:: LOPBAN_3 semantic presentation

REAL is V4() V47() V52() V53() V54() V58() set
NAT is V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() Element of K32(REAL)
K32(REAL) is V4() set
COMPLEX is V4() V47() V52() V58() set
RAT is V4() V47() V52() V53() V54() V55() V58() set
INT is V4() V47() V52() V53() V54() V55() V56() V58() set
K33(COMPLEX,COMPLEX) is V4() complex-valued set
K32(K33(COMPLEX,COMPLEX)) is V4() set
K33(K33(COMPLEX,COMPLEX),COMPLEX) is V4() complex-valued set
K32(K33(K33(COMPLEX,COMPLEX),COMPLEX)) is V4() set
K33(REAL,REAL) is V4() complex-valued ext-real-valued real-valued set
K32(K33(REAL,REAL)) is V4() set
K33(K33(REAL,REAL),REAL) is V4() complex-valued ext-real-valued real-valued set
K32(K33(K33(REAL,REAL),REAL)) is V4() set
K33(RAT,RAT) is V4() RAT -valued complex-valued ext-real-valued real-valued set
K32(K33(RAT,RAT)) is V4() set
K33(K33(RAT,RAT),RAT) is V4() RAT -valued complex-valued ext-real-valued real-valued set
K32(K33(K33(RAT,RAT),RAT)) is V4() set
K33(INT,INT) is V4() RAT -valued INT -valued complex-valued ext-real-valued real-valued set
K32(K33(INT,INT)) is V4() set
K33(K33(INT,INT),INT) is V4() RAT -valued INT -valued complex-valued ext-real-valued real-valued set
K32(K33(K33(INT,INT),INT)) is V4() set
K33(NAT,NAT) is V4() RAT -valued INT -valued complex-valued ext-real-valued real-valued natural-valued set
K33(K33(NAT,NAT),NAT) is V4() RAT -valued INT -valued complex-valued ext-real-valued real-valued natural-valued set
K32(K33(K33(NAT,NAT),NAT)) is V4() set
K33(NAT,REAL) is V4() complex-valued ext-real-valued real-valued set
K32(K33(NAT,REAL)) is V4() set
omega is V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() set
K32(omega) is V4() set
K32(NAT) is V4() set
K390() is V4() set
K33(K390(),K390()) is V4() set
K33(K33(K390(),K390()),K390()) is V4() set
K32(K33(K33(K390(),K390()),K390())) is V4() set
K33(REAL,K390()) is V4() set
K33(K33(REAL,K390()),K390()) is V4() set
K32(K33(K33(REAL,K390()),K390())) is V4() set
K396() is RLSStruct
the carrier of K396() is set
K32( the carrier of K396()) is V4() set
K400() is Element of K32( the carrier of K396())
K33(K400(),K400()) is set
K33(K33(K400(),K400()),REAL) is complex-valued ext-real-valued real-valued set
K32(K33(K33(K400(),K400()),REAL)) is V4() set
the_set_of_l1RealSequences is Element of K32( the carrier of K396())
K33(the_set_of_l1RealSequences,REAL) is complex-valued ext-real-valued real-valued set
K32(K33(the_set_of_l1RealSequences,REAL)) is V4() set
0c is V4() epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural V24() real ext-real non positive non negative V52() V53() V54() V55() V56() V57() V58() set
1 is V4() epsilon-transitive epsilon-connected ordinal natural V24() real ext-real positive non negative V45() V46() V52() V53() V54() V55() V56() V57() Element of NAT
0 is V4() epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural V24() real ext-real non positive non negative V45() V46() V52() V53() V54() V55() V56() V57() V58() Element of NAT
- 1 is V24() real ext-real non positive Element of REAL
K32(K33(NAT,NAT)) is V4() set
2 is V4() epsilon-transitive epsilon-connected ordinal natural V24() real ext-real positive non negative V45() V46() V52() V53() V54() V55() V56() V57() Element of NAT
X is non empty right_complementable add-associative right_zeroed NORMSTR
the carrier of X is V4() set
K33(NAT, the carrier of X) is V4() set
K32(K33(NAT, the carrier of X)) is V4() set
0. X is V79(X) right_complementable Element of the carrier of X
x is V4() Relation-like NAT -defined the carrier of X -valued Function-like V29( NAT ) V30( NAT , the carrier of X) Element of K32(K33(NAT, the carrier of X))
Partial_Sums x is V4() Relation-like NAT -defined the carrier of X -valued Function-like V29( NAT ) V30( NAT , the carrier of X) Element of K32(K33(NAT, the carrier of X))
seq is epsilon-transitive epsilon-connected ordinal natural V24() real ext-real V45() V46() V52() V53() V54() V55() V56() V57() Element of NAT
(Partial_Sums x) . seq is right_complementable Element of the carrier of X
y is epsilon-transitive epsilon-connected ordinal natural V24() real ext-real V45() V46() V52() V53() V54() V55() V56() V57() Element of NAT
x . y is right_complementable Element of the carrier of X
(Partial_Sums x) . y is right_complementable Element of the carrier of X
y + 1 is epsilon-transitive epsilon-connected ordinal natural V24() real ext-real V45() V46() V52() V53() V54() V55() V56() V57() Element of NAT
x . (y + 1) is right_complementable Element of the carrier of X
(Partial_Sums x) . (y + 1) is right_complementable Element of the carrier of X
(0. X) + (x . (y + 1)) is right_complementable Element of the carrier of X
the addF of X is V4() Relation-like K33( the carrier of X, the carrier of X) -defined the carrier of X -valued Function-like V29(K33( the carrier of X, the carrier of X)) V30(K33( the carrier of X, the carrier of X), the carrier of X) Element of K32(K33(K33( the carrier of X, the carrier of X), the carrier of X))
K33( the carrier of X, the carrier of X) is V4() set
K33(K33( the carrier of X, the carrier of X), the carrier of X) is V4() set
K32(K33(K33( the carrier of X, the carrier of X), the carrier of X)) is V4() set
K411( the carrier of X, the addF of X,(0. X),(x . (y + 1))) is right_complementable Element of the carrier of X
[(0. X),(x . (y + 1))] is set
the addF of X . [(0. X),(x . (y + 1))] is set
((Partial_Sums x) . y) + (x . (y + 1)) is right_complementable Element of the carrier of X
K411( the carrier of X, the addF of X,((Partial_Sums x) . y),(x . (y + 1))) is right_complementable Element of the carrier of X
[((Partial_Sums x) . y),(x . (y + 1))] is set
the addF of X . [((Partial_Sums x) . y),(x . (y + 1))] is set
x . 0 is right_complementable Element of the carrier of X
(Partial_Sums x) . 0 is right_complementable Element of the carrier of X
X is non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like NORMSTR
the carrier of X is V4() set
K33(NAT, the carrier of X) is V4() set
K32(K33(NAT, the carrier of X)) is V4() set
X is non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like NORMSTR
the carrier of X is V4() set
K33(NAT, the carrier of X) is V4() set
K32(K33(NAT, the carrier of X)) is V4() set
0. X is V79(X) right_complementable Element of the carrier of X
NAT --> (0. X) is V4() Relation-like NAT -defined the carrier of X -valued Function-like constant V29( NAT ) V30( NAT , the carrier of X) Element of K32(K33(NAT, the carrier of X))
x is V4() Relation-like NAT -defined the carrier of X -valued Function-like V29( NAT ) V30( NAT , the carrier of X) Element of K32(K33(NAT, the carrier of X))
Partial_Sums x is V4() Relation-like NAT -defined the carrier of X -valued Function-like V29( NAT ) V30( NAT , the carrier of X) Element of K32(K33(NAT, the carrier of X))
seq is V24() real ext-real Element of REAL
y is epsilon-transitive epsilon-connected ordinal natural V24() real ext-real V45() V46() V52() V53() V54() V55() V56() V57() Element of NAT
(Partial_Sums x) . y is right_complementable Element of the carrier of X
((Partial_Sums x) . y) - (0. X) is right_complementable Element of the carrier of X
- (0. X) is right_complementable Element of the carrier of X
((Partial_Sums x) . y) + (- (0. X)) is right_complementable Element of the carrier of X
the addF of X is V4() Relation-like K33( the carrier of X, the carrier of X) -defined the carrier of X -valued Function-like V29(K33( the carrier of X, the carrier of X)) V30(K33( the carrier of X, the carrier of X), the carrier of X) Element of K32(K33(K33( the carrier of X, the carrier of X), the carrier of X))
K33( the carrier of X, the carrier of X) is V4() set
K33(K33( the carrier of X, the carrier of X), the carrier of X) is V4() set
K32(K33(K33( the carrier of X, the carrier of X), the carrier of X)) is V4() set
K411( the carrier of X, the addF of X,((Partial_Sums x) . y),(- (0. X))) is right_complementable Element of the carrier of X
[((Partial_Sums x) . y),(- (0. X))] is set
the addF of X . [((Partial_Sums x) . y),(- (0. X))] is set
||.(((Partial_Sums x) . y) - (0. X)).|| is V24() real ext-real non negative Element of REAL
n is epsilon-transitive epsilon-connected ordinal natural V24() real ext-real V45() V46() V52() V53() V54() V55() V56() V57() Element of NAT
x . n is right_complementable Element of the carrier of X
(0. X) - (0. X) is right_complementable Element of the carrier of X
(0. X) + (- (0. X)) is right_complementable Element of the carrier of X
K411( the carrier of X, the addF of X,(0. X),(- (0. X))) is right_complementable Element of the carrier of X
[(0. X),(- (0. X))] is set
the addF of X . [(0. X),(- (0. X))] is set
||.((0. X) - (0. X)).|| is V24() real ext-real non negative Element of REAL
X is non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like NORMSTR
the carrier of X is V4() set
K33(NAT, the carrier of X) is V4() set
K32(K33(NAT, the carrier of X)) is V4() set
x is V4() Relation-like NAT -defined the carrier of X -valued Function-like V29( NAT ) V30( NAT , the carrier of X) Element of K32(K33(NAT, the carrier of X))
Partial_Sums x is V4() Relation-like NAT -defined the carrier of X -valued Function-like V29( NAT ) V30( NAT , the carrier of X) Element of K32(K33(NAT, the carrier of X))
lim (Partial_Sums x) is right_complementable Element of the carrier of X
X is non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like NORMSTR
the carrier of X is V4() set
K33(NAT, the carrier of X) is V4() set
K32(K33(NAT, the carrier of X)) is V4() set
X is non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like NORMSTR
the carrier of X is V4() set
K33(NAT, the carrier of X) is V4() set
K32(K33(NAT, the carrier of X)) is V4() set
x is V4() Relation-like NAT -defined the carrier of X -valued Function-like V29( NAT ) V30( NAT , the carrier of X) Element of K32(K33(NAT, the carrier of X))
||.x.|| is V4() Relation-like NAT -defined REAL -valued Function-like V29( NAT ) V30( NAT , REAL ) complex-valued ext-real-valued real-valued Element of K32(K33(NAT,REAL))
seq is epsilon-transitive epsilon-connected ordinal natural V24() real ext-real V45() V46() V52() V53() V54() V55() V56() V57() Element of NAT
||.x.|| . seq is V24() real ext-real Element of REAL
x . seq is right_complementable Element of the carrier of X
||.(x . seq).|| is V24() real ext-real non negative Element of REAL
X is non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like NORMSTR
the carrier of X is V4() set
x is right_complementable Element of the carrier of X
seq is right_complementable Element of the carrier of X
x - seq is right_complementable Element of the carrier of X
- seq is right_complementable Element of the carrier of X
x + (- seq) is right_complementable Element of the carrier of X
the addF of X is V4() Relation-like K33( the carrier of X, the carrier of X) -defined the carrier of X -valued Function-like V29(K33( the carrier of X, the carrier of X)) V30(K33( the carrier of X, the carrier of X), the carrier of X) Element of K32(K33(K33( the carrier of X, the carrier of X), the carrier of X))
K33( the carrier of X, the carrier of X) is V4() set
K33(K33( the carrier of X, the carrier of X), the carrier of X) is V4() set
K32(K33(K33( the carrier of X, the carrier of X), the carrier of X)) is V4() set
K411( the carrier of X, the addF of X,x,(- seq)) is right_complementable Element of the carrier of X
[x,(- seq)] is set
the addF of X . [x,(- seq)] is set
||.(x - seq).|| is V24() real ext-real non negative Element of REAL
y is right_complementable Element of the carrier of X
x - y is right_complementable Element of the carrier of X
- y is right_complementable Element of the carrier of X
x + (- y) is right_complementable Element of the carrier of X
K411( the carrier of X, the addF of X,x,(- y)) is right_complementable Element of the carrier of X
[x,(- y)] is set
the addF of X . [x,(- y)] is set
y - seq is right_complementable Element of the carrier of X
y + (- seq) is right_complementable Element of the carrier of X
K411( the carrier of X, the addF of X,y,(- seq)) is right_complementable Element of the carrier of X
[y,(- seq)] is set
the addF of X . [y,(- seq)] is set
(x - y) + (y - seq) is right_complementable Element of the carrier of X
K411( the carrier of X, the addF of X,(x - y),(y - seq)) is right_complementable Element of the carrier of X
[(x - y),(y - seq)] is set
the addF of X . [(x - y),(y - seq)] is set
||.((x - y) + (y - seq)).|| is V24() real ext-real non negative Element of REAL
0. X is V79(X) right_complementable Element of the carrier of X
x - (0. X) is right_complementable Element of the carrier of X
- (0. X) is right_complementable Element of the carrier of X
x + (- (0. X)) is right_complementable Element of the carrier of X
K411( the carrier of X, the addF of X,x,(- (0. X))) is right_complementable Element of the carrier of X
[x,(- (0. X))] is set
the addF of X . [x,(- (0. X))] is set
(x - (0. X)) - seq is right_complementable Element of the carrier of X
(x - (0. X)) + (- seq) is right_complementable Element of the carrier of X
K411( the carrier of X, the addF of X,(x - (0. X)),(- seq)) is right_complementable Element of the carrier of X
[(x - (0. X)),(- seq)] is set
the addF of X . [(x - (0. X)),(- seq)] is set
||.((x - (0. X)) - seq).|| is V24() real ext-real non negative Element of REAL
y - y is right_complementable Element of the carrier of X
y + (- y) is right_complementable Element of the carrier of X
K411( the carrier of X, the addF of X,y,(- y)) is right_complementable Element of the carrier of X
[y,(- y)] is set
the addF of X . [y,(- y)] is set
x - (y - y) is right_complementable Element of the carrier of X
- (y - y) is right_complementable Element of the carrier of X
x + (- (y - y)) is right_complementable Element of the carrier of X
K411( the carrier of X, the addF of X,x,(- (y - y))) is right_complementable Element of the carrier of X
[x,(- (y - y))] is set
the addF of X . [x,(- (y - y))] is set
(x - (y - y)) - seq is right_complementable Element of the carrier of X
(x - (y - y)) + (- seq) is right_complementable Element of the carrier of X
K411( the carrier of X, the addF of X,(x - (y - y)),(- seq)) is right_complementable Element of the carrier of X
[(x - (y - y)),(- seq)] is set
the addF of X . [(x - (y - y)),(- seq)] is set
||.((x - (y - y)) - seq).|| is V24() real ext-real non negative Element of REAL
(x - y) + y is right_complementable Element of the carrier of X
K411( the carrier of X, the addF of X,(x - y),y) is right_complementable Element of the carrier of X
[(x - y),y] is set
the addF of X . [(x - y),y] is set
((x - y) + y) - seq is right_complementable Element of the carrier of X
((x - y) + y) + (- seq) is right_complementable Element of the carrier of X
K411( the carrier of X, the addF of X,((x - y) + y),(- seq)) is right_complementable Element of the carrier of X
[((x - y) + y),(- seq)] is set
the addF of X . [((x - y) + y),(- seq)] is set
||.(((x - y) + y) - seq).|| is V24() real ext-real non negative Element of REAL
X is non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like NORMSTR
the carrier of X is V4() set
K33(NAT, the carrier of X) is V4() set
K32(K33(NAT, the carrier of X)) is V4() set
x is V4() Relation-like NAT -defined the carrier of X -valued Function-like V29( NAT ) V30( NAT , the carrier of X) Element of K32(K33(NAT, the carrier of X))
seq is right_complementable Element of the carrier of X
y is V24() real ext-real Element of REAL
y / 2 is V24() real ext-real Element of REAL
n is epsilon-transitive epsilon-connected ordinal natural V24() real ext-real V45() V46() V52() V53() V54() V55() V56() V57() Element of NAT
yn is epsilon-transitive epsilon-connected ordinal natural V24() real ext-real V45() V46() V52() V53() V54() V55() V56() V57() Element of NAT
x . yn is right_complementable Element of the carrier of X
(x . yn) - seq is right_complementable Element of the carrier of X
- seq is right_complementable Element of the carrier of X
(x . yn) + (- seq) is right_complementable Element of the carrier of X
the addF of X is V4() Relation-like K33( the carrier of X, the carrier of X) -defined the carrier of X -valued Function-like V29(K33( the carrier of X, the carrier of X)) V30(K33( the carrier of X, the carrier of X), the carrier of X) Element of K32(K33(K33( the carrier of X, the carrier of X), the carrier of X))
K33( the carrier of X, the carrier of X) is V4() set
K33(K33( the carrier of X, the carrier of X), the carrier of X) is V4() set
K32(K33(K33( the carrier of X, the carrier of X), the carrier of X)) is V4() set
K411( the carrier of X, the addF of X,(x . yn),(- seq)) is right_complementable Element of the carrier of X
[(x . yn),(- seq)] is set
the addF of X . [(x . yn),(- seq)] is set
||.((x . yn) - seq).|| is V24() real ext-real non negative Element of REAL
x . n is right_complementable Element of the carrier of X
seq - (x . n) is right_complementable Element of the carrier of X
- (x . n) is right_complementable Element of the carrier of X
seq + (- (x . n)) is right_complementable Element of the carrier of X
K411( the carrier of X, the addF of X,seq,(- (x . n))) is right_complementable Element of the carrier of X
[seq,(- (x . n))] is set
the addF of X . [seq,(- (x . n))] is set
((x . yn) - seq) + (seq - (x . n)) is right_complementable Element of the carrier of X
K411( the carrier of X, the addF of X,((x . yn) - seq),(seq - (x . n))) is right_complementable Element of the carrier of X
[((x . yn) - seq),(seq - (x . n))] is set
the addF of X . [((x . yn) - seq),(seq - (x . n))] is set
||.(((x . yn) - seq) + (seq - (x . n))).|| is V24() real ext-real non negative Element of REAL
||.(seq - (x . n)).|| is V24() real ext-real non negative Element of REAL
||.((x . yn) - seq).|| + ||.(seq - (x . n)).|| is V24() real ext-real non negative Element of REAL
(x . yn) - (x . n) is right_complementable Element of the carrier of X
(x . yn) + (- (x . n)) is right_complementable Element of the carrier of X
K411( the carrier of X, the addF of X,(x . yn),(- (x . n))) is right_complementable Element of the carrier of X
[(x . yn),(- (x . n))] is set
the addF of X . [(x . yn),(- (x . n))] is set
||.((x . yn) - (x . n)).|| is V24() real ext-real non negative Element of REAL
(x . n) - seq is right_complementable Element of the carrier of X
(x . n) + (- seq) is right_complementable Element of the carrier of X
K411( the carrier of X, the addF of X,(x . n),(- seq)) is right_complementable Element of the carrier of X
[(x . n),(- seq)] is set
the addF of X . [(x . n),(- seq)] is set
||.((x . n) - seq).|| is V24() real ext-real non negative Element of REAL
(y / 2) + (y / 2) is V24() real ext-real Element of REAL
X is non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like NORMSTR
the carrier of X is V4() set
K33(NAT, the carrier of X) is V4() set
K32(K33(NAT, the carrier of X)) is V4() set
x is V4() Relation-like NAT -defined the carrier of X -valued Function-like V29( NAT ) V30( NAT , the carrier of X) Element of K32(K33(NAT, the carrier of X))
seq is V24() real ext-real Element of REAL
seq / 2 is V24() real ext-real Element of REAL
y is epsilon-transitive epsilon-connected ordinal natural V24() real ext-real V45() V46() V52() V53() V54() V55() V56() V57() Element of NAT
x . y is right_complementable Element of the carrier of X
n is epsilon-transitive epsilon-connected ordinal natural V24() real ext-real V45() V46() V52() V53() V54() V55() V56() V57() Element of NAT
yn is epsilon-transitive epsilon-connected ordinal natural V24() real ext-real V45() V46() V52() V53() V54() V55() V56() V57() Element of NAT
x . yn is right_complementable Element of the carrier of X
(x . yn) - (x . y) is right_complementable Element of the carrier of X
- (x . y) is right_complementable Element of the carrier of X
(x . yn) + (- (x . y)) is right_complementable Element of the carrier of X
the addF of X is V4() Relation-like K33( the carrier of X, the carrier of X) -defined the carrier of X -valued Function-like V29(K33( the carrier of X, the carrier of X)) V30(K33( the carrier of X, the carrier of X), the carrier of X) Element of K32(K33(K33( the carrier of X, the carrier of X), the carrier of X))
K33( the carrier of X, the carrier of X) is V4() set
K33(K33( the carrier of X, the carrier of X), the carrier of X) is V4() set
K32(K33(K33( the carrier of X, the carrier of X), the carrier of X)) is V4() set
K411( the carrier of X, the addF of X,(x . yn),(- (x . y))) is right_complementable Element of the carrier of X
[(x . yn),(- (x . y))] is set
the addF of X . [(x . yn),(- (x . y))] is set
||.((x . yn) - (x . y)).|| is V24() real ext-real non negative Element of REAL
(x . y) - (x . yn) is right_complementable Element of the carrier of X
- (x . yn) is right_complementable Element of the carrier of X
(x . y) + (- (x . yn)) is right_complementable Element of the carrier of X
K411( the carrier of X, the addF of X,(x . y),(- (x . yn))) is right_complementable Element of the carrier of X
[(x . y),(- (x . yn))] is set
the addF of X . [(x . y),(- (x . yn))] is set
||.((x . y) - (x . yn)).|| is V24() real ext-real non negative Element of REAL
x . n is right_complementable Element of the carrier of X
(x . n) - (x . y) is right_complementable Element of the carrier of X
(x . n) + (- (x . y)) is right_complementable Element of the carrier of X
K411( the carrier of X, the addF of X,(x . n),(- (x . y))) is right_complementable Element of the carrier of X
[(x . n),(- (x . y))] is set
the addF of X . [(x . n),(- (x . y))] is set
||.((x . n) - (x . y)).|| is V24() real ext-real non negative Element of REAL
(seq / 2) + (seq / 2) is V24() real ext-real Element of REAL
||.((x . n) - (x . y)).|| + ||.((x . y) - (x . yn)).|| is V24() real ext-real non negative Element of REAL
((x . n) - (x . y)) + ((x . y) - (x . yn)) is right_complementable Element of the carrier of X
K411( the carrier of X, the addF of X,((x . n) - (x . y)),((x . y) - (x . yn))) is right_complementable Element of the carrier of X
[((x . n) - (x . y)),((x . y) - (x . yn))] is set
the addF of X . [((x . n) - (x . y)),((x . y) - (x . yn))] is set
||.(((x . n) - (x . y)) + ((x . y) - (x . yn))).|| is V24() real ext-real non negative Element of REAL
(x . n) - (x . yn) is right_complementable Element of the carrier of X
(x . n) + (- (x . yn)) is right_complementable Element of the carrier of X
K411( the carrier of X, the addF of X,(x . n),(- (x . yn))) is right_complementable Element of the carrier of X
[(x . n),(- (x . yn))] is set
the addF of X . [(x . n),(- (x . yn))] is set
||.((x . n) - (x . yn)).|| is V24() real ext-real non negative Element of REAL
seq is V24() real ext-real Element of REAL
y is epsilon-transitive epsilon-connected ordinal natural V24() real ext-real V45() V46() V52() V53() V54() V55() V56() V57() Element of NAT
x . y is right_complementable Element of the carrier of X
n is epsilon-transitive epsilon-connected ordinal natural V24() real ext-real V45() V46() V52() V53() V54() V55() V56() V57() Element of NAT
x . n is right_complementable Element of the carrier of X
(x . n) - (x . y) is right_complementable Element of the carrier of X
- (x . y) is right_complementable Element of the carrier of X
(x . n) + (- (x . y)) is right_complementable Element of the carrier of X
K411( the carrier of X, the addF of X,(x . n),(- (x . y))) is right_complementable Element of the carrier of X
[(x . n),(- (x . y))] is set
the addF of X . [(x . n),(- (x . y))] is set
||.((x . n) - (x . y)).|| is V24() real ext-real non negative Element of REAL
seq is V24() real ext-real Element of REAL
y is V24() real ext-real Element of REAL
X is non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like NORMSTR
the carrier of X is V4() set
K33(NAT, the carrier of X) is V4() set
K32(K33(NAT, the carrier of X)) is V4() set
0. X is V79(X) right_complementable Element of the carrier of X
x is V4() Relation-like NAT -defined the carrier of X -valued Function-like V29( NAT ) V30( NAT , the carrier of X) Element of K32(K33(NAT, the carrier of X))
||.x.|| is V4() Relation-like NAT -defined REAL -valued Function-like V29( NAT ) V30( NAT , REAL ) complex-valued ext-real-valued real-valued Element of K32(K33(NAT,REAL))
Partial_Sums ||.x.|| is V4() Relation-like NAT -defined REAL -valued Function-like V29( NAT ) V30( NAT , REAL ) complex-valued ext-real-valued real-valued Element of K32(K33(NAT,REAL))
seq is epsilon-transitive epsilon-connected ordinal natural V24() real ext-real V45() V46() V52() V53() V54() V55() V56() V57() Element of NAT
(Partial_Sums ||.x.||) . seq is V24() real ext-real Element of REAL
y is epsilon-transitive epsilon-connected ordinal natural V24() real ext-real V45() V46() V52() V53() V54() V55() V56() V57() Element of NAT
||.x.|| . y is V24() real ext-real Element of REAL
(Partial_Sums ||.x.||) . y is V24() real ext-real Element of REAL
y + 1 is epsilon-transitive epsilon-connected ordinal natural V24() real ext-real V45() V46() V52() V53() V54() V55() V56() V57() Element of NAT
||.x.|| . (y + 1) is V24() real ext-real Element of REAL
(Partial_Sums ||.x.||) . (y + 1) is V24() real ext-real Element of REAL
||.(0. X).|| is V4() epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural V24() real ext-real non positive non negative V52() V53() V54() V55() V56() V57() V58() Element of REAL
||.(0. X).|| + (||.x.|| . (y + 1)) is V24() real ext-real Element of REAL
x . y is right_complementable Element of the carrier of X
||.(x . y).|| is V24() real ext-real non negative Element of REAL
||.(x . y).|| + (||.x.|| . (y + 1)) is V24() real ext-real Element of REAL
((Partial_Sums ||.x.||) . y) + (||.x.|| . (y + 1)) is V24() real ext-real Element of REAL
||.x.|| . 0 is V24() real ext-real Element of REAL
(Partial_Sums ||.x.||) . 0 is V24() real ext-real Element of REAL
||.x.|| . seq is V24() real ext-real Element of REAL
x . seq is right_complementable Element of the carrier of X
||.(x . seq).|| is V24() real ext-real non negative Element of REAL
||.(0. X).|| is V4() epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural V24() real ext-real non positive non negative V52() V53() V54() V55() V56() V57() V58() Element of REAL
X is non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like NORMSTR
the carrier of X is V4() set
K33(NAT, the carrier of X) is V4() set
K32(K33(NAT, the carrier of X)) is V4() set
seq is V4() Relation-like NAT -defined the carrier of X -valued Function-like V29( NAT ) V30( NAT , the carrier of X) Element of K32(K33(NAT, the carrier of X))
x is V4() Relation-like NAT -defined the carrier of X -valued Function-like V29( NAT ) V30( NAT , the carrier of X) Element of K32(K33(NAT, the carrier of X))
y is right_complementable Element of the carrier of X
n is V24() real ext-real Element of REAL
yn is epsilon-transitive epsilon-connected ordinal natural V24() real ext-real V45() V46() V52() V53() V54() V55() V56() V57() Element of NAT
k is epsilon-transitive epsilon-connected ordinal natural V24() real ext-real V45() V46() V52() V53() V54() V55() V56() V57() Element of NAT
k is epsilon-transitive epsilon-connected ordinal natural V24() real ext-real V45() V46() V52() V53() V54() V55() V56() V57() Element of NAT
seq . k is right_complementable Element of the carrier of X
(seq . k) - y is right_complementable Element of the carrier of X
- y is right_complementable Element of the carrier of X
(seq . k) + (- y) is right_complementable Element of the carrier of X
the addF of X is V4() Relation-like K33( the carrier of X, the carrier of X) -defined the carrier of X -valued Function-like V29(K33( the carrier of X, the carrier of X)) V30(K33( the carrier of X, the carrier of X), the carrier of X) Element of K32(K33(K33( the carrier of X, the carrier of X), the carrier of X))
K33( the carrier of X, the carrier of X) is V4() set
K33(K33( the carrier of X, the carrier of X), the carrier of X) is V4() set
K32(K33(K33( the carrier of X, the carrier of X), the carrier of X)) is V4() set
K411( the carrier of X, the addF of X,(seq . k),(- y)) is right_complementable Element of the carrier of X
[(seq . k),(- y)] is set
the addF of X . [(seq . k),(- y)] is set
||.((seq . k) - y).|| is V24() real ext-real non negative Element of REAL
l is V4() Relation-like NAT -defined NAT -valued Function-like V29( NAT ) V30( NAT , NAT ) complex-valued ext-real-valued real-valued natural-valued increasing non-decreasing Element of K32(K33(NAT,NAT))
x * l is V4() Relation-like NAT -defined the carrier of X -valued Function-like V29( NAT ) V30( NAT , the carrier of X) subsequence of x
l . k is epsilon-transitive epsilon-connected ordinal natural V24() real ext-real V45() V46() V52() V53() V54() V55() V56() V57() Element of NAT
x . (l . k) is right_complementable Element of the carrier of X
X is non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like NORMSTR
the carrier of X is V4() set
K33(NAT, the carrier of X) is V4() set
K32(K33(NAT, the carrier of X)) is V4() set
seq is V4() Relation-like NAT -defined the carrier of X -valued Function-like V29( NAT ) V30( NAT , the carrier of X) Element of K32(K33(NAT, the carrier of X))
x is V4() Relation-like NAT -defined the carrier of X -valued Function-like V29( NAT ) V30( NAT , the carrier of X) Element of K32(K33(NAT, the carrier of X))
lim seq is right_complementable Element of the carrier of X
lim x is right_complementable Element of the carrier of X
y is V4() Relation-like NAT -defined NAT -valued Function-like V29( NAT ) V30( NAT , NAT ) complex-valued ext-real-valued real-valued natural-valued increasing non-decreasing Element of K32(K33(NAT,NAT))
x * y is V4() Relation-like NAT -defined the carrier of X -valued Function-like V29( NAT ) V30( NAT , the carrier of X) subsequence of x
n is V24() real ext-real Element of REAL
yn is epsilon-transitive epsilon-connected ordinal natural V24() real ext-real V45() V46() V52() V53() V54() V55() V56() V57() Element of NAT
k is epsilon-transitive epsilon-connected ordinal natural V24() real ext-real V45() V46() V52() V53() V54() V55() V56() V57() Element of NAT
k is epsilon-transitive epsilon-connected ordinal natural V24() real ext-real V45() V46() V52() V53() V54() V55() V56() V57() Element of NAT
y . k is epsilon-transitive epsilon-connected ordinal natural V24() real ext-real V45() V46() V52() V53() V54() V55() V56() V57() Element of NAT
seq . k is right_complementable Element of the carrier of X
x . (y . k) is right_complementable Element of the carrier of X
(seq . k) - (lim x) is right_complementable Element of the carrier of X
- (lim x) is right_complementable Element of the carrier of X
(seq . k) + (- (lim x)) is right_complementable Element of the carrier of X
the addF of X is V4() Relation-like K33( the carrier of X, the carrier of X) -defined the carrier of X -valued Function-like V29(K33( the carrier of X, the carrier of X)) V30(K33( the carrier of X, the carrier of X), the carrier of X) Element of K32(K33(K33( the carrier of X, the carrier of X), the carrier of X))
K33( the carrier of X, the carrier of X) is V4() set
K33(K33( the carrier of X, the carrier of X), the carrier of X) is V4() set
K32(K33(K33( the carrier of X, the carrier of X), the carrier of X)) is V4() set
K411( the carrier of X, the addF of X,(seq . k),(- (lim x))) is right_complementable Element of the carrier of X
[(seq . k),(- (lim x))] is set
the addF of X . [(seq . k),(- (lim x))] is set
||.((seq . k) - (lim x)).|| is V24() real ext-real non negative Element of REAL
X is non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like NORMSTR
the carrier of X is V4() set
K33(NAT, the carrier of X) is V4() set
K32(K33(NAT, the carrier of X)) is V4() set
x is V4() Relation-like NAT -defined the carrier of X -valued Function-like V29( NAT ) V30( NAT , the carrier of X) Element of K32(K33(NAT, the carrier of X))
y is epsilon-transitive epsilon-connected ordinal natural V24() real ext-real V45() V46() V52() V53() V54() V55() V56() V57() Element of NAT
x ^\ y is V4() Relation-like NAT -defined the carrier of X -valued Function-like V29( NAT ) V30( NAT , the carrier of X) subsequence of x
lim (x ^\ y) is right_complementable Element of the carrier of X
lim x is right_complementable Element of the carrier of X
X is non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like NORMSTR
the carrier of X is V4() set
K33(NAT, the carrier of X) is V4() set
K32(K33(NAT, the carrier of X)) is V4() set
x is V4() Relation-like NAT -defined the carrier of X -valued Function-like V29( NAT ) V30( NAT , the carrier of X) Element of K32(K33(NAT, the carrier of X))
seq is V4() Relation-like NAT -defined the carrier of X -valued Function-like V29( NAT ) V30( NAT , the carrier of X) Element of K32(K33(NAT, the carrier of X))
y is epsilon-transitive epsilon-connected ordinal natural V24() real ext-real V45() V46() V52() V53() V54() V55() V56() V57() Element of NAT
seq ^\ y is V4() Relation-like NAT -defined the carrier of X -valued Function-like V29( NAT ) V30( NAT , the carrier of X) subsequence of seq
y is epsilon-transitive epsilon-connected ordinal natural V24() real ext-real V45() V46() V52() V53() V54() V55() V56() V57() Element of NAT
seq ^\ y is V4() Relation-like NAT -defined the carrier of X -valued Function-like V29( NAT ) V30( NAT , the carrier of X) subsequence of seq
n is right_complementable Element of the carrier of X
yn is V24() real ext-real Element of REAL
k is epsilon-transitive epsilon-connected ordinal natural V24() real ext-real V45() V46() V52() V53() V54() V55() V56() V57() Element of NAT
k + y is epsilon-transitive epsilon-connected ordinal natural V24() real ext-real V45() V46() V52() V53() V54() V55() V56() V57() Element of NAT
k is epsilon-transitive epsilon-connected ordinal natural V24() real ext-real V45() V46() V52() V53() V54() V55() V56() V57() Element of NAT
l is epsilon-transitive epsilon-connected ordinal natural V24() real ext-real V45() V46() V52() V53() V54() V55() V56() V57() Element of NAT
seq . l is right_complementable Element of the carrier of X
(seq . l) - n is right_complementable Element of the carrier of X
- n is right_complementable Element of the carrier of X
(seq . l) + (- n) is right_complementable Element of the carrier of X
the addF of X is V4() Relation-like K33( the carrier of X, the carrier of X) -defined the carrier of X -valued Function-like V29(K33( the carrier of X, the carrier of X)) V30(K33( the carrier of X, the carrier of X), the carrier of X) Element of K32(K33(K33( the carrier of X, the carrier of X), the carrier of X))
K33( the carrier of X, the carrier of X) is V4() set
K33(K33( the carrier of X, the carrier of X), the carrier of X) is V4() set
K32(K33(K33( the carrier of X, the carrier of X), the carrier of X)) is V4() set
K411( the carrier of X, the addF of X,(seq . l),(- n)) is right_complementable Element of the carrier of X
[(seq . l),(- n)] is set
the addF of X . [(seq . l),(- n)] is set
||.((seq . l) - n).|| is V24() real ext-real non negative Element of REAL
