REAL is non empty V34() V144() V145() V146() V150() set
NAT is non empty epsilon-transitive epsilon-connected ordinal V144() V145() V146() V147() V148() V149() V150() Element of bool REAL
bool REAL is non empty set
{} is empty Function-like functional ext-real non positive non negative epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural V31() real V144() V145() V146() V147() V148() V149() V150() set
1 is non empty ext-real positive non negative epsilon-transitive epsilon-connected ordinal natural V31() real V113() V114() V144() V145() V146() V147() V148() V149() Element of NAT
{{},1} is non empty V144() V145() V146() V147() V148() V149() set
INT is non empty V34() V144() V145() V146() V147() V148() V150() set
COMPLEX is non empty V34() V144() V150() set
omega is non empty epsilon-transitive epsilon-connected ordinal V144() V145() V146() V147() V148() V149() V150() set
bool omega is non empty set
bool NAT is non empty set
RAT is non empty V34() V144() V145() V146() V147() V150() set
[:REAL,REAL:] is non empty V134() V135() V136() set
bool [:REAL,REAL:] is non empty set
K392() is non empty strict multMagma
the carrier of K392() is non empty set
<REAL,+> is non empty strict V115() V116() associative commutative left-invertible right-invertible invertible left-cancelable right-cancelable V171() multMagma
K398() is non empty strict associative commutative left-cancelable right-cancelable V171() SubStr of <REAL,+>
<NAT,+> is non empty strict V115() associative commutative left-cancelable right-cancelable V171() uniquely-decomposable SubStr of K398()
<REAL,*> is non empty strict V115() associative commutative multMagma
<NAT,*> is non empty strict V115() associative commutative uniquely-decomposable SubStr of <REAL,*>
ExtREAL is non empty V145() set
[:NAT,REAL:] is non empty V134() V135() V136() set
bool [:NAT,REAL:] is non empty set
the_set_of_RealSequences is non empty set
[:the_set_of_RealSequences,the_set_of_RealSequences:] is non empty set
[:[:the_set_of_RealSequences,the_set_of_RealSequences:],the_set_of_RealSequences:] is non empty set
bool [:[:the_set_of_RealSequences,the_set_of_RealSequences:],the_set_of_RealSequences:] is non empty set
[:REAL,the_set_of_RealSequences:] is non empty set
[:[:REAL,the_set_of_RealSequences:],the_set_of_RealSequences:] is non empty set
bool [:[:REAL,the_set_of_RealSequences:],the_set_of_RealSequences:] is non empty set
Linear_Space_of_RealSequences is RLSStruct
the carrier of Linear_Space_of_RealSequences is set
bool the carrier of Linear_Space_of_RealSequences is non empty set
the_set_of_l2RealSequences is Element of bool the carrier of Linear_Space_of_RealSequences
[:the_set_of_l2RealSequences,the_set_of_l2RealSequences:] is set
[:[:the_set_of_l2RealSequences,the_set_of_l2RealSequences:],REAL:] is V134() V135() V136() set
bool [:[:the_set_of_l2RealSequences,the_set_of_l2RealSequences:],REAL:] is non empty set
the_set_of_l1RealSequences is non empty linearly-closed Element of bool the carrier of Linear_Space_of_RealSequences
[:the_set_of_l1RealSequences,REAL:] is non empty V134() V135() V136() set
bool [:the_set_of_l1RealSequences,REAL:] is non empty set
[:COMPLEX,COMPLEX:] is non empty V134() set
bool [:COMPLEX,COMPLEX:] is non empty set
[:COMPLEX,REAL:] is non empty V134() V135() V136() set
bool [:COMPLEX,REAL:] is non empty set
[:[:COMPLEX,COMPLEX:],COMPLEX:] is non empty V134() set
bool [:[:COMPLEX,COMPLEX:],COMPLEX:] is non empty set
[:[:REAL,REAL:],REAL:] is non empty V134() V135() V136() set
bool [:[:REAL,REAL:],REAL:] is non empty set
[:RAT,RAT:] is non empty RAT -valued V134() V135() V136() set
bool [:RAT,RAT:] is non empty set
[:[:RAT,RAT:],RAT:] is non empty RAT -valued V134() V135() V136() set
bool [:[:RAT,RAT:],RAT:] is non empty set
[:INT,INT:] is non empty RAT -valued INT -valued V134() V135() V136() set
bool [:INT,INT:] is non empty set
[:[:INT,INT:],INT:] is non empty RAT -valued INT -valued V134() V135() V136() set
bool [:[:INT,INT:],INT:] is non empty set
[:NAT,NAT:] is non empty RAT -valued INT -valued V134() V135() V136() V137() set
[:[:NAT,NAT:],NAT:] is non empty RAT -valued INT -valued V134() V135() V136() V137() set
bool [:[:NAT,NAT:],NAT:] is non empty set
0 is empty Function-like functional ext-real non positive non negative epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural V31() real V113() V114() V144() V145() V146() V147() V148() V149() V150() Element of NAT
- 1 is ext-real non positive V31() real Element of REAL
l1_Space is non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() discerning reflexive RealNormSpace-like NORMSTR
the carrier of l1_Space is non empty set
[:NAT, the carrier of l1_Space:] is non empty set
bool [:NAT, the carrier of l1_Space:] is non empty set
2 is non empty ext-real positive non negative epsilon-transitive epsilon-connected ordinal natural V31() real V113() V114() V144() V145() V146() V147() V148() V149() Element of NAT
Y is non empty set
[:REAL,Y:] is non empty set
[:[:REAL,Y:],Y:] is non empty set
bool [:[:REAL,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 [:REAL,Y:] -defined Y -valued Function-like total quasi_total Element of bool [:[:REAL,Y:],Y:]
f is ext-real V31() real 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 V134() V135() V136() Element of bool [:(dom tseq),{f}:]
{f} is non empty V144() V145() V146() set
[:(dom tseq),{f}:] is V134() V135() V136() set
bool [:(dom tseq),{f}:] is non empty set
[:X,REAL:] is V134() V135() V136() set
bool [:X,REAL:] is non empty set
[:X,[:REAL,Y:]:] is set
bool [:X,[:REAL,Y:]:] is non empty set
<:((dom tseq) --> f),tseq:> is Relation-like Function-like set
tseq is Relation-like X -defined [:REAL,Y:] -valued Function-like total quasi_total Element of bool [:X,[:REAL,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 addLoopStr
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
[:(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
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
vseq 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 V110() Element of bool [:[:(Funcs (X, the carrier of Y)),(Funcs (X, the carrier of Y)):],(Funcs (X, the carrier of Y)):]
f 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 V110() Element of bool [:[:(Funcs (X, the carrier of Y)),(Funcs (X, the carrier of Y)):],(Funcs (X, 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)
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 . (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,tseq] is set
vseq . [tseq,tseq] is Relation-like Function-like set
the addF 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)
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] is Relation-like Function-like set
X is non empty set
Y is non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() RLSStruct
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
[:REAL,(Funcs (X, the carrier of Y)):] is non empty set
[:[:REAL,(Funcs (X, the carrier of Y)):],(Funcs (X, the carrier of Y)):] is non empty set
bool [:[:REAL,(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 [:REAL, the carrier of Y:] -defined the carrier of Y -valued Function-like total quasi_total Element of bool [:[:REAL, the carrier of Y:], the carrier of Y:]
[:REAL, the carrier of Y:] is non empty set
[:[:REAL, the carrier of Y:], the carrier of Y:] is non empty set
bool [:[:REAL, the carrier of Y:], the carrier of Y:] is non empty set
vseq is non empty Relation-like [:REAL,(Funcs (X, the carrier of Y)):] -defined Funcs (X, the carrier of Y) -valued Function-like total quasi_total Function-yielding V110() Element of bool [:[:REAL,(Funcs (X, the carrier of Y)):],(Funcs (X, the carrier of Y)):]
f is ext-real V31() real Element of REAL
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 Element of [:REAL,(Funcs (X, the carrier of 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)
tseq is Element of X
(vseq . [f,tseq]) . tseq is Element of the carrier of Y
tseq . tseq is Element of the carrier of Y
f * (tseq . tseq) is Element of 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
dom (vseq . [f,tseq]) is set
vseq . (f,tseq) is Relation-like Function-like Element of Funcs (X, the carrier of Y)
[f,tseq] is set
vseq . [f,tseq] is Relation-like Function-like set
(X, the carrier of Y, the Mult of 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 is non empty Relation-like [:REAL,(Funcs (X, the carrier of Y)):] -defined Funcs (X, the carrier of Y) -valued Function-like total quasi_total Function-yielding V110() Element of bool [:[:REAL,(Funcs (X, the carrier of Y)):],(Funcs (X, the carrier of Y)):]
f is non empty Relation-like [:REAL,(Funcs (X, the carrier of Y)):] -defined Funcs (X, the carrier of Y) -valued Function-like total quasi_total Function-yielding V110() Element of bool [:[:REAL,(Funcs (X, the carrier of Y)):],(Funcs (X, the carrier of Y)):]
tseq is ext-real V31() real Element of REAL
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 [:REAL,(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
the Mult of Y is non empty Relation-like [:REAL, the carrier of Y:] -defined the carrier of Y -valued Function-like total quasi_total Element of bool [:[:REAL, the carrier of Y:], the carrier of Y:]
[:REAL, the carrier of Y:] is non empty set
[:[:REAL, the carrier of Y:], the carrier of Y:] is non empty set
bool [:[:REAL, the carrier of Y:], the carrier of Y:] is non empty set
the Mult of Y . (tseq,(tseq . tseq)) is set
[tseq,(tseq . tseq)] is set
the Mult of Y . [tseq,(tseq . tseq)] is set
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
Y is non empty ZeroStr
the carrier of Y is non empty set
X is set
0. Y is V53(Y) Element of the carrier of Y
the ZeroF of Y is Element of the carrier of Y
X --> (0. Y) is 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 set
bool [:X, 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 is non empty set
Y is non empty set
[:X,Y:] is non empty set
bool [:X,Y:] is non empty set
vseq is Element of X
f is non empty Relation-like X -defined Y -valued Function-like total quasi_total Element of bool [:X,Y:]
dom f is set
X is non empty set
Y is non empty addLoopStr
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 [:(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 V110() 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
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 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)
(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
(X,Y) . [vseq,f] is Relation-like Function-like set
tseq is Element of X
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) is Relation-like X -defined the carrier of Y -valued Function-like total quasi_total Element of Funcs (X, the carrier of Y)
dom ( the addF of Y .: (vseq,f)) is set
tseq . tseq is Element of the carrier of Y
( the addF of Y .: (vseq,f)) . tseq is Element of the carrier of Y
vseq . tseq is Element of the carrier of Y
f . tseq is Element of the carrier of Y
(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 is Element of X
((X,Y) . (vseq,f)) . tseq is Element of the carrier of Y
( the addF of Y .: (vseq,f)) . tseq is Element of the carrier of Y
vseq . tseq is Element of the carrier of Y
f . tseq is Element of the carrier of Y
(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 . tseq is Element of the carrier of Y
X is non empty set
Y is non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() RLSStruct
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 [:REAL,(Funcs (X, the carrier of Y)):] -defined Funcs (X, the carrier of Y) -valued Function-like total quasi_total Function-yielding V110() Element of bool [:[:REAL,(Funcs (X, the carrier of Y)):],(Funcs (X, the carrier of Y)):]
[:REAL,(Funcs (X, the carrier of Y)):] is non empty set
[:[:REAL,(Funcs (X, the carrier of Y)):],(Funcs (X, the carrier of Y)):] is non empty set
bool [:[:REAL,(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 ext-real V31() real Element of REAL
[tseq,f] is Element of [:REAL,(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
f . tseq is Element of the carrier of Y
tseq * (f . tseq) is Element of the carrier of Y
the Mult of Y is non empty Relation-like [:REAL, the carrier of Y:] -defined the carrier of Y -valued Function-like total quasi_total Element of bool [:[:REAL, the carrier of Y:], the carrier of Y:]
[:REAL, the carrier of Y:] is non empty set
[:[:REAL, the carrier of Y:], the carrier of Y:] is non empty set
bool [:[:REAL, the carrier of Y:], the carrier of Y:] is non empty set
the Mult of Y . (tseq,(f . tseq)) is set
[tseq,(f . tseq)] is set
the Mult of Y . [tseq,(f . tseq)] is set
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
the Mult of Y is non empty Relation-like [:REAL, the carrier of Y:] -defined the carrier of Y -valued Function-like total quasi_total Element of bool [:[:REAL, the carrier of Y:], the carrier of Y:]
[:REAL, the carrier of Y:] is non empty set
[:[:REAL, the carrier of Y:], the carrier of Y:] is non empty set
bool [:[:REAL, the carrier of Y:], the carrier of Y:] is non empty set
the Mult of Y . (tseq,(f . tseq)) is set
[tseq,(f . tseq)] is set
the Mult of Y . [tseq,(f . tseq)] is 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
the Mult of Y is non empty Relation-like [:REAL, the carrier of Y:] -defined the carrier of Y -valued Function-like total quasi_total Element of bool [:[:REAL, the carrier of Y:], the carrier of Y:]
[:REAL, the carrier of Y:] is non empty set
[:[:REAL, the carrier of Y:], the carrier of Y:] is non empty set
bool [:[:REAL, the carrier of Y:], the carrier of Y:] is non empty set
the Mult of Y . (tseq,(f . tseq)) is set
[tseq,(f . tseq)] is set
the Mult of Y . [tseq,(f . tseq)] is set
X is non empty set
Y is non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() RLSStruct
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 [:(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 V110() 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)
(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
(X,Y) . [vseq,f] is Relation-like Function-like set
(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
(X,Y) . [f,vseq] is Relation-like Function-like set
tseq is Element of X
((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
((X,Y) . (f,vseq)) . tseq is Element of the carrier of Y
X is non empty set
Y is non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() RLSStruct
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 [:(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 V110() 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)
(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
(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)
(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
(X,Y) . [f,tseq] is Relation-like Function-like set
(X,Y) . (vseq,((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,((X,Y) . (f,tseq))] is set
(X,Y) . [vseq,((X,Y) . (f,tseq))] is Relation-like Function-like set
(X,Y) . (((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)
[((X,Y) . (vseq,f)),tseq] is set
(X,Y) . [((X,Y) . (vseq,f)),tseq] is Relation-like Function-like set
tseq is Element of X
((X,Y) . (vseq,((X,Y) . (f,tseq)))) . tseq is Element of the carrier of Y
vseq . tseq is Element of the carrier of Y
((X,Y) . (f,tseq)) . tseq is Element of the carrier of Y
(vseq . tseq) + (((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),(((X,Y) . (f,tseq)) . tseq)) is Element of the carrier of Y
[(vseq . tseq),(((X,Y) . (f,tseq)) . tseq)] is set
the addF of Y . [(vseq . tseq),(((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
((X,Y) . (vseq,f)) . tseq is Element of the carrier of Y
(((X,Y) . (vseq,f)) . tseq) + (tseq . tseq) is Element of the carrier of Y
the addF of Y . ((((X,Y) . (vseq,f)) . tseq),(tseq . tseq)) is Element of the carrier of Y
[(((X,Y) . (vseq,f)) . tseq),(tseq . tseq)] is set
the addF of Y . [(((X,Y) . (vseq,f)) . tseq),(tseq . tseq)] is set
((X,Y) . (((X,Y) . (vseq,f)),tseq)) . tseq is Element of the carrier of Y
X is non empty set
Y is non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() RLSStruct
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 [:(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 V110() 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 Relation-like X -defined the carrier of Y -valued Function-like total quasi_total Element of Funcs (X, the carrier of Y)
0. Y is V53(Y) Element of the carrier of Y
the ZeroF of Y 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
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) . ((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)
[(X,Y),vseq] is set
(X,Y) . [(X,Y),vseq] is Relation-like Function-like set
f is Element of X
((X,Y) . ((X,Y),vseq)) . f is Element of the carrier of Y
(X,Y) . f is Element of the carrier of Y
vseq . f is Element of the carrier of Y
((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 . (((X,Y) . f),(vseq . f)) is Element of the carrier of Y
[((X,Y) . f),(vseq . f)] is set
the addF of Y . [((X,Y) . f),(vseq . f)] is set
(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 left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() RLSStruct
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 [:(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 V110() 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 [:REAL,(Funcs (X, the carrier of Y)):] -defined Funcs (X, the carrier of Y) -valued Function-like total quasi_total Function-yielding V110() Element of bool [:[:REAL,(Funcs (X, the carrier of Y)):],(Funcs (X, the carrier of Y)):]
[:REAL,(Funcs (X, the carrier of Y)):] is non empty set
[:[:REAL,(Funcs (X, the carrier of Y)):],(Funcs (X, the carrier of Y)):] is non empty set
bool [:[:REAL,(Funcs (X, the carrier of Y)):],(Funcs (X, the carrier of Y)):] is non empty set
(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)
0. Y is V53(Y) Element of the carrier of Y
the ZeroF of Y 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
vseq is Relation-like X -defined the carrier of Y -valued Function-like total quasi_total Element of Funcs (X, the carrier of Y)
[(- 1),vseq] is Element of [:REAL,(Funcs (X, the carrier of Y)):]
(X,Y) . [(- 1),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) . (vseq,((X,Y) . [(- 1),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) . [(- 1),vseq])] is set
(X,Y) . [vseq,((X,Y) . [(- 1),vseq])] is Relation-like Function-like set
f is Element of X
vseq . f is Element of the carrier of Y
((X,Y) . (vseq,((X,Y) . [(- 1),vseq]))) . f is Element of the carrier of Y
((X,Y) . [(- 1),vseq]) . f is Element of the carrier of Y
(vseq . f) + (((X,Y) . [(- 1),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) . [(- 1),vseq]) . f)) is Element of the carrier of Y
[(vseq . f),(((X,Y) . [(- 1),vseq]) . f)] is set
the addF of Y . [(vseq . f),(((X,Y) . [(- 1),vseq]) . f)] is set
(- 1) * (vseq . f) is Element of the carrier of Y
the Mult of Y is non empty Relation-like [:REAL, the carrier of Y:] -defined the carrier of Y -valued Function-like total quasi_total Element of bool [:[:REAL, the carrier of Y:], the carrier of Y:]
[:REAL, the carrier of Y:] is non empty set
[:[:REAL, the carrier of Y:], the carrier of Y:] is non empty set
bool [:[:REAL, the carrier of Y:], the carrier of Y:] is non empty set
the Mult of Y . ((- 1),(vseq . f)) is set
[(- 1),(vseq . f)] is set
the Mult of Y . [(- 1),(vseq . f)] is set
(vseq . f) + ((- 1) * (vseq . f)) is Element of the carrier of Y
the addF of Y . ((vseq . f),((- 1) * (vseq . f))) is Element of the carrier of Y
[(vseq . f),((- 1) * (vseq . f))] is set
the addF of Y . [(vseq . f),((- 1) * (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
(X,Y) . f is Element of the carrier of Y
X is non empty set
Y is non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() RLSStruct
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 [:REAL,(Funcs (X, the carrier of Y)):] -defined Funcs (X, the carrier of Y) -valued Function-like total quasi_total Function-yielding V110() Element of bool [:[:REAL,(Funcs (X, the carrier of Y)):],(Funcs (X, the carrier of Y)):]
[:REAL,(Funcs (X, the carrier of Y)):] is non empty set
[:[:REAL,(Funcs (X, the carrier of Y)):],(Funcs (X, the carrier of Y)):] is non empty set
bool [:[:REAL,(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)
[1,vseq] is Element of [:NAT,(Funcs (X, the carrier of Y)):]
[:NAT,(Funcs (X, the carrier of Y)):] is non empty set
(X,Y) . [1,vseq] is Relation-like Function-like set
[1,vseq] is Element of [:REAL,(Funcs (X, the carrier of Y)):]
(X,Y) . [1,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) . [1,vseq]) . f is Element of the carrier of Y
vseq . f is Element of the carrier of Y
1 * (vseq . f) is Element of the carrier of Y
the Mult of Y is non empty Relation-like [:REAL, the carrier of Y:] -defined the carrier of Y -valued Function-like total quasi_total Element of bool [:[:REAL, the carrier of Y:], the carrier of Y:]
[:REAL, the carrier of Y:] is non empty set
[:[:REAL, the carrier of Y:], the carrier of Y:] is non empty set
bool [:[:REAL, the carrier of Y:], the carrier of Y:] is non empty set
the Mult of Y . (1,(vseq . f)) is set
[1,(vseq . f)] is set
the Mult of Y . [1,(vseq . f)] is set
X is non empty set
Y is non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() RLSStruct
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 [:REAL,(Funcs (X, the carrier of Y)):] -defined Funcs (X, the carrier of Y) -valued Function-like total quasi_total Function-yielding V110() Element of bool [:[:REAL,(Funcs (X, the carrier of Y)):],(Funcs (X, the carrier of Y)):]
[:REAL,(Funcs (X, the carrier of Y)):] is non empty set
[:[:REAL,(Funcs (X, the carrier of Y)):],(Funcs (X, the carrier of Y)):] is non empty set
bool [:[:REAL,(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 ext-real V31() real Element of REAL
tseq is ext-real V31() real Element of REAL
[tseq,vseq] is Element of [:REAL,(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)
[f,((X,Y) . [tseq,vseq])] is Element of [:REAL,(Funcs (X, the carrier of Y)):]
(X,Y) . [f,((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)
f * tseq is ext-real V31() real Element of REAL
[(f * tseq),vseq] is Element of [:REAL,(Funcs (X, the carrier of Y)):]
(X,Y) . [(f * 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 Element of X
((X,Y) . [f,((X,Y) . [tseq,vseq])]) . tseq is Element of the carrier of Y
((X,Y) . [tseq,vseq]) . tseq is Element of the carrier of Y
f * (((X,Y) . [tseq,vseq]) . tseq) is Element of the carrier of Y
the Mult of Y is non empty Relation-like [:REAL, the carrier of Y:] -defined the carrier of Y -valued Function-like total quasi_total Element of bool [:[:REAL, the carrier of Y:], the carrier of Y:]
[:REAL, the carrier of Y:] is non empty set
[:[:REAL, the carrier of Y:], the carrier of Y:] is non empty set
bool [:[:REAL, the carrier of Y:], the carrier of Y:] is non empty set
the Mult of Y . (f,(((X,Y) . [tseq,vseq]) . tseq)) is set
[f,(((X,Y) . [tseq,vseq]) . tseq)] is set
the Mult of Y . [f,(((X,Y) . [tseq,vseq]) . tseq)] is set
vseq . tseq is Element of the carrier of Y
tseq * (vseq . tseq) is Element of the carrier of Y
the Mult of Y . (tseq,(vseq . tseq)) is set
[tseq,(vseq . tseq)] is set
the Mult of Y . [tseq,(vseq . tseq)] is set
f * (tseq * (vseq . tseq)) is Element of the carrier of Y
the Mult of Y . (f,(tseq * (vseq . tseq))) is set
[f,(tseq * (vseq . tseq))] is set
the Mult of Y . [f,(tseq * (vseq . tseq))] is set
(f * tseq) * (vseq . tseq) is Element of the carrier of Y
the Mult of Y . ((f * tseq),(vseq . tseq)) is set
[(f * tseq),(vseq . tseq)] is set
the Mult of Y . [(f * tseq),(vseq . tseq)] is set
((X,Y) . [(f * tseq),vseq]) . tseq is Element of the carrier of Y
X is non empty set
Y is non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() RLSStruct
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 [:(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 V110() 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 [:REAL,(Funcs (X, the carrier of Y)):] -defined Funcs (X, the carrier of Y) -valued Function-like total quasi_total Function-yielding V110() Element of bool [:[:REAL,(Funcs (X, the carrier of Y)):],(Funcs (X, the carrier of Y)):]
[:REAL,(Funcs (X, the carrier of Y)):] is non empty set
[:[:REAL,(Funcs (X, the carrier of Y)):],(Funcs (X, the carrier of Y)):] is non empty set
bool [:[:REAL,(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 ext-real V31() real Element of REAL
[f,vseq] is Element of [:REAL,(Funcs (X, the carrier of Y)):]
(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)
tseq is ext-real V31() real Element of REAL
[tseq,vseq] is Element of [:REAL,(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)
(X,Y) . (((X,Y) . [f,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) . [f,vseq]),((X,Y) . [tseq,vseq])] is set
(X,Y) . [((X,Y) . [f,vseq]),((X,Y) . [tseq,vseq])] is Relation-like Function-like set
f + tseq is ext-real V31() real Element of REAL
[(f + tseq),vseq] is Element of [:REAL,(Funcs (X, the carrier of Y)):]
(X,Y) . [(f + 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 Element of X
((X,Y) . (((X,Y) . [f,vseq]),((X,Y) . [tseq,vseq]))) . tseq is Element of the carrier of Y
((X,Y) . [f,vseq]) . tseq is Element of the carrier of Y
((X,Y) . [tseq,vseq]) . tseq is Element of the carrier of Y
(((X,Y) . [f,vseq]) . tseq) + (((X,Y) . [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 . ((((X,Y) . [f,vseq]) . tseq),(((X,Y) . [tseq,vseq]) . tseq)) is Element of the carrier of Y
[(((X,Y) . [f,vseq]) . tseq),(((X,Y) . [tseq,vseq]) . tseq)] is set
the addF of Y . [(((X,Y) . [f,vseq]) . tseq),(((X,Y) . [tseq,vseq]) . tseq)] is set
vseq . tseq is Element of the carrier of Y
f * (vseq . tseq) is Element of the carrier of Y
the Mult of Y is non empty Relation-like [:REAL, the carrier of Y:] -defined the carrier of Y -valued Function-like total quasi_total Element of bool [:[:REAL, the carrier of Y:], the carrier of Y:]
[:REAL, the carrier of Y:] is non empty set
[:[:REAL, the carrier of Y:], the carrier of Y:] is non empty set
bool [:[:REAL, the carrier of Y:], the carrier of Y:] is non empty set
the Mult of Y . (f,(vseq . tseq)) is set
[f,(vseq . tseq)] is set
the Mult of Y . [f,(vseq . tseq)] is set
(f * (vseq . tseq)) + (((X,Y) . [tseq,vseq]) . tseq) is Element of the carrier of Y
the addF of Y . ((f * (vseq . tseq)),(((X,Y) . [tseq,vseq]) . tseq)) is Element of the carrier of Y
[(f * (vseq . tseq)),(((X,Y) . [tseq,vseq]) . tseq)] is set
the addF of Y . [(f * (vseq . tseq)),(((X,Y) . [tseq,vseq]) . tseq)] is set
tseq * (vseq . tseq) is Element of the carrier of Y
the Mult of Y . (tseq,(vseq . tseq)) is set
[tseq,(vseq . tseq)] is set
the Mult of Y . [tseq,(vseq . tseq)] is set
(f * (vseq . tseq)) + (tseq * (vseq . tseq)) is Element of the carrier of Y
the addF of Y . ((f * (vseq . tseq)),(tseq * (vseq . tseq))) is Element of the carrier of Y
[(f * (vseq . tseq)),(tseq * (vseq . tseq))] is set
the addF of Y . [(f * (vseq . tseq)),(tseq * (vseq . tseq))] is set
(f + tseq) * (vseq . tseq) is Element of the carrier of Y
the Mult of Y . ((f + tseq),(vseq . tseq)) is set
[(f + tseq),(vseq . tseq)] is set
the Mult of Y . [(f + tseq),(vseq . tseq)] is set
((X,Y) . [(f + tseq),vseq]) . tseq is Element of the carrier of Y
X is non empty set
Y is non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() RLSStruct
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 [:(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 V110() 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 [:REAL,(Funcs (X, the carrier of Y)):] -defined Funcs (X, the carrier of Y) -valued Function-like total quasi_total Function-yielding V110() Element of bool [:[:REAL,(Funcs (X, the carrier of Y)):],(Funcs (X, the carrier of Y)):]
[:REAL,(Funcs (X, the carrier of Y)):] is non empty set
[:[:REAL,(Funcs (X, the carrier of Y)):],(Funcs (X, the carrier of Y)):] is non empty set
bool [:[:REAL,(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)
(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
(X,Y) . [vseq,f] is Relation-like Function-like set
tseq is ext-real V31() real Element of REAL
[tseq,vseq] is Element of [:REAL,(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 [:REAL,(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)
(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
(X,Y) . [((X,Y) . [tseq,vseq]),((X,Y) . [tseq,f])] is Relation-like Function-like set
[tseq,((X,Y) . (vseq,f))] is Element of [:REAL,(Funcs (X, the carrier of Y)):]
(X,Y) . [tseq,((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)
tseq is Element of X
((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
the Mult of Y is non empty Relation-like [:REAL, the carrier of Y:] -defined the carrier of Y -valued Function-like total quasi_total Element of bool [:[:REAL, the carrier of Y:], the carrier of Y:]
[:REAL, the carrier of Y:] is non empty set
[:[:REAL, the carrier of Y:], the carrier of Y:] is non empty set
bool [:[:REAL, the carrier of Y:], the carrier of Y:] is non empty set
the Mult of Y . (tseq,(vseq . tseq)) is set
[tseq,(vseq . tseq)] is set
the Mult of Y . [tseq,(vseq . tseq)] is set
(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
f . tseq is Element of the carrier of Y
tseq * (f . tseq) is Element of the carrier of Y
the Mult of Y . (tseq,(f . tseq)) is set
[tseq,(f . tseq)] is set
the Mult of Y . [tseq,(f . tseq)] is set
(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
the Mult of Y . (tseq,((vseq . tseq) + (f . tseq))) is set
[tseq,((vseq . tseq) + (f . tseq))] is set
the Mult of Y . [tseq,((vseq . tseq) + (f . tseq))] is set
((X,Y) . (vseq,f)) . tseq is Element of the carrier of Y
tseq * (((X,Y) . (vseq,f)) . tseq) is Element of the carrier of Y
the Mult of Y . (tseq,(((X,Y) . (vseq,f)) . tseq)) is set
[tseq,(((X,Y) . (vseq,f)) . tseq)] is set
the Mult of Y . [tseq,(((X,Y) . (vseq,f)) . tseq)] is set
((X,Y) . [tseq,((X,Y) . (vseq,f))]) . tseq is Element of the carrier of Y
X is non empty set
Y is non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() RLSStruct
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 Relation-like X -defined the carrier of Y -valued Function-like total quasi_total Element of Funcs (X, the carrier of Y)
0. Y is V53(Y) Element of the carrier of Y
the ZeroF of Y 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,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 V110() 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 [:REAL,(Funcs (X, the carrier of Y)):] -defined Funcs (X, the carrier of Y) -valued Function-like total quasi_total Function-yielding V110() Element of bool [:[:REAL,(Funcs (X, the carrier of Y)):],(Funcs (X, the carrier of Y)):]
[:REAL,(Funcs (X, the carrier of Y)):] is non empty set
[:[:REAL,(Funcs (X, the carrier of Y)):],(Funcs (X, the carrier of Y)):] is non empty set
bool [:[:REAL,(Funcs (X, the carrier of Y)):],(Funcs (X, the carrier of Y)):] is non empty set
RLSStruct(# (Funcs (X, the carrier of Y)),(X,Y),(X,Y),(X,Y) #) is non empty strict RLSStruct
the carrier of RLSStruct(# (Funcs (X, the carrier of Y)),(X,Y),(X,Y),(X,Y) #) is non empty set
vseq is ext-real V31() real set
f is ext-real V31() real set
vseq * f is ext-real V31() real set
tseq is ext-real V31() real Element of REAL
tseq is ext-real V31() real Element of REAL
tseq * tseq is ext-real V31() real Element of REAL
tseq is Element of the carrier of RLSStruct(# (Funcs (X, the carrier of Y)),(X,Y),(X,Y),(X,Y) #)
(tseq * tseq) * tseq is Element of the carrier of RLSStruct(# (Funcs (X, the carrier of Y)),(X,Y),(X,Y),(X,Y) #)
the Mult of RLSStruct(# (Funcs (X, the carrier of Y)),(X,Y),(X,Y),(X,Y) #) is non empty Relation-like [:REAL, the carrier of RLSStruct(# (Funcs (X, the carrier of Y)),(X,Y),(X,Y),(X,Y) #):] -defined the carrier of RLSStruct(# (Funcs (X, the carrier of Y)),(X,Y),(X,Y),(X,Y) #) -valued Function-like total quasi_total Element of bool [:[:REAL, the carrier of RLSStruct(# (Funcs (X, the carrier of Y)),(X,Y),(X,Y),(X,Y) #):], the carrier of RLSStruct(# (Funcs (X, the carrier of Y)),(X,Y),(X,Y),(X,Y) #):]
[:REAL, the carrier of RLSStruct(# (Funcs (X, the carrier of Y)),(X,Y),(X,Y),(X,Y) #):] is non empty set
[:[:REAL, the carrier of RLSStruct(# (Funcs (X, the carrier of Y)),(X,Y),(X,Y),(X,Y) #):], the carrier of RLSStruct(# (Funcs (X, the carrier of Y)),(X,Y),(X,Y),(X,Y) #):] is non empty set
bool [:[:REAL, the carrier of RLSStruct(# (Funcs (X, the carrier of Y)),(X,Y),(X,Y),(X,Y) #):], the carrier of RLSStruct(# (Funcs (X, the carrier of Y)),(X,Y),(X,Y),(X,Y) #):] is non empty set
the Mult of RLSStruct(# (Funcs (X, the carrier of Y)),(X,Y),(X,Y),(X,Y) #) . ((tseq * tseq),tseq) is set
[(tseq * tseq),tseq] is set
the Mult of RLSStruct(# (Funcs (X, the carrier of Y)),(X,Y),(X,Y),(X,Y) #) . [(tseq * tseq),tseq] is set
tseq * tseq is Element of the carrier of RLSStruct(# (Funcs (X, the carrier of Y)),(X,Y),(X,Y),(X,Y) #)
the Mult of RLSStruct(# (Funcs (X, the carrier of Y)),(X,Y),(X,Y),(X,Y) #) . (tseq,tseq) is set
[tseq,tseq] is set
the Mult of RLSStruct(# (Funcs (X, the carrier of Y)),(X,Y),(X,Y),(X,Y) #) . [tseq,tseq] is set
tseq * (tseq * tseq) is Element of the carrier of RLSStruct(# (Funcs (X, the carrier of Y)),(X,Y),(X,Y),(X,Y) #)
the Mult of RLSStruct(# (Funcs (X, the carrier of Y)),(X,Y),(X,Y),(X,Y) #) . (tseq,(tseq * tseq)) is set
[tseq,(tseq * tseq)] is set
the Mult of RLSStruct(# (Funcs (X, the carrier of Y)),(X,Y),(X,Y),(X,Y) #) . [tseq,(tseq * tseq)] is set
tseq is Element of the carrier of RLSStruct(# (Funcs (X, the carrier of Y)),(X,Y),(X,Y),(X,Y) #)
(vseq * f) * tseq is Element of the carrier of RLSStruct(# (Funcs (X, the carrier of Y)),(X,Y),(X,Y),(X,Y) #)
the Mult of RLSStruct(# (Funcs (X, the carrier of Y)),(X,Y),(X,Y),(X,Y) #) is non empty Relation-like [:REAL, the carrier of RLSStruct(# (Funcs (X, the carrier of Y)),(X,Y),(X,Y),(X,Y) #):] -defined the carrier of RLSStruct(# (Funcs (X, the carrier of Y)),(X,Y),(X,Y),(X,Y) #) -valued Function-like total quasi_total Element of bool [:[:REAL, the carrier of RLSStruct(# (Funcs (X, the carrier of Y)),(X,Y),(X,Y),(X,Y) #):], the carrier of RLSStruct(# (Funcs (X, the carrier of Y)),(X,Y),(X,Y),(X,Y) #):]
[:REAL, the carrier of RLSStruct(# (Funcs (X, the carrier of Y)),(X,Y),(X,Y),(X,Y) #):] is non empty set
[:[:REAL, the carrier of RLSStruct(# (Funcs (X, the carrier of Y)),(X,Y),(X,Y),(X,Y) #):], the carrier of RLSStruct(# (Funcs (X, the carrier of Y)),(X,Y),(X,Y),(X,Y) #):] is non empty set
bool [:[:REAL, the carrier of RLSStruct(# (Funcs (X, the carrier of Y)),(X,Y),(X,Y),(X,Y) #):], the carrier of RLSStruct(# (Funcs (X, the carrier of Y)),(X,Y),(X,Y),(X,Y) #):] is non empty set
the Mult of RLSStruct(# (Funcs (X, the carrier of Y)),(X,Y),(X,Y),(X,Y) #) . ((vseq * f),tseq) is set
[(vseq * f),tseq] is set
the Mult of RLSStruct(# (Funcs (X, the carrier of Y)),(X,Y),(X,Y),(X,Y) #) . [(vseq * f),tseq] is set
f * tseq is Element of the carrier of RLSStruct(# (Funcs (X, the carrier of Y)),(X,Y),(X,Y),(X,Y) #)
the Mult of RLSStruct(# (Funcs (X, the carrier of Y)),(X,Y),(X,Y),(X,Y) #) . (f,tseq) is set
[f,tseq] is set
