:: CLOPBAN1 semantic presentation

REAL is non empty V35() V142() V143() V144() V148() set
NAT is non empty epsilon-transitive epsilon-connected ordinal V142() V143() V144() V145() V146() V147() V148() Element of bool REAL
bool REAL is non empty set
COMPLEX is non empty V35() V142() V148() set
omega is non empty epsilon-transitive epsilon-connected ordinal V142() V143() V144() V145() V146() V147() V148() set
bool omega is non empty set
bool NAT is non empty set
RAT is non empty V35() V142() V143() V144() V145() V148() set
INT is non empty V35() V142() V143() V144() V145() V146() V148() set
[:REAL,REAL:] is non empty V132() V133() V134() set
bool [:REAL,REAL:] is non empty set
K373() is non empty strict multMagma
the carrier of K373() is non empty set
<REAL,+> is non empty strict V110() V111() V112() V114() left-invertible right-invertible invertible left-cancelable right-cancelable V169() multMagma
K379() is non empty strict V112() V114() left-cancelable right-cancelable V169() SubStr of <REAL,+>
<NAT,+> is non empty strict V110() V112() V114() left-cancelable right-cancelable V169() uniquely-decomposable SubStr of K379()
<REAL,*> is non empty strict V110() V112() V114() multMagma
<NAT,*> is non empty strict V110() V112() V114() uniquely-decomposable SubStr of <REAL,*>
[:NAT,REAL:] is non empty V132() V133() V134() set
bool [:NAT,REAL:] is non empty set
[:NAT,COMPLEX:] is non empty V132() set
bool [:NAT,COMPLEX:] is non empty set
K428() is non empty set
[:K428(),K428():] is non empty set
[:[:K428(),K428():],K428():] is non empty set
bool [:[:K428(),K428():],K428():] is non empty set
[:COMPLEX,K428():] is non empty set
[:[:COMPLEX,K428():],K428():] is non empty set
bool [:[:COMPLEX,K428():],K428():] is non empty set
K434() is non empty strict CLSStruct
the carrier of K434() is non empty set
bool the carrier of K434() is non empty set
K438() is Element of bool the carrier of K434()
[:K438(),K438():] is set
[:[:K438(),K438():],COMPLEX:] is V132() set
bool [:[:K438(),K438():],COMPLEX:] is non empty set
the_set_of_l1ComplexSequences is non empty linearly-closed Element of bool the carrier of K434()
[:the_set_of_l1ComplexSequences,REAL:] is non empty V132() V133() V134() set
bool [:the_set_of_l1ComplexSequences,REAL:] is non empty set
[:COMPLEX,COMPLEX:] is non empty V132() set
bool [:COMPLEX,COMPLEX:] is non empty set
[:COMPLEX,REAL:] is non empty V132() V133() V134() set
bool [:COMPLEX,REAL:] is non empty set
[:[:COMPLEX,COMPLEX:],COMPLEX:] is non empty V132() set
bool [:[:COMPLEX,COMPLEX:],COMPLEX:] is non empty set
[:[:REAL,REAL:],REAL:] is non empty V132() V133() V134() set
bool [:[:REAL,REAL:],REAL:] is non empty set
[:RAT,RAT:] is non empty RAT -valued V132() V133() V134() set
bool [:RAT,RAT:] is non empty set
[:[:RAT,RAT:],RAT:] is non empty RAT -valued V132() V133() V134() set
bool [:[:RAT,RAT:],RAT:] is non empty set
[:INT,INT:] is non empty RAT -valued INT -valued V132() V133() V134() set
bool [:INT,INT:] is non empty set
[:[:INT,INT:],INT:] is non empty RAT -valued INT -valued V132() V133() V134() set
bool [:[:INT,INT:],INT:] is non empty set
[:NAT,NAT:] is non empty RAT -valued INT -valued V132() V133() V134() V135() set
[:[:NAT,NAT:],NAT:] is non empty RAT -valued INT -valued V132() V133() V134() V135() set
bool [:[:NAT,NAT:],NAT:] is non empty set
{} is empty Function-like functional epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural complex real ext-real non positive non negative V142() V143() V144() V145() V146() V147() V148() set
1 is non empty epsilon-transitive epsilon-connected ordinal natural complex real V108() V109() ext-real positive non negative V142() V143() V144() V145() V146() V147() Element of NAT
{{},1} is non empty V142() V143() V144() V145() V146() V147() set
0 is empty Function-like functional epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural complex real V108() V109() ext-real non positive non negative V142() V143() V144() V145() V146() V147() V148() Element of NAT
1r is complex Element of COMPLEX
- 1r is complex Element of COMPLEX
|.0.| is complex real V109() ext-real Element of REAL
Complex_l1_Space is non empty right_complementable Abelian add-associative right_zeroed discerning reflexive vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like CNORMSTR
the carrier of Complex_l1_Space is non empty set
[:NAT, the carrier of Complex_l1_Space:] is non empty set
bool [:NAT, the carrier of Complex_l1_Space:] is non empty set
2 is non empty epsilon-transitive epsilon-connected ordinal natural complex real V108() V109() ext-real positive non negative V142() V143() V144() V145() V146() V147() Element of NAT
Y is non empty set
[:COMPLEX,Y:] is non empty set
[:[:COMPLEX,Y:],Y:] is non empty set
bool [:[:COMPLEX,Y:],Y:] is non empty set
X is set
[:X,Y:] is set
bool [:X,Y:] is non empty set
vseq is non empty Relation-like [:COMPLEX,Y:] -defined Y -valued Function-like total quasi_total Element of bool [:[:COMPLEX,Y:],Y:]
f is complex set
tseq is Relation-like X -defined Y -valued Function-like total quasi_total Element of bool [:X,Y:]
vseq [;] (f,tseq) is Relation-like Function-like set
Funcs (X,Y) is non empty functional FUNCTION_DOMAIN of X,Y
dom tseq is set
(dom tseq) --> f is Relation-like dom tseq -defined {f} -valued Function-like constant total quasi_total V132() Element of bool [:(dom tseq),{f}:]
{f} is non empty V142() set
[:(dom tseq),{f}:] is V132() set
bool [:(dom tseq),{f}:] is non empty set
[:X,COMPLEX:] is V132() set
bool [:X,COMPLEX:] is non empty set
[:X,[:COMPLEX,Y:]:] is set
bool [:X,[:COMPLEX,Y:]:] is non empty set
<:((dom tseq) --> f),tseq:> is Relation-like Function-like set
tseq is Relation-like X -defined [:COMPLEX,Y:] -valued Function-like total quasi_total Element of bool [:X,[:COMPLEX,Y:]:]
vseq * tseq is Relation-like X -defined Y -valued Function-like total quasi_total Element of bool [:X,Y:]
X is non empty set
Y is non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital CLSStruct
the carrier of Y is non empty set
Funcs (X, the carrier of Y) is non empty functional FUNCTION_DOMAIN of X, the carrier of Y
[:COMPLEX,(Funcs (X, the carrier of Y)):] is non empty set
[:[:COMPLEX,(Funcs (X, the carrier of Y)):],(Funcs (X, the carrier of Y)):] is non empty set
bool [:[:COMPLEX,(Funcs (X, the carrier of Y)):],(Funcs (X, the carrier of Y)):] is non empty set
the Mult of Y is non empty Relation-like [:COMPLEX, the carrier of Y:] -defined the carrier of Y -valued Function-like total quasi_total Element of bool [:[:COMPLEX, the carrier of Y:], the carrier of Y:]
[:COMPLEX, the carrier of Y:] is non empty set
[:[:COMPLEX, the carrier of Y:], the carrier of Y:] is non empty set
bool [:[:COMPLEX, the carrier of Y:], the carrier of Y:] is non empty set
vseq is non empty Relation-like [:COMPLEX,(Funcs (X, the carrier of Y)):] -defined Funcs (X, the carrier of Y) -valued Function-like total quasi_total Function-yielding V34() Element of bool [:[:COMPLEX,(Funcs (X, the carrier of Y)):],(Funcs (X, the carrier of Y)):]
f is complex set
tseq is Relation-like X -defined the carrier of Y -valued Function-like total quasi_total Element of Funcs (X, the carrier of Y)
[f,tseq] is set
vseq . [f,tseq] is Relation-like Function-like set
tseq is Element of X
(vseq . [f,tseq]) . tseq is set
tseq . tseq is Element of the carrier of Y
f * (tseq . tseq) is Element of the carrier of Y
tseq is complex Element of COMPLEX
[tseq,tseq] is Element of [:COMPLEX,(Funcs (X, the carrier of Y)):]
vseq . [tseq,tseq] is Relation-like X -defined the carrier of Y -valued Function-like total quasi_total Element of Funcs (X, the carrier of Y)
dom (vseq . [tseq,tseq]) is set
vseq . (tseq,tseq) is Relation-like Function-like Element of Funcs (X, the carrier of Y)
[tseq,tseq] is set
vseq . [tseq,tseq] is Relation-like Function-like set
(X, the carrier of Y, the Mult of Y,tseq,tseq) is Relation-like X -defined the carrier of Y -valued Function-like total quasi_total Element of Funcs (X, the carrier of Y)
the Mult of Y . (f,(tseq . tseq)) is set
[f,(tseq . tseq)] is set
the Mult of Y . [f,(tseq . tseq)] is set
vseq is non empty Relation-like [:COMPLEX,(Funcs (X, the carrier of Y)):] -defined Funcs (X, the carrier of Y) -valued Function-like total quasi_total Function-yielding V34() Element of bool [:[:COMPLEX,(Funcs (X, the carrier of Y)):],(Funcs (X, the carrier of Y)):]
f is non empty Relation-like [:COMPLEX,(Funcs (X, the carrier of Y)):] -defined Funcs (X, the carrier of Y) -valued Function-like total quasi_total Function-yielding V34() Element of bool [:[:COMPLEX,(Funcs (X, the carrier of Y)):],(Funcs (X, the carrier of Y)):]
tseq is complex Element of COMPLEX
tseq is Relation-like X -defined the carrier of Y -valued Function-like total quasi_total Element of Funcs (X, the carrier of Y)
[tseq,tseq] is Element of [:COMPLEX,(Funcs (X, the carrier of Y)):]
vseq . [tseq,tseq] is Relation-like X -defined the carrier of Y -valued Function-like total quasi_total Element of Funcs (X, the carrier of Y)
tseq is Element of X
(vseq . [tseq,tseq]) . tseq is Element of the carrier of Y
tseq . tseq is Element of the carrier of Y
tseq * (tseq . tseq) is Element of the carrier of Y
f . [tseq,tseq] is Relation-like X -defined the carrier of Y -valued Function-like total quasi_total Element of Funcs (X, the carrier of Y)
(f . [tseq,tseq]) . tseq is Element of the carrier of Y
vseq . (tseq,tseq) is Relation-like Function-like Element of Funcs (X, the carrier of Y)
[tseq,tseq] is set
vseq . [tseq,tseq] is Relation-like Function-like set
f . (tseq,tseq) is Relation-like Function-like Element of Funcs (X, the carrier of Y)
f . [tseq,tseq] is Relation-like Function-like set
X is non empty set
Y is non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital CLSStruct
the carrier of Y is non empty set
FuncZero (X,Y) is Relation-like X -defined the carrier of Y -valued Function-like total quasi_total Element of Funcs (X, the carrier of Y)
Funcs (X, the carrier of Y) is non empty functional FUNCTION_DOMAIN of X, the carrier of Y
0. Y is zero Element of the carrier of Y
the ZeroF of Y is Element of the carrier of Y
vseq is Element of X
(FuncZero (X,Y)) . vseq is Element of the carrier of Y
X --> (0. Y) is non empty Relation-like X -defined the carrier of Y -valued Function-like constant total quasi_total Element of bool [:X, the carrier of Y:]
[:X, the carrier of Y:] is non empty set
bool [:X, the carrier of Y:] is non empty set
(X --> (0. Y)) . vseq is Element of the carrier of Y
X is non empty set
Y is non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital CLSStruct
the carrier of Y is non empty set
Funcs (X, the carrier of Y) is non empty functional FUNCTION_DOMAIN of X, the carrier of Y
(X,Y) is non empty Relation-like [:COMPLEX,(Funcs (X, the carrier of Y)):] -defined Funcs (X, the carrier of Y) -valued Function-like total quasi_total Function-yielding V34() Element of bool [:[:COMPLEX,(Funcs (X, the carrier of Y)):],(Funcs (X, the carrier of Y)):]
[:COMPLEX,(Funcs (X, the carrier of Y)):] is non empty set
[:[:COMPLEX,(Funcs (X, the carrier of Y)):],(Funcs (X, the carrier of Y)):] is non empty set
bool [:[:COMPLEX,(Funcs (X, the carrier of Y)):],(Funcs (X, the carrier of Y)):] is non empty set
vseq is Relation-like X -defined the carrier of Y -valued Function-like total quasi_total Element of Funcs (X, the carrier of Y)
f is Relation-like X -defined the carrier of Y -valued Function-like total quasi_total Element of Funcs (X, the carrier of Y)
tseq is complex set
[tseq,f] is set
(X,Y) . [tseq,f] is Relation-like Function-like set
tseq is Element of X
vseq . tseq is Element of the carrier of Y
f . tseq is Element of the carrier of Y
tseq * (f . tseq) is Element of the carrier of Y
tseq is complex Element of COMPLEX
[tseq,f] is Element of [:COMPLEX,(Funcs (X, the carrier of Y)):]
(X,Y) . [tseq,f] is Relation-like X -defined the carrier of Y -valued Function-like total quasi_total Element of Funcs (X, the carrier of Y)
tseq is Element of X
vseq . tseq is Element of the carrier of Y
((X,Y) . [tseq,f]) . tseq is Element of the carrier of Y
f . tseq is Element of the carrier of Y
tseq * (f . tseq) is Element of the carrier of Y
tseq is Element of X
vseq . tseq is Element of the carrier of Y
f . tseq is Element of the carrier of Y
tseq * (f . tseq) is Element of the carrier of Y
X is non empty set
Y is non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital CLSStruct
the carrier of Y is non empty set
Funcs (X, the carrier of Y) is non empty functional FUNCTION_DOMAIN of X, the carrier of Y
FuncAdd (X,Y) is non empty Relation-like [:(Funcs (X, the carrier of Y)),(Funcs (X, the carrier of Y)):] -defined Funcs (X, the carrier of Y) -valued Function-like total quasi_total Function-yielding V34() Element of bool [:[:(Funcs (X, the carrier of Y)),(Funcs (X, the carrier of Y)):],(Funcs (X, the carrier of Y)):]
[:(Funcs (X, the carrier of Y)),(Funcs (X, the carrier of Y)):] is non empty set
[:[:(Funcs (X, the carrier of Y)),(Funcs (X, the carrier of Y)):],(Funcs (X, the carrier of Y)):] is non empty set
bool [:[:(Funcs (X, the carrier of Y)),(Funcs (X, the carrier of Y)):],(Funcs (X, the carrier of Y)):] is non empty set
vseq is Relation-like X -defined the carrier of Y -valued Function-like total quasi_total Element of Funcs (X, the carrier of Y)
f is Relation-like X -defined the carrier of Y -valued Function-like total quasi_total Element of Funcs (X, the carrier of Y)
(FuncAdd (X,Y)) . (vseq,f) is Relation-like X -defined the carrier of Y -valued Function-like total quasi_total Element of Funcs (X, the carrier of Y)
[vseq,f] is set
(FuncAdd (X,Y)) . [vseq,f] is Relation-like Function-like set
(FuncAdd (X,Y)) . (f,vseq) is Relation-like X -defined the carrier of Y -valued Function-like total quasi_total Element of Funcs (X, the carrier of Y)
[f,vseq] is set
(FuncAdd (X,Y)) . [f,vseq] is Relation-like Function-like set
tseq is Element of X
((FuncAdd (X,Y)) . (vseq,f)) . tseq is Element of the carrier of Y
f . tseq is Element of the carrier of Y
vseq . tseq is Element of the carrier of Y
(f . tseq) + (vseq . tseq) is Element of the carrier of Y
the addF of Y is non empty Relation-like [: the carrier of Y, the carrier of Y:] -defined the carrier of Y -valued Function-like total quasi_total Element of bool [:[: the carrier of Y, the carrier of Y:], the carrier of Y:]
[: the carrier of Y, the carrier of Y:] is non empty set
[:[: the carrier of Y, the carrier of Y:], the carrier of Y:] is non empty set
bool [:[: the carrier of Y, the carrier of Y:], the carrier of Y:] is non empty set
the addF of Y . ((f . tseq),(vseq . tseq)) is Element of the carrier of Y
[(f . tseq),(vseq . tseq)] is set
the addF of Y . [(f . tseq),(vseq . tseq)] is set
((FuncAdd (X,Y)) . (f,vseq)) . tseq is Element of the carrier of Y
X is non empty set
Y is non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital CLSStruct
the carrier of Y is non empty set
Funcs (X, the carrier of Y) is non empty functional FUNCTION_DOMAIN of X, the carrier of Y
FuncAdd (X,Y) is non empty Relation-like [:(Funcs (X, the carrier of Y)),(Funcs (X, the carrier of Y)):] -defined Funcs (X, the carrier of Y) -valued Function-like total quasi_total Function-yielding V34() Element of bool [:[:(Funcs (X, the carrier of Y)),(Funcs (X, the carrier of Y)):],(Funcs (X, the carrier of Y)):]
[:(Funcs (X, the carrier of Y)),(Funcs (X, the carrier of Y)):] is non empty set
[:[:(Funcs (X, the carrier of Y)),(Funcs (X, the carrier of Y)):],(Funcs (X, the carrier of Y)):] is non empty set
bool [:[:(Funcs (X, the carrier of Y)),(Funcs (X, the carrier of Y)):],(Funcs (X, the carrier of Y)):] is non empty set
vseq is Relation-like X -defined the carrier of Y -valued Function-like total quasi_total Element of Funcs (X, the carrier of Y)
f is Relation-like X -defined the carrier of Y -valued Function-like total quasi_total Element of Funcs (X, the carrier of Y)
(FuncAdd (X,Y)) . (vseq,f) is Relation-like X -defined the carrier of Y -valued Function-like total quasi_total Element of Funcs (X, the carrier of Y)
[vseq,f] is set
(FuncAdd (X,Y)) . [vseq,f] is Relation-like Function-like set
tseq is Relation-like X -defined the carrier of Y -valued Function-like total quasi_total Element of Funcs (X, the carrier of Y)
(FuncAdd (X,Y)) . (f,tseq) is Relation-like X -defined the carrier of Y -valued Function-like total quasi_total Element of Funcs (X, the carrier of Y)
[f,tseq] is set
(FuncAdd (X,Y)) . [f,tseq] is Relation-like Function-like set
(FuncAdd (X,Y)) . (vseq,((FuncAdd (X,Y)) . (f,tseq))) is Relation-like X -defined the carrier of Y -valued Function-like total quasi_total Element of Funcs (X, the carrier of Y)
[vseq,((FuncAdd (X,Y)) . (f,tseq))] is set
(FuncAdd (X,Y)) . [vseq,((FuncAdd (X,Y)) . (f,tseq))] is Relation-like Function-like set
(FuncAdd (X,Y)) . (((FuncAdd (X,Y)) . (vseq,f)),tseq) is Relation-like X -defined the carrier of Y -valued Function-like total quasi_total Element of Funcs (X, the carrier of Y)
[((FuncAdd (X,Y)) . (vseq,f)),tseq] is set
(FuncAdd (X,Y)) . [((FuncAdd (X,Y)) . (vseq,f)),tseq] is Relation-like Function-like set
tseq is Element of X
((FuncAdd (X,Y)) . (vseq,((FuncAdd (X,Y)) . (f,tseq)))) . tseq is Element of the carrier of Y
vseq . tseq is Element of the carrier of Y
((FuncAdd (X,Y)) . (f,tseq)) . tseq is Element of the carrier of Y
(vseq . tseq) + (((FuncAdd (X,Y)) . (f,tseq)) . tseq) is Element of the carrier of Y
the addF of Y is non empty Relation-like [: the carrier of Y, the carrier of Y:] -defined the carrier of Y -valued Function-like total quasi_total Element of bool [:[: the carrier of Y, the carrier of Y:], the carrier of Y:]
[: the carrier of Y, the carrier of Y:] is non empty set
[:[: the carrier of Y, the carrier of Y:], the carrier of Y:] is non empty set
bool [:[: the carrier of Y, the carrier of Y:], the carrier of Y:] is non empty set
the addF of Y . ((vseq . tseq),(((FuncAdd (X,Y)) . (f,tseq)) . tseq)) is Element of the carrier of Y
[(vseq . tseq),(((FuncAdd (X,Y)) . (f,tseq)) . tseq)] is set
the addF of Y . [(vseq . tseq),(((FuncAdd (X,Y)) . (f,tseq)) . tseq)] is set
f . tseq is Element of the carrier of Y
tseq . tseq is Element of the carrier of Y
(f . tseq) + (tseq . tseq) is Element of the carrier of Y
the addF of Y . ((f . tseq),(tseq . tseq)) is Element of the carrier of Y
[(f . tseq),(tseq . tseq)] is set
the addF of Y . [(f . tseq),(tseq . tseq)] is set
(vseq . tseq) + ((f . tseq) + (tseq . tseq)) is Element of the carrier of Y
the addF of Y . ((vseq . tseq),((f . tseq) + (tseq . tseq))) is Element of the carrier of Y
[(vseq . tseq),((f . tseq) + (tseq . tseq))] is set
the addF of Y . [(vseq . tseq),((f . tseq) + (tseq . tseq))] is set
(vseq . tseq) + (f . tseq) is Element of the carrier of Y
the addF of Y . ((vseq . tseq),(f . tseq)) is Element of the carrier of Y
[(vseq . tseq),(f . tseq)] is set
the addF of Y . [(vseq . tseq),(f . tseq)] is set
((vseq . tseq) + (f . tseq)) + (tseq . tseq) is Element of the carrier of Y
the addF of Y . (((vseq . tseq) + (f . tseq)),(tseq . tseq)) is Element of the carrier of Y
[((vseq . tseq) + (f . tseq)),(tseq . tseq)] is set
the addF of Y . [((vseq . tseq) + (f . tseq)),(tseq . tseq)] is set
((FuncAdd (X,Y)) . (vseq,f)) . tseq is Element of the carrier of Y
(((FuncAdd (X,Y)) . (vseq,f)) . tseq) + (tseq . tseq) is Element of the carrier of Y
the addF of Y . ((((FuncAdd (X,Y)) . (vseq,f)) . tseq),(tseq . tseq)) is Element of the carrier of Y
[(((FuncAdd (X,Y)) . (vseq,f)) . tseq),(tseq . tseq)] is set
the addF of Y . [(((FuncAdd (X,Y)) . (vseq,f)) . tseq),(tseq . tseq)] is set
((FuncAdd (X,Y)) . (((FuncAdd (X,Y)) . (vseq,f)),tseq)) . tseq is Element of the carrier of Y
X is non empty set
Y is non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital CLSStruct
the carrier of Y is non empty set
Funcs (X, the carrier of Y) is non empty functional FUNCTION_DOMAIN of X, the carrier of Y
FuncAdd (X,Y) is non empty Relation-like [:(Funcs (X, the carrier of Y)),(Funcs (X, the carrier of Y)):] -defined Funcs (X, the carrier of Y) -valued Function-like total quasi_total Function-yielding V34() Element of bool [:[:(Funcs (X, the carrier of Y)),(Funcs (X, the carrier of Y)):],(Funcs (X, the carrier of Y)):]
[:(Funcs (X, the carrier of Y)),(Funcs (X, the carrier of Y)):] is non empty set
[:[:(Funcs (X, the carrier of Y)),(Funcs (X, the carrier of Y)):],(Funcs (X, the carrier of Y)):] is non empty set
bool [:[:(Funcs (X, the carrier of Y)),(Funcs (X, the carrier of Y)):],(Funcs (X, the carrier of Y)):] is non empty set
FuncZero (X,Y) is Relation-like X -defined the carrier of Y -valued Function-like total quasi_total Element of Funcs (X, the carrier of Y)
vseq is Relation-like X -defined the carrier of Y -valued Function-like total quasi_total Element of Funcs (X, the carrier of Y)
(FuncAdd (X,Y)) . ((FuncZero (X,Y)),vseq) is Relation-like X -defined the carrier of Y -valued Function-like total quasi_total Element of Funcs (X, the carrier of Y)
[(FuncZero (X,Y)),vseq] is set
(FuncAdd (X,Y)) . [(FuncZero (X,Y)),vseq] is Relation-like Function-like set
f is Element of X
((FuncAdd (X,Y)) . ((FuncZero (X,Y)),vseq)) . f is Element of the carrier of Y
(FuncZero (X,Y)) . f is Element of the carrier of Y
vseq . f is Element of the carrier of Y
((FuncZero (X,Y)) . f) + (vseq . f) is Element of the carrier of Y
the addF of Y is non empty Relation-like [: the carrier of Y, the carrier of Y:] -defined the carrier of Y -valued Function-like total quasi_total Element of bool [:[: the carrier of Y, the carrier of Y:], the carrier of Y:]
[: the carrier of Y, the carrier of Y:] is non empty set
[:[: the carrier of Y, the carrier of Y:], the carrier of Y:] is non empty set
bool [:[: the carrier of Y, the carrier of Y:], the carrier of Y:] is non empty set
the addF of Y . (((FuncZero (X,Y)) . f),(vseq . f)) is Element of the carrier of Y
[((FuncZero (X,Y)) . f),(vseq . f)] is set
the addF of Y . [((FuncZero (X,Y)) . f),(vseq . f)] is set
0. Y is zero Element of the carrier of Y
the ZeroF of Y is Element of the carrier of Y
(0. Y) + (vseq . f) is Element of the carrier of Y
the addF of Y . ((0. Y),(vseq . f)) is Element of the carrier of Y
[(0. Y),(vseq . f)] is set
the addF of Y . [(0. Y),(vseq . f)] is set
X is non empty set
Y is non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital CLSStruct
the carrier of Y is non empty set
Funcs (X, the carrier of Y) is non empty functional FUNCTION_DOMAIN of X, the carrier of Y
FuncAdd (X,Y) is non empty Relation-like [:(Funcs (X, the carrier of Y)),(Funcs (X, the carrier of Y)):] -defined Funcs (X, the carrier of Y) -valued Function-like total quasi_total Function-yielding V34() Element of bool [:[:(Funcs (X, the carrier of Y)),(Funcs (X, the carrier of Y)):],(Funcs (X, the carrier of Y)):]
[:(Funcs (X, the carrier of Y)),(Funcs (X, the carrier of Y)):] is non empty set
[:[:(Funcs (X, the carrier of Y)),(Funcs (X, the carrier of Y)):],(Funcs (X, the carrier of Y)):] is non empty set
bool [:[:(Funcs (X, the carrier of Y)),(Funcs (X, the carrier of Y)):],(Funcs (X, the carrier of Y)):] is non empty set
(X,Y) is non empty Relation-like [:COMPLEX,(Funcs (X, the carrier of Y)):] -defined Funcs (X, the carrier of Y) -valued Function-like total quasi_total Function-yielding V34() Element of bool [:[:COMPLEX,(Funcs (X, the carrier of Y)):],(Funcs (X, the carrier of Y)):]
[:COMPLEX,(Funcs (X, the carrier of Y)):] is non empty set
[:[:COMPLEX,(Funcs (X, the carrier of Y)):],(Funcs (X, the carrier of Y)):] is non empty set
bool [:[:COMPLEX,(Funcs (X, the carrier of Y)):],(Funcs (X, the carrier of Y)):] is non empty set
FuncZero (X,Y) is Relation-like X -defined the carrier of Y -valued Function-like total quasi_total Element of Funcs (X, the carrier of Y)
vseq is Relation-like X -defined the carrier of Y -valued Function-like total quasi_total Element of Funcs (X, the carrier of Y)
[(- 1r),vseq] is Element of [:COMPLEX,(Funcs (X, the carrier of Y)):]
(X,Y) . [(- 1r),vseq] is Relation-like X -defined the carrier of Y -valued Function-like total quasi_total Element of Funcs (X, the carrier of Y)
(FuncAdd (X,Y)) . (vseq,((X,Y) . [(- 1r),vseq])) is Relation-like X -defined the carrier of Y -valued Function-like total quasi_total Element of Funcs (X, the carrier of Y)
[vseq,((X,Y) . [(- 1r),vseq])] is set
(FuncAdd (X,Y)) . [vseq,((X,Y) . [(- 1r),vseq])] is Relation-like Function-like set
f is Element of X
vseq . f is Element of the carrier of Y
((FuncAdd (X,Y)) . (vseq,((X,Y) . [(- 1r),vseq]))) . f is Element of the carrier of Y
((X,Y) . [(- 1r),vseq]) . f is Element of the carrier of Y
(vseq . f) + (((X,Y) . [(- 1r),vseq]) . f) is Element of the carrier of Y
the addF of Y is non empty Relation-like [: the carrier of Y, the carrier of Y:] -defined the carrier of Y -valued Function-like total quasi_total Element of bool [:[: the carrier of Y, the carrier of Y:], the carrier of Y:]
[: the carrier of Y, the carrier of Y:] is non empty set
[:[: the carrier of Y, the carrier of Y:], the carrier of Y:] is non empty set
bool [:[: the carrier of Y, the carrier of Y:], the carrier of Y:] is non empty set
the addF of Y . ((vseq . f),(((X,Y) . [(- 1r),vseq]) . f)) is Element of the carrier of Y
[(vseq . f),(((X,Y) . [(- 1r),vseq]) . f)] is set
the addF of Y . [(vseq . f),(((X,Y) . [(- 1r),vseq]) . f)] is set
(- 1r) * (vseq . f) is Element of the carrier of Y
(vseq . f) + ((- 1r) * (vseq . f)) is Element of the carrier of Y
the addF of Y . ((vseq . f),((- 1r) * (vseq . f))) is Element of the carrier of Y
[(vseq . f),((- 1r) * (vseq . f))] is set
the addF of Y . [(vseq . f),((- 1r) * (vseq . f))] is set
- (vseq . f) is Element of the carrier of Y
(vseq . f) + (- (vseq . f)) is Element of the carrier of Y
the addF of Y . ((vseq . f),(- (vseq . f))) is Element of the carrier of Y
[(vseq . f),(- (vseq . f))] is set
the addF of Y . [(vseq . f),(- (vseq . f))] is set
0. Y is zero Element of the carrier of Y
the ZeroF of Y is Element of the carrier of Y
(FuncZero (X,Y)) . f is Element of the carrier of Y
X is non empty set
Y is non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital CLSStruct
the carrier of Y is non empty set
Funcs (X, the carrier of Y) is non empty functional FUNCTION_DOMAIN of X, the carrier of Y
(X,Y) is non empty Relation-like [:COMPLEX,(Funcs (X, the carrier of Y)):] -defined Funcs (X, the carrier of Y) -valued Function-like total quasi_total Function-yielding V34() Element of bool [:[:COMPLEX,(Funcs (X, the carrier of Y)):],(Funcs (X, the carrier of Y)):]
[:COMPLEX,(Funcs (X, the carrier of Y)):] is non empty set
[:[:COMPLEX,(Funcs (X, the carrier of Y)):],(Funcs (X, the carrier of Y)):] is non empty set
bool [:[:COMPLEX,(Funcs (X, the carrier of Y)):],(Funcs (X, the carrier of Y)):] is non empty set
vseq is Relation-like X -defined the carrier of Y -valued Function-like total quasi_total Element of Funcs (X, the carrier of Y)
[1r,vseq] is Element of [:COMPLEX,(Funcs (X, the carrier of Y)):]
(X,Y) . [1r,vseq] is Relation-like X -defined the carrier of Y -valued Function-like total quasi_total Element of Funcs (X, the carrier of Y)
f is Element of X
((X,Y) . [1r,vseq]) . f is Element of the carrier of Y
vseq . f is Element of the carrier of Y
1r * (vseq . f) is Element of the carrier of Y
X is non empty set
Y is non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital CLSStruct
the carrier of Y is non empty set
Funcs (X, the carrier of Y) is non empty functional FUNCTION_DOMAIN of X, the carrier of Y
(X,Y) is non empty Relation-like [:COMPLEX,(Funcs (X, the carrier of Y)):] -defined Funcs (X, the carrier of Y) -valued Function-like total quasi_total Function-yielding V34() Element of bool [:[:COMPLEX,(Funcs (X, the carrier of Y)):],(Funcs (X, the carrier of Y)):]
[:COMPLEX,(Funcs (X, the carrier of Y)):] is non empty set
[:[:COMPLEX,(Funcs (X, the carrier of Y)):],(Funcs (X, the carrier of Y)):] is non empty set
bool [:[:COMPLEX,(Funcs (X, the carrier of Y)):],(Funcs (X, the carrier of Y)):] is non empty set
vseq is Relation-like X -defined the carrier of Y -valued Function-like total quasi_total Element of Funcs (X, the carrier of Y)
f is complex set
tseq is complex set
[tseq,vseq] is set
(X,Y) . [tseq,vseq] is Relation-like Function-like set
[f,((X,Y) . [tseq,vseq])] is set
(X,Y) . [f,((X,Y) . [tseq,vseq])] is Relation-like Function-like set
f * tseq is complex set
[(f * tseq),vseq] is set
(X,Y) . [(f * tseq),vseq] is Relation-like Function-like set
tseq is complex Element of COMPLEX
tseq is complex Element of COMPLEX
[tseq,vseq] is Element of [:COMPLEX,(Funcs (X, the carrier of Y)):]
(X,Y) . [tseq,vseq] is Relation-like X -defined the carrier of Y -valued Function-like total quasi_total Element of Funcs (X, the carrier of Y)
[tseq,((X,Y) . [tseq,vseq])] is Element of [:COMPLEX,(Funcs (X, the carrier of Y)):]
(X,Y) . [tseq,((X,Y) . [tseq,vseq])] is Relation-like X -defined the carrier of Y -valued Function-like total quasi_total Element of Funcs (X, the carrier of Y)
tv is Element of X
((X,Y) . [tseq,((X,Y) . [tseq,vseq])]) . tv is Element of the carrier of Y
((X,Y) . [tseq,vseq]) . tv is Element of the carrier of Y
tseq * (((X,Y) . [tseq,vseq]) . tv) is Element of the carrier of Y
vseq . tv is Element of the carrier of Y
tseq * (vseq . tv) is Element of the carrier of Y
f * (tseq * (vseq . tv)) is Element of the carrier of Y
(f * tseq) * (vseq . tv) is Element of the carrier of Y
tseq * tseq is complex Element of COMPLEX
[(tseq * tseq),vseq] is Element of [:COMPLEX,(Funcs (X, the carrier of Y)):]
(X,Y) . [(tseq * tseq),vseq] is Relation-like X -defined the carrier of Y -valued Function-like total quasi_total Element of Funcs (X, the carrier of Y)
((X,Y) . [(tseq * tseq),vseq]) . tv is Element of the carrier of Y
X is non empty set
Y is non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital CLSStruct
the carrier of Y is non empty set
Funcs (X, the carrier of Y) is non empty functional FUNCTION_DOMAIN of X, the carrier of Y
FuncAdd (X,Y) is non empty Relation-like [:(Funcs (X, the carrier of Y)),(Funcs (X, the carrier of Y)):] -defined Funcs (X, the carrier of Y) -valued Function-like total quasi_total Function-yielding V34() Element of bool [:[:(Funcs (X, the carrier of Y)),(Funcs (X, the carrier of Y)):],(Funcs (X, the carrier of Y)):]
[:(Funcs (X, the carrier of Y)),(Funcs (X, the carrier of Y)):] is non empty set
[:[:(Funcs (X, the carrier of Y)),(Funcs (X, the carrier of Y)):],(Funcs (X, the carrier of Y)):] is non empty set
bool [:[:(Funcs (X, the carrier of Y)),(Funcs (X, the carrier of Y)):],(Funcs (X, the carrier of Y)):] is non empty set
(X,Y) is non empty Relation-like [:COMPLEX,(Funcs (X, the carrier of Y)):] -defined Funcs (X, the carrier of Y) -valued Function-like total quasi_total Function-yielding V34() Element of bool [:[:COMPLEX,(Funcs (X, the carrier of Y)):],(Funcs (X, the carrier of Y)):]
[:COMPLEX,(Funcs (X, the carrier of Y)):] is non empty set
[:[:COMPLEX,(Funcs (X, the carrier of Y)):],(Funcs (X, the carrier of Y)):] is non empty set
bool [:[:COMPLEX,(Funcs (X, the carrier of Y)):],(Funcs (X, the carrier of Y)):] is non empty set
vseq is Relation-like X -defined the carrier of Y -valued Function-like total quasi_total Element of Funcs (X, the carrier of Y)
f is complex set
[f,vseq] is set
(X,Y) . [f,vseq] is Relation-like Function-like set
tseq is complex set
[tseq,vseq] is set
(X,Y) . [tseq,vseq] is Relation-like Function-like set
(FuncAdd (X,Y)) . (((X,Y) . [f,vseq]),((X,Y) . [tseq,vseq])) is set
[((X,Y) . [f,vseq]),((X,Y) . [tseq,vseq])] is set
(FuncAdd (X,Y)) . [((X,Y) . [f,vseq]),((X,Y) . [tseq,vseq])] is Relation-like Function-like set
f + tseq is complex set
[(f + tseq),vseq] is set
(X,Y) . [(f + tseq),vseq] is Relation-like Function-like set
tseq is complex Element of COMPLEX
[tseq,vseq] is Element of [:COMPLEX,(Funcs (X, the carrier of Y)):]
(X,Y) . [tseq,vseq] is Relation-like X -defined the carrier of Y -valued Function-like total quasi_total Element of Funcs (X, the carrier of Y)
tseq is complex Element of COMPLEX
[tseq,vseq] is Element of [:COMPLEX,(Funcs (X, the carrier of Y)):]
(X,Y) . [tseq,vseq] is Relation-like X -defined the carrier of Y -valued Function-like total quasi_total Element of Funcs (X, the carrier of Y)
(FuncAdd (X,Y)) . (((X,Y) . [tseq,vseq]),((X,Y) . [tseq,vseq])) is Relation-like X -defined the carrier of Y -valued Function-like total quasi_total Element of Funcs (X, the carrier of Y)
[((X,Y) . [tseq,vseq]),((X,Y) . [tseq,vseq])] is set
(FuncAdd (X,Y)) . [((X,Y) . [tseq,vseq]),((X,Y) . [tseq,vseq])] is Relation-like Function-like set
tv is Element of X
((FuncAdd (X,Y)) . (((X,Y) . [tseq,vseq]),((X,Y) . [tseq,vseq]))) . tv is Element of the carrier of Y
((X,Y) . [tseq,vseq]) . tv is Element of the carrier of Y
((X,Y) . [tseq,vseq]) . tv is Element of the carrier of Y
(((X,Y) . [tseq,vseq]) . tv) + (((X,Y) . [tseq,vseq]) . tv) is Element of the carrier of Y
the addF of Y is non empty Relation-like [: the carrier of Y, the carrier of Y:] -defined the carrier of Y -valued Function-like total quasi_total Element of bool [:[: the carrier of Y, the carrier of Y:], the carrier of Y:]
[: the carrier of Y, the carrier of Y:] is non empty set
[:[: the carrier of Y, the carrier of Y:], the carrier of Y:] is non empty set
bool [:[: the carrier of Y, the carrier of Y:], the carrier of Y:] is non empty set
the addF of Y . ((((X,Y) . [tseq,vseq]) . tv),(((X,Y) . [tseq,vseq]) . tv)) is Element of the carrier of Y
[(((X,Y) . [tseq,vseq]) . tv),(((X,Y) . [tseq,vseq]) . tv)] is set
the addF of Y . [(((X,Y) . [tseq,vseq]) . tv),(((X,Y) . [tseq,vseq]) . tv)] is set
vseq . tv is Element of the carrier of Y
tseq * (vseq . tv) is Element of the carrier of Y
(tseq * (vseq . tv)) + (((X,Y) . [tseq,vseq]) . tv) is Element of the carrier of Y
the addF of Y . ((tseq * (vseq . tv)),(((X,Y) . [tseq,vseq]) . tv)) is Element of the carrier of Y
[(tseq * (vseq . tv)),(((X,Y) . [tseq,vseq]) . tv)] is set
the addF of Y . [(tseq * (vseq . tv)),(((X,Y) . [tseq,vseq]) . tv)] is set
f * (vseq . tv) is Element of the carrier of Y
tseq * (vseq . tv) is Element of the carrier of Y
(f * (vseq . tv)) + (tseq * (vseq . tv)) is Element of the carrier of Y
the addF of Y . ((f * (vseq . tv)),(tseq * (vseq . tv))) is Element of the carrier of Y
[(f * (vseq . tv)),(tseq * (vseq . tv))] is set
the addF of Y . [(f * (vseq . tv)),(tseq * (vseq . tv))] is set
(f + tseq) * (vseq . tv) is Element of the carrier of Y
tseq + tseq is complex Element of COMPLEX
[(tseq + tseq),vseq] is Element of [:COMPLEX,(Funcs (X, the carrier of Y)):]
(X,Y) . [(tseq + tseq),vseq] is Relation-like X -defined the carrier of Y -valued Function-like total quasi_total Element of Funcs (X, the carrier of Y)
((X,Y) . [(tseq + tseq),vseq]) . tv is Element of the carrier of Y
X is non empty set
Y is non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital CLSStruct
the carrier of Y is non empty set
Funcs (X, the carrier of Y) is non empty functional FUNCTION_DOMAIN of X, the carrier of Y
FuncAdd (X,Y) is non empty Relation-like [:(Funcs (X, the carrier of Y)),(Funcs (X, the carrier of Y)):] -defined Funcs (X, the carrier of Y) -valued Function-like total quasi_total Function-yielding V34() Element of bool [:[:(Funcs (X, the carrier of Y)),(Funcs (X, the carrier of Y)):],(Funcs (X, the carrier of Y)):]
[:(Funcs (X, the carrier of Y)),(Funcs (X, the carrier of Y)):] is non empty set
[:[:(Funcs (X, the carrier of Y)),(Funcs (X, the carrier of Y)):],(Funcs (X, the carrier of Y)):] is non empty set
bool [:[:(Funcs (X, the carrier of Y)),(Funcs (X, the carrier of Y)):],(Funcs (X, the carrier of Y)):] is non empty set
(X,Y) is non empty Relation-like [:COMPLEX,(Funcs (X, the carrier of Y)):] -defined Funcs (X, the carrier of Y) -valued Function-like total quasi_total Function-yielding V34() Element of bool [:[:COMPLEX,(Funcs (X, the carrier of Y)):],(Funcs (X, the carrier of Y)):]
[:COMPLEX,(Funcs (X, the carrier of Y)):] is non empty set
[:[:COMPLEX,(Funcs (X, the carrier of Y)):],(Funcs (X, the carrier of Y)):] is non empty set
bool [:[:COMPLEX,(Funcs (X, the carrier of Y)):],(Funcs (X, the carrier of Y)):] is non empty set
vseq is Relation-like X -defined the carrier of Y -valued Function-like total quasi_total Element of Funcs (X, the carrier of Y)
f is Relation-like X -defined the carrier of Y -valued Function-like total quasi_total Element of Funcs (X, the carrier of Y)
(FuncAdd (X,Y)) . (vseq,f) is Relation-like X -defined the carrier of Y -valued Function-like total quasi_total Element of Funcs (X, the carrier of Y)
[vseq,f] is set
(FuncAdd (X,Y)) . [vseq,f] is Relation-like Function-like set
tseq is complex set
[tseq,vseq] is set
(X,Y) . [tseq,vseq] is Relation-like Function-like set
[tseq,f] is set
(X,Y) . [tseq,f] is Relation-like Function-like set
(FuncAdd (X,Y)) . (((X,Y) . [tseq,vseq]),((X,Y) . [tseq,f])) is set
[((X,Y) . [tseq,vseq]),((X,Y) . [tseq,f])] is set
(FuncAdd (X,Y)) . [((X,Y) . [tseq,vseq]),((X,Y) . [tseq,f])] is Relation-like Function-like set
[tseq,((FuncAdd (X,Y)) . (vseq,f))] is set
(X,Y) . [tseq,((FuncAdd (X,Y)) . (vseq,f))] is Relation-like Function-like set
tseq is complex Element of COMPLEX
[tseq,vseq] is Element of [:COMPLEX,(Funcs (X, the carrier of Y)):]
(X,Y) . [tseq,vseq] is Relation-like X -defined the carrier of Y -valued Function-like total quasi_total Element of Funcs (X, the carrier of Y)
[tseq,f] is Element of [:COMPLEX,(Funcs (X, the carrier of Y)):]
(X,Y) . [tseq,f] is Relation-like X -defined the carrier of Y -valued Function-like total quasi_total Element of Funcs (X, the carrier of Y)
(FuncAdd (X,Y)) . (((X,Y) . [tseq,vseq]),((X,Y) . [tseq,f])) is Relation-like X -defined the carrier of Y -valued Function-like total quasi_total Element of Funcs (X, the carrier of Y)
[((X,Y) . [tseq,vseq]),((X,Y) . [tseq,f])] is set
(FuncAdd (X,Y)) . [((X,Y) . [tseq,vseq]),((X,Y) . [tseq,f])] is Relation-like Function-like set
tseq is Element of X
((FuncAdd (X,Y)) . (((X,Y) . [tseq,vseq]),((X,Y) . [tseq,f]))) . tseq is Element of the carrier of Y
((X,Y) . [tseq,vseq]) . tseq is Element of the carrier of Y
((X,Y) . [tseq,f]) . tseq is Element of the carrier of Y
(((X,Y) . [tseq,vseq]) . tseq) + (((X,Y) . [tseq,f]) . tseq) is Element of the carrier of Y
the addF of Y is non empty Relation-like [: the carrier of Y, the carrier of Y:] -defined the carrier of Y -valued Function-like total quasi_total Element of bool [:[: the carrier of Y, the carrier of Y:], the carrier of Y:]
[: the carrier of Y, the carrier of Y:] is non empty set
[:[: the carrier of Y, the carrier of Y:], the carrier of Y:] is non empty set
bool [:[: the carrier of Y, the carrier of Y:], the carrier of Y:] is non empty set
the addF of Y . ((((X,Y) . [tseq,vseq]) . tseq),(((X,Y) . [tseq,f]) . tseq)) is Element of the carrier of Y
[(((X,Y) . [tseq,vseq]) . tseq),(((X,Y) . [tseq,f]) . tseq)] is set
the addF of Y . [(((X,Y) . [tseq,vseq]) . tseq),(((X,Y) . [tseq,f]) . tseq)] is set
vseq . tseq is Element of the carrier of Y
tseq * (vseq . tseq) is Element of the carrier of Y
(tseq * (vseq . tseq)) + (((X,Y) . [tseq,f]) . tseq) is Element of the carrier of Y
the addF of Y . ((tseq * (vseq . tseq)),(((X,Y) . [tseq,f]) . tseq)) is Element of the carrier of Y
[(tseq * (vseq . tseq)),(((X,Y) . [tseq,f]) . tseq)] is set
the addF of Y . [(tseq * (vseq . tseq)),(((X,Y) . [tseq,f]) . tseq)] is set
tseq * (vseq . tseq) is Element of the carrier of Y
f . tseq is Element of the carrier of Y
tseq * (f . tseq) is Element of the carrier of Y
(tseq * (vseq . tseq)) + (tseq * (f . tseq)) is Element of the carrier of Y
the addF of Y . ((tseq * (vseq . tseq)),(tseq * (f . tseq))) is Element of the carrier of Y
[(tseq * (vseq . tseq)),(tseq * (f . tseq))] is set
the addF of Y . [(tseq * (vseq . tseq)),(tseq * (f . tseq))] is set
(vseq . tseq) + (f . tseq) is Element of the carrier of Y
the addF of Y . ((vseq . tseq),(f . tseq)) is Element of the carrier of Y
[(vseq . tseq),(f . tseq)] is set
the addF of Y . [(vseq . tseq),(f . tseq)] is set
tseq * ((vseq . tseq) + (f . tseq)) is Element of the carrier of Y
((FuncAdd (X,Y)) . (vseq,f)) . tseq is Element of the carrier of Y
tseq * (((FuncAdd (X,Y)) . (vseq,f)) . tseq) is Element of the carrier of Y
[tseq,((FuncAdd (X,Y)) . (vseq,f))] is Element of [:COMPLEX,(Funcs (X, the carrier of Y)):]
(X,Y) . [tseq,((FuncAdd (X,Y)) . (vseq,f))] is Relation-like X -defined the carrier of Y -valued Function-like total quasi_total Element of Funcs (X, the carrier of Y)
((X,Y) . [tseq,((FuncAdd (X,Y)) . (vseq,f))]) . tseq is Element of the carrier of Y
X is non empty set
Y is non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital CLSStruct
the carrier of Y is non empty set
Funcs (X, the carrier of Y) is non empty functional FUNCTION_DOMAIN of X, the carrier of Y
FuncZero (X,Y) is Relation-like X -defined the carrier of Y -valued Function-like total quasi_total Element of Funcs (X, the carrier of Y)
FuncAdd (X,Y) is non empty Relation-like [:(Funcs (X, the carrier of Y)),(Funcs (X, the carrier of Y)):] -defined Funcs (X, the carrier of Y) -valued Function-like total quasi_total Function-yielding V34() Element of bool [:[:(Funcs (X, the carrier of Y)),(Funcs (X, the carrier of Y)):],(Funcs (X, the carrier of Y)):]
[:(Funcs (X, the carrier of Y)),(Funcs (X, the carrier of Y)):] is non empty set
[:[:(Funcs (X, the carrier of Y)),(Funcs (X, the carrier of Y)):],(Funcs (X, the carrier of Y)):] is non empty set
bool [:[:(Funcs (X, the carrier of Y)),(Funcs (X, the carrier of Y)):],(Funcs (X, the carrier of Y)):] is non empty set
(X,Y) is non empty Relation-like [:COMPLEX,(Funcs (X, the carrier of Y)):] -defined Funcs (X, the carrier of Y) -valued Function-like total quasi_total Function-yielding V34() Element of bool [:[:COMPLEX,(Funcs (X, the carrier of Y)):],(Funcs (X, the carrier of Y)):]
[:COMPLEX,(Funcs (X, the carrier of Y)):] is non empty set
[:[:COMPLEX,(Funcs (X, the carrier of Y)):],(Funcs (X, the carrier of Y)):] is non empty set
bool [:[:COMPLEX,(Funcs (X, the carrier of Y)):],(Funcs (X, the carrier of Y)):] is non empty set
CLSStruct(# (Funcs (X, the carrier of Y)),(FuncZero (X,Y)),(FuncAdd (X,Y)),(X,Y) #) is non empty strict CLSStruct
the carrier of CLSStruct(# (Funcs (X, the carrier of Y)),(FuncZero (X,Y)),(FuncAdd (X,Y)),(X,Y) #) is non empty set
f is Element of the carrier of CLSStruct(# (Funcs (X, the carrier of Y)),(FuncZero (X,Y)),(FuncAdd (X,Y)),(X,Y) #)
tseq is Element of the carrier of CLSStruct(# (Funcs (X, the carrier of Y)),(FuncZero (X,Y)),(FuncAdd (X,Y)),(X,Y) #)
f + tseq is Element of the carrier of CLSStruct(# (Funcs (X, the carrier of Y)),(FuncZero (X,Y)),(FuncAdd (X,Y)),(X,Y) #)
the addF of CLSStruct(# (Funcs (X, the carrier of Y)),(FuncZero (X,Y)),(FuncAdd (X,Y)),(X,Y) #) is non empty Relation-like [: the carrier of CLSStruct(# (Funcs (X, the carrier of Y)),(FuncZero (X,Y)),(FuncAdd (X,Y)),(X,Y) #), the carrier of CLSStruct(# (Funcs (X, the carrier of Y)),(FuncZero (X,Y)),(FuncAdd (X,Y)),(X,Y) #):] -defined the carrier of CLSStruct(# (Funcs (X, the carrier of Y)),(FuncZero (X,Y)),(FuncAdd (X,Y)),(X,Y) #) -valued Function-like total quasi_total Element of bool [:[: the carrier of CLSStruct(# (Funcs (X, the carrier of Y)),(FuncZero (X,Y)),(FuncAdd (X,Y)),(X,Y) #), the carrier of CLSStruct(# (Funcs (X, the carrier of Y)),(FuncZero (X,Y)),(FuncAdd (X,Y)),(X,Y) #):], the carrier of CLSStruct(# (Funcs (X, the carrier of Y)),(FuncZero (X,Y)),(FuncAdd (X,Y)),(X,Y) #):]
[: the carrier of CLSStruct(# (Funcs (X, the carrier of Y)),(FuncZero (X,Y)),(FuncAdd (X,Y)),(X,Y) #), the carrier of CLSStruct(# (Funcs (X, the carrier of Y)),(FuncZero (X,Y)),(FuncAdd (X,Y)),(X,Y) #):] is non empty set
[:[: the carrier of CLSStruct(# (Funcs (X, the carrier of Y)),(FuncZero (X,Y)),(FuncAdd (X,Y)),(X,Y) #), the carrier of CLSStruct(# (Funcs (X, the carrier of Y)),(FuncZero (X,Y)),(FuncAdd (X,Y)),(X,Y) #):], the carrier of CLSStruct(# (Funcs (X, the carrier of Y)),(FuncZero (X,Y)),(FuncAdd (X,Y)),(X,Y) #):] is non empty set
bool [:[: the carrier of CLSStruct(# (Funcs (X, the carrier of Y)),(FuncZero (X,Y)),(FuncAdd (X,Y)),(X,Y) #), the carrier of CLSStruct(# (Funcs (X, the carrier of Y)),(FuncZero (X,Y)),(FuncAdd (X,Y)),(X,Y) #):], the carrier of CLSStruct(# (Funcs (X, the carrier of Y)),(FuncZero (X,Y)),(FuncAdd (X,Y)),(X,Y) #):] is non empty set
the addF of CLSStruct(# (Funcs (X, the carrier of Y)),(FuncZero (X,Y)),(FuncAdd (X,Y)),(X,Y) #) . (f,tseq) is Element of the carrier of CLSStruct(# (Funcs (X, the carrier of Y)),(FuncZero (X,Y)),(FuncAdd (X,Y)),(X,Y) #)
[f,tseq] is set
the addF of CLSStruct(# (Funcs (X, the carrier of Y)),(FuncZero (X,Y)),(FuncAdd (X,Y)),(X,Y) #) . [f,tseq] is set
tseq is Element of the carrier of CLSStruct(# (Funcs (X, the carrier of Y)),(FuncZero (X,Y)),(FuncAdd (X,Y)),(X,Y) #)
(f + tseq) + tseq is Element of the carrier of CLSStruct(# (Funcs (X, the carrier of Y)),(FuncZero (X,Y)),(FuncAdd (X,Y)),(X,Y) #)
the addF of CLSStruct(# (Funcs (X, the carrier of Y)),(FuncZero (X,Y)),(FuncAdd (X,Y)),(X,Y) #) . ((f + tseq),tseq) is Element of the carrier of CLSStruct(# (Funcs (X, the carrier of Y)),(FuncZero (X,Y)),(FuncAdd (X,Y)),(X,Y) #)
[(f + tseq),tseq] is set
the addF of CLSStruct(# (Funcs (X, the carrier of Y)),(FuncZero (X,Y)),(FuncAdd (X,Y)),(X,Y) #) . [(f + tseq),tseq] is set
tseq + tseq is Element of the carrier of CLSStruct(# (Funcs (X, the carrier of Y)),(FuncZero (X,Y)),(FuncAdd (X,Y)),(X,Y) #)
the addF of CLSStruct(# (Funcs (X, the carrier of Y)),(FuncZero (X,Y)),(FuncAdd (X,Y)),(X,Y) #) . (tseq,tseq) is Element of the carrier of CLSStruct(# (Funcs (X, the carrier of Y)),(FuncZero (X,Y)),(FuncAdd (X,Y)),(X,Y) #)
[tseq,tseq] is set
the addF of CLSStruct(# (Funcs (X, the carrier of Y)),(FuncZero (X,Y)),(FuncAdd (X,Y)),(X,Y) #) . [tseq,tseq] is set
f + (tseq + tseq) is Element of the carrier of CLSStruct(# (Funcs (X, the carrier of Y)),(FuncZero (X,Y)),(FuncAdd (X,Y)),(X,Y) #)
the addF of CLSStruct(# (Funcs (X, the carrier of Y)),(FuncZero (X,Y)),(FuncAdd (X,Y)),(X,Y) #) . (f,(tseq + tseq)) is Element of the carrier of CLSStruct(# (Funcs (X, the carrier of Y)),(FuncZero (X,Y)),(FuncAdd (X,Y)),(X,Y) #)
[f,(tseq + tseq)] is set
the addF of CLSStruct(# (Funcs (X, the carrier of Y)),(FuncZero (X,Y)),(FuncAdd (X,Y)),(X,Y) #) . [f,(tseq + tseq)] is set
f is Element of the carrier of CLSStruct(# (Funcs (X, the carrier of Y)),(FuncZero (X,Y)),(FuncAdd (X,Y)),(X,Y) #)
tseq is Relation-like X -defined the carrier of Y -valued Function-like total quasi_total Element of Funcs (X, the carrier of Y)
[(- 1r),tseq] is Element of [:COMPLEX,(Funcs (X, the carrier of Y)):]
(X,Y) . [(- 1r),tseq] is Relation-like X -defined the carrier of Y -valued Function-like total quasi_total Element of Funcs (X, the carrier of Y)
tseq is Element of the carrier of CLSStruct(# (Funcs (X, the carrier of Y)),(FuncZero (X,Y)),(FuncAdd (X,Y)),(X,Y) #)
f + tseq is Element of the carrier of CLSStruct(# (Funcs (X, the carrier of Y)),(FuncZero (X,Y)),(FuncAdd (X,Y)),(X,Y) #)
the addF of CLSStruct(# (Funcs (X, the carrier of Y)),(FuncZero (X,Y)),(FuncAdd (X,Y)),(X,Y) #) is non empty Relation-like [: the carrier of CLSStruct(# (Funcs (X, the carrier of Y)),(FuncZero (X,Y)),(FuncAdd (X,Y)),(X,Y) #), the carrier of CLSStruct(# (Funcs (X, the carrier of Y)),(FuncZero (X,Y)),(FuncAdd (X,Y)),(X,Y) #):] -defined the carrier of CLSStruct(# (Funcs (X, the carrier of Y)),(FuncZero (X,Y)),(FuncAdd (X,Y)),(X,Y) #) -valued Function-like total quasi_total Element of bool [:[: the carrier of CLSStruct(# (Funcs (X, the carrier of Y)),(FuncZero (X,Y)),(FuncAdd (X,Y)),(X,Y) #), the carrier of CLSStruct(# (Funcs (X, the carrier of Y)),(FuncZero (X,Y)),(FuncAdd (X,Y)),(X,Y) #):], the carrier of CLSStruct(# (Funcs (X, the carrier of Y)),(FuncZero (X,Y)),(FuncAdd (X,Y)),(X,Y) #):]
[: the carrier of CLSStruct(# (Funcs (X, the carrier of Y)),(FuncZero (X,Y)),(FuncAdd (X,Y)),(X,Y) #), the carrier of CLSStruct(# (Funcs (X, the carrier of Y)),(FuncZero (X,Y)),(FuncAdd (X,Y)),(X,Y) #):] is non empty set
[:[: the carrier of CLSStruct(# (Funcs (X, the carrier of Y)),(FuncZero (X,Y)),(FuncAdd (X,Y)),(X,Y) #), the carrier of CLSStruct(# (Funcs (X, the carrier of Y)),(FuncZero (X,Y)),(FuncAdd (X,Y)),(X,Y) #):], the carrier of CLSStruct(# (Funcs (X, the carrier of Y)),(FuncZero (X,Y)),(FuncAdd (X,Y)),(X,Y) #):] is non empty set
bool [:[: the carrier of CLSStruct(# (Funcs (X, the carrier of Y)),(FuncZero (X,Y)),(FuncAdd (X,Y)),(X,Y) #), the carrier of CLSStruct(# (Funcs (X, the carrier of Y)),(FuncZero (X,Y)),(FuncAdd (X,Y)),(X,Y) #):], the carrier of CLSStruct(# (Funcs (X, the carrier of Y)),(FuncZero (X,Y)),(FuncAdd (X,Y)),(X,Y) #):] is non empty set
the addF of CLSStruct(# (Funcs (X, the carrier of Y)),(FuncZero (X,Y)),(FuncAdd (X,Y)),(X,Y) #) . (f,tseq) is Element of the carrier of CLSStruct(# (Funcs (X, the carrier of Y)),(FuncZero (X,Y)),(FuncAdd (X,Y)),(X,Y) #)
[f,tseq] is set
the addF of CLSStruct(# (Funcs (X, the carrier of Y)),(FuncZero (X,Y)),(FuncAdd (X,Y)),(X,Y) #) . [f,tseq] is set
0. CLSStruct(# (Funcs (X, the carrier of Y)),(FuncZero (X,Y)),(FuncAdd (X,Y)),(X,Y) #) is zero Element of the carrier of CLSStruct(# (Funcs (X, the carrier of Y)),(FuncZero (X,Y)),(FuncAdd (X,Y)),(X,Y) #)
the ZeroF of CLSStruct(# (Funcs (X, the carrier of Y)),(FuncZero (X,Y)),(FuncAdd (X,Y)),(X,Y) #) is Element of the carrier of CLSStruct(# (Funcs (X, the carrier of Y)),(FuncZero (X,Y)),(FuncAdd (X,Y)),(X,Y) #)
f is complex set
tseq is Element of the carrier of CLSStruct(# (Funcs (X, the carrier of Y)),(FuncZero (X,Y)),(FuncAdd (X,Y)),(X,Y) #)
tseq is Element of the carrier of CLSStruct(# (Funcs (X, the carrier of Y)),(FuncZero (X,Y)),(FuncAdd (X,Y)),(X,Y) #)
tseq + tseq is Element of the carrier of CLSStruct(# (Funcs (X, the carrier of Y)),(FuncZero (X,Y)),(FuncAdd (X,Y)),(X,Y) #)
the addF of CLSStruct(# (Funcs (X, the carrier of Y)),(FuncZero (X,Y)),(FuncAdd (X,Y)),(X,Y) #) is non empty Relation-like [: the carrier of CLSStruct(# (Funcs (X, the carrier of Y)),(FuncZero (X,Y)),(FuncAdd (X,Y)),(X,Y) #), the carrier of CLSStruct(# (Funcs (X, the carrier of Y)),(FuncZero (X,Y)),(FuncAdd (X,Y)),(X,Y) #):] -defined the carrier of CLSStruct(# (Funcs (X, the carrier of Y)),(FuncZero (X,Y)),(FuncAdd (X,Y)),(X,Y) #) -valued Function-like total quasi_total Element of bool [:[: the carrier of CLSStruct(# (Funcs (X, the carrier of Y)),(FuncZero (X,Y)),(FuncAdd (X,Y)),(X,Y) #), the carrier of CLSStruct(# (Funcs (X, the carrier of Y)),(FuncZero (X,Y)),(FuncAdd (X,Y)),(X,Y) #):], the carrier of CLSStruct(# (Funcs (X, the carrier of Y)),(FuncZero (X,Y)),(FuncAdd (X,Y)),(X,Y) #):]
[: the carrier of CLSStruct(# (Funcs (X, the carrier of Y)),(FuncZero (X,Y)),(FuncAdd (X,Y)),(X,Y) #), the carrier of CLSStruct(# (Funcs (X, the carrier of Y)),(FuncZero (X,Y)),(FuncAdd (X,Y)),(X,Y) #):] is non empty set
[:[: the carrier of CLSStruct(# (Funcs (X, the carrier of Y)),(FuncZero (X,Y)),(FuncAdd (X,Y)),(X,Y) #), the carrier of CLSStruct(# (Funcs (X, the carrier of Y)),(FuncZero (X,Y)),(FuncAdd (X,Y)),(X,Y) #):], the carrier of CLSStruct(# (Funcs (X, the carrier of Y)),(FuncZero (X,Y)),(FuncAdd (X,Y)),(X,Y) #):] is non empty set
bool [:[: the carrier of CLSStruct(# (Funcs (X, the carrier of Y)),(FuncZero (X,Y)),(FuncAdd (X,Y)),(X,Y) #), the carrier of CLSStruct(# (Funcs (X, the carrier of Y)),(FuncZero (X,Y)),(FuncAdd (X,Y)),(X,Y) #):], the carrier of CLSStruct(# (Funcs (X, the carrier of Y)),(FuncZero (X,Y)),(FuncAdd (X,Y)),(X,Y) #):] is non empty set
the addF of CLSStruct(# (Funcs (X, the carrier of Y)),(FuncZero (X,Y)),(FuncAdd (X,Y)),(X,Y) #) . (tseq,tseq) is Element of the carrier of CLSStruct(# (Funcs (X, the carrier of Y)),(FuncZero (X,Y)),(FuncAdd (X,Y)),(X,Y) #)
[tseq,tseq] is set
the addF of CLSStruct(# (Funcs (X, the carrier of Y)),(FuncZero (X,Y)),(FuncAdd (X,Y)),(X,Y) #) . [tseq,tseq] is set
f * (tseq + tseq) is Element of the carrier of CLSStruct(# (Funcs (X, the carrier of Y)),(FuncZero (X,Y)),(FuncAdd (X,Y)),(X,Y) #)
f * tseq is Element of the carrier of CLSStruct(# (Funcs (X, the carrier of Y)),(FuncZero (X,Y)),(FuncAdd (X,Y)),(X,Y) #)
f * tseq is Element of the carrier of CLSStruct(# (Funcs (X, the carrier of Y)),(FuncZero (X,Y)),(FuncAdd (X,Y)),(X,Y) #)
(f * tseq) + (f * tseq) is Element of the carrier of CLSStruct(# (Funcs (X, the carrier of Y)),(FuncZero (X,Y)),(FuncAdd (X,Y)),(X,Y) #)
the addF of CLSStruct(# (Funcs (X, the carrier of Y)),(FuncZero (X,Y)),(FuncAdd (X,Y)),(X,Y) #) . ((f * tseq),(f * tseq)) is Element of the carrier of CLSStruct(# (Funcs (X, the carrier of Y)),(FuncZero (X,Y)),(FuncAdd (X,Y)),(X,Y) #)
[(f * tseq),(f * tseq)] is set
the addF of CLSStruct(# (Funcs (X, the carrier of Y)),(FuncZero (X,Y)),(FuncAdd (X,Y)),(X,Y) #) . [(f * tseq),(f * tseq)] is set
tseq is complex Element of COMPLEX
tseq * (tseq + tseq) is Element of the carrier of CLSStruct(# (Funcs (X, the carrier of Y)),(FuncZero (X,Y)),(FuncAdd (X,Y)),(X,Y) #)
tseq * tseq is Element of the carrier of CLSStruct(# (Funcs (X, the carrier of Y)),(FuncZero (X,Y)),(FuncAdd (X,Y)),(X,Y) #)
tseq * tseq is Element of the carrier of CLSStruct(# (Funcs (X, the carrier of Y)),(FuncZero (X,Y)),(FuncAdd (X,Y)),(X,Y) #)
(tseq * tseq) + (tseq * tseq) is Element of the carrier of CLSStruct(# (Funcs (X, the carrier of Y)),(FuncZero (X,Y)),(FuncAdd (X,Y)),(X,Y) #)
the addF of CLSStruct(# (Funcs (X, the carrier of Y)),(FuncZero (X,Y)),(FuncAdd (X,Y)),(X,Y) #) . ((tseq * tseq),(tseq * tseq)) is Element of the carrier of CLSStruct(# (Funcs (X, the carrier of Y)),(FuncZero (X,Y)),(FuncAdd (X,Y)),(X,Y) #)
[(tseq * tseq),(tseq * tseq)] is set
the addF of CLSStruct(# (Funcs (X, the carrier of Y)),(FuncZero (X,Y)),(FuncAdd (X,Y)),(X,Y) #) . [(tseq * tseq),(tseq * tseq)] is set
e is Relation-like X -defined the carrier of Y -valued Function-like total quasi_total Element of Funcs (X, the carrier of Y)
[tseq,e] is Element of [:COMPLEX,(Funcs (X, the carrier of Y)):]
(X,Y) . [tseq,e] is Relation-like X -defined the carrier of Y -valued Function-like total quasi_total Element of Funcs (X, the carrier of Y)
tv is Relation-like X -defined the carrier of Y -valued Function-like total quasi_total Element of Funcs (X, the carrier of Y)
[tseq,tv] is Element of [:COMPLEX,(Funcs (X, the carrier of Y)):]
(X,Y) . [tseq,tv] is Relation-like X -defined the carrier of Y -valued Function-like total quasi_total Element of Funcs (X, the carrier of Y)
(FuncAdd (X,Y)) . (e,tv) is Relation-like X -defined the carrier of Y -valued Function-like total quasi_total Element of Funcs (X, the carrier of Y)
[e,tv] is set
(FuncAdd (X,Y)) . [e,tv] is Relation-like Function-like set
[tseq,((FuncAdd (X,Y)) . (e,tv))] is Element of [:COMPLEX,(Funcs (X, the carrier of Y)):]
(X,Y) . [tseq,((FuncAdd (X,Y)) . (e,tv))] is Relation-like X -defined the carrier of Y -valued Function-like total quasi_total Element of Funcs (X, the carrier of Y)
m is Element of the carrier of CLSStruct(# (Funcs (X, the carrier of Y)),(FuncZero (X,Y)),(FuncAdd (X,Y)),(X,Y) #)
n is Element of the carrier of CLSStruct(# (Funcs (X, the carrier of Y)),(FuncZero (X,Y)),(FuncAdd (X,Y)),(X,Y) #)
(FuncAdd (X,Y)) . (m,n) is set
[m,n] is set
(FuncAdd (X,Y)) . [m,n] is Relation-like Function-like set
m + (tseq * tseq) is Element of the carrier of CLSStruct(# (Funcs (X, the carrier of Y)),(FuncZero (X,Y)),(FuncAdd (X,Y)),(X,Y) #)
the addF of CLSStruct(# (Funcs (X, the carrier of Y)),(FuncZero (X,Y)),(FuncAdd (X,Y)),(X,Y) #) . (m,(tseq * tseq)) is Element of the carrier of CLSStruct(# (Funcs (X, the carrier of Y)),(FuncZero (X,Y)),(FuncAdd (X,Y)),(X,Y) #)
[m,(tseq * tseq)] is set
the addF of CLSStruct(# (Funcs (X, the carrier of Y)),(FuncZero (X,Y)),(FuncAdd (X,Y)),(X,Y) #) . [m,(tseq * tseq)] is set
f is complex set
tseq is complex set
f * tseq is complex set
tseq is Element of the carrier of CLSStruct(# (Funcs (X, the carrier of Y)),(FuncZero (X,Y)),(FuncAdd (X,Y)),(X,Y) #)
(f * tseq) * tseq is Element of the carrier of CLSStruct(# (Funcs (X, the carrier of Y)),(FuncZero (X,Y)),(FuncAdd (X,Y)),(X,Y) #)
tseq * tseq is Element of the carrier of CLSStruct(# (Funcs (X, the carrier of Y)),(FuncZero (X,Y)),(FuncAdd (X,Y)),(X,Y) #)
f * (tseq * tseq) is Element of the carrier of CLSStruct(# (Funcs (X, the carrier of Y)),(FuncZero (X,Y)),(FuncAdd (X,Y)),(X,Y) #)
tseq is Relation-like X -defined the carrier of Y -valued Function-like total quasi_total Element of Funcs (X, the carrier of Y)
[(f * tseq),tseq] is set
(X,Y) . [(f * tseq),tseq] is Relation-like Function-like set
[tseq,tseq] is set
(X,Y) . [tseq,tseq] is Relation-like Function-like set
[f,((X,Y) . [tseq,tseq])] is set
(X,Y) . [f,((X,Y) . [tseq,tseq])] is Relation-like Function-like set