l is epsilon-transitive epsilon-connected ordinal natural V24() real ext-real set
(k + y) + l is epsilon-transitive epsilon-connected ordinal natural V24() real ext-real V45() V46() V52() V53() V54() V55() V56() V57() Element of NAT
l - y is V24() real ext-real Element of REAL
m1 is epsilon-transitive epsilon-connected ordinal natural V24() real ext-real V45() V46() V52() V53() V54() V55() V56() V57() Element of NAT
k + m1 is epsilon-transitive epsilon-connected ordinal natural V24() real ext-real V45() V46() V52() V53() V54() V55() V56() V57() Element of NAT
(k + m1) + y is epsilon-transitive epsilon-connected ordinal natural V24() real ext-real V45() V46() V52() V53() V54() V55() V56() V57() Element of NAT
- y is V24() real ext-real Element of REAL
((k + m1) + y) + (- y) is V24() real ext-real Element of REAL
m1 is epsilon-transitive epsilon-connected ordinal natural V24() real ext-real V45() V46() V52() V53() V54() V55() V56() V57() Element of NAT
m1 + y is epsilon-transitive epsilon-connected ordinal natural V24() real ext-real V45() V46() V52() V53() V54() V55() V56() V57() Element of NAT
x . m1 is right_complementable Element of the carrier of X
(x . m1) - n is right_complementable Element of the carrier of X
(x . m1) + (- n) is right_complementable Element of the carrier of X
K411( the carrier of X, the addF of X,(x . m1),(- n)) is right_complementable Element of the carrier of X
[(x . m1),(- n)] is set
the addF of X . [(x . m1),(- n)] is set
||.((x . m1) - n).|| is V24() real ext-real non negative Element of REAL
X is non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like NORMSTR
the carrier of X is V4() set
K33(NAT, the carrier of X) is V4() set
K32(K33(NAT, the carrier of X)) is V4() set
x is V4() Relation-like NAT -defined the carrier of X -valued Function-like V29( NAT ) V30( NAT , the carrier of X) Element of K32(K33(NAT, the carrier of X))
seq is V4() Relation-like NAT -defined the carrier of X -valued Function-like V29( NAT ) V30( NAT , the carrier of X) Element of K32(K33(NAT, the carrier of X))
lim seq is right_complementable Element of the carrier of X
lim x is right_complementable Element of the carrier of X
y is epsilon-transitive epsilon-connected ordinal natural V24() real ext-real V45() V46() V52() V53() V54() V55() V56() V57() Element of NAT
seq ^\ y is V4() Relation-like NAT -defined the carrier of X -valued Function-like V29( NAT ) V30( NAT , the carrier of X) subsequence of seq
y is epsilon-transitive epsilon-connected ordinal natural V24() real ext-real V45() V46() V52() V53() V54() V55() V56() V57() Element of NAT
seq ^\ y is V4() Relation-like NAT -defined the carrier of X -valued Function-like V29( NAT ) V30( NAT , the carrier of X) subsequence of seq
n is V24() real ext-real Element of REAL
yn is epsilon-transitive epsilon-connected ordinal natural V24() real ext-real V45() V46() V52() V53() V54() V55() V56() V57() Element of NAT
yn + y is epsilon-transitive epsilon-connected ordinal natural V24() real ext-real V45() V46() V52() V53() V54() V55() V56() V57() Element of NAT
k is epsilon-transitive epsilon-connected ordinal natural V24() real ext-real V45() V46() V52() V53() V54() V55() V56() V57() Element of NAT
k is epsilon-transitive epsilon-connected ordinal natural V24() real ext-real V45() V46() V52() V53() V54() V55() V56() V57() Element of NAT
l is epsilon-transitive epsilon-connected ordinal natural V24() real ext-real set
(yn + y) + l is epsilon-transitive epsilon-connected ordinal natural V24() real ext-real V45() V46() V52() V53() V54() V55() V56() V57() Element of NAT
k - y is V24() real ext-real Element of REAL
l is epsilon-transitive epsilon-connected ordinal natural V24() real ext-real V45() V46() V52() V53() V54() V55() V56() V57() Element of NAT
yn + l is epsilon-transitive epsilon-connected ordinal natural V24() real ext-real V45() V46() V52() V53() V54() V55() V56() V57() Element of NAT
(yn + l) + y is epsilon-transitive epsilon-connected ordinal natural V24() real ext-real V45() V46() V52() V53() V54() V55() V56() V57() Element of NAT
- y is V24() real ext-real Element of REAL
((yn + l) + y) + (- y) is V24() real ext-real Element of REAL
m1 is epsilon-transitive epsilon-connected ordinal natural V24() real ext-real V45() V46() V52() V53() V54() V55() V56() V57() Element of NAT
m1 + y is epsilon-transitive epsilon-connected ordinal natural V24() real ext-real V45() V46() V52() V53() V54() V55() V56() V57() Element of NAT
x . m1 is right_complementable Element of the carrier of X
(x . m1) - (lim x) is right_complementable Element of the carrier of X
- (lim x) is right_complementable Element of the carrier of X
(x . m1) + (- (lim x)) is right_complementable Element of the carrier of X
the addF of X is V4() Relation-like K33( the carrier of X, the carrier of X) -defined the carrier of X -valued Function-like V29(K33( the carrier of X, the carrier of X)) V30(K33( the carrier of X, the carrier of X), the carrier of X) Element of K32(K33(K33( the carrier of X, the carrier of X), the carrier of X))
K33( the carrier of X, the carrier of X) is V4() set
K33(K33( the carrier of X, the carrier of X), the carrier of X) is V4() set
K32(K33(K33( the carrier of X, the carrier of X), the carrier of X)) is V4() set
K411( the carrier of X, the addF of X,(x . m1),(- (lim x))) is right_complementable Element of the carrier of X
[(x . m1),(- (lim x))] is set
the addF of X . [(x . m1),(- (lim x))] is set
||.((x . m1) - (lim x)).|| is V24() real ext-real non negative Element of REAL
seq . k is right_complementable Element of the carrier of X
(seq . k) - (lim x) is right_complementable Element of the carrier of X
(seq . k) + (- (lim x)) is right_complementable Element of the carrier of X
K411( the carrier of X, the addF of X,(seq . k),(- (lim x))) is right_complementable Element of the carrier of X
[(seq . k),(- (lim x))] is set
the addF of X . [(seq . k),(- (lim x))] is set
||.((seq . k) - (lim x)).|| is V24() real ext-real non negative Element of REAL
n is epsilon-transitive epsilon-connected ordinal natural V24() real ext-real V45() V46() V52() V53() V54() V55() V56() V57() Element of NAT
seq ^\ n is V4() Relation-like NAT -defined the carrier of X -valued Function-like V29( NAT ) V30( NAT , the carrier of X) subsequence of seq
X is non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like NORMSTR
the carrier of X is V4() set
K33(NAT, the carrier of X) is V4() set
K32(K33(NAT, the carrier of X)) is V4() set
x is V4() Relation-like NAT -defined the carrier of X -valued Function-like V29( NAT ) V30( NAT , the carrier of X) Element of K32(K33(NAT, the carrier of X))
seq is right_complementable Element of the carrier of X
y is right_complementable Element of the carrier of X
n is V24() real ext-real Element of REAL
yn is V4() epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural V24() real ext-real non positive non negative V45() V46() V52() V53() V54() V55() V56() V57() V58() Element of NAT
k is epsilon-transitive epsilon-connected ordinal natural V24() real ext-real V45() V46() V52() V53() V54() V55() V56() V57() Element of NAT
x . k is right_complementable Element of the carrier of X
(x . k) - y is right_complementable Element of the carrier of X
- y is right_complementable Element of the carrier of X
(x . k) + (- y) is right_complementable Element of the carrier of X
the addF of X is V4() Relation-like K33( the carrier of X, the carrier of X) -defined the carrier of X -valued Function-like V29(K33( the carrier of X, the carrier of X)) V30(K33( the carrier of X, the carrier of X), the carrier of X) Element of K32(K33(K33( the carrier of X, the carrier of X), the carrier of X))
K33( the carrier of X, the carrier of X) is V4() set
K33(K33( the carrier of X, the carrier of X), the carrier of X) is V4() set
K32(K33(K33( the carrier of X, the carrier of X), the carrier of X)) is V4() set
K411( the carrier of X, the addF of X,(x . k),(- y)) is right_complementable Element of the carrier of X
[(x . k),(- y)] is set
the addF of X . [(x . k),(- y)] is set
||.((x . k) - y).|| is V24() real ext-real non negative Element of REAL
seq - y is right_complementable Element of the carrier of X
seq + (- y) is right_complementable Element of the carrier of X
K411( the carrier of X, the addF of X,seq,(- y)) is right_complementable Element of the carrier of X
[seq,(- y)] is set
the addF of X . [seq,(- y)] is set
||.(seq - y).|| is V24() real ext-real non negative Element of REAL
0. X is V79(X) right_complementable Element of the carrier of X
||.(0. X).|| is V4() epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural V24() real ext-real non positive non negative V52() V53() V54() V55() V56() V57() V58() Element of REAL
X is non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like NORMSTR
the carrier of X is V4() set
K33(NAT, the carrier of X) is V4() set
K32(K33(NAT, the carrier of X)) is V4() set
0. X is V79(X) right_complementable Element of the carrier of X
x is V4() Relation-like NAT -defined the carrier of X -valued Function-like V29( NAT ) V30( NAT , the carrier of X) Element of K32(K33(NAT, the carrier of X))
||.x.|| is V4() Relation-like NAT -defined REAL -valued Function-like V29( NAT ) V30( NAT , REAL ) complex-valued ext-real-valued real-valued Element of K32(K33(NAT,REAL))
Partial_Sums ||.x.|| is V4() Relation-like NAT -defined REAL -valued Function-like V29( NAT ) V30( NAT , REAL ) complex-valued ext-real-valued real-valued Element of K32(K33(NAT,REAL))
seq is V24() real ext-real set
y is epsilon-transitive epsilon-connected ordinal natural V24() real ext-real V45() V46() V52() V53() V54() V55() V56() V57() Element of NAT
(Partial_Sums ||.x.||) . y is V24() real ext-real Element of REAL
((Partial_Sums ||.x.||) . y) - 0 is V24() real ext-real Element of REAL
abs (((Partial_Sums ||.x.||) . y) - 0) is V24() real ext-real Element of REAL
0 - 0 is V4() epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural V24() real ext-real non positive non negative V52() V53() V54() V55() V56() V57() V58() Element of REAL
abs (0 - 0) is V24() real ext-real Element of REAL
X is non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like NORMSTR
the carrier of X is V4() set
K33(NAT, the carrier of X) is V4() set
K32(K33(NAT, the carrier of X)) is V4() set
0. X is V79(X) right_complementable Element of the carrier of X
NAT --> (0. X) is V4() Relation-like NAT -defined the carrier of X -valued Function-like constant V29( NAT ) V30( NAT , the carrier of X) Element of K32(K33(NAT, the carrier of X))
x is V4() Relation-like NAT -defined the carrier of X -valued Function-like V29( NAT ) V30( NAT , the carrier of X) Element of K32(K33(NAT, the carrier of X))
seq is epsilon-transitive epsilon-connected ordinal natural V24() real ext-real V45() V46() V52() V53() V54() V55() V56() V57() Element of NAT
x . seq is right_complementable Element of the carrier of X
X is non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like NORMSTR
the carrier of X is V4() set
K33(NAT, the carrier of X) is V4() set
K32(K33(NAT, the carrier of X)) is V4() set
0. X is V79(X) right_complementable Element of the carrier of X
x is V4() Relation-like NAT -defined the carrier of X -valued Function-like V29( NAT ) V30( NAT , the carrier of X) Element of K32(K33(NAT, the carrier of X))
lim x is right_complementable Element of the carrier of X
Partial_Sums x is V4() Relation-like NAT -defined the carrier of X -valued Function-like V29( NAT ) V30( NAT , the carrier of X) Element of K32(K33(NAT, the carrier of X))
(Partial_Sums x) ^\ 1 is V4() Relation-like NAT -defined the carrier of X -valued Function-like V29( NAT ) V30( NAT , the carrier of X) subsequence of Partial_Sums x
lim ((Partial_Sums x) ^\ 1) is right_complementable Element of the carrier of X
lim (Partial_Sums x) is right_complementable Element of the carrier of X
((Partial_Sums x) ^\ 1) - (Partial_Sums x) is V4() Relation-like NAT -defined the carrier of X -valued Function-like V29( NAT ) V30( NAT , the carrier of X) Element of K32(K33(NAT, the carrier of X))
lim (((Partial_Sums x) ^\ 1) - (Partial_Sums x)) is right_complementable Element of the carrier of X
(lim (Partial_Sums x)) - (lim (Partial_Sums x)) is right_complementable Element of the carrier of X
- (lim (Partial_Sums x)) is right_complementable Element of the carrier of X
(lim (Partial_Sums x)) + (- (lim (Partial_Sums x))) is right_complementable Element of the carrier of X
the addF of X is V4() Relation-like K33( the carrier of X, the carrier of X) -defined the carrier of X -valued Function-like V29(K33( the carrier of X, the carrier of X)) V30(K33( the carrier of X, the carrier of X), the carrier of X) Element of K32(K33(K33( the carrier of X, the carrier of X), the carrier of X))
K33( the carrier of X, the carrier of X) is V4() set
K33(K33( the carrier of X, the carrier of X), the carrier of X) is V4() set
K32(K33(K33( the carrier of X, the carrier of X), the carrier of X)) is V4() set
K411( the carrier of X, the addF of X,(lim (Partial_Sums x)),(- (lim (Partial_Sums x)))) is right_complementable Element of the carrier of X
[(lim (Partial_Sums x)),(- (lim (Partial_Sums x)))] is set
the addF of X . [(lim (Partial_Sums x)),(- (lim (Partial_Sums x)))] is set
seq is epsilon-transitive epsilon-connected ordinal natural V24() real ext-real V45() V46() V52() V53() V54() V55() V56() V57() Element of NAT
seq + 1 is epsilon-transitive epsilon-connected ordinal natural V24() real ext-real V45() V46() V52() V53() V54() V55() V56() V57() Element of NAT
(Partial_Sums x) . (seq + 1) is right_complementable Element of the carrier of X
(Partial_Sums x) . seq is right_complementable Element of the carrier of X
x . (seq + 1) is right_complementable Element of the carrier of X
((Partial_Sums x) . seq) + (x . (seq + 1)) is right_complementable Element of the carrier of X
K411( the carrier of X, the addF of X,((Partial_Sums x) . seq),(x . (seq + 1))) is right_complementable Element of the carrier of X
[((Partial_Sums x) . seq),(x . (seq + 1))] is set
the addF of X . [((Partial_Sums x) . seq),(x . (seq + 1))] is set
x ^\ 1 is V4() Relation-like NAT -defined the carrier of X -valued Function-like V29( NAT ) V30( NAT , the carrier of X) subsequence of x
(x ^\ 1) . seq is right_complementable Element of the carrier of X
((Partial_Sums x) . seq) + ((x ^\ 1) . seq) is right_complementable Element of the carrier of X
K411( the carrier of X, the addF of X,((Partial_Sums x) . seq),((x ^\ 1) . seq)) is right_complementable Element of the carrier of X
[((Partial_Sums x) . seq),((x ^\ 1) . seq)] is set
the addF of X . [((Partial_Sums x) . seq),((x ^\ 1) . seq)] is set
((Partial_Sums x) ^\ 1) . seq is right_complementable Element of the carrier of X
((x ^\ 1) . seq) + ((Partial_Sums x) . seq) is right_complementable Element of the carrier of X
K411( the carrier of X, the addF of X,((x ^\ 1) . seq),((Partial_Sums x) . seq)) is right_complementable Element of the carrier of X
[((x ^\ 1) . seq),((Partial_Sums x) . seq)] is set
the addF of X . [((x ^\ 1) . seq),((Partial_Sums x) . seq)] is set
(((Partial_Sums x) ^\ 1) . seq) - ((Partial_Sums x) . seq) is right_complementable Element of the carrier of X
- ((Partial_Sums x) . seq) is right_complementable Element of the carrier of X
(((Partial_Sums x) ^\ 1) . seq) + (- ((Partial_Sums x) . seq)) is right_complementable Element of the carrier of X
K411( the carrier of X, the addF of X,(((Partial_Sums x) ^\ 1) . seq),(- ((Partial_Sums x) . seq))) is right_complementable Element of the carrier of X
[(((Partial_Sums x) ^\ 1) . seq),(- ((Partial_Sums x) . seq))] is set
the addF of X . [(((Partial_Sums x) ^\ 1) . seq),(- ((Partial_Sums x) . seq))] is set
((Partial_Sums x) . seq) - ((Partial_Sums x) . seq) is right_complementable Element of the carrier of X
((Partial_Sums x) . seq) + (- ((Partial_Sums x) . seq)) is right_complementable Element of the carrier of X
K411( the carrier of X, the addF of X,((Partial_Sums x) . seq),(- ((Partial_Sums x) . seq))) is right_complementable Element of the carrier of X
[((Partial_Sums x) . seq),(- ((Partial_Sums x) . seq))] is set
the addF of X . [((Partial_Sums x) . seq),(- ((Partial_Sums x) . seq))] is set
((x ^\ 1) . seq) + (((Partial_Sums x) . seq) - ((Partial_Sums x) . seq)) is right_complementable Element of the carrier of X
K411( the carrier of X, the addF of X,((x ^\ 1) . seq),(((Partial_Sums x) . seq) - ((Partial_Sums x) . seq))) is right_complementable Element of the carrier of X
[((x ^\ 1) . seq),(((Partial_Sums x) . seq) - ((Partial_Sums x) . seq))] is set
the addF of X . [((x ^\ 1) . seq),(((Partial_Sums x) . seq) - ((Partial_Sums x) . seq))] is set
((x ^\ 1) . seq) + (0. X) is right_complementable Element of the carrier of X
K411( the carrier of X, the addF of X,((x ^\ 1) . seq),(0. X)) is right_complementable Element of the carrier of X
[((x ^\ 1) . seq),(0. X)] is set
the addF of X . [((x ^\ 1) . seq),(0. X)] is set
X is non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like NORMSTR
the carrier of X is V4() set
K33(NAT, the carrier of X) is V4() set
K32(K33(NAT, the carrier of X)) is V4() set
x is V4() Relation-like NAT -defined the carrier of X -valued Function-like V29( NAT ) V30( NAT , the carrier of X) Element of K32(K33(NAT, the carrier of X))
Partial_Sums x is V4() Relation-like NAT -defined the carrier of X -valued Function-like V29( NAT ) V30( NAT , the carrier of X) Element of K32(K33(NAT, the carrier of X))
seq is V4() Relation-like NAT -defined the carrier of X -valued Function-like V29( NAT ) V30( NAT , the carrier of X) Element of K32(K33(NAT, the carrier of X))
Partial_Sums seq is V4() Relation-like NAT -defined the carrier of X -valued Function-like V29( NAT ) V30( NAT , the carrier of X) Element of K32(K33(NAT, the carrier of X))
(Partial_Sums x) + (Partial_Sums seq) is V4() Relation-like NAT -defined the carrier of X -valued Function-like V29( NAT ) V30( NAT , the carrier of X) Element of K32(K33(NAT, the carrier of X))
x + seq is V4() Relation-like NAT -defined the carrier of X -valued Function-like V29( NAT ) V30( NAT , the carrier of X) Element of K32(K33(NAT, the carrier of X))
Partial_Sums (x + seq) is V4() Relation-like NAT -defined the carrier of X -valued Function-like V29( NAT ) V30( NAT , the carrier of X) Element of K32(K33(NAT, the carrier of X))
y is epsilon-transitive epsilon-connected ordinal natural V24() real ext-real V45() V46() V52() V53() V54() V55() V56() V57() Element of NAT
y + 1 is epsilon-transitive epsilon-connected ordinal natural V24() real ext-real V45() V46() V52() V53() V54() V55() V56() V57() Element of NAT
((Partial_Sums x) + (Partial_Sums seq)) . (y + 1) is right_complementable Element of the carrier of X
(Partial_Sums x) . (y + 1) is right_complementable Element of the carrier of X
(Partial_Sums seq) . (y + 1) is right_complementable Element of the carrier of X
((Partial_Sums x) . (y + 1)) + ((Partial_Sums seq) . (y + 1)) is right_complementable Element of the carrier of X
the addF of X is V4() Relation-like K33( the carrier of X, the carrier of X) -defined the carrier of X -valued Function-like V29(K33( the carrier of X, the carrier of X)) V30(K33( the carrier of X, the carrier of X), the carrier of X) Element of K32(K33(K33( the carrier of X, the carrier of X), the carrier of X))
K33( the carrier of X, the carrier of X) is V4() set
K33(K33( the carrier of X, the carrier of X), the carrier of X) is V4() set
K32(K33(K33( the carrier of X, the carrier of X), the carrier of X)) is V4() set
K411( the carrier of X, the addF of X,((Partial_Sums x) . (y + 1)),((Partial_Sums seq) . (y + 1))) is right_complementable Element of the carrier of X
[((Partial_Sums x) . (y + 1)),((Partial_Sums seq) . (y + 1))] is set
the addF of X . [((Partial_Sums x) . (y + 1)),((Partial_Sums seq) . (y + 1))] is set
(Partial_Sums x) . y is right_complementable Element of the carrier of X
x . (y + 1) is right_complementable Element of the carrier of X
((Partial_Sums x) . y) + (x . (y + 1)) is right_complementable Element of the carrier of X
K411( the carrier of X, the addF of X,((Partial_Sums x) . y),(x . (y + 1))) is right_complementable Element of the carrier of X
[((Partial_Sums x) . y),(x . (y + 1))] is set
the addF of X . [((Partial_Sums x) . y),(x . (y + 1))] is set
(((Partial_Sums x) . y) + (x . (y + 1))) + ((Partial_Sums seq) . (y + 1)) is right_complementable Element of the carrier of X
K411( the carrier of X, the addF of X,(((Partial_Sums x) . y) + (x . (y + 1))),((Partial_Sums seq) . (y + 1))) is right_complementable Element of the carrier of X
[(((Partial_Sums x) . y) + (x . (y + 1))),((Partial_Sums seq) . (y + 1))] is set
the addF of X . [(((Partial_Sums x) . y) + (x . (y + 1))),((Partial_Sums seq) . (y + 1))] is set
seq . (y + 1) is right_complementable Element of the carrier of X
(Partial_Sums seq) . y is right_complementable Element of the carrier of X
(seq . (y + 1)) + ((Partial_Sums seq) . y) is right_complementable Element of the carrier of X
K411( the carrier of X, the addF of X,(seq . (y + 1)),((Partial_Sums seq) . y)) is right_complementable Element of the carrier of X
[(seq . (y + 1)),((Partial_Sums seq) . y)] is set
the addF of X . [(seq . (y + 1)),((Partial_Sums seq) . y)] is set
(((Partial_Sums x) . y) + (x . (y + 1))) + ((seq . (y + 1)) + ((Partial_Sums seq) . y)) is right_complementable Element of the carrier of X
K411( the carrier of X, the addF of X,(((Partial_Sums x) . y) + (x . (y + 1))),((seq . (y + 1)) + ((Partial_Sums seq) . y))) is right_complementable Element of the carrier of X
[(((Partial_Sums x) . y) + (x . (y + 1))),((seq . (y + 1)) + ((Partial_Sums seq) . y))] is set
the addF of X . [(((Partial_Sums x) . y) + (x . (y + 1))),((seq . (y + 1)) + ((Partial_Sums seq) . y))] is set
(((Partial_Sums x) . y) + (x . (y + 1))) + (seq . (y + 1)) is right_complementable Element of the carrier of X
K411( the carrier of X, the addF of X,(((Partial_Sums x) . y) + (x . (y + 1))),(seq . (y + 1))) is right_complementable Element of the carrier of X
[(((Partial_Sums x) . y) + (x . (y + 1))),(seq . (y + 1))] is set
the addF of X . [(((Partial_Sums x) . y) + (x . (y + 1))),(seq . (y + 1))] is set
((((Partial_Sums x) . y) + (x . (y + 1))) + (seq . (y + 1))) + ((Partial_Sums seq) . y) is right_complementable Element of the carrier of X
K411( the carrier of X, the addF of X,((((Partial_Sums x) . y) + (x . (y + 1))) + (seq . (y + 1))),((Partial_Sums seq) . y)) is right_complementable Element of the carrier of X
[((((Partial_Sums x) . y) + (x . (y + 1))) + (seq . (y + 1))),((Partial_Sums seq) . y)] is set
the addF of X . [((((Partial_Sums x) . y) + (x . (y + 1))) + (seq . (y + 1))),((Partial_Sums seq) . y)] is set
(x . (y + 1)) + (seq . (y + 1)) is right_complementable Element of the carrier of X
K411( the carrier of X, the addF of X,(x . (y + 1)),(seq . (y + 1))) is right_complementable Element of the carrier of X
[(x . (y + 1)),(seq . (y + 1))] is set
the addF of X . [(x . (y + 1)),(seq . (y + 1))] is set
((Partial_Sums x) . y) + ((x . (y + 1)) + (seq . (y + 1))) is right_complementable Element of the carrier of X
K411( the carrier of X, the addF of X,((Partial_Sums x) . y),((x . (y + 1)) + (seq . (y + 1)))) is right_complementable Element of the carrier of X
[((Partial_Sums x) . y),((x . (y + 1)) + (seq . (y + 1)))] is set
the addF of X . [((Partial_Sums x) . y),((x . (y + 1)) + (seq . (y + 1)))] is set
(((Partial_Sums x) . y) + ((x . (y + 1)) + (seq . (y + 1)))) + ((Partial_Sums seq) . y) is right_complementable Element of the carrier of X
K411( the carrier of X, the addF of X,(((Partial_Sums x) . y) + ((x . (y + 1)) + (seq . (y + 1)))),((Partial_Sums seq) . y)) is right_complementable Element of the carrier of X
[(((Partial_Sums x) . y) + ((x . (y + 1)) + (seq . (y + 1)))),((Partial_Sums seq) . y)] is set
the addF of X . [(((Partial_Sums x) . y) + ((x . (y + 1)) + (seq . (y + 1)))),((Partial_Sums seq) . y)] is set
(x + seq) . (y + 1) is right_complementable Element of the carrier of X
((Partial_Sums x) . y) + ((x + seq) . (y + 1)) is right_complementable Element of the carrier of X
K411( the carrier of X, the addF of X,((Partial_Sums x) . y),((x + seq) . (y + 1))) is right_complementable Element of the carrier of X
[((Partial_Sums x) . y),((x + seq) . (y + 1))] is set
the addF of X . [((Partial_Sums x) . y),((x + seq) . (y + 1))] is set
(((Partial_Sums x) . y) + ((x + seq) . (y + 1))) + ((Partial_Sums seq) . y) is right_complementable Element of the carrier of X
K411( the carrier of X, the addF of X,(((Partial_Sums x) . y) + ((x + seq) . (y + 1))),((Partial_Sums seq) . y)) is right_complementable Element of the carrier of X
[(((Partial_Sums x) . y) + ((x + seq) . (y + 1))),((Partial_Sums seq) . y)] is set
the addF of X . [(((Partial_Sums x) . y) + ((x + seq) . (y + 1))),((Partial_Sums seq) . y)] is set
((Partial_Sums x) . y) + ((Partial_Sums seq) . y) is right_complementable Element of the carrier of X
K411( the carrier of X, the addF of X,((Partial_Sums x) . y),((Partial_Sums seq) . y)) is right_complementable Element of the carrier of X
[((Partial_Sums x) . y),((Partial_Sums seq) . y)] is set
the addF of X . [((Partial_Sums x) . y),((Partial_Sums seq) . y)] is set
(((Partial_Sums x) . y) + ((Partial_Sums seq) . y)) + ((x + seq) . (y + 1)) is right_complementable Element of the carrier of X
K411( the carrier of X, the addF of X,(((Partial_Sums x) . y) + ((Partial_Sums seq) . y)),((x + seq) . (y + 1))) is right_complementable Element of the carrier of X
[(((Partial_Sums x) . y) + ((Partial_Sums seq) . y)),((x + seq) . (y + 1))] is set
the addF of X . [(((Partial_Sums x) . y) + ((Partial_Sums seq) . y)),((x + seq) . (y + 1))] is set
((Partial_Sums x) + (Partial_Sums seq)) . y is right_complementable Element of the carrier of X
(((Partial_Sums x) + (Partial_Sums seq)) . y) + ((x + seq) . (y + 1)) is right_complementable Element of the carrier of X
K411( the carrier of X, the addF of X,(((Partial_Sums x) + (Partial_Sums seq)) . y),((x + seq) . (y + 1))) is right_complementable Element of the carrier of X
[(((Partial_Sums x) + (Partial_Sums seq)) . y),((x + seq) . (y + 1))] is set
the addF of X . [(((Partial_Sums x) + (Partial_Sums seq)) . y),((x + seq) . (y + 1))] is set
((Partial_Sums x) + (Partial_Sums seq)) . 0 is right_complementable Element of the carrier of X
(Partial_Sums x) . 0 is right_complementable Element of the carrier of X
(Partial_Sums seq) . 0 is right_complementable Element of the carrier of X
((Partial_Sums x) . 0) + ((Partial_Sums seq) . 0) is right_complementable Element of the carrier of X
K411( the carrier of X, the addF of X,((Partial_Sums x) . 0),((Partial_Sums seq) . 0)) is right_complementable Element of the carrier of X
[((Partial_Sums x) . 0),((Partial_Sums seq) . 0)] is set
the addF of X . [((Partial_Sums x) . 0),((Partial_Sums seq) . 0)] is set
x . 0 is right_complementable Element of the carrier of X
(x . 0) + ((Partial_Sums seq) . 0) is right_complementable Element of the carrier of X
K411( the carrier of X, the addF of X,(x . 0),((Partial_Sums seq) . 0)) is right_complementable Element of the carrier of X
[(x . 0),((Partial_Sums seq) . 0)] is set
the addF of X . [(x . 0),((Partial_Sums seq) . 0)] is set
seq . 0 is right_complementable Element of the carrier of X
(x . 0) + (seq . 0) is right_complementable Element of the carrier of X
K411( the carrier of X, the addF of X,(x . 0),(seq . 0)) is right_complementable Element of the carrier of X
[(x . 0),(seq . 0)] is set
the addF of X . [(x . 0),(seq . 0)] is set
(x + seq) . 0 is right_complementable Element of the carrier of X
X is non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like NORMSTR
the carrier of X is V4() set
K33(NAT, the carrier of X) is V4() set
K32(K33(NAT, the carrier of X)) is V4() set
x is V4() Relation-like NAT -defined the carrier of X -valued Function-like V29( NAT ) V30( NAT , the carrier of X) Element of K32(K33(NAT, the carrier of X))
Partial_Sums x is V4() Relation-like NAT -defined the carrier of X -valued Function-like V29( NAT ) V30( NAT , the carrier of X) Element of K32(K33(NAT, the carrier of X))
seq is V4() Relation-like NAT -defined the carrier of X -valued Function-like V29( NAT ) V30( NAT , the carrier of X) Element of K32(K33(NAT, the carrier of X))
Partial_Sums seq is V4() Relation-like NAT -defined the carrier of X -valued Function-like V29( NAT ) V30( NAT , the carrier of X) Element of K32(K33(NAT, the carrier of X))
(Partial_Sums x) - (Partial_Sums seq) is V4() Relation-like NAT -defined the carrier of X -valued Function-like V29( NAT ) V30( NAT , the carrier of X) Element of K32(K33(NAT, the carrier of X))
x - seq is V4() Relation-like NAT -defined the carrier of X -valued Function-like V29( NAT ) V30( NAT , the carrier of X) Element of K32(K33(NAT, the carrier of X))
Partial_Sums (x - seq) is V4() Relation-like NAT -defined the carrier of X -valued Function-like V29( NAT ) V30( NAT , the carrier of X) Element of K32(K33(NAT, the carrier of X))
y is epsilon-transitive epsilon-connected ordinal natural V24() real ext-real V45() V46() V52() V53() V54() V55() V56() V57() Element of NAT
y + 1 is epsilon-transitive epsilon-connected ordinal natural V24() real ext-real V45() V46() V52() V53() V54() V55() V56() V57() Element of NAT
((Partial_Sums x) - (Partial_Sums seq)) . (y + 1) is right_complementable Element of the carrier of X
(Partial_Sums x) . (y + 1) is right_complementable Element of the carrier of X
(Partial_Sums seq) . (y + 1) is right_complementable Element of the carrier of X
((Partial_Sums x) . (y + 1)) - ((Partial_Sums seq) . (y + 1)) is right_complementable Element of the carrier of X
- ((Partial_Sums seq) . (y + 1)) is right_complementable Element of the carrier of X
((Partial_Sums x) . (y + 1)) + (- ((Partial_Sums seq) . (y + 1))) is right_complementable Element of the carrier of X
the addF of X is V4() Relation-like K33( the carrier of X, the carrier of X) -defined the carrier of X -valued Function-like V29(K33( the carrier of X, the carrier of X)) V30(K33( the carrier of X, the carrier of X), the carrier of X) Element of K32(K33(K33( the carrier of X, the carrier of X), the carrier of X))
K33( the carrier of X, the carrier of X) is V4() set
K33(K33( the carrier of X, the carrier of X), the carrier of X) is V4() set
K32(K33(K33( the carrier of X, the carrier of X), the carrier of X)) is V4() set
K411( the carrier of X, the addF of X,((Partial_Sums x) . (y + 1)),(- ((Partial_Sums seq) . (y + 1)))) is right_complementable Element of the carrier of X
[((Partial_Sums x) . (y + 1)),(- ((Partial_Sums seq) . (y + 1)))] is set
the addF of X . [((Partial_Sums x) . (y + 1)),(- ((Partial_Sums seq) . (y + 1)))] is set
(Partial_Sums x) . y is right_complementable Element of the carrier of X
x . (y + 1) is right_complementable Element of the carrier of X
((Partial_Sums x) . y) + (x . (y + 1)) is right_complementable Element of the carrier of X
K411( the carrier of X, the addF of X,((Partial_Sums x) . y),(x . (y + 1))) is right_complementable Element of the carrier of X
[((Partial_Sums x) . y),(x . (y + 1))] is set
the addF of X . [((Partial_Sums x) . y),(x . (y + 1))] is set
(((Partial_Sums x) . y) + (x . (y + 1))) - ((Partial_Sums seq) . (y + 1)) is right_complementable Element of the carrier of X
(((Partial_Sums x) . y) + (x . (y + 1))) + (- ((Partial_Sums seq) . (y + 1))) is right_complementable Element of the carrier of X
K411( the carrier of X, the addF of X,(((Partial_Sums x) . y) + (x . (y + 1))),(- ((Partial_Sums seq) . (y + 1)))) is right_complementable Element of the carrier of X
[(((Partial_Sums x) . y) + (x . (y + 1))),(- ((Partial_Sums seq) . (y + 1)))] is set
the addF of X . [(((Partial_Sums x) . y) + (x . (y + 1))),(- ((Partial_Sums seq) . (y + 1)))] is set
seq . (y + 1) is right_complementable Element of the carrier of X
(Partial_Sums seq) . y is right_complementable Element of the carrier of X
(seq . (y + 1)) + ((Partial_Sums seq) . y) is right_complementable Element of the carrier of X
K411( the carrier of X, the addF of X,(seq . (y + 1)),((Partial_Sums seq) . y)) is right_complementable Element of the carrier of X
[(seq . (y + 1)),((Partial_Sums seq) . y)] is set
the addF of X . [(seq . (y + 1)),((Partial_Sums seq) . y)] is set
(((Partial_Sums x) . y) + (x . (y + 1))) - ((seq . (y + 1)) + ((Partial_Sums seq) . y)) is right_complementable Element of the carrier of X
- ((seq . (y + 1)) + ((Partial_Sums seq) . y)) is right_complementable Element of the carrier of X
(((Partial_Sums x) . y) + (x . (y + 1))) + (- ((seq . (y + 1)) + ((Partial_Sums seq) . y))) is right_complementable Element of the carrier of X
K411( the carrier of X, the addF of X,(((Partial_Sums x) . y) + (x . (y + 1))),(- ((seq . (y + 1)) + ((Partial_Sums seq) . y)))) is right_complementable Element of the carrier of X
[(((Partial_Sums x) . y) + (x . (y + 1))),(- ((seq . (y + 1)) + ((Partial_Sums seq) . y)))] is set
the addF of X . [(((Partial_Sums x) . y) + (x . (y + 1))),(- ((seq . (y + 1)) + ((Partial_Sums seq) . y)))] is set
(((Partial_Sums x) . y) + (x . (y + 1))) - (seq . (y + 1)) is right_complementable Element of the carrier of X
- (seq . (y + 1)) is right_complementable Element of the carrier of X
(((Partial_Sums x) . y) + (x . (y + 1))) + (- (seq . (y + 1))) is right_complementable Element of the carrier of X
K411( the carrier of X, the addF of X,(((Partial_Sums x) . y) + (x . (y + 1))),(- (seq . (y + 1)))) is right_complementable Element of the carrier of X
[(((Partial_Sums x) . y) + (x . (y + 1))),(- (seq . (y + 1)))] is set
the addF of X . [(((Partial_Sums x) . y) + (x . (y + 1))),(- (seq . (y + 1)))] is set
((((Partial_Sums x) . y) + (x . (y + 1))) - (seq . (y + 1))) - ((Partial_Sums seq) . y) is right_complementable Element of the carrier of X
- ((Partial_Sums seq) . y) is right_complementable Element of the carrier of X
((((Partial_Sums x) . y) + (x . (y + 1))) - (seq . (y + 1))) + (- ((Partial_Sums seq) . y)) is right_complementable Element of the carrier of X
K411( the carrier of X, the addF of X,((((Partial_Sums x) . y) + (x . (y + 1))) - (seq . (y + 1))),(- ((Partial_Sums seq) . y))) is right_complementable Element of the carrier of X
[((((Partial_Sums x) . y) + (x . (y + 1))) - (seq . (y + 1))),(- ((Partial_Sums seq) . y))] is set
the addF of X . [((((Partial_Sums x) . y) + (x . (y + 1))) - (seq . (y + 1))),(- ((Partial_Sums seq) . y))] is set
(x . (y + 1)) - (seq . (y + 1)) is right_complementable Element of the carrier of X
(x . (y + 1)) + (- (seq . (y + 1))) is right_complementable Element of the carrier of X
K411( the carrier of X, the addF of X,(x . (y + 1)),(- (seq . (y + 1)))) is right_complementable Element of the carrier of X
[(x . (y + 1)),(- (seq . (y + 1)))] is set
the addF of X . [(x . (y + 1)),(- (seq . (y + 1)))] is set
((Partial_Sums x) . y) + ((x . (y + 1)) - (seq . (y + 1))) is right_complementable Element of the carrier of X
K411( the carrier of X, the addF of X,((Partial_Sums x) . y),((x . (y + 1)) - (seq . (y + 1)))) is right_complementable Element of the carrier of X
