrier of
b2 : ( ( non
empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non
empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like )
RealUnitarySpace) : ( ( ) (
V1() )
set ) )
V21()
V26(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) )
V30(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) , the
carrier of
b2 : ( ( non
empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non
empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like )
RealUnitarySpace) : ( ( ) (
V1() )
set ) ) )
sequence of ( ( ) (
V1() )
set ) ) is
bounded holds
Rseq : ( (
V21()
V30(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) ) (
V1()
V16()
V19(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) )
V20(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) )
V21()
V26(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) )
V30(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) )
complex-valued ext-real-valued real-valued )
Real_Sequence)
* seq : ( (
V21()
V30(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) , the
carrier of
b2 : ( ( non
empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non
empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like )
RealUnitarySpace) : ( ( ) (
V1() )
set ) ) ) (
V1()
V16()
V19(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) )
V20( the
carrier of
b2 : ( ( non
empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non
empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like )
RealUnitarySpace) : ( ( ) (
V1() )
set ) )
V21()
V26(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) )
V30(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) , the
carrier of
b2 : ( ( non
empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non
empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like )
RealUnitarySpace) : ( ( ) (
V1() )
set ) ) )
sequence of ( ( ) (
V1() )
set ) ) : ( (
V21()
V30(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) , the
carrier of
b2 : ( ( non
empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non
empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like )
RealUnitarySpace) : ( ( ) (
V1() )
set ) ) ) (
V1()
V16()
V19(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) )
V20( the
carrier of
b2 : ( ( non
empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non
empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like )
RealUnitarySpace) : ( ( ) (
V1() )
set ) )
V21()
V26(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) )
V30(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) , the
carrier of
b2 : ( ( non
empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non
empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like )
RealUnitarySpace) : ( ( ) (
V1() )
set ) ) )
sequence of ( ( ) (
V1() )
set ) ) is
bounded ;
theorem
for
Rseq being ( (
V21()
V30(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) ) (
V1()
V16()
V19(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) )
V20(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) )
V21()
V26(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) )
V30(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) )
complex-valued ext-real-valued real-valued )
Real_Sequence)
for
X being ( ( non
empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non
empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like )
RealUnitarySpace)
for
seq being ( (
V21()
V30(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) , the
carrier of
b2 : ( ( non
empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non
empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like )
RealUnitarySpace) : ( ( ) (
V1() )
set ) ) ) (
V1()
V16()
V19(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) )
V20( the
carrier of
b2 : ( ( non
empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non
empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like )
RealUnitarySpace) : ( ( ) (
V1() )
set ) )
V21()
V26(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) )
V30(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) , the
carrier of
b2 : ( ( non
empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non
empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like )
RealUnitarySpace) : ( ( ) (
V1() )
set ) ) )
sequence of ( ( ) (
V1() )
set ) ) st
Rseq : ( (
V21()
V30(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) ) (
V1()
V16()
V19(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) )
V20(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) )
V21()
V26(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) )
V30(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) )
complex-valued ext-real-valued real-valued )
Real_Sequence) is
convergent &
seq : ( (
V21()
V30(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) , the
carrier of
b2 : ( ( non
empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non
empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like )
RealUnitarySpace) : ( ( ) (
V1() )
set ) ) ) (
V1()
V16()
V19(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) )
V20( the
carrier of
b2 : ( ( non
empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non
empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like )
RealUnitarySpace) : ( ( ) (
V1() )
set ) )
V21()
V26(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) )
V30(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) , the
carrier of
b2 : ( ( non
empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non
empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like )
RealUnitarySpace) : ( ( ) (
V1() )
set ) ) )
sequence of ( ( ) (
V1() )
set ) ) is
convergent holds
(
Rseq : ( (
V21()
V30(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) ) (
V1()
V16()
V19(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) )
V20(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) )
V21()
V26(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) )
V30(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) )
complex-valued ext-real-valued real-valued )
Real_Sequence)
* seq : ( (
V21()
V30(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) , the
carrier of
b2 : ( ( non
empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non
empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like )
RealUnitarySpace) : ( ( ) (
V1() )
set ) ) ) (