the Mult of RLSStruct(# (Funcs (X, the carrier of Y)),(X,Y),(X,Y),(X,Y) #) . [f,tseq] is set
vseq * (f * tseq) is Element of the carrier of RLSStruct(# (Funcs (X, the carrier of Y)),(X,Y),(X,Y),(X,Y) #)
the Mult of RLSStruct(# (Funcs (X, the carrier of Y)),(X,Y),(X,Y),(X,Y) #) . (vseq,(f * tseq)) is set
[vseq,(f * tseq)] is set
the Mult of RLSStruct(# (Funcs (X, the carrier of Y)),(X,Y),(X,Y),(X,Y) #) . [vseq,(f * tseq)] is set
vseq is ext-real V31() real set
f is ext-real V31() real set
vseq + f is ext-real V31() real set
tseq is ext-real V31() real Element of REAL
tseq is ext-real V31() real Element of REAL
tseq + tseq is ext-real V31() real Element of REAL
tseq is Element of the carrier of RLSStruct(# (Funcs (X, the carrier of Y)),(X,Y),(X,Y),(X,Y) #)
(tseq + tseq) * tseq is Element of the carrier of RLSStruct(# (Funcs (X, the carrier of Y)),(X,Y),(X,Y),(X,Y) #)
the Mult of RLSStruct(# (Funcs (X, the carrier of Y)),(X,Y),(X,Y),(X,Y) #) is non empty Relation-like [:REAL, the carrier of RLSStruct(# (Funcs (X, the carrier of Y)),(X,Y),(X,Y),(X,Y) #):] -defined the carrier of RLSStruct(# (Funcs (X, the carrier of Y)),(X,Y),(X,Y),(X,Y) #) -valued Function-like total quasi_total Element of bool [:[:REAL, the carrier of RLSStruct(# (Funcs (X, the carrier of Y)),(X,Y),(X,Y),(X,Y) #):], the carrier of RLSStruct(# (Funcs (X, the carrier of Y)),(X,Y),(X,Y),(X,Y) #):]
[:REAL, the carrier of RLSStruct(# (Funcs (X, the carrier of Y)),(X,Y),(X,Y),(X,Y) #):] is non empty set
[:[:REAL, the carrier of RLSStruct(# (Funcs (X, the carrier of Y)),(X,Y),(X,Y),(X,Y) #):], the carrier of RLSStruct(# (Funcs (X, the carrier of Y)),(X,Y),(X,Y),(X,Y) #):] is non empty set
bool [:[:REAL, the carrier of RLSStruct(# (Funcs (X, the carrier of Y)),(X,Y),(X,Y),(X,Y) #):], the carrier of RLSStruct(# (Funcs (X, the carrier of Y)),(X,Y),(X,Y),(X,Y) #):] is non empty set
the Mult of RLSStruct(# (Funcs (X, the carrier of Y)),(X,Y),(X,Y),(X,Y) #) . ((tseq + tseq),tseq) is set
[(tseq + tseq),tseq] is set
the Mult of RLSStruct(# (Funcs (X, the carrier of Y)),(X,Y),(X,Y),(X,Y) #) . [(tseq + tseq),tseq] is set
tseq * tseq is Element of the carrier of RLSStruct(# (Funcs (X, the carrier of Y)),(X,Y),(X,Y),(X,Y) #)
the Mult of RLSStruct(# (Funcs (X, the carrier of Y)),(X,Y),(X,Y),(X,Y) #) . (tseq,tseq) is set
[tseq,tseq] is set
the Mult of RLSStruct(# (Funcs (X, the carrier of Y)),(X,Y),(X,Y),(X,Y) #) . [tseq,tseq] is set
tseq * tseq is Element of the carrier of RLSStruct(# (Funcs (X, the carrier of Y)),(X,Y),(X,Y),(X,Y) #)
the Mult of RLSStruct(# (Funcs (X, the carrier of Y)),(X,Y),(X,Y),(X,Y) #) . (tseq,tseq) is set
[tseq,tseq] is set
the Mult of RLSStruct(# (Funcs (X, the carrier of Y)),(X,Y),(X,Y),(X,Y) #) . [tseq,tseq] is set
(tseq * tseq) + (tseq * tseq) is Element of the carrier of RLSStruct(# (Funcs (X, the carrier of Y)),(X,Y),(X,Y),(X,Y) #)
the addF of RLSStruct(# (Funcs (X, the carrier of Y)),(X,Y),(X,Y),(X,Y) #) is non empty Relation-like [: the carrier of RLSStruct(# (Funcs (X, the carrier of Y)),(X,Y),(X,Y),(X,Y) #), the carrier of RLSStruct(# (Funcs (X, the carrier of Y)),(X,Y),(X,Y),(X,Y) #):] -defined the carrier of RLSStruct(# (Funcs (X, the carrier of Y)),(X,Y),(X,Y),(X,Y) #) -valued Function-like total quasi_total Element of bool [:[: the carrier of RLSStruct(# (Funcs (X, the carrier of Y)),(X,Y),(X,Y),(X,Y) #), the carrier of RLSStruct(# (Funcs (X, the carrier of Y)),(X,Y),(X,Y),(X,Y) #):], the carrier of RLSStruct(# (Funcs (X, the carrier of Y)),(X,Y),(X,Y),(X,Y) #):]
[: the carrier of RLSStruct(# (Funcs (X, the carrier of Y)),(X,Y),(X,Y),(X,Y) #), the carrier of RLSStruct(# (Funcs (X, the carrier of Y)),(X,Y),(X,Y),(X,Y) #):] is non empty set
[:[: the carrier of RLSStruct(# (Funcs (X, the carrier of Y)),(X,Y),(X,Y),(X,Y) #), the carrier of RLSStruct(# (Funcs (X, the carrier of Y)),(X,Y),(X,Y),(X,Y) #):], the carrier of RLSStruct(# (Funcs (X, the carrier of Y)),(X,Y),(X,Y),(X,Y) #):] is non empty set
bool [:[: the carrier of RLSStruct(# (Funcs (X, the carrier of Y)),(X,Y),(X,Y),(X,Y) #), the carrier of RLSStruct(# (Funcs (X, the carrier of Y)),(X,Y),(X,Y),(X,Y) #):], the carrier of RLSStruct(# (Funcs (X, the carrier of Y)),(X,Y),(X,Y),(X,Y) #):] is non empty set
the addF of RLSStruct(# (Funcs (X, the carrier of Y)),(X,Y),(X,Y),(X,Y) #) . ((tseq * tseq),(tseq * tseq)) is Element of the carrier of RLSStruct(# (Funcs (X, the carrier of Y)),(X,Y),(X,Y),(X,Y) #)
[(tseq * tseq),(tseq * tseq)] is set
the addF of RLSStruct(# (Funcs (X, the carrier of Y)),(X,Y),(X,Y),(X,Y) #) . [(tseq * tseq),(tseq * tseq)] is set
tseq is Element of the carrier of RLSStruct(# (Funcs (X, the carrier of Y)),(X,Y),(X,Y),(X,Y) #)
(vseq + f) * tseq is Element of the carrier of RLSStruct(# (Funcs (X, the carrier of Y)),(X,Y),(X,Y),(X,Y) #)
the Mult of RLSStruct(# (Funcs (X, the carrier of Y)),(X,Y),(X,Y),(X,Y) #) is non empty Relation-like [:REAL, the carrier of RLSStruct(# (Funcs (X, the carrier of Y)),(X,Y),(X,Y),(X,Y) #):] -defined the carrier of RLSStruct(# (Funcs (X, the carrier of Y)),(X,Y),(X,Y),(X,Y) #) -valued Function-like total quasi_total Element of bool [:[:REAL, the carrier of RLSStruct(# (Funcs (X, the carrier of Y)),(X,Y),(X,Y),(X,Y) #):], the carrier of RLSStruct(# (Funcs (X, the carrier of Y)),(X,Y),(X,Y),(X,Y) #):]
[:REAL, the carrier of RLSStruct(# (Funcs (X, the carrier of Y)),(X,Y),(X,Y),(X,Y) #):] is non empty set
[:[:REAL, the carrier of RLSStruct(# (Funcs (X, the carrier of Y)),(X,Y),(X,Y),(X,Y) #):], the carrier of RLSStruct(# (Funcs (X, the carrier of Y)),(X,Y),(X,Y),(X,Y) #):] is non empty set
bool [:[:REAL, the carrier of RLSStruct(# (Funcs (X, the carrier of Y)),(X,Y),(X,Y),(X,Y) #):], the carrier of RLSStruct(# (Funcs (X, the carrier of Y)),(X,Y),(X,Y),(X,Y) #):] is non empty set
the Mult of RLSStruct(# (Funcs (X, the carrier of Y)),(X,Y),(X,Y),(X,Y) #) . ((vseq + f),tseq) is set
[(vseq + f),tseq] is set
the Mult of RLSStruct(# (Funcs (X, the carrier of Y)),(X,Y),(X,Y),(X,Y) #) . [(vseq + f),tseq] is set
vseq * tseq is Element of the carrier of RLSStruct(# (Funcs (X, the carrier of Y)),(X,Y),(X,Y),(X,Y) #)
the Mult of RLSStruct(# (Funcs (X, the carrier of Y)),(X,Y),(X,Y),(X,Y) #) . (vseq,tseq) is set
[vseq,tseq] is set
the Mult of RLSStruct(# (Funcs (X, the carrier of Y)),(X,Y),(X,Y),(X,Y) #) . [vseq,tseq] is set
f * tseq is Element of the carrier of RLSStruct(# (Funcs (X, the carrier of Y)),(X,Y),(X,Y),(X,Y) #)
the Mult of RLSStruct(# (Funcs (X, the carrier of Y)),(X,Y),(X,Y),(X,Y) #) . (f,tseq) is set
[f,tseq] is set
the Mult of RLSStruct(# (Funcs (X, the carrier of Y)),(X,Y),(X,Y),(X,Y) #) . [f,tseq] is set
(vseq * tseq) + (f * tseq) is Element of the carrier of RLSStruct(# (Funcs (X, the carrier of Y)),(X,Y),(X,Y),(X,Y) #)
the addF of RLSStruct(# (Funcs (X, the carrier of Y)),(X,Y),(X,Y),(X,Y) #) is non empty Relation-like [: the carrier of RLSStruct(# (Funcs (X, the carrier of Y)),(X,Y),(X,Y),(X,Y) #), the carrier of RLSStruct(# (Funcs (X, the carrier of Y)),(X,Y),(X,Y),(X,Y) #):] -defined the carrier of RLSStruct(# (Funcs (X, the carrier of Y)),(X,Y),(X,Y),(X,Y) #) -valued Function-like total quasi_total Element of bool [:[: the carrier of RLSStruct(# (Funcs (X, the carrier of Y)),(X,Y),(X,Y),(X,Y) #), the carrier of RLSStruct(# (Funcs (X, the carrier of Y)),(X,Y),(X,Y),(X,Y) #):], the carrier of RLSStruct(# (Funcs (X, the carrier of Y)),(X,Y),(X,Y),(X,Y) #):]
[: the carrier of RLSStruct(# (Funcs (X, the carrier of Y)),(X,Y),(X,Y),(X,Y) #), the carrier of RLSStruct(# (Funcs (X, the carrier of Y)),(X,Y),(X,Y),(X,Y) #):] is non empty set
[:[: the carrier of RLSStruct(# (Funcs (X, the carrier of Y)),(X,Y),(X,Y),(X,Y) #), the carrier of RLSStruct(# (Funcs (X, the carrier of Y)),(X,Y),(X,Y),(X,Y) #):], the carrier of RLSStruct(# (Funcs (X, the carrier of Y)),(X,Y),(X,Y),(X,Y) #):] is non empty set
bool [:[: the carrier of RLSStruct(# (Funcs (X, the carrier of Y)),(X,Y),(X,Y),(X,Y) #), the carrier of RLSStruct(# (Funcs (X, the carrier of Y)),(X,Y),(X,Y),(X,Y) #):], the carrier of RLSStruct(# (Funcs (X, the carrier of Y)),(X,Y),(X,Y),(X,Y) #):] is non empty set
the addF of RLSStruct(# (Funcs (X, the carrier of Y)),(X,Y),(X,Y),(X,Y) #) . ((vseq * tseq),(f * tseq)) is Element of the carrier of RLSStruct(# (Funcs (X, the carrier of Y)),(X,Y),(X,Y),(X,Y) #)
[(vseq * tseq),(f * tseq)] is set
the addF of RLSStruct(# (Funcs (X, the carrier of Y)),(X,Y),(X,Y),(X,Y) #) . [(vseq * tseq),(f * tseq)] is set
f is Element of the carrier of RLSStruct(# (Funcs (X, the carrier of Y)),(X,Y),(X,Y),(X,Y) #)
tseq is Element of the carrier of RLSStruct(# (Funcs (X, the carrier of Y)),(X,Y),(X,Y),(X,Y) #)
f + tseq is Element of the carrier of RLSStruct(# (Funcs (X, the carrier of Y)),(X,Y),(X,Y),(X,Y) #)
the addF of RLSStruct(# (Funcs (X, the carrier of Y)),(X,Y),(X,Y),(X,Y) #) is non empty Relation-like [: the carrier of RLSStruct(# (Funcs (X, the carrier of Y)),(X,Y),(X,Y),(X,Y) #), the carrier of RLSStruct(# (Funcs (X, the carrier of Y)),(X,Y),(X,Y),(X,Y) #):] -defined the carrier of RLSStruct(# (Funcs (X, the carrier of Y)),(X,Y),(X,Y),(X,Y) #) -valued Function-like total quasi_total Element of bool [:[: the carrier of RLSStruct(# (Funcs (X, the carrier of Y)),(X,Y),(X,Y),(X,Y) #), the carrier of RLSStruct(# (Funcs (X, the carrier of Y)),(X,Y),(X,Y),(X,Y) #):], the carrier of RLSStruct(# (Funcs (X, the carrier of Y)),(X,Y),(X,Y),(X,Y) #):]
[: the carrier of RLSStruct(# (Funcs (X, the carrier of Y)),(X,Y),(X,Y),(X,Y) #), the carrier of RLSStruct(# (Funcs (X, the carrier of Y)),(X,Y),(X,Y),(X,Y) #):] is non empty set
[:[: the carrier of RLSStruct(# (Funcs (X, the carrier of Y)),(X,Y),(X,Y),(X,Y) #), the carrier of RLSStruct(# (Funcs (X, the carrier of Y)),(X,Y),(X,Y),(X,Y) #):], the carrier of RLSStruct(# (Funcs (X, the carrier of Y)),(X,Y),(X,Y),(X,Y) #):] is non empty set
bool [:[: the carrier of RLSStruct(# (Funcs (X, the carrier of Y)),(X,Y),(X,Y),(X,Y) #), the carrier of RLSStruct(# (Funcs (X, the carrier of Y)),(X,Y),(X,Y),(X,Y) #):], the carrier of RLSStruct(# (Funcs (X, the carrier of Y)),(X,Y),(X,Y),(X,Y) #):] is non empty set
the addF of RLSStruct(# (Funcs (X, the carrier of Y)),(X,Y),(X,Y),(X,Y) #) . (f,tseq) is Element of the carrier of RLSStruct(# (Funcs (X, the carrier of Y)),(X,Y),(X,Y),(X,Y) #)
[f,tseq] is set
the addF of RLSStruct(# (Funcs (X, the carrier of Y)),(X,Y),(X,Y),(X,Y) #) . [f,tseq] is set
tseq is Element of the carrier of RLSStruct(# (Funcs (X, the carrier of Y)),(X,Y),(X,Y),(X,Y) #)
(f + tseq) + tseq is Element of the carrier of RLSStruct(# (Funcs (X, the carrier of Y)),(X,Y),(X,Y),(X,Y) #)
the addF of RLSStruct(# (Funcs (X, the carrier of Y)),(X,Y),(X,Y),(X,Y) #) . ((f + tseq),tseq) is Element of the carrier of RLSStruct(# (Funcs (X, the carrier of Y)),(X,Y),(X,Y),(X,Y) #)
[(f + tseq),tseq] is set
the addF of RLSStruct(# (Funcs (X, the carrier of Y)),(X,Y),(X,Y),(X,Y) #) . [(f + tseq),tseq] is set
tseq + tseq is Element of the carrier of RLSStruct(# (Funcs (X, the carrier of Y)),(X,Y),(X,Y),(X,Y) #)
the addF of RLSStruct(# (Funcs (X, the carrier of Y)),(X,Y),(X,Y),(X,Y) #) . (tseq,tseq) is Element of the carrier of RLSStruct(# (Funcs (X, the carrier of Y)),(X,Y),(X,Y),(X,Y) #)
[tseq,tseq] is set
the addF of RLSStruct(# (Funcs (X, the carrier of Y)),(X,Y),(X,Y),(X,Y) #) . [tseq,tseq] is set
f + (tseq + tseq) is Element of the carrier of RLSStruct(# (Funcs (X, the carrier of Y)),(X,Y),(X,Y),(X,Y) #)
the addF of RLSStruct(# (Funcs (X, the carrier of Y)),(X,Y),(X,Y),(X,Y) #) . (f,(tseq + tseq)) is Element of the carrier of RLSStruct(# (Funcs (X, the carrier of Y)),(X,Y),(X,Y),(X,Y) #)
[f,(tseq + tseq)] is set
the addF of RLSStruct(# (Funcs (X, the carrier of Y)),(X,Y),(X,Y),(X,Y) #) . [f,(tseq + tseq)] is set
f is Element of the carrier of RLSStruct(# (Funcs (X, the carrier of Y)),(X,Y),(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)
[(- 1),tseq] is Element of [:REAL,(Funcs (X, the carrier of Y)):]
(X,Y) . [(- 1),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 RLSStruct(# (Funcs (X, the carrier of Y)),(X,Y),(X,Y),(X,Y) #)
f + tseq is Element of the carrier of RLSStruct(# (Funcs (X, the carrier of Y)),(X,Y),(X,Y),(X,Y) #)
the addF of RLSStruct(# (Funcs (X, the carrier of Y)),(X,Y),(X,Y),(X,Y) #) is non empty Relation-like [: the carrier of RLSStruct(# (Funcs (X, the carrier of Y)),(X,Y),(X,Y),(X,Y) #), the carrier of RLSStruct(# (Funcs (X, the carrier of Y)),(X,Y),(X,Y),(X,Y) #):] -defined the carrier of RLSStruct(# (Funcs (X, the carrier of Y)),(X,Y),(X,Y),(X,Y) #) -valued Function-like total quasi_total Element of bool [:[: the carrier of RLSStruct(# (Funcs (X, the carrier of Y)),(X,Y),(X,Y),(X,Y) #), the carrier of RLSStruct(# (Funcs (X, the carrier of Y)),(X,Y),(X,Y),(X,Y) #):], the carrier of RLSStruct(# (Funcs (X, the carrier of Y)),(X,Y),(X,Y),(X,Y) #):]
[: the carrier of RLSStruct(# (Funcs (X, the carrier of Y)),(X,Y),(X,Y),(X,Y) #), the carrier of RLSStruct(# (Funcs (X, the carrier of Y)),(X,Y),(X,Y),(X,Y) #):] is non empty set
[:[: the carrier of RLSStruct(# (Funcs (X, the carrier of Y)),(X,Y),(X,Y),(X,Y) #), the carrier of RLSStruct(# (Funcs (X, the carrier of Y)),(X,Y),(X,Y),(X,Y) #):], the carrier of RLSStruct(# (Funcs (X, the carrier of Y)),(X,Y),(X,Y),(X,Y) #):] is non empty set
bool [:[: the carrier of RLSStruct(# (Funcs (X, the carrier of Y)),(X,Y),(X,Y),(X,Y) #), the carrier of RLSStruct(# (Funcs (X, the carrier of Y)),(X,Y),(X,Y),(X,Y) #):], the carrier of RLSStruct(# (Funcs (X, the carrier of Y)),(X,Y),(X,Y),(X,Y) #):] is non empty set
the addF of RLSStruct(# (Funcs (X, the carrier of Y)),(X,Y),(X,Y),(X,Y) #) . (f,tseq) is Element of the carrier of RLSStruct(# (Funcs (X, the carrier of Y)),(X,Y),(X,Y),(X,Y) #)
[f,tseq] is set
the addF of RLSStruct(# (Funcs (X, the carrier of Y)),(X,Y),(X,Y),(X,Y) #) . [f,tseq] is set
0. RLSStruct(# (Funcs (X, the carrier of Y)),(X,Y),(X,Y),(X,Y) #) is V53( RLSStruct(# (Funcs (X, the carrier of Y)),(X,Y),(X,Y),(X,Y) #)) Element of the carrier of RLSStruct(# (Funcs (X, the carrier of Y)),(X,Y),(X,Y),(X,Y) #)
the ZeroF of RLSStruct(# (Funcs (X, the carrier of Y)),(X,Y),(X,Y),(X,Y) #) is Element of the carrier of RLSStruct(# (Funcs (X, the carrier of Y)),(X,Y),(X,Y),(X,Y) #)
f is ext-real V31() real set
tseq is ext-real V31() real Element of REAL
tseq is Element of the carrier of RLSStruct(# (Funcs (X, the carrier of Y)),(X,Y),(X,Y),(X,Y) #)
tseq is Element of the carrier of RLSStruct(# (Funcs (X, the carrier of Y)),(X,Y),(X,Y),(X,Y) #)
tseq + tseq is Element of the carrier of RLSStruct(# (Funcs (X, the carrier of Y)),(X,Y),(X,Y),(X,Y) #)
the addF of RLSStruct(# (Funcs (X, the carrier of Y)),(X,Y),(X,Y),(X,Y) #) is non empty Relation-like [: the carrier of RLSStruct(# (Funcs (X, the carrier of Y)),(X,Y),(X,Y),(X,Y) #), the carrier of RLSStruct(# (Funcs (X, the carrier of Y)),(X,Y),(X,Y),(X,Y) #):] -defined the carrier of RLSStruct(# (Funcs (X, the carrier of Y)),(X,Y),(X,Y),(X,Y) #) -valued Function-like total quasi_total Element of bool [:[: the carrier of RLSStruct(# (Funcs (X, the carrier of Y)),(X,Y),(X,Y),(X,Y) #), the carrier of RLSStruct(# (Funcs (X, the carrier of Y)),(X,Y),(X,Y),(X,Y) #):], the carrier of RLSStruct(# (Funcs (X, the carrier of Y)),(X,Y),(X,Y),(X,Y) #):]
[: the carrier of RLSStruct(# (Funcs (X, the carrier of Y)),(X,Y),(X,Y),(X,Y) #), the carrier of RLSStruct(# (Funcs (X, the carrier of Y)),(X,Y),(X,Y),(X,Y) #):] is non empty set
[:[: the carrier of RLSStruct(# (Funcs (X, the carrier of Y)),(X,Y),(X,Y),(X,Y) #), the carrier of RLSStruct(# (Funcs (X, the carrier of Y)),(X,Y),(X,Y),(X,Y) #):], the carrier of RLSStruct(# (Funcs (X, the carrier of Y)),(X,Y),(X,Y),(X,Y) #):] is non empty set
bool [:[: the carrier of RLSStruct(# (Funcs (X, the carrier of Y)),(X,Y),(X,Y),(X,Y) #), the carrier of RLSStruct(# (Funcs (X, the carrier of Y)),(X,Y),(X,Y),(X,Y) #):], the carrier of RLSStruct(# (Funcs (X, the carrier of Y)),(X,Y),(X,Y),(X,Y) #):] is non empty set
the addF of RLSStruct(# (Funcs (X, the carrier of Y)),(X,Y),(X,Y),(X,Y) #) . (tseq,tseq) is Element of the carrier of RLSStruct(# (Funcs (X, the carrier of Y)),(X,Y),(X,Y),(X,Y) #)
[tseq,tseq] is set
the addF of RLSStruct(# (Funcs (X, the carrier of Y)),(X,Y),(X,Y),(X,Y) #) . [tseq,tseq] is set
tseq * (tseq + tseq) is Element of the carrier of RLSStruct(# (Funcs (X, the carrier of Y)),(X,Y),(X,Y),(X,Y) #)
the Mult of RLSStruct(# (Funcs (X, the carrier of Y)),(X,Y),(X,Y),(X,Y) #) is non empty Relation-like [:REAL, the carrier of RLSStruct(# (Funcs (X, the carrier of Y)),(X,Y),(X,Y),(X,Y) #):] -defined the carrier of RLSStruct(# (Funcs (X, the carrier of Y)),(X,Y),(X,Y),(X,Y) #) -valued Function-like total quasi_total Element of bool [:[:REAL, the carrier of RLSStruct(# (Funcs (X, the carrier of Y)),(X,Y),(X,Y),(X,Y) #):], the carrier of RLSStruct(# (Funcs (X, the carrier of Y)),(X,Y),(X,Y),(X,Y) #):]
[:REAL, the carrier of RLSStruct(# (Funcs (X, the carrier of Y)),(X,Y),(X,Y),(X,Y) #):] is non empty set
[:[:REAL, the carrier of RLSStruct(# (Funcs (X, the carrier of Y)),(X,Y),(X,Y),(X,Y) #):], the carrier of RLSStruct(# (Funcs (X, the carrier of Y)),(X,Y),(X,Y),(X,Y) #):] is non empty set
bool [:[:REAL, the carrier of RLSStruct(# (Funcs (X, the carrier of Y)),(X,Y),(X,Y),(X,Y) #):], the carrier of RLSStruct(# (Funcs (X, the carrier of Y)),(X,Y),(X,Y),(X,Y) #):] is non empty set
the Mult of RLSStruct(# (Funcs (X, the carrier of Y)),(X,Y),(X,Y),(X,Y) #) . (tseq,(tseq + tseq)) is set
[tseq,(tseq + tseq)] is set
the Mult of RLSStruct(# (Funcs (X, the carrier of Y)),(X,Y),(X,Y),(X,Y) #) . [tseq,(tseq + tseq)] is set
tseq * tseq is Element of the carrier of RLSStruct(# (Funcs (X, the carrier of Y)),(X,Y),(X,Y),(X,Y) #)
the Mult of RLSStruct(# (Funcs (X, the carrier of Y)),(X,Y),(X,Y),(X,Y) #) . (tseq,tseq) is set
[tseq,tseq] is set
the Mult of RLSStruct(# (Funcs (X, the carrier of Y)),(X,Y),(X,Y),(X,Y) #) . [tseq,tseq] is set
tseq * tseq is Element of the carrier of RLSStruct(# (Funcs (X, the carrier of Y)),(X,Y),(X,Y),(X,Y) #)
the Mult of RLSStruct(# (Funcs (X, the carrier of Y)),(X,Y),(X,Y),(X,Y) #) . (tseq,tseq) is set
[tseq,tseq] is set
the Mult of RLSStruct(# (Funcs (X, the carrier of Y)),(X,Y),(X,Y),(X,Y) #) . [tseq,tseq] is set
(tseq * tseq) + (tseq * tseq) is Element of the carrier of RLSStruct(# (Funcs (X, the carrier of Y)),(X,Y),(X,Y),(X,Y) #)
the addF of RLSStruct(# (Funcs (X, the carrier of Y)),(X,Y),(X,Y),(X,Y) #) . ((tseq * tseq),(tseq * tseq)) is Element of the carrier of RLSStruct(# (Funcs (X, the carrier of Y)),(X,Y),(X,Y),(X,Y) #)
[(tseq * tseq),(tseq * tseq)] is set
the addF of RLSStruct(# (Funcs (X, the carrier of Y)),(X,Y),(X,Y),(X,Y) #) . [(tseq * tseq),(tseq * tseq)] is set
tseq is Element of the carrier of RLSStruct(# (Funcs (X, the carrier of Y)),(X,Y),(X,Y),(X,Y) #)
tseq is Element of the carrier of RLSStruct(# (Funcs (X, the carrier of Y)),(X,Y),(X,Y),(X,Y) #)
tseq + tseq is Element of the carrier of RLSStruct(# (Funcs (X, the carrier of Y)),(X,Y),(X,Y),(X,Y) #)
the addF of RLSStruct(# (Funcs (X, the carrier of Y)),(X,Y),(X,Y),(X,Y) #) is non empty Relation-like [: the carrier of RLSStruct(# (Funcs (X, the carrier of Y)),(X,Y),(X,Y),(X,Y) #), the carrier of RLSStruct(# (Funcs (X, the carrier of Y)),(X,Y),(X,Y),(X,Y) #):] -defined the carrier of RLSStruct(# (Funcs (X, the carrier of Y)),(X,Y),(X,Y),(X,Y) #) -valued Function-like total quasi_total Element of bool [:[: the carrier of RLSStruct(# (Funcs (X, the carrier of Y)),(X,Y),(X,Y),(X,Y) #), the carrier of RLSStruct(# (Funcs (X, the carrier of Y)),(X,Y),(X,Y),(X,Y) #):], the carrier of RLSStruct(# (Funcs (X, the carrier of Y)),(X,Y),(X,Y),(X,Y) #):]
[: the carrier of RLSStruct(# (Funcs (X, the carrier of Y)),(X,Y),(X,Y),(X,Y) #), the carrier of RLSStruct(# (Funcs (X, the carrier of Y)),(X,Y),(X,Y),(X,Y) #):] is non empty set
[:[: the carrier of RLSStruct(# (Funcs (X, the carrier of Y)),(X,Y),(X,Y),(X,Y) #), the carrier of RLSStruct(# (Funcs (X, the carrier of Y)),(X,Y),(X,Y),(X,Y) #):], the carrier of RLSStruct(# (Funcs (X, the carrier of Y)),(X,Y),(X,Y),(X,Y) #):] is non empty set
bool [:[: the carrier of RLSStruct(# (Funcs (X, the carrier of Y)),(X,Y),(X,Y),(X,Y) #), the carrier of RLSStruct(# (Funcs (X, the carrier of Y)),(X,Y),(X,Y),(X,Y) #):], the carrier of RLSStruct(# (Funcs (X, the carrier of Y)),(X,Y),(X,Y),(X,Y) #):] is non empty set
the addF of RLSStruct(# (Funcs (X, the carrier of Y)),(X,Y),(X,Y),(X,Y) #) . (tseq,tseq) is Element of the carrier of RLSStruct(# (Funcs (X, the carrier of Y)),(X,Y),(X,Y),(X,Y) #)
[tseq,tseq] is set
the addF of RLSStruct(# (Funcs (X, the carrier of Y)),(X,Y),(X,Y),(X,Y) #) . [tseq,tseq] is set
f * (tseq + tseq) is Element of the carrier of RLSStruct(# (Funcs (X, the carrier of Y)),(X,Y),(X,Y),(X,Y) #)
the Mult of RLSStruct(# (Funcs (X, the carrier of Y)),(X,Y),(X,Y),(X,Y) #) is non empty Relation-like [:REAL, the carrier of RLSStruct(# (Funcs (X, the carrier of Y)),(X,Y),(X,Y),(X,Y) #):] -defined the carrier of RLSStruct(# (Funcs (X, the carrier of Y)),(X,Y),(X,Y),(X,Y) #) -valued Function-like total quasi_total Element of bool [:[:REAL, the carrier of RLSStruct(# (Funcs (X, the carrier of Y)),(X,Y),(X,Y),(X,Y) #):], the carrier of RLSStruct(# (Funcs (X, the carrier of Y)),(X,Y),(X,Y),(X,Y) #):]
[:REAL, the carrier of RLSStruct(# (Funcs (X, the carrier of Y)),(X,Y),(X,Y),(X,Y) #):] is non empty set
[:[:REAL, the carrier of RLSStruct(# (Funcs (X, the carrier of Y)),(X,Y),(X,Y),(X,Y) #):], the carrier of RLSStruct(# (Funcs (X, the carrier of Y)),(X,Y),(X,Y),(X,Y) #):] is non empty set
bool [:[:REAL, the carrier of RLSStruct(# (Funcs (X, the carrier of Y)),(X,Y),(X,Y),(X,Y) #):], the carrier of RLSStruct(# (Funcs (X, the carrier of Y)),(X,Y),(X,Y),(X,Y) #):] is non empty set
the Mult of RLSStruct(# (Funcs (X, the carrier of Y)),(X,Y),(X,Y),(X,Y) #) . (f,(tseq + tseq)) is set
[f,(tseq + tseq)] is set
the Mult of RLSStruct(# (Funcs (X, the carrier of Y)),(X,Y),(X,Y),(X,Y) #) . [f,(tseq + tseq)] is set
f * tseq is Element of the carrier of RLSStruct(# (Funcs (X, the carrier of Y)),(X,Y),(X,Y),(X,Y) #)
the Mult of RLSStruct(# (Funcs (X, the carrier of Y)),(X,Y),(X,Y),(X,Y) #) . (f,tseq) is set
[f,tseq] is set
the Mult of RLSStruct(# (Funcs (X, the carrier of Y)),(X,Y),(X,Y),(X,Y) #) . [f,tseq] is set
f * tseq is Element of the carrier of RLSStruct(# (Funcs (X, the carrier of Y)),(X,Y),(X,Y),(X,Y) #)
the Mult of RLSStruct(# (Funcs (X, the carrier of Y)),(X,Y),(X,Y),(X,Y) #) . (f,tseq) is set
[f,tseq] is set
the Mult of RLSStruct(# (Funcs (X, the carrier of Y)),(X,Y),(X,Y),(X,Y) #) . [f,tseq] is set
(f * tseq) + (f * tseq) is Element of the carrier of RLSStruct(# (Funcs (X, the carrier of Y)),(X,Y),(X,Y),(X,Y) #)
the addF of RLSStruct(# (Funcs (X, the carrier of Y)),(X,Y),(X,Y),(X,Y) #) . ((f * tseq),(f * tseq)) is Element of the carrier of RLSStruct(# (Funcs (X, the carrier of Y)),(X,Y),(X,Y),(X,Y) #)
[(f * tseq),(f * tseq)] is set
the addF of RLSStruct(# (Funcs (X, the carrier of Y)),(X,Y),(X,Y),(X,Y) #) . [(f * tseq),(f * tseq)] is set
f is Element of the carrier of RLSStruct(# (Funcs (X, the carrier of Y)),(X,Y),(X,Y),(X,Y) #)
1 * f is Element of the carrier of RLSStruct(# (Funcs (X, the carrier of Y)),(X,Y),(X,Y),(X,Y) #)
the Mult of RLSStruct(# (Funcs (X, the carrier of Y)),(X,Y),(X,Y),(X,Y) #) is non empty Relation-like [:REAL, the carrier of RLSStruct(# (Funcs (X, the carrier of Y)),(X,Y),(X,Y),(X,Y) #):] -defined the carrier of RLSStruct(# (Funcs (X, the carrier of Y)),(X,Y),(X,Y),(X,Y) #) -valued Function-like total quasi_total Element of bool [:[:REAL, the carrier of RLSStruct(# (Funcs (X, the carrier of Y)),(X,Y),(X,Y),(X,Y) #):], the carrier of RLSStruct(# (Funcs (X, the carrier of Y)),(X,Y),(X,Y),(X,Y) #):]
[:REAL, the carrier of RLSStruct(# (Funcs (X, the carrier of Y)),(X,Y),(X,Y),(X,Y) #):] is non empty set
[:[:REAL, the carrier of RLSStruct(# (Funcs (X, the carrier of Y)),(X,Y),(X,Y),(X,Y) #):], the carrier of RLSStruct(# (Funcs (X, the carrier of Y)),(X,Y),(X,Y),(X,Y) #):] is non empty set
bool [:[:REAL, the carrier of RLSStruct(# (Funcs (X, the carrier of Y)),(X,Y),(X,Y),(X,Y) #):], the carrier of RLSStruct(# (Funcs (X, the carrier of Y)),(X,Y),(X,Y),(X,Y) #):] is non empty set
the Mult of RLSStruct(# (Funcs (X, the carrier of Y)),(X,Y),(X,Y),(X,Y) #) . (1,f) is set
[1,f] is set
the Mult of RLSStruct(# (Funcs (X, the carrier of Y)),(X,Y),(X,Y),(X,Y) #) . [1,f] is set
0. RLSStruct(# (Funcs (X, the carrier of Y)),(X,Y),(X,Y),(X,Y) #) is V53( RLSStruct(# (Funcs (X, the carrier of Y)),(X,Y),(X,Y),(X,Y) #)) Element of the carrier of RLSStruct(# (Funcs (X, the carrier of Y)),(X,Y),(X,Y),(X,Y) #)
the ZeroF of RLSStruct(# (Funcs (X, the carrier of Y)),(X,Y),(X,Y),(X,Y) #) is Element of the carrier of RLSStruct(# (Funcs (X, the carrier of Y)),(X,Y),(X,Y),(X,Y) #)
f is Element of the carrier of RLSStruct(# (Funcs (X, the carrier of Y)),(X,Y),(X,Y),(X,Y) #)
f + (0. RLSStruct(# (Funcs (X, the carrier of Y)),(X,Y),(X,Y),(X,Y) #)) is Element of the carrier of RLSStruct(# (Funcs (X, the carrier of Y)),(X,Y),(X,Y),(X,Y) #)
the addF of RLSStruct(# (Funcs (X, the carrier of Y)),(X,Y),(X,Y),(X,Y) #) is non empty Relation-like [: the carrier of RLSStruct(# (Funcs (X, the carrier of Y)),(X,Y),(X,Y),(X,Y) #), the carrier of RLSStruct(# (Funcs (X, the carrier of Y)),(X,Y),(X,Y),(X,Y) #):] -defined the carrier of RLSStruct(# (Funcs (X, the carrier of Y)),(X,Y),(X,Y),(X,Y) #) -valued Function-like total quasi_total Element of bool [:[: the carrier of RLSStruct(# (Funcs (X, the carrier of Y)),(X,Y),(X,Y),(X,Y) #), the carrier of RLSStruct(# (Funcs (X, the carrier of Y)),(X,Y),(X,Y),(X,Y) #):], the carrier of RLSStruct(# (Funcs (X, the carrier of Y)),(X,Y),(X,Y),(X,Y) #):]
[: the carrier of RLSStruct(# (Funcs (X, the carrier of Y)),(X,Y),(X,Y),(X,Y) #), the carrier of RLSStruct(# (Funcs (X, the carrier of Y)),(X,Y),(X,Y),(X,Y) #):] is non empty set
[:[: the carrier of RLSStruct(# (Funcs (X, the carrier of Y)),(X,Y),(X,Y),(X,Y) #), the carrier of RLSStruct(# (Funcs (X, the carrier of Y)),(X,Y),(X,Y),(X,Y) #):], the carrier of RLSStruct(# (Funcs (X, the carrier of Y)),(X,Y),(X,Y),(X,Y) #):] is non empty set
bool [:[: the carrier of RLSStruct(# (Funcs (X, the carrier of Y)),(X,Y),(X,Y),(X,Y) #), the carrier of RLSStruct(# (Funcs (X, the carrier of Y)),(X,Y),(X,Y),(X,Y) #):], the carrier of RLSStruct(# (Funcs (X, the carrier of Y)),(X,Y),(X,Y),(X,Y) #):] is non empty set
the addF of RLSStruct(# (Funcs (X, the carrier of Y)),(X,Y),(X,Y),(X,Y) #) . (f,(0. RLSStruct(# (Funcs (X, the carrier of Y)),(X,Y),(X,Y),(X,Y) #))) is Element of the carrier of RLSStruct(# (Funcs (X, the carrier of Y)),(X,Y),(X,Y),(X,Y) #)
[f,(0. RLSStruct(# (Funcs (X, the carrier of Y)),(X,Y),(X,Y),(X,Y) #))] is set
the addF of RLSStruct(# (Funcs (X, the carrier of Y)),(X,Y),(X,Y),(X,Y) #) . [f,(0. RLSStruct(# (Funcs (X, the carrier of Y)),(X,Y),(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)
(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)
[(X,Y),tseq] is set
(X,Y) . [(X,Y),tseq] is Relation-like Function-like set
f is Element of the carrier of RLSStruct(# (Funcs (X, the carrier of Y)),(X,Y),(X,Y),(X,Y) #)
tseq is Element of the carrier of RLSStruct(# (Funcs (X, the carrier of Y)),(X,Y),(X,Y),(X,Y) #)
f + tseq is Element of the carrier of RLSStruct(# (Funcs (X, the carrier of Y)),(X,Y),(X,Y),(X,Y) #)
the addF of RLSStruct(# (Funcs (X, the carrier of Y)),(X,Y),(X,Y),(X,Y) #) is non empty Relation-like [: the carrier of RLSStruct(# (Funcs (X, the carrier of Y)),(X,Y),(X,Y),(X,Y) #), the carrier of RLSStruct(# (Funcs (X, the carrier of Y)),(X,Y),(X,Y),(X,Y) #):] -defined the carrier of RLSStruct(# (Funcs (X, the carrier of Y)),(X,Y),(X,Y),(X,Y) #) -valued Function-like total quasi_total Element of bool [:[: the carrier of RLSStruct(# (Funcs (X, the carrier of Y)),(X,Y),(X,Y),(X,Y) #), the carrier of RLSStruct(# (Funcs (X, the carrier of Y)),(X,Y),(X,Y),(X,Y) #):], the carrier of RLSStruct(# (Funcs (X, the carrier of Y)),(X,Y),(X,Y),(X,Y) #):]
[: the carrier of RLSStruct(# (Funcs (X, the carrier of Y)),(X,Y),(X,Y),(X,Y) #), the carrier of RLSStruct(# (Funcs (X, the carrier of Y)),(X,Y),(X,Y),(X,Y) #):] is non empty set
[:[: the carrier of RLSStruct(# (Funcs (X, the carrier of Y)),(X,Y),(X,Y),(X,Y) #), the carrier of RLSStruct(# (Funcs (X, the carrier of Y)),(X,Y),(X,Y),(X,Y) #):], the carrier of RLSStruct(# (Funcs (X, the carrier of Y)),(X,Y),(X,Y),(X,Y) #):] is non empty set
bool [:[: the carrier of RLSStruct(# (Funcs (X, the carrier of Y)),(X,Y),(X,Y),(X,Y) #), the carrier of RLSStruct(# (Funcs (X, the carrier of Y)),(X,Y),(X,Y),(X,Y) #):], the carrier of RLSStruct(# (Funcs (X, the carrier of Y)),(X,Y),(X,Y),(X,Y) #):] is non empty set
the addF of RLSStruct(# (Funcs (X, the carrier of Y)),(X,Y),(X,Y),(X,Y) #) . (f,tseq) is Element of the carrier of RLSStruct(# (Funcs (X, the carrier of Y)),(X,Y),(X,Y),(X,Y) #)
[f,tseq] is set
the addF of RLSStruct(# (Funcs (X, the carrier of Y)),(X,Y),(X,Y),(X,Y) #) . [f,tseq] is set
tseq + f is Element of the carrier of RLSStruct(# (Funcs (X, the carrier of Y)),(X,Y),(X,Y),(X,Y) #)
the addF of RLSStruct(# (Funcs (X, the carrier of Y)),(X,Y),(X,Y),(X,Y) #) . (tseq,f) is Element of the carrier of RLSStruct(# (Funcs (X, the carrier of Y)),(X,Y),(X,Y),(X,Y) #)
[tseq,f] is set
the addF of RLSStruct(# (Funcs (X, the carrier of Y)),(X,Y),(X,Y),(X,Y) #) . [tseq,f] is set
X is non empty set
Y is non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() RLSStruct
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 Relation-like X -defined the carrier of Y -valued Function-like total quasi_total Element of Funcs (X, the carrier of Y)
0. Y is V53(Y) Element of the carrier of Y
the ZeroF of Y 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,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 V110() 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 [:REAL,(Funcs (X, the carrier of Y)):] -defined Funcs (X, the carrier of Y) -valued Function-like total quasi_total Function-yielding V110() Element of bool [:[:REAL,(Funcs (X, the carrier of Y)):],(Funcs (X, the carrier of Y)):]
[:REAL,(Funcs (X, the carrier of Y)):] is non empty set
[:[:REAL,(Funcs (X, the carrier of Y)):],(Funcs (X, the carrier of Y)):] is non empty set
bool [:[:REAL,(Funcs (X, the carrier of Y)):],(Funcs (X, the carrier of Y)):] is non empty set
RLSStruct(# (Funcs (X, the carrier of Y)),(X,Y),(X,Y),(X,Y) #) is non empty strict RLSStruct
X is non empty set
Y is non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() RLSStruct
(X,Y) is non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() RLSStruct
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 Relation-like X -defined the carrier of Y -valued Function-like total quasi_total Element of Funcs (X, the carrier of Y)
0. Y is V53(Y) Element of the carrier of Y
the ZeroF of Y 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,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 V110() 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 [:REAL,(Funcs (X, the carrier of Y)):] -defined Funcs (X, the carrier of Y) -valued Function-like total quasi_total Function-yielding V110() Element of bool [:[:REAL,(Funcs (X, the carrier of Y)):],(Funcs (X, the carrier of Y)):]
[:REAL,(Funcs (X, the carrier of Y)):] is non empty set
[:[:REAL,(Funcs (X, the carrier of Y)):],(Funcs (X, the carrier of Y)):] is non empty set
bool [:[:REAL,(Funcs (X, the carrier of Y)):],(Funcs (X, the carrier of Y)):] is non empty set
RLSStruct(# (Funcs (X, the carrier of Y)),(X,Y),(X,Y),(X,Y) #) is non empty strict RLSStruct
X is non empty set
Y is non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() RLSStruct
(X,Y) is non empty left_complementable right_complementable strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() RLSStruct
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 Relation-like X -defined the carrier of Y -valued Function-like total quasi_total Element of Funcs (X, the carrier of Y)
0. Y is V53(Y) Element of the carrier of Y
the ZeroF of Y 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,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 V110() 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 [:REAL,(Funcs (X, the carrier of Y)):] -defined Funcs (X, the carrier of Y) -valued Function-like total quasi_total Function-yielding V110() Element of bool [:[:REAL,(Funcs (X, the carrier of Y)):],(Funcs (X, the carrier of Y)):]
[:REAL,(Funcs (X, the carrier of Y)):] is non empty set
[:[:REAL,(Funcs (X, the carrier of Y)):],(Funcs (X, the carrier of Y)):] is non empty set
bool [:[:REAL,(Funcs (X, the carrier of Y)):],(Funcs (X, the carrier of Y)):] is non empty set
RLSStruct(# (Funcs (X, the carrier of Y)),(X,Y),(X,Y),(X,Y) #) is non empty strict RLSStruct
X is non empty set
Y is non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() RLSStruct
(X,Y) is non empty left_complementable right_complementable strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() constituted-Functions RLSStruct
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 Relation-like X -defined the carrier of Y -valued Function-like total quasi_total Element of Funcs (X, the carrier of Y)
0. Y is V53(Y) Element of the carrier of Y
the ZeroF of Y 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,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 V110() 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 [:REAL,(Funcs (X, the carrier of Y)):] -defined Funcs (X, the carrier of Y) -valued Function-like total quasi_total Function-yielding V110() Element of bool [:[:REAL,(Funcs (X, the carrier of Y)):],(Funcs (X, the carrier of Y)):]
[:REAL,(Funcs (X, the carrier of Y)):] is non empty set
[:[:REAL,(Funcs (X, the carrier of Y)):],(Funcs (X, the carrier of Y)):] is non empty set
bool [:[:REAL,(Funcs (X, the carrier of Y)):],(Funcs (X, the carrier of Y)):] is non empty set
RLSStruct(# (Funcs (X, the carrier of Y)),(X,Y),(X,Y),(X,Y) #) is non empty strict RLSStruct
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 left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() RLSStruct
(X,Y) is non empty left_complementable right_complementable strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() constituted-Functions RLSStruct
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 Relation-like X -defined the carrier of Y -valued Function-like total quasi_total Element of Funcs (X, the carrier of Y)
0. Y is V53(Y) Element of the carrier of Y
the ZeroF of Y 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,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 V110() 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 [:REAL,(Funcs (X, the carrier of Y)):] -defined Funcs (X, the carrier of Y) -valued Function-like total quasi_total Function-yielding V110() Element of bool [:[:REAL,(Funcs (X, the carrier of Y)):],(Funcs (X, the carrier of Y)):]
[:REAL,(Funcs (X, the carrier of Y)):] is non empty set
[:[:REAL,(Funcs (X, the carrier of Y)):],(Funcs (X, the carrier of Y)):] is non empty set
bool [:[:REAL,(Funcs (X, the carrier of Y)):],(Funcs (X, the carrier of Y)):] is non empty set