[((Partial_Sums x) . y),((x . (y + 1)) - (seq . (y + 1)))] is set
the addF of X . [((Partial_Sums x) . y),((x . (y + 1)) - (seq . (y + 1)))] is set
(((Partial_Sums x) . y) + ((x . (y + 1)) - (seq . (y + 1)))) - ((Partial_Sums seq) . y) is right_complementable Element of the carrier of X
(((Partial_Sums x) . y) + ((x . (y + 1)) - (seq . (y + 1)))) + (- ((Partial_Sums seq) . y)) is right_complementable Element of the carrier of X
K411( the carrier of X, the addF of X,(((Partial_Sums x) . y) + ((x . (y + 1)) - (seq . (y + 1)))),(- ((Partial_Sums seq) . y))) is right_complementable Element of the carrier of X
[(((Partial_Sums x) . y) + ((x . (y + 1)) - (seq . (y + 1)))),(- ((Partial_Sums seq) . y))] is set
the addF of X . [(((Partial_Sums x) . y) + ((x . (y + 1)) - (seq . (y + 1)))),(- ((Partial_Sums seq) . y))] is set
(x - seq) . (y + 1) is right_complementable Element of the carrier of X
((x - seq) . (y + 1)) + ((Partial_Sums x) . y) is right_complementable Element of the carrier of X
K411( the carrier of X, the addF of X,((x - seq) . (y + 1)),((Partial_Sums x) . y)) is right_complementable Element of the carrier of X
[((x - seq) . (y + 1)),((Partial_Sums x) . y)] is set
the addF of X . [((x - seq) . (y + 1)),((Partial_Sums x) . y)] is set
(((x - seq) . (y + 1)) + ((Partial_Sums x) . y)) - ((Partial_Sums seq) . y) is right_complementable Element of the carrier of X
(((x - seq) . (y + 1)) + ((Partial_Sums x) . y)) + (- ((Partial_Sums seq) . y)) is right_complementable Element of the carrier of X
K411( the carrier of X, the addF of X,(((x - seq) . (y + 1)) + ((Partial_Sums x) . y)),(- ((Partial_Sums seq) . y))) is right_complementable Element of the carrier of X
[(((x - seq) . (y + 1)) + ((Partial_Sums x) . y)),(- ((Partial_Sums seq) . y))] is set
the addF of X . [(((x - seq) . (y + 1)) + ((Partial_Sums x) . y)),(- ((Partial_Sums seq) . y))] is set
((Partial_Sums x) . y) - ((Partial_Sums seq) . y) is right_complementable Element of the carrier of X
((Partial_Sums x) . y) + (- ((Partial_Sums seq) . y)) is right_complementable Element of the carrier of X
K411( the carrier of X, the addF of X,((Partial_Sums x) . y),(- ((Partial_Sums seq) . y))) is right_complementable Element of the carrier of X
[((Partial_Sums x) . y),(- ((Partial_Sums seq) . y))] is set
the addF of X . [((Partial_Sums x) . y),(- ((Partial_Sums seq) . y))] is set
((x - seq) . (y + 1)) + (((Partial_Sums x) . y) - ((Partial_Sums seq) . y)) is right_complementable Element of the carrier of X
K411( the carrier of X, the addF of X,((x - seq) . (y + 1)),(((Partial_Sums x) . y) - ((Partial_Sums seq) . y))) is right_complementable Element of the carrier of X
[((x - seq) . (y + 1)),(((Partial_Sums x) . y) - ((Partial_Sums seq) . y))] is set
the addF of X . [((x - seq) . (y + 1)),(((Partial_Sums x) . y) - ((Partial_Sums seq) . y))] is set
((Partial_Sums x) - (Partial_Sums seq)) . y is right_complementable Element of the carrier of X
(((Partial_Sums x) - (Partial_Sums seq)) . y) + ((x - seq) . (y + 1)) is right_complementable Element of the carrier of X
K411( the carrier of X, the addF of X,(((Partial_Sums x) - (Partial_Sums seq)) . y),((x - seq) . (y + 1))) is right_complementable Element of the carrier of X
[(((Partial_Sums x) - (Partial_Sums seq)) . y),((x - seq) . (y + 1))] is set
the addF of X . [(((Partial_Sums x) - (Partial_Sums seq)) . y),((x - seq) . (y + 1))] is set
((Partial_Sums x) - (Partial_Sums seq)) . 0 is right_complementable Element of the carrier of X
(Partial_Sums x) . 0 is right_complementable Element of the carrier of X
(Partial_Sums seq) . 0 is right_complementable Element of the carrier of X
((Partial_Sums x) . 0) - ((Partial_Sums seq) . 0) is right_complementable Element of the carrier of X
- ((Partial_Sums seq) . 0) is right_complementable Element of the carrier of X
((Partial_Sums x) . 0) + (- ((Partial_Sums seq) . 0)) is right_complementable Element of the carrier of X
K411( the carrier of X, the addF of X,((Partial_Sums x) . 0),(- ((Partial_Sums seq) . 0))) is right_complementable Element of the carrier of X
[((Partial_Sums x) . 0),(- ((Partial_Sums seq) . 0))] is set
the addF of X . [((Partial_Sums x) . 0),(- ((Partial_Sums seq) . 0))] is set
x . 0 is right_complementable Element of the carrier of X
(x . 0) - ((Partial_Sums seq) . 0) is right_complementable Element of the carrier of X
(x . 0) + (- ((Partial_Sums seq) . 0)) is right_complementable Element of the carrier of X
K411( the carrier of X, the addF of X,(x . 0),(- ((Partial_Sums seq) . 0))) is right_complementable Element of the carrier of X
[(x . 0),(- ((Partial_Sums seq) . 0))] is set
the addF of X . [(x . 0),(- ((Partial_Sums seq) . 0))] is set
seq . 0 is right_complementable Element of the carrier of X
(x . 0) - (seq . 0) is right_complementable Element of the carrier of X
- (seq . 0) is right_complementable Element of the carrier of X
(x . 0) + (- (seq . 0)) is right_complementable Element of the carrier of X
K411( the carrier of X, the addF of X,(x . 0),(- (seq . 0))) is right_complementable Element of the carrier of X
[(x . 0),(- (seq . 0))] is set
the addF of X . [(x . 0),(- (seq . 0))] is set
(x - seq) . 0 is right_complementable Element of the carrier of X
X is non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like NORMSTR
the carrier of X is V4() set
K33(NAT, the carrier of X) is V4() set
K32(K33(NAT, the carrier of X)) is V4() set
x is V4() Relation-like NAT -defined the carrier of X -valued Function-like V29( NAT ) V30( NAT , the carrier of X) (X) Element of K32(K33(NAT, the carrier of X))
||.x.|| is V4() Relation-like NAT -defined REAL -valued Function-like V29( NAT ) V30( NAT , REAL ) complex-valued ext-real-valued real-valued Element of K32(K33(NAT,REAL))
seq is V4() Relation-like NAT -defined REAL -valued Function-like V29( NAT ) V30( NAT , REAL ) complex-valued ext-real-valued real-valued Element of K32(K33(NAT,REAL))
X is non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like NORMSTR
the carrier of X is V4() set
K33(NAT, the carrier of X) is V4() set
K32(K33(NAT, the carrier of X)) is V4() set
x is V4() Relation-like NAT -defined the carrier of X -valued Function-like V29( NAT ) V30( NAT , the carrier of X) Element of K32(K33(NAT, the carrier of X))
X is non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like NORMSTR
the carrier of X is V4() set
K33(NAT, the carrier of X) is V4() set
K32(K33(NAT, the carrier of X)) is V4() set
x is V4() Relation-like NAT -defined the carrier of X -valued Function-like V29( NAT ) V30( NAT , the carrier of X) Element of K32(K33(NAT, the carrier of X))
seq is V4() Relation-like NAT -defined the carrier of X -valued Function-like V29( NAT ) V30( NAT , the carrier of X) Element of K32(K33(NAT, the carrier of X))
x + seq is V4() Relation-like NAT -defined the carrier of X -valued Function-like V29( NAT ) V30( NAT , the carrier of X) Element of K32(K33(NAT, the carrier of X))
(X,(x + seq)) is right_complementable Element of the carrier of X
Partial_Sums (x + seq) is V4() Relation-like NAT -defined the carrier of X -valued Function-like V29( NAT ) V30( NAT , the carrier of X) Element of K32(K33(NAT, the carrier of X))
lim (Partial_Sums (x + seq)) is right_complementable Element of the carrier of X
(X,x) is right_complementable Element of the carrier of X
Partial_Sums x is V4() Relation-like NAT -defined the carrier of X -valued Function-like V29( NAT ) V30( NAT , the carrier of X) Element of K32(K33(NAT, the carrier of X))
lim (Partial_Sums x) is right_complementable Element of the carrier of X
(X,seq) is right_complementable Element of the carrier of X
Partial_Sums seq is V4() Relation-like NAT -defined the carrier of X -valued Function-like V29( NAT ) V30( NAT , the carrier of X) Element of K32(K33(NAT, the carrier of X))
lim (Partial_Sums seq) is right_complementable Element of the carrier of X
(X,x) + (X,seq) is right_complementable Element of the carrier of X
the addF of X is V4() Relation-like K33( the carrier of X, the carrier of X) -defined the carrier of X -valued Function-like V29(K33( the carrier of X, the carrier of X)) V30(K33( the carrier of X, the carrier of X), the carrier of X) Element of K32(K33(K33( the carrier of X, the carrier of X), the carrier of X))
K33( the carrier of X, the carrier of X) is V4() set
K33(K33( the carrier of X, the carrier of X), the carrier of X) is V4() set
K32(K33(K33( the carrier of X, the carrier of X), the carrier of X)) is V4() set
K411( the carrier of X, the addF of X,(X,x),(X,seq)) is right_complementable Element of the carrier of X
[(X,x),(X,seq)] is set
the addF of X . [(X,x),(X,seq)] is set
(Partial_Sums x) + (Partial_Sums seq) is V4() Relation-like NAT -defined the carrier of X -valued Function-like V29( NAT ) V30( NAT , the carrier of X) Element of K32(K33(NAT, the carrier of X))
lim ((Partial_Sums x) + (Partial_Sums seq)) is right_complementable Element of the carrier of X
(lim (Partial_Sums x)) + (lim (Partial_Sums seq)) is right_complementable Element of the carrier of X
K411( the carrier of X, the addF of X,(lim (Partial_Sums x)),(lim (Partial_Sums seq))) is right_complementable Element of the carrier of X
[(lim (Partial_Sums x)),(lim (Partial_Sums seq))] is set
the addF of X . [(lim (Partial_Sums x)),(lim (Partial_Sums seq))] is set
X is non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like NORMSTR
the carrier of X is V4() set
K33(NAT, the carrier of X) is V4() set
K32(K33(NAT, the carrier of X)) is V4() set
x is V4() Relation-like NAT -defined the carrier of X -valued Function-like V29( NAT ) V30( NAT , the carrier of X) Element of K32(K33(NAT, the carrier of X))
seq is V4() Relation-like NAT -defined the carrier of X -valued Function-like V29( NAT ) V30( NAT , the carrier of X) Element of K32(K33(NAT, the carrier of X))
x - seq is V4() Relation-like NAT -defined the carrier of X -valued Function-like V29( NAT ) V30( NAT , the carrier of X) Element of K32(K33(NAT, the carrier of X))
(X,(x - seq)) is right_complementable Element of the carrier of X
Partial_Sums (x - seq) is V4() Relation-like NAT -defined the carrier of X -valued Function-like V29( NAT ) V30( NAT , the carrier of X) Element of K32(K33(NAT, the carrier of X))
lim (Partial_Sums (x - seq)) is right_complementable Element of the carrier of X
(X,x) is right_complementable Element of the carrier of X
Partial_Sums x is V4() Relation-like NAT -defined the carrier of X -valued Function-like V29( NAT ) V30( NAT , the carrier of X) Element of K32(K33(NAT, the carrier of X))
lim (Partial_Sums x) is right_complementable Element of the carrier of X
(X,seq) is right_complementable Element of the carrier of X
Partial_Sums seq is V4() Relation-like NAT -defined the carrier of X -valued Function-like V29( NAT ) V30( NAT , the carrier of X) Element of K32(K33(NAT, the carrier of X))
lim (Partial_Sums seq) is right_complementable Element of the carrier of X
(X,x) - (X,seq) is right_complementable Element of the carrier of X
- (X,seq) is right_complementable Element of the carrier of X
(X,x) + (- (X,seq)) is right_complementable Element of the carrier of X
the addF of X is V4() Relation-like K33( the carrier of X, the carrier of X) -defined the carrier of X -valued Function-like V29(K33( the carrier of X, the carrier of X)) V30(K33( the carrier of X, the carrier of X), the carrier of X) Element of K32(K33(K33( the carrier of X, the carrier of X), the carrier of X))
K33( the carrier of X, the carrier of X) is V4() set
K33(K33( the carrier of X, the carrier of X), the carrier of X) is V4() set
K32(K33(K33( the carrier of X, the carrier of X), the carrier of X)) is V4() set
K411( the carrier of X, the addF of X,(X,x),(- (X,seq))) is right_complementable Element of the carrier of X
[(X,x),(- (X,seq))] is set
the addF of X . [(X,x),(- (X,seq))] is set
(Partial_Sums x) - (Partial_Sums seq) is V4() Relation-like NAT -defined the carrier of X -valued Function-like V29( NAT ) V30( NAT , the carrier of X) Element of K32(K33(NAT, the carrier of X))
lim ((Partial_Sums x) - (Partial_Sums seq)) is right_complementable Element of the carrier of X
(lim (Partial_Sums x)) - (lim (Partial_Sums seq)) is right_complementable Element of the carrier of X
- (lim (Partial_Sums seq)) is right_complementable Element of the carrier of X
(lim (Partial_Sums x)) + (- (lim (Partial_Sums seq))) is right_complementable Element of the carrier of X
K411( the carrier of X, the addF of X,(lim (Partial_Sums x)),(- (lim (Partial_Sums seq)))) is right_complementable Element of the carrier of X
[(lim (Partial_Sums x)),(- (lim (Partial_Sums seq)))] is set
the addF of X . [(lim (Partial_Sums x)),(- (lim (Partial_Sums seq)))] is set
X is non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like NORMSTR
the carrier of X is V4() set
K33(NAT, the carrier of X) is V4() set
K32(K33(NAT, the carrier of X)) is V4() set
x is V4() Relation-like NAT -defined the carrier of X -valued Function-like V29( NAT ) V30( NAT , the carrier of X) convergent (X) Element of K32(K33(NAT, the carrier of X))
seq is V4() Relation-like NAT -defined the carrier of X -valued Function-like V29( NAT ) V30( NAT , the carrier of X) convergent (X) Element of K32(K33(NAT, the carrier of X))
x + seq is V4() Relation-like NAT -defined the carrier of X -valued Function-like V29( NAT ) V30( NAT , the carrier of X) Element of K32(K33(NAT, the carrier of X))
x - seq is V4() Relation-like NAT -defined the carrier of X -valued Function-like V29( NAT ) V30( NAT , the carrier of X) Element of K32(K33(NAT, the carrier of X))
X is non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like NORMSTR
the carrier of X is V4() set
K33(NAT, the carrier of X) is V4() set
K32(K33(NAT, the carrier of X)) is V4() set
x is V4() Relation-like NAT -defined the carrier of X -valued Function-like V29( NAT ) V30( NAT , the carrier of X) Element of K32(K33(NAT, the carrier of X))
Partial_Sums x is V4() Relation-like NAT -defined the carrier of X -valued Function-like V29( NAT ) V30( NAT , the carrier of X) Element of K32(K33(NAT, the carrier of X))
seq is V24() real ext-real Element of REAL
seq * x is V4() Relation-like NAT -defined the carrier of X -valued Function-like V29( NAT ) V30( NAT , the carrier of X) Element of K32(K33(NAT, the carrier of X))
Partial_Sums (seq * x) is V4() Relation-like NAT -defined the carrier of X -valued Function-like V29( NAT ) V30( NAT , the carrier of X) Element of K32(K33(NAT, the carrier of X))
seq * (Partial_Sums x) is V4() Relation-like NAT -defined the carrier of X -valued Function-like V29( NAT ) V30( NAT , the carrier of X) Element of K32(K33(NAT, the carrier of X))
y is epsilon-transitive epsilon-connected ordinal natural V24() real ext-real V45() V46() V52() V53() V54() V55() V56() V57() Element of NAT
(Partial_Sums (seq * x)) . y is right_complementable Element of the carrier of X
(seq * (Partial_Sums x)) . y is right_complementable Element of the carrier of X
y + 1 is epsilon-transitive epsilon-connected ordinal natural V24() real ext-real V45() V46() V52() V53() V54() V55() V56() V57() Element of NAT
(Partial_Sums (seq * x)) . (y + 1) is right_complementable Element of the carrier of X
(seq * x) . (y + 1) is right_complementable Element of the carrier of X
((Partial_Sums (seq * x)) . y) + ((seq * x) . (y + 1)) is right_complementable Element of the carrier of X
the addF of X is V4() Relation-like K33( the carrier of X, the carrier of X) -defined the carrier of X -valued Function-like V29(K33( the carrier of X, the carrier of X)) V30(K33( the carrier of X, the carrier of X), the carrier of X) Element of K32(K33(K33( the carrier of X, the carrier of X), the carrier of X))
K33( the carrier of X, the carrier of X) is V4() set
K33(K33( the carrier of X, the carrier of X), the carrier of X) is V4() set
K32(K33(K33( the carrier of X, the carrier of X), the carrier of X)) is V4() set
K411( the carrier of X, the addF of X,((Partial_Sums (seq * x)) . y),((seq * x) . (y + 1))) is right_complementable Element of the carrier of X
[((Partial_Sums (seq * x)) . y),((seq * x) . (y + 1))] is set
the addF of X . [((Partial_Sums (seq * x)) . y),((seq * x) . (y + 1))] is set
(Partial_Sums x) . y is right_complementable Element of the carrier of X
seq * ((Partial_Sums x) . y) is right_complementable Element of the carrier of X
(seq * ((Partial_Sums x) . y)) + ((seq * x) . (y + 1)) is right_complementable Element of the carrier of X
K411( the carrier of X, the addF of X,(seq * ((Partial_Sums x) . y)),((seq * x) . (y + 1))) is right_complementable Element of the carrier of X
[(seq * ((Partial_Sums x) . y)),((seq * x) . (y + 1))] is set
the addF of X . [(seq * ((Partial_Sums x) . y)),((seq * x) . (y + 1))] is set
x . (y + 1) is right_complementable Element of the carrier of X
seq * (x . (y + 1)) is right_complementable Element of the carrier of X
(seq * ((Partial_Sums x) . y)) + (seq * (x . (y + 1))) is right_complementable Element of the carrier of X
K411( the carrier of X, the addF of X,(seq * ((Partial_Sums x) . y)),(seq * (x . (y + 1)))) is right_complementable Element of the carrier of X
[(seq * ((Partial_Sums x) . y)),(seq * (x . (y + 1)))] is set
the addF of X . [(seq * ((Partial_Sums x) . y)),(seq * (x . (y + 1)))] is set
((Partial_Sums x) . y) + (x . (y + 1)) is right_complementable Element of the carrier of X
K411( the carrier of X, the addF of X,((Partial_Sums x) . y),(x . (y + 1))) is right_complementable Element of the carrier of X
[((Partial_Sums x) . y),(x . (y + 1))] is set
the addF of X . [((Partial_Sums x) . y),(x . (y + 1))] is set
seq * (((Partial_Sums x) . y) + (x . (y + 1))) is right_complementable Element of the carrier of X
(Partial_Sums x) . (y + 1) is right_complementable Element of the carrier of X
seq * ((Partial_Sums x) . (y + 1)) is right_complementable Element of the carrier of X
(seq * (Partial_Sums x)) . (y + 1) is right_complementable Element of the carrier of X
(Partial_Sums (seq * x)) . 0 is right_complementable Element of the carrier of X
(seq * x) . 0 is right_complementable Element of the carrier of X
x . 0 is right_complementable Element of the carrier of X
seq * (x . 0) is right_complementable Element of the carrier of X
(Partial_Sums x) . 0 is right_complementable Element of the carrier of X
seq * ((Partial_Sums x) . 0) is right_complementable Element of the carrier of X
(seq * (Partial_Sums x)) . 0 is right_complementable Element of the carrier of X
X is non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like NORMSTR
the carrier of X is V4() set
K33(NAT, the carrier of X) is V4() set
K32(K33(NAT, the carrier of X)) is V4() set
x is V4() Relation-like NAT -defined the carrier of X -valued Function-like V29( NAT ) V30( NAT , the carrier of X) convergent (X) Element of K32(K33(NAT, the carrier of X))
(X,x) is right_complementable Element of the carrier of X
Partial_Sums x is V4() Relation-like NAT -defined the carrier of X -valued Function-like V29( NAT ) V30( NAT , the carrier of X) Element of K32(K33(NAT, the carrier of X))
lim (Partial_Sums x) is right_complementable Element of the carrier of X
seq is V24() real ext-real Element of REAL
seq * x is V4() Relation-like NAT -defined the carrier of X -valued Function-like V29( NAT ) V30( NAT , the carrier of X) Element of K32(K33(NAT, the carrier of X))
(X,(seq * x)) is right_complementable Element of the carrier of X
Partial_Sums (seq * x) is V4() Relation-like NAT -defined the carrier of X -valued Function-like V29( NAT ) V30( NAT , the carrier of X) Element of K32(K33(NAT, the carrier of X))
lim (Partial_Sums (seq * x)) is right_complementable Element of the carrier of X
seq * (X,x) is right_complementable Element of the carrier of X
seq * (Partial_Sums x) is V4() Relation-like NAT -defined the carrier of X -valued Function-like V29( NAT ) V30( NAT , the carrier of X) Element of K32(K33(NAT, the carrier of X))
lim (seq * (Partial_Sums x)) is right_complementable Element of the carrier of X
X is non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like NORMSTR
the carrier of X is V4() set
K33(NAT, the carrier of X) is V4() set
K32(K33(NAT, the carrier of X)) is V4() set
seq is V4() Relation-like NAT -defined the carrier of X -valued Function-like V29( NAT ) V30( NAT , the carrier of X) convergent (X) Element of K32(K33(NAT, the carrier of X))
x is V24() real ext-real Element of REAL
x * seq is V4() Relation-like NAT -defined the carrier of X -valued Function-like V29( NAT ) V30( NAT , the carrier of X) Element of K32(K33(NAT, the carrier of X))
X is non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like NORMSTR
the carrier of X is V4() set
K33(NAT, the carrier of X) is V4() set
K32(K33(NAT, the carrier of X)) is V4() set
seq is V4() Relation-like NAT -defined the carrier of X -valued Function-like V29( NAT ) V30( NAT , the carrier of X) Element of K32(K33(NAT, the carrier of X))
x is V4() Relation-like NAT -defined the carrier of X -valued Function-like V29( NAT ) V30( NAT , the carrier of X) Element of K32(K33(NAT, the carrier of X))
x . 0 is right_complementable Element of the carrier of X
x ^\ 1 is V4() Relation-like NAT -defined the carrier of X -valued Function-like V29( NAT ) V30( NAT , the carrier of X) subsequence of x
Partial_Sums (x ^\ 1) is V4() Relation-like NAT -defined the carrier of X -valued Function-like V29( NAT ) V30( NAT , the carrier of X) Element of K32(K33(NAT, the carrier of X))
Partial_Sums x is V4() Relation-like NAT -defined the carrier of X -valued Function-like V29( NAT ) V30( NAT , the carrier of X) Element of K32(K33(NAT, the carrier of X))
(Partial_Sums x) ^\ 1 is V4() Relation-like NAT -defined the carrier of X -valued Function-like V29( NAT ) V30( NAT , the carrier of X) subsequence of Partial_Sums x
((Partial_Sums x) ^\ 1) - seq is V4() Relation-like NAT -defined the carrier of X -valued Function-like V29( NAT ) V30( NAT , the carrier of X) Element of K32(K33(NAT, the carrier of X))
y is epsilon-transitive epsilon-connected ordinal natural V24() real ext-real V45() V46() V52() V53() V54() V55() V56() V57() Element of NAT
y + 1 is epsilon-transitive epsilon-connected ordinal natural V24() real ext-real V45() V46() V52() V53() V54() V55() V56() V57() Element of NAT
(((Partial_Sums x) ^\ 1) - seq) . (y + 1) is right_complementable Element of the carrier of X
((Partial_Sums x) ^\ 1) . (y + 1) is right_complementable Element of the carrier of X
seq . (y + 1) is right_complementable Element of the carrier of X
(((Partial_Sums x) ^\ 1) . (y + 1)) - (seq . (y + 1)) is right_complementable Element of the carrier of X
- (seq . (y + 1)) is right_complementable Element of the carrier of X
(((Partial_Sums x) ^\ 1) . (y + 1)) + (- (seq . (y + 1))) is right_complementable Element of the carrier of X
the addF of X is V4() Relation-like K33( the carrier of X, the carrier of X) -defined the carrier of X -valued Function-like V29(K33( the carrier of X, the carrier of X)) V30(K33( the carrier of X, the carrier of X), the carrier of X) Element of K32(K33(K33( the carrier of X, the carrier of X), the carrier of X))
K33( the carrier of X, the carrier of X) is V4() set
K33(K33( the carrier of X, the carrier of X), the carrier of X) is V4() set
K32(K33(K33( the carrier of X, the carrier of X), the carrier of X)) is V4() set
K411( the carrier of X, the addF of X,(((Partial_Sums x) ^\ 1) . (y + 1)),(- (seq . (y + 1)))) is right_complementable Element of the carrier of X
[(((Partial_Sums x) ^\ 1) . (y + 1)),(- (seq . (y + 1)))] is set
the addF of X . [(((Partial_Sums x) ^\ 1) . (y + 1)),(- (seq . (y + 1)))] is set
(((Partial_Sums x) ^\ 1) . (y + 1)) - (x . 0) is right_complementable Element of the carrier of X
- (x . 0) is right_complementable Element of the carrier of X
(((Partial_Sums x) ^\ 1) . (y + 1)) + (- (x . 0)) is right_complementable Element of the carrier of X
K411( the carrier of X, the addF of X,(((Partial_Sums x) ^\ 1) . (y + 1)),(- (x . 0))) is right_complementable Element of the carrier of X
[(((Partial_Sums x) ^\ 1) . (y + 1)),(- (x . 0))] is set
the addF of X . [(((Partial_Sums x) ^\ 1) . (y + 1)),(- (x . 0))] is set
(y + 1) + 1 is epsilon-transitive epsilon-connected ordinal natural V24() real ext-real V45() V46() V52() V53() V54() V55() V56() V57() Element of NAT
(Partial_Sums x) . ((y + 1) + 1) is right_complementable Element of the carrier of X
((Partial_Sums x) . ((y + 1) + 1)) - (x . 0) is right_complementable Element of the carrier of X
((Partial_Sums x) . ((y + 1) + 1)) + (- (x . 0)) is right_complementable Element of the carrier of X
K411( the carrier of X, the addF of X,((Partial_Sums x) . ((y + 1) + 1)),(- (x . 0))) is right_complementable Element of the carrier of X
[((Partial_Sums x) . ((y + 1) + 1)),(- (x . 0))] is set
the addF of X . [((Partial_Sums x) . ((y + 1) + 1)),(- (x . 0))] is set
x . ((y + 1) + 1) is right_complementable Element of the carrier of X
(Partial_Sums x) . (y + 1) is right_complementable Element of the carrier of X
(x . ((y + 1) + 1)) + ((Partial_Sums x) . (y + 1)) is right_complementable Element of the carrier of X
K411( the carrier of X, the addF of X,(x . ((y + 1) + 1)),((Partial_Sums x) . (y + 1))) is right_complementable Element of the carrier of X
[(x . ((y + 1) + 1)),((Partial_Sums x) . (y + 1))] is set
the addF of X . [(x . ((y + 1) + 1)),((Partial_Sums x) . (y + 1))] is set
((x . ((y + 1) + 1)) + ((Partial_Sums x) . (y + 1))) - (x . 0) is right_complementable Element of the carrier of X
((x . ((y + 1) + 1)) + ((Partial_Sums x) . (y + 1))) + (- (x . 0)) is right_complementable Element of the carrier of X
K411( the carrier of X, the addF of X,((x . ((y + 1) + 1)) + ((Partial_Sums x) . (y + 1))),(- (x . 0))) is right_complementable Element of the carrier of X
[((x . ((y + 1) + 1)) + ((Partial_Sums x) . (y + 1))),(- (x . 0))] is set
the addF of X . [((x . ((y + 1) + 1)) + ((Partial_Sums x) . (y + 1))),(- (x . 0))] is set
seq . y is right_complementable Element of the carrier of X
((x . ((y + 1) + 1)) + ((Partial_Sums x) . (y + 1))) - (seq . y) is right_complementable Element of the carrier of X
- (seq . y) is right_complementable Element of the carrier of X
((x . ((y + 1) + 1)) + ((Partial_Sums x) . (y + 1))) + (- (seq . y)) is right_complementable Element of the carrier of X
K411( the carrier of X, the addF of X,((x . ((y + 1) + 1)) + ((Partial_Sums x) . (y + 1))),(- (seq . y))) is right_complementable Element of the carrier of X
[((x . ((y + 1) + 1)) + ((Partial_Sums x) . (y + 1))),(- (seq . y))] is set
the addF of X . [((x . ((y + 1) + 1)) + ((Partial_Sums x) . (y + 1))),(- (seq . y))] is set
((Partial_Sums x) . (y + 1)) - (seq . y) is right_complementable Element of the carrier of X
((Partial_Sums x) . (y + 1)) + (- (seq . y)) is right_complementable Element of the carrier of X
K411( the carrier of X, the addF of X,((Partial_Sums x) . (y + 1)),(- (seq . y))) is right_complementable Element of the carrier of X
[((Partial_Sums x) . (y + 1)),(- (seq . y))] is set
the addF of X . [((Partial_Sums x) . (y + 1)),(- (seq . y))] is set
(x . ((y + 1) + 1)) + (((Partial_Sums x) . (y + 1)) - (seq . y)) is right_complementable Element of the carrier of X
K411( the carrier of X, the addF of X,(x . ((y + 1) + 1)),(((Partial_Sums x) . (y + 1)) - (seq . y))) is right_complementable Element of the carrier of X
[(x . ((y + 1) + 1)),(((Partial_Sums x) . (y + 1)) - (seq . y))] is set
the addF of X . [(x . ((y + 1) + 1)),(((Partial_Sums x) . (y + 1)) - (seq . y))] is set
((Partial_Sums x) ^\ 1) . y is right_complementable Element of the carrier of X
(((Partial_Sums x) ^\ 1) . y) - (seq . y) is right_complementable Element of the carrier of X
(((Partial_Sums x) ^\ 1) . y) + (- (seq . y)) is right_complementable Element of the carrier of X
K411( the carrier of X, the addF of X,(((Partial_Sums x) ^\ 1) . y),(- (seq . y))) is right_complementable Element of the carrier of X
[(((Partial_Sums x) ^\ 1) . y),(- (seq . y))] is set
the addF of X . [(((Partial_Sums x) ^\ 1) . y),(- (seq . y))] is set
(x . ((y + 1) + 1)) + ((((Partial_Sums x) ^\ 1) . y) - (seq . y)) is right_complementable Element of the carrier of X
K411( the carrier of X, the addF of X,(x . ((y + 1) + 1)),((((Partial_Sums x) ^\ 1) . y) - (seq . y))) is right_complementable Element of the carrier of X
[(x . ((y + 1) + 1)),((((Partial_Sums x) ^\ 1) . y) - (seq . y))] is set
the addF of X . [(x . ((y + 1) + 1)),((((Partial_Sums x) ^\ 1) . y) - (seq . y))] is set
(((Partial_Sums x) ^\ 1) - seq) . y is right_complementable Element of the carrier of X
(x . ((y + 1) + 1)) + ((((Partial_Sums x) ^\ 1) - seq) . y) is right_complementable Element of the carrier of X
K411( the carrier of X, the addF of X,(x . ((y + 1) + 1)),((((Partial_Sums x) ^\ 1) - seq) . y)) is right_complementable Element of the carrier of X
[(x . ((y + 1) + 1)),((((Partial_Sums x) ^\ 1) - seq) . y)] is set
the addF of X . [(x . ((y + 1) + 1)),((((Partial_Sums x) ^\ 1) - seq) . y)] is set
(x ^\ 1) . (y + 1) is right_complementable Element of the carrier of X
((((Partial_Sums x) ^\ 1) - seq) . y) + ((x ^\ 1) . (y + 1)) is right_complementable Element of the carrier of X
K411( the carrier of X, the addF of X,((((Partial_Sums x) ^\ 1) - seq) . y),((x ^\ 1) . (y + 1))) is right_complementable Element of the carrier of X
[((((Partial_Sums x) ^\ 1) - seq) . y),((x ^\ 1) . (y + 1))] is set
the addF of X . [((((Partial_Sums x) ^\ 1) - seq) . y),((x ^\ 1) . (y + 1))] is set
(((Partial_Sums x) ^\ 1) - seq) . 0 is right_complementable Element of the carrier of X
((Partial_Sums x) ^\ 1) . 0 is right_complementable Element of the carrier of X
seq . 0 is right_complementable Element of the carrier of X
(((Partial_Sums x) ^\ 1) . 0) - (seq . 0) is right_complementable Element of the carrier of X
- (seq . 0) is right_complementable Element of the carrier of X
(((Partial_Sums x) ^\ 1) . 0) + (- (seq . 0)) is right_complementable Element of the carrier of X
K411( the carrier of X, the addF of X,(((Partial_Sums x) ^\ 1) . 0),(- (seq . 0))) is right_complementable Element of the carrier of X
[(((Partial_Sums x) ^\ 1) . 0),(- (seq . 0))] is set
the addF of X . [(((Partial_Sums x) ^\ 1) . 0),(- (seq . 0))] is set
(((Partial_Sums x) ^\ 1) . 0) - (x . 0) is right_complementable Element of the carrier of X
(((Partial_Sums x) ^\ 1) . 0) + (- (x . 0)) is right_complementable Element of the carrier of X
K411( the carrier of X, the addF of X,(((Partial_Sums x) ^\ 1) . 0),(- (x . 0))) is right_complementable Element of the carrier of X
[(((Partial_Sums x) ^\ 1) . 0),(- (x . 0))] is set
the addF of X . [(((Partial_Sums x) ^\ 1) . 0),(- (x . 0))] is set
0 + 1 is V4() epsilon-transitive epsilon-connected ordinal natural V24() real ext-real positive non negative V45() V46() V52() V53() V54() V55() V56() V57() Element of NAT
(Partial_Sums x) . (0 + 1) is right_complementable Element of the carrier of X
((Partial_Sums x) . (0 + 1)) - (x . 0) is right_complementable Element of the carrier of X
((Partial_Sums x) . (0 + 1)) + (- (x . 0)) is right_complementable Element of the carrier of X
K411( the carrier of X, the addF of X,((Partial_Sums x) . (0 + 1)),(- (x . 0))) is right_complementable Element of the carrier of X
[((Partial_Sums x) . (0 + 1)),(- (x . 0))] is set
the addF of X . [((Partial_Sums x) . (0 + 1)),(- (x . 0))] is set
(Partial_Sums x) . 0 is right_complementable Element of the carrier of X
x . (0 + 1) is right_complementable Element of the carrier of X
((Partial_Sums x) . 0) + (x . (0 + 1)) is right_complementable Element of the carrier of X
K411( the carrier of X, the addF of X,((Partial_Sums x) . 0),(x . (0 + 1))) is right_complementable Element of the carrier of X
[((Partial_Sums x) . 0),(x . (0 + 1))] is set
the addF of X . [((Partial_Sums x) . 0),(x . (0 + 1))] is set
(((Partial_Sums x) . 0) + (x . (0 + 1))) - (x . 0) is right_complementable Element of the carrier of X
(((Partial_Sums x) . 0) + (x . (0 + 1))) + (- (x . 0)) is right_complementable Element of the carrier of X
K411( the carrier of X, the addF of X,(((Partial_Sums x) . 0) + (x . (0 + 1))),(- (x . 0))) is right_complementable Element of the carrier of X
[(((Partial_Sums x) . 0) + (x . (0 + 1))),(- (x . 0))] is set
the addF of X . [(((Partial_Sums x) . 0) + (x . (0 + 1))),(- (x . 0))] is set
(x . (0 + 1)) + (x . 0) is right_complementable Element of the carrier of X
K411( the carrier of X, the addF of X,(x . (0 + 1)),(x . 0)) is right_complementable Element of the carrier of X
[(x . (0 + 1)),(x . 0)] is set
the addF of X . [(x . (0 + 1)),(x . 0)] is set
((x . (0 + 1)) + (x . 0)) - (x . 0) is right_complementable Element of the carrier of X
((x . (0 + 1)) + (x . 0)) + (- (x . 0)) is right_complementable Element of the carrier of X
K411( the carrier of X, the addF of X,((x . (0 + 1)) + (x . 0)),(- (x . 0))) is right_complementable Element of the carrier of X
[((x . (0 + 1)) + (x . 0)),(- (x . 0))] is set
the addF of X . [((x . (0 + 1)) + (x . 0)),(- (x . 0))] is set
(x . 0) - (x . 0) is right_complementable Element of the carrier of X
(x . 0) + (- (x . 0)) is right_complementable Element of the carrier of X
K411( the carrier of X, the addF of X,(x . 0),(- (x . 0))) is right_complementable Element of the carrier of X
[(x . 0),(- (x . 0))] is set
the addF of X . [(x . 0),(- (x . 0))] is set
(x . (0 + 1)) + ((x . 0) - (x . 0)) is right_complementable Element of the carrier of X
K411( the carrier of X, the addF of X,(x . (0 + 1)),((x . 0) - (x . 0))) is right_complementable Element of the carrier of X
[(x . (0 + 1)),((x . 0) - (x . 0))] is set
the addF of X . [(x . (0 + 1)),((x . 0) - (x . 0))] is set
0. X is V79(X) right_complementable Element of the carrier of X
(x . (0 + 1)) + (0. X) is right_complementable Element of the carrier of X
K411( the carrier of X, the addF of X,(x . (0 + 1)),(0. X)) is right_complementable Element of the carrier of X
[(x . (0 + 1)),(0. X)] is set
the addF of X . [(x . (0 + 1)),(0. X)] is set
(x ^\ 1) . 0 is right_complementable Element of the carrier of X
X is non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like NORMSTR
the carrier of X is V4() set
K33(NAT, the carrier of X) is V4() set
K32(K33(NAT, the carrier of X)) is V4() set
x is V4() Relation-like NAT -defined the carrier of X -valued Function-like V29( NAT ) V30( NAT , the carrier of X) Element of K32(K33(NAT, the carrier of X))
seq is epsilon-transitive epsilon-connected ordinal natural V24() real ext-real V45() V46() V52() V53() V54() V55() V56() V57() Element of NAT
x ^\ seq is V4() Relation-like NAT -defined the carrier of X -valued Function-like V29( NAT ) V30( NAT , the carrier of X) subsequence of x
seq + 1 is epsilon-transitive epsilon-connected ordinal natural V24() real ext-real V45() V46() V52() V53() V54() V55() V56() V57() Element of NAT
x ^\ (seq + 1) is V4() Relation-like NAT -defined the carrier of X -valued Function-like V29( NAT ) V30( NAT , the carrier of X) subsequence of x
(x ^\ seq) . 0 is right_complementable Element of the carrier of X
NAT --> ((x ^\ seq) . 0) is V4() Relation-like NAT -defined the carrier of X -valued Function-like constant V29( NAT ) V30( NAT , the carrier of X) Element of K32(K33(NAT, the carrier of X))