V1()
V16()
V19(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) )
V20( the
carrier of
b2 : ( ( non
empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non
empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like )
RealUnitarySpace) : ( ( ) (
V1() )
set ) )
V21()
V26(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) )
V30(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) , the
carrier of
b2 : ( ( non
empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non
empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like )
RealUnitarySpace) : ( ( ) (
V1() )
set ) ) )
sequence of ( ( ) (
V1() )
set ) ) : ( (
V21()
V30(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) , the
carrier of
b2 : ( ( non
empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non
empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like )
RealUnitarySpace) : ( ( ) (
V1() )
set ) ) ) (
V1()
V16()
V19(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) )
V20( the
carrier of
b2 : ( ( non
empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non
empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like )
RealUnitarySpace) : ( ( ) (
V1() )
set ) )
V21()
V26(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) )
V30(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) , the
carrier of
b2 : ( ( non
empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non
empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like )
RealUnitarySpace) : ( ( ) (
V1() )
set ) ) )
sequence of ( ( ) (
V1() )
set ) ) is
convergent &
lim (Rseq : ( ( V21() V30( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) ) ( V1() V16() V19( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) V21() V26( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) * seq : ( ( V21() V30( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V1() ) set ) ) ) ( V1() V16() V19( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) ) V20( the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V1() ) set ) ) V21() V26( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V1() ) set ) ) ) sequence of ( ( ) ( V1() ) set ) ) ) : ( (
V21()
V30(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) , the
carrier of
b2 : ( ( non
empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non
empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like )
RealUnitarySpace) : ( ( ) (
V1() )
set ) ) ) (
V1()
V16()
V19(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) )
V20( the
carrier of
b2 : ( ( non
empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non
empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like )
RealUnitarySpace) : ( ( ) (
V1() )
set ) )
V21()
V26(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) )
V30(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) , the
carrier of
b2 : ( ( non
empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non
empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like )
RealUnitarySpace) : ( ( ) (
V1() )
set ) ) )
sequence of ( ( ) (
V1() )
set ) ) : ( ( ) ( )
Element of the
carrier of
b2 : ( ( non
empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non
empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like )
RealUnitarySpace) : ( ( ) (
V1() )
set ) )
= (lim Rseq : ( ( V21() V30( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) ) ( V1() V16() V19( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) V21() V26( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) ) : ( ( ) (
V11()
real ext-real )
Element of
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) )
* (lim seq : ( ( V21() V30( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V1() ) set ) ) ) ( V1() V16() V19( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) ) V20( the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V1() ) set ) ) V21() V26( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V1() ) set ) ) ) sequence of ( ( ) ( V1() ) set ) ) ) : ( ( ) ( )
Element of the
carrier of
b2 : ( ( non
empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non
empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like )
RealUnitarySpace) : ( ( ) (
V1() )
set ) ) : ( ( ) ( )
Element of the
carrier of
b2 : ( ( non
empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non
empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like )
RealUnitarySpace) : ( ( ) (
V1() )
set ) ) ) ;
theorem
for
Rseq being ( (
V21()
V30(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) ) (
V1()
V16()
V19(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) )
V20(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) )
V21()
V26(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) )
V30(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) )
complex-valued ext-real-valued real-valued )
Real_Sequence)
for
X being ( ( non
empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like V144() ) ( non
empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like V144() )
RealHilbertSpace)
for
seq being ( (
V21()
V30(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) , the
carrier of
b2 : ( ( non
empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like V144() ) ( non
empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like V144() )
RealHilbertSpace) : ( ( ) (
V1() )
set ) ) ) (
V1()
V16()
V19(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) )
V20( the
carrier of
b2 : ( ( non
empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like V144() ) ( non
empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like V144() )
RealHilbertSpace) : ( ( ) (
V1() )
set ) )
V21()
V26(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) )
V30(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) , the
carrier of
b2 : ( ( non
empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like V144() ) ( non
empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like V144() )
RealHilbertSpace) : ( ( ) (
V1() )
set ) ) )
sequence of ( ( ) (
V1() )
set ) ) st
seq : ( (
V21()
V30(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) , the
carrier of
b2 : ( ( non
empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like V144() ) ( non
empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like V144() )
RealHilbertSpace) : ( ( ) (
V1() )
set ) ) ) (
V1()
V16()
V19(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) )
