:: CSSPACE4 semantic presentation

REAL is non empty V45() V46() V47() V51() V52() set
NAT is non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() Element of bool REAL
bool REAL is non empty set
COMPLEX is non empty V45() V51() V52() set
omega is non empty epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() set
bool omega is non empty set
bool NAT is non empty set
ExtREAL is non empty V46() set
[:NAT,REAL:] is non empty Relation-like V33() V34() V35() set
bool [:NAT,REAL:] is non empty set
[:NAT,COMPLEX:] is non empty Relation-like V33() set
bool [:NAT,COMPLEX:] is non empty set
the_set_of_ComplexSequences is non empty set
[:the_set_of_ComplexSequences,the_set_of_ComplexSequences:] is non empty Relation-like set
[:[:the_set_of_ComplexSequences,the_set_of_ComplexSequences:],the_set_of_ComplexSequences:] is non empty Relation-like set
bool [:[:the_set_of_ComplexSequences,the_set_of_ComplexSequences:],the_set_of_ComplexSequences:] is non empty set
[:COMPLEX,the_set_of_ComplexSequences:] is non empty Relation-like set
[:[:COMPLEX,the_set_of_ComplexSequences:],the_set_of_ComplexSequences:] is non empty Relation-like set
bool [:[:COMPLEX,the_set_of_ComplexSequences:],the_set_of_ComplexSequences:] is non empty set
Linear_Space_of_ComplexSequences is non empty right_complementable Abelian add-associative right_zeroed strict vector-distributive scalar-distributive scalar-associative scalar-unital CLSStruct
the carrier of Linear_Space_of_ComplexSequences is non empty set
bool the carrier of Linear_Space_of_ComplexSequences is non empty set
the_set_of_l2ComplexSequences is Element of bool the carrier of Linear_Space_of_ComplexSequences
[:the_set_of_l2ComplexSequences,the_set_of_l2ComplexSequences:] is Relation-like set
[:[:the_set_of_l2ComplexSequences,the_set_of_l2ComplexSequences:],COMPLEX:] is Relation-like V33() set
bool [:[:the_set_of_l2ComplexSequences,the_set_of_l2ComplexSequences:],COMPLEX:] is non empty set
K379() is non empty linearly-closed Element of bool the carrier of Linear_Space_of_ComplexSequences
[:K379(),REAL:] is non empty Relation-like V33() V34() V35() set
bool [:K379(),REAL:] is non empty set
RAT is non empty V45() V46() V47() V48() V51() V52() set
INT is non empty V45() V46() V47() V48() V49() V51() V52() set
[:REAL,REAL:] is non empty Relation-like V33() V34() V35() set
bool [:REAL,REAL:] is non empty set
[:COMPLEX,COMPLEX:] is non empty Relation-like V33() set
bool [:COMPLEX,COMPLEX:] is non empty set
[:COMPLEX,REAL:] is non empty Relation-like V33() V34() V35() set
bool [:COMPLEX,REAL:] is non empty set
[:[:COMPLEX,COMPLEX:],COMPLEX:] is non empty Relation-like V33() set
bool [:[:COMPLEX,COMPLEX:],COMPLEX:] is non empty set
[:[:REAL,REAL:],REAL:] is non empty Relation-like V33() V34() V35() set
bool [:[:REAL,REAL:],REAL:] is non empty set
[:RAT,RAT:] is non empty Relation-like RAT -valued V33() V34() V35() set
bool [:RAT,RAT:] is non empty set
[:[:RAT,RAT:],RAT:] is non empty Relation-like RAT -valued V33() V34() V35() set
bool [:[:RAT,RAT:],RAT:] is non empty set
[:INT,INT:] is non empty Relation-like RAT -valued INT -valued V33() V34() V35() set
bool [:INT,INT:] is non empty set
[:[:INT,INT:],INT:] is non empty Relation-like RAT -valued INT -valued V33() V34() V35() set
bool [:[:INT,INT:],INT:] is non empty set
[:NAT,NAT:] is non empty Relation-like RAT -valued INT -valued V33() V34() V35() V36() set
[:[:NAT,NAT:],NAT:] is non empty Relation-like RAT -valued INT -valued V33() V34() V35() V36() set
bool [:[:NAT,NAT:],NAT:] is non empty set
{} is set
the empty epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural V11() real Relation-like non-empty empty-yielding RAT -valued ext-real non positive non negative V33() V34() V35() V36() V45() V46() V47() V48() V49() V50() V51() set is empty epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural V11() real Relation-like non-empty empty-yielding RAT -valued ext-real non positive non negative V33() V34() V35() V36() V45() V46() V47() V48() V49() V50() V51() set
1 is non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V45() V46() V47() V48() V49() V50() V136() V181() Element of NAT
0 is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real V45() V46() V47() V48() V49() V50() V136() V181() Element of NAT
CZeroseq is Element of the_set_of_ComplexSequences
0c is V11() Element of COMPLEX
l_add is non empty Relation-like [:the_set_of_ComplexSequences,the_set_of_ComplexSequences:] -defined the_set_of_ComplexSequences -valued Function-like V23([:the_set_of_ComplexSequences,the_set_of_ComplexSequences:]) quasi_total Element of bool [:[:the_set_of_ComplexSequences,the_set_of_ComplexSequences:],the_set_of_ComplexSequences:]
l_mult is non empty Relation-like [:COMPLEX,the_set_of_ComplexSequences:] -defined the_set_of_ComplexSequences -valued Function-like V23([:COMPLEX,the_set_of_ComplexSequences:]) quasi_total Element of bool [:[:COMPLEX,the_set_of_ComplexSequences:],the_set_of_ComplexSequences:]
CLSStruct(# the_set_of_ComplexSequences,CZeroseq,l_add,l_mult #) is non empty Abelian strict CLSStruct
the carrier of CLSStruct(# the_set_of_ComplexSequences,CZeroseq,l_add,l_mult #) is non empty set
1r is V11() Element of COMPLEX
|.0.| is V11() real ext-real V181() Element of REAL
|.1r.| is V11() real ext-real Element of REAL
- 1r is V11() Element of COMPLEX
2 is non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V45() V46() V47() V48() V49() V50() V136() V181() Element of NAT
X is non empty Relation-like NAT -defined REAL -valued Function-like V23( NAT ) quasi_total V33() V34() V35() Element of bool [:NAT,REAL:]
rng X is non empty V45() V46() V47() Element of bool REAL
upper_bound (rng X) is V11() real ext-real Element of REAL
Y is V11() real ext-real set
vseq is V11() real ext-real set
f is set
X . f is V11() real ext-real Element of REAL
X is non empty Relation-like NAT -defined REAL -valued Function-like V23( NAT ) quasi_total V33() V34() V35() Element of bool [:NAT,REAL:]
rng X is non empty V45() V46() V47() Element of bool REAL
upper_bound (rng X) is V11() real ext-real Element of REAL
Y is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real V45() V46() V47() V48() V49() V50() V136() V181() Element of NAT
X . Y is V11() real ext-real Element of REAL
dom X is non empty V45() V46() V47() V48() V49() V50() Element of bool NAT
X is set
Y is set
bool the_set_of_ComplexSequences is non empty set
X is Element of bool the carrier of Linear_Space_of_ComplexSequences
Y is Element of bool the carrier of Linear_Space_of_ComplexSequences
vseq is set
seq_id vseq is non empty Relation-like NAT -defined COMPLEX -valued Function-like V23( NAT ) quasi_total V33() Element of bool [:NAT,COMPLEX:]
vseq is set
seq_id vseq is non empty Relation-like NAT -defined COMPLEX -valued Function-like V23( NAT ) quasi_total V33() Element of bool [:NAT,COMPLEX:]
() is Element of bool the carrier of Linear_Space_of_ComplexSequences
X is non empty Relation-like NAT -defined COMPLEX -valued Function-like V23( NAT ) quasi_total V33() Element of bool [:NAT,COMPLEX:]
Y is non empty Relation-like NAT -defined COMPLEX -valued Function-like V23( NAT ) quasi_total V33() Element of bool [:NAT,COMPLEX:]
X + Y is non empty Relation-like NAT -defined COMPLEX -valued Function-like V23( NAT ) quasi_total V33() Element of bool [:NAT,COMPLEX:]
vseq is V11() real ext-real Element of REAL
f is V11() real ext-real Element of REAL
f + vseq is V11() real ext-real Element of REAL
tseq is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real V45() V46() V47() V48() V49() V50() V136() V181() Element of NAT
(X + Y) . tseq is V11() Element of COMPLEX
|.((X + Y) . tseq).| is V11() real ext-real Element of REAL
X . tseq is V11() Element of COMPLEX
Y . tseq is V11() Element of COMPLEX
(X . tseq) + (Y . tseq) is V11() Element of COMPLEX
|.((X . tseq) + (Y . tseq)).| is V11() real ext-real Element of REAL
|.(X . tseq).| is V11() real ext-real Element of REAL
|.(Y . tseq).| is V11() real ext-real Element of REAL
|.(X . tseq).| + |.(Y . tseq).| is V11() real ext-real Element of REAL
X is V11() set
Y is non empty Relation-like NAT -defined COMPLEX -valued Function-like V23( NAT ) quasi_total V33() Element of bool [:NAT,COMPLEX:]
X (#) Y is non empty Relation-like NAT -defined COMPLEX -valued Function-like V23( NAT ) quasi_total V33() Element of bool [:NAT,COMPLEX:]
vseq is V11() real ext-real Element of REAL
tseq is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real V45() V46() V47() V48() V49() V50() V136() V181() Element of NAT
(X (#) Y) . tseq is V11() Element of COMPLEX
|.((X (#) Y) . tseq).| is V11() real ext-real Element of REAL
Y . tseq is V11() Element of COMPLEX
X * (Y . tseq) is V11() set
|.X.| is V11() real ext-real Element of REAL
|.X.| * vseq is V11() real ext-real Element of REAL
tseq is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real V45() V46() V47() V48() V49() V50() V136() V181() Element of NAT
(X (#) Y) . tseq is V11() Element of COMPLEX
|.((X (#) Y) . tseq).| is V11() real ext-real Element of REAL
Y . tseq is V11() Element of COMPLEX
X * (Y . tseq) is V11() set
|.(X * (Y . tseq)).| is V11() real ext-real Element of REAL
|.(Y . tseq).| is V11() real ext-real Element of REAL
|.X.| * |.(Y . tseq).| is V11() real ext-real Element of REAL
tseq is V11() real ext-real Element of REAL
tv is V11() real ext-real Element of REAL
f is V11() real ext-real Element of REAL
seq_id CZeroseq is non empty Relation-like NAT -defined COMPLEX -valued Function-like V23( NAT ) quasi_total V33() Element of bool [:NAT,COMPLEX:]
X is non empty Relation-like NAT -defined COMPLEX -valued Function-like V23( NAT ) quasi_total V33() Element of bool [:NAT,COMPLEX:]
Y is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real V45() V46() V47() V48() V49() V50() V136() V181() Element of NAT
X . Y is V11() Element of COMPLEX
|.(X . Y).| is V11() real ext-real Element of REAL
Y is V11() set
vseq is Element of the carrier of Linear_Space_of_ComplexSequences
Y * vseq is Element of the carrier of Linear_Space_of_ComplexSequences
seq_id vseq is non empty Relation-like NAT -defined COMPLEX -valued Function-like V23( NAT ) quasi_total V33() Element of bool [:NAT,COMPLEX:]
seq_id (Y * vseq) is non empty Relation-like NAT -defined COMPLEX -valued Function-like V23( NAT ) quasi_total V33() Element of bool [:NAT,COMPLEX:]
Y (#) (seq_id vseq) is non empty Relation-like NAT -defined COMPLEX -valued Function-like V23( NAT ) quasi_total V33() Element of bool [:NAT,COMPLEX:]
seq_id (Y (#) (seq_id vseq)) is non empty Relation-like NAT -defined COMPLEX -valued Function-like V23( NAT ) quasi_total V33() Element of bool [:NAT,COMPLEX:]
Y is Element of the carrier of Linear_Space_of_ComplexSequences
vseq is Element of the carrier of Linear_Space_of_ComplexSequences
Y + vseq is Element of the carrier of Linear_Space_of_ComplexSequences
the addF of Linear_Space_of_ComplexSequences is non empty Relation-like [: the carrier of Linear_Space_of_ComplexSequences, the carrier of Linear_Space_of_ComplexSequences:] -defined the carrier of Linear_Space_of_ComplexSequences -valued Function-like V23([: the carrier of Linear_Space_of_ComplexSequences, the carrier of Linear_Space_of_ComplexSequences:]) quasi_total Element of bool [:[: the carrier of Linear_Space_of_ComplexSequences, the carrier of Linear_Space_of_ComplexSequences:], the carrier of Linear_Space_of_ComplexSequences:]
[: the carrier of Linear_Space_of_ComplexSequences, the carrier of Linear_Space_of_ComplexSequences:] is non empty Relation-like set
[:[: the carrier of Linear_Space_of_ComplexSequences, the carrier of Linear_Space_of_ComplexSequences:], the carrier of Linear_Space_of_ComplexSequences:] is non empty Relation-like set
bool [:[: the carrier of Linear_Space_of_ComplexSequences, the carrier of Linear_Space_of_ComplexSequences:], the carrier of Linear_Space_of_ComplexSequences:] is non empty set
K602( the carrier of Linear_Space_of_ComplexSequences, the addF of Linear_Space_of_ComplexSequences,Y,vseq) is Element of the carrier of Linear_Space_of_ComplexSequences
[Y,vseq] is set
{Y,vseq} is set
{Y} is set
{{Y,vseq},{Y}} is set
the addF of Linear_Space_of_ComplexSequences . [Y,vseq] is set
seq_id Y is non empty Relation-like NAT -defined COMPLEX -valued Function-like V23( NAT ) quasi_total V33() Element of bool [:NAT,COMPLEX:]
seq_id vseq is non empty Relation-like NAT -defined COMPLEX -valued Function-like V23( NAT ) quasi_total V33() Element of bool [:NAT,COMPLEX:]
seq_id (Y + vseq) is non empty Relation-like NAT -defined COMPLEX -valued Function-like V23( NAT ) quasi_total V33() Element of bool [:NAT,COMPLEX:]
(seq_id Y) + (seq_id vseq) is non empty Relation-like NAT -defined COMPLEX -valued Function-like V23( NAT ) quasi_total V33() Element of bool [:NAT,COMPLEX:]