RLSStruct(# (Funcs (X, the carrier of Y)),(X,Y),(X,Y),(X,Y) #) is non empty strict RLSStruct
the carrier of (X,Y) is non empty set
tseq is non empty set
tv is non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() RLSStruct
(tseq,tv) is non empty left_complementable right_complementable strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() constituted-Functions RLSStruct
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
(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)
0. tv is V53(tv) Element of the carrier of tv
the ZeroF of tv is Element of the carrier of tv
tseq --> (0. tv) is non empty Relation-like tseq -defined the carrier of tv -valued Function-like constant total quasi_total Element of bool [:tseq, the carrier of tv:]
[:tseq, the carrier of tv:] is non empty set
bool [:tseq, the carrier of tv:] is non empty set
(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 V110() 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 [:REAL,(Funcs (tseq, the carrier of tv)):] -defined Funcs (tseq, the carrier of tv) -valued Function-like total quasi_total Function-yielding V110() Element of bool [:[:REAL,(Funcs (tseq, the carrier of tv)):],(Funcs (tseq, the carrier of tv)):]
[:REAL,(Funcs (tseq, the carrier of tv)):] is non empty set
[:[:REAL,(Funcs (tseq, the carrier of tv)):],(Funcs (tseq, the carrier of tv)):] is non empty set
bool [:[:REAL,(Funcs (tseq, the carrier of tv)):],(Funcs (tseq, the carrier of tv)):] is non empty set
RLSStruct(# (Funcs (tseq, the carrier of tv)),(tseq,tv),(tseq,tv),(tseq,tv) #) is non empty strict RLSStruct
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 left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() RLSStruct
(X,Y) is non empty left_complementable right_complementable strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() constituted-Functions RLSStruct
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 Relation-like X -defined the carrier of Y -valued Function-like total quasi_total Element of Funcs (X, the carrier of Y)
0. Y is V53(Y) Element of the carrier of Y
the ZeroF of Y 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,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 V110() 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 [:REAL,(Funcs (X, the carrier of Y)):] -defined Funcs (X, the carrier of Y) -valued Function-like total quasi_total Function-yielding V110() Element of bool [:[:REAL,(Funcs (X, the carrier of Y)):],(Funcs (X, the carrier of Y)):]
[:REAL,(Funcs (X, the carrier of Y)):] is non empty set
[:[:REAL,(Funcs (X, the carrier of Y)):],(Funcs (X, the carrier of Y)):] is non empty set
bool [:[:REAL,(Funcs (X, the carrier of Y)):],(Funcs (X, the carrier of Y)):] is non empty set
RLSStruct(# (Funcs (X, the carrier of Y)),(X,Y),(X,Y),(X,Y) #) is non empty strict RLSStruct
the carrier of (X,Y) is non empty set
tseq is non empty set
tv is non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() RLSStruct
(tseq,tv) is non empty left_complementable right_complementable strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() constituted-Functions RLSStruct
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
(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)
0. tv is V53(tv) Element of the carrier of tv
the ZeroF of tv is Element of the carrier of tv
tseq --> (0. tv) is non empty Relation-like tseq -defined the carrier of tv -valued Function-like constant total quasi_total Element of bool [:tseq, the carrier of tv:]
[:tseq, the carrier of tv:] is non empty set
bool [:tseq, the carrier of tv:] is non empty set
(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 V110() 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 [:REAL,(Funcs (tseq, the carrier of tv)):] -defined Funcs (tseq, the carrier of tv) -valued Function-like total quasi_total Function-yielding V110() Element of bool [:[:REAL,(Funcs (tseq, the carrier of tv)):],(Funcs (tseq, the carrier of tv)):]
[:REAL,(Funcs (tseq, the carrier of tv)):] is non empty set
[:[:REAL,(Funcs (tseq, the carrier of tv)):],(Funcs (tseq, the carrier of tv)):] is non empty set
bool [:[:REAL,(Funcs (tseq, the carrier of tv)):],(Funcs (tseq, the carrier of tv)):] is non empty set
RLSStruct(# (Funcs (tseq, the carrier of tv)),(tseq,tv),(tseq,tv),(tseq,tv) #) is non empty strict RLSStruct
the carrier of (tseq,tv) 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 ext-real V31() real Element of REAL
tseq * vseq is Relation-like Function-like Element of the carrier of (X,Y)
the Mult of (X,Y) is non empty Relation-like [:REAL, the carrier of (X,Y):] -defined the carrier of (X,Y) -valued Function-like total quasi_total Element of bool [:[:REAL, the carrier of (X,Y):], the carrier of (X,Y):]
[:REAL, the carrier of (X,Y):] is non empty set
[:[:REAL, the carrier of (X,Y):], the carrier of (X,Y):] is non empty set
bool [:[:REAL, the carrier of (X,Y):], the carrier of (X,Y):] is non empty set
the Mult of (X,Y) . (tseq,vseq) is set
[tseq,vseq] is set
the Mult of (X,Y) . [tseq,vseq] is set
tseq is Element of X
(X,Y,f,tseq) is Element of the carrier of Y
(X,Y,vseq,tseq) is Element of the carrier of Y
tseq * (X,Y,vseq,tseq) is Element of the carrier of Y
the Mult of Y is non empty Relation-like [:REAL, the carrier of Y:] -defined the carrier of Y -valued Function-like total quasi_total Element of bool [:[:REAL, the carrier of Y:], the carrier of Y:]
[:REAL, the carrier of Y:] is non empty set
[:[:REAL, the carrier of Y:], the carrier of Y:] is non empty set
bool [:[:REAL, the carrier of Y:], the carrier of Y:] is non empty set
the Mult of Y . (tseq,(X,Y,vseq,tseq)) is set
[tseq,(X,Y,vseq,tseq)] is set
the Mult of Y . [tseq,(X,Y,vseq,tseq)] is set
m 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)
n is ext-real V31() real Element of REAL
n * e is Relation-like Function-like Element of the carrier of (tseq,tv)
the Mult of (tseq,tv) is non empty Relation-like [:REAL, the carrier of (tseq,tv):] -defined the carrier of (tseq,tv) -valued Function-like total quasi_total Element of bool [:[:REAL, the carrier of (tseq,tv):], the carrier of (tseq,tv):]
[:REAL, the carrier of (tseq,tv):] is non empty set
[:[:REAL, the carrier of (tseq,tv):], the carrier of (tseq,tv):] is non empty set
bool [:[:REAL, the carrier of (tseq,tv):], the carrier of (tseq,tv):] is non empty set
the Mult of (tseq,tv) . (n,e) is set
[n,e] is set
the Mult of (tseq,tv) . [n,e] is set
X is non empty set
Y is non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() RLSStruct
(X,Y) is non empty left_complementable right_complementable strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() constituted-Functions RLSStruct
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 Relation-like X -defined the carrier of Y -valued Function-like total quasi_total Element of Funcs (X, the carrier of Y)
0. Y is V53(Y) Element of the carrier of Y
the ZeroF of Y 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,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 V110() 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 [:REAL,(Funcs (X, the carrier of Y)):] -defined Funcs (X, the carrier of Y) -valued Function-like total quasi_total Function-yielding V110() Element of bool [:[:REAL,(Funcs (X, the carrier of Y)):],(Funcs (X, the carrier of Y)):]
[:REAL,(Funcs (X, the carrier of Y)):] is non empty set
[:[:REAL,(Funcs (X, the carrier of Y)):],(Funcs (X, the carrier of Y)):] is non empty set
bool [:[:REAL,(Funcs (X, the carrier of Y)):],(Funcs (X, the carrier of Y)):] is non empty set
RLSStruct(# (Funcs (X, the carrier of Y)),(X,Y),(X,Y),(X,Y) #) is non empty strict RLSStruct
0. (X,Y) is Relation-like Function-like V53((X,Y)) 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 RLSStruct
the carrier of X is non empty set
Y is non empty RLSStruct
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 RLSStruct
the carrier of X is non empty set
Y is non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() RLSStruct
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 V53(Y) 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 ext-real V31() real Element of REAL
tseq * tseq is Element of the carrier of X
the Mult of X is non empty Relation-like [:REAL, the carrier of X:] -defined the carrier of X -valued Function-like total quasi_total Element of bool [:[:REAL, the carrier of X:], the carrier of X:]
[:REAL, the carrier of X:] is non empty set
[:[:REAL, the carrier of X:], the carrier of X:] is non empty set
bool [:[:REAL, the carrier of X:], the carrier of X:] is non empty set
the Mult of X . (tseq,tseq) is set
[tseq,tseq] is set
the Mult of X . [tseq,tseq] is set
f . (tseq * tseq) is Element of the carrier of Y
tseq * (0. Y) is Element of the carrier of Y
the Mult of Y is non empty Relation-like [:REAL, the carrier of Y:] -defined the carrier of Y -valued Function-like total quasi_total Element of bool [:[:REAL, the carrier of Y:], the carrier of Y:]
[:REAL, the carrier of Y:] is non empty set
[:[:REAL, the carrier of Y:], the carrier of Y:] is non empty set
bool [:[:REAL, the carrier of Y:], the carrier of Y:] is non empty set
the Mult of Y . (tseq,(0. Y)) is set
[tseq,(0. Y)] is set
the Mult of Y . [tseq,(0. Y)] is set
f . tseq is Element of the carrier of Y
tseq * (f . tseq) is Element of the carrier of Y
the Mult of Y . (tseq,(f . tseq)) is set
[tseq,(f . tseq)] is set
the Mult of Y . [tseq,(f . tseq)] is set
X is non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() RLSStruct
the carrier of X is non empty set
Y is non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() RLSStruct
( the carrier of X,Y) is non empty left_complementable right_complementable strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() constituted-Functions RLSStruct
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
( 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)
0. Y is V53(Y) 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
( 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 V110() 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 [:REAL,(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 V110() Element of bool [:[:REAL,(Funcs ( the carrier of X, the carrier of Y)):],(Funcs ( the carrier of X, the carrier of Y)):]
[:REAL,(Funcs ( the carrier of X, the carrier of Y)):] is non empty set
[:[:REAL,(Funcs ( the carrier of X, the carrier of Y)):],(Funcs ( the carrier of X, the carrier of Y)):] is non empty set
bool [:[:REAL,(Funcs ( the carrier of X, the carrier of Y)):],(Funcs ( the carrier of X, the carrier of Y)):] is non empty set
RLSStruct(# (Funcs ( the carrier of X, the carrier of Y)),( the carrier of X,Y),( the carrier of X,Y),( the carrier of X,Y) #) is non empty strict RLSStruct
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
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 left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() RLSStruct
Y is non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() RLSStruct
(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 left_complementable right_complementable strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() constituted-Functions RLSStruct
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
( 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)
0. Y is V53(Y) 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
( 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 V110() 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 [:REAL,(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 V110() Element of bool [:[:REAL,(Funcs ( the carrier of X, the carrier of Y)):],(Funcs ( the carrier of X, the carrier of Y)):]
[:REAL,(Funcs ( the carrier of X, the carrier of Y)):] is non empty set
[:[:REAL,(Funcs ( the carrier of X, the carrier of Y)):],(Funcs ( the carrier of X, the carrier of Y)):] is non empty set
bool [:[:REAL,(Funcs ( the carrier of X, the carrier of Y)):],(Funcs ( the carrier of X, the carrier of Y)):] is non empty set
RLSStruct(# (Funcs ( the carrier of X, the carrier of Y)),( the carrier of X,Y),( the carrier of X,Y),( the carrier of X,Y) #) is non empty strict RLSStruct
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 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 ext-real V31() real Element of REAL
tseq * tseq is Element of the carrier of X
the Mult of X is non empty Relation-like [:REAL, the carrier of X:] -defined the carrier of X -valued Function-like total quasi_total Element of bool [:[:REAL, the carrier of X:], the carrier of X:]
[:REAL, the carrier of X:] is non empty set
[:[:REAL, the carrier of X:], the carrier of X:] is non empty set
bool [:[:REAL, the carrier of X:], the carrier of X:] is non empty set
the Mult of X . (tseq,tseq) is set
[tseq,tseq] is set
the Mult of X . [tseq,tseq] is set
f . (tseq * tseq) is Element of the carrier of Y
tseq * (f . tseq) is Element of the carrier of Y
the Mult of Y is non empty Relation-like [:REAL, the carrier of Y:] -defined the carrier of Y -valued Function-like total quasi_total Element of bool [:[:REAL, the carrier of Y:], the carrier of Y:]
[:REAL, the carrier of Y:] is non empty set
[:[:REAL, the carrier of Y:], the carrier of Y:] is non empty set
bool [:[:REAL, the carrier of Y:], the carrier of Y:] is non empty set
the Mult of Y . (tseq,(f . tseq)) is set
[tseq,(f . tseq)] is set
the Mult of Y . [tseq,(f . tseq)] is set
tseq * (0. Y) is Element of the carrier of Y
the Mult of Y . (tseq,(0. Y)) is set
[tseq,(0. Y)] is set
the Mult of Y . [tseq,(0. Y)] is set
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
tseq is set
X is non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() RLSStruct
the carrier of X is non empty set
Y is non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() RLSStruct
( the carrier of X,Y) is non empty left_complementable right_complementable strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() constituted-Functions RLSStruct
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
( 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)
0. Y is V53(Y) 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
( 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 V110() 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 [:REAL,(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 V110() Element of bool [:[:REAL,(Funcs ( the carrier of X, the carrier of Y)):],(Funcs ( the carrier of X, the carrier of Y)):]
[:REAL,(Funcs ( the carrier of X, the carrier of Y)):] is non empty set
[:[:REAL,(Funcs ( the carrier of X, the carrier of Y)):],(Funcs ( the carrier of X, the carrier of Y)):] is non empty set
bool [:[:REAL,(Funcs ( the carrier of X, the carrier of Y)):],(Funcs ( the carrier of X, the carrier of Y)):] is non empty set
RLSStruct(# (Funcs ( the carrier of X, the carrier of Y)),( the carrier of X,Y),( the carrier of X,Y),( the carrier of X,Y) #) is non empty strict RLSStruct
(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 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
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 ext-real V31() real Element of REAL
m * e is Element of the carrier of X
the Mult of X is non empty Relation-like [:REAL, the carrier of X:] -defined the carrier of X -valued Function-like total quasi_total Element of bool [:[:REAL, the carrier of X:], the carrier of X:]
[:REAL, the carrier of X:] is non empty set
[:[:REAL, the carrier of X:], the carrier of X:] is non empty set
bool [:[:REAL, the carrier of X:], the carrier of X:] is non empty set
the Mult of X . (m,e) is set
[m,e] is set
the Mult of X . [m,e] is set
tseq . (m * e) is Element of the carrier of Y
m * (tseq . e) is Element of the carrier of Y
the Mult of Y is non empty Relation-like [:REAL, the carrier of Y:] -defined the carrier of Y -valued Function-like total quasi_total Element of bool [:[:REAL, the carrier of Y:], the carrier of Y:]
[:REAL, the carrier of Y:] is non empty set
[:[:REAL, the carrier of Y:], the carrier of Y:] is non empty set
bool [:[:REAL, the carrier of Y:], the carrier of Y:] is non empty set
the Mult of Y . (m,(tseq . e)) is set
[m,(tseq . e)] is set
the Mult of Y . [m,(tseq . e)] 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)
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
the Mult of Y . (m,(tv . e)) is set
[m,(tv . e)] is set
the Mult of Y . [m,(tv . e)] is set
(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
the Mult of Y . (m,(tseq . e)) is set
[m,(tseq . e)] is set
the Mult of Y . [m,(tseq . e)] is set
(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
the Mult of Y . (m,((tv . e) + (tseq . e))) is set
[m,((tv . e) + (tseq . e))] is set
the Mult of Y . [m,((tv . e) + (tseq . e))] is set
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
f is ext-real V31() real Element of REAL
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 Mult of ( the carrier of X,Y) is non empty Relation-like [:REAL, 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 [:[:REAL, the carrier of ( the carrier of X,Y):], the carrier of ( the carrier of X,Y):]
[:REAL, the carrier of ( the carrier of X,Y):] is non empty set
[:[:REAL, the carrier of ( the carrier of X,Y):], the carrier of ( the carrier of X,Y):] is non empty set
bool [:[:REAL, the carrier of ( the carrier of X,Y):], the carrier of ( the carrier of X,Y):] is non empty set
the Mult of ( the carrier of X,Y) . (f,tseq) is set
[f,tseq] is set
the Mult of ( the carrier of X,Y) . [f,tseq] is 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 ext-real V31() real Element of REAL
e * tv is Element of the carrier of X
the Mult of X is non empty Relation-like [:REAL, the carrier of X:] -defined the carrier of X -valued Function-like total quasi_total Element of bool [:[:REAL, the carrier of X:], the carrier of X:]
[:REAL, the carrier of X:] is non empty set
[:[:REAL, the carrier of X:], the carrier of X:] is non empty set
bool [:[:REAL, the carrier of X:], the carrier of X:] is non empty set
the Mult of X . (e,tv) is set
[e,tv] is set
the Mult of X . [e,tv] is set
tseq . (e * tv) is Element of the carrier of Y
e * (tseq . tv) is Element of the carrier of Y
the Mult of Y is non empty Relation-like [:REAL, the carrier of Y:] -defined the carrier of Y -valued Function-like total quasi_total Element of bool [:[:REAL, the carrier of Y:], the carrier of Y:]
[:REAL, the carrier of Y:] is non empty set
[:[:REAL, the carrier of Y:], the carrier of Y:] is non empty set
bool [:[:REAL, the carrier of Y:], the carrier of Y:] is non empty set
the Mult of Y . (e,(tseq . tv)) is set
[e,(tseq . tv)] is set
the Mult of Y . [e,(tseq . tv)] 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)
tseq . (e * tv) is Element of the carrier of Y
f * (tseq . (e * tv)) is Element of the carrier of Y
the Mult of Y . (f,(tseq . (e * tv))) is set
[f,(tseq . (e * tv))] is set
the Mult of Y . [f,(tseq . (e * tv))] is set
tseq . tv is Element of the carrier of Y
e * (tseq . tv) is Element of the carrier of Y
the Mult of Y . (e,(tseq . tv)) is set
[e,(tseq . tv)] is set
the Mult of Y . [e,(tseq . tv)] is set
f * (e * (tseq . tv)) is Element of the carrier of Y
the Mult of Y . (f,(e * (tseq . tv))) is set
[f,(e * (tseq . tv))] is set
the Mult of Y . [f,(e * (tseq . tv))] is set
f * e is ext-real V31() real Element of REAL
(f * e) * (tseq . tv) is Element of the carrier of Y
the Mult of Y . ((f * e),(tseq . tv)) is set
[(f * e),(tseq . tv)] is set
the Mult of Y . [(f * e),(tseq . tv)] is set
f * (tseq . tv) is Element of the carrier of Y
the Mult of Y . (f,(tseq . tv)) is set
[f,(tseq . tv)] is set
the Mult of Y . [f,(tseq . tv)] is set
e * (f * (tseq . tv)) is Element of the carrier of Y
the Mult of Y . (e,(f * (tseq . tv))) is set
[e,(f * (tseq . tv))] is set
the Mult of Y . [e,(f * (tseq . tv))] is set
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)
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
the Mult of Y is non empty Relation-like [:REAL, the carrier of Y:] -defined the carrier of Y -valued Function-like total quasi_total Element of bool [:[:REAL, the carrier of Y:], the carrier of Y:]
[:REAL, the carrier of Y:] is non empty set
[:[:REAL, the carrier of Y:], the carrier of Y:] is non empty set
bool [:[:REAL, the carrier of Y:], the carrier of Y:] is non empty set
the Mult of Y . (f,(tseq . (tv + e))) is set
[f,(tseq . (tv + e))] is set
the Mult of Y . [f,(tseq . (tv + e))] is set
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
the Mult of Y . (f,((tseq . tv) + (tseq . e))) is set
[f,((tseq . tv) + (tseq . e))] is set
the Mult of Y . [f,((tseq . tv) + (tseq . e))] is set
f * (tseq . tv) is Element of the carrier of Y
the Mult of Y . (f,(tseq . tv)) is set
[f,(tseq . tv)] is set
the Mult of Y . [f,(tseq . tv)] is set
f * (tseq . e) is Element of the carrier of Y
the Mult of Y . (f,(tseq . e)) is set
[f,(tseq . e)] is set
the Mult of Y . [f,(tseq . e)] is set
(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
X is non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() RLSStruct
Y is non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() RLSStruct
(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 left_complementable right_complementable strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() constituted-Functions RLSStruct
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
( 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)
0. Y is V53(Y) 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
( 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 V110() 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 [:REAL,(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 V110() Element of bool [:[:REAL,(Funcs ( the carrier of X, the carrier of Y)):],(Funcs ( the carrier of X, the carrier of Y)):]
[:REAL,(Funcs ( the carrier of X, the carrier of Y)):] is non empty set
[:[:REAL,(Funcs ( the carrier of X, the carrier of Y)):],(Funcs ( the carrier of X, the carrier of Y)):] is non empty set
bool [:[:REAL,(Funcs ( the carrier of X, the carrier of Y)):],(Funcs ( the carrier of X, the carrier of Y)):] is non empty set
RLSStruct(# (Funcs ( the carrier of X, the carrier of Y)),( the carrier of X,Y),( the carrier of X,Y),( the carrier of X,Y) #) is non empty strict RLSStruct
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 V110() 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 [:REAL,(X,Y):] -defined (X,Y) -valued Function-like total quasi_total Function-yielding V110() Element of bool [:[:REAL,(X,Y):],(X,Y):]
[:REAL,(X,Y):] is non empty set
[:[:REAL,(X,Y):],(X,Y):] is non empty set
bool [:[:REAL,(X,Y):],(X,Y):] is non empty set
RLSStruct(# (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 RLSStruct
X is non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() RLSStruct
Y is non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() RLSStruct
(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 left_complementable right_complementable strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() constituted-Functions RLSStruct
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
( 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)
0. Y is V53(Y) 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
( 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 V110() 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 [:REAL,(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 V110() Element of bool [:[:REAL,(Funcs ( the carrier of X, the carrier of Y)):],(Funcs ( the carrier of X, the carrier of Y)):]
[:REAL,(Funcs ( the carrier of X, the carrier of Y)):] is non empty set
[:[:REAL,(Funcs ( the carrier of X, the carrier of Y)):],(Funcs ( the carrier of X, the carrier of Y)):] is non empty set
bool [:[:REAL,(Funcs ( the carrier of X, the carrier of Y)):],(Funcs ( the carrier of X, the carrier of Y)):] is non empty set
RLSStruct(# (Funcs ( the carrier of X, the carrier of Y)),( the carrier of X,Y),( the carrier of X,Y),( the carrier of X,Y) #) is non empty strict RLSStruct
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 V110() 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 [:REAL,(X,Y):] -defined (X,Y) -valued Function-like total quasi_total Function-yielding V110() Element of bool [:[:REAL,(X,Y):],(X,Y):]
[:REAL,(X,Y):] is non empty set
[:[:REAL,(X,Y):],(X,Y):] is non empty set
bool [:[:REAL,(X,Y):],(X,Y):] is non empty set
RLSStruct(# (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 RLSStruct
X is non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() RLSStruct
Y is non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() RLSStruct
(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 left_complementable right_complementable strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() constituted-Functions RLSStruct
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
( 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)
0. Y is V53(Y) 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
( 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 V110() 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 [:REAL,(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 V110() Element of bool [:[:REAL,(Funcs ( the carrier of X, the carrier of Y)):],(Funcs ( the carrier of X, the carrier of Y)):]
[:REAL,(Funcs ( the carrier of X, the carrier of Y)):] is non empty set
[:[:REAL,(Funcs ( the carrier of X, the carrier of Y)):],(Funcs ( the carrier of X, the carrier of Y)):] is non empty set
bool [:[:REAL,(Funcs ( the carrier of X, the carrier of Y)):],(Funcs ( the carrier of X, the carrier of Y)):] is non empty set
RLSStruct(# (Funcs ( the carrier of X, the carrier of Y)),( the carrier of X,Y),( the carrier of X,Y),( the carrier of X,Y) #) is non empty strict RLSStruct
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 V110() 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 [:REAL,(X,Y):] -defined (X,Y) -valued Function-like total quasi_total Function-yielding V110() Element of bool [:[:REAL,(X,Y):],(X,Y):]
[:REAL,(X,Y):] is non empty set
[:[:REAL,(X,Y):],(X,Y):] is non empty set
bool [:[:REAL,(X,Y):],(X,Y):] is non empty set
RLSStruct(# (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 left_complementable right_complementable strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() RLSStruct
X is non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() RLSStruct
Y is non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() RLSStruct
(X,Y) is non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() RLSStruct
(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 left_complementable right_complementable strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() constituted-Functions RLSStruct
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
( 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)
0. Y is V53(Y) 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
( 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 V110() 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 [:REAL,(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 V110() Element of bool [:[:REAL,(Funcs ( the carrier of X, the carrier of Y)):],(Funcs ( the carrier of X, the carrier of Y)):]
[:REAL,(Funcs ( the carrier of X, the carrier of Y)):] is non empty set
[:[:REAL,(Funcs ( the carrier of X, the carrier of Y)):],(Funcs ( the carrier of X, the carrier of Y)):] is non empty set
bool [:[:REAL,(Funcs ( the carrier of X, the carrier of Y)):],(Funcs ( the carrier of X, the carrier of Y)):] is non empty set
RLSStruct(# (Funcs ( the carrier of X, the carrier of Y)),( the carrier of X,Y),( the carrier of X,Y),( the carrier of X,Y) #) is non empty strict RLSStruct
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 V110() 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 [:REAL,(X,Y):] -defined (X,Y) -valued Function-like total quasi_total Function-yielding V110() Element of bool [:[:REAL,(X,Y):],(X,Y):]
[:REAL,(X,Y):] is non empty set
[:[:REAL,(X,Y):],(X,Y):] is non empty set
bool [:[:REAL,(X,Y):],(X,Y):] is non empty set
RLSStruct(# (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 left_complementable right_complementable strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() RLSStruct
X is non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() RLSStruct
Y is non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() RLSStruct
(X,Y) is non empty left_complementable right_complementable strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() RLSStruct
(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 left_complementable right_complementable strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() constituted-Functions RLSStruct
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
( 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)
0. Y is V53(Y) 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
( 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 V110() 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 [:REAL,(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 V110() Element of bool [:[:REAL,(Funcs ( the carrier of X, the carrier of Y)):],(Funcs ( the carrier of X, the carrier of Y)):]
[:REAL,(Funcs ( the carrier of X, the carrier of Y)):] is non empty set
[:[:REAL,(Funcs ( the carrier of X, the carrier of Y)):],(Funcs ( the carrier of X, the carrier of Y)):] is non empty set
bool [:[:REAL,(Funcs ( the carrier of X, the carrier of Y)):],(Funcs ( the carrier of X, the carrier of Y)):] is non empty set
RLSStruct(# (Funcs ( the carrier of X, the carrier of Y)),( the carrier of X,Y),( the carrier of X,Y),( the carrier of X,Y) #) is non empty strict RLSStruct
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 V110() 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 [:REAL,(X,Y):] -defined (X,Y) -valued Function-like total quasi_total Function-yielding V110() Element of bool [:[:REAL,(X,Y):],(X,Y):]
[:REAL,(X,Y):] is non empty set
[:[:REAL,(X,Y):],(X,Y):] is non empty set
bool [:[:REAL,(X,Y):],(X,Y):] is non empty set
RLSStruct(# (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 left_complementable right_complementable strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() RLSStruct
X is non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() RLSStruct
Y is non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() RLSStruct
(X,Y) is non empty left_complementable right_complementable strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() constituted-Functions RLSStruct
(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 left_complementable right_complementable strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() constituted-Functions RLSStruct
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
( 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)
0. Y is V53(Y) 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
( 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 V110() 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 [:REAL,(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 V110() Element of bool [:[:REAL,(Funcs ( the carrier of X, the carrier of Y)):],(Funcs ( the carrier of X, the carrier of Y)):]
[:REAL,(Funcs ( the carrier of X, the carrier of Y)):] is non empty set
[:[:REAL,(Funcs ( the carrier of X, the carrier of Y)):],(Funcs ( the carrier of X, the carrier of Y)):] is non empty set
bool [:[:REAL,(Funcs ( the carrier of X, the carrier of Y)):],(Funcs ( the carrier of X, the carrier of Y)):] is non empty set
RLSStruct(# (Funcs ( the carrier of X, the carrier of Y)),( the carrier of X,Y),( the carrier of X,Y),( the carrier of X,Y) #) is non empty strict RLSStruct
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 V110() 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 [:REAL,(X,Y):] -defined (X,Y) -valued Function-like total quasi_total Function-yielding V110() Element of bool [:[:REAL,(X,Y):],(X,Y):]
[:REAL,(X,Y):] is non empty set
[:[:REAL,(X,Y):],(X,Y):] is non empty set
bool [:[:REAL,(X,Y):],(X,Y):] is non empty set
RLSStruct(# (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 left_complementable right_complementable strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() RLSStruct
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
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 left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() RLSStruct
Y is non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() RLSStruct
(X,Y) is non empty left_complementable right_complementable strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() constituted-Functions RLSStruct
(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 left_complementable right_complementable strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() constituted-Functions RLSStruct
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
( 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)
0. Y is V53(Y) 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
( 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 V110() 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 [:REAL,(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 V110() Element of bool [:[:REAL,(Funcs ( the carrier of X, the carrier of Y)):],(Funcs ( the carrier of X, the carrier of Y)):]
[:REAL,(Funcs ( the carrier of X, the carrier of Y)):] is non empty set
[:[:REAL,(Funcs ( the carrier of X, the carrier of Y)):],(Funcs ( the carrier of X, the carrier of Y)):] is non empty set
bool [:[:REAL,(Funcs ( the carrier of X, the carrier of Y)):],(Funcs ( the carrier of X, the carrier of Y)):] is non empty set
RLSStruct(# (Funcs ( the carrier of X, the carrier of Y)),( the carrier of X,Y),( the carrier of X,Y),( the carrier of X,Y) #) is non empty strict RLSStruct
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 V110() 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 [:REAL,(X,Y):] -defined (X,Y) -valued Function-like total quasi_total Function-yielding V110() Element of bool [:[:REAL,(X,Y):],(X,Y):]
[:REAL,(X,Y):] is non empty set
[:[:REAL,(X,Y):],(X,Y):] is non empty set
bool [:[:REAL,(X,Y):],(X,Y):] is non empty set
RLSStruct(# (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 left_complementable right_complementable strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() RLSStruct
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
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 left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() RLSStruct
Y is non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() RLSStruct
(X,Y) is non empty left_complementable right_complementable strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() constituted-Functions RLSStruct
(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 left_complementable right_complementable strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() constituted-Functions RLSStruct
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
( 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)
0. Y is V53(Y) 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