V20( the
carrier of
b2 : ( ( non
empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like V144() ) ( non
empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like V144() )
RealHilbertSpace) : ( ( ) (
V1() )
set ) )
V21()
V26(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) )
V30(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) , the
carrier of
b2 : ( ( non
empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like V144() ) ( non
empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like V144() )
RealHilbertSpace) : ( ( ) (
V1() )
set ) ) )
sequence of ( ( ) (
V1() )
set ) ) is
Cauchy &
Rseq : ( (
V21()
V30(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) ) (
V1()
V16()
V19(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) )
V20(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) )
V21()
V26(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) )
V30(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) )
complex-valued ext-real-valued real-valued )
Real_Sequence) is
Cauchy holds
Rseq : ( (
V21()
V30(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) ) (
V1()
V16()
V19(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) )
V20(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) )
V21()
V26(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) )
V30(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) )
complex-valued ext-real-valued real-valued )
Real_Sequence)
* seq : ( (
V21()
V30(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) , the
carrier of
b2 : ( ( non
empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like V144() ) ( non
empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like V144() )
RealHilbertSpace) : ( ( ) (
V1() )
set ) ) ) (
V1()
V16()
V19(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) )
V20( the
carrier of
b2 : ( ( non
empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like V144() ) ( non
empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like V144() )
RealHilbertSpace) : ( ( ) (
V1() )
set ) )
V21()
V26(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) )
V30(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) , the
carrier of
b2 : ( ( non
empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like V144() ) ( non
empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like V144() )
RealHilbertSpace) : ( ( ) (
V1() )
set ) ) )
sequence of ( ( ) (
V1() )
set ) ) : ( (
V21()
V30(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) , the
carrier of
b2 : ( ( non
empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like V144() ) ( non
empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like V144() )
RealHilbertSpace) : ( ( ) (
V1() )
set ) ) ) (
V1()
V16()
V19(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) )
V20( the
carrier of
b2 : ( ( non
empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like V144() ) ( non
empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like V144() )
RealHilbertSpace) : ( ( ) (
V1() )
set ) )
V21()
V26(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) )
V30(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) , the
carrier of
b2 : ( ( non
empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like V144() ) ( non
empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like V144() )
RealHilbertSpace) : ( ( ) (
V1() )
set ) ) )
sequence of ( ( ) (
V1() )
set ) ) is
Cauchy ;
theorem
for
Rseq being ( (
V21()
V30(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) ) (
V1()
V16()
V19(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) )
V20(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) )
V21()
V26(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) )
V30(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) )
complex-valued ext-real-valued real-valued )
Real_Sequence)
for
X being ( ( non
empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non
empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like )
RealUnitarySpace)
for
seq being ( (
V21()
V30(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) , the
carrier of
b2 : ( ( non
empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non
empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like )
RealUnitarySpace) : ( ( ) (
V1() )
set ) ) ) (
V1()
V16()
V19(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) )
V20( the
carrier of
b2 : ( ( non
empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non
empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like )
RealUnitarySpace) : ( ( ) (
V1() )
set ) )
V21()
V26(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) )
V30(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) , the
carrier of
b2 : ( ( non
empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non
empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like )
RealUnitarySpace) : ( ( ) (
V1() )
set ) ) )
sequence of ( ( ) (
V1() )
set ) )
for
n being ( ( ) (
epsilon-transitive epsilon-connected ordinal natural V11()
real ext-real integer V58()
V59()
V60()
V61()
V62()
V63()
V64() )
Element of
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) ) holds
(Partial_Sums ((Rseq : ( ( V21() V30( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) ) ( V1() V16() V19( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) V21() V26( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) - (Rseq : ( ( V21() V30( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) ) ( V1() V16() V19( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) V21() V26( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) ^\ 1 : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative integer V58() V59() V60() V61() V62() V63() V64() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V1() V16() V19( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) V21() V26( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) complex-valued ext-real-valued real-valued ) subsequence of b1 : ( ( V21() V30( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) ) ( V1() V16() V19( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) V21() V26( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) ) ) : ( ( V21() ) ( V1() V16() V19( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) V21() V26( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) complex-valued