y is V4() Relation-like NAT -defined the carrier of X -valued Function-like V29( NAT ) V30( NAT , the carrier of X) Element of K32(K33(NAT, the carrier of X))
n is epsilon-transitive epsilon-connected ordinal natural V24() real ext-real V45() V46() V52() V53() V54() V55() V56() V57() Element of NAT
y . n is right_complementable Element of the carrier of X
(x ^\ seq) ^\ 1 is V4() Relation-like NAT -defined the carrier of X -valued Function-like V29( NAT ) V30( NAT , the carrier of X) subsequence of x ^\ seq
Partial_Sums ((x ^\ seq) ^\ 1) is V4() Relation-like NAT -defined the carrier of X -valued Function-like V29( NAT ) V30( NAT , the carrier of X) Element of K32(K33(NAT, the carrier of X))
Partial_Sums (x ^\ seq) is V4() Relation-like NAT -defined the carrier of X -valued Function-like V29( NAT ) V30( NAT , the carrier of X) Element of K32(K33(NAT, the carrier of X))
(Partial_Sums (x ^\ seq)) ^\ 1 is V4() Relation-like NAT -defined the carrier of X -valued Function-like V29( NAT ) V30( NAT , the carrier of X) subsequence of Partial_Sums (x ^\ seq)
((Partial_Sums (x ^\ seq)) ^\ 1) - y is V4() Relation-like NAT -defined the carrier of X -valued Function-like V29( NAT ) V30( NAT , the carrier of X) Element of K32(K33(NAT, the carrier of X))
x ^\ 0 is V4() Relation-like NAT -defined the carrier of X -valued Function-like V29( NAT ) V30( NAT , the carrier of X) subsequence of x
X is non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like NORMSTR
the carrier of X is V4() set
K33(NAT, the carrier of X) is V4() set
K32(K33(NAT, the carrier of X)) is V4() set
x is V4() Relation-like NAT -defined the carrier of X -valued Function-like V29( NAT ) V30( NAT , the carrier of X) convergent (X) Element of K32(K33(NAT, the carrier of X))
seq is epsilon-transitive epsilon-connected ordinal natural V24() real ext-real V45() V46() V52() V53() V54() V55() V56() V57() Element of NAT
x ^\ seq is V4() Relation-like NAT -defined the carrier of X -valued Function-like V29( NAT ) V30( NAT , the carrier of X) subsequence of x
y is V4() Relation-like NAT -defined the carrier of X -valued Function-like V29( NAT ) V30( NAT , the carrier of X) Element of K32(K33(NAT, the carrier of X))
X is non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like NORMSTR
the carrier of X is V4() set
K33(NAT, the carrier of X) is V4() set
K32(K33(NAT, the carrier of X)) is V4() set
x is V4() Relation-like NAT -defined the carrier of X -valued Function-like V29( NAT ) V30( NAT , the carrier of X) Element of K32(K33(NAT, the carrier of X))
||.x.|| is V4() Relation-like NAT -defined REAL -valued Function-like V29( NAT ) V30( NAT , REAL ) complex-valued ext-real-valued real-valued Element of K32(K33(NAT,REAL))
Partial_Sums ||.x.|| is V4() Relation-like NAT -defined REAL -valued Function-like V29( NAT ) V30( NAT , REAL ) complex-valued ext-real-valued real-valued Element of K32(K33(NAT,REAL))
seq is epsilon-transitive epsilon-connected ordinal natural V24() real ext-real V45() V46() V52() V53() V54() V55() V56() V57() Element of NAT
||.x.|| . seq is V24() real ext-real Element of REAL
X is non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like NORMSTR
the carrier of X is V4() set
K33(NAT, the carrier of X) is V4() set
K32(K33(NAT, the carrier of X)) is V4() set
x is V4() Relation-like NAT -defined the carrier of X -valued Function-like V29( NAT ) V30( NAT , the carrier of X) (X) Element of K32(K33(NAT, the carrier of X))
||.x.|| is V4() Relation-like NAT -defined REAL -valued Function-like V29( NAT ) V30( NAT , REAL ) complex-valued ext-real-valued real-valued summable Element of K32(K33(NAT,REAL))
Partial_Sums ||.x.|| is V4() Relation-like NAT -defined REAL -valued Function-like V29( NAT ) V30( NAT , REAL ) complex-valued ext-real-valued real-valued Element of K32(K33(NAT,REAL))
seq is V4() Relation-like NAT -defined REAL -valued Function-like V29( NAT ) V30( NAT , REAL ) complex-valued ext-real-valued real-valued Element of K32(K33(NAT,REAL))
X is non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like complete NORMSTR
the carrier of X is V4() set
K33(NAT, the carrier of X) is V4() set
K32(K33(NAT, the carrier of X)) is V4() set
x is V4() Relation-like NAT -defined the carrier of X -valued Function-like V29( NAT ) V30( NAT , the carrier of X) Element of K32(K33(NAT, the carrier of X))
Partial_Sums x is V4() Relation-like NAT -defined the carrier of X -valued Function-like V29( NAT ) V30( NAT , the carrier of X) Element of K32(K33(NAT, the carrier of X))
seq is V24() real ext-real Element of REAL
seq is V24() real ext-real Element of REAL
y is V24() real ext-real Element of REAL
X is non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like NORMSTR
the carrier of X is V4() set
K33(NAT, the carrier of X) is V4() set
K32(K33(NAT, the carrier of X)) is V4() set
x is V4() Relation-like NAT -defined the carrier of X -valued Function-like V29( NAT ) V30( NAT , the carrier of X) Element of K32(K33(NAT, the carrier of X))
Partial_Sums x is V4() Relation-like NAT -defined the carrier of X -valued Function-like V29( NAT ) V30( NAT , the carrier of X) Element of K32(K33(NAT, the carrier of X))
||.x.|| is V4() Relation-like NAT -defined REAL -valued Function-like V29( NAT ) V30( NAT , REAL ) complex-valued ext-real-valued real-valued Element of K32(K33(NAT,REAL))
Partial_Sums ||.x.|| is V4() Relation-like NAT -defined REAL -valued Function-like V29( NAT ) V30( NAT , REAL ) complex-valued ext-real-valued real-valued Element of K32(K33(NAT,REAL))
n is epsilon-transitive epsilon-connected ordinal natural V24() real ext-real V45() V46() V52() V53() V54() V55() V56() V57() Element of NAT
yn is epsilon-transitive epsilon-connected ordinal natural V24() real ext-real V45() V46() V52() V53() V54() V55() V56() V57() Element of NAT
(Partial_Sums x) . yn is right_complementable Element of the carrier of X
(Partial_Sums x) . n is right_complementable Element of the carrier of X
((Partial_Sums x) . yn) - ((Partial_Sums x) . n) is right_complementable Element of the carrier of X
- ((Partial_Sums x) . n) is right_complementable Element of the carrier of X
((Partial_Sums x) . yn) + (- ((Partial_Sums x) . n)) is right_complementable Element of the carrier of X
the addF of X is V4() Relation-like K33( the carrier of X, the carrier of X) -defined the carrier of X -valued Function-like V29(K33( the carrier of X, the carrier of X)) V30(K33( the carrier of X, the carrier of X), the carrier of X) Element of K32(K33(K33( the carrier of X, the carrier of X), the carrier of X))
K33( the carrier of X, the carrier of X) is V4() set
K33(K33( the carrier of X, the carrier of X), the carrier of X) is V4() set
K32(K33(K33( the carrier of X, the carrier of X), the carrier of X)) is V4() set
K411( the carrier of X, the addF of X,((Partial_Sums x) . yn),(- ((Partial_Sums x) . n))) is right_complementable Element of the carrier of X
[((Partial_Sums x) . yn),(- ((Partial_Sums x) . n))] is set
the addF of X . [((Partial_Sums x) . yn),(- ((Partial_Sums x) . n))] is set
||.(((Partial_Sums x) . yn) - ((Partial_Sums x) . n)).|| is V24() real ext-real non negative Element of REAL
(Partial_Sums ||.x.||) . yn is V24() real ext-real Element of REAL
(Partial_Sums ||.x.||) . n is V24() real ext-real Element of REAL
((Partial_Sums ||.x.||) . yn) - ((Partial_Sums ||.x.||) . n) is V24() real ext-real Element of REAL
abs (((Partial_Sums ||.x.||) . yn) - ((Partial_Sums ||.x.||) . n)) is V24() real ext-real Element of REAL
yn - n is V24() real ext-real Element of REAL
k is epsilon-transitive epsilon-connected ordinal natural V24() real ext-real V45() V46() V52() V53() V54() V55() V56() V57() Element of NAT
n + k is epsilon-transitive epsilon-connected ordinal natural V24() real ext-real V45() V46() V52() V53() V54() V55() V56() V57() Element of NAT
k is epsilon-transitive epsilon-connected ordinal natural V24() real ext-real V45() V46() V52() V53() V54() V55() V56() V57() Element of NAT
x . k is right_complementable Element of the carrier of X
||.(x . k).|| is V24() real ext-real non negative Element of REAL
||.x.|| . k is V24() real ext-real Element of REAL
k is epsilon-transitive epsilon-connected ordinal natural V24() real ext-real V45() V46() V52() V53() V54() V55() V56() V57() Element of NAT
n + k is epsilon-transitive epsilon-connected ordinal natural V24() real ext-real V45() V46() V52() V53() V54() V55() V56() V57() Element of NAT
(Partial_Sums x) . (n + k) is right_complementable Element of the carrier of X
((Partial_Sums x) . (n + k)) - ((Partial_Sums x) . n) is right_complementable Element of the carrier of X
((Partial_Sums x) . (n + k)) + (- ((Partial_Sums x) . n)) is right_complementable Element of the carrier of X
K411( the carrier of X, the addF of X,((Partial_Sums x) . (n + k)),(- ((Partial_Sums x) . n))) is right_complementable Element of the carrier of X
[((Partial_Sums x) . (n + k)),(- ((Partial_Sums x) . n))] is set
the addF of X . [((Partial_Sums x) . (n + k)),(- ((Partial_Sums x) . n))] is set
||.(((Partial_Sums x) . (n + k)) - ((Partial_Sums x) . n)).|| is V24() real ext-real non negative Element of REAL
(Partial_Sums ||.x.||) . (n + k) is V24() real ext-real Element of REAL
((Partial_Sums ||.x.||) . (n + k)) - ((Partial_Sums ||.x.||) . n) is V24() real ext-real Element of REAL
abs (((Partial_Sums ||.x.||) . (n + k)) - ((Partial_Sums ||.x.||) . n)) is V24() real ext-real Element of REAL
k + 1 is epsilon-transitive epsilon-connected ordinal natural V24() real ext-real V45() V46() V52() V53() V54() V55() V56() V57() Element of NAT
n + (k + 1) is epsilon-transitive epsilon-connected ordinal natural V24() real ext-real V45() V46() V52() V53() V54() V55() V56() V57() Element of NAT
(Partial_Sums x) . (n + (k + 1)) is right_complementable Element of the carrier of X
((Partial_Sums x) . (n + (k + 1))) - ((Partial_Sums x) . n) is right_complementable Element of the carrier of X
((Partial_Sums x) . (n + (k + 1))) + (- ((Partial_Sums x) . n)) is right_complementable Element of the carrier of X
K411( the carrier of X, the addF of X,((Partial_Sums x) . (n + (k + 1))),(- ((Partial_Sums x) . n))) is right_complementable Element of the carrier of X
[((Partial_Sums x) . (n + (k + 1))),(- ((Partial_Sums x) . n))] is set
the addF of X . [((Partial_Sums x) . (n + (k + 1))),(- ((Partial_Sums x) . n))] is set
||.(((Partial_Sums x) . (n + (k + 1))) - ((Partial_Sums x) . n)).|| is V24() real ext-real non negative Element of REAL
(Partial_Sums ||.x.||) . (n + (k + 1)) is V24() real ext-real Element of REAL
((Partial_Sums ||.x.||) . (n + (k + 1))) - ((Partial_Sums ||.x.||) . n) is V24() real ext-real Element of REAL
abs (((Partial_Sums ||.x.||) . (n + (k + 1))) - ((Partial_Sums ||.x.||) . n)) is V24() real ext-real Element of REAL
(n + k) + 1 is epsilon-transitive epsilon-connected ordinal natural V24() real ext-real V45() V46() V52() V53() V54() V55() V56() V57() Element of NAT
||.x.|| . ((n + k) + 1) is V24() real ext-real Element of REAL
((Partial_Sums ||.x.||) . (n + k)) + (||.x.|| . ((n + k) + 1)) is V24() real ext-real Element of REAL
(((Partial_Sums ||.x.||) . (n + k)) + (||.x.|| . ((n + k) + 1))) - ((Partial_Sums ||.x.||) . n) is V24() real ext-real Element of REAL
abs ((((Partial_Sums ||.x.||) . (n + k)) + (||.x.|| . ((n + k) + 1))) - ((Partial_Sums ||.x.||) . n)) is V24() real ext-real Element of REAL
x . ((n + k) + 1) is right_complementable Element of the carrier of X
||.(x . ((n + k) + 1)).|| is V24() real ext-real non negative Element of REAL
((Partial_Sums ||.x.||) . (n + k)) + ||.(x . ((n + k) + 1)).|| is V24() real ext-real Element of REAL
(((Partial_Sums ||.x.||) . (n + k)) + ||.(x . ((n + k) + 1)).||) - ((Partial_Sums ||.x.||) . n) is V24() real ext-real Element of REAL
abs ((((Partial_Sums ||.x.||) . (n + k)) + ||.(x . ((n + k) + 1)).||) - ((Partial_Sums ||.x.||) . n)) is V24() real ext-real Element of REAL
||.(x . ((n + k) + 1)).|| + (((Partial_Sums ||.x.||) . (n + k)) - ((Partial_Sums ||.x.||) . n)) is V24() real ext-real Element of REAL
abs (||.(x . ((n + k) + 1)).|| + (((Partial_Sums ||.x.||) . (n + k)) - ((Partial_Sums ||.x.||) . n))) is V24() real ext-real Element of REAL
(x . ((n + k) + 1)) + ((Partial_Sums x) . (n + k)) is right_complementable Element of the carrier of X
K411( the carrier of X, the addF of X,(x . ((n + k) + 1)),((Partial_Sums x) . (n + k))) is right_complementable Element of the carrier of X
[(x . ((n + k) + 1)),((Partial_Sums x) . (n + k))] is set
the addF of X . [(x . ((n + k) + 1)),((Partial_Sums x) . (n + k))] is set
((x . ((n + k) + 1)) + ((Partial_Sums x) . (n + k))) - ((Partial_Sums x) . n) is right_complementable Element of the carrier of X
((x . ((n + k) + 1)) + ((Partial_Sums x) . (n + k))) + (- ((Partial_Sums x) . n)) is right_complementable Element of the carrier of X
K411( the carrier of X, the addF of X,((x . ((n + k) + 1)) + ((Partial_Sums x) . (n + k))),(- ((Partial_Sums x) . n))) is right_complementable Element of the carrier of X
[((x . ((n + k) + 1)) + ((Partial_Sums x) . (n + k))),(- ((Partial_Sums x) . n))] is set
the addF of X . [((x . ((n + k) + 1)) + ((Partial_Sums x) . (n + k))),(- ((Partial_Sums x) . n))] is set
||.(((x . ((n + k) + 1)) + ((Partial_Sums x) . (n + k))) - ((Partial_Sums x) . n)).|| is V24() real ext-real non negative Element of REAL
(x . ((n + k) + 1)) + (((Partial_Sums x) . (n + k)) - ((Partial_Sums x) . n)) is right_complementable Element of the carrier of X
K411( the carrier of X, the addF of X,(x . ((n + k) + 1)),(((Partial_Sums x) . (n + k)) - ((Partial_Sums x) . n))) is right_complementable Element of the carrier of X
[(x . ((n + k) + 1)),(((Partial_Sums x) . (n + k)) - ((Partial_Sums x) . n))] is set
the addF of X . [(x . ((n + k) + 1)),(((Partial_Sums x) . (n + k)) - ((Partial_Sums x) . n))] is set
||.((x . ((n + k) + 1)) + (((Partial_Sums x) . (n + k)) - ((Partial_Sums x) . n))).|| is V24() real ext-real non negative Element of REAL
||.(x . ((n + k) + 1)).|| + ||.(((Partial_Sums x) . (n + k)) - ((Partial_Sums x) . n)).|| is V24() real ext-real non negative Element of REAL
||.(x . ((n + k) + 1)).|| + (abs (((Partial_Sums ||.x.||) . (n + k)) - ((Partial_Sums ||.x.||) . n))) is V24() real ext-real Element of REAL
n + 0 is epsilon-transitive epsilon-connected ordinal natural V24() real ext-real V45() V46() V52() V53() V54() V55() V56() V57() Element of NAT
(Partial_Sums x) . (n + 0) is right_complementable Element of the carrier of X
((Partial_Sums x) . (n + 0)) - ((Partial_Sums x) . n) is right_complementable Element of the carrier of X
((Partial_Sums x) . (n + 0)) + (- ((Partial_Sums x) . n)) is right_complementable Element of the carrier of X
K411( the carrier of X, the addF of X,((Partial_Sums x) . (n + 0)),(- ((Partial_Sums x) . n))) is right_complementable Element of the carrier of X
[((Partial_Sums x) . (n + 0)),(- ((Partial_Sums x) . n))] is set
the addF of X . [((Partial_Sums x) . (n + 0)),(- ((Partial_Sums x) . n))] is set
||.(((Partial_Sums x) . (n + 0)) - ((Partial_Sums x) . n)).|| is V24() real ext-real non negative Element of REAL
0. X is V79(X) right_complementable Element of the carrier of X
||.(0. X).|| is V4() epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural V24() real ext-real non positive non negative V52() V53() V54() V55() V56() V57() V58() Element of REAL
(Partial_Sums ||.x.||) . (n + 0) is V24() real ext-real Element of REAL
((Partial_Sums ||.x.||) . (n + 0)) - ((Partial_Sums ||.x.||) . n) is V24() real ext-real Element of REAL
abs (((Partial_Sums ||.x.||) . (n + 0)) - ((Partial_Sums ||.x.||) . n)) is V24() real ext-real Element of REAL
X is non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like complete NORMSTR
the carrier of X is V4() set
K33(NAT, the carrier of X) is V4() set
K32(K33(NAT, the carrier of X)) is V4() set
x is V4() Relation-like NAT -defined the carrier of X -valued Function-like V29( NAT ) V30( NAT , the carrier of X) Element of K32(K33(NAT, the carrier of X))
||.x.|| is V4() Relation-like NAT -defined REAL -valued Function-like V29( NAT ) V30( NAT , REAL ) complex-valued ext-real-valued real-valued Element of K32(K33(NAT,REAL))
Partial_Sums ||.x.|| is V4() Relation-like NAT -defined REAL -valued Function-like V29( NAT ) V30( NAT , REAL ) complex-valued ext-real-valued real-valued Element of K32(K33(NAT,REAL))
seq is V24() real ext-real Element of REAL
y is epsilon-transitive epsilon-connected ordinal natural V24() real ext-real V45() V46() V52() V53() V54() V55() V56() V57() Element of NAT
(Partial_Sums ||.x.||) . y is V24() real ext-real Element of REAL
n is epsilon-transitive epsilon-connected ordinal natural V24() real ext-real V45() V46() V52() V53() V54() V55() V56() V57() Element of NAT
yn is epsilon-transitive epsilon-connected ordinal natural V24() real ext-real V45() V46() V52() V53() V54() V55() V56() V57() Element of NAT
Partial_Sums x is V4() Relation-like NAT -defined the carrier of X -valued Function-like V29( NAT ) V30( NAT , the carrier of X) Element of K32(K33(NAT, the carrier of X))
(Partial_Sums x) . yn is right_complementable Element of the carrier of X
(Partial_Sums x) . n is right_complementable Element of the carrier of X
((Partial_Sums x) . yn) - ((Partial_Sums x) . n) is right_complementable Element of the carrier of X
- ((Partial_Sums x) . n) is right_complementable Element of the carrier of X
((Partial_Sums x) . yn) + (- ((Partial_Sums x) . n)) is right_complementable Element of the carrier of X
the addF of X is V4() Relation-like K33( the carrier of X, the carrier of X) -defined the carrier of X -valued Function-like V29(K33( the carrier of X, the carrier of X)) V30(K33( the carrier of X, the carrier of X), the carrier of X) Element of K32(K33(K33( the carrier of X, the carrier of X), the carrier of X))
K33( the carrier of X, the carrier of X) is V4() set
K33(K33( the carrier of X, the carrier of X), the carrier of X) is V4() set
K32(K33(K33( the carrier of X, the carrier of X), the carrier of X)) is V4() set
K411( the carrier of X, the addF of X,((Partial_Sums x) . yn),(- ((Partial_Sums x) . n))) is right_complementable Element of the carrier of X
[((Partial_Sums x) . yn),(- ((Partial_Sums x) . n))] is set
the addF of X . [((Partial_Sums x) . yn),(- ((Partial_Sums x) . n))] is set
||.(((Partial_Sums x) . yn) - ((Partial_Sums x) . n)).|| is V24() real ext-real non negative Element of REAL
(Partial_Sums ||.x.||) . yn is V24() real ext-real Element of REAL
(Partial_Sums ||.x.||) . n is V24() real ext-real Element of REAL
((Partial_Sums ||.x.||) . yn) - ((Partial_Sums ||.x.||) . n) is V24() real ext-real Element of REAL
abs (((Partial_Sums ||.x.||) . yn) - ((Partial_Sums ||.x.||) . n)) is V24() real ext-real Element of REAL
X is non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like NORMSTR
the carrier of X is V4() set
K33(NAT, the carrier of X) is V4() set
K32(K33(NAT, the carrier of X)) is V4() set
x is V4() Relation-like NAT -defined REAL -valued Function-like V29( NAT ) V30( NAT , REAL ) complex-valued ext-real-valued real-valued Element of K32(K33(NAT,REAL))
seq is V4() Relation-like NAT -defined the carrier of X -valued Function-like V29( NAT ) V30( NAT , the carrier of X) Element of K32(K33(NAT, the carrier of X))
y is epsilon-transitive epsilon-connected ordinal natural V24() real ext-real V45() V46() V52() V53() V54() V55() V56() V57() Element of NAT
y is epsilon-transitive epsilon-connected ordinal natural V24() real ext-real V45() V46() V52() V53() V54() V55() V56() V57() Element of NAT
n is epsilon-transitive epsilon-connected ordinal natural V24() real ext-real V45() V46() V52() V53() V54() V55() V56() V57() Element of NAT
seq . n is right_complementable Element of the carrier of X
||.(seq . n).|| is V24() real ext-real non negative Element of REAL
||.seq.|| is V4() Relation-like NAT -defined REAL -valued Function-like V29( NAT ) V30( NAT , REAL ) complex-valued ext-real-valued real-valued Element of K32(K33(NAT,REAL))
||.seq.|| . n is V24() real ext-real Element of REAL
n is epsilon-transitive epsilon-connected ordinal natural V24() real ext-real V45() V46() V52() V53() V54() V55() V56() V57() Element of NAT
seq . n is right_complementable Element of the carrier of X
||.(seq . n).|| is V24() real ext-real non negative Element of REAL
x . n is V24() real ext-real Element of REAL
||.seq.|| . n is V24() real ext-real Element of REAL
X is non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like NORMSTR
the carrier of X is V4() set
K33(NAT, the carrier of X) is V4() set
K32(K33(NAT, the carrier of X)) is V4() set
x is V4() Relation-like NAT -defined the carrier of X -valued Function-like V29( NAT ) V30( NAT , the carrier of X) Element of K32(K33(NAT, the carrier of X))
||.x.|| is V4() Relation-like NAT -defined REAL -valued Function-like V29( NAT ) V30( NAT , REAL ) complex-valued ext-real-valued real-valued Element of K32(K33(NAT,REAL))
seq is V4() Relation-like NAT -defined the carrier of X -valued Function-like V29( NAT ) V30( NAT , the carrier of X) Element of K32(K33(NAT, the carrier of X))
||.seq.|| is V4() Relation-like NAT -defined REAL -valued Function-like V29( NAT ) V30( NAT , REAL ) complex-valued ext-real-valued real-valued Element of K32(K33(NAT,REAL))
Sum ||.x.|| is V24() real ext-real Element of REAL
Partial_Sums ||.x.|| is V4() Relation-like NAT -defined REAL -valued Function-like V29( NAT ) V30( NAT , REAL ) complex-valued ext-real-valued real-valued Element of K32(K33(NAT,REAL))
lim (Partial_Sums ||.x.||) is V24() real ext-real Element of REAL
Sum ||.seq.|| is V24() real ext-real Element of REAL
Partial_Sums ||.seq.|| is V4() Relation-like NAT -defined REAL -valued Function-like V29( NAT ) V30( NAT , REAL ) complex-valued ext-real-valued real-valued Element of K32(K33(NAT,REAL))
lim (Partial_Sums ||.seq.||) is V24() real ext-real Element of REAL
X is non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like NORMSTR
the carrier of X is V4() set
K33(NAT, the carrier of X) is V4() set
K32(K33(NAT, the carrier of X)) is V4() set
x is V4() Relation-like NAT -defined the carrier of X -valued Function-like V29( NAT ) V30( NAT , the carrier of X) Element of K32(K33(NAT, the carrier of X))
||.x.|| is V4() Relation-like NAT -defined REAL -valued Function-like V29( NAT ) V30( NAT , REAL ) complex-valued ext-real-valued real-valued Element of K32(K33(NAT,REAL))
seq is epsilon-transitive epsilon-connected ordinal natural V24() real ext-real V45() V46() V52() V53() V54() V55() V56() V57() Element of NAT
X is non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like NORMSTR
the carrier of X is V4() set
K33(NAT, the carrier of X) is V4() set
K32(K33(NAT, the carrier of X)) is V4() set
x is V4() Relation-like NAT -defined the carrier of X -valued Function-like V29( NAT ) V30( NAT , the carrier of X) Element of K32(K33(NAT, the carrier of X))
||.x.|| is V4() Relation-like NAT -defined REAL -valued Function-like V29( NAT ) V30( NAT , REAL ) complex-valued ext-real-valued real-valued Element of K32(K33(NAT,REAL))
seq is V4() Relation-like NAT -defined REAL -valued Function-like V29( NAT ) V30( NAT , REAL ) complex-valued ext-real-valued real-valued Element of K32(K33(NAT,REAL))
lim seq is V24() real ext-real Element of REAL
y is epsilon-transitive epsilon-connected ordinal natural V24() real ext-real V45() V46() V52() V53() V54() V55() V56() V57() Element of NAT
||.x.|| . y is V24() real ext-real Element of REAL
X is non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like NORMSTR
the carrier of X is V4() set
K33(NAT, the carrier of X) is V4() set
K32(K33(NAT, the carrier of X)) is V4() set
x is V4() Relation-like NAT -defined the carrier of X -valued Function-like V29( NAT ) V30( NAT , the carrier of X) Element of K32(K33(NAT, the carrier of X))
||.x.|| is V4() Relation-like NAT -defined REAL -valued Function-like V29( NAT ) V30( NAT , REAL ) complex-valued ext-real-valued real-valued Element of K32(K33(NAT,REAL))
seq is V4() Relation-like NAT -defined REAL -valued Function-like V29( NAT ) V30( NAT , REAL ) complex-valued ext-real-valued real-valued Element of K32(K33(NAT,REAL))
y is epsilon-transitive epsilon-connected ordinal natural V24() real ext-real V45() V46() V52() V53() V54() V55() V56() V57() Element of NAT
||.x.|| . y is V24() real ext-real Element of REAL
y is epsilon-transitive epsilon-connected ordinal natural V24() real ext-real V45() V46() V52() V53() V54() V55() V56() V57() Element of NAT
X is non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like NORMSTR
the carrier of X is V4() set
K33(NAT, the carrier of X) is V4() set
K32(K33(NAT, the carrier of X)) is V4() set
x is V4() Relation-like NAT -defined the carrier of X -valued Function-like V29( NAT ) V30( NAT , the carrier of X) Element of K32(K33(NAT, the carrier of X))
||.x.|| is V4() Relation-like NAT -defined REAL -valued Function-like V29( NAT ) V30( NAT , REAL ) complex-valued ext-real-valued real-valued Element of K32(K33(NAT,REAL))
seq is V4() Relation-like NAT -defined REAL -valued Function-like V29( NAT ) V30( NAT , REAL ) complex-valued ext-real-valued real-valued Element of K32(K33(NAT,REAL))
lim seq is V24() real ext-real Element of REAL
y is epsilon-transitive epsilon-connected ordinal natural V24() real ext-real V45() V46() V52() V53() V54() V55() V56() V57() Element of NAT
||.x.|| . y is V24() real ext-real Element of REAL
X is non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like NORMSTR
the carrier of X is V4() set
K33(NAT, the carrier of X) is V4() set
K32(K33(NAT, the carrier of X)) is V4() set
x is V4() Relation-like NAT -defined the carrier of X -valued Function-like V29( NAT ) V30( NAT , the carrier of X) Element of K32(K33(NAT, the carrier of X))
||.x.|| is V4() Relation-like NAT -defined REAL -valued Function-like V29( NAT ) V30( NAT , REAL ) complex-valued ext-real-valued real-valued Element of K32(K33(NAT,REAL))
seq is V4() Relation-like NAT -defined REAL -valued Function-like V29( NAT ) V30( NAT , REAL ) complex-valued ext-real-valued real-valued Element of K32(K33(NAT,REAL))
y is epsilon-transitive epsilon-connected ordinal natural V24() real ext-real V45() V46() V52() V53() V54() V55() V56() V57() Element of NAT
||.x.|| . y is V24() real ext-real Element of REAL
n is epsilon-transitive epsilon-connected ordinal natural V24() real ext-real V45() V46() V52() V53() V54() V55() V56() V57() Element of NAT
seq . n is V24() real ext-real Element of REAL
2 to_power n is V24() real ext-real Element of REAL
||.x.|| . (2 to_power n) is V24() real ext-real Element of REAL
(2 to_power n) * (||.x.|| . (2 to_power n)) is V24() real ext-real Element of REAL
X is non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like NORMSTR
the carrier of X is V4() set
K33(NAT, the carrier of X) is V4() set
K32(K33(NAT, the carrier of X)) is V4() set
x is V4() Relation-like NAT -defined the carrier of X -valued Function-like V29( NAT ) V30( NAT , the carrier of X) Element of K32(K33(NAT, the carrier of X))
||.x.|| is V4() Relation-like NAT -defined REAL -valued Function-like V29( NAT ) V30( NAT , REAL ) complex-valued ext-real-valued real-valued Element of K32(K33(NAT,REAL))
seq is V24() real ext-real Element of REAL
X is non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like NORMSTR
the carrier of X is V4() set
K33(NAT, the carrier of X) is V4() set
K32(K33(NAT, the carrier of X)) is V4() set
seq is V24() real ext-real Element of REAL
x is V4() Relation-like NAT -defined the carrier of X -valued Function-like V29( NAT ) V30( NAT , the carrier of X) Element of K32(K33(NAT, the carrier of X))
||.x.|| is V4() Relation-like NAT -defined REAL -valued Function-like V29( NAT ) V30( NAT , REAL ) complex-valued ext-real-valued real-valued Element of K32(K33(NAT,REAL))
X is non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like NORMSTR
the carrier of X is V4() set
K33(NAT, the carrier of X) is V4() set
K32(K33(NAT, the carrier of X)) is V4() set
0. X is V79(X) right_complementable Element of the carrier of X
x is V4() Relation-like NAT -defined the carrier of X -valued Function-like V29( NAT ) V30( NAT , the carrier of X) Element of K32(K33(NAT, the carrier of X))
||.x.|| is V4() Relation-like NAT -defined REAL -valued Function-like V29( NAT ) V30( NAT , REAL ) complex-valued ext-real-valued real-valued Element of K32(K33(NAT,REAL))
seq is V4() Relation-like NAT -defined REAL -valued Function-like V29( NAT ) V30( NAT , REAL ) complex-valued ext-real-valued real-valued Element of K32(K33(NAT,REAL))
lim seq is V24() real ext-real Element of REAL
y is epsilon-transitive epsilon-connected ordinal natural V24() real ext-real V45() V46() V52() V53() V54() V55() V56() V57() Element of NAT
||.x.|| . y is V24() real ext-real Element of REAL
y + 1 is epsilon-transitive epsilon-connected ordinal natural V24() real ext-real V45() V46() V52() V53() V54() V55() V56() V57() Element of NAT
||.x.|| . (y + 1) is V24() real ext-real Element of REAL
abs ||.x.|| is V4() Relation-like NAT -defined REAL -valued Function-like V29( NAT ) V30( NAT , REAL ) complex-valued ext-real-valued real-valued Element of K32(K33(NAT,REAL))
(abs ||.x.||) . (y + 1) is V24() real ext-real Element of REAL
abs (||.x.|| . (y + 1)) is V24() real ext-real Element of REAL
x . y is right_complementable Element of the carrier of X
||.(x . y).|| is V24() real ext-real non negative Element of REAL
(abs ||.x.||) . y is V24() real ext-real Element of REAL
abs (||.x.|| . y) is V24() real ext-real Element of REAL
seq . y is V24() real ext-real Element of REAL
((abs ||.x.||) . (y + 1)) / ((abs ||.x.||) . y) is V24() real ext-real Element of REAL
y is epsilon-transitive epsilon-connected ordinal natural V24() real ext-real V45() V46() V52() V53() V54() V55() V56() V57() Element of NAT
||.x.|| . y is V24() real ext-real Element of REAL
(abs ||.x.||) . y is V24() real ext-real Element of REAL
abs (||.x.|| . y) is V24() real ext-real Element of REAL
X is non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like NORMSTR
the carrier of X is V4() set
K33(NAT, the carrier of X) is V4() set
K32(K33(NAT, the carrier of X)) is V4() set
0. X is V79(X) right_complementable Element of the carrier of X
x is V4() Relation-like NAT -defined the carrier of X -valued Function-like V29( NAT ) V30( NAT , the carrier of X) Element of K32(K33(NAT, the carrier of X))
||.x.|| is V4() Relation-like NAT -defined REAL -valued Function-like V29( NAT ) V30( NAT , REAL ) complex-valued ext-real-valued real-valued Element of K32(K33(NAT,REAL))
seq is epsilon-transitive epsilon-connected ordinal natural V24() real ext-real V45() V46() V52() V53() V54() V55() V56() V57() Element of NAT
seq is epsilon-transitive epsilon-connected ordinal natural V24() real ext-real V45() V46() V52() V53() V54() V55() V56() V57() Element of NAT
y is epsilon-transitive epsilon-connected ordinal natural V24() real ext-real V45() V46() V52() V53() V54() V55() V56() V57() Element of NAT
||.x.|| . y is V24() real ext-real Element of REAL
y + 1 is epsilon-transitive epsilon-connected ordinal natural V24() real ext-real V45() V46() V52() V53() V54() V55() V56() V57() Element of NAT
||.x.|| . (y + 1) is V24() real ext-real Element of REAL
abs ||.x.|| is V4() Relation-like NAT -defined REAL -valued Function-like V29( NAT ) V30( NAT , REAL ) complex-valued ext-real-valued real-valued Element of K32(K33(NAT,REAL))
(abs ||.x.||) . (y + 1) is V24() real ext-real Element of REAL
abs (||.x.|| . (y + 1)) is V24() real ext-real Element of REAL
(abs ||.x.||) . y is V24() real ext-real Element of REAL
abs (||.x.|| . y) is V24() real ext-real Element of REAL
((abs ||.x.||) . (y + 1)) / ((abs ||.x.||) . y) is V24() real ext-real Element of REAL
y is epsilon-transitive epsilon-connected ordinal natural V24() real ext-real V45() V46() V52() V53() V54() V55() V56() V57() Element of NAT
x . y is right_complementable Element of the carrier of X
||.(x . y).|| is V24() real ext-real non negative Element of REAL
||.x.|| . y is V24() real ext-real Element of REAL
X is non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like complete NORMSTR
the carrier of X is V4() set
K33(NAT, the carrier of X) is V4() set
K32(K33(NAT, the carrier of X)) is V4() set
x is V4() Relation-like NAT -defined the carrier of X -valued Function-like V29( NAT ) V30( NAT , the carrier of X) Element of K32(K33(NAT, the carrier of X))
X is non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like V150() associative right-distributive left-distributive right_unital left_unital complete Banach_Algebra-like_1 Banach_Algebra-like_2 Banach_Algebra-like_3 Banach_Algebra-like Normed_AlgebraStr
the carrier of X is V4() set
0. X is V79(X) right_complementable Element of the carrier of X
1. X is right_complementable Element of the carrier of X
||.(1. X).|| is V24() real ext-real non negative Element of REAL
x is right_complementable Element of the carrier of X
seq is right_complementable Element of the carrier of X
x + seq is right_complementable Element of the carrier of X
the addF of X is V4() Relation-like K33( the carrier of X, the carrier of X) -defined the carrier of X -valued Function-like V29(K33( the carrier of X, the carrier of X)) V30(K33( the carrier of X, the carrier of X), the carrier of X) Element of K32(K33(K33( the carrier of X, the carrier of X), the carrier of X))
K33( the carrier of X, the carrier of X) is V4() set
K33(K33( the carrier of X, the carrier of X), the carrier of X) is V4() set
K32(K33(K33( the carrier of X, the carrier of X), the carrier of X)) is V4() set
K411( the carrier of X, the addF of X,x,seq) is right_complementable Element of the carrier of X
[x,seq] is set
the addF of X . [x,seq] is set
seq + x is right_complementable Element of the carrier of X
K411( the carrier of X, the addF of X,seq,x) is right_complementable Element of the carrier of X
[seq,x] is set
the addF of X . [seq,x] is set
y is right_complementable Element of the carrier of X
(x + seq) + y is right_complementable Element of the carrier of X
K411( the carrier of X, the addF of X,(x + seq),y) is right_complementable Element of the carrier of X
[(x + seq),y] is set
the addF of X . [(x + seq),y] is set
seq + y is right_complementable Element of the carrier of X
K411( the carrier of X, the addF of X,seq,y) is right_complementable Element of the carrier of X
[seq,y] is set
the addF of X . [seq,y] is set
x + (seq + y) is right_complementable Element of the carrier of X
K411( the carrier of X, the addF of X,x,(seq + y)) is right_complementable Element of the carrier of X
[x,(seq + y)] is set
the addF of X . [x,(seq + y)] is set
x + (0. X) is right_complementable Element of the carrier of X
K411( the carrier of X, the addF of X,x,(0. X)) is right_complementable Element of the carrier of X
[x,(0. X)] is set
the addF of X . [x,(0. X)] is set
x * seq is right_complementable Element of the carrier of X