[f,(tseq * tseq)] is set
(X,Y) . [f,(tseq * tseq)] is Relation-like Function-like set
f is complex set
tseq is complex set
f + tseq is complex set
tseq is Element of the carrier of CLSStruct(# (Funcs (X, the carrier of Y)),(FuncZero (X,Y)),(FuncAdd (X,Y)),(X,Y) #)
(f + tseq) * tseq is Element of the carrier of CLSStruct(# (Funcs (X, the carrier of Y)),(FuncZero (X,Y)),(FuncAdd (X,Y)),(X,Y) #)
f * tseq is Element of the carrier of CLSStruct(# (Funcs (X, the carrier of Y)),(FuncZero (X,Y)),(FuncAdd (X,Y)),(X,Y) #)
tseq * tseq is Element of the carrier of CLSStruct(# (Funcs (X, the carrier of Y)),(FuncZero (X,Y)),(FuncAdd (X,Y)),(X,Y) #)
(f * tseq) + (tseq * tseq) is Element of the carrier of CLSStruct(# (Funcs (X, the carrier of Y)),(FuncZero (X,Y)),(FuncAdd (X,Y)),(X,Y) #)
the addF of CLSStruct(# (Funcs (X, the carrier of Y)),(FuncZero (X,Y)),(FuncAdd (X,Y)),(X,Y) #) is non empty Relation-like [: the carrier of CLSStruct(# (Funcs (X, the carrier of Y)),(FuncZero (X,Y)),(FuncAdd (X,Y)),(X,Y) #), the carrier of CLSStruct(# (Funcs (X, the carrier of Y)),(FuncZero (X,Y)),(FuncAdd (X,Y)),(X,Y) #):] -defined the carrier of CLSStruct(# (Funcs (X, the carrier of Y)),(FuncZero (X,Y)),(FuncAdd (X,Y)),(X,Y) #) -valued Function-like total quasi_total Element of bool [:[: the carrier of CLSStruct(# (Funcs (X, the carrier of Y)),(FuncZero (X,Y)),(FuncAdd (X,Y)),(X,Y) #), the carrier of CLSStruct(# (Funcs (X, the carrier of Y)),(FuncZero (X,Y)),(FuncAdd (X,Y)),(X,Y) #):], the carrier of CLSStruct(# (Funcs (X, the carrier of Y)),(FuncZero (X,Y)),(FuncAdd (X,Y)),(X,Y) #):]
[: the carrier of CLSStruct(# (Funcs (X, the carrier of Y)),(FuncZero (X,Y)),(FuncAdd (X,Y)),(X,Y) #), the carrier of CLSStruct(# (Funcs (X, the carrier of Y)),(FuncZero (X,Y)),(FuncAdd (X,Y)),(X,Y) #):] is non empty set
[:[: the carrier of CLSStruct(# (Funcs (X, the carrier of Y)),(FuncZero (X,Y)),(FuncAdd (X,Y)),(X,Y) #), the carrier of CLSStruct(# (Funcs (X, the carrier of Y)),(FuncZero (X,Y)),(FuncAdd (X,Y)),(X,Y) #):], the carrier of CLSStruct(# (Funcs (X, the carrier of Y)),(FuncZero (X,Y)),(FuncAdd (X,Y)),(X,Y) #):] is non empty set
bool [:[: the carrier of CLSStruct(# (Funcs (X, the carrier of Y)),(FuncZero (X,Y)),(FuncAdd (X,Y)),(X,Y) #), the carrier of CLSStruct(# (Funcs (X, the carrier of Y)),(FuncZero (X,Y)),(FuncAdd (X,Y)),(X,Y) #):], the carrier of CLSStruct(# (Funcs (X, the carrier of Y)),(FuncZero (X,Y)),(FuncAdd (X,Y)),(X,Y) #):] is non empty set
the addF of CLSStruct(# (Funcs (X, the carrier of Y)),(FuncZero (X,Y)),(FuncAdd (X,Y)),(X,Y) #) . ((f * tseq),(tseq * tseq)) is Element of the carrier of CLSStruct(# (Funcs (X, the carrier of Y)),(FuncZero (X,Y)),(FuncAdd (X,Y)),(X,Y) #)
[(f * tseq),(tseq * tseq)] is set
the addF of CLSStruct(# (Funcs (X, the carrier of Y)),(FuncZero (X,Y)),(FuncAdd (X,Y)),(X,Y) #) . [(f * tseq),(tseq * tseq)] is set
tseq is complex Element of COMPLEX
e is Relation-like X -defined the carrier of Y -valued Function-like total quasi_total Element of Funcs (X, the carrier of Y)
[tseq,e] is Element of [:COMPLEX,(Funcs (X, the carrier of Y)):]
(X,Y) . [tseq,e] is Relation-like X -defined the carrier of Y -valued Function-like total quasi_total Element of Funcs (X, the carrier of Y)
tv is complex Element of COMPLEX
[tv,e] is Element of [:COMPLEX,(Funcs (X, the carrier of Y)):]
(X,Y) . [tv,e] is Relation-like X -defined the carrier of Y -valued Function-like total quasi_total Element of Funcs (X, the carrier of Y)
tseq + tv is complex Element of COMPLEX
(tseq + tv) * tseq is Element of the carrier of CLSStruct(# (Funcs (X, the carrier of Y)),(FuncZero (X,Y)),(FuncAdd (X,Y)),(X,Y) #)
[(tseq + tv),e] is Element of [:COMPLEX,(Funcs (X, the carrier of Y)):]
(X,Y) . [(tseq + tv),e] is Relation-like X -defined the carrier of Y -valued Function-like total quasi_total Element of Funcs (X, the carrier of Y)
m is Element of the carrier of CLSStruct(# (Funcs (X, the carrier of Y)),(FuncZero (X,Y)),(FuncAdd (X,Y)),(X,Y) #)
n is Element of the carrier of CLSStruct(# (Funcs (X, the carrier of Y)),(FuncZero (X,Y)),(FuncAdd (X,Y)),(X,Y) #)
(FuncAdd (X,Y)) . (m,n) is set
[m,n] is set
(FuncAdd (X,Y)) . [m,n] is Relation-like Function-like set
tv * tseq is Element of the carrier of CLSStruct(# (Funcs (X, the carrier of Y)),(FuncZero (X,Y)),(FuncAdd (X,Y)),(X,Y) #)
m + (tv * tseq) is Element of the carrier of CLSStruct(# (Funcs (X, the carrier of Y)),(FuncZero (X,Y)),(FuncAdd (X,Y)),(X,Y) #)
the addF of CLSStruct(# (Funcs (X, the carrier of Y)),(FuncZero (X,Y)),(FuncAdd (X,Y)),(X,Y) #) . (m,(tv * tseq)) is Element of the carrier of CLSStruct(# (Funcs (X, the carrier of Y)),(FuncZero (X,Y)),(FuncAdd (X,Y)),(X,Y) #)
[m,(tv * tseq)] is set
the addF of CLSStruct(# (Funcs (X, the carrier of Y)),(FuncZero (X,Y)),(FuncAdd (X,Y)),(X,Y) #) . [m,(tv * tseq)] is set
tseq * tseq is Element of the carrier of CLSStruct(# (Funcs (X, the carrier of Y)),(FuncZero (X,Y)),(FuncAdd (X,Y)),(X,Y) #)
(tseq * tseq) + (tv * tseq) is Element of the carrier of CLSStruct(# (Funcs (X, the carrier of Y)),(FuncZero (X,Y)),(FuncAdd (X,Y)),(X,Y) #)
the addF of CLSStruct(# (Funcs (X, the carrier of Y)),(FuncZero (X,Y)),(FuncAdd (X,Y)),(X,Y) #) . ((tseq * tseq),(tv * tseq)) is Element of the carrier of CLSStruct(# (Funcs (X, the carrier of Y)),(FuncZero (X,Y)),(FuncAdd (X,Y)),(X,Y) #)
[(tseq * tseq),(tv * tseq)] is set
the addF of CLSStruct(# (Funcs (X, the carrier of Y)),(FuncZero (X,Y)),(FuncAdd (X,Y)),(X,Y) #) . [(tseq * tseq),(tv * tseq)] is set
0. CLSStruct(# (Funcs (X, the carrier of Y)),(FuncZero (X,Y)),(FuncAdd (X,Y)),(X,Y) #) is zero Element of the carrier of CLSStruct(# (Funcs (X, the carrier of Y)),(FuncZero (X,Y)),(FuncAdd (X,Y)),(X,Y) #)
the ZeroF of CLSStruct(# (Funcs (X, the carrier of Y)),(FuncZero (X,Y)),(FuncAdd (X,Y)),(X,Y) #) is Element of the carrier of CLSStruct(# (Funcs (X, the carrier of Y)),(FuncZero (X,Y)),(FuncAdd (X,Y)),(X,Y) #)
f is Element of the carrier of CLSStruct(# (Funcs (X, the carrier of Y)),(FuncZero (X,Y)),(FuncAdd (X,Y)),(X,Y) #)
f + (0. CLSStruct(# (Funcs (X, the carrier of Y)),(FuncZero (X,Y)),(FuncAdd (X,Y)),(X,Y) #)) is Element of the carrier of CLSStruct(# (Funcs (X, the carrier of Y)),(FuncZero (X,Y)),(FuncAdd (X,Y)),(X,Y) #)
the addF of CLSStruct(# (Funcs (X, the carrier of Y)),(FuncZero (X,Y)),(FuncAdd (X,Y)),(X,Y) #) is non empty Relation-like [: the carrier of CLSStruct(# (Funcs (X, the carrier of Y)),(FuncZero (X,Y)),(FuncAdd (X,Y)),(X,Y) #), the carrier of CLSStruct(# (Funcs (X, the carrier of Y)),(FuncZero (X,Y)),(FuncAdd (X,Y)),(X,Y) #):] -defined the carrier of CLSStruct(# (Funcs (X, the carrier of Y)),(FuncZero (X,Y)),(FuncAdd (X,Y)),(X,Y) #) -valued Function-like total quasi_total Element of bool [:[: the carrier of CLSStruct(# (Funcs (X, the carrier of Y)),(FuncZero (X,Y)),(FuncAdd (X,Y)),(X,Y) #), the carrier of CLSStruct(# (Funcs (X, the carrier of Y)),(FuncZero (X,Y)),(FuncAdd (X,Y)),(X,Y) #):], the carrier of CLSStruct(# (Funcs (X, the carrier of Y)),(FuncZero (X,Y)),(FuncAdd (X,Y)),(X,Y) #):]
[: the carrier of CLSStruct(# (Funcs (X, the carrier of Y)),(FuncZero (X,Y)),(FuncAdd (X,Y)),(X,Y) #), the carrier of CLSStruct(# (Funcs (X, the carrier of Y)),(FuncZero (X,Y)),(FuncAdd (X,Y)),(X,Y) #):] is non empty set
[:[: the carrier of CLSStruct(# (Funcs (X, the carrier of Y)),(FuncZero (X,Y)),(FuncAdd (X,Y)),(X,Y) #), the carrier of CLSStruct(# (Funcs (X, the carrier of Y)),(FuncZero (X,Y)),(FuncAdd (X,Y)),(X,Y) #):], the carrier of CLSStruct(# (Funcs (X, the carrier of Y)),(FuncZero (X,Y)),(FuncAdd (X,Y)),(X,Y) #):] is non empty set
bool [:[: the carrier of CLSStruct(# (Funcs (X, the carrier of Y)),(FuncZero (X,Y)),(FuncAdd (X,Y)),(X,Y) #), the carrier of CLSStruct(# (Funcs (X, the carrier of Y)),(FuncZero (X,Y)),(FuncAdd (X,Y)),(X,Y) #):], the carrier of CLSStruct(# (Funcs (X, the carrier of Y)),(FuncZero (X,Y)),(FuncAdd (X,Y)),(X,Y) #):] is non empty set
the addF of CLSStruct(# (Funcs (X, the carrier of Y)),(FuncZero (X,Y)),(FuncAdd (X,Y)),(X,Y) #) . (f,(0. CLSStruct(# (Funcs (X, the carrier of Y)),(FuncZero (X,Y)),(FuncAdd (X,Y)),(X,Y) #))) is Element of the carrier of CLSStruct(# (Funcs (X, the carrier of Y)),(FuncZero (X,Y)),(FuncAdd (X,Y)),(X,Y) #)
[f,(0. CLSStruct(# (Funcs (X, the carrier of Y)),(FuncZero (X,Y)),(FuncAdd (X,Y)),(X,Y) #))] is set
the addF of CLSStruct(# (Funcs (X, the carrier of Y)),(FuncZero (X,Y)),(FuncAdd (X,Y)),(X,Y) #) . [f,(0. CLSStruct(# (Funcs (X, the carrier of Y)),(FuncZero (X,Y)),(FuncAdd (X,Y)),(X,Y) #))] is set
tseq is Relation-like X -defined the carrier of Y -valued Function-like total quasi_total Element of Funcs (X, the carrier of Y)
(FuncAdd (X,Y)) . ((FuncZero (X,Y)),tseq) is Relation-like X -defined the carrier of Y -valued Function-like total quasi_total Element of Funcs (X, the carrier of Y)
[(FuncZero (X,Y)),tseq] is set
(FuncAdd (X,Y)) . [(FuncZero (X,Y)),tseq] is Relation-like Function-like set
f is Element of the carrier of CLSStruct(# (Funcs (X, the carrier of Y)),(FuncZero (X,Y)),(FuncAdd (X,Y)),(X,Y) #)
1r * f is Element of the carrier of CLSStruct(# (Funcs (X, the carrier of Y)),(FuncZero (X,Y)),(FuncAdd (X,Y)),(X,Y) #)
tseq is Relation-like X -defined the carrier of Y -valued Function-like total quasi_total Element of Funcs (X, the carrier of Y)
[1r,tseq] is Element of [:COMPLEX,(Funcs (X, the carrier of Y)):]
(X,Y) . [1r,tseq] is Relation-like X -defined the carrier of Y -valued Function-like total quasi_total Element of Funcs (X, the carrier of Y)
f is Element of the carrier of CLSStruct(# (Funcs (X, the carrier of Y)),(FuncZero (X,Y)),(FuncAdd (X,Y)),(X,Y) #)
tseq is Element of the carrier of CLSStruct(# (Funcs (X, the carrier of Y)),(FuncZero (X,Y)),(FuncAdd (X,Y)),(X,Y) #)
f + tseq is Element of the carrier of CLSStruct(# (Funcs (X, the carrier of Y)),(FuncZero (X,Y)),(FuncAdd (X,Y)),(X,Y) #)
the addF of CLSStruct(# (Funcs (X, the carrier of Y)),(FuncZero (X,Y)),(FuncAdd (X,Y)),(X,Y) #) is non empty Relation-like [: the carrier of CLSStruct(# (Funcs (X, the carrier of Y)),(FuncZero (X,Y)),(FuncAdd (X,Y)),(X,Y) #), the carrier of CLSStruct(# (Funcs (X, the carrier of Y)),(FuncZero (X,Y)),(FuncAdd (X,Y)),(X,Y) #):] -defined the carrier of CLSStruct(# (Funcs (X, the carrier of Y)),(FuncZero (X,Y)),(FuncAdd (X,Y)),(X,Y) #) -valued Function-like total quasi_total Element of bool [:[: the carrier of CLSStruct(# (Funcs (X, the carrier of Y)),(FuncZero (X,Y)),(FuncAdd (X,Y)),(X,Y) #), the carrier of CLSStruct(# (Funcs (X, the carrier of Y)),(FuncZero (X,Y)),(FuncAdd (X,Y)),(X,Y) #):], the carrier of CLSStruct(# (Funcs (X, the carrier of Y)),(FuncZero (X,Y)),(FuncAdd (X,Y)),(X,Y) #):]
[: the carrier of CLSStruct(# (Funcs (X, the carrier of Y)),(FuncZero (X,Y)),(FuncAdd (X,Y)),(X,Y) #), the carrier of CLSStruct(# (Funcs (X, the carrier of Y)),(FuncZero (X,Y)),(FuncAdd (X,Y)),(X,Y) #):] is non empty set
[:[: the carrier of CLSStruct(# (Funcs (X, the carrier of Y)),(FuncZero (X,Y)),(FuncAdd (X,Y)),(X,Y) #), the carrier of CLSStruct(# (Funcs (X, the carrier of Y)),(FuncZero (X,Y)),(FuncAdd (X,Y)),(X,Y) #):], the carrier of CLSStruct(# (Funcs (X, the carrier of Y)),(FuncZero (X,Y)),(FuncAdd (X,Y)),(X,Y) #):] is non empty set
bool [:[: the carrier of CLSStruct(# (Funcs (X, the carrier of Y)),(FuncZero (X,Y)),(FuncAdd (X,Y)),(X,Y) #), the carrier of CLSStruct(# (Funcs (X, the carrier of Y)),(FuncZero (X,Y)),(FuncAdd (X,Y)),(X,Y) #):], the carrier of CLSStruct(# (Funcs (X, the carrier of Y)),(FuncZero (X,Y)),(FuncAdd (X,Y)),(X,Y) #):] is non empty set
the addF of CLSStruct(# (Funcs (X, the carrier of Y)),(FuncZero (X,Y)),(FuncAdd (X,Y)),(X,Y) #) . (f,tseq) is Element of the carrier of CLSStruct(# (Funcs (X, the carrier of Y)),(FuncZero (X,Y)),(FuncAdd (X,Y)),(X,Y) #)
[f,tseq] is set
the addF of CLSStruct(# (Funcs (X, the carrier of Y)),(FuncZero (X,Y)),(FuncAdd (X,Y)),(X,Y) #) . [f,tseq] is set
tseq + f is Element of the carrier of CLSStruct(# (Funcs (X, the carrier of Y)),(FuncZero (X,Y)),(FuncAdd (X,Y)),(X,Y) #)
the addF of CLSStruct(# (Funcs (X, the carrier of Y)),(FuncZero (X,Y)),(FuncAdd (X,Y)),(X,Y) #) . (tseq,f) is Element of the carrier of CLSStruct(# (Funcs (X, the carrier of Y)),(FuncZero (X,Y)),(FuncAdd (X,Y)),(X,Y) #)
[tseq,f] is set
the addF of CLSStruct(# (Funcs (X, the carrier of Y)),(FuncZero (X,Y)),(FuncAdd (X,Y)),(X,Y) #) . [tseq,f] is set
X is non empty set
Y is non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital CLSStruct
the carrier of Y is non empty set
Funcs (X, the carrier of Y) is non empty functional FUNCTION_DOMAIN of X, the carrier of Y
FuncZero (X,Y) is Relation-like X -defined the carrier of Y -valued Function-like total quasi_total Element of Funcs (X, the carrier of Y)
FuncAdd (X,Y) is non empty Relation-like [:(Funcs (X, the carrier of Y)),(Funcs (X, the carrier of Y)):] -defined Funcs (X, the carrier of Y) -valued Function-like total quasi_total Function-yielding V34() Element of bool [:[:(Funcs (X, the carrier of Y)),(Funcs (X, the carrier of Y)):],(Funcs (X, the carrier of Y)):]
[:(Funcs (X, the carrier of Y)),(Funcs (X, the carrier of Y)):] is non empty set
[:[:(Funcs (X, the carrier of Y)),(Funcs (X, the carrier of Y)):],(Funcs (X, the carrier of Y)):] is non empty set
bool [:[:(Funcs (X, the carrier of Y)),(Funcs (X, the carrier of Y)):],(Funcs (X, the carrier of Y)):] is non empty set
(X,Y) is non empty Relation-like [:COMPLEX,(Funcs (X, the carrier of Y)):] -defined Funcs (X, the carrier of Y) -valued Function-like total quasi_total Function-yielding V34() Element of bool [:[:COMPLEX,(Funcs (X, the carrier of Y)):],(Funcs (X, the carrier of Y)):]
[:COMPLEX,(Funcs (X, the carrier of Y)):] is non empty set
[:[:COMPLEX,(Funcs (X, the carrier of Y)):],(Funcs (X, the carrier of Y)):] is non empty set
bool [:[:COMPLEX,(Funcs (X, the carrier of Y)):],(Funcs (X, the carrier of Y)):] is non empty set
CLSStruct(# (Funcs (X, the carrier of Y)),(FuncZero (X,Y)),(FuncAdd (X,Y)),(X,Y) #) is non empty strict CLSStruct
X is non empty set
Y is non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital CLSStruct
(X,Y) is non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital CLSStruct
the carrier of Y is non empty set
Funcs (X, the carrier of Y) is non empty functional FUNCTION_DOMAIN of X, the carrier of Y
FuncZero (X,Y) is Relation-like X -defined the carrier of Y -valued Function-like total quasi_total Element of Funcs (X, the carrier of Y)
FuncAdd (X,Y) is non empty Relation-like [:(Funcs (X, the carrier of Y)),(Funcs (X, the carrier of Y)):] -defined Funcs (X, the carrier of Y) -valued Function-like total quasi_total Function-yielding V34() Element of bool [:[:(Funcs (X, the carrier of Y)),(Funcs (X, the carrier of Y)):],(Funcs (X, the carrier of Y)):]
[:(Funcs (X, the carrier of Y)),(Funcs (X, the carrier of Y)):] is non empty set
[:[:(Funcs (X, the carrier of Y)),(Funcs (X, the carrier of Y)):],(Funcs (X, the carrier of Y)):] is non empty set
bool [:[:(Funcs (X, the carrier of Y)),(Funcs (X, the carrier of Y)):],(Funcs (X, the carrier of Y)):] is non empty set
(X,Y) is non empty Relation-like [:COMPLEX,(Funcs (X, the carrier of Y)):] -defined Funcs (X, the carrier of Y) -valued Function-like total quasi_total Function-yielding V34() Element of bool [:[:COMPLEX,(Funcs (X, the carrier of Y)):],(Funcs (X, the carrier of Y)):]
[:COMPLEX,(Funcs (X, the carrier of Y)):] is non empty set
[:[:COMPLEX,(Funcs (X, the carrier of Y)):],(Funcs (X, the carrier of Y)):] is non empty set
bool [:[:COMPLEX,(Funcs (X, the carrier of Y)):],(Funcs (X, the carrier of Y)):] is non empty set
CLSStruct(# (Funcs (X, the carrier of Y)),(FuncZero (X,Y)),(FuncAdd (X,Y)),(X,Y) #) is non empty strict CLSStruct
X is non empty set
Y is non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital CLSStruct
(X,Y) is non empty right_complementable Abelian add-associative right_zeroed strict vector-distributive scalar-distributive scalar-associative scalar-unital CLSStruct
the carrier of Y is non empty set
Funcs (X, the carrier of Y) is non empty functional FUNCTION_DOMAIN of X, the carrier of Y
FuncZero (X,Y) is Relation-like X -defined the carrier of Y -valued Function-like total quasi_total Element of Funcs (X, the carrier of Y)
FuncAdd (X,Y) is non empty Relation-like [:(Funcs (X, the carrier of Y)),(Funcs (X, the carrier of Y)):] -defined Funcs (X, the carrier of Y) -valued Function-like total quasi_total Function-yielding V34() Element of bool [:[:(Funcs (X, the carrier of Y)),(Funcs (X, the carrier of Y)):],(Funcs (X, the carrier of Y)):]
[:(Funcs (X, the carrier of Y)),(Funcs (X, the carrier of Y)):] is non empty set
[:[:(Funcs (X, the carrier of Y)),(Funcs (X, the carrier of Y)):],(Funcs (X, the carrier of Y)):] is non empty set
bool [:[:(Funcs (X, the carrier of Y)),(Funcs (X, the carrier of Y)):],(Funcs (X, the carrier of Y)):] is non empty set
(X,Y) is non empty Relation-like [:COMPLEX,(Funcs (X, the carrier of Y)):] -defined Funcs (X, the carrier of Y) -valued Function-like total quasi_total Function-yielding V34() Element of bool [:[:COMPLEX,(Funcs (X, the carrier of Y)):],(Funcs (X, the carrier of Y)):]
[:COMPLEX,(Funcs (X, the carrier of Y)):] is non empty set
[:[:COMPLEX,(Funcs (X, the carrier of Y)):],(Funcs (X, the carrier of Y)):] is non empty set
bool [:[:COMPLEX,(Funcs (X, the carrier of Y)):],(Funcs (X, the carrier of Y)):] is non empty set
CLSStruct(# (Funcs (X, the carrier of Y)),(FuncZero (X,Y)),(FuncAdd (X,Y)),(X,Y) #) is non empty strict CLSStruct
X is non empty set
Y is non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital CLSStruct
(X,Y) is non empty right_complementable Abelian add-associative right_zeroed constituted-Functions strict vector-distributive scalar-distributive scalar-associative scalar-unital CLSStruct
the carrier of Y is non empty set
Funcs (X, the carrier of Y) is non empty functional FUNCTION_DOMAIN of X, the carrier of Y
FuncZero (X,Y) is Relation-like X -defined the carrier of Y -valued Function-like total quasi_total Element of Funcs (X, the carrier of Y)
FuncAdd (X,Y) is non empty Relation-like [:(Funcs (X, the carrier of Y)),(Funcs (X, the carrier of Y)):] -defined Funcs (X, the carrier of Y) -valued Function-like total quasi_total Function-yielding V34() Element of bool [:[:(Funcs (X, the carrier of Y)),(Funcs (X, the carrier of Y)):],(Funcs (X, the carrier of Y)):]
[:(Funcs (X, the carrier of Y)),(Funcs (X, the carrier of Y)):] is non empty set
[:[:(Funcs (X, the carrier of Y)),(Funcs (X, the carrier of Y)):],(Funcs (X, the carrier of Y)):] is non empty set
bool [:[:(Funcs (X, the carrier of Y)),(Funcs (X, the carrier of Y)):],(Funcs (X, the carrier of Y)):] is non empty set
(X,Y) is non empty Relation-like [:COMPLEX,(Funcs (X, the carrier of Y)):] -defined Funcs (X, the carrier of Y) -valued Function-like total quasi_total Function-yielding V34() Element of bool [:[:COMPLEX,(Funcs (X, the carrier of Y)):],(Funcs (X, the carrier of Y)):]
[:COMPLEX,(Funcs (X, the carrier of Y)):] is non empty set
[:[:COMPLEX,(Funcs (X, the carrier of Y)):],(Funcs (X, the carrier of Y)):] is non empty set
bool [:[:COMPLEX,(Funcs (X, the carrier of Y)):],(Funcs (X, the carrier of Y)):] is non empty set
CLSStruct(# (Funcs (X, the carrier of Y)),(FuncZero (X,Y)),(FuncAdd (X,Y)),(X,Y) #) is non empty strict CLSStruct
the carrier of (X,Y) is non empty set
vseq is Relation-like Function-like Element of the carrier of (X,Y)
f is Element of X
vseq . f is set
tseq is Relation-like Function-like set
dom tseq is set
rng tseq is set
X is non empty set
Y is non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital CLSStruct
(X,Y) is non empty right_complementable Abelian add-associative right_zeroed constituted-Functions strict vector-distributive scalar-distributive scalar-associative scalar-unital CLSStruct
the carrier of Y is non empty set
Funcs (X, the carrier of Y) is non empty functional FUNCTION_DOMAIN of X, the carrier of Y
FuncZero (X,Y) is Relation-like X -defined the carrier of Y -valued Function-like total quasi_total Element of Funcs (X, the carrier of Y)
FuncAdd (X,Y) is non empty Relation-like [:(Funcs (X, the carrier of Y)),(Funcs (X, the carrier of Y)):] -defined Funcs (X, the carrier of Y) -valued Function-like total quasi_total Function-yielding V34() Element of bool [:[:(Funcs (X, the carrier of Y)),(Funcs (X, the carrier of Y)):],(Funcs (X, the carrier of Y)):]
[:(Funcs (X, the carrier of Y)),(Funcs (X, the carrier of Y)):] is non empty set
[:[:(Funcs (X, the carrier of Y)),(Funcs (X, the carrier of Y)):],(Funcs (X, the carrier of Y)):] is non empty set
bool [:[:(Funcs (X, the carrier of Y)),(Funcs (X, the carrier of Y)):],(Funcs (X, the carrier of Y)):] is non empty set
(X,Y) is non empty Relation-like [:COMPLEX,(Funcs (X, the carrier of Y)):] -defined Funcs (X, the carrier of Y) -valued Function-like total quasi_total Function-yielding V34() Element of bool [:[:COMPLEX,(Funcs (X, the carrier of Y)):],(Funcs (X, the carrier of Y)):]
[:COMPLEX,(Funcs (X, the carrier of Y)):] is non empty set
[:[:COMPLEX,(Funcs (X, the carrier of Y)):],(Funcs (X, the carrier of Y)):] is non empty set
bool [:[:COMPLEX,(Funcs (X, the carrier of Y)):],(Funcs (X, the carrier of Y)):] is non empty set
CLSStruct(# (Funcs (X, the carrier of Y)),(FuncZero (X,Y)),(FuncAdd (X,Y)),(X,Y) #) is non empty strict CLSStruct
the carrier of (X,Y) is non empty set
tseq is non empty set
tv is non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital CLSStruct
(tseq,tv) is non empty right_complementable Abelian add-associative right_zeroed constituted-Functions strict vector-distributive scalar-distributive scalar-associative scalar-unital CLSStruct
the carrier of tv is non empty set
Funcs (tseq, the carrier of tv) is non empty functional FUNCTION_DOMAIN of tseq, the carrier of tv
FuncZero (tseq,tv) is Relation-like tseq -defined the carrier of tv -valued Function-like total quasi_total Element of Funcs (tseq, the carrier of tv)
FuncAdd (tseq,tv) is non empty Relation-like [:(Funcs (tseq, the carrier of tv)),(Funcs (tseq, the carrier of tv)):] -defined Funcs (tseq, the carrier of tv) -valued Function-like total quasi_total Function-yielding V34() Element of bool [:[:(Funcs (tseq, the carrier of tv)),(Funcs (tseq, the carrier of tv)):],(Funcs (tseq, the carrier of tv)):]
[:(Funcs (tseq, the carrier of tv)),(Funcs (tseq, the carrier of tv)):] is non empty set
[:[:(Funcs (tseq, the carrier of tv)),(Funcs (tseq, the carrier of tv)):],(Funcs (tseq, the carrier of tv)):] is non empty set
bool [:[:(Funcs (tseq, the carrier of tv)),(Funcs (tseq, the carrier of tv)):],(Funcs (tseq, the carrier of tv)):] is non empty set
(tseq,tv) is non empty Relation-like [:COMPLEX,(Funcs (tseq, the carrier of tv)):] -defined Funcs (tseq, the carrier of tv) -valued Function-like total quasi_total Function-yielding V34() Element of bool [:[:COMPLEX,(Funcs (tseq, the carrier of tv)):],(Funcs (tseq, the carrier of tv)):]
[:COMPLEX,(Funcs (tseq, the carrier of tv)):] is non empty set
[:[:COMPLEX,(Funcs (tseq, the carrier of tv)):],(Funcs (tseq, the carrier of tv)):] is non empty set
bool [:[:COMPLEX,(Funcs (tseq, the carrier of tv)):],(Funcs (tseq, the carrier of tv)):] is non empty set
CLSStruct(# (Funcs (tseq, the carrier of tv)),(FuncZero (tseq,tv)),(FuncAdd (tseq,tv)),(tseq,tv) #) is non empty strict CLSStruct
the carrier of (tseq,tv) is non empty set
tseq is Relation-like Function-like Element of the carrier of (X,Y)
vseq is Relation-like Function-like Element of the carrier of (X,Y)
f is Relation-like Function-like Element of the carrier of (X,Y)
vseq + f is Relation-like Function-like Element of the carrier of (X,Y)
the addF of (X,Y) is non empty Relation-like [: the carrier of (X,Y), the carrier of (X,Y):] -defined the carrier of (X,Y) -valued Function-like total quasi_total Element of bool [:[: the carrier of (X,Y), the carrier of (X,Y):], the carrier of (X,Y):]
[: the carrier of (X,Y), the carrier of (X,Y):] is non empty set
[:[: the carrier of (X,Y), the carrier of (X,Y):], the carrier of (X,Y):] is non empty set
bool [:[: the carrier of (X,Y), the carrier of (X,Y):], the carrier of (X,Y):] is non empty set
the addF of (X,Y) . (vseq,f) is Relation-like Function-like Element of the carrier of (X,Y)
[vseq,f] is set
the addF of (X,Y) . [vseq,f] is set
tseq is Element of X
(X,Y,tseq,tseq) is Element of the carrier of Y
(X,Y,vseq,tseq) is Element of the carrier of Y
(X,Y,f,tseq) is Element of the carrier of Y
(X,Y,vseq,tseq) + (X,Y,f,tseq) is Element of the carrier of Y
the addF of Y is non empty Relation-like [: the carrier of Y, the carrier of Y:] -defined the carrier of Y -valued Function-like total quasi_total Element of bool [:[: the carrier of Y, the carrier of Y:], the carrier of Y:]
[: the carrier of Y, the carrier of Y:] is non empty set
[:[: the carrier of Y, the carrier of Y:], the carrier of Y:] is non empty set
bool [:[: the carrier of Y, the carrier of Y:], the carrier of Y:] is non empty set
the addF of Y . ((X,Y,vseq,tseq),(X,Y,f,tseq)) is Element of the carrier of Y
[(X,Y,vseq,tseq),(X,Y,f,tseq)] is set
the addF of Y . [(X,Y,vseq,tseq),(X,Y,f,tseq)] is set
n is Relation-like Function-like Element of the carrier of (tseq,tv)
e is Relation-like Function-like Element of the carrier of (tseq,tv)
m is Relation-like Function-like Element of the carrier of (tseq,tv)
e + m is Relation-like Function-like Element of the carrier of (tseq,tv)
the addF of (tseq,tv) is non empty Relation-like [: the carrier of (tseq,tv), the carrier of (tseq,tv):] -defined the carrier of (tseq,tv) -valued Function-like total quasi_total Element of bool [:[: the carrier of (tseq,tv), the carrier of (tseq,tv):], the carrier of (tseq,tv):]
[: the carrier of (tseq,tv), the carrier of (tseq,tv):] is non empty set
[:[: the carrier of (tseq,tv), the carrier of (tseq,tv):], the carrier of (tseq,tv):] is non empty set
bool [:[: the carrier of (tseq,tv), the carrier of (tseq,tv):], the carrier of (tseq,tv):] is non empty set
the addF of (tseq,tv) . (e,m) is Relation-like Function-like Element of the carrier of (tseq,tv)
[e,m] is set
the addF of (tseq,tv) . [e,m] is set
X is non empty set
Y is non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital CLSStruct
(X,Y) is non empty right_complementable Abelian add-associative right_zeroed constituted-Functions strict vector-distributive scalar-distributive scalar-associative scalar-unital CLSStruct
the carrier of Y is non empty set
Funcs (X, the carrier of Y) is non empty functional FUNCTION_DOMAIN of X, the carrier of Y
FuncZero (X,Y) is Relation-like X -defined the carrier of Y -valued Function-like total quasi_total Element of Funcs (X, the carrier of Y)
FuncAdd (X,Y) is non empty Relation-like [:(Funcs (X, the carrier of Y)),(Funcs (X, the carrier of Y)):] -defined Funcs (X, the carrier of Y) -valued Function-like total quasi_total Function-yielding V34() Element of bool [:[:(Funcs (X, the carrier of Y)),(Funcs (X, the carrier of Y)):],(Funcs (X, the carrier of Y)):]
[:(Funcs (X, the carrier of Y)),(Funcs (X, the carrier of Y)):] is non empty set
[:[:(Funcs (X, the carrier of Y)),(Funcs (X, the carrier of Y)):],(Funcs (X, the carrier of Y)):] is non empty set
bool [:[:(Funcs (X, the carrier of Y)),(Funcs (X, the carrier of Y)):],(Funcs (X, the carrier of Y)):] is non empty set
(X,Y) is non empty Relation-like [:COMPLEX,(Funcs (X, the carrier of Y)):] -defined Funcs (X, the carrier of Y) -valued Function-like total quasi_total Function-yielding V34() Element of bool [:[:COMPLEX,(Funcs (X, the carrier of Y)):],(Funcs (X, the carrier of Y)):]
[:COMPLEX,(Funcs (X, the carrier of Y)):] is non empty set
[:[:COMPLEX,(Funcs (X, the carrier of Y)):],(Funcs (X, the carrier of Y)):] is non empty set
bool [:[:COMPLEX,(Funcs (X, the carrier of Y)):],(Funcs (X, the carrier of Y)):] is non empty set
CLSStruct(# (Funcs (X, the carrier of Y)),(FuncZero (X,Y)),(FuncAdd (X,Y)),(X,Y) #) is non empty strict CLSStruct
the carrier of (X,Y) is non empty set
f is Relation-like Function-like Element of the carrier of (X,Y)
vseq is Relation-like Function-like Element of the carrier of (X,Y)
tseq is complex set
tseq * vseq is Relation-like Function-like Element of the carrier of (X,Y)
tseq is Relation-like X -defined the carrier of Y -valued Function-like total quasi_total Element of Funcs (X, the carrier of Y)
[tseq,tseq] is set
(X,Y) . [tseq,tseq] is Relation-like Function-like set
tv is Element of X
(X,Y,f,tv) is Element of the carrier of Y
(X,Y,vseq,tv) is Element of the carrier of Y
tseq * (X,Y,vseq,tv) is Element of the carrier of Y
tseq is Relation-like X -defined the carrier of Y -valued Function-like total quasi_total Element of Funcs (X, the carrier of Y)
X is non empty CLSStruct
the carrier of X is non empty set
Y is non empty CLSStruct
the carrier of Y is non empty set
[: the carrier of X, the carrier of Y:] is non empty set
bool [: the carrier of X, the carrier of Y:] is non empty set
X is non empty CLSStruct
the carrier of X is non empty set
Y is non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital CLSStruct
the carrier of Y is non empty set
[: the carrier of X, the carrier of Y:] is non empty set
bool [: the carrier of X, the carrier of Y:] is non empty set
0. Y is zero Element of the carrier of Y
the ZeroF of Y is Element of the carrier of Y
the carrier of X --> (0. Y) is non empty Relation-like the carrier of X -defined the carrier of Y -valued Function-like constant total quasi_total Element of bool [: the carrier of X, the carrier of Y:]
f is non empty Relation-like the carrier of X -defined the carrier of Y -valued Function-like total quasi_total Element of bool [: the carrier of X, the carrier of Y:]
tseq is Element of the carrier of X
tseq is Element of the carrier of X
tseq + tseq is Element of the carrier of X
the addF of X is non empty Relation-like [: the carrier of X, the carrier of X:] -defined the carrier of X -valued Function-like total quasi_total Element of bool [:[: the carrier of X, the carrier of X:], the carrier of X:]
[: the carrier of X, the carrier of X:] is non empty set
[:[: the carrier of X, the carrier of X:], the carrier of X:] is non empty set
bool [:[: the carrier of X, the carrier of X:], the carrier of X:] is non empty set
the addF of X . (tseq,tseq) is Element of the carrier of X
[tseq,tseq] is set
the addF of X . [tseq,tseq] is set
f . (tseq + tseq) is Element of the carrier of Y
(0. Y) + (0. Y) is Element of the carrier of Y
the addF of Y is non empty Relation-like [: the carrier of Y, the carrier of Y:] -defined the carrier of Y -valued Function-like total quasi_total Element of bool [:[: the carrier of Y, the carrier of Y:], the carrier of Y:]
[: the carrier of Y, the carrier of Y:] is non empty set
[:[: the carrier of Y, the carrier of Y:], the carrier of Y:] is non empty set
bool [:[: the carrier of Y, the carrier of Y:], the carrier of Y:] is non empty set
the addF of Y . ((0. Y),(0. Y)) is Element of the carrier of Y
[(0. Y),(0. Y)] is set
the addF of Y . [(0. Y),(0. Y)] is set
f . tseq is Element of the carrier of Y
(f . tseq) + (0. Y) is Element of the carrier of Y
the addF of Y . ((f . tseq),(0. Y)) is Element of the carrier of Y
[(f . tseq),(0. Y)] is set
the addF of Y . [(f . tseq),(0. Y)] is set
f . tseq is Element of the carrier of Y
(f . tseq) + (f . tseq) is Element of the carrier of Y
the addF of Y . ((f . tseq),(f . tseq)) is Element of the carrier of Y
[(f . tseq),(f . tseq)] is set
the addF of Y . [(f . tseq),(f . tseq)] is set
tseq is Element of the carrier of X
tseq is complex set
tseq * tseq is Element of the carrier of X
f . (tseq * tseq) is Element of the carrier of Y
tseq * (0. Y) is Element of the carrier of Y
f . tseq is Element of the carrier of Y
tseq * (f . tseq) is Element of the carrier of Y
X is non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital CLSStruct
the carrier of X is non empty set
Y is non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital CLSStruct
( the carrier of X,Y) is non empty right_complementable Abelian add-associative right_zeroed constituted-Functions strict vector-distributive scalar-distributive scalar-associative scalar-unital CLSStruct
the carrier of Y is non empty set
Funcs ( the carrier of X, the carrier of Y) is non empty functional FUNCTION_DOMAIN of the carrier of X, the carrier of Y
FuncZero ( the carrier of X,Y) is Relation-like the carrier of X -defined the carrier of Y -valued Function-like total quasi_total Element of Funcs ( the carrier of X, the carrier of Y)
FuncAdd ( the carrier of X,Y) is non empty Relation-like [:(Funcs ( the carrier of X, the carrier of Y)),(Funcs ( the carrier of X, the carrier of Y)):] -defined Funcs ( the carrier of X, the carrier of Y) -valued Function-like total quasi_total Function-yielding V34() Element of bool [:[:(Funcs ( the carrier of X, the carrier of Y)),(Funcs ( the carrier of X, the carrier of Y)):],(Funcs ( the carrier of X, the carrier of Y)):]
[:(Funcs ( the carrier of X, the carrier of Y)),(Funcs ( the carrier of X, the carrier of Y)):] is non empty set
[:[:(Funcs ( the carrier of X, the carrier of Y)),(Funcs ( the carrier of X, the carrier of Y)):],(Funcs ( the carrier of X, the carrier of Y)):] is non empty set
bool [:[:(Funcs ( the carrier of X, the carrier of Y)),(Funcs ( the carrier of X, the carrier of Y)):],(Funcs ( the carrier of X, the carrier of Y)):] is non empty set
( the carrier of X,Y) is non empty Relation-like [:COMPLEX,(Funcs ( the carrier of X, the carrier of Y)):] -defined Funcs ( the carrier of X, the carrier of Y) -valued Function-like total quasi_total Function-yielding V34() Element of bool [:[:COMPLEX,(Funcs ( the carrier of X, the carrier of Y)):],(Funcs ( the carrier of X, the carrier of Y)):]
[:COMPLEX,(Funcs ( the carrier of X, the carrier of Y)):] is non empty set
[:[:COMPLEX,(Funcs ( the carrier of X, the carrier of Y)):],(Funcs ( the carrier of X, the carrier of Y)):] is non empty set
bool [:[:COMPLEX,(Funcs ( the carrier of X, the carrier of Y)):],(Funcs ( the carrier of X, the carrier of Y)):] is non empty set
CLSStruct(# (Funcs ( the carrier of X, the carrier of Y)),(FuncZero ( the carrier of X,Y)),(FuncAdd ( the carrier of X,Y)),( the carrier of X,Y) #) is non empty strict CLSStruct
the carrier of ( the carrier of X,Y) is non empty set
bool the carrier of ( the carrier of X,Y) is non empty set
[: the carrier of X, the carrier of Y:] is non empty set
bool [: the carrier of X, the carrier of Y:] is non empty set
vseq is set
f is set
f is set
vseq is Element of bool the carrier of ( the carrier of X,Y)
f is Element of bool the carrier of ( the carrier of X,Y)
tseq is set
tseq is set
X is non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital CLSStruct
Y is non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital CLSStruct
(X,Y) is Element of bool the carrier of ( the carrier of X,Y)
the carrier of X is non empty set
( the carrier of X,Y) is non empty right_complementable Abelian add-associative right_zeroed constituted-Functions strict vector-distributive scalar-distributive scalar-associative scalar-unital CLSStruct
the carrier of Y is non empty set
Funcs ( the carrier of X, the carrier of Y) is non empty functional FUNCTION_DOMAIN of the carrier of X, the carrier of Y
FuncZero ( the carrier of X,Y) is Relation-like the carrier of X -defined the carrier of Y -valued Function-like total quasi_total Element of Funcs ( the carrier of X, the carrier of Y)
FuncAdd ( the carrier of X,Y) is non empty Relation-like [:(Funcs ( the carrier of X, the carrier of Y)),(Funcs ( the carrier of X, the carrier of Y)):] -defined Funcs ( the carrier of X, the carrier of Y) -valued Function-like total quasi_total Function-yielding V34() Element of bool [:[:(Funcs ( the carrier of X, the carrier of Y)),(Funcs ( the carrier of X, the carrier of Y)):],(Funcs ( the carrier of X, the carrier of Y)):]
[:(Funcs ( the carrier of X, the carrier of Y)),(Funcs ( the carrier of X, the carrier of Y)):] is non empty set
[:[:(Funcs ( the carrier of X, the carrier of Y)),(Funcs ( the carrier of X, the carrier of Y)):],(Funcs ( the carrier of X, the carrier of Y)):] is non empty set
bool [:[:(Funcs ( the carrier of X, the carrier of Y)),(Funcs ( the carrier of X, the carrier of Y)):],(Funcs ( the carrier of X, the carrier of Y)):] is non empty set
( the carrier of X,Y) is non empty Relation-like [:COMPLEX,(Funcs ( the carrier of X, the carrier of Y)):] -defined Funcs ( the carrier of X, the carrier of Y) -valued Function-like total quasi_total Function-yielding V34() Element of bool [:[:COMPLEX,(Funcs ( the carrier of X, the carrier of Y)):],(Funcs ( the carrier of X, the carrier of Y)):]
[:COMPLEX,(Funcs ( the carrier of X, the carrier of Y)):] is non empty set
[:[:COMPLEX,(Funcs ( the carrier of X, the carrier of Y)):],(Funcs ( the carrier of X, the carrier of Y)):] is non empty set
bool [:[:COMPLEX,(Funcs ( the carrier of X, the carrier of Y)):],(Funcs ( the carrier of X, the carrier of Y)):] is non empty set
CLSStruct(# (Funcs ( the carrier of X, the carrier of Y)),(FuncZero ( the carrier of X,Y)),(FuncAdd ( the carrier of X,Y)),( the carrier of X,Y) #) is non empty strict CLSStruct
the carrier of ( the carrier of X,Y) is non empty set
bool the carrier of ( the carrier of X,Y) is non empty set
0. Y is zero Element of the carrier of Y
the ZeroF of Y is Element of the carrier of Y
the carrier of X --> (0. Y) is non empty Relation-like the carrier of X -defined the carrier of Y -valued Function-like constant total quasi_total Element of bool [: the carrier of X, the carrier of Y:]
[: the carrier of X, the carrier of Y:] is non empty set
bool [: the carrier of X, the carrier of Y:] is non empty set
f is Element of the carrier of X
( the carrier of X --> (0. Y)) . f is Element of the carrier of Y
tseq is complex set
tseq * f is Element of the carrier of X
( the carrier of X --> (0. Y)) . (tseq * f) is Element of the carrier of Y
tseq * (( the carrier of X --> (0. Y)) . f) is Element of the carrier of Y
tseq * (0. Y) is Element of the carrier of Y
f is Element of the carrier of X
tseq is Element of the carrier of X
f + tseq is Element of the carrier of X
the addF of X is non empty Relation-like [: the carrier of X, the carrier of X:] -defined the carrier of X -valued Function-like total quasi_total Element of bool [:[: the carrier of X, the carrier of X:], the carrier of X:]
[: the carrier of X, the carrier of X:] is non empty set
[:[: the carrier of X, the carrier of X:], the carrier of X:] is non empty set
bool [:[: the carrier of X, the carrier of X:], the carrier of X:] is non empty set
the addF of X . (f,tseq) is Element of the carrier of X
[f,tseq] is set
the addF of X . [f,tseq] is set
( the carrier of X --> (0. Y)) . (f + tseq) is Element of the carrier of Y
( the carrier of X --> (0. Y)) . f is Element of the carrier of Y
( the carrier of X --> (0. Y)) . tseq is Element of the carrier of Y
(( the carrier of X --> (0. Y)) . f) + (( the carrier of X --> (0. Y)) . tseq) is Element of the carrier of Y
the addF of Y is non empty Relation-like [: the carrier of Y, the carrier of Y:] -defined the carrier of Y -valued Function-like total quasi_total Element of bool [:[: the carrier of Y, the carrier of Y:], the carrier of Y:]
[: the carrier of Y, the carrier of Y:] is non empty set
[:[: the carrier of Y, the carrier of Y:], the carrier of Y:] is non empty set
bool [:[: the carrier of Y, the carrier of Y:], the carrier of Y:] is non empty set
the addF of Y . ((( the carrier of X --> (0. Y)) . f),(( the carrier of X --> (0. Y)) . tseq)) is Element of the carrier of Y
[(( the carrier of X --> (0. Y)) . f),(( the carrier of X --> (0. Y)) . tseq)] is set
the addF of Y . [(( the carrier of X --> (0. Y)) . f),(( the carrier of X --> (0. Y)) . tseq)] is set
f + tseq is Element of the carrier of X
( the carrier of X --> (0. Y)) . (f + tseq) is Element of the carrier of Y
(0. Y) + (0. Y) is Element of the carrier of Y
the addF of Y . ((0. Y),(0. Y)) is Element of the carrier of Y
[(0. Y),(0. Y)] is set
the addF of Y . [(0. Y),(0. Y)] is set
(( the carrier of X --> (0. Y)) . f) + (0. Y) is Element of the carrier of Y
the addF of Y . ((( the carrier of X --> (0. Y)) . f),(0. Y)) is Element of the carrier of Y
[(( the carrier of X --> (0. Y)) . f),(0. Y)] is set
the addF of Y . [(( the carrier of X --> (0. Y)) . f),(0. Y)] is set
(( the carrier of X --> (0. Y)) . f) + (( the carrier of X --> (0. Y)) . tseq) is Element of the carrier of Y
f is set
X is non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital CLSStruct
the carrier of X is non empty set
Y is non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital CLSStruct
( the carrier of X,Y) is non empty right_complementable Abelian add-associative right_zeroed constituted-Functions strict vector-distributive scalar-distributive scalar-associative scalar-unital CLSStruct
the carrier of Y is non empty set
Funcs ( the carrier of X, the carrier of Y) is non empty functional FUNCTION_DOMAIN of the carrier of X, the carrier of Y
FuncZero ( the carrier of X,Y) is Relation-like the carrier of X -defined the carrier of Y -valued Function-like total quasi_total Element of Funcs ( the carrier of X, the carrier of Y)
FuncAdd ( the carrier of X,Y) is non empty Relation-like [:(Funcs ( the carrier of X, the carrier of Y)),(Funcs ( the carrier of X, the carrier of Y)):] -defined Funcs ( the carrier of X, the carrier of Y) -valued Function-like total quasi_total Function-yielding V34() Element of bool [:[:(Funcs ( the carrier of X, the carrier of Y)),(Funcs ( the carrier of X, the carrier of Y)):],(Funcs ( the carrier of X, the carrier of Y)):]
[:(Funcs ( the carrier of X, the carrier of Y)),(Funcs ( the carrier of X, the carrier of Y)):] is non empty set
[:[:(Funcs ( the carrier of X, the carrier of Y)),(Funcs ( the carrier of X, the carrier of Y)):],(Funcs ( the carrier of X, the carrier of Y)):] is non empty set
bool [:[:(Funcs ( the carrier of X, the carrier of Y)),(Funcs ( the carrier of X, the carrier of Y)):],(Funcs ( the carrier of X, the carrier of Y)):] is non empty set
( the carrier of X,Y) is non empty Relation-like [:COMPLEX,(Funcs ( the carrier of X, the carrier of Y)):] -defined Funcs ( the carrier of X, the carrier of Y) -valued Function-like total quasi_total Function-yielding V34() Element of bool [:[:COMPLEX,(Funcs ( the carrier of X, the carrier of Y)):],(Funcs ( the carrier of X, the carrier of Y)):]
[:COMPLEX,(Funcs ( the carrier of X, the carrier of Y)):] is non empty set
[:[:COMPLEX,(Funcs ( the carrier of X, the carrier of Y)):],(Funcs ( the carrier of X, the carrier of Y)):] is non empty set
bool [:[:COMPLEX,(Funcs ( the carrier of X, the carrier of Y)):],(Funcs ( the carrier of X, the carrier of Y)):] is non empty set
CLSStruct(# (Funcs ( the carrier of X, the carrier of Y)),(FuncZero ( the carrier of X,Y)),(FuncAdd ( the carrier of X,Y)),( the carrier of X,Y) #) is non empty strict CLSStruct
(X,Y) is non empty functional Element of bool the carrier of ( the carrier of X,Y)
the carrier of ( the carrier of X,Y) is non empty set
bool the carrier of ( the carrier of X,Y) is non empty set
f is complex set
tseq is Relation-like Function-like Element of the carrier of ( the carrier of X,Y)
f * tseq is Relation-like Function-like Element of the carrier of ( the carrier of X,Y)
[: the carrier of X, the carrier of Y:] is non empty set
bool [: the carrier of X, the carrier of Y:] is non empty set
tseq is non empty Relation-like the carrier of X -defined the carrier of Y -valued Function-like total quasi_total Element of bool [: the carrier of X, the carrier of Y:]
tv is Element of the carrier of X
tseq . tv is Element of the carrier of Y
e is complex set
e * tv is Element of the carrier of X
tseq . (e * tv) is Element of the carrier of Y
e * (tseq . tv) is Element of the carrier of Y
tseq is Relation-like the carrier of X -defined the carrier of Y -valued Function-like total quasi_total Element of Funcs ( the carrier of X, the carrier of Y)
[f,tseq] is set
( the carrier of X,Y) . [f,tseq] is Relation-like Function-like set
tseq . (e * tv) is Element of the carrier of Y
f * (tseq . (e * tv)) is Element of the carrier of Y
tseq . tv is Element of the carrier of Y
e * (tseq . tv) is Element of the carrier of Y
f * (e * (tseq . tv)) is Element of the carrier of Y
f * e is complex set
(f * e) * (tseq . tv) is Element of the carrier of Y
f * (tseq . tv) is Element of the carrier of Y
e * (f * (tseq . tv)) is Element of the carrier of Y
tv is Element of the carrier of X
e is Element of the carrier of X
tv + e is Element of the carrier of X
the addF of X is non empty Relation-like [: the carrier of X, the carrier of X:] -defined the carrier of X -valued Function-like total quasi_total Element of bool [:[: the carrier of X, the carrier of X:], the carrier of X:]
[: the carrier of X, the carrier of X:] is non empty set
[:[: the carrier of X, the carrier of X:], the carrier of X:] is non empty set
bool [:[: the carrier of X, the carrier of X:], the carrier of X:] is non empty set
the addF of X . (tv,e) is Element of the carrier of X
[tv,e] is set
the addF of X . [tv,e] is set
tseq . (tv + e) is Element of the carrier of Y
tseq . tv is Element of the carrier of Y
tseq . e is Element of the carrier of Y
(tseq . tv) + (tseq . e) is Element of the carrier of Y
the addF of Y is non empty Relation-like [: the carrier of Y, the carrier of Y:] -defined the carrier of Y -valued Function-like total quasi_total Element of bool [:[: the carrier of Y, the carrier of Y:], the carrier of Y:]
[: the carrier of Y, the carrier of Y:] is non empty set
[:[: the carrier of Y, the carrier of Y:], the carrier of Y:] is non empty set
bool [:[: the carrier of Y, the carrier of Y:], the carrier of Y:] is non empty set
the addF of Y . ((tseq . tv),(tseq . e)) is Element of the carrier of Y
[(tseq . tv),(tseq . e)] is set
the addF of Y . [(tseq . tv),(tseq . e)] is set
tseq is Relation-like the carrier of X -defined the carrier of Y -valued Function-like total quasi_total Element of Funcs ( the carrier of X, the carrier of Y)
[f,tseq] is set
( the carrier of X,Y) . [f,tseq] is Relation-like Function-like set
tv + e is Element of the carrier of X
tseq . (tv + e) is Element of the carrier of Y
tseq . (tv + e) is Element of the carrier of Y
f * (tseq . (tv + e)) is Element of the carrier of Y
tseq . tv is Element of the carrier of Y
tseq . e is Element of the carrier of Y
(tseq . tv) + (tseq . e) is Element of the carrier of Y
the addF of Y . ((tseq . tv),(tseq . e)) is Element of the carrier of Y
[(tseq . tv),(tseq . e)] is set
the addF of Y . [(tseq . tv),(tseq . e)] is set
f * ((tseq . tv) + (tseq . e)) is Element of the carrier of Y
f * (tseq . tv) is Element of the carrier of Y
f * (tseq . e) is Element of the carrier of Y
(f * (tseq . tv)) + (f * (tseq . e)) is Element of the carrier of Y
the addF of Y . ((f * (tseq . tv)),(f * (tseq . e))) is Element of the carrier of Y
[(f * (tseq . tv)),(f * (tseq . e))] is set
the addF of Y . [(f * (tseq . tv)),(f * (tseq . e))] is set
(tseq . tv) + (f * (tseq . e)) is Element of the carrier of Y
the addF of Y . ((tseq . tv),(f * (tseq . e))) is Element of the carrier of Y
[(tseq . tv),(f * (tseq . e))] is set
the addF of Y . [(tseq . tv),(f * (tseq . e))] is set
(tseq . tv) + (tseq . e) is Element of the carrier of Y
f is Relation-like Function-like Element of the carrier of ( the carrier of X,Y)
tseq is Relation-like Function-like Element of the carrier of ( the carrier of X,Y)
f + tseq is Relation-like Function-like Element of the carrier of ( the carrier of X,Y)
the addF of ( the carrier of X,Y) is non empty Relation-like [: the carrier of ( the carrier of X,Y), the carrier of ( the carrier of X,Y):] -defined the carrier of ( the carrier of X,Y) -valued Function-like total quasi_total Element of bool [:[: the carrier of ( the carrier of X,Y), the carrier of ( the carrier of X,Y):], the carrier of ( the carrier of X,Y):]
[: the carrier of ( the carrier of X,Y), the carrier of ( the carrier of X,Y):] is non empty set
[:[: the carrier of ( the carrier of X,Y), the carrier of ( the carrier of X,Y):], the carrier of ( the carrier of X,Y):] is non empty set
bool [:[: the carrier of ( the carrier of X,Y), the carrier of ( the carrier of X,Y):], the carrier of ( the carrier of X,Y):] is non empty set
the addF of ( the carrier of X,Y) . (f,tseq) is Relation-like Function-like Element of the carrier of ( the carrier of X,Y)
[f,tseq] is set
the addF of ( the carrier of X,Y) . [f,tseq] is set
[: the carrier of X, the carrier of Y:] is non empty set
bool [: the carrier of X, the carrier of Y:] is non empty set
tseq is non empty Relation-like the carrier of X -defined the carrier of Y -valued Function-like total quasi_total Element of bool [: the carrier of X, the carrier of Y:]
e is Element of the carrier of X
tseq . e is Element of the carrier of Y
m is complex set
m * e is Element of the carrier of X
tseq . (m * e) is Element of the carrier of Y
m * (tseq . e) is Element of the carrier of Y
tv is Relation-like the carrier of X -defined the carrier of Y -valued Function-like total quasi_total Element of Funcs ( the carrier of X, the carrier of Y)
tseq is Relation-like the carrier of X -defined the carrier of Y -valued Function-like total quasi_total Element of Funcs ( the carrier of X, the carrier of Y)
tv . (m * e) is Element of the carrier of Y
tseq . (m * e) is Element of the carrier of Y
(tv . (m * e)) + (tseq . (m * e)) is Element of the carrier of Y
the addF of Y is non empty Relation-like [: the carrier of Y, the carrier of Y:] -defined the carrier of Y -valued Function-like total quasi_total Element of bool [:[: the carrier of Y, the carrier of Y:], the carrier of Y:]
[: the carrier of Y, the carrier of Y:] is non empty set
[:[: the carrier of Y, the carrier of Y:], the carrier of Y:] is non empty set
bool [:[: the carrier of Y, the carrier of Y:], the carrier of Y:] is non empty set
the addF of Y . ((tv . (m * e)),(tseq . (m * e))) is Element of the carrier of Y
[(tv . (m * e)),(tseq . (m * e))] is set
the addF of Y . [(tv . (m * e)),(tseq . (m * e))] is set
tv . e is Element of the carrier of Y
m * (tv . e) is Element of the carrier of Y
(m * (tv . e)) + (tseq . (m * e)) is Element of the carrier of Y
the addF of Y . ((m * (tv . e)),(tseq . (m * e))) is Element of the carrier of Y
[(m * (tv . e)),(tseq . (m * e))] is set
the addF of Y . [(m * (tv . e)),(tseq . (m * e))] is set
tseq . e is Element of the carrier of Y
m * (tseq . e) is Element of the carrier of Y
(m * (tv . e)) + (m * (tseq . e)) is Element of the carrier of Y
the addF of Y . ((m * (tv . e)),(m * (tseq . e))) is Element of the carrier of Y
[(m * (tv . e)),(m * (tseq . e))] is set
the addF of Y . [(m * (tv . e)),(m * (tseq . e))] is set
(tv . e) + (tseq . e) is Element of the carrier of Y
the addF of Y . ((tv . e),(tseq . e)) is Element of the carrier of Y
[(tv . e),(tseq . e)] is set
the addF of Y . [(tv . e),(tseq . e)] is set
m * ((tv . e) + (tseq . e)) is Element of the carrier of Y
e is Element of the carrier of X
m is Element of the carrier of X
e + m is Element of the carrier of X
the addF of X is non empty Relation-like [: the carrier of X, the carrier of X:] -defined the carrier of X -valued Function-like total quasi_total Element of bool [:[: the carrier of X, the carrier of X:], the carrier of X:]
[: the carrier of X, the carrier of X:] is non empty set
[:[: the carrier of X, the carrier of X:], the carrier of X:] is non empty set
bool [:[: the carrier of X, the carrier of X:], the carrier of X:] is non empty set
the addF of X . (e,m) is Element of the carrier of X
[e,m] is set
the addF of X . [e,m] is set
tseq . (e + m) is Element of the carrier of Y
tseq . e is Element of the carrier of Y
tseq . m is Element of the carrier of Y
(tseq . e) + (tseq . m) is Element of the carrier of Y
the addF of Y is non empty Relation-like [: the carrier of Y, the carrier of Y:] -defined the carrier of Y -valued Function-like total quasi_total Element of bool [:[: the carrier of Y, the carrier of Y:], the carrier of Y:]
[: the carrier of Y, the carrier of Y:] is non empty set
[:[: the carrier of Y, the carrier of Y:], the carrier of Y:] is non empty set
bool [:[: the carrier of Y, the carrier of Y:], the carrier of Y:] is non empty set
the addF of Y . ((tseq . e),(tseq . m)) is Element of the carrier of Y
[(tseq . e),(tseq . m)] is set
the addF of Y . [(tseq . e),(tseq . m)] is set
tv is Relation-like the carrier of X -defined the carrier of Y -valued Function-like total quasi_total Element of Funcs ( the carrier of X, the carrier of Y)
tseq is Relation-like the carrier of X -defined the carrier of Y -valued Function-like total quasi_total Element of Funcs ( the carrier of X, the carrier of Y)
e + m is Element of the carrier of X
tseq . (e + m) is Element of the carrier of Y
tv . (e + m) is Element of the carrier of Y
tseq . (e + m) is Element of the carrier of Y
(tv . (e + m)) + (tseq . (e + m)) is Element of the carrier of Y
the addF of Y . ((tv . (e + m)),(tseq . (e + m))) is Element of the carrier of Y
[(tv . (e + m)),(tseq . (e + m))] is set
the addF of Y . [(tv . (e + m)),(tseq . (e + m))] is set
tv . e is Element of the carrier of Y
tv . m is Element of the carrier of Y
(tv . e) + (tv . m) is Element of the carrier of Y
the addF of Y . ((tv . e),(tv . m)) is Element of the carrier of Y
[(tv . e),(tv . m)] is set
the addF of Y . [(tv . e),(tv . m)] is set
((tv . e) + (tv . m)) + (tseq . (e + m)) is Element of the carrier of Y
the addF of Y . (((tv . e) + (tv . m)),(tseq . (e + m))) is Element of the carrier of Y
[((tv . e) + (tv . m)),(tseq . (e + m))] is set
the addF of Y . [((tv . e) + (tv . m)),(tseq . (e + m))] is set
tseq . e is Element of the carrier of Y
tseq . m is Element of the carrier of Y
(tseq . e) + (tseq . m) is Element of the carrier of Y
the addF of Y . ((tseq . e),(tseq . m)) is Element of the carrier of Y
[(tseq . e),(tseq . m)] is set
the addF of Y . [(tseq . e),(tseq . m)] is set
((tv . e) + (tv . m)) + ((tseq . e) + (tseq . m)) is Element of the carrier of Y
the addF of Y . (((tv . e) + (tv . m)),((tseq . e) + (tseq . m))) is Element of the carrier of Y
[((tv . e) + (tv . m)),((tseq . e) + (tseq . m))] is set
the addF of Y . [((tv . e) + (tv . m)),((tseq . e) + (tseq . m))] is set
((tv . e) + (tv . m)) + (tseq . e) is Element of the carrier of Y
the addF of Y . (((tv . e) + (tv . m)),(tseq . e)) is Element of the carrier of Y
[((tv . e) + (tv . m)),(tseq . e)] is set
the addF of Y . [((tv . e) + (tv . m)),(tseq . e)] is set
(((tv . e) + (tv . m)) + (tseq . e)) + (tseq . m) is Element of the carrier of Y
the addF of Y . ((((tv . e) + (tv . m)) + (tseq . e)),(tseq . m)) is Element of the carrier of Y
[(((tv . e) + (tv . m)) + (tseq . e)),(tseq . m)] is set
the addF of Y . [(((tv . e) + (tv . m)) + (tseq . e)),(tseq . m)] is set
(tv . e) + (tseq . e) is Element of the carrier of Y
the addF of Y . ((tv . e),(tseq . e)) is Element of the carrier of Y
[(tv . e),(tseq . e)] is set
the addF of Y . [(tv . e),(tseq . e)] is set
((tv . e) + (tseq . e)) + (tv . m) is Element of the carrier of Y
the addF of Y . (((tv . e) + (tseq . e)),(tv . m)) is Element of the carrier of Y
[((tv . e) + (tseq . e)),(tv . m)] is set
the addF of Y . [((tv . e) + (tseq . e)),(tv . m)] is set
(((tv . e) + (tseq . e)) + (tv . m)) + (tseq . m) is Element of the carrier of Y
the addF of Y . ((((tv . e) + (tseq . e)) + (tv . m)),(tseq . m)) is Element of the carrier of Y
[(((tv . e) + (tseq . e)) + (tv . m)),(tseq . m)] is set
the addF of Y . [(((tv . e) + (tseq . e)) + (tv . m)),(tseq . m)] is set
(tseq . e) + (tv . m) is Element of the carrier of Y
the addF of Y . ((tseq . e),(tv . m)) is Element of the carrier of Y
[(tseq . e),(tv . m)] is set
the addF of Y . [(tseq . e),(tv . m)] is set
((tseq . e) + (tv . m)) + (tseq . m) is Element of the carrier of Y
the addF of Y . (((tseq . e) + (tv . m)),(tseq . m)) is Element of the carrier of Y
[((tseq . e) + (tv . m)),(tseq . m)] is set
the addF of Y . [((tseq . e) + (tv . m)),(tseq . m)] is set
(tv . m) + (tseq . m) is Element of the carrier of Y
the addF of Y . ((tv . m),(tseq . m)) is Element of the carrier of Y
[(tv . m),(tseq . m)] is set
the addF of Y . [(tv . m),(tseq . m)] is set
(tseq . e) + ((tv . m) + (tseq . m)) is Element of the carrier of Y
the addF of Y . ((tseq . e),((tv . m) + (tseq . m))) is Element of the carrier of Y
[(tseq . e),((tv . m) + (tseq . m))] is set
the addF of Y . [(tseq . e),((tv . m) + (tseq . m))] is set
(tseq . e) + (tseq . m) is Element of the carrier of Y
X is non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital CLSStruct