seq_id ((seq_id Y) + (seq_id vseq)) is non empty Relation-like NAT -defined COMPLEX -valued Function-like V23( NAT ) quasi_total V33() Element of bool [:NAT,COMPLEX:]
Zero_ ((),Linear_Space_of_ComplexSequences) is Element of ()
Add_ ((),Linear_Space_of_ComplexSequences) is non empty Relation-like [:(),():] -defined () -valued Function-like V23([:(),():]) quasi_total Element of bool [:[:(),():],():]
[:(),():] is non empty Relation-like set
[:[:(),():],():] is non empty Relation-like set
bool [:[:(),():],():] is non empty set
Mult_ ((),Linear_Space_of_ComplexSequences) is non empty Relation-like [:COMPLEX,():] -defined () -valued Function-like V23([:COMPLEX,():]) quasi_total Element of bool [:[:COMPLEX,():],():]
[:COMPLEX,():] is non empty Relation-like set
[:[:COMPLEX,():],():] is non empty Relation-like set
bool [:[:COMPLEX,():],():] is non empty set
CLSStruct(# (),(Zero_ ((),Linear_Space_of_ComplexSequences)),(Add_ ((),Linear_Space_of_ComplexSequences)),(Mult_ ((),Linear_Space_of_ComplexSequences)) #) is non empty strict CLSStruct
[:(),REAL:] is non empty Relation-like V33() V34() V35() set
bool [:(),REAL:] is non empty set
X is set
seq_id X is non empty Relation-like NAT -defined COMPLEX -valued Function-like V23( NAT ) quasi_total V33() Element of bool [:NAT,COMPLEX:]
|.(seq_id X).| is non empty Relation-like NAT -defined REAL -valued Function-like V23( NAT ) quasi_total V33() V34() V35() Element of bool [:NAT,REAL:]
rng |.(seq_id X).| is non empty V45() V46() V47() Element of bool REAL
upper_bound (rng |.(seq_id X).|) is V11() real ext-real Element of REAL
X is non empty Relation-like () -defined REAL -valued Function-like V23(()) quasi_total V33() V34() V35() Element of bool [:(),REAL:]
X is non empty Relation-like () -defined REAL -valued Function-like V23(()) quasi_total V33() V34() V35() Element of bool [:(),REAL:]
Y is non empty Relation-like () -defined REAL -valued Function-like V23(()) quasi_total V33() V34() V35() Element of bool [:(),REAL:]
vseq is set
X . vseq is V11() real ext-real Element of REAL
Y . vseq is V11() real ext-real Element of REAL
seq_id vseq is non empty Relation-like NAT -defined COMPLEX -valued Function-like V23( NAT ) quasi_total V33() Element of bool [:NAT,COMPLEX:]
|.(seq_id vseq).| is non empty Relation-like NAT -defined REAL -valued Function-like V23( NAT ) quasi_total V33() V34() V35() Element of bool [:NAT,REAL:]
rng |.(seq_id vseq).| is non empty V45() V46() V47() Element of bool REAL
upper_bound (rng |.(seq_id vseq).|) is V11() real ext-real Element of REAL
dom X is non empty Element of bool ()
bool () is non empty set
dom Y is non empty Element of bool ()
() is non empty Relation-like () -defined REAL -valued Function-like V23(()) quasi_total V33() V34() V35() Element of bool [:(),REAL:]
X is non empty Relation-like NAT -defined COMPLEX -valued Function-like V23( NAT ) quasi_total V33() Element of bool [:NAT,COMPLEX:]
|.X.| is non empty Relation-like NAT -defined REAL -valued Function-like V23( NAT ) quasi_total V33() V34() V35() Element of bool [:NAT,REAL:]
rng |.X.| is non empty V45() V46() V47() Element of bool REAL
upper_bound (rng |.X.|) is V11() real ext-real Element of REAL
Y is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real V45() V46() V47() V48() V49() V50() V136() V181() Element of NAT
X . Y is V11() Element of COMPLEX
|.(X . Y).| is V11() real ext-real Element of REAL
{0} is V45() V46() V47() V48() V49() V50() Element of bool REAL
Y is set
vseq is set
|.X.| . vseq is V11() real ext-real Element of REAL
f is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real V45() V46() V47() V48() V49() V50() V136() V181() Element of NAT
|.X.| . f is V11() real ext-real Element of REAL
X . f is V11() Element of COMPLEX
|.(X . f).| is V11() real ext-real Element of REAL
|.0c.| is V11() real ext-real Element of REAL
X is non empty Relation-like NAT -defined COMPLEX -valued Function-like V23( NAT ) quasi_total V33() Element of bool [:NAT,COMPLEX:]
|.X.| is non empty Relation-like NAT -defined REAL -valued Function-like V23( NAT ) quasi_total V33() V34() V35() Element of bool [:NAT,REAL:]
X . 0 is V11() Element of COMPLEX
|.(X . 0).| is V11() real ext-real Element of REAL
Y is V11() real ext-real Element of REAL
vseq is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real V45() V46() V47() V48() V49() V50() V136() V181() Element of NAT
|.X.| . vseq is V11() real ext-real Element of REAL
abs (|.X.| . vseq) is V11() real ext-real Element of REAL
X . vseq is V11() Element of COMPLEX
|.(X . vseq).| is V11() real ext-real Element of REAL
X is non empty Relation-like NAT -defined COMPLEX -valued Function-like V23( NAT ) quasi_total V33() Element of bool [:NAT,COMPLEX:]
|.X.| is non empty Relation-like NAT -defined REAL -valued Function-like V23( NAT ) quasi_total V33() V34() V35() Element of bool [:NAT,REAL:]
Y is V11() real ext-real set
vseq is V11() real ext-real Element of REAL
f is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real V45() V46() V47() V48() V49() V50() V136() V181() Element of NAT
X . f is V11() Element of COMPLEX
|.(X . f).| is V11() real ext-real Element of REAL
|.X.| . f is V11() real ext-real Element of REAL
abs (|.X.| . f) is V11() real ext-real Element of REAL
X is non empty Relation-like NAT -defined COMPLEX -valued Function-like V23( NAT ) quasi_total V33() Element of bool [:NAT,COMPLEX:]
|.X.| is non empty Relation-like NAT -defined REAL -valued Function-like V23( NAT ) quasi_total V33() V34() V35() Element of bool [:NAT,REAL:]
rng |.X.| is non empty V45() V46() V47() Element of bool REAL
upper_bound (rng |.X.|) is V11() real ext-real Element of REAL
Y is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real V45() V46() V47() V48() V49() V50() V136() V181() Element of NAT
X . Y is V11() Element of COMPLEX
|.(X . Y).| is V11() real ext-real Element of REAL
|.X.| . Y is V11() real ext-real Element of REAL
X is non empty Relation-like NAT -defined COMPLEX -valued Function-like V23( NAT ) quasi_total V33() Element of bool [:NAT,COMPLEX:]
|.X.| is non empty Relation-like NAT -defined REAL -valued Function-like V23( NAT ) quasi_total V33() V34() V35() Element of bool [:NAT,REAL:]
rng |.X.| is non empty V45() V46() V47() Element of bool REAL
upper_bound (rng |.X.|) is V11() real ext-real Element of REAL
Y is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real V45() V46() V47() V48() V49() V50() V136() V181() Element of NAT
X . Y is V11() Element of COMPLEX
vseq is non empty Relation-like NAT -defined COMPLEX -valued Function-like V23( NAT ) quasi_total V33() Element of bool [:NAT,COMPLEX:]
|.vseq.| is non empty Relation-like NAT -defined REAL -valued Function-like V23( NAT ) quasi_total V33() V34() V35() Element of bool [:NAT,REAL:]
rng |.vseq.| is non empty V45() V46() V47() Element of bool REAL
upper_bound (rng |.vseq.|) is V11() real ext-real Element of REAL
CNORMSTR(# (),(Zero_ ((),Linear_Space_of_ComplexSequences)),(Add_ ((),Linear_Space_of_ComplexSequences)),(Mult_ ((),Linear_Space_of_ComplexSequences)),() #) is non empty strict CNORMSTR
() is non empty CNORMSTR
the carrier of () is non empty set
0. () is V82(()) Element of the carrier of ()
the ZeroF of () is Element of the carrier of ()
Y is set
seq_id Y is non empty Relation-like NAT -defined COMPLEX -valued Function-like V23( NAT ) quasi_total V33() Element of bool [:NAT,COMPLEX:]
Y is Element of the carrier of ()
vseq is Element of the carrier of ()
Y + vseq is Element of the carrier of ()
the addF of () is non empty Relation-like [: the carrier of (), the carrier of ():] -defined the carrier of () -valued Function-like V23([: the carrier of (), the carrier of ():]) quasi_total Element of bool [:[: the carrier of (), the carrier of ():], the carrier of ():]
[: the carrier of (), the carrier of ():] is non empty Relation-like set
[:[: the carrier of (), the carrier of ():], the carrier of ():] is non empty Relation-like set
bool [:[: the carrier of (), the carrier of ():], the carrier of ():] is non empty set
K602( the carrier of (), the addF of (),Y,vseq) is Element of the carrier of ()
[Y,vseq] is set
{Y,vseq} is set
{Y} is set
{{Y,vseq},{Y}} is set
the addF of () . [Y,vseq] is set
seq_id Y is non empty Relation-like NAT -defined COMPLEX -valued Function-like V23( NAT ) quasi_total V33() Element of bool [:NAT,COMPLEX:]
seq_id vseq is non empty Relation-like NAT -defined COMPLEX -valued Function-like V23( NAT ) quasi_total V33() Element of bool [:NAT,COMPLEX:]
(seq_id Y) + (seq_id vseq) is non empty Relation-like NAT -defined COMPLEX -valued Function-like V23( NAT ) quasi_total V33() Element of bool [:NAT,COMPLEX:]
[: the carrier of Linear_Space_of_ComplexSequences, the carrier of Linear_Space_of_ComplexSequences:] is non empty Relation-like set
the addF of Linear_Space_of_ComplexSequences is non empty Relation-like [: the carrier of Linear_Space_of_ComplexSequences, the carrier of Linear_Space_of_ComplexSequences:] -defined the carrier of Linear_Space_of_ComplexSequences -valued Function-like V23([: the carrier of Linear_Space_of_ComplexSequences, the carrier of Linear_Space_of_ComplexSequences:]) quasi_total Element of bool [:[: the carrier of Linear_Space_of_ComplexSequences, the carrier of Linear_Space_of_ComplexSequences:], the carrier of Linear_Space_of_ComplexSequences:]
[:[: the carrier of Linear_Space_of_ComplexSequences, the carrier of Linear_Space_of_ComplexSequences:], the carrier of Linear_Space_of_ComplexSequences:] is non empty Relation-like set
bool [:[: the carrier of Linear_Space_of_ComplexSequences, the carrier of Linear_Space_of_ComplexSequences:], the carrier of Linear_Space_of_ComplexSequences:] is non empty set
dom the addF of Linear_Space_of_ComplexSequences is non empty Relation-like the carrier of Linear_Space_of_ComplexSequences -defined the carrier of Linear_Space_of_ComplexSequences -valued Element of bool [: the carrier of Linear_Space_of_ComplexSequences, the carrier of Linear_Space_of_ComplexSequences:]
bool [: the carrier of Linear_Space_of_ComplexSequences, the carrier of Linear_Space_of_ComplexSequences:] is non empty set
the addF of Linear_Space_of_ComplexSequences || () is Relation-like Function-like set
the addF of Linear_Space_of_ComplexSequences | [:(),():] is Relation-like [:(),():] -defined [: the carrier of Linear_Space_of_ComplexSequences, the carrier of Linear_Space_of_ComplexSequences:] -defined the carrier of Linear_Space_of_ComplexSequences -valued set
dom ( the addF of Linear_Space_of_ComplexSequences || ()) is set
[:(),():] is non empty Relation-like the carrier of Linear_Space_of_ComplexSequences -defined the carrier of Linear_Space_of_ComplexSequences -valued Element of bool [: the carrier of Linear_Space_of_ComplexSequences, the carrier of Linear_Space_of_ComplexSequences:]
[Y,vseq] is Element of [: the carrier of (), the carrier of ():]
( the addF of Linear_Space_of_ComplexSequences || ()) . [Y,vseq] is set
f is Element of the carrier of Linear_Space_of_ComplexSequences
tseq is Element of the carrier of Linear_Space_of_ComplexSequences
f + tseq is Element of the carrier of Linear_Space_of_ComplexSequences
K602( the carrier of Linear_Space_of_ComplexSequences, the addF of Linear_Space_of_ComplexSequences,f,tseq) is Element of the carrier of Linear_Space_of_ComplexSequences
[f,tseq] is set
{f,tseq} is set
{f} is set
{{f,tseq},{f}} is set
the addF of Linear_Space_of_ComplexSequences . [f,tseq] is set
Y is V11() set
vseq is Element of the carrier of ()
Y * vseq is Element of the carrier of ()
seq_id vseq is non empty Relation-like NAT -defined COMPLEX -valued Function-like V23( NAT ) quasi_total V33() Element of bool [:NAT,COMPLEX:]
Y (#) (seq_id vseq) is non empty Relation-like NAT -defined COMPLEX -valued Function-like V23( NAT ) quasi_total V33() Element of bool [:NAT,COMPLEX:]
[:COMPLEX, the carrier of Linear_Space_of_ComplexSequences:] is non empty Relation-like set
the Mult of Linear_Space_of_ComplexSequences is non empty Relation-like [:COMPLEX, the carrier of Linear_Space_of_ComplexSequences:] -defined the carrier of Linear_Space_of_ComplexSequences -valued Function-like V23([:COMPLEX, the carrier of Linear_Space_of_ComplexSequences:]) quasi_total Element of bool [:[:COMPLEX, the carrier of Linear_Space_of_ComplexSequences:], the carrier of Linear_Space_of_ComplexSequences:]
[:[:COMPLEX, the carrier of Linear_Space_of_ComplexSequences:], the carrier of Linear_Space_of_ComplexSequences:] is non empty Relation-like set
bool [:[:COMPLEX, the carrier of Linear_Space_of_ComplexSequences:], the carrier of Linear_Space_of_ComplexSequences:] is non empty set