the multF of X is V4() Relation-like K33( the carrier of X, the carrier of X) -defined the carrier of X -valued Function-like V29(K33( the carrier of X, the carrier of X)) V30(K33( the carrier of X, the carrier of X), the carrier of X) Element of K32(K33(K33( the carrier of X, the carrier of X), the carrier of X))
K411( the carrier of X, the multF of X,x,seq) is right_complementable Element of the carrier of X
the multF of X . [x,seq] is set
(x * seq) * y is right_complementable Element of the carrier of X
K411( the carrier of X, the multF of X,(x * seq),y) is right_complementable Element of the carrier of X
[(x * seq),y] is set
the multF of X . [(x * seq),y] is set
seq * y is right_complementable Element of the carrier of X
K411( the carrier of X, the multF of X,seq,y) is right_complementable Element of the carrier of X
the multF of X . [seq,y] is set
x * (seq * y) is right_complementable Element of the carrier of X
K411( the carrier of X, the multF of X,x,(seq * y)) is right_complementable Element of the carrier of X
[x,(seq * y)] is set
the multF of X . [x,(seq * y)] is set
1 * x is right_complementable Element of the carrier of X
0 * x is right_complementable Element of the carrier of X
(- 1) * x is right_complementable Element of the carrier of X
- x is right_complementable Element of the carrier of X
x * (1. X) is right_complementable Element of the carrier of X
K411( the carrier of X, the multF of X,x,(1. X)) is right_complementable Element of the carrier of X
[x,(1. X)] is set
the multF of X . [x,(1. X)] is set
(1. X) * x is right_complementable Element of the carrier of X
K411( the carrier of X, the multF of X,(1. X),x) is right_complementable Element of the carrier of X
[(1. X),x] is set
the multF of X . [(1. X),x] is set
x * (seq + y) is right_complementable Element of the carrier of X
K411( the carrier of X, the multF of X,x,(seq + y)) is right_complementable Element of the carrier of X
the multF of X . [x,(seq + y)] is set
x * y is right_complementable Element of the carrier of X
K411( the carrier of X, the multF of X,x,y) is right_complementable Element of the carrier of X
[x,y] is set
the multF of X . [x,y] is set
(x * seq) + (x * y) is right_complementable Element of the carrier of X
K411( the carrier of X, the addF of X,(x * seq),(x * y)) is right_complementable Element of the carrier of X
[(x * seq),(x * y)] is set
the addF of X . [(x * seq),(x * y)] is set
(seq + y) * x is right_complementable Element of the carrier of X
K411( the carrier of X, the multF of X,(seq + y),x) is right_complementable Element of the carrier of X
[(seq + y),x] is set
the multF of X . [(seq + y),x] is set
seq * x is right_complementable Element of the carrier of X
K411( the carrier of X, the multF of X,seq,x) is right_complementable Element of the carrier of X
the multF of X . [seq,x] is set
y * x is right_complementable Element of the carrier of X
K411( the carrier of X, the multF of X,y,x) is right_complementable Element of the carrier of X
[y,x] is set
the multF of X . [y,x] is set
(seq * x) + (y * x) is right_complementable Element of the carrier of X
K411( the carrier of X, the addF of X,(seq * x),(y * x)) is right_complementable Element of the carrier of X
[(seq * x),(y * x)] is set
the addF of X . [(seq * x),(y * x)] is set
(0. X) * x is right_complementable Element of the carrier of X
K411( the carrier of X, the multF of X,(0. X),x) is right_complementable Element of the carrier of X
[(0. X),x] is set
the multF of X . [(0. X),x] is set
x * (0. X) is right_complementable Element of the carrier of X
K411( the carrier of X, the multF of X,x,(0. X)) is right_complementable Element of the carrier of X
the multF of X . [x,(0. X)] is set
seq - y is right_complementable Element of the carrier of X
- y is right_complementable Element of the carrier of X
seq + (- y) is right_complementable Element of the carrier of X
K411( the carrier of X, the addF of X,seq,(- y)) is right_complementable Element of the carrier of X
[seq,(- y)] is set
the addF of X . [seq,(- y)] is set
x * (seq - y) is right_complementable Element of the carrier of X
K411( the carrier of X, the multF of X,x,(seq - y)) is right_complementable Element of the carrier of X
[x,(seq - y)] is set
the multF of X . [x,(seq - y)] is set
(x * seq) - (x * y) is right_complementable Element of the carrier of X
- (x * y) is right_complementable Element of the carrier of X
(x * seq) + (- (x * y)) is right_complementable Element of the carrier of X
K411( the carrier of X, the addF of X,(x * seq),(- (x * y))) is right_complementable Element of the carrier of X
[(x * seq),(- (x * y))] is set
the addF of X . [(x * seq),(- (x * y))] is set
(seq - y) * x is right_complementable Element of the carrier of X
K411( the carrier of X, the multF of X,(seq - y),x) is right_complementable Element of the carrier of X
[(seq - y),x] is set
the multF of X . [(seq - y),x] is set
(seq * x) - (y * x) is right_complementable Element of the carrier of X
- (y * x) is right_complementable Element of the carrier of X
(seq * x) + (- (y * x)) is right_complementable Element of the carrier of X
K411( the carrier of X, the addF of X,(seq * x),(- (y * x))) is right_complementable Element of the carrier of X
[(seq * x),(- (y * x))] is set
the addF of X . [(seq * x),(- (y * x))] is set
(x + seq) - y is right_complementable Element of the carrier of X
(x + seq) + (- y) is right_complementable Element of the carrier of X
K411( the carrier of X, the addF of X,(x + seq),(- y)) is right_complementable Element of the carrier of X
[(x + seq),(- y)] is set
the addF of X . [(x + seq),(- y)] is set
x + (seq - y) is right_complementable Element of the carrier of X
K411( the carrier of X, the addF of X,x,(seq - y)) is right_complementable Element of the carrier of X
the addF of X . [x,(seq - y)] is set
x - seq is right_complementable Element of the carrier of X
- seq is right_complementable Element of the carrier of X
x + (- seq) is right_complementable Element of the carrier of X
K411( the carrier of X, the addF of X,x,(- seq)) is right_complementable Element of the carrier of X
[x,(- seq)] is set
the addF of X . [x,(- seq)] is set
(x - seq) + y is right_complementable Element of the carrier of X
K411( the carrier of X, the addF of X,(x - seq),y) is right_complementable Element of the carrier of X
[(x - seq),y] is set
the addF of X . [(x - seq),y] is set
x - (seq - y) is right_complementable Element of the carrier of X
- (seq - y) is right_complementable Element of the carrier of X
x + (- (seq - y)) is right_complementable Element of the carrier of X
K411( the carrier of X, the addF of X,x,(- (seq - y))) is right_complementable Element of the carrier of X
[x,(- (seq - y))] is set
the addF of X . [x,(- (seq - y))] is set
(x - seq) - y is right_complementable Element of the carrier of X
(x - seq) + (- y) is right_complementable Element of the carrier of X
K411( the carrier of X, the addF of X,(x - seq),(- y)) is right_complementable Element of the carrier of X
[(x - seq),(- y)] is set
the addF of X . [(x - seq),(- y)] is set
x - (seq + y) is right_complementable Element of the carrier of X
- (seq + y) is right_complementable Element of the carrier of X
x + (- (seq + y)) is right_complementable Element of the carrier of X
K411( the carrier of X, the addF of X,x,(- (seq + y))) is right_complementable Element of the carrier of X
[x,(- (seq + y))] is set
the addF of X . [x,(- (seq + y))] is set
x - y is right_complementable Element of the carrier of X
x + (- y) is right_complementable Element of the carrier of X
K411( the carrier of X, the addF of X,x,(- y)) is right_complementable Element of the carrier of X
[x,(- y)] is set
the addF of X . [x,(- y)] is set
y + seq is right_complementable Element of the carrier of X
K411( the carrier of X, the addF of X,y,seq) is right_complementable Element of the carrier of X
[y,seq] is set
the addF of X . [y,seq] is set
(x - y) + (y + seq) is right_complementable Element of the carrier of X
K411( the carrier of X, the addF of X,(x - y),(y + seq)) is right_complementable Element of the carrier of X
[(x - y),(y + seq)] is set
the addF of X . [(x - y),(y + seq)] is set
y - seq is right_complementable Element of the carrier of X
y + (- seq) is right_complementable Element of the carrier of X
K411( the carrier of X, the addF of X,y,(- seq)) is right_complementable Element of the carrier of X
[y,(- seq)] is set
the addF of X . [y,(- seq)] is set
(x - y) + (y - seq) is right_complementable Element of the carrier of X
K411( the carrier of X, the addF of X,(x - y),(y - seq)) is right_complementable Element of the carrier of X
[(x - y),(y - seq)] is set
the addF of X . [(x - y),(y - seq)] is set
(x - seq) + seq is right_complementable Element of the carrier of X
K411( the carrier of X, the addF of X,(x - seq),seq) is right_complementable Element of the carrier of X
[(x - seq),seq] is set
the addF of X . [(x - seq),seq] is set
seq - x is right_complementable Element of the carrier of X
seq + (- x) is right_complementable Element of the carrier of X
K411( the carrier of X, the addF of X,seq,(- x)) is right_complementable Element of the carrier of X
[seq,(- x)] is set
the addF of X . [seq,(- x)] is set
seq - (seq - x) is right_complementable Element of the carrier of X
- (seq - x) is right_complementable Element of the carrier of X
seq + (- (seq - x)) is right_complementable Element of the carrier of X
K411( the carrier of X, the addF of X,seq,(- (seq - x))) is right_complementable Element of the carrier of X
[seq,(- (seq - x))] is set
the addF of X . [seq,(- (seq - x))] is set
||.x.|| is V24() real ext-real non negative Element of REAL
||.(x + seq).|| is V24() real ext-real non negative Element of REAL
||.seq.|| is V24() real ext-real non negative Element of REAL
||.x.|| + ||.seq.|| is V24() real ext-real non negative Element of REAL
||.(x * seq).|| is V24() real ext-real non negative Element of REAL
||.x.|| * ||.seq.|| is V24() real ext-real non negative Element of REAL
n is V24() real ext-real Element of REAL
n * (0. X) is right_complementable Element of the carrier of X
n * (x * seq) is right_complementable Element of the carrier of X
n * x is right_complementable Element of the carrier of X
(n * x) * seq is right_complementable Element of the carrier of X
K411( the carrier of X, the multF of X,(n * x),seq) is right_complementable Element of the carrier of X
[(n * x),seq] is set
the multF of X . [(n * x),seq] is set
n * (x + seq) is right_complementable Element of the carrier of X
n * seq is right_complementable Element of the carrier of X
(n * x) + (n * seq) is right_complementable Element of the carrier of X
K411( the carrier of X, the addF of X,(n * x),(n * seq)) is right_complementable Element of the carrier of X
[(n * x),(n * seq)] is set
the addF of X . [(n * x),(n * seq)] is set
yn is V24() real ext-real Element of REAL
n + yn is V24() real ext-real Element of REAL
(n + yn) * x is right_complementable Element of the carrier of X
yn * x is right_complementable Element of the carrier of X
(n * x) + (yn * x) is right_complementable Element of the carrier of X
K411( the carrier of X, the addF of X,(n * x),(yn * x)) is right_complementable Element of the carrier of X
[(n * x),(yn * x)] is set
the addF of X . [(n * x),(yn * x)] is set
n * yn is V24() real ext-real Element of REAL
(n * yn) * x is right_complementable Element of the carrier of X
n * (yn * x) is right_complementable Element of the carrier of X
(n * yn) * (x * seq) is right_complementable Element of the carrier of X
yn * seq is right_complementable Element of the carrier of X
(n * x) * (yn * seq) is right_complementable Element of the carrier of X
K411( the carrier of X, the multF of X,(n * x),(yn * seq)) is right_complementable Element of the carrier of X
[(n * x),(yn * seq)] is set
the multF of X . [(n * x),(yn * seq)] is set
x * (n * seq) is right_complementable Element of the carrier of X
K411( the carrier of X, the multF of X,x,(n * seq)) is right_complementable Element of the carrier of X
[x,(n * seq)] is set
the multF of X . [x,(n * seq)] is set
||.(n * x).|| is V24() real ext-real non negative Element of REAL
abs n is V24() real ext-real Element of REAL
(abs n) * ||.x.|| is V24() real ext-real Element of REAL
yn * (x * seq) is right_complementable Element of the carrier of X
n * (yn * (x * seq)) is right_complementable Element of the carrier of X
x * (yn * seq) is right_complementable Element of the carrier of X
K411( the carrier of X, the multF of X,x,(yn * seq)) is right_complementable Element of the carrier of X
[x,(yn * seq)] is set
the multF of X . [x,(yn * seq)] is set
n * (x * (yn * seq)) is right_complementable Element of the carrier of X
(x * (seq - y)) + (x * y) is right_complementable Element of the carrier of X
K411( the carrier of X, the addF of X,(x * (seq - y)),(x * y)) is right_complementable Element of the carrier of X
[(x * (seq - y)),(x * y)] is set
the addF of X . [(x * (seq - y)),(x * y)] is set
(seq - y) + y is right_complementable Element of the carrier of X
K411( the carrier of X, the addF of X,(seq - y),y) is right_complementable Element of the carrier of X
[(seq - y),y] is set
the addF of X . [(seq - y),y] is set
x * ((seq - y) + y) is right_complementable Element of the carrier of X
K411( the carrier of X, the multF of X,x,((seq - y) + y)) is right_complementable Element of the carrier of X
[x,((seq - y) + y)] is set
the multF of X . [x,((seq - y) + y)] is set
y - y is right_complementable Element of the carrier of X
y + (- y) is right_complementable Element of the carrier of X
K411( the carrier of X, the addF of X,y,(- y)) is right_complementable Element of the carrier of X
[y,(- y)] is set
the addF of X . [y,(- y)] is set
seq - (y - y) is right_complementable Element of the carrier of X
- (y - y) is right_complementable Element of the carrier of X
seq + (- (y - y)) is right_complementable Element of the carrier of X
K411( the carrier of X, the addF of X,seq,(- (y - y))) is right_complementable Element of the carrier of X
[seq,(- (y - y))] is set
the addF of X . [seq,(- (y - y))] is set
x * (seq - (y - y)) is right_complementable Element of the carrier of X
K411( the carrier of X, the multF of X,x,(seq - (y - y))) is right_complementable Element of the carrier of X
[x,(seq - (y - y))] is set
the multF of X . [x,(seq - (y - y))] is set
seq - (0. X) is right_complementable Element of the carrier of X
- (0. X) is right_complementable Element of the carrier of X
seq + (- (0. X)) is right_complementable Element of the carrier of X
K411( the carrier of X, the addF of X,seq,(- (0. X))) is right_complementable Element of the carrier of X
[seq,(- (0. X))] is set
the addF of X . [seq,(- (0. X))] is set
x * (seq - (0. X)) is right_complementable Element of the carrier of X
K411( the carrier of X, the multF of X,x,(seq - (0. X))) is right_complementable Element of the carrier of X
[x,(seq - (0. X))] is set
the multF of X . [x,(seq - (0. X))] is set
(0. X) + (0. X) is right_complementable Element of the carrier of X
K411( the carrier of X, the addF of X,(0. X),(0. X)) is right_complementable Element of the carrier of X
[(0. X),(0. X)] is set
the addF of X . [(0. X),(0. X)] is set
x * ((0. X) + (0. X)) is right_complementable Element of the carrier of X
K411( the carrier of X, the multF of X,x,((0. X) + (0. X))) is right_complementable Element of the carrier of X
[x,((0. X) + (0. X))] is set
the multF of X . [x,((0. X) + (0. X))] is set
(x * (0. X)) + (x * (0. X)) is right_complementable Element of the carrier of X
K411( the carrier of X, the addF of X,(x * (0. X)),(x * (0. X))) is right_complementable Element of the carrier of X
[(x * (0. X)),(x * (0. X))] is set
the addF of X . [(x * (0. X)),(x * (0. X))] is set
((x * (0. X)) + (x * (0. X))) - (x * (0. X)) is right_complementable Element of the carrier of X
- (x * (0. X)) is right_complementable Element of the carrier of X
((x * (0. X)) + (x * (0. X))) + (- (x * (0. X))) is right_complementable Element of the carrier of X
K411( the carrier of X, the addF of X,((x * (0. X)) + (x * (0. X))),(- (x * (0. X)))) is right_complementable Element of the carrier of X
[((x * (0. X)) + (x * (0. X))),(- (x * (0. X)))] is set
the addF of X . [((x * (0. X)) + (x * (0. X))),(- (x * (0. X)))] is set
(x * (0. X)) - (x * (0. X)) is right_complementable Element of the carrier of X
(x * (0. X)) + (- (x * (0. X))) is right_complementable Element of the carrier of X
K411( the carrier of X, the addF of X,(x * (0. X)),(- (x * (0. X)))) is right_complementable Element of the carrier of X
[(x * (0. X)),(- (x * (0. X)))] is set
the addF of X . [(x * (0. X)),(- (x * (0. X)))] is set
(x * (0. X)) + ((x * (0. X)) - (x * (0. X))) is right_complementable Element of the carrier of X
K411( the carrier of X, the addF of X,(x * (0. X)),((x * (0. X)) - (x * (0. X)))) is right_complementable Element of the carrier of X
[(x * (0. X)),((x * (0. X)) - (x * (0. X)))] is set
the addF of X . [(x * (0. X)),((x * (0. X)) - (x * (0. X)))] is set
(x * (0. X)) + (0. X) is right_complementable Element of the carrier of X
K411( the carrier of X, the addF of X,(x * (0. X)),(0. X)) is right_complementable Element of the carrier of X
[(x * (0. X)),(0. X)] is set
the addF of X . [(x * (0. X)),(0. X)] is set
((0. X) + (0. X)) * x is right_complementable Element of the carrier of X
K411( the carrier of X, the multF of X,((0. X) + (0. X)),x) is right_complementable Element of the carrier of X
[((0. X) + (0. X)),x] is set
the multF of X . [((0. X) + (0. X)),x] is set
((0. X) * x) + ((0. X) * x) is right_complementable Element of the carrier of X
K411( the carrier of X, the addF of X,((0. X) * x),((0. X) * x)) is right_complementable Element of the carrier of X
[((0. X) * x),((0. X) * x)] is set
the addF of X . [((0. X) * x),((0. X) * x)] is set
(((0. X) * x) + ((0. X) * x)) - ((0. X) * x) is right_complementable Element of the carrier of X
- ((0. X) * x) is right_complementable Element of the carrier of X
(((0. X) * x) + ((0. X) * x)) + (- ((0. X) * x)) is right_complementable Element of the carrier of X
K411( the carrier of X, the addF of X,(((0. X) * x) + ((0. X) * x)),(- ((0. X) * x))) is right_complementable Element of the carrier of X
[(((0. X) * x) + ((0. X) * x)),(- ((0. X) * x))] is set
the addF of X . [(((0. X) * x) + ((0. X) * x)),(- ((0. X) * x))] is set
((0. X) * x) - ((0. X) * x) is right_complementable Element of the carrier of X
((0. X) * x) + (- ((0. X) * x)) is right_complementable Element of the carrier of X
K411( the carrier of X, the addF of X,((0. X) * x),(- ((0. X) * x))) is right_complementable Element of the carrier of X
[((0. X) * x),(- ((0. X) * x))] is set
the addF of X . [((0. X) * x),(- ((0. X) * x))] is set
((0. X) * x) + (((0. X) * x) - ((0. X) * x)) is right_complementable Element of the carrier of X
K411( the carrier of X, the addF of X,((0. X) * x),(((0. X) * x) - ((0. X) * x))) is right_complementable Element of the carrier of X
[((0. X) * x),(((0. X) * x) - ((0. X) * x))] is set
the addF of X . [((0. X) * x),(((0. X) * x) - ((0. X) * x))] is set
((0. X) * x) + (0. X) is right_complementable Element of the carrier of X
K411( the carrier of X, the addF of X,((0. X) * x),(0. X)) is right_complementable Element of the carrier of X
[((0. X) * x),(0. X)] is set
the addF of X . [((0. X) * x),(0. X)] is set
(x * (seq - y)) + (0. X) is right_complementable Element of the carrier of X
K411( the carrier of X, the addF of X,(x * (seq - y)),(0. X)) is right_complementable Element of the carrier of X
[(x * (seq - y)),(0. X)] is set
the addF of X . [(x * (seq - y)),(0. X)] is set
(x * y) - (x * y) is right_complementable Element of the carrier of X
(x * y) + (- (x * y)) is right_complementable Element of the carrier of X
K411( the carrier of X, the addF of X,(x * y),(- (x * y))) is right_complementable Element of the carrier of X
[(x * y),(- (x * y))] is set
the addF of X . [(x * y),(- (x * y))] is set
(x * (seq - y)) + ((x * y) - (x * y)) is right_complementable Element of the carrier of X
K411( the carrier of X, the addF of X,(x * (seq - y)),((x * y) - (x * y))) is right_complementable Element of the carrier of X
[(x * (seq - y)),((x * y) - (x * y))] is set
the addF of X . [(x * (seq - y)),((x * y) - (x * y))] is set
((seq - y) * x) + (y * x) is right_complementable Element of the carrier of X
K411( the carrier of X, the addF of X,((seq - y) * x),(y * x)) is right_complementable Element of the carrier of X
[((seq - y) * x),(y * x)] is set
the addF of X . [((seq - y) * x),(y * x)] is set
((seq - y) + y) * x is right_complementable Element of the carrier of X
K411( the carrier of X, the multF of X,((seq - y) + y),x) is right_complementable Element of the carrier of X
[((seq - y) + y),x] is set
the multF of X . [((seq - y) + y),x] is set
(seq - (y - y)) * x is right_complementable Element of the carrier of X
K411( the carrier of X, the multF of X,(seq - (y - y)),x) is right_complementable Element of the carrier of X
[(seq - (y - y)),x] is set
the multF of X . [(seq - (y - y)),x] is set
(seq - (0. X)) * x is right_complementable Element of the carrier of X
K411( the carrier of X, the multF of X,(seq - (0. X)),x) is right_complementable Element of the carrier of X
[(seq - (0. X)),x] is set
the multF of X . [(seq - (0. X)),x] is set
((seq - y) * x) + (0. X) is right_complementable Element of the carrier of X
K411( the carrier of X, the addF of X,((seq - y) * x),(0. X)) is right_complementable Element of the carrier of X
[((seq - y) * x),(0. X)] is set
the addF of X . [((seq - y) * x),(0. X)] is set
(y * x) - (y * x) is right_complementable Element of the carrier of X
(y * x) + (- (y * x)) is right_complementable Element of the carrier of X
K411( the carrier of X, the addF of X,(y * x),(- (y * x))) is right_complementable Element of the carrier of X
[(y * x),(- (y * x))] is set
the addF of X . [(y * x),(- (y * x))] is set
((seq - y) * x) + ((y * x) - (y * x)) is right_complementable Element of the carrier of X
K411( the carrier of X, the addF of X,((seq - y) * x),((y * x) - (y * x))) is right_complementable Element of the carrier of X
[((seq - y) * x),((y * x) - (y * x))] is set
the addF of X . [((seq - y) * x),((y * x) - (y * x))] is set
(x - y) + y is right_complementable Element of the carrier of X
K411( the carrier of X, the addF of X,(x - y),y) is right_complementable Element of the carrier of X
[(x - y),y] is set
the addF of X . [(x - y),y] is set
((x - y) + y) + seq is right_complementable Element of the carrier of X
K411( the carrier of X, the addF of X,((x - y) + y),seq) is right_complementable Element of the carrier of X
[((x - y) + y),seq] is set
the addF of X . [((x - y) + y),seq] is set
x - (y - y) is right_complementable Element of the carrier of X
x + (- (y - y)) is right_complementable Element of the carrier of X
K411( the carrier of X, the addF of X,x,(- (y - y))) is right_complementable Element of the carrier of X
[x,(- (y - y))] is set
the addF of X . [x,(- (y - y))] is set
(x - (y - y)) + seq is right_complementable Element of the carrier of X
K411( the carrier of X, the addF of X,(x - (y - y)),seq) is right_complementable Element of the carrier of X
[(x - (y - y)),seq] is set
the addF of X . [(x - (y - y)),seq] is set
x - (0. X) is right_complementable Element of the carrier of X
x + (- (0. X)) is right_complementable Element of the carrier of X
K411( the carrier of X, the addF of X,x,(- (0. X))) is right_complementable Element of the carrier of X
[x,(- (0. X))] is set
the addF of X . [x,(- (0. X))] is set
(x - (0. X)) + seq is right_complementable Element of the carrier of X
K411( the carrier of X, the addF of X,(x - (0. X)),seq) is right_complementable Element of the carrier of X
[(x - (0. X)),seq] is set
the addF of X . [(x - (0. X)),seq] is set
((x - y) + y) - seq is right_complementable Element of the carrier of X
((x - y) + y) + (- seq) is right_complementable Element of the carrier of X
K411( the carrier of X, the addF of X,((x - y) + y),(- seq)) is right_complementable Element of the carrier of X
[((x - y) + y),(- seq)] is set
the addF of X . [((x - y) + y),(- seq)] is set
(x - (y - y)) - seq is right_complementable Element of the carrier of X
(x - (y - y)) + (- seq) is right_complementable Element of the carrier of X
K411( the carrier of X, the addF of X,(x - (y - y)),(- seq)) is right_complementable Element of the carrier of X
[(x - (y - y)),(- seq)] is set
the addF of X . [(x - (y - y)),(- seq)] is set
(x - (0. X)) - seq is right_complementable Element of the carrier of X
(x - (0. X)) + (- seq) is right_complementable Element of the carrier of X
K411( the carrier of X, the addF of X,(x - (0. X)),(- seq)) is right_complementable Element of the carrier of X
[(x - (0. X)),(- seq)] is set
the addF of X . [(x - (0. X)),(- seq)] is set
seq - seq is right_complementable Element of the carrier of X
seq + (- seq) is right_complementable Element of the carrier of X
K411( the carrier of X, the addF of X,seq,(- seq)) is right_complementable Element of the carrier of X
[seq,(- seq)] is set
the addF of X . [seq,(- seq)] is set
x - (seq - seq) is right_complementable Element of the carrier of X
- (seq - seq) is right_complementable Element of the carrier of X
x + (- (seq - seq)) is right_complementable Element of the carrier of X
K411( the carrier of X, the addF of X,x,(- (seq - seq))) is right_complementable Element of the carrier of X
[x,(- (seq - seq))] is set
the addF of X . [x,(- (seq - seq))] is set
(seq - seq) + x is right_complementable Element of the carrier of X
K411( the carrier of X, the addF of X,(seq - seq),x) is right_complementable Element of the carrier of X
[(seq - seq),x] is set
the addF of X . [(seq - seq),x] is set
(0. X) + x is right_complementable Element of the carrier of X
K411( the carrier of X, the addF of X,(0. X),x) is right_complementable Element of the carrier of X
the addF of X . [(0. X),x] is set
X is non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like V150() associative right-distributive left-distributive right_unital left_unital complete Banach_Algebra-like_1 Banach_Algebra-like_2 Banach_Algebra-like_3 Banach_Algebra-like Normed_AlgebraStr
the carrier of X is V4() set
x is right_complementable Element of the carrier of X
1. X is right_complementable Element of the carrier of X
x * (1. X) is right_complementable Element of the carrier of X
the multF of X is V4() Relation-like K33( the carrier of X, the carrier of X) -defined the carrier of X -valued Function-like V29(K33( the carrier of X, the carrier of X)) V30(K33( the carrier of X, the carrier of X), the carrier of X) Element of K32(K33(K33( the carrier of X, the carrier of X), the carrier of X))
K33( the carrier of X, the carrier of X) is V4() set
K33(K33( the carrier of X, the carrier of X), the carrier of X) is V4() set
K32(K33(K33( the carrier of X, the carrier of X), the carrier of X)) is V4() set
K411( the carrier of X, the multF of X,x,(1. X)) is right_complementable Element of the carrier of X
[x,(1. X)] is set
the multF of X . [x,(1. X)] is set
(1. X) * x is right_complementable Element of the carrier of X
K411( the carrier of X, the multF of X,(1. X),x) is right_complementable Element of the carrier of X
[(1. X),x] is set
the multF of X . [(1. X),x] is set
X is non empty associative well-unital multLoopStr
the carrier of X is V4() set
x is Element of the carrier of X
1. X is Element of the carrier of X
seq is Element of the carrier of X
x * seq is Element of the carrier of X
the multF of X is V4() Relation-like K33( the carrier of X, the carrier of X) -defined the carrier of X -valued Function-like V29(K33( the carrier of X, the carrier of X)) V30(K33( the carrier of X, the carrier of X), the carrier of X) Element of K32(K33(K33( the carrier of X, the carrier of X), the carrier of X))
K33( the carrier of X, the carrier of X) is V4() set
K33(K33( the carrier of X, the carrier of X), the carrier of X) is V4() set
K32(K33(K33( the carrier of X, the carrier of X), the carrier of X)) is V4() set
K411( the carrier of X, the multF of X,x,seq) is Element of the carrier of X
[x,seq] is set
the multF of X . [x,seq] is set
seq * x is Element of the carrier of X
K411( the carrier of X, the multF of X,seq,x) is Element of the carrier of X
[seq,x] is set
the multF of X . [seq,x] is set
y is Element of the carrier of X
y * x is Element of the carrier of X
K411( the carrier of X, the multF of X,y,x) is Element of the carrier of X
[y,x] is set
the multF of X . [y,x] is set
y * (1. X) is Element of the carrier of X
K411( the carrier of X, the multF of X,y,(1. X)) is Element of the carrier of X
[y,(1. X)] is set
the multF of X . [y,(1. X)] is set
(1. X) * seq is Element of the carrier of X
K411( the carrier of X, the multF of X,(1. X),seq) is Element of the carrier of X
[(1. X),seq] is set
the multF of X . [(1. X),seq] is set
seq is Element of the carrier of X
x * seq is Element of the carrier of X
the multF of X is V4() Relation-like K33( the carrier of X, the carrier of X) -defined the carrier of X -valued Function-like V29(K33( the carrier of X, the carrier of X)) V30(K33( the carrier of X, the carrier of X), the carrier of X) Element of K32(K33(K33( the carrier of X, the carrier of X), the carrier of X))
K33( the carrier of X, the carrier of X) is V4() set
K33(K33( the carrier of X, the carrier of X), the carrier of X) is V4() set
K32(K33(K33( the carrier of X, the carrier of X), the carrier of X)) is V4() set
K411( the carrier of X, the multF of X,x,seq) is Element of the carrier of X
[x,seq] is set
the multF of X . [x,seq] is set
seq * x is Element of the carrier of X
K411( the carrier of X, the multF of X,seq,x) is Element of the carrier of X
[seq,x] is set
the multF of X . [seq,x] is set
X is non empty Normed_AlgebraStr
the carrier of X is V4() set
K33(NAT, the carrier of X) is V4() set
K32(K33(NAT, the carrier of X)) is V4() set
seq is Element of the carrier of X
x is V4() Relation-like NAT -defined the carrier of X -valued Function-like V29( NAT ) V30( NAT , the carrier of X) Element of K32(K33(NAT, the carrier of X))
y is V4() Relation-like NAT -defined the carrier of X -valued Function-like V29( NAT ) V30( NAT , the carrier of X) Element of K32(K33(NAT, the carrier of X))
y is V4() Relation-like NAT -defined the carrier of X -valued Function-like V29( NAT ) V30( NAT , the carrier of X) Element of K32(K33(NAT, the carrier of X))
n is V4() Relation-like NAT -defined the carrier of X -valued Function-like V29( NAT ) V30( NAT , the carrier of X) Element of K32(K33(NAT, the carrier of X))
yn is epsilon-transitive epsilon-connected ordinal natural V24() real ext-real V45() V46() V52() V53() V54() V55() V56() V57() Element of NAT
y . yn is Element of the carrier of X
n . yn is Element of the carrier of X
x . yn is Element of the carrier of X
seq * (x . yn) is Element of the carrier of X
the multF of X is V4() Relation-like K33( the carrier of X, the carrier of X) -defined the carrier of X -valued Function-like V29(K33( the carrier of X, the carrier of X)) V30(K33( the carrier of X, the carrier of X), the carrier of X) Element of K32(K33(K33( the carrier of X, the carrier of X), the carrier of X))
K33( the carrier of X, the carrier of X) is V4() set
K33(K33( the carrier of X, the carrier of X), the carrier of X) is V4() set
K32(K33(K33( the carrier of X, the carrier of X), the carrier of X)) is V4() set
K411( the carrier of X, the multF of X,seq,(x . yn)) is Element of the carrier of X
[seq,(x . yn)] is set
the multF of X . [seq,(x . yn)] is set
X is non empty Normed_AlgebraStr
the carrier of X is V4() set
K33(NAT, the carrier of X) is V4() set
K32(K33(NAT, the carrier of X)) is V4() set
x is V4() Relation-like NAT -defined the carrier of X -valued Function-like V29( NAT ) V30( NAT , the carrier of X) Element of K32(K33(NAT, the carrier of X))
seq is Element of the carrier of X
y is V4() Relation-like NAT -defined the carrier of X -valued Function-like V29( NAT ) V30( NAT , the carrier of X) Element of K32(K33(NAT, the carrier of X))
y is V4() Relation-like NAT -defined the carrier of X -valued Function-like V29( NAT ) V30( NAT , the carrier of X) Element of K32(K33(NAT, the carrier of X))
n is V4() Relation-like NAT -defined the carrier of X -valued Function-like V29( NAT ) V30( NAT , the carrier of X) Element of K32(K33(NAT, the carrier of X))
yn is epsilon-transitive epsilon-connected ordinal natural V24() real ext-real V45() V46() V52() V53() V54() V55() V56() V57() Element of NAT
y . yn is Element of the carrier of X
n . yn is Element of the carrier of X
x . yn is Element of the carrier of X
(x . yn) * seq is Element of the carrier of X
the multF of X is V4() Relation-like K33( the carrier of X, the carrier of X) -defined the carrier of X -valued Function-like V29(K33( the carrier of X, the carrier of X)) V30(K33( the carrier of X, the carrier of X), the carrier of X) Element of K32(K33(K33( the carrier of X, the carrier of X), the carrier of X))
K33( the carrier of X, the carrier of X) is V4() set
K33(K33( the carrier of X, the carrier of X), the carrier of X) is V4() set
K32(K33(K33( the carrier of X, the carrier of X), the carrier of X)) is V4() set
K411( the carrier of X, the multF of X,(x . yn),seq) is Element of the carrier of X
[(x . yn),seq] is set
the multF of X . [(x . yn),seq] is set
X is non empty Normed_AlgebraStr
the carrier of X is V4() set
K33(NAT, the carrier of X) is V4() set
K32(K33(NAT, the carrier of X)) is V4() set
x is V4() Relation-like NAT -defined the carrier of X -valued Function-like V29( NAT ) V30( NAT , the carrier of X) Element of K32(K33(NAT, the carrier of X))
seq is V4() Relation-like NAT -defined the carrier of X -valued Function-like V29( NAT ) V30( NAT , the carrier of X) Element of K32(K33(NAT, the carrier of X))
y is V4() Relation-like NAT -defined the carrier of X -valued Function-like V29( NAT ) V30( NAT , the carrier of X) Element of K32(K33(NAT, the carrier of X))
y is V4() Relation-like NAT -defined the carrier of X -valued Function-like V29( NAT ) V30( NAT , the carrier of X) Element of K32(K33(NAT, the carrier of X))
n is V4() Relation-like NAT -defined the carrier of X -valued Function-like V29( NAT ) V30( NAT , the carrier of X) Element of K32(K33(NAT, the carrier of X))
yn is epsilon-transitive epsilon-connected ordinal natural V24() real ext-real V45() V46() V52() V53() V54() V55() V56() V57() Element of NAT
y . yn is Element of the carrier of X
n . yn is Element of the carrier of X
x . yn is Element of the carrier of X
seq . yn is Element of the carrier of X
(x . yn) * (seq . yn) is Element of the carrier of X
the multF of X is V4() Relation-like K33( the carrier of X, the carrier of X) -defined the carrier of X -valued Function-like V29(K33( the carrier of X, the carrier of X)) V30(K33( the carrier of X, the carrier of X), the carrier of X) Element of K32(K33(K33( the carrier of X, the carrier of X), the carrier of X))
K33( the carrier of X, the carrier of X) is V4() set
K33(K33( the carrier of X, the carrier of X), the carrier of X) is V4() set
K32(K33(K33( the carrier of X, the carrier of X), the carrier of X)) is V4() set