ext-real-valued real-valued ) Element of K6(K7(NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) ,REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) * (Partial_Sums seq : ( ( V21() V30( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V1() ) set ) ) ) ( V1() V16() V19( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) ) V20( the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V1() ) set ) ) V21() V26( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V1() ) set ) ) ) sequence of ( ( ) ( V1() ) set ) ) ) : ( ( V21() V30( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V1() ) set ) ) ) ( V1() V16() V19( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) ) V20( the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V1() ) set ) ) V21() V26( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V1() ) set ) ) ) sequence of ( ( ) ( V1() ) set ) ) ) : ( ( V21() V30( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V1() ) set ) ) ) ( V1() V16() V19( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) ) V20( the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V1() ) set ) ) V21() V26( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V1() ) set ) ) ) sequence of ( ( ) ( V1() ) set ) ) ) : ( (
V21()
V30(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) , the
carrier of
b2 : ( ( non
empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non
empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like )
RealUnitarySpace) : ( ( ) (
V1() )
set ) ) ) (
V1()
V16()
V19(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) )
V20( the
carrier of
b2 : ( ( non
empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non
empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like )
RealUnitarySpace) : ( ( ) (
V1() )
set ) )
V21()
V26(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) )
V30(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) , the
carrier of
b2 : ( ( non
empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non
empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like )
RealUnitarySpace) : ( ( ) (
V1() )
set ) ) )
sequence of ( ( ) (
V1() )
set ) )
. n : ( ( ) (
epsilon-transitive epsilon-connected ordinal natural V11()
real ext-real integer V58()
V59()
V60()
V61()
V62()
V63()
V64() )
Element of
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( )
Element of the
carrier of
b2 : ( ( non
empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non
empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like )
RealUnitarySpace) : ( ( ) (
V1() )
set ) )
= ((Partial_Sums (Rseq : ( ( V21() V30( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) ) ( V1() V16() V19( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) V21() V26( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) * seq : ( ( V21() V30( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V1() ) set ) ) ) ( V1() V16() V19( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) ) V20( the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V1() ) set ) ) V21() V26( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V1() ) set ) ) ) sequence of ( ( ) ( V1() ) set ) ) ) : ( ( V21() V30( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V1() ) set ) ) ) ( V1() V16() V19( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) ) V20( the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V1() ) set ) ) V21() V26( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V1() ) set ) ) ) sequence of ( ( ) ( V1() ) set ) ) ) : ( ( V21() V30( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V1() ) set ) ) ) ( V1() V16() V19( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) ) V20( the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V1() ) set ) ) V21() V26( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V1() ) set ) ) ) sequence of ( ( ) ( V1() ) set ) ) . (n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real integer V58() V59() V60() V61() V62() V63() V64() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) ) + 1 : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative integer V58() V59() V60() V61() V62() V63() V64() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real integer V58() V59() V60() V61() V62() V63() V64() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( )
Element of the
carrier of
b2 : ( ( non
empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non
empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like )
RealUnitarySpace) : ( ( ) (
V1() )
set ) )
- ((Rseq : ( ( V21() V30( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) ) ( V1() V16() V19( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) V21() V26( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) * (Partial_Sums seq : ( ( V21() V30( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V1() ) set ) ) ) ( V1() V16() V19( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) ) V20( the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V1() ) set ) ) V21() V26( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V1() ) set ) ) ) sequence of ( ( ) ( V1() ) set ) ) ) : ( ( V21() V30( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V1() ) set ) ) ) ( V1() V16() V19( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) ) V20( the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V1() ) set ) ) V21() V26( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V1() ) set ) ) ) sequence of ( ( ) ( V1() ) set ) ) ) : ( ( V21() V30( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V1() ) set ) ) ) ( V1() V16() V19( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) ) V20( the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V1() ) set ) ) V21() V26( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V1() ) set ) ) ) sequence of ( ( ) ( V1() ) set ) ) . (n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real integer V58() V59() V60() V61() V62() V63() V64() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) ) + 1 : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative integer V58() V59() V60() V61() V62() V63() V64() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real integer V58() V59() V60() V61() V62() V63() V64() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( )
Element of the
carrier of
b2 : ( ( non
empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non
empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like )
RealUnitarySpace) : ( ( ) (
V1() )
set ) ) : ( ( ) ( )
Element of the
carrier of
b2 : ( ( non
empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non
empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like )
RealUnitarySpace) : ( ( ) (
V1() )
set ) ) ;
theorem
for
Rseq being ( (
V21()
V30(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) ) (
V1()
V16()
V19(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) )
V20(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) )
V21()
V26(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) )
V30(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) )
complex-valued ext-real-valued real-valued )
Real_Sequence)
for
X being ( ( non
empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non
empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like )
RealUnitarySpace)
for
seq being ( (
V21()
V30(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) , the
carrier of
b2 : ( ( non
empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non
empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like )
RealUnitarySpace) : ( ( ) (
V1() )
set ) ) ) (
V1()
V16()
V19(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) )
V20( the
carrier of
b2 : ( ( non
empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non
empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like )
RealUnitarySpace) : ( ( ) (
V1() )
set ) )
V21()
V26(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) )
V30(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) , the
carrier of
b2 : ( ( non
empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non
empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like )
RealUnitarySpace) : ( ( ) (
V1() )
set ) ) )
sequence of ( ( ) (
V1() )
set ) )
for
n being ( ( ) (
epsilon-transitive epsilon-connected ordinal natural V11()
real ext-real integer V58()
V59()
V60()
V61()
V62()
V63()
V64() )
Element of
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) ) holds
(Partial_Sums (Rseq : ( ( V21() V30( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) ) ( V1() V16() V19( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) V21() V26( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) * seq : ( ( V21() V30( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V1() ) set ) ) ) ( V1() V16() V19( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) ) V20( the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V1() ) set ) ) V21() V26( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V1() ) set ) ) ) sequence of ( ( ) ( V1() ) set ) ) ) : ( ( V21() V30( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V1() ) set ) ) ) ( V1() V16() V19( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) ) V20( the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V1() ) set ) ) V21() V26( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V1() ) set ) ) ) sequence of ( ( ) ( V1() ) set ) ) ) : ( (
V21()
V30(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) , the
carrier of
b2 : ( ( non
empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non
empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like )
RealUnitarySpace) : ( ( ) (
V1() )
set ) ) ) (
V1()
V16()
V19(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) )
V20( the
carrier of
b2 : ( ( non
empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non
empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like )
RealUnitarySpace) : ( ( ) (
V1() )
set ) )
V21()
V26(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) )
V30(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) , the
carrier of
b2 : ( ( non
empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non
empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like )
RealUnitarySpace) : ( ( ) (
V1() )
set ) ) )
sequence of ( ( ) (
V1() )
set ) )
. (n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real integer V58() V59() V60() V61() V62() V63() V64() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) ) + 1 : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative integer V58() V59() V60() V61() V62() V63() V64() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) (
epsilon-transitive epsilon-connected ordinal natural V11()
real ext-real integer V58()
V59()
V60()
V61()
V62()
V63()
V64() )
Element of
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( )
Element of the
carrier of
b2 : ( ( non
empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non
empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like )
RealUnitarySpace) : ( ( ) (
V1() )
set ) )
= ((Rseq : ( ( V21() V30( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) ) ( V1() V16() V19( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) V21() V26( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) * (Partial_Sums seq : ( ( V21() V30( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V1() ) set ) ) ) ( V1() V16() V19( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) ) V20( the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V1() ) set ) ) V21() V26( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V1() ) set ) ) ) sequence of ( ( ) ( V1() ) set ) ) ) : ( ( V21() V30( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V1() ) set ) ) ) ( V1() V16() V19( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) ) V20( the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V1() ) set ) ) V21() V26( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V1() ) set ) ) ) sequence of ( ( ) ( V1() ) set ) ) ) : ( ( V21() V30( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V1() ) set ) ) ) ( V1() V16() V19( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) ) V20( the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V1() ) set ) ) V21() V26( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V1() ) set ) ) ) sequence of ( ( ) ( V1() ) set ) ) . (n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real integer V58() V59() V60() V61() V62() V63() V64() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) ) + 1 : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative integer V58() V59() V60() V61() V62() V63() V64() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real integer V58() V59() V60() V61() V62() V63() V64() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( )