dom the Mult of Linear_Space_of_ComplexSequences is non empty Relation-like COMPLEX -defined the carrier of Linear_Space_of_ComplexSequences -valued Element of bool [:COMPLEX, the carrier of Linear_Space_of_ComplexSequences:]
bool [:COMPLEX, the carrier of Linear_Space_of_ComplexSequences:] is non empty set
the Mult of Linear_Space_of_ComplexSequences | [:COMPLEX,():] is Relation-like [:COMPLEX,():] -defined [:COMPLEX, the carrier of Linear_Space_of_ComplexSequences:] -defined the carrier of Linear_Space_of_ComplexSequences -valued Function-like Element of bool [:[:COMPLEX, the carrier of Linear_Space_of_ComplexSequences:], the carrier of Linear_Space_of_ComplexSequences:]
dom ( the Mult of Linear_Space_of_ComplexSequences | [:COMPLEX,():]) is Relation-like COMPLEX -defined () -valued Element of bool [:COMPLEX,():]
bool [:COMPLEX,():] is non empty set
tv is V11() Element of COMPLEX
tv * vseq is Element of the carrier of ()
[:COMPLEX, the carrier of ():] is non empty Relation-like set
the Mult of () is non empty Relation-like [:COMPLEX, the carrier of ():] -defined the carrier of () -valued Function-like V23([:COMPLEX, the carrier of ():]) quasi_total Element of bool [:[:COMPLEX, the carrier of ():], the carrier of ():]
[:[:COMPLEX, the carrier of ():], the carrier of ():] is non empty Relation-like set
bool [:[:COMPLEX, the carrier of ():], the carrier of ():] is non empty set
[tv,vseq] is Element of [:COMPLEX, the carrier of ():]
{tv,vseq} is set
{tv} is V45() set
{{tv,vseq},{tv}} is set
the Mult of () . [tv,vseq] is Element of the carrier of ()
( the Mult of Linear_Space_of_ComplexSequences | [:COMPLEX,():]) . [tv,vseq] is set
the Mult of Linear_Space_of_ComplexSequences . [tv,vseq] is set
f is Element of the carrier of Linear_Space_of_ComplexSequences
tv * f is Element of the carrier of Linear_Space_of_ComplexSequences
Y is Element of the carrier of ()
seq_id Y is non empty Relation-like NAT -defined COMPLEX -valued Function-like V23( NAT ) quasi_total V33() Element of bool [:NAT,COMPLEX:]
Y is Element of the carrier of ()
- Y is Element of the carrier of ()
seq_id Y is non empty Relation-like NAT -defined COMPLEX -valued Function-like V23( NAT ) quasi_total V33() Element of bool [:NAT,COMPLEX:]
- (seq_id Y) is non empty Relation-like NAT -defined COMPLEX -valued Function-like V23( NAT ) quasi_total V33() Element of bool [:NAT,COMPLEX:]
- 1 is non empty V11() real ext-real non positive negative set
(- 1) (#) (seq_id Y) is Relation-like NAT -defined Function-like V23( NAT ) V33() set
seq_id (- Y) is non empty Relation-like NAT -defined COMPLEX -valued Function-like V23( NAT ) quasi_total V33() Element of bool [:NAT,COMPLEX:]
(- 1r) * Y is Element of the carrier of ()
(- 1r) (#) (seq_id Y) is non empty Relation-like NAT -defined COMPLEX -valued Function-like V23( NAT ) quasi_total V33() Element of bool [:NAT,COMPLEX:]
Y is Element of the carrier of ()
vseq is Element of the carrier of ()
Y - vseq is Element of the carrier of ()
- vseq is Element of the carrier of ()
Y + (- vseq) is Element of the carrier of ()
the addF of () is non empty Relation-like [: the carrier of (), the carrier of ():] -defined the carrier of () -valued Function-like V23([: the carrier of (), the carrier of ():]) quasi_total Element of bool [:[: the carrier of (), the carrier of ():], the carrier of ():]
[: the carrier of (), the carrier of ():] is non empty Relation-like set
[:[: the carrier of (), the carrier of ():], the carrier of ():] is non empty Relation-like set
bool [:[: the carrier of (), the carrier of ():], the carrier of ():] is non empty set
K602( the carrier of (), the addF of (),Y,(- vseq)) is Element of the carrier of ()
[Y,(- vseq)] is set
{Y,(- vseq)} is set
{Y} is set
{{Y,(- vseq)},{Y}} is set
the addF of () . [Y,(- vseq)] is set
seq_id Y is non empty Relation-like NAT -defined COMPLEX -valued Function-like V23( NAT ) quasi_total V33() Element of bool [:NAT,COMPLEX:]
seq_id vseq is non empty Relation-like NAT -defined COMPLEX -valued Function-like V23( NAT ) quasi_total V33() Element of bool [:NAT,COMPLEX:]
(seq_id Y) - (seq_id vseq) is non empty Relation-like NAT -defined COMPLEX -valued Function-like V23( NAT ) quasi_total V33() Element of bool [:NAT,COMPLEX:]
- (seq_id vseq) is Relation-like NAT -defined Function-like V23( NAT ) V33() set
- 1 is non empty V11() real ext-real non positive negative set
(- 1) (#) (seq_id vseq) is Relation-like NAT -defined Function-like V23( NAT ) V33() set
(seq_id Y) + (- (seq_id vseq)) is Relation-like NAT -defined Function-like V23( NAT ) V33() set
seq_id (- vseq) is non empty Relation-like NAT -defined COMPLEX -valued Function-like V23( NAT ) quasi_total V33() Element of bool [:NAT,COMPLEX:]
(seq_id Y) + (seq_id (- vseq)) is non empty Relation-like NAT -defined COMPLEX -valued Function-like V23( NAT ) quasi_total V33() Element of bool [:NAT,COMPLEX:]
Y is Element of the carrier of ()
||.Y.|| is V11() real ext-real Element of REAL
the normF of () is non empty Relation-like the carrier of () -defined REAL -valued Function-like V23( the carrier of ()) quasi_total V33() V34() V35() Element of bool [: the carrier of (),REAL:]
[: the carrier of (),REAL:] is non empty Relation-like V33() V34() V35() set
bool [: the carrier of (),REAL:] is non empty set
the normF of () . Y is V11() real ext-real Element of REAL
seq_id Y is non empty Relation-like NAT -defined COMPLEX -valued Function-like V23( NAT ) quasi_total V33() Element of bool [:NAT,COMPLEX:]
|.(seq_id Y).| is non empty Relation-like NAT -defined REAL -valued Function-like V23( NAT ) quasi_total V33() V34() V35() Element of bool [:NAT,REAL:]
rng |.(seq_id Y).| is non empty V45() V46() V47() Element of bool REAL
upper_bound (rng |.(seq_id Y).|) is V11() real ext-real Element of REAL
0. Linear_Space_of_ComplexSequences is V82( Linear_Space_of_ComplexSequences ) Element of the carrier of Linear_Space_of_ComplexSequences
the ZeroF of Linear_Space_of_ComplexSequences is Element of the carrier of Linear_Space_of_ComplexSequences
Y is set
seq_id Y is non empty Relation-like NAT -defined COMPLEX -valued Function-like V23( NAT ) quasi_total V33() Element of bool [:NAT,COMPLEX:]
vseq is set
seq_id vseq is non empty Relation-like NAT -defined COMPLEX -valued Function-like V23( NAT ) quasi_total V33() Element of bool [:NAT,COMPLEX:]
f is Element of the carrier of ()
seq_id f is non empty Relation-like NAT -defined COMPLEX -valued Function-like V23( NAT ) quasi_total V33() Element of bool [:NAT,COMPLEX:]
tseq is Element of the carrier of ()
tseq is Element of the carrier of ()
tseq + tseq is Element of the carrier of ()
the addF of () is non empty Relation-like [: the carrier of (), the carrier of ():] -defined the carrier of () -valued Function-like V23([: the carrier of (), the carrier of ():]) quasi_total Element of bool [:[: the carrier of (), the carrier of ():], the carrier of ():]
[: the carrier of (), the carrier of ():] is non empty Relation-like set
[:[: the carrier of (), the carrier of ():], the carrier of ():] is non empty Relation-like set
bool [:[: the carrier of (), the carrier of ():], the carrier of ():] is non empty set
K602( the carrier of (), the addF of (),tseq,tseq) is Element of the carrier of ()
[tseq,tseq] is set
{tseq,tseq} is set
{tseq} is set
{{tseq,tseq},{tseq}} is set
the addF of () . [tseq,tseq] is set
seq_id tseq is non empty Relation-like NAT -defined COMPLEX -valued Function-like V23( NAT ) quasi_total V33() Element of bool [:NAT,COMPLEX:]
seq_id tseq is non empty Relation-like NAT -defined COMPLEX -valued Function-like V23( NAT ) quasi_total V33() Element of bool [:NAT,COMPLEX:]
(seq_id tseq) + (seq_id tseq) is non empty Relation-like NAT -defined COMPLEX -valued Function-like V23( NAT ) quasi_total V33() Element of bool [:NAT,COMPLEX:]
e is Element of the carrier of ()
tv is V11() set
tv * e is Element of the carrier of ()
seq_id e is non empty Relation-like NAT -defined COMPLEX -valued Function-like V23( NAT ) quasi_total V33() Element of bool [:NAT,COMPLEX:]
tv (#) (seq_id e) is non empty Relation-like NAT -defined COMPLEX -valued Function-like V23( NAT ) quasi_total V33() Element of bool [:NAT,COMPLEX:]
m is Element of the carrier of ()
- m is Element of the carrier of ()
seq_id m is non empty Relation-like NAT -defined COMPLEX -valued Function-like V23( NAT ) quasi_total V33() Element of bool [:NAT,COMPLEX:]
- (seq_id m) is non empty Relation-like NAT -defined COMPLEX -valued Function-like V23( NAT ) quasi_total V33() Element of bool [:NAT,COMPLEX:]
- 1 is non empty V11() real ext-real non positive negative set
(- 1) (#) (seq_id m) is Relation-like NAT -defined Function-like V23( NAT ) V33() set
n is Element of the carrier of ()
- n is Element of the carrier of ()
seq_id (- n) is non empty Relation-like NAT -defined COMPLEX -valued Function-like V23( NAT ) quasi_total V33() Element of bool [:NAT,COMPLEX:]
seq_id n is non empty Relation-like NAT -defined COMPLEX -valued Function-like V23( NAT ) quasi_total V33() Element of bool [:NAT,COMPLEX:]
- (seq_id n) is non empty Relation-like NAT -defined COMPLEX -valued Function-like V23( NAT ) quasi_total V33() Element of bool [:NAT,COMPLEX:]
(- 1) (#) (seq_id n) is Relation-like NAT -defined Function-like V23( NAT ) V33() set
n is Element of the carrier of ()
h1 is Element of the carrier of ()
n - h1 is Element of the carrier of ()
- h1 is Element of the carrier of ()
n + (- h1) is Element of the carrier of ()
K602( the carrier of (), the addF of (),n,(- h1)) is Element of the carrier of ()
[n,(- h1)] is set
{n,(- h1)} is set
{n} is set
{{n,(- h1)},{n}} is set
the addF of () . [n,(- h1)] is set
seq_id n is non empty Relation-like NAT -defined COMPLEX -valued Function-like V23( NAT ) quasi_total V33() Element of bool [:NAT,COMPLEX:]
seq_id h1 is non empty Relation-like NAT -defined COMPLEX -valued Function-like V23( NAT ) quasi_total V33() Element of bool [:NAT,COMPLEX:]
(seq_id n) - (seq_id h1) is non empty Relation-like NAT -defined COMPLEX -valued Function-like V23( NAT ) quasi_total V33() Element of bool [:NAT,COMPLEX:]
- (seq_id h1) is Relation-like NAT -defined Function-like V23( NAT ) V33() set
(- 1) (#) (seq_id h1) is Relation-like NAT -defined Function-like V23( NAT ) V33() set
(seq_id n) + (- (seq_id h1)) is Relation-like NAT -defined Function-like V23( NAT ) V33() set
f1 is Element of the carrier of ()
seq_id f1 is non empty Relation-like NAT -defined COMPLEX -valued Function-like V23( NAT ) quasi_total V33() Element of bool [:NAT,COMPLEX:]
r is Element of the carrier of ()
||.r.|| is V11() real ext-real Element of REAL
the normF of () is non empty Relation-like the carrier of () -defined REAL -valued Function-like V23( the carrier of ()) quasi_total V33() V34() V35() Element of bool [: the carrier of (),REAL:]
[: the carrier of (),REAL:] is non empty Relation-like V33() V34() V35() set
bool [: the carrier of (),REAL:] is non empty set
the normF of () . r is V11() real ext-real Element of REAL
seq_id r is non empty Relation-like NAT -defined COMPLEX -valued Function-like V23( NAT ) quasi_total V33() Element of bool [:NAT,COMPLEX:]
|.(seq_id r).| is non empty Relation-like NAT -defined REAL -valued Function-like V23( NAT ) quasi_total V33() V34() V35() Element of bool [:NAT,REAL:]
rng |.(seq_id r).| is non empty V45() V46() V47() Element of bool REAL
upper_bound (rng |.(seq_id r).|) is V11() real ext-real Element of REAL
X is Element of the carrier of ()
||.X.|| is V11() real ext-real Element of REAL
the normF of () is non empty Relation-like the carrier of () -defined REAL -valued Function-like V23( the carrier of ()) quasi_total V33() V34() V35() Element of bool [: the carrier of (),REAL:]
[: the carrier of (),REAL:] is non empty Relation-like V33() V34() V35() set
bool [: the carrier of (),REAL:] is non empty set
the normF of () . X is V11() real ext-real Element of REAL
Y is Element of the carrier of ()
X + Y is Element of the carrier of ()
the addF of () is non empty Relation-like [: the carrier of (), the carrier of ():] -defined the carrier of () -valued Function-like V23([: the carrier of (), the carrier of ():]) quasi_total Element of bool [:[: the carrier of (), the carrier of ():], the carrier of ():]
[: the carrier of (), the carrier of ():] is non empty Relation-like set
[:[: the carrier of (), the carrier of ():], the carrier of ():] is non empty Relation-like set
bool [:[: the carrier of (), the carrier of ():], the carrier of ():] is non empty set
K602( the carrier of (), the addF of (),X,Y) is Element of the carrier of ()
[X,Y] is set
{X,Y} is set
{X} is set
{{X,Y},{X}} is set
the addF of () . [X,Y] is set
||.(X + Y).|| is V11() real ext-real Element of REAL
the normF of () . (X + Y) is V11() real ext-real Element of REAL
||.Y.|| is V11() real ext-real Element of REAL
the normF of () . Y is V11() real ext-real Element of REAL