K411( the carrier of X, the multF of X,(x . yn),(seq . yn)) is Element of the carrier of X
[(x . yn),(seq . yn)] is set
the multF of X . [(x . yn),(seq . yn)] is set
X is non empty associative well-unital multLoopStr
the carrier of X is V4() set
X is non empty associative well-unital multLoopStr
the carrier of X is V4() set
x is Element of the carrier of X
/ x is Element of the carrier of X
1. X is Element of the carrier of X
seq is Element of the carrier of X
x * seq is Element of the carrier of X
the multF of X is V4() Relation-like K33( the carrier of X, the carrier of X) -defined the carrier of X -valued Function-like V29(K33( the carrier of X, the carrier of X)) V30(K33( the carrier of X, the carrier of X), the carrier of X) Element of K32(K33(K33( the carrier of X, the carrier of X), the carrier of X))
K33( the carrier of X, the carrier of X) is V4() set
K33(K33( the carrier of X, the carrier of X), the carrier of X) is V4() set
K32(K33(K33( the carrier of X, the carrier of X), the carrier of X)) is V4() set
K411( the carrier of X, the multF of X,x,seq) is Element of the carrier of X
[x,seq] is set
the multF of X . [x,seq] is set
seq * x is Element of the carrier of X
K411( the carrier of X, the multF of X,seq,x) is Element of the carrier of X
[seq,x] is set
the multF of X . [seq,x] is set
y is Element of the carrier of X
x * y is Element of the carrier of X
K411( the carrier of X, the multF of X,x,y) is Element of the carrier of X
[x,y] is set
the multF of X . [x,y] is set
n is Element of the carrier of X
n * x is Element of the carrier of X
K411( the carrier of X, the multF of X,n,x) is Element of the carrier of X
[n,x] is set
the multF of X . [n,x] is set
yn is Element of the carrier of X
yn * x is Element of the carrier of X
K411( the carrier of X, the multF of X,yn,x) is Element of the carrier of X
[yn,x] is set
the multF of X . [yn,x] is set
n * (1. X) is Element of the carrier of X
K411( the carrier of X, the multF of X,n,(1. X)) is Element of the carrier of X
[n,(1. X)] is set
the multF of X . [n,(1. X)] is set
(yn * x) * y is Element of the carrier of X
K411( the carrier of X, the multF of X,(yn * x),y) is Element of the carrier of X
[(yn * x),y] is set
the multF of X . [(yn * x),y] is set
yn * (1. X) is Element of the carrier of X
K411( the carrier of X, the multF of X,yn,(1. X)) is Element of the carrier of X
[yn,(1. X)] is set
the multF of X . [yn,(1. X)] is set
seq * (1. X) is Element of the carrier of X
K411( the carrier of X, the multF of X,seq,(1. X)) is Element of the carrier of X
[seq,(1. X)] is set
the multF of X . [seq,(1. X)] is set
(seq * x) * y is Element of the carrier of X
K411( the carrier of X, the multF of X,(seq * x),y) is Element of the carrier of X
[(seq * x),y] is set
the multF of X . [(seq * x),y] is set
(1. X) * y is Element of the carrier of X
K411( the carrier of X, the multF of X,(1. X),y) is Element of the carrier of X
[(1. X),y] is set
the multF of X . [(1. X),y] is set
X is non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like V150() associative right-distributive left-distributive right_unital well-unital left_unital complete Banach_Algebra-like_1 Banach_Algebra-like_2 Banach_Algebra-like_3 Banach_Algebra-like Normed_AlgebraStr
the carrier of X is V4() set
K33(NAT, the carrier of X) is V4() set
K32(K33(NAT, the carrier of X)) is V4() set
1. X is right_complementable Element of the carrier of X
x is right_complementable Element of the carrier of X
the multF of X is V4() Relation-like K33( the carrier of X, the carrier of X) -defined the carrier of X -valued Function-like V29(K33( the carrier of X, the carrier of X)) V30(K33( the carrier of X, the carrier of X), the carrier of X) Element of K32(K33(K33( the carrier of X, the carrier of X), the carrier of X))
K33( the carrier of X, the carrier of X) is V4() set
K33(K33( the carrier of X, the carrier of X), the carrier of X) is V4() set
K32(K33(K33( the carrier of X, the carrier of X), the carrier of X)) is V4() set
seq is Relation-like Function-like set
dom seq is set
seq . 0 is set
y is epsilon-transitive epsilon-connected ordinal natural V24() real ext-real V45() V46() V52() V53() V54() V55() V56() V57() Element of NAT
seq . y is set
y + 1 is epsilon-transitive epsilon-connected ordinal natural V24() real ext-real V45() V46() V52() V53() V54() V55() V56() V57() Element of NAT
seq . (y + 1) is set
n is right_complementable Element of the carrier of X
[n,x] is Element of K33( the carrier of X, the carrier of X)
the multF of X . [n,x] is right_complementable Element of the carrier of X
y is set
seq . y is set
y is V4() Relation-like NAT -defined the carrier of X -valued Function-like V29( NAT ) V30( NAT , the carrier of X) Element of K32(K33(NAT, the carrier of X))
y . 0 is right_complementable Element of the carrier of X
n is epsilon-transitive epsilon-connected ordinal natural V24() real ext-real V45() V46() V52() V53() V54() V55() V56() V57() Element of NAT
n + 1 is epsilon-transitive epsilon-connected ordinal natural V24() real ext-real V45() V46() V52() V53() V54() V55() V56() V57() Element of NAT
y . (n + 1) is right_complementable Element of the carrier of X
y . n is right_complementable Element of the carrier of X
(y . n) * x is right_complementable Element of the carrier of X
K411( the carrier of X, the multF of X,(y . n),x) is right_complementable Element of the carrier of X
[(y . n),x] is set
the multF of X . [(y . n),x] is set
seq is V4() Relation-like NAT -defined the carrier of X -valued Function-like V29( NAT ) V30( NAT , the carrier of X) Element of K32(K33(NAT, the carrier of X))
seq . 0 is right_complementable Element of the carrier of X
y is V4() Relation-like NAT -defined the carrier of X -valued Function-like V29( NAT ) V30( NAT , the carrier of X) Element of K32(K33(NAT, the carrier of X))
y . 0 is right_complementable Element of the carrier of X
n is epsilon-transitive epsilon-connected ordinal natural V24() real ext-real V45() V46() V52() V53() V54() V55() V56() V57() Element of NAT
seq . n is right_complementable Element of the carrier of X
y . n is right_complementable Element of the carrier of X
n + 1 is epsilon-transitive epsilon-connected ordinal natural V24() real ext-real V45() V46() V52() V53() V54() V55() V56() V57() Element of NAT
seq . (n + 1) is right_complementable Element of the carrier of X
y . (n + 1) is right_complementable Element of the carrier of X
(y . n) * x is right_complementable Element of the carrier of X
the multF of X is V4() Relation-like K33( the carrier of X, the carrier of X) -defined the carrier of X -valued Function-like V29(K33( the carrier of X, the carrier of X)) V30(K33( the carrier of X, the carrier of X), the carrier of X) Element of K32(K33(K33( the carrier of X, the carrier of X), the carrier of X))
K33( the carrier of X, the carrier of X) is V4() set
K33(K33( the carrier of X, the carrier of X), the carrier of X) is V4() set
K32(K33(K33( the carrier of X, the carrier of X), the carrier of X)) is V4() set
K411( the carrier of X, the multF of X,(y . n),x) is right_complementable Element of the carrier of X
[(y . n),x] is set
the multF of X . [(y . n),x] is set
X is non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like V150() associative right-distributive left-distributive right_unital well-unital left_unital complete Banach_Algebra-like_1 Banach_Algebra-like_2 Banach_Algebra-like_3 Banach_Algebra-like Normed_AlgebraStr
the carrier of X is V4() set
x is right_complementable Element of the carrier of X
(X,x) is V4() Relation-like NAT -defined the carrier of X -valued Function-like V29( NAT ) V30( NAT , the carrier of X) Element of K32(K33(NAT, the carrier of X))
K33(NAT, the carrier of X) is V4() set
K32(K33(NAT, the carrier of X)) is V4() set
seq is epsilon-transitive epsilon-connected ordinal natural V24() real ext-real V45() V46() V52() V53() V54() V55() V56() V57() Element of NAT
(X,x) . seq is right_complementable Element of the carrier of X
X is non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like V150() associative right-distributive left-distributive right_unital well-unital left_unital complete Banach_Algebra-like_1 Banach_Algebra-like_2 Banach_Algebra-like_3 Banach_Algebra-like Normed_AlgebraStr
the carrier of X is V4() set
x is right_complementable Element of the carrier of X
(X,x,0) is right_complementable Element of the carrier of X
(X,x) is V4() Relation-like NAT -defined the carrier of X -valued Function-like V29( NAT ) V30( NAT , the carrier of X) Element of K32(K33(NAT, the carrier of X))
K33(NAT, the carrier of X) is V4() set
K32(K33(NAT, the carrier of X)) is V4() set
(X,x) . 0 is right_complementable Element of the carrier of X
1. X is right_complementable Element of the carrier of X
X is non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like V150() associative right-distributive left-distributive right_unital well-unital left_unital complete Banach_Algebra-like_1 Banach_Algebra-like_2 Banach_Algebra-like_3 Banach_Algebra-like Normed_AlgebraStr
the carrier of X is V4() set
x is right_complementable Element of the carrier of X
||.x.|| is V24() real ext-real non negative Element of REAL
(X,x) is V4() Relation-like NAT -defined the carrier of X -valued Function-like V29( NAT ) V30( NAT , the carrier of X) Element of K32(K33(NAT, the carrier of X))
K33(NAT, the carrier of X) is V4() set
K32(K33(NAT, the carrier of X)) is V4() set
||.(X,x).|| is V4() Relation-like NAT -defined REAL -valued Function-like V29( NAT ) V30( NAT , REAL ) complex-valued ext-real-valued real-valued Element of K32(K33(NAT,REAL))
||.x.|| GeoSeq is V4() Relation-like NAT -defined REAL -valued Function-like V29( NAT ) V30( NAT , REAL ) complex-valued ext-real-valued real-valued Element of K32(K33(NAT,REAL))
||.(X,x).|| . 0 is V24() real ext-real Element of REAL
(X,x) . 0 is right_complementable Element of the carrier of X
||.((X,x) . 0).|| is V24() real ext-real non negative Element of REAL
seq is epsilon-transitive epsilon-connected ordinal natural V24() real ext-real V45() V46() V52() V53() V54() V55() V56() V57() Element of NAT
||.(X,x).|| . seq is V24() real ext-real Element of REAL
(||.x.|| GeoSeq) . seq is V24() real ext-real Element of REAL
seq + 1 is epsilon-transitive epsilon-connected ordinal natural V24() real ext-real V45() V46() V52() V53() V54() V55() V56() V57() Element of NAT
||.(X,x).|| . (seq + 1) is V24() real ext-real Element of REAL
(||.x.|| GeoSeq) . (seq + 1) is V24() real ext-real Element of REAL
(X,x) . seq is right_complementable Element of the carrier of X
((X,x) . seq) * x is right_complementable Element of the carrier of X
the multF of X is V4() Relation-like K33( the carrier of X, the carrier of X) -defined the carrier of X -valued Function-like V29(K33( the carrier of X, the carrier of X)) V30(K33( the carrier of X, the carrier of X), the carrier of X) Element of K32(K33(K33( the carrier of X, the carrier of X), the carrier of X))
K33( the carrier of X, the carrier of X) is V4() set
K33(K33( the carrier of X, the carrier of X), the carrier of X) is V4() set
K32(K33(K33( the carrier of X, the carrier of X), the carrier of X)) is V4() set
K411( the carrier of X, the multF of X,((X,x) . seq),x) is right_complementable Element of the carrier of X
[((X,x) . seq),x] is set
the multF of X . [((X,x) . seq),x] is set
||.(((X,x) . seq) * x).|| is V24() real ext-real non negative Element of REAL
||.((X,x) . seq).|| is V24() real ext-real non negative Element of REAL
||.((X,x) . seq).|| * ||.x.|| is V24() real ext-real non negative Element of REAL
(||.(X,x).|| . seq) * ||.x.|| is V24() real ext-real Element of REAL
((||.x.|| GeoSeq) . seq) * ||.x.|| is V24() real ext-real Element of REAL
(X,x) . (seq + 1) is right_complementable Element of the carrier of X
||.((X,x) . (seq + 1)).|| is V24() real ext-real non negative Element of REAL
1. X is right_complementable Element of the carrier of X
||.(1. X).|| is V24() real ext-real non negative Element of REAL
(||.x.|| GeoSeq) . 0 is V24() real ext-real Element of REAL
seq is epsilon-transitive epsilon-connected ordinal natural V24() real ext-real V45() V46() V52() V53() V54() V55() V56() V57() Element of NAT
||.(X,x).|| . seq is V24() real ext-real Element of REAL
y is epsilon-transitive epsilon-connected ordinal natural V24() real ext-real V45() V46() V52() V53() V54() V55() V56() V57() Element of NAT
||.(X,x).|| . y is V24() real ext-real Element of REAL
(||.x.|| GeoSeq) . y is V24() real ext-real Element of REAL
abs ||.x.|| is V24() real ext-real Element of REAL
X is non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like V150() associative right-distributive left-distributive right_unital well-unital left_unital complete Banach_Algebra-like_1 Banach_Algebra-like_2 Banach_Algebra-like_3 Banach_Algebra-like Normed_AlgebraStr
the carrier of X is V4() set
1. X is right_complementable Element of the carrier of X
x is right_complementable Element of the carrier of X
(1. X) - x is right_complementable Element of the carrier of X
- x is right_complementable Element of the carrier of X
(1. X) + (- x) is right_complementable Element of the carrier of X
the addF of X is V4() Relation-like K33( the carrier of X, the carrier of X) -defined the carrier of X -valued Function-like V29(K33( the carrier of X, the carrier of X)) V30(K33( the carrier of X, the carrier of X), the carrier of X) Element of K32(K33(K33( the carrier of X, the carrier of X), the carrier of X))
K33( the carrier of X, the carrier of X) is V4() set
K33(K33( the carrier of X, the carrier of X), the carrier of X) is V4() set
K32(K33(K33( the carrier of X, the carrier of X), the carrier of X)) is V4() set
K411( the carrier of X, the addF of X,(1. X),(- x)) is right_complementable Element of the carrier of X
[(1. X),(- x)] is set
the addF of X . [(1. X),(- x)] is set
||.((1. X) - x).|| is V24() real ext-real non negative Element of REAL
(X,((1. X) - x)) is V4() Relation-like NAT -defined the carrier of X -valued Function-like V29( NAT ) V30( NAT , the carrier of X) Element of K32(K33(NAT, the carrier of X))
K33(NAT, the carrier of X) is V4() set
K32(K33(NAT, the carrier of X)) is V4() set
X is non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like NORMSTR
the carrier of X is V4() set
0. X is V79(X) right_complementable Element of the carrier of X
x is right_complementable Element of the carrier of X
||.x.|| is V24() real ext-real non negative Element of REAL
X is non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like NORMSTR
the carrier of X is V4() set
x is right_complementable Element of the carrier of X
seq is right_complementable Element of the carrier of X
x - seq is right_complementable Element of the carrier of X
- seq is right_complementable Element of the carrier of X
x + (- seq) is right_complementable Element of the carrier of X
the addF of X is V4() Relation-like K33( the carrier of X, the carrier of X) -defined the carrier of X -valued Function-like V29(K33( the carrier of X, the carrier of X)) V30(K33( the carrier of X, the carrier of X), the carrier of X) Element of K32(K33(K33( the carrier of X, the carrier of X), the carrier of X))
K33( the carrier of X, the carrier of X) is V4() set
K33(K33( the carrier of X, the carrier of X), the carrier of X) is V4() set
K32(K33(K33( the carrier of X, the carrier of X), the carrier of X)) is V4() set
K411( the carrier of X, the addF of X,x,(- seq)) is right_complementable Element of the carrier of X
[x,(- seq)] is set
the addF of X . [x,(- seq)] is set
||.(x - seq).|| is V24() real ext-real non negative Element of REAL
0. X is V79(X) right_complementable Element of the carrier of X
X is non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like NORMSTR
the carrier of X is V4() set
x is right_complementable Element of the carrier of X
seq is right_complementable Element of the carrier of X
x - seq is right_complementable Element of the carrier of X
- seq is right_complementable Element of the carrier of X
x + (- seq) is right_complementable Element of the carrier of X
the addF of X is V4() Relation-like K33( the carrier of X, the carrier of X) -defined the carrier of X -valued Function-like V29(K33( the carrier of X, the carrier of X)) V30(K33( the carrier of X, the carrier of X), the carrier of X) Element of K32(K33(K33( the carrier of X, the carrier of X), the carrier of X))
K33( the carrier of X, the carrier of X) is V4() set
K33(K33( the carrier of X, the carrier of X), the carrier of X) is V4() set
K32(K33(K33( the carrier of X, the carrier of X), the carrier of X)) is V4() set
K411( the carrier of X, the addF of X,x,(- seq)) is right_complementable Element of the carrier of X
[x,(- seq)] is set
the addF of X . [x,(- seq)] is set
||.(x - seq).|| is V24() real ext-real non negative Element of REAL
y is V4() Relation-like NAT -defined REAL -valued Function-like V29( NAT ) V30( NAT , REAL ) complex-valued ext-real-valued real-valued Element of K32(K33(NAT,REAL))
lim y is V24() real ext-real Element of REAL
n is V24() real ext-real Element of REAL
yn is epsilon-transitive epsilon-connected ordinal natural V24() real ext-real V45() V46() V52() V53() V54() V55() V56() V57() Element of NAT
y . yn is V24() real ext-real Element of REAL
abs (y . yn) is V24() real ext-real Element of REAL
(y . yn) - 0 is V24() real ext-real Element of REAL
abs ((y . yn) - 0) is V24() real ext-real Element of REAL
X is non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like V150() associative right-distributive left-distributive right_unital well-unital left_unital complete Banach_Algebra-like_1 Banach_Algebra-like_2 Banach_Algebra-like_3 Banach_Algebra-like Normed_AlgebraStr
the carrier of X is V4() set
1. X is right_complementable Element of the carrier of X
x is right_complementable Element of the carrier of X
(1. X) - x is right_complementable Element of the carrier of X
- x is right_complementable Element of the carrier of X
(1. X) + (- x) is right_complementable Element of the carrier of X
the addF of X is V4() Relation-like K33( the carrier of X, the carrier of X) -defined the carrier of X -valued Function-like V29(K33( the carrier of X, the carrier of X)) V30(K33( the carrier of X, the carrier of X), the carrier of X) Element of K32(K33(K33( the carrier of X, the carrier of X), the carrier of X))
K33( the carrier of X, the carrier of X) is V4() set
K33(K33( the carrier of X, the carrier of X), the carrier of X) is V4() set
K32(K33(K33( the carrier of X, the carrier of X), the carrier of X)) is V4() set
K411( the carrier of X, the addF of X,(1. X),(- x)) is right_complementable Element of the carrier of X
[(1. X),(- x)] is set
the addF of X . [(1. X),(- x)] is set
||.((1. X) - x).|| is V24() real ext-real non negative Element of REAL
/ x is right_complementable Element of the carrier of X
(X,((1. X) - x)) is V4() Relation-like NAT -defined the carrier of X -valued Function-like V29( NAT ) V30( NAT , the carrier of X) Element of K32(K33(NAT, the carrier of X))
K33(NAT, the carrier of X) is V4() set
K32(K33(NAT, the carrier of X)) is V4() set
(X,(X,((1. X) - x))) is right_complementable Element of the carrier of X
Partial_Sums (X,((1. X) - x)) is V4() Relation-like NAT -defined the carrier of X -valued Function-like V29( NAT ) V30( NAT , the carrier of X) Element of K32(K33(NAT, the carrier of X))
lim (Partial_Sums (X,((1. X) - x))) is right_complementable Element of the carrier of X
(X,((1. X) - x)) ^\ 1 is V4() Relation-like NAT -defined the carrier of X -valued Function-like V29( NAT ) V30( NAT , the carrier of X) subsequence of (X,((1. X) - x))
||.((X,((1. X) - x)) ^\ 1).|| is V4() Relation-like NAT -defined REAL -valued Function-like V29( NAT ) V30( NAT , REAL ) complex-valued ext-real-valued real-valued Element of K32(K33(NAT,REAL))
y is right_complementable Element of the carrier of X
(Partial_Sums (X,((1. X) - x))) - y is V4() Relation-like NAT -defined the carrier of X -valued Function-like V29( NAT ) V30( NAT , the carrier of X) Element of K32(K33(NAT, the carrier of X))
||.((Partial_Sums (X,((1. X) - x))) - y).|| is V4() Relation-like NAT -defined REAL -valued Function-like V29( NAT ) V30( NAT , REAL ) complex-valued ext-real-valued real-valued Element of K32(K33(NAT,REAL))
lim ||.((Partial_Sums (X,((1. X) - x))) - y).|| is V24() real ext-real Element of REAL
||.x.|| is V24() real ext-real non negative Element of REAL
||.x.|| (#) ||.((Partial_Sums (X,((1. X) - x))) - y).|| is V4() Relation-like NAT -defined REAL -valued Function-like V29( NAT ) V30( NAT , REAL ) complex-valued ext-real-valued real-valued Element of K32(K33(NAT,REAL))
lim (||.x.|| (#) ||.((Partial_Sums (X,((1. X) - x))) - y).||) is V24() real ext-real Element of REAL
||.x.|| * 0 is V4() epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural V24() real ext-real non positive non negative V52() V53() V54() V55() V56() V57() V58() Element of REAL
lim ((X,((1. X) - x)) ^\ 1) is right_complementable Element of the carrier of X
0. X is V79(X) right_complementable Element of the carrier of X
lim ||.((X,((1. X) - x)) ^\ 1).|| is V24() real ext-real Element of REAL
||.(0. X).|| is V4() epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural V24() real ext-real non positive non negative V52() V53() V54() V55() V56() V57() V58() Element of REAL
||.((X,((1. X) - x)) ^\ 1).|| + (||.x.|| (#) ||.((Partial_Sums (X,((1. X) - x))) - y).||) is V4() Relation-like NAT -defined REAL -valued Function-like V29( NAT ) V30( NAT , REAL ) complex-valued ext-real-valued real-valued Element of K32(K33(NAT,REAL))
lim (||.((X,((1. X) - x)) ^\ 1).|| + (||.x.|| (#) ||.((Partial_Sums (X,((1. X) - x))) - y).||)) is V24() real ext-real Element of REAL
(lim ||.((X,((1. X) - x)) ^\ 1).||) + (lim (||.x.|| (#) ||.((Partial_Sums (X,((1. X) - x))) - y).||)) is V24() real ext-real Element of REAL
(X,((1. X) - x)) . 0 is right_complementable Element of the carrier of X
((1. X) - x) * ((X,((1. X) - x)) . 0) is right_complementable Element of the carrier of X
the multF of X is V4() Relation-like K33( the carrier of X, the carrier of X) -defined the carrier of X -valued Function-like V29(K33( the carrier of X, the carrier of X)) V30(K33( the carrier of X, the carrier of X), the carrier of X) Element of K32(K33(K33( the carrier of X, the carrier of X), the carrier of X))
K411( the carrier of X, the multF of X,((1. X) - x),((X,((1. X) - x)) . 0)) is right_complementable Element of the carrier of X
[((1. X) - x),((X,((1. X) - x)) . 0)] is set
the multF of X . [((1. X) - x),((X,((1. X) - x)) . 0)] is set
((1. X) - x) * (1. X) is right_complementable Element of the carrier of X
K411( the carrier of X, the multF of X,((1. X) - x),(1. X)) is right_complementable Element of the carrier of X
[((1. X) - x),(1. X)] is set
the multF of X . [((1. X) - x),(1. X)] is set
n is epsilon-transitive epsilon-connected ordinal natural V24() real ext-real V45() V46() V52() V53() V54() V55() V56() V57() Element of NAT
(X,((1. X) - x)) . n is right_complementable Element of the carrier of X
((1. X) - x) * ((X,((1. X) - x)) . n) is right_complementable Element of the carrier of X
K411( the carrier of X, the multF of X,((1. X) - x),((X,((1. X) - x)) . n)) is right_complementable Element of the carrier of X
[((1. X) - x),((X,((1. X) - x)) . n)] is set
the multF of X . [((1. X) - x),((X,((1. X) - x)) . n)] is set
n + 1 is epsilon-transitive epsilon-connected ordinal natural V24() real ext-real V45() V46() V52() V53() V54() V55() V56() V57() Element of NAT
(X,((1. X) - x)) . (n + 1) is right_complementable Element of the carrier of X
((1. X) - x) * ((X,((1. X) - x)) . (n + 1)) is right_complementable Element of the carrier of X
K411( the carrier of X, the multF of X,((1. X) - x),((X,((1. X) - x)) . (n + 1))) is right_complementable Element of the carrier of X
[((1. X) - x),((X,((1. X) - x)) . (n + 1))] is set
the multF of X . [((1. X) - x),((X,((1. X) - x)) . (n + 1))] is set
(n + 1) + 1 is epsilon-transitive epsilon-connected ordinal natural V24() real ext-real V45() V46() V52() V53() V54() V55() V56() V57() Element of NAT
(X,((1. X) - x)) . ((n + 1) + 1) is right_complementable Element of the carrier of X
((X,((1. X) - x)) . n) * ((1. X) - x) is right_complementable Element of the carrier of X
K411( the carrier of X, the multF of X,((X,((1. X) - x)) . n),((1. X) - x)) is right_complementable Element of the carrier of X
[((X,((1. X) - x)) . n),((1. X) - x)] is set
the multF of X . [((X,((1. X) - x)) . n),((1. X) - x)] is set
((1. X) - x) * (((X,((1. X) - x)) . n) * ((1. X) - x)) is right_complementable Element of the carrier of X
K411( the carrier of X, the multF of X,((1. X) - x),(((X,((1. X) - x)) . n) * ((1. X) - x))) is right_complementable Element of the carrier of X
[((1. X) - x),(((X,((1. X) - x)) . n) * ((1. X) - x))] is set
the multF of X . [((1. X) - x),(((X,((1. X) - x)) . n) * ((1. X) - x))] is set
(((1. X) - x) * ((X,((1. X) - x)) . n)) * ((1. X) - x) is right_complementable Element of the carrier of X
K411( the carrier of X, the multF of X,(((1. X) - x) * ((X,((1. X) - x)) . n)),((1. X) - x)) is right_complementable Element of the carrier of X
[(((1. X) - x) * ((X,((1. X) - x)) . n)),((1. X) - x)] is set
the multF of X . [(((1. X) - x) * ((X,((1. X) - x)) . n)),((1. X) - x)] is set
0 + 1 is V4() epsilon-transitive epsilon-connected ordinal natural V24() real ext-real positive non negative V45() V46() V52() V53() V54() V55() V56() V57() Element of NAT
(X,((1. X) - x)) . (0 + 1) is right_complementable Element of the carrier of X
((X,((1. X) - x)) . 0) * ((1. X) - x) is right_complementable Element of the carrier of X
K411( the carrier of X, the multF of X,((X,((1. X) - x)) . 0),((1. X) - x)) is right_complementable Element of the carrier of X
[((X,((1. X) - x)) . 0),((1. X) - x)] is set
the multF of X . [((X,((1. X) - x)) . 0),((1. X) - x)] is set
(1. X) * ((1. X) - x) is right_complementable Element of the carrier of X
K411( the carrier of X, the multF of X,(1. X),((1. X) - x)) is right_complementable Element of the carrier of X
[(1. X),((1. X) - x)] is set
the multF of X . [(1. X),((1. X) - x)] is set
n is epsilon-transitive epsilon-connected ordinal natural V24() real ext-real V45() V46() V52() V53() V54() V55() V56() V57() Element of NAT
(X,((1. X) - x)) . n is right_complementable Element of the carrier of X
((1. X) - x) * ((X,((1. X) - x)) . n) is right_complementable Element of the carrier of X
K411( the carrier of X, the multF of X,((1. X) - x),((X,((1. X) - x)) . n)) is right_complementable Element of the carrier of X
[((1. X) - x),((X,((1. X) - x)) . n)] is set
the multF of X . [((1. X) - x),((X,((1. X) - x)) . n)] is set
n + 1 is epsilon-transitive epsilon-connected ordinal natural V24() real ext-real V45() V46() V52() V53() V54() V55() V56() V57() Element of NAT
(X,((1. X) - x)) . (n + 1) is right_complementable Element of the carrier of X
(Partial_Sums (X,((1. X) - x))) ^\ 1 is V4() Relation-like NAT -defined the carrier of X -valued Function-like V29( NAT ) V30( NAT , the carrier of X) subsequence of Partial_Sums (X,((1. X) - x))
(X,((1. X) - x)) . 0 is right_complementable Element of the carrier of X
(Partial_Sums (X,((1. X) - x))) . 0 is right_complementable Element of the carrier of X
((1. X) - x) * ((Partial_Sums (X,((1. X) - x))) . 0) is right_complementable Element of the carrier of X
the multF of X is V4() Relation-like K33( the carrier of X, the carrier of X) -defined the carrier of X -valued Function-like V29(K33( the carrier of X, the carrier of X)) V30(K33( the carrier of X, the carrier of X), the carrier of X) Element of K32(K33(K33( the carrier of X, the carrier of X), the carrier of X))
K411( the carrier of X, the multF of X,((1. X) - x),((Partial_Sums (X,((1. X) - x))) . 0)) is right_complementable Element of the carrier of X
[((1. X) - x),((Partial_Sums (X,((1. X) - x))) . 0)] is set
the multF of X . [((1. X) - x),((Partial_Sums (X,((1. X) - x))) . 0)] is set
((1. X) - x) * ((X,((1. X) - x)) . 0) is right_complementable Element of the carrier of X
K411( the carrier of X, the multF of X,((1. X) - x),((X,((1. X) - x)) . 0)) is right_complementable Element of the carrier of X
[((1. X) - x),((X,((1. X) - x)) . 0)] is set
the multF of X . [((1. X) - x),((X,((1. X) - x)) . 0)] is set
((1. X) - x) * (1. X) is right_complementable Element of the carrier of X
K411( the carrier of X, the multF of X,((1. X) - x),(1. X)) is right_complementable Element of the carrier of X
[((1. X) - x),(1. X)] is set
the multF of X . [((1. X) - x),(1. X)] is set
n is epsilon-transitive epsilon-connected ordinal natural V24() real ext-real V45() V46() V52() V53() V54() V55() V56() V57() Element of NAT
(Partial_Sums (X,((1. X) - x))) . n is right_complementable Element of the carrier of X
((1. X) - x) * ((Partial_Sums (X,((1. X) - x))) . n) is right_complementable Element of the carrier of X
K411( the carrier of X, the multF of X,((1. X) - x),((Partial_Sums (X,((1. X) - x))) . n)) is right_complementable Element of the carrier of X
[((1. X) - x),((Partial_Sums (X,((1. X) - x))) . n)] is set
the multF of X . [((1. X) - x),((Partial_Sums (X,((1. X) - x))) . n)] is set
((Partial_Sums (X,((1. X) - x))) ^\ 1) . n is right_complementable Element of the carrier of X
(((Partial_Sums (X,((1. X) - x))) ^\ 1) . n) - ((X,((1. X) - x)) . 0) is right_complementable Element of the carrier of X
- ((X,((1. X) - x)) . 0) is right_complementable Element of the carrier of X
(((Partial_Sums (X,((1. X) - x))) ^\ 1) . n) + (- ((X,((1. X) - x)) . 0)) is right_complementable Element of the carrier of X
K411( the carrier of X, the addF of X,(((Partial_Sums (X,((1. X) - x))) ^\ 1) . n),(- ((X,((1. X) - x)) . 0))) is right_complementable Element of the carrier of X
[(((Partial_Sums (X,((1. X) - x))) ^\ 1) . n),(- ((X,((1. X) - x)) . 0))] is set
the addF of X . [(((Partial_Sums (X,((1. X) - x))) ^\ 1) . n),(- ((X,((1. X) - x)) . 0))] is set
n + 1 is epsilon-transitive epsilon-connected ordinal natural V24() real ext-real V45() V46() V52() V53() V54() V55() V56() V57() Element of NAT
(Partial_Sums (X,((1. X) - x))) . (n + 1) is right_complementable Element of the carrier of X
((1. X) - x) * ((Partial_Sums (X,((1. X) - x))) . (n + 1)) is right_complementable Element of the carrier of X
K411( the carrier of X, the multF of X,((1. X) - x),((Partial_Sums (X,((1. X) - x))) . (n + 1))) is right_complementable Element of the carrier of X
[((1. X) - x),((Partial_Sums (X,((1. X) - x))) . (n + 1))] is set
the multF of X . [((1. X) - x),((Partial_Sums (X,((1. X) - x))) . (n + 1))] is set
((Partial_Sums (X,((1. X) - x))) ^\ 1) . (n + 1) is right_complementable Element of the carrier of X
(((Partial_Sums (X,((1. X) - x))) ^\ 1) . (n + 1)) - ((X,((1. X) - x)) . 0) is right_complementable Element of the carrier of X
(((Partial_Sums (X,((1. X) - x))) ^\ 1) . (n + 1)) + (- ((X,((1. X) - x)) . 0)) is right_complementable Element of the carrier of X
K411( the carrier of X, the addF of X,(((Partial_Sums (X,((1. X) - x))) ^\ 1) . (n + 1)),(- ((X,((1. X) - x)) . 0))) is right_complementable Element of the carrier of X
[(((Partial_Sums (X,((1. X) - x))) ^\ 1) . (n + 1)),(- ((X,((1. X) - x)) . 0))] is set
the addF of X . [(((Partial_Sums (X,((1. X) - x))) ^\ 1) . (n + 1)),(- ((X,((1. X) - x)) . 0))] is set
(n + 1) + 1 is epsilon-transitive epsilon-connected ordinal natural V24() real ext-real V45() V46() V52() V53() V54() V55() V56() V57() Element of NAT
(Partial_Sums (X,((1. X) - x))) . ((n + 1) + 1) is right_complementable Element of the carrier of X
((Partial_Sums (X,((1. X) - x))) . ((n + 1) + 1)) - ((X,((1. X) - x)) . 0) is right_complementable Element of the carrier of X
((Partial_Sums (X,((1. X) - x))) . ((n + 1) + 1)) + (- ((X,((1. X) - x)) . 0)) is right_complementable Element of the carrier of X
K411( the carrier of X, the addF of X,((Partial_Sums (X,((1. X) - x))) . ((n + 1) + 1)),(- ((X,((1. X) - x)) . 0))) is right_complementable Element of the carrier of X
[((Partial_Sums (X,((1. X) - x))) . ((n + 1) + 1)),(- ((X,((1. X) - x)) . 0))] is set
the addF of X . [((Partial_Sums (X,((1. X) - x))) . ((n + 1) + 1)),(- ((X,((1. X) - x)) . 0))] is set
(X,((1. X) - x)) . ((n + 1) + 1) is right_complementable Element of the carrier of X
((Partial_Sums (X,((1. X) - x))) . (n + 1)) + ((X,((1. X) - x)) . ((n + 1) + 1)) is right_complementable Element of the carrier of X
K411( the carrier of X, the addF of X,((Partial_Sums (X,((1. X) - x))) . (n + 1)),((X,((1. X) - x)) . ((n + 1) + 1))) is right_complementable Element of the carrier of X
[((Partial_Sums (X,((1. X) - x))) . (n + 1)),((X,((1. X) - x)) . ((n + 1) + 1))] is set
the addF of X . [((Partial_Sums (X,((1. X) - x))) . (n + 1)),((X,((1. X) - x)) . ((n + 1) + 1))] is set
(((Partial_Sums (X,((1. X) - x))) . (n + 1)) + ((X,((1. X) - x)) . ((n + 1) + 1))) - ((X,((1. X) - x)) . 0) is right_complementable Element of the carrier of X
(((Partial_Sums (X,((1. X) - x))) . (n + 1)) + ((X,((1. X) - x)) . ((n + 1) + 1))) + (- ((X,((1. X) - x)) . 0)) is right_complementable Element of the carrier of X
K411( the carrier of X, the addF of X,(((Partial_Sums (X,((1. X) - x))) . (n + 1)) + ((X,((1. X) - x)) . ((n + 1) + 1))),(- ((X,((1. X) - x)) . 0))) is right_complementable Element of the carrier of X
[(((Partial_Sums (X,((1. X) - x))) . (n + 1)) + ((X,((1. X) - x)) . ((n + 1) + 1))),(- ((X,((1. X) - x)) . 0))] is set
the addF of X . [(((Partial_Sums (X,((1. X) - x))) . (n + 1)) + ((X,((1. X) - x)) . ((n + 1) + 1))),(- ((X,((1. X) - x)) . 0))] is set
(((Partial_Sums (X,((1. X) - x))) ^\ 1) . n) + ((X,((1. X) - x)) . ((n + 1) + 1)) is right_complementable Element of the carrier of X
K411( the carrier of X, the addF of X,(((Partial_Sums (X,((1. X) - x))) ^\ 1) . n),((X,((1. X) - x)) . ((n + 1) + 1))) is right_complementable Element of the carrier of X
[(((Partial_Sums (X,((1. X) - x))) ^\ 1) . n),((X,((1. X) - x)) . ((n + 1) + 1))] is set
the addF of X . [(((Partial_Sums (X,((1. X) - x))) ^\ 1) . n),((X,((1. X) - x)) . ((n + 1) + 1))] is set
((((Partial_Sums (X,((1. X) - x))) ^\ 1) . n) + ((X,((1. X) - x)) . ((n + 1) + 1))) + (- ((X,((1. X) - x)) . 0)) is right_complementable Element of the carrier of X
K411( the carrier of X, the addF of X,((((Partial_Sums (X,((1. X) - x))) ^\ 1) . n) + ((X,((1. X) - x)) . ((n + 1) + 1))),(- ((X,((1. X) - x)) . 0))) is right_complementable Element of the carrier of X
[((((Partial_Sums (X,((1. X) - x))) ^\ 1) . n) + ((X,((1. X) - x)) . ((n + 1) + 1))),(- ((X,((1. X) - x)) . 0))] is set
the addF of X . [((((Partial_Sums (X,((1. X) - x))) ^\ 1) . n) + ((X,((1. X) - x)) . ((n + 1) + 1))),(- ((X,((1. X) - x)) . 0))] is set
((((Partial_Sums (X,((1. X) - x))) ^\ 1) . n) - ((X,((1. X) - x)) . 0)) + ((X,((1. X) - x)) . ((n + 1) + 1)) is right_complementable Element of the carrier of X
K411( the carrier of X, the addF of X,((((Partial_Sums (X,((1. X) - x))) ^\ 1) . n) - ((X,((1. X) - x)) . 0)),((X,((1. X) - x)) . ((n + 1) + 1))) is right_complementable Element of the carrier of X