Element of the
carrier of
b2 : ( ( non
empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non
empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like )
RealUnitarySpace) : ( ( ) (
V1() )
set ) )
- ((Partial_Sums (((Rseq : ( ( V21() V30( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) ) ( V1() V16() V19( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) V21() V26( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) ^\ 1 : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative integer V58() V59() V60() V61() V62() V63() V64() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V1() V16() V19( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) V21() V26( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) complex-valued ext-real-valued real-valued ) subsequence of b1 : ( ( V21() V30( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) ) ( V1() V16() V19( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) V21() V26( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) ) - Rseq : ( ( V21() V30( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) ) ( V1() V16() V19( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) V21() V26( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) ) : ( ( V21() ) ( V1() V16() V19( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) V21() V26( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) complex-valued ext-real-valued real-valued ) Element of K6(K7(NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) ,REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) * (Partial_Sums seq : ( ( V21() V30( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V1() ) set ) ) ) ( V1() V16() V19( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) ) V20( the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V1() ) set ) ) V21() V26( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V1() ) set ) ) ) sequence of ( ( ) ( V1() ) set ) ) ) : ( ( V21() V30( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V1() ) set ) ) ) ( V1() V16() V19( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) ) V20( the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V1() ) set ) ) V21() V26( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V1() ) set ) ) ) sequence of ( ( ) ( V1() ) set ) ) ) : ( ( V21() V30( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V1() ) set ) ) ) ( V1() V16() V19( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) ) V20( the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V1() ) set ) ) V21() V26( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V1() ) set ) ) ) sequence of ( ( ) ( V1() ) set ) ) ) : ( ( V21() V30( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V1() ) set ) ) ) ( V1() V16() V19( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) ) V20( the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V1() ) set ) ) V21() V26( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V1() ) set ) ) ) sequence of ( ( ) ( V1() ) set ) ) . n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real integer V58() V59() V60() V61() V62() V63() V64() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( )
Element of the
carrier of
b2 : ( ( non
empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non
empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like )
RealUnitarySpace) : ( ( ) (
V1() )
set ) ) : ( ( ) ( )
Element of the
carrier of
b2 : ( ( non
empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non
empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like )
RealUnitarySpace) : ( ( ) (
V1() )
set ) ) ;
theorem
for
Rseq being ( (
V21()
V30(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) ) (
V1()
V16()
V19(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) )
V20(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) )
V21()
V26(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) )
V30(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) )
complex-valued ext-real-valued real-valued )
Real_Sequence)
for
X being ( ( non
empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non
empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like )
RealUnitarySpace)
for
seq being ( (
V21()
V30(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) , the
carrier of
b2 : ( ( non
empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non
empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like )
RealUnitarySpace) : ( ( ) (
V1() )
set ) ) ) (
V1()
V16()
V19(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) )
V20( the
carrier of
b2 : ( ( non
empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non
empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like )
RealUnitarySpace) : ( ( ) (
V1() )
set ) )
V21()
V26(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) )
V30(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) , the
carrier of
b2 : ( ( non
empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non
empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like )
RealUnitarySpace) : ( ( ) (
V1() )
set ) ) )
sequence of ( ( ) (
V1() )
set ) )
for
n being ( ( ) (
epsilon-transitive epsilon-connected ordinal natural V11()
real ext-real integer V58()
V59()
V60()
V61()
V62()
V63()
V64() )
Element of
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) ) holds
Sum (
(Rseq : ( ( V21() V30( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) ) ( V1() V16() V19( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) V21() V26( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) * seq : ( ( V21() V30( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V1() ) set ) ) ) ( V1() V16() V19( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) ) V20( the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V1() ) set ) ) V21() V26( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V1() ) set ) ) ) sequence of ( ( ) ( V1() ) set ) ) ) : ( (
V21()
V30(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) , the
carrier of
b2 : ( ( non
empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non
empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like )
RealUnitarySpace) : ( ( ) (
V1() )
set ) ) ) (
V1()
V16()