||.X.|| + ||.Y.|| is V11() real ext-real Element of REAL
vseq is V11() set
vseq * X is Element of the carrier of ()
||.(vseq * X).|| is V11() real ext-real Element of REAL
the normF of () . (vseq * X) is V11() real ext-real Element of REAL
|.vseq.| is V11() real ext-real Element of REAL
|.vseq.| * ||.X.|| is V11() real ext-real Element of REAL
seq_id X is non empty Relation-like NAT -defined COMPLEX -valued Function-like V23( NAT ) quasi_total V33() Element of bool [:NAT,COMPLEX:]
vseq (#) (seq_id X) is non empty Relation-like NAT -defined COMPLEX -valued Function-like V23( NAT ) quasi_total V33() Element of bool [:NAT,COMPLEX:]
|.(vseq (#) (seq_id X)).| is non empty Relation-like NAT -defined REAL -valued Function-like V23( NAT ) quasi_total V33() V34() V35() Element of bool [:NAT,REAL:]
|.(seq_id X).| is non empty Relation-like NAT -defined REAL -valued Function-like V23( NAT ) quasi_total V33() V34() V35() Element of bool [:NAT,REAL:]
f is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real V45() V46() V47() V48() V49() V50() V136() V181() Element of NAT
|.(vseq (#) (seq_id X)).| . f is V11() real ext-real Element of REAL
|.(seq_id X).| . f is V11() real ext-real Element of REAL
|.vseq.| * (|.(seq_id X).| . f) is V11() real ext-real Element of REAL
(vseq (#) (seq_id X)) . f is V11() Element of COMPLEX
|.((vseq (#) (seq_id X)) . f).| is V11() real ext-real Element of REAL
(seq_id X) . f is V11() Element of COMPLEX
vseq * ((seq_id X) . f) is V11() set
|.(vseq * ((seq_id X) . f)).| is V11() real ext-real Element of REAL
|.((seq_id X) . f).| is V11() real ext-real Element of REAL
|.vseq.| * |.((seq_id X) . f).| is V11() real ext-real Element of REAL
|.(seq_id X).| . 1 is V11() real ext-real Element of REAL
(seq_id X) . 1 is V11() Element of COMPLEX
|.((seq_id X) . 1).| is V11() real ext-real Element of REAL
seq_id (X + Y) is non empty Relation-like NAT -defined COMPLEX -valued Function-like V23( NAT ) quasi_total V33() Element of bool [:NAT,COMPLEX:]
|.(seq_id (X + Y)).| is non empty Relation-like NAT -defined REAL -valued Function-like V23( NAT ) quasi_total V33() V34() V35() Element of bool [:NAT,REAL:]
seq_id Y is non empty Relation-like NAT -defined COMPLEX -valued Function-like V23( NAT ) quasi_total V33() Element of bool [:NAT,COMPLEX:]
f is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real V45() V46() V47() V48() V49() V50() V136() V181() Element of NAT
|.(seq_id (X + Y)).| . f is V11() real ext-real Element of REAL
(seq_id X) . f is V11() Element of COMPLEX
(seq_id Y) . f is V11() Element of COMPLEX
((seq_id X) . f) + ((seq_id Y) . f) is V11() Element of COMPLEX
|.(((seq_id X) . f) + ((seq_id Y) . f)).| is V11() real ext-real Element of REAL
(seq_id X) + (seq_id Y) is non empty Relation-like NAT -defined COMPLEX -valued Function-like V23( NAT ) quasi_total V33() Element of bool [:NAT,COMPLEX:]
seq_id ((seq_id X) + (seq_id Y)) is non empty Relation-like NAT -defined COMPLEX -valued Function-like V23( NAT ) quasi_total V33() Element of bool [:NAT,COMPLEX:]
|.(seq_id ((seq_id X) + (seq_id Y))).| is non empty Relation-like NAT -defined REAL -valued Function-like V23( NAT ) quasi_total V33() V34() V35() Element of bool [:NAT,REAL:]
|.(seq_id ((seq_id X) + (seq_id Y))).| . f is V11() real ext-real Element of REAL
|.((seq_id X) + (seq_id Y)).| is non empty Relation-like NAT -defined REAL -valued Function-like V23( NAT ) quasi_total V33() V34() V35() Element of bool [:NAT,REAL:]
|.((seq_id X) + (seq_id Y)).| . f is V11() real ext-real Element of REAL
((seq_id X) + (seq_id Y)) . f is V11() Element of COMPLEX
|.(((seq_id X) + (seq_id Y)) . f).| is V11() real ext-real Element of REAL
|.(seq_id Y).| is non empty Relation-like NAT -defined REAL -valued Function-like V23( NAT ) quasi_total V33() V34() V35() Element of bool [:NAT,REAL:]
f is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real V45() V46() V47() V48() V49() V50() V136() V181() Element of NAT
|.(seq_id (X + Y)).| . f is V11() real ext-real Element of REAL
|.(seq_id X).| . f is V11() real ext-real Element of REAL
|.(seq_id Y).| . f is V11() real ext-real Element of REAL
(|.(seq_id X).| . f) + (|.(seq_id Y).| . f) is V11() real ext-real Element of REAL
(seq_id X) . f is V11() Element of COMPLEX
(seq_id Y) . f is V11() Element of COMPLEX
((seq_id X) . f) + ((seq_id Y) . f) is V11() Element of COMPLEX
|.(((seq_id X) . f) + ((seq_id Y) . f)).| is V11() real ext-real Element of REAL
|.((seq_id X) . f).| is V11() real ext-real Element of REAL
|.((seq_id Y) . f).| is V11() real ext-real Element of REAL
|.((seq_id X) . f).| + |.((seq_id Y) . f).| is V11() real ext-real Element of REAL
(|.(seq_id X).| . f) + |.((seq_id Y) . f).| is V11() real ext-real Element of REAL
|.(seq_id X).| + |.(seq_id Y).| is non empty Relation-like NAT -defined REAL -valued Function-like V23( NAT ) quasi_total V33() V34() V35() Element of bool [:NAT,REAL:]
f is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real V45() V46() V47() V48() V49() V50() V136() V181() Element of NAT
|.(seq_id (X + Y)).| . f is V11() real ext-real Element of REAL
(|.(seq_id X).| + |.(seq_id Y).|) . f is V11() real ext-real Element of REAL
|.(seq_id X).| . f is V11() real ext-real Element of REAL
|.(seq_id Y).| . f is V11() real ext-real Element of REAL
(|.(seq_id X).| . f) + (|.(seq_id Y).| . f) is V11() real ext-real Element of REAL
rng |.(seq_id X).| is non empty V45() V46() V47() Element of bool REAL
upper_bound (rng |.(seq_id X).|) is V11() real ext-real Element of REAL
f is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real V45() V46() V47() V48() V49() V50() V136() V181() Element of NAT
(seq_id X) . f is V11() Element of COMPLEX
f is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real V45() V46() V47() V48() V49() V50() V136() V181() Element of NAT
(seq_id X) . f is V11() Element of COMPLEX
f is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real V45() V46() V47() V48() V49() V50() V136() V181() Element of NAT
|.(seq_id Y).| . f is V11() real ext-real Element of REAL
rng |.(seq_id Y).| is non empty V45() V46() V47() Element of bool REAL
upper_bound (rng |.(seq_id Y).|) is V11() real ext-real Element of REAL
(|.(seq_id X).| + |.(seq_id Y).|) . f is V11() real ext-real Element of REAL
|.(seq_id X).| . f is V11() real ext-real Element of REAL
(|.(seq_id X).| . f) + (|.(seq_id Y).| . f) is V11() real ext-real Element of REAL
(upper_bound (rng |.(seq_id X).|)) + (upper_bound (rng |.(seq_id Y).|)) is V11() real ext-real Element of REAL
|.(seq_id (X + Y)).| . f is V11() real ext-real Element of REAL
rng |.(seq_id (X + Y)).| is non empty V45() V46() V47() Element of bool REAL
upper_bound (rng |.(seq_id (X + Y)).|) is V11() real ext-real Element of REAL
seq_id (vseq * X) is non empty Relation-like NAT -defined COMPLEX -valued Function-like V23( NAT ) quasi_total V33() Element of bool [:NAT,COMPLEX:]
|.(seq_id (vseq * X)).| is non empty Relation-like NAT -defined REAL -valued Function-like V23( NAT ) quasi_total V33() V34() V35() Element of bool [:NAT,REAL:]
rng |.(seq_id (vseq * X)).| is non empty V45() V46() V47() Element of bool REAL
upper_bound (rng |.(seq_id (vseq * X)).|) is V11() real ext-real Element of REAL
seq_id (vseq (#) (seq_id X)) is non empty Relation-like NAT -defined COMPLEX -valued Function-like V23( NAT ) quasi_total V33() Element of bool [:NAT,COMPLEX:]
|.(seq_id (vseq (#) (seq_id X))).| is non empty Relation-like NAT -defined REAL -valued Function-like V23( NAT ) quasi_total V33() V34() V35() Element of bool [:NAT,REAL:]
rng |.(seq_id (vseq (#) (seq_id X))).| is non empty V45() V46() V47() Element of bool REAL
upper_bound (rng |.(seq_id (vseq (#) (seq_id X))).|) is V11() real ext-real Element of REAL
rng |.(vseq (#) (seq_id X)).| is non empty V45() V46() V47() Element of bool REAL
upper_bound (rng |.(vseq (#) (seq_id X)).|) is V11() real ext-real Element of REAL
|.vseq.| (#) |.(seq_id X).| is non empty Relation-like NAT -defined REAL -valued Function-like V23( NAT ) quasi_total V33() V34() V35() Element of bool [:NAT,REAL:]
rng (|.vseq.| (#) |.(seq_id X).|) is non empty V45() V46() V47() Element of bool REAL
upper_bound (rng (|.vseq.| (#) |.(seq_id X).|)) is V11() real ext-real Element of REAL
|.vseq.| ** (rng |.(seq_id X).|) is V45() V46() V47() Element of bool REAL
upper_bound (|.vseq.| ** (rng |.(seq_id X).|)) is V11() real ext-real Element of REAL
|.vseq.| * (upper_bound (rng |.(seq_id X).|)) is V11() real ext-real Element of REAL
||.(0. ()).|| is V11() real ext-real Element of REAL
the normF of () is non empty Relation-like the carrier of () -defined REAL -valued Function-like V23( the carrier of ()) quasi_total V33() V34() V35() Element of bool [: the carrier of (),REAL:]
[: the carrier of (),REAL:] is non empty Relation-like V33() V34() V35() set
bool [: the carrier of (),REAL:] is non empty set
the normF of () . (0. ()) is V11() real ext-real Element of REAL
X is non empty Relation-like NAT -defined COMPLEX -valued Function-like V23( NAT ) quasi_total V33() Element of bool [:NAT,COMPLEX:]
Y is non empty Relation-like NAT -defined COMPLEX -valued Function-like V23( NAT ) quasi_total V33() Element of bool [:NAT,COMPLEX:]
vseq is non empty Relation-like NAT -defined COMPLEX -valued Function-like V23( NAT ) quasi_total V33() Element of bool [:NAT,COMPLEX:]
Y - vseq is non empty Relation-like NAT -defined COMPLEX -valued Function-like V23( NAT ) quasi_total V33() Element of bool [:NAT,COMPLEX:]
- vseq is Relation-like NAT -defined Function-like V23( NAT ) V33() set
- 1 is non empty V11() real ext-real non positive negative set
(- 1) (#) vseq is Relation-like NAT -defined Function-like V23( NAT ) V33() set
Y + (- vseq) is Relation-like NAT -defined Function-like V23( NAT ) V33() set
f is set
X . f is V11() set
(Y - vseq) . f is V11() set
tseq is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real V45() V46() V47() V48() V49() V50() V136() V181() Element of NAT
X . tseq is V11() Element of COMPLEX
Y . tseq is V11() Element of COMPLEX
vseq . tseq is V11() Element of COMPLEX
(Y . tseq) - (vseq . tseq) is V11() Element of COMPLEX
- (vseq . tseq) is V11() Element of COMPLEX
(Y . tseq) + (- (vseq . tseq)) is V11() Element of COMPLEX
- vseq is non empty Relation-like NAT -defined COMPLEX -valued Function-like V23( NAT ) quasi_total V33() Element of bool [:NAT,COMPLEX:]
(- vseq) . tseq is V11() Element of COMPLEX
(Y . tseq) + ((- vseq) . tseq) is V11() Element of COMPLEX
Y + (- vseq) is non empty Relation-like NAT -defined COMPLEX -valued Function-like V23( NAT ) quasi_total V33() Element of bool [:NAT,COMPLEX:]
(Y + (- vseq)) . tseq is V11() Element of COMPLEX
f is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real V45() V46() V47() V48() V49() V50() V136() V181() Element of NAT
X . f is V11() Element of COMPLEX
Y . f is V11() Element of COMPLEX
- vseq is non empty Relation-like NAT -defined COMPLEX -valued Function-like V23( NAT ) quasi_total V33() Element of bool [:NAT,COMPLEX:]
(- vseq) . f is V11() Element of COMPLEX
(Y . f) + ((- vseq) . f) is V11() Element of COMPLEX
vseq . f is V11() Element of COMPLEX
- (vseq . f) is V11() Element of COMPLEX
(Y . f) + (- (vseq . f)) is V11() Element of COMPLEX
(Y . f) - (vseq . f) is V11() Element of COMPLEX
f is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real V45() V46() V47() V48() V49() V50() V136() V181() Element of NAT
X . f is V11() Element of COMPLEX
Y . f is V11() Element of COMPLEX
vseq . f is V11() Element of COMPLEX
(Y . f) - (vseq . f) is V11() Element of COMPLEX
f is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real V45() V46() V47() V48() V49() V50() V136() V181() Element of NAT
X . f is V11() Element of COMPLEX
Y . f is V11() Element of COMPLEX
vseq . f is V11() Element of COMPLEX
(Y . f) - (vseq . f) is V11() Element of COMPLEX
tseq is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real V45() V46() V47() V48() V49() V50() V136() V181() Element of NAT
X . tseq is V11() Element of COMPLEX
Y . tseq is V11() Element of COMPLEX
vseq . tseq is V11() Element of COMPLEX
(Y . tseq) - (vseq . tseq) is V11() Element of COMPLEX
[:NAT, the carrier of ():] is non empty Relation-like set
bool [:NAT, the carrier of ():] is non empty set
X is non empty Relation-like NAT -defined the carrier of () -valued Function-like V23( NAT ) quasi_total Element of bool [:NAT, the carrier of ():]
Y is set
vseq is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real V45() V46() V47() V48() V49() V50() V136() V181() Element of NAT
f is non empty Relation-like NAT -defined COMPLEX -valued Function-like V23( NAT ) quasi_total V33() Element of bool [:NAT,COMPLEX:]
lim f is V11() Element of COMPLEX
tseq is V11() real ext-real Element of REAL