Y is non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital CLSStruct
(X,Y) is non empty functional Element of bool the carrier of ( the carrier of X,Y)
the carrier of X is non empty set
( the carrier of X,Y) is non empty right_complementable Abelian add-associative right_zeroed constituted-Functions strict vector-distributive scalar-distributive scalar-associative scalar-unital CLSStruct
the carrier of Y is non empty set
Funcs ( the carrier of X, the carrier of Y) is non empty functional FUNCTION_DOMAIN of the carrier of X, the carrier of Y
FuncZero ( the carrier of X,Y) is Relation-like the carrier of X -defined the carrier of Y -valued Function-like total quasi_total Element of Funcs ( the carrier of X, the carrier of Y)
FuncAdd ( the carrier of X,Y) is non empty Relation-like [:(Funcs ( the carrier of X, the carrier of Y)),(Funcs ( the carrier of X, the carrier of Y)):] -defined Funcs ( the carrier of X, the carrier of Y) -valued Function-like total quasi_total Function-yielding V34() Element of bool [:[:(Funcs ( the carrier of X, the carrier of Y)),(Funcs ( the carrier of X, the carrier of Y)):],(Funcs ( the carrier of X, the carrier of Y)):]
[:(Funcs ( the carrier of X, the carrier of Y)),(Funcs ( the carrier of X, the carrier of Y)):] is non empty set
[:[:(Funcs ( the carrier of X, the carrier of Y)),(Funcs ( the carrier of X, the carrier of Y)):],(Funcs ( the carrier of X, the carrier of Y)):] is non empty set
bool [:[:(Funcs ( the carrier of X, the carrier of Y)),(Funcs ( the carrier of X, the carrier of Y)):],(Funcs ( the carrier of X, the carrier of Y)):] is non empty set
( the carrier of X,Y) is non empty Relation-like [:COMPLEX,(Funcs ( the carrier of X, the carrier of Y)):] -defined Funcs ( the carrier of X, the carrier of Y) -valued Function-like total quasi_total Function-yielding V34() Element of bool [:[:COMPLEX,(Funcs ( the carrier of X, the carrier of Y)):],(Funcs ( the carrier of X, the carrier of Y)):]
[:COMPLEX,(Funcs ( the carrier of X, the carrier of Y)):] is non empty set
[:[:COMPLEX,(Funcs ( the carrier of X, the carrier of Y)):],(Funcs ( the carrier of X, the carrier of Y)):] is non empty set
bool [:[:COMPLEX,(Funcs ( the carrier of X, the carrier of Y)):],(Funcs ( the carrier of X, the carrier of Y)):] is non empty set
CLSStruct(# (Funcs ( the carrier of X, the carrier of Y)),(FuncZero ( the carrier of X,Y)),(FuncAdd ( the carrier of X,Y)),( the carrier of X,Y) #) is non empty strict CLSStruct
the carrier of ( the carrier of X,Y) is non empty set
bool the carrier of ( the carrier of X,Y) is non empty set
Zero_ ((X,Y),( the carrier of X,Y)) is Relation-like Function-like Element of (X,Y)
Add_ ((X,Y),( the carrier of X,Y)) is non empty Relation-like [:(X,Y),(X,Y):] -defined (X,Y) -valued Function-like total quasi_total Function-yielding V34() Element of bool [:[:(X,Y),(X,Y):],(X,Y):]
[:(X,Y),(X,Y):] is non empty set
[:[:(X,Y),(X,Y):],(X,Y):] is non empty set
bool [:[:(X,Y),(X,Y):],(X,Y):] is non empty set
Mult_ ((X,Y),( the carrier of X,Y)) is non empty Relation-like [:COMPLEX,(X,Y):] -defined (X,Y) -valued Function-like total quasi_total Function-yielding V34() Element of bool [:[:COMPLEX,(X,Y):],(X,Y):]
[:COMPLEX,(X,Y):] is non empty set
[:[:COMPLEX,(X,Y):],(X,Y):] is non empty set
bool [:[:COMPLEX,(X,Y):],(X,Y):] is non empty set
CLSStruct(# (X,Y),(Zero_ ((X,Y),( the carrier of X,Y))),(Add_ ((X,Y),( the carrier of X,Y))),(Mult_ ((X,Y),( the carrier of X,Y))) #) is non empty strict CLSStruct
X is non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital CLSStruct
Y is non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital CLSStruct
(X,Y) is non empty functional Element of bool the carrier of ( the carrier of X,Y)
the carrier of X is non empty set
( the carrier of X,Y) is non empty right_complementable Abelian add-associative right_zeroed constituted-Functions strict vector-distributive scalar-distributive scalar-associative scalar-unital CLSStruct
the carrier of Y is non empty set
Funcs ( the carrier of X, the carrier of Y) is non empty functional FUNCTION_DOMAIN of the carrier of X, the carrier of Y
FuncZero ( the carrier of X,Y) is Relation-like the carrier of X -defined the carrier of Y -valued Function-like total quasi_total Element of Funcs ( the carrier of X, the carrier of Y)
FuncAdd ( the carrier of X,Y) is non empty Relation-like [:(Funcs ( the carrier of X, the carrier of Y)),(Funcs ( the carrier of X, the carrier of Y)):] -defined Funcs ( the carrier of X, the carrier of Y) -valued Function-like total quasi_total Function-yielding V34() Element of bool [:[:(Funcs ( the carrier of X, the carrier of Y)),(Funcs ( the carrier of X, the carrier of Y)):],(Funcs ( the carrier of X, the carrier of Y)):]
[:(Funcs ( the carrier of X, the carrier of Y)),(Funcs ( the carrier of X, the carrier of Y)):] is non empty set
[:[:(Funcs ( the carrier of X, the carrier of Y)),(Funcs ( the carrier of X, the carrier of Y)):],(Funcs ( the carrier of X, the carrier of Y)):] is non empty set
bool [:[:(Funcs ( the carrier of X, the carrier of Y)),(Funcs ( the carrier of X, the carrier of Y)):],(Funcs ( the carrier of X, the carrier of Y)):] is non empty set
( the carrier of X,Y) is non empty Relation-like [:COMPLEX,(Funcs ( the carrier of X, the carrier of Y)):] -defined Funcs ( the carrier of X, the carrier of Y) -valued Function-like total quasi_total Function-yielding V34() Element of bool [:[:COMPLEX,(Funcs ( the carrier of X, the carrier of Y)):],(Funcs ( the carrier of X, the carrier of Y)):]
[:COMPLEX,(Funcs ( the carrier of X, the carrier of Y)):] is non empty set
[:[:COMPLEX,(Funcs ( the carrier of X, the carrier of Y)):],(Funcs ( the carrier of X, the carrier of Y)):] is non empty set
bool [:[:COMPLEX,(Funcs ( the carrier of X, the carrier of Y)):],(Funcs ( the carrier of X, the carrier of Y)):] is non empty set
CLSStruct(# (Funcs ( the carrier of X, the carrier of Y)),(FuncZero ( the carrier of X,Y)),(FuncAdd ( the carrier of X,Y)),( the carrier of X,Y) #) is non empty strict CLSStruct
the carrier of ( the carrier of X,Y) is non empty set
bool the carrier of ( the carrier of X,Y) is non empty set
Zero_ ((X,Y),( the carrier of X,Y)) is Relation-like Function-like Element of (X,Y)
Add_ ((X,Y),( the carrier of X,Y)) is non empty Relation-like [:(X,Y),(X,Y):] -defined (X,Y) -valued Function-like total quasi_total Function-yielding V34() Element of bool [:[:(X,Y),(X,Y):],(X,Y):]
[:(X,Y),(X,Y):] is non empty set
[:[:(X,Y),(X,Y):],(X,Y):] is non empty set
bool [:[:(X,Y),(X,Y):],(X,Y):] is non empty set
Mult_ ((X,Y),( the carrier of X,Y)) is non empty Relation-like [:COMPLEX,(X,Y):] -defined (X,Y) -valued Function-like total quasi_total Function-yielding V34() Element of bool [:[:COMPLEX,(X,Y):],(X,Y):]
[:COMPLEX,(X,Y):] is non empty set
[:[:COMPLEX,(X,Y):],(X,Y):] is non empty set
bool [:[:COMPLEX,(X,Y):],(X,Y):] is non empty set
CLSStruct(# (X,Y),(Zero_ ((X,Y),( the carrier of X,Y))),(Add_ ((X,Y),( the carrier of X,Y))),(Mult_ ((X,Y),( the carrier of X,Y))) #) is non empty strict CLSStruct
X is non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital CLSStruct
Y is non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital CLSStruct
(X,Y) is non empty functional Element of bool the carrier of ( the carrier of X,Y)
the carrier of X is non empty set
( the carrier of X,Y) is non empty right_complementable Abelian add-associative right_zeroed constituted-Functions strict vector-distributive scalar-distributive scalar-associative scalar-unital CLSStruct
the carrier of Y is non empty set
Funcs ( the carrier of X, the carrier of Y) is non empty functional FUNCTION_DOMAIN of the carrier of X, the carrier of Y
FuncZero ( the carrier of X,Y) is Relation-like the carrier of X -defined the carrier of Y -valued Function-like total quasi_total Element of Funcs ( the carrier of X, the carrier of Y)
FuncAdd ( the carrier of X,Y) is non empty Relation-like [:(Funcs ( the carrier of X, the carrier of Y)),(Funcs ( the carrier of X, the carrier of Y)):] -defined Funcs ( the carrier of X, the carrier of Y) -valued Function-like total quasi_total Function-yielding V34() Element of bool [:[:(Funcs ( the carrier of X, the carrier of Y)),(Funcs ( the carrier of X, the carrier of Y)):],(Funcs ( the carrier of X, the carrier of Y)):]
[:(Funcs ( the carrier of X, the carrier of Y)),(Funcs ( the carrier of X, the carrier of Y)):] is non empty set
[:[:(Funcs ( the carrier of X, the carrier of Y)),(Funcs ( the carrier of X, the carrier of Y)):],(Funcs ( the carrier of X, the carrier of Y)):] is non empty set
bool [:[:(Funcs ( the carrier of X, the carrier of Y)),(Funcs ( the carrier of X, the carrier of Y)):],(Funcs ( the carrier of X, the carrier of Y)):] is non empty set
( the carrier of X,Y) is non empty Relation-like [:COMPLEX,(Funcs ( the carrier of X, the carrier of Y)):] -defined Funcs ( the carrier of X, the carrier of Y) -valued Function-like total quasi_total Function-yielding V34() Element of bool [:[:COMPLEX,(Funcs ( the carrier of X, the carrier of Y)):],(Funcs ( the carrier of X, the carrier of Y)):]
[:COMPLEX,(Funcs ( the carrier of X, the carrier of Y)):] is non empty set
[:[:COMPLEX,(Funcs ( the carrier of X, the carrier of Y)):],(Funcs ( the carrier of X, the carrier of Y)):] is non empty set
bool [:[:COMPLEX,(Funcs ( the carrier of X, the carrier of Y)):],(Funcs ( the carrier of X, the carrier of Y)):] is non empty set
CLSStruct(# (Funcs ( the carrier of X, the carrier of Y)),(FuncZero ( the carrier of X,Y)),(FuncAdd ( the carrier of X,Y)),( the carrier of X,Y) #) is non empty strict CLSStruct
the carrier of ( the carrier of X,Y) is non empty set
bool the carrier of ( the carrier of X,Y) is non empty set
Zero_ ((X,Y),( the carrier of X,Y)) is Relation-like Function-like Element of (X,Y)
Add_ ((X,Y),( the carrier of X,Y)) is non empty Relation-like [:(X,Y),(X,Y):] -defined (X,Y) -valued Function-like total quasi_total Function-yielding V34() Element of bool [:[:(X,Y),(X,Y):],(X,Y):]
[:(X,Y),(X,Y):] is non empty set
[:[:(X,Y),(X,Y):],(X,Y):] is non empty set
bool [:[:(X,Y),(X,Y):],(X,Y):] is non empty set
Mult_ ((X,Y),( the carrier of X,Y)) is non empty Relation-like [:COMPLEX,(X,Y):] -defined (X,Y) -valued Function-like total quasi_total Function-yielding V34() Element of bool [:[:COMPLEX,(X,Y):],(X,Y):]
[:COMPLEX,(X,Y):] is non empty set
[:[:COMPLEX,(X,Y):],(X,Y):] is non empty set
bool [:[:COMPLEX,(X,Y):],(X,Y):] is non empty set
CLSStruct(# (X,Y),(Zero_ ((X,Y),( the carrier of X,Y))),(Add_ ((X,Y),( the carrier of X,Y))),(Mult_ ((X,Y),( the carrier of X,Y))) #) is non empty right_complementable Abelian add-associative right_zeroed strict vector-distributive scalar-distributive scalar-associative scalar-unital CLSStruct
X is non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital CLSStruct
Y is non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital CLSStruct
(X,Y) is non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital CLSStruct
(X,Y) is non empty functional Element of bool the carrier of ( the carrier of X,Y)
the carrier of X is non empty set
( the carrier of X,Y) is non empty right_complementable Abelian add-associative right_zeroed constituted-Functions strict vector-distributive scalar-distributive scalar-associative scalar-unital CLSStruct
the carrier of Y is non empty set
Funcs ( the carrier of X, the carrier of Y) is non empty functional FUNCTION_DOMAIN of the carrier of X, the carrier of Y
FuncZero ( the carrier of X,Y) is Relation-like the carrier of X -defined the carrier of Y -valued Function-like total quasi_total Element of Funcs ( the carrier of X, the carrier of Y)
FuncAdd ( the carrier of X,Y) is non empty Relation-like [:(Funcs ( the carrier of X, the carrier of Y)),(Funcs ( the carrier of X, the carrier of Y)):] -defined Funcs ( the carrier of X, the carrier of Y) -valued Function-like total quasi_total Function-yielding V34() Element of bool [:[:(Funcs ( the carrier of X, the carrier of Y)),(Funcs ( the carrier of X, the carrier of Y)):],(Funcs ( the carrier of X, the carrier of Y)):]
[:(Funcs ( the carrier of X, the carrier of Y)),(Funcs ( the carrier of X, the carrier of Y)):] is non empty set
[:[:(Funcs ( the carrier of X, the carrier of Y)),(Funcs ( the carrier of X, the carrier of Y)):],(Funcs ( the carrier of X, the carrier of Y)):] is non empty set
bool [:[:(Funcs ( the carrier of X, the carrier of Y)),(Funcs ( the carrier of X, the carrier of Y)):],(Funcs ( the carrier of X, the carrier of Y)):] is non empty set
( the carrier of X,Y) is non empty Relation-like [:COMPLEX,(Funcs ( the carrier of X, the carrier of Y)):] -defined Funcs ( the carrier of X, the carrier of Y) -valued Function-like total quasi_total Function-yielding V34() Element of bool [:[:COMPLEX,(Funcs ( the carrier of X, the carrier of Y)):],(Funcs ( the carrier of X, the carrier of Y)):]
[:COMPLEX,(Funcs ( the carrier of X, the carrier of Y)):] is non empty set
[:[:COMPLEX,(Funcs ( the carrier of X, the carrier of Y)):],(Funcs ( the carrier of X, the carrier of Y)):] is non empty set
bool [:[:COMPLEX,(Funcs ( the carrier of X, the carrier of Y)):],(Funcs ( the carrier of X, the carrier of Y)):] is non empty set
CLSStruct(# (Funcs ( the carrier of X, the carrier of Y)),(FuncZero ( the carrier of X,Y)),(FuncAdd ( the carrier of X,Y)),( the carrier of X,Y) #) is non empty strict CLSStruct
the carrier of ( the carrier of X,Y) is non empty set
bool the carrier of ( the carrier of X,Y) is non empty set
Zero_ ((X,Y),( the carrier of X,Y)) is Relation-like Function-like Element of (X,Y)
Add_ ((X,Y),( the carrier of X,Y)) is non empty Relation-like [:(X,Y),(X,Y):] -defined (X,Y) -valued Function-like total quasi_total Function-yielding V34() Element of bool [:[:(X,Y),(X,Y):],(X,Y):]
[:(X,Y),(X,Y):] is non empty set
[:[:(X,Y),(X,Y):],(X,Y):] is non empty set
bool [:[:(X,Y),(X,Y):],(X,Y):] is non empty set
Mult_ ((X,Y),( the carrier of X,Y)) is non empty Relation-like [:COMPLEX,(X,Y):] -defined (X,Y) -valued Function-like total quasi_total Function-yielding V34() Element of bool [:[:COMPLEX,(X,Y):],(X,Y):]
[:COMPLEX,(X,Y):] is non empty set
[:[:COMPLEX,(X,Y):],(X,Y):] is non empty set
bool [:[:COMPLEX,(X,Y):],(X,Y):] is non empty set
CLSStruct(# (X,Y),(Zero_ ((X,Y),( the carrier of X,Y))),(Add_ ((X,Y),( the carrier of X,Y))),(Mult_ ((X,Y),( the carrier of X,Y))) #) is non empty right_complementable Abelian add-associative right_zeroed strict vector-distributive scalar-distributive scalar-associative scalar-unital CLSStruct
X is non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital CLSStruct
Y is non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital CLSStruct
(X,Y) is non empty right_complementable Abelian add-associative right_zeroed strict vector-distributive scalar-distributive scalar-associative scalar-unital CLSStruct
(X,Y) is non empty functional Element of bool the carrier of ( the carrier of X,Y)
the carrier of X is non empty set
( the carrier of X,Y) is non empty right_complementable Abelian add-associative right_zeroed constituted-Functions strict vector-distributive scalar-distributive scalar-associative scalar-unital CLSStruct
the carrier of Y is non empty set
Funcs ( the carrier of X, the carrier of Y) is non empty functional FUNCTION_DOMAIN of the carrier of X, the carrier of Y
FuncZero ( the carrier of X,Y) is Relation-like the carrier of X -defined the carrier of Y -valued Function-like total quasi_total Element of Funcs ( the carrier of X, the carrier of Y)
FuncAdd ( the carrier of X,Y) is non empty Relation-like [:(Funcs ( the carrier of X, the carrier of Y)),(Funcs ( the carrier of X, the carrier of Y)):] -defined Funcs ( the carrier of X, the carrier of Y) -valued Function-like total quasi_total Function-yielding V34() Element of bool [:[:(Funcs ( the carrier of X, the carrier of Y)),(Funcs ( the carrier of X, the carrier of Y)):],(Funcs ( the carrier of X, the carrier of Y)):]
[:(Funcs ( the carrier of X, the carrier of Y)),(Funcs ( the carrier of X, the carrier of Y)):] is non empty set
[:[:(Funcs ( the carrier of X, the carrier of Y)),(Funcs ( the carrier of X, the carrier of Y)):],(Funcs ( the carrier of X, the carrier of Y)):] is non empty set
bool [:[:(Funcs ( the carrier of X, the carrier of Y)),(Funcs ( the carrier of X, the carrier of Y)):],(Funcs ( the carrier of X, the carrier of Y)):] is non empty set
( the carrier of X,Y) is non empty Relation-like [:COMPLEX,(Funcs ( the carrier of X, the carrier of Y)):] -defined Funcs ( the carrier of X, the carrier of Y) -valued Function-like total quasi_total Function-yielding V34() Element of bool [:[:COMPLEX,(Funcs ( the carrier of X, the carrier of Y)):],(Funcs ( the carrier of X, the carrier of Y)):]
[:COMPLEX,(Funcs ( the carrier of X, the carrier of Y)):] is non empty set
[:[:COMPLEX,(Funcs ( the carrier of X, the carrier of Y)):],(Funcs ( the carrier of X, the carrier of Y)):] is non empty set
bool [:[:COMPLEX,(Funcs ( the carrier of X, the carrier of Y)):],(Funcs ( the carrier of X, the carrier of Y)):] is non empty set
CLSStruct(# (Funcs ( the carrier of X, the carrier of Y)),(FuncZero ( the carrier of X,Y)),(FuncAdd ( the carrier of X,Y)),( the carrier of X,Y) #) is non empty strict CLSStruct
the carrier of ( the carrier of X,Y) is non empty set
bool the carrier of ( the carrier of X,Y) is non empty set
Zero_ ((X,Y),( the carrier of X,Y)) is Relation-like Function-like Element of (X,Y)
Add_ ((X,Y),( the carrier of X,Y)) is non empty Relation-like [:(X,Y),(X,Y):] -defined (X,Y) -valued Function-like total quasi_total Function-yielding V34() Element of bool [:[:(X,Y),(X,Y):],(X,Y):]
[:(X,Y),(X,Y):] is non empty set
[:[:(X,Y),(X,Y):],(X,Y):] is non empty set
bool [:[:(X,Y),(X,Y):],(X,Y):] is non empty set
Mult_ ((X,Y),( the carrier of X,Y)) is non empty Relation-like [:COMPLEX,(X,Y):] -defined (X,Y) -valued Function-like total quasi_total Function-yielding V34() Element of bool [:[:COMPLEX,(X,Y):],(X,Y):]
[:COMPLEX,(X,Y):] is non empty set
[:[:COMPLEX,(X,Y):],(X,Y):] is non empty set
bool [:[:COMPLEX,(X,Y):],(X,Y):] is non empty set
CLSStruct(# (X,Y),(Zero_ ((X,Y),( the carrier of X,Y))),(Add_ ((X,Y),( the carrier of X,Y))),(Mult_ ((X,Y),( the carrier of X,Y))) #) is non empty right_complementable Abelian add-associative right_zeroed strict vector-distributive scalar-distributive scalar-associative scalar-unital CLSStruct
X is non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital CLSStruct
Y is non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital CLSStruct
(X,Y) is non empty right_complementable Abelian add-associative right_zeroed constituted-Functions strict vector-distributive scalar-distributive scalar-associative scalar-unital CLSStruct
(X,Y) is non empty functional Element of bool the carrier of ( the carrier of X,Y)
the carrier of X is non empty set
( the carrier of X,Y) is non empty right_complementable Abelian add-associative right_zeroed constituted-Functions strict vector-distributive scalar-distributive scalar-associative scalar-unital CLSStruct
the carrier of Y is non empty set
Funcs ( the carrier of X, the carrier of Y) is non empty functional FUNCTION_DOMAIN of the carrier of X, the carrier of Y
FuncZero ( the carrier of X,Y) is Relation-like the carrier of X -defined the carrier of Y -valued Function-like total quasi_total Element of Funcs ( the carrier of X, the carrier of Y)
FuncAdd ( the carrier of X,Y) is non empty Relation-like [:(Funcs ( the carrier of X, the carrier of Y)),(Funcs ( the carrier of X, the carrier of Y)):] -defined Funcs ( the carrier of X, the carrier of Y) -valued Function-like total quasi_total Function-yielding V34() Element of bool [:[:(Funcs ( the carrier of X, the carrier of Y)),(Funcs ( the carrier of X, the carrier of Y)):],(Funcs ( the carrier of X, the carrier of Y)):]
[:(Funcs ( the carrier of X, the carrier of Y)),(Funcs ( the carrier of X, the carrier of Y)):] is non empty set
[:[:(Funcs ( the carrier of X, the carrier of Y)),(Funcs ( the carrier of X, the carrier of Y)):],(Funcs ( the carrier of X, the carrier of Y)):] is non empty set
bool [:[:(Funcs ( the carrier of X, the carrier of Y)),(Funcs ( the carrier of X, the carrier of Y)):],(Funcs ( the carrier of X, the carrier of Y)):] is non empty set
( the carrier of X,Y) is non empty Relation-like [:COMPLEX,(Funcs ( the carrier of X, the carrier of Y)):] -defined Funcs ( the carrier of X, the carrier of Y) -valued Function-like total quasi_total Function-yielding V34() Element of bool [:[:COMPLEX,(Funcs ( the carrier of X, the carrier of Y)):],(Funcs ( the carrier of X, the carrier of Y)):]
[:COMPLEX,(Funcs ( the carrier of X, the carrier of Y)):] is non empty set
[:[:COMPLEX,(Funcs ( the carrier of X, the carrier of Y)):],(Funcs ( the carrier of X, the carrier of Y)):] is non empty set
bool [:[:COMPLEX,(Funcs ( the carrier of X, the carrier of Y)):],(Funcs ( the carrier of X, the carrier of Y)):] is non empty set
CLSStruct(# (Funcs ( the carrier of X, the carrier of Y)),(FuncZero ( the carrier of X,Y)),(FuncAdd ( the carrier of X,Y)),( the carrier of X,Y) #) is non empty strict CLSStruct
the carrier of ( the carrier of X,Y) is non empty set
bool the carrier of ( the carrier of X,Y) is non empty set
Zero_ ((X,Y),( the carrier of X,Y)) is Relation-like Function-like Element of (X,Y)
Add_ ((X,Y),( the carrier of X,Y)) is non empty Relation-like [:(X,Y),(X,Y):] -defined (X,Y) -valued Function-like total quasi_total Function-yielding V34() Element of bool [:[:(X,Y),(X,Y):],(X,Y):]
[:(X,Y),(X,Y):] is non empty set
[:[:(X,Y),(X,Y):],(X,Y):] is non empty set
bool [:[:(X,Y),(X,Y):],(X,Y):] is non empty set
Mult_ ((X,Y),( the carrier of X,Y)) is non empty Relation-like [:COMPLEX,(X,Y):] -defined (X,Y) -valued Function-like total quasi_total Function-yielding V34() Element of bool [:[:COMPLEX,(X,Y):],(X,Y):]
[:COMPLEX,(X,Y):] is non empty set
[:[:COMPLEX,(X,Y):],(X,Y):] is non empty set
bool [:[:COMPLEX,(X,Y):],(X,Y):] is non empty set
CLSStruct(# (X,Y),(Zero_ ((X,Y),( the carrier of X,Y))),(Add_ ((X,Y),( the carrier of X,Y))),(Mult_ ((X,Y),( the carrier of X,Y))) #) is non empty right_complementable Abelian add-associative right_zeroed strict vector-distributive scalar-distributive scalar-associative scalar-unital CLSStruct
the carrier of (X,Y) is non empty set
vseq is Relation-like Function-like Element of the carrier of (X,Y)
f is Element of the carrier of X
vseq . f is set
[: the carrier of X, the carrier of Y:] is non empty set
bool [: the carrier of X, the carrier of Y:] is non empty set
tseq is non empty Relation-like the carrier of X -defined the carrier of Y -valued Function-like total quasi_total additive (X,Y) Element of bool [: the carrier of X, the carrier of Y:]
tseq . f is Element of the carrier of Y
X is non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital CLSStruct
Y is non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital CLSStruct
(X,Y) is non empty right_complementable Abelian add-associative right_zeroed constituted-Functions strict vector-distributive scalar-distributive scalar-associative scalar-unital CLSStruct
(X,Y) is non empty functional Element of bool the carrier of ( the carrier of X,Y)
the carrier of X is non empty set
( the carrier of X,Y) is non empty right_complementable Abelian add-associative right_zeroed constituted-Functions strict vector-distributive scalar-distributive scalar-associative scalar-unital CLSStruct
the carrier of Y is non empty set
Funcs ( the carrier of X, the carrier of Y) is non empty functional FUNCTION_DOMAIN of the carrier of X, the carrier of Y
FuncZero ( the carrier of X,Y) is Relation-like the carrier of X -defined the carrier of Y -valued Function-like total quasi_total Element of Funcs ( the carrier of X, the carrier of Y)
FuncAdd ( the carrier of X,Y) is non empty Relation-like [:(Funcs ( the carrier of X, the carrier of Y)),(Funcs ( the carrier of X, the carrier of Y)):] -defined Funcs ( the carrier of X, the carrier of Y) -valued Function-like total quasi_total Function-yielding V34() Element of bool [:[:(Funcs ( the carrier of X, the carrier of Y)),(Funcs ( the carrier of X, the carrier of Y)):],(Funcs ( the carrier of X, the carrier of Y)):]
[:(Funcs ( the carrier of X, the carrier of Y)),(Funcs ( the carrier of X, the carrier of Y)):] is non empty set
[:[:(Funcs ( the carrier of X, the carrier of Y)),(Funcs ( the carrier of X, the carrier of Y)):],(Funcs ( the carrier of X, the carrier of Y)):] is non empty set
bool [:[:(Funcs ( the carrier of X, the carrier of Y)),(Funcs ( the carrier of X, the carrier of Y)):],(Funcs ( the carrier of X, the carrier of Y)):] is non empty set
( the carrier of X,Y) is non empty Relation-like [:COMPLEX,(Funcs ( the carrier of X, the carrier of Y)):] -defined Funcs ( the carrier of X, the carrier of Y) -valued Function-like total quasi_total Function-yielding V34() Element of bool [:[:COMPLEX,(Funcs ( the carrier of X, the carrier of Y)):],(Funcs ( the carrier of X, the carrier of Y)):]
[:COMPLEX,(Funcs ( the carrier of X, the carrier of Y)):] is non empty set
[:[:COMPLEX,(Funcs ( the carrier of X, the carrier of Y)):],(Funcs ( the carrier of X, the carrier of Y)):] is non empty set
bool [:[:COMPLEX,(Funcs ( the carrier of X, the carrier of Y)):],(Funcs ( the carrier of X, the carrier of Y)):] is non empty set
CLSStruct(# (Funcs ( the carrier of X, the carrier of Y)),(FuncZero ( the carrier of X,Y)),(FuncAdd ( the carrier of X,Y)),( the carrier of X,Y) #) is non empty strict CLSStruct
the carrier of ( the carrier of X,Y) is non empty set
bool the carrier of ( the carrier of X,Y) is non empty set
Zero_ ((X,Y),( the carrier of X,Y)) is Relation-like Function-like Element of (X,Y)
Add_ ((X,Y),( the carrier of X,Y)) is non empty Relation-like [:(X,Y),(X,Y):] -defined (X,Y) -valued Function-like total quasi_total Function-yielding V34() Element of bool [:[:(X,Y),(X,Y):],(X,Y):]
[:(X,Y),(X,Y):] is non empty set
[:[:(X,Y),(X,Y):],(X,Y):] is non empty set
bool [:[:(X,Y),(X,Y):],(X,Y):] is non empty set
Mult_ ((X,Y),( the carrier of X,Y)) is non empty Relation-like [:COMPLEX,(X,Y):] -defined (X,Y) -valued Function-like total quasi_total Function-yielding V34() Element of bool [:[:COMPLEX,(X,Y):],(X,Y):]
[:COMPLEX,(X,Y):] is non empty set
[:[:COMPLEX,(X,Y):],(X,Y):] is non empty set
bool [:[:COMPLEX,(X,Y):],(X,Y):] is non empty set
CLSStruct(# (X,Y),(Zero_ ((X,Y),( the carrier of X,Y))),(Add_ ((X,Y),( the carrier of X,Y))),(Mult_ ((X,Y),( the carrier of X,Y))) #) is non empty right_complementable Abelian add-associative right_zeroed strict vector-distributive scalar-distributive scalar-associative scalar-unital CLSStruct
the carrier of (X,Y) is non empty set
tseq is Relation-like Function-like Element of the carrier of (X,Y)
vseq is Relation-like Function-like Element of the carrier of (X,Y)
f is Relation-like Function-like Element of the carrier of (X,Y)
vseq + f is Relation-like Function-like Element of the carrier of (X,Y)
the addF of (X,Y) is non empty Relation-like [: the carrier of (X,Y), the carrier of (X,Y):] -defined the carrier of (X,Y) -valued Function-like total quasi_total Element of bool [:[: the carrier of (X,Y), the carrier of (X,Y):], the carrier of (X,Y):]
[: the carrier of (X,Y), the carrier of (X,Y):] is non empty set
[:[: the carrier of (X,Y), the carrier of (X,Y):], the carrier of (X,Y):] is non empty set
bool [:[: the carrier of (X,Y), the carrier of (X,Y):], the carrier of (X,Y):] is non empty set
the addF of (X,Y) . (vseq,f) is Relation-like Function-like Element of the carrier of (X,Y)
[vseq,f] is set
the addF of (X,Y) . [vseq,f] is set
[: the carrier of X, the carrier of Y:] is non empty set
bool [: the carrier of X, the carrier of Y:] is non empty set
m is Relation-like Function-like Element of the carrier of ( the carrier of X,Y)
e is Relation-like Function-like Element of the carrier of ( the carrier of X,Y)
n is Relation-like Function-like Element of the carrier of ( the carrier of X,Y)
e + n is Relation-like Function-like Element of the carrier of ( the carrier of X,Y)
the addF of ( the carrier of X,Y) is non empty Relation-like [: the carrier of ( the carrier of X,Y), the carrier of ( the carrier of X,Y):] -defined the carrier of ( the carrier of X,Y) -valued Function-like total quasi_total Element of bool [:[: the carrier of ( the carrier of X,Y), the carrier of ( the carrier of X,Y):], the carrier of ( the carrier of X,Y):]
[: the carrier of ( the carrier of X,Y), the carrier of ( the carrier of X,Y):] is non empty set
[:[: the carrier of ( the carrier of X,Y), the carrier of ( the carrier of X,Y):], the carrier of ( the carrier of X,Y):] is non empty set
bool [:[: the carrier of ( the carrier of X,Y), the carrier of ( the carrier of X,Y):], the carrier of ( the carrier of X,Y):] is non empty set
the addF of ( the carrier of X,Y) . (e,n) is Relation-like Function-like Element of the carrier of ( the carrier of X,Y)
[e,n] is set
the addF of ( the carrier of X,Y) . [e,n] is set
tv is non empty Relation-like the carrier of X -defined the carrier of Y -valued Function-like total quasi_total additive (X,Y) Element of bool [: the carrier of X, the carrier of Y:]
n is Element of the carrier of X
tv . n is Element of the carrier of Y
tseq is non empty Relation-like the carrier of X -defined the carrier of Y -valued Function-like total quasi_total additive (X,Y) Element of bool [: the carrier of X, the carrier of Y:]
tseq . n is Element of the carrier of Y
tseq is non empty Relation-like the carrier of X -defined the carrier of Y -valued Function-like total quasi_total additive (X,Y) Element of bool [: the carrier of X, the carrier of Y:]
tseq . n is Element of the carrier of Y
(tseq . n) + (tseq . n) is Element of the carrier of Y
the addF of Y is non empty Relation-like [: the carrier of Y, the carrier of Y:] -defined the carrier of Y -valued Function-like total quasi_total Element of bool [:[: the carrier of Y, the carrier of Y:], the carrier of Y:]
[: the carrier of Y, the carrier of Y:] is non empty set
[:[: the carrier of Y, the carrier of Y:], the carrier of Y:] is non empty set
bool [:[: the carrier of Y, the carrier of Y:], the carrier of Y:] is non empty set
the addF of Y . ((tseq . n),(tseq . n)) is Element of the carrier of Y
[(tseq . n),(tseq . n)] is set
the addF of Y . [(tseq . n),(tseq . n)] is set
n is Element of the carrier of X
(X,Y,tseq,n) is Element of the carrier of Y
(X,Y,vseq,n) is Element of the carrier of Y
(X,Y,f,n) is Element of the carrier of Y
(X,Y,vseq,n) + (X,Y,f,n) is Element of the carrier of Y
the addF of Y . ((X,Y,vseq,n),(X,Y,f,n)) is Element of the carrier of Y
[(X,Y,vseq,n),(X,Y,f,n)] is set
the addF of Y . [(X,Y,vseq,n),(X,Y,f,n)] is set
h1 is Element of the carrier of X
tv . h1 is Element of the carrier of Y
tseq . h1 is Element of the carrier of Y
tseq . h1 is Element of the carrier of Y
(tseq . h1) + (tseq . h1) is Element of the carrier of Y
the addF of Y . ((tseq . h1),(tseq . h1)) is Element of the carrier of Y
[(tseq . h1),(tseq . h1)] is set
the addF of Y . [(tseq . h1),(tseq . h1)] is set
X is non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital CLSStruct
Y is non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital CLSStruct
(X,Y) is non empty right_complementable Abelian add-associative right_zeroed constituted-Functions strict vector-distributive scalar-distributive scalar-associative scalar-unital CLSStruct
(X,Y) is non empty functional Element of bool the carrier of ( the carrier of X,Y)
the carrier of X is non empty set
( the carrier of X,Y) is non empty right_complementable Abelian add-associative right_zeroed constituted-Functions strict vector-distributive scalar-distributive scalar-associative scalar-unital CLSStruct
the carrier of Y is non empty set
Funcs ( the carrier of X, the carrier of Y) is non empty functional FUNCTION_DOMAIN of the carrier of X, the carrier of Y
FuncZero ( the carrier of X,Y) is Relation-like the carrier of X -defined the carrier of Y -valued Function-like total quasi_total Element of Funcs ( the carrier of X, the carrier of Y)
FuncAdd ( the carrier of X,Y) is non empty Relation-like [:(Funcs ( the carrier of X, the carrier of Y)),(Funcs ( the carrier of X, the carrier of Y)):] -defined Funcs ( the carrier of X, the carrier of Y) -valued Function-like total quasi_total Function-yielding V34() Element of bool [:[:(Funcs ( the carrier of X, the carrier of Y)),(Funcs ( the carrier of X, the carrier of Y)):],(Funcs ( the carrier of X, the carrier of Y)):]
[:(Funcs ( the carrier of X, the carrier of Y)),(Funcs ( the carrier of X, the carrier of Y)):] is non empty set
[:[:(Funcs ( the carrier of X, the carrier of Y)),(Funcs ( the carrier of X, the carrier of Y)):],(Funcs ( the carrier of X, the carrier of Y)):] is non empty set
bool [:[:(Funcs ( the carrier of X, the carrier of Y)),(Funcs ( the carrier of X, the carrier of Y)):],(Funcs ( the carrier of X, the carrier of Y)):] is non empty set
( the carrier of X,Y) is non empty Relation-like [:COMPLEX,(Funcs ( the carrier of X, the carrier of Y)):] -defined Funcs ( the carrier of X, the carrier of Y) -valued Function-like total quasi_total Function-yielding V34() Element of bool [:[:COMPLEX,(Funcs ( the carrier of X, the carrier of Y)):],(Funcs ( the carrier of X, the carrier of Y)):]
[:COMPLEX,(Funcs ( the carrier of X, the carrier of Y)):] is non empty set
[:[:COMPLEX,(Funcs ( the carrier of X, the carrier of Y)):],(Funcs ( the carrier of X, the carrier of Y)):] is non empty set
bool [:[:COMPLEX,(Funcs ( the carrier of X, the carrier of Y)):],(Funcs ( the carrier of X, the carrier of Y)):] is non empty set
CLSStruct(# (Funcs ( the carrier of X, the carrier of Y)),(FuncZero ( the carrier of X,Y)),(FuncAdd ( the carrier of X,Y)),( the carrier of X,Y) #) is non empty strict CLSStruct
the carrier of ( the carrier of X,Y) is non empty set
bool the carrier of ( the carrier of X,Y) is non empty set
Zero_ ((X,Y),( the carrier of X,Y)) is Relation-like Function-like Element of (X,Y)
Add_ ((X,Y),( the carrier of X,Y)) is non empty Relation-like [:(X,Y),(X,Y):] -defined (X,Y) -valued Function-like total quasi_total Function-yielding V34() Element of bool [:[:(X,Y),(X,Y):],(X,Y):]
[:(X,Y),(X,Y):] is non empty set
[:[:(X,Y),(X,Y):],(X,Y):] is non empty set
bool [:[:(X,Y),(X,Y):],(X,Y):] is non empty set
Mult_ ((X,Y),( the carrier of X,Y)) is non empty Relation-like [:COMPLEX,(X,Y):] -defined (X,Y) -valued Function-like total quasi_total Function-yielding V34() Element of bool [:[:COMPLEX,(X,Y):],(X,Y):]
[:COMPLEX,(X,Y):] is non empty set
[:[:COMPLEX,(X,Y):],(X,Y):] is non empty set
bool [:[:COMPLEX,(X,Y):],(X,Y):] is non empty set
CLSStruct(# (X,Y),(Zero_ ((X,Y),( the carrier of X,Y))),(Add_ ((X,Y),( the carrier of X,Y))),(Mult_ ((X,Y),( the carrier of X,Y))) #) is non empty right_complementable Abelian add-associative right_zeroed strict vector-distributive scalar-distributive scalar-associative scalar-unital CLSStruct
the carrier of (X,Y) is non empty set
f is Relation-like Function-like Element of the carrier of (X,Y)
vseq is Relation-like Function-like Element of the carrier of (X,Y)
[: the carrier of X, the carrier of Y:] is non empty set
bool [: the carrier of X, the carrier of Y:] is non empty set
tseq is complex set
tseq * vseq is Relation-like Function-like Element of the carrier of (X,Y)
e is Relation-like Function-like Element of the carrier of ( the carrier of X,Y)
tv is Relation-like Function-like Element of the carrier of ( the carrier of X,Y)
tseq * tv is Relation-like Function-like Element of the carrier of ( the carrier of X,Y)
tseq is non empty Relation-like the carrier of X -defined the carrier of Y -valued Function-like total quasi_total additive (X,Y) Element of bool [: the carrier of X, the carrier of Y:]
m is Element of the carrier of X
tseq . m is Element of the carrier of Y
tseq is non empty Relation-like the carrier of X -defined the carrier of Y -valued Function-like total quasi_total additive (X,Y) Element of bool [: the carrier of X, the carrier of Y:]
tseq . m is Element of the carrier of Y
tseq * (tseq . m) is Element of the carrier of Y
m is Element of the carrier of X
(X,Y,f,m) is Element of the carrier of Y
(X,Y,vseq,m) is Element of the carrier of Y
tseq * (X,Y,vseq,m) is Element of the carrier of Y
n is Element of the carrier of X
tseq . n is Element of the carrier of Y
tseq . n is Element of the carrier of Y
tseq * (tseq . n) is Element of the carrier of Y
X is non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital CLSStruct
Y is non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital CLSStruct
(X,Y) is non empty right_complementable Abelian add-associative right_zeroed constituted-Functions strict vector-distributive scalar-distributive scalar-associative scalar-unital CLSStruct
(X,Y) is non empty functional Element of bool the carrier of ( the carrier of X,Y)
the carrier of X is non empty set
( the carrier of X,Y) is non empty right_complementable Abelian add-associative right_zeroed constituted-Functions strict vector-distributive scalar-distributive scalar-associative scalar-unital CLSStruct
the carrier of Y is non empty set
Funcs ( the carrier of X, the carrier of Y) is non empty functional FUNCTION_DOMAIN of the carrier of X, the carrier of Y
FuncZero ( the carrier of X,Y) is Relation-like the carrier of X -defined the carrier of Y -valued Function-like total quasi_total Element of Funcs ( the carrier of X, the carrier of Y)
FuncAdd ( the carrier of X,Y) is non empty Relation-like [:(Funcs ( the carrier of X, the carrier of Y)),(Funcs ( the carrier of X, the carrier of Y)):] -defined Funcs ( the carrier of X, the carrier of Y) -valued Function-like total quasi_total Function-yielding V34() Element of bool [:[:(Funcs ( the carrier of X, the carrier of Y)),(Funcs ( the carrier of X, the carrier of Y)):],(Funcs ( the carrier of X, the carrier of Y)):]
[:(Funcs ( the carrier of X, the carrier of Y)),(Funcs ( the carrier of X, the carrier of Y)):] is non empty set
[:[:(Funcs ( the carrier of X, the carrier of Y)),(Funcs ( the carrier of X, the carrier of Y)):],(Funcs ( the carrier of X, the carrier of Y)):] is non empty set
bool [:[:(Funcs ( the carrier of X, the carrier of Y)),(Funcs ( the carrier of X, the carrier of Y)):],(Funcs ( the carrier of X, the carrier of Y)):] is non empty set
( the carrier of X,Y) is non empty Relation-like [:COMPLEX,(Funcs ( the carrier of X, the carrier of Y)):] -defined Funcs ( the carrier of X, the carrier of Y) -valued Function-like total quasi_total Function-yielding V34() Element of bool [:[:COMPLEX,(Funcs ( the carrier of X, the carrier of Y)):],(Funcs ( the carrier of X, the carrier of Y)):]
[:COMPLEX,(Funcs ( the carrier of X, the carrier of Y)):] is non empty set
[:[:COMPLEX,(Funcs ( the carrier of X, the carrier of Y)):],(Funcs ( the carrier of X, the carrier of Y)):] is non empty set
bool [:[:COMPLEX,(Funcs ( the carrier of X, the carrier of Y)):],(Funcs ( the carrier of X, the carrier of Y)):] is non empty set
CLSStruct(# (Funcs ( the carrier of X, the carrier of Y)),(FuncZero ( the carrier of X,Y)),(FuncAdd ( the carrier of X,Y)),( the carrier of X,Y) #) is non empty strict CLSStruct
the carrier of ( the carrier of X,Y) is non empty set
bool the carrier of ( the carrier of X,Y) is non empty set
Zero_ ((X,Y),( the carrier of X,Y)) is Relation-like Function-like Element of (X,Y)
Add_ ((X,Y),( the carrier of X,Y)) is non empty Relation-like [:(X,Y),(X,Y):] -defined (X,Y) -valued Function-like total quasi_total Function-yielding V34() Element of bool [:[:(X,Y),(X,Y):],(X,Y):]
[:(X,Y),(X,Y):] is non empty set
[:[:(X,Y),(X,Y):],(X,Y):] is non empty set
bool [:[:(X,Y),(X,Y):],(X,Y):] is non empty set
Mult_ ((X,Y),( the carrier of X,Y)) is non empty Relation-like [:COMPLEX,(X,Y):] -defined (X,Y) -valued Function-like total quasi_total Function-yielding V34() Element of bool [:[:COMPLEX,(X,Y):],(X,Y):]
[:COMPLEX,(X,Y):] is non empty set
[:[:COMPLEX,(X,Y):],(X,Y):] is non empty set
bool [:[:COMPLEX,(X,Y):],(X,Y):] is non empty set
CLSStruct(# (X,Y),(Zero_ ((X,Y),( the carrier of X,Y))),(Add_ ((X,Y),( the carrier of X,Y))),(Mult_ ((X,Y),( the carrier of X,Y))) #) is non empty right_complementable Abelian add-associative right_zeroed strict vector-distributive scalar-distributive scalar-associative scalar-unital CLSStruct
0. (X,Y) is Relation-like Function-like zero Element of the carrier of (X,Y)
the carrier of (X,Y) is non empty set
the ZeroF of (X,Y) is Relation-like Function-like Element of the carrier of (X,Y)
0. Y is zero Element of the carrier of Y
the ZeroF of Y is Element of the carrier of Y
the carrier of X --> (0. Y) is non empty Relation-like the carrier of X -defined the carrier of Y -valued Function-like constant total quasi_total Element of bool [: the carrier of X, the carrier of Y:]
[: the carrier of X, the carrier of Y:] is non empty set
bool [: the carrier of X, the carrier of Y:] is non empty set
0. ( the carrier of X,Y) is Relation-like Function-like zero Element of the carrier of ( the carrier of X,Y)
the ZeroF of ( the carrier of X,Y) is Relation-like Function-like Element of the carrier of ( the carrier of X,Y)
Y is non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital CLSStruct
the carrier of Y is non empty set
X is non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital CLSStruct
the carrier of X is non empty set
0. Y is zero Element of the carrier of Y
the ZeroF of Y is Element of the carrier of Y
the carrier of X --> (0. Y) is non empty Relation-like the carrier of X -defined the carrier of Y -valued Function-like constant total quasi_total Element of bool [: the carrier of X, the carrier of Y:]
[: the carrier of X, the carrier of Y:] is non empty set
bool [: the carrier of X, the carrier of Y:] is non empty set
f is non empty Relation-like the carrier of X -defined the carrier of Y -valued Function-like total quasi_total Element of bool [: the carrier of X, the carrier of Y:]
tseq is Element of the carrier of X
f . tseq is Element of the carrier of Y
tseq is complex set
tseq * tseq is Element of the carrier of X
f . (tseq * tseq) is Element of the carrier of Y
tseq * (f . tseq) is Element of the carrier of Y
tseq * (0. Y) is Element of the carrier of Y
tseq is Element of the carrier of X
tseq is Element of the carrier of X
tseq + tseq is Element of the carrier of X
the addF of X is non empty Relation-like [: the carrier of X, the carrier of X:] -defined the carrier of X -valued Function-like total quasi_total Element of bool [:[: the carrier of X, the carrier of X:], the carrier of X:]
[: the carrier of X, the carrier of X:] is non empty set
[:[: the carrier of X, the carrier of X:], the carrier of X:] is non empty set
bool [:[: the carrier of X, the carrier of X:], the carrier of X:] is non empty set
the addF of X . (tseq,tseq) is Element of the carrier of X
[tseq,tseq] is set
the addF of X . [tseq,tseq] is set
f . (tseq + tseq) is Element of the carrier of Y
f . tseq is Element of the carrier of Y
f . tseq is Element of the carrier of Y
(f . tseq) + (f . tseq) is Element of the carrier of Y
the addF of Y is non empty Relation-like [: the carrier of Y, the carrier of Y:] -defined the carrier of Y -valued Function-like total quasi_total Element of bool [:[: the carrier of Y, the carrier of Y:], the carrier of Y:]
[: the carrier of Y, the carrier of Y:] is non empty set
[:[: the carrier of Y, the carrier of Y:], the carrier of Y:] is non empty set
bool [:[: the carrier of Y, the carrier of Y:], the carrier of Y:] is non empty set
the addF of Y . ((f . tseq),(f . tseq)) is Element of the carrier of Y
[(f . tseq),(f . tseq)] is set
the addF of Y . [(f . tseq),(f . tseq)] is set
tseq + tseq is Element of the carrier of X
f . (tseq + tseq) is Element of the carrier of Y
(0. Y) + (0. Y) is Element of the carrier of Y
the addF of Y . ((0. Y),(0. Y)) is Element of the carrier of Y
[(0. Y),(0. Y)] is set
the addF of Y . [(0. Y),(0. Y)] is set
(f . tseq) + (0. Y) is Element of the carrier of Y
the addF of Y . ((f . tseq),(0. Y)) is Element of the carrier of Y
[(f . tseq),(0. Y)] is set
the addF of Y . [(f . tseq),(0. Y)] is set
(f . tseq) + (f . tseq) is Element of the carrier of Y
X is non empty right_complementable Abelian add-associative right_zeroed discerning reflexive vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like CNORMSTR
the carrier of X is non empty set
[:NAT, the carrier of X:] is non empty set
bool [:NAT, the carrier of X:] is non empty set
Y is non empty Relation-like NAT -defined the carrier of X -valued Function-like total quasi_total Element of bool [:NAT, the carrier of X:]
lim Y is Element of the carrier of X
||.Y.|| is non empty Relation-like NAT -defined REAL -valued Function-like total quasi_total V132() V133() V134() Element of bool [:NAT,REAL:]
lim ||.Y.|| is complex real ext-real Element of REAL
vseq is Element of the carrier of X
||.vseq.|| is complex real ext-real Element of REAL
the U9 of X is non empty Relation-like the carrier of X -defined REAL -valued Function-like total quasi_total V132() V133() V134() Element of bool [: the carrier of X,REAL:]
[: the carrier of X,REAL:] is non empty V132() V133() V134() set
bool [: the carrier of X,REAL:] is non empty set
the U9 of X . vseq is complex real ext-real Element of REAL
f is complex real ext-real set
tseq is epsilon-transitive epsilon-connected ordinal natural complex real V108() V109() ext-real V142() V143() V144() V145() V146() V147() Element of NAT
tseq is epsilon-transitive epsilon-connected ordinal natural complex real V108() V109() ext-real V142() V143() V144() V145() V146() V147() Element of NAT
tseq is epsilon-transitive epsilon-connected ordinal natural complex real V108() V109() ext-real V142() V143() V144() V145() V146() V147() Element of NAT
Y . tseq is Element of the carrier of X
(Y . tseq) - vseq is Element of the carrier of X
- vseq is Element of the carrier of X
(Y . tseq) + (- vseq) is Element of the carrier of X
the addF of X is non empty Relation-like [: the carrier of X, the carrier of X:] -defined the carrier of X -valued Function-like total quasi_total Element of bool [:[: the carrier of X, the carrier of X:], the carrier of X:]
[: the carrier of X, the carrier of X:] is non empty set
[:[: the carrier of X, the carrier of X:], the carrier of X:] is non empty set
bool [:[: the carrier of X, the carrier of X:], the carrier of X:] is non empty set
the addF of X . ((Y . tseq),(- vseq)) is Element of the carrier of X
[(Y . tseq),(- vseq)] is set
the addF of X . [(Y . tseq),(- vseq)] is set
||.((Y . tseq) - vseq).|| is complex real ext-real Element of REAL
the U9 of X . ((Y . tseq) - vseq) is complex real ext-real Element of REAL
||.(Y . tseq).|| is complex real ext-real Element of REAL
the U9 of X . (Y . tseq) is complex real ext-real Element of REAL
||.(Y . tseq).|| - ||.vseq.|| is complex real ext-real Element of REAL
abs (||.(Y . tseq).|| - ||.vseq.||) is complex real ext-real Element of REAL
||.Y.|| . tseq is complex real ext-real Element of REAL
(||.Y.|| . tseq) - ||.vseq.|| is complex real ext-real Element of REAL
abs ((||.Y.|| . tseq) - ||.vseq.||) is complex real ext-real Element of REAL
tseq is epsilon-transitive epsilon-connected ordinal natural complex real V108() V109() ext-real V142() V143() V144() V145() V146() V147() Element of NAT
||.Y.|| . tseq is complex real ext-real Element of REAL
(||.Y.|| . tseq) - ||.vseq.|| is complex real ext-real Element of REAL
abs ((||.Y.|| . tseq) - ||.vseq.||) is complex real ext-real Element of REAL
X is non empty right_complementable Abelian add-associative right_zeroed discerning reflexive vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like CNORMSTR
the carrier of X is non empty set
Y is non empty right_complementable Abelian add-associative right_zeroed discerning reflexive vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like CNORMSTR
the carrier of Y is non empty set
[: the carrier of X, the carrier of Y:] is non empty set
bool [: the carrier of X, the carrier of Y:] is non empty set
X is non empty right_complementable Abelian add-associative right_zeroed discerning reflexive vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like CNORMSTR
the carrier of X is non empty set
Y is non empty right_complementable Abelian add-associative right_zeroed discerning reflexive vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like CNORMSTR
the carrier of Y is non empty set
[: the carrier of X, the carrier of Y:] is non empty set
bool [: the carrier of X, the carrier of Y:] is non empty set
0. Y is zero Element of the carrier of Y
the ZeroF of Y is Element of the carrier of Y
vseq is non empty Relation-like the carrier of X -defined the carrier of Y -valued Function-like total quasi_total additive (X,Y) Element of bool [: the carrier of X, the carrier of Y:]
f is Element of the carrier of X
vseq . f is Element of the carrier of Y
||.(vseq . f).|| is complex real ext-real Element of REAL
the U9 of Y is non empty Relation-like the carrier of Y -defined REAL -valued Function-like total quasi_total V132() V133() V134() Element of bool [: the carrier of Y,REAL:]
[: the carrier of Y,REAL:] is non empty V132() V133() V134() set
bool [: the carrier of Y,REAL:] is non empty set
the U9 of Y . (vseq . f) is complex real ext-real Element of REAL
||.(0. Y).|| is complex real ext-real Element of REAL
the U9 of Y . (0. Y) is complex real ext-real Element of REAL
||.f.|| is complex real ext-real Element of REAL
the U9 of X is non empty Relation-like the carrier of X -defined REAL -valued Function-like total quasi_total V132() V133() V134() Element of bool [: the carrier of X,REAL:]
[: the carrier of X,REAL:] is non empty V132() V133() V134() set
bool [: the carrier of X,REAL:] is non empty set
the U9 of X . f is complex real ext-real Element of REAL
0 * ||.f.|| is complex real ext-real Element of REAL
f is Element of the carrier of X
vseq . f is Element of the carrier of Y
||.(vseq . f).|| is complex real ext-real Element of REAL
the U9 of Y . (vseq . f) is complex real ext-real Element of REAL
||.f.|| is complex real ext-real Element of REAL
the U9 of X . f is complex real ext-real Element of REAL
0 * ||.f.|| is complex real ext-real Element of REAL
X is non empty right_complementable Abelian add-associative right_zeroed discerning reflexive vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like CNORMSTR
the carrier of X is non empty set
Y is non empty right_complementable Abelian add-associative right_zeroed discerning reflexive vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like CNORMSTR
the carrier of Y is non empty set
[: the carrier of X, the carrier of Y:] is non empty set
bool [: the carrier of X, the carrier of Y:] is non empty set
0. Y is zero Element of the carrier of Y
the ZeroF of Y is Element of the carrier of Y
the carrier of X --> (0. Y) is non empty Relation-like the carrier of X -defined the carrier of Y -valued Function-like constant total quasi_total Element of bool [: the carrier of X, the carrier of Y:]
f is non empty Relation-like the carrier of X -defined the carrier of Y -valued Function-like total quasi_total additive (X,Y) Element of bool [: the carrier of X, the carrier of Y:]
tseq is Element of the carrier of X
f . tseq is Element of the carrier of Y
X is non empty right_complementable Abelian add-associative right_zeroed discerning reflexive vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like CNORMSTR
Y is non empty right_complementable Abelian add-associative right_zeroed discerning reflexive vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like CNORMSTR
(X,Y) is non empty right_complementable Abelian add-associative right_zeroed constituted-Functions strict vector-distributive scalar-distributive scalar-associative scalar-unital CLSStruct
(X,Y) is non empty functional Element of bool the carrier of ( the carrier of X,Y)
the carrier of X is non empty set
( the carrier of X,Y) is non empty right_complementable Abelian add-associative right_zeroed constituted-Functions strict vector-distributive scalar-distributive scalar-associative scalar-unital CLSStruct
the carrier of Y is non empty set
Funcs ( the carrier of X, the carrier of Y) is non empty functional FUNCTION_DOMAIN of the carrier of X, the carrier of Y
FuncZero ( the carrier of X,Y) is Relation-like the carrier of X -defined the carrier of Y -valued Function-like total quasi_total Element of Funcs ( the carrier of X, the carrier of Y)
FuncAdd ( the carrier of X,Y) is non empty Relation-like [:(Funcs ( the carrier of X, the carrier of Y)),(Funcs ( the carrier of X, the carrier of Y)):] -defined Funcs ( the carrier of X, the carrier of Y) -valued Function-like total quasi_total Function-yielding V34() Element of bool [:[:(Funcs ( the carrier of X, the carrier of Y)),(Funcs ( the carrier of X, the carrier of Y)):],(Funcs ( the carrier of X, the carrier of Y)):]
[:(Funcs ( the carrier of X, the carrier of Y)),(Funcs ( the carrier of X, the carrier of Y)):] is non empty set
[:[:(Funcs ( the carrier of X, the carrier of Y)),(Funcs ( the carrier of X, the carrier of Y)):],(Funcs ( the carrier of X, the carrier of Y)):] is non empty set
bool [:[:(Funcs ( the carrier of X, the carrier of Y)),(Funcs ( the carrier of X, the carrier of Y)):],(Funcs ( the carrier of X, the carrier of Y)):] is non empty set
( the carrier of X,Y) is non empty Relation-like [:COMPLEX,(Funcs ( the carrier of X, the carrier of Y)):] -defined Funcs ( the carrier of X, the carrier of Y) -valued Function-like total quasi_total Function-yielding V34() Element of bool [:[:COMPLEX,(Funcs ( the carrier of X, the carrier of Y)):],(Funcs ( the carrier of X, the carrier of Y)):]
[:COMPLEX,(Funcs ( the carrier of X, the carrier of Y)):] is non empty set
[:[:COMPLEX,(Funcs ( the carrier of X, the carrier of Y)):],(Funcs ( the carrier of X, the carrier of Y)):] is non empty set
bool [:[:COMPLEX,(Funcs ( the carrier of X, the carrier of Y)):],(Funcs ( the carrier of X, the carrier of Y)):] is non empty set
CLSStruct(# (Funcs ( the carrier of X, the carrier of Y)),(FuncZero ( the carrier of X,Y)),(FuncAdd ( the carrier of X,Y)),( the carrier of X,Y) #) is non empty strict CLSStruct
the carrier of ( the carrier of X,Y) is non empty set
bool the carrier of ( the carrier of X,Y) is non empty set
Zero_ ((X,Y),( the carrier of X,Y)) is Relation-like Function-like Element of (X,Y)
Add_ ((X,Y),( the carrier of X,Y)) is non empty Relation-like [:(X,Y),(X,Y):] -defined (X,Y) -valued Function-like total quasi_total Function-yielding V34() Element of bool [:[:(X,Y),(X,Y):],(X,Y):]
[:(X,Y),(X,Y):] is non empty set
[:[:(X,Y),(X,Y):],(X,Y):] is non empty set
bool [:[:(X,Y),(X,Y):],(X,Y):] is non empty set
Mult_ ((X,Y),( the carrier of X,Y)) is non empty Relation-like [:COMPLEX,(X,Y):] -defined (X,Y) -valued Function-like total quasi_total Function-yielding V34() Element of bool [:[:COMPLEX,(X,Y):],(X,Y):]
[:COMPLEX,(X,Y):] is non empty set
[:[:COMPLEX,(X,Y):],(X,Y):] is non empty set
bool [:[:COMPLEX,(X,Y):],(X,Y):] is non empty set
CLSStruct(# (X,Y),(Zero_ ((X,Y),( the carrier of X,Y))),(Add_ ((X,Y),( the carrier of X,Y))),(Mult_ ((X,Y),( the carrier of X,Y))) #) is non empty right_complementable Abelian add-associative right_zeroed strict vector-distributive scalar-distributive scalar-associative scalar-unital CLSStruct
the carrier of (X,Y) is non empty set
bool the carrier of (X,Y) is non empty set
[: the carrier of X, the carrier of Y:] is non empty set
bool [: the carrier of X, the carrier of Y:] is non empty set
vseq is set
f is set
f is set
vseq is Element of bool the carrier of (X,Y)
f is Element of bool the carrier of (X,Y)
tseq is set
tseq is set
X is non empty right_complementable Abelian add-associative right_zeroed discerning reflexive vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like CNORMSTR
Y is non empty right_complementable Abelian add-associative right_zeroed discerning reflexive vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like CNORMSTR
(X,Y) is Element of bool the carrier of (X,Y)
(X,Y) is non empty right_complementable Abelian add-associative right_zeroed constituted-Functions strict vector-distributive scalar-distributive scalar-associative scalar-unital CLSStruct
(X,Y) is non empty functional Element of bool the carrier of ( the carrier of X,Y)
the carrier of X is non empty set
( the carrier of X,Y) is non empty right_complementable Abelian add-associative right_zeroed constituted-Functions strict vector-distributive scalar-distributive scalar-associative scalar-unital CLSStruct
the carrier of Y is non empty set