V19(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) )
V20( the
carrier of
b2 : ( ( non
empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non
empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like )
RealUnitarySpace) : ( ( ) (
V1() )
set ) )
V21()
V26(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) )
V30(
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) , the
carrier of
b2 : ( ( non
empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non
empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like )
RealUnitarySpace) : ( ( ) (
V1() )
set ) ) )
sequence of ( ( ) (
V1() )
set ) ) ,
(n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real integer V58() V59() V60() V61() V62() V63() V64() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) ) + 1 : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative integer V58() V59() V60() V61() V62() V63() V64() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) (
epsilon-transitive epsilon-connected ordinal natural V11()
real ext-real integer V58()
V59()
V60()
V61()
V62()
V63()
V64() )
Element of
NAT : ( ( ) (
V1()
epsilon-transitive epsilon-connected ordinal V59()
V60()
V61()
V62()
V63()
V64()
V65() )
Element of
K6(
REAL : ( ( ) (
V1()
V47()
V59()
V60()
V61()
V65() )
set ) ) : ( ( ) ( )
set ) ) ) ) : ( ( ) ( )
Point of ( ( ) (
V1() )
set ) )
= ((Rseq : ( ( V21() V30( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) ) ( V1() V16() V19( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) V21() V26( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) * (Partial_Sums seq : ( ( V21() V30( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V1() ) set ) ) ) ( V1() V16() V19( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) ) V20( the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V1() ) set ) ) V21() V26( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V1() ) set ) ) ) sequence of ( ( ) ( V1() ) set ) ) ) : ( ( V21() V30( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V1() ) set ) ) ) ( V1() V16() V19( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) ) V20( the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V1() ) set ) ) V21() V26( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V1() ) set ) ) ) sequence of ( ( ) ( V1() ) set ) ) ) : ( ( V21() V30( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V1() ) set ) ) ) ( V1() V16() V19( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) ) V20( the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V1() ) set ) ) V21() V26( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V1() ) set ) ) ) sequence of ( ( ) ( V1() ) set ) ) . (n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real integer V58() V59() V60() V61() V62() V63() V64() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) ) + 1 : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative integer V58() V59() V60() V61() V62() V63() V64() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real integer V58() V59() V60() V61() V62() V63() V64() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( )
Element of the
carrier of
b2 : ( ( non
empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non
empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like )
RealUnitarySpace) : ( ( ) (
V1() )
set ) )
- (Sum ((((Rseq : ( ( V21() V30( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) ) ( V1() V16() V19( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) V21() V26( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) ^\ 1 : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative integer V58() V59() V60() V61() V62() V63() V64() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V1() V16() V19( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) V21() V26( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) complex-valued ext-real-valued real-valued ) subsequence of b1 : ( ( V21() V30( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) ) ( V1() V16() V19( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) V21() V26( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) ) - Rseq : ( ( V21() V30( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) ) ( V1() V16() V19( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) V21() V26( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) ) : ( ( V21() ) ( V1() V16() V19( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) V21() V26( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) complex-valued ext-real-valued real-valued ) Element of K6(K7(NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) ,REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) * (Partial_Sums seq : ( ( V21() V30( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V1() ) set ) ) ) ( V1() V16() V19( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) ) V20( the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V1() ) set ) ) V21() V26( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V1() ) set ) ) ) sequence of ( ( ) ( V1() ) set ) ) ) : ( ( V21() V30( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V1() ) set ) ) ) ( V1() V16() V19( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) ) V20( the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V1() ) set ) ) V21() V26( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V1() ) set ) ) ) sequence of ( ( ) ( V1() ) set ) ) ) : ( ( V21() V30( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V1() ) set ) ) ) ( V1() V16() V19( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) ) V20( the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V1() ) set ) ) V21() V26( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V1() ) set ) ) ) sequence of ( ( ) ( V1() ) set ) ) ,n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real integer V58() V59() V60() V61() V62() V63() V64() ) Element of NAT : ( ( ) ( V1() epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of K6(REAL : ( ( ) ( V1() V47() V59() V60() V61() V65() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( ) ( )
Point of ( ( ) (
V1() )
set ) ) : ( ( ) ( )
Element of the
carrier of
b2 : ( ( non
empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RealUnitarySpace-like ) ( non
empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealUnitarySpace-like )
RealUnitarySpace) : ( ( ) (
V1() )
set ) ) ;