tseq is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real V45() V46() V47() V48() V49() V50() V136() V181() Element of NAT
f . tseq is V11() Element of COMPLEX
tv is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real V45() V46() V47() V48() V49() V50() V136() V181() Element of NAT
f . tv is V11() Element of COMPLEX
(f . tv) - (f . tseq) is V11() Element of COMPLEX
|.((f . tv) - (f . tseq)).| is V11() real ext-real Element of REAL
X . tv is Element of the carrier of ()
X . tseq is Element of the carrier of ()
(X . tv) - (X . tseq) is Element of the carrier of ()
- (X . tseq) is Element of the carrier of ()
(X . tv) + (- (X . tseq)) is Element of the carrier of ()
the addF of () is non empty Relation-like [: the carrier of (), the carrier of ():] -defined the carrier of () -valued Function-like V23([: the carrier of (), the carrier of ():]) quasi_total Element of bool [:[: the carrier of (), the carrier of ():], the carrier of ():]
[: the carrier of (), the carrier of ():] is non empty Relation-like set
[:[: the carrier of (), the carrier of ():], the carrier of ():] is non empty Relation-like set
bool [:[: the carrier of (), the carrier of ():], the carrier of ():] is non empty set
K602( the carrier of (), the addF of (),(X . tv),(- (X . tseq))) is Element of the carrier of ()
[(X . tv),(- (X . tseq))] is set
{(X . tv),(- (X . tseq))} is set
{(X . tv)} is set
{{(X . tv),(- (X . tseq))},{(X . tv)}} is set
the addF of () . [(X . tv),(- (X . tseq))] is set
seq_id ((X . tv) - (X . tseq)) is non empty Relation-like NAT -defined COMPLEX -valued Function-like V23( NAT ) quasi_total V33() Element of bool [:NAT,COMPLEX:]
seq_id (X . tv) is non empty Relation-like NAT -defined COMPLEX -valued Function-like V23( NAT ) quasi_total V33() Element of bool [:NAT,COMPLEX:]
seq_id (X . tseq) is non empty Relation-like NAT -defined COMPLEX -valued Function-like V23( NAT ) quasi_total V33() Element of bool [:NAT,COMPLEX:]
(seq_id (X . tv)) - (seq_id (X . tseq)) is non empty Relation-like NAT -defined COMPLEX -valued Function-like V23( NAT ) quasi_total V33() Element of bool [:NAT,COMPLEX:]
- (seq_id (X . tseq)) is Relation-like NAT -defined Function-like V23( NAT ) V33() set
- 1 is non empty V11() real ext-real non positive negative set
(- 1) (#) (seq_id (X . tseq)) is Relation-like NAT -defined Function-like V23( NAT ) V33() set
(seq_id (X . tv)) + (- (seq_id (X . tseq))) is Relation-like NAT -defined Function-like V23( NAT ) V33() set
seq_id ((seq_id (X . tv)) - (seq_id (X . tseq))) is non empty Relation-like NAT -defined COMPLEX -valued Function-like V23( NAT ) quasi_total V33() Element of bool [:NAT,COMPLEX:]
- (seq_id (X . tseq)) is non empty Relation-like NAT -defined COMPLEX -valued Function-like V23( NAT ) quasi_total V33() Element of bool [:NAT,COMPLEX:]
(seq_id (X . tv)) + (- (seq_id (X . tseq))) is non empty Relation-like NAT -defined COMPLEX -valued Function-like V23( NAT ) quasi_total V33() Element of bool [:NAT,COMPLEX:]
(seq_id ((X . tv) - (X . tseq))) . vseq is V11() Element of COMPLEX
(seq_id (X . tv)) . vseq is V11() Element of COMPLEX
(- (seq_id (X . tseq))) . vseq is V11() Element of COMPLEX
((seq_id (X . tv)) . vseq) + ((- (seq_id (X . tseq))) . vseq) is V11() Element of COMPLEX
(seq_id (X . tseq)) . vseq is V11() Element of COMPLEX
- ((seq_id (X . tseq)) . vseq) is V11() Element of COMPLEX
((seq_id (X . tv)) . vseq) + (- ((seq_id (X . tseq)) . vseq)) is V11() Element of COMPLEX
((seq_id (X . tv)) . vseq) - ((seq_id (X . tseq)) . vseq) is V11() Element of COMPLEX
(f . tv) - ((seq_id (X . tseq)) . vseq) is V11() Element of COMPLEX
|.(seq_id ((X . tv) - (X . tseq))).| is non empty Relation-like NAT -defined REAL -valued Function-like V23( NAT ) quasi_total V33() V34() V35() Element of bool [:NAT,REAL:]
|.(seq_id ((X . tv) - (X . tseq))).| . vseq is V11() real ext-real Element of REAL
rng |.(seq_id ((X . tv) - (X . tseq))).| is non empty V45() V46() V47() Element of bool REAL
upper_bound (rng |.(seq_id ((X . tv) - (X . tseq))).|) is V11() real ext-real Element of REAL
||.((X . tv) - (X . tseq)).|| is V11() real ext-real Element of REAL
the normF of () is non empty Relation-like the carrier of () -defined REAL -valued Function-like V23( the carrier of ()) quasi_total V33() V34() V35() Element of bool [: the carrier of (),REAL:]
[: the carrier of (),REAL:] is non empty Relation-like V33() V34() V35() set
bool [: the carrier of (),REAL:] is non empty set
the normF of () . ((X . tv) - (X . tseq)) is V11() real ext-real Element of REAL
Y is non empty Relation-like NAT -defined COMPLEX -valued Function-like V23( NAT ) quasi_total V33() Element of bool [:NAT,COMPLEX:]
f is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real V45() V46() V47() V48() V49() V50() V136() V181() Element of NAT
tseq is set
Y . tseq is V11() set
vseq is non empty Relation-like NAT -defined COMPLEX -valued Function-like V23( NAT ) quasi_total V33() Element of bool [:NAT,COMPLEX:]
vseq . f is V11() Element of COMPLEX
tseq is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real V45() V46() V47() V48() V49() V50() V136() V181() Element of NAT
tv is non empty Relation-like NAT -defined COMPLEX -valued Function-like V23( NAT ) quasi_total V33() Element of bool [:NAT,COMPLEX:]
lim tv is V11() Element of COMPLEX
seq_id vseq is non empty Relation-like NAT -defined COMPLEX -valued Function-like V23( NAT ) quasi_total V33() Element of bool [:NAT,COMPLEX:]
f is V11() real ext-real Element of REAL
tseq is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real V45() V46() V47() V48() V49() V50() V136() V181() Element of NAT
tv is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real V45() V46() V47() V48() V49() V50() V136() V181() Element of NAT
tseq is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real V45() V46() V47() V48() V49() V50() V136() V181() Element of NAT
X . tv is Element of the carrier of ()
X . tseq is Element of the carrier of ()
(X . tv) - (X . tseq) is Element of the carrier of ()
- (X . tseq) is Element of the carrier of ()
(X . tv) + (- (X . tseq)) is Element of the carrier of ()
the addF of () is non empty Relation-like [: the carrier of (), the carrier of ():] -defined the carrier of () -valued Function-like V23([: the carrier of (), the carrier of ():]) quasi_total Element of bool [:[: the carrier of (), the carrier of ():], the carrier of ():]
[: the carrier of (), the carrier of ():] is non empty Relation-like set
[:[: the carrier of (), the carrier of ():], the carrier of ():] is non empty Relation-like set
bool [:[: the carrier of (), the carrier of ():], the carrier of ():] is non empty set
K602( the carrier of (), the addF of (),(X . tv),(- (X . tseq))) is Element of the carrier of ()
[(X . tv),(- (X . tseq))] is set
{(X . tv),(- (X . tseq))} is set
{(X . tv)} is set
{{(X . tv),(- (X . tseq))},{(X . tv)}} is set
the addF of () . [(X . tv),(- (X . tseq))] is set
seq_id ((X . tv) - (X . tseq)) is non empty Relation-like NAT -defined COMPLEX -valued Function-like V23( NAT ) quasi_total V33() Element of bool [:NAT,COMPLEX:]
|.(seq_id ((X . tv) - (X . tseq))).| is non empty Relation-like NAT -defined REAL -valued Function-like V23( NAT ) quasi_total V33() V34() V35() Element of bool [:NAT,REAL:]
rng |.(seq_id ((X . tv) - (X . tseq))).| is non empty V45() V46() V47() Element of bool REAL
upper_bound (rng |.(seq_id ((X . tv) - (X . tseq))).|) is V11() real ext-real Element of REAL
||.((X . tv) - (X . tseq)).|| is V11() real ext-real Element of REAL
the normF of () is non empty Relation-like the carrier of () -defined REAL -valued Function-like V23( the carrier of ()) quasi_total V33() V34() V35() Element of bool [: the carrier of (),REAL:]
[: the carrier of (),REAL:] is non empty Relation-like V33() V34() V35() set
bool [: the carrier of (),REAL:] is non empty set
the normF of () . ((X . tv) - (X . tseq)) is V11() real ext-real Element of REAL
the normF of () . ((X . tv) - (X . tseq)) is V11() real ext-real Element of REAL
tseq is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real V45() V46() V47() V48() V49() V50() V136() V181() Element of NAT
X . tseq is Element of the carrier of ()
seq_id (X . tseq) is non empty Relation-like NAT -defined COMPLEX -valued Function-like V23( NAT ) quasi_total V33() Element of bool [:NAT,COMPLEX:]
(seq_id vseq) - (seq_id (X . tseq)) is non empty Relation-like NAT -defined COMPLEX -valued Function-like V23( NAT ) quasi_total V33() Element of bool [:NAT,COMPLEX:]
- (seq_id (X . tseq)) is Relation-like NAT -defined Function-like V23( NAT ) V33() set
- 1 is non empty V11() real ext-real non positive negative set
(- 1) (#) (seq_id (X . tseq)) is Relation-like NAT -defined Function-like V23( NAT ) V33() set
(seq_id vseq) + (- (seq_id (X . tseq))) is Relation-like NAT -defined Function-like V23( NAT ) V33() set
|.((seq_id vseq) - (seq_id (X . tseq))).| is non empty Relation-like NAT -defined REAL -valued Function-like V23( NAT ) quasi_total V33() V34() V35() Element of bool [:NAT,REAL:]
tv is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real V45() V46() V47() V48() V49() V50() V136() V181() Element of NAT
X . tv is Element of the carrier of ()
e is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real V45() V46() V47() V48() V49() V50() V136() V181() Element of NAT
X . e is Element of the carrier of ()
(X . tv) - (X . e) is Element of the carrier of ()
- (X . e) is Element of the carrier of ()
(X . tv) + (- (X . e)) is Element of the carrier of ()
the addF of () is non empty Relation-like [: the carrier of (), the carrier of ():] -defined the carrier of () -valued Function-like V23([: the carrier of (), the carrier of ():]) quasi_total Element of bool [:[: the carrier of (), the carrier of ():], the carrier of ():]
[: the carrier of (), the carrier of ():] is non empty Relation-like set
[:[: the carrier of (), the carrier of ():], the carrier of ():] is non empty Relation-like set
bool [:[: the carrier of (), the carrier of ():], the carrier of ():] is non empty set
K602( the carrier of (), the addF of (),(X . tv),(- (X . e))) is Element of the carrier of ()
[(X . tv),(- (X . e))] is set
{(X . tv),(- (X . e))} is set
{(X . tv)} is set
{{(X . tv),(- (X . e))},{(X . tv)}} is set
the addF of () . [(X . tv),(- (X . e))] is set
seq_id ((X . tv) - (X . e)) is non empty Relation-like NAT -defined COMPLEX -valued Function-like V23( NAT ) quasi_total V33() Element of bool [:NAT,COMPLEX:]
seq_id (X . tv) is non empty Relation-like NAT -defined COMPLEX -valued Function-like V23( NAT ) quasi_total V33() Element of bool [:NAT,COMPLEX:]
seq_id (X . e) is non empty Relation-like NAT -defined COMPLEX -valued Function-like V23( NAT ) quasi_total V33() Element of bool [:NAT,COMPLEX:]
(seq_id (X . tv)) - (seq_id (X . e)) is non empty Relation-like NAT -defined COMPLEX -valued Function-like V23( NAT ) quasi_total V33() Element of bool [:NAT,COMPLEX:]
- (seq_id (X . e)) is Relation-like NAT -defined Function-like V23( NAT ) V33() set
(- 1) (#) (seq_id (X . e)) is Relation-like NAT -defined Function-like V23( NAT ) V33() set
(seq_id (X . tv)) + (- (seq_id (X . e))) is Relation-like NAT -defined Function-like V23( NAT ) V33() set
seq_id ((seq_id (X . tv)) - (seq_id (X . e))) is non empty Relation-like NAT -defined COMPLEX -valued Function-like V23( NAT ) quasi_total V33() Element of bool [:NAT,COMPLEX:]
tv is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real V45() V46() V47() V48() V49() V50() V136() V181() Element of NAT
vseq . tv is V11() Element of COMPLEX
e is non empty Relation-like NAT -defined COMPLEX -valued Function-like V23( NAT ) quasi_total V33() Element of bool [:NAT,COMPLEX:]
lim e is V11() Element of COMPLEX
m is non empty Relation-like NAT -defined REAL -valued Function-like V23( NAT ) quasi_total V33() V34() V35() Element of bool [:NAT,REAL:]
n is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real V45() V46() V47() V48() V49() V50() V136() V181() Element of NAT
X . n is Element of the carrier of ()
(X . n) - (X . tseq) is Element of the carrier of ()
- (X . tseq) is Element of the carrier of ()
(X . n) + (- (X . tseq)) is Element of the carrier of ()
the addF of () is non empty Relation-like [: the carrier of (), the carrier of ():] -defined the carrier of () -valued Function-like V23([: the carrier of (), the carrier of ():]) quasi_total Element of bool [:[: the carrier of (), the carrier of ():], the carrier of ():]
[: the carrier of (), the carrier of ():] is non empty Relation-like set
[:[: the carrier of (), the carrier of ():], the carrier of ():] is non empty Relation-like set
bool [:[: the carrier of (), the carrier of ():], the carrier of ():] is non empty set
K602( the carrier of (), the addF of (),(X . n),(- (X . tseq))) is Element of the carrier of ()
[(X . n),(- (X . tseq))] is set