Funcs ( the carrier of X, the carrier of Y) is non empty functional FUNCTION_DOMAIN of the carrier of X, the carrier of Y
FuncZero ( the carrier of X,Y) is Relation-like the carrier of X -defined the carrier of Y -valued Function-like total quasi_total Element of Funcs ( the carrier of X, the carrier of Y)
FuncAdd ( the carrier of X,Y) is non empty Relation-like [:(Funcs ( the carrier of X, the carrier of Y)),(Funcs ( the carrier of X, the carrier of Y)):] -defined Funcs ( the carrier of X, the carrier of Y) -valued Function-like total quasi_total Function-yielding V34() Element of bool [:[:(Funcs ( the carrier of X, the carrier of Y)),(Funcs ( the carrier of X, the carrier of Y)):],(Funcs ( the carrier of X, the carrier of Y)):]
[:(Funcs ( the carrier of X, the carrier of Y)),(Funcs ( the carrier of X, the carrier of Y)):] is non empty set
[:[:(Funcs ( the carrier of X, the carrier of Y)),(Funcs ( the carrier of X, the carrier of Y)):],(Funcs ( the carrier of X, the carrier of Y)):] is non empty set
bool [:[:(Funcs ( the carrier of X, the carrier of Y)),(Funcs ( the carrier of X, the carrier of Y)):],(Funcs ( the carrier of X, the carrier of Y)):] is non empty set
( the carrier of X,Y) is non empty Relation-like [:COMPLEX,(Funcs ( the carrier of X, the carrier of Y)):] -defined Funcs ( the carrier of X, the carrier of Y) -valued Function-like total quasi_total Function-yielding V34() Element of bool [:[:COMPLEX,(Funcs ( the carrier of X, the carrier of Y)):],(Funcs ( the carrier of X, the carrier of Y)):]
[:COMPLEX,(Funcs ( the carrier of X, the carrier of Y)):] is non empty set
[:[:COMPLEX,(Funcs ( the carrier of X, the carrier of Y)):],(Funcs ( the carrier of X, the carrier of Y)):] is non empty set
bool [:[:COMPLEX,(Funcs ( the carrier of X, the carrier of Y)):],(Funcs ( the carrier of X, the carrier of Y)):] is non empty set
CLSStruct(# (Funcs ( the carrier of X, the carrier of Y)),(FuncZero ( the carrier of X,Y)),(FuncAdd ( the carrier of X,Y)),( the carrier of X,Y) #) is non empty strict CLSStruct
the carrier of ( the carrier of X,Y) is non empty set
bool the carrier of ( the carrier of X,Y) is non empty set
Zero_ ((X,Y),( the carrier of X,Y)) is Relation-like Function-like Element of (X,Y)
Add_ ((X,Y),( the carrier of X,Y)) is non empty Relation-like [:(X,Y),(X,Y):] -defined (X,Y) -valued Function-like total quasi_total Function-yielding V34() Element of bool [:[:(X,Y),(X,Y):],(X,Y):]
[:(X,Y),(X,Y):] is non empty set
[:[:(X,Y),(X,Y):],(X,Y):] is non empty set
bool [:[:(X,Y),(X,Y):],(X,Y):] is non empty set
Mult_ ((X,Y),( the carrier of X,Y)) is non empty Relation-like [:COMPLEX,(X,Y):] -defined (X,Y) -valued Function-like total quasi_total Function-yielding V34() Element of bool [:[:COMPLEX,(X,Y):],(X,Y):]
[:COMPLEX,(X,Y):] is non empty set
[:[:COMPLEX,(X,Y):],(X,Y):] is non empty set
bool [:[:COMPLEX,(X,Y):],(X,Y):] is non empty set
CLSStruct(# (X,Y),(Zero_ ((X,Y),( the carrier of X,Y))),(Add_ ((X,Y),( the carrier of X,Y))),(Mult_ ((X,Y),( the carrier of X,Y))) #) is non empty right_complementable Abelian add-associative right_zeroed strict vector-distributive scalar-distributive scalar-associative scalar-unital CLSStruct
the carrier of (X,Y) is non empty set
bool the carrier of (X,Y) is non empty set
0. Y is zero Element of the carrier of Y
the ZeroF of Y is Element of the carrier of Y
the carrier of X --> (0. Y) is non empty Relation-like the carrier of X -defined the carrier of Y -valued Function-like constant total quasi_total Element of bool [: the carrier of X, the carrier of Y:]
[: the carrier of X, the carrier of Y:] is non empty set
bool [: the carrier of X, the carrier of Y:] is non empty set
f is non empty Relation-like the carrier of X -defined the carrier of Y -valued Function-like total quasi_total additive (X,Y) Element of bool [: the carrier of X, the carrier of Y:]
tseq is Element of the carrier of X
f . tseq is Element of the carrier of Y
X is non empty right_complementable Abelian add-associative right_zeroed discerning reflexive vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like CNORMSTR
Y is non empty right_complementable Abelian add-associative right_zeroed discerning reflexive vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like CNORMSTR
(X,Y) is non empty right_complementable Abelian add-associative right_zeroed constituted-Functions strict vector-distributive scalar-distributive scalar-associative scalar-unital CLSStruct
(X,Y) is non empty functional Element of bool the carrier of ( the carrier of X,Y)
the carrier of X is non empty set
( the carrier of X,Y) is non empty right_complementable Abelian add-associative right_zeroed constituted-Functions strict vector-distributive scalar-distributive scalar-associative scalar-unital CLSStruct
the carrier of Y is non empty set
Funcs ( the carrier of X, the carrier of Y) is non empty functional FUNCTION_DOMAIN of the carrier of X, the carrier of Y
FuncZero ( the carrier of X,Y) is Relation-like the carrier of X -defined the carrier of Y -valued Function-like total quasi_total Element of Funcs ( the carrier of X, the carrier of Y)
FuncAdd ( the carrier of X,Y) is non empty Relation-like [:(Funcs ( the carrier of X, the carrier of Y)),(Funcs ( the carrier of X, the carrier of Y)):] -defined Funcs ( the carrier of X, the carrier of Y) -valued Function-like total quasi_total Function-yielding V34() Element of bool [:[:(Funcs ( the carrier of X, the carrier of Y)),(Funcs ( the carrier of X, the carrier of Y)):],(Funcs ( the carrier of X, the carrier of Y)):]
[:(Funcs ( the carrier of X, the carrier of Y)),(Funcs ( the carrier of X, the carrier of Y)):] is non empty set
[:[:(Funcs ( the carrier of X, the carrier of Y)),(Funcs ( the carrier of X, the carrier of Y)):],(Funcs ( the carrier of X, the carrier of Y)):] is non empty set
bool [:[:(Funcs ( the carrier of X, the carrier of Y)),(Funcs ( the carrier of X, the carrier of Y)):],(Funcs ( the carrier of X, the carrier of Y)):] is non empty set
( the carrier of X,Y) is non empty Relation-like [:COMPLEX,(Funcs ( the carrier of X, the carrier of Y)):] -defined Funcs ( the carrier of X, the carrier of Y) -valued Function-like total quasi_total Function-yielding V34() Element of bool [:[:COMPLEX,(Funcs ( the carrier of X, the carrier of Y)):],(Funcs ( the carrier of X, the carrier of Y)):]
[:COMPLEX,(Funcs ( the carrier of X, the carrier of Y)):] is non empty set
[:[:COMPLEX,(Funcs ( the carrier of X, the carrier of Y)):],(Funcs ( the carrier of X, the carrier of Y)):] is non empty set
bool [:[:COMPLEX,(Funcs ( the carrier of X, the carrier of Y)):],(Funcs ( the carrier of X, the carrier of Y)):] is non empty set
CLSStruct(# (Funcs ( the carrier of X, the carrier of Y)),(FuncZero ( the carrier of X,Y)),(FuncAdd ( the carrier of X,Y)),( the carrier of X,Y) #) is non empty strict CLSStruct
the carrier of ( the carrier of X,Y) is non empty set
bool the carrier of ( the carrier of X,Y) is non empty set
Zero_ ((X,Y),( the carrier of X,Y)) is Relation-like Function-like Element of (X,Y)
Add_ ((X,Y),( the carrier of X,Y)) is non empty Relation-like [:(X,Y),(X,Y):] -defined (X,Y) -valued Function-like total quasi_total Function-yielding V34() Element of bool [:[:(X,Y),(X,Y):],(X,Y):]
[:(X,Y),(X,Y):] is non empty set
[:[:(X,Y),(X,Y):],(X,Y):] is non empty set
bool [:[:(X,Y),(X,Y):],(X,Y):] is non empty set
Mult_ ((X,Y),( the carrier of X,Y)) is non empty Relation-like [:COMPLEX,(X,Y):] -defined (X,Y) -valued Function-like total quasi_total Function-yielding V34() Element of bool [:[:COMPLEX,(X,Y):],(X,Y):]
[:COMPLEX,(X,Y):] is non empty set
[:[:COMPLEX,(X,Y):],(X,Y):] is non empty set
bool [:[:COMPLEX,(X,Y):],(X,Y):] is non empty set
CLSStruct(# (X,Y),(Zero_ ((X,Y),( the carrier of X,Y))),(Add_ ((X,Y),( the carrier of X,Y))),(Mult_ ((X,Y),( the carrier of X,Y))) #) is non empty right_complementable Abelian add-associative right_zeroed strict vector-distributive scalar-distributive scalar-associative scalar-unital CLSStruct
(X,Y) is non empty Element of bool the carrier of (X,Y)
the carrier of (X,Y) is non empty set
bool the carrier of (X,Y) is non empty set
f is Relation-like Function-like Element of the carrier of (X,Y)
tseq is Relation-like Function-like Element of the carrier of (X,Y)
f + tseq is Relation-like Function-like Element of the carrier of (X,Y)
the addF of (X,Y) is non empty Relation-like [: the carrier of (X,Y), the carrier of (X,Y):] -defined the carrier of (X,Y) -valued Function-like total quasi_total Element of bool [:[: the carrier of (X,Y), the carrier of (X,Y):], the carrier of (X,Y):]
[: the carrier of (X,Y), the carrier of (X,Y):] is non empty set
[:[: the carrier of (X,Y), the carrier of (X,Y):], the carrier of (X,Y):] is non empty set
bool [:[: the carrier of (X,Y), the carrier of (X,Y):], the carrier of (X,Y):] is non empty set
the addF of (X,Y) . (f,tseq) is Relation-like Function-like Element of the carrier of (X,Y)
[f,tseq] is set
the addF of (X,Y) . [f,tseq] is set
[: the carrier of X, the carrier of Y:] is non empty set
bool [: the carrier of X, the carrier of Y:] is non empty set
tseq is non empty Relation-like the carrier of X -defined the carrier of Y -valued Function-like total quasi_total additive (X,Y) Element of bool [: the carrier of X, the carrier of Y:]
tseq is non empty Relation-like the carrier of X -defined the carrier of Y -valued Function-like total quasi_total additive (X,Y) (X,Y) Element of bool [: the carrier of X, the carrier of Y:]
tv is complex real ext-real Element of REAL
e is non empty Relation-like the carrier of X -defined the carrier of Y -valued Function-like total quasi_total additive (X,Y) (X,Y) Element of bool [: the carrier of X, the carrier of Y:]
m is complex real ext-real Element of REAL
m + tv is complex real ext-real Element of REAL
n is complex real ext-real Element of REAL
n is Element of the carrier of X
e . n is Element of the carrier of Y
tseq . n is Element of the carrier of Y
(e . n) + (tseq . n) is Element of the carrier of Y
the addF of Y is non empty Relation-like [: the carrier of Y, the carrier of Y:] -defined the carrier of Y -valued Function-like total quasi_total Element of bool [:[: the carrier of Y, the carrier of Y:], the carrier of Y:]
[: the carrier of Y, the carrier of Y:] is non empty set
[:[: the carrier of Y, the carrier of Y:], the carrier of Y:] is non empty set
bool [:[: the carrier of Y, the carrier of Y:], the carrier of Y:] is non empty set
the addF of Y . ((e . n),(tseq . n)) is Element of the carrier of Y
[(e . n),(tseq . n)] is set
the addF of Y . [(e . n),(tseq . n)] is set
||.((e . n) + (tseq . n)).|| is complex real ext-real Element of REAL
the U9 of Y is non empty Relation-like the carrier of Y -defined REAL -valued Function-like total quasi_total V132() V133() V134() Element of bool [: the carrier of Y,REAL:]
[: the carrier of Y,REAL:] is non empty V132() V133() V134() set
bool [: the carrier of Y,REAL:] is non empty set
the U9 of Y . ((e . n) + (tseq . n)) is complex real ext-real Element of REAL
||.(e . n).|| is complex real ext-real Element of REAL
the U9 of Y . (e . n) is complex real ext-real Element of REAL
||.(tseq . n).|| is complex real ext-real Element of REAL
the U9 of Y . (tseq . n) is complex real ext-real Element of REAL
||.(e . n).|| + ||.(tseq . n).|| is complex real ext-real Element of REAL
||.n.|| is complex real ext-real Element of REAL
the U9 of X is non empty Relation-like the carrier of X -defined REAL -valued Function-like total quasi_total V132() V133() V134() Element of bool [: the carrier of X,REAL:]
[: the carrier of X,REAL:] is non empty V132() V133() V134() set
bool [: the carrier of X,REAL:] is non empty set
the U9 of X . n is complex real ext-real Element of REAL
tv * ||.n.|| is complex real ext-real Element of REAL
m * ||.n.|| is complex real ext-real Element of REAL
(m * ||.n.||) + (tv * ||.n.||) is complex real ext-real Element of REAL
tseq . n is Element of the carrier of Y
||.(tseq . n).|| is complex real ext-real Element of REAL
the U9 of Y . (tseq . n) is complex real ext-real Element of REAL
n * ||.n.|| is complex real ext-real Element of REAL
n is Element of the carrier of X
tseq . n is Element of the carrier of Y
||.(tseq . n).|| is complex real ext-real Element of REAL
the U9 of Y . (tseq . n) is complex real ext-real Element of REAL
||.n.|| is complex real ext-real Element of REAL
the U9 of X . n is complex real ext-real Element of REAL
n * ||.n.|| is complex real ext-real Element of REAL
f is complex set
tseq is Relation-like Function-like Element of the carrier of (X,Y)
f * tseq is Relation-like Function-like Element of the carrier of (X,Y)
[: the carrier of X, the carrier of Y:] is non empty set
bool [: the carrier of X, the carrier of Y:] is non empty set
tseq is non empty Relation-like the carrier of X -defined the carrier of Y -valued Function-like total quasi_total additive (X,Y) Element of bool [: the carrier of X, the carrier of Y:]
tseq is non empty Relation-like the carrier of X -defined the carrier of Y -valued Function-like total quasi_total additive (X,Y) (X,Y) Element of bool [: the carrier of X, the carrier of Y:]
tv is complex real ext-real Element of REAL
|.f.| is complex real ext-real Element of REAL
|.f.| * tv is complex real ext-real Element of REAL
e is Element of the carrier of X
tseq . e is Element of the carrier of Y
||.(tseq . e).|| is complex real ext-real Element of REAL
the U9 of Y is non empty Relation-like the carrier of Y -defined REAL -valued Function-like total quasi_total V132() V133() V134() Element of bool [: the carrier of Y,REAL:]
[: the carrier of Y,REAL:] is non empty V132() V133() V134() set
bool [: the carrier of Y,REAL:] is non empty set
the U9 of Y . (tseq . e) is complex real ext-real Element of REAL
|.f.| * ||.(tseq . e).|| is complex real ext-real Element of REAL
||.e.|| is complex real ext-real Element of REAL
the U9 of X is non empty Relation-like the carrier of X -defined REAL -valued Function-like total quasi_total V132() V133() V134() Element of bool [: the carrier of X,REAL:]
[: the carrier of X,REAL:] is non empty V132() V133() V134() set
bool [: the carrier of X,REAL:] is non empty set
the U9 of X . e is complex real ext-real Element of REAL
tv * ||.e.|| is complex real ext-real Element of REAL
|.f.| * (tv * ||.e.||) is complex real ext-real Element of REAL
f * (tseq . e) is Element of the carrier of Y
||.(f * (tseq . e)).|| is complex real ext-real Element of REAL
the U9 of Y . (f * (tseq . e)) is complex real ext-real Element of REAL
tseq . e is Element of the carrier of Y
||.(tseq . e).|| is complex real ext-real Element of REAL
the U9 of Y . (tseq . e) is complex real ext-real Element of REAL
(|.f.| * tv) * ||.e.|| is complex real ext-real Element of REAL
e is Element of the carrier of X
tseq . e is Element of the carrier of Y
||.(tseq . e).|| is complex real ext-real Element of REAL
the U9 of Y . (tseq . e) is complex real ext-real Element of REAL
||.e.|| is complex real ext-real Element of REAL
the U9 of X . e is complex real ext-real Element of REAL
(|.f.| * tv) * ||.e.|| is complex real ext-real Element of REAL
X is non empty right_complementable Abelian add-associative right_zeroed discerning reflexive vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like CNORMSTR
Y is non empty right_complementable Abelian add-associative right_zeroed discerning reflexive vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like CNORMSTR
(X,Y) is non empty Element of bool the carrier of (X,Y)
(X,Y) is non empty right_complementable Abelian add-associative right_zeroed constituted-Functions strict vector-distributive scalar-distributive scalar-associative scalar-unital CLSStruct
(X,Y) is non empty functional Element of bool the carrier of ( the carrier of X,Y)
the carrier of X is non empty set
( the carrier of X,Y) is non empty right_complementable Abelian add-associative right_zeroed constituted-Functions strict vector-distributive scalar-distributive scalar-associative scalar-unital CLSStruct
the carrier of Y is non empty set
Funcs ( the carrier of X, the carrier of Y) is non empty functional FUNCTION_DOMAIN of the carrier of X, the carrier of Y
FuncZero ( the carrier of X,Y) is Relation-like the carrier of X -defined the carrier of Y -valued Function-like total quasi_total Element of Funcs ( the carrier of X, the carrier of Y)
FuncAdd ( the carrier of X,Y) is non empty Relation-like [:(Funcs ( the carrier of X, the carrier of Y)),(Funcs ( the carrier of X, the carrier of Y)):] -defined Funcs ( the carrier of X, the carrier of Y) -valued Function-like total quasi_total Function-yielding V34() Element of bool [:[:(Funcs ( the carrier of X, the carrier of Y)),(Funcs ( the carrier of X, the carrier of Y)):],(Funcs ( the carrier of X, the carrier of Y)):]
[:(Funcs ( the carrier of X, the carrier of Y)),(Funcs ( the carrier of X, the carrier of Y)):] is non empty set
[:[:(Funcs ( the carrier of X, the carrier of Y)),(Funcs ( the carrier of X, the carrier of Y)):],(Funcs ( the carrier of X, the carrier of Y)):] is non empty set
bool [:[:(Funcs ( the carrier of X, the carrier of Y)),(Funcs ( the carrier of X, the carrier of Y)):],(Funcs ( the carrier of X, the carrier of Y)):] is non empty set
( the carrier of X,Y) is non empty Relation-like [:COMPLEX,(Funcs ( the carrier of X, the carrier of Y)):] -defined Funcs ( the carrier of X, the carrier of Y) -valued Function-like total quasi_total Function-yielding V34() Element of bool [:[:COMPLEX,(Funcs ( the carrier of X, the carrier of Y)):],(Funcs ( the carrier of X, the carrier of Y)):]
[:COMPLEX,(Funcs ( the carrier of X, the carrier of Y)):] is non empty set
[:[:COMPLEX,(Funcs ( the carrier of X, the carrier of Y)):],(Funcs ( the carrier of X, the carrier of Y)):] is non empty set
bool [:[:COMPLEX,(Funcs ( the carrier of X, the carrier of Y)):],(Funcs ( the carrier of X, the carrier of Y)):] is non empty set
CLSStruct(# (Funcs ( the carrier of X, the carrier of Y)),(FuncZero ( the carrier of X,Y)),(FuncAdd ( the carrier of X,Y)),( the carrier of X,Y) #) is non empty strict CLSStruct
the carrier of ( the carrier of X,Y) is non empty set
bool the carrier of ( the carrier of X,Y) is non empty set
Zero_ ((X,Y),( the carrier of X,Y)) is Relation-like Function-like Element of (X,Y)
Add_ ((X,Y),( the carrier of X,Y)) is non empty Relation-like [:(X,Y),(X,Y):] -defined (X,Y) -valued Function-like total quasi_total Function-yielding V34() Element of bool [:[:(X,Y),(X,Y):],(X,Y):]
[:(X,Y),(X,Y):] is non empty set
[:[:(X,Y),(X,Y):],(X,Y):] is non empty set
bool [:[:(X,Y),(X,Y):],(X,Y):] is non empty set
Mult_ ((X,Y),( the carrier of X,Y)) is non empty Relation-like [:COMPLEX,(X,Y):] -defined (X,Y) -valued Function-like total quasi_total Function-yielding V34() Element of bool [:[:COMPLEX,(X,Y):],(X,Y):]
[:COMPLEX,(X,Y):] is non empty set
[:[:COMPLEX,(X,Y):],(X,Y):] is non empty set
bool [:[:COMPLEX,(X,Y):],(X,Y):] is non empty set
CLSStruct(# (X,Y),(Zero_ ((X,Y),( the carrier of X,Y))),(Add_ ((X,Y),( the carrier of X,Y))),(Mult_ ((X,Y),( the carrier of X,Y))) #) is non empty right_complementable Abelian add-associative right_zeroed strict vector-distributive scalar-distributive scalar-associative scalar-unital CLSStruct
the carrier of (X,Y) is non empty set
bool the carrier of (X,Y) is non empty set
Zero_ ((X,Y),(X,Y)) is Element of (X,Y)
Add_ ((X,Y),(X,Y)) is non empty Relation-like [:(X,Y),(X,Y):] -defined (X,Y) -valued Function-like total quasi_total Element of bool [:[:(X,Y),(X,Y):],(X,Y):]
[:(X,Y),(X,Y):] is non empty set
[:[:(X,Y),(X,Y):],(X,Y):] is non empty set
bool [:[:(X,Y),(X,Y):],(X,Y):] is non empty set
Mult_ ((X,Y),(X,Y)) is non empty Relation-like [:COMPLEX,(X,Y):] -defined (X,Y) -valued Function-like total quasi_total Element of bool [:[:COMPLEX,(X,Y):],(X,Y):]
[:COMPLEX,(X,Y):] is non empty set
[:[:COMPLEX,(X,Y):],(X,Y):] is non empty set
bool [:[:COMPLEX,(X,Y):],(X,Y):] is non empty set
CLSStruct(# (X,Y),(Zero_ ((X,Y),(X,Y))),(Add_ ((X,Y),(X,Y))),(Mult_ ((X,Y),(X,Y))) #) is non empty strict CLSStruct
X is non empty right_complementable Abelian add-associative right_zeroed discerning reflexive vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like CNORMSTR
Y is non empty right_complementable Abelian add-associative right_zeroed discerning reflexive vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like CNORMSTR
(X,Y) is non empty Element of bool the carrier of (X,Y)
(X,Y) is non empty right_complementable Abelian add-associative right_zeroed constituted-Functions strict vector-distributive scalar-distributive scalar-associative scalar-unital CLSStruct
(X,Y) is non empty functional Element of bool the carrier of ( the carrier of X,Y)
the carrier of X is non empty set
( the carrier of X,Y) is non empty right_complementable Abelian add-associative right_zeroed constituted-Functions strict vector-distributive scalar-distributive scalar-associative scalar-unital CLSStruct
the carrier of Y is non empty set
Funcs ( the carrier of X, the carrier of Y) is non empty functional FUNCTION_DOMAIN of the carrier of X, the carrier of Y
FuncZero ( the carrier of X,Y) is Relation-like the carrier of X -defined the carrier of Y -valued Function-like total quasi_total Element of Funcs ( the carrier of X, the carrier of Y)
FuncAdd ( the carrier of X,Y) is non empty Relation-like [:(Funcs ( the carrier of X, the carrier of Y)),(Funcs ( the carrier of X, the carrier of Y)):] -defined Funcs ( the carrier of X, the carrier of Y) -valued Function-like total quasi_total Function-yielding V34() Element of bool [:[:(Funcs ( the carrier of X, the carrier of Y)),(Funcs ( the carrier of X, the carrier of Y)):],(Funcs ( the carrier of X, the carrier of Y)):]
[:(Funcs ( the carrier of X, the carrier of Y)),(Funcs ( the carrier of X, the carrier of Y)):] is non empty set
[:[:(Funcs ( the carrier of X, the carrier of Y)),(Funcs ( the carrier of X, the carrier of Y)):],(Funcs ( the carrier of X, the carrier of Y)):] is non empty set
bool [:[:(Funcs ( the carrier of X, the carrier of Y)),(Funcs ( the carrier of X, the carrier of Y)):],(Funcs ( the carrier of X, the carrier of Y)):] is non empty set
( the carrier of X,Y) is non empty Relation-like [:COMPLEX,(Funcs ( the carrier of X, the carrier of Y)):] -defined Funcs ( the carrier of X, the carrier of Y) -valued Function-like total quasi_total Function-yielding V34() Element of bool [:[:COMPLEX,(Funcs ( the carrier of X, the carrier of Y)):],(Funcs ( the carrier of X, the carrier of Y)):]
[:COMPLEX,(Funcs ( the carrier of X, the carrier of Y)):] is non empty set
[:[:COMPLEX,(Funcs ( the carrier of X, the carrier of Y)):],(Funcs ( the carrier of X, the carrier of Y)):] is non empty set
bool [:[:COMPLEX,(Funcs ( the carrier of X, the carrier of Y)):],(Funcs ( the carrier of X, the carrier of Y)):] is non empty set
CLSStruct(# (Funcs ( the carrier of X, the carrier of Y)),(FuncZero ( the carrier of X,Y)),(FuncAdd ( the carrier of X,Y)),( the carrier of X,Y) #) is non empty strict CLSStruct
the carrier of ( the carrier of X,Y) is non empty set
bool the carrier of ( the carrier of X,Y) is non empty set
Zero_ ((X,Y),( the carrier of X,Y)) is Relation-like Function-like Element of (X,Y)
Add_ ((X,Y),( the carrier of X,Y)) is non empty Relation-like [:(X,Y),(X,Y):] -defined (X,Y) -valued Function-like total quasi_total Function-yielding V34() Element of bool [:[:(X,Y),(X,Y):],(X,Y):]
[:(X,Y),(X,Y):] is non empty set
[:[:(X,Y),(X,Y):],(X,Y):] is non empty set
bool [:[:(X,Y),(X,Y):],(X,Y):] is non empty set
Mult_ ((X,Y),( the carrier of X,Y)) is non empty Relation-like [:COMPLEX,(X,Y):] -defined (X,Y) -valued Function-like total quasi_total Function-yielding V34() Element of bool [:[:COMPLEX,(X,Y):],(X,Y):]
[:COMPLEX,(X,Y):] is non empty set
[:[:COMPLEX,(X,Y):],(X,Y):] is non empty set
bool [:[:COMPLEX,(X,Y):],(X,Y):] is non empty set
CLSStruct(# (X,Y),(Zero_ ((X,Y),( the carrier of X,Y))),(Add_ ((X,Y),( the carrier of X,Y))),(Mult_ ((X,Y),( the carrier of X,Y))) #) is non empty right_complementable Abelian add-associative right_zeroed strict vector-distributive scalar-distributive scalar-associative scalar-unital CLSStruct
the carrier of (X,Y) is non empty set
bool the carrier of (X,Y) is non empty set
Zero_ ((X,Y),(X,Y)) is Element of (X,Y)
Add_ ((X,Y),(X,Y)) is non empty Relation-like [:(X,Y),(X,Y):] -defined (X,Y) -valued Function-like total quasi_total Element of bool [:[:(X,Y),(X,Y):],(X,Y):]
[:(X,Y),(X,Y):] is non empty set
[:[:(X,Y),(X,Y):],(X,Y):] is non empty set
bool [:[:(X,Y),(X,Y):],(X,Y):] is non empty set
Mult_ ((X,Y),(X,Y)) is non empty Relation-like [:COMPLEX,(X,Y):] -defined (X,Y) -valued Function-like total quasi_total Element of bool [:[:COMPLEX,(X,Y):],(X,Y):]
[:COMPLEX,(X,Y):] is non empty set
[:[:COMPLEX,(X,Y):],(X,Y):] is non empty set
bool [:[:COMPLEX,(X,Y):],(X,Y):] is non empty set
CLSStruct(# (X,Y),(Zero_ ((X,Y),(X,Y))),(Add_ ((X,Y),(X,Y))),(Mult_ ((X,Y),(X,Y))) #) is non empty strict CLSStruct
X is non empty right_complementable Abelian add-associative right_zeroed discerning reflexive vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like CNORMSTR
Y is non empty right_complementable Abelian add-associative right_zeroed discerning reflexive vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like CNORMSTR
(X,Y) is non empty Element of bool the carrier of (X,Y)
(X,Y) is non empty right_complementable Abelian add-associative right_zeroed constituted-Functions strict vector-distributive scalar-distributive scalar-associative scalar-unital CLSStruct
(X,Y) is non empty functional Element of bool the carrier of ( the carrier of X,Y)
the carrier of X is non empty set
( the carrier of X,Y) is non empty right_complementable Abelian add-associative right_zeroed constituted-Functions strict vector-distributive scalar-distributive scalar-associative scalar-unital CLSStruct
the carrier of Y is non empty set
Funcs ( the carrier of X, the carrier of Y) is non empty functional FUNCTION_DOMAIN of the carrier of X, the carrier of Y
FuncZero ( the carrier of X,Y) is Relation-like the carrier of X -defined the carrier of Y -valued Function-like total quasi_total Element of Funcs ( the carrier of X, the carrier of Y)
FuncAdd ( the carrier of X,Y) is non empty Relation-like [:(Funcs ( the carrier of X, the carrier of Y)),(Funcs ( the carrier of X, the carrier of Y)):] -defined Funcs ( the carrier of X, the carrier of Y) -valued Function-like total quasi_total Function-yielding V34() Element of bool [:[:(Funcs ( the carrier of X, the carrier of Y)),(Funcs ( the carrier of X, the carrier of Y)):],(Funcs ( the carrier of X, the carrier of Y)):]
[:(Funcs ( the carrier of X, the carrier of Y)),(Funcs ( the carrier of X, the carrier of Y)):] is non empty set
[:[:(Funcs ( the carrier of X, the carrier of Y)),(Funcs ( the carrier of X, the carrier of Y)):],(Funcs ( the carrier of X, the carrier of Y)):] is non empty set
bool [:[:(Funcs ( the carrier of X, the carrier of Y)),(Funcs ( the carrier of X, the carrier of Y)):],(Funcs ( the carrier of X, the carrier of Y)):] is non empty set
( the carrier of X,Y) is non empty Relation-like [:COMPLEX,(Funcs ( the carrier of X, the carrier of Y)):] -defined Funcs ( the carrier of X, the carrier of Y) -valued Function-like total quasi_total Function-yielding V34() Element of bool [:[:COMPLEX,(Funcs ( the carrier of X, the carrier of Y)):],(Funcs ( the carrier of X, the carrier of Y)):]
[:COMPLEX,(Funcs ( the carrier of X, the carrier of Y)):] is non empty set
[:[:COMPLEX,(Funcs ( the carrier of X, the carrier of Y)):],(Funcs ( the carrier of X, the carrier of Y)):] is non empty set
bool [:[:COMPLEX,(Funcs ( the carrier of X, the carrier of Y)):],(Funcs ( the carrier of X, the carrier of Y)):] is non empty set
CLSStruct(# (Funcs ( the carrier of X, the carrier of Y)),(FuncZero ( the carrier of X,Y)),(FuncAdd ( the carrier of X,Y)),( the carrier of X,Y) #) is non empty strict CLSStruct
the carrier of ( the carrier of X,Y) is non empty set
bool the carrier of ( the carrier of X,Y) is non empty set
Zero_ ((X,Y),( the carrier of X,Y)) is Relation-like Function-like Element of (X,Y)
Add_ ((X,Y),( the carrier of X,Y)) is non empty Relation-like [:(X,Y),(X,Y):] -defined (X,Y) -valued Function-like total quasi_total Function-yielding V34() Element of bool [:[:(X,Y),(X,Y):],(X,Y):]
[:(X,Y),(X,Y):] is non empty set
[:[:(X,Y),(X,Y):],(X,Y):] is non empty set
bool [:[:(X,Y),(X,Y):],(X,Y):] is non empty set
Mult_ ((X,Y),( the carrier of X,Y)) is non empty Relation-like [:COMPLEX,(X,Y):] -defined (X,Y) -valued Function-like total quasi_total Function-yielding V34() Element of bool [:[:COMPLEX,(X,Y):],(X,Y):]
[:COMPLEX,(X,Y):] is non empty set
[:[:COMPLEX,(X,Y):],(X,Y):] is non empty set
bool [:[:COMPLEX,(X,Y):],(X,Y):] is non empty set
CLSStruct(# (X,Y),(Zero_ ((X,Y),( the carrier of X,Y))),(Add_ ((X,Y),( the carrier of X,Y))),(Mult_ ((X,Y),( the carrier of X,Y))) #) is non empty right_complementable Abelian add-associative right_zeroed strict vector-distributive scalar-distributive scalar-associative scalar-unital CLSStruct
the carrier of (X,Y) is non empty set
bool the carrier of (X,Y) is non empty set
Zero_ ((X,Y),(X,Y)) is Element of (X,Y)
Add_ ((X,Y),(X,Y)) is non empty Relation-like [:(X,Y),(X,Y):] -defined (X,Y) -valued Function-like total quasi_total Element of bool [:[:(X,Y),(X,Y):],(X,Y):]
[:(X,Y),(X,Y):] is non empty set
[:[:(X,Y),(X,Y):],(X,Y):] is non empty set
bool [:[:(X,Y),(X,Y):],(X,Y):] is non empty set
Mult_ ((X,Y),(X,Y)) is non empty Relation-like [:COMPLEX,(X,Y):] -defined (X,Y) -valued Function-like total quasi_total Element of bool [:[:COMPLEX,(X,Y):],(X,Y):]
[:COMPLEX,(X,Y):] is non empty set
[:[:COMPLEX,(X,Y):],(X,Y):] is non empty set
bool [:[:COMPLEX,(X,Y):],(X,Y):] is non empty set
CLSStruct(# (X,Y),(Zero_ ((X,Y),(X,Y))),(Add_ ((X,Y),(X,Y))),(Mult_ ((X,Y),(X,Y))) #) is non empty right_complementable Abelian add-associative right_zeroed strict vector-distributive scalar-distributive scalar-associative scalar-unital CLSStruct
X is non empty right_complementable Abelian add-associative right_zeroed discerning reflexive vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like CNORMSTR
Y is non empty right_complementable Abelian add-associative right_zeroed discerning reflexive vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like CNORMSTR
(X,Y) is non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital CLSStruct
(X,Y) is non empty Element of bool the carrier of (X,Y)
(X,Y) is non empty right_complementable Abelian add-associative right_zeroed constituted-Functions strict vector-distributive scalar-distributive scalar-associative scalar-unital CLSStruct
(X,Y) is non empty functional Element of bool the carrier of ( the carrier of X,Y)
the carrier of X is non empty set
( the carrier of X,Y) is non empty right_complementable Abelian add-associative right_zeroed constituted-Functions strict vector-distributive scalar-distributive scalar-associative scalar-unital CLSStruct
the carrier of Y is non empty set
Funcs ( the carrier of X, the carrier of Y) is non empty functional FUNCTION_DOMAIN of the carrier of X, the carrier of Y
FuncZero ( the carrier of X,Y) is Relation-like the carrier of X -defined the carrier of Y -valued Function-like total quasi_total Element of Funcs ( the carrier of X, the carrier of Y)
FuncAdd ( the carrier of X,Y) is non empty Relation-like [:(Funcs ( the carrier of X, the carrier of Y)),(Funcs ( the carrier of X, the carrier of Y)):] -defined Funcs ( the carrier of X, the carrier of Y) -valued Function-like total quasi_total Function-yielding V34() Element of bool [:[:(Funcs ( the carrier of X, the carrier of Y)),(Funcs ( the carrier of X, the carrier of Y)):],(Funcs ( the carrier of X, the carrier of Y)):]
[:(Funcs ( the carrier of X, the carrier of Y)),(Funcs ( the carrier of X, the carrier of Y)):] is non empty set
[:[:(Funcs ( the carrier of X, the carrier of Y)),(Funcs ( the carrier of X, the carrier of Y)):],(Funcs ( the carrier of X, the carrier of Y)):] is non empty set
bool [:[:(Funcs ( the carrier of X, the carrier of Y)),(Funcs ( the carrier of X, the carrier of Y)):],(Funcs ( the carrier of X, the carrier of Y)):] is non empty set
( the carrier of X,Y) is non empty Relation-like [:COMPLEX,(Funcs ( the carrier of X, the carrier of Y)):] -defined Funcs ( the carrier of X, the carrier of Y) -valued Function-like total quasi_total Function-yielding V34() Element of bool [:[:COMPLEX,(Funcs ( the carrier of X, the carrier of Y)):],(Funcs ( the carrier of X, the carrier of Y)):]
[:COMPLEX,(Funcs ( the carrier of X, the carrier of Y)):] is non empty set
[:[:COMPLEX,(Funcs ( the carrier of X, the carrier of Y)):],(Funcs ( the carrier of X, the carrier of Y)):] is non empty set
bool [:[:COMPLEX,(Funcs ( the carrier of X, the carrier of Y)):],(Funcs ( the carrier of X, the carrier of Y)):] is non empty set
CLSStruct(# (Funcs ( the carrier of X, the carrier of Y)),(FuncZero ( the carrier of X,Y)),(FuncAdd ( the carrier of X,Y)),( the carrier of X,Y) #) is non empty strict CLSStruct
the carrier of ( the carrier of X,Y) is non empty set
bool the carrier of ( the carrier of X,Y) is non empty set
Zero_ ((X,Y),( the carrier of X,Y)) is Relation-like Function-like Element of (X,Y)
Add_ ((X,Y),( the carrier of X,Y)) is non empty Relation-like [:(X,Y),(X,Y):] -defined (X,Y) -valued Function-like total quasi_total Function-yielding V34() Element of bool [:[:(X,Y),(X,Y):],(X,Y):]
[:(X,Y),(X,Y):] is non empty set
[:[:(X,Y),(X,Y):],(X,Y):] is non empty set
bool [:[:(X,Y),(X,Y):],(X,Y):] is non empty set
Mult_ ((X,Y),( the carrier of X,Y)) is non empty Relation-like [:COMPLEX,(X,Y):] -defined (X,Y) -valued Function-like total quasi_total Function-yielding V34() Element of bool [:[:COMPLEX,(X,Y):],(X,Y):]
[:COMPLEX,(X,Y):] is non empty set
[:[:COMPLEX,(X,Y):],(X,Y):] is non empty set
bool [:[:COMPLEX,(X,Y):],(X,Y):] is non empty set
CLSStruct(# (X,Y),(Zero_ ((X,Y),( the carrier of X,Y))),(Add_ ((X,Y),( the carrier of X,Y))),(Mult_ ((X,Y),( the carrier of X,Y))) #) is non empty right_complementable Abelian add-associative right_zeroed strict vector-distributive scalar-distributive scalar-associative scalar-unital CLSStruct
the carrier of (X,Y) is non empty set
bool the carrier of (X,Y) is non empty set
Zero_ ((X,Y),(X,Y)) is Element of (X,Y)
Add_ ((X,Y),(X,Y)) is non empty Relation-like [:(X,Y),(X,Y):] -defined (X,Y) -valued Function-like total quasi_total Element of bool [:[:(X,Y),(X,Y):],(X,Y):]
[:(X,Y),(X,Y):] is non empty set
[:[:(X,Y),(X,Y):],(X,Y):] is non empty set
bool [:[:(X,Y),(X,Y):],(X,Y):] is non empty set
Mult_ ((X,Y),(X,Y)) is non empty Relation-like [:COMPLEX,(X,Y):] -defined (X,Y) -valued Function-like total quasi_total Element of bool [:[:COMPLEX,(X,Y):],(X,Y):]
[:COMPLEX,(X,Y):] is non empty set
[:[:COMPLEX,(X,Y):],(X,Y):] is non empty set
bool [:[:COMPLEX,(X,Y):],(X,Y):] is non empty set
CLSStruct(# (X,Y),(Zero_ ((X,Y),(X,Y))),(Add_ ((X,Y),(X,Y))),(Mult_ ((X,Y),(X,Y))) #) is non empty right_complementable Abelian add-associative right_zeroed strict vector-distributive scalar-distributive scalar-associative scalar-unital CLSStruct
X is non empty right_complementable Abelian add-associative right_zeroed discerning reflexive vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like CNORMSTR
Y is non empty right_complementable Abelian add-associative right_zeroed discerning reflexive vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like CNORMSTR
(X,Y) is non empty right_complementable Abelian add-associative right_zeroed strict vector-distributive scalar-distributive scalar-associative scalar-unital CLSStruct
(X,Y) is non empty Element of bool the carrier of (X,Y)
(X,Y) is non empty right_complementable Abelian add-associative right_zeroed constituted-Functions strict vector-distributive scalar-distributive scalar-associative scalar-unital CLSStruct
(X,Y) is non empty functional Element of bool the carrier of ( the carrier of X,Y)
the carrier of X is non empty set
( the carrier of X,Y) is non empty right_complementable Abelian add-associative right_zeroed constituted-Functions strict vector-distributive scalar-distributive scalar-associative scalar-unital CLSStruct
the carrier of Y is non empty set
Funcs ( the carrier of X, the carrier of Y) is non empty functional FUNCTION_DOMAIN of the carrier of X, the carrier of Y
FuncZero ( the carrier of X,Y) is Relation-like the carrier of X -defined the carrier of Y -valued Function-like total quasi_total Element of Funcs ( the carrier of X, the carrier of Y)
FuncAdd ( the carrier of X,Y) is non empty Relation-like [:(Funcs ( the carrier of X, the carrier of Y)),(Funcs ( the carrier of X, the carrier of Y)):] -defined Funcs ( the carrier of X, the carrier of Y) -valued Function-like total quasi_total Function-yielding V34() Element of bool [:[:(Funcs ( the carrier of X, the carrier of Y)),(Funcs ( the carrier of X, the carrier of Y)):],(Funcs ( the carrier of X, the carrier of Y)):]
[:(Funcs ( the carrier of X, the carrier of Y)),(Funcs ( the carrier of X, the carrier of Y)):] is non empty set
[:[:(Funcs ( the carrier of X, the carrier of Y)),(Funcs ( the carrier of X, the carrier of Y)):],(Funcs ( the carrier of X, the carrier of Y)):] is non empty set
bool [:[:(Funcs ( the carrier of X, the carrier of Y)),(Funcs ( the carrier of X, the carrier of Y)):],(Funcs ( the carrier of X, the carrier of Y)):] is non empty set
( the carrier of X,Y) is non empty Relation-like [:COMPLEX,(Funcs ( the carrier of X, the carrier of Y)):] -defined Funcs ( the carrier of X, the carrier of Y) -valued Function-like total quasi_total Function-yielding V34() Element of bool [:[:COMPLEX,(Funcs ( the carrier of X, the carrier of Y)):],(Funcs ( the carrier of X, the carrier of Y)):]
[:COMPLEX,(Funcs ( the carrier of X, the carrier of Y)):] is non empty set
[:[:COMPLEX,(Funcs ( the carrier of X, the carrier of Y)):],(Funcs ( the carrier of X, the carrier of Y)):] is non empty set
bool [:[:COMPLEX,(Funcs ( the carrier of X, the carrier of Y)):],(Funcs ( the carrier of X, the carrier of Y)):] is non empty set
CLSStruct(# (Funcs ( the carrier of X, the carrier of Y)),(FuncZero ( the carrier of X,Y)),(FuncAdd ( the carrier of X,Y)),( the carrier of X,Y) #) is non empty strict CLSStruct
the carrier of ( the carrier of X,Y) is non empty set
bool the carrier of ( the carrier of X,Y) is non empty set
Zero_ ((X,Y),( the carrier of X,Y)) is Relation-like Function-like Element of (X,Y)
Add_ ((X,Y),( the carrier of X,Y)) is non empty Relation-like [:(X,Y),(X,Y):] -defined (X,Y) -valued Function-like total quasi_total Function-yielding V34() Element of bool [:[:(X,Y),(X,Y):],(X,Y):]
[:(X,Y),(X,Y):] is non empty set
[:[:(X,Y),(X,Y):],(X,Y):] is non empty set
bool [:[:(X,Y),(X,Y):],(X,Y):] is non empty set
Mult_ ((X,Y),( the carrier of X,Y)) is non empty Relation-like [:COMPLEX,(X,Y):] -defined (X,Y) -valued Function-like total quasi_total Function-yielding V34() Element of bool [:[:COMPLEX,(X,Y):],(X,Y):]
[:COMPLEX,(X,Y):] is non empty set
[:[:COMPLEX,(X,Y):],(X,Y):] is non empty set
bool [:[:COMPLEX,(X,Y):],(X,Y):] is non empty set
CLSStruct(# (X,Y),(Zero_ ((X,Y),( the carrier of X,Y))),(Add_ ((X,Y),( the carrier of X,Y))),(Mult_ ((X,Y),( the carrier of X,Y))) #) is non empty right_complementable Abelian add-associative right_zeroed strict vector-distributive scalar-distributive scalar-associative scalar-unital CLSStruct
the carrier of (X,Y) is non empty set
bool the carrier of (X,Y) is non empty set
Zero_ ((X,Y),(X,Y)) is Element of (X,Y)
Add_ ((X,Y),(X,Y)) is non empty Relation-like [:(X,Y),(X,Y):] -defined (X,Y) -valued Function-like total quasi_total Element of bool [:[:(X,Y),(X,Y):],(X,Y):]
[:(X,Y),(X,Y):] is non empty set
[:[:(X,Y),(X,Y):],(X,Y):] is non empty set
bool [:[:(X,Y),(X,Y):],(X,Y):] is non empty set
Mult_ ((X,Y),(X,Y)) is non empty Relation-like [:COMPLEX,(X,Y):] -defined (X,Y) -valued Function-like total quasi_total Element of bool [:[:COMPLEX,(X,Y):],(X,Y):]
[:COMPLEX,(X,Y):] is non empty set
[:[:COMPLEX,(X,Y):],(X,Y):] is non empty set
bool [:[:COMPLEX,(X,Y):],(X,Y):] is non empty set
CLSStruct(# (X,Y),(Zero_ ((X,Y),(X,Y))),(Add_ ((X,Y),(X,Y))),(Mult_ ((X,Y),(X,Y))) #) is non empty right_complementable Abelian add-associative right_zeroed strict vector-distributive scalar-distributive scalar-associative scalar-unital CLSStruct
the carrier of (X,Y) is non empty set
vseq is Element of the carrier of (X,Y)
f is Element of the carrier of (X,Y)
X is non empty right_complementable Abelian add-associative right_zeroed discerning reflexive vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like CNORMSTR
Y is non empty right_complementable Abelian add-associative right_zeroed discerning reflexive vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like CNORMSTR
(X,Y) is non empty right_complementable Abelian add-associative right_zeroed strict vector-distributive scalar-distributive scalar-associative scalar-unital CLSStruct
(X,Y) is non empty Element of bool the carrier of (X,Y)
(X,Y) is non empty right_complementable Abelian add-associative right_zeroed constituted-Functions strict vector-distributive scalar-distributive scalar-associative scalar-unital CLSStruct
(X,Y) is non empty functional Element of bool the carrier of ( the carrier of X,Y)
the carrier of X is non empty set
( the carrier of X,Y) is non empty right_complementable Abelian add-associative right_zeroed constituted-Functions strict vector-distributive scalar-distributive scalar-associative scalar-unital CLSStruct
the carrier of Y is non empty set
Funcs ( the carrier of X, the carrier of Y) is non empty functional FUNCTION_DOMAIN of the carrier of X, the carrier of Y
FuncZero ( the carrier of X,Y) is Relation-like the carrier of X -defined the carrier of Y -valued Function-like total quasi_total Element of Funcs ( the carrier of X, the carrier of Y)
FuncAdd ( the carrier of X,Y) is non empty Relation-like [:(Funcs ( the carrier of X, the carrier of Y)),(Funcs ( the carrier of X, the carrier of Y)):] -defined Funcs ( the carrier of X, the carrier of Y) -valued Function-like total quasi_total Function-yielding V34() Element of bool [:[:(Funcs ( the carrier of X, the carrier of Y)),(Funcs ( the carrier of X, the carrier of Y)):],(Funcs ( the carrier of X, the carrier of Y)):]
[:(Funcs ( the carrier of X, the carrier of Y)),(Funcs ( the carrier of X, the carrier of Y)):] is non empty set
[:[:(Funcs ( the carrier of X, the carrier of Y)),(Funcs ( the carrier of X, the carrier of Y)):],(Funcs ( the carrier of X, the carrier of Y)):] is non empty set
bool [:[:(Funcs ( the carrier of X, the carrier of Y)),(Funcs ( the carrier of X, the carrier of Y)):],(Funcs ( the carrier of X, the carrier of Y)):] is non empty set
( the carrier of X,Y) is non empty Relation-like [:COMPLEX,(Funcs ( the carrier of X, the carrier of Y)):] -defined Funcs ( the carrier of X, the carrier of Y) -valued Function-like total quasi_total Function-yielding V34() Element of bool [:[:COMPLEX,(Funcs ( the carrier of X, the carrier of Y)):],(Funcs ( the carrier of X, the carrier of Y)):]
[:COMPLEX,(Funcs ( the carrier of X, the carrier of Y)):] is non empty set
[:[:COMPLEX,(Funcs ( the carrier of X, the carrier of Y)):],(Funcs ( the carrier of X, the carrier of Y)):] is non empty set
bool [:[:COMPLEX,(Funcs ( the carrier of X, the carrier of Y)):],(Funcs ( the carrier of X, the carrier of Y)):] is non empty set
CLSStruct(# (Funcs ( the carrier of X, the carrier of Y)),(FuncZero ( the carrier of X,Y)),(FuncAdd ( the carrier of X,Y)),( the carrier of X,Y) #) is non empty strict CLSStruct
the carrier of ( the carrier of X,Y) is non empty set
bool the carrier of ( the carrier of X,Y) is non empty set
Zero_ ((X,Y),( the carrier of X,Y)) is Relation-like Function-like Element of (X,Y)
Add_ ((X,Y),( the carrier of X,Y)) is non empty Relation-like [:(X,Y),(X,Y):] -defined (X,Y) -valued Function-like total quasi_total Function-yielding V34() Element of bool [:[:(X,Y),(X,Y):],(X,Y):]
[:(X,Y),(X,Y):] is non empty set
[:[:(X,Y),(X,Y):],(X,Y):] is non empty set
bool [:[:(X,Y),(X,Y):],(X,Y):] is non empty set
Mult_ ((X,Y),( the carrier of X,Y)) is non empty Relation-like [:COMPLEX,(X,Y):] -defined (X,Y) -valued Function-like total quasi_total Function-yielding V34() Element of bool [:[:COMPLEX,(X,Y):],(X,Y):]
[:COMPLEX,(X,Y):] is non empty set
[:[:COMPLEX,(X,Y):],(X,Y):] is non empty set
bool [:[:COMPLEX,(X,Y):],(X,Y):] is non empty set
CLSStruct(# (X,Y),(Zero_ ((X,Y),( the carrier of X,Y))),(Add_ ((X,Y),( the carrier of X,Y))),(Mult_ ((X,Y),( the carrier of X,Y))) #) is non empty right_complementable Abelian add-associative right_zeroed strict vector-distributive scalar-distributive scalar-associative scalar-unital CLSStruct
the carrier of (X,Y) is non empty set
bool the carrier of (X,Y) is non empty set
Zero_ ((X,Y),(X,Y)) is Element of (X,Y)
Add_ ((X,Y),(X,Y)) is non empty Relation-like [:(X,Y),(X,Y):] -defined (X,Y) -valued Function-like total quasi_total Element of bool [:[:(X,Y),(X,Y):],(X,Y):]
[:(X,Y),(X,Y):] is non empty set
[:[:(X,Y),(X,Y):],(X,Y):] is non empty set
bool [:[:(X,Y),(X,Y):],(X,Y):] is non empty set
Mult_ ((X,Y),(X,Y)) is non empty Relation-like [:COMPLEX,(X,Y):] -defined (X,Y) -valued Function-like total quasi_total Element of bool [:[:COMPLEX,(X,Y):],(X,Y):]
[:COMPLEX,(X,Y):] is non empty set
[:[:COMPLEX,(X,Y):],(X,Y):] is non empty set
bool [:[:COMPLEX,(X,Y):],(X,Y):] is non empty set
CLSStruct(# (X,Y),(Zero_ ((X,Y),(X,Y))),(Add_ ((X,Y),(X,Y))),(Mult_ ((X,Y),(X,Y))) #) is non empty right_complementable Abelian add-associative right_zeroed strict vector-distributive scalar-distributive scalar-associative scalar-unital CLSStruct
the carrier of (X,Y) is non empty set
vseq is Relation-like Function-like Element of the carrier of (X,Y)
f is Element of the carrier of X
vseq . f is set
[: the carrier of X, the carrier of Y:] is non empty set
bool [: the carrier of X, the carrier of Y:] is non empty set
tseq is non empty Relation-like the carrier of X -defined the carrier of Y -valued Function-like total quasi_total additive (X,Y) Element of bool [: the carrier of X, the carrier of Y:]
tseq . f is Element of the carrier of Y
X is non empty right_complementable Abelian add-associative right_zeroed discerning reflexive vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like CNORMSTR
Y is non empty right_complementable Abelian add-associative right_zeroed discerning reflexive vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like CNORMSTR
(X,Y) is non empty right_complementable Abelian add-associative right_zeroed strict vector-distributive scalar-distributive scalar-associative scalar-unital CLSStruct
(X,Y) is non empty Element of bool the carrier of (X,Y)
(X,Y) is non empty right_complementable Abelian add-associative right_zeroed constituted-Functions strict vector-distributive scalar-distributive scalar-associative scalar-unital CLSStruct
(X,Y) is non empty functional Element of bool the carrier of ( the carrier of X,Y)
the carrier of X is non empty set
( the carrier of X,Y) is non empty right_complementable Abelian add-associative right_zeroed constituted-Functions strict vector-distributive scalar-distributive scalar-associative scalar-unital CLSStruct
the carrier of Y is non empty set
Funcs ( the carrier of X, the carrier of Y) is non empty functional FUNCTION_DOMAIN of the carrier of X, the carrier of Y
FuncZero ( the carrier of X,Y) is Relation-like the carrier of X -defined the carrier of Y -valued Function-like total quasi_total Element of Funcs ( the carrier of X, the carrier of Y)
FuncAdd ( the carrier of X,Y) is non empty Relation-like [:(Funcs ( the carrier of X, the carrier of Y)),(Funcs ( the carrier of X, the carrier of Y)):] -defined Funcs ( the carrier of X, the carrier of Y) -valued Function-like total quasi_total Function-yielding V34() Element of bool [:[:(Funcs ( the carrier of X, the carrier of Y)),(Funcs ( the carrier of X, the carrier of Y)):],(Funcs ( the carrier of X, the carrier of Y)):]
[:(Funcs ( the carrier of X, the carrier of Y)),(Funcs ( the carrier of X, the carrier of Y)):] is non empty set
[:[:(Funcs ( the carrier of X, the carrier of Y)),(Funcs ( the carrier of X, the carrier of Y)):],(Funcs ( the carrier of X, the carrier of Y)):] is non empty set
bool [:[:(Funcs ( the carrier of X, the carrier of Y)),(Funcs ( the carrier of X, the carrier of Y)):],(Funcs ( the carrier of X, the carrier of Y)):] is non empty set
( the carrier of X,Y) is non empty Relation-like [:COMPLEX,(Funcs ( the carrier of X, the carrier of Y)):] -defined Funcs ( the carrier of X, the carrier of Y) -valued Function-like total quasi_total Function-yielding V34() Element of bool [:[:COMPLEX,(Funcs ( the carrier of X, the carrier of Y)):],(Funcs ( the carrier of X, the carrier of Y)):]
[:COMPLEX,(Funcs ( the carrier of X, the carrier of Y)):] is non empty set
[:[:COMPLEX,(Funcs ( the carrier of X, the carrier of Y)):],(Funcs ( the carrier of X, the carrier of Y)):] is non empty set
bool [:[:COMPLEX,(Funcs ( the carrier of X, the carrier of Y)):],(Funcs ( the carrier of X, the carrier of Y)):] is non empty set
CLSStruct(# (Funcs ( the carrier of X, the carrier of Y)),(FuncZero ( the carrier of X,Y)),(FuncAdd ( the carrier of X,Y)),( the carrier of X,Y) #) is non empty strict CLSStruct
the carrier of ( the carrier of X,Y) is non empty set
bool the carrier of ( the carrier of X,Y) is non empty set
Zero_ ((X,Y),( the carrier of X,Y)) is Relation-like Function-like Element of (X,Y)
Add_ ((X,Y),( the carrier of X,Y)) is non empty Relation-like [:(X,Y),(X,Y):] -defined (X,Y) -valued Function-like total quasi_total Function-yielding V34() Element of bool [:[:(X,Y),(X,Y):],(X,Y):]
[:(X,Y),(X,Y):] is non empty set
[:[:(X,Y),(X,Y):],(X,Y):] is non empty set
bool [:[:(X,Y),(X,Y):],(X,Y):] is non empty set
Mult_ ((X,Y),( the carrier of X,Y)) is non empty Relation-like [:COMPLEX,(X,Y):] -defined (X,Y) -valued Function-like total quasi_total Function-yielding V34() Element of bool [:[:COMPLEX,(X,Y):],(X,Y):]
[:COMPLEX,(X,Y):] is non empty set
[:[:COMPLEX,(X,Y):],(X,Y):] is non empty set
bool [:[:COMPLEX,(X,Y):],(X,Y):] is non empty set
CLSStruct(# (X,Y),(Zero_ ((X,Y),( the carrier of X,Y))),(Add_ ((X,Y),( the carrier of X,Y))),(Mult_ ((X,Y),( the carrier of X,Y))) #) is non empty right_complementable Abelian add-associative right_zeroed strict vector-distributive scalar-distributive scalar-associative scalar-unital CLSStruct
the carrier of (X,Y) is non empty set
bool the carrier of (X,Y) is non empty set
Zero_ ((X,Y),(X,Y)) is Element of (X,Y)
Add_ ((X,Y),(X,Y)) is non empty Relation-like [:(X,Y),(X,Y):] -defined (X,Y) -valued Function-like total quasi_total Element of bool [:[:(X,Y),(X,Y):],(X,Y):]
[:(X,Y),(X,Y):] is non empty set
[:[:(X,Y),(X,Y):],(X,Y):] is non empty set
bool [:[:(X,Y),(X,Y):],(X,Y):] is non empty set
Mult_ ((X,Y),(X,Y)) is non empty Relation-like [:COMPLEX,(X,Y):] -defined (X,Y) -valued Function-like total quasi_total Element of bool [:[:COMPLEX,(X,Y):],(X,Y):]
[:COMPLEX,(X,Y):] is non empty set
[:[:COMPLEX,(X,Y):],(X,Y):] is non empty set
bool [:[:COMPLEX,(X,Y):],(X,Y):] is non empty set
CLSStruct(# (X,Y),(Zero_ ((X,Y),(X,Y))),(Add_ ((X,Y),(X,Y))),(Mult_ ((X,Y),(X,Y))) #) is non empty right_complementable Abelian add-associative right_zeroed strict vector-distributive scalar-distributive scalar-associative scalar-unital CLSStruct
the carrier of (X,Y) is non empty set
tseq is Relation-like Function-like Element of the carrier of (X,Y)
vseq is Relation-like Function-like Element of the carrier of (X,Y)
f is Relation-like Function-like Element of the carrier of (X,Y)
vseq + f is Relation-like Function-like Element of the carrier of (X,Y)
the addF of (X,Y) is non empty Relation-like [: the carrier of (X,Y), the carrier of (X,Y):] -defined the carrier of (X,Y) -valued Function-like total quasi_total Element of bool [:[: the carrier of (X,Y), the carrier of (X,Y):], the carrier of (X,Y):]
[: the carrier of (X,Y), the carrier of (X,Y):] is non empty set
[:[: the carrier of (X,Y), the carrier of (X,Y):], the carrier of (X,Y):] is non empty set
bool [:[: the carrier of (X,Y), the carrier of (X,Y):], the carrier of (X,Y):] is non empty set
the addF of (X,Y) . (vseq,f) is Relation-like Function-like Element of the carrier of (X,Y)
[vseq,f] is set
the addF of (X,Y) . [vseq,f] is set
tseq is Relation-like Function-like Element of the carrier of (X,Y)
tseq is Relation-like Function-like Element of the carrier of (X,Y)
tv is Relation-like Function-like Element of the carrier of (X,Y)
tseq + tv is Relation-like Function-like Element of the carrier of (X,Y)
the addF of (X,Y) is non empty Relation-like [: the carrier of (X,Y), the carrier of (X,Y):] -defined the carrier of (X,Y) -valued Function-like total quasi_total Element of bool [:[: the carrier of (X,Y), the carrier of (X,Y):], the carrier of (X,Y):]
[: the carrier of (X,Y), the carrier of (X,Y):] is non empty set
[:[: the carrier of (X,Y), the carrier of (X,Y):], the carrier of (X,Y):] is non empty set
bool [:[: the carrier of (X,Y), the carrier of (X,Y):], the carrier of (X,Y):] is non empty set
the addF of (X,Y) . (tseq,tv) is Relation-like Function-like Element of the carrier of (X,Y)
[tseq,tv] is set
the addF of (X,Y) . [tseq,tv] is set
e is Element of the carrier of X
(X,Y,tseq,e) is Element of the carrier of Y
(X,Y,vseq,e) is Element of the carrier of Y
(X,Y,f,e) is Element of the carrier of Y
(X,Y,vseq,e) + (X,Y,f,e) is Element of the carrier of Y
the addF of Y is non empty Relation-like [: the carrier of Y, the carrier of Y:] -defined the carrier of Y -valued Function-like total quasi_total Element of bool [:[: the carrier of Y, the carrier of Y:], the carrier of Y:]
[: the carrier of Y, the carrier of Y:] is non empty set
[:[: the carrier of Y, the carrier of Y:], the carrier of Y:] is non empty set
bool [:[: the carrier of Y, the carrier of Y:], the carrier of Y:] is non empty set
the addF of Y . ((X,Y,vseq,e),(X,Y,f,e)) is Element of the carrier of Y
[(X,Y,vseq,e),(X,Y,f,e)] is set
the addF of Y . [(X,Y,vseq,e),(X,Y,f,e)] is set
X is non empty right_complementable Abelian add-associative right_zeroed discerning reflexive vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like CNORMSTR
Y is non empty right_complementable Abelian add-associative right_zeroed discerning reflexive vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like CNORMSTR
(X,Y) is non empty right_complementable Abelian add-associative right_zeroed strict vector-distributive scalar-distributive scalar-associative scalar-unital CLSStruct
(X,Y) is non empty Element of bool the carrier of (X,Y)
(X,Y) is non empty right_complementable Abelian add-associative right_zeroed constituted-Functions strict vector-distributive scalar-distributive scalar-associative scalar-unital CLSStruct
(X,Y) is non empty functional Element of bool the carrier of ( the carrier of X,Y)
the carrier of X is non empty set
( the carrier of X,Y) is non empty right_complementable Abelian add-associative right_zeroed constituted-Functions strict vector-distributive scalar-distributive scalar-associative scalar-unital CLSStruct
the carrier of Y is non empty set
Funcs ( the carrier of X, the carrier of Y) is non empty functional FUNCTION_DOMAIN of the carrier of X, the carrier of Y
FuncZero ( the carrier of X,Y) is Relation-like the carrier of X -defined the carrier of Y -valued Function-like total quasi_total Element of Funcs ( the carrier of X, the carrier of Y)
FuncAdd ( the carrier of X,Y) is non empty Relation-like [:(Funcs ( the carrier of X, the carrier of Y)),(Funcs ( the carrier of X, the carrier of Y)):] -defined Funcs ( the carrier of X, the carrier of Y) -valued Function-like total quasi_total Function-yielding V34() Element of bool [:[:(Funcs ( the carrier of X, the carrier of Y)),(Funcs ( the carrier of X, the carrier of Y)):],(Funcs ( the carrier of X, the carrier of Y)):]
[:(Funcs ( the carrier of X, the carrier of Y)),(Funcs ( the carrier of X, the carrier of Y)):] is non empty set
[:[:(Funcs ( the carrier of X, the carrier of Y)),(Funcs ( the carrier of X, the carrier of Y)):],(Funcs ( the carrier of X, the carrier of Y)):] is non empty set
bool [:[:(Funcs ( the carrier of X, the carrier of Y)),(Funcs ( the carrier of X, the carrier of Y)):],(Funcs ( the carrier of X, the carrier of Y)):] is non empty set
( the carrier of X,Y) is non empty Relation-like [:COMPLEX,(Funcs ( the carrier of X, the carrier of Y)):] -defined Funcs ( the carrier of X, the carrier of Y) -valued Function-like total quasi_total Function-yielding V34() Element of bool [:[:COMPLEX,(Funcs ( the carrier of X, the carrier of Y)):],(Funcs ( the carrier of X, the carrier of Y)):]