( 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 V110() 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 [:REAL,(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 V110() Element of bool [:[:REAL,(Funcs ( the carrier of X, the carrier of Y)):],(Funcs ( the carrier of X, the carrier of Y)):]
[:REAL,(Funcs ( the carrier of X, the carrier of Y)):] is non empty set
[:[:REAL,(Funcs ( the carrier of X, the carrier of Y)):],(Funcs ( the carrier of X, the carrier of Y)):] is non empty set
bool [:[:REAL,(Funcs ( the carrier of X, the carrier of Y)):],(Funcs ( the carrier of X, the carrier of Y)):] is non empty set
RLSStruct(# (Funcs ( the carrier of X, the carrier of Y)),( the carrier of X,Y),( the carrier of X,Y),( the carrier of X,Y) #) is non empty strict RLSStruct
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 V110() 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 [:REAL,(X,Y):] -defined (X,Y) -valued Function-like total quasi_total Function-yielding V110() Element of bool [:[:REAL,(X,Y):],(X,Y):]
[:REAL,(X,Y):] is non empty set
[:[:REAL,(X,Y):],(X,Y):] is non empty set
bool [:[:REAL,(X,Y):],(X,Y):] is non empty set
RLSStruct(# (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 left_complementable right_complementable strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() RLSStruct
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 ext-real V31() real Element of REAL
tseq * vseq is Relation-like Function-like Element of the carrier of (X,Y)
the Mult of (X,Y) is non empty Relation-like [:REAL, the carrier of (X,Y):] -defined the carrier of (X,Y) -valued Function-like total quasi_total Element of bool [:[:REAL, the carrier of (X,Y):], the carrier of (X,Y):]
[:REAL, the carrier of (X,Y):] is non empty set
[:[:REAL, the carrier of (X,Y):], the carrier of (X,Y):] is non empty set
bool [:[:REAL, the carrier of (X,Y):], the carrier of (X,Y):] is non empty set
the Mult of (X,Y) . (tseq,vseq) is set
[tseq,vseq] is set
the Mult of (X,Y) . [tseq,vseq] is set
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)
the Mult of ( the carrier of X,Y) is non empty Relation-like [:REAL, 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 [:[:REAL, the carrier of ( the carrier of X,Y):], the carrier of ( the carrier of X,Y):]
[:REAL, the carrier of ( the carrier of X,Y):] is non empty set
[:[:REAL, the carrier of ( the carrier of X,Y):], the carrier of ( the carrier of X,Y):] is non empty set
bool [:[:REAL, the carrier of ( the carrier of X,Y):], the carrier of ( the carrier of X,Y):] is non empty set
the Mult of ( the carrier of X,Y) . (tseq,tv) is set
[tseq,tv] is set
the Mult of ( the carrier of X,Y) . [tseq,tv] 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) 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
the Mult of Y is non empty Relation-like [:REAL, the carrier of Y:] -defined the carrier of Y -valued Function-like total quasi_total Element of bool [:[:REAL, the carrier of Y:], the carrier of Y:]
[:REAL, the carrier of Y:] is non empty set
[:[:REAL, the carrier of Y:], the carrier of Y:] is non empty set
bool [:[:REAL, the carrier of Y:], the carrier of Y:] is non empty set
the Mult of Y . (tseq,(tseq . m)) is set
[tseq,(tseq . m)] is set
the Mult of Y . [tseq,(tseq . m)] is set
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
the Mult of Y . (tseq,(X,Y,vseq,m)) is set
[tseq,(X,Y,vseq,m)] is set
the Mult of Y . [tseq,(X,Y,vseq,m)] is set
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
the Mult of Y . (tseq,(tseq . n)) is set
[tseq,(tseq . n)] is set
the Mult of Y . [tseq,(tseq . n)] is set
X is non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() RLSStruct
Y is non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() RLSStruct
(X,Y) is non empty left_complementable right_complementable strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() constituted-Functions RLSStruct
(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 left_complementable right_complementable strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() constituted-Functions RLSStruct
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
( 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)
0. Y is V53(Y) 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
( 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 V110() 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 [:REAL,(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 V110() Element of bool [:[:REAL,(Funcs ( the carrier of X, the carrier of Y)):],(Funcs ( the carrier of X, the carrier of Y)):]
[:REAL,(Funcs ( the carrier of X, the carrier of Y)):] is non empty set
[:[:REAL,(Funcs ( the carrier of X, the carrier of Y)):],(Funcs ( the carrier of X, the carrier of Y)):] is non empty set
bool [:[:REAL,(Funcs ( the carrier of X, the carrier of Y)):],(Funcs ( the carrier of X, the carrier of Y)):] is non empty set
RLSStruct(# (Funcs ( the carrier of X, the carrier of Y)),( the carrier of X,Y),( the carrier of X,Y),( the carrier of X,Y) #) is non empty strict RLSStruct
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 V110() 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 [:REAL,(X,Y):] -defined (X,Y) -valued Function-like total quasi_total Function-yielding V110() Element of bool [:[:REAL,(X,Y):],(X,Y):]
[:REAL,(X,Y):] is non empty set
[:[:REAL,(X,Y):],(X,Y):] is non empty set
bool [:[:REAL,(X,Y):],(X,Y):] is non empty set
RLSStruct(# (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 left_complementable right_complementable strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() RLSStruct
0. (X,Y) is Relation-like Function-like V53((X,Y)) 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. ( the carrier of X,Y) is Relation-like Function-like V53(( the carrier of X,Y)) 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 left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() RLSStruct
the carrier of Y is non empty set
X is non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() RLSStruct
the carrier of X is non empty set
0. Y is V53(Y) 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 ext-real V31() real Element of REAL
tseq * tseq is Element of the carrier of X
the Mult of X is non empty Relation-like [:REAL, the carrier of X:] -defined the carrier of X -valued Function-like total quasi_total Element of bool [:[:REAL, the carrier of X:], the carrier of X:]
[:REAL, the carrier of X:] is non empty set
[:[:REAL, the carrier of X:], the carrier of X:] is non empty set
bool [:[:REAL, the carrier of X:], the carrier of X:] is non empty set
the Mult of X . (tseq,tseq) is set
[tseq,tseq] is set
the Mult of X . [tseq,tseq] is set
f . (tseq * tseq) is Element of the carrier of Y
tseq * (f . tseq) is Element of the carrier of Y
the Mult of Y is non empty Relation-like [:REAL, the carrier of Y:] -defined the carrier of Y -valued Function-like total quasi_total Element of bool [:[:REAL, the carrier of Y:], the carrier of Y:]
[:REAL, the carrier of Y:] is non empty set
[:[:REAL, the carrier of Y:], the carrier of Y:] is non empty set
bool [:[:REAL, the carrier of Y:], the carrier of Y:] is non empty set
the Mult of Y . (tseq,(f . tseq)) is set
[tseq,(f . tseq)] is set
the Mult of Y . [tseq,(f . tseq)] is set
tseq * (0. Y) is Element of the carrier of Y
the Mult of Y . (tseq,(0. Y)) is set
[tseq,(0. Y)] is set
the Mult of Y . [tseq,(0. Y)] is set
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 left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() discerning reflexive RealNormSpace-like NORMSTR
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 V134() V135() V136() Element of bool [:NAT,REAL:]
lim ||.Y.|| is ext-real V31() real Element of REAL
vseq is Element of the carrier of X
||.vseq.|| is ext-real non negative V31() 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 V134() V135() V136() Element of bool [: the carrier of X,REAL:]
[: the carrier of X,REAL:] is non empty V134() V135() V136() set
bool [: the carrier of X,REAL:] is non empty set
the U9 of X . vseq is ext-real V31() real Element of REAL
f is ext-real V31() real set
tseq is ext-real epsilon-transitive epsilon-connected ordinal natural V31() real V113() V114() V144() V145() V146() V147() V148() V149() Element of NAT
tseq is ext-real epsilon-transitive epsilon-connected ordinal natural V31() real V113() V114() V144() V145() V146() V147() V148() V149() Element of NAT
tseq is ext-real epsilon-transitive epsilon-connected ordinal natural V31() real V113() V114() V144() V145() V146() V147() V148() V149() 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 ext-real non negative V31() real Element of REAL
the U9 of X . ((Y . tseq) - vseq) is ext-real V31() real Element of REAL
||.(Y . tseq).|| is ext-real non negative V31() real Element of REAL
the U9 of X . (Y . tseq) is ext-real V31() real Element of REAL
||.(Y . tseq).|| - ||.vseq.|| is ext-real V31() real Element of REAL
abs (||.(Y . tseq).|| - ||.vseq.||) is ext-real V31() real Element of REAL
||.Y.|| . tseq is ext-real V31() real Element of REAL
(||.Y.|| . tseq) - ||.vseq.|| is ext-real V31() real Element of REAL
abs ((||.Y.|| . tseq) - ||.vseq.||) is ext-real V31() real Element of REAL
tseq is ext-real epsilon-transitive epsilon-connected ordinal natural V31() real V113() V114() V144() V145() V146() V147() V148() V149() Element of NAT
||.Y.|| . tseq is ext-real V31() real Element of REAL
(||.Y.|| . tseq) - ||.vseq.|| is ext-real V31() real Element of REAL
abs ((||.Y.|| . tseq) - ||.vseq.||) is ext-real V31() real Element of REAL
X is non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() discerning reflexive RealNormSpace-like NORMSTR
the carrier of X is non empty set
Y is non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() discerning reflexive RealNormSpace-like NORMSTR
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 left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() discerning reflexive RealNormSpace-like NORMSTR
the carrier of X is non empty set
Y is non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() discerning reflexive RealNormSpace-like NORMSTR
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 V53(Y) 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 ext-real non negative V31() 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 V134() V135() V136() Element of bool [: the carrier of Y,REAL:]
[: the carrier of Y,REAL:] is non empty V134() V135() V136() set
bool [: the carrier of Y,REAL:] is non empty set
the U9 of Y . (vseq . f) is ext-real V31() real Element of REAL
||.(0. Y).|| is empty Function-like functional ext-real non positive non negative epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural V31() real V144() V145() V146() V147() V148() V149() V150() Element of REAL
the U9 of Y . (0. Y) is ext-real V31() real Element of REAL
||.f.|| is ext-real non negative V31() 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 V134() V135() V136() Element of bool [: the carrier of X,REAL:]
[: the carrier of X,REAL:] is non empty V134() V135() V136() set
bool [: the carrier of X,REAL:] is non empty set
the U9 of X . f is ext-real V31() real Element of REAL
0 * ||.f.|| is empty Function-like functional ext-real non positive non negative epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural V31() real V144() V145() V146() V147() V148() V149() V150() Element of REAL
f is Element of the carrier of X
vseq . f is Element of the carrier of Y
||.(vseq . f).|| is ext-real non negative V31() real Element of REAL
the U9 of Y . (vseq . f) is ext-real V31() real Element of REAL
||.f.|| is ext-real non negative V31() real Element of REAL
the U9 of X . f is ext-real V31() real Element of REAL
0 * ||.f.|| is empty Function-like functional ext-real non positive non negative epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural V31() real V144() V145() V146() V147() V148() V149() V150() Element of REAL
X is non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() discerning reflexive RealNormSpace-like NORMSTR
the carrier of X is non empty set
Y is non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() discerning reflexive RealNormSpace-like NORMSTR
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 V53(Y) 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 left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() discerning reflexive RealNormSpace-like NORMSTR
Y is non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() discerning reflexive RealNormSpace-like NORMSTR
(X,Y) is non empty left_complementable right_complementable strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() constituted-Functions RLSStruct
(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 left_complementable right_complementable strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() constituted-Functions RLSStruct
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
( 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)
0. Y is V53(Y) 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
( 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 V110() 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 [:REAL,(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 V110() Element of bool [:[:REAL,(Funcs ( the carrier of X, the carrier of Y)):],(Funcs ( the carrier of X, the carrier of Y)):]
[:REAL,(Funcs ( the carrier of X, the carrier of Y)):] is non empty set
[:[:REAL,(Funcs ( the carrier of X, the carrier of Y)):],(Funcs ( the carrier of X, the carrier of Y)):] is non empty set
bool [:[:REAL,(Funcs ( the carrier of X, the carrier of Y)):],(Funcs ( the carrier of X, the carrier of Y)):] is non empty set
RLSStruct(# (Funcs ( the carrier of X, the carrier of Y)),( the carrier of X,Y),( the carrier of X,Y),( the carrier of X,Y) #) is non empty strict RLSStruct
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 V110() 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 [:REAL,(X,Y):] -defined (X,Y) -valued Function-like total quasi_total Function-yielding V110() Element of bool [:[:REAL,(X,Y):],(X,Y):]
[:REAL,(X,Y):] is non empty set
[:[:REAL,(X,Y):],(X,Y):] is non empty set
bool [:[:REAL,(X,Y):],(X,Y):] is non empty set
RLSStruct(# (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 left_complementable right_complementable strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() RLSStruct
the carrier of (X,Y) is non empty set
bool the carrier of (X,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 left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() discerning reflexive RealNormSpace-like NORMSTR
Y is non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() discerning reflexive RealNormSpace-like NORMSTR
(X,Y) is Element of bool the carrier of (X,Y)
(X,Y) is non empty left_complementable right_complementable strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() constituted-Functions RLSStruct
(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 left_complementable right_complementable strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() constituted-Functions RLSStruct
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
( 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)
0. Y is V53(Y) 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
( 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 V110() 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 [:REAL,(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 V110() Element of bool [:[:REAL,(Funcs ( the carrier of X, the carrier of Y)):],(Funcs ( the carrier of X, the carrier of Y)):]
[:REAL,(Funcs ( the carrier of X, the carrier of Y)):] is non empty set
[:[:REAL,(Funcs ( the carrier of X, the carrier of Y)):],(Funcs ( the carrier of X, the carrier of Y)):] is non empty set
bool [:[:REAL,(Funcs ( the carrier of X, the carrier of Y)):],(Funcs ( the carrier of X, the carrier of Y)):] is non empty set
RLSStruct(# (Funcs ( the carrier of X, the carrier of Y)),( the carrier of X,Y),( the carrier of X,Y),( the carrier of X,Y) #) is non empty strict RLSStruct
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 V110() 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 [:REAL,(X,Y):] -defined (X,Y) -valued Function-like total quasi_total Function-yielding V110() Element of bool [:[:REAL,(X,Y):],(X,Y):]
[:REAL,(X,Y):] is non empty set
[:[:REAL,(X,Y):],(X,Y):] is non empty set
bool [:[:REAL,(X,Y):],(X,Y):] is non empty set
RLSStruct(# (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 left_complementable right_complementable strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() RLSStruct
the carrier of (X,Y) is non empty set
bool the carrier of (X,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 left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() discerning reflexive RealNormSpace-like NORMSTR
Y is non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() discerning reflexive RealNormSpace-like NORMSTR
(X,Y) is non empty left_complementable right_complementable strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() constituted-Functions RLSStruct
(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 left_complementable right_complementable strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() constituted-Functions RLSStruct
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
( 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)
0. Y is V53(Y) 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
( 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 V110() 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 [:REAL,(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 V110() Element of bool [:[:REAL,(Funcs ( the carrier of X, the carrier of Y)):],(Funcs ( the carrier of X, the carrier of Y)):]
[:REAL,(Funcs ( the carrier of X, the carrier of Y)):] is non empty set
[:[:REAL,(Funcs ( the carrier of X, the carrier of Y)):],(Funcs ( the carrier of X, the carrier of Y)):] is non empty set
bool [:[:REAL,(Funcs ( the carrier of X, the carrier of Y)):],(Funcs ( the carrier of X, the carrier of Y)):] is non empty set
RLSStruct(# (Funcs ( the carrier of X, the carrier of Y)),( the carrier of X,Y),( the carrier of X,Y),( the carrier of X,Y) #) is non empty strict RLSStruct
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 V110() 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 [:REAL,(X,Y):] -defined (X,Y) -valued Function-like total quasi_total Function-yielding V110() Element of bool [:[:REAL,(X,Y):],(X,Y):]
[:REAL,(X,Y):] is non empty set
[:[:REAL,(X,Y):],(X,Y):] is non empty set
bool [:[:REAL,(X,Y):],(X,Y):] is non empty set
RLSStruct(# (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 left_complementable right_complementable strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() RLSStruct
(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
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 ext-real V31() 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 ext-real V31() real Element of REAL
m + tv is ext-real V31() real Element of REAL
n is ext-real V31() 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 ext-real non negative V31() 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 V134() V135() V136() Element of bool [: the carrier of Y,REAL:]
[: the carrier of Y,REAL:] is non empty V134() V135() V136() set
bool [: the carrier of Y,REAL:] is non empty set
the U9 of Y . ((e . n) + (tseq . n)) is ext-real V31() real Element of REAL
||.(e . n).|| is ext-real non negative V31() real Element of REAL
the U9 of Y . (e . n) is ext-real V31() real Element of REAL
||.(tseq . n).|| is ext-real non negative V31() real Element of REAL
the U9 of Y . (tseq . n) is ext-real V31() real Element of REAL
||.(e . n).|| + ||.(tseq . n).|| is ext-real non negative V31() real Element of REAL
||.n.|| is ext-real non negative V31() 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 V134() V135() V136() Element of bool [: the carrier of X,REAL:]
[: the carrier of X,REAL:] is non empty V134() V135() V136() set
bool [: the carrier of X,REAL:] is non empty set
the U9 of X . n is ext-real V31() real Element of REAL
tv * ||.n.|| is ext-real V31() real Element of REAL
m * ||.n.|| is ext-real V31() real Element of REAL
(m * ||.n.||) + (tv * ||.n.||) is ext-real V31() real Element of REAL
tseq . n is Element of the carrier of Y
||.(tseq . n).|| is ext-real non negative V31() real Element of REAL
the U9 of Y . (tseq . n) is ext-real V31() real Element of REAL
n * ||.n.|| is ext-real V31() real Element of REAL
n is Element of the carrier of X
tseq . n is Element of the carrier of Y
||.(tseq . n).|| is ext-real non negative V31() real Element of REAL
the U9 of Y . (tseq . n) is ext-real V31() real Element of REAL
||.n.|| is ext-real non negative V31() real Element of REAL
the U9 of X . n is ext-real V31() real Element of REAL
n * ||.n.|| is ext-real V31() real Element of REAL
f is ext-real V31() real Element of REAL
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 Mult of (X,Y) is non empty Relation-like [:REAL, the carrier of (X,Y):] -defined the carrier of (X,Y) -valued Function-like total quasi_total Element of bool [:[:REAL, the carrier of (X,Y):], the carrier of (X,Y):]
[:REAL, the carrier of (X,Y):] is non empty set
[:[:REAL, the carrier of (X,Y):], the carrier of (X,Y):] is non empty set
bool [:[:REAL, the carrier of (X,Y):], the carrier of (X,Y):] is non empty set
the Mult of (X,Y) . (f,tseq) is set
[f,tseq] is set
the Mult of (X,Y) . [f,tseq] 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) 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 ext-real V31() real Element of REAL
abs f is ext-real V31() real Element of REAL
(abs f) * tv is ext-real V31() real Element of REAL
e is Element of the carrier of X
tseq . e is Element of the carrier of Y
||.(tseq . e).|| is ext-real non negative V31() 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 V134() V135() V136() Element of bool [: the carrier of Y,REAL:]
[: the carrier of Y,REAL:] is non empty V134() V135() V136() set
bool [: the carrier of Y,REAL:] is non empty set
the U9 of Y . (tseq . e) is ext-real V31() real Element of REAL
(abs f) * ||.(tseq . e).|| is ext-real V31() real Element of REAL
||.e.|| is ext-real non negative V31() 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 V134() V135() V136() Element of bool [: the carrier of X,REAL:]
[: the carrier of X,REAL:] is non empty V134() V135() V136() set
bool [: the carrier of X,REAL:] is non empty set
the U9 of X . e is ext-real V31() real Element of REAL
tv * ||.e.|| is ext-real V31() real Element of REAL
(abs f) * (tv * ||.e.||) is ext-real V31() real Element of REAL
f * (tseq . e) is Element of the carrier of Y
the Mult of Y is non empty Relation-like [:REAL, the carrier of Y:] -defined the carrier of Y -valued Function-like total quasi_total Element of bool [:[:REAL, the carrier of Y:], the carrier of Y:]
[:REAL, the carrier of Y:] is non empty set
[:[:REAL, the carrier of Y:], the carrier of Y:] is non empty set
bool [:[:REAL, the carrier of Y:], the carrier of Y:] is non empty set
the Mult of Y . (f,(tseq . e)) is set
[f,(tseq . e)] is set
the Mult of Y . [f,(tseq . e)] is set
||.(f * (tseq . e)).|| is ext-real non negative V31() real Element of REAL
the U9 of Y . (f * (tseq . e)) is ext-real V31() real Element of REAL
tseq . e is Element of the carrier of Y
||.(tseq . e).|| is ext-real non negative V31() real Element of REAL
the U9 of Y . (tseq . e) is ext-real V31() real Element of REAL
((abs f) * tv) * ||.e.|| is ext-real V31() real Element of REAL
e is Element of the carrier of X
tseq . e is Element of the carrier of Y
||.(tseq . e).|| is ext-real non negative V31() real Element of REAL
the U9 of Y . (tseq . e) is ext-real V31() real Element of REAL
||.e.|| is ext-real non negative V31() real Element of REAL
the U9 of X . e is ext-real V31() real Element of REAL
((abs f) * tv) * ||.e.|| is ext-real V31() real Element of REAL
X is non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() discerning reflexive RealNormSpace-like NORMSTR
Y is non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() discerning reflexive RealNormSpace-like NORMSTR
(X,Y) is non empty Element of bool the carrier of (X,Y)
(X,Y) is non empty left_complementable right_complementable strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() constituted-Functions RLSStruct
(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 left_complementable right_complementable strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() constituted-Functions RLSStruct
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
( 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)
0. Y is V53(Y) 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
( 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 V110() 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 [:REAL,(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 V110() Element of bool [:[:REAL,(Funcs ( the carrier of X, the carrier of Y)):],(Funcs ( the carrier of X, the carrier of Y)):]
[:REAL,(Funcs ( the carrier of X, the carrier of Y)):] is non empty set
[:[:REAL,(Funcs ( the carrier of X, the carrier of Y)):],(Funcs ( the carrier of X, the carrier of Y)):] is non empty set
bool [:[:REAL,(Funcs ( the carrier of X, the carrier of Y)):],(Funcs ( the carrier of X, the carrier of Y)):] is non empty set
RLSStruct(# (Funcs ( the carrier of X, the carrier of Y)),( the carrier of X,Y),( the carrier of X,Y),( the carrier of X,Y) #) is non empty strict RLSStruct
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 V110() 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 [:REAL,(X,Y):] -defined (X,Y) -valued Function-like total quasi_total Function-yielding V110() Element of bool [:[:REAL,(X,Y):],(X,Y):]
[:REAL,(X,Y):] is non empty set
[:[:REAL,(X,Y):],(X,Y):] is non empty set
bool [:[:REAL,(X,Y):],(X,Y):] is non empty set
RLSStruct(# (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 left_complementable right_complementable strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() RLSStruct
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 [:REAL,(X,Y):] -defined (X,Y) -valued Function-like total quasi_total Element of bool [:[:REAL,(X,Y):],(X,Y):]
[:REAL,(X,Y):] is non empty set
[:[:REAL,(X,Y):],(X,Y):] is non empty set
bool [:[:REAL,(X,Y):],(X,Y):] is non empty set
RLSStruct(# (X,Y),(Zero_ ((X,Y),(X,Y))),(Add_ ((X,Y),(X,Y))),(Mult_ ((X,Y),(X,Y))) #) is non empty strict RLSStruct
X is non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() discerning reflexive RealNormSpace-like NORMSTR
Y is non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() discerning reflexive RealNormSpace-like NORMSTR
(X,Y) is non empty Element of bool the carrier of (X,Y)
(X,Y) is non empty left_complementable right_complementable strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() constituted-Functions RLSStruct
(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 left_complementable right_complementable strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() constituted-Functions RLSStruct
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
( 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)
0. Y is V53(Y) 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
( 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 V110() 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 [:REAL,(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 V110() Element of bool [:[:REAL,(Funcs ( the carrier of X, the carrier of Y)):],(Funcs ( the carrier of X, the carrier of Y)):]
[:REAL,(Funcs ( the carrier of X, the carrier of Y)):] is non empty set
[:[:REAL,(Funcs ( the carrier of X, the carrier of Y)):],(Funcs ( the carrier of X, the carrier of Y)):] is non empty set
bool [:[:REAL,(Funcs ( the carrier of X, the carrier of Y)):],(Funcs ( the carrier of X, the carrier of Y)):] is non empty set
RLSStruct(# (Funcs ( the carrier of X, the carrier of Y)),( the carrier of X,Y),( the carrier of X,Y),( the carrier of X,Y) #) is non empty strict RLSStruct
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 V110() 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 [:REAL,(X,Y):] -defined (X,Y) -valued Function-like total quasi_total Function-yielding V110() Element of bool [:[:REAL,(X,Y):],(X,Y):]
[:REAL,(X,Y):] is non empty set
[:[:REAL,(X,Y):],(X,Y):] is non empty set
bool [:[:REAL,(X,Y):],(X,Y):] is non empty set
RLSStruct(# (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 left_complementable right_complementable strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() RLSStruct
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 [:REAL,(X,Y):] -defined (X,Y) -valued Function-like total quasi_total Element of bool [:[:REAL,(X,Y):],(X,Y):]
[:REAL,(X,Y):] is non empty set
[:[:REAL,(X,Y):],(X,Y):] is non empty set
bool [:[:REAL,(X,Y):],(X,Y):] is non empty set
RLSStruct(# (X,Y),(Zero_ ((X,Y),(X,Y))),(Add_ ((X,Y),(X,Y))),(Mult_ ((X,Y),(X,Y))) #) is non empty strict RLSStruct
X is non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() discerning reflexive RealNormSpace-like NORMSTR
Y is non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() discerning reflexive RealNormSpace-like NORMSTR
(X,Y) is non empty Element of bool the carrier of (X,Y)
(X,Y) is non empty left_complementable right_complementable strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() constituted-Functions RLSStruct
(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 left_complementable right_complementable strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() constituted-Functions RLSStruct
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
( 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)
0. Y is V53(Y) 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
( 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 V110() 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 [:REAL,(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 V110() Element of bool [:[:REAL,(Funcs ( the carrier of X, the carrier of Y)):],(Funcs ( the carrier of X, the carrier of Y)):]
[:REAL,(Funcs ( the carrier of X, the carrier of Y)):] is non empty set
[:[:REAL,(Funcs ( the carrier of X, the carrier of Y)):],(Funcs ( the carrier of X, the carrier of Y)):] is non empty set
bool [:[:REAL,(Funcs ( the carrier of X, the carrier of Y)):],(Funcs ( the carrier of X, the carrier of Y)):] is non empty set
RLSStruct(# (Funcs ( the carrier of X, the carrier of Y)),( the carrier of X,Y),( the carrier of X,Y),( the carrier of X,Y) #) is non empty strict RLSStruct
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 V110() 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 [:REAL,(X,Y):] -defined (X,Y) -valued Function-like total quasi_total Function-yielding V110() Element of bool [:[:REAL,(X,Y):],(X,Y):]
[:REAL,(X,Y):] is non empty set
[:[:REAL,(X,Y):],(X,Y):] is non empty set
bool [:[:REAL,(X,Y):],(X,Y):] is non empty set
RLSStruct(# (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 left_complementable right_complementable strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() RLSStruct
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 [:REAL,(X,Y):] -defined (X,Y) -valued Function-like total quasi_total Element of bool [:[:REAL,(X,Y):],(X,Y):]
[:REAL,(X,Y):] is non empty set
[:[:REAL,(X,Y):],(X,Y):] is non empty set
bool [:[:REAL,(X,Y):],(X,Y):] is non empty set
RLSStruct(# (X,Y),(Zero_ ((X,Y),(X,Y))),(Add_ ((X,Y),(X,Y))),(Mult_ ((X,Y),(X,Y))) #) is non empty left_complementable right_complementable strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() RLSStruct
X is non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() discerning reflexive RealNormSpace-like NORMSTR
Y is non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() discerning reflexive RealNormSpace-like NORMSTR
(X,Y) is non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() RLSStruct
(X,Y) is non empty Element of bool the carrier of (X,Y)
(X,Y) is non empty left_complementable right_complementable strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() constituted-Functions RLSStruct
(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 left_complementable right_complementable strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() constituted-Functions RLSStruct
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
( 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)
0. Y is V53(Y) 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
( 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 V110() 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 [:REAL,(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 V110() Element of bool [:[:REAL,(Funcs ( the carrier of X, the carrier of Y)):],(Funcs ( the carrier of X, the carrier of Y)):]
[:REAL,(Funcs ( the carrier of X, the carrier of Y)):] is non empty set
[:[:REAL,(Funcs ( the carrier of X, the carrier of Y)):],(Funcs ( the carrier of X, the carrier of Y)):] is non empty set
bool [:[:REAL,(Funcs ( the carrier of X, the carrier of Y)):],(Funcs ( the carrier of X, the carrier of Y)):] is non empty set
RLSStruct(# (Funcs ( the carrier of X, the carrier of Y)),( the carrier of X,Y),( the carrier of X,Y),( the carrier of X,Y) #) is non empty strict RLSStruct
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 V110() 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 [:REAL,(X,Y):] -defined (X,Y) -valued Function-like total quasi_total Function-yielding V110() Element of bool [:[:REAL,(X,Y):],(X,Y):]
[:REAL,(X,Y):] is non empty set
[:[:REAL,(X,Y):],(X,Y):] is non empty set
bool [:[:REAL,(X,Y):],(X,Y):] is non empty set
RLSStruct(# (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 left_complementable right_complementable strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() RLSStruct
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 [:REAL,(X,Y):] -defined (X,Y) -valued Function-like total quasi_total Element of bool [:[:REAL,(X,Y):],(X,Y):]
[:REAL,(X,Y):] is non empty set
[:[:REAL,(X,Y):],(X,Y):] is non empty set
bool [:[:REAL,(X,Y):],(X,Y):] is non empty set
RLSStruct(# (X,Y),(Zero_ ((X,Y),(X,Y))),(Add_ ((X,Y),(X,Y))),(Mult_ ((X,Y),(X,Y))) #) is non empty left_complementable right_complementable strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() RLSStruct
X is non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() discerning reflexive RealNormSpace-like NORMSTR
Y is non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() discerning reflexive RealNormSpace-like NORMSTR
(X,Y) is non empty left_complementable right_complementable strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() RLSStruct
(X,Y) is non empty Element of bool the carrier of (X,Y)
(X,Y) is non empty left_complementable right_complementable strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() constituted-Functions RLSStruct
(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 left_complementable right_complementable strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() constituted-Functions RLSStruct
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
( 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)
0. Y is V53(Y) 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