{(X . n),(- (X . tseq))} is set
{(X . n)} is set
{{(X . n),(- (X . tseq))},{(X . n)}} is set
the addF of () . [(X . n),(- (X . tseq))] is set
seq_id ((X . n) - (X . tseq)) is non empty Relation-like NAT -defined COMPLEX -valued Function-like V23( NAT ) quasi_total V33() Element of bool [:NAT,COMPLEX:]
seq_id (X . n) is non empty Relation-like NAT -defined COMPLEX -valued Function-like V23( NAT ) quasi_total V33() Element of bool [:NAT,COMPLEX:]
(seq_id (X . n)) - (seq_id (X . tseq)) is non empty Relation-like NAT -defined COMPLEX -valued Function-like V23( NAT ) quasi_total V33() Element of bool [:NAT,COMPLEX:]
(seq_id (X . n)) + (- (seq_id (X . tseq))) is Relation-like NAT -defined Function-like V23( NAT ) V33() set
m . n is V11() real ext-real Element of REAL
|.(seq_id ((X . n) - (X . tseq))).| is non empty Relation-like NAT -defined REAL -valued Function-like V23( NAT ) quasi_total V33() V34() V35() Element of bool [:NAT,REAL:]
|.(seq_id ((X . n) - (X . tseq))).| . tv is V11() real ext-real Element of REAL
(seq_id ((X . n) - (X . tseq))) . tv is V11() Element of COMPLEX
|.((seq_id ((X . n) - (X . tseq))) . tv).| is V11() real ext-real Element of REAL
(seq_id (X . n)) . tv is V11() Element of COMPLEX
(seq_id (X . tseq)) . tv is V11() Element of COMPLEX
((seq_id (X . n)) . tv) - ((seq_id (X . tseq)) . tv) is V11() Element of COMPLEX
|.(((seq_id (X . n)) . tv) - ((seq_id (X . tseq)) . tv)).| is V11() real ext-real Element of REAL
e . n is V11() Element of COMPLEX
(e . n) - ((seq_id (X . tseq)) . tv) is V11() Element of COMPLEX
|.((e . n) - ((seq_id (X . tseq)) . tv)).| is V11() real ext-real Element of REAL
(vseq . tv) - ((seq_id (X . tseq)) . tv) is V11() Element of COMPLEX
|.((vseq . tv) - ((seq_id (X . tseq)) . tv)).| is V11() real ext-real Element of REAL
vseq - (seq_id (X . tseq)) is non empty Relation-like NAT -defined COMPLEX -valued Function-like V23( NAT ) quasi_total V33() Element of bool [:NAT,COMPLEX:]
vseq + (- (seq_id (X . tseq))) is Relation-like NAT -defined Function-like V23( NAT ) V33() set
(vseq - (seq_id (X . tseq))) . tv is V11() Element of COMPLEX
|.((vseq - (seq_id (X . tseq))) . tv).| is V11() real ext-real Element of REAL
((seq_id vseq) - (seq_id (X . tseq))) . tv is V11() Element of COMPLEX
|.(((seq_id vseq) - (seq_id (X . tseq))) . tv).| is V11() real ext-real Element of REAL
|.((seq_id vseq) - (seq_id (X . tseq))).| . tv is V11() real ext-real Element of REAL
lim m is V11() real ext-real Element of REAL
m is non empty Relation-like NAT -defined REAL -valued Function-like V23( NAT ) quasi_total V33() V34() V35() Element of bool [:NAT,REAL:]
lim m is V11() real ext-real Element of REAL
e is non empty Relation-like NAT -defined REAL -valued Function-like V23( NAT ) quasi_total V33() V34() V35() Element of bool [:NAT,REAL:]
tv is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real V45() V46() V47() V48() V49() V50() V136() V181() Element of NAT
lim e is V11() real ext-real Element of REAL
|.((seq_id vseq) - (seq_id (X . tseq))).| . tv is V11() real ext-real Element of REAL
tseq is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real V45() V46() V47() V48() V49() V50() V136() V181() Element of NAT
X . tseq is Element of the carrier of ()
seq_id (X . tseq) is non empty Relation-like NAT -defined COMPLEX -valued Function-like V23( NAT ) quasi_total V33() Element of bool [:NAT,COMPLEX:]
(seq_id vseq) - (seq_id (X . tseq)) is non empty Relation-like NAT -defined COMPLEX -valued Function-like V23( NAT ) quasi_total V33() Element of bool [:NAT,COMPLEX:]
- (seq_id (X . tseq)) is Relation-like NAT -defined Function-like V23( NAT ) V33() set
- 1 is non empty V11() real ext-real non positive negative set
(- 1) (#) (seq_id (X . tseq)) is Relation-like NAT -defined Function-like V23( NAT ) V33() set
(seq_id vseq) + (- (seq_id (X . tseq))) is Relation-like NAT -defined Function-like V23( NAT ) V33() set
|.((seq_id vseq) - (seq_id (X . tseq))).| is non empty Relation-like NAT -defined REAL -valued Function-like V23( NAT ) quasi_total V33() V34() V35() Element of bool [:NAT,REAL:]
rng |.((seq_id vseq) - (seq_id (X . tseq))).| is non empty V45() V46() V47() Element of bool REAL
upper_bound (rng |.((seq_id vseq) - (seq_id (X . tseq))).|) is V11() real ext-real Element of REAL
tv is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real V45() V46() V47() V48() V49() V50() V136() V181() Element of NAT
|.((seq_id vseq) - (seq_id (X . tseq))).| . tv is V11() real ext-real Element of REAL
e is non empty Relation-like NAT -defined REAL -valued Function-like V23( NAT ) quasi_total V33() V34() V35() Element of bool [:NAT,REAL:]
m is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real V45() V46() V47() V48() V49() V50() V136() V181() Element of NAT
e . m is V11() real ext-real Element of REAL
X . m is Element of the carrier of ()
(X . m) - (X . tseq) is Element of the carrier of ()
- (X . tseq) is Element of the carrier of ()
(X . m) + (- (X . tseq)) is Element of the carrier of ()
the addF of () is non empty Relation-like [: the carrier of (), the carrier of ():] -defined the carrier of () -valued Function-like V23([: the carrier of (), the carrier of ():]) quasi_total Element of bool [:[: the carrier of (), the carrier of ():], the carrier of ():]
[: the carrier of (), the carrier of ():] is non empty Relation-like set
[:[: the carrier of (), the carrier of ():], the carrier of ():] is non empty Relation-like set
bool [:[: the carrier of (), the carrier of ():], the carrier of ():] is non empty set
K602( the carrier of (), the addF of (),(X . m),(- (X . tseq))) is Element of the carrier of ()
[(X . m),(- (X . tseq))] is set
{(X . m),(- (X . tseq))} is set
{(X . m)} is set
{{(X . m),(- (X . tseq))},{(X . m)}} is set
the addF of () . [(X . m),(- (X . tseq))] is set
seq_id ((X . m) - (X . tseq)) is non empty Relation-like NAT -defined COMPLEX -valued Function-like V23( NAT ) quasi_total V33() Element of bool [:NAT,COMPLEX:]
|.(seq_id ((X . m) - (X . tseq))).| is non empty Relation-like NAT -defined REAL -valued Function-like V23( NAT ) quasi_total V33() V34() V35() Element of bool [:NAT,REAL:]
|.(seq_id ((X . m) - (X . tseq))).| . tv is V11() real ext-real Element of REAL
rng |.(seq_id ((X . m) - (X . tseq))).| is non empty V45() V46() V47() Element of bool REAL
upper_bound (rng |.(seq_id ((X . m) - (X . tseq))).|) is V11() real ext-real Element of REAL
lim e is V11() real ext-real Element of REAL
1 + f is V11() real ext-real Element of REAL
0 + f is V11() real ext-real Element of REAL
tv is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real V45() V46() V47() V48() V49() V50() V136() V181() Element of NAT
|.((seq_id vseq) - (seq_id (X . tseq))).| . tv is V11() real ext-real Element of REAL
((seq_id vseq) - (seq_id (X . tseq))) . tv is V11() Element of COMPLEX
|.(((seq_id vseq) - (seq_id (X . tseq))) . tv).| is V11() real ext-real Element of REAL
f + 1 is V11() real ext-real Element of REAL
f is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real V45() V46() V47() V48() V49() V50() V136() V181() Element of NAT
X . f is Element of the carrier of ()
seq_id (X . f) is non empty Relation-like NAT -defined COMPLEX -valued Function-like V23( NAT ) quasi_total V33() Element of bool [:NAT,COMPLEX:]
(seq_id vseq) - (seq_id (X . f)) is non empty Relation-like NAT -defined COMPLEX -valued Function-like V23( NAT ) quasi_total V33() Element of bool [:NAT,COMPLEX:]
- (seq_id (X . f)) is Relation-like NAT -defined Function-like V23( NAT ) V33() set
- 1 is non empty V11() real ext-real non positive negative set
(- 1) (#) (seq_id (X . f)) is Relation-like NAT -defined Function-like V23( NAT ) V33() set
(seq_id vseq) + (- (seq_id (X . f))) is Relation-like NAT -defined Function-like V23( NAT ) V33() set
|.((seq_id vseq) - (seq_id (X . f))).| is non empty Relation-like NAT -defined REAL -valued Function-like V23( NAT ) quasi_total V33() V34() V35() Element of bool [:NAT,REAL:]
|.(seq_id vseq).| is non empty Relation-like NAT -defined REAL -valued Function-like V23( NAT ) quasi_total V33() V34() V35() Element of bool [:NAT,REAL:]
|.(seq_id (X . f)).| is non empty Relation-like NAT -defined REAL -valued Function-like V23( NAT ) quasi_total V33() V34() V35() Element of bool [:NAT,REAL:]
|.((seq_id vseq) - (seq_id (X . f))).| + |.(seq_id (X . f)).| is non empty Relation-like NAT -defined REAL -valued Function-like V23( NAT ) quasi_total V33() V34() V35() Element of bool [:NAT,REAL:]
e is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real V45() V46() V47() V48() V49() V50() V136() V181() Element of NAT
|.(seq_id vseq).| . e is V11() real ext-real Element of REAL
(|.((seq_id vseq) - (seq_id (X . f))).| + |.(seq_id (X . f)).|) . e is V11() real ext-real Element of REAL
|.(seq_id (X . f)).| . e is V11() real ext-real Element of REAL
(seq_id (X . f)) . e is V11() Element of COMPLEX
|.((seq_id (X . f)) . e).| is V11() real ext-real Element of REAL
(seq_id vseq) . e is V11() Element of COMPLEX
|.((seq_id vseq) . e).| is V11() real ext-real Element of REAL
|.((seq_id vseq) - (seq_id (X . f))).| . e is V11() real ext-real Element of REAL
- (seq_id (X . f)) is non empty Relation-like NAT -defined COMPLEX -valued Function-like V23( NAT ) quasi_total V33() Element of bool [:NAT,COMPLEX:]
(seq_id vseq) + (- (seq_id (X . f))) is non empty Relation-like NAT -defined COMPLEX -valued Function-like V23( NAT ) quasi_total V33() Element of bool [:NAT,COMPLEX:]
((seq_id vseq) + (- (seq_id (X . f)))) . e is V11() Element of COMPLEX
|.(((seq_id vseq) + (- (seq_id (X . f)))) . e).| is V11() real ext-real Element of REAL
(- (seq_id (X . f))) . e is V11() Element of COMPLEX
((seq_id vseq) . e) + ((- (seq_id (X . f))) . e) is V11() Element of COMPLEX
|.(((seq_id vseq) . e) + ((- (seq_id (X . f))) . e)).| is V11() real ext-real Element of REAL
- ((seq_id (X . f)) . e) is V11() Element of COMPLEX
((seq_id vseq) . e) + (- ((seq_id (X . f)) . e)) is V11() Element of COMPLEX
|.(((seq_id vseq) . e) + (- ((seq_id (X . f)) . e))).| is V11() real ext-real Element of REAL
((seq_id vseq) . e) - ((seq_id (X . f)) . e) is V11() Element of COMPLEX
|.(((seq_id vseq) . e) - ((seq_id (X . f)) . e)).| is V11() real ext-real Element of REAL
(|.(seq_id vseq).| . e) - (|.(seq_id (X . f)).| . e) is V11() real ext-real Element of REAL
((|.(seq_id vseq).| . e) - (|.(seq_id (X . f)).| . e)) + (|.(seq_id (X . f)).| . e) is V11() real ext-real Element of REAL
(|.((seq_id vseq) - (seq_id (X . f))).| . e) + (|.(seq_id (X . f)).| . e) is V11() real ext-real Element of REAL
rng (|.((seq_id vseq) - (seq_id (X . f))).| + |.(seq_id (X . f)).|) is non empty V45() V46() V47() Element of bool REAL
upper_bound (rng (|.((seq_id vseq) - (seq_id (X . f))).| + |.(seq_id (X . f)).|)) is V11() real ext-real Element of REAL
(upper_bound (rng (|.((seq_id vseq) - (seq_id (X . f))).| + |.(seq_id (X . f)).|))) + 1 is V11() real ext-real Element of REAL
|.(seq_id (X . f)).| . 1 is V11() real ext-real Element of REAL
(seq_id (X . f)) . 1 is V11() Element of COMPLEX
|.((seq_id (X . f)) . 1).| is V11() real ext-real Element of REAL
(upper_bound (rng (|.((seq_id vseq) - (seq_id (X . f))).| + |.(seq_id (X . f)).|))) + 0 is V11() real ext-real Element of REAL
m is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real V45() V46() V47() V48() V49() V50() V136() V181() Element of NAT
|.(seq_id vseq).| . m is V11() real ext-real Element of REAL
(|.((seq_id vseq) - (seq_id (X . f))).| + |.(seq_id (X . f)).|) . m is V11() real ext-real Element of REAL
(seq_id vseq) . m is V11() Element of COMPLEX
|.((seq_id vseq) . m).| is V11() real ext-real Element of REAL
e is V11() real ext-real set
|.((seq_id vseq) - (seq_id (X . f))).| . 1 is V11() real ext-real Element of REAL
((seq_id vseq) - (seq_id (X . f))) . 1 is V11() Element of COMPLEX
|.(((seq_id vseq) - (seq_id (X . f))) . 1).| is V11() real ext-real Element of REAL
(|.((seq_id vseq) - (seq_id (X . f))).| + |.(seq_id (X . f)).|) . 1 is V11() real ext-real Element of REAL
(|.((seq_id vseq) - (seq_id (X . f))).| . 1) + (|.(seq_id (X . f)).| . 1) is V11() real ext-real Element of REAL
f is Element of the carrier of ()
tseq is V11() real ext-real Element of REAL
tseq / 2 is V11() real ext-real Element of REAL
tv is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real V45() V46() V47() V48() V49() V50() V136() V181() Element of NAT
m is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real V45() V46() V47() V48() V49() V50() V136() V181() Element of NAT
X . m is Element of the carrier of ()
seq_id (X . m) is non empty Relation-like NAT -defined COMPLEX -valued Function-like V23( NAT ) quasi_total V33() Element of bool [:NAT,COMPLEX:]