[:COMPLEX,(Funcs ( the carrier of X, the carrier of Y)):] is non empty set
[:[:COMPLEX,(Funcs ( the carrier of X, the carrier of Y)):],(Funcs ( the carrier of X, the carrier of Y)):] is non empty set
bool [:[:COMPLEX,(Funcs ( the carrier of X, the carrier of Y)):],(Funcs ( the carrier of X, the carrier of Y)):] is non empty set
CLSStruct(# (Funcs ( the carrier of X, the carrier of Y)),(FuncZero ( the carrier of X,Y)),(FuncAdd ( the carrier of X,Y)),( the carrier of X,Y) #) is non empty strict CLSStruct
the carrier of ( the carrier of X,Y) is non empty set
bool the carrier of ( the carrier of X,Y) is non empty set
Zero_ ((X,Y),( the carrier of X,Y)) is Relation-like Function-like Element of (X,Y)
Add_ ((X,Y),( the carrier of X,Y)) is non empty Relation-like [:(X,Y),(X,Y):] -defined (X,Y) -valued Function-like total quasi_total Function-yielding V34() Element of bool [:[:(X,Y),(X,Y):],(X,Y):]
[:(X,Y),(X,Y):] is non empty set
[:[:(X,Y),(X,Y):],(X,Y):] is non empty set
bool [:[:(X,Y),(X,Y):],(X,Y):] is non empty set
Mult_ ((X,Y),( the carrier of X,Y)) is non empty Relation-like [:COMPLEX,(X,Y):] -defined (X,Y) -valued Function-like total quasi_total Function-yielding V34() Element of bool [:[:COMPLEX,(X,Y):],(X,Y):]
[:COMPLEX,(X,Y):] is non empty set
[:[:COMPLEX,(X,Y):],(X,Y):] is non empty set
bool [:[:COMPLEX,(X,Y):],(X,Y):] is non empty set
CLSStruct(# (X,Y),(Zero_ ((X,Y),( the carrier of X,Y))),(Add_ ((X,Y),( the carrier of X,Y))),(Mult_ ((X,Y),( the carrier of X,Y))) #) is non empty right_complementable Abelian add-associative right_zeroed strict vector-distributive scalar-distributive scalar-associative scalar-unital CLSStruct
the carrier of (X,Y) is non empty set
bool the carrier of (X,Y) is non empty set
Zero_ ((X,Y),(X,Y)) is Element of (X,Y)
Add_ ((X,Y),(X,Y)) is non empty Relation-like [:(X,Y),(X,Y):] -defined (X,Y) -valued Function-like total quasi_total Element of bool [:[:(X,Y),(X,Y):],(X,Y):]
[:(X,Y),(X,Y):] is non empty set
[:[:(X,Y),(X,Y):],(X,Y):] is non empty set
bool [:[:(X,Y),(X,Y):],(X,Y):] is non empty set
Mult_ ((X,Y),(X,Y)) is non empty Relation-like [:COMPLEX,(X,Y):] -defined (X,Y) -valued Function-like total quasi_total Element of bool [:[:COMPLEX,(X,Y):],(X,Y):]
[:COMPLEX,(X,Y):] is non empty set
[:[:COMPLEX,(X,Y):],(X,Y):] is non empty set
bool [:[:COMPLEX,(X,Y):],(X,Y):] is non empty set
CLSStruct(# (X,Y),(Zero_ ((X,Y),(X,Y))),(Add_ ((X,Y),(X,Y))),(Mult_ ((X,Y),(X,Y))) #) is non empty right_complementable Abelian add-associative right_zeroed strict vector-distributive scalar-distributive scalar-associative scalar-unital CLSStruct
the carrier of (X,Y) is non empty set
f is Relation-like Function-like Element of the carrier of (X,Y)
vseq is Relation-like Function-like Element of the carrier of (X,Y)
tseq is complex set
tseq * vseq is Relation-like Function-like Element of the carrier of (X,Y)
tseq is Relation-like Function-like Element of the carrier of (X,Y)
tseq is Relation-like Function-like Element of the carrier of (X,Y)
tseq * tseq is Relation-like Function-like Element of the carrier of (X,Y)
tv is Element of the carrier of X
(X,Y,f,tv) is Element of the carrier of Y
(X,Y,vseq,tv) is Element of the carrier of Y
tseq * (X,Y,vseq,tv) is Element of the carrier of Y
X is non empty right_complementable Abelian add-associative right_zeroed discerning reflexive vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like CNORMSTR
Y is non empty right_complementable Abelian add-associative right_zeroed discerning reflexive vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like CNORMSTR
(X,Y) is non empty right_complementable Abelian add-associative right_zeroed strict vector-distributive scalar-distributive scalar-associative scalar-unital CLSStruct
(X,Y) is non empty Element of bool the carrier of (X,Y)
(X,Y) is non empty right_complementable Abelian add-associative right_zeroed constituted-Functions strict vector-distributive scalar-distributive scalar-associative scalar-unital CLSStruct
(X,Y) is non empty functional Element of bool the carrier of ( the carrier of X,Y)
the carrier of X is non empty set
( the carrier of X,Y) is non empty right_complementable Abelian add-associative right_zeroed constituted-Functions strict vector-distributive scalar-distributive scalar-associative scalar-unital CLSStruct
the carrier of Y is non empty set
Funcs ( the carrier of X, the carrier of Y) is non empty functional FUNCTION_DOMAIN of the carrier of X, the carrier of Y
FuncZero ( the carrier of X,Y) is Relation-like the carrier of X -defined the carrier of Y -valued Function-like total quasi_total Element of Funcs ( the carrier of X, the carrier of Y)
FuncAdd ( the carrier of X,Y) is non empty Relation-like [:(Funcs ( the carrier of X, the carrier of Y)),(Funcs ( the carrier of X, the carrier of Y)):] -defined Funcs ( the carrier of X, the carrier of Y) -valued Function-like total quasi_total Function-yielding V34() Element of bool [:[:(Funcs ( the carrier of X, the carrier of Y)),(Funcs ( the carrier of X, the carrier of Y)):],(Funcs ( the carrier of X, the carrier of Y)):]
[:(Funcs ( the carrier of X, the carrier of Y)),(Funcs ( the carrier of X, the carrier of Y)):] is non empty set
[:[:(Funcs ( the carrier of X, the carrier of Y)),(Funcs ( the carrier of X, the carrier of Y)):],(Funcs ( the carrier of X, the carrier of Y)):] is non empty set
bool [:[:(Funcs ( the carrier of X, the carrier of Y)),(Funcs ( the carrier of X, the carrier of Y)):],(Funcs ( the carrier of X, the carrier of Y)):] is non empty set
( the carrier of X,Y) is non empty Relation-like [:COMPLEX,(Funcs ( the carrier of X, the carrier of Y)):] -defined Funcs ( the carrier of X, the carrier of Y) -valued Function-like total quasi_total Function-yielding V34() Element of bool [:[:COMPLEX,(Funcs ( the carrier of X, the carrier of Y)):],(Funcs ( the carrier of X, the carrier of Y)):]
[:COMPLEX,(Funcs ( the carrier of X, the carrier of Y)):] is non empty set
[:[:COMPLEX,(Funcs ( the carrier of X, the carrier of Y)):],(Funcs ( the carrier of X, the carrier of Y)):] is non empty set
bool [:[:COMPLEX,(Funcs ( the carrier of X, the carrier of Y)):],(Funcs ( the carrier of X, the carrier of Y)):] is non empty set
CLSStruct(# (Funcs ( the carrier of X, the carrier of Y)),(FuncZero ( the carrier of X,Y)),(FuncAdd ( the carrier of X,Y)),( the carrier of X,Y) #) is non empty strict CLSStruct
the carrier of ( the carrier of X,Y) is non empty set
bool the carrier of ( the carrier of X,Y) is non empty set
Zero_ ((X,Y),( the carrier of X,Y)) is Relation-like Function-like Element of (X,Y)
Add_ ((X,Y),( the carrier of X,Y)) is non empty Relation-like [:(X,Y),(X,Y):] -defined (X,Y) -valued Function-like total quasi_total Function-yielding V34() Element of bool [:[:(X,Y),(X,Y):],(X,Y):]
[:(X,Y),(X,Y):] is non empty set
[:[:(X,Y),(X,Y):],(X,Y):] is non empty set
bool [:[:(X,Y),(X,Y):],(X,Y):] is non empty set
Mult_ ((X,Y),( the carrier of X,Y)) is non empty Relation-like [:COMPLEX,(X,Y):] -defined (X,Y) -valued Function-like total quasi_total Function-yielding V34() Element of bool [:[:COMPLEX,(X,Y):],(X,Y):]
[:COMPLEX,(X,Y):] is non empty set
[:[:COMPLEX,(X,Y):],(X,Y):] is non empty set
bool [:[:COMPLEX,(X,Y):],(X,Y):] is non empty set
CLSStruct(# (X,Y),(Zero_ ((X,Y),( the carrier of X,Y))),(Add_ ((X,Y),( the carrier of X,Y))),(Mult_ ((X,Y),( the carrier of X,Y))) #) is non empty right_complementable Abelian add-associative right_zeroed strict vector-distributive scalar-distributive scalar-associative scalar-unital CLSStruct
the carrier of (X,Y) is non empty set
bool the carrier of (X,Y) is non empty set
Zero_ ((X,Y),(X,Y)) is Element of (X,Y)
Add_ ((X,Y),(X,Y)) is non empty Relation-like [:(X,Y),(X,Y):] -defined (X,Y) -valued Function-like total quasi_total Element of bool [:[:(X,Y),(X,Y):],(X,Y):]
[:(X,Y),(X,Y):] is non empty set
[:[:(X,Y),(X,Y):],(X,Y):] is non empty set
bool [:[:(X,Y),(X,Y):],(X,Y):] is non empty set
Mult_ ((X,Y),(X,Y)) is non empty Relation-like [:COMPLEX,(X,Y):] -defined (X,Y) -valued Function-like total quasi_total Element of bool [:[:COMPLEX,(X,Y):],(X,Y):]
[:COMPLEX,(X,Y):] is non empty set
[:[:COMPLEX,(X,Y):],(X,Y):] is non empty set
bool [:[:COMPLEX,(X,Y):],(X,Y):] is non empty set
CLSStruct(# (X,Y),(Zero_ ((X,Y),(X,Y))),(Add_ ((X,Y),(X,Y))),(Mult_ ((X,Y),(X,Y))) #) is non empty right_complementable Abelian add-associative right_zeroed strict vector-distributive scalar-distributive scalar-associative scalar-unital CLSStruct
0. (X,Y) is Relation-like Function-like zero Element of the carrier of (X,Y)
the carrier of (X,Y) is non empty set
the ZeroF of (X,Y) is Relation-like Function-like Element of the carrier of (X,Y)
0. Y is zero Element of the carrier of Y
the ZeroF of Y is Element of the carrier of Y
the carrier of X --> (0. Y) is non empty Relation-like the carrier of X -defined the carrier of Y -valued Function-like constant total quasi_total Element of bool [: the carrier of X, the carrier of Y:]
[: the carrier of X, the carrier of Y:] is non empty set
bool [: the carrier of X, the carrier of Y:] is non empty set
0. (X,Y) is Relation-like Function-like zero Element of the carrier of (X,Y)
the ZeroF of (X,Y) is Relation-like Function-like Element of the carrier of (X,Y)
vseq is set
X is non empty right_complementable Abelian add-associative right_zeroed discerning reflexive vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like CNORMSTR
Y is non empty right_complementable Abelian add-associative right_zeroed discerning reflexive vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like CNORMSTR
(X,Y) is non empty Element of bool the carrier of (X,Y)
(X,Y) is non empty right_complementable Abelian add-associative right_zeroed constituted-Functions strict vector-distributive scalar-distributive scalar-associative scalar-unital CLSStruct
(X,Y) is non empty functional Element of bool the carrier of ( the carrier of X,Y)
the carrier of X is non empty set
( the carrier of X,Y) is non empty right_complementable Abelian add-associative right_zeroed constituted-Functions strict vector-distributive scalar-distributive scalar-associative scalar-unital CLSStruct
the carrier of Y is non empty set
Funcs ( the carrier of X, the carrier of Y) is non empty functional FUNCTION_DOMAIN of the carrier of X, the carrier of Y
FuncZero ( the carrier of X,Y) is Relation-like the carrier of X -defined the carrier of Y -valued Function-like total quasi_total Element of Funcs ( the carrier of X, the carrier of Y)
FuncAdd ( the carrier of X,Y) is non empty Relation-like [:(Funcs ( the carrier of X, the carrier of Y)),(Funcs ( the carrier of X, the carrier of Y)):] -defined Funcs ( the carrier of X, the carrier of Y) -valued Function-like total quasi_total Function-yielding V34() Element of bool [:[:(Funcs ( the carrier of X, the carrier of Y)),(Funcs ( the carrier of X, the carrier of Y)):],(Funcs ( the carrier of X, the carrier of Y)):]
[:(Funcs ( the carrier of X, the carrier of Y)),(Funcs ( the carrier of X, the carrier of Y)):] is non empty set
[:[:(Funcs ( the carrier of X, the carrier of Y)),(Funcs ( the carrier of X, the carrier of Y)):],(Funcs ( the carrier of X, the carrier of Y)):] is non empty set
bool [:[:(Funcs ( the carrier of X, the carrier of Y)),(Funcs ( the carrier of X, the carrier of Y)):],(Funcs ( the carrier of X, the carrier of Y)):] is non empty set
( the carrier of X,Y) is non empty Relation-like [:COMPLEX,(Funcs ( the carrier of X, the carrier of Y)):] -defined Funcs ( the carrier of X, the carrier of Y) -valued Function-like total quasi_total Function-yielding V34() Element of bool [:[:COMPLEX,(Funcs ( the carrier of X, the carrier of Y)):],(Funcs ( the carrier of X, the carrier of Y)):]
[:COMPLEX,(Funcs ( the carrier of X, the carrier of Y)):] is non empty set
[:[:COMPLEX,(Funcs ( the carrier of X, the carrier of Y)):],(Funcs ( the carrier of X, the carrier of Y)):] is non empty set
bool [:[:COMPLEX,(Funcs ( the carrier of X, the carrier of Y)):],(Funcs ( the carrier of X, the carrier of Y)):] is non empty set
CLSStruct(# (Funcs ( the carrier of X, the carrier of Y)),(FuncZero ( the carrier of X,Y)),(FuncAdd ( the carrier of X,Y)),( the carrier of X,Y) #) is non empty strict CLSStruct
the carrier of ( the carrier of X,Y) is non empty set
bool the carrier of ( the carrier of X,Y) is non empty set
Zero_ ((X,Y),( the carrier of X,Y)) is Relation-like Function-like Element of (X,Y)
Add_ ((X,Y),( the carrier of X,Y)) is non empty Relation-like [:(X,Y),(X,Y):] -defined (X,Y) -valued Function-like total quasi_total Function-yielding V34() Element of bool [:[:(X,Y),(X,Y):],(X,Y):]
[:(X,Y),(X,Y):] is non empty set
[:[:(X,Y),(X,Y):],(X,Y):] is non empty set
bool [:[:(X,Y),(X,Y):],(X,Y):] is non empty set
Mult_ ((X,Y),( the carrier of X,Y)) is non empty Relation-like [:COMPLEX,(X,Y):] -defined (X,Y) -valued Function-like total quasi_total Function-yielding V34() Element of bool [:[:COMPLEX,(X,Y):],(X,Y):]
[:COMPLEX,(X,Y):] is non empty set
[:[:COMPLEX,(X,Y):],(X,Y):] is non empty set
bool [:[:COMPLEX,(X,Y):],(X,Y):] is non empty set
CLSStruct(# (X,Y),(Zero_ ((X,Y),( the carrier of X,Y))),(Add_ ((X,Y),( the carrier of X,Y))),(Mult_ ((X,Y),( the carrier of X,Y))) #) is non empty right_complementable Abelian add-associative right_zeroed strict vector-distributive scalar-distributive scalar-associative scalar-unital CLSStruct
the carrier of (X,Y) is non empty set
bool the carrier of (X,Y) is non empty set
[: the carrier of X, the carrier of Y:] is non empty set
bool [: the carrier of X, the carrier of Y:] is non empty set
X is non empty right_complementable Abelian add-associative right_zeroed discerning reflexive vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like CNORMSTR
the carrier of X is non empty set
Y is non empty right_complementable Abelian add-associative right_zeroed discerning reflexive vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like CNORMSTR
the carrier of Y is non empty set
[: the carrier of X, the carrier of Y:] is non empty set
bool [: the carrier of X, the carrier of Y:] is non empty set
vseq is non empty Relation-like the carrier of X -defined the carrier of Y -valued Function-like total quasi_total additive (X,Y) Element of bool [: the carrier of X, the carrier of Y:]
{ ||.(vseq . b₁).|| where b₁ is Element of the carrier of X : ||.b₁.|| <= 1 } is set
f is set
tseq is Element of the carrier of X
vseq . tseq is Element of the carrier of Y
||.(vseq . tseq).|| is complex real ext-real Element of REAL
the U9 of Y is non empty Relation-like the carrier of Y -defined REAL -valued Function-like total quasi_total V132() V133() V134() Element of bool [: the carrier of Y,REAL:]
[: the carrier of Y,REAL:] is non empty V132() V133() V134() set
bool [: the carrier of Y,REAL:] is non empty set
the U9 of Y . (vseq . tseq) is complex real ext-real Element of REAL
||.tseq.|| is complex real ext-real Element of REAL
the U9 of X is non empty Relation-like the carrier of X -defined REAL -valued Function-like total quasi_total V132() V133() V134() Element of bool [: the carrier of X,REAL:]
[: the carrier of X,REAL:] is non empty V132() V133() V134() set
bool [: the carrier of X,REAL:] is non empty set
the U9 of X . tseq is complex real ext-real Element of REAL
0. X is zero Element of the carrier of X
the ZeroF of X is Element of the carrier of X
||.(0. X).|| is complex real ext-real Element of REAL
the U9 of X is non empty Relation-like the carrier of X -defined REAL -valued Function-like total quasi_total V132() V133() V134() Element of bool [: the carrier of X,REAL:]
[: the carrier of X,REAL:] is non empty V132() V133() V134() set
bool [: the carrier of X,REAL:] is non empty set
the U9 of X . (0. X) is complex real ext-real Element of REAL
vseq . (0. X) is Element of the carrier of Y
||.(vseq . (0. X)).|| is complex real ext-real Element of REAL
the U9 of Y is non empty Relation-like the carrier of Y -defined REAL -valued Function-like total quasi_total V132() V133() V134() Element of bool [: the carrier of Y,REAL:]
[: the carrier of Y,REAL:] is non empty V132() V133() V134() set
bool [: the carrier of Y,REAL:] is non empty set
the U9 of Y . (vseq . (0. X)) is complex real ext-real Element of REAL
X is non empty right_complementable Abelian add-associative right_zeroed discerning reflexive vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like CNORMSTR
the carrier of X is non empty set
Y is non empty right_complementable Abelian add-associative right_zeroed discerning reflexive vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like CNORMSTR
the carrier of Y is non empty set
[: the carrier of X, the carrier of Y:] is non empty set
bool [: the carrier of X, the carrier of Y:] is non empty set
vseq is non empty Relation-like the carrier of X -defined the carrier of Y -valued Function-like total quasi_total additive (X,Y) (X,Y) Element of bool [: the carrier of X, the carrier of Y:]
(X,Y,vseq) is non empty V142() V143() V144() Element of bool REAL
{ ||.(vseq . b₁).|| where b₁ is Element of the carrier of X : ||.b₁.|| <= 1 } is set
f is complex real ext-real Element of REAL
tseq is ext-real set
tseq is Element of the carrier of X
vseq . tseq is Element of the carrier of Y
||.(vseq . tseq).|| is complex real ext-real Element of REAL
the U9 of Y is non empty Relation-like the carrier of Y -defined REAL -valued Function-like total quasi_total V132() V133() V134() Element of bool [: the carrier of Y,REAL:]
[: the carrier of Y,REAL:] is non empty V132() V133() V134() set
bool [: the carrier of Y,REAL:] is non empty set
the U9 of Y . (vseq . tseq) is complex real ext-real Element of REAL
||.tseq.|| is complex real ext-real Element of REAL
the U9 of X is non empty Relation-like the carrier of X -defined REAL -valued Function-like total quasi_total V132() V133() V134() Element of bool [: the carrier of X,REAL:]
[: the carrier of X,REAL:] is non empty V132() V133() V134() set
bool [: the carrier of X,REAL:] is non empty set
the U9 of X . tseq is complex real ext-real Element of REAL
f * ||.tseq.|| is complex real ext-real Element of REAL
f * 1 is complex real ext-real Element of REAL
X is non empty right_complementable Abelian add-associative right_zeroed discerning reflexive vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like CNORMSTR
the carrier of X is non empty set
Y is non empty right_complementable Abelian add-associative right_zeroed discerning reflexive vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like CNORMSTR
the carrier of Y is non empty set
[: the carrier of X, the carrier of Y:] is non empty set
bool [: the carrier of X, the carrier of Y:] is non empty set
vseq is non empty Relation-like the carrier of X -defined the carrier of Y -valued Function-like total quasi_total additive (X,Y) Element of bool [: the carrier of X, the carrier of Y:]
(X,Y,vseq) is non empty V142() V143() V144() Element of bool REAL
{ ||.(vseq . b₁).|| where b₁ is Element of the carrier of X : ||.b₁.|| <= 1 } is set
upper_bound (X,Y,vseq) is complex real ext-real Element of REAL
tseq is Element of the carrier of X
0. X is zero Element of the carrier of X
the ZeroF of X is Element of the carrier of X
||.tseq.|| is complex real ext-real Element of REAL
the U9 of X is non empty Relation-like the carrier of X -defined REAL -valued Function-like total quasi_total V132() V133() V134() Element of bool [: the carrier of X,REAL:]
[: the carrier of X,REAL:] is non empty V132() V133() V134() set
bool [: the carrier of X,REAL:] is non empty set
the U9 of X . tseq is complex real ext-real Element of REAL
vseq . tseq is Element of the carrier of Y
0c is empty Function-like functional epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural complex real ext-real non positive non negative V142() V143() V144() V145() V146() V147() V148() Element of COMPLEX
0c * (0. X) is Element of the carrier of X
vseq . (0c * (0. X)) is Element of the carrier of Y
vseq . (0. X) is Element of the carrier of Y
0c * (vseq . (0. X)) is Element of the carrier of Y
0. Y is zero Element of the carrier of Y
the ZeroF of Y is Element of the carrier of Y
||.(vseq . tseq).|| is complex real ext-real Element of REAL
the U9 of Y is non empty Relation-like the carrier of Y -defined REAL -valued Function-like total quasi_total V132() V133() V134() Element of bool [: the carrier of Y,REAL:]
[: the carrier of Y,REAL:] is non empty V132() V133() V134() set
bool [: the carrier of Y,REAL:] is non empty set
the U9 of Y . (vseq . tseq) is complex real ext-real Element of REAL
f is complex real ext-real Element of REAL
f * ||.tseq.|| is complex real ext-real Element of REAL
tseq is Element of the carrier of X
0. X is zero Element of the carrier of X
the ZeroF of X is Element of the carrier of X
||.tseq.|| is complex real ext-real Element of REAL
the U9 of X is non empty Relation-like the carrier of X -defined REAL -valued Function-like total quasi_total V132() V133() V134() Element of bool [: the carrier of X,REAL:]
[: the carrier of X,REAL:] is non empty V132() V133() V134() set
bool [: the carrier of X,REAL:] is non empty set
the U9 of X . tseq is complex real ext-real Element of REAL
||.tseq.|| " is complex real ext-real Element of REAL
<i> is complex Element of COMPLEX
0 * <i> is complex set
(||.tseq.|| ") + (0 * <i>) is complex set
tseq is complex Element of COMPLEX
tseq * tseq is Element of the carrier of X
|.((||.tseq.|| ") + (0 * <i>)).| is complex real ext-real Element of REAL
1 * (||.tseq.|| ") is complex real ext-real Element of REAL
abs (1 * (||.tseq.|| ")) is complex real ext-real Element of REAL
1 / ||.tseq.|| is complex real ext-real Element of REAL
abs (1 / ||.tseq.||) is complex real ext-real Element of REAL
abs ||.tseq.|| is complex real ext-real Element of REAL
1 / (abs ||.tseq.||) is complex real ext-real Element of REAL
vseq . tseq is Element of the carrier of Y
||.(vseq . tseq).|| is complex real ext-real Element of REAL
the U9 of Y is non empty Relation-like the carrier of Y -defined REAL -valued Function-like total quasi_total V132() V133() V134() Element of bool [: the carrier of Y,REAL:]
[: the carrier of Y,REAL:] is non empty V132() V133() V134() set
bool [: the carrier of Y,REAL:] is non empty set
the U9 of Y . (vseq . tseq) is complex real ext-real Element of REAL
||.(vseq . tseq).|| / ||.tseq.|| is complex real ext-real Element of REAL
|.tseq.| is complex real ext-real Element of REAL
||.(vseq . tseq).|| * |.tseq.| is complex real ext-real Element of REAL
tseq * (vseq . tseq) is Element of the carrier of Y
||.(tseq * (vseq . tseq)).|| is complex real ext-real Element of REAL
the U9 of Y . (tseq * (vseq . tseq)) is complex real ext-real Element of REAL
tseq is Element of the carrier of X
vseq . tseq is Element of the carrier of Y
||.(vseq . tseq).|| is complex real ext-real Element of REAL
the U9 of Y . (vseq . tseq) is complex real ext-real Element of REAL
||.tseq.|| is complex real ext-real Element of REAL
the U9 of X . tseq is complex real ext-real Element of REAL
|.tseq.| * ||.tseq.|| is complex real ext-real Element of REAL
f is complex real ext-real Element of REAL
(||.(vseq . tseq).|| / ||.tseq.||) * ||.tseq.|| is complex real ext-real Element of REAL
||.(vseq . tseq).|| * (||.tseq.|| ") is complex real ext-real Element of REAL
(||.(vseq . tseq).|| * (||.tseq.|| ")) * ||.tseq.|| is complex real ext-real Element of REAL
(||.tseq.|| ") * ||.tseq.|| is complex real ext-real Element of REAL
||.(vseq . tseq).|| * ((||.tseq.|| ") * ||.tseq.||) is complex real ext-real Element of REAL
||.(vseq . tseq).|| * 1 is complex real ext-real Element of REAL
f * ||.tseq.|| is complex real ext-real Element of REAL
tseq is Element of the carrier of X
0. X is zero Element of the carrier of X
the ZeroF of X is Element of the carrier of X
vseq . tseq is Element of the carrier of Y
||.(vseq . tseq).|| is complex real ext-real Element of REAL
the U9 of Y is non empty Relation-like the carrier of Y -defined REAL -valued Function-like total quasi_total V132() V133() V134() Element of bool [: the carrier of Y,REAL:]
[: the carrier of Y,REAL:] is non empty V132() V133() V134() set
bool [: the carrier of Y,REAL:] is non empty set
the U9 of Y . (vseq . tseq) is complex real ext-real Element of REAL
f is complex real ext-real Element of REAL
||.tseq.|| is complex real ext-real Element of REAL
the U9 of X is non empty Relation-like the carrier of X -defined REAL -valued Function-like total quasi_total V132() V133() V134() Element of bool [: the carrier of X,REAL:]
[: the carrier of X,REAL:] is non empty V132() V133() V134() set
bool [: the carrier of X,REAL:] is non empty set
the U9 of X . tseq is complex real ext-real Element of REAL
f * ||.tseq.|| is complex real ext-real Element of REAL
tseq is Element of the carrier of X
0. X is zero Element of the carrier of X
the ZeroF of X is Element of the carrier of X
vseq . tseq is Element of the carrier of Y
||.(vseq . tseq).|| is complex real ext-real Element of REAL
the U9 of Y is non empty Relation-like the carrier of Y -defined REAL -valued Function-like total quasi_total V132() V133() V134() Element of bool [: the carrier of Y,REAL:]
[: the carrier of Y,REAL:] is non empty V132() V133() V134() set
bool [: the carrier of Y,REAL:] is non empty set
the U9 of Y . (vseq . tseq) is complex real ext-real Element of REAL
f is complex real ext-real Element of REAL
||.tseq.|| is complex real ext-real Element of REAL
the U9 of X is non empty Relation-like the carrier of X -defined REAL -valued Function-like total quasi_total V132() V133() V134() Element of bool [: the carrier of X,REAL:]
[: the carrier of X,REAL:] is non empty V132() V133() V134() set
bool [: the carrier of X,REAL:] is non empty set
the U9 of X . tseq is complex real ext-real Element of REAL
f * ||.tseq.|| is complex real ext-real Element of REAL
tseq is complex real ext-real Element of REAL
tseq is set
tv is complex real ext-real Element of REAL
e is Element of the carrier of X
vseq . e is Element of the carrier of Y
||.(vseq . e).|| is complex real ext-real Element of REAL
the U9 of Y . (vseq . e) is complex real ext-real Element of REAL
||.e.|| is complex real ext-real Element of REAL
the U9 of X . e is complex real ext-real Element of REAL
tseq is complex real ext-real Element of REAL
X is non empty right_complementable Abelian add-associative right_zeroed discerning reflexive vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like CNORMSTR
Y is non empty right_complementable Abelian add-associative right_zeroed discerning reflexive vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like CNORMSTR
(X,Y) is non empty Element of bool the carrier of (X,Y)
(X,Y) is non empty right_complementable Abelian add-associative right_zeroed constituted-Functions strict vector-distributive scalar-distributive scalar-associative scalar-unital CLSStruct
(X,Y) is non empty functional Element of bool the carrier of ( the carrier of X,Y)
the carrier of X is non empty set
( the carrier of X,Y) is non empty right_complementable Abelian add-associative right_zeroed constituted-Functions strict vector-distributive scalar-distributive scalar-associative scalar-unital CLSStruct
the carrier of Y is non empty set
Funcs ( the carrier of X, the carrier of Y) is non empty functional FUNCTION_DOMAIN of the carrier of X, the carrier of Y
FuncZero ( the carrier of X,Y) is Relation-like the carrier of X -defined the carrier of Y -valued Function-like total quasi_total Element of Funcs ( the carrier of X, the carrier of Y)
FuncAdd ( the carrier of X,Y) is non empty Relation-like [:(Funcs ( the carrier of X, the carrier of Y)),(Funcs ( the carrier of X, the carrier of Y)):] -defined Funcs ( the carrier of X, the carrier of Y) -valued Function-like total quasi_total Function-yielding V34() Element of bool [:[:(Funcs ( the carrier of X, the carrier of Y)),(Funcs ( the carrier of X, the carrier of Y)):],(Funcs ( the carrier of X, the carrier of Y)):]
[:(Funcs ( the carrier of X, the carrier of Y)),(Funcs ( the carrier of X, the carrier of Y)):] is non empty set
[:[:(Funcs ( the carrier of X, the carrier of Y)),(Funcs ( the carrier of X, the carrier of Y)):],(Funcs ( the carrier of X, the carrier of Y)):] is non empty set
bool [:[:(Funcs ( the carrier of X, the carrier of Y)),(Funcs ( the carrier of X, the carrier of Y)):],(Funcs ( the carrier of X, the carrier of Y)):] is non empty set
( the carrier of X,Y) is non empty Relation-like [:COMPLEX,(Funcs ( the carrier of X, the carrier of Y)):] -defined Funcs ( the carrier of X, the carrier of Y) -valued Function-like total quasi_total Function-yielding V34() Element of bool [:[:COMPLEX,(Funcs ( the carrier of X, the carrier of Y)):],(Funcs ( the carrier of X, the carrier of Y)):]
[:COMPLEX,(Funcs ( the carrier of X, the carrier of Y)):] is non empty set
[:[:COMPLEX,(Funcs ( the carrier of X, the carrier of Y)):],(Funcs ( the carrier of X, the carrier of Y)):] is non empty set
bool [:[:COMPLEX,(Funcs ( the carrier of X, the carrier of Y)):],(Funcs ( the carrier of X, the carrier of Y)):] is non empty set
CLSStruct(# (Funcs ( the carrier of X, the carrier of Y)),(FuncZero ( the carrier of X,Y)),(FuncAdd ( the carrier of X,Y)),( the carrier of X,Y) #) is non empty strict CLSStruct
the carrier of ( the carrier of X,Y) is non empty set
bool the carrier of ( the carrier of X,Y) is non empty set
Zero_ ((X,Y),( the carrier of X,Y)) is Relation-like Function-like Element of (X,Y)
Add_ ((X,Y),( the carrier of X,Y)) is non empty Relation-like [:(X,Y),(X,Y):] -defined (X,Y) -valued Function-like total quasi_total Function-yielding V34() Element of bool [:[:(X,Y),(X,Y):],(X,Y):]
[:(X,Y),(X,Y):] is non empty set
[:[:(X,Y),(X,Y):],(X,Y):] is non empty set
bool [:[:(X,Y),(X,Y):],(X,Y):] is non empty set
Mult_ ((X,Y),( the carrier of X,Y)) is non empty Relation-like [:COMPLEX,(X,Y):] -defined (X,Y) -valued Function-like total quasi_total Function-yielding V34() Element of bool [:[:COMPLEX,(X,Y):],(X,Y):]
[:COMPLEX,(X,Y):] is non empty set
[:[:COMPLEX,(X,Y):],(X,Y):] is non empty set
bool [:[:COMPLEX,(X,Y):],(X,Y):] is non empty set
CLSStruct(# (X,Y),(Zero_ ((X,Y),( the carrier of X,Y))),(Add_ ((X,Y),( the carrier of X,Y))),(Mult_ ((X,Y),( the carrier of X,Y))) #) is non empty right_complementable Abelian add-associative right_zeroed strict vector-distributive scalar-distributive scalar-associative scalar-unital CLSStruct
the carrier of (X,Y) is non empty set
bool the carrier of (X,Y) is non empty set
[:(X,Y),REAL:] is non empty V132() V133() V134() set
bool [:(X,Y),REAL:] is non empty set
vseq is set
(X,Y,vseq) is non empty Relation-like the carrier of X -defined the carrier of Y -valued Function-like total quasi_total additive (X,Y) (X,Y) Element of bool [: the carrier of X, the carrier of Y:]
[: the carrier of X, the carrier of Y:] is non empty set
bool [: the carrier of X, the carrier of Y:] is non empty set
(X,Y,(X,Y,vseq)) is non empty V142() V143() V144() Element of bool REAL
{ ||.((X,Y,vseq) . b₁).|| where b₁ is Element of the carrier of X : ||.b₁.|| <= 1 } is set
upper_bound (X,Y,(X,Y,vseq)) is complex real ext-real Element of REAL
vseq is non empty Relation-like (X,Y) -defined REAL -valued Function-like total quasi_total V132() V133() V134() Element of bool [:(X,Y),REAL:]
vseq is non empty Relation-like (X,Y) -defined REAL -valued Function-like total quasi_total V132() V133() V134() Element of bool [:(X,Y),REAL:]
f is non empty Relation-like (X,Y) -defined REAL -valued Function-like total quasi_total V132() V133() V134() Element of bool [:(X,Y),REAL:]
tseq is set
vseq . tseq is complex real ext-real Element of REAL
f . tseq is complex real ext-real Element of REAL
(X,Y,tseq) is non empty Relation-like the carrier of X -defined the carrier of Y -valued Function-like total quasi_total additive (X,Y) (X,Y) Element of bool [: the carrier of X, the carrier of Y:]
[: the carrier of X, the carrier of Y:] is non empty set
bool [: the carrier of X, the carrier of Y:] is non empty set
(X,Y,(X,Y,tseq)) is non empty V142() V143() V144() Element of bool REAL
{ ||.((X,Y,tseq) . b₁).|| where b₁ is Element of the carrier of X : ||.b₁.|| <= 1 } is set
upper_bound (X,Y,(X,Y,tseq)) is complex real ext-real Element of REAL
dom f is set
dom vseq is set
X is non empty right_complementable Abelian add-associative right_zeroed discerning reflexive vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like CNORMSTR
the carrier of X is non empty set
Y is non empty right_complementable Abelian add-associative right_zeroed discerning reflexive vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like CNORMSTR
the carrier of Y is non empty set
[: the carrier of X, the carrier of Y:] is non empty set
bool [: the carrier of X, the carrier of Y:] is non empty set
vseq is non empty Relation-like the carrier of X -defined the carrier of Y -valued Function-like total quasi_total additive (X,Y) (X,Y) Element of bool [: the carrier of X, the carrier of Y:]
(X,Y,vseq) is non empty Relation-like the carrier of X -defined the carrier of Y -valued Function-like total quasi_total additive (X,Y) (X,Y) Element of bool [: the carrier of X, the carrier of Y:]
(X,Y) is non empty Element of bool the carrier of (X,Y)
(X,Y) is non empty right_complementable Abelian add-associative right_zeroed constituted-Functions strict vector-distributive scalar-distributive scalar-associative scalar-unital CLSStruct
(X,Y) is non empty functional Element of bool the carrier of ( the carrier of X,Y)
( the carrier of X,Y) is non empty right_complementable Abelian add-associative right_zeroed constituted-Functions strict vector-distributive scalar-distributive scalar-associative scalar-unital CLSStruct
Funcs ( the carrier of X, the carrier of Y) is non empty functional FUNCTION_DOMAIN of the carrier of X, the carrier of Y
FuncZero ( the carrier of X,Y) is Relation-like the carrier of X -defined the carrier of Y -valued Function-like total quasi_total Element of Funcs ( the carrier of X, the carrier of Y)
FuncAdd ( the carrier of X,Y) is non empty Relation-like [:(Funcs ( the carrier of X, the carrier of Y)),(Funcs ( the carrier of X, the carrier of Y)):] -defined Funcs ( the carrier of X, the carrier of Y) -valued Function-like total quasi_total Function-yielding V34() Element of bool [:[:(Funcs ( the carrier of X, the carrier of Y)),(Funcs ( the carrier of X, the carrier of Y)):],(Funcs ( the carrier of X, the carrier of Y)):]
[:(Funcs ( the carrier of X, the carrier of Y)),(Funcs ( the carrier of X, the carrier of Y)):] is non empty set
[:[:(Funcs ( the carrier of X, the carrier of Y)),(Funcs ( the carrier of X, the carrier of Y)):],(Funcs ( the carrier of X, the carrier of Y)):] is non empty set
bool [:[:(Funcs ( the carrier of X, the carrier of Y)),(Funcs ( the carrier of X, the carrier of Y)):],(Funcs ( the carrier of X, the carrier of Y)):] is non empty set
( the carrier of X,Y) is non empty Relation-like [:COMPLEX,(Funcs ( the carrier of X, the carrier of Y)):] -defined Funcs ( the carrier of X, the carrier of Y) -valued Function-like total quasi_total Function-yielding V34() Element of bool [:[:COMPLEX,(Funcs ( the carrier of X, the carrier of Y)):],(Funcs ( the carrier of X, the carrier of Y)):]
[:COMPLEX,(Funcs ( the carrier of X, the carrier of Y)):] is non empty set
[:[:COMPLEX,(Funcs ( the carrier of X, the carrier of Y)):],(Funcs ( the carrier of X, the carrier of Y)):] is non empty set
bool [:[:COMPLEX,(Funcs ( the carrier of X, the carrier of Y)):],(Funcs ( the carrier of X, the carrier of Y)):] is non empty set
CLSStruct(# (Funcs ( the carrier of X, the carrier of Y)),(FuncZero ( the carrier of X,Y)),(FuncAdd ( the carrier of X,Y)),( the carrier of X,Y) #) is non empty strict CLSStruct
the carrier of ( the carrier of X,Y) is non empty set
bool the carrier of ( the carrier of X,Y) is non empty set
Zero_ ((X,Y),( the carrier of X,Y)) is Relation-like Function-like Element of (X,Y)
Add_ ((X,Y),( the carrier of X,Y)) is non empty Relation-like [:(X,Y),(X,Y):] -defined (X,Y) -valued Function-like total quasi_total Function-yielding V34() Element of bool [:[:(X,Y),(X,Y):],(X,Y):]
[:(X,Y),(X,Y):] is non empty set
[:[:(X,Y),(X,Y):],(X,Y):] is non empty set
bool [:[:(X,Y),(X,Y):],(X,Y):] is non empty set
Mult_ ((X,Y),( the carrier of X,Y)) is non empty Relation-like [:COMPLEX,(X,Y):] -defined (X,Y) -valued Function-like total quasi_total Function-yielding V34() Element of bool [:[:COMPLEX,(X,Y):],(X,Y):]
[:COMPLEX,(X,Y):] is non empty set
[:[:COMPLEX,(X,Y):],(X,Y):] is non empty set
bool [:[:COMPLEX,(X,Y):],(X,Y):] is non empty set
CLSStruct(# (X,Y),(Zero_ ((X,Y),( the carrier of X,Y))),(Add_ ((X,Y),( the carrier of X,Y))),(Mult_ ((X,Y),( the carrier of X,Y))) #) is non empty right_complementable Abelian add-associative right_zeroed strict vector-distributive scalar-distributive scalar-associative scalar-unital CLSStruct
the carrier of (X,Y) is non empty set
bool the carrier of (X,Y) is non empty set
X is non empty right_complementable Abelian add-associative right_zeroed discerning reflexive vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like CNORMSTR
the carrier of X is non empty set
Y is non empty right_complementable Abelian add-associative right_zeroed discerning reflexive vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like CNORMSTR
the carrier of Y is non empty set
[: the carrier of X, the carrier of Y:] is non empty set
bool [: the carrier of X, the carrier of Y:] is non empty set
(X,Y) is non empty Relation-like (X,Y) -defined REAL -valued Function-like total quasi_total V132() V133() V134() Element of bool [:(X,Y),REAL:]
(X,Y) is non empty Element of bool the carrier of (X,Y)
(X,Y) is non empty right_complementable Abelian add-associative right_zeroed constituted-Functions strict vector-distributive scalar-distributive scalar-associative scalar-unital CLSStruct
(X,Y) is non empty functional Element of bool the carrier of ( the carrier of X,Y)
( the carrier of X,Y) is non empty right_complementable Abelian add-associative right_zeroed constituted-Functions strict vector-distributive scalar-distributive scalar-associative scalar-unital CLSStruct
Funcs ( the carrier of X, the carrier of Y) is non empty functional FUNCTION_DOMAIN of the carrier of X, the carrier of Y
FuncZero ( the carrier of X,Y) is Relation-like the carrier of X -defined the carrier of Y -valued Function-like total quasi_total Element of Funcs ( the carrier of X, the carrier of Y)
FuncAdd ( the carrier of X,Y) is non empty Relation-like [:(Funcs ( the carrier of X, the carrier of Y)),(Funcs ( the carrier of X, the carrier of Y)):] -defined Funcs ( the carrier of X, the carrier of Y) -valued Function-like total quasi_total Function-yielding V34() Element of bool [:[:(Funcs ( the carrier of X, the carrier of Y)),(Funcs ( the carrier of X, the carrier of Y)):],(Funcs ( the carrier of X, the carrier of Y)):]
[:(Funcs ( the carrier of X, the carrier of Y)),(Funcs ( the carrier of X, the carrier of Y)):] is non empty set
[:[:(Funcs ( the carrier of X, the carrier of Y)),(Funcs ( the carrier of X, the carrier of Y)):],(Funcs ( the carrier of X, the carrier of Y)):] is non empty set
bool [:[:(Funcs ( the carrier of X, the carrier of Y)),(Funcs ( the carrier of X, the carrier of Y)):],(Funcs ( the carrier of X, the carrier of Y)):] is non empty set
( the carrier of X,Y) is non empty Relation-like [:COMPLEX,(Funcs ( the carrier of X, the carrier of Y)):] -defined Funcs ( the carrier of X, the carrier of Y) -valued Function-like total quasi_total Function-yielding V34() Element of bool [:[:COMPLEX,(Funcs ( the carrier of X, the carrier of Y)):],(Funcs ( the carrier of X, the carrier of Y)):]
[:COMPLEX,(Funcs ( the carrier of X, the carrier of Y)):] is non empty set
[:[:COMPLEX,(Funcs ( the carrier of X, the carrier of Y)):],(Funcs ( the carrier of X, the carrier of Y)):] is non empty set
bool [:[:COMPLEX,(Funcs ( the carrier of X, the carrier of Y)):],(Funcs ( the carrier of X, the carrier of Y)):] is non empty set
CLSStruct(# (Funcs ( the carrier of X, the carrier of Y)),(FuncZero ( the carrier of X,Y)),(FuncAdd ( the carrier of X,Y)),( the carrier of X,Y) #) is non empty strict CLSStruct
the carrier of ( the carrier of X,Y) is non empty set
bool the carrier of ( the carrier of X,Y) is non empty set
Zero_ ((X,Y),( the carrier of X,Y)) is Relation-like Function-like Element of (X,Y)
Add_ ((X,Y),( the carrier of X,Y)) is non empty Relation-like [:(X,Y),(X,Y):] -defined (X,Y) -valued Function-like total quasi_total Function-yielding V34() Element of bool [:[:(X,Y),(X,Y):],(X,Y):]
[:(X,Y),(X,Y):] is non empty set
[:[:(X,Y),(X,Y):],(X,Y):] is non empty set
bool [:[:(X,Y),(X,Y):],(X,Y):] is non empty set
Mult_ ((X,Y),( the carrier of X,Y)) is non empty Relation-like [:COMPLEX,(X,Y):] -defined (X,Y) -valued Function-like total quasi_total Function-yielding V34() Element of bool [:[:COMPLEX,(X,Y):],(X,Y):]
[:COMPLEX,(X,Y):] is non empty set
[:[:COMPLEX,(X,Y):],(X,Y):] is non empty set
bool [:[:COMPLEX,(X,Y):],(X,Y):] is non empty set
CLSStruct(# (X,Y),(Zero_ ((X,Y),( the carrier of X,Y))),(Add_ ((X,Y),( the carrier of X,Y))),(Mult_ ((X,Y),( the carrier of X,Y))) #) is non empty right_complementable Abelian add-associative right_zeroed strict vector-distributive scalar-distributive scalar-associative scalar-unital CLSStruct
the carrier of (X,Y) is non empty set
bool the carrier of (X,Y) is non empty set
[:(X,Y),REAL:] is non empty V132() V133() V134() set
bool [:(X,Y),REAL:] is non empty set
vseq is non empty Relation-like the carrier of X -defined the carrier of Y -valued Function-like total quasi_total additive (X,Y) (X,Y) Element of bool [: the carrier of X, the carrier of Y:]
(X,Y) . vseq is complex real ext-real Element of REAL
(X,Y,vseq) is non empty V142() V143() V144() Element of bool REAL
{ ||.(vseq . b₁).|| where b₁ is Element of the carrier of X : ||.b₁.|| <= 1 } is set
upper_bound (X,Y,vseq) is complex real ext-real Element of REAL
f is set
(X,Y,f) is non empty Relation-like the carrier of X -defined the carrier of Y -valued Function-like total quasi_total additive (X,Y) (X,Y) Element of bool [: the carrier of X, the carrier of Y:]
(X,Y,(X,Y,f)) is non empty V142() V143() V144() Element of bool REAL
{ ||.((X,Y,f) . b₁).|| where b₁ is Element of the carrier of X : ||.b₁.|| <= 1 } is set
upper_bound (X,Y,(X,Y,f)) is complex real ext-real Element of REAL
X is non empty right_complementable Abelian add-associative right_zeroed discerning reflexive vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like CNORMSTR
Y is non empty right_complementable Abelian add-associative right_zeroed discerning reflexive vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like CNORMSTR
(X,Y) is non empty Element of bool the carrier of (X,Y)
(X,Y) is non empty right_complementable Abelian add-associative right_zeroed constituted-Functions strict vector-distributive scalar-distributive scalar-associative scalar-unital CLSStruct
(X,Y) is non empty functional Element of bool the carrier of ( the carrier of X,Y)
the carrier of X is non empty set
( the carrier of X,Y) is non empty right_complementable Abelian add-associative right_zeroed constituted-Functions strict vector-distributive scalar-distributive scalar-associative scalar-unital CLSStruct
the carrier of Y is non empty set
Funcs ( the carrier of X, the carrier of Y) is non empty functional FUNCTION_DOMAIN of the carrier of X, the carrier of Y
FuncZero ( the carrier of X,Y) is Relation-like the carrier of X -defined the carrier of Y -valued Function-like total quasi_total Element of Funcs ( the carrier of X, the carrier of Y)
FuncAdd ( the carrier of X,Y) is non empty Relation-like [:(Funcs ( the carrier of X, the carrier of Y)),(Funcs ( the carrier of X, the carrier of Y)):] -defined Funcs ( the carrier of X, the carrier of Y) -valued Function-like total quasi_total Function-yielding V34() Element of bool [:[:(Funcs ( the carrier of X, the carrier of Y)),(Funcs ( the carrier of X, the carrier of Y)):],(Funcs ( the carrier of X, the carrier of Y)):]
[:(Funcs ( the carrier of X, the carrier of Y)),(Funcs ( the carrier of X, the carrier of Y)):] is non empty set
[:[:(Funcs ( the carrier of X, the carrier of Y)),(Funcs ( the carrier of X, the carrier of Y)):],(Funcs ( the carrier of X, the carrier of Y)):] is non empty set
bool [:[:(Funcs ( the carrier of X, the carrier of Y)),(Funcs ( the carrier of X, the carrier of Y)):],(Funcs ( the carrier of X, the carrier of Y)):] is non empty set
( the carrier of X,Y) is non empty Relation-like [:COMPLEX,(Funcs ( the carrier of X, the carrier of Y)):] -defined Funcs ( the carrier of X, the carrier of Y) -valued Function-like total quasi_total Function-yielding V34() Element of bool [:[:COMPLEX,(Funcs ( the carrier of X, the carrier of Y)):],(Funcs ( the carrier of X, the carrier of Y)):]
[:COMPLEX,(Funcs ( the carrier of X, the carrier of Y)):] is non empty set
[:[:COMPLEX,(Funcs ( the carrier of X, the carrier of Y)):],(Funcs ( the carrier of X, the carrier of Y)):] is non empty set
bool [:[:COMPLEX,(Funcs ( the carrier of X, the carrier of Y)):],(Funcs ( the carrier of X, the carrier of Y)):] is non empty set
CLSStruct(# (Funcs ( the carrier of X, the carrier of Y)),(FuncZero ( the carrier of X,Y)),(FuncAdd ( the carrier of X,Y)),( the carrier of X,Y) #) is non empty strict CLSStruct
the carrier of ( the carrier of X,Y) is non empty set
bool the carrier of ( the carrier of X,Y) is non empty set
Zero_ ((X,Y),( the carrier of X,Y)) is Relation-like Function-like Element of (X,Y)
Add_ ((X,Y),( the carrier of X,Y)) is non empty Relation-like [:(X,Y),(X,Y):] -defined (X,Y) -valued Function-like total quasi_total Function-yielding V34() Element of bool [:[:(X,Y),(X,Y):],(X,Y):]
[:(X,Y),(X,Y):] is non empty set
[:[:(X,Y),(X,Y):],(X,Y):] is non empty set
bool [:[:(X,Y),(X,Y):],(X,Y):] is non empty set
Mult_ ((X,Y),( the carrier of X,Y)) is non empty Relation-like [:COMPLEX,(X,Y):] -defined (X,Y) -valued Function-like total quasi_total Function-yielding V34() Element of bool [:[:COMPLEX,(X,Y):],(X,Y):]
[:COMPLEX,(X,Y):] is non empty set
[:[:COMPLEX,(X,Y):],(X,Y):] is non empty set
bool [:[:COMPLEX,(X,Y):],(X,Y):] is non empty set
CLSStruct(# (X,Y),(Zero_ ((X,Y),( the carrier of X,Y))),(Add_ ((X,Y),( the carrier of X,Y))),(Mult_ ((X,Y),( the carrier of X,Y))) #) is non empty right_complementable Abelian add-associative right_zeroed strict vector-distributive scalar-distributive scalar-associative scalar-unital CLSStruct
the carrier of (X,Y) is non empty set
bool the carrier of (X,Y) is non empty set
Zero_ ((X,Y),(X,Y)) is Element of (X,Y)
Add_ ((X,Y),(X,Y)) is non empty Relation-like [:(X,Y),(X,Y):] -defined (X,Y) -valued Function-like total quasi_total Element of bool [:[:(X,Y),(X,Y):],(X,Y):]
[:(X,Y),(X,Y):] is non empty set
[:[:(X,Y),(X,Y):],(X,Y):] is non empty set
bool [:[:(X,Y),(X,Y):],(X,Y):] is non empty set
Mult_ ((X,Y),(X,Y)) is non empty Relation-like [:COMPLEX,(X,Y):] -defined (X,Y) -valued Function-like total quasi_total Element of bool [:[:COMPLEX,(X,Y):],(X,Y):]
[:COMPLEX,(X,Y):] is non empty set
[:[:COMPLEX,(X,Y):],(X,Y):] is non empty set
bool [:[:COMPLEX,(X,Y):],(X,Y):] is non empty set
(X,Y) is non empty Relation-like (X,Y) -defined REAL -valued Function-like total quasi_total V132() V133() V134() Element of bool [:(X,Y),REAL:]
[:(X,Y),REAL:] is non empty V132() V133() V134() set
bool [:(X,Y),REAL:] is non empty set
CNORMSTR(# (X,Y),(Zero_ ((X,Y),(X,Y))),(Add_ ((X,Y),(X,Y))),(Mult_ ((X,Y),(X,Y))),(X,Y) #) is non empty strict CNORMSTR
Y is non empty right_complementable Abelian add-associative right_zeroed discerning reflexive vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like CNORMSTR
the carrier of Y is non empty set
X is non empty right_complementable Abelian add-associative right_zeroed discerning reflexive vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like CNORMSTR
the carrier of X is non empty set
0. Y is zero Element of the carrier of Y
the ZeroF of Y is Element of the carrier of Y
the carrier of X --> (0. Y) is non empty Relation-like the carrier of X -defined the carrier of Y -valued Function-like constant total quasi_total Element of bool [: the carrier of X, the carrier of Y:]
[: the carrier of X, the carrier of Y:] is non empty set
bool [: the carrier of X, the carrier of Y:] is non empty set
(X,Y) is non empty CNORMSTR
(X,Y) is non empty Element of bool the carrier of (X,Y)
(X,Y) is non empty right_complementable Abelian add-associative right_zeroed constituted-Functions strict vector-distributive scalar-distributive scalar-associative scalar-unital CLSStruct
(X,Y) is non empty functional Element of bool the carrier of ( the carrier of X,Y)
( the carrier of X,Y) is non empty right_complementable Abelian add-associative right_zeroed constituted-Functions strict vector-distributive scalar-distributive scalar-associative scalar-unital CLSStruct
Funcs ( the carrier of X, the carrier of Y) is non empty functional FUNCTION_DOMAIN of the carrier of X, the carrier of Y
FuncZero ( the carrier of X,Y) is Relation-like the carrier of X -defined the carrier of Y -valued Function-like total quasi_total Element of Funcs ( the carrier of X, the carrier of Y)
FuncAdd ( the carrier of X,Y) is non empty Relation-like [:(Funcs ( the carrier of X, the carrier of Y)),(Funcs ( the carrier of X, the carrier of Y)):] -defined Funcs ( the carrier of X, the carrier of Y) -valued Function-like total quasi_total Function-yielding V34() Element of bool [:[:(Funcs ( the carrier of X, the carrier of Y)),(Funcs ( the carrier of X, the carrier of Y)):],(Funcs ( the carrier of X, the carrier of Y)):]
[:(Funcs ( the carrier of X, the carrier of Y)),(Funcs ( the carrier of X, the carrier of Y)):] is non empty set
[:[:(Funcs ( the carrier of X, the carrier of Y)),(Funcs ( the carrier of X, the carrier of Y)):],(Funcs ( the carrier of X, the carrier of Y)):] is non empty set
bool [:[:(Funcs ( the carrier of X, the carrier of Y)),(Funcs ( the carrier of X, the carrier of Y)):],(Funcs ( the carrier of X, the carrier of Y)):] is non empty set
( the carrier of X,Y) is non empty Relation-like [:COMPLEX,(Funcs ( the carrier of X, the carrier of Y)):] -defined Funcs ( the carrier of X, the carrier of Y) -valued Function-like total quasi_total Function-yielding V34() Element of bool [:[:COMPLEX,(Funcs ( the carrier of X, the carrier of Y)):],(Funcs ( the carrier of X, the carrier of Y)):]
[:COMPLEX,(Funcs ( the carrier of X, the carrier of Y)):] is non empty set
[:[:COMPLEX,(Funcs ( the carrier of X, the carrier of Y)):],(Funcs ( the carrier of X, the carrier of Y)):] is non empty set
bool [:[:COMPLEX,(Funcs ( the carrier of X, the carrier of Y)):],(Funcs ( the carrier of X, the carrier of Y)):] is non empty set
CLSStruct(# (Funcs ( the carrier of X, the carrier of Y)),(FuncZero ( the carrier of X,Y)),(FuncAdd ( the carrier of X,Y)),( the carrier of X,Y) #) is non empty strict CLSStruct
the carrier of ( the carrier of X,Y) is non empty set
bool the carrier of ( the carrier of X,Y) is non empty set
Zero_ ((X,Y),( the carrier of X,Y)) is Relation-like Function-like Element of (X,Y)
Add_ ((X,Y),( the carrier of X,Y)) is non empty Relation-like [:(X,Y),(X,Y):] -defined (X,Y) -valued Function-like total quasi_total Function-yielding V34() Element of bool [:[:(X,Y),(X,Y):],(X,Y):]
[:(X,Y),(X,Y):] is non empty set
[:[:(X,Y),(X,Y):],(X,Y):] is non empty set
bool [:[:(X,Y),(X,Y):],(X,Y):] is non empty set
Mult_ ((X,Y),( the carrier of X,Y)) is non empty Relation-like [:COMPLEX,(X,Y):] -defined (X,Y) -valued Function-like total quasi_total Function-yielding V34() Element of bool [:[:COMPLEX,(X,Y):],(X,Y):]
[:COMPLEX,(X,Y):] is non empty set
[:[:COMPLEX,(X,Y):],(X,Y):] is non empty set
bool [:[:COMPLEX,(X,Y):],(X,Y):] is non empty set
CLSStruct(# (X,Y),(Zero_ ((X,Y),( the carrier of X,Y))),(Add_ ((X,Y),( the carrier of X,Y))),(Mult_ ((X,Y),( the carrier of X,Y))) #) is non empty right_complementable Abelian add-associative right_zeroed strict vector-distributive scalar-distributive scalar-associative scalar-unital CLSStruct
the carrier of (X,Y) is non empty set
bool the carrier of (X,Y) is non empty set
Zero_ ((X,Y),(X,Y)) is Element of (X,Y)
Add_ ((X,Y),(X,Y)) is non empty Relation-like [:(X,Y),(X,Y):] -defined (X,Y) -valued Function-like total quasi_total Element of bool [:[:(X,Y),(X,Y):],(X,Y):]
[:(X,Y),(X,Y):] is non empty set
[:[:(X,Y),(X,Y):],(X,Y):] is non empty set
bool [:[:(X,Y),(X,Y):],(X,Y):] is non empty set
Mult_ ((X,Y),(X,Y)) is non empty Relation-like [:COMPLEX,(X,Y):] -defined (X,Y) -valued Function-like total quasi_total Element of bool [:[:COMPLEX,(X,Y):],(X,Y):]
[:COMPLEX,(X,Y):] is non empty set
[:[:COMPLEX,(X,Y):],(X,Y):] is non empty set
bool [:[:COMPLEX,(X,Y):],(X,Y):] is non empty set
(X,Y) is non empty Relation-like (X,Y) -defined REAL -valued Function-like total quasi_total V132() V133() V134() Element of bool [:(X,Y),REAL:]
[:(X,Y),REAL:] is non empty V132() V133() V134() set
bool [:(X,Y),REAL:] is non empty set
CNORMSTR(# (X,Y),(Zero_ ((X,Y),(X,Y))),(Add_ ((X,Y),(X,Y))),(Mult_ ((X,Y),(X,Y))),(X,Y) #) is non empty strict CNORMSTR
0. (X,Y) is zero Element of the carrier of (X,Y)
the carrier of (X,Y) is non empty set
the ZeroF of (X,Y) is Element of the carrier of (X,Y)
(X,Y) is non empty right_complementable Abelian add-associative right_zeroed strict vector-distributive scalar-distributive scalar-associative scalar-unital CLSStruct
CLSStruct(# (X,Y),(Zero_ ((X,Y),(X,Y))),(Add_ ((X,Y),(X,Y))),(Mult_ ((X,Y),(X,Y))) #) is non empty right_complementable Abelian add-associative right_zeroed strict vector-distributive scalar-distributive scalar-associative scalar-unital CLSStruct
0. (X,Y) is Relation-like Function-like zero Element of the carrier of (X,Y)
the carrier of (X,Y) is non empty set
the ZeroF of (X,Y) is Relation-like Function-like Element of the carrier of (X,Y)
X is non empty right_complementable Abelian add-associative right_zeroed discerning reflexive vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like CNORMSTR
Y is non empty right_complementable Abelian add-associative right_zeroed discerning reflexive vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like CNORMSTR
(X,Y) is non empty CNORMSTR
(X,Y) is non empty Element of bool the carrier of (X,Y)
(X,Y) is non empty right_complementable Abelian add-associative right_zeroed constituted-Functions strict vector-distributive scalar-distributive scalar-associative scalar-unital CLSStruct
(X,Y) is non empty functional Element of bool the carrier of ( the carrier of X,Y)
the carrier of X is non empty set
( the carrier of X,Y) is non empty right_complementable Abelian add-associative right_zeroed constituted-Functions strict vector-distributive scalar-distributive scalar-associative scalar-unital CLSStruct
the carrier of Y is non empty set
Funcs ( the carrier of X, the carrier of Y) is non empty functional FUNCTION_DOMAIN of the carrier of X, the carrier of Y
FuncZero ( the carrier of X,Y) is Relation-like the carrier of X -defined the carrier of Y -valued Function-like total quasi_total Element of Funcs ( the carrier of X, the carrier of Y)
FuncAdd ( the carrier of X,Y) is non empty Relation-like [:(Funcs ( the carrier of X, the carrier of Y)),(Funcs ( the carrier of X, the carrier of Y)):] -defined Funcs ( the carrier of X, the carrier of Y) -valued Function-like total quasi_total Function-yielding V34() Element of bool [:[:(Funcs ( the carrier of X, the carrier of Y)),(Funcs ( the carrier of X, the carrier of Y)):],(Funcs ( the carrier of X, the carrier of Y)):]
[:(Funcs ( the carrier of X, the carrier of Y)),(Funcs ( the carrier of X, the carrier of Y)):] is non empty set
[:[:(Funcs ( the carrier of X, the carrier of Y)),(Funcs ( the carrier of X, the carrier of Y)):],(Funcs ( the carrier of X, the carrier of Y)):] is non empty set
bool [:[:(Funcs ( the carrier of X, the carrier of Y)),(Funcs ( the carrier of X, the carrier of Y)):],(Funcs ( the carrier of X, the carrier of Y)):] is non empty set
( the carrier of X,Y) is non empty Relation-like [:COMPLEX,(Funcs ( the carrier of X, the carrier of Y)):] -defined Funcs ( the carrier of X, the carrier of Y) -valued Function-like total quasi_total Function-yielding V34() Element of bool [:[:COMPLEX,(Funcs ( the carrier of X, the carrier of Y)):],(Funcs ( the carrier of X, the carrier of Y)):]
[:COMPLEX,(Funcs ( the carrier of X, the carrier of Y)):] is non empty set
[:[:COMPLEX,(Funcs ( the carrier of X, the carrier of Y)):],(Funcs ( the carrier of X, the carrier of Y)):] is non empty set