[((((Partial_Sums (X,((1. X) - x))) ^\ 1) . n) - ((X,((1. X) - x)) . 0)),((X,((1. X) - x)) . ((n + 1) + 1))] is set
the addF of X . [((((Partial_Sums (X,((1. X) - x))) ^\ 1) . n) - ((X,((1. X) - x)) . 0)),((X,((1. X) - x)) . ((n + 1) + 1))] is set
(X,((1. X) - x)) . (n + 1) is right_complementable Element of the carrier of X
((Partial_Sums (X,((1. X) - x))) . n) + ((X,((1. X) - x)) . (n + 1)) is right_complementable Element of the carrier of X
K411( the carrier of X, the addF of X,((Partial_Sums (X,((1. X) - x))) . n),((X,((1. X) - x)) . (n + 1))) is right_complementable Element of the carrier of X
[((Partial_Sums (X,((1. X) - x))) . n),((X,((1. X) - x)) . (n + 1))] is set
the addF of X . [((Partial_Sums (X,((1. X) - x))) . n),((X,((1. X) - x)) . (n + 1))] is set
((1. X) - x) * (((Partial_Sums (X,((1. X) - x))) . n) + ((X,((1. X) - x)) . (n + 1))) is right_complementable Element of the carrier of X
K411( the carrier of X, the multF of X,((1. X) - x),(((Partial_Sums (X,((1. X) - x))) . n) + ((X,((1. X) - x)) . (n + 1)))) is right_complementable Element of the carrier of X
[((1. X) - x),(((Partial_Sums (X,((1. X) - x))) . n) + ((X,((1. X) - x)) . (n + 1)))] is set
the multF of X . [((1. X) - x),(((Partial_Sums (X,((1. X) - x))) . n) + ((X,((1. X) - x)) . (n + 1)))] is set
((1. X) - x) * ((X,((1. X) - x)) . (n + 1)) is right_complementable Element of the carrier of X
K411( the carrier of X, the multF of X,((1. X) - x),((X,((1. X) - x)) . (n + 1))) is right_complementable Element of the carrier of X
[((1. X) - x),((X,((1. X) - x)) . (n + 1))] is set
the multF of X . [((1. X) - x),((X,((1. X) - x)) . (n + 1))] is set
(((1. X) - x) * ((Partial_Sums (X,((1. X) - x))) . n)) + (((1. X) - x) * ((X,((1. X) - x)) . (n + 1))) is right_complementable Element of the carrier of X
K411( the carrier of X, the addF of X,(((1. X) - x) * ((Partial_Sums (X,((1. X) - x))) . n)),(((1. X) - x) * ((X,((1. X) - x)) . (n + 1)))) is right_complementable Element of the carrier of X
[(((1. X) - x) * ((Partial_Sums (X,((1. X) - x))) . n)),(((1. X) - x) * ((X,((1. X) - x)) . (n + 1)))] is set
the addF of X . [(((1. X) - x) * ((Partial_Sums (X,((1. X) - x))) . n)),(((1. X) - x) * ((X,((1. X) - x)) . (n + 1)))] is set
((Partial_Sums (X,((1. X) - x))) ^\ 1) . 0 is right_complementable Element of the carrier of X
(((Partial_Sums (X,((1. X) - x))) ^\ 1) . 0) - ((X,((1. X) - x)) . 0) is right_complementable Element of the carrier of X
- ((X,((1. X) - x)) . 0) is right_complementable Element of the carrier of X
(((Partial_Sums (X,((1. X) - x))) ^\ 1) . 0) + (- ((X,((1. X) - x)) . 0)) is right_complementable Element of the carrier of X
K411( the carrier of X, the addF of X,(((Partial_Sums (X,((1. X) - x))) ^\ 1) . 0),(- ((X,((1. X) - x)) . 0))) is right_complementable Element of the carrier of X
[(((Partial_Sums (X,((1. X) - x))) ^\ 1) . 0),(- ((X,((1. X) - x)) . 0))] is set
the addF of X . [(((Partial_Sums (X,((1. X) - x))) ^\ 1) . 0),(- ((X,((1. X) - x)) . 0))] is set
0 + 1 is V4() epsilon-transitive epsilon-connected ordinal natural V24() real ext-real positive non negative V45() V46() V52() V53() V54() V55() V56() V57() Element of NAT
(Partial_Sums (X,((1. X) - x))) . (0 + 1) is right_complementable Element of the carrier of X
((Partial_Sums (X,((1. X) - x))) . (0 + 1)) - ((X,((1. X) - x)) . 0) is right_complementable Element of the carrier of X
((Partial_Sums (X,((1. X) - x))) . (0 + 1)) + (- ((X,((1. X) - x)) . 0)) is right_complementable Element of the carrier of X
K411( the carrier of X, the addF of X,((Partial_Sums (X,((1. X) - x))) . (0 + 1)),(- ((X,((1. X) - x)) . 0))) is right_complementable Element of the carrier of X
[((Partial_Sums (X,((1. X) - x))) . (0 + 1)),(- ((X,((1. X) - x)) . 0))] is set
the addF of X . [((Partial_Sums (X,((1. X) - x))) . (0 + 1)),(- ((X,((1. X) - x)) . 0))] is set
(X,((1. X) - x)) . 1 is right_complementable Element of the carrier of X
((Partial_Sums (X,((1. X) - x))) . 0) + ((X,((1. X) - x)) . 1) is right_complementable Element of the carrier of X
K411( the carrier of X, the addF of X,((Partial_Sums (X,((1. X) - x))) . 0),((X,((1. X) - x)) . 1)) is right_complementable Element of the carrier of X
[((Partial_Sums (X,((1. X) - x))) . 0),((X,((1. X) - x)) . 1)] is set
the addF of X . [((Partial_Sums (X,((1. X) - x))) . 0),((X,((1. X) - x)) . 1)] is set
(((Partial_Sums (X,((1. X) - x))) . 0) + ((X,((1. X) - x)) . 1)) - ((X,((1. X) - x)) . 0) is right_complementable Element of the carrier of X
(((Partial_Sums (X,((1. X) - x))) . 0) + ((X,((1. X) - x)) . 1)) + (- ((X,((1. X) - x)) . 0)) is right_complementable Element of the carrier of X
K411( the carrier of X, the addF of X,(((Partial_Sums (X,((1. X) - x))) . 0) + ((X,((1. X) - x)) . 1)),(- ((X,((1. X) - x)) . 0))) is right_complementable Element of the carrier of X
[(((Partial_Sums (X,((1. X) - x))) . 0) + ((X,((1. X) - x)) . 1)),(- ((X,((1. X) - x)) . 0))] is set
the addF of X . [(((Partial_Sums (X,((1. X) - x))) . 0) + ((X,((1. X) - x)) . 1)),(- ((X,((1. X) - x)) . 0))] is set
((X,((1. X) - x)) . 0) + ((X,((1. X) - x)) . 1) is right_complementable Element of the carrier of X
K411( the carrier of X, the addF of X,((X,((1. X) - x)) . 0),((X,((1. X) - x)) . 1)) is right_complementable Element of the carrier of X
[((X,((1. X) - x)) . 0),((X,((1. X) - x)) . 1)] is set
the addF of X . [((X,((1. X) - x)) . 0),((X,((1. X) - x)) . 1)] is set
(((X,((1. X) - x)) . 0) + ((X,((1. X) - x)) . 1)) - ((X,((1. X) - x)) . 0) is right_complementable Element of the carrier of X
(((X,((1. X) - x)) . 0) + ((X,((1. X) - x)) . 1)) + (- ((X,((1. X) - x)) . 0)) is right_complementable Element of the carrier of X
K411( the carrier of X, the addF of X,(((X,((1. X) - x)) . 0) + ((X,((1. X) - x)) . 1)),(- ((X,((1. X) - x)) . 0))) is right_complementable Element of the carrier of X
[(((X,((1. X) - x)) . 0) + ((X,((1. X) - x)) . 1)),(- ((X,((1. X) - x)) . 0))] is set
the addF of X . [(((X,((1. X) - x)) . 0) + ((X,((1. X) - x)) . 1)),(- ((X,((1. X) - x)) . 0))] is set
((X,((1. X) - x)) . 0) - ((X,((1. X) - x)) . 0) is right_complementable Element of the carrier of X
((X,((1. X) - x)) . 0) + (- ((X,((1. X) - x)) . 0)) is right_complementable Element of the carrier of X
K411( the carrier of X, the addF of X,((X,((1. X) - x)) . 0),(- ((X,((1. X) - x)) . 0))) is right_complementable Element of the carrier of X
[((X,((1. X) - x)) . 0),(- ((X,((1. X) - x)) . 0))] is set
the addF of X . [((X,((1. X) - x)) . 0),(- ((X,((1. X) - x)) . 0))] is set
((X,((1. X) - x)) . 1) + (((X,((1. X) - x)) . 0) - ((X,((1. X) - x)) . 0)) is right_complementable Element of the carrier of X
K411( the carrier of X, the addF of X,((X,((1. X) - x)) . 1),(((X,((1. X) - x)) . 0) - ((X,((1. X) - x)) . 0))) is right_complementable Element of the carrier of X
[((X,((1. X) - x)) . 1),(((X,((1. X) - x)) . 0) - ((X,((1. X) - x)) . 0))] is set
the addF of X . [((X,((1. X) - x)) . 1),(((X,((1. X) - x)) . 0) - ((X,((1. X) - x)) . 0))] is set
((X,((1. X) - x)) . 1) + (0. X) is right_complementable Element of the carrier of X
K411( the carrier of X, the addF of X,((X,((1. X) - x)) . 1),(0. X)) is right_complementable Element of the carrier of X
[((X,((1. X) - x)) . 1),(0. X)] is set
the addF of X . [((X,((1. X) - x)) . 1),(0. X)] is set
(X,((1. X) - x)) . (0 + 1) is right_complementable Element of the carrier of X
((X,((1. X) - x)) . 0) * ((1. X) - x) is right_complementable Element of the carrier of X
K411( the carrier of X, the multF of X,((X,((1. X) - x)) . 0),((1. X) - x)) is right_complementable Element of the carrier of X
[((X,((1. X) - x)) . 0),((1. X) - x)] is set
the multF of X . [((X,((1. X) - x)) . 0),((1. X) - x)] is set
(1. X) * ((1. X) - x) is right_complementable Element of the carrier of X
K411( the carrier of X, the multF of X,(1. X),((1. X) - x)) is right_complementable Element of the carrier of X
[(1. X),((1. X) - x)] is set
the multF of X . [(1. X),((1. X) - x)] is set
n is epsilon-transitive epsilon-connected ordinal natural V24() real ext-real V45() V46() V52() V53() V54() V55() V56() V57() Element of NAT
(Partial_Sums (X,((1. X) - x))) . n is right_complementable Element of the carrier of X
((1. X) - x) * ((Partial_Sums (X,((1. X) - x))) . n) is right_complementable Element of the carrier of X
K411( the carrier of X, the multF of X,((1. X) - x),((Partial_Sums (X,((1. X) - x))) . n)) is right_complementable Element of the carrier of X
[((1. X) - x),((Partial_Sums (X,((1. X) - x))) . n)] is set
the multF of X . [((1. X) - x),((Partial_Sums (X,((1. X) - x))) . n)] is set
((Partial_Sums (X,((1. X) - x))) ^\ 1) . n is right_complementable Element of the carrier of X
(((Partial_Sums (X,((1. X) - x))) ^\ 1) . n) - ((X,((1. X) - x)) . 0) is right_complementable Element of the carrier of X
(((Partial_Sums (X,((1. X) - x))) ^\ 1) . n) + (- ((X,((1. X) - x)) . 0)) is right_complementable Element of the carrier of X
K411( the carrier of X, the addF of X,(((Partial_Sums (X,((1. X) - x))) ^\ 1) . n),(- ((X,((1. X) - x)) . 0))) is right_complementable Element of the carrier of X
[(((Partial_Sums (X,((1. X) - x))) ^\ 1) . n),(- ((X,((1. X) - x)) . 0))] is set
the addF of X . [(((Partial_Sums (X,((1. X) - x))) ^\ 1) . n),(- ((X,((1. X) - x)) . 0))] is set
n is epsilon-transitive epsilon-connected ordinal natural V24() real ext-real V45() V46() V52() V53() V54() V55() V56() V57() Element of NAT
(Partial_Sums (X,((1. X) - x))) . n is right_complementable Element of the carrier of X
n + 1 is epsilon-transitive epsilon-connected ordinal natural V24() real ext-real V45() V46() V52() V53() V54() V55() V56() V57() Element of NAT
(Partial_Sums (X,((1. X) - x))) . (n + 1) is right_complementable Element of the carrier of X
(X,((1. X) - x)) . (n + 1) is right_complementable Element of the carrier of X
((Partial_Sums (X,((1. X) - x))) . n) + ((X,((1. X) - x)) . (n + 1)) is right_complementable Element of the carrier of X
K411( the carrier of X, the addF of X,((Partial_Sums (X,((1. X) - x))) . n),((X,((1. X) - x)) . (n + 1))) is right_complementable Element of the carrier of X
[((Partial_Sums (X,((1. X) - x))) . n),((X,((1. X) - x)) . (n + 1))] is set
the addF of X . [((Partial_Sums (X,((1. X) - x))) . n),((X,((1. X) - x)) . (n + 1))] is set
((X,((1. X) - x)) ^\ 1) . n is right_complementable Element of the carrier of X
((Partial_Sums (X,((1. X) - x))) . n) + (((X,((1. X) - x)) ^\ 1) . n) is right_complementable Element of the carrier of X
K411( the carrier of X, the addF of X,((Partial_Sums (X,((1. X) - x))) . n),(((X,((1. X) - x)) ^\ 1) . n)) is right_complementable Element of the carrier of X
[((Partial_Sums (X,((1. X) - x))) . n),(((X,((1. X) - x)) ^\ 1) . n)] is set
the addF of X . [((Partial_Sums (X,((1. X) - x))) . n),(((X,((1. X) - x)) ^\ 1) . n)] is set
(1. X) - ((1. X) - x) is right_complementable Element of the carrier of X
- ((1. X) - x) is right_complementable Element of the carrier of X
(1. X) + (- ((1. X) - x)) is right_complementable Element of the carrier of X
K411( the carrier of X, the addF of X,(1. X),(- ((1. X) - x))) is right_complementable Element of the carrier of X
[(1. X),(- ((1. X) - x))] is set
the addF of X . [(1. X),(- ((1. X) - x))] is set
yn is right_complementable Element of the carrier of X
((1. X) - ((1. X) - x)) * yn is right_complementable Element of the carrier of X
the multF of X is V4() Relation-like K33( the carrier of X, the carrier of X) -defined the carrier of X -valued Function-like V29(K33( the carrier of X, the carrier of X)) V30(K33( the carrier of X, the carrier of X), the carrier of X) Element of K32(K33(K33( the carrier of X, the carrier of X), the carrier of X))
K411( the carrier of X, the multF of X,((1. X) - ((1. X) - x)),yn) is right_complementable Element of the carrier of X
[((1. X) - ((1. X) - x)),yn] is set
the multF of X . [((1. X) - ((1. X) - x)),yn] is set
(1. X) * yn is right_complementable Element of the carrier of X
K411( the carrier of X, the multF of X,(1. X),yn) is right_complementable Element of the carrier of X
[(1. X),yn] is set
the multF of X . [(1. X),yn] is set
((1. X) - x) * yn is right_complementable Element of the carrier of X
K411( the carrier of X, the multF of X,((1. X) - x),yn) is right_complementable Element of the carrier of X
[((1. X) - x),yn] is set
the multF of X . [((1. X) - x),yn] is set
((1. X) * yn) - (((1. X) - x) * yn) is right_complementable Element of the carrier of X
- (((1. X) - x) * yn) is right_complementable Element of the carrier of X
((1. X) * yn) + (- (((1. X) - x) * yn)) is right_complementable Element of the carrier of X
K411( the carrier of X, the addF of X,((1. X) * yn),(- (((1. X) - x) * yn))) is right_complementable Element of the carrier of X
[((1. X) * yn),(- (((1. X) - x) * yn))] is set
the addF of X . [((1. X) * yn),(- (((1. X) - x) * yn))] is set
yn - (((1. X) - x) * yn) is right_complementable Element of the carrier of X
yn + (- (((1. X) - x) * yn)) is right_complementable Element of the carrier of X
K411( the carrier of X, the addF of X,yn,(- (((1. X) - x) * yn))) is right_complementable Element of the carrier of X
[yn,(- (((1. X) - x) * yn))] is set
the addF of X . [yn,(- (((1. X) - x) * yn))] is set
((Partial_Sums (X,((1. X) - x))) ^\ 1) . n is right_complementable Element of the carrier of X
(((Partial_Sums (X,((1. X) - x))) ^\ 1) . n) - ((X,((1. X) - x)) . 0) is right_complementable Element of the carrier of X
- ((X,((1. X) - x)) . 0) is right_complementable Element of the carrier of X
(((Partial_Sums (X,((1. X) - x))) ^\ 1) . n) + (- ((X,((1. X) - x)) . 0)) is right_complementable Element of the carrier of X
K411( the carrier of X, the addF of X,(((Partial_Sums (X,((1. X) - x))) ^\ 1) . n),(- ((X,((1. X) - x)) . 0))) is right_complementable Element of the carrier of X
[(((Partial_Sums (X,((1. X) - x))) ^\ 1) . n),(- ((X,((1. X) - x)) . 0))] is set
the addF of X . [(((Partial_Sums (X,((1. X) - x))) ^\ 1) . n),(- ((X,((1. X) - x)) . 0))] is set
((Partial_Sums (X,((1. X) - x))) . n) - ((((Partial_Sums (X,((1. X) - x))) ^\ 1) . n) - ((X,((1. X) - x)) . 0)) is right_complementable Element of the carrier of X
- ((((Partial_Sums (X,((1. X) - x))) ^\ 1) . n) - ((X,((1. X) - x)) . 0)) is right_complementable Element of the carrier of X
((Partial_Sums (X,((1. X) - x))) . n) + (- ((((Partial_Sums (X,((1. X) - x))) ^\ 1) . n) - ((X,((1. X) - x)) . 0))) is right_complementable Element of the carrier of X
K411( the carrier of X, the addF of X,((Partial_Sums (X,((1. X) - x))) . n),(- ((((Partial_Sums (X,((1. X) - x))) ^\ 1) . n) - ((X,((1. X) - x)) . 0)))) is right_complementable Element of the carrier of X
[((Partial_Sums (X,((1. X) - x))) . n),(- ((((Partial_Sums (X,((1. X) - x))) ^\ 1) . n) - ((X,((1. X) - x)) . 0)))] is set
the addF of X . [((Partial_Sums (X,((1. X) - x))) . n),(- ((((Partial_Sums (X,((1. X) - x))) ^\ 1) . n) - ((X,((1. X) - x)) . 0)))] is set
((Partial_Sums (X,((1. X) - x))) . n) - (((Partial_Sums (X,((1. X) - x))) ^\ 1) . n) is right_complementable Element of the carrier of X
- (((Partial_Sums (X,((1. X) - x))) ^\ 1) . n) is right_complementable Element of the carrier of X
((Partial_Sums (X,((1. X) - x))) . n) + (- (((Partial_Sums (X,((1. X) - x))) ^\ 1) . n)) is right_complementable Element of the carrier of X
K411( the carrier of X, the addF of X,((Partial_Sums (X,((1. X) - x))) . n),(- (((Partial_Sums (X,((1. X) - x))) ^\ 1) . n))) is right_complementable Element of the carrier of X
[((Partial_Sums (X,((1. X) - x))) . n),(- (((Partial_Sums (X,((1. X) - x))) ^\ 1) . n))] is set
the addF of X . [((Partial_Sums (X,((1. X) - x))) . n),(- (((Partial_Sums (X,((1. X) - x))) ^\ 1) . n))] is set
(((Partial_Sums (X,((1. X) - x))) . n) - (((Partial_Sums (X,((1. X) - x))) ^\ 1) . n)) + ((X,((1. X) - x)) . 0) is right_complementable Element of the carrier of X
K411( the carrier of X, the addF of X,(((Partial_Sums (X,((1. X) - x))) . n) - (((Partial_Sums (X,((1. X) - x))) ^\ 1) . n)),((X,((1. X) - x)) . 0)) is right_complementable Element of the carrier of X
[(((Partial_Sums (X,((1. X) - x))) . n) - (((Partial_Sums (X,((1. X) - x))) ^\ 1) . n)),((X,((1. X) - x)) . 0)] is set
the addF of X . [(((Partial_Sums (X,((1. X) - x))) . n) - (((Partial_Sums (X,((1. X) - x))) ^\ 1) . n)),((X,((1. X) - x)) . 0)] is set
((Partial_Sums (X,((1. X) - x))) . n) - (((Partial_Sums (X,((1. X) - x))) . n) + (((X,((1. X) - x)) ^\ 1) . n)) is right_complementable Element of the carrier of X
- (((Partial_Sums (X,((1. X) - x))) . n) + (((X,((1. X) - x)) ^\ 1) . n)) is right_complementable Element of the carrier of X
((Partial_Sums (X,((1. X) - x))) . n) + (- (((Partial_Sums (X,((1. X) - x))) . n) + (((X,((1. X) - x)) ^\ 1) . n))) is right_complementable Element of the carrier of X
K411( the carrier of X, the addF of X,((Partial_Sums (X,((1. X) - x))) . n),(- (((Partial_Sums (X,((1. X) - x))) . n) + (((X,((1. X) - x)) ^\ 1) . n)))) is right_complementable Element of the carrier of X
[((Partial_Sums (X,((1. X) - x))) . n),(- (((Partial_Sums (X,((1. X) - x))) . n) + (((X,((1. X) - x)) ^\ 1) . n)))] is set
the addF of X . [((Partial_Sums (X,((1. X) - x))) . n),(- (((Partial_Sums (X,((1. X) - x))) . n) + (((X,((1. X) - x)) ^\ 1) . n)))] is set
(((Partial_Sums (X,((1. X) - x))) . n) - (((Partial_Sums (X,((1. X) - x))) . n) + (((X,((1. X) - x)) ^\ 1) . n))) + ((X,((1. X) - x)) . 0) is right_complementable Element of the carrier of X
K411( the carrier of X, the addF of X,(((Partial_Sums (X,((1. X) - x))) . n) - (((Partial_Sums (X,((1. X) - x))) . n) + (((X,((1. X) - x)) ^\ 1) . n))),((X,((1. X) - x)) . 0)) is right_complementable Element of the carrier of X
[(((Partial_Sums (X,((1. X) - x))) . n) - (((Partial_Sums (X,((1. X) - x))) . n) + (((X,((1. X) - x)) ^\ 1) . n))),((X,((1. X) - x)) . 0)] is set
the addF of X . [(((Partial_Sums (X,((1. X) - x))) . n) - (((Partial_Sums (X,((1. X) - x))) . n) + (((X,((1. X) - x)) ^\ 1) . n))),((X,((1. X) - x)) . 0)] is set
((Partial_Sums (X,((1. X) - x))) . n) - ((Partial_Sums (X,((1. X) - x))) . n) is right_complementable Element of the carrier of X
- ((Partial_Sums (X,((1. X) - x))) . n) is right_complementable Element of the carrier of X
((Partial_Sums (X,((1. X) - x))) . n) + (- ((Partial_Sums (X,((1. X) - x))) . n)) is right_complementable Element of the carrier of X
K411( the carrier of X, the addF of X,((Partial_Sums (X,((1. X) - x))) . n),(- ((Partial_Sums (X,((1. X) - x))) . n))) is right_complementable Element of the carrier of X
[((Partial_Sums (X,((1. X) - x))) . n),(- ((Partial_Sums (X,((1. X) - x))) . n))] is set
the addF of X . [((Partial_Sums (X,((1. X) - x))) . n),(- ((Partial_Sums (X,((1. X) - x))) . n))] is set
(((Partial_Sums (X,((1. X) - x))) . n) - ((Partial_Sums (X,((1. X) - x))) . n)) - (((X,((1. X) - x)) ^\ 1) . n) is right_complementable Element of the carrier of X
- (((X,((1. X) - x)) ^\ 1) . n) is right_complementable Element of the carrier of X
(((Partial_Sums (X,((1. X) - x))) . n) - ((Partial_Sums (X,((1. X) - x))) . n)) + (- (((X,((1. X) - x)) ^\ 1) . n)) is right_complementable Element of the carrier of X
K411( the carrier of X, the addF of X,(((Partial_Sums (X,((1. X) - x))) . n) - ((Partial_Sums (X,((1. X) - x))) . n)),(- (((X,((1. X) - x)) ^\ 1) . n))) is right_complementable Element of the carrier of X
[(((Partial_Sums (X,((1. X) - x))) . n) - ((Partial_Sums (X,((1. X) - x))) . n)),(- (((X,((1. X) - x)) ^\ 1) . n))] is set
the addF of X . [(((Partial_Sums (X,((1. X) - x))) . n) - ((Partial_Sums (X,((1. X) - x))) . n)),(- (((X,((1. X) - x)) ^\ 1) . n))] is set
((((Partial_Sums (X,((1. X) - x))) . n) - ((Partial_Sums (X,((1. X) - x))) . n)) - (((X,((1. X) - x)) ^\ 1) . n)) + ((X,((1. X) - x)) . 0) is right_complementable Element of the carrier of X
K411( the carrier of X, the addF of X,((((Partial_Sums (X,((1. X) - x))) . n) - ((Partial_Sums (X,((1. X) - x))) . n)) - (((X,((1. X) - x)) ^\ 1) . n)),((X,((1. X) - x)) . 0)) is right_complementable Element of the carrier of X
[((((Partial_Sums (X,((1. X) - x))) . n) - ((Partial_Sums (X,((1. X) - x))) . n)) - (((X,((1. X) - x)) ^\ 1) . n)),((X,((1. X) - x)) . 0)] is set
the addF of X . [((((Partial_Sums (X,((1. X) - x))) . n) - ((Partial_Sums (X,((1. X) - x))) . n)) - (((X,((1. X) - x)) ^\ 1) . n)),((X,((1. X) - x)) . 0)] is set
(0. X) - (((X,((1. X) - x)) ^\ 1) . n) is right_complementable Element of the carrier of X
(0. X) + (- (((X,((1. X) - x)) ^\ 1) . n)) is right_complementable Element of the carrier of X
K411( the carrier of X, the addF of X,(0. X),(- (((X,((1. X) - x)) ^\ 1) . n))) is right_complementable Element of the carrier of X
[(0. X),(- (((X,((1. X) - x)) ^\ 1) . n))] is set
the addF of X . [(0. X),(- (((X,((1. X) - x)) ^\ 1) . n))] is set
((0. X) - (((X,((1. X) - x)) ^\ 1) . n)) + ((X,((1. X) - x)) . 0) is right_complementable Element of the carrier of X
K411( the carrier of X, the addF of X,((0. X) - (((X,((1. X) - x)) ^\ 1) . n)),((X,((1. X) - x)) . 0)) is right_complementable Element of the carrier of X
[((0. X) - (((X,((1. X) - x)) ^\ 1) . n)),((X,((1. X) - x)) . 0)] is set
the addF of X . [((0. X) - (((X,((1. X) - x)) ^\ 1) . n)),((X,((1. X) - x)) . 0)] is set
(((X,((1. X) - x)) ^\ 1) . n) - ((X,((1. X) - x)) . 0) is right_complementable Element of the carrier of X
(((X,((1. X) - x)) ^\ 1) . n) + (- ((X,((1. X) - x)) . 0)) is right_complementable Element of the carrier of X
K411( the carrier of X, the addF of X,(((X,((1. X) - x)) ^\ 1) . n),(- ((X,((1. X) - x)) . 0))) is right_complementable Element of the carrier of X
[(((X,((1. X) - x)) ^\ 1) . n),(- ((X,((1. X) - x)) . 0))] is set
the addF of X . [(((X,((1. X) - x)) ^\ 1) . n),(- ((X,((1. X) - x)) . 0))] is set
(0. X) - ((((X,((1. X) - x)) ^\ 1) . n) - ((X,((1. X) - x)) . 0)) is right_complementable Element of the carrier of X
- ((((X,((1. X) - x)) ^\ 1) . n) - ((X,((1. X) - x)) . 0)) is right_complementable Element of the carrier of X
(0. X) + (- ((((X,((1. X) - x)) ^\ 1) . n) - ((X,((1. X) - x)) . 0))) is right_complementable Element of the carrier of X
K411( the carrier of X, the addF of X,(0. X),(- ((((X,((1. X) - x)) ^\ 1) . n) - ((X,((1. X) - x)) . 0)))) is right_complementable Element of the carrier of X
[(0. X),(- ((((X,((1. X) - x)) ^\ 1) . n) - ((X,((1. X) - x)) . 0)))] is set
the addF of X . [(0. X),(- ((((X,((1. X) - x)) ^\ 1) . n) - ((X,((1. X) - x)) . 0)))] is set
((X,((1. X) - x)) . 0) - (((X,((1. X) - x)) ^\ 1) . n) is right_complementable Element of the carrier of X
((X,((1. X) - x)) . 0) + (- (((X,((1. X) - x)) ^\ 1) . n)) is right_complementable Element of the carrier of X
K411( the carrier of X, the addF of X,((X,((1. X) - x)) . 0),(- (((X,((1. X) - x)) ^\ 1) . n))) is right_complementable Element of the carrier of X
[((X,((1. X) - x)) . 0),(- (((X,((1. X) - x)) ^\ 1) . n))] is set
the addF of X . [((X,((1. X) - x)) . 0),(- (((X,((1. X) - x)) ^\ 1) . n))] is set
((X,((1. X) - x)) . 0) - ((X,((1. X) - x)) . (n + 1)) is right_complementable Element of the carrier of X
- ((X,((1. X) - x)) . (n + 1)) is right_complementable Element of the carrier of X
((X,((1. X) - x)) . 0) + (- ((X,((1. X) - x)) . (n + 1))) is right_complementable Element of the carrier of X
K411( the carrier of X, the addF of X,((X,((1. X) - x)) . 0),(- ((X,((1. X) - x)) . (n + 1)))) is right_complementable Element of the carrier of X
[((X,((1. X) - x)) . 0),(- ((X,((1. X) - x)) . (n + 1)))] is set
the addF of X . [((X,((1. X) - x)) . 0),(- ((X,((1. X) - x)) . (n + 1)))] is set
(1. X) - ((X,((1. X) - x)) . (n + 1)) is right_complementable Element of the carrier of X
(1. X) + (- ((X,((1. X) - x)) . (n + 1))) is right_complementable Element of the carrier of X
K411( the carrier of X, the addF of X,(1. X),(- ((X,((1. X) - x)) . (n + 1)))) is right_complementable Element of the carrier of X
[(1. X),(- ((X,((1. X) - x)) . (n + 1)))] is set
the addF of X . [(1. X),(- ((X,((1. X) - x)) . (n + 1)))] is set
x * yn is right_complementable Element of the carrier of X
K411( the carrier of X, the multF of X,x,yn) is right_complementable Element of the carrier of X
[x,yn] is set
the multF of X . [x,yn] is set
(1. X) - (x * yn) is right_complementable Element of the carrier of X
- (x * yn) is right_complementable Element of the carrier of X
(1. X) + (- (x * yn)) is right_complementable Element of the carrier of X
K411( the carrier of X, the addF of X,(1. X),(- (x * yn))) is right_complementable Element of the carrier of X
[(1. X),(- (x * yn))] is set
the addF of X . [(1. X),(- (x * yn))] is set
(1. X) - ((1. X) - ((X,((1. X) - x)) . (n + 1))) is right_complementable Element of the carrier of X
- ((1. X) - ((X,((1. X) - x)) . (n + 1))) is right_complementable Element of the carrier of X
(1. X) + (- ((1. X) - ((X,((1. X) - x)) . (n + 1)))) is right_complementable Element of the carrier of X
K411( the carrier of X, the addF of X,(1. X),(- ((1. X) - ((X,((1. X) - x)) . (n + 1))))) is right_complementable Element of the carrier of X
[(1. X),(- ((1. X) - ((X,((1. X) - x)) . (n + 1))))] is set
the addF of X . [(1. X),(- ((1. X) - ((X,((1. X) - x)) . (n + 1))))] is set
yn - y is right_complementable Element of the carrier of X
- y is right_complementable Element of the carrier of X
yn + (- y) is right_complementable Element of the carrier of X
K411( the carrier of X, the addF of X,yn,(- y)) is right_complementable Element of the carrier of X
[yn,(- y)] is set
the addF of X . [yn,(- y)] is set
x * (yn - y) is right_complementable Element of the carrier of X
K411( the carrier of X, the multF of X,x,(yn - y)) is right_complementable Element of the carrier of X
[x,(yn - y)] is set
the multF of X . [x,(yn - y)] is set
||.(x * (yn - y)).|| is V24() real ext-real non negative Element of REAL
||.(yn - y).|| is V24() real ext-real non negative Element of REAL
||.x.|| * ||.(yn - y).|| is V24() real ext-real non negative Element of REAL
||.((X,((1. X) - x)) ^\ 1).|| . n is V24() real ext-real Element of REAL
(||.((X,((1. X) - x)) ^\ 1).|| . n) + ||.(x * (yn - y)).|| is V24() real ext-real Element of REAL
(||.((X,((1. X) - x)) ^\ 1).|| . n) + (||.x.|| * ||.(yn - y).||) is V24() real ext-real Element of REAL
(((X,((1. X) - x)) ^\ 1) . n) + (x * (yn - y)) is right_complementable Element of the carrier of X
K411( the carrier of X, the addF of X,(((X,((1. X) - x)) ^\ 1) . n),(x * (yn - y))) is right_complementable Element of the carrier of X
[(((X,((1. X) - x)) ^\ 1) . n),(x * (yn - y))] is set
the addF of X . [(((X,((1. X) - x)) ^\ 1) . n),(x * (yn - y))] is set
||.((((X,((1. X) - x)) ^\ 1) . n) + (x * (yn - y))).|| is V24() real ext-real non negative Element of REAL
||.(((X,((1. X) - x)) ^\ 1) . n).|| is V24() real ext-real non negative Element of REAL
||.(((X,((1. X) - x)) ^\ 1) . n).|| + ||.(x * (yn - y)).|| is V24() real ext-real non negative Element of REAL
x * y is right_complementable Element of the carrier of X
K411( the carrier of X, the multF of X,x,y) is right_complementable Element of the carrier of X
[x,y] is set
the multF of X . [x,y] is set
(1. X) - (x * y) is right_complementable Element of the carrier of X
- (x * y) is right_complementable Element of the carrier of X
(1. X) + (- (x * y)) is right_complementable Element of the carrier of X
K411( the carrier of X, the addF of X,(1. X),(- (x * y))) is right_complementable Element of the carrier of X
[(1. X),(- (x * y))] is set
the addF of X . [(1. X),(- (x * y))] is set
(1. X) - (0. X) is right_complementable Element of the carrier of X
- (0. X) is right_complementable Element of the carrier of X
(1. X) + (- (0. X)) is right_complementable Element of the carrier of X
K411( the carrier of X, the addF of X,(1. X),(- (0. X))) is right_complementable Element of the carrier of X
[(1. X),(- (0. X))] is set
the addF of X . [(1. X),(- (0. X))] is set
((1. X) - (0. X)) - (x * y) is right_complementable Element of the carrier of X
((1. X) - (0. X)) + (- (x * y)) is right_complementable Element of the carrier of X
K411( the carrier of X, the addF of X,((1. X) - (0. X)),(- (x * y))) is right_complementable Element of the carrier of X
[((1. X) - (0. X)),(- (x * y))] is set
the addF of X . [((1. X) - (0. X)),(- (x * y))] is set
(x * yn) - (x * yn) is right_complementable Element of the carrier of X
(x * yn) + (- (x * yn)) is right_complementable Element of the carrier of X
K411( the carrier of X, the addF of X,(x * yn),(- (x * yn))) is right_complementable Element of the carrier of X
[(x * yn),(- (x * yn))] is set
the addF of X . [(x * yn),(- (x * yn))] is set
(1. X) - ((x * yn) - (x * yn)) is right_complementable Element of the carrier of X
- ((x * yn) - (x * yn)) is right_complementable Element of the carrier of X
(1. X) + (- ((x * yn) - (x * yn))) is right_complementable Element of the carrier of X
K411( the carrier of X, the addF of X,(1. X),(- ((x * yn) - (x * yn)))) is right_complementable Element of the carrier of X
[(1. X),(- ((x * yn) - (x * yn)))] is set
the addF of X . [(1. X),(- ((x * yn) - (x * yn)))] is set
((1. X) - ((x * yn) - (x * yn))) - (x * y) is right_complementable Element of the carrier of X
((1. X) - ((x * yn) - (x * yn))) + (- (x * y)) is right_complementable Element of the carrier of X
K411( the carrier of X, the addF of X,((1. X) - ((x * yn) - (x * yn))),(- (x * y))) is right_complementable Element of the carrier of X
[((1. X) - ((x * yn) - (x * yn))),(- (x * y))] is set
the addF of X . [((1. X) - ((x * yn) - (x * yn))),(- (x * y))] is set
((1. X) - (x * yn)) + (x * yn) is right_complementable Element of the carrier of X
K411( the carrier of X, the addF of X,((1. X) - (x * yn)),(x * yn)) is right_complementable Element of the carrier of X
[((1. X) - (x * yn)),(x * yn)] is set
the addF of X . [((1. X) - (x * yn)),(x * yn)] is set
(((1. X) - (x * yn)) + (x * yn)) - (x * y) is right_complementable Element of the carrier of X
(((1. X) - (x * yn)) + (x * yn)) + (- (x * y)) is right_complementable Element of the carrier of X
K411( the carrier of X, the addF of X,(((1. X) - (x * yn)) + (x * yn)),(- (x * y))) is right_complementable Element of the carrier of X
[(((1. X) - (x * yn)) + (x * yn)),(- (x * y))] is set
the addF of X . [(((1. X) - (x * yn)) + (x * yn)),(- (x * y))] is set
(x * yn) - (x * y) is right_complementable Element of the carrier of X
(x * yn) + (- (x * y)) is right_complementable Element of the carrier of X
K411( the carrier of X, the addF of X,(x * yn),(- (x * y))) is right_complementable Element of the carrier of X
[(x * yn),(- (x * y))] is set
the addF of X . [(x * yn),(- (x * y))] is set
((1. X) - (x * yn)) + ((x * yn) - (x * y)) is right_complementable Element of the carrier of X
K411( the carrier of X, the addF of X,((1. X) - (x * yn)),((x * yn) - (x * y))) is right_complementable Element of the carrier of X
[((1. X) - (x * yn)),((x * yn) - (x * y))] is set
the addF of X . [((1. X) - (x * yn)),((x * yn) - (x * y))] is set
((1. X) - (x * yn)) + (x * (yn - y)) is right_complementable Element of the carrier of X
K411( the carrier of X, the addF of X,((1. X) - (x * yn)),(x * (yn - y))) is right_complementable Element of the carrier of X
[((1. X) - (x * yn)),(x * (yn - y))] is set
the addF of X . [((1. X) - (x * yn)),(x * (yn - y))] is set
||.((1. X) - (x * y)).|| is V24() real ext-real non negative Element of REAL
((Partial_Sums (X,((1. X) - x))) - y) . n is right_complementable Element of the carrier of X
||.(((Partial_Sums (X,((1. X) - x))) - y) . n).|| is V24() real ext-real non negative Element of REAL
||.x.|| * ||.(((Partial_Sums (X,((1. X) - x))) - y) . n).|| is V24() real ext-real non negative Element of REAL
(||.((X,((1. X) - x)) ^\ 1).|| . n) + (||.x.|| * ||.(((Partial_Sums (X,((1. X) - x))) - y) . n).||) is V24() real ext-real Element of REAL
||.((Partial_Sums (X,((1. X) - x))) - y).|| . n is V24() real ext-real Element of REAL
||.x.|| * (||.((Partial_Sums (X,((1. X) - x))) - y).|| . n) is V24() real ext-real Element of REAL
(||.((X,((1. X) - x)) ^\ 1).|| . n) + (||.x.|| * (||.((Partial_Sums (X,((1. X) - x))) - y).|| . n)) is V24() real ext-real Element of REAL
(||.x.|| (#) ||.((Partial_Sums (X,((1. X) - x))) - y).||) . n is V24() real ext-real Element of REAL
(||.((X,((1. X) - x)) ^\ 1).|| . n) + ((||.x.|| (#) ||.((Partial_Sums (X,((1. X) - x))) - y).||) . n) is V24() real ext-real Element of REAL
(||.((X,((1. X) - x)) ^\ 1).|| + (||.x.|| (#) ||.((Partial_Sums (X,((1. X) - x))) - y).||)) . n is V24() real ext-real Element of REAL
(Partial_Sums (X,((1. X) - x))) . 0 is right_complementable Element of the carrier of X
((Partial_Sums (X,((1. X) - x))) . 0) * ((1. X) - x) is right_complementable Element of the carrier of X
K411( the carrier of X, the multF of X,((Partial_Sums (X,((1. X) - x))) . 0),((1. X) - x)) is right_complementable Element of the carrier of X
[((Partial_Sums (X,((1. X) - x))) . 0),((1. X) - x)] is set
the multF of X . [((Partial_Sums (X,((1. X) - x))) . 0),((1. X) - x)] is set
((X,((1. X) - x)) . 0) * ((1. X) - x) is right_complementable Element of the carrier of X
K411( the carrier of X, the multF of X,((X,((1. X) - x)) . 0),((1. X) - x)) is right_complementable Element of the carrier of X
[((X,((1. X) - x)) . 0),((1. X) - x)] is set
the multF of X . [((X,((1. X) - x)) . 0),((1. X) - x)] is set
(1. X) * ((1. X) - x) is right_complementable Element of the carrier of X
K411( the carrier of X, the multF of X,(1. X),((1. X) - x)) is right_complementable Element of the carrier of X
[(1. X),((1. X) - x)] is set
the multF of X . [(1. X),((1. X) - x)] is set
n is epsilon-transitive epsilon-connected ordinal natural V24() real ext-real V45() V46() V52() V53() V54() V55() V56() V57() Element of NAT
(Partial_Sums (X,((1. X) - x))) . n is right_complementable Element of the carrier of X
((Partial_Sums (X,((1. X) - x))) . n) * ((1. X) - x) is right_complementable Element of the carrier of X
K411( the carrier of X, the multF of X,((Partial_Sums (X,((1. X) - x))) . n),((1. X) - x)) is right_complementable Element of the carrier of X
[((Partial_Sums (X,((1. X) - x))) . n),((1. X) - x)] is set
the multF of X . [((Partial_Sums (X,((1. X) - x))) . n),((1. X) - x)] is set
((Partial_Sums (X,((1. X) - x))) ^\ 1) . n is right_complementable Element of the carrier of X
(((Partial_Sums (X,((1. X) - x))) ^\ 1) . n) - ((X,((1. X) - x)) . 0) is right_complementable Element of the carrier of X
(((Partial_Sums (X,((1. X) - x))) ^\ 1) . n) + (- ((X,((1. X) - x)) . 0)) is right_complementable Element of the carrier of X
K411( the carrier of X, the addF of X,(((Partial_Sums (X,((1. X) - x))) ^\ 1) . n),(- ((X,((1. X) - x)) . 0))) is right_complementable Element of the carrier of X
[(((Partial_Sums (X,((1. X) - x))) ^\ 1) . n),(- ((X,((1. X) - x)) . 0))] is set
the addF of X . [(((Partial_Sums (X,((1. X) - x))) ^\ 1) . n),(- ((X,((1. X) - x)) . 0))] is set
n + 1 is epsilon-transitive epsilon-connected ordinal natural V24() real ext-real V45() V46() V52() V53() V54() V55() V56() V57() Element of NAT
(Partial_Sums (X,((1. X) - x))) . (n + 1) is right_complementable Element of the carrier of X
((Partial_Sums (X,((1. X) - x))) . (n + 1)) * ((1. X) - x) is right_complementable Element of the carrier of X