( 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 V110() 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 [:REAL,(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 V110() Element of bool [:[:REAL,(Funcs ( the carrier of X, the carrier of Y)):],(Funcs ( the carrier of X, the carrier of Y)):]
[:REAL,(Funcs ( the carrier of X, the carrier of Y)):] is non empty set
[:[:REAL,(Funcs ( the carrier of X, the carrier of Y)):],(Funcs ( the carrier of X, the carrier of Y)):] is non empty set
bool [:[:REAL,(Funcs ( the carrier of X, the carrier of Y)):],(Funcs ( the carrier of X, the carrier of Y)):] is non empty set
RLSStruct(# (Funcs ( the carrier of X, the carrier of Y)),( the carrier of X,Y),( the carrier of X,Y),( the carrier of X,Y) #) is non empty strict RLSStruct
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 V110() 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 [:REAL,(X,Y):] -defined (X,Y) -valued Function-like total quasi_total Function-yielding V110() Element of bool [:[:REAL,(X,Y):],(X,Y):]
[:REAL,(X,Y):] is non empty set
[:[:REAL,(X,Y):],(X,Y):] is non empty set
bool [:[:REAL,(X,Y):],(X,Y):] is non empty set
RLSStruct(# (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 left_complementable right_complementable strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() RLSStruct
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 [:REAL,(X,Y):] -defined (X,Y) -valued Function-like total quasi_total Element of bool [:[:REAL,(X,Y):],(X,Y):]
[:REAL,(X,Y):] is non empty set
[:[:REAL,(X,Y):],(X,Y):] is non empty set
bool [:[:REAL,(X,Y):],(X,Y):] is non empty set
RLSStruct(# (X,Y),(Zero_ ((X,Y),(X,Y))),(Add_ ((X,Y),(X,Y))),(Mult_ ((X,Y),(X,Y))) #) is non empty left_complementable right_complementable strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() RLSStruct
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 left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() discerning reflexive RealNormSpace-like NORMSTR
Y is non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() discerning reflexive RealNormSpace-like NORMSTR
(X,Y) is non empty left_complementable right_complementable strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() RLSStruct
(X,Y) is non empty Element of bool the carrier of (X,Y)
(X,Y) is non empty left_complementable right_complementable strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() constituted-Functions RLSStruct
(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 left_complementable right_complementable strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() constituted-Functions RLSStruct
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
( 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)
0. Y is V53(Y) 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
( 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 V110() 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 [:REAL,(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 V110() Element of bool [:[:REAL,(Funcs ( the carrier of X, the carrier of Y)):],(Funcs ( the carrier of X, the carrier of Y)):]
[:REAL,(Funcs ( the carrier of X, the carrier of Y)):] is non empty set
[:[:REAL,(Funcs ( the carrier of X, the carrier of Y)):],(Funcs ( the carrier of X, the carrier of Y)):] is non empty set
bool [:[:REAL,(Funcs ( the carrier of X, the carrier of Y)):],(Funcs ( the carrier of X, the carrier of Y)):] is non empty set
RLSStruct(# (Funcs ( the carrier of X, the carrier of Y)),( the carrier of X,Y),( the carrier of X,Y),( the carrier of X,Y) #) is non empty strict RLSStruct
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 V110() 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 [:REAL,(X,Y):] -defined (X,Y) -valued Function-like total quasi_total Function-yielding V110() Element of bool [:[:REAL,(X,Y):],(X,Y):]
[:REAL,(X,Y):] is non empty set
[:[:REAL,(X,Y):],(X,Y):] is non empty set
bool [:[:REAL,(X,Y):],(X,Y):] is non empty set
RLSStruct(# (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 left_complementable right_complementable strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() RLSStruct
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 [:REAL,(X,Y):] -defined (X,Y) -valued Function-like total quasi_total Element of bool [:[:REAL,(X,Y):],(X,Y):]
[:REAL,(X,Y):] is non empty set
[:[:REAL,(X,Y):],(X,Y):] is non empty set
bool [:[:REAL,(X,Y):],(X,Y):] is non empty set
RLSStruct(# (X,Y),(Zero_ ((X,Y),(X,Y))),(Add_ ((X,Y),(X,Y))),(Mult_ ((X,Y),(X,Y))) #) is non empty left_complementable right_complementable strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() RLSStruct
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
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 left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() discerning reflexive RealNormSpace-like NORMSTR
Y is non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() discerning reflexive RealNormSpace-like NORMSTR
(X,Y) is non empty left_complementable right_complementable strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() RLSStruct
(X,Y) is non empty Element of bool the carrier of (X,Y)
(X,Y) is non empty left_complementable right_complementable strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() constituted-Functions RLSStruct
(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 left_complementable right_complementable strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() constituted-Functions RLSStruct
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
( 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)
0. Y is V53(Y) 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
( 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 V110() 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 [:REAL,(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 V110() Element of bool [:[:REAL,(Funcs ( the carrier of X, the carrier of Y)):],(Funcs ( the carrier of X, the carrier of Y)):]
[:REAL,(Funcs ( the carrier of X, the carrier of Y)):] is non empty set
[:[:REAL,(Funcs ( the carrier of X, the carrier of Y)):],(Funcs ( the carrier of X, the carrier of Y)):] is non empty set
bool [:[:REAL,(Funcs ( the carrier of X, the carrier of Y)):],(Funcs ( the carrier of X, the carrier of Y)):] is non empty set
RLSStruct(# (Funcs ( the carrier of X, the carrier of Y)),( the carrier of X,Y),( the carrier of X,Y),( the carrier of X,Y) #) is non empty strict RLSStruct
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 V110() 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 [:REAL,(X,Y):] -defined (X,Y) -valued Function-like total quasi_total Function-yielding V110() Element of bool [:[:REAL,(X,Y):],(X,Y):]
[:REAL,(X,Y):] is non empty set
[:[:REAL,(X,Y):],(X,Y):] is non empty set
bool [:[:REAL,(X,Y):],(X,Y):] is non empty set
RLSStruct(# (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 left_complementable right_complementable strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() RLSStruct
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 [:REAL,(X,Y):] -defined (X,Y) -valued Function-like total quasi_total Element of bool [:[:REAL,(X,Y):],(X,Y):]
[:REAL,(X,Y):] is non empty set
[:[:REAL,(X,Y):],(X,Y):] is non empty set
bool [:[:REAL,(X,Y):],(X,Y):] is non empty set
RLSStruct(# (X,Y),(Zero_ ((X,Y),(X,Y))),(Add_ ((X,Y),(X,Y))),(Mult_ ((X,Y),(X,Y))) #) is non empty left_complementable right_complementable strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() RLSStruct
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 left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() discerning reflexive RealNormSpace-like NORMSTR
Y is non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() discerning reflexive RealNormSpace-like NORMSTR
(X,Y) is non empty left_complementable right_complementable strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() RLSStruct
(X,Y) is non empty Element of bool the carrier of (X,Y)
(X,Y) is non empty left_complementable right_complementable strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() constituted-Functions RLSStruct
(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 left_complementable right_complementable strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() constituted-Functions RLSStruct
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
( 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)
0. Y is V53(Y) 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
( 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 V110() 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 [:REAL,(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 V110() Element of bool [:[:REAL,(Funcs ( the carrier of X, the carrier of Y)):],(Funcs ( the carrier of X, the carrier of Y)):]
[:REAL,(Funcs ( the carrier of X, the carrier of Y)):] is non empty set
[:[:REAL,(Funcs ( the carrier of X, the carrier of Y)):],(Funcs ( the carrier of X, the carrier of Y)):] is non empty set
bool [:[:REAL,(Funcs ( the carrier of X, the carrier of Y)):],(Funcs ( the carrier of X, the carrier of Y)):] is non empty set
RLSStruct(# (Funcs ( the carrier of X, the carrier of Y)),( the carrier of X,Y),( the carrier of X,Y),( the carrier of X,Y) #) is non empty strict RLSStruct
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 V110() 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 [:REAL,(X,Y):] -defined (X,Y) -valued Function-like total quasi_total Function-yielding V110() Element of bool [:[:REAL,(X,Y):],(X,Y):]
[:REAL,(X,Y):] is non empty set
[:[:REAL,(X,Y):],(X,Y):] is non empty set
bool [:[:REAL,(X,Y):],(X,Y):] is non empty set
RLSStruct(# (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 left_complementable right_complementable strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() RLSStruct
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 [:REAL,(X,Y):] -defined (X,Y) -valued Function-like total quasi_total Element of bool [:[:REAL,(X,Y):],(X,Y):]
[:REAL,(X,Y):] is non empty set
[:[:REAL,(X,Y):],(X,Y):] is non empty set
bool [:[:REAL,(X,Y):],(X,Y):] is non empty set
RLSStruct(# (X,Y),(Zero_ ((X,Y),(X,Y))),(Add_ ((X,Y),(X,Y))),(Mult_ ((X,Y),(X,Y))) #) is non empty left_complementable right_complementable strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() RLSStruct
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 ext-real V31() real Element of REAL
tseq * vseq is Relation-like Function-like Element of the carrier of (X,Y)
the Mult of (X,Y) is non empty Relation-like [:REAL, the carrier of (X,Y):] -defined the carrier of (X,Y) -valued Function-like total quasi_total Element of bool [:[:REAL, the carrier of (X,Y):], the carrier of (X,Y):]
[:REAL, the carrier of (X,Y):] is non empty set
[:[:REAL, the carrier of (X,Y):], the carrier of (X,Y):] is non empty set
bool [:[:REAL, the carrier of (X,Y):], the carrier of (X,Y):] is non empty set
the Mult of (X,Y) . (tseq,vseq) is set
[tseq,vseq] is set
the Mult of (X,Y) . [tseq,vseq] 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)
tseq * tseq is Relation-like Function-like Element of the carrier of (X,Y)
the Mult of (X,Y) is non empty Relation-like [:REAL, the carrier of (X,Y):] -defined the carrier of (X,Y) -valued Function-like total quasi_total Element of bool [:[:REAL, the carrier of (X,Y):], the carrier of (X,Y):]
[:REAL, the carrier of (X,Y):] is non empty set
[:[:REAL, the carrier of (X,Y):], the carrier of (X,Y):] is non empty set
bool [:[:REAL, the carrier of (X,Y):], the carrier of (X,Y):] is non empty set
the Mult of (X,Y) . (tseq,tseq) is set
[tseq,tseq] is set
the Mult of (X,Y) . [tseq,tseq] 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
the Mult of Y is non empty Relation-like [:REAL, the carrier of Y:] -defined the carrier of Y -valued Function-like total quasi_total Element of bool [:[:REAL, the carrier of Y:], the carrier of Y:]
[:REAL, the carrier of Y:] is non empty set
[:[:REAL, the carrier of Y:], the carrier of Y:] is non empty set
bool [:[:REAL, the carrier of Y:], the carrier of Y:] is non empty set
the Mult of Y . (tseq,(X,Y,vseq,tv)) is set
[tseq,(X,Y,vseq,tv)] is set
the Mult of Y . [tseq,(X,Y,vseq,tv)] is set
X is non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() discerning reflexive RealNormSpace-like NORMSTR
Y is non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() discerning reflexive RealNormSpace-like NORMSTR
(X,Y) is non empty left_complementable right_complementable strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() RLSStruct
(X,Y) is non empty Element of bool the carrier of (X,Y)
(X,Y) is non empty left_complementable right_complementable strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() constituted-Functions RLSStruct
(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 left_complementable right_complementable strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() constituted-Functions RLSStruct
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
( 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)
0. Y is V53(Y) 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
( 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 V110() 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 [:REAL,(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 V110() Element of bool [:[:REAL,(Funcs ( the carrier of X, the carrier of Y)):],(Funcs ( the carrier of X, the carrier of Y)):]
[:REAL,(Funcs ( the carrier of X, the carrier of Y)):] is non empty set
[:[:REAL,(Funcs ( the carrier of X, the carrier of Y)):],(Funcs ( the carrier of X, the carrier of Y)):] is non empty set
bool [:[:REAL,(Funcs ( the carrier of X, the carrier of Y)):],(Funcs ( the carrier of X, the carrier of Y)):] is non empty set
RLSStruct(# (Funcs ( the carrier of X, the carrier of Y)),( the carrier of X,Y),( the carrier of X,Y),( the carrier of X,Y) #) is non empty strict RLSStruct
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 V110() 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 [:REAL,(X,Y):] -defined (X,Y) -valued Function-like total quasi_total Function-yielding V110() Element of bool [:[:REAL,(X,Y):],(X,Y):]
[:REAL,(X,Y):] is non empty set
[:[:REAL,(X,Y):],(X,Y):] is non empty set
bool [:[:REAL,(X,Y):],(X,Y):] is non empty set
RLSStruct(# (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 left_complementable right_complementable strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() RLSStruct
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 [:REAL,(X,Y):] -defined (X,Y) -valued Function-like total quasi_total Element of bool [:[:REAL,(X,Y):],(X,Y):]
[:REAL,(X,Y):] is non empty set
[:[:REAL,(X,Y):],(X,Y):] is non empty set
bool [:[:REAL,(X,Y):],(X,Y):] is non empty set
RLSStruct(# (X,Y),(Zero_ ((X,Y),(X,Y))),(Add_ ((X,Y),(X,Y))),(Mult_ ((X,Y),(X,Y))) #) is non empty left_complementable right_complementable strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() RLSStruct
0. (X,Y) is Relation-like Function-like V53((X,Y)) 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 Relation-like Function-like V53((X,Y)) 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 left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() discerning reflexive RealNormSpace-like NORMSTR
Y is non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() discerning reflexive RealNormSpace-like NORMSTR
(X,Y) is non empty Element of bool the carrier of (X,Y)
(X,Y) is non empty left_complementable right_complementable strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() constituted-Functions RLSStruct
(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 left_complementable right_complementable strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() constituted-Functions RLSStruct
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
( 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)
0. Y is V53(Y) 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
( 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 V110() 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 [:REAL,(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 V110() Element of bool [:[:REAL,(Funcs ( the carrier of X, the carrier of Y)):],(Funcs ( the carrier of X, the carrier of Y)):]
[:REAL,(Funcs ( the carrier of X, the carrier of Y)):] is non empty set
[:[:REAL,(Funcs ( the carrier of X, the carrier of Y)):],(Funcs ( the carrier of X, the carrier of Y)):] is non empty set
bool [:[:REAL,(Funcs ( the carrier of X, the carrier of Y)):],(Funcs ( the carrier of X, the carrier of Y)):] is non empty set
RLSStruct(# (Funcs ( the carrier of X, the carrier of Y)),( the carrier of X,Y),( the carrier of X,Y),( the carrier of X,Y) #) is non empty strict RLSStruct
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 V110() 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 [:REAL,(X,Y):] -defined (X,Y) -valued Function-like total quasi_total Function-yielding V110() Element of bool [:[:REAL,(X,Y):],(X,Y):]
[:REAL,(X,Y):] is non empty set
[:[:REAL,(X,Y):],(X,Y):] is non empty set
bool [:[:REAL,(X,Y):],(X,Y):] is non empty set
RLSStruct(# (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 left_complementable right_complementable strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() RLSStruct
the carrier of (X,Y) is non empty set
bool the carrier of (X,Y) is non empty set
X is non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() discerning reflexive RealNormSpace-like NORMSTR
the carrier of X is non empty set
Y is non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() discerning reflexive RealNormSpace-like NORMSTR
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 ext-real non negative V31() 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 V134() V135() V136() Element of bool [: the carrier of Y,REAL:]
[: the carrier of Y,REAL:] is non empty V134() V135() V136() set
bool [: the carrier of Y,REAL:] is non empty set
the U9 of Y . (vseq . tseq) is ext-real V31() real Element of REAL
||.tseq.|| is ext-real non negative V31() 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 V134() V135() V136() Element of bool [: the carrier of X,REAL:]
[: the carrier of X,REAL:] is non empty V134() V135() V136() set
bool [: the carrier of X,REAL:] is non empty set
the U9 of X . tseq is ext-real V31() real Element of REAL
0. X is V53(X) Element of the carrier of X
the ZeroF of X is Element of the carrier of X
||.(0. X).|| is empty Function-like functional ext-real non positive non negative epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural V31() real V144() V145() V146() V147() V148() V149() V150() Element of REAL
the U9 of X is non empty Relation-like the carrier of X -defined REAL -valued Function-like total quasi_total V134() V135() V136() Element of bool [: the carrier of X,REAL:]
[: the carrier of X,REAL:] is non empty V134() V135() V136() set
bool [: the carrier of X,REAL:] is non empty set
the U9 of X . (0. X) is ext-real V31() real Element of REAL
vseq . (0. X) is Element of the carrier of Y
||.(vseq . (0. X)).|| is ext-real non negative V31() 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 V134() V135() V136() Element of bool [: the carrier of Y,REAL:]
[: the carrier of Y,REAL:] is non empty V134() V135() V136() set
bool [: the carrier of Y,REAL:] is non empty set
the U9 of Y . (vseq . (0. X)) is ext-real V31() real Element of REAL
X is non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() discerning reflexive RealNormSpace-like NORMSTR
the carrier of X is non empty set
Y is non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() discerning reflexive RealNormSpace-like NORMSTR
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 V144() V145() V146() Element of bool REAL
{ ||.(vseq . b1).|| where b1 is Element of the carrier of X : ||.b1.|| <= 1 } is set
f is ext-real V31() 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 ext-real non negative V31() 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 V134() V135() V136() Element of bool [: the carrier of Y,REAL:]
[: the carrier of Y,REAL:] is non empty V134() V135() V136() set
bool [: the carrier of Y,REAL:] is non empty set
the U9 of Y . (vseq . tseq) is ext-real V31() real Element of REAL
||.tseq.|| is ext-real non negative V31() 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 V134() V135() V136() Element of bool [: the carrier of X,REAL:]
[: the carrier of X,REAL:] is non empty V134() V135() V136() set
bool [: the carrier of X,REAL:] is non empty set
the U9 of X . tseq is ext-real V31() real Element of REAL
f * ||.tseq.|| is ext-real V31() real Element of REAL
f * 1 is ext-real V31() real Element of REAL
X is non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() discerning reflexive RealNormSpace-like NORMSTR
the carrier of X is non empty set
Y is non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() discerning reflexive RealNormSpace-like NORMSTR
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 V144() V145() V146() 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 ext-real V31() real Element of REAL
tseq is Element of the carrier of X
0. X is V53(X) Element of the carrier of X
the ZeroF of X is Element of the carrier of X
||.tseq.|| is ext-real non negative V31() 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 V134() V135() V136() Element of bool [: the carrier of X,REAL:]
[: the carrier of X,REAL:] is non empty V134() V135() V136() set
bool [: the carrier of X,REAL:] is non empty set
the U9 of X . tseq is ext-real V31() real Element of REAL
vseq . tseq is Element of the carrier of Y
0 * (0. X) is Element of the carrier of X
the Mult of X is non empty Relation-like [:REAL, the carrier of X:] -defined the carrier of X -valued Function-like total quasi_total Element of bool [:[:REAL, the carrier of X:], the carrier of X:]
[:REAL, the carrier of X:] is non empty set
[:[:REAL, the carrier of X:], the carrier of X:] is non empty set
bool [:[:REAL, the carrier of X:], the carrier of X:] is non empty set
the Mult of X . (0,(0. X)) is set
[0,(0. X)] is set
the Mult of X . [0,(0. X)] is set
vseq . (0 * (0. X)) is Element of the carrier of Y
vseq . (0. X) is Element of the carrier of Y
0 * (vseq . (0. X)) is Element of the carrier of Y
the Mult of Y is non empty Relation-like [:REAL, the carrier of Y:] -defined the carrier of Y -valued Function-like total quasi_total Element of bool [:[:REAL, the carrier of Y:], the carrier of Y:]
[:REAL, the carrier of Y:] is non empty set
[:[:REAL, the carrier of Y:], the carrier of Y:] is non empty set
bool [:[:REAL, the carrier of Y:], the carrier of Y:] is non empty set
the Mult of Y . (0,(vseq . (0. X))) is set
[0,(vseq . (0. X))] is set
the Mult of Y . [0,(vseq . (0. X))] is set
0. Y is V53(Y) Element of the carrier of Y
the ZeroF of Y is Element of the carrier of Y
||.(vseq . tseq).|| is ext-real non negative V31() 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 V134() V135() V136() Element of bool [: the carrier of Y,REAL:]
[: the carrier of Y,REAL:] is non empty V134() V135() V136() set
bool [: the carrier of Y,REAL:] is non empty set
the U9 of Y . (vseq . tseq) is ext-real V31() real Element of REAL
f is ext-real V31() real Element of REAL
f * ||.tseq.|| is ext-real V31() real Element of REAL
tseq is Element of the carrier of X
0. X is V53(X) Element of the carrier of X
the ZeroF of X is Element of the carrier of X
||.tseq.|| is ext-real non negative V31() 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 V134() V135() V136() Element of bool [: the carrier of X,REAL:]
[: the carrier of X,REAL:] is non empty V134() V135() V136() set
bool [: the carrier of X,REAL:] is non empty set
the U9 of X . tseq is ext-real V31() real Element of REAL
||.tseq.|| " is ext-real non negative V31() real Element of REAL
(||.tseq.|| ") * tseq is Element of the carrier of X
the Mult of X is non empty Relation-like [:REAL, the carrier of X:] -defined the carrier of X -valued Function-like total quasi_total Element of bool [:[:REAL, the carrier of X:], the carrier of X:]
[:REAL, the carrier of X:] is non empty set
[:[:REAL, the carrier of X:], the carrier of X:] is non empty set
bool [:[:REAL, the carrier of X:], the carrier of X:] is non empty set
the Mult of X . ((||.tseq.|| "),tseq) is set
[(||.tseq.|| "),tseq] is set
the Mult of X . [(||.tseq.|| "),tseq] is set
vseq . tseq is Element of the carrier of Y
||.(vseq . tseq).|| is ext-real non negative V31() 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 V134() V135() V136() Element of bool [: the carrier of Y,REAL:]
[: the carrier of Y,REAL:] is non empty V134() V135() V136() set
bool [: the carrier of Y,REAL:] is non empty set
the U9 of Y . (vseq . tseq) is ext-real V31() real Element of REAL
||.(vseq . tseq).|| / ||.tseq.|| is ext-real non negative V31() real Element of REAL
(||.(vseq . tseq).|| / ||.tseq.||) * ||.tseq.|| is ext-real non negative V31() real Element of REAL
||.(vseq . tseq).|| * (||.tseq.|| ") is ext-real non negative V31() real Element of REAL
(||.(vseq . tseq).|| * (||.tseq.|| ")) * ||.tseq.|| is ext-real non negative V31() real Element of REAL
(||.tseq.|| ") * ||.tseq.|| is ext-real non negative V31() real Element of REAL
||.(vseq . tseq).|| * ((||.tseq.|| ") * ||.tseq.||) is ext-real non negative V31() real Element of REAL
||.(vseq . tseq).|| * 1 is ext-real non negative V31() real Element of REAL
abs (||.tseq.|| ") is ext-real V31() real Element of REAL
1 * (||.tseq.|| ") is ext-real non negative V31() real Element of REAL
abs (1 * (||.tseq.|| ")) is ext-real V31() real Element of REAL
1 / ||.tseq.|| is ext-real non negative V31() real Element of REAL
abs (1 / ||.tseq.||) is ext-real V31() real Element of REAL
abs ||.tseq.|| is ext-real V31() real Element of REAL
1 / (abs ||.tseq.||) is ext-real V31() real Element of REAL
tseq is Element of the carrier of X
||.tseq.|| is ext-real non negative V31() real Element of REAL
the U9 of X . tseq is ext-real V31() real Element of REAL
(abs (||.tseq.|| ")) * ||.tseq.|| is ext-real V31() real Element of REAL
vseq . tseq is Element of the carrier of Y
||.(vseq . tseq).|| is ext-real non negative V31() real Element of REAL
the U9 of Y . (vseq . tseq) is ext-real V31() real Element of REAL
(||.tseq.|| ") * (vseq . tseq) is Element of the carrier of Y
the Mult of Y is non empty Relation-like [:REAL, the carrier of Y:] -defined the carrier of Y -valued Function-like total quasi_total Element of bool [:[:REAL, the carrier of Y:], the carrier of Y:]
[:REAL, the carrier of Y:] is non empty set
[:[:REAL, the carrier of Y:], the carrier of Y:] is non empty set
bool [:[:REAL, the carrier of Y:], the carrier of Y:] is non empty set
the Mult of Y . ((||.tseq.|| "),(vseq . tseq)) is set
[(||.tseq.|| "),(vseq . tseq)] is set
the Mult of Y . [(||.tseq.|| "),(vseq . tseq)] is set
||.((||.tseq.|| ") * (vseq . tseq)).|| is ext-real non negative V31() real Element of REAL
the U9 of Y . ((||.tseq.|| ") * (vseq . tseq)) is ext-real V31() real Element of REAL
f is ext-real V31() real Element of REAL
f * ||.tseq.|| is ext-real V31() real Element of REAL
tseq is Element of the carrier of X
0. X is V53(X) 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 ext-real non negative V31() 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 V134() V135() V136() Element of bool [: the carrier of Y,REAL:]
[: the carrier of Y,REAL:] is non empty V134() V135() V136() set
bool [: the carrier of Y,REAL:] is non empty set
the U9 of Y . (vseq . tseq) is ext-real V31() real Element of REAL
f is ext-real V31() real Element of REAL
||.tseq.|| is ext-real non negative V31() 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 V134() V135() V136() Element of bool [: the carrier of X,REAL:]
[: the carrier of X,REAL:] is non empty V134() V135() V136() set
bool [: the carrier of X,REAL:] is non empty set
the U9 of X . tseq is ext-real V31() real Element of REAL
f * ||.tseq.|| is ext-real V31() real Element of REAL
tseq is Element of the carrier of X
0. X is V53(X) 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 ext-real non negative V31() 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 V134() V135() V136() Element of bool [: the carrier of Y,REAL:]
[: the carrier of Y,REAL:] is non empty V134() V135() V136() set
bool [: the carrier of Y,REAL:] is non empty set
the U9 of Y . (vseq . tseq) is ext-real V31() real Element of REAL
f is ext-real V31() real Element of REAL
||.tseq.|| is ext-real non negative V31() 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 V134() V135() V136() Element of bool [: the carrier of X,REAL:]
[: the carrier of X,REAL:] is non empty V134() V135() V136() set
bool [: the carrier of X,REAL:] is non empty set
the U9 of X . tseq is ext-real V31() real Element of REAL
f * ||.tseq.|| is ext-real V31() real Element of REAL
tseq is ext-real V31() real Element of REAL
tseq is set
tv is ext-real V31() real Element of REAL
e is Element of the carrier of X
vseq . e is Element of the carrier of Y
||.(vseq . e).|| is ext-real non negative V31() real Element of REAL
the U9 of Y . (vseq . e) is ext-real V31() real Element of REAL
||.e.|| is ext-real non negative V31() real Element of REAL
the U9 of X . e is ext-real V31() real Element of REAL
tseq is ext-real V31() real Element of REAL
X is non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() discerning reflexive RealNormSpace-like NORMSTR
Y is non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() discerning reflexive RealNormSpace-like NORMSTR
(X,Y) is non empty Element of bool the carrier of (X,Y)
(X,Y) is non empty left_complementable right_complementable strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() constituted-Functions RLSStruct
(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 left_complementable right_complementable strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() constituted-Functions RLSStruct
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
( 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)
0. Y is V53(Y) 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
( 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 V110() 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 [:REAL,(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 V110() Element of bool [:[:REAL,(Funcs ( the carrier of X, the carrier of Y)):],(Funcs ( the carrier of X, the carrier of Y)):]
[:REAL,(Funcs ( the carrier of X, the carrier of Y)):] is non empty set
[:[:REAL,(Funcs ( the carrier of X, the carrier of Y)):],(Funcs ( the carrier of X, the carrier of Y)):] is non empty set
bool [:[:REAL,(Funcs ( the carrier of X, the carrier of Y)):],(Funcs ( the carrier of X, the carrier of Y)):] is non empty set
RLSStruct(# (Funcs ( the carrier of X, the carrier of Y)),( the carrier of X,Y),( the carrier of X,Y),( the carrier of X,Y) #) is non empty strict RLSStruct
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 V110() 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 [:REAL,(X,Y):] -defined (X,Y) -valued Function-like total quasi_total Function-yielding V110() Element of bool [:[:REAL,(X,Y):],(X,Y):]
[:REAL,(X,Y):] is non empty set
[:[:REAL,(X,Y):],(X,Y):] is non empty set
bool [:[:REAL,(X,Y):],(X,Y):] is non empty set
RLSStruct(# (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 left_complementable right_complementable strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() RLSStruct
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 V134() V135() V136() 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:]
(X,Y,(X,Y,vseq)) is non empty V144() V145() V146() 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 ext-real V31() real Element of REAL
vseq is non empty Relation-like (X,Y) -defined REAL -valued Function-like total quasi_total V134() V135() V136() Element of bool [:(X,Y),REAL:]
f is non empty Relation-like (X,Y) -defined REAL -valued Function-like total quasi_total V134() V135() V136() Element of bool [:(X,Y),REAL:]
tseq is set
vseq . tseq is ext-real V31() real Element of REAL
f . tseq is ext-real V31() 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:]
(X,Y,(X,Y,tseq)) is non empty V144() V145() V146() 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 ext-real V31() real Element of REAL
dom f is set
dom vseq is set
X is non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() discerning reflexive RealNormSpace-like NORMSTR
the carrier of X is non empty set
Y is non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() discerning reflexive RealNormSpace-like NORMSTR
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 left_complementable right_complementable strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() constituted-Functions RLSStruct
(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 left_complementable right_complementable strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() constituted-Functions RLSStruct
Funcs ( the carrier of X, the carrier of Y) is non empty functional FUNCTION_DOMAIN of the carrier of X, the carrier of Y
( 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)
0. Y is V53(Y) 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,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 V110() 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 [:REAL,(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 V110() Element of bool [:[:REAL,(Funcs ( the carrier of X, the carrier of Y)):],(Funcs ( the carrier of X, the carrier of Y)):]
[:REAL,(Funcs ( the carrier of X, the carrier of Y)):] is non empty set
[:[:REAL,(Funcs ( the carrier of X, the carrier of Y)):],(Funcs ( the carrier of X, the carrier of Y)):] is non empty set
bool [:[:REAL,(Funcs ( the carrier of X, the carrier of Y)):],(Funcs ( the carrier of X, the carrier of Y)):] is non empty set
RLSStruct(# (Funcs ( the carrier of X, the carrier of Y)),( the carrier of X,Y),( the carrier of X,Y),( the carrier of X,Y) #) is non empty strict RLSStruct
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 V110() 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 [:REAL,(X,Y):] -defined (X,Y) -valued Function-like total quasi_total Function-yielding V110() Element of bool [:[:REAL,(X,Y):],(X,Y):]
[:REAL,(X,Y):] is non empty set
[:[:REAL,(X,Y):],(X,Y):] is non empty set
bool [:[:REAL,(X,Y):],(X,Y):] is non empty set
RLSStruct(# (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 left_complementable right_complementable strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() RLSStruct
the carrier of (X,Y) is non empty set
bool the carrier of (X,Y) is non empty set
X is non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() discerning reflexive RealNormSpace-like NORMSTR
the carrier of X is non empty set
Y is non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() discerning reflexive RealNormSpace-like NORMSTR
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 V134() V135() V136() Element of bool [:(X,Y),REAL:]
(X,Y) is non empty Element of bool the carrier of (X,Y)
(X,Y) is non empty left_complementable right_complementable strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() constituted-Functions RLSStruct