bool [:[:COMPLEX,(Funcs ( the carrier of X, the carrier of Y)):],(Funcs ( the carrier of X, the carrier of Y)):] is non empty set
CLSStruct(# (Funcs ( the carrier of X, the carrier of Y)),(FuncZero ( the carrier of X,Y)),(FuncAdd ( the carrier of X,Y)),( the carrier of X,Y) #) is non empty strict CLSStruct
the carrier of ( the carrier of X,Y) is non empty set
bool the carrier of ( the carrier of X,Y) is non empty set
Zero_ ((X,Y),( the carrier of X,Y)) is Relation-like Function-like Element of (X,Y)
Add_ ((X,Y),( the carrier of X,Y)) is non empty Relation-like [:(X,Y),(X,Y):] -defined (X,Y) -valued Function-like total quasi_total Function-yielding V34() Element of bool [:[:(X,Y),(X,Y):],(X,Y):]
[:(X,Y),(X,Y):] is non empty set
[:[:(X,Y),(X,Y):],(X,Y):] is non empty set
bool [:[:(X,Y),(X,Y):],(X,Y):] is non empty set
Mult_ ((X,Y),( the carrier of X,Y)) is non empty Relation-like [:COMPLEX,(X,Y):] -defined (X,Y) -valued Function-like total quasi_total Function-yielding V34() Element of bool [:[:COMPLEX,(X,Y):],(X,Y):]
[:COMPLEX,(X,Y):] is non empty set
[:[:COMPLEX,(X,Y):],(X,Y):] is non empty set
bool [:[:COMPLEX,(X,Y):],(X,Y):] is non empty set
CLSStruct(# (X,Y),(Zero_ ((X,Y),( the carrier of X,Y))),(Add_ ((X,Y),( the carrier of X,Y))),(Mult_ ((X,Y),( the carrier of X,Y))) #) is non empty right_complementable Abelian add-associative right_zeroed strict vector-distributive scalar-distributive scalar-associative scalar-unital CLSStruct
the carrier of (X,Y) is non empty set
bool the carrier of (X,Y) is non empty set
Zero_ ((X,Y),(X,Y)) is Element of (X,Y)
Add_ ((X,Y),(X,Y)) is non empty Relation-like [:(X,Y),(X,Y):] -defined (X,Y) -valued Function-like total quasi_total Element of bool [:[:(X,Y),(X,Y):],(X,Y):]
[:(X,Y),(X,Y):] is non empty set
[:[:(X,Y),(X,Y):],(X,Y):] is non empty set
bool [:[:(X,Y),(X,Y):],(X,Y):] is non empty set
Mult_ ((X,Y),(X,Y)) is non empty Relation-like [:COMPLEX,(X,Y):] -defined (X,Y) -valued Function-like total quasi_total Element of bool [:[:COMPLEX,(X,Y):],(X,Y):]
[:COMPLEX,(X,Y):] is non empty set
[:[:COMPLEX,(X,Y):],(X,Y):] is non empty set
bool [:[:COMPLEX,(X,Y):],(X,Y):] is non empty set
(X,Y) is non empty Relation-like (X,Y) -defined REAL -valued Function-like total quasi_total V132() V133() V134() Element of bool [:(X,Y),REAL:]
[:(X,Y),REAL:] is non empty V132() V133() V134() set
bool [:(X,Y),REAL:] is non empty set
CNORMSTR(# (X,Y),(Zero_ ((X,Y),(X,Y))),(Add_ ((X,Y),(X,Y))),(Mult_ ((X,Y),(X,Y))),(X,Y) #) is non empty strict CNORMSTR
the carrier of (X,Y) is non empty set
[: the carrier of X, the carrier of Y:] is non empty set
bool [: the carrier of X, the carrier of Y:] is non empty set
vseq is Element of the carrier of (X,Y)
||.vseq.|| is complex real ext-real Element of REAL
the U9 of (X,Y) is non empty Relation-like the carrier of (X,Y) -defined REAL -valued Function-like total quasi_total V132() V133() V134() Element of bool [: the carrier of (X,Y),REAL:]
[: the carrier of (X,Y),REAL:] is non empty V132() V133() V134() set
bool [: the carrier of (X,Y),REAL:] is non empty set
the U9 of (X,Y) . vseq is complex real ext-real Element of REAL
f is non empty Relation-like the carrier of X -defined the carrier of Y -valued Function-like total quasi_total additive (X,Y) (X,Y) Element of bool [: the carrier of X, the carrier of Y:]
(X,Y,f) is non empty V142() V143() V144() Element of bool REAL
{ ||.(f . b₁).|| where b₁ is Element of the carrier of X : ||.b₁.|| <= 1 } is set
tseq is Element of the carrier of X
0. X is zero Element of the carrier of X
the ZeroF of X is Element of the carrier of X
||.tseq.|| is complex real ext-real Element of REAL
the U9 of X is non empty Relation-like the carrier of X -defined REAL -valued Function-like total quasi_total V132() V133() V134() Element of bool [: the carrier of X,REAL:]
[: the carrier of X,REAL:] is non empty V132() V133() V134() set
bool [: the carrier of X,REAL:] is non empty set
the U9 of X . tseq is complex real ext-real Element of REAL
f . tseq is Element of the carrier of Y
0c is empty Function-like functional epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural complex real ext-real non positive non negative V142() V143() V144() V145() V146() V147() V148() Element of COMPLEX
0c * (0. X) is Element of the carrier of X
f . (0c * (0. X)) is Element of the carrier of Y
f . (0. X) is Element of the carrier of Y
0c * (f . (0. X)) is Element of the carrier of Y
0. Y is zero Element of the carrier of Y
the ZeroF of Y is Element of the carrier of Y
||.(f . tseq).|| is complex real ext-real Element of REAL
the U9 of Y is non empty Relation-like the carrier of Y -defined REAL -valued Function-like total quasi_total V132() V133() V134() Element of bool [: the carrier of Y,REAL:]
[: the carrier of Y,REAL:] is non empty V132() V133() V134() set
bool [: the carrier of Y,REAL:] is non empty set
the U9 of Y . (f . tseq) is complex real ext-real Element of REAL
||.vseq.|| * ||.tseq.|| is complex real ext-real Element of REAL
tseq is Element of the carrier of X
0. X is zero Element of the carrier of X
the ZeroF of X is Element of the carrier of X
||.tseq.|| is complex real ext-real Element of REAL
the U9 of X is non empty Relation-like the carrier of X -defined REAL -valued Function-like total quasi_total V132() V133() V134() Element of bool [: the carrier of X,REAL:]
[: the carrier of X,REAL:] is non empty V132() V133() V134() set
bool [: the carrier of X,REAL:] is non empty set
the U9 of X . tseq is complex real ext-real Element of REAL
||.tseq.|| " is complex real ext-real Element of REAL
<i> is complex Element of COMPLEX
0 * <i> is complex set
(||.tseq.|| ") + (0 * <i>) is complex set
tseq is complex Element of COMPLEX
tseq * tseq is Element of the carrier of X
|.tseq.| is complex real ext-real Element of REAL
1 * (||.tseq.|| ") is complex real ext-real Element of REAL
abs (1 * (||.tseq.|| ")) is complex real ext-real Element of REAL
1 / ||.tseq.|| is complex real ext-real Element of REAL
abs (1 / ||.tseq.||) is complex real ext-real Element of REAL
abs ||.tseq.|| is complex real ext-real Element of REAL
1 / (abs ||.tseq.||) is complex real ext-real Element of REAL
f . tseq is Element of the carrier of Y
||.(f . tseq).|| is complex real ext-real Element of REAL
the U9 of Y is non empty Relation-like the carrier of Y -defined REAL -valued Function-like total quasi_total V132() V133() V134() Element of bool [: the carrier of Y,REAL:]
[: the carrier of Y,REAL:] is non empty V132() V133() V134() set
bool [: the carrier of Y,REAL:] is non empty set
the U9 of Y . (f . tseq) is complex real ext-real Element of REAL
||.(f . tseq).|| / ||.tseq.|| is complex real ext-real Element of REAL
||.(f . tseq).|| * |.tseq.| is complex real ext-real Element of REAL
tseq * (f . tseq) is Element of the carrier of Y
||.(tseq * (f . tseq)).|| is complex real ext-real Element of REAL
the U9 of Y . (tseq * (f . tseq)) is complex real ext-real Element of REAL
tseq is Element of the carrier of X
f . tseq is Element of the carrier of Y
||.(f . tseq).|| is complex real ext-real Element of REAL
the U9 of Y . (f . tseq) is complex real ext-real Element of REAL
||.tseq.|| is complex real ext-real Element of REAL
the U9 of X . tseq is complex real ext-real Element of REAL
|.tseq.| * ||.tseq.|| is complex real ext-real Element of REAL
upper_bound (X,Y,f) is complex real ext-real Element of REAL
(X,Y) . f is complex real ext-real Element of REAL
(||.(f . tseq).|| / ||.tseq.||) * ||.tseq.|| is complex real ext-real Element of REAL
||.(f . tseq).|| * (||.tseq.|| ") is complex real ext-real Element of REAL
(||.(f . tseq).|| * (||.tseq.|| ")) * ||.tseq.|| is complex real ext-real Element of REAL
(||.tseq.|| ") * ||.tseq.|| is complex real ext-real Element of REAL
||.(f . tseq).|| * ((||.tseq.|| ") * ||.tseq.||) is complex real ext-real Element of REAL
||.(f . tseq).|| * 1 is complex real ext-real Element of REAL
||.vseq.|| * ||.tseq.|| is complex real ext-real Element of REAL
tseq is Element of the carrier of X
0. X is zero Element of the carrier of X
the ZeroF of X is Element of the carrier of X
f . tseq is Element of the carrier of Y
||.(f . tseq).|| is complex real ext-real Element of REAL
the U9 of Y is non empty Relation-like the carrier of Y -defined REAL -valued Function-like total quasi_total V132() V133() V134() Element of bool [: the carrier of Y,REAL:]
[: the carrier of Y,REAL:] is non empty V132() V133() V134() set
bool [: the carrier of Y,REAL:] is non empty set
the U9 of Y . (f . tseq) is complex real ext-real Element of REAL
||.tseq.|| is complex real ext-real Element of REAL
the U9 of X is non empty Relation-like the carrier of X -defined REAL -valued Function-like total quasi_total V132() V133() V134() Element of bool [: the carrier of X,REAL:]
[: the carrier of X,REAL:] is non empty V132() V133() V134() set
bool [: the carrier of X,REAL:] is non empty set
the U9 of X . tseq is complex real ext-real Element of REAL
||.vseq.|| * ||.tseq.|| is complex real ext-real Element of REAL
tseq is Element of the carrier of X
0. X is zero Element of the carrier of X
the ZeroF of X is Element of the carrier of X
f . tseq is Element of the carrier of Y
||.(f . tseq).|| is complex real ext-real Element of REAL
the U9 of Y is non empty Relation-like the carrier of Y -defined REAL -valued Function-like total quasi_total V132() V133() V134() Element of bool [: the carrier of Y,REAL:]
[: the carrier of Y,REAL:] is non empty V132() V133() V134() set
bool [: the carrier of Y,REAL:] is non empty set
the U9 of Y . (f . tseq) is complex real ext-real Element of REAL
||.tseq.|| is complex real ext-real Element of REAL
the U9 of X is non empty Relation-like the carrier of X -defined REAL -valued Function-like total quasi_total V132() V133() V134() Element of bool [: the carrier of X,REAL:]
[: the carrier of X,REAL:] is non empty V132() V133() V134() set
bool [: the carrier of X,REAL:] is non empty set
the U9 of X . tseq is complex real ext-real Element of REAL
||.vseq.|| * ||.tseq.|| is complex real ext-real Element of REAL
tseq is Element of the carrier of X
f . tseq is Element of the carrier of Y
||.(f . tseq).|| is complex real ext-real Element of REAL
the U9 of Y . (f . tseq) is complex real ext-real Element of REAL
||.tseq.|| is complex real ext-real Element of REAL
the U9 of X . tseq is complex real ext-real Element of REAL
||.vseq.|| * ||.tseq.|| is complex real ext-real Element of REAL
X is non empty right_complementable Abelian add-associative right_zeroed discerning reflexive vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like CNORMSTR
Y is non empty right_complementable Abelian add-associative right_zeroed discerning reflexive vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like CNORMSTR
(X,Y) is non empty CNORMSTR
(X,Y) is non empty Element of bool the carrier of (X,Y)
(X,Y) is non empty right_complementable Abelian add-associative right_zeroed constituted-Functions strict vector-distributive scalar-distributive scalar-associative scalar-unital CLSStruct
(X,Y) is non empty functional Element of bool the carrier of ( the carrier of X,Y)
the carrier of X is non empty set
( the carrier of X,Y) is non empty right_complementable Abelian add-associative right_zeroed constituted-Functions strict vector-distributive scalar-distributive scalar-associative scalar-unital CLSStruct
the carrier of Y is non empty set
Funcs ( the carrier of X, the carrier of Y) is non empty functional FUNCTION_DOMAIN of the carrier of X, the carrier of Y
FuncZero ( the carrier of X,Y) is Relation-like the carrier of X -defined the carrier of Y -valued Function-like total quasi_total Element of Funcs ( the carrier of X, the carrier of Y)
FuncAdd ( the carrier of X,Y) is non empty Relation-like [:(Funcs ( the carrier of X, the carrier of Y)),(Funcs ( the carrier of X, the carrier of Y)):] -defined Funcs ( the carrier of X, the carrier of Y) -valued Function-like total quasi_total Function-yielding V34() Element of bool [:[:(Funcs ( the carrier of X, the carrier of Y)),(Funcs ( the carrier of X, the carrier of Y)):],(Funcs ( the carrier of X, the carrier of Y)):]
[:(Funcs ( the carrier of X, the carrier of Y)),(Funcs ( the carrier of X, the carrier of Y)):] is non empty set
[:[:(Funcs ( the carrier of X, the carrier of Y)),(Funcs ( the carrier of X, the carrier of Y)):],(Funcs ( the carrier of X, the carrier of Y)):] is non empty set
bool [:[:(Funcs ( the carrier of X, the carrier of Y)),(Funcs ( the carrier of X, the carrier of Y)):],(Funcs ( the carrier of X, the carrier of Y)):] is non empty set
( the carrier of X,Y) is non empty Relation-like [:COMPLEX,(Funcs ( the carrier of X, the carrier of Y)):] -defined Funcs ( the carrier of X, the carrier of Y) -valued Function-like total quasi_total Function-yielding V34() Element of bool [:[:COMPLEX,(Funcs ( the carrier of X, the carrier of Y)):],(Funcs ( the carrier of X, the carrier of Y)):]
[:COMPLEX,(Funcs ( the carrier of X, the carrier of Y)):] is non empty set
[:[:COMPLEX,(Funcs ( the carrier of X, the carrier of Y)):],(Funcs ( the carrier of X, the carrier of Y)):] is non empty set
bool [:[:COMPLEX,(Funcs ( the carrier of X, the carrier of Y)):],(Funcs ( the carrier of X, the carrier of Y)):] is non empty set
CLSStruct(# (Funcs ( the carrier of X, the carrier of Y)),(FuncZero ( the carrier of X,Y)),(FuncAdd ( the carrier of X,Y)),( the carrier of X,Y) #) is non empty strict CLSStruct
the carrier of ( the carrier of X,Y) is non empty set
bool the carrier of ( the carrier of X,Y) is non empty set
Zero_ ((X,Y),( the carrier of X,Y)) is Relation-like Function-like Element of (X,Y)
Add_ ((X,Y),( the carrier of X,Y)) is non empty Relation-like [:(X,Y),(X,Y):] -defined (X,Y) -valued Function-like total quasi_total Function-yielding V34() Element of bool [:[:(X,Y),(X,Y):],(X,Y):]
[:(X,Y),(X,Y):] is non empty set
[:[:(X,Y),(X,Y):],(X,Y):] is non empty set
bool [:[:(X,Y),(X,Y):],(X,Y):] is non empty set
Mult_ ((X,Y),( the carrier of X,Y)) is non empty Relation-like [:COMPLEX,(X,Y):] -defined (X,Y) -valued Function-like total quasi_total Function-yielding V34() Element of bool [:[:COMPLEX,(X,Y):],(X,Y):]
[:COMPLEX,(X,Y):] is non empty set
[:[:COMPLEX,(X,Y):],(X,Y):] is non empty set
bool [:[:COMPLEX,(X,Y):],(X,Y):] is non empty set
CLSStruct(# (X,Y),(Zero_ ((X,Y),( the carrier of X,Y))),(Add_ ((X,Y),( the carrier of X,Y))),(Mult_ ((X,Y),( the carrier of X,Y))) #) is non empty right_complementable Abelian add-associative right_zeroed strict vector-distributive scalar-distributive scalar-associative scalar-unital CLSStruct
the carrier of (X,Y) is non empty set
bool the carrier of (X,Y) is non empty set
Zero_ ((X,Y),(X,Y)) is Element of (X,Y)
Add_ ((X,Y),(X,Y)) is non empty Relation-like [:(X,Y),(X,Y):] -defined (X,Y) -valued Function-like total quasi_total Element of bool [:[:(X,Y),(X,Y):],(X,Y):]
[:(X,Y),(X,Y):] is non empty set
[:[:(X,Y),(X,Y):],(X,Y):] is non empty set
bool [:[:(X,Y),(X,Y):],(X,Y):] is non empty set
Mult_ ((X,Y),(X,Y)) is non empty Relation-like [:COMPLEX,(X,Y):] -defined (X,Y) -valued Function-like total quasi_total Element of bool [:[:COMPLEX,(X,Y):],(X,Y):]
[:COMPLEX,(X,Y):] is non empty set
[:[:COMPLEX,(X,Y):],(X,Y):] is non empty set
bool [:[:COMPLEX,(X,Y):],(X,Y):] is non empty set
(X,Y) is non empty Relation-like (X,Y) -defined REAL -valued Function-like total quasi_total V132() V133() V134() Element of bool [:(X,Y),REAL:]
[:(X,Y),REAL:] is non empty V132() V133() V134() set
bool [:(X,Y),REAL:] is non empty set
CNORMSTR(# (X,Y),(Zero_ ((X,Y),(X,Y))),(Add_ ((X,Y),(X,Y))),(Mult_ ((X,Y),(X,Y))),(X,Y) #) is non empty strict CNORMSTR
the carrier of (X,Y) is non empty set
vseq is Element of the carrier of (X,Y)
||.vseq.|| is complex real ext-real Element of REAL
the U9 of (X,Y) is non empty Relation-like the carrier of (X,Y) -defined REAL -valued Function-like total quasi_total V132() V133() V134() Element of bool [: the carrier of (X,Y),REAL:]
[: the carrier of (X,Y),REAL:] is non empty V132() V133() V134() set
bool [: the carrier of (X,Y),REAL:] is non empty set
the U9 of (X,Y) . vseq is complex real ext-real Element of REAL
[: the carrier of X, the carrier of Y:] is non empty set
bool [: the carrier of X, the carrier of Y:] is non empty set
f is non empty Relation-like the carrier of X -defined the carrier of Y -valued Function-like total quasi_total additive (X,Y) (X,Y) Element of bool [: the carrier of X, the carrier of Y:]
(X,Y,f) is non empty V142() V143() V144() Element of bool REAL
{ ||.(f . b₁).|| where b₁ is Element of the carrier of X : ||.b₁.|| <= 1 } is set
tseq is set
(X,Y) . vseq is complex real ext-real Element of REAL
upper_bound (X,Y,f) is complex real ext-real Element of REAL
tseq is complex real ext-real Element of REAL
tv is Element of the carrier of X
f . tv is Element of the carrier of Y
||.(f . tv).|| is complex real ext-real Element of REAL
the U9 of Y is non empty Relation-like the carrier of Y -defined REAL -valued Function-like total quasi_total V132() V133() V134() Element of bool [: the carrier of Y,REAL:]
[: the carrier of Y,REAL:] is non empty V132() V133() V134() set
bool [: the carrier of Y,REAL:] is non empty set
the U9 of Y . (f . tv) is complex real ext-real Element of REAL
||.tv.|| is complex real ext-real Element of REAL
the U9 of X is non empty Relation-like the carrier of X -defined REAL -valued Function-like total quasi_total V132() V133() V134() Element of bool [: the carrier of X,REAL:]
[: the carrier of X,REAL:] is non empty V132() V133() V134() set
bool [: the carrier of X,REAL:] is non empty set
the U9 of X . tv is complex real ext-real Element of REAL
tseq is complex real ext-real Element of REAL
X is non empty right_complementable Abelian add-associative right_zeroed discerning reflexive vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like CNORMSTR
Y is non empty right_complementable Abelian add-associative right_zeroed discerning reflexive vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like CNORMSTR
(X,Y) is non empty CNORMSTR
(X,Y) is non empty Element of bool the carrier of (X,Y)
(X,Y) is non empty right_complementable Abelian add-associative right_zeroed constituted-Functions strict vector-distributive scalar-distributive scalar-associative scalar-unital CLSStruct
(X,Y) is non empty functional Element of bool the carrier of ( the carrier of X,Y)
the carrier of X is non empty set
( the carrier of X,Y) is non empty right_complementable Abelian add-associative right_zeroed constituted-Functions strict vector-distributive scalar-distributive scalar-associative scalar-unital CLSStruct
the carrier of Y is non empty set
Funcs ( the carrier of X, the carrier of Y) is non empty functional FUNCTION_DOMAIN of the carrier of X, the carrier of Y
FuncZero ( the carrier of X,Y) is Relation-like the carrier of X -defined the carrier of Y -valued Function-like total quasi_total Element of Funcs ( the carrier of X, the carrier of Y)
FuncAdd ( the carrier of X,Y) is non empty Relation-like [:(Funcs ( the carrier of X, the carrier of Y)),(Funcs ( the carrier of X, the carrier of Y)):] -defined Funcs ( the carrier of X, the carrier of Y) -valued Function-like total quasi_total Function-yielding V34() Element of bool [:[:(Funcs ( the carrier of X, the carrier of Y)),(Funcs ( the carrier of X, the carrier of Y)):],(Funcs ( the carrier of X, the carrier of Y)):]
[:(Funcs ( the carrier of X, the carrier of Y)),(Funcs ( the carrier of X, the carrier of Y)):] is non empty set
[:[:(Funcs ( the carrier of X, the carrier of Y)),(Funcs ( the carrier of X, the carrier of Y)):],(Funcs ( the carrier of X, the carrier of Y)):] is non empty set
bool [:[:(Funcs ( the carrier of X, the carrier of Y)),(Funcs ( the carrier of X, the carrier of Y)):],(Funcs ( the carrier of X, the carrier of Y)):] is non empty set
( the carrier of X,Y) is non empty Relation-like [:COMPLEX,(Funcs ( the carrier of X, the carrier of Y)):] -defined Funcs ( the carrier of X, the carrier of Y) -valued Function-like total quasi_total Function-yielding V34() Element of bool [:[:COMPLEX,(Funcs ( the carrier of X, the carrier of Y)):],(Funcs ( the carrier of X, the carrier of Y)):]
[:COMPLEX,(Funcs ( the carrier of X, the carrier of Y)):] is non empty set
[:[:COMPLEX,(Funcs ( the carrier of X, the carrier of Y)):],(Funcs ( the carrier of X, the carrier of Y)):] is non empty set
bool [:[:COMPLEX,(Funcs ( the carrier of X, the carrier of Y)):],(Funcs ( the carrier of X, the carrier of Y)):] is non empty set
CLSStruct(# (Funcs ( the carrier of X, the carrier of Y)),(FuncZero ( the carrier of X,Y)),(FuncAdd ( the carrier of X,Y)),( the carrier of X,Y) #) is non empty strict CLSStruct
the carrier of ( the carrier of X,Y) is non empty set
bool the carrier of ( the carrier of X,Y) is non empty set
Zero_ ((X,Y),( the carrier of X,Y)) is Relation-like Function-like Element of (X,Y)
Add_ ((X,Y),( the carrier of X,Y)) is non empty Relation-like [:(X,Y),(X,Y):] -defined (X,Y) -valued Function-like total quasi_total Function-yielding V34() Element of bool [:[:(X,Y),(X,Y):],(X,Y):]
[:(X,Y),(X,Y):] is non empty set
[:[:(X,Y),(X,Y):],(X,Y):] is non empty set
bool [:[:(X,Y),(X,Y):],(X,Y):] is non empty set
Mult_ ((X,Y),( the carrier of X,Y)) is non empty Relation-like [:COMPLEX,(X,Y):] -defined (X,Y) -valued Function-like total quasi_total Function-yielding V34() Element of bool [:[:COMPLEX,(X,Y):],(X,Y):]
[:COMPLEX,(X,Y):] is non empty set
[:[:COMPLEX,(X,Y):],(X,Y):] is non empty set
bool [:[:COMPLEX,(X,Y):],(X,Y):] is non empty set
CLSStruct(# (X,Y),(Zero_ ((X,Y),( the carrier of X,Y))),(Add_ ((X,Y),( the carrier of X,Y))),(Mult_ ((X,Y),( the carrier of X,Y))) #) is non empty right_complementable Abelian add-associative right_zeroed strict vector-distributive scalar-distributive scalar-associative scalar-unital CLSStruct
the carrier of (X,Y) is non empty set
bool the carrier of (X,Y) is non empty set
Zero_ ((X,Y),(X,Y)) is Element of (X,Y)
Add_ ((X,Y),(X,Y)) is non empty Relation-like [:(X,Y),(X,Y):] -defined (X,Y) -valued Function-like total quasi_total Element of bool [:[:(X,Y),(X,Y):],(X,Y):]
[:(X,Y),(X,Y):] is non empty set
[:[:(X,Y),(X,Y):],(X,Y):] is non empty set
bool [:[:(X,Y),(X,Y):],(X,Y):] is non empty set
Mult_ ((X,Y),(X,Y)) is non empty Relation-like [:COMPLEX,(X,Y):] -defined (X,Y) -valued Function-like total quasi_total Element of bool [:[:COMPLEX,(X,Y):],(X,Y):]
[:COMPLEX,(X,Y):] is non empty set
[:[:COMPLEX,(X,Y):],(X,Y):] is non empty set
bool [:[:COMPLEX,(X,Y):],(X,Y):] is non empty set
(X,Y) is non empty Relation-like (X,Y) -defined REAL -valued Function-like total quasi_total V132() V133() V134() Element of bool [:(X,Y),REAL:]
[:(X,Y),REAL:] is non empty V132() V133() V134() set
bool [:(X,Y),REAL:] is non empty set
CNORMSTR(# (X,Y),(Zero_ ((X,Y),(X,Y))),(Add_ ((X,Y),(X,Y))),(Mult_ ((X,Y),(X,Y))),(X,Y) #) is non empty strict CNORMSTR
the carrier of (X,Y) is non empty set
0. (X,Y) is zero Element of the carrier of (X,Y)
the ZeroF of (X,Y) is Element of the carrier of (X,Y)
vseq is Element of the carrier of (X,Y)
||.vseq.|| is complex real ext-real Element of REAL
the U9 of (X,Y) is non empty Relation-like the carrier of (X,Y) -defined REAL -valued Function-like total quasi_total V132() V133() V134() Element of bool [: the carrier of (X,Y),REAL:]
[: the carrier of (X,Y),REAL:] is non empty V132() V133() V134() set
bool [: the carrier of (X,Y),REAL:] is non empty set
the U9 of (X,Y) . vseq is complex real ext-real Element of REAL
[: the carrier of X, the carrier of Y:] is non empty set
bool [: the carrier of X, the carrier of Y:] is non empty set
0. Y is zero Element of the carrier of Y
the ZeroF of Y is Element of the carrier of Y
the carrier of X --> (0. Y) is non empty Relation-like the carrier of X -defined the carrier of Y -valued Function-like constant total quasi_total Element of bool [: the carrier of X, the carrier of Y:]
f is non empty Relation-like the carrier of X -defined the carrier of Y -valued Function-like total quasi_total additive (X,Y) (X,Y) Element of bool [: the carrier of X, the carrier of Y:]
(X,Y,f) is non empty V142() V143() V144() Element of bool REAL
{ ||.(f . b₁).|| where b₁ is Element of the carrier of X : ||.b₁.|| <= 1 } is set
tseq is set
upper_bound (X,Y,f) is complex real ext-real Element of REAL
tseq is non empty Relation-like the carrier of X -defined the carrier of Y -valued Function-like total quasi_total Element of bool [: the carrier of X, the carrier of Y:]
e is complex real ext-real Element of REAL
m is Element of the carrier of X
f . m is Element of the carrier of Y
||.(f . m).|| is complex real ext-real Element of REAL
the U9 of Y is non empty Relation-like the carrier of Y -defined REAL -valued Function-like total quasi_total V132() V133() V134() Element of bool [: the carrier of Y,REAL:]
[: the carrier of Y,REAL:] is non empty V132() V133() V134() set
bool [: the carrier of Y,REAL:] is non empty set
the U9 of Y . (f . m) is complex real ext-real Element of REAL
||.m.|| is complex real ext-real Element of REAL
the U9 of X is non empty Relation-like the carrier of X -defined REAL -valued Function-like total quasi_total V132() V133() V134() Element of bool [: the carrier of X,REAL:]
[: the carrier of X,REAL:] is non empty V132() V133() V134() set
bool [: the carrier of X,REAL:] is non empty set
the U9 of X . m is complex real ext-real Element of REAL
||.(0. Y).|| is complex real ext-real Element of REAL
the U9 of Y . (0. Y) is complex real ext-real Element of REAL
tv is complex real ext-real Element of REAL
e is complex real ext-real set
(X,Y) . vseq is complex real ext-real Element of REAL
X is non empty right_complementable Abelian add-associative right_zeroed discerning reflexive vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like CNORMSTR
Y is non empty right_complementable Abelian add-associative right_zeroed discerning reflexive vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like CNORMSTR
(X,Y) is non empty CNORMSTR
(X,Y) is non empty Element of bool the carrier of (X,Y)
(X,Y) is non empty right_complementable Abelian add-associative right_zeroed constituted-Functions strict vector-distributive scalar-distributive scalar-associative scalar-unital CLSStruct
(X,Y) is non empty functional Element of bool the carrier of ( the carrier of X,Y)
the carrier of X is non empty set
( the carrier of X,Y) is non empty right_complementable Abelian add-associative right_zeroed constituted-Functions strict vector-distributive scalar-distributive scalar-associative scalar-unital CLSStruct
the carrier of Y is non empty set
Funcs ( the carrier of X, the carrier of Y) is non empty functional FUNCTION_DOMAIN of the carrier of X, the carrier of Y
FuncZero ( the carrier of X,Y) is Relation-like the carrier of X -defined the carrier of Y -valued Function-like total quasi_total Element of Funcs ( the carrier of X, the carrier of Y)
FuncAdd ( the carrier of X,Y) is non empty Relation-like [:(Funcs ( the carrier of X, the carrier of Y)),(Funcs ( the carrier of X, the carrier of Y)):] -defined Funcs ( the carrier of X, the carrier of Y) -valued Function-like total quasi_total Function-yielding V34() Element of bool [:[:(Funcs ( the carrier of X, the carrier of Y)),(Funcs ( the carrier of X, the carrier of Y)):],(Funcs ( the carrier of X, the carrier of Y)):]
[:(Funcs ( the carrier of X, the carrier of Y)),(Funcs ( the carrier of X, the carrier of Y)):] is non empty set
[:[:(Funcs ( the carrier of X, the carrier of Y)),(Funcs ( the carrier of X, the carrier of Y)):],(Funcs ( the carrier of X, the carrier of Y)):] is non empty set
bool [:[:(Funcs ( the carrier of X, the carrier of Y)),(Funcs ( the carrier of X, the carrier of Y)):],(Funcs ( the carrier of X, the carrier of Y)):] is non empty set
( the carrier of X,Y) is non empty Relation-like [:COMPLEX,(Funcs ( the carrier of X, the carrier of Y)):] -defined Funcs ( the carrier of X, the carrier of Y) -valued Function-like total quasi_total Function-yielding V34() Element of bool [:[:COMPLEX,(Funcs ( the carrier of X, the carrier of Y)):],(Funcs ( the carrier of X, the carrier of Y)):]
[:COMPLEX,(Funcs ( the carrier of X, the carrier of Y)):] is non empty set
[:[:COMPLEX,(Funcs ( the carrier of X, the carrier of Y)):],(Funcs ( the carrier of X, the carrier of Y)):] is non empty set
bool [:[:COMPLEX,(Funcs ( the carrier of X, the carrier of Y)):],(Funcs ( the carrier of X, the carrier of Y)):] is non empty set
CLSStruct(# (Funcs ( the carrier of X, the carrier of Y)),(FuncZero ( the carrier of X,Y)),(FuncAdd ( the carrier of X,Y)),( the carrier of X,Y) #) is non empty strict CLSStruct
the carrier of ( the carrier of X,Y) is non empty set
bool the carrier of ( the carrier of X,Y) is non empty set
Zero_ ((X,Y),( the carrier of X,Y)) is Relation-like Function-like Element of (X,Y)
Add_ ((X,Y),( the carrier of X,Y)) is non empty Relation-like [:(X,Y),(X,Y):] -defined (X,Y) -valued Function-like total quasi_total Function-yielding V34() Element of bool [:[:(X,Y),(X,Y):],(X,Y):]
[:(X,Y),(X,Y):] is non empty set
[:[:(X,Y),(X,Y):],(X,Y):] is non empty set
bool [:[:(X,Y),(X,Y):],(X,Y):] is non empty set
Mult_ ((X,Y),( the carrier of X,Y)) is non empty Relation-like [:COMPLEX,(X,Y):] -defined (X,Y) -valued Function-like total quasi_total Function-yielding V34() Element of bool [:[:COMPLEX,(X,Y):],(X,Y):]
[:COMPLEX,(X,Y):] is non empty set
[:[:COMPLEX,(X,Y):],(X,Y):] is non empty set
bool [:[:COMPLEX,(X,Y):],(X,Y):] is non empty set
CLSStruct(# (X,Y),(Zero_ ((X,Y),( the carrier of X,Y))),(Add_ ((X,Y),( the carrier of X,Y))),(Mult_ ((X,Y),( the carrier of X,Y))) #) is non empty right_complementable Abelian add-associative right_zeroed strict vector-distributive scalar-distributive scalar-associative scalar-unital CLSStruct
the carrier of (X,Y) is non empty set
bool the carrier of (X,Y) is non empty set
Zero_ ((X,Y),(X,Y)) is Element of (X,Y)
Add_ ((X,Y),(X,Y)) is non empty Relation-like [:(X,Y),(X,Y):] -defined (X,Y) -valued Function-like total quasi_total Element of bool [:[:(X,Y),(X,Y):],(X,Y):]
[:(X,Y),(X,Y):] is non empty set
[:[:(X,Y),(X,Y):],(X,Y):] is non empty set
bool [:[:(X,Y),(X,Y):],(X,Y):] is non empty set
Mult_ ((X,Y),(X,Y)) is non empty Relation-like [:COMPLEX,(X,Y):] -defined (X,Y) -valued Function-like total quasi_total Element of bool [:[:COMPLEX,(X,Y):],(X,Y):]
[:COMPLEX,(X,Y):] is non empty set
[:[:COMPLEX,(X,Y):],(X,Y):] is non empty set
bool [:[:COMPLEX,(X,Y):],(X,Y):] is non empty set
(X,Y) is non empty Relation-like (X,Y) -defined REAL -valued Function-like total quasi_total V132() V133() V134() Element of bool [:(X,Y),REAL:]
[:(X,Y),REAL:] is non empty V132() V133() V134() set
bool [:(X,Y),REAL:] is non empty set
CNORMSTR(# (X,Y),(Zero_ ((X,Y),(X,Y))),(Add_ ((X,Y),(X,Y))),(Mult_ ((X,Y),(X,Y))),(X,Y) #) is non empty strict CNORMSTR
the carrier of (X,Y) is non empty set
vseq is Element of the carrier of (X,Y)
f is Element of the carrier of (X,Y)
X is non empty right_complementable Abelian add-associative right_zeroed discerning reflexive vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like CNORMSTR
Y is non empty right_complementable Abelian add-associative right_zeroed discerning reflexive vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like CNORMSTR
(X,Y) is non empty CNORMSTR
(X,Y) is non empty Element of bool the carrier of (X,Y)
(X,Y) is non empty right_complementable Abelian add-associative right_zeroed constituted-Functions strict vector-distributive scalar-distributive scalar-associative scalar-unital CLSStruct
(X,Y) is non empty functional Element of bool the carrier of ( the carrier of X,Y)
the carrier of X is non empty set
( the carrier of X,Y) is non empty right_complementable Abelian add-associative right_zeroed constituted-Functions strict vector-distributive scalar-distributive scalar-associative scalar-unital CLSStruct
the carrier of Y is non empty set
Funcs ( the carrier of X, the carrier of Y) is non empty functional FUNCTION_DOMAIN of the carrier of X, the carrier of Y
FuncZero ( the carrier of X,Y) is Relation-like the carrier of X -defined the carrier of Y -valued Function-like total quasi_total Element of Funcs ( the carrier of X, the carrier of Y)
FuncAdd ( the carrier of X,Y) is non empty Relation-like [:(Funcs ( the carrier of X, the carrier of Y)),(Funcs ( the carrier of X, the carrier of Y)):] -defined Funcs ( the carrier of X, the carrier of Y) -valued Function-like total quasi_total Function-yielding V34() Element of bool [:[:(Funcs ( the carrier of X, the carrier of Y)),(Funcs ( the carrier of X, the carrier of Y)):],(Funcs ( the carrier of X, the carrier of Y)):]
[:(Funcs ( the carrier of X, the carrier of Y)),(Funcs ( the carrier of X, the carrier of Y)):] is non empty set
[:[:(Funcs ( the carrier of X, the carrier of Y)),(Funcs ( the carrier of X, the carrier of Y)):],(Funcs ( the carrier of X, the carrier of Y)):] is non empty set
bool [:[:(Funcs ( the carrier of X, the carrier of Y)),(Funcs ( the carrier of X, the carrier of Y)):],(Funcs ( the carrier of X, the carrier of Y)):] is non empty set
( the carrier of X,Y) is non empty Relation-like [:COMPLEX,(Funcs ( the carrier of X, the carrier of Y)):] -defined Funcs ( the carrier of X, the carrier of Y) -valued Function-like total quasi_total Function-yielding V34() Element of bool [:[:COMPLEX,(Funcs ( the carrier of X, the carrier of Y)):],(Funcs ( the carrier of X, the carrier of Y)):]
[:COMPLEX,(Funcs ( the carrier of X, the carrier of Y)):] is non empty set
[:[:COMPLEX,(Funcs ( the carrier of X, the carrier of Y)):],(Funcs ( the carrier of X, the carrier of Y)):] is non empty set
bool [:[:COMPLEX,(Funcs ( the carrier of X, the carrier of Y)):],(Funcs ( the carrier of X, the carrier of Y)):] is non empty set
CLSStruct(# (Funcs ( the carrier of X, the carrier of Y)),(FuncZero ( the carrier of X,Y)),(FuncAdd ( the carrier of X,Y)),( the carrier of X,Y) #) is non empty strict CLSStruct
the carrier of ( the carrier of X,Y) is non empty set
bool the carrier of ( the carrier of X,Y) is non empty set
Zero_ ((X,Y),( the carrier of X,Y)) is Relation-like Function-like Element of (X,Y)
Add_ ((X,Y),( the carrier of X,Y)) is non empty Relation-like [:(X,Y),(X,Y):] -defined (X,Y) -valued Function-like total quasi_total Function-yielding V34() Element of bool [:[:(X,Y),(X,Y):],(X,Y):]
[:(X,Y),(X,Y):] is non empty set
[:[:(X,Y),(X,Y):],(X,Y):] is non empty set
bool [:[:(X,Y),(X,Y):],(X,Y):] is non empty set
Mult_ ((X,Y),( the carrier of X,Y)) is non empty Relation-like [:COMPLEX,(X,Y):] -defined (X,Y) -valued Function-like total quasi_total Function-yielding V34() Element of bool [:[:COMPLEX,(X,Y):],(X,Y):]
[:COMPLEX,(X,Y):] is non empty set
[:[:COMPLEX,(X,Y):],(X,Y):] is non empty set
bool [:[:COMPLEX,(X,Y):],(X,Y):] is non empty set
CLSStruct(# (X,Y),(Zero_ ((X,Y),( the carrier of X,Y))),(Add_ ((X,Y),( the carrier of X,Y))),(Mult_ ((X,Y),( the carrier of X,Y))) #) is non empty right_complementable Abelian add-associative right_zeroed strict vector-distributive scalar-distributive scalar-associative scalar-unital CLSStruct
the carrier of (X,Y) is non empty set
bool the carrier of (X,Y) is non empty set
Zero_ ((X,Y),(X,Y)) is Element of (X,Y)
Add_ ((X,Y),(X,Y)) is non empty Relation-like [:(X,Y),(X,Y):] -defined (X,Y) -valued Function-like total quasi_total Element of bool [:[:(X,Y),(X,Y):],(X,Y):]
[:(X,Y),(X,Y):] is non empty set
[:[:(X,Y),(X,Y):],(X,Y):] is non empty set
bool [:[:(X,Y),(X,Y):],(X,Y):] is non empty set
Mult_ ((X,Y),(X,Y)) is non empty Relation-like [:COMPLEX,(X,Y):] -defined (X,Y) -valued Function-like total quasi_total Element of bool [:[:COMPLEX,(X,Y):],(X,Y):]
[:COMPLEX,(X,Y):] is non empty set
[:[:COMPLEX,(X,Y):],(X,Y):] is non empty set
bool [:[:COMPLEX,(X,Y):],(X,Y):] is non empty set
(X,Y) is non empty Relation-like (X,Y) -defined REAL -valued Function-like total quasi_total V132() V133() V134() Element of bool [:(X,Y),REAL:]
[:(X,Y),REAL:] is non empty V132() V133() V134() set
bool [:(X,Y),REAL:] is non empty set
CNORMSTR(# (X,Y),(Zero_ ((X,Y),(X,Y))),(Add_ ((X,Y),(X,Y))),(Mult_ ((X,Y),(X,Y))),(X,Y) #) is non empty strict CNORMSTR
the carrier of (X,Y) is non empty set
vseq is Relation-like Function-like Element of the carrier of (X,Y)
f is Element of the carrier of X
vseq . f is set
[: the carrier of X, the carrier of Y:] is non empty set
bool [: the carrier of X, the carrier of Y:] is non empty set
tseq is non empty Relation-like the carrier of X -defined the carrier of Y -valued Function-like total quasi_total additive (X,Y) Element of bool [: the carrier of X, the carrier of Y:]
tseq . f is Element of the carrier of Y
X is non empty right_complementable Abelian add-associative right_zeroed discerning reflexive vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like CNORMSTR
Y is non empty right_complementable Abelian add-associative right_zeroed discerning reflexive vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like CNORMSTR
(X,Y) is non empty CNORMSTR
(X,Y) is non empty Element of bool the carrier of (X,Y)
(X,Y) is non empty right_complementable Abelian add-associative right_zeroed constituted-Functions strict vector-distributive scalar-distributive scalar-associative scalar-unital CLSStruct
(X,Y) is non empty functional Element of bool the carrier of ( the carrier of X,Y)
the carrier of X is non empty set
( the carrier of X,Y) is non empty right_complementable Abelian add-associative right_zeroed constituted-Functions strict vector-distributive scalar-distributive scalar-associative scalar-unital CLSStruct
the carrier of Y is non empty set
Funcs ( the carrier of X, the carrier of Y) is non empty functional FUNCTION_DOMAIN of the carrier of X, the carrier of Y
FuncZero ( the carrier of X,Y) is Relation-like the carrier of X -defined the carrier of Y -valued Function-like total quasi_total Element of Funcs ( the carrier of X, the carrier of Y)
FuncAdd ( the carrier of X,Y) is non empty Relation-like [:(Funcs ( the carrier of X, the carrier of Y)),(Funcs ( the carrier of X, the carrier of Y)):] -defined Funcs ( the carrier of X, the carrier of Y) -valued Function-like total quasi_total Function-yielding V34() Element of bool [:[:(Funcs ( the carrier of X, the carrier of Y)),(Funcs ( the carrier of X, the carrier of Y)):],(Funcs ( the carrier of X, the carrier of Y)):]
[:(Funcs ( the carrier of X, the carrier of Y)),(Funcs ( the carrier of X, the carrier of Y)):] is non empty set
[:[:(Funcs ( the carrier of X, the carrier of Y)),(Funcs ( the carrier of X, the carrier of Y)):],(Funcs ( the carrier of X, the carrier of Y)):] is non empty set
bool [:[:(Funcs ( the carrier of X, the carrier of Y)),(Funcs ( the carrier of X, the carrier of Y)):],(Funcs ( the carrier of X, the carrier of Y)):] is non empty set
( the carrier of X,Y) is non empty Relation-like [:COMPLEX,(Funcs ( the carrier of X, the carrier of Y)):] -defined Funcs ( the carrier of X, the carrier of Y) -valued Function-like total quasi_total Function-yielding V34() Element of bool [:[:COMPLEX,(Funcs ( the carrier of X, the carrier of Y)):],(Funcs ( the carrier of X, the carrier of Y)):]
[:COMPLEX,(Funcs ( the carrier of X, the carrier of Y)):] is non empty set
[:[:COMPLEX,(Funcs ( the carrier of X, the carrier of Y)):],(Funcs ( the carrier of X, the carrier of Y)):] is non empty set
bool [:[:COMPLEX,(Funcs ( the carrier of X, the carrier of Y)):],(Funcs ( the carrier of X, the carrier of Y)):] is non empty set
CLSStruct(# (Funcs ( the carrier of X, the carrier of Y)),(FuncZero ( the carrier of X,Y)),(FuncAdd ( the carrier of X,Y)),( the carrier of X,Y) #) is non empty strict CLSStruct
the carrier of ( the carrier of X,Y) is non empty set
bool the carrier of ( the carrier of X,Y) is non empty set
Zero_ ((X,Y),( the carrier of X,Y)) is Relation-like Function-like Element of (X,Y)
Add_ ((X,Y),( the carrier of X,Y)) is non empty Relation-like [:(X,Y),(X,Y):] -defined (X,Y) -valued Function-like total quasi_total Function-yielding V34() Element of bool [:[:(X,Y),(X,Y):],(X,Y):]
[:(X,Y),(X,Y):] is non empty set
[:[:(X,Y),(X,Y):],(X,Y):] is non empty set
bool [:[:(X,Y),(X,Y):],(X,Y):] is non empty set
Mult_ ((X,Y),( the carrier of X,Y)) is non empty Relation-like [:COMPLEX,(X,Y):] -defined (X,Y) -valued Function-like total quasi_total Function-yielding V34() Element of bool [:[:COMPLEX,(X,Y):],(X,Y):]
[:COMPLEX,(X,Y):] is non empty set
[:[:COMPLEX,(X,Y):],(X,Y):] is non empty set
bool [:[:COMPLEX,(X,Y):],(X,Y):] is non empty set
CLSStruct(# (X,Y),(Zero_ ((X,Y),( the carrier of X,Y))),(Add_ ((X,Y),( the carrier of X,Y))),(Mult_ ((X,Y),( the carrier of X,Y))) #) is non empty right_complementable Abelian add-associative right_zeroed strict vector-distributive scalar-distributive scalar-associative scalar-unital CLSStruct
the carrier of (X,Y) is non empty set
bool the carrier of (X,Y) is non empty set
Zero_ ((X,Y),(X,Y)) is Element of (X,Y)
Add_ ((X,Y),(X,Y)) is non empty Relation-like [:(X,Y),(X,Y):] -defined (X,Y) -valued Function-like total quasi_total Element of bool [:[:(X,Y),(X,Y):],(X,Y):]
[:(X,Y),(X,Y):] is non empty set
[:[:(X,Y),(X,Y):],(X,Y):] is non empty set
bool [:[:(X,Y),(X,Y):],(X,Y):] is non empty set
Mult_ ((X,Y),(X,Y)) is non empty Relation-like [:COMPLEX,(X,Y):] -defined (X,Y) -valued Function-like total quasi_total Element of bool [:[:COMPLEX,(X,Y):],(X,Y):]
[:COMPLEX,(X,Y):] is non empty set
[:[:COMPLEX,(X,Y):],(X,Y):] is non empty set
bool [:[:COMPLEX,(X,Y):],(X,Y):] is non empty set
(X,Y) is non empty Relation-like (X,Y) -defined REAL -valued Function-like total quasi_total V132() V133() V134() Element of bool [:(X,Y),REAL:]
[:(X,Y),REAL:] is non empty V132() V133() V134() set
bool [:(X,Y),REAL:] is non empty set
CNORMSTR(# (X,Y),(Zero_ ((X,Y),(X,Y))),(Add_ ((X,Y),(X,Y))),(Mult_ ((X,Y),(X,Y))),(X,Y) #) is non empty strict CNORMSTR
the carrier of (X,Y) is non empty set
tseq is Relation-like Function-like Element of the carrier of (X,Y)
vseq is Relation-like Function-like Element of the carrier of (X,Y)
f is Relation-like Function-like Element of the carrier of (X,Y)
vseq + f is Relation-like Function-like Element of the carrier of (X,Y)
the addF of (X,Y) is non empty Relation-like [: the carrier of (X,Y), the carrier of (X,Y):] -defined the carrier of (X,Y) -valued Function-like total quasi_total Element of bool [:[: the carrier of (X,Y), the carrier of (X,Y):], the carrier of (X,Y):]
[: the carrier of (X,Y), the carrier of (X,Y):] is non empty set
[:[: the carrier of (X,Y), the carrier of (X,Y):], the carrier of (X,Y):] is non empty set
bool [:[: the carrier of (X,Y), the carrier of (X,Y):], the carrier of (X,Y):] is non empty set
the addF of (X,Y) . (vseq,f) is Relation-like Function-like Element of the carrier of (X,Y)
[vseq,f] is set
the addF of (X,Y) . [vseq,f] is set
(X,Y) is non empty right_complementable Abelian add-associative right_zeroed strict vector-distributive scalar-distributive scalar-associative scalar-unital CLSStruct
CLSStruct(# (X,Y),(Zero_ ((X,Y),(X,Y))),(Add_ ((X,Y),(X,Y))),(Mult_ ((X,Y),(X,Y))) #) is non empty right_complementable Abelian add-associative right_zeroed strict vector-distributive scalar-distributive scalar-associative scalar-unital CLSStruct
the carrier of (X,Y) is non empty set
tv is Relation-like Function-like Element of the carrier of (X,Y)
tseq is Relation-like Function-like Element of the carrier of (X,Y)
tseq is Relation-like Function-like Element of the carrier of (X,Y)
tseq + tseq is Relation-like Function-like Element of the carrier of (X,Y)
the addF of (X,Y) is non empty Relation-like [: the carrier of (X,Y), the carrier of (X,Y):] -defined the carrier of (X,Y) -valued Function-like total quasi_total Element of bool [:[: the carrier of (X,Y), the carrier of (X,Y):], the carrier of (X,Y):]
[: the carrier of (X,Y), the carrier of (X,Y):] is non empty set
[:[: the carrier of (X,Y), the carrier of (X,Y):], the carrier of (X,Y):] is non empty set
bool [:[: the carrier of (X,Y), the carrier of (X,Y):], the carrier of (X,Y):] is non empty set
the addF of (X,Y) . (tseq,tseq) is Relation-like Function-like Element of the carrier of (X,Y)
[tseq,tseq] is set
the addF of (X,Y) . [tseq,tseq] is set
e is Element of the carrier of X
(X,Y,tseq,e) is Element of the carrier of Y
(X,Y,vseq,e) is Element of the carrier of Y
(X,Y,f,e) is Element of the carrier of Y
(X,Y,vseq,e) + (X,Y,f,e) is Element of the carrier of Y
the addF of Y is non empty Relation-like [: the carrier of Y, the carrier of Y:] -defined the carrier of Y -valued Function-like total quasi_total Element of bool [:[: the carrier of Y, the carrier of Y:], the carrier of Y:]
[: the carrier of Y, the carrier of Y:] is non empty set
[:[: the carrier of Y, the carrier of Y:], the carrier of Y:] is non empty set
bool [:[: the carrier of Y, the carrier of Y:], the carrier of Y:] is non empty set
the addF of Y . ((X,Y,vseq,e),(X,Y,f,e)) is Element of the carrier of Y
[(X,Y,vseq,e),(X,Y,f,e)] is set
the addF of Y . [(X,Y,vseq,e),(X,Y,f,e)] is set
X is non empty right_complementable Abelian add-associative right_zeroed discerning reflexive vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like CNORMSTR
Y is non empty right_complementable Abelian add-associative right_zeroed discerning reflexive vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like CNORMSTR
(X,Y) is non empty CNORMSTR
(X,Y) is non empty Element of bool the carrier of (X,Y)
(X,Y) is non empty right_complementable Abelian add-associative right_zeroed constituted-Functions strict vector-distributive scalar-distributive scalar-associative scalar-unital CLSStruct
(X,Y) is non empty functional Element of bool the carrier of ( the carrier of X,Y)
the carrier of X is non empty set
( the carrier of X,Y) is non empty right_complementable Abelian add-associative right_zeroed constituted-Functions strict vector-distributive scalar-distributive scalar-associative scalar-unital CLSStruct
the carrier of Y is non empty set
Funcs ( the carrier of X, the carrier of Y) is non empty functional FUNCTION_DOMAIN of the carrier of X, the carrier of Y
FuncZero ( the carrier of X,Y) is Relation-like the carrier of X -defined the carrier of Y -valued Function-like total quasi_total Element of Funcs ( the carrier of X, the carrier of Y)
FuncAdd ( the carrier of X,Y) is non empty Relation-like [:(Funcs ( the carrier of X, the carrier of Y)),(Funcs ( the carrier of X, the carrier of Y)):] -defined Funcs ( the carrier of X, the carrier of Y) -valued Function-like total quasi_total Function-yielding V34() Element of bool [:[:(Funcs ( the carrier of X, the carrier of Y)),(Funcs ( the carrier of X, the carrier of Y)):],(Funcs ( the carrier of X, the carrier of Y)):]
[:(Funcs ( the carrier of X, the carrier of Y)),(Funcs ( the carrier of X, the carrier of Y)):] is non empty set
[:[:(Funcs ( the carrier of X, the carrier of Y)),(Funcs ( the carrier of X, the carrier of Y)):],(Funcs ( the carrier of X, the carrier of Y)):] is non empty set
bool [:[:(Funcs ( the carrier of X, the carrier of Y)),(Funcs ( the carrier of X, the carrier of Y)):],(Funcs ( the carrier of X, the carrier of Y)):] is non empty set
( the carrier of X,Y) is non empty Relation-like [:COMPLEX,(Funcs ( the carrier of X, the carrier of Y)):] -defined Funcs ( the carrier of X, the carrier of Y) -valued Function-like total quasi_total Function-yielding V34() Element of bool [:[:COMPLEX,(Funcs ( the carrier of X, the carrier of Y)):],(Funcs ( the carrier of X, the carrier of Y)):]
[:COMPLEX,(Funcs ( the carrier of X, the carrier of Y)):] is non empty set
[:[:COMPLEX,(Funcs ( the carrier of X, the carrier of Y)):],(Funcs ( the carrier of X, the carrier of Y)):] is non empty set
bool [:[:COMPLEX,(Funcs ( the carrier of X, the carrier of Y)):],(Funcs ( the carrier of X, the carrier of Y)):] is non empty set
CLSStruct(# (Funcs ( the carrier of X, the carrier of Y)),(FuncZero ( the carrier of X,Y)),(FuncAdd ( the carrier of X,Y)),( the carrier of X,Y) #) is non empty strict CLSStruct
the carrier of ( the carrier of X,Y) is non empty set
bool the carrier of ( the carrier of X,Y) is non empty set
Zero_ ((X,Y),( the carrier of X,Y)) is Relation-like Function-like Element of (X,Y)
Add_ ((X,Y),( the carrier of X,Y)) is non empty Relation-like [:(X,Y),(X,Y):] -defined (X,Y) -valued Function-like total quasi_total Function-yielding V34() Element of bool [:[:(X,Y),(X,Y):],(X,Y):]
[:(X,Y),(X,Y):] is non empty set
[:[:(X,Y),(X,Y):],(X,Y):] is non empty set
bool [:[:(X,Y),(X,Y):],(X,Y):] is non empty set
Mult_ ((X,Y),( the carrier of X,Y)) is non empty Relation-like [:COMPLEX,(X,Y):] -defined (X,Y) -valued Function-like total quasi_total Function-yielding V34() Element of bool [:[:COMPLEX,(X,Y):],(X,Y):]
[:COMPLEX,(X,Y):] is non empty set
[:[:COMPLEX,(X,Y):],(X,Y):] is non empty set
bool [:[:COMPLEX,(X,Y):],(X,Y):] is non empty set
CLSStruct(# (X,Y),(Zero_ ((X,Y),( the carrier of X,Y))),(Add_ ((X,Y),( the carrier of X,Y))),(Mult_ ((X,Y),( the carrier of X,Y))) #) is non empty right_complementable Abelian add-associative right_zeroed strict vector-distributive scalar-distributive scalar-associative scalar-unital CLSStruct
the carrier of (X,Y) is non empty set
bool the carrier of (X,Y) is non empty set
Zero_ ((X,Y),(X,Y)) is Element of (X,Y)
Add_ ((X,Y),(X,Y)) is non empty Relation-like [:(X,Y),(X,Y):] -defined (X,Y) -valued Function-like total quasi_total Element of bool [:[:(X,Y),(X,Y):],(X,Y):]
[:(X,Y),(X,Y):] is non empty set
[:[:(X,Y),(X,Y):],(X,Y):] is non empty set
bool [:[:(X,Y),(X,Y):],(X,Y):] is non empty set
Mult_ ((X,Y),(X,Y)) is non empty Relation-like [:COMPLEX,(X,Y):] -defined (X,Y) -valued Function-like total quasi_total Element of bool [:[:COMPLEX,(X,Y):],(X,Y):]
[:COMPLEX,(X,Y):] is non empty set
[:[:COMPLEX,(X,Y):],(X,Y):] is non empty set
bool [:[:COMPLEX,(X,Y):],(X,Y):] is non empty set
(X,Y) is non empty Relation-like (X,Y) -defined REAL -valued Function-like total quasi_total V132() V133() V134() Element of bool [:(X,Y),REAL:]
[:(X,Y),REAL:] is non empty V132() V133() V134() set
bool [:(X,Y),REAL:] is non empty set
CNORMSTR(# (X,Y),(Zero_ ((X,Y),(X,Y))),(Add_ ((X,Y),(X,Y))),(Mult_ ((X,Y),(X,Y))),(X,Y) #) is non empty strict CNORMSTR
the carrier of (X,Y) is non empty set
f is Relation-like Function-like Element of the carrier of (X,Y)
vseq is Relation-like Function-like Element of the carrier of (X,Y)
tseq is complex set
tseq * vseq is Relation-like Function-like Element of the carrier of (X,Y)
(X,Y) is non empty right_complementable Abelian add-associative right_zeroed strict vector-distributive scalar-distributive scalar-associative scalar-unital CLSStruct
CLSStruct(# (X,Y),(Zero_ ((X,Y),(X,Y))),(Add_ ((X,Y),(X,Y))),(Mult_ ((X,Y),(X,Y))) #) is non empty right_complementable Abelian add-associative right_zeroed strict vector-distributive scalar-distributive scalar-associative scalar-unital CLSStruct
the carrier of (X,Y) is non empty set
tseq is Relation-like Function-like Element of the carrier of (X,Y)
tseq is Relation-like Function-like Element of the carrier of (X,Y)
tseq * tseq is Relation-like Function-like Element of the carrier of (X,Y)
[tseq,tseq] is set
(Mult_ ((X,Y),(X,Y))) . [tseq,tseq] is set
[tseq,vseq] is set
(Mult_ ((X,Y),(X,Y))) . [tseq,vseq] is set
tv is Element of the carrier of X
(X,Y,f,tv) is Element of the carrier of Y
(X,Y,vseq,tv) is Element of the carrier of Y
tseq * (X,Y,vseq,tv) is Element of the carrier of Y
X is non empty right_complementable Abelian add-associative right_zeroed discerning reflexive vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like CNORMSTR
Y is non empty right_complementable Abelian add-associative right_zeroed discerning reflexive vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like CNORMSTR
(X,Y) is non empty CNORMSTR
(X,Y) is non empty Element of bool the carrier of (X,Y)
(X,Y) is non empty right_complementable Abelian add-associative right_zeroed constituted-Functions strict vector-distributive scalar-distributive scalar-associative scalar-unital CLSStruct
(X,Y) is non empty functional Element of bool the carrier of ( the carrier of X,Y)
the carrier of X is non empty set
( the carrier of X,Y) is non empty right_complementable Abelian add-associative right_zeroed constituted-Functions strict vector-distributive scalar-distributive scalar-associative scalar-unital CLSStruct
the carrier of Y is non empty set
Funcs ( the carrier of X, the carrier of Y) is non empty functional FUNCTION_DOMAIN of the carrier of X, the carrier of Y
FuncZero ( the carrier of X,Y) is Relation-like the carrier of X -defined the carrier of Y -valued Function-like total quasi_total Element of Funcs ( the carrier of X, the carrier of Y)
FuncAdd ( the carrier of X,Y) is non empty Relation-like [:(Funcs ( the carrier of X, the carrier of Y)),(Funcs ( the carrier of X, the carrier of Y)):] -defined Funcs ( the carrier of X, the carrier of Y) -valued Function-like total quasi_total Function-yielding V34() Element of bool [:[:(Funcs ( the carrier of X, the carrier of Y)),(Funcs ( the carrier of X, the carrier of Y)):],(Funcs ( the carrier of X, the carrier of Y)):]
[:(Funcs ( the carrier of X, the carrier of Y)),(Funcs ( the carrier of X, the carrier of Y)):] is non empty set
[:[:(Funcs ( the carrier of X, the carrier of Y)),(Funcs ( the carrier of X, the carrier of Y)):],(Funcs ( the carrier of X, the carrier of Y)):] is non empty set
bool [:[:(Funcs ( the carrier of X, the carrier of Y)),(Funcs ( the carrier of X, the carrier of Y)):],(Funcs ( the carrier of X, the carrier of Y)):] is non empty set
( the carrier of X,Y) is non empty Relation-like [:COMPLEX,(Funcs ( the carrier of X, the carrier of Y)):] -defined Funcs ( the carrier of X, the carrier of Y) -valued Function-like total quasi_total Function-yielding V34() Element of bool [:[:COMPLEX,(Funcs ( the carrier of X, the carrier of Y)):],(Funcs ( the carrier of X, the carrier of Y)):]
[:COMPLEX,(Funcs ( the carrier of X, the carrier of Y)):] is non empty set
[:[:COMPLEX,(Funcs ( the carrier of X, the carrier of Y)):],(Funcs ( the carrier of X, the carrier of Y)):] is non empty set
bool [:[:COMPLEX,(Funcs ( the carrier of X, the carrier of Y)):],(Funcs ( the carrier of X, the carrier of Y)):] is non empty set
CLSStruct(# (Funcs ( the carrier of X, the carrier of Y)),(FuncZero ( the carrier of X,Y)),(FuncAdd ( the carrier of X,Y)),( the carrier of X,Y) #) is non empty strict CLSStruct
the carrier of ( the carrier of X,Y) is non empty set
bool the carrier of ( the carrier of X,Y) is non empty set
Zero_ ((X,Y),( the carrier of X,Y)) is Relation-like Function-like Element of (X,Y)
Add_ ((X,Y),( the carrier of X,Y)) is non empty Relation-like [:(X,Y),(X,Y):] -defined (X,Y) -valued Function-like total quasi_total Function-yielding V34() Element of bool [:[:(X,Y),(X,Y):],(X,Y):]
[:(X,Y),(X,Y):] is non empty set
[:[:(X,Y),(X,Y):],(X,Y):] is non empty set
bool [:[:(X,Y),(X,Y):],(X,Y):] is non empty set
Mult_ ((X,Y),( the carrier of X,Y)) is non empty Relation-like [:COMPLEX,(X,Y):] -defined (X,Y) -valued Function-like total quasi_total Function-yielding V34() Element of bool [:[:COMPLEX,(X,Y):],(X,Y):]
[:COMPLEX,(X,Y):] is non empty set
[:[:COMPLEX,(X,Y):],(X,Y):] is non empty set
bool [:[:COMPLEX,(X,Y):],(X,Y):] is non empty set
CLSStruct(# (X,Y),(Zero_ ((X,Y),( the carrier of X,Y))),(Add_ ((X,Y),( the carrier of X,Y))),(Mult_ ((X,Y),( the carrier of X,Y))) #) is non empty right_complementable Abelian add-associative right_zeroed strict vector-distributive scalar-distributive scalar-associative scalar-unital CLSStruct
the carrier of (X,Y) is non empty set
bool the carrier of (X,Y) is non empty set
Zero_ ((X,Y),(X,Y)) is Element of (X,Y)
Add_ ((X,Y),(X,Y)) is non empty Relation-like [:(X,Y),(X,Y):] -defined (X,Y) -valued Function-like total quasi_total Element of bool [:[:(X,Y),(X,Y):],(X,Y):]
[:(X,Y),(X,Y):] is non empty set
[:[:(X,Y),(X,Y):],(X,Y):] is non empty set
bool [:[:(X,Y),(X,Y):],(X,Y):] is non empty set
Mult_ ((X,Y),(X,Y)) is non empty Relation-like [:COMPLEX,(X,Y):] -defined (X,Y) -valued Function-like total quasi_total Element of bool [:[:COMPLEX,(X,Y):],(X,Y):]
[:COMPLEX,(X,Y):] is non empty set
[:[:COMPLEX,(X,Y):],(X,Y):] is non empty set
bool [:[:COMPLEX,(X,Y):],(X,Y):] is non empty set
(X,Y) is non empty Relation-like (X,Y) -defined REAL -valued Function-like total quasi_total V132() V133() V134() Element of bool [:(X,Y),REAL:]
[:(X,Y),REAL:] is non empty V132() V133() V134() set
bool [:(X,Y),REAL:] is non empty set
CNORMSTR(# (X,Y),(Zero_ ((X,Y),(X,Y))),(Add_ ((X,Y),(X,Y))),(Mult_ ((X,Y),(X,Y))),(X,Y) #) is non empty strict CNORMSTR