K411( the carrier of X, the multF of X,((Partial_Sums (X,((1. X) - x))) . (n + 1)),((1. X) - x)) is right_complementable Element of the carrier of X
[((Partial_Sums (X,((1. X) - x))) . (n + 1)),((1. X) - x)] is set
the multF of X . [((Partial_Sums (X,((1. X) - x))) . (n + 1)),((1. X) - x)] is set
((Partial_Sums (X,((1. X) - x))) ^\ 1) . (n + 1) is right_complementable Element of the carrier of X
(((Partial_Sums (X,((1. X) - x))) ^\ 1) . (n + 1)) - ((X,((1. X) - x)) . 0) is right_complementable Element of the carrier of X
(((Partial_Sums (X,((1. X) - x))) ^\ 1) . (n + 1)) + (- ((X,((1. X) - x)) . 0)) is right_complementable Element of the carrier of X
K411( the carrier of X, the addF of X,(((Partial_Sums (X,((1. X) - x))) ^\ 1) . (n + 1)),(- ((X,((1. X) - x)) . 0))) is right_complementable Element of the carrier of X
[(((Partial_Sums (X,((1. X) - x))) ^\ 1) . (n + 1)),(- ((X,((1. X) - x)) . 0))] is set
the addF of X . [(((Partial_Sums (X,((1. X) - x))) ^\ 1) . (n + 1)),(- ((X,((1. X) - x)) . 0))] is set
(n + 1) + 1 is epsilon-transitive epsilon-connected ordinal natural V24() real ext-real V45() V46() V52() V53() V54() V55() V56() V57() Element of NAT
(Partial_Sums (X,((1. X) - x))) . ((n + 1) + 1) is right_complementable Element of the carrier of X
((Partial_Sums (X,((1. X) - x))) . ((n + 1) + 1)) - ((X,((1. X) - x)) . 0) is right_complementable Element of the carrier of X
((Partial_Sums (X,((1. X) - x))) . ((n + 1) + 1)) + (- ((X,((1. X) - x)) . 0)) is right_complementable Element of the carrier of X
K411( the carrier of X, the addF of X,((Partial_Sums (X,((1. X) - x))) . ((n + 1) + 1)),(- ((X,((1. X) - x)) . 0))) is right_complementable Element of the carrier of X
[((Partial_Sums (X,((1. X) - x))) . ((n + 1) + 1)),(- ((X,((1. X) - x)) . 0))] is set
the addF of X . [((Partial_Sums (X,((1. X) - x))) . ((n + 1) + 1)),(- ((X,((1. X) - x)) . 0))] is set
(X,((1. X) - x)) . ((n + 1) + 1) is right_complementable Element of the carrier of X
((Partial_Sums (X,((1. X) - x))) . (n + 1)) + ((X,((1. X) - x)) . ((n + 1) + 1)) is right_complementable Element of the carrier of X
K411( the carrier of X, the addF of X,((Partial_Sums (X,((1. X) - x))) . (n + 1)),((X,((1. X) - x)) . ((n + 1) + 1))) is right_complementable Element of the carrier of X
[((Partial_Sums (X,((1. X) - x))) . (n + 1)),((X,((1. X) - x)) . ((n + 1) + 1))] is set
the addF of X . [((Partial_Sums (X,((1. X) - x))) . (n + 1)),((X,((1. X) - x)) . ((n + 1) + 1))] is set
(((Partial_Sums (X,((1. X) - x))) . (n + 1)) + ((X,((1. X) - x)) . ((n + 1) + 1))) - ((X,((1. X) - x)) . 0) is right_complementable Element of the carrier of X
(((Partial_Sums (X,((1. X) - x))) . (n + 1)) + ((X,((1. X) - x)) . ((n + 1) + 1))) + (- ((X,((1. X) - x)) . 0)) is right_complementable Element of the carrier of X
K411( the carrier of X, the addF of X,(((Partial_Sums (X,((1. X) - x))) . (n + 1)) + ((X,((1. X) - x)) . ((n + 1) + 1))),(- ((X,((1. X) - x)) . 0))) is right_complementable Element of the carrier of X
[(((Partial_Sums (X,((1. X) - x))) . (n + 1)) + ((X,((1. X) - x)) . ((n + 1) + 1))),(- ((X,((1. X) - x)) . 0))] is set
the addF of X . [(((Partial_Sums (X,((1. X) - x))) . (n + 1)) + ((X,((1. X) - x)) . ((n + 1) + 1))),(- ((X,((1. X) - x)) . 0))] is set
(((Partial_Sums (X,((1. X) - x))) ^\ 1) . n) + ((X,((1. X) - x)) . ((n + 1) + 1)) is right_complementable Element of the carrier of X
K411( the carrier of X, the addF of X,(((Partial_Sums (X,((1. X) - x))) ^\ 1) . n),((X,((1. X) - x)) . ((n + 1) + 1))) is right_complementable Element of the carrier of X
[(((Partial_Sums (X,((1. X) - x))) ^\ 1) . n),((X,((1. X) - x)) . ((n + 1) + 1))] is set
the addF of X . [(((Partial_Sums (X,((1. X) - x))) ^\ 1) . n),((X,((1. X) - x)) . ((n + 1) + 1))] is set
((((Partial_Sums (X,((1. X) - x))) ^\ 1) . n) + ((X,((1. X) - x)) . ((n + 1) + 1))) + (- ((X,((1. X) - x)) . 0)) is right_complementable Element of the carrier of X
K411( the carrier of X, the addF of X,((((Partial_Sums (X,((1. X) - x))) ^\ 1) . n) + ((X,((1. X) - x)) . ((n + 1) + 1))),(- ((X,((1. X) - x)) . 0))) is right_complementable Element of the carrier of X
[((((Partial_Sums (X,((1. X) - x))) ^\ 1) . n) + ((X,((1. X) - x)) . ((n + 1) + 1))),(- ((X,((1. X) - x)) . 0))] is set
the addF of X . [((((Partial_Sums (X,((1. X) - x))) ^\ 1) . n) + ((X,((1. X) - x)) . ((n + 1) + 1))),(- ((X,((1. X) - x)) . 0))] is set
((((Partial_Sums (X,((1. X) - x))) ^\ 1) . n) - ((X,((1. X) - x)) . 0)) + ((X,((1. X) - x)) . ((n + 1) + 1)) is right_complementable Element of the carrier of X
K411( the carrier of X, the addF of X,((((Partial_Sums (X,((1. X) - x))) ^\ 1) . n) - ((X,((1. X) - x)) . 0)),((X,((1. X) - x)) . ((n + 1) + 1))) is right_complementable Element of the carrier of X
[((((Partial_Sums (X,((1. X) - x))) ^\ 1) . n) - ((X,((1. X) - x)) . 0)),((X,((1. X) - x)) . ((n + 1) + 1))] is set
the addF of X . [((((Partial_Sums (X,((1. X) - x))) ^\ 1) . n) - ((X,((1. X) - x)) . 0)),((X,((1. X) - x)) . ((n + 1) + 1))] is set
(X,((1. X) - x)) . (n + 1) is right_complementable Element of the carrier of X
((Partial_Sums (X,((1. X) - x))) . n) + ((X,((1. X) - x)) . (n + 1)) is right_complementable Element of the carrier of X
K411( the carrier of X, the addF of X,((Partial_Sums (X,((1. X) - x))) . n),((X,((1. X) - x)) . (n + 1))) is right_complementable Element of the carrier of X
[((Partial_Sums (X,((1. X) - x))) . n),((X,((1. X) - x)) . (n + 1))] is set
the addF of X . [((Partial_Sums (X,((1. X) - x))) . n),((X,((1. X) - x)) . (n + 1))] is set
(((Partial_Sums (X,((1. X) - x))) . n) + ((X,((1. X) - x)) . (n + 1))) * ((1. X) - x) is right_complementable Element of the carrier of X
K411( the carrier of X, the multF of X,(((Partial_Sums (X,((1. X) - x))) . n) + ((X,((1. X) - x)) . (n + 1))),((1. X) - x)) is right_complementable Element of the carrier of X
[(((Partial_Sums (X,((1. X) - x))) . n) + ((X,((1. X) - x)) . (n + 1))),((1. X) - x)] is set
the multF of X . [(((Partial_Sums (X,((1. X) - x))) . n) + ((X,((1. X) - x)) . (n + 1))),((1. X) - x)] is set
((X,((1. X) - x)) . (n + 1)) * ((1. X) - x) is right_complementable Element of the carrier of X
K411( the carrier of X, the multF of X,((X,((1. X) - x)) . (n + 1)),((1. X) - x)) is right_complementable Element of the carrier of X
[((X,((1. X) - x)) . (n + 1)),((1. X) - x)] is set
the multF of X . [((X,((1. X) - x)) . (n + 1)),((1. X) - x)] is set
(((Partial_Sums (X,((1. X) - x))) . n) * ((1. X) - x)) + (((X,((1. X) - x)) . (n + 1)) * ((1. X) - x)) is right_complementable Element of the carrier of X
K411( the carrier of X, the addF of X,(((Partial_Sums (X,((1. X) - x))) . n) * ((1. X) - x)),(((X,((1. X) - x)) . (n + 1)) * ((1. X) - x))) is right_complementable Element of the carrier of X
[(((Partial_Sums (X,((1. X) - x))) . n) * ((1. X) - x)),(((X,((1. X) - x)) . (n + 1)) * ((1. X) - x))] is set
the addF of X . [(((Partial_Sums (X,((1. X) - x))) . n) * ((1. X) - x)),(((X,((1. X) - x)) . (n + 1)) * ((1. X) - x))] is set
((Partial_Sums (X,((1. X) - x))) ^\ 1) . 0 is right_complementable Element of the carrier of X
(((Partial_Sums (X,((1. X) - x))) ^\ 1) . 0) - ((X,((1. X) - x)) . 0) is right_complementable Element of the carrier of X
(((Partial_Sums (X,((1. X) - x))) ^\ 1) . 0) + (- ((X,((1. X) - x)) . 0)) is right_complementable Element of the carrier of X
K411( the carrier of X, the addF of X,(((Partial_Sums (X,((1. X) - x))) ^\ 1) . 0),(- ((X,((1. X) - x)) . 0))) is right_complementable Element of the carrier of X
[(((Partial_Sums (X,((1. X) - x))) ^\ 1) . 0),(- ((X,((1. X) - x)) . 0))] is set
the addF of X . [(((Partial_Sums (X,((1. X) - x))) ^\ 1) . 0),(- ((X,((1. X) - x)) . 0))] is set
0 + 1 is V4() epsilon-transitive epsilon-connected ordinal natural V24() real ext-real positive non negative V45() V46() V52() V53() V54() V55() V56() V57() Element of NAT
(Partial_Sums (X,((1. X) - x))) . (0 + 1) is right_complementable Element of the carrier of X
((Partial_Sums (X,((1. X) - x))) . (0 + 1)) - ((X,((1. X) - x)) . 0) is right_complementable Element of the carrier of X
((Partial_Sums (X,((1. X) - x))) . (0 + 1)) + (- ((X,((1. X) - x)) . 0)) is right_complementable Element of the carrier of X
K411( the carrier of X, the addF of X,((Partial_Sums (X,((1. X) - x))) . (0 + 1)),(- ((X,((1. X) - x)) . 0))) is right_complementable Element of the carrier of X
[((Partial_Sums (X,((1. X) - x))) . (0 + 1)),(- ((X,((1. X) - x)) . 0))] is set
the addF of X . [((Partial_Sums (X,((1. X) - x))) . (0 + 1)),(- ((X,((1. X) - x)) . 0))] is set
(X,((1. X) - x)) . 1 is right_complementable Element of the carrier of X
((Partial_Sums (X,((1. X) - x))) . 0) + ((X,((1. X) - x)) . 1) is right_complementable Element of the carrier of X
K411( the carrier of X, the addF of X,((Partial_Sums (X,((1. X) - x))) . 0),((X,((1. X) - x)) . 1)) is right_complementable Element of the carrier of X
[((Partial_Sums (X,((1. X) - x))) . 0),((X,((1. X) - x)) . 1)] is set
the addF of X . [((Partial_Sums (X,((1. X) - x))) . 0),((X,((1. X) - x)) . 1)] is set
(((Partial_Sums (X,((1. X) - x))) . 0) + ((X,((1. X) - x)) . 1)) - ((X,((1. X) - x)) . 0) is right_complementable Element of the carrier of X
(((Partial_Sums (X,((1. X) - x))) . 0) + ((X,((1. X) - x)) . 1)) + (- ((X,((1. X) - x)) . 0)) is right_complementable Element of the carrier of X
K411( the carrier of X, the addF of X,(((Partial_Sums (X,((1. X) - x))) . 0) + ((X,((1. X) - x)) . 1)),(- ((X,((1. X) - x)) . 0))) is right_complementable Element of the carrier of X
[(((Partial_Sums (X,((1. X) - x))) . 0) + ((X,((1. X) - x)) . 1)),(- ((X,((1. X) - x)) . 0))] is set
the addF of X . [(((Partial_Sums (X,((1. X) - x))) . 0) + ((X,((1. X) - x)) . 1)),(- ((X,((1. X) - x)) . 0))] is set
((X,((1. X) - x)) . 0) + ((X,((1. X) - x)) . 1) is right_complementable Element of the carrier of X
K411( the carrier of X, the addF of X,((X,((1. X) - x)) . 0),((X,((1. X) - x)) . 1)) is right_complementable Element of the carrier of X
[((X,((1. X) - x)) . 0),((X,((1. X) - x)) . 1)] is set
the addF of X . [((X,((1. X) - x)) . 0),((X,((1. X) - x)) . 1)] is set
(((X,((1. X) - x)) . 0) + ((X,((1. X) - x)) . 1)) - ((X,((1. X) - x)) . 0) is right_complementable Element of the carrier of X
(((X,((1. X) - x)) . 0) + ((X,((1. X) - x)) . 1)) + (- ((X,((1. X) - x)) . 0)) is right_complementable Element of the carrier of X
K411( the carrier of X, the addF of X,(((X,((1. X) - x)) . 0) + ((X,((1. X) - x)) . 1)),(- ((X,((1. X) - x)) . 0))) is right_complementable Element of the carrier of X
[(((X,((1. X) - x)) . 0) + ((X,((1. X) - x)) . 1)),(- ((X,((1. X) - x)) . 0))] is set
the addF of X . [(((X,((1. X) - x)) . 0) + ((X,((1. X) - x)) . 1)),(- ((X,((1. X) - x)) . 0))] is set
((X,((1. X) - x)) . 0) - ((X,((1. X) - x)) . 0) is right_complementable Element of the carrier of X
((X,((1. X) - x)) . 0) + (- ((X,((1. X) - x)) . 0)) is right_complementable Element of the carrier of X
K411( the carrier of X, the addF of X,((X,((1. X) - x)) . 0),(- ((X,((1. X) - x)) . 0))) is right_complementable Element of the carrier of X
[((X,((1. X) - x)) . 0),(- ((X,((1. X) - x)) . 0))] is set
the addF of X . [((X,((1. X) - x)) . 0),(- ((X,((1. X) - x)) . 0))] is set
((X,((1. X) - x)) . 1) + (((X,((1. X) - x)) . 0) - ((X,((1. X) - x)) . 0)) is right_complementable Element of the carrier of X
K411( the carrier of X, the addF of X,((X,((1. X) - x)) . 1),(((X,((1. X) - x)) . 0) - ((X,((1. X) - x)) . 0))) is right_complementable Element of the carrier of X
[((X,((1. X) - x)) . 1),(((X,((1. X) - x)) . 0) - ((X,((1. X) - x)) . 0))] is set
the addF of X . [((X,((1. X) - x)) . 1),(((X,((1. X) - x)) . 0) - ((X,((1. X) - x)) . 0))] is set
((X,((1. X) - x)) . 1) + (0. X) is right_complementable Element of the carrier of X
K411( the carrier of X, the addF of X,((X,((1. X) - x)) . 1),(0. X)) is right_complementable Element of the carrier of X
[((X,((1. X) - x)) . 1),(0. X)] is set
the addF of X . [((X,((1. X) - x)) . 1),(0. X)] is set
(X,((1. X) - x)) . (0 + 1) is right_complementable Element of the carrier of X
n is epsilon-transitive epsilon-connected ordinal natural V24() real ext-real V45() V46() V52() V53() V54() V55() V56() V57() Element of NAT
(Partial_Sums (X,((1. X) - x))) . n is right_complementable Element of the carrier of X
((Partial_Sums (X,((1. X) - x))) . n) * ((1. X) - x) is right_complementable Element of the carrier of X
K411( the carrier of X, the multF of X,((Partial_Sums (X,((1. X) - x))) . n),((1. X) - x)) is right_complementable Element of the carrier of X
[((Partial_Sums (X,((1. X) - x))) . n),((1. X) - x)] is set
the multF of X . [((Partial_Sums (X,((1. X) - x))) . n),((1. X) - x)] is set
((Partial_Sums (X,((1. X) - x))) ^\ 1) . n is right_complementable Element of the carrier of X
(((Partial_Sums (X,((1. X) - x))) ^\ 1) . n) - ((X,((1. X) - x)) . 0) is right_complementable Element of the carrier of X
(((Partial_Sums (X,((1. X) - x))) ^\ 1) . n) + (- ((X,((1. X) - x)) . 0)) is right_complementable Element of the carrier of X
K411( the carrier of X, the addF of X,(((Partial_Sums (X,((1. X) - x))) ^\ 1) . n),(- ((X,((1. X) - x)) . 0))) is right_complementable Element of the carrier of X
[(((Partial_Sums (X,((1. X) - x))) ^\ 1) . n),(- ((X,((1. X) - x)) . 0))] is set
the addF of X . [(((Partial_Sums (X,((1. X) - x))) ^\ 1) . n),(- ((X,((1. X) - x)) . 0))] is set
n is epsilon-transitive epsilon-connected ordinal natural V24() real ext-real V45() V46() V52() V53() V54() V55() V56() V57() Element of NAT
(Partial_Sums (X,((1. X) - x))) . n is right_complementable Element of the carrier of X
n + 1 is epsilon-transitive epsilon-connected ordinal natural V24() real ext-real V45() V46() V52() V53() V54() V55() V56() V57() Element of NAT
(Partial_Sums (X,((1. X) - x))) . (n + 1) is right_complementable Element of the carrier of X
(X,((1. X) - x)) . (n + 1) is right_complementable Element of the carrier of X
((Partial_Sums (X,((1. X) - x))) . n) + ((X,((1. X) - x)) . (n + 1)) is right_complementable Element of the carrier of X
K411( the carrier of X, the addF of X,((Partial_Sums (X,((1. X) - x))) . n),((X,((1. X) - x)) . (n + 1))) is right_complementable Element of the carrier of X
[((Partial_Sums (X,((1. X) - x))) . n),((X,((1. X) - x)) . (n + 1))] is set
the addF of X . [((Partial_Sums (X,((1. X) - x))) . n),((X,((1. X) - x)) . (n + 1))] is set
((X,((1. X) - x)) ^\ 1) . n is right_complementable Element of the carrier of X
((Partial_Sums (X,((1. X) - x))) . n) + (((X,((1. X) - x)) ^\ 1) . n) is right_complementable Element of the carrier of X
K411( the carrier of X, the addF of X,((Partial_Sums (X,((1. X) - x))) . n),(((X,((1. X) - x)) ^\ 1) . n)) is right_complementable Element of the carrier of X
[((Partial_Sums (X,((1. X) - x))) . n),(((X,((1. X) - x)) ^\ 1) . n)] is set
the addF of X . [((Partial_Sums (X,((1. X) - x))) . n),(((X,((1. X) - x)) ^\ 1) . n)] is set
yn is right_complementable Element of the carrier of X
yn * ((1. X) - ((1. X) - x)) is right_complementable Element of the carrier of X
K411( the carrier of X, the multF of X,yn,((1. X) - ((1. X) - x))) is right_complementable Element of the carrier of X
[yn,((1. X) - ((1. X) - x))] is set
the multF of X . [yn,((1. X) - ((1. X) - x))] is set
yn * (1. X) is right_complementable Element of the carrier of X
K411( the carrier of X, the multF of X,yn,(1. X)) is right_complementable Element of the carrier of X
[yn,(1. X)] is set
the multF of X . [yn,(1. X)] is set
yn * ((1. X) - x) is right_complementable Element of the carrier of X
K411( the carrier of X, the multF of X,yn,((1. X) - x)) is right_complementable Element of the carrier of X
[yn,((1. X) - x)] is set
the multF of X . [yn,((1. X) - x)] is set
(yn * (1. X)) - (yn * ((1. X) - x)) is right_complementable Element of the carrier of X
- (yn * ((1. X) - x)) is right_complementable Element of the carrier of X
(yn * (1. X)) + (- (yn * ((1. X) - x))) is right_complementable Element of the carrier of X
K411( the carrier of X, the addF of X,(yn * (1. X)),(- (yn * ((1. X) - x)))) is right_complementable Element of the carrier of X
[(yn * (1. X)),(- (yn * ((1. X) - x)))] is set
the addF of X . [(yn * (1. X)),(- (yn * ((1. X) - x)))] is set
yn - (yn * ((1. X) - x)) is right_complementable Element of the carrier of X
yn + (- (yn * ((1. X) - x))) is right_complementable Element of the carrier of X
K411( the carrier of X, the addF of X,yn,(- (yn * ((1. X) - x)))) is right_complementable Element of the carrier of X
[yn,(- (yn * ((1. X) - x)))] is set
the addF of X . [yn,(- (yn * ((1. X) - x)))] is set
((Partial_Sums (X,((1. X) - x))) ^\ 1) . n is right_complementable Element of the carrier of X
(((Partial_Sums (X,((1. X) - x))) ^\ 1) . n) - ((X,((1. X) - x)) . 0) is right_complementable Element of the carrier of X
(((Partial_Sums (X,((1. X) - x))) ^\ 1) . n) + (- ((X,((1. X) - x)) . 0)) is right_complementable Element of the carrier of X
K411( the carrier of X, the addF of X,(((Partial_Sums (X,((1. X) - x))) ^\ 1) . n),(- ((X,((1. X) - x)) . 0))) is right_complementable Element of the carrier of X
[(((Partial_Sums (X,((1. X) - x))) ^\ 1) . n),(- ((X,((1. X) - x)) . 0))] is set
the addF of X . [(((Partial_Sums (X,((1. X) - x))) ^\ 1) . n),(- ((X,((1. X) - x)) . 0))] is set
((Partial_Sums (X,((1. X) - x))) . n) - ((((Partial_Sums (X,((1. X) - x))) ^\ 1) . n) - ((X,((1. X) - x)) . 0)) is right_complementable Element of the carrier of X
- ((((Partial_Sums (X,((1. X) - x))) ^\ 1) . n) - ((X,((1. X) - x)) . 0)) is right_complementable Element of the carrier of X
((Partial_Sums (X,((1. X) - x))) . n) + (- ((((Partial_Sums (X,((1. X) - x))) ^\ 1) . n) - ((X,((1. X) - x)) . 0))) is right_complementable Element of the carrier of X
K411( the carrier of X, the addF of X,((Partial_Sums (X,((1. X) - x))) . n),(- ((((Partial_Sums (X,((1. X) - x))) ^\ 1) . n) - ((X,((1. X) - x)) . 0)))) is right_complementable Element of the carrier of X
[((Partial_Sums (X,((1. X) - x))) . n),(- ((((Partial_Sums (X,((1. X) - x))) ^\ 1) . n) - ((X,((1. X) - x)) . 0)))] is set
the addF of X . [((Partial_Sums (X,((1. X) - x))) . n),(- ((((Partial_Sums (X,((1. X) - x))) ^\ 1) . n) - ((X,((1. X) - x)) . 0)))] is set
((Partial_Sums (X,((1. X) - x))) . n) - (((Partial_Sums (X,((1. X) - x))) ^\ 1) . n) is right_complementable Element of the carrier of X
- (((Partial_Sums (X,((1. X) - x))) ^\ 1) . n) is right_complementable Element of the carrier of X
((Partial_Sums (X,((1. X) - x))) . n) + (- (((Partial_Sums (X,((1. X) - x))) ^\ 1) . n)) is right_complementable Element of the carrier of X
K411( the carrier of X, the addF of X,((Partial_Sums (X,((1. X) - x))) . n),(- (((Partial_Sums (X,((1. X) - x))) ^\ 1) . n))) is right_complementable Element of the carrier of X
[((Partial_Sums (X,((1. X) - x))) . n),(- (((Partial_Sums (X,((1. X) - x))) ^\ 1) . n))] is set
the addF of X . [((Partial_Sums (X,((1. X) - x))) . n),(- (((Partial_Sums (X,((1. X) - x))) ^\ 1) . n))] is set
(((Partial_Sums (X,((1. X) - x))) . n) - (((Partial_Sums (X,((1. X) - x))) ^\ 1) . n)) + ((X,((1. X) - x)) . 0) is right_complementable Element of the carrier of X
K411( the carrier of X, the addF of X,(((Partial_Sums (X,((1. X) - x))) . n) - (((Partial_Sums (X,((1. X) - x))) ^\ 1) . n)),((X,((1. X) - x)) . 0)) is right_complementable Element of the carrier of X
[(((Partial_Sums (X,((1. X) - x))) . n) - (((Partial_Sums (X,((1. X) - x))) ^\ 1) . n)),((X,((1. X) - x)) . 0)] is set
the addF of X . [(((Partial_Sums (X,((1. X) - x))) . n) - (((Partial_Sums (X,((1. X) - x))) ^\ 1) . n)),((X,((1. X) - x)) . 0)] is set
((Partial_Sums (X,((1. X) - x))) . n) - (((Partial_Sums (X,((1. X) - x))) . n) + (((X,((1. X) - x)) ^\ 1) . n)) is right_complementable Element of the carrier of X
- (((Partial_Sums (X,((1. X) - x))) . n) + (((X,((1. X) - x)) ^\ 1) . n)) is right_complementable Element of the carrier of X
((Partial_Sums (X,((1. X) - x))) . n) + (- (((Partial_Sums (X,((1. X) - x))) . n) + (((X,((1. X) - x)) ^\ 1) . n))) is right_complementable Element of the carrier of X
K411( the carrier of X, the addF of X,((Partial_Sums (X,((1. X) - x))) . n),(- (((Partial_Sums (X,((1. X) - x))) . n) + (((X,((1. X) - x)) ^\ 1) . n)))) is right_complementable Element of the carrier of X
[((Partial_Sums (X,((1. X) - x))) . n),(- (((Partial_Sums (X,((1. X) - x))) . n) + (((X,((1. X) - x)) ^\ 1) . n)))] is set
the addF of X . [((Partial_Sums (X,((1. X) - x))) . n),(- (((Partial_Sums (X,((1. X) - x))) . n) + (((X,((1. X) - x)) ^\ 1) . n)))] is set
(((Partial_Sums (X,((1. X) - x))) . n) - (((Partial_Sums (X,((1. X) - x))) . n) + (((X,((1. X) - x)) ^\ 1) . n))) + ((X,((1. X) - x)) . 0) is right_complementable Element of the carrier of X
K411( the carrier of X, the addF of X,(((Partial_Sums (X,((1. X) - x))) . n) - (((Partial_Sums (X,((1. X) - x))) . n) + (((X,((1. X) - x)) ^\ 1) . n))),((X,((1. X) - x)) . 0)) is right_complementable Element of the carrier of X
[(((Partial_Sums (X,((1. X) - x))) . n) - (((Partial_Sums (X,((1. X) - x))) . n) + (((X,((1. X) - x)) ^\ 1) . n))),((X,((1. X) - x)) . 0)] is set
the addF of X . [(((Partial_Sums (X,((1. X) - x))) . n) - (((Partial_Sums (X,((1. X) - x))) . n) + (((X,((1. X) - x)) ^\ 1) . n))),((X,((1. X) - x)) . 0)] is set
((Partial_Sums (X,((1. X) - x))) . n) - ((Partial_Sums (X,((1. X) - x))) . n) is right_complementable Element of the carrier of X
- ((Partial_Sums (X,((1. X) - x))) . n) is right_complementable Element of the carrier of X
((Partial_Sums (X,((1. X) - x))) . n) + (- ((Partial_Sums (X,((1. X) - x))) . n)) is right_complementable Element of the carrier of X
K411( the carrier of X, the addF of X,((Partial_Sums (X,((1. X) - x))) . n),(- ((Partial_Sums (X,((1. X) - x))) . n))) is right_complementable Element of the carrier of X
[((Partial_Sums (X,((1. X) - x))) . n),(- ((Partial_Sums (X,((1. X) - x))) . n))] is set
the addF of X . [((Partial_Sums (X,((1. X) - x))) . n),(- ((Partial_Sums (X,((1. X) - x))) . n))] is set
(((Partial_Sums (X,((1. X) - x))) . n) - ((Partial_Sums (X,((1. X) - x))) . n)) - (((X,((1. X) - x)) ^\ 1) . n) is right_complementable Element of the carrier of X
- (((X,((1. X) - x)) ^\ 1) . n) is right_complementable Element of the carrier of X
(((Partial_Sums (X,((1. X) - x))) . n) - ((Partial_Sums (X,((1. X) - x))) . n)) + (- (((X,((1. X) - x)) ^\ 1) . n)) is right_complementable Element of the carrier of X
K411( the carrier of X, the addF of X,(((Partial_Sums (X,((1. X) - x))) . n) - ((Partial_Sums (X,((1. X) - x))) . n)),(- (((X,((1. X) - x)) ^\ 1) . n))) is right_complementable Element of the carrier of X
[(((Partial_Sums (X,((1. X) - x))) . n) - ((Partial_Sums (X,((1. X) - x))) . n)),(- (((X,((1. X) - x)) ^\ 1) . n))] is set
the addF of X . [(((Partial_Sums (X,((1. X) - x))) . n) - ((Partial_Sums (X,((1. X) - x))) . n)),(- (((X,((1. X) - x)) ^\ 1) . n))] is set
((((Partial_Sums (X,((1. X) - x))) . n) - ((Partial_Sums (X,((1. X) - x))) . n)) - (((X,((1. X) - x)) ^\ 1) . n)) + ((X,((1. X) - x)) . 0) is right_complementable Element of the carrier of X
K411( the carrier of X, the addF of X,((((Partial_Sums (X,((1. X) - x))) . n) - ((Partial_Sums (X,((1. X) - x))) . n)) - (((X,((1. X) - x)) ^\ 1) . n)),((X,((1. X) - x)) . 0)) is right_complementable Element of the carrier of X
[((((Partial_Sums (X,((1. X) - x))) . n) - ((Partial_Sums (X,((1. X) - x))) . n)) - (((X,((1. X) - x)) ^\ 1) . n)),((X,((1. X) - x)) . 0)] is set
the addF of X . [((((Partial_Sums (X,((1. X) - x))) . n) - ((Partial_Sums (X,((1. X) - x))) . n)) - (((X,((1. X) - x)) ^\ 1) . n)),((X,((1. X) - x)) . 0)] is set
(0. X) - (((X,((1. X) - x)) ^\ 1) . n) is right_complementable Element of the carrier of X
(0. X) + (- (((X,((1. X) - x)) ^\ 1) . n)) is right_complementable Element of the carrier of X
K411( the carrier of X, the addF of X,(0. X),(- (((X,((1. X) - x)) ^\ 1) . n))) is right_complementable Element of the carrier of X
[(0. X),(- (((X,((1. X) - x)) ^\ 1) . n))] is set
the addF of X . [(0. X),(- (((X,((1. X) - x)) ^\ 1) . n))] is set
((0. X) - (((X,((1. X) - x)) ^\ 1) . n)) + ((X,((1. X) - x)) . 0) is right_complementable Element of the carrier of X
K411( the carrier of X, the addF of X,((0. X) - (((X,((1. X) - x)) ^\ 1) . n)),((X,((1. X) - x)) . 0)) is right_complementable Element of the carrier of X
[((0. X) - (((X,((1. X) - x)) ^\ 1) . n)),((X,((1. X) - x)) . 0)] is set
the addF of X . [((0. X) - (((X,((1. X) - x)) ^\ 1) . n)),((X,((1. X) - x)) . 0)] is set
(((X,((1. X) - x)) ^\ 1) . n) - ((X,((1. X) - x)) . 0) is right_complementable Element of the carrier of X
(((X,((1. X) - x)) ^\ 1) . n) + (- ((X,((1. X) - x)) . 0)) is right_complementable Element of the carrier of X
K411( the carrier of X, the addF of X,(((X,((1. X) - x)) ^\ 1) . n),(- ((X,((1. X) - x)) . 0))) is right_complementable Element of the carrier of X
[(((X,((1. X) - x)) ^\ 1) . n),(- ((X,((1. X) - x)) . 0))] is set
the addF of X . [(((X,((1. X) - x)) ^\ 1) . n),(- ((X,((1. X) - x)) . 0))] is set
(0. X) - ((((X,((1. X) - x)) ^\ 1) . n) - ((X,((1. X) - x)) . 0)) is right_complementable Element of the carrier of X
- ((((X,((1. X) - x)) ^\ 1) . n) - ((X,((1. X) - x)) . 0)) is right_complementable Element of the carrier of X
(0. X) + (- ((((X,((1. X) - x)) ^\ 1) . n) - ((X,((1. X) - x)) . 0))) is right_complementable Element of the carrier of X
K411( the carrier of X, the addF of X,(0. X),(- ((((X,((1. X) - x)) ^\ 1) . n) - ((X,((1. X) - x)) . 0)))) is right_complementable Element of the carrier of X
[(0. X),(- ((((X,((1. X) - x)) ^\ 1) . n) - ((X,((1. X) - x)) . 0)))] is set
the addF of X . [(0. X),(- ((((X,((1. X) - x)) ^\ 1) . n) - ((X,((1. X) - x)) . 0)))] is set
((X,((1. X) - x)) . 0) - (((X,((1. X) - x)) ^\ 1) . n) is right_complementable Element of the carrier of X
((X,((1. X) - x)) . 0) + (- (((X,((1. X) - x)) ^\ 1) . n)) is right_complementable Element of the carrier of X
K411( the carrier of X, the addF of X,((X,((1. X) - x)) . 0),(- (((X,((1. X) - x)) ^\ 1) . n))) is right_complementable Element of the carrier of X
[((X,((1. X) - x)) . 0),(- (((X,((1. X) - x)) ^\ 1) . n))] is set
the addF of X . [((X,((1. X) - x)) . 0),(- (((X,((1. X) - x)) ^\ 1) . n))] is set
((X,((1. X) - x)) . 0) - ((X,((1. X) - x)) . (n + 1)) is right_complementable Element of the carrier of X
- ((X,((1. X) - x)) . (n + 1)) is right_complementable Element of the carrier of X
((X,((1. X) - x)) . 0) + (- ((X,((1. X) - x)) . (n + 1))) is right_complementable Element of the carrier of X
K411( the carrier of X, the addF of X,((X,((1. X) - x)) . 0),(- ((X,((1. X) - x)) . (n + 1)))) is right_complementable Element of the carrier of X
[((X,((1. X) - x)) . 0),(- ((X,((1. X) - x)) . (n + 1)))] is set
the addF of X . [((X,((1. X) - x)) . 0),(- ((X,((1. X) - x)) . (n + 1)))] is set
(1. X) - ((X,((1. X) - x)) . (n + 1)) is right_complementable Element of the carrier of X
(1. X) + (- ((X,((1. X) - x)) . (n + 1))) is right_complementable Element of the carrier of X
K411( the carrier of X, the addF of X,(1. X),(- ((X,((1. X) - x)) . (n + 1)))) is right_complementable Element of the carrier of X
[(1. X),(- ((X,((1. X) - x)) . (n + 1)))] is set
the addF of X . [(1. X),(- ((X,((1. X) - x)) . (n + 1)))] is set
yn * x is right_complementable Element of the carrier of X
K411( the carrier of X, the multF of X,yn,x) is right_complementable Element of the carrier of X
[yn,x] is set
the multF of X . [yn,x] is set
(1. X) - (yn * x) is right_complementable Element of the carrier of X
- (yn * x) is right_complementable Element of the carrier of X
(1. X) + (- (yn * x)) is right_complementable Element of the carrier of X
K411( the carrier of X, the addF of X,(1. X),(- (yn * x))) is right_complementable Element of the carrier of X
[(1. X),(- (yn * x))] is set
the addF of X . [(1. X),(- (yn * x))] is set
(1. X) - ((1. X) - ((X,((1. X) - x)) . (n + 1))) is right_complementable Element of the carrier of X
- ((1. X) - ((X,((1. X) - x)) . (n + 1))) is right_complementable Element of the carrier of X
(1. X) + (- ((1. X) - ((X,((1. X) - x)) . (n + 1)))) is right_complementable Element of the carrier of X
K411( the carrier of X, the addF of X,(1. X),(- ((1. X) - ((X,((1. X) - x)) . (n + 1))))) is right_complementable Element of the carrier of X
[(1. X),(- ((1. X) - ((X,((1. X) - x)) . (n + 1))))] is set
the addF of X . [(1. X),(- ((1. X) - ((X,((1. X) - x)) . (n + 1))))] is set
yn - y is right_complementable Element of the carrier of X
yn + (- y) is right_complementable Element of the carrier of X
K411( the carrier of X, the addF of X,yn,(- y)) is right_complementable Element of the carrier of X
[yn,(- y)] is set
the addF of X . [yn,(- y)] is set
(yn - y) * x is right_complementable Element of the carrier of X
K411( the carrier of X, the multF of X,(yn - y),x) is right_complementable Element of the carrier of X
[(yn - y),x] is set
the multF of X . [(yn - y),x] is set
||.((yn - y) * x).|| is V24() real ext-real non negative Element of REAL
||.(yn - y).|| is V24() real ext-real non negative Element of REAL
||.(yn - y).|| * ||.x.|| is V24() real ext-real non negative Element of REAL
||.((X,((1. X) - x)) ^\ 1).|| . n is V24() real ext-real Element of REAL
(||.((X,((1. X) - x)) ^\ 1).|| . n) + ||.((yn - y) * x).|| is V24() real ext-real Element of REAL
(||.((X,((1. X) - x)) ^\ 1).|| . n) + (||.(yn - y).|| * ||.x.||) is V24() real ext-real Element of REAL
(((X,((1. X) - x)) ^\ 1) . n) + ((yn - y) * x) is right_complementable Element of the carrier of X
K411( the carrier of X, the addF of X,(((X,((1. X) - x)) ^\ 1) . n),((yn - y) * x)) is right_complementable Element of the carrier of X
[(((X,((1. X) - x)) ^\ 1) . n),((yn - y) * x)] is set
the addF of X . [(((X,((1. X) - x)) ^\ 1) . n),((yn - y) * x)] is set
||.((((X,((1. X) - x)) ^\ 1) . n) + ((yn - y) * x)).|| is V24() real ext-real non negative Element of REAL
||.(((X,((1. X) - x)) ^\ 1) . n).|| is V24() real ext-real non negative Element of REAL
||.(((X,((1. X) - x)) ^\ 1) . n).|| + ||.((yn - y) * x).|| is V24() real ext-real non negative Element of REAL
y * x is right_complementable Element of the carrier of X
K411( the carrier of X, the multF of X,y,x) is right_complementable Element of the carrier of X
[y,x] is set
the multF of X . [y,x] is set
(1. X) - (y * x) is right_complementable Element of the carrier of X
- (y * x) is right_complementable Element of the carrier of X
(1. X) + (- (y * x)) is right_complementable Element of the carrier of X
K411( the carrier of X, the addF of X,(1. X),(- (y * x))) is right_complementable Element of the carrier of X
[(1. X),(- (y * x))] is set
the addF of X . [(1. X),(- (y * x))] is set
((1. X) - (0. X)) - (y * x) is right_complementable Element of the carrier of X
((1. X) - (0. X)) + (- (y * x)) is right_complementable Element of the carrier of X
K411( the carrier of X, the addF of X,((1. X) - (0. X)),(- (y * x))) is right_complementable Element of the carrier of X
[((1. X) - (0. X)),(- (y * x))] is set
the addF of X . [((1. X) - (0. X)),(- (y * x))] is set
(yn * x) - (yn * x) is right_complementable Element of the carrier of X
(yn * x) + (- (yn * x)) is right_complementable Element of the carrier of X
K411( the carrier of X, the addF of X,(yn * x),(- (yn * x))) is right_complementable Element of the carrier of X
[(yn * x),(- (yn * x))] is set
the addF of X . [(yn * x),(- (yn * x))] is set
(1. X) - ((yn * x) - (yn * x)) is right_complementable Element of the carrier of X
- ((yn * x) - (yn * x)) is right_complementable Element of the carrier of X
(1. X) + (- ((yn * x) - (yn * x))) is right_complementable Element of the carrier of X
K411( the carrier of X, the addF of X,(1. X),(- ((yn * x) - (yn * x)))) is right_complementable Element of the carrier of X
[(1. X),(- ((yn * x) - (yn * x)))] is set
the addF of X . [(1. X),(- ((yn * x) - (yn * x)))] is set
((1. X) - ((yn * x) - (yn * x))) - (y * x) is right_complementable Element of the carrier of X
((1. X) - ((yn * x) - (yn * x))) + (- (y * x)) is right_complementable Element of the carrier of X
K411( the carrier of X, the addF of X,((1. X) - ((yn * x) - (yn * x))),(- (y * x))) is right_complementable Element of the carrier of X
[((1. X) - ((yn * x) - (yn * x))),(- (y * x))] is set
the addF of X . [((1. X) - ((yn * x) - (yn * x))),(- (y * x))] is set
((1. X) - (yn * x)) + (yn * x) is right_complementable Element of the carrier of X
K411( the carrier of X, the addF of X,((1. X) - (yn * x)),(yn * x)) is right_complementable Element of the carrier of X
[((1. X) - (yn * x)),(yn * x)] is set
the addF of X . [((1. X) - (yn * x)),(yn * x)] is set
(((1. X) - (yn * x)) + (yn * x)) - (y * x) is right_complementable Element of the carrier of X
(((1. X) - (yn * x)) + (yn * x)) + (- (y * x)) is right_complementable Element of the carrier of X
K411( the carrier of X, the addF of X,(((1. X) - (yn * x)) + (yn * x)),(- (y * x))) is right_complementable Element of the carrier of X
[(((1. X) - (yn * x)) + (yn * x)),(- (y * x))] is set
the addF of X . [(((1. X) - (yn * x)) + (yn * x)),(- (y * x))] is set
(yn * x) - (y * x) is right_complementable Element of the carrier of X
(yn * x) + (- (y * x)) is right_complementable Element of the carrier of X
K411( the carrier of X, the addF of X,(yn * x),(- (y * x))) is right_complementable Element of the carrier of X
[(yn * x),(- (y * x))] is set
the addF of X . [(yn * x),(- (y * x))] is set
((1. X) - (yn * x)) + ((yn * x) - (y * x)) is right_complementable Element of the carrier of X
K411( the carrier of X, the addF of X,((1. X) - (yn * x)),((yn * x) - (y * x))) is right_complementable Element of the carrier of X
[((1. X) - (yn * x)),((yn * x) - (y * x))] is set
the addF of X . [((1. X) - (yn * x)),((yn * x) - (y * x))] is set
((1. X) - (yn * x)) + ((yn - y) * x) is right_complementable Element of the carrier of X
K411( the carrier of X, the addF of X,((1. X) - (yn * x)),((yn - y) * x)) is right_complementable Element of the carrier of X
[((1. X) - (yn * x)),((yn - y) * x)] is set
the addF of X . [((1. X) - (yn * x)),((yn - y) * x)] is set
||.((1. X) - (y * x)).|| is V24() real ext-real non negative Element of REAL
((Partial_Sums (X,((1. X) - x))) - y) . n is right_complementable Element of the carrier of X
||.(((Partial_Sums (X,((1. X) - x))) - y) . n).|| is V24() real ext-real non negative Element of REAL
||.(((Partial_Sums (X,((1. X) - x))) - y) . n).|| * ||.x.|| is V24() real ext-real non negative Element of REAL
(||.((X,((1. X) - x)) ^\ 1).|| . n) + (||.(((Partial_Sums (X,((1. X) - x))) - y) . n).|| * ||.x.||) is V24() real ext-real Element of REAL
||.((Partial_Sums (X,((1. X) - x))) - y).|| . n is V24() real ext-real Element of REAL
(||.((Partial_Sums (X,((1. X) - x))) - y).|| . n) * ||.x.|| is V24() real ext-real Element of REAL
(||.((X,((1. X) - x)) ^\ 1).|| . n) + ((||.((Partial_Sums (X,((1. X) - x))) - y).|| . n) * ||.x.||) is V24() real ext-real Element of REAL
(||.x.|| (#) ||.((Partial_Sums (X,((1. X) - x))) - y).||) . n is V24() real ext-real Element of REAL
(||.((X,((1. X) - x)) ^\ 1).|| . n) + ((||.x.|| (#) ||.((Partial_Sums (X,((1. X) - x))) - y).||) . n) is V24() real ext-real Element of REAL
(||.((X,((1. X) - x)) ^\ 1).|| + (||.x.|| (#) ||.((Partial_Sums (X,((1. X) - x))) - y).||)) . n is V24() real ext-real Element of REAL