(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 left_complementable right_complementable strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() constituted-Functions RLSStruct
Funcs ( the carrier of X, the carrier of Y) is non empty functional FUNCTION_DOMAIN of the carrier of X, the carrier of Y
( 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)
0. Y is V53(Y) 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,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 V110() 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 [:REAL,(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 V110() Element of bool [:[:REAL,(Funcs ( the carrier of X, the carrier of Y)):],(Funcs ( the carrier of X, the carrier of Y)):]
[:REAL,(Funcs ( the carrier of X, the carrier of Y)):] is non empty set
[:[:REAL,(Funcs ( the carrier of X, the carrier of Y)):],(Funcs ( the carrier of X, the carrier of Y)):] is non empty set
bool [:[:REAL,(Funcs ( the carrier of X, the carrier of Y)):],(Funcs ( the carrier of X, the carrier of Y)):] is non empty set
RLSStruct(# (Funcs ( the carrier of X, the carrier of Y)),( the carrier of X,Y),( the carrier of X,Y),( the carrier of X,Y) #) is non empty strict RLSStruct
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 V110() 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 [:REAL,(X,Y):] -defined (X,Y) -valued Function-like total quasi_total Function-yielding V110() Element of bool [:[:REAL,(X,Y):],(X,Y):]
[:REAL,(X,Y):] is non empty set
[:[:REAL,(X,Y):],(X,Y):] is non empty set
bool [:[:REAL,(X,Y):],(X,Y):] is non empty set
RLSStruct(# (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 left_complementable right_complementable strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() RLSStruct
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 V134() V135() V136() 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 ext-real V31() real Element of REAL
(X,Y,vseq) is non empty V144() V145() V146() 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 ext-real V31() 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 V144() V145() V146() 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 ext-real V31() real Element of REAL
X is non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() discerning reflexive RealNormSpace-like NORMSTR
Y is non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() discerning reflexive RealNormSpace-like NORMSTR
(X,Y) is non empty Element of bool the carrier of (X,Y)
(X,Y) is non empty left_complementable right_complementable strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() constituted-Functions RLSStruct
(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 left_complementable right_complementable strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() constituted-Functions RLSStruct
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
( 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)
0. Y is V53(Y) 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
( 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 V110() 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 [:REAL,(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 V110() Element of bool [:[:REAL,(Funcs ( the carrier of X, the carrier of Y)):],(Funcs ( the carrier of X, the carrier of Y)):]
[:REAL,(Funcs ( the carrier of X, the carrier of Y)):] is non empty set
[:[:REAL,(Funcs ( the carrier of X, the carrier of Y)):],(Funcs ( the carrier of X, the carrier of Y)):] is non empty set
bool [:[:REAL,(Funcs ( the carrier of X, the carrier of Y)):],(Funcs ( the carrier of X, the carrier of Y)):] is non empty set
RLSStruct(# (Funcs ( the carrier of X, the carrier of Y)),( the carrier of X,Y),( the carrier of X,Y),( the carrier of X,Y) #) is non empty strict RLSStruct
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 V110() 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 [:REAL,(X,Y):] -defined (X,Y) -valued Function-like total quasi_total Function-yielding V110() Element of bool [:[:REAL,(X,Y):],(X,Y):]
[:REAL,(X,Y):] is non empty set
[:[:REAL,(X,Y):],(X,Y):] is non empty set
bool [:[:REAL,(X,Y):],(X,Y):] is non empty set
RLSStruct(# (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 left_complementable right_complementable strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() RLSStruct
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 [:REAL,(X,Y):] -defined (X,Y) -valued Function-like total quasi_total Element of bool [:[:REAL,(X,Y):],(X,Y):]
[:REAL,(X,Y):] is non empty set
[:[:REAL,(X,Y):],(X,Y):] is non empty set
bool [:[:REAL,(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 V134() V135() V136() Element of bool [:(X,Y),REAL:]
[:(X,Y),REAL:] is non empty V134() V135() V136() set
bool [:(X,Y),REAL:] is non empty set
NORMSTR(# (X,Y),(Zero_ ((X,Y),(X,Y))),(Add_ ((X,Y),(X,Y))),(Mult_ ((X,Y),(X,Y))),(X,Y) #) is non empty strict NORMSTR
Y is non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() discerning reflexive RealNormSpace-like NORMSTR
the carrier of Y is non empty set
X is non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() discerning reflexive RealNormSpace-like NORMSTR
the carrier of X is non empty set
0. Y is V53(Y) 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 NORMSTR
(X,Y) is non empty Element of bool the carrier of (X,Y)
(X,Y) is non empty left_complementable right_complementable strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() constituted-Functions RLSStruct
(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 left_complementable right_complementable strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() constituted-Functions RLSStruct
Funcs ( the carrier of X, the carrier of Y) is non empty functional FUNCTION_DOMAIN of the carrier of X, the carrier of Y
( 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)
( 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 V110() 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 [:REAL,(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 V110() Element of bool [:[:REAL,(Funcs ( the carrier of X, the carrier of Y)):],(Funcs ( the carrier of X, the carrier of Y)):]
[:REAL,(Funcs ( the carrier of X, the carrier of Y)):] is non empty set
[:[:REAL,(Funcs ( the carrier of X, the carrier of Y)):],(Funcs ( the carrier of X, the carrier of Y)):] is non empty set
bool [:[:REAL,(Funcs ( the carrier of X, the carrier of Y)):],(Funcs ( the carrier of X, the carrier of Y)):] is non empty set
RLSStruct(# (Funcs ( the carrier of X, the carrier of Y)),( the carrier of X,Y),( the carrier of X,Y),( the carrier of X,Y) #) is non empty strict RLSStruct
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 V110() 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 [:REAL,(X,Y):] -defined (X,Y) -valued Function-like total quasi_total Function-yielding V110() Element of bool [:[:REAL,(X,Y):],(X,Y):]
[:REAL,(X,Y):] is non empty set
[:[:REAL,(X,Y):],(X,Y):] is non empty set
bool [:[:REAL,(X,Y):],(X,Y):] is non empty set
RLSStruct(# (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 left_complementable right_complementable strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() RLSStruct
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 [:REAL,(X,Y):] -defined (X,Y) -valued Function-like total quasi_total Element of bool [:[:REAL,(X,Y):],(X,Y):]
[:REAL,(X,Y):] is non empty set
[:[:REAL,(X,Y):],(X,Y):] is non empty set
bool [:[:REAL,(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 V134() V135() V136() Element of bool [:(X,Y),REAL:]
[:(X,Y),REAL:] is non empty V134() V135() V136() set
bool [:(X,Y),REAL:] is non empty set
NORMSTR(# (X,Y),(Zero_ ((X,Y),(X,Y))),(Add_ ((X,Y),(X,Y))),(Mult_ ((X,Y),(X,Y))),(X,Y) #) is non empty strict NORMSTR
0. (X,Y) is V53((X,Y)) 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 left_complementable right_complementable strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() RLSStruct
RLSStruct(# (X,Y),(Zero_ ((X,Y),(X,Y))),(Add_ ((X,Y),(X,Y))),(Mult_ ((X,Y),(X,Y))) #) is non empty left_complementable right_complementable strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() RLSStruct
0. (X,Y) is Relation-like Function-like V53((X,Y)) 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 left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() discerning reflexive RealNormSpace-like NORMSTR
Y is non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() discerning reflexive RealNormSpace-like NORMSTR
(X,Y) is non empty NORMSTR
(X,Y) is non empty Element of bool the carrier of (X,Y)
(X,Y) is non empty left_complementable right_complementable strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() constituted-Functions RLSStruct
(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 left_complementable right_complementable strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() constituted-Functions RLSStruct
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
( 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)
0. Y is V53(Y) 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
( 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 V110() 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 [:REAL,(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 V110() Element of bool [:[:REAL,(Funcs ( the carrier of X, the carrier of Y)):],(Funcs ( the carrier of X, the carrier of Y)):]
[:REAL,(Funcs ( the carrier of X, the carrier of Y)):] is non empty set
[:[:REAL,(Funcs ( the carrier of X, the carrier of Y)):],(Funcs ( the carrier of X, the carrier of Y)):] is non empty set
bool [:[:REAL,(Funcs ( the carrier of X, the carrier of Y)):],(Funcs ( the carrier of X, the carrier of Y)):] is non empty set
RLSStruct(# (Funcs ( the carrier of X, the carrier of Y)),( the carrier of X,Y),( the carrier of X,Y),( the carrier of X,Y) #) is non empty strict RLSStruct
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 V110() 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 [:REAL,(X,Y):] -defined (X,Y) -valued Function-like total quasi_total Function-yielding V110() Element of bool [:[:REAL,(X,Y):],(X,Y):]
[:REAL,(X,Y):] is non empty set
[:[:REAL,(X,Y):],(X,Y):] is non empty set
bool [:[:REAL,(X,Y):],(X,Y):] is non empty set
RLSStruct(# (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 left_complementable right_complementable strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() RLSStruct
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 [:REAL,(X,Y):] -defined (X,Y) -valued Function-like total quasi_total Element of bool [:[:REAL,(X,Y):],(X,Y):]
[:REAL,(X,Y):] is non empty set
[:[:REAL,(X,Y):],(X,Y):] is non empty set
bool [:[:REAL,(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 V134() V135() V136() Element of bool [:(X,Y),REAL:]
[:(X,Y),REAL:] is non empty V134() V135() V136() set
bool [:(X,Y),REAL:] is non empty set
NORMSTR(# (X,Y),(Zero_ ((X,Y),(X,Y))),(Add_ ((X,Y),(X,Y))),(Mult_ ((X,Y),(X,Y))),(X,Y) #) is non empty strict NORMSTR
the carrier of (X,Y) is non empty set
vseq is Element of the carrier of (X,Y)
||.vseq.|| is ext-real V31() 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 V134() V135() V136() Element of bool [: the carrier of (X,Y),REAL:]
[: the carrier of (X,Y),REAL:] is non empty V134() V135() V136() set
bool [: the carrier of (X,Y),REAL:] is non empty set
the U9 of (X,Y) . vseq is ext-real V31() 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 V144() V145() V146() 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 V53(X) Element of the carrier of X
the ZeroF of X is Element of the carrier of X
||.tseq.|| is ext-real non negative V31() 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 V134() V135() V136() Element of bool [: the carrier of X,REAL:]
[: the carrier of X,REAL:] is non empty V134() V135() V136() set
bool [: the carrier of X,REAL:] is non empty set
the U9 of X . tseq is ext-real V31() real Element of REAL
f . tseq is Element of the carrier of Y
0 * (0. X) is Element of the carrier of X
the Mult of X is non empty Relation-like [:REAL, the carrier of X:] -defined the carrier of X -valued Function-like total quasi_total Element of bool [:[:REAL, the carrier of X:], the carrier of X:]
[:REAL, the carrier of X:] is non empty set
[:[:REAL, the carrier of X:], the carrier of X:] is non empty set
bool [:[:REAL, the carrier of X:], the carrier of X:] is non empty set
the Mult of X . (0,(0. X)) is set
[0,(0. X)] is set
the Mult of X . [0,(0. X)] is set
f . (0 * (0. X)) is Element of the carrier of Y
f . (0. X) is Element of the carrier of Y
0 * (f . (0. X)) is Element of the carrier of Y
the Mult of Y is non empty Relation-like [:REAL, the carrier of Y:] -defined the carrier of Y -valued Function-like total quasi_total Element of bool [:[:REAL, the carrier of Y:], the carrier of Y:]
[:REAL, the carrier of Y:] is non empty set
[:[:REAL, the carrier of Y:], the carrier of Y:] is non empty set
bool [:[:REAL, the carrier of Y:], the carrier of Y:] is non empty set
the Mult of Y . (0,(f . (0. X))) is set
[0,(f . (0. X))] is set
the Mult of Y . [0,(f . (0. X))] is set
||.(f . tseq).|| is ext-real non negative V31() 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 V134() V135() V136() Element of bool [: the carrier of Y,REAL:]
[: the carrier of Y,REAL:] is non empty V134() V135() V136() set
bool [: the carrier of Y,REAL:] is non empty set
the U9 of Y . (f . tseq) is ext-real V31() real Element of REAL
||.vseq.|| * ||.tseq.|| is ext-real V31() real Element of REAL
tseq is Element of the carrier of X
0. X is V53(X) Element of the carrier of X
the ZeroF of X is Element of the carrier of X
||.tseq.|| is ext-real non negative V31() 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 V134() V135() V136() Element of bool [: the carrier of X,REAL:]
[: the carrier of X,REAL:] is non empty V134() V135() V136() set
bool [: the carrier of X,REAL:] is non empty set
the U9 of X . tseq is ext-real V31() real Element of REAL
||.tseq.|| " is ext-real non negative V31() real Element of REAL
(||.tseq.|| ") * tseq is Element of the carrier of X
the Mult of X is non empty Relation-like [:REAL, the carrier of X:] -defined the carrier of X -valued Function-like total quasi_total Element of bool [:[:REAL, the carrier of X:], the carrier of X:]
[:REAL, the carrier of X:] is non empty set
[:[:REAL, the carrier of X:], the carrier of X:] is non empty set
bool [:[:REAL, the carrier of X:], the carrier of X:] is non empty set
the Mult of X . ((||.tseq.|| "),tseq) is set
[(||.tseq.|| "),tseq] is set
the Mult of X . [(||.tseq.|| "),tseq] is set
abs (||.tseq.|| ") is ext-real V31() real Element of REAL
1 * (||.tseq.|| ") is ext-real non negative V31() real Element of REAL
abs (1 * (||.tseq.|| ")) is ext-real V31() real Element of REAL
1 / ||.tseq.|| is ext-real non negative V31() real Element of REAL
abs (1 / ||.tseq.||) is ext-real V31() real Element of REAL
abs ||.tseq.|| is ext-real V31() real Element of REAL
1 / (abs ||.tseq.||) is ext-real V31() real Element of REAL
f . tseq is Element of the carrier of Y
||.(f . tseq).|| is ext-real non negative V31() 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 V134() V135() V136() Element of bool [: the carrier of Y,REAL:]
[: the carrier of Y,REAL:] is non empty V134() V135() V136() set
bool [: the carrier of Y,REAL:] is non empty set
the U9 of Y . (f . tseq) is ext-real V31() real Element of REAL
||.(f . tseq).|| / ||.tseq.|| is ext-real non negative V31() real Element of REAL
||.(f . tseq).|| * (||.tseq.|| ") is ext-real non negative V31() real Element of REAL
(||.tseq.|| ") * (f . tseq) is Element of the carrier of Y
the Mult of Y is non empty Relation-like [:REAL, the carrier of Y:] -defined the carrier of Y -valued Function-like total quasi_total Element of bool [:[:REAL, the carrier of Y:], the carrier of Y:]
[:REAL, the carrier of Y:] is non empty set
[:[:REAL, the carrier of Y:], the carrier of Y:] is non empty set
bool [:[:REAL, the carrier of Y:], the carrier of Y:] is non empty set
the Mult of Y . ((||.tseq.|| "),(f . tseq)) is set
[(||.tseq.|| "),(f . tseq)] is set
the Mult of Y . [(||.tseq.|| "),(f . tseq)] is set
||.((||.tseq.|| ") * (f . tseq)).|| is ext-real non negative V31() real Element of REAL
the U9 of Y . ((||.tseq.|| ") * (f . tseq)) is ext-real V31() real Element of REAL
tseq is Element of the carrier of X
f . tseq is Element of the carrier of Y
||.(f . tseq).|| is ext-real non negative V31() real Element of REAL
the U9 of Y . (f . tseq) is ext-real V31() real Element of REAL
||.tseq.|| is ext-real non negative V31() real Element of REAL
the U9 of X . tseq is ext-real V31() real Element of REAL
(abs (||.tseq.|| ")) * ||.tseq.|| is ext-real V31() real Element of REAL
upper_bound (X,Y,f) is ext-real V31() real Element of REAL
(X,Y) . f is ext-real V31() real Element of REAL
(||.(f . tseq).|| / ||.tseq.||) * ||.tseq.|| is ext-real non negative V31() real Element of REAL
(||.(f . tseq).|| * (||.tseq.|| ")) * ||.tseq.|| is ext-real non negative V31() real Element of REAL
(||.tseq.|| ") * ||.tseq.|| is ext-real non negative V31() real Element of REAL
||.(f . tseq).|| * ((||.tseq.|| ") * ||.tseq.||) is ext-real non negative V31() real Element of REAL
||.(f . tseq).|| * 1 is ext-real non negative V31() real Element of REAL
||.vseq.|| * ||.tseq.|| is ext-real V31() real Element of REAL
tseq is Element of the carrier of X
0. X is V53(X) 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 ext-real non negative V31() 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 V134() V135() V136() Element of bool [: the carrier of Y,REAL:]
[: the carrier of Y,REAL:] is non empty V134() V135() V136() set
bool [: the carrier of Y,REAL:] is non empty set
the U9 of Y . (f . tseq) is ext-real V31() real Element of REAL
||.tseq.|| is ext-real non negative V31() 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 V134() V135() V136() Element of bool [: the carrier of X,REAL:]
[: the carrier of X,REAL:] is non empty V134() V135() V136() set
bool [: the carrier of X,REAL:] is non empty set
the U9 of X . tseq is ext-real V31() real Element of REAL
||.vseq.|| * ||.tseq.|| is ext-real V31() real Element of REAL
tseq is Element of the carrier of X
0. X is V53(X) 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 ext-real non negative V31() 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 V134() V135() V136() Element of bool [: the carrier of Y,REAL:]
[: the carrier of Y,REAL:] is non empty V134() V135() V136() set
bool [: the carrier of Y,REAL:] is non empty set
the U9 of Y . (f . tseq) is ext-real V31() real Element of REAL
||.tseq.|| is ext-real non negative V31() 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 V134() V135() V136() Element of bool [: the carrier of X,REAL:]
[: the carrier of X,REAL:] is non empty V134() V135() V136() set
bool [: the carrier of X,REAL:] is non empty set
the U9 of X . tseq is ext-real V31() real Element of REAL
||.vseq.|| * ||.tseq.|| is ext-real V31() real Element of REAL
tseq is Element of the carrier of X
f . tseq is Element of the carrier of Y
||.(f . tseq).|| is ext-real non negative V31() real Element of REAL
the U9 of Y . (f . tseq) is ext-real V31() real Element of REAL
||.tseq.|| is ext-real non negative V31() real Element of REAL
the U9 of X . tseq is ext-real V31() real Element of REAL
||.vseq.|| * ||.tseq.|| is ext-real V31() real Element of REAL
X is non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() discerning reflexive RealNormSpace-like NORMSTR
Y is non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() discerning reflexive RealNormSpace-like NORMSTR
(X,Y) is non empty NORMSTR
(X,Y) is non empty Element of bool the carrier of (X,Y)
(X,Y) is non empty left_complementable right_complementable strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() constituted-Functions RLSStruct
(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 left_complementable right_complementable strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() constituted-Functions RLSStruct
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
( 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)
0. Y is V53(Y) 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
( 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 V110() 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 [:REAL,(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 V110() Element of bool [:[:REAL,(Funcs ( the carrier of X, the carrier of Y)):],(Funcs ( the carrier of X, the carrier of Y)):]
[:REAL,(Funcs ( the carrier of X, the carrier of Y)):] is non empty set
[:[:REAL,(Funcs ( the carrier of X, the carrier of Y)):],(Funcs ( the carrier of X, the carrier of Y)):] is non empty set
bool [:[:REAL,(Funcs ( the carrier of X, the carrier of Y)):],(Funcs ( the carrier of X, the carrier of Y)):] is non empty set
RLSStruct(# (Funcs ( the carrier of X, the carrier of Y)),( the carrier of X,Y),( the carrier of X,Y),( the carrier of X,Y) #) is non empty strict RLSStruct
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 V110() 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 [:REAL,(X,Y):] -defined (X,Y) -valued Function-like total quasi_total Function-yielding V110() Element of bool [:[:REAL,(X,Y):],(X,Y):]
[:REAL,(X,Y):] is non empty set
[:[:REAL,(X,Y):],(X,Y):] is non empty set
bool [:[:REAL,(X,Y):],(X,Y):] is non empty set
RLSStruct(# (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 left_complementable right_complementable strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() RLSStruct
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 [:REAL,(X,Y):] -defined (X,Y) -valued Function-like total quasi_total Element of bool [:[:REAL,(X,Y):],(X,Y):]
[:REAL,(X,Y):] is non empty set
[:[:REAL,(X,Y):],(X,Y):] is non empty set
bool [:[:REAL,(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 V134() V135() V136() Element of bool [:(X,Y),REAL:]
[:(X,Y),REAL:] is non empty V134() V135() V136() set
bool [:(X,Y),REAL:] is non empty set
NORMSTR(# (X,Y),(Zero_ ((X,Y),(X,Y))),(Add_ ((X,Y),(X,Y))),(Mult_ ((X,Y),(X,Y))),(X,Y) #) is non empty strict NORMSTR
the carrier of (X,Y) is non empty set
vseq is Element of the carrier of (X,Y)
||.vseq.|| is ext-real V31() 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 V134() V135() V136() Element of bool [: the carrier of (X,Y),REAL:]
[: the carrier of (X,Y),REAL:] is non empty V134() V135() V136() set
bool [: the carrier of (X,Y),REAL:] is non empty set
the U9 of (X,Y) . vseq is ext-real V31() 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 V144() V145() V146() 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 ext-real V31() real Element of REAL
upper_bound (X,Y,f) is ext-real V31() real Element of REAL
tseq is ext-real V31() real Element of REAL
tv is Element of the carrier of X
f . tv is Element of the carrier of Y
||.(f . tv).|| is ext-real non negative V31() 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 V134() V135() V136() Element of bool [: the carrier of Y,REAL:]
[: the carrier of Y,REAL:] is non empty V134() V135() V136() set
bool [: the carrier of Y,REAL:] is non empty set
the U9 of Y . (f . tv) is ext-real V31() real Element of REAL
||.tv.|| is ext-real non negative V31() 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 V134() V135() V136() Element of bool [: the carrier of X,REAL:]
[: the carrier of X,REAL:] is non empty V134() V135() V136() set
bool [: the carrier of X,REAL:] is non empty set
the U9 of X . tv is ext-real V31() real Element of REAL
tseq is ext-real V31() real Element of REAL
X is non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() discerning reflexive RealNormSpace-like NORMSTR
Y is non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() discerning reflexive RealNormSpace-like NORMSTR
(X,Y) is non empty NORMSTR
(X,Y) is non empty Element of bool the carrier of (X,Y)
(X,Y) is non empty left_complementable right_complementable strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() constituted-Functions RLSStruct
(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 left_complementable right_complementable strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() constituted-Functions RLSStruct
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
( 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)
0. Y is V53(Y) 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
( 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 V110() 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 [:REAL,(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 V110() Element of bool [:[:REAL,(Funcs ( the carrier of X, the carrier of Y)):],(Funcs ( the carrier of X, the carrier of Y)):]
[:REAL,(Funcs ( the carrier of X, the carrier of Y)):] is non empty set
[:[:REAL,(Funcs ( the carrier of X, the carrier of Y)):],(Funcs ( the carrier of X, the carrier of Y)):] is non empty set
bool [:[:REAL,(Funcs ( the carrier of X, the carrier of Y)):],(Funcs ( the carrier of X, the carrier of Y)):] is non empty set
RLSStruct(# (Funcs ( the carrier of X, the carrier of Y)),( the carrier of X,Y),( the carrier of X,Y),( the carrier of X,Y) #) is non empty strict RLSStruct
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 V110() 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 [:REAL,(X,Y):] -defined (X,Y) -valued Function-like total quasi_total Function-yielding V110() Element of bool [:[:REAL,(X,Y):],(X,Y):]
[:REAL,(X,Y):] is non empty set
[:[:REAL,(X,Y):],(X,Y):] is non empty set
bool [:[:REAL,(X,Y):],(X,Y):] is non empty set
RLSStruct(# (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 left_complementable right_complementable strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() RLSStruct
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 [:REAL,(X,Y):] -defined (X,Y) -valued Function-like total quasi_total Element of bool [:[:REAL,(X,Y):],(X,Y):]
[:REAL,(X,Y):] is non empty set
[:[:REAL,(X,Y):],(X,Y):] is non empty set
bool [:[:REAL,(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 V134() V135() V136() Element of bool [:(X,Y),REAL:]
[:(X,Y),REAL:] is non empty V134() V135() V136() set
bool [:(X,Y),REAL:] is non empty set
NORMSTR(# (X,Y),(Zero_ ((X,Y),(X,Y))),(Add_ ((X,Y),(X,Y))),(Mult_ ((X,Y),(X,Y))),(X,Y) #) is non empty strict NORMSTR
the carrier of (X,Y) is non empty set
0. (X,Y) is V53((X,Y)) 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 ext-real V31() 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 V134() V135() V136() Element of bool [: the carrier of (X,Y),REAL:]
[: the carrier of (X,Y),REAL:] is non empty V134() V135() V136() set
bool [: the carrier of (X,Y),REAL:] is non empty set
the U9 of (X,Y) . vseq is ext-real V31() 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 V144() V145() V146() 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 ext-real V31() 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 ext-real V31() real Element of REAL
m is Element of the carrier of X
f . m is Element of the carrier of Y
||.(f . m).|| is ext-real non negative V31() 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 V134() V135() V136() Element of bool [: the carrier of Y,REAL:]
[: the carrier of Y,REAL:] is non empty V134() V135() V136() set
bool [: the carrier of Y,REAL:] is non empty set
the U9 of Y . (f . m) is ext-real V31() real Element of REAL
||.m.|| is ext-real non negative V31() 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 V134() V135() V136() Element of bool [: the carrier of X,REAL:]
[: the carrier of X,REAL:] is non empty V134() V135() V136() set
bool [: the carrier of X,REAL:] is non empty set
the U9 of X . m is ext-real V31() real Element of REAL
||.(0. Y).|| is empty Function-like functional ext-real non positive non negative epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural V31() real V144() V145() V146() V147() V148() V149() V150() Element of REAL
the U9 of Y . (0. Y) is ext-real V31() real Element of REAL
tv is ext-real V31() real Element of REAL
e is ext-real V31() real set
(X,Y) . vseq is ext-real V31() real Element of REAL
X is non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() discerning reflexive RealNormSpace-like NORMSTR
Y is non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() discerning reflexive RealNormSpace-like NORMSTR
(X,Y) is non empty NORMSTR
(X,Y) is non empty Element of bool the carrier of (X,Y)
(X,Y) is non empty left_complementable right_complementable strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() constituted-Functions RLSStruct
(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 left_complementable right_complementable strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() constituted-Functions RLSStruct
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
( 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)
0. Y is V53(Y) 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
( 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 V110() 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 [:REAL,(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 V110() Element of bool [:[:REAL,(Funcs ( the carrier of X, the carrier of Y)):],(Funcs ( the carrier of X, the carrier of Y)):]
[:REAL,(Funcs ( the carrier of X, the carrier of Y)):] is non empty set
[:[:REAL,(Funcs ( the carrier of X, the carrier of Y)):],(Funcs ( the carrier of X, the carrier of Y)):] is non empty set
bool [:[:REAL,(Funcs ( the carrier of X, the carrier of Y)):],(Funcs ( the carrier of X, the carrier of Y)):] is non empty set
RLSStruct(# (Funcs ( the carrier of X, the carrier of Y)),( the carrier of X,Y),( the carrier of X,Y),( the carrier of X,Y) #) is non empty strict RLSStruct
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 V110() 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 [:REAL,(X,Y):] -defined (X,Y) -valued Function-like total quasi_total Function-yielding V110() Element of bool [:[:REAL,(X,Y):],(X,Y):]
[:REAL,(X,Y):] is non empty set
[:[:REAL,(X,Y):],(X,Y):] is non empty set
bool [:[:REAL,(X,Y):],(X,Y):] is non empty set
RLSStruct(# (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 left_complementable right_complementable strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() RLSStruct
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 [:REAL,(X,Y):] -defined (X,Y) -valued Function-like total quasi_total Element of bool [:[:REAL,(X,Y):],(X,Y):]
[:REAL,(X,Y):] is non empty set
[:[:REAL,(X,Y):],(X,Y):] is non empty set
bool [:[:REAL,(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 V134() V135() V136() Element of bool [:(X,Y),REAL:]
[:(X,Y),REAL:] is non empty V134() V135() V136() set
bool [:(X,Y),REAL:] is non empty set
NORMSTR(# (X,Y),(Zero_ ((X,Y),(X,Y))),(Add_ ((X,Y),(X,Y))),(Mult_ ((X,Y),(X,Y))),(X,Y) #) is non empty strict NORMSTR
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 left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() discerning reflexive RealNormSpace-like NORMSTR
Y is non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() discerning reflexive RealNormSpace-like NORMSTR
(X,Y) is non empty NORMSTR
(X,Y) is non empty Element of bool the carrier of (X,Y)
(X,Y) is non empty left_complementable right_complementable strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() constituted-Functions RLSStruct
(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 left_complementable right_complementable strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() constituted-Functions RLSStruct
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
( 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)
0. Y is V53(Y) 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
( 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 V110() 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 [:REAL,(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 V110() Element of bool [:[:REAL,(Funcs ( the carrier of X, the carrier of Y)):],(Funcs ( the carrier of X, the carrier of Y)):]
[:REAL,(Funcs ( the carrier of X, the carrier of Y)):] is non empty set
[:[:REAL,(Funcs ( the carrier of X, the carrier of Y)):],(Funcs ( the carrier of X, the carrier of Y)):] is non empty set
bool [:[:REAL,(Funcs ( the carrier of X, the carrier of Y)):],(Funcs ( the carrier of X, the carrier of Y)):] is non empty set
RLSStruct(# (Funcs ( the carrier of X, the carrier of Y)),( the carrier of X,Y),( the carrier of X,Y),( the carrier of X,Y) #) is non empty strict RLSStruct
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 V110() 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 [:REAL,(X,Y):] -defined (X,Y) -valued Function-like total quasi_total Function-yielding V110() Element of bool [:[:REAL,(X,Y):],(X,Y):]
[:REAL,(X,Y):] is non empty set
[:[:REAL,(X,Y):],(X,Y):] is non empty set
bool [:[:REAL,(X,Y):],(X,Y):] is non empty set
RLSStruct(# (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 left_complementable right_complementable strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() RLSStruct
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 [:REAL,(X,Y):] -defined (X,Y) -valued Function-like total quasi_total Element of bool [:[:REAL,(X,Y):],(X,Y):]
[:REAL,(X,Y):] is non empty set
[:[:REAL,(X,Y):],(X,Y):] is non empty set
bool [:[:REAL,(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 V134() V135() V136() Element of bool [:(X,Y),REAL:]
[:(X,Y),REAL:] is non empty V134() V135() V136() set
bool [:(X,Y),REAL:] is non empty set
NORMSTR(# (X,Y),(Zero_ ((X,Y),(X,Y))),(Add_ ((X,Y),(X,Y))),(Mult_ ((X,Y),(X,Y))),(X,Y) #) is non empty strict NORMSTR
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
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 left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() discerning reflexive RealNormSpace-like NORMSTR
Y is non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() discerning reflexive RealNormSpace-like NORMSTR
(X,Y) is non empty NORMSTR
(X,Y) is non empty Element of bool the carrier of (X,Y)
(X,Y) is non empty left_complementable right_complementable strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() constituted-Functions RLSStruct
(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 left_complementable right_complementable strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() constituted-Functions RLSStruct
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
( 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)
0. Y is V53(Y) 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
( 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 V110() 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 [:REAL,(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 V110() Element of bool [:[:REAL,(Funcs ( the carrier of X, the carrier of Y)):],(Funcs ( the carrier of X, the carrier of Y)):]
[:REAL,(Funcs ( the carrier of X, the carrier of Y)):] is non empty set
[:[:REAL,(Funcs ( the carrier of X, the carrier of Y)):],(Funcs ( the carrier of X, the carrier of Y)):] is non empty set
bool [:[:REAL,(Funcs ( the carrier of X, the carrier of Y)):],(Funcs ( the carrier of X, the carrier of Y)):] is non empty set