the carrier of (X,Y) is non empty set
0. (X,Y) is Relation-like Function-like zero Element of the carrier of (X,Y)
the ZeroF of (X,Y) is Relation-like Function-like Element of the carrier of (X,Y)
vseq is Relation-like Function-like Element of the carrier of (X,Y)
||.vseq.|| is complex real ext-real Element of REAL
the U9 of (X,Y) is non empty Relation-like the carrier of (X,Y) -defined REAL -valued Function-like total quasi_total V132() V133() V134() Element of bool [: the carrier of (X,Y),REAL:]
[: the carrier of (X,Y),REAL:] is non empty V132() V133() V134() set
bool [: the carrier of (X,Y),REAL:] is non empty set
the U9 of (X,Y) . vseq is complex real ext-real Element of REAL
f is Relation-like Function-like Element of the carrier of (X,Y)
vseq + f is Relation-like Function-like Element of the carrier of (X,Y)
the addF of (X,Y) is non empty Relation-like [: the carrier of (X,Y), the carrier of (X,Y):] -defined the carrier of (X,Y) -valued Function-like total quasi_total Element of bool [:[: the carrier of (X,Y), the carrier of (X,Y):], the carrier of (X,Y):]
[: the carrier of (X,Y), the carrier of (X,Y):] is non empty set
[:[: the carrier of (X,Y), the carrier of (X,Y):], the carrier of (X,Y):] is non empty set
bool [:[: the carrier of (X,Y), the carrier of (X,Y):], the carrier of (X,Y):] is non empty set
the addF of (X,Y) . (vseq,f) is Relation-like Function-like Element of the carrier of (X,Y)
[vseq,f] is set
the addF of (X,Y) . [vseq,f] is set
||.(vseq + f).|| is complex real ext-real Element of REAL
the U9 of (X,Y) . (vseq + f) is complex real ext-real Element of REAL
||.f.|| is complex real ext-real Element of REAL
the U9 of (X,Y) . f is complex real ext-real Element of REAL
||.vseq.|| + ||.f.|| is complex real ext-real Element of REAL
tseq is complex set
tseq * vseq is Relation-like Function-like Element of the carrier of (X,Y)
||.(tseq * vseq).|| is complex real ext-real Element of REAL
the U9 of (X,Y) . (tseq * vseq) is complex real ext-real Element of REAL
|.tseq.| is complex real ext-real Element of REAL
|.tseq.| * ||.vseq.|| is complex real ext-real Element of REAL
[: the carrier of X, the carrier of Y:] is non empty set
bool [: the carrier of X, the carrier of Y:] is non empty set
0. Y is zero Element of the carrier of Y
the ZeroF of Y is Element of the carrier of Y
the carrier of X --> (0. Y) is non empty Relation-like the carrier of X -defined the carrier of Y -valued Function-like constant total quasi_total Element of bool [: the carrier of X, the carrier of Y:]
tseq is non empty Relation-like the carrier of X -defined the carrier of Y -valued Function-like total quasi_total additive (X,Y) (X,Y) Element of bool [: the carrier of X, the carrier of Y:]
(X,Y,tseq) is non empty V142() V143() V144() Element of bool REAL
{ ||.(tseq . b₁).|| where b₁ is Element of the carrier of X : ||.b₁.|| <= 1 } is set
e is set
upper_bound (X,Y,tseq) is complex real ext-real Element of REAL
tv is non empty Relation-like the carrier of X -defined the carrier of Y -valued Function-like total quasi_total Element of bool [: the carrier of X, the carrier of Y:]
n is complex real ext-real Element of REAL
n is Element of the carrier of X
tseq . n is Element of the carrier of Y
||.(tseq . n).|| is complex real ext-real Element of REAL
the U9 of Y is non empty Relation-like the carrier of Y -defined REAL -valued Function-like total quasi_total V132() V133() V134() Element of bool [: the carrier of Y,REAL:]
[: the carrier of Y,REAL:] is non empty V132() V133() V134() set
bool [: the carrier of Y,REAL:] is non empty set
the U9 of Y . (tseq . n) is complex real ext-real Element of REAL
||.n.|| is complex real ext-real Element of REAL
the U9 of X is non empty Relation-like the carrier of X -defined REAL -valued Function-like total quasi_total V132() V133() V134() Element of bool [: the carrier of X,REAL:]
[: the carrier of X,REAL:] is non empty V132() V133() V134() set
bool [: the carrier of X,REAL:] is non empty set
the U9 of X . n is complex real ext-real Element of REAL
||.(0. Y).|| is complex real ext-real Element of REAL
the U9 of Y . (0. Y) is complex real ext-real Element of REAL
m is complex real ext-real Element of REAL
n is complex real ext-real set
(X,Y) . vseq is complex real ext-real Element of REAL
[: the carrier of X, the carrier of Y:] is non empty set
bool [: the carrier of X, the carrier of Y:] is non empty set
tv is non empty Relation-like the carrier of X -defined the carrier of Y -valued Function-like total quasi_total additive (X,Y) (X,Y) Element of bool [: the carrier of X, the carrier of Y:]
(X,Y,tv) is non empty V142() V143() V144() Element of bool REAL
{ ||.(tv . b₁).|| where b₁ is Element of the carrier of X : ||.b₁.|| <= 1 } is set
upper_bound (X,Y,tv) is complex real ext-real Element of REAL
e is Element of the carrier of X
||.e.|| is complex real ext-real Element of REAL
the U9 of X is non empty Relation-like the carrier of X -defined REAL -valued Function-like total quasi_total V132() V133() V134() Element of bool [: the carrier of X,REAL:]
[: the carrier of X,REAL:] is non empty V132() V133() V134() set
bool [: the carrier of X,REAL:] is non empty set
the U9 of X . e is complex real ext-real Element of REAL
||.f.|| * ||.e.|| is complex real ext-real Element of REAL
||.f.|| * 1 is complex real ext-real Element of REAL
||.vseq.|| * ||.e.|| is complex real ext-real Element of REAL
||.vseq.|| * 1 is complex real ext-real Element of REAL
(||.vseq.|| * ||.e.||) + (||.f.|| * ||.e.||) is complex real ext-real Element of REAL
(||.vseq.|| * 1) + (||.f.|| * 1) is complex real ext-real Element of REAL
tseq is non empty Relation-like the carrier of X -defined the carrier of Y -valued Function-like total quasi_total additive (X,Y) (X,Y) Element of bool [: the carrier of X, the carrier of Y:]
tseq . e is Element of the carrier of Y
tseq is non empty Relation-like the carrier of X -defined the carrier of Y -valued Function-like total quasi_total additive (X,Y) (X,Y) Element of bool [: the carrier of X, the carrier of Y:]
tseq . e is Element of the carrier of Y
(tseq . e) + (tseq . e) is Element of the carrier of Y
the addF of Y is non empty Relation-like [: the carrier of Y, the carrier of Y:] -defined the carrier of Y -valued Function-like total quasi_total Element of bool [:[: the carrier of Y, the carrier of Y:], the carrier of Y:]
[: the carrier of Y, the carrier of Y:] is non empty set
[:[: the carrier of Y, the carrier of Y:], the carrier of Y:] is non empty set
bool [:[: the carrier of Y, the carrier of Y:], the carrier of Y:] is non empty set
the addF of Y . ((tseq . e),(tseq . e)) is Element of the carrier of Y
[(tseq . e),(tseq . e)] is set
the addF of Y . [(tseq . e),(tseq . e)] is set
||.((tseq . e) + (tseq . e)).|| is complex real ext-real Element of REAL
the U9 of Y is non empty Relation-like the carrier of Y -defined REAL -valued Function-like total quasi_total V132() V133() V134() Element of bool [: the carrier of Y,REAL:]
[: the carrier of Y,REAL:] is non empty V132() V133() V134() set
bool [: the carrier of Y,REAL:] is non empty set
the U9 of Y . ((tseq . e) + (tseq . e)) is complex real ext-real Element of REAL
||.(tseq . e).|| is complex real ext-real Element of REAL
the U9 of Y . (tseq . e) is complex real ext-real Element of REAL
||.(tseq . e).|| is complex real ext-real Element of REAL
the U9 of Y . (tseq . e) is complex real ext-real Element of REAL
||.(tseq . e).|| + ||.(tseq . e).|| is complex real ext-real Element of REAL
tv . e is Element of the carrier of Y
||.(tv . e).|| is complex real ext-real Element of REAL
the U9 of Y . (tv . e) is complex real ext-real Element of REAL
e is complex real ext-real Element of REAL
m is Element of the carrier of X
tv . m is Element of the carrier of Y
||.(tv . m).|| is complex real ext-real Element of REAL
the U9 of Y . (tv . m) is complex real ext-real Element of REAL
||.m.|| is complex real ext-real Element of REAL
the U9 of X . m is complex real ext-real Element of REAL
(X,Y) . (vseq + f) is complex real ext-real Element of REAL
e is complex real ext-real set
[: the carrier of X, the carrier of Y:] is non empty set
bool [: the carrier of X, the carrier of Y:] is non empty set
tseq is non empty Relation-like the carrier of X -defined the carrier of Y -valued Function-like total quasi_total additive (X,Y) (X,Y) Element of bool [: the carrier of X, the carrier of Y:]
(X,Y,tseq) is non empty V142() V143() V144() Element of bool REAL
{ ||.(tseq . b₁).|| where b₁ is Element of the carrier of X : ||.b₁.|| <= 1 } is set
upper_bound (X,Y,tseq) is complex real ext-real Element of REAL
tv is Element of the carrier of X
||.tv.|| is complex real ext-real Element of REAL
the U9 of X is non empty Relation-like the carrier of X -defined REAL -valued Function-like total quasi_total V132() V133() V134() Element of bool [: the carrier of X,REAL:]
[: the carrier of X,REAL:] is non empty V132() V133() V134() set
bool [: the carrier of X,REAL:] is non empty set
the U9 of X . tv is complex real ext-real Element of REAL
||.vseq.|| * ||.tv.|| is complex real ext-real Element of REAL
||.vseq.|| * 1 is complex real ext-real Element of REAL
tseq is non empty Relation-like the carrier of X -defined the carrier of Y -valued Function-like total quasi_total additive (X,Y) (X,Y) Element of bool [: the carrier of X, the carrier of Y:]
tseq . tv is Element of the carrier of Y
||.(tseq . tv).|| is complex real ext-real Element of REAL
the U9 of Y is non empty Relation-like the carrier of Y -defined REAL -valued Function-like total quasi_total V132() V133() V134() Element of bool [: the carrier of Y,REAL:]
[: the carrier of Y,REAL:] is non empty V132() V133() V134() set
bool [: the carrier of Y,REAL:] is non empty set
the U9 of Y . (tseq . tv) is complex real ext-real Element of REAL
tseq * (tseq . tv) is Element of the carrier of Y
||.(tseq * (tseq . tv)).|| is complex real ext-real Element of REAL
the U9 of Y . (tseq * (tseq . tv)) is complex real ext-real Element of REAL
|.tseq.| * ||.(tseq . tv).|| is complex real ext-real Element of REAL
tseq . tv is Element of the carrier of Y
||.(tseq . tv).|| is complex real ext-real Element of REAL
the U9 of Y . (tseq . tv) is complex real ext-real Element of REAL
tv is complex real ext-real Element of REAL
e is Element of the carrier of X
tseq . e is Element of the carrier of Y
||.(tseq . e).|| is complex real ext-real Element of REAL
the U9 of Y . (tseq . e) is complex real ext-real Element of REAL
||.e.|| is complex real ext-real Element of REAL
the U9 of X . e is complex real ext-real Element of REAL
0c is empty Function-like functional epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural complex real ext-real non positive non negative V142() V143() V144() V145() V146() V147() V148() Element of COMPLEX
tv is Element of the carrier of X
||.tv.|| is complex real ext-real Element of REAL
the U9 of X . tv is complex real ext-real Element of REAL
||.(tseq * vseq).|| * ||.tv.|| is complex real ext-real Element of REAL
||.(tseq * vseq).|| * 1 is complex real ext-real Element of REAL
tseq . tv is Element of the carrier of Y
||.(tseq . tv).|| is complex real ext-real Element of REAL
the U9 of Y . (tseq . tv) is complex real ext-real Element of REAL
tseq . tv is Element of the carrier of Y
tseq * (tseq . tv) is Element of the carrier of Y
tseq " is complex set
(tseq ") * (tseq . tv) is Element of the carrier of Y
(tseq ") * tseq is complex set
((tseq ") * tseq) * (tseq . tv) is Element of the carrier of Y
1r * (tseq . tv) is Element of the carrier of Y
|.(tseq ").| is complex real ext-real Element of REAL
|.tseq.| " is complex real ext-real Element of REAL
||.((tseq ") * (tseq . tv)).|| is complex real ext-real Element of REAL
the U9 of Y . ((tseq ") * (tseq . tv)) is complex real ext-real Element of REAL
|.(tseq ").| * ||.(tseq . tv).|| is complex real ext-real Element of REAL
||.(tseq . tv).|| is complex real ext-real Element of REAL
the U9 of Y . (tseq . tv) is complex real ext-real Element of REAL
(|.tseq.| ") * ||.(tseq * vseq).|| is complex real ext-real Element of REAL
tv is complex real ext-real Element of REAL
(X,Y,tseq) is non empty V142() V143() V144() Element of bool REAL
{ ||.(tseq . b₁).|| where b₁ is Element of the carrier of X : ||.b₁.|| <= 1 } is set
e is Element of the carrier of X
tseq . e is Element of the carrier of Y
||.(tseq . e).|| is complex real ext-real Element of REAL
the U9 of Y . (tseq . e) is complex real ext-real Element of REAL
||.e.|| is complex real ext-real Element of REAL
the U9 of X . e is complex real ext-real Element of REAL
upper_bound (X,Y,tseq) is complex real ext-real Element of REAL
(X,Y) . vseq is complex real ext-real Element of REAL
tv is complex real ext-real set
|.tseq.| * ((|.tseq.| ") * ||.(tseq * vseq).||) is complex real ext-real Element of REAL
|.tseq.| * (|.tseq.| ") is complex real ext-real Element of REAL
(|.tseq.| * (|.tseq.| ")) * ||.(tseq * vseq).|| is complex real ext-real Element of REAL
1 * ||.(tseq * vseq).|| is complex real ext-real Element of REAL
0c is empty Function-like functional epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural complex real ext-real non positive non negative V142() V143() V144() V145() V146() V147() V148() Element of COMPLEX
(X,Y) is non empty right_complementable Abelian add-associative right_zeroed strict vector-distributive scalar-distributive scalar-associative scalar-unital CLSStruct
CLSStruct(# (X,Y),(Zero_ ((X,Y),(X,Y))),(Add_ ((X,Y),(X,Y))),(Mult_ ((X,Y),(X,Y))) #) is non empty right_complementable Abelian add-associative right_zeroed strict vector-distributive scalar-distributive scalar-associative scalar-unital CLSStruct
the carrier of (X,Y) is non empty set
[tseq,vseq] is set
(Mult_ ((X,Y),(X,Y))) . [tseq,vseq] is set
tv is Relation-like Function-like Element of the carrier of (X,Y)
tseq * tv is Relation-like Function-like Element of the carrier of (X,Y)
0. (X,Y) is Relation-like Function-like zero Element of the carrier of (X,Y)
the ZeroF of (X,Y) is Relation-like Function-like Element of the carrier of (X,Y)
0c is empty Function-like functional epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural complex real ext-real non positive non negative V142() V143() V144() V145() V146() V147() V148() Element of COMPLEX
0c is empty Function-like functional epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural complex real ext-real non positive non negative V142() V143() V144() V145() V146() V147() V148() Element of COMPLEX
(X,Y) . (tseq * vseq) is complex real ext-real Element of REAL
tv is complex real ext-real set
[: the carrier of X, the carrier of Y:] is non empty set
bool [: the carrier of X, the carrier of Y:] is non empty set
0. Y is zero Element of the carrier of Y
the ZeroF of Y is Element of the carrier of Y
the carrier of X --> (0. Y) is non empty Relation-like the carrier of X -defined the carrier of Y -valued Function-like constant total quasi_total Element of bool [: the carrier of X, the carrier of Y:]
tseq is non empty Relation-like the carrier of X -defined the carrier of Y -valued Function-like total quasi_total additive (X,Y) (X,Y) Element of bool [: the carrier of X, the carrier of Y:]
e is Element of the carrier of X
tseq . e is Element of the carrier of Y
||.(tseq . e).|| is complex real ext-real Element of REAL
the U9 of Y is non empty Relation-like the carrier of Y -defined REAL -valued Function-like total quasi_total V132() V133() V134() Element of bool [: the carrier of Y,REAL:]
[: the carrier of Y,REAL:] is non empty V132() V133() V134() set
bool [: the carrier of Y,REAL:] is non empty set
the U9 of Y . (tseq . e) is complex real ext-real Element of REAL
||.e.|| is complex real ext-real Element of REAL
the U9 of X is non empty Relation-like the carrier of X -defined REAL -valued Function-like total quasi_total V132() V133() V134() Element of bool [: the carrier of X,REAL:]
[: the carrier of X,REAL:] is non empty V132() V133() V134() set
bool [: the carrier of X,REAL:] is non empty set
the U9 of X . e is complex real ext-real Element of REAL
||.vseq.|| * ||.e.|| is complex real ext-real Element of REAL
tv is non empty Relation-like the carrier of X -defined the carrier of Y -valued Function-like total quasi_total Element of bool [: the carrier of X, the carrier of Y:]
tv . e is Element of the carrier of Y
X is non empty right_complementable Abelian add-associative right_zeroed discerning reflexive vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like CNORMSTR
Y is non empty right_complementable Abelian add-associative right_zeroed discerning reflexive vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like CNORMSTR
(X,Y) is non empty CNORMSTR
(X,Y) is non empty Element of bool the carrier of (X,Y)
(X,Y) is non empty right_complementable Abelian add-associative right_zeroed constituted-Functions strict vector-distributive scalar-distributive scalar-associative scalar-unital CLSStruct
(X,Y) is non empty functional Element of bool the carrier of ( the carrier of X,Y)
the carrier of X is non empty set
( the carrier of X,Y) is non empty right_complementable Abelian add-associative right_zeroed constituted-Functions strict vector-distributive scalar-distributive scalar-associative scalar-unital CLSStruct
the carrier of Y is non empty set
Funcs ( the carrier of X, the carrier of Y) is non empty functional FUNCTION_DOMAIN of the carrier of X, the carrier of Y
FuncZero ( the carrier of X,Y) is Relation-like the carrier of X -defined the carrier of Y -valued Function-like total quasi_total Element of Funcs ( the carrier of X, the carrier of Y)
FuncAdd ( the carrier of X,Y) is non empty Relation-like [:(Funcs ( the carrier of X, the carrier of Y)),(Funcs ( the carrier of X, the carrier of Y)):] -defined Funcs ( the carrier of X, the carrier of Y) -valued Function-like total quasi_total Function-yielding V34() Element of bool [:[:(Funcs ( the carrier of X, the carrier of Y)),(Funcs ( the carrier of X, the carrier of Y)):],(Funcs ( the carrier of X, the carrier of Y)):]
[:(Funcs ( the carrier of X, the carrier of Y)),(Funcs ( the carrier of X, the carrier of Y)):] is non empty set
[:[:(Funcs ( the carrier of X, the carrier of Y)),(Funcs ( the carrier of X, the carrier of Y)):],(Funcs ( the carrier of X, the carrier of Y)):] is non empty set
bool [:[:(Funcs ( the carrier of X, the carrier of Y)),(Funcs ( the carrier of X, the carrier of Y)):],(Funcs ( the carrier of X, the carrier of Y)):] is non empty set
( the carrier of X,Y) is non empty Relation-like [:COMPLEX,(Funcs ( the carrier of X, the carrier of Y)):] -defined Funcs ( the carrier of X, the carrier of Y) -valued Function-like total quasi_total Function-yielding V34() Element of bool [:[:COMPLEX,(Funcs ( the carrier of X, the carrier of Y)):],(Funcs ( the carrier of X, the carrier of Y)):]
[:COMPLEX,(Funcs ( the carrier of X, the carrier of Y)):] is non empty set
[:[:COMPLEX,(Funcs ( the carrier of X, the carrier of Y)):],(Funcs ( the carrier of X, the carrier of Y)):] is non empty set
bool [:[:COMPLEX,(Funcs ( the carrier of X, the carrier of Y)):],(Funcs ( the carrier of X, the carrier of Y)):] is non empty set
CLSStruct(# (Funcs ( the carrier of X, the carrier of Y)),(FuncZero ( the carrier of X,Y)),(FuncAdd ( the carrier of X,Y)),( the carrier of X,Y) #) is non empty strict CLSStruct
the carrier of ( the carrier of X,Y) is non empty set
bool the carrier of ( the carrier of X,Y) is non empty set
Zero_ ((X,Y),( the carrier of X,Y)) is Relation-like Function-like Element of (X,Y)
Add_ ((X,Y),( the carrier of X,Y)) is non empty Relation-like [:(X,Y),(X,Y):] -defined (X,Y) -valued Function-like total quasi_total Function-yielding V34() Element of bool [:[:(X,Y),(X,Y):],(X,Y):]
[:(X,Y),(X,Y):] is non empty set
[:[:(X,Y),(X,Y):],(X,Y):] is non empty set
bool [:[:(X,Y),(X,Y):],(X,Y):] is non empty set
Mult_ ((X,Y),( the carrier of X,Y)) is non empty Relation-like [:COMPLEX,(X,Y):] -defined (X,Y) -valued Function-like total quasi_total Function-yielding V34() Element of bool [:[:COMPLEX,(X,Y):],(X,Y):]
[:COMPLEX,(X,Y):] is non empty set
[:[:COMPLEX,(X,Y):],(X,Y):] is non empty set
bool [:[:COMPLEX,(X,Y):],(X,Y):] is non empty set
CLSStruct(# (X,Y),(Zero_ ((X,Y),( the carrier of X,Y))),(Add_ ((X,Y),( the carrier of X,Y))),(Mult_ ((X,Y),( the carrier of X,Y))) #) is non empty right_complementable Abelian add-associative right_zeroed strict vector-distributive scalar-distributive scalar-associative scalar-unital CLSStruct
the carrier of (X,Y) is non empty set
bool the carrier of (X,Y) is non empty set
Zero_ ((X,Y),(X,Y)) is Element of (X,Y)
Add_ ((X,Y),(X,Y)) is non empty Relation-like [:(X,Y),(X,Y):] -defined (X,Y) -valued Function-like total quasi_total Element of bool [:[:(X,Y),(X,Y):],(X,Y):]
[:(X,Y),(X,Y):] is non empty set
[:[:(X,Y),(X,Y):],(X,Y):] is non empty set
bool [:[:(X,Y),(X,Y):],(X,Y):] is non empty set
Mult_ ((X,Y),(X,Y)) is non empty Relation-like [:COMPLEX,(X,Y):] -defined (X,Y) -valued Function-like total quasi_total Element of bool [:[:COMPLEX,(X,Y):],(X,Y):]
[:COMPLEX,(X,Y):] is non empty set
[:[:COMPLEX,(X,Y):],(X,Y):] is non empty set
bool [:[:COMPLEX,(X,Y):],(X,Y):] is non empty set
(X,Y) is non empty Relation-like (X,Y) -defined REAL -valued Function-like total quasi_total V132() V133() V134() Element of bool [:(X,Y),REAL:]
[:(X,Y),REAL:] is non empty V132() V133() V134() set
bool [:(X,Y),REAL:] is non empty set
CNORMSTR(# (X,Y),(Zero_ ((X,Y),(X,Y))),(Add_ ((X,Y),(X,Y))),(Mult_ ((X,Y),(X,Y))),(X,Y) #) is non empty strict CNORMSTR
0. (X,Y) is Relation-like Function-like zero Element of the carrier of (X,Y)
the carrier of (X,Y) is non empty set
the ZeroF of (X,Y) is Relation-like Function-like Element of the carrier of (X,Y)
||.(0. (X,Y)).|| is complex real ext-real Element of REAL
the U9 of (X,Y) is non empty Relation-like the carrier of (X,Y) -defined REAL -valued Function-like total quasi_total V132() V133() V134() Element of bool [: the carrier of (X,Y),REAL:]
[: the carrier of (X,Y),REAL:] is non empty V132() V133() V134() set
bool [: the carrier of (X,Y),REAL:] is non empty set
the U9 of (X,Y) . (0. (X,Y)) is complex real ext-real Element of REAL
the carrier of (X,Y) is non empty set
0. (X,Y) is Relation-like Function-like zero Element of the carrier of (X,Y)
the ZeroF of (X,Y) is Relation-like Function-like Element of the carrier of (X,Y)
vseq is Relation-like Function-like Element of the carrier of (X,Y)
||.vseq.|| is complex real ext-real Element of REAL
the U9 of (X,Y) is non empty Relation-like the carrier of (X,Y) -defined REAL -valued Function-like total quasi_total V132() V133() V134() Element of bool [: the carrier of (X,Y),REAL:]
[: the carrier of (X,Y),REAL:] is non empty V132() V133() V134() set
bool [: the carrier of (X,Y),REAL:] is non empty set
the U9 of (X,Y) . vseq is complex real ext-real Element of REAL
f is Relation-like Function-like Element of the carrier of (X,Y)
||.f.|| is complex real ext-real Element of REAL
the U9 of (X,Y) . f is complex real ext-real Element of REAL
tseq is Relation-like Function-like Element of the carrier of (X,Y)
tseq is complex set
tseq * tseq is Relation-like Function-like Element of the carrier of (X,Y)
||.(tseq * tseq).|| is complex real ext-real Element of REAL
the U9 of (X,Y) . (tseq * tseq) is complex real ext-real Element of REAL
|.tseq.| is complex real ext-real Element of REAL
||.tseq.|| is complex real ext-real Element of REAL
the U9 of (X,Y) . tseq is complex real ext-real Element of REAL
|.tseq.| * ||.tseq.|| is complex real ext-real Element of REAL
tseq is Relation-like Function-like Element of the carrier of (X,Y)
tv is Relation-like Function-like Element of the carrier of (X,Y)
tseq + tv is Relation-like Function-like Element of the carrier of (X,Y)
the addF of (X,Y) is non empty Relation-like [: the carrier of (X,Y), the carrier of (X,Y):] -defined the carrier of (X,Y) -valued Function-like total quasi_total Element of bool [:[: the carrier of (X,Y), the carrier of (X,Y):], the carrier of (X,Y):]
[: the carrier of (X,Y), the carrier of (X,Y):] is non empty set
[:[: the carrier of (X,Y), the carrier of (X,Y):], the carrier of (X,Y):] is non empty set
bool [:[: the carrier of (X,Y), the carrier of (X,Y):], the carrier of (X,Y):] is non empty set
the addF of (X,Y) . (tseq,tv) is Relation-like Function-like Element of the carrier of (X,Y)
[tseq,tv] is set
the addF of (X,Y) . [tseq,tv] is set
||.(tseq + tv).|| is complex real ext-real Element of REAL
the U9 of (X,Y) . (tseq + tv) is complex real ext-real Element of REAL
||.tseq.|| is complex real ext-real Element of REAL
the U9 of (X,Y) . tseq is complex real ext-real Element of REAL
||.tv.|| is complex real ext-real Element of REAL
the U9 of (X,Y) . tv is complex real ext-real Element of REAL
||.tseq.|| + ||.tv.|| is complex real ext-real Element of REAL
X is non empty right_complementable Abelian add-associative right_zeroed discerning reflexive vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like CNORMSTR
Y is non empty right_complementable Abelian add-associative right_zeroed discerning reflexive vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like CNORMSTR
(X,Y) is non empty CNORMSTR
(X,Y) is non empty Element of bool the carrier of (X,Y)
(X,Y) is non empty right_complementable Abelian add-associative right_zeroed constituted-Functions strict vector-distributive scalar-distributive scalar-associative scalar-unital CLSStruct
(X,Y) is non empty functional Element of bool the carrier of ( the carrier of X,Y)
the carrier of X is non empty set
( the carrier of X,Y) is non empty right_complementable Abelian add-associative right_zeroed constituted-Functions strict vector-distributive scalar-distributive scalar-associative scalar-unital CLSStruct
the carrier of Y is non empty set
Funcs ( the carrier of X, the carrier of Y) is non empty functional FUNCTION_DOMAIN of the carrier of X, the carrier of Y
FuncZero ( the carrier of X,Y) is Relation-like the carrier of X -defined the carrier of Y -valued Function-like total quasi_total Element of Funcs ( the carrier of X, the carrier of Y)
FuncAdd ( the carrier of X,Y) is non empty Relation-like [:(Funcs ( the carrier of X, the carrier of Y)),(Funcs ( the carrier of X, the carrier of Y)):] -defined Funcs ( the carrier of X, the carrier of Y) -valued Function-like total quasi_total Function-yielding V34() Element of bool [:[:(Funcs ( the carrier of X, the carrier of Y)),(Funcs ( the carrier of X, the carrier of Y)):],(Funcs ( the carrier of X, the carrier of Y)):]
[:(Funcs ( the carrier of X, the carrier of Y)),(Funcs ( the carrier of X, the carrier of Y)):] is non empty set
[:[:(Funcs ( the carrier of X, the carrier of Y)),(Funcs ( the carrier of X, the carrier of Y)):],(Funcs ( the carrier of X, the carrier of Y)):] is non empty set
bool [:[:(Funcs ( the carrier of X, the carrier of Y)),(Funcs ( the carrier of X, the carrier of Y)):],(Funcs ( the carrier of X, the carrier of Y)):] is non empty set
( the carrier of X,Y) is non empty Relation-like [:COMPLEX,(Funcs ( the carrier of X, the carrier of Y)):] -defined Funcs ( the carrier of X, the carrier of Y) -valued Function-like total quasi_total Function-yielding V34() Element of bool [:[:COMPLEX,(Funcs ( the carrier of X, the carrier of Y)):],(Funcs ( the carrier of X, the carrier of Y)):]
[:COMPLEX,(Funcs ( the carrier of X, the carrier of Y)):] is non empty set
[:[:COMPLEX,(Funcs ( the carrier of X, the carrier of Y)):],(Funcs ( the carrier of X, the carrier of Y)):] is non empty set
bool [:[:COMPLEX,(Funcs ( the carrier of X, the carrier of Y)):],(Funcs ( the carrier of X, the carrier of Y)):] is non empty set
CLSStruct(# (Funcs ( the carrier of X, the carrier of Y)),(FuncZero ( the carrier of X,Y)),(FuncAdd ( the carrier of X,Y)),( the carrier of X,Y) #) is non empty strict CLSStruct
the carrier of ( the carrier of X,Y) is non empty set
bool the carrier of ( the carrier of X,Y) is non empty set
Zero_ ((X,Y),( the carrier of X,Y)) is Relation-like Function-like Element of (X,Y)
Add_ ((X,Y),( the carrier of X,Y)) is non empty Relation-like [:(X,Y),(X,Y):] -defined (X,Y) -valued Function-like total quasi_total Function-yielding V34() Element of bool [:[:(X,Y),(X,Y):],(X,Y):]
[:(X,Y),(X,Y):] is non empty set
[:[:(X,Y),(X,Y):],(X,Y):] is non empty set
bool [:[:(X,Y),(X,Y):],(X,Y):] is non empty set
Mult_ ((X,Y),( the carrier of X,Y)) is non empty Relation-like [:COMPLEX,(X,Y):] -defined (X,Y) -valued Function-like total quasi_total Function-yielding V34() Element of bool [:[:COMPLEX,(X,Y):],(X,Y):]
[:COMPLEX,(X,Y):] is non empty set
[:[:COMPLEX,(X,Y):],(X,Y):] is non empty set
bool [:[:COMPLEX,(X,Y):],(X,Y):] is non empty set
CLSStruct(# (X,Y),(Zero_ ((X,Y),( the carrier of X,Y))),(Add_ ((X,Y),( the carrier of X,Y))),(Mult_ ((X,Y),( the carrier of X,Y))) #) is non empty right_complementable Abelian add-associative right_zeroed strict vector-distributive scalar-distributive scalar-associative scalar-unital CLSStruct
the carrier of (X,Y) is non empty set
bool the carrier of (X,Y) is non empty set
Zero_ ((X,Y),(X,Y)) is Element of (X,Y)
Add_ ((X,Y),(X,Y)) is non empty Relation-like [:(X,Y),(X,Y):] -defined (X,Y) -valued Function-like total quasi_total Element of bool [:[:(X,Y),(X,Y):],(X,Y):]
[:(X,Y),(X,Y):] is non empty set
[:[:(X,Y),(X,Y):],(X,Y):] is non empty set
bool [:[:(X,Y),(X,Y):],(X,Y):] is non empty set
Mult_ ((X,Y),(X,Y)) is non empty Relation-like [:COMPLEX,(X,Y):] -defined (X,Y) -valued Function-like total quasi_total Element of bool [:[:COMPLEX,(X,Y):],(X,Y):]
[:COMPLEX,(X,Y):] is non empty set
[:[:COMPLEX,(X,Y):],(X,Y):] is non empty set
bool [:[:COMPLEX,(X,Y):],(X,Y):] is non empty set
(X,Y) is non empty Relation-like (X,Y) -defined REAL -valued Function-like total quasi_total V132() V133() V134() Element of bool [:(X,Y),REAL:]
[:(X,Y),REAL:] is non empty V132() V133() V134() set
bool [:(X,Y),REAL:] is non empty set
CNORMSTR(# (X,Y),(Zero_ ((X,Y),(X,Y))),(Add_ ((X,Y),(X,Y))),(Mult_ ((X,Y),(X,Y))),(X,Y) #) is non empty strict CNORMSTR
CLSStruct(# (X,Y),(Zero_ ((X,Y),(X,Y))),(Add_ ((X,Y),(X,Y))),(Mult_ ((X,Y),(X,Y))) #) is non empty right_complementable Abelian add-associative right_zeroed strict vector-distributive scalar-distributive scalar-associative scalar-unital CLSStruct
X is non empty right_complementable Abelian add-associative right_zeroed discerning reflexive vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like CNORMSTR
Y is non empty right_complementable Abelian add-associative right_zeroed discerning reflexive vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like CNORMSTR
(X,Y) is non empty CNORMSTR
(X,Y) is non empty Element of bool the carrier of (X,Y)
(X,Y) is non empty right_complementable Abelian add-associative right_zeroed constituted-Functions strict vector-distributive scalar-distributive scalar-associative scalar-unital CLSStruct
(X,Y) is non empty functional Element of bool the carrier of ( the carrier of X,Y)
the carrier of X is non empty set
( the carrier of X,Y) is non empty right_complementable Abelian add-associative right_zeroed constituted-Functions strict vector-distributive scalar-distributive scalar-associative scalar-unital CLSStruct
the carrier of Y is non empty set
Funcs ( the carrier of X, the carrier of Y) is non empty functional FUNCTION_DOMAIN of the carrier of X, the carrier of Y
FuncZero ( the carrier of X,Y) is Relation-like the carrier of X -defined the carrier of Y -valued Function-like total quasi_total Element of Funcs ( the carrier of X, the carrier of Y)
FuncAdd ( the carrier of X,Y) is non empty Relation-like [:(Funcs ( the carrier of X, the carrier of Y)),(Funcs ( the carrier of X, the carrier of Y)):] -defined Funcs ( the carrier of X, the carrier of Y) -valued Function-like total quasi_total Function-yielding V34() Element of bool [:[:(Funcs ( the carrier of X, the carrier of Y)),(Funcs ( the carrier of X, the carrier of Y)):],(Funcs ( the carrier of X, the carrier of Y)):]
[:(Funcs ( the carrier of X, the carrier of Y)),(Funcs ( the carrier of X, the carrier of Y)):] is non empty set
[:[:(Funcs ( the carrier of X, the carrier of Y)),(Funcs ( the carrier of X, the carrier of Y)):],(Funcs ( the carrier of X, the carrier of Y)):] is non empty set
bool [:[:(Funcs ( the carrier of X, the carrier of Y)),(Funcs ( the carrier of X, the carrier of Y)):],(Funcs ( the carrier of X, the carrier of Y)):] is non empty set
( the carrier of X,Y) is non empty Relation-like [:COMPLEX,(Funcs ( the carrier of X, the carrier of Y)):] -defined Funcs ( the carrier of X, the carrier of Y) -valued Function-like total quasi_total Function-yielding V34() Element of bool [:[:COMPLEX,(Funcs ( the carrier of X, the carrier of Y)):],(Funcs ( the carrier of X, the carrier of Y)):]
[:COMPLEX,(Funcs ( the carrier of X, the carrier of Y)):] is non empty set
[:[:COMPLEX,(Funcs ( the carrier of X, the carrier of Y)):],(Funcs ( the carrier of X, the carrier of Y)):] is non empty set
bool [:[:COMPLEX,(Funcs ( the carrier of X, the carrier of Y)):],(Funcs ( the carrier of X, the carrier of Y)):] is non empty set
CLSStruct(# (Funcs ( the carrier of X, the carrier of Y)),(FuncZero ( the carrier of X,Y)),(FuncAdd ( the carrier of X,Y)),( the carrier of X,Y) #) is non empty strict CLSStruct
the carrier of ( the carrier of X,Y) is non empty set
bool the carrier of ( the carrier of X,Y) is non empty set
Zero_ ((X,Y),( the carrier of X,Y)) is Relation-like Function-like Element of (X,Y)
Add_ ((X,Y),( the carrier of X,Y)) is non empty Relation-like [:(X,Y),(X,Y):] -defined (X,Y) -valued Function-like total quasi_total Function-yielding V34() Element of bool [:[:(X,Y),(X,Y):],(X,Y):]
[:(X,Y),(X,Y):] is non empty set
[:[:(X,Y),(X,Y):],(X,Y):] is non empty set
bool [:[:(X,Y),(X,Y):],(X,Y):] is non empty set
Mult_ ((X,Y),( the carrier of X,Y)) is non empty Relation-like [:COMPLEX,(X,Y):] -defined (X,Y) -valued Function-like total quasi_total Function-yielding V34() Element of bool [:[:COMPLEX,(X,Y):],(X,Y):]
[:COMPLEX,(X,Y):] is non empty set
[:[:COMPLEX,(X,Y):],(X,Y):] is non empty set
bool [:[:COMPLEX,(X,Y):],(X,Y):] is non empty set
CLSStruct(# (X,Y),(Zero_ ((X,Y),( the carrier of X,Y))),(Add_ ((X,Y),( the carrier of X,Y))),(Mult_ ((X,Y),( the carrier of X,Y))) #) is non empty right_complementable Abelian add-associative right_zeroed strict vector-distributive scalar-distributive scalar-associative scalar-unital CLSStruct
the carrier of (X,Y) is non empty set
bool the carrier of (X,Y) is non empty set
Zero_ ((X,Y),(X,Y)) is Element of (X,Y)
Add_ ((X,Y),(X,Y)) is non empty Relation-like [:(X,Y),(X,Y):] -defined (X,Y) -valued Function-like total quasi_total Element of bool [:[:(X,Y),(X,Y):],(X,Y):]
[:(X,Y),(X,Y):] is non empty set
[:[:(X,Y),(X,Y):],(X,Y):] is non empty set
bool [:[:(X,Y),(X,Y):],(X,Y):] is non empty set
Mult_ ((X,Y),(X,Y)) is non empty Relation-like [:COMPLEX,(X,Y):] -defined (X,Y) -valued Function-like total quasi_total Element of bool [:[:COMPLEX,(X,Y):],(X,Y):]
[:COMPLEX,(X,Y):] is non empty set
[:[:COMPLEX,(X,Y):],(X,Y):] is non empty set
bool [:[:COMPLEX,(X,Y):],(X,Y):] is non empty set
(X,Y) is non empty Relation-like (X,Y) -defined REAL -valued Function-like total quasi_total V132() V133() V134() Element of bool [:(X,Y),REAL:]
[:(X,Y),REAL:] is non empty V132() V133() V134() set
bool [:(X,Y),REAL:] is non empty set
CNORMSTR(# (X,Y),(Zero_ ((X,Y),(X,Y))),(Add_ ((X,Y),(X,Y))),(Mult_ ((X,Y),(X,Y))),(X,Y) #) is non empty strict CNORMSTR
X is non empty right_complementable Abelian add-associative right_zeroed discerning reflexive vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like CNORMSTR
Y is non empty right_complementable Abelian add-associative right_zeroed discerning reflexive vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like CNORMSTR
(X,Y) is non empty right_complementable Abelian add-associative right_zeroed discerning reflexive vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like CNORMSTR
(X,Y) is non empty Element of bool the carrier of (X,Y)
(X,Y) is non empty right_complementable Abelian add-associative right_zeroed constituted-Functions strict vector-distributive scalar-distributive scalar-associative scalar-unital CLSStruct
(X,Y) is non empty functional Element of bool the carrier of ( the carrier of X,Y)
the carrier of X is non empty set
( the carrier of X,Y) is non empty right_complementable Abelian add-associative right_zeroed constituted-Functions strict vector-distributive scalar-distributive scalar-associative scalar-unital CLSStruct
the carrier of Y is non empty set
Funcs ( the carrier of X, the carrier of Y) is non empty functional FUNCTION_DOMAIN of the carrier of X, the carrier of Y
FuncZero ( the carrier of X,Y) is Relation-like the carrier of X -defined the carrier of Y -valued Function-like total quasi_total Element of Funcs ( the carrier of X, the carrier of Y)
FuncAdd ( the carrier of X,Y) is non empty Relation-like [:(Funcs ( the carrier of X, the carrier of Y)),(Funcs ( the carrier of X, the carrier of Y)):] -defined Funcs ( the carrier of X, the carrier of Y) -valued Function-like total quasi_total Function-yielding V34() Element of bool [:[:(Funcs ( the carrier of X, the carrier of Y)),(Funcs ( the carrier of X, the carrier of Y)):],(Funcs ( the carrier of X, the carrier of Y)):]
[:(Funcs ( the carrier of X, the carrier of Y)),(Funcs ( the carrier of X, the carrier of Y)):] is non empty set
[:[:(Funcs ( the carrier of X, the carrier of Y)),(Funcs ( the carrier of X, the carrier of Y)):],(Funcs ( the carrier of X, the carrier of Y)):] is non empty set
bool [:[:(Funcs ( the carrier of X, the carrier of Y)),(Funcs ( the carrier of X, the carrier of Y)):],(Funcs ( the carrier of X, the carrier of Y)):] is non empty set
( the carrier of X,Y) is non empty Relation-like [:COMPLEX,(Funcs ( the carrier of X, the carrier of Y)):] -defined Funcs ( the carrier of X, the carrier of Y) -valued Function-like total quasi_total Function-yielding V34() Element of bool [:[:COMPLEX,(Funcs ( the carrier of X, the carrier of Y)):],(Funcs ( the carrier of X, the carrier of Y)):]
[:COMPLEX,(Funcs ( the carrier of X, the carrier of Y)):] is non empty set
[:[:COMPLEX,(Funcs ( the carrier of X, the carrier of Y)):],(Funcs ( the carrier of X, the carrier of Y)):] is non empty set
bool [:[:COMPLEX,(Funcs ( the carrier of X, the carrier of Y)):],(Funcs ( the carrier of X, the carrier of Y)):] is non empty set
CLSStruct(# (Funcs ( the carrier of X, the carrier of Y)),(FuncZero ( the carrier of X,Y)),(FuncAdd ( the carrier of X,Y)),( the carrier of X,Y) #) is non empty strict CLSStruct
the carrier of ( the carrier of X,Y) is non empty set
bool the carrier of ( the carrier of X,Y) is non empty set
Zero_ ((X,Y),( the carrier of X,Y)) is Relation-like Function-like Element of (X,Y)
Add_ ((X,Y),( the carrier of X,Y)) is non empty Relation-like [:(X,Y),(X,Y):] -defined (X,Y) -valued Function-like total quasi_total Function-yielding V34() Element of bool [:[:(X,Y),(X,Y):],(X,Y):]
[:(X,Y),(X,Y):] is non empty set
[:[:(X,Y),(X,Y):],(X,Y):] is non empty set
bool [:[:(X,Y),(X,Y):],(X,Y):] is non empty set
Mult_ ((X,Y),( the carrier of X,Y)) is non empty Relation-like [:COMPLEX,(X,Y):] -defined (X,Y) -valued Function-like total quasi_total Function-yielding V34() Element of bool [:[:COMPLEX,(X,Y):],(X,Y):]
[:COMPLEX,(X,Y):] is non empty set
[:[:COMPLEX,(X,Y):],(X,Y):] is non empty set
bool [:[:COMPLEX,(X,Y):],(X,Y):] is non empty set
CLSStruct(# (X,Y),(Zero_ ((X,Y),( the carrier of X,Y))),(Add_ ((X,Y),( the carrier of X,Y))),(Mult_ ((X,Y),( the carrier of X,Y))) #) is non empty right_complementable Abelian add-associative right_zeroed strict vector-distributive scalar-distributive scalar-associative scalar-unital CLSStruct
the carrier of (X,Y) is non empty set
bool the carrier of (X,Y) is non empty set
Zero_ ((X,Y),(X,Y)) is Element of (X,Y)
Add_ ((X,Y),(X,Y)) is non empty Relation-like [:(X,Y),(X,Y):] -defined (X,Y) -valued Function-like total quasi_total Element of bool [:[:(X,Y),(X,Y):],(X,Y):]
[:(X,Y),(X,Y):] is non empty set
[:[:(X,Y),(X,Y):],(X,Y):] is non empty set
bool [:[:(X,Y),(X,Y):],(X,Y):] is non empty set
Mult_ ((X,Y),(X,Y)) is non empty Relation-like [:COMPLEX,(X,Y):] -defined (X,Y) -valued Function-like total quasi_total Element of bool [:[:COMPLEX,(X,Y):],(X,Y):]
[:COMPLEX,(X,Y):] is non empty set
[:[:COMPLEX,(X,Y):],(X,Y):] is non empty set
bool [:[:COMPLEX,(X,Y):],(X,Y):] is non empty set
(X,Y) is non empty Relation-like (X,Y) -defined REAL -valued Function-like total quasi_total V132() V133() V134() Element of bool [:(X,Y),REAL:]
[:(X,Y),REAL:] is non empty V132() V133() V134() set
bool [:(X,Y),REAL:] is non empty set
CNORMSTR(# (X,Y),(Zero_ ((X,Y),(X,Y))),(Add_ ((X,Y),(X,Y))),(Mult_ ((X,Y),(X,Y))),(X,Y) #) is non empty strict CNORMSTR
the carrier of (X,Y) is non empty set
tseq is Relation-like Function-like Element of the carrier of (X,Y)
vseq is Relation-like Function-like Element of the carrier of (X,Y)
f is Relation-like Function-like Element of the carrier of (X,Y)
vseq - f is Relation-like Function-like Element of the carrier of (X,Y)
- f is Relation-like Function-like Element of the carrier of (X,Y)
vseq + (- f) is Relation-like Function-like Element of the carrier of (X,Y)
the addF of (X,Y) is non empty Relation-like [: the carrier of (X,Y), the carrier of (X,Y):] -defined the carrier of (X,Y) -valued Function-like total quasi_total Element of bool [:[: the carrier of (X,Y), the carrier of (X,Y):], the carrier of (X,Y):]
[: the carrier of (X,Y), the carrier of (X,Y):] is non empty set
[:[: the carrier of (X,Y), the carrier of (X,Y):], the carrier of (X,Y):] is non empty set
bool [:[: the carrier of (X,Y), the carrier of (X,Y):], the carrier of (X,Y):] is non empty set
the addF of (X,Y) . (vseq,(- f)) is Relation-like Function-like Element of the carrier of (X,Y)
[vseq,(- f)] is set
the addF of (X,Y) . [vseq,(- f)] is set
[: the carrier of X, the carrier of Y:] is non empty set
bool [: the carrier of X, the carrier of Y:] is non empty set
tseq + f is Relation-like Function-like Element of the carrier of (X,Y)
the addF of (X,Y) . (tseq,f) is Relation-like Function-like Element of the carrier of (X,Y)
[tseq,f] is set
the addF of (X,Y) . [tseq,f] is set
f - f is Relation-like Function-like Element of the carrier of (X,Y)
f + (- f) is Relation-like Function-like Element of the carrier of (X,Y)
the addF of (X,Y) . (f,(- f)) is Relation-like Function-like Element of the carrier of (X,Y)
[f,(- f)] is set
the addF of (X,Y) . [f,(- f)] is set
vseq - (f - f) is Relation-like Function-like Element of the carrier of (X,Y)
- (f - f) is Relation-like Function-like Element of the carrier of (X,Y)
vseq + (- (f - f)) is Relation-like Function-like Element of the carrier of (X,Y)
the addF of (X,Y) . (vseq,(- (f - f))) is Relation-like Function-like Element of the carrier of (X,Y)
[vseq,(- (f - f))] is set
the addF of (X,Y) . [vseq,(- (f - f))] is set
0. (X,Y) is Relation-like Function-like zero Element of the carrier of (X,Y)
the ZeroF of (X,Y) is Relation-like Function-like Element of the carrier of (X,Y)
vseq - (0. (X,Y)) is Relation-like Function-like Element of the carrier of (X,Y)
- (0. (X,Y)) is Relation-like Function-like Element of the carrier of (X,Y)
vseq + (- (0. (X,Y))) is Relation-like Function-like Element of the carrier of (X,Y)
the addF of (X,Y) . (vseq,(- (0. (X,Y)))) is Relation-like Function-like Element of the carrier of (X,Y)
[vseq,(- (0. (X,Y)))] is set
the addF of (X,Y) . [vseq,(- (0. (X,Y)))] is set
tseq is non empty Relation-like the carrier of X -defined the carrier of Y -valued Function-like total quasi_total additive (X,Y) (X,Y) Element of bool [: the carrier of X, the carrier of Y:]
e is Element of the carrier of X
tseq . e is Element of the carrier of Y
tv is non empty Relation-like the carrier of X -defined the carrier of Y -valued Function-like total quasi_total additive (X,Y) (X,Y) Element of bool [: the carrier of X, the carrier of Y:]
tv . e is Element of the carrier of Y
tseq is non empty Relation-like the carrier of X -defined the carrier of Y -valued Function-like total quasi_total additive (X,Y) (X,Y) Element of bool [: the carrier of X, the carrier of Y:]
tseq . e is Element of the carrier of Y
(tv . e) + (tseq . e) is Element of the carrier of Y
the addF of Y is non empty Relation-like [: the carrier of Y, the carrier of Y:] -defined the carrier of Y -valued Function-like total quasi_total Element of bool [:[: the carrier of Y, the carrier of Y:], the carrier of Y:]
[: the carrier of Y, the carrier of Y:] is non empty set
[:[: the carrier of Y, the carrier of Y:], the carrier of Y:] is non empty set
bool [:[: the carrier of Y, the carrier of Y:], the carrier of Y:] is non empty set
the addF of Y . ((tv . e),(tseq . e)) is Element of the carrier of Y
[(tv . e),(tseq . e)] is set
the addF of Y . [(tv . e),(tseq . e)] is set
(tseq . e) - (tseq . e) is Element of the carrier of Y
- (tseq . e) is Element of the carrier of Y
(tseq . e) + (- (tseq . e)) is Element of the carrier of Y
the addF of Y . ((tseq . e),(- (tseq . e))) is Element of the carrier of Y
[(tseq . e),(- (tseq . e))] is set
the addF of Y . [(tseq . e),(- (tseq . e))] is set
(tseq . e) - (tseq . e) is Element of the carrier of Y
(tseq . e) + (- (tseq . e)) is Element of the carrier of Y
the addF of Y . ((tseq . e),(- (tseq . e))) is Element of the carrier of Y
[(tseq . e),(- (tseq . e))] is set
the addF of Y . [(tseq . e),(- (tseq . e))] is set
(tv . e) + ((tseq . e) - (tseq . e)) is Element of the carrier of Y
the addF of Y . ((tv . e),((tseq . e) - (tseq . e))) is Element of the carrier of Y
[(tv . e),((tseq . e) - (tseq . e))] is set
the addF of Y . [(tv . e),((tseq . e) - (tseq . e))] is set
0. Y is zero Element of the carrier of Y
the ZeroF of Y is Element of the carrier of Y
(tv . e) + (0. Y) is Element of the carrier of Y
the addF of Y . ((tv . e),(0. Y)) is Element of the carrier of Y
[(tv . e),(0. Y)] is set
the addF of Y . [(tv . e),(0. Y)] is set
e is Element of the carrier of X
(X,Y,tseq,e) is Element of the carrier of Y
(X,Y,vseq,e) is Element of the carrier of Y
(X,Y,f,e) is Element of the carrier of Y
(X,Y,vseq,e) - (X,Y,f,e) is Element of the carrier of Y
- (X,Y,f,e) is Element of the carrier of Y
(X,Y,vseq,e) + (- (X,Y,f,e)) is Element of the carrier of Y
the addF of Y . ((X,Y,vseq,e),(- (X,Y,f,e))) is Element of the carrier of Y
[(X,Y,vseq,e),(- (X,Y,f,e))] is set
the addF of Y . [(X,Y,vseq,e),(- (X,Y,f,e))] is set
e is Element of the carrier of X
tv . e is Element of the carrier of Y
tseq . e is Element of the carrier of Y
tseq . e is Element of the carrier of Y
(tseq . e) - (tseq . e) is Element of the carrier of Y
- (tseq . e) is Element of the carrier of Y
(tseq . e) + (- (tseq . e)) is Element of the carrier of Y
the addF of Y . ((tseq . e),(- (tseq . e))) is Element of the carrier of Y
[(tseq . e),(- (tseq . e))] is set
the addF of Y . [(tseq . e),(- (tseq . e))] is set
(tv . e) + (tseq . e) is Element of the carrier of Y
the addF of Y . ((tv . e),(tseq . e)) is Element of the carrier of Y
[(tv . e),(tseq . e)] is set
the addF of Y . [(tv . e),(tseq . e)] is set
(tseq . e) - (tseq . e) is Element of the carrier of Y
(tseq . e) + (- (tseq . e)) is Element of the carrier of Y
the addF of Y . ((tseq . e),(- (tseq . e))) is Element of the carrier of Y
[(tseq . e),(- (tseq . e))] is set
the addF of Y . [(tseq . e),(- (tseq . e))] is set
(tseq . e) - ((tseq . e) - (tseq . e)) is Element of the carrier of Y
- ((tseq . e) - (tseq . e)) is Element of the carrier of Y
(tseq . e) + (- ((tseq . e) - (tseq . e))) is Element of the carrier of Y
the addF of Y . ((tseq . e),(- ((tseq . e) - (tseq . e)))) is Element of the carrier of Y
[(tseq . e),(- ((tseq . e) - (tseq . e)))] is set
the addF of Y . [(tseq . e),(- ((tseq . e) - (tseq . e)))] is set
(tseq . e) - (0. Y) is Element of the carrier of Y
- (0. Y) is Element of the carrier of Y
(tseq . e) + (- (0. Y)) is Element of the carrier of Y
the addF of Y . ((tseq . e),(- (0. Y))) is Element of the carrier of Y
[(tseq . e),(- (0. Y))] is set
the addF of Y . [(tseq . e),(- (0. Y))] is set
tseq + (f - f) is Relation-like Function-like Element of the carrier of (X,Y)
the addF of (X,Y) . (tseq,(f - f)) is Relation-like Function-like Element of the carrier of (X,Y)
[tseq,(f - f)] is set
the addF of (X,Y) . [tseq,(f - f)] is set
tseq + (0. (X,Y)) is Relation-like Function-like Element of the carrier of (X,Y)
the addF of (X,Y) . (tseq,(0. (X,Y))) is Relation-like Function-like Element of the carrier of (X,Y)
[tseq,(0. (X,Y))] is set
the addF of (X,Y) . [tseq,(0. (X,Y))] is set
X is complex real ext-real Element of REAL
Y is non empty Relation-like NAT -defined REAL -valued Function-like total quasi_total V132() V133() V134() Element of bool [:NAT,REAL:]
lim Y is complex real ext-real Element of REAL
vseq is epsilon-transitive epsilon-connected ordinal natural complex real V108() V109() ext-real V142() V143() V144() V145() V146() V147() Element of NAT
vseq is epsilon-transitive epsilon-connected ordinal natural complex real V108() V109() ext-real V142() V143() V144() V145() V146() V147() Element of NAT
NAT --> X is non empty Relation-like NAT -defined REAL -valued Function-like constant total quasi_total V132() V133() V134() convergent Element of bool [:NAT,REAL:]
f is non empty Relation-like NAT -defined REAL -valued Function-like total quasi_total V132() V133() V134() Element of bool [:NAT,REAL:]
lim f is complex real ext-real Element of REAL
f . 0 is complex real ext-real Element of REAL
Y ^\ vseq is non empty Relation-like NAT -defined REAL -valued Function-like total quasi_total V132() V133() V134() Element of bool [:NAT,REAL:]
tseq is epsilon-transitive epsilon-connected ordinal natural complex real V108() V109() ext-real V142() V143() V144() V145() V146() V147() Element of NAT
(Y ^\ vseq) . tseq is complex real ext-real Element of REAL
vseq + tseq is epsilon-transitive epsilon-connected ordinal natural complex real V108() V109() ext-real V142() V143() V144() V145() V146() V147() Element of NAT
Y . (vseq + tseq) is complex real ext-real Element of REAL
f . tseq is complex real ext-real Element of REAL
lim (Y ^\ vseq) is complex real ext-real Element of REAL
X is non empty right_complementable Abelian add-associative right_zeroed discerning reflexive vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like CNORMSTR
the carrier of X is non empty set
[:NAT, the carrier of X:] is non empty set
bool [:NAT, the carrier of X:] is non empty set
Y is non empty Relation-like NAT -defined the carrier of X -valued Function-like total quasi_total Element of bool [:NAT, the carrier of X:]
||.Y.|| is non empty Relation-like NAT -defined REAL -valued Function-like total quasi_total V132() V133() V134() Element of bool [:NAT,REAL:]
lim ||.Y.|| is complex real ext-real Element of REAL
lim Y is Element of the carrier of X
||.(lim Y).|| is complex real ext-real Element of REAL
the U9 of X is non empty Relation-like the carrier of X -defined REAL -valued Function-like total quasi_total V132() V133() V134() Element of bool [: the carrier of X,REAL:]
[: the carrier of X,REAL:] is non empty V132() V133() V134() set
bool [: the carrier of X,REAL:] is non empty set
the U9 of X . (lim Y) is complex real ext-real Element of REAL
vseq is complex real ext-real set
f is complex real ext-real Element of REAL
tseq is epsilon-transitive epsilon-connected ordinal natural complex real V108() V109() ext-real V142() V143() V144() V145() V146() V147() Element of NAT
tseq is epsilon-transitive epsilon-connected ordinal natural complex real V108() V109() ext-real V142() V143() V144() V145() V146() V147() Element of NAT
tseq is epsilon-transitive epsilon-connected ordinal natural complex real V108() V109() ext-real V142() V143() V144() V145() V146() V147() Element of NAT
Y . tseq is Element of the carrier of X
(Y . tseq) - (lim Y) is Element of the carrier of X
- (lim Y) is Element of the carrier of X
(Y . tseq) + (- (lim Y)) is Element of the carrier of X
the addF of X is non empty Relation-like [: the carrier of X, the carrier of X:] -defined the carrier of X -valued Function-like total quasi_total Element of bool [:[: the carrier of X, the carrier of X:], the carrier of X:]
[: the carrier of X, the carrier of X:] is non empty set
[:[: the carrier of X, the carrier of X:], the carrier of X:] is non empty set
bool [:[: the carrier of X, the carrier of X:], the carrier of X:] is non empty set
the addF of X . ((Y . tseq),(- (lim Y))) is Element of the carrier of X
[(Y . tseq),(- (lim Y))] is set
the addF of X . [(Y . tseq),(- (lim Y))] is set
||.((Y . tseq) - (lim Y)).|| is complex real ext-real Element of REAL
the U9 of X . ((Y . tseq) - (lim Y)) is complex real ext-real Element of REAL
||.(Y . tseq).|| is complex real ext-real Element of REAL
the U9 of X . (Y . tseq) is complex real ext-real Element of REAL
||.Y.|| . tseq is complex real ext-real Element of REAL
(||.Y.|| . tseq) - ||.(lim Y).|| is complex real ext-real Element of REAL
abs ((||.Y.|| . tseq) - ||.(lim Y).||) is complex real ext-real Element of REAL
tseq is epsilon-transitive epsilon-connected ordinal natural complex real V108() V109() ext-real V142() V143() V144() V145() V146() V147() Element of NAT
||.Y.|| . tseq is complex real ext-real Element of REAL
(||.Y.|| . tseq) - ||.(lim Y).|| is complex real ext-real Element of REAL
abs ((||.Y.|| . tseq) - ||.(lim Y).||) is complex real ext-real Element of REAL
Y is non empty right_complementable Abelian add-associative right_zeroed discerning reflexive vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like CNORMSTR
X is non empty right_complementable Abelian add-associative right_zeroed discerning reflexive vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like CNORMSTR
(X,Y) is non empty right_complementable Abelian add-associative right_zeroed discerning reflexive vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like CNORMSTR
(X,Y) is non empty Element of bool the carrier of (X,Y)
(X,Y) is non empty right_complementable Abelian add-associative right_zeroed constituted-Functions strict vector-distributive scalar-distributive scalar-associative scalar-unital CLSStruct
(X,Y) is non empty functional Element of bool the carrier of ( the carrier of X,Y)
the carrier of X is non empty set
( the carrier of X,Y) is non empty right_complementable Abelian add-associative right_zeroed constituted-Functions strict vector-distributive scalar-distributive scalar-associative scalar-unital CLSStruct
the carrier of Y is non empty set
Funcs ( the carrier of X, the carrier of Y) is non empty functional FUNCTION_DOMAIN of the carrier of X, the carrier of Y
FuncZero ( the carrier of X,Y) is Relation-like the carrier of X -defined the carrier of Y -valued Function-like total quasi_total Element of Funcs ( the carrier of X, the carrier of Y)
FuncAdd ( the carrier of X,Y) is non empty Relation-like [:(Funcs ( the carrier of X, the carrier of Y)),(Funcs ( the carrier of X, the carrier of Y)):] -defined Funcs ( the carrier of X, the carrier of Y) -valued Function-like total quasi_total Function-yielding V34() Element of bool [:[:(Funcs ( the carrier of X, the carrier of Y)),(Funcs ( the carrier of X, the carrier of Y)):],(Funcs ( the carrier of X, the carrier of Y)):]
[:(Funcs ( the carrier of X, the carrier of Y)),(Funcs ( the carrier of X, the carrier of Y)):] is non empty set
[:[:(Funcs ( the carrier of X, the carrier of Y)),(Funcs ( the carrier of X, the carrier of Y)):],(Funcs ( the carrier of X, the carrier of Y)):] is non empty set
bool [:[:(Funcs ( the carrier of X, the carrier of Y)),(Funcs ( the carrier of X, the carrier of Y)):],(Funcs ( the carrier of X, the carrier of Y)):] is non empty set
( the carrier of X,Y) is non empty Relation-like [:COMPLEX,(Funcs ( the carrier of X, the carrier of Y)):] -defined Funcs ( the carrier of X, the carrier of Y) -valued Function-like total quasi_total Function-yielding V34() Element of bool [:[:COMPLEX,(Funcs ( the carrier of X, the carrier of Y)):],(Funcs ( the carrier of X, the carrier of Y)):]
[:COMPLEX,(Funcs ( the carrier of X, the carrier of Y)):] is non empty set
[:[:COMPLEX,(Funcs ( the carrier of X, the carrier of Y)):],(Funcs ( the carrier of X, the carrier of Y)):] is non empty set
bool [:[:COMPLEX,(Funcs ( the carrier of X, the carrier of Y)):],(Funcs ( the carrier of X, the carrier of Y)):] is non empty set
CLSStruct(# (Funcs ( the carrier of X, the carrier of Y)),(FuncZero ( the carrier of X,Y)),(FuncAdd ( the carrier of X,Y)),( the carrier of X,Y) #) is non empty strict CLSStruct
the carrier of ( the carrier of X,Y) is non empty set