(seq_id vseq) - (seq_id (X . m)) is non empty Relation-like NAT -defined COMPLEX -valued Function-like V23( NAT ) quasi_total V33() Element of bool [:NAT,COMPLEX:]
- (seq_id (X . m)) is Relation-like NAT -defined Function-like V23( NAT ) V33() set
- 1 is non empty V11() real ext-real non positive negative set
(- 1) (#) (seq_id (X . m)) is Relation-like NAT -defined Function-like V23( NAT ) V33() set
(seq_id vseq) + (- (seq_id (X . m))) is Relation-like NAT -defined Function-like V23( NAT ) V33() set
|.((seq_id vseq) - (seq_id (X . m))).| is non empty Relation-like NAT -defined REAL -valued Function-like V23( NAT ) quasi_total V33() V34() V35() Element of bool [:NAT,REAL:]
rng |.((seq_id vseq) - (seq_id (X . m))).| is non empty V45() V46() V47() Element of bool REAL
upper_bound (rng |.((seq_id vseq) - (seq_id (X . m))).|) is V11() real ext-real Element of REAL
e is Element of the carrier of ()
n is Element of the carrier of ()
e - n is Element of the carrier of ()
- n is Element of the carrier of ()
e + (- n) is Element of the carrier of ()
the addF of () is non empty Relation-like [: the carrier of (), the carrier of ():] -defined the carrier of () -valued Function-like V23([: the carrier of (), the carrier of ():]) quasi_total Element of bool [:[: the carrier of (), the carrier of ():], the carrier of ():]
[: the carrier of (), the carrier of ():] is non empty Relation-like set
[:[: the carrier of (), the carrier of ():], the carrier of ():] is non empty Relation-like set
bool [:[: the carrier of (), the carrier of ():], the carrier of ():] is non empty set
K602( the carrier of (), the addF of (),e,(- n)) is Element of the carrier of ()
[e,(- n)] is set
{e,(- n)} is set
{e} is set
{{e,(- n)},{e}} is set
the addF of () . [e,(- n)] is set
seq_id (e - n) is non empty Relation-like NAT -defined COMPLEX -valued Function-like V23( NAT ) quasi_total V33() Element of bool [:NAT,COMPLEX:]
|.(seq_id (e - n)).| is non empty Relation-like NAT -defined REAL -valued Function-like V23( NAT ) quasi_total V33() V34() V35() Element of bool [:NAT,REAL:]
rng |.(seq_id (e - n)).| is non empty V45() V46() V47() Element of bool REAL
upper_bound (rng |.(seq_id (e - n)).|) is V11() real ext-real Element of REAL
the normF of () is non empty Relation-like the carrier of () -defined REAL -valued Function-like V23( the carrier of ()) quasi_total V33() V34() V35() Element of bool [: the carrier of (),REAL:]
[: the carrier of (),REAL:] is non empty Relation-like V33() V34() V35() set
bool [: the carrier of (),REAL:] is non empty set
the normF of () . (e - n) is V11() real ext-real Element of REAL
(X . m) - f is Element of the carrier of ()
- f is Element of the carrier of ()
(X . m) + (- f) is Element of the carrier of ()
K602( the carrier of (), the addF of (),(X . m),(- f)) is Element of the carrier of ()
[(X . m),(- f)] is set
{(X . m),(- f)} is set
{(X . m)} is set
{{(X . m),(- f)},{(X . m)}} is set
the addF of () . [(X . m),(- f)] is set
||.((X . m) - f).|| is V11() real ext-real Element of REAL
the normF of () . ((X . m) - f) is V11() real ext-real Element of REAL
f - (X . m) is Element of the carrier of ()
- (X . m) is Element of the carrier of ()
f + (- (X . m)) is Element of the carrier of ()
K602( the carrier of (), the addF of (),f,(- (X . m))) is Element of the carrier of ()
[f,(- (X . m))] is set
{f,(- (X . m))} is set
{f} is set
{{f,(- (X . m))},{f}} is set
the addF of () . [f,(- (X . m))] is set
- (f - (X . m)) is Element of the carrier of ()
||.(- (f - (X . m))).|| is V11() real ext-real Element of REAL
the normF of () . (- (f - (X . m))) is V11() real ext-real Element of REAL
||.(f - (X . m)).|| is V11() real ext-real Element of REAL
the normF of () . (f - (X . m)) is V11() real ext-real Element of REAL
tseq is Element of the carrier of ()
X is non empty set
Y is non empty right_complementable Abelian add-associative right_zeroed discerning reflexive vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like CNORMSTR
the carrier of Y is non empty set
[:X, the carrier of Y:] is non empty Relation-like set
bool [:X, the carrier of Y:] is non empty set
X is non empty set
Y is non empty right_complementable Abelian add-associative right_zeroed discerning reflexive vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like CNORMSTR
the carrier of Y is non empty set
[:X, the carrier of Y:] is non empty Relation-like set
bool [:X, the carrier of Y:] is non empty set
0. Y is V82(Y) Element of the carrier of Y
the ZeroF of Y is Element of the carrier of Y
vseq is non empty Relation-like X -defined the carrier of Y -valued Function-like V23(X) quasi_total Element of bool [:X, the carrier of Y:]
f is Element of X
vseq . f is Element of the carrier of Y
||.(vseq . f).|| is V11() real ext-real Element of REAL
the normF of Y is non empty Relation-like the carrier of Y -defined REAL -valued Function-like V23( the carrier of Y) quasi_total V33() V34() V35() Element of bool [: the carrier of Y,REAL:]
[: the carrier of Y,REAL:] is non empty Relation-like V33() V34() V35() set
bool [: the carrier of Y,REAL:] is non empty set
the normF of Y . (vseq . f) is V11() real ext-real Element of REAL
||.(0. Y).|| is empty epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural V11() real Relation-like non-empty empty-yielding RAT -valued ext-real non positive non negative V33() V34() V35() V36() V45() V46() V47() V48() V49() V50() V51() Element of REAL
the normF of Y . (0. Y) is V11() real ext-real Element of REAL
f is Element of X
vseq . f is Element of the carrier of Y
||.(vseq . f).|| is V11() real ext-real Element of REAL
the normF of Y . (vseq . f) is V11() real ext-real Element of REAL
X is non empty set
Y is non empty right_complementable Abelian add-associative right_zeroed discerning reflexive vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like CNORMSTR
the carrier of Y is non empty set
[:X, the carrier of Y:] is non empty Relation-like set
bool [:X, the carrier of Y:] is non empty set
0. Y is V82(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 V23(X) quasi_total Element of bool [:X, the carrier of Y:]
f is non empty Relation-like X -defined the carrier of Y -valued Function-like V23(X) quasi_total Element of bool [:X, the carrier of Y:]
tseq is Element of X
f . tseq is Element of the carrier of Y
X is non empty set
Y is non empty right_complementable Abelian add-associative right_zeroed discerning reflexive vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like CNORMSTR
ComplexVectSpace (X,Y) is non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital CLSStruct
the carrier of Y is non empty set
Funcs (X, the carrier of Y) is non empty functional FUNCTION_DOMAIN of X, the carrier of Y
K340(X,Y) is Relation-like X -defined the carrier of Y -valued Function-like V23(X) quasi_total Element of Funcs (X, the carrier of Y)
K338(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 V23([:(Funcs (X, the carrier of Y)),(Funcs (X, the carrier of Y)):]) quasi_total Function-yielding V135() 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 Relation-like set
[:[:(Funcs (X, the carrier of Y)),(Funcs (X, the carrier of Y)):],(Funcs (X, the carrier of Y)):] is non empty Relation-like set
bool [:[:(Funcs (X, the carrier of Y)),(Funcs (X, the carrier of Y)):],(Funcs (X, the carrier of Y)):] is non empty set
FuncExtMult (X,Y) is non empty Relation-like [:COMPLEX,(Funcs (X, the carrier of Y)):] -defined Funcs (X, the carrier of Y) -valued Function-like V23([:COMPLEX,(Funcs (X, the carrier of Y)):]) quasi_total Function-yielding V135() Element of bool [:[:COMPLEX,(Funcs (X, the carrier of Y)):],(Funcs (X, the carrier of Y)):]
[:COMPLEX,(Funcs (X, the carrier of Y)):] is non empty Relation-like set
[:[:COMPLEX,(Funcs (X, the carrier of Y)):],(Funcs (X, the carrier of Y)):] is non empty Relation-like set
bool [:[:COMPLEX,(Funcs (X, the carrier of Y)):],(Funcs (X, the carrier of Y)):] is non empty set
CLSStruct(# (Funcs (X, the carrier of Y)),K340(X,Y),K338(X,Y),(FuncExtMult (X,Y)) #) is non empty strict CLSStruct
the carrier of (ComplexVectSpace (X,Y)) is non empty set
bool the carrier of (ComplexVectSpace (X,Y)) is non empty set
[:X, the carrier of Y:] is non empty Relation-like set
bool [:X, the carrier of Y:] is non empty set
vseq is set
f is set
f is set
vseq is Element of bool the carrier of (ComplexVectSpace (X,Y))
f is Element of bool the carrier of (ComplexVectSpace (X,Y))
tseq is set
tseq is set
X is non empty set
Y is non empty right_complementable Abelian add-associative right_zeroed discerning reflexive vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like CNORMSTR
(X,Y) is Element of bool the carrier of (ComplexVectSpace (X,Y))
ComplexVectSpace (X,Y) is non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital CLSStruct
the carrier of Y is non empty set
Funcs (X, the carrier of Y) is non empty functional FUNCTION_DOMAIN of X, the carrier of Y
K340(X,Y) is Relation-like X -defined the carrier of Y -valued Function-like V23(X) quasi_total Element of Funcs (X, the carrier of Y)
K338(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 V23([:(Funcs (X, the carrier of Y)),(Funcs (X, the carrier of Y)):]) quasi_total Function-yielding V135() 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 Relation-like set
[:[:(Funcs (X, the carrier of Y)),(Funcs (X, the carrier of Y)):],(Funcs (X, the carrier of Y)):] is non empty Relation-like set
bool [:[:(Funcs (X, the carrier of Y)),(Funcs (X, the carrier of Y)):],(Funcs (X, the carrier of Y)):] is non empty set
FuncExtMult (X,Y) is non empty Relation-like [:COMPLEX,(Funcs (X, the carrier of Y)):] -defined Funcs (X, the carrier of Y) -valued Function-like V23([:COMPLEX,(Funcs (X, the carrier of Y)):]) quasi_total Function-yielding V135() Element of bool [:[:COMPLEX,(Funcs (X, the carrier of Y)):],(Funcs (X, the carrier of Y)):]
[:COMPLEX,(Funcs (X, the carrier of Y)):] is non empty Relation-like set
[:[:COMPLEX,(Funcs (X, the carrier of Y)):],(Funcs (X, the carrier of Y)):] is non empty Relation-like set
bool [:[:COMPLEX,(Funcs (X, the carrier of Y)):],(Funcs (X, the carrier of Y)):] is non empty set
CLSStruct(# (Funcs (X, the carrier of Y)),K340(X,Y),K338(X,Y),(FuncExtMult (X,Y)) #) is non empty strict CLSStruct
the carrier of (ComplexVectSpace (X,Y)) is non empty set
bool the carrier of (ComplexVectSpace (X,Y)) is non empty set
0. Y is V82(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 V23(X) quasi_total Element of bool [:X, the carrier of Y:]
[:X, the carrier of Y:] is non empty Relation-like set
bool [:X, the carrier of Y:] is non empty set
f is non empty Relation-like X -defined the carrier of Y -valued Function-like V23(X) quasi_total Element of bool [:X, the carrier of Y:]
tseq is Element of X
f . tseq is Element of the carrier of Y
X is non empty set
Y is non empty right_complementable Abelian add-associative right_zeroed discerning reflexive vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like CNORMSTR
ComplexVectSpace (X,Y) is non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital CLSStruct
the carrier of Y is non empty set
Funcs (X, the carrier of Y) is non empty functional FUNCTION_DOMAIN of X, the carrier of Y
K340(X,Y) is Relation-like X -defined the carrier of Y -valued Function-like V23(X) quasi_total Element of Funcs (X, the carrier of Y)
K338(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 V23([:(Funcs (X, the carrier of Y)),(Funcs (X, the carrier of Y)):]) quasi_total Function-yielding V135() 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 Relation-like set
[:[:(Funcs (X, the carrier of Y)),(Funcs (X, the carrier of Y)):],(Funcs (X, the carrier of Y)):] is non empty Relation-like set
bool [:[:(Funcs (X, the carrier of Y)),(Funcs (X, the carrier of Y)):],(Funcs (X, the carrier of Y)):] is non empty set
FuncExtMult (X,Y) is non empty Relation-like [:COMPLEX,(Funcs (X, the carrier of Y)):] -defined Funcs (X, the carrier of Y) -valued Function-like V23([:COMPLEX,(Funcs (X, the carrier of Y)):]) quasi_total Function-yielding V135() Element of bool [:[:COMPLEX,(Funcs (X, the carrier of Y)):],(Funcs (X, the carrier of Y)):]
[:COMPLEX,(Funcs (X, the carrier of Y)):] is non empty Relation-like set
[:[:COMPLEX,(Funcs (X, the carrier of Y)):],(Funcs (X, the carrier of Y)):] is non empty Relation-like set
bool [:[:COMPLEX,(Funcs (X, the carrier of Y)):],(Funcs (X, the carrier of Y)):] is non empty set
CLSStruct(# (Funcs (X, the carrier of Y)),K340(X,Y),K338(X,Y),(FuncExtMult (X,Y)) #) is non empty strict CLSStruct
(X,Y) is non empty Element of bool the carrier of (ComplexVectSpace (X,Y))
the carrier of (ComplexVectSpace (X,Y)) is non empty set
bool the carrier of (ComplexVectSpace (X,Y)) is non empty set
f is Element of the carrier of (ComplexVectSpace (X,Y))
tseq is Element of the carrier of (ComplexVectSpace (X,Y))