RLSStruct(# (Funcs ( the carrier of X, the carrier of Y)),( the carrier of X,Y),( the carrier of X,Y),( the carrier of X,Y) #) is non empty strict RLSStruct
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 V110() 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 [:REAL,(X,Y):] -defined (X,Y) -valued Function-like total quasi_total Function-yielding V110() Element of bool [:[:REAL,(X,Y):],(X,Y):]
[:REAL,(X,Y):] is non empty set
[:[:REAL,(X,Y):],(X,Y):] is non empty set
bool [:[:REAL,(X,Y):],(X,Y):] is non empty set
RLSStruct(# (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 left_complementable right_complementable strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() RLSStruct
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 [:REAL,(X,Y):] -defined (X,Y) -valued Function-like total quasi_total Element of bool [:[:REAL,(X,Y):],(X,Y):]
[:REAL,(X,Y):] is non empty set
[:[:REAL,(X,Y):],(X,Y):] is non empty set
bool [:[:REAL,(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 V134() V135() V136() Element of bool [:(X,Y),REAL:]
[:(X,Y),REAL:] is non empty V134() V135() V136() set
bool [:(X,Y),REAL:] is non empty set
NORMSTR(# (X,Y),(Zero_ ((X,Y),(X,Y))),(Add_ ((X,Y),(X,Y))),(Mult_ ((X,Y),(X,Y))),(X,Y) #) is non empty strict NORMSTR
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 left_complementable right_complementable strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() RLSStruct
RLSStruct(# (X,Y),(Zero_ ((X,Y),(X,Y))),(Add_ ((X,Y),(X,Y))),(Mult_ ((X,Y),(X,Y))) #) is non empty left_complementable right_complementable strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() RLSStruct
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 left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() discerning reflexive RealNormSpace-like NORMSTR
Y is non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() discerning reflexive RealNormSpace-like NORMSTR
(X,Y) is non empty NORMSTR
(X,Y) is non empty Element of bool the carrier of (X,Y)
(X,Y) is non empty left_complementable right_complementable strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() constituted-Functions RLSStruct
(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 left_complementable right_complementable strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() constituted-Functions RLSStruct
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
( 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)
0. Y is V53(Y) 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
( 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 V110() 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 [:REAL,(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 V110() Element of bool [:[:REAL,(Funcs ( the carrier of X, the carrier of Y)):],(Funcs ( the carrier of X, the carrier of Y)):]
[:REAL,(Funcs ( the carrier of X, the carrier of Y)):] is non empty set
[:[:REAL,(Funcs ( the carrier of X, the carrier of Y)):],(Funcs ( the carrier of X, the carrier of Y)):] is non empty set
bool [:[:REAL,(Funcs ( the carrier of X, the carrier of Y)):],(Funcs ( the carrier of X, the carrier of Y)):] is non empty set
RLSStruct(# (Funcs ( the carrier of X, the carrier of Y)),( the carrier of X,Y),( the carrier of X,Y),( the carrier of X,Y) #) is non empty strict RLSStruct
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 V110() 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 [:REAL,(X,Y):] -defined (X,Y) -valued Function-like total quasi_total Function-yielding V110() Element of bool [:[:REAL,(X,Y):],(X,Y):]
[:REAL,(X,Y):] is non empty set
[:[:REAL,(X,Y):],(X,Y):] is non empty set
bool [:[:REAL,(X,Y):],(X,Y):] is non empty set
RLSStruct(# (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 left_complementable right_complementable strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() RLSStruct
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 [:REAL,(X,Y):] -defined (X,Y) -valued Function-like total quasi_total Element of bool [:[:REAL,(X,Y):],(X,Y):]
[:REAL,(X,Y):] is non empty set
[:[:REAL,(X,Y):],(X,Y):] is non empty set
bool [:[:REAL,(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 V134() V135() V136() Element of bool [:(X,Y),REAL:]
[:(X,Y),REAL:] is non empty V134() V135() V136() set
bool [:(X,Y),REAL:] is non empty set
NORMSTR(# (X,Y),(Zero_ ((X,Y),(X,Y))),(Add_ ((X,Y),(X,Y))),(Mult_ ((X,Y),(X,Y))),(X,Y) #) is non empty strict NORMSTR
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 ext-real V31() real Element of REAL
tseq * vseq is Relation-like Function-like Element of the carrier of (X,Y)
the Mult of (X,Y) is non empty Relation-like [:REAL, the carrier of (X,Y):] -defined the carrier of (X,Y) -valued Function-like total quasi_total Element of bool [:[:REAL, the carrier of (X,Y):], the carrier of (X,Y):]
[:REAL, the carrier of (X,Y):] is non empty set
[:[:REAL, the carrier of (X,Y):], the carrier of (X,Y):] is non empty set
bool [:[:REAL, the carrier of (X,Y):], the carrier of (X,Y):] is non empty set
the Mult of (X,Y) . (tseq,vseq) is set
[tseq,vseq] is set
the Mult of (X,Y) . [tseq,vseq] is set
(X,Y) is non empty left_complementable right_complementable strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() RLSStruct
RLSStruct(# (X,Y),(Zero_ ((X,Y),(X,Y))),(Add_ ((X,Y),(X,Y))),(Mult_ ((X,Y),(X,Y))) #) is non empty left_complementable right_complementable strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() RLSStruct
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)
the Mult of (X,Y) is non empty Relation-like [:REAL, the carrier of (X,Y):] -defined the carrier of (X,Y) -valued Function-like total quasi_total Element of bool [:[:REAL, the carrier of (X,Y):], the carrier of (X,Y):]
[:REAL, the carrier of (X,Y):] is non empty set
[:[:REAL, the carrier of (X,Y):], the carrier of (X,Y):] is non empty set
bool [:[:REAL, the carrier of (X,Y):], the carrier of (X,Y):] is non empty set
the Mult of (X,Y) . (tseq,tseq) is set
[tseq,tseq] is set
the Mult of (X,Y) . [tseq,tseq] 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
the Mult of Y is non empty Relation-like [:REAL, the carrier of Y:] -defined the carrier of Y -valued Function-like total quasi_total Element of bool [:[:REAL, the carrier of Y:], the carrier of Y:]
[:REAL, the carrier of Y:] is non empty set
[:[:REAL, the carrier of Y:], the carrier of Y:] is non empty set
bool [:[:REAL, the carrier of Y:], the carrier of Y:] is non empty set
the Mult of Y . (tseq,(X,Y,vseq,tv)) is set
[tseq,(X,Y,vseq,tv)] is set
the Mult of Y . [tseq,(X,Y,vseq,tv)] is set
X is non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() discerning reflexive RealNormSpace-like NORMSTR
Y is non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() discerning reflexive RealNormSpace-like NORMSTR
(X,Y) is non empty NORMSTR
(X,Y) is non empty Element of bool the carrier of (X,Y)
(X,Y) is non empty left_complementable right_complementable strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() constituted-Functions RLSStruct
(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 left_complementable right_complementable strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() constituted-Functions RLSStruct
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
( 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)
0. Y is V53(Y) 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
( 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 V110() 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 [:REAL,(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 V110() Element of bool [:[:REAL,(Funcs ( the carrier of X, the carrier of Y)):],(Funcs ( the carrier of X, the carrier of Y)):]
[:REAL,(Funcs ( the carrier of X, the carrier of Y)):] is non empty set
[:[:REAL,(Funcs ( the carrier of X, the carrier of Y)):],(Funcs ( the carrier of X, the carrier of Y)):] is non empty set
bool [:[:REAL,(Funcs ( the carrier of X, the carrier of Y)):],(Funcs ( the carrier of X, the carrier of Y)):] is non empty set
RLSStruct(# (Funcs ( the carrier of X, the carrier of Y)),( the carrier of X,Y),( the carrier of X,Y),( the carrier of X,Y) #) is non empty strict RLSStruct
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 V110() 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 [:REAL,(X,Y):] -defined (X,Y) -valued Function-like total quasi_total Function-yielding V110() Element of bool [:[:REAL,(X,Y):],(X,Y):]
[:REAL,(X,Y):] is non empty set
[:[:REAL,(X,Y):],(X,Y):] is non empty set
bool [:[:REAL,(X,Y):],(X,Y):] is non empty set
RLSStruct(# (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 left_complementable right_complementable strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() RLSStruct
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 [:REAL,(X,Y):] -defined (X,Y) -valued Function-like total quasi_total Element of bool [:[:REAL,(X,Y):],(X,Y):]
[:REAL,(X,Y):] is non empty set
[:[:REAL,(X,Y):],(X,Y):] is non empty set
bool [:[:REAL,(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 V134() V135() V136() Element of bool [:(X,Y),REAL:]
[:(X,Y),REAL:] is non empty V134() V135() V136() set
bool [:(X,Y),REAL:] is non empty set
NORMSTR(# (X,Y),(Zero_ ((X,Y),(X,Y))),(Add_ ((X,Y),(X,Y))),(Mult_ ((X,Y),(X,Y))),(X,Y) #) is non empty strict NORMSTR
the carrier of (X,Y) is non empty set
0. (X,Y) is Relation-like Function-like V53((X,Y)) 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 ext-real V31() 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 V134() V135() V136() Element of bool [: the carrier of (X,Y),REAL:]
[: the carrier of (X,Y),REAL:] is non empty V134() V135() V136() set
bool [: the carrier of (X,Y),REAL:] is non empty set
the U9 of (X,Y) . vseq is ext-real V31() 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 ext-real V31() real Element of REAL
the U9 of (X,Y) . (vseq + f) is ext-real V31() real Element of REAL
||.f.|| is ext-real V31() real Element of REAL
the U9 of (X,Y) . f is ext-real V31() real Element of REAL
||.vseq.|| + ||.f.|| is ext-real V31() real Element of REAL
tseq is ext-real V31() real Element of REAL
tseq * vseq is Relation-like Function-like Element of the carrier of (X,Y)
the Mult of (X,Y) is non empty Relation-like [:REAL, the carrier of (X,Y):] -defined the carrier of (X,Y) -valued Function-like total quasi_total Element of bool [:[:REAL, the carrier of (X,Y):], the carrier of (X,Y):]
[:REAL, the carrier of (X,Y):] is non empty set
[:[:REAL, the carrier of (X,Y):], the carrier of (X,Y):] is non empty set
bool [:[:REAL, the carrier of (X,Y):], the carrier of (X,Y):] is non empty set
the Mult of (X,Y) . (tseq,vseq) is set
[tseq,vseq] is set
the Mult of (X,Y) . [tseq,vseq] is set
||.(tseq * vseq).|| is ext-real V31() real Element of REAL
the U9 of (X,Y) . (tseq * vseq) is ext-real V31() real Element of REAL
abs tseq is ext-real V31() real Element of REAL
(abs tseq) * ||.vseq.|| is ext-real V31() 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:]
(X,Y,tseq) is non empty V144() V145() V146() 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 ext-real V31() 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 ext-real V31() real Element of REAL
n is Element of the carrier of X
tseq . n is Element of the carrier of Y
||.(tseq . n).|| is ext-real non negative V31() 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 V134() V135() V136() Element of bool [: the carrier of Y,REAL:]
[: the carrier of Y,REAL:] is non empty V134() V135() V136() set
bool [: the carrier of Y,REAL:] is non empty set
the U9 of Y . (tseq . n) is ext-real V31() real Element of REAL
||.n.|| is ext-real non negative V31() 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 V134() V135() V136() Element of bool [: the carrier of X,REAL:]
[: the carrier of X,REAL:] is non empty V134() V135() V136() set
bool [: the carrier of X,REAL:] is non empty set
the U9 of X . n is ext-real V31() real Element of REAL
||.(0. Y).|| is empty Function-like functional ext-real non positive non negative epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural V31() real V144() V145() V146() V147() V148() V149() V150() Element of REAL
the U9 of Y . (0. Y) is ext-real V31() real Element of REAL
m is ext-real V31() real Element of REAL
n is ext-real V31() real set
(X,Y) . vseq is ext-real V31() 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 additive (X,Y) (X,Y) Element of bool [: the carrier of X, the carrier of Y:]
(X,Y,tv) is non empty V144() V145() V146() 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 ext-real V31() real Element of REAL
e is Element of the carrier of X
||.e.|| is ext-real non negative V31() 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 V134() V135() V136() Element of bool [: the carrier of X,REAL:]
[: the carrier of X,REAL:] is non empty V134() V135() V136() set
bool [: the carrier of X,REAL:] is non empty set
the U9 of X . e is ext-real V31() real Element of REAL
||.f.|| * ||.e.|| is ext-real V31() real Element of REAL
||.f.|| * 1 is ext-real V31() real Element of REAL
||.vseq.|| * ||.e.|| is ext-real V31() real Element of REAL
||.vseq.|| * 1 is ext-real V31() real Element of REAL
(||.vseq.|| * ||.e.||) + (||.f.|| * ||.e.||) is ext-real V31() real Element of REAL
(||.vseq.|| * 1) + (||.f.|| * 1) is ext-real V31() 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 ext-real non negative V31() 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 V134() V135() V136() Element of bool [: the carrier of Y,REAL:]
[: the carrier of Y,REAL:] is non empty V134() V135() V136() set
bool [: the carrier of Y,REAL:] is non empty set
the U9 of Y . ((tseq . e) + (tseq . e)) is ext-real V31() real Element of REAL
||.(tseq . e).|| is ext-real non negative V31() real Element of REAL
the U9 of Y . (tseq . e) is ext-real V31() real Element of REAL
||.(tseq . e).|| is ext-real non negative V31() real Element of REAL
the U9 of Y . (tseq . e) is ext-real V31() real Element of REAL
||.(tseq . e).|| + ||.(tseq . e).|| is ext-real non negative V31() real Element of REAL
tv . e is Element of the carrier of Y
||.(tv . e).|| is ext-real non negative V31() real Element of REAL
the U9 of Y . (tv . e) is ext-real V31() real Element of REAL
e is ext-real V31() real Element of REAL
m is Element of the carrier of X
tv . m is Element of the carrier of Y
||.(tv . m).|| is ext-real non negative V31() real Element of REAL
the U9 of Y . (tv . m) is ext-real V31() real Element of REAL
||.m.|| is ext-real non negative V31() real Element of REAL
the U9 of X . m is ext-real V31() real Element of REAL
(X,Y) . (vseq + f) is ext-real V31() real Element of REAL
e is ext-real V31() real 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 V144() V145() V146() 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 ext-real V31() real Element of REAL
tv is Element of the carrier of X
||.tv.|| is ext-real non negative V31() 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 V134() V135() V136() Element of bool [: the carrier of X,REAL:]
[: the carrier of X,REAL:] is non empty V134() V135() V136() set
bool [: the carrier of X,REAL:] is non empty set
the U9 of X . tv is ext-real V31() real Element of REAL
||.vseq.|| * ||.tv.|| is ext-real V31() real Element of REAL
||.vseq.|| * 1 is ext-real V31() 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 ext-real non negative V31() 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 V134() V135() V136() Element of bool [: the carrier of Y,REAL:]
[: the carrier of Y,REAL:] is non empty V134() V135() V136() set
bool [: the carrier of Y,REAL:] is non empty set
the U9 of Y . (tseq . tv) is ext-real V31() real Element of REAL
tseq * (tseq . tv) is Element of the carrier of Y
the Mult of Y is non empty Relation-like [:REAL, the carrier of Y:] -defined the carrier of Y -valued Function-like total quasi_total Element of bool [:[:REAL, the carrier of Y:], the carrier of Y:]
[:REAL, the carrier of Y:] is non empty set
[:[:REAL, the carrier of Y:], the carrier of Y:] is non empty set
bool [:[:REAL, the carrier of Y:], the carrier of Y:] is non empty set
the Mult of Y . (tseq,(tseq . tv)) is set
[tseq,(tseq . tv)] is set
the Mult of Y . [tseq,(tseq . tv)] is set
||.(tseq * (tseq . tv)).|| is ext-real non negative V31() real Element of REAL
the U9 of Y . (tseq * (tseq . tv)) is ext-real V31() real Element of REAL
(abs tseq) * ||.(tseq . tv).|| is ext-real V31() real Element of REAL
tseq . tv is Element of the carrier of Y
||.(tseq . tv).|| is ext-real non negative V31() real Element of REAL
the U9 of Y . (tseq . tv) is ext-real V31() real Element of REAL
tv is ext-real V31() real Element of REAL
e is Element of the carrier of X
tseq . e is Element of the carrier of Y
||.(tseq . e).|| is ext-real non negative V31() real Element of REAL
the U9 of Y . (tseq . e) is ext-real V31() real Element of REAL
||.e.|| is ext-real non negative V31() real Element of REAL
the U9 of X . e is ext-real V31() real Element of REAL
tv is Element of the carrier of X
||.tv.|| is ext-real non negative V31() real Element of REAL
the U9 of X . tv is ext-real V31() real Element of REAL
||.(tseq * vseq).|| * ||.tv.|| is ext-real V31() real Element of REAL
||.(tseq * vseq).|| * 1 is ext-real V31() real Element of REAL
tseq . tv is Element of the carrier of Y
||.(tseq . tv).|| is ext-real non negative V31() real Element of REAL
the U9 of Y . (tseq . tv) is ext-real V31() real Element of REAL
tseq . tv is Element of the carrier of Y
tseq * (tseq . tv) is Element of the carrier of Y
the Mult of Y . (tseq,(tseq . tv)) is set
[tseq,(tseq . tv)] is set
the Mult of Y . [tseq,(tseq . tv)] is set
tseq " is ext-real V31() real Element of REAL
(tseq ") * (tseq . tv) is Element of the carrier of Y
the Mult of Y . ((tseq "),(tseq . tv)) is set
[(tseq "),(tseq . tv)] is set
the Mult of Y . [(tseq "),(tseq . tv)] is set
(tseq ") * tseq is ext-real V31() real Element of REAL
((tseq ") * tseq) * (tseq . tv) is Element of the carrier of Y
the Mult of Y . (((tseq ") * tseq),(tseq . tv)) is set
[((tseq ") * tseq),(tseq . tv)] is set
the Mult of Y . [((tseq ") * tseq),(tseq . tv)] is set
1 * (tseq . tv) is Element of the carrier of Y
the Mult of Y . (1,(tseq . tv)) is set
[1,(tseq . tv)] is set
the Mult of Y . [1,(tseq . tv)] is set
abs (tseq ") is ext-real V31() real Element of REAL
1 * (tseq ") is ext-real V31() real Element of REAL
abs (1 * (tseq ")) is ext-real V31() real Element of REAL
1 / tseq is ext-real V31() real Element of REAL
abs (1 / tseq) is ext-real V31() real Element of REAL
1 / (abs tseq) is ext-real V31() real Element of REAL
(abs tseq) " is ext-real V31() real Element of REAL
1 * ((abs tseq) ") is ext-real V31() real Element of REAL
||.((tseq ") * (tseq . tv)).|| is ext-real non negative V31() real Element of REAL
the U9 of Y . ((tseq ") * (tseq . tv)) is ext-real V31() real Element of REAL
(abs (tseq ")) * ||.(tseq . tv).|| is ext-real V31() real Element of REAL
||.(tseq . tv).|| is ext-real non negative V31() real Element of REAL
the U9 of Y . (tseq . tv) is ext-real V31() real Element of REAL
((abs tseq) ") * ||.(tseq * vseq).|| is ext-real V31() real Element of REAL
tv is ext-real V31() real Element of REAL
(X,Y,tseq) is non empty V144() V145() V146() 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 ext-real non negative V31() real Element of REAL
the U9 of Y . (tseq . e) is ext-real V31() real Element of REAL
||.e.|| is ext-real non negative V31() real Element of REAL
the U9 of X . e is ext-real V31() real Element of REAL
upper_bound (X,Y,tseq) is ext-real V31() real Element of REAL
(X,Y) . vseq is ext-real V31() real Element of REAL
tv is ext-real V31() real set
(abs tseq) * (((abs tseq) ") * ||.(tseq * vseq).||) is ext-real V31() real Element of REAL
(abs tseq) * ((abs tseq) ") is ext-real V31() real Element of REAL
((abs tseq) * ((abs tseq) ")) * ||.(tseq * vseq).|| is ext-real V31() real Element of REAL
1 * ||.(tseq * vseq).|| is ext-real V31() real Element of REAL
(X,Y) is non empty left_complementable right_complementable strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() RLSStruct
RLSStruct(# (X,Y),(Zero_ ((X,Y),(X,Y))),(Add_ ((X,Y),(X,Y))),(Mult_ ((X,Y),(X,Y))) #) is non empty left_complementable right_complementable strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() RLSStruct
the carrier of (X,Y) is non empty 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)
the Mult of (X,Y) is non empty Relation-like [:REAL, the carrier of (X,Y):] -defined the carrier of (X,Y) -valued Function-like total quasi_total Element of bool [:[:REAL, the carrier of (X,Y):], the carrier of (X,Y):]
[:REAL, the carrier of (X,Y):] is non empty set
[:[:REAL, the carrier of (X,Y):], the carrier of (X,Y):] is non empty set
bool [:[:REAL, the carrier of (X,Y):], the carrier of (X,Y):] is non empty set
the Mult of (X,Y) . (tseq,tv) is set
[tseq,tv] is set
the Mult of (X,Y) . [tseq,tv] is set
0. (X,Y) is Relation-like Function-like V53((X,Y)) Element of the carrier of (X,Y)
the ZeroF of (X,Y) is Relation-like Function-like Element of the carrier of (X,Y)
0 * ||.vseq.|| is ext-real V31() real Element of REAL
(X,Y) . (tseq * vseq) is ext-real V31() real Element of REAL
tv is ext-real V31() real 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
||.(tseq . e).|| is ext-real non negative V31() 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 V134() V135() V136() Element of bool [: the carrier of Y,REAL:]
[: the carrier of Y,REAL:] is non empty V134() V135() V136() set
bool [: the carrier of Y,REAL:] is non empty set
the U9 of Y . (tseq . e) is ext-real V31() real Element of REAL
||.e.|| is ext-real non negative V31() 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 V134() V135() V136() Element of bool [: the carrier of X,REAL:]
[: the carrier of X,REAL:] is non empty V134() V135() V136() set
bool [: the carrier of X,REAL:] is non empty set
the U9 of X . e is ext-real V31() real Element of REAL
||.vseq.|| * ||.e.|| is ext-real V31() 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 left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() discerning reflexive RealNormSpace-like NORMSTR
Y is non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() discerning reflexive RealNormSpace-like NORMSTR
(X,Y) is non empty NORMSTR
(X,Y) is non empty Element of bool the carrier of (X,Y)
(X,Y) is non empty left_complementable right_complementable strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() constituted-Functions RLSStruct
(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 left_complementable right_complementable strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() constituted-Functions RLSStruct
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
( 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)
0. Y is V53(Y) 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
( 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 V110() 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 [:REAL,(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 V110() Element of bool [:[:REAL,(Funcs ( the carrier of X, the carrier of Y)):],(Funcs ( the carrier of X, the carrier of Y)):]
[:REAL,(Funcs ( the carrier of X, the carrier of Y)):] is non empty set
[:[:REAL,(Funcs ( the carrier of X, the carrier of Y)):],(Funcs ( the carrier of X, the carrier of Y)):] is non empty set
bool [:[:REAL,(Funcs ( the carrier of X, the carrier of Y)):],(Funcs ( the carrier of X, the carrier of Y)):] is non empty set
RLSStruct(# (Funcs ( the carrier of X, the carrier of Y)),( the carrier of X,Y),( the carrier of X,Y),( the carrier of X,Y) #) is non empty strict RLSStruct
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 V110() 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 [:REAL,(X,Y):] -defined (X,Y) -valued Function-like total quasi_total Function-yielding V110() Element of bool [:[:REAL,(X,Y):],(X,Y):]
[:REAL,(X,Y):] is non empty set
[:[:REAL,(X,Y):],(X,Y):] is non empty set
bool [:[:REAL,(X,Y):],(X,Y):] is non empty set
RLSStruct(# (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 left_complementable right_complementable strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() RLSStruct
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 [:REAL,(X,Y):] -defined (X,Y) -valued Function-like total quasi_total Element of bool [:[:REAL,(X,Y):],(X,Y):]
[:REAL,(X,Y):] is non empty set
[:[:REAL,(X,Y):],(X,Y):] is non empty set
bool [:[:REAL,(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 V134() V135() V136() Element of bool [:(X,Y),REAL:]
[:(X,Y),REAL:] is non empty V134() V135() V136() set
bool [:(X,Y),REAL:] is non empty set
NORMSTR(# (X,Y),(Zero_ ((X,Y),(X,Y))),(Add_ ((X,Y),(X,Y))),(Mult_ ((X,Y),(X,Y))),(X,Y) #) is non empty strict NORMSTR
0. (X,Y) is Relation-like Function-like V53((X,Y)) 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 ext-real V31() 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 V134() V135() V136() Element of bool [: the carrier of (X,Y),REAL:]
[: the carrier of (X,Y),REAL:] is non empty V134() V135() V136() set
bool [: the carrier of (X,Y),REAL:] is non empty set
the U9 of (X,Y) . (0. (X,Y)) is ext-real V31() real Element of REAL
vseq is Relation-like Function-like Element of the carrier of (X,Y)
||.vseq.|| is ext-real V31() real Element of REAL
the U9 of (X,Y) . vseq is ext-real V31() real Element of REAL
f is Relation-like Function-like Element of the carrier of (X,Y)
||.f.|| is ext-real V31() real Element of REAL
the U9 of (X,Y) . f is ext-real V31() real Element of REAL
tseq is Relation-like Function-like Element of the carrier of (X,Y)
tseq is ext-real V31() real Element of REAL
tseq * tseq is Relation-like Function-like Element of the carrier of (X,Y)
the Mult of (X,Y) is non empty Relation-like [:REAL, the carrier of (X,Y):] -defined the carrier of (X,Y) -valued Function-like total quasi_total Element of bool [:[:REAL, the carrier of (X,Y):], the carrier of (X,Y):]
[:REAL, the carrier of (X,Y):] is non empty set
[:[:REAL, the carrier of (X,Y):], the carrier of (X,Y):] is non empty set
bool [:[:REAL, the carrier of (X,Y):], the carrier of (X,Y):] is non empty set
the Mult of (X,Y) . (tseq,tseq) is set
[tseq,tseq] is set
the Mult of (X,Y) . [tseq,tseq] is set
||.(tseq * tseq).|| is ext-real V31() real Element of REAL
the U9 of (X,Y) . (tseq * tseq) is ext-real V31() real Element of REAL
abs tseq is ext-real V31() real Element of REAL
||.tseq.|| is ext-real V31() real Element of REAL
the U9 of (X,Y) . tseq is ext-real V31() real Element of REAL
(abs tseq) * ||.tseq.|| is ext-real V31() 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 ext-real V31() real Element of REAL
the U9 of (X,Y) . (tseq + tv) is ext-real V31() real Element of REAL
||.tseq.|| is ext-real V31() real Element of REAL
the U9 of (X,Y) . tseq is ext-real V31() real Element of REAL
||.tv.|| is ext-real V31() real Element of REAL
the U9 of (X,Y) . tv is ext-real V31() real Element of REAL
||.tseq.|| + ||.tv.|| is ext-real V31() real Element of REAL
X is non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() discerning reflexive RealNormSpace-like NORMSTR
Y is non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() discerning reflexive RealNormSpace-like NORMSTR
(X,Y) is non empty NORMSTR
(X,Y) is non empty Element of bool the carrier of (X,Y)
(X,Y) is non empty left_complementable right_complementable strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() constituted-Functions RLSStruct
(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 left_complementable right_complementable strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() constituted-Functions RLSStruct
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
( 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)
0. Y is V53(Y) 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
( 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 V110() 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 [:REAL,(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 V110() Element of bool [:[:REAL,(Funcs ( the carrier of X, the carrier of Y)):],(Funcs ( the carrier of X, the carrier of Y)):]
[:REAL,(Funcs ( the carrier of X, the carrier of Y)):] is non empty set
[:[:REAL,(Funcs ( the carrier of X, the carrier of Y)):],(Funcs ( the carrier of X, the carrier of Y)):] is non empty set
bool [:[:REAL,(Funcs ( the carrier of X, the carrier of Y)):],(Funcs ( the carrier of X, the carrier of Y)):] is non empty set
RLSStruct(# (Funcs ( the carrier of X, the carrier of Y)),( the carrier of X,Y),( the carrier of X,Y),( the carrier of X,Y) #) is non empty strict RLSStruct
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 V110() 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 [:REAL,(X,Y):] -defined (X,Y) -valued Function-like total quasi_total Function-yielding V110() Element of bool [:[:REAL,(X,Y):],(X,Y):]
[:REAL,(X,Y):] is non empty set
[:[:REAL,(X,Y):],(X,Y):] is non empty set
bool [:[:REAL,(X,Y):],(X,Y):] is non empty set
RLSStruct(# (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 left_complementable right_complementable strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() RLSStruct
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 [:REAL,(X,Y):] -defined (X,Y) -valued Function-like total quasi_total Element of bool [:[:REAL,(X,Y):],(X,Y):]
[:REAL,(X,Y):] is non empty set
[:[:REAL,(X,Y):],(X,Y):] is non empty set
bool [:[:REAL,(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 V134() V135() V136() Element of bool [:(X,Y),REAL:]
[:(X,Y),REAL:] is non empty V134() V135() V136() set
bool [:(X,Y),REAL:] is non empty set
NORMSTR(# (X,Y),(Zero_ ((X,Y),(X,Y))),(Add_ ((X,Y),(X,Y))),(Mult_ ((X,Y),(X,Y))),(X,Y) #) is non empty strict NORMSTR
RLSStruct(# (X,Y),(Zero_ ((X,Y),(X,Y))),(Add_ ((X,Y),(X,Y))),(Mult_ ((X,Y),(X,Y))) #) is non empty left_complementable right_complementable strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() RLSStruct
X is non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() discerning reflexive RealNormSpace-like NORMSTR
Y is non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() discerning reflexive RealNormSpace-like NORMSTR
(X,Y) is non empty NORMSTR
(X,Y) is non empty Element of bool the carrier of (X,Y)
(X,Y) is non empty left_complementable right_complementable strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() constituted-Functions RLSStruct
(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 left_complementable right_complementable strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() constituted-Functions RLSStruct
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
( 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)
0. Y is V53(Y) 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
( 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 V110() 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 [:REAL,(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 V110() Element of bool [:[:REAL,(Funcs ( the carrier of X, the carrier of Y)):],(Funcs ( the carrier of X, the carrier of Y)):]
[:REAL,(Funcs ( the carrier of X, the carrier of Y)):] is non empty set
[:[:REAL,(Funcs ( the carrier of X, the carrier of Y)):],(Funcs ( the carrier of X, the carrier of Y)):] is non empty set
bool [:[:REAL,(Funcs ( the carrier of X, the carrier of Y)):],(Funcs ( the carrier of X, the carrier of Y)):] is non empty set
RLSStruct(# (Funcs ( the carrier of X, the carrier of Y)),( the carrier of X,Y),( the carrier of X,Y),( the carrier of X,Y) #) is non empty strict RLSStruct
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 V110() 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 [:REAL,(X,Y):] -defined (X,Y) -valued Function-like total quasi_total Function-yielding V110() Element of bool [:[:REAL,(X,Y):],(X,Y):]
[:REAL,(X,Y):] is non empty set
[:[:REAL,(X,Y):],(X,Y):] is non empty set
bool [:[:REAL,(X,Y):],(X,Y):] is non empty set
RLSStruct(# (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 left_complementable right_complementable strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() RLSStruct
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 [:REAL,(X,Y):] -defined (X,Y) -valued Function-like total quasi_total Element of bool [:[:REAL,(X,Y):],(X,Y):]
[:REAL,(X,Y):] is non empty set
[:[:REAL,(X,Y):],(X,Y):] is non empty set
bool [:[:REAL,(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 V134() V135() V136() Element of bool [:(X,Y),REAL:]
[:(X,Y),REAL:] is non empty V134() V135() V136() set
bool [:(X,Y),REAL:] is non empty set
NORMSTR(# (X,Y),(Zero_ ((X,Y),(X,Y))),(Add_ ((X,Y),(X,Y))),(Mult_ ((X,Y),(X,Y))),(X,Y) #) is non empty strict NORMSTR
X is non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() discerning reflexive RealNormSpace-like NORMSTR
Y is non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() discerning reflexive RealNormSpace-like NORMSTR
(X,Y) is non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() discerning reflexive RealNormSpace-like NORMSTR
(X,Y) is non empty Element of bool the carrier of (X,Y)
(X,Y) is non empty left_complementable right_complementable strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() constituted-Functions RLSStruct
(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 left_complementable right_complementable strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() constituted-Functions RLSStruct
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
( 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)
0. Y is V53(Y) 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
( 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 V110() 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 [:REAL,(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 V110() Element of bool [:[:REAL,(Funcs ( the carrier of X, the carrier of Y)):],(Funcs ( the carrier of X, the carrier of Y)):]
[:REAL,(Funcs ( the carrier of X, the carrier of Y)):] is non empty set
[:[:REAL,(Funcs ( the carrier of X, the carrier of Y)):],(Funcs ( the carrier of X, the carrier of Y)):] is non empty set
bool [:[:REAL,(Funcs ( the carrier of X, the carrier of Y)):],(Funcs ( the carrier of X, the carrier of Y)):] is non empty set
RLSStruct(# (Funcs ( the carrier of X, the carrier of Y)),( the carrier of X,Y),( the carrier of X,Y),( the carrier of X,Y) #) is non empty strict RLSStruct
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 V110() 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 [:REAL,(X,Y):] -defined (X,Y) -valued Function-like total quasi_total Function-yielding V110() Element of bool [:[:REAL,(X,Y):],(X,Y):]
[:REAL,(X,Y):] is non empty set
[:[:REAL,(X,Y):],(X,Y):] is non empty set
bool [:[:REAL,(X,Y):],(X,Y):] is non empty set
RLSStruct(# (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 left_complementable right_complementable strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() RLSStruct
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 [:REAL,(X,Y):] -defined (X,Y) -valued Function-like total quasi_total Element of bool [:[:REAL,(X,Y):],(X,Y):]
[:REAL,(X,Y):] is non empty set
[:[:REAL,(X,Y):],(X,Y):] is non empty set
bool [:[:REAL,(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 V134() V135() V136() Element of bool [:(X,Y),REAL:]