bool the carrier of ( the carrier of X,Y) is non empty set
Zero_ ((X,Y),( the carrier of X,Y)) is Relation-like Function-like Element of (X,Y)
Add_ ((X,Y),( the carrier of X,Y)) is non empty Relation-like [:(X,Y),(X,Y):] -defined (X,Y) -valued Function-like total quasi_total Function-yielding V34() Element of bool [:[:(X,Y),(X,Y):],(X,Y):]
[:(X,Y),(X,Y):] is non empty set
[:[:(X,Y),(X,Y):],(X,Y):] is non empty set
bool [:[:(X,Y),(X,Y):],(X,Y):] is non empty set
Mult_ ((X,Y),( the carrier of X,Y)) is non empty Relation-like [:COMPLEX,(X,Y):] -defined (X,Y) -valued Function-like total quasi_total Function-yielding V34() Element of bool [:[:COMPLEX,(X,Y):],(X,Y):]
[:COMPLEX,(X,Y):] is non empty set
[:[:COMPLEX,(X,Y):],(X,Y):] is non empty set
bool [:[:COMPLEX,(X,Y):],(X,Y):] is non empty set
CLSStruct(# (X,Y),(Zero_ ((X,Y),( the carrier of X,Y))),(Add_ ((X,Y),( the carrier of X,Y))),(Mult_ ((X,Y),( the carrier of X,Y))) #) is non empty right_complementable Abelian add-associative right_zeroed strict vector-distributive scalar-distributive scalar-associative scalar-unital CLSStruct
the carrier of (X,Y) is non empty set
bool the carrier of (X,Y) is non empty set
Zero_ ((X,Y),(X,Y)) is Element of (X,Y)
Add_ ((X,Y),(X,Y)) is non empty Relation-like [:(X,Y),(X,Y):] -defined (X,Y) -valued Function-like total quasi_total Element of bool [:[:(X,Y),(X,Y):],(X,Y):]
[:(X,Y),(X,Y):] is non empty set
[:[:(X,Y),(X,Y):],(X,Y):] is non empty set
bool [:[:(X,Y),(X,Y):],(X,Y):] is non empty set
Mult_ ((X,Y),(X,Y)) is non empty Relation-like [:COMPLEX,(X,Y):] -defined (X,Y) -valued Function-like total quasi_total Element of bool [:[:COMPLEX,(X,Y):],(X,Y):]
[:COMPLEX,(X,Y):] is non empty set
[:[:COMPLEX,(X,Y):],(X,Y):] is non empty set
bool [:[:COMPLEX,(X,Y):],(X,Y):] is non empty set
(X,Y) is non empty Relation-like (X,Y) -defined REAL -valued Function-like total quasi_total V132() V133() V134() Element of bool [:(X,Y),REAL:]
[:(X,Y),REAL:] is non empty V132() V133() V134() set
bool [:(X,Y),REAL:] is non empty set
CNORMSTR(# (X,Y),(Zero_ ((X,Y),(X,Y))),(Add_ ((X,Y),(X,Y))),(Mult_ ((X,Y),(X,Y))),(X,Y) #) is non empty strict CNORMSTR
the carrier of (X,Y) is non empty set
[:NAT, the carrier of (X,Y):] is non empty set
bool [:NAT, the carrier of (X,Y):] is non empty set
vseq is non empty Relation-like NAT -defined the carrier of (X,Y) -valued Function-like total quasi_total Element of bool [:NAT, the carrier of (X,Y):]
[:NAT, the carrier of Y:] is non empty set
bool [:NAT, the carrier of Y:] is non empty set
f is Element of the carrier of X
tseq is non empty Relation-like NAT -defined the carrier of Y -valued Function-like total quasi_total Element of bool [:NAT, the carrier of Y:]
lim tseq is Element of the carrier of Y
||.f.|| is complex real ext-real Element of REAL
the U9 of X is non empty Relation-like the carrier of X -defined REAL -valued Function-like total quasi_total V132() V133() V134() Element of bool [: the carrier of X,REAL:]
[: the carrier of X,REAL:] is non empty V132() V133() V134() set
bool [: the carrier of X,REAL:] is non empty set
the U9 of X . f is complex real ext-real Element of REAL
tseq is epsilon-transitive epsilon-connected ordinal natural complex real V108() V109() ext-real V142() V143() V144() V145() V146() V147() Element of NAT
tseq . tseq is Element of the carrier of Y
tseq is epsilon-transitive epsilon-connected ordinal natural complex real V108() V109() ext-real V142() V143() V144() V145() V146() V147() Element of NAT
tseq . tseq is Element of the carrier of Y
(tseq . tseq) - (tseq . tseq) is Element of the carrier of Y
- (tseq . tseq) is Element of the carrier of Y
(tseq . tseq) + (- (tseq . tseq)) is Element of the carrier of Y
the addF of Y is non empty Relation-like [: the carrier of Y, the carrier of Y:] -defined the carrier of Y -valued Function-like total quasi_total Element of bool [:[: the carrier of Y, the carrier of Y:], the carrier of Y:]
[: the carrier of Y, the carrier of Y:] is non empty set
[:[: the carrier of Y, the carrier of Y:], the carrier of Y:] is non empty set
bool [:[: the carrier of Y, the carrier of Y:], the carrier of Y:] is non empty set
the addF of Y . ((tseq . tseq),(- (tseq . tseq))) is Element of the carrier of Y
[(tseq . tseq),(- (tseq . tseq))] is set
the addF of Y . [(tseq . tseq),(- (tseq . tseq))] is set
||.((tseq . tseq) - (tseq . tseq)).|| is complex real ext-real Element of REAL
the U9 of Y is non empty Relation-like the carrier of Y -defined REAL -valued Function-like total quasi_total V132() V133() V134() Element of bool [: the carrier of Y,REAL:]
[: the carrier of Y,REAL:] is non empty V132() V133() V134() set
bool [: the carrier of Y,REAL:] is non empty set
the U9 of Y . ((tseq . tseq) - (tseq . tseq)) is complex real ext-real Element of REAL
vseq . tseq is Relation-like Function-like Element of the carrier of (X,Y)
vseq . tseq is Relation-like Function-like Element of the carrier of (X,Y)
(vseq . tseq) - (vseq . tseq) is Relation-like Function-like Element of the carrier of (X,Y)
- (vseq . tseq) is Relation-like Function-like Element of the carrier of (X,Y)
(vseq . tseq) + (- (vseq . tseq)) is Relation-like Function-like Element of the carrier of (X,Y)
the addF of (X,Y) is non empty Relation-like [: the carrier of (X,Y), the carrier of (X,Y):] -defined the carrier of (X,Y) -valued Function-like total quasi_total Element of bool [:[: the carrier of (X,Y), the carrier of (X,Y):], the carrier of (X,Y):]
[: the carrier of (X,Y), the carrier of (X,Y):] is non empty set
[:[: the carrier of (X,Y), the carrier of (X,Y):], the carrier of (X,Y):] is non empty set
bool [:[: the carrier of (X,Y), the carrier of (X,Y):], the carrier of (X,Y):] is non empty set
the addF of (X,Y) . ((vseq . tseq),(- (vseq . tseq))) is Relation-like Function-like Element of the carrier of (X,Y)
[(vseq . tseq),(- (vseq . tseq))] is set
the addF of (X,Y) . [(vseq . tseq),(- (vseq . tseq))] is set
||.((vseq . tseq) - (vseq . tseq)).|| is complex real ext-real Element of REAL
the U9 of (X,Y) is non empty Relation-like the carrier of (X,Y) -defined REAL -valued Function-like total quasi_total V132() V133() V134() Element of bool [: the carrier of (X,Y),REAL:]
[: the carrier of (X,Y),REAL:] is non empty V132() V133() V134() set
bool [: the carrier of (X,Y),REAL:] is non empty set
the U9 of (X,Y) . ((vseq . tseq) - (vseq . tseq)) is complex real ext-real Element of REAL
||.((vseq . tseq) - (vseq . tseq)).|| * ||.f.|| is complex real ext-real Element of REAL
[: the carrier of X, the carrier of Y:] is non empty set
bool [: the carrier of X, the carrier of Y:] is non empty set
(X,Y,(vseq . tseq)) is non empty Relation-like the carrier of X -defined the carrier of Y -valued Function-like total quasi_total additive (X,Y) (X,Y) Element of bool [: the carrier of X, the carrier of Y:]
(X,Y,(vseq . tseq)) . f is Element of the carrier of Y
(X,Y,(vseq . tseq)) is non empty Relation-like the carrier of X -defined the carrier of Y -valued Function-like total quasi_total additive (X,Y) (X,Y) Element of bool [: the carrier of X, the carrier of Y:]
(X,Y,(vseq . tseq)) . f is Element of the carrier of Y
tv is non empty Relation-like the carrier of X -defined the carrier of Y -valued Function-like total quasi_total additive (X,Y) (X,Y) Element of bool [: the carrier of X, the carrier of Y:]
tv . f is Element of the carrier of Y
tseq is complex real ext-real Element of REAL
0. X is zero Element of the carrier of X
the ZeroF of X is Element of the carrier of X
tseq is empty Function-like functional epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural complex real V108() V109() ext-real non positive non negative V142() V143() V144() V145() V146() V147() V148() Element of NAT
tv is epsilon-transitive epsilon-connected ordinal natural complex real V108() V109() ext-real V142() V143() V144() V145() V146() V147() Element of NAT
e is epsilon-transitive epsilon-connected ordinal natural complex real V108() V109() ext-real V142() V143() V144() V145() V146() V147() Element of NAT
tseq . tv is Element of the carrier of Y
tseq . e is Element of the carrier of Y
(tseq . tv) - (tseq . e) is Element of the carrier of Y
- (tseq . e) is Element of the carrier of Y
(tseq . tv) + (- (tseq . e)) is Element of the carrier of Y
the addF of Y is non empty Relation-like [: the carrier of Y, the carrier of Y:] -defined the carrier of Y -valued Function-like total quasi_total Element of bool [:[: the carrier of Y, the carrier of Y:], the carrier of Y:]
[: the carrier of Y, the carrier of Y:] is non empty set
[:[: the carrier of Y, the carrier of Y:], the carrier of Y:] is non empty set
bool [:[: the carrier of Y, the carrier of Y:], the carrier of Y:] is non empty set
the addF of Y . ((tseq . tv),(- (tseq . e))) is Element of the carrier of Y
[(tseq . tv),(- (tseq . e))] is set
the addF of Y . [(tseq . tv),(- (tseq . e))] is set
||.((tseq . tv) - (tseq . e)).|| is complex real ext-real Element of REAL
the U9 of Y is non empty Relation-like the carrier of Y -defined REAL -valued Function-like total quasi_total V132() V133() V134() Element of bool [: the carrier of Y,REAL:]
[: the carrier of Y,REAL:] is non empty V132() V133() V134() set
bool [: the carrier of Y,REAL:] is non empty set
the U9 of Y . ((tseq . tv) - (tseq . e)) is complex real ext-real Element of REAL
vseq . e is Relation-like Function-like Element of the carrier of (X,Y)
(X,Y,(vseq . e)) is non empty Relation-like the carrier of X -defined the carrier of Y -valued Function-like total quasi_total additive (X,Y) (X,Y) Element of bool [: the carrier of X, the carrier of Y:]
[: the carrier of X, the carrier of Y:] is non empty set
bool [: the carrier of X, the carrier of Y:] is non empty set
(X,Y,(vseq . e)) . f is Element of the carrier of Y
0c is empty Function-like functional epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural complex real ext-real non positive non negative V142() V143() V144() V145() V146() V147() V148() Element of COMPLEX
0c * f is Element of the carrier of X
(X,Y,(vseq . e)) . (0c * f) is Element of the carrier of Y
0c * ((X,Y,(vseq . e)) . f) is Element of the carrier of Y
0. Y is zero Element of the carrier of Y
the ZeroF of Y is Element of the carrier of Y
vseq . tv is Relation-like Function-like Element of the carrier of (X,Y)
(X,Y,(vseq . tv)) is non empty Relation-like the carrier of X -defined the carrier of Y -valued Function-like total quasi_total additive (X,Y) (X,Y) Element of bool [: the carrier of X, the carrier of Y:]
(X,Y,(vseq . tv)) . f is Element of the carrier of Y
(X,Y,(vseq . tv)) . (0c * f) is Element of the carrier of Y
0c * ((X,Y,(vseq . tv)) . f) is Element of the carrier of Y
||.(0. Y).|| is complex real ext-real Element of REAL
the U9 of Y . (0. Y) is complex real ext-real Element of REAL
0. X is zero Element of the carrier of X
the ZeroF of X is Element of the carrier of X
tseq / ||.f.|| is complex real ext-real Element of REAL
tseq is epsilon-transitive epsilon-connected ordinal natural complex real V108() V109() ext-real V142() V143() V144() V145() V146() V147() Element of NAT
tv is epsilon-transitive epsilon-connected ordinal natural complex real V108() V109() ext-real V142() V143() V144() V145() V146() V147() Element of NAT
e is epsilon-transitive epsilon-connected ordinal natural complex real V108() V109() ext-real V142() V143() V144() V145() V146() V147() Element of NAT
m is epsilon-transitive epsilon-connected ordinal natural complex real V108() V109() ext-real V142() V143() V144() V145() V146() V147() Element of NAT
tseq . e is Element of the carrier of Y
tseq . m is Element of the carrier of Y
(tseq . e) - (tseq . m) is Element of the carrier of Y
- (tseq . m) is Element of the carrier of Y
(tseq . e) + (- (tseq . m)) is Element of the carrier of Y
the addF of Y is non empty Relation-like [: the carrier of Y, the carrier of Y:] -defined the carrier of Y -valued Function-like total quasi_total Element of bool [:[: the carrier of Y, the carrier of Y:], the carrier of Y:]
[: the carrier of Y, the carrier of Y:] is non empty set
[:[: the carrier of Y, the carrier of Y:], the carrier of Y:] is non empty set
bool [:[: the carrier of Y, the carrier of Y:], the carrier of Y:] is non empty set
the addF of Y . ((tseq . e),(- (tseq . m))) is Element of the carrier of Y
[(tseq . e),(- (tseq . m))] is set
the addF of Y . [(tseq . e),(- (tseq . m))] is set
||.((tseq . e) - (tseq . m)).|| is complex real ext-real Element of REAL
the U9 of Y is non empty Relation-like the carrier of Y -defined REAL -valued Function-like total quasi_total V132() V133() V134() Element of bool [: the carrier of Y,REAL:]
[: the carrier of Y,REAL:] is non empty V132() V133() V134() set
bool [: the carrier of Y,REAL:] is non empty set
the U9 of Y . ((tseq . e) - (tseq . m)) is complex real ext-real Element of REAL
vseq . e is Relation-like Function-like Element of the carrier of (X,Y)
vseq . m is Relation-like Function-like Element of the carrier of (X,Y)
(vseq . e) - (vseq . m) is Relation-like Function-like Element of the carrier of (X,Y)
- (vseq . m) is Relation-like Function-like Element of the carrier of (X,Y)
(vseq . e) + (- (vseq . m)) is Relation-like Function-like Element of the carrier of (X,Y)
the addF of (X,Y) is non empty Relation-like [: the carrier of (X,Y), the carrier of (X,Y):] -defined the carrier of (X,Y) -valued Function-like total quasi_total Element of bool [:[: the carrier of (X,Y), the carrier of (X,Y):], the carrier of (X,Y):]
[: the carrier of (X,Y), the carrier of (X,Y):] is non empty set
[:[: the carrier of (X,Y), the carrier of (X,Y):], the carrier of (X,Y):] is non empty set
bool [:[: the carrier of (X,Y), the carrier of (X,Y):], the carrier of (X,Y):] is non empty set
the addF of (X,Y) . ((vseq . e),(- (vseq . m))) is Relation-like Function-like Element of the carrier of (X,Y)
[(vseq . e),(- (vseq . m))] is set
the addF of (X,Y) . [(vseq . e),(- (vseq . m))] is set
||.((vseq . e) - (vseq . m)).|| is complex real ext-real Element of REAL
the U9 of (X,Y) is non empty Relation-like the carrier of (X,Y) -defined REAL -valued Function-like total quasi_total V132() V133() V134() Element of bool [: the carrier of (X,Y),REAL:]
[: the carrier of (X,Y),REAL:] is non empty V132() V133() V134() set
bool [: the carrier of (X,Y),REAL:] is non empty set
the U9 of (X,Y) . ((vseq . e) - (vseq . m)) is complex real ext-real Element of REAL
(tseq / ||.f.||) * ||.f.|| is complex real ext-real Element of REAL
||.((vseq . e) - (vseq . m)).|| * ||.f.|| is complex real ext-real Element of REAL
||.f.|| " is complex real ext-real Element of REAL
tseq * (||.f.|| ") is complex real ext-real Element of REAL
(tseq * (||.f.|| ")) * ||.f.|| is complex real ext-real Element of REAL
(||.f.|| ") * ||.f.|| is complex real ext-real Element of REAL
tseq * ((||.f.|| ") * ||.f.||) is complex real ext-real Element of REAL
tseq * 1 is complex real ext-real Element of REAL
0. X is zero Element of the carrier of X
the ZeroF of X is Element of the carrier of X
0. X is zero Element of the carrier of X
the ZeroF of X is Element of the carrier of X
tseq is empty Function-like functional epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural complex real V108() V109() ext-real non positive non negative V142() V143() V144() V145() V146() V147() V148() Element of NAT
tv is epsilon-transitive epsilon-connected ordinal natural complex real V108() V109() ext-real V142() V143() V144() V145() V146() V147() Element of NAT
[: the carrier of X, the carrier of Y:] is non empty set
bool [: the carrier of X, the carrier of Y:] is non empty set
f is non empty Relation-like the carrier of X -defined the carrier of Y -valued Function-like total quasi_total Element of bool [: the carrier of X, the carrier of Y:]
tseq is Element of the carrier of X
tseq is non empty Relation-like the carrier of X -defined the carrier of Y -valued Function-like total quasi_total Element of bool [: the carrier of X, the carrier of Y:]
tseq . tseq is Element of the carrier of Y
tv is non empty Relation-like NAT -defined the carrier of Y -valued Function-like total quasi_total Element of bool [:NAT, the carrier of Y:]
lim tv is Element of the carrier of Y
tseq is Element of the carrier of X
tseq + tseq is Element of the carrier of X
the addF of X is non empty Relation-like [: the carrier of X, the carrier of X:] -defined the carrier of X -valued Function-like total quasi_total Element of bool [:[: the carrier of X, the carrier of X:], the carrier of X:]
[: the carrier of X, the carrier of X:] is non empty set
[:[: the carrier of X, the carrier of X:], the carrier of X:] is non empty set
bool [:[: the carrier of X, the carrier of X:], the carrier of X:] is non empty set
the addF of X . (tseq,tseq) is Element of the carrier of X
[tseq,tseq] is set
the addF of X . [tseq,tseq] is set
tseq . (tseq + tseq) is Element of the carrier of Y
e is non empty Relation-like NAT -defined the carrier of Y -valued Function-like total quasi_total Element of bool [:NAT, the carrier of Y:]
lim e is Element of the carrier of Y
tseq . tseq is Element of the carrier of Y
m is non empty Relation-like NAT -defined the carrier of Y -valued Function-like total quasi_total Element of bool [:NAT, the carrier of Y:]
lim m is Element of the carrier of Y
n is epsilon-transitive epsilon-connected ordinal natural complex real V108() V109() ext-real V142() V143() V144() V145() V146() V147() Element of NAT
e . n is Element of the carrier of Y
vseq . n is Relation-like Function-like Element of the carrier of (X,Y)
(X,Y,(vseq . n)) is non empty Relation-like the carrier of X -defined the carrier of Y -valued Function-like total quasi_total additive (X,Y) (X,Y) Element of bool [: the carrier of X, the carrier of Y:]
(X,Y,(vseq . n)) . (tseq + tseq) is Element of the carrier of Y
(X,Y,(vseq . n)) . tseq is Element of the carrier of Y
(X,Y,(vseq . n)) . tseq is Element of the carrier of Y
((X,Y,(vseq . n)) . tseq) + ((X,Y,(vseq . n)) . tseq) is Element of the carrier of Y
the addF of Y is non empty Relation-like [: the carrier of Y, the carrier of Y:] -defined the carrier of Y -valued Function-like total quasi_total Element of bool [:[: the carrier of Y, the carrier of Y:], the carrier of Y:]
[: the carrier of Y, the carrier of Y:] is non empty set
[:[: the carrier of Y, the carrier of Y:], the carrier of Y:] is non empty set
bool [:[: the carrier of Y, the carrier of Y:], the carrier of Y:] is non empty set
the addF of Y . (((X,Y,(vseq . n)) . tseq),((X,Y,(vseq . n)) . tseq)) is Element of the carrier of Y
[((X,Y,(vseq . n)) . tseq),((X,Y,(vseq . n)) . tseq)] is set
the addF of Y . [((X,Y,(vseq . n)) . tseq),((X,Y,(vseq . n)) . tseq)] is set
tv . n is Element of the carrier of Y
(tv . n) + ((X,Y,(vseq . n)) . tseq) is Element of the carrier of Y
the addF of Y . ((tv . n),((X,Y,(vseq . n)) . tseq)) is Element of the carrier of Y
[(tv . n),((X,Y,(vseq . n)) . tseq)] is set
the addF of Y . [(tv . n),((X,Y,(vseq . n)) . tseq)] is set
m . n is Element of the carrier of Y
(tv . n) + (m . n) is Element of the carrier of Y
the addF of Y . ((tv . n),(m . n)) is Element of the carrier of Y
[(tv . n),(m . n)] is set
the addF of Y . [(tv . n),(m . n)] is set
tv + m is non empty Relation-like NAT -defined the carrier of Y -valued Function-like total quasi_total Element of bool [:NAT, the carrier of Y:]
(tseq . tseq) + (tseq . tseq) is Element of the carrier of Y
the addF of Y . ((tseq . tseq),(tseq . tseq)) is Element of the carrier of Y
[(tseq . tseq),(tseq . tseq)] is set
the addF of Y . [(tseq . tseq),(tseq . tseq)] is set
tseq is Element of the carrier of X
tseq . tseq is Element of the carrier of Y
tv is non empty Relation-like NAT -defined the carrier of Y -valued Function-like total quasi_total Element of bool [:NAT, the carrier of Y:]
lim tv is Element of the carrier of Y
tseq is complex set
tseq * tseq is Element of the carrier of X
tseq . (tseq * tseq) is Element of the carrier of Y
e is non empty Relation-like NAT -defined the carrier of Y -valued Function-like total quasi_total Element of bool [:NAT, the carrier of Y:]
lim e is Element of the carrier of Y
m is epsilon-transitive epsilon-connected ordinal natural complex real V108() V109() ext-real V142() V143() V144() V145() V146() V147() Element of NAT
e . m is Element of the carrier of Y
vseq . m is Relation-like Function-like Element of the carrier of (X,Y)
(X,Y,(vseq . m)) is non empty Relation-like the carrier of X -defined the carrier of Y -valued Function-like total quasi_total additive (X,Y) (X,Y) Element of bool [: the carrier of X, the carrier of Y:]
(X,Y,(vseq . m)) . (tseq * tseq) is Element of the carrier of Y
(X,Y,(vseq . m)) . tseq is Element of the carrier of Y
tseq * ((X,Y,(vseq . m)) . tseq) is Element of the carrier of Y
tv . m is Element of the carrier of Y
tseq * (tv . m) is Element of the carrier of Y
tseq * tv is non empty Relation-like NAT -defined the carrier of Y -valued Function-like total quasi_total Element of bool [:NAT, the carrier of Y:]
tseq * (tseq . tseq) is Element of the carrier of Y
tseq is complex real ext-real set
tv is complex real ext-real Element of REAL
e is epsilon-transitive epsilon-connected ordinal natural complex real V108() V109() ext-real V142() V143() V144() V145() V146() V147() Element of NAT
m is epsilon-transitive epsilon-connected ordinal natural complex real V108() V109() ext-real V142() V143() V144() V145() V146() V147() Element of NAT
n is epsilon-transitive epsilon-connected ordinal natural complex real V108() V109() ext-real V142() V143() V144() V145() V146() V147() Element of NAT
vseq . n is Relation-like Function-like Element of the carrier of (X,Y)
vseq . m is Relation-like Function-like Element of the carrier of (X,Y)
(vseq . n) - (vseq . m) is Relation-like Function-like Element of the carrier of (X,Y)
- (vseq . m) is Relation-like Function-like Element of the carrier of (X,Y)
(vseq . n) + (- (vseq . m)) is Relation-like Function-like Element of the carrier of (X,Y)
the addF of (X,Y) is non empty Relation-like [: the carrier of (X,Y), the carrier of (X,Y):] -defined the carrier of (X,Y) -valued Function-like total quasi_total Element of bool [:[: the carrier of (X,Y), the carrier of (X,Y):], the carrier of (X,Y):]
[: the carrier of (X,Y), the carrier of (X,Y):] is non empty set
[:[: the carrier of (X,Y), the carrier of (X,Y):], the carrier of (X,Y):] is non empty set
bool [:[: the carrier of (X,Y), the carrier of (X,Y):], the carrier of (X,Y):] is non empty set
the addF of (X,Y) . ((vseq . n),(- (vseq . m))) is Relation-like Function-like Element of the carrier of (X,Y)
[(vseq . n),(- (vseq . m))] is set
the addF of (X,Y) . [(vseq . n),(- (vseq . m))] is set
||.((vseq . n) - (vseq . m)).|| is complex real ext-real Element of REAL
the U9 of (X,Y) is non empty Relation-like the carrier of (X,Y) -defined REAL -valued Function-like total quasi_total V132() V133() V134() Element of bool [: the carrier of (X,Y),REAL:]
[: the carrier of (X,Y),REAL:] is non empty V132() V133() V134() set
bool [: the carrier of (X,Y),REAL:] is non empty set
the U9 of (X,Y) . ((vseq . n) - (vseq . m)) is complex real ext-real Element of REAL
||.(vseq . n).|| is complex real ext-real Element of REAL
the U9 of (X,Y) . (vseq . n) is complex real ext-real Element of REAL
||.vseq.|| is non empty Relation-like NAT -defined REAL -valued Function-like total quasi_total V132() V133() V134() Element of bool [:NAT,REAL:]
||.vseq.|| . n is complex real ext-real Element of REAL
||.(vseq . m).|| is complex real ext-real Element of REAL
the U9 of (X,Y) . (vseq . m) is complex real ext-real Element of REAL
||.vseq.|| . m is complex real ext-real Element of REAL
||.(vseq . n).|| - ||.(vseq . m).|| is complex real ext-real Element of REAL
abs (||.(vseq . n).|| - ||.(vseq . m).||) is complex real ext-real Element of REAL
(||.vseq.|| . n) - (||.vseq.|| . m) is complex real ext-real Element of REAL
abs ((||.vseq.|| . n) - (||.vseq.|| . m)) is complex real ext-real Element of REAL
n is epsilon-transitive epsilon-connected ordinal natural complex real V108() V109() ext-real V142() V143() V144() V145() V146() V147() Element of NAT
||.vseq.|| . n is complex real ext-real Element of REAL
(||.vseq.|| . n) - (||.vseq.|| . m) is complex real ext-real Element of REAL
abs ((||.vseq.|| . n) - (||.vseq.|| . m)) is complex real ext-real Element of REAL
tseq is non empty Relation-like the carrier of X -defined the carrier of Y -valued Function-like total quasi_total additive (X,Y) Element of bool [: the carrier of X, the carrier of Y:]
lim ||.vseq.|| is complex real ext-real Element of REAL
tseq is Element of the carrier of X
tseq . tseq is Element of the carrier of Y
tv is non empty Relation-like NAT -defined the carrier of Y -valued Function-like total quasi_total Element of bool [:NAT, the carrier of Y:]
lim tv is Element of the carrier of Y
||.(tseq . tseq).|| is complex real ext-real Element of REAL
the U9 of Y is non empty Relation-like the carrier of Y -defined REAL -valued Function-like total quasi_total V132() V133() V134() Element of bool [: the carrier of Y,REAL:]
[: the carrier of Y,REAL:] is non empty V132() V133() V134() set
bool [: the carrier of Y,REAL:] is non empty set
the U9 of Y . (tseq . tseq) is complex real ext-real Element of REAL
||.tv.|| is non empty Relation-like NAT -defined REAL -valued Function-like total quasi_total V132() V133() V134() Element of bool [:NAT,REAL:]
lim ||.tv.|| is complex real ext-real Element of REAL
||.tseq.|| is complex real ext-real Element of REAL
the U9 of X is non empty Relation-like the carrier of X -defined REAL -valued Function-like total quasi_total V132() V133() V134() Element of bool [: the carrier of X,REAL:]
[: the carrier of X,REAL:] is non empty V132() V133() V134() set
bool [: the carrier of X,REAL:] is non empty set
the U9 of X . tseq is complex real ext-real Element of REAL
e is epsilon-transitive epsilon-connected ordinal natural complex real V108() V109() ext-real V142() V143() V144() V145() V146() V147() Element of NAT
tv . e is Element of the carrier of Y
||.(tv . e).|| is complex real ext-real Element of REAL
the U9 of Y . (tv . e) is complex real ext-real Element of REAL
vseq . e is Relation-like Function-like Element of the carrier of (X,Y)
||.(vseq . e).|| is complex real ext-real Element of REAL
the U9 of (X,Y) . (vseq . e) is complex real ext-real Element of REAL
||.(vseq . e).|| * ||.tseq.|| is complex real ext-real Element of REAL
(X,Y,(vseq . e)) is non empty Relation-like the carrier of X -defined the carrier of Y -valued Function-like total quasi_total additive (X,Y) (X,Y) Element of bool [: the carrier of X, the carrier of Y:]
(X,Y,(vseq . e)) . tseq is Element of the carrier of Y
||.tseq.|| (#) ||.vseq.|| is non empty Relation-like NAT -defined REAL -valued Function-like total quasi_total V132() V133() V134() Element of bool [:NAT,REAL:]
e is epsilon-transitive epsilon-connected ordinal natural complex real V108() V109() ext-real V142() V143() V144() V145() V146() V147() Element of NAT
||.tv.|| . e is complex real ext-real Element of REAL
(||.tseq.|| (#) ||.vseq.||) . e is complex real ext-real Element of REAL
tv . e is Element of the carrier of Y
||.(tv . e).|| is complex real ext-real Element of REAL
the U9 of Y . (tv . e) is complex real ext-real Element of REAL
vseq . e is Relation-like Function-like Element of the carrier of (X,Y)
||.(vseq . e).|| is complex real ext-real Element of REAL
the U9 of (X,Y) . (vseq . e) is complex real ext-real Element of REAL
||.vseq.|| . e is complex real ext-real Element of REAL
||.(vseq . e).|| * ||.tseq.|| is complex real ext-real Element of REAL
lim (||.tseq.|| (#) ||.vseq.||) is complex real ext-real Element of REAL
(lim ||.vseq.||) * ||.tseq.|| is complex real ext-real Element of REAL
tseq is epsilon-transitive epsilon-connected ordinal natural complex real V108() V109() ext-real V142() V143() V144() V145() V146() V147() Element of NAT
vseq . tseq is Relation-like Function-like Element of the carrier of (X,Y)
||.(vseq . tseq).|| is complex real ext-real Element of REAL
the U9 of (X,Y) . (vseq . tseq) is complex real ext-real Element of REAL
||.vseq.|| . tseq is complex real ext-real Element of REAL
tseq is Element of the carrier of X
tseq . tseq is Element of the carrier of Y
||.(tseq . tseq).|| is complex real ext-real Element of REAL
the U9 of Y . (tseq . tseq) is complex real ext-real Element of REAL
||.tseq.|| is complex real ext-real Element of REAL
the U9 of X . tseq is complex real ext-real Element of REAL
(lim ||.vseq.||) * ||.tseq.|| is complex real ext-real Element of REAL
tseq is complex real ext-real Element of REAL
tv is epsilon-transitive epsilon-connected ordinal natural complex real V108() V109() ext-real V142() V143() V144() V145() V146() V147() Element of NAT
e is epsilon-transitive epsilon-connected ordinal natural complex real V108() V109() ext-real V142() V143() V144() V145() V146() V147() Element of NAT
m is Element of the carrier of X
tseq . m is Element of the carrier of Y
n is non empty Relation-like NAT -defined the carrier of Y -valued Function-like total quasi_total Element of bool [:NAT, the carrier of Y:]
lim n is Element of the carrier of Y
||.m.|| is complex real ext-real Element of REAL
the U9 of X is non empty Relation-like the carrier of X -defined REAL -valued Function-like total quasi_total V132() V133() V134() Element of bool [: the carrier of X,REAL:]
[: the carrier of X,REAL:] is non empty V132() V133() V134() set
bool [: the carrier of X,REAL:] is non empty set
the U9 of X . m is complex real ext-real Element of REAL
n is epsilon-transitive epsilon-connected ordinal natural complex real V108() V109() ext-real V142() V143() V144() V145() V146() V147() Element of NAT
n . n is Element of the carrier of Y
h1 is epsilon-transitive epsilon-connected ordinal natural complex real V108() V109() ext-real V142() V143() V144() V145() V146() V147() Element of NAT
n . h1 is Element of the carrier of Y
(n . n) - (n . h1) is Element of the carrier of Y
- (n . h1) is Element of the carrier of Y
(n . n) + (- (n . h1)) is Element of the carrier of Y
the addF of Y . ((n . n),(- (n . h1))) is Element of the carrier of Y
[(n . n),(- (n . h1))] is set
the addF of Y . [(n . n),(- (n . h1))] is set
||.((n . n) - (n . h1)).|| is complex real ext-real Element of REAL
the U9 of Y is non empty Relation-like the carrier of Y -defined REAL -valued Function-like total quasi_total V132() V133() V134() Element of bool [: the carrier of Y,REAL:]
[: the carrier of Y,REAL:] is non empty V132() V133() V134() set
bool [: the carrier of Y,REAL:] is non empty set
the U9 of Y . ((n . n) - (n . h1)) is complex real ext-real Element of REAL
vseq . n is Relation-like Function-like Element of the carrier of (X,Y)
vseq . h1 is Relation-like Function-like Element of the carrier of (X,Y)
(vseq . n) - (vseq . h1) is Relation-like Function-like Element of the carrier of (X,Y)
- (vseq . h1) is Relation-like Function-like Element of the carrier of (X,Y)
(vseq . n) + (- (vseq . h1)) is Relation-like Function-like Element of the carrier of (X,Y)
the addF of (X,Y) . ((vseq . n),(- (vseq . h1))) is Relation-like Function-like Element of the carrier of (X,Y)
[(vseq . n),(- (vseq . h1))] is set
the addF of (X,Y) . [(vseq . n),(- (vseq . h1))] is set
||.((vseq . n) - (vseq . h1)).|| is complex real ext-real Element of REAL
the U9 of (X,Y) . ((vseq . n) - (vseq . h1)) is complex real ext-real Element of REAL
||.((vseq . n) - (vseq . h1)).|| * ||.m.|| is complex real ext-real Element of REAL
(X,Y,(vseq . h1)) is non empty Relation-like the carrier of X -defined the carrier of Y -valued Function-like total quasi_total additive (X,Y) (X,Y) Element of bool [: the carrier of X, the carrier of Y:]
(X,Y,(vseq . h1)) . m is Element of the carrier of Y
(X,Y,(vseq . n)) is non empty Relation-like the carrier of X -defined the carrier of Y -valued Function-like total quasi_total additive (X,Y) (X,Y) Element of bool [: the carrier of X, the carrier of Y:]
(X,Y,(vseq . n)) . m is Element of the carrier of Y
f1 is non empty Relation-like the carrier of X -defined the carrier of Y -valued Function-like total quasi_total additive (X,Y) (X,Y) Element of bool [: the carrier of X, the carrier of Y:]
f1 . m is Element of the carrier of Y
n . e is Element of the carrier of Y
tseq * ||.m.|| is complex real ext-real Element of REAL
n is epsilon-transitive epsilon-connected ordinal natural complex real V108() V109() ext-real V142() V143() V144() V145() V146() V147() Element of NAT
n . n is Element of the carrier of Y
(n . e) - (n . n) is Element of the carrier of Y
- (n . n) is Element of the carrier of Y
(n . e) + (- (n . n)) is Element of the carrier of Y
the addF of Y . ((n . e),(- (n . n))) is Element of the carrier of Y
[(n . e),(- (n . n))] is set
the addF of Y . [(n . e),(- (n . n))] is set
||.((n . e) - (n . n)).|| is complex real ext-real Element of REAL
the U9 of Y is non empty Relation-like the carrier of Y -defined REAL -valued Function-like total quasi_total V132() V133() V134() Element of bool [: the carrier of Y,REAL:]
[: the carrier of Y,REAL:] is non empty V132() V133() V134() set
bool [: the carrier of Y,REAL:] is non empty set
the U9 of Y . ((n . e) - (n . n)) is complex real ext-real Element of REAL
vseq . e is Relation-like Function-like Element of the carrier of (X,Y)
vseq . n is Relation-like Function-like Element of the carrier of (X,Y)
(vseq . e) - (vseq . n) is Relation-like Function-like Element of the carrier of (X,Y)
- (vseq . n) is Relation-like Function-like Element of the carrier of (X,Y)
(vseq . e) + (- (vseq . n)) is Relation-like Function-like Element of the carrier of (X,Y)
the addF of (X,Y) . ((vseq . e),(- (vseq . n))) is Relation-like Function-like Element of the carrier of (X,Y)
[(vseq . e),(- (vseq . n))] is set
the addF of (X,Y) . [(vseq . e),(- (vseq . n))] is set
||.((vseq . e) - (vseq . n)).|| is complex real ext-real Element of REAL
the U9 of (X,Y) . ((vseq . e) - (vseq . n)) is complex real ext-real Element of REAL
||.((vseq . e) - (vseq . n)).|| * ||.m.|| is complex real ext-real Element of REAL
(n . e) - (tseq . m) is Element of the carrier of Y
- (tseq . m) is Element of the carrier of Y
(n . e) + (- (tseq . m)) is Element of the carrier of Y
the addF of Y . ((n . e),(- (tseq . m))) is Element of the carrier of Y
[(n . e),(- (tseq . m))] is set
the addF of Y . [(n . e),(- (tseq . m))] is set
||.((n . e) - (tseq . m)).|| is complex real ext-real Element of REAL
the U9 of Y is non empty Relation-like the carrier of Y -defined REAL -valued Function-like total quasi_total V132() V133() V134() Element of bool [: the carrier of Y,REAL:]
[: the carrier of Y,REAL:] is non empty V132() V133() V134() set
bool [: the carrier of Y,REAL:] is non empty set
the U9 of Y . ((n . e) - (tseq . m)) is complex real ext-real Element of REAL
n is non empty Relation-like NAT -defined REAL -valued Function-like total quasi_total V132() V133() V134() Element of bool [:NAT,REAL:]
h1 is set
n . h1 is complex real ext-real Element of REAL
f1 is epsilon-transitive epsilon-connected ordinal natural complex real V108() V109() ext-real V142() V143() V144() V145() V146() V147() Element of NAT
n . f1 is Element of the carrier of Y
(n . f1) - (n . e) is Element of the carrier of Y
- (n . e) is Element of the carrier of Y
(n . f1) + (- (n . e)) is Element of the carrier of Y
the addF of Y . ((n . f1),(- (n . e))) is Element of the carrier of Y
[(n . f1),(- (n . e))] is set
the addF of Y . [(n . f1),(- (n . e))] is set
||.((n . f1) - (n . e)).|| is complex real ext-real Element of REAL
the U9 of Y . ((n . f1) - (n . e)) is complex real ext-real Element of REAL
n - (n . e) is non empty Relation-like NAT -defined the carrier of Y -valued Function-like total quasi_total Element of bool [:NAT, the carrier of Y:]
(n - (n . e)) . f1 is Element of the carrier of Y
||.((n - (n . e)) . f1).|| is complex real ext-real Element of REAL
the U9 of Y . ((n - (n . e)) . f1) is complex real ext-real Element of REAL
||.(n - (n . e)).|| is non empty Relation-like NAT -defined REAL -valued Function-like total quasi_total V132() V133() V134() Element of bool [:NAT,REAL:]
||.(n - (n . e)).|| . h1 is complex real ext-real Element of REAL
lim (n - (n . e)) is Element of the carrier of Y
(tseq . m) - (n . e) is Element of the carrier of Y
(tseq . m) + (- (n . e)) is Element of the carrier of Y
the addF of Y . ((tseq . m),(- (n . e))) is Element of the carrier of Y
[(tseq . m),(- (n . e))] is set
the addF of Y . [(tseq . m),(- (n . e))] is set
lim n is complex real ext-real Element of REAL
||.((tseq . m) - (n . e)).|| is complex real ext-real Element of REAL
the U9 of Y . ((tseq . m) - (n . e)) is complex real ext-real Element of REAL
h1 is epsilon-transitive epsilon-connected ordinal natural complex real V108() V109() ext-real V142() V143() V144() V145() V146() V147() Element of NAT
n . h1 is complex real ext-real Element of REAL
n . h1 is Element of the carrier of Y
(n . h1) - (n . e) is Element of the carrier of Y
(n . h1) + (- (n . e)) is Element of the carrier of Y
the addF of Y . ((n . h1),(- (n . e))) is Element of the carrier of Y
[(n . h1),(- (n . e))] is set
the addF of Y . [(n . h1),(- (n . e))] is set
||.((n . h1) - (n . e)).|| is complex real ext-real Element of REAL
the U9 of Y . ((n . h1) - (n . e)) is complex real ext-real Element of REAL
(n . e) - (n . h1) is Element of the carrier of Y
- (n . h1) is Element of the carrier of Y
(n . e) + (- (n . h1)) is Element of the carrier of Y
the addF of Y . ((n . e),(- (n . h1))) is Element of the carrier of Y
[(n . e),(- (n . h1))] is set
the addF of Y . [(n . e),(- (n . h1))] is set
||.((n . e) - (n . h1)).|| is complex real ext-real Element of REAL
the U9 of Y . ((n . e) - (n . h1)) is complex real ext-real Element of REAL
vseq . e is Relation-like Function-like Element of the carrier of (X,Y)
(X,Y,(vseq . e)) is non empty Relation-like the carrier of X -defined the carrier of Y -valued Function-like total quasi_total additive (X,Y) (X,Y) Element of bool [: the carrier of X, the carrier of Y:]
(X,Y,(vseq . e)) . m is Element of the carrier of Y
((X,Y,(vseq . e)) . m) - (tseq . m) is Element of the carrier of Y
((X,Y,(vseq . e)) . m) + (- (tseq . m)) is Element of the carrier of Y
the addF of Y . (((X,Y,(vseq . e)) . m),(- (tseq . m))) is Element of the carrier of Y
[((X,Y,(vseq . e)) . m),(- (tseq . m))] is set
the addF of Y . [((X,Y,(vseq . e)) . m),(- (tseq . m))] is set
||.(((X,Y,(vseq . e)) . m) - (tseq . m)).|| is complex real ext-real Element of REAL
the U9 of Y . (((X,Y,(vseq . e)) . m) - (tseq . m)) is complex real ext-real Element of REAL
m is Element of the carrier of X
(X,Y,(vseq . e)) . m is Element of the carrier of Y
tseq . m is Element of the carrier of Y
((X,Y,(vseq . e)) . m) - (tseq . m) is Element of the carrier of Y
- (tseq . m) is Element of the carrier of Y
((X,Y,(vseq . e)) . m) + (- (tseq . m)) is Element of the carrier of Y
the addF of Y . (((X,Y,(vseq . e)) . m),(- (tseq . m))) is Element of the carrier of Y
[((X,Y,(vseq . e)) . m),(- (tseq . m))] is set
the addF of Y . [((X,Y,(vseq . e)) . m),(- (tseq . m))] is set
||.(((X,Y,(vseq . e)) . m) - (tseq . m)).|| is complex real ext-real Element of REAL
the U9 of Y . (((X,Y,(vseq . e)) . m) - (tseq . m)) is complex real ext-real Element of REAL
||.m.|| is complex real ext-real Element of REAL
the U9 of X . m is complex real ext-real Element of REAL
tseq * ||.m.|| is complex real ext-real Element of REAL
e is epsilon-transitive epsilon-connected ordinal natural complex real V108() V109() ext-real V142() V143() V144() V145() V146() V147() Element of NAT
vseq . e is Relation-like Function-like Element of the carrier of (X,Y)
(X,Y,(vseq . e)) is non empty Relation-like the carrier of X -defined the carrier of Y -valued Function-like total quasi_total additive (X,Y) (X,Y) Element of bool [: the carrier of X, the carrier of Y:]
m is Element of the carrier of X
(X,Y,(vseq . e)) . m is Element of the carrier of Y
tseq . m is Element of the carrier of Y
((X,Y,(vseq . e)) . m) - (tseq . m) is Element of the carrier of Y
- (tseq . m) is Element of the carrier of Y
((X,Y,(vseq . e)) . m) + (- (tseq . m)) is Element of the carrier of Y
the addF of Y . (((X,Y,(vseq . e)) . m),(- (tseq . m))) is Element of the carrier of Y
[((X,Y,(vseq . e)) . m),(- (tseq . m))] is set
the addF of Y . [((X,Y,(vseq . e)) . m),(- (tseq . m))] is set
||.(((X,Y,(vseq . e)) . m) - (tseq . m)).|| is complex real ext-real Element of REAL
the U9 of Y . (((X,Y,(vseq . e)) . m) - (tseq . m)) is complex real ext-real Element of REAL
||.m.|| is complex real ext-real Element of REAL
the U9 of X . m is complex real ext-real Element of REAL
tseq * ||.m.|| is complex real ext-real Element of REAL
tseq is non empty Relation-like the carrier of X -defined the carrier of Y -valued Function-like total quasi_total additive (X,Y) (X,Y) Element of bool [: the carrier of X, the carrier of Y:]
tv is Relation-like Function-like Element of the carrier of (X,Y)
e is complex real ext-real Element of REAL
m is epsilon-transitive epsilon-connected ordinal natural complex real V108() V109() ext-real V142() V143() V144() V145() V146() V147() Element of NAT
n is epsilon-transitive epsilon-connected ordinal natural complex real V108() V109() ext-real V142() V143() V144() V145() V146() V147() Element of NAT
vseq . n is Relation-like Function-like Element of the carrier of (X,Y)
(vseq . n) - tv is Relation-like Function-like Element of the carrier of (X,Y)
- tv is Relation-like Function-like Element of the carrier of (X,Y)
(vseq . n) + (- tv) is Relation-like Function-like Element of the carrier of (X,Y)
the addF of (X,Y) . ((vseq . n),(- tv)) is Relation-like Function-like Element of the carrier of (X,Y)
[(vseq . n),(- tv)] is set
the addF of (X,Y) . [(vseq . n),(- tv)] is set
(X,Y,(vseq . n)) is non empty Relation-like the carrier of X -defined the carrier of Y -valued Function-like total quasi_total additive (X,Y) (X,Y) Element of bool [: the carrier of X, the carrier of Y:]
r is Element of the carrier of X
||.r.|| is complex real ext-real Element of REAL
the U9 of X is non empty Relation-like the carrier of X -defined REAL -valued Function-like total quasi_total V132() V133() V134() Element of bool [: the carrier of X,REAL:]
[: the carrier of X,REAL:] is non empty V132() V133() V134() set
bool [: the carrier of X,REAL:] is non empty set
the U9 of X . r is complex real ext-real Element of REAL
e * ||.r.|| is complex real ext-real Element of REAL
e * 1 is complex real ext-real Element of REAL
(X,Y,(vseq . n)) . r is Element of the carrier of Y
tseq . r is Element of the carrier of Y
((X,Y,(vseq . n)) . r) - (tseq . r) is Element of the carrier of Y
- (tseq . r) is Element of the carrier of Y
((X,Y,(vseq . n)) . r) + (- (tseq . r)) is Element of the carrier of Y
the addF of Y . (((X,Y,(vseq . n)) . r),(- (tseq . r))) is Element of the carrier of Y
[((X,Y,(vseq . n)) . r),(- (tseq . r))] is set
the addF of Y . [((X,Y,(vseq . n)) . r),(- (tseq . r))] is set
||.(((X,Y,(vseq . n)) . r) - (tseq . r)).|| is complex real ext-real Element of REAL
the U9 of Y is non empty Relation-like the carrier of Y -defined REAL -valued Function-like total quasi_total V132() V133() V134() Element of bool [: the carrier of Y,REAL:]
[: the carrier of Y,REAL:] is non empty V132() V133() V134() set
bool [: the carrier of Y,REAL:] is non empty set
the U9 of Y . (((X,Y,(vseq . n)) . r) - (tseq . r)) is complex real ext-real Element of REAL
h1 is non empty Relation-like the carrier of X -defined the carrier of Y -valued Function-like total quasi_total additive (X,Y) (X,Y) Element of bool [: the carrier of X, the carrier of Y:]
h1 . r is Element of the carrier of Y
||.(h1 . r).|| is complex real ext-real Element of REAL
the U9 of Y . (h1 . r) is complex real ext-real Element of REAL
r is complex real ext-real Element of REAL
(X,Y,h1) is non empty V142() V143() V144() Element of bool REAL
{ ||.(h1 . b₁).|| where b₁ is Element of the carrier of X : ||.b₁.|| <= 1 } is set
c₁₆ is Element of the carrier of X
h1 . c₁₆ is Element of the carrier of Y
||.(h1 . c₁₆).|| is complex real ext-real Element of REAL
the U9 of Y . (h1 . c₁₆) is complex real ext-real Element of REAL
||.c₁₆.|| is complex real ext-real Element of REAL
the U9 of X . c₁₆ is complex real ext-real Element of REAL
upper_bound (X,Y,h1) is complex real ext-real Element of REAL
(X,Y) . ((vseq . n) - tv) is complex real ext-real Element of REAL
||.((vseq . n) - tv).|| is complex real ext-real Element of REAL
the U9 of (X,Y) . ((vseq . n) - tv) is complex real ext-real Element of REAL
r is complex real ext-real set
e is complex real ext-real Element of REAL
e / 2 is complex real ext-real Element of REAL
m is epsilon-transitive epsilon-connected ordinal natural complex real V108() V109() ext-real V142() V143() V144() V145() V146() V147() Element of NAT
n is epsilon-transitive epsilon-connected ordinal natural complex real V108() V109() ext-real V142() V143() V144() V145() V146() V147() Element of NAT
vseq . n is Relation-like Function-like Element of the carrier of (X,Y)
(vseq . n) - tv is Relation-like Function-like Element of the carrier of (X,Y)
- tv is Relation-like Function-like Element of the carrier of (X,Y)
(vseq . n) + (- tv) is Relation-like Function-like Element of the carrier of (X,Y)
the addF of (X,Y) . ((vseq . n),(- tv)) is Relation-like Function-like Element of the carrier of (X,Y)
[(vseq . n),(- tv)] is set
the addF of (X,Y) . [(vseq . n),(- tv)] is set
||.((vseq . n) - tv).|| is complex real ext-real Element of REAL
the U9 of (X,Y) . ((vseq . n) - tv) is complex real ext-real Element of REAL
X is non empty right_complementable Abelian add-associative right_zeroed discerning reflexive vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like CNORMSTR
Y is non empty right_complementable Abelian add-associative right_zeroed discerning reflexive vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like () CNORMSTR
(X,Y) is non empty right_complementable Abelian add-associative right_zeroed discerning reflexive vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like CNORMSTR
(X,Y) is non empty Element of bool the carrier of (X,Y)
(X,Y) is non empty right_complementable Abelian add-associative right_zeroed constituted-Functions strict vector-distributive scalar-distributive scalar-associative scalar-unital CLSStruct
(X,Y) is non empty functional Element of bool the carrier of ( the carrier of X,Y)
the carrier of X is non empty set
( the carrier of X,Y) is non empty right_complementable Abelian add-associative right_zeroed constituted-Functions strict vector-distributive scalar-distributive scalar-associative scalar-unital CLSStruct
the carrier of Y is non empty set
Funcs ( the carrier of X, the carrier of Y) is non empty functional FUNCTION_DOMAIN of the carrier of X, the carrier of Y
FuncZero ( the carrier of X,Y) is Relation-like the carrier of X -defined the carrier of Y -valued Function-like total quasi_total Element of Funcs ( the carrier of X, the carrier of Y)
FuncAdd ( the carrier of X,Y) is non empty Relation-like [:(Funcs ( the carrier of X, the carrier of Y)),(Funcs ( the carrier of X, the carrier of Y)):] -defined Funcs ( the carrier of X, the carrier of Y) -valued Function-like total quasi_total Function-yielding V34() Element of bool [:[:(Funcs ( the carrier of X, the carrier of Y)),(Funcs ( the carrier of X, the carrier of Y)):],(Funcs ( the carrier of X, the carrier of Y)):]
[:(Funcs ( the carrier of X, the carrier of Y)),(Funcs ( the carrier of X, the carrier of Y)):] is non empty set
[:[:(Funcs ( the carrier of X, the carrier of Y)),(Funcs ( the carrier of X, the carrier of Y)):],(Funcs ( the carrier of X, the carrier of Y)):] is non empty set
bool [:[:(Funcs ( the carrier of X, the carrier of Y)),(Funcs ( the carrier of X, the carrier of Y)):],(Funcs ( the carrier of X, the carrier of Y)):] is non empty set
( the carrier of X,Y) is non empty Relation-like [:COMPLEX,(Funcs ( the carrier of X, the carrier of Y)):] -defined Funcs ( the carrier of X, the carrier of Y) -valued Function-like total quasi_total Function-yielding V34() Element of bool [:[:COMPLEX,(Funcs ( the carrier of X, the carrier of Y)):],(Funcs ( the carrier of X, the carrier of Y)):]
[:COMPLEX,(Funcs ( the carrier of X, the carrier of Y)):] is non empty set
[:[:COMPLEX,(Funcs ( the carrier of X, the carrier of Y)):],(Funcs ( the carrier of X, the carrier of Y)):] is non empty set
bool [:[:COMPLEX,(Funcs ( the carrier of X, the carrier of Y)):],(Funcs ( the carrier of X, the carrier of Y)):] is non empty set
CLSStruct(# (Funcs ( the carrier of X, the carrier of Y)),(FuncZero ( the carrier of X,Y)),(FuncAdd ( the carrier of X,Y)),( the carrier of X,Y) #) is non empty strict CLSStruct
the carrier of ( the carrier of X,Y) is non empty set
bool the carrier of ( the carrier of X,Y) is non empty set
Zero_ ((X,Y),( the carrier of X,Y)) is Relation-like Function-like Element of (X,Y)
Add_ ((X,Y),( the carrier of X,Y)) is non empty Relation-like [:(X,Y),(X,Y):] -defined (X,Y) -valued Function-like total quasi_total Function-yielding V34() Element of bool [:[:(X,Y),(X,Y):],(X,Y):]
[:(X,Y),(X,Y):] is non empty set
[:[:(X,Y),(X,Y):],(X,Y):] is non empty set
bool [:[:(X,Y),(X,Y):],(X,Y):] is non empty set
Mult_ ((X,Y),( the carrier of X,Y)) is non empty Relation-like [:COMPLEX,(X,Y):] -defined (X,Y) -valued Function-like total quasi_total Function-yielding V34() Element of bool [:[:COMPLEX,(X,Y):],(X,Y):]
[:COMPLEX,(X,Y):] is non empty set
[:[:COMPLEX,(X,Y):],(X,Y):] is non empty set
bool [:[:COMPLEX,(X,Y):],(X,Y):] is non empty set
CLSStruct(# (X,Y),(Zero_ ((X,Y),( the carrier of X,Y))),(Add_ ((X,Y),( the carrier of X,Y))),(Mult_ ((X,Y),( the carrier of X,Y))) #) is non empty right_complementable Abelian add-associative right_zeroed strict vector-distributive scalar-distributive scalar-associative scalar-unital CLSStruct
the carrier of (X,Y) is non empty set
bool the carrier of (X,Y) is non empty set
Zero_ ((X,Y),(X,Y)) is Element of (X,Y)
Add_ ((X,Y),(X,Y)) is non empty Relation-like [:(X,Y),(X,Y):] -defined (X,Y) -valued Function-like total quasi_total Element of bool [:[:(X,Y),(X,Y):],(X,Y):]
[:(X,Y),(X,Y):] is non empty set
[:[:(X,Y),(X,Y):],(X,Y):] is non empty set
bool [:[:(X,Y),(X,Y):],(X,Y):] is non empty set
Mult_ ((X,Y),(X,Y)) is non empty Relation-like [:COMPLEX,(X,Y):] -defined (X,Y) -valued Function-like total quasi_total Element of bool [:[:COMPLEX,(X,Y):],(X,Y):]
[:COMPLEX,(X,Y):] is non empty set
[:[:COMPLEX,(X,Y):],(X,Y):] is non empty set
bool [:[:COMPLEX,(X,Y):],(X,Y):] is non empty set
(X,Y) is non empty Relation-like (X,Y) -defined REAL -valued Function-like total quasi_total V132() V133() V134() Element of bool [:(X,Y),REAL:]
[:(X,Y),REAL:] is non empty V132() V133() V134() set
bool [:(X,Y),REAL:] is non empty set
CNORMSTR(# (X,Y),(Zero_ ((X,Y),(X,Y))),(Add_ ((X,Y),(X,Y))),(Mult_ ((X,Y),(X,Y))),(X,Y) #) is non empty strict CNORMSTR
the carrier of (X,Y) is non empty set
[:NAT, the carrier of (X,Y):] is non empty set
bool [:NAT, the carrier of (X,Y):] is non empty set
vseq is non empty Relation-like NAT -defined the carrier of (X,Y) -valued Function-like total quasi_total Element of bool [:NAT, the carrier of (X,Y):]
X is non empty right_complementable Abelian add-associative right_zeroed discerning reflexive vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like CNORMSTR
Y is non empty right_complementable Abelian add-associative right_zeroed discerning reflexive vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like () CNORMSTR
(X,Y) is non empty right_complementable Abelian add-associative right_zeroed discerning reflexive vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like CNORMSTR
(X,Y) is non empty Element of bool the carrier of (X,Y)
(X,Y) is non empty right_complementable Abelian add-associative right_zeroed constituted-Functions strict vector-distributive scalar-distributive scalar-associative scalar-unital CLSStruct
(X,Y) is non empty functional Element of bool the carrier of ( the carrier of X,Y)
the carrier of X is non empty set
( the carrier of X,Y) is non empty right_complementable Abelian add-associative right_zeroed constituted-Functions strict vector-distributive scalar-distributive scalar-associative scalar-unital CLSStruct
the carrier of Y is non empty set
Funcs ( the carrier of X, the carrier of Y) is non empty functional FUNCTION_DOMAIN of the carrier of X, the carrier of Y
FuncZero ( the carrier of X,Y) is Relation-like the carrier of X -defined the carrier of Y -valued Function-like total quasi_total Element of Funcs ( the carrier of X, the carrier of Y)
FuncAdd ( the carrier of X,Y) is non empty Relation-like [:(Funcs ( the carrier of X, the carrier of Y)),(Funcs ( the carrier of X, the carrier of Y)):] -defined Funcs ( the carrier of X, the carrier of Y) -valued Function-like total quasi_total Function-yielding V34() Element of bool [:[:(Funcs ( the carrier of X, the carrier of Y)),(Funcs ( the carrier of X, the carrier of Y)):],(Funcs ( the carrier of X, the carrier of Y)):]
[:(Funcs ( the carrier of X, the carrier of Y)),(Funcs ( the carrier of X, the carrier of Y)):] is non empty set
[:[:(Funcs ( the carrier of X, the carrier of Y)),(Funcs ( the carrier of X, the carrier of Y)):],(Funcs ( the carrier of X, the carrier of Y)):] is non empty set
bool [:[:(Funcs ( the carrier of X, the carrier of Y)),(Funcs ( the carrier of X, the carrier of Y)):],(Funcs ( the carrier of X, the carrier of Y)):] is non empty set
( the carrier of X,Y) is non empty Relation-like [:COMPLEX,(Funcs ( the carrier of X, the carrier of Y)):] -defined Funcs ( the carrier of X, the carrier of Y) -valued Function-like total quasi_total Function-yielding V34() Element of bool [:[:COMPLEX,(Funcs ( the carrier of X, the carrier of Y)):],(Funcs ( the carrier of X, the carrier of Y)):]
[:COMPLEX,(Funcs ( the carrier of X, the carrier of Y)):] is non empty set
[:[:COMPLEX,(Funcs ( the carrier of X, the carrier of Y)):],(Funcs ( the carrier of X, the carrier of Y)):] is non empty set
bool [:[:COMPLEX,(Funcs ( the carrier of X, the carrier of Y)):],(Funcs ( the carrier of X, the carrier of Y)):] is non empty set
CLSStruct(# (Funcs ( the carrier of X, the carrier of Y)),(FuncZero ( the carrier of X,Y)),(FuncAdd ( the carrier of X,Y)),( the carrier of X,Y) #) is non empty strict CLSStruct
the carrier of ( the carrier of X,Y) is non empty set
bool the carrier of ( the carrier of X,Y) is non empty set
Zero_ ((X,Y),( the carrier of X,Y)) is Relation-like Function-like Element of (X,Y)
Add_ ((X,Y),( the carrier of X,Y)) is non empty Relation-like [:(X,Y),(X,Y):] -defined (X,Y) -valued Function-like total quasi_total Function-yielding V34() Element of bool [:[:(X,Y),(X,Y):],(X,Y):]
[:(X,Y),(X,Y):] is non empty set
[:[:(X,Y),(X,Y):],(X,Y):] is non empty set
bool [:[:(X,Y),(X,Y):],(X,Y):] is non empty set
Mult_ ((X,Y),( the carrier of X,Y)) is non empty Relation-like [:COMPLEX,(X,Y):] -defined (X,Y) -valued Function-like total quasi_total Function-yielding V34() Element of bool [:[:COMPLEX,(X,Y):],(X,Y):]
[:COMPLEX,(X,Y):] is non empty set
[:[:COMPLEX,(X,Y):],(X,Y):] is non empty set
bool [:[:COMPLEX,(X,Y):],(X,Y):] is non empty set
CLSStruct(# (X,Y),(Zero_ ((X,Y),( the carrier of X,Y))),(Add_ ((X,Y),( the carrier of X,Y))),(Mult_ ((X,Y),( the carrier of X,Y))) #) is non empty right_complementable Abelian add-associative right_zeroed strict vector-distributive scalar-distributive scalar-associative scalar-unital CLSStruct
the carrier of (X,Y) is non empty set
bool the carrier of (X,Y) is non empty set
Zero_ ((X,Y),(X,Y)) is Element of (X,Y)
Add_ ((X,Y),(X,Y)) is non empty Relation-like [:(X,Y),(X,Y):] -defined (X,Y) -valued Function-like total quasi_total Element of bool [:[:(X,Y),(X,Y):],(X,Y):]
[:(X,Y),(X,Y):] is non empty set
[:[:(X,Y),(X,Y):],(X,Y):] is non empty set
bool [:[:(X,Y),(X,Y):],(X,Y):] is non empty set
Mult_ ((X,Y),(X,Y)) is non empty Relation-like [:COMPLEX,(X,Y):] -defined (X,Y) -valued Function-like total quasi_total Element of bool [:[:COMPLEX,(X,Y):],(X,Y):]
[:COMPLEX,(X,Y):] is non empty set
[:[:COMPLEX,(X,Y):],(X,Y):] is non empty set
bool [:[:COMPLEX,(X,Y):],(X,Y):] is non empty set
(X,Y) is non empty Relation-like (X,Y) -defined REAL -valued Function-like total quasi_total V132() V133() V134() Element of bool [:(X,Y),REAL:]
[:(X,Y),REAL:] is non empty V132() V133() V134() set
bool [:(X,Y),REAL:] is non empty set
CNORMSTR(# (X,Y),(Zero_ ((X,Y),(X,Y))),(Add_ ((X,Y),(X,Y))),(Mult_ ((X,Y),(X,Y))),(X,Y) #) is non empty strict CNORMSTR