[:(X,Y),REAL:] is non empty V134() V135() V136() set
bool [:(X,Y),REAL:] is non empty set
NORMSTR(# (X,Y),(Zero_ ((X,Y),(X,Y))),(Add_ ((X,Y),(X,Y))),(Mult_ ((X,Y),(X,Y))),(X,Y) #) is non empty strict NORMSTR
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
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 V53((X,Y)) 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
(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 ext-real V31() real Element of REAL
Y is non empty Relation-like NAT -defined REAL -valued Function-like total quasi_total V134() V135() V136() Element of bool [:NAT,REAL:]
lim Y is ext-real V31() real Element of REAL
vseq is ext-real epsilon-transitive epsilon-connected ordinal natural V31() real V113() V114() V144() V145() V146() V147() V148() V149() Element of NAT
vseq is ext-real epsilon-transitive epsilon-connected ordinal natural V31() real V113() V114() V144() V145() V146() V147() V148() V149() Element of NAT
NAT --> X is non empty Relation-like NAT -defined REAL -valued Function-like constant total quasi_total V134() V135() V136() convergent Element of bool [:NAT,REAL:]
f is non empty Relation-like NAT -defined REAL -valued Function-like total quasi_total V134() V135() V136() Element of bool [:NAT,REAL:]
lim f is ext-real V31() real Element of REAL
f . 0 is ext-real V31() real Element of REAL
Y ^\ vseq is non empty Relation-like NAT -defined REAL -valued Function-like total quasi_total V134() V135() V136() Element of bool [:NAT,REAL:]
tseq is ext-real epsilon-transitive epsilon-connected ordinal natural V31() real V113() V114() V144() V145() V146() V147() V148() V149() Element of NAT
(Y ^\ vseq) . tseq is ext-real V31() real Element of REAL
vseq + tseq is ext-real epsilon-transitive epsilon-connected ordinal natural V31() real V113() V114() V144() V145() V146() V147() V148() V149() Element of NAT
Y . (vseq + tseq) is ext-real V31() real Element of REAL
f . tseq is ext-real V31() real Element of REAL
lim (Y ^\ vseq) is ext-real V31() real Element of REAL
X is non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() discerning reflexive RealNormSpace-like NORMSTR
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 V134() V135() V136() Element of bool [:NAT,REAL:]
lim ||.Y.|| is ext-real V31() real Element of REAL
lim Y is Element of the carrier of X
||.(lim Y).|| is ext-real non negative V31() 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 V134() V135() V136() Element of bool [: the carrier of X,REAL:]
[: the carrier of X,REAL:] is non empty V134() V135() V136() set
bool [: the carrier of X,REAL:] is non empty set
the U9 of X . (lim Y) is ext-real V31() real Element of REAL
vseq is ext-real V31() real set
f is ext-real V31() real Element of REAL
tseq is ext-real epsilon-transitive epsilon-connected ordinal natural V31() real V113() V114() V144() V145() V146() V147() V148() V149() Element of NAT
tseq is ext-real epsilon-transitive epsilon-connected ordinal natural V31() real V113() V114() V144() V145() V146() V147() V148() V149() Element of NAT
tseq is ext-real epsilon-transitive epsilon-connected ordinal natural V31() real V113() V114() V144() V145() V146() V147() V148() V149() 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 ext-real non negative V31() real Element of REAL
the U9 of X . ((Y . tseq) - (lim Y)) is ext-real V31() real Element of REAL
||.(Y . tseq).|| is ext-real non negative V31() real Element of REAL
the U9 of X . (Y . tseq) is ext-real V31() real Element of REAL
||.Y.|| . tseq is ext-real V31() real Element of REAL
(||.Y.|| . tseq) - ||.(lim Y).|| is ext-real V31() real Element of REAL
abs ((||.Y.|| . tseq) - ||.(lim Y).||) is ext-real V31() real Element of REAL
tseq is ext-real epsilon-transitive epsilon-connected ordinal natural V31() real V113() V114() V144() V145() V146() V147() V148() V149() Element of NAT
||.Y.|| . tseq is ext-real V31() real Element of REAL
(||.Y.|| . tseq) - ||.(lim Y).|| is ext-real V31() real Element of REAL
abs ((||.Y.|| . tseq) - ||.(lim Y).||) is ext-real V31() real Element of REAL
Y is non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() discerning reflexive RealNormSpace-like NORMSTR
X is non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() discerning reflexive RealNormSpace-like NORMSTR
(X,Y) is non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() discerning reflexive RealNormSpace-like NORMSTR
(X,Y) is non empty Element of bool the carrier of (X,Y)
(X,Y) is non empty left_complementable right_complementable strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() constituted-Functions RLSStruct
(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 left_complementable right_complementable strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() constituted-Functions RLSStruct
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
( 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)
0. Y is V53(Y) 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
( 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 V110() 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 [:REAL,(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 V110() Element of bool [:[:REAL,(Funcs ( the carrier of X, the carrier of Y)):],(Funcs ( the carrier of X, the carrier of Y)):]
[:REAL,(Funcs ( the carrier of X, the carrier of Y)):] is non empty set
[:[:REAL,(Funcs ( the carrier of X, the carrier of Y)):],(Funcs ( the carrier of X, the carrier of Y)):] is non empty set
bool [:[:REAL,(Funcs ( the carrier of X, the carrier of Y)):],(Funcs ( the carrier of X, the carrier of Y)):] is non empty set
RLSStruct(# (Funcs ( the carrier of X, the carrier of Y)),( the carrier of X,Y),( the carrier of X,Y),( the carrier of X,Y) #) is non empty strict RLSStruct
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 V110() 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 [:REAL,(X,Y):] -defined (X,Y) -valued Function-like total quasi_total Function-yielding V110() Element of bool [:[:REAL,(X,Y):],(X,Y):]
[:REAL,(X,Y):] is non empty set
[:[:REAL,(X,Y):],(X,Y):] is non empty set
bool [:[:REAL,(X,Y):],(X,Y):] is non empty set
RLSStruct(# (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 left_complementable right_complementable strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() RLSStruct
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 [:REAL,(X,Y):] -defined (X,Y) -valued Function-like total quasi_total Element of bool [:[:REAL,(X,Y):],(X,Y):]
[:REAL,(X,Y):] is non empty set
[:[:REAL,(X,Y):],(X,Y):] is non empty set
bool [:[:REAL,(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 V134() V135() V136() Element of bool [:(X,Y),REAL:]
[:(X,Y),REAL:] is non empty V134() V135() V136() set
bool [:(X,Y),REAL:] is non empty set
NORMSTR(# (X,Y),(Zero_ ((X,Y),(X,Y))),(Add_ ((X,Y),(X,Y))),(Mult_ ((X,Y),(X,Y))),(X,Y) #) is non empty strict NORMSTR
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 ext-real non negative V31() 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 V134() V135() V136() Element of bool [: the carrier of X,REAL:]
[: the carrier of X,REAL:] is non empty V134() V135() V136() set
bool [: the carrier of X,REAL:] is non empty set
the U9 of X . f is ext-real V31() real Element of REAL
tseq is ext-real epsilon-transitive epsilon-connected ordinal natural V31() real V113() V114() V144() V145() V146() V147() V148() V149() Element of NAT
tseq . tseq is Element of the carrier of Y
tseq is ext-real epsilon-transitive epsilon-connected ordinal natural V31() real V113() V114() V144() V145() V146() V147() V148() V149() 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 ext-real non negative V31() 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 V134() V135() V136() Element of bool [: the carrier of Y,REAL:]
[: the carrier of Y,REAL:] is non empty V134() V135() V136() set
bool [: the carrier of Y,REAL:] is non empty set
the U9 of Y . ((tseq . tseq) - (tseq . tseq)) is ext-real V31() 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 ext-real non negative V31() 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 V134() V135() V136() Element of bool [: the carrier of (X,Y),REAL:]
[: the carrier of (X,Y),REAL:] is non empty V134() V135() V136() set
bool [: the carrier of (X,Y),REAL:] is non empty set
the U9 of (X,Y) . ((vseq . tseq) - (vseq . tseq)) is ext-real V31() real Element of REAL
||.((vseq . tseq) - (vseq . tseq)).|| * ||.f.|| is ext-real non negative V31() real Element of REAL
(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 ext-real V31() real Element of REAL
0. X is V53(X) Element of the carrier of X
the ZeroF of X is Element of the carrier of X
tseq is empty Function-like functional ext-real non positive non negative epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural V31() real V113() V114() V144() V145() V146() V147() V148() V149() V150() Element of NAT
tv is ext-real epsilon-transitive epsilon-connected ordinal natural V31() real V113() V114() V144() V145() V146() V147() V148() V149() Element of NAT
e is ext-real epsilon-transitive epsilon-connected ordinal natural V31() real V113() V114() V144() V145() V146() V147() V148() V149() 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 ext-real non negative V31() 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 V134() V135() V136() Element of bool [: the carrier of Y,REAL:]
[: the carrier of Y,REAL:] is non empty V134() V135() V136() set
bool [: the carrier of Y,REAL:] is non empty set
the U9 of Y . ((tseq . tv) - (tseq . e)) is ext-real V31() 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)) . f is Element of the carrier of Y
0 * f is Element of the carrier of X
the Mult of X is non empty Relation-like [:REAL, the carrier of X:] -defined the carrier of X -valued Function-like total quasi_total Element of bool [:[:REAL, the carrier of X:], the carrier of X:]
[:REAL, the carrier of X:] is non empty set
[:[:REAL, the carrier of X:], the carrier of X:] is non empty set
bool [:[:REAL, the carrier of X:], the carrier of X:] is non empty set
the Mult of X . (0,f) is set
[0,f] is set
the Mult of X . [0,f] is set
(X,Y,(vseq . e)) . (0 * f) is Element of the carrier of Y
0 * ((X,Y,(vseq . e)) . f) is Element of the carrier of Y
the Mult of Y is non empty Relation-like [:REAL, the carrier of Y:] -defined the carrier of Y -valued Function-like total quasi_total Element of bool [:[:REAL, the carrier of Y:], the carrier of Y:]
[:REAL, the carrier of Y:] is non empty set
[:[:REAL, the carrier of Y:], the carrier of Y:] is non empty set
bool [:[:REAL, the carrier of Y:], the carrier of Y:] is non empty set
the Mult of Y . (0,((X,Y,(vseq . e)) . f)) is set
[0,((X,Y,(vseq . e)) . f)] is set
the Mult of Y . [0,((X,Y,(vseq . e)) . f)] is set
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)) . (0 * f) is Element of the carrier of Y
0 * ((X,Y,(vseq . tv)) . f) is Element of the carrier of Y
the Mult of Y . (0,((X,Y,(vseq . tv)) . f)) is set
[0,((X,Y,(vseq . tv)) . f)] is set
the Mult of Y . [0,((X,Y,(vseq . tv)) . f)] is set
||.(0. Y).|| is empty Function-like functional ext-real non positive non negative epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural V31() real V144() V145() V146() V147() V148() V149() V150() Element of REAL
the U9 of Y . (0. Y) is ext-real V31() real Element of REAL
0. X is V53(X) Element of the carrier of X
the ZeroF of X is Element of the carrier of X
tseq / ||.f.|| is ext-real V31() real Element of REAL
tseq is ext-real epsilon-transitive epsilon-connected ordinal natural V31() real V113() V114() V144() V145() V146() V147() V148() V149() Element of NAT
tv is ext-real epsilon-transitive epsilon-connected ordinal natural V31() real V113() V114() V144() V145() V146() V147() V148() V149() Element of NAT
e is ext-real epsilon-transitive epsilon-connected ordinal natural V31() real V113() V114() V144() V145() V146() V147() V148() V149() Element of NAT
m is ext-real epsilon-transitive epsilon-connected ordinal natural V31() real V113() V114() V144() V145() V146() V147() V148() V149() 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 ext-real non negative V31() 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 V134() V135() V136() Element of bool [: the carrier of Y,REAL:]
[: the carrier of Y,REAL:] is non empty V134() V135() V136() set
bool [: the carrier of Y,REAL:] is non empty set
the U9 of Y . ((tseq . e) - (tseq . m)) is ext-real V31() 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 ext-real non negative V31() 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 V134() V135() V136() Element of bool [: the carrier of (X,Y),REAL:]
[: the carrier of (X,Y),REAL:] is non empty V134() V135() V136() set
bool [: the carrier of (X,Y),REAL:] is non empty set
the U9 of (X,Y) . ((vseq . e) - (vseq . m)) is ext-real V31() real Element of REAL
(tseq / ||.f.||) * ||.f.|| is ext-real V31() real Element of REAL
||.((vseq . e) - (vseq . m)).|| * ||.f.|| is ext-real non negative V31() real Element of REAL
||.f.|| " is ext-real non negative V31() real Element of REAL
tseq * (||.f.|| ") is ext-real V31() real Element of REAL
(tseq * (||.f.|| ")) * ||.f.|| is ext-real V31() real Element of REAL
(||.f.|| ") * ||.f.|| is ext-real non negative V31() real Element of REAL
tseq * ((||.f.|| ") * ||.f.||) is ext-real V31() real Element of REAL
tseq * 1 is ext-real V31() real Element of REAL
0. X is V53(X) Element of the carrier of X
the ZeroF of X is Element of the carrier of X
0. X is V53(X) Element of the carrier of X
the ZeroF of X is Element of the carrier of X
tseq is empty Function-like functional ext-real non positive non negative epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural V31() real V113() V114() V144() V145() V146() V147() V148() V149() V150() Element of NAT
tv is ext-real epsilon-transitive epsilon-connected ordinal natural V31() real V113() V114() V144() V145() V146() V147() V148() V149() Element of NAT
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 ext-real epsilon-transitive epsilon-connected ordinal natural V31() real V113() V114() V144() V145() V146() V147() V148() V149() 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 ext-real V31() real Element of REAL
tseq * tseq is Element of the carrier of X
the Mult of X is non empty Relation-like [:REAL, the carrier of X:] -defined the carrier of X -valued Function-like total quasi_total Element of bool [:[:REAL, the carrier of X:], the carrier of X:]
[:REAL, the carrier of X:] is non empty set
[:[:REAL, the carrier of X:], the carrier of X:] is non empty set
bool [:[:REAL, the carrier of X:], the carrier of X:] is non empty set
the Mult of X . (tseq,tseq) is set
[tseq,tseq] is set
the Mult 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
m is ext-real epsilon-transitive epsilon-connected ordinal natural V31() real V113() V114() V144() V145() V146() V147() V148() V149() 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
the Mult of Y is non empty Relation-like [:REAL, the carrier of Y:] -defined the carrier of Y -valued Function-like total quasi_total Element of bool [:[:REAL, the carrier of Y:], the carrier of Y:]
[:REAL, the carrier of Y:] is non empty set
[:[:REAL, the carrier of Y:], the carrier of Y:] is non empty set
bool [:[:REAL, the carrier of Y:], the carrier of Y:] is non empty set
the Mult of Y . (tseq,((X,Y,(vseq . m)) . tseq)) is set
[tseq,((X,Y,(vseq . m)) . tseq)] is set
the Mult of Y . [tseq,((X,Y,(vseq . m)) . tseq)] is set
tv . m is Element of the carrier of Y
tseq * (tv . m) is Element of the carrier of Y
the Mult of Y . (tseq,(tv . m)) is set
[tseq,(tv . m)] is set
the Mult of Y . [tseq,(tv . m)] is set
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
the Mult of Y . (tseq,(tseq . tseq)) is set
[tseq,(tseq . tseq)] is set
the Mult of Y . [tseq,(tseq . tseq)] is set
tseq is ext-real V31() real set
tv is ext-real V31() real Element of REAL
e is ext-real epsilon-transitive epsilon-connected ordinal natural V31() real V113() V114() V144() V145() V146() V147() V148() V149() Element of NAT
m is ext-real epsilon-transitive epsilon-connected ordinal natural V31() real V113() V114() V144() V145() V146() V147() V148() V149() Element of NAT
n is ext-real epsilon-transitive epsilon-connected ordinal natural V31() real V113() V114() V144() V145() V146() V147() V148() V149() 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 ext-real non negative V31() 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 V134() V135() V136() Element of bool [: the carrier of (X,Y),REAL:]
[: the carrier of (X,Y),REAL:] is non empty V134() V135() V136() set
bool [: the carrier of (X,Y),REAL:] is non empty set
the U9 of (X,Y) . ((vseq . n) - (vseq . m)) is ext-real V31() real Element of REAL
||.(vseq . n).|| is ext-real non negative V31() real Element of REAL
the U9 of (X,Y) . (vseq . n) is ext-real V31() real Element of REAL
||.vseq.|| is non empty Relation-like NAT -defined REAL -valued Function-like total quasi_total V134() V135() V136() Element of bool [:NAT,REAL:]
||.vseq.|| . n is ext-real V31() real Element of REAL
||.(vseq . m).|| is ext-real non negative V31() real Element of REAL
the U9 of (X,Y) . (vseq . m) is ext-real V31() real Element of REAL
||.vseq.|| . m is ext-real V31() real Element of REAL
||.(vseq . n).|| - ||.(vseq . m).|| is ext-real V31() real Element of REAL
abs (||.(vseq . n).|| - ||.(vseq . m).||) is ext-real V31() real Element of REAL
(||.vseq.|| . n) - (||.vseq.|| . m) is ext-real V31() real Element of REAL
abs ((||.vseq.|| . n) - (||.vseq.|| . m)) is ext-real V31() real Element of REAL
n is ext-real epsilon-transitive epsilon-connected ordinal natural V31() real V113() V114() V144() V145() V146() V147() V148() V149() Element of NAT
||.vseq.|| . n is ext-real V31() real Element of REAL
(||.vseq.|| . n) - (||.vseq.|| . m) is ext-real V31() real Element of REAL
abs ((||.vseq.|| . n) - (||.vseq.|| . m)) is ext-real V31() 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 ext-real V31() 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 ext-real non negative V31() 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 V134() V135() V136() Element of bool [: the carrier of Y,REAL:]
[: the carrier of Y,REAL:] is non empty V134() V135() V136() set
bool [: the carrier of Y,REAL:] is non empty set
the U9 of Y . (tseq . tseq) is ext-real V31() real Element of REAL
||.tv.|| is non empty Relation-like NAT -defined REAL -valued Function-like total quasi_total V134() V135() V136() Element of bool [:NAT,REAL:]
lim ||.tv.|| is ext-real V31() real Element of REAL
||.tseq.|| is ext-real non negative V31() 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 V134() V135() V136() Element of bool [: the carrier of X,REAL:]
[: the carrier of X,REAL:] is non empty V134() V135() V136() set
bool [: the carrier of X,REAL:] is non empty set
the U9 of X . tseq is ext-real V31() real Element of REAL
e is ext-real epsilon-transitive epsilon-connected ordinal natural V31() real V113() V114() V144() V145() V146() V147() V148() V149() Element of NAT
tv . e is Element of the carrier of Y
||.(tv . e).|| is ext-real non negative V31() real Element of REAL
the U9 of Y . (tv . e) is ext-real V31() real Element of REAL
vseq . e is Relation-like Function-like Element of the carrier of (X,Y)
||.(vseq . e).|| is ext-real non negative V31() real Element of REAL
the U9 of (X,Y) . (vseq . e) is ext-real V31() real Element of REAL
||.(vseq . e).|| * ||.tseq.|| is ext-real non negative V31() 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 V134() V135() V136() Element of bool [:NAT,REAL:]
e is ext-real epsilon-transitive epsilon-connected ordinal natural V31() real V113() V114() V144() V145() V146() V147() V148() V149() Element of NAT
||.tv.|| . e is ext-real V31() real Element of REAL
(||.tseq.|| (#) ||.vseq.||) . e is ext-real V31() real Element of REAL
tv . e is Element of the carrier of Y
||.(tv . e).|| is ext-real non negative V31() real Element of REAL
the U9 of Y . (tv . e) is ext-real V31() real Element of REAL
vseq . e is Relation-like Function-like Element of the carrier of (X,Y)
||.(vseq . e).|| is ext-real non negative V31() real Element of REAL
the U9 of (X,Y) . (vseq . e) is ext-real V31() real Element of REAL
||.vseq.|| . e is ext-real V31() real Element of REAL
||.(vseq . e).|| * ||.tseq.|| is ext-real non negative V31() real Element of REAL
lim (||.tseq.|| (#) ||.vseq.||) is ext-real V31() real Element of REAL
(lim ||.vseq.||) * ||.tseq.|| is ext-real V31() real Element of REAL
tseq is ext-real epsilon-transitive epsilon-connected ordinal natural V31() real V113() V114() V144() V145() V146() V147() V148() V149() Element of NAT
vseq . tseq is Relation-like Function-like Element of the carrier of (X,Y)
||.(vseq . tseq).|| is ext-real non negative V31() real Element of REAL
the U9 of (X,Y) . (vseq . tseq) is ext-real V31() real Element of REAL
||.vseq.|| . tseq is ext-real V31() real Element of REAL
tseq is Element of the carrier of X
tseq . tseq is Element of the carrier of Y
||.(tseq . tseq).|| is ext-real non negative V31() real Element of REAL
the U9 of Y . (tseq . tseq) is ext-real V31() real Element of REAL
||.tseq.|| is ext-real non negative V31() real Element of REAL
the U9 of X . tseq is ext-real V31() real Element of REAL
(lim ||.vseq.||) * ||.tseq.|| is ext-real V31() real Element of REAL
tseq is ext-real V31() real Element of REAL
tv is ext-real epsilon-transitive epsilon-connected ordinal natural V31() real V113() V114() V144() V145() V146() V147() V148() V149() Element of NAT
e is ext-real epsilon-transitive epsilon-connected ordinal natural V31() real V113() V114() V144() V145() V146() V147() V148() V149() 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 ext-real non negative V31() 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 V134() V135() V136() Element of bool [: the carrier of X,REAL:]
[: the carrier of X,REAL:] is non empty V134() V135() V136() set
bool [: the carrier of X,REAL:] is non empty set
the U9 of X . m is ext-real V31() real Element of REAL
n is ext-real epsilon-transitive epsilon-connected ordinal natural V31() real V113() V114() V144() V145() V146() V147() V148() V149() Element of NAT
n . n is Element of the carrier of Y
h1 is ext-real epsilon-transitive epsilon-connected ordinal natural V31() real V113() V114() V144() V145() V146() V147() V148() V149() 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 ext-real non negative V31() 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 V134() V135() V136() Element of bool [: the carrier of Y,REAL:]
[: the carrier of Y,REAL:] is non empty V134() V135() V136() set
bool [: the carrier of Y,REAL:] is non empty set
the U9 of Y . ((n . n) - (n . h1)) is ext-real V31() 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 ext-real non negative V31() real Element of REAL
the U9 of (X,Y) . ((vseq . n) - (vseq . h1)) is ext-real V31() real Element of REAL
||.((vseq . n) - (vseq . h1)).|| * ||.m.|| is ext-real non negative V31() 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 ext-real V31() real Element of REAL
n is ext-real epsilon-transitive epsilon-connected ordinal natural V31() real V113() V114() V144() V145() V146() V147() V148() V149() 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 ext-real non negative V31() 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 V134() V135() V136() Element of bool [: the carrier of Y,REAL:]
[: the carrier of Y,REAL:] is non empty V134() V135() V136() set
bool [: the carrier of Y,REAL:] is non empty set
the U9 of Y . ((n . e) - (n . n)) is ext-real V31() 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 ext-real non negative V31() real Element of REAL
the U9 of (X,Y) . ((vseq . e) - (vseq . n)) is ext-real V31() real Element of REAL
||.((vseq . e) - (vseq . n)).|| * ||.m.|| is ext-real non negative V31() 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 ext-real non negative V31() 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 V134() V135() V136() Element of bool [: the carrier of Y,REAL:]
[: the carrier of Y,REAL:] is non empty V134() V135() V136() set
bool [: the carrier of Y,REAL:] is non empty set
the U9 of Y . ((n . e) - (tseq . m)) is ext-real V31() real Element of REAL
n is non empty Relation-like NAT -defined REAL -valued Function-like total quasi_total V134() V135() V136() Element of bool [:NAT,REAL:]
h1 is set
n . h1 is ext-real V31() real Element of REAL
f1 is ext-real epsilon-transitive epsilon-connected ordinal natural V31() real V113() V114() V144() V145() V146() V147() V148() V149() 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 ext-real non negative V31() real Element of REAL
the U9 of Y . ((n . f1) - (n . e)) is ext-real V31() 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 ext-real non negative V31() real Element of REAL
the U9 of Y . ((n - (n . e)) . f1) is ext-real V31() real Element of REAL
||.(n - (n . e)).|| is non empty Relation-like NAT -defined REAL -valued Function-like total quasi_total V134() V135() V136() Element of bool [:NAT,REAL:]
||.(n - (n . e)).|| . h1 is ext-real V31() 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 ext-real V31() real Element of REAL
||.((tseq . m) - (n . e)).|| is ext-real non negative V31() real Element of REAL
the U9 of Y . ((tseq . m) - (n . e)) is ext-real V31() real Element of REAL
h1 is ext-real epsilon-transitive epsilon-connected ordinal natural V31() real V113() V114() V144() V145() V146() V147() V148() V149() Element of NAT
n . h1 is ext-real V31() 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 ext-real non negative V31() real Element of REAL
the U9 of Y . ((n . h1) - (n . e)) is ext-real V31() 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 ext-real non negative V31() real Element of REAL
the U9 of Y . ((n . e) - (n . h1)) is ext-real V31() 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 ext-real non negative V31() real Element of REAL
the U9 of Y . (((X,Y,(vseq . e)) . m) - (tseq . m)) is ext-real V31() 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 ext-real non negative V31() real Element of REAL
the U9 of Y . (((X,Y,(vseq . e)) . m) - (tseq . m)) is ext-real V31() real Element of REAL
||.m.|| is ext-real non negative V31() real Element of REAL
the U9 of X . m is ext-real V31() real Element of REAL
tseq * ||.m.|| is ext-real V31() real Element of REAL
e is ext-real epsilon-transitive epsilon-connected ordinal natural V31() real V113() V114() V144() V145() V146() V147() V148() V149() 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 ext-real non negative V31() real Element of REAL
the U9 of Y . (((X,Y,(vseq . e)) . m) - (tseq . m)) is ext-real V31() real Element of REAL
||.m.|| is ext-real non negative V31() real Element of REAL
the U9 of X . m is ext-real V31() real Element of REAL
tseq * ||.m.|| is ext-real V31() 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 ext-real V31() real Element of REAL
m is ext-real epsilon-transitive epsilon-connected ordinal natural V31() real V113() V114() V144() V145() V146() V147() V148() V149() Element of NAT
n is ext-real epsilon-transitive epsilon-connected ordinal natural V31() real V113() V114() V144() V145() V146() V147() V148() V149() 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 ext-real non negative V31() 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 V134() V135() V136() Element of bool [: the carrier of X,REAL:]
[: the carrier of X,REAL:] is non empty V134() V135() V136() set
bool [: the carrier of X,REAL:] is non empty set
the U9 of X . r is ext-real V31() real Element of REAL
e * ||.r.|| is ext-real V31() real Element of REAL
e * 1 is ext-real V31() 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 ext-real non negative V31() 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 V134() V135() V136() Element of bool [: the carrier of Y,REAL:]
[: the carrier of Y,REAL:] is non empty V134() V135() V136() set
bool [: the carrier of Y,REAL:] is non empty set
the U9 of Y . (((X,Y,(vseq . n)) . r) - (tseq . r)) is ext-real V31() 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 ext-real non negative V31() real Element of REAL
the U9 of Y . (h1 . r) is ext-real V31() real Element of REAL
r is ext-real V31() real Element of REAL
(X,Y,h1) is non empty V144() V145() V146() 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 ext-real non negative V31() real Element of REAL
the U9 of Y . (h1 . c16) is ext-real V31() real Element of REAL
||.c16.|| is ext-real non negative V31() real Element of REAL
the U9 of X . c16 is ext-real V31() real Element of REAL
upper_bound (X,Y,h1) is ext-real V31() real Element of REAL
(X,Y) . ((vseq . n) - tv) is ext-real V31() real Element of REAL
||.((vseq . n) - tv).|| is ext-real non negative V31() real Element of REAL
the U9 of (X,Y) . ((vseq . n) - tv) is ext-real V31() real Element of REAL
r is ext-real V31() real set
e is ext-real V31() real Element of REAL
e / 2 is ext-real V31() real Element of REAL
m is ext-real epsilon-transitive epsilon-connected ordinal natural V31() real V113() V114() V144() V145() V146() V147() V148() V149() Element of NAT
n is ext-real epsilon-transitive epsilon-connected ordinal natural V31() real V113() V114() V144() V145() V146() V147() V148() V149() 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 ext-real non negative V31() real Element of REAL
the U9 of (X,Y) . ((vseq . n) - tv) is ext-real V31() real Element of REAL
X is non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() discerning reflexive RealNormSpace-like NORMSTR
Y is non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() discerning reflexive RealNormSpace-like () NORMSTR
(X,Y) is non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() discerning reflexive RealNormSpace-like NORMSTR
(X,Y) is non empty Element of bool the carrier of (X,Y)
(X,Y) is non empty left_complementable right_complementable strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() constituted-Functions RLSStruct
(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 left_complementable right_complementable strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() constituted-Functions RLSStruct
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
( 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)
0. Y is V53(Y) 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
( 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 V110() 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 [:REAL,(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 V110() Element of bool [:[:REAL,(Funcs ( the carrier of X, the carrier of Y)):],(Funcs ( the carrier of X, the carrier of Y)):]
[:REAL,(Funcs ( the carrier of X, the carrier of Y)):] is non empty set
[:[:REAL,(Funcs ( the carrier of X, the carrier of Y)):],(Funcs ( the carrier of X, the carrier of Y)):] is non empty set
bool [:[:REAL,(Funcs ( the carrier of X, the carrier of Y)):],(Funcs ( the carrier of X, the carrier of Y)):] is non empty set
RLSStruct(# (Funcs ( the carrier of X, the carrier of Y)),( the carrier of X,Y),( the carrier of X,Y),( the carrier of X,Y) #) is non empty strict RLSStruct
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 V110() 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 [:REAL,(X,Y):] -defined (X,Y) -valued Function-like total quasi_total Function-yielding V110() Element of bool [:[:REAL,(X,Y):],(X,Y):]
[:REAL,(X,Y):] is non empty set
[:[:REAL,(X,Y):],(X,Y):] is non empty set
bool [:[:REAL,(X,Y):],(X,Y):] is non empty set
RLSStruct(# (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 left_complementable right_complementable strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() RLSStruct
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 [:REAL,(X,Y):] -defined (X,Y) -valued Function-like total quasi_total Element of bool [:[:REAL,(X,Y):],(X,Y):]
[:REAL,(X,Y):] is non empty set
[:[:REAL,(X,Y):],(X,Y):] is non empty set
bool [:[:REAL,(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 V134() V135() V136() Element of bool [:(X,Y),REAL:]
[:(X,Y),REAL:] is non empty V134() V135() V136() set
bool [:(X,Y),REAL:] is non empty set
NORMSTR(# (X,Y),(Zero_ ((X,Y),(X,Y))),(Add_ ((X,Y),(X,Y))),(Mult_ ((X,Y),(X,Y))),(X,Y) #) is non empty strict NORMSTR
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 left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() discerning reflexive RealNormSpace-like NORMSTR
Y is non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() discerning reflexive RealNormSpace-like () NORMSTR
(X,Y) is non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() discerning reflexive RealNormSpace-like NORMSTR
(X,Y) is non empty Element of bool the carrier of (X,Y)
(X,Y) is non empty left_complementable right_complementable strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() constituted-Functions RLSStruct
(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 left_complementable right_complementable strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() constituted-Functions RLSStruct
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
( 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)
0. Y is V53(Y) 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
( 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 V110() 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 [:REAL,(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 V110() Element of bool [:[:REAL,(Funcs ( the carrier of X, the carrier of Y)):],(Funcs ( the carrier of X, the carrier of Y)):]
[:REAL,(Funcs ( the carrier of X, the carrier of Y)):] is non empty set
[:[:REAL,(Funcs ( the carrier of X, the carrier of Y)):],(Funcs ( the carrier of X, the carrier of Y)):] is non empty set
bool [:[:REAL,(Funcs ( the carrier of X, the carrier of Y)):],(Funcs ( the carrier of X, the carrier of Y)):] is non empty set
RLSStruct(# (Funcs ( the carrier of X, the carrier of Y)),( the carrier of X,Y),( the carrier of X,Y),( the carrier of X,Y) #) is non empty strict RLSStruct
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 V110() 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 [:REAL,(X,Y):] -defined (X,Y) -valued Function-like total quasi_total Function-yielding V110() Element of bool [:[:REAL,(X,Y):],(X,Y):]
[:REAL,(X,Y):] is non empty set
[:[:REAL,(X,Y):],(X,Y):] is non empty set
bool [:[:REAL,(X,Y):],(X,Y):] is non empty set
RLSStruct(# (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 left_complementable right_complementable strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() RLSStruct
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 [:REAL,(X,Y):] -defined (X,Y) -valued Function-like total quasi_total Element of bool [:[:REAL,(X,Y):],(X,Y):]
[:REAL,(X,Y):] is non empty set
[:[:REAL,(X,Y):],(X,Y):] is non empty set
bool [:[:REAL,(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 V134() V135() V136() Element of bool [:(X,Y),REAL:]
[:(X,Y),REAL:] is non empty V134() V135() V136() set
bool [:(X,Y),REAL:] is non empty set
NORMSTR(# (X,Y),(Zero_ ((X,Y),(X,Y))),(Add_ ((X,Y),(X,Y))),(Mult_ ((X,Y),(X,Y))),(X,Y) #) is non empty strict NORMSTR