f + tseq is Element of the carrier of (ComplexVectSpace (X,Y))
the addF of (ComplexVectSpace (X,Y)) is non empty Relation-like [: the carrier of (ComplexVectSpace (X,Y)), the carrier of (ComplexVectSpace (X,Y)):] -defined the carrier of (ComplexVectSpace (X,Y)) -valued Function-like V23([: the carrier of (ComplexVectSpace (X,Y)), the carrier of (ComplexVectSpace (X,Y)):]) quasi_total Element of bool [:[: the carrier of (ComplexVectSpace (X,Y)), the carrier of (ComplexVectSpace (X,Y)):], the carrier of (ComplexVectSpace (X,Y)):]
[: the carrier of (ComplexVectSpace (X,Y)), the carrier of (ComplexVectSpace (X,Y)):] is non empty Relation-like set
[:[: the carrier of (ComplexVectSpace (X,Y)), the carrier of (ComplexVectSpace (X,Y)):], the carrier of (ComplexVectSpace (X,Y)):] is non empty Relation-like set
bool [:[: the carrier of (ComplexVectSpace (X,Y)), the carrier of (ComplexVectSpace (X,Y)):], the carrier of (ComplexVectSpace (X,Y)):] is non empty set
K602( the carrier of (ComplexVectSpace (X,Y)), the addF of (ComplexVectSpace (X,Y)),f,tseq) is Element of the carrier of (ComplexVectSpace (X,Y))
[f,tseq] is set
{f,tseq} is set
{f} is set
{{f,tseq},{f}} is set
the addF of (ComplexVectSpace (X,Y)) . [f,tseq] is set
[:X, the carrier of Y:] is non empty Relation-like set
bool [:X, the carrier of Y:] is non empty set
tseq is non empty Relation-like X -defined the carrier of Y -valued Function-like V23(X) quasi_total Element of bool [:X, the carrier of Y:]
tv is non empty Relation-like X -defined the carrier of Y -valued Function-like V23(X) quasi_total (X,Y) Element of bool [:X, the carrier of Y:]
e is V11() real ext-real Element of REAL
m is non empty Relation-like X -defined the carrier of Y -valued Function-like V23(X) quasi_total (X,Y) Element of bool [:X, the carrier of Y:]
n is V11() real ext-real Element of REAL
n + e is V11() real ext-real Element of REAL
n is V11() real ext-real Element of REAL
h1 is Element of X
m . h1 is Element of the carrier of Y
||.(m . h1).|| is V11() real ext-real Element of REAL
the normF of Y is non empty Relation-like the carrier of Y -defined REAL -valued Function-like V23( the carrier of Y) quasi_total V33() V34() V35() Element of bool [: the carrier of Y,REAL:]
[: the carrier of Y,REAL:] is non empty Relation-like V33() V34() V35() set
bool [: the carrier of Y,REAL:] is non empty set
the normF of Y . (m . h1) is V11() real ext-real Element of REAL
tv . h1 is Element of the carrier of Y
||.(tv . h1).|| is V11() real ext-real Element of REAL
the normF of Y . (tv . h1) is V11() real ext-real Element of REAL
||.(m . h1).|| + ||.(tv . h1).|| is V11() real ext-real Element of REAL
tseq . h1 is Element of the carrier of Y
||.(tseq . h1).|| is V11() real ext-real Element of REAL
the normF of Y . (tseq . h1) is V11() real ext-real Element of REAL
(tv . h1) + (m . h1) 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 V23([: the carrier of Y, the carrier of Y:]) 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 Relation-like set
[:[: the carrier of Y, the carrier of Y:], the carrier of Y:] is non empty Relation-like set
bool [:[: the carrier of Y, the carrier of Y:], the carrier of Y:] is non empty set
K602( the carrier of Y, the addF of Y,(tv . h1),(m . h1)) is Element of the carrier of Y
[(tv . h1),(m . h1)] is set
{(tv . h1),(m . h1)} is set
{(tv . h1)} is set
{{(tv . h1),(m . h1)},{(tv . h1)}} is set
the addF of Y . [(tv . h1),(m . h1)] is set
||.((tv . h1) + (m . h1)).|| is V11() real ext-real Element of REAL
the normF of Y . ((tv . h1) + (m . h1)) is V11() real ext-real Element of REAL
(m . h1) + (tv . h1) is Element of the carrier of Y
K602( the carrier of Y, the addF of Y,(m . h1),(tv . h1)) is Element of the carrier of Y
[(m . h1),(tv . h1)] is set
{(m . h1),(tv . h1)} is set
{(m . h1)} is set
{{(m . h1),(tv . h1)},{(m . h1)}} is set
the addF of Y . [(m . h1),(tv . h1)] is set
||.((m . h1) + (tv . h1)).|| is V11() real ext-real Element of REAL
the normF of Y . ((m . h1) + (tv . h1)) is V11() real ext-real Element of REAL
h1 is Element of X
tseq . h1 is Element of the carrier of Y
||.(tseq . h1).|| is V11() real ext-real Element of REAL
the normF of Y . (tseq . h1) is V11() real ext-real Element of REAL
f is V11() set
tseq is Element of the carrier of (ComplexVectSpace (X,Y))
f * tseq is Element of the carrier of (ComplexVectSpace (X,Y))
[:X, the carrier of Y:] is non empty Relation-like set
bool [:X, the carrier of Y:] is non empty set
tseq is non empty Relation-like X -defined the carrier of Y -valued Function-like V23(X) quasi_total Element of bool [:X, the carrier of Y:]
tv is non empty Relation-like X -defined the carrier of Y -valued Function-like V23(X) quasi_total (X,Y) Element of bool [:X, the carrier of Y:]
e is V11() real ext-real Element of REAL
|.f.| is V11() real ext-real Element of REAL
|.f.| * e is V11() real ext-real Element of REAL
m is Element of X
tseq . m is Element of the carrier of Y
||.(tseq . m).|| is V11() real ext-real Element of REAL
the normF of Y is non empty Relation-like the carrier of Y -defined REAL -valued Function-like V23( the carrier of Y) quasi_total V33() V34() V35() Element of bool [: the carrier of Y,REAL:]
[: the carrier of Y,REAL:] is non empty Relation-like V33() V34() V35() set
bool [: the carrier of Y,REAL:] is non empty set
the normF of Y . (tseq . m) is V11() real ext-real Element of REAL
tv . m is Element of the carrier of Y
f * (tv . m) is Element of the carrier of Y
||.(f * (tv . m)).|| is V11() real ext-real Element of REAL
the normF of Y . (f * (tv . m)) is V11() real ext-real Element of REAL
||.(tv . m).|| is V11() real ext-real Element of REAL
the normF of Y . (tv . m) is V11() real ext-real Element of REAL
|.f.| * ||.(tv . m).|| is V11() real ext-real Element of REAL
m is Element of X
tseq . m is Element of the carrier of Y
||.(tseq . m).|| is V11() real ext-real Element of REAL
the normF of Y . (tseq . m) is V11() real ext-real Element of REAL
X is non empty set
Y is non empty right_complementable Abelian add-associative right_zeroed discerning reflexive vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like CNORMSTR
(X,Y) is non empty Element of bool the carrier of (ComplexVectSpace (X,Y))
ComplexVectSpace (X,Y) is non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital CLSStruct
the carrier of Y is non empty set
Funcs (X, the carrier of Y) is non empty functional FUNCTION_DOMAIN of X, the carrier of Y
K340(X,Y) is Relation-like X -defined the carrier of Y -valued Function-like V23(X) quasi_total Element of Funcs (X, the carrier of Y)
K338(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 V23([:(Funcs (X, the carrier of Y)),(Funcs (X, the carrier of Y)):]) quasi_total Function-yielding V135() 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 Relation-like set
[:[:(Funcs (X, the carrier of Y)),(Funcs (X, the carrier of Y)):],(Funcs (X, the carrier of Y)):] is non empty Relation-like set
bool [:[:(Funcs (X, the carrier of Y)),(Funcs (X, the carrier of Y)):],(Funcs (X, the carrier of Y)):] is non empty set
FuncExtMult (X,Y) is non empty Relation-like [:COMPLEX,(Funcs (X, the carrier of Y)):] -defined Funcs (X, the carrier of Y) -valued Function-like V23([:COMPLEX,(Funcs (X, the carrier of Y)):]) quasi_total Function-yielding V135() Element of bool [:[:COMPLEX,(Funcs (X, the carrier of Y)):],(Funcs (X, the carrier of Y)):]
[:COMPLEX,(Funcs (X, the carrier of Y)):] is non empty Relation-like set
[:[:COMPLEX,(Funcs (X, the carrier of Y)):],(Funcs (X, the carrier of Y)):] is non empty Relation-like set
bool [:[:COMPLEX,(Funcs (X, the carrier of Y)):],(Funcs (X, the carrier of Y)):] is non empty set
CLSStruct(# (Funcs (X, the carrier of Y)),K340(X,Y),K338(X,Y),(FuncExtMult (X,Y)) #) is non empty strict CLSStruct
the carrier of (ComplexVectSpace (X,Y)) is non empty set
bool the carrier of (ComplexVectSpace (X,Y)) is non empty set
Zero_ ((X,Y),(ComplexVectSpace (X,Y))) is Element of (X,Y)
Add_ ((X,Y),(ComplexVectSpace (X,Y))) is non empty Relation-like [:(X,Y),(X,Y):] -defined (X,Y) -valued Function-like V23([:(X,Y),(X,Y):]) quasi_total Element of bool [:[:(X,Y),(X,Y):],(X,Y):]
[:(X,Y),(X,Y):] is non empty Relation-like set
[:[:(X,Y),(X,Y):],(X,Y):] is non empty Relation-like set
bool [:[:(X,Y),(X,Y):],(X,Y):] is non empty set
Mult_ ((X,Y),(ComplexVectSpace (X,Y))) is non empty Relation-like [:COMPLEX,(X,Y):] -defined (X,Y) -valued Function-like V23([:COMPLEX,(X,Y):]) quasi_total Element of bool [:[:COMPLEX,(X,Y):],(X,Y):]
[:COMPLEX,(X,Y):] is non empty Relation-like set
[:[:COMPLEX,(X,Y):],(X,Y):] is non empty Relation-like set
bool [:[:COMPLEX,(X,Y):],(X,Y):] is non empty set
CLSStruct(# (X,Y),(Zero_ ((X,Y),(ComplexVectSpace (X,Y)))),(Add_ ((X,Y),(ComplexVectSpace (X,Y)))),(Mult_ ((X,Y),(ComplexVectSpace (X,Y)))) #) is non empty strict CLSStruct
X is non empty set
Y is non empty right_complementable Abelian add-associative right_zeroed discerning reflexive vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like CNORMSTR
(X,Y) is non empty Element of bool the carrier of (ComplexVectSpace (X,Y))
ComplexVectSpace (X,Y) is non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital CLSStruct
the carrier of Y is non empty set
Funcs (X, the carrier of Y) is non empty functional FUNCTION_DOMAIN of X, the carrier of Y
K340(X,Y) is Relation-like X -defined the carrier of Y -valued Function-like V23(X) quasi_total Element of Funcs (X, the carrier of Y)
K338(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 V23([:(Funcs (X, the carrier of Y)),(Funcs (X, the carrier of Y)):]) quasi_total Function-yielding V135() 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 Relation-like set
[:[:(Funcs (X, the carrier of Y)),(Funcs (X, the carrier of Y)):],(Funcs (X, the carrier of Y)):] is non empty Relation-like set
bool [:[:(Funcs (X, the carrier of Y)),(Funcs (X, the carrier of Y)):],(Funcs (X, the carrier of Y)):] is non empty set
FuncExtMult (X,Y) is non empty Relation-like [:COMPLEX,(Funcs (X, the carrier of Y)):] -defined Funcs (X, the carrier of Y) -valued Function-like V23([:COMPLEX,(Funcs (X, the carrier of Y)):]) quasi_total Function-yielding V135() Element of bool [:[:COMPLEX,(Funcs (X, the carrier of Y)):],(Funcs (X, the carrier of Y)):]
[:COMPLEX,(Funcs (X, the carrier of Y)):] is non empty Relation-like set
[:[:COMPLEX,(Funcs (X, the carrier of Y)):],(Funcs (X, the carrier of Y)):] is non empty Relation-like set
bool [:[:COMPLEX,(Funcs (X, the carrier of Y)):],(Funcs (X, the carrier of Y)):] is non empty set
CLSStruct(# (Funcs (X, the carrier of Y)),K340(X,Y),K338(X,Y),(FuncExtMult (X,Y)) #) is non empty strict CLSStruct
the carrier of (ComplexVectSpace (X,Y)) is non empty set
bool the carrier of (ComplexVectSpace (X,Y)) is non empty set
Zero_ ((X,Y),(ComplexVectSpace (X,Y))) is Element of (X,Y)
Add_ ((X,Y),(ComplexVectSpace (X,Y))) is non empty Relation-like [:(X,Y),(X,Y):] -defined (X,Y) -valued Function-like V23([:(X,Y),(X,Y):]) quasi_total Element of bool [:[:(X,Y),(X,Y):],(X,Y):]
[:(X,Y),(X,Y):] is non empty Relation-like set
[:[:(X,Y),(X,Y):],(X,Y):] is non empty Relation-like set
bool [:[:(X,Y),(X,Y):],(X,Y):] is non empty set
Mult_ ((X,Y),(ComplexVectSpace (X,Y))) is non empty Relation-like [:COMPLEX,(X,Y):] -defined (X,Y) -valued Function-like V23([:COMPLEX,(X,Y):]) quasi_total Element of bool [:[:COMPLEX,(X,Y):],(X,Y):]
[:COMPLEX,(X,Y):] is non empty Relation-like set
[:[:COMPLEX,(X,Y):],(X,Y):] is non empty Relation-like set
bool [:[:COMPLEX,(X,Y):],(X,Y):] is non empty set
CLSStruct(# (X,Y),(Zero_ ((X,Y),(ComplexVectSpace (X,Y)))),(Add_ ((X,Y),(ComplexVectSpace (X,Y)))),(Mult_ ((X,Y),(ComplexVectSpace (X,Y)))) #) is non empty strict CLSStruct
X is non empty set
Y is non empty right_complementable Abelian add-associative right_zeroed discerning reflexive vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like CNORMSTR
(X,Y) is non empty Element of bool the carrier of (ComplexVectSpace (X,Y))
ComplexVectSpace (X,Y) is non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital CLSStruct
the carrier of Y is non empty set
Funcs (X, the carrier of Y) is non empty functional FUNCTION_DOMAIN of X, the carrier of Y
K340(X,Y) is Relation-like X -defined the carrier of Y -valued Function-like V23(X) quasi_total Element of Funcs (X, the carrier of Y)
K338(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 V23([:(Funcs (X, the carrier of Y)),(Funcs (X, the carrier of Y)):]) quasi_total Function-yielding V135() 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 Relation-like set
[:[:(Funcs (X, the carrier of Y)),(Funcs (X, the carrier of Y)):],(Funcs (X, the carrier of Y)):] is non empty Relation-like set
bool [:[:(Funcs (X, the carrier of Y)),(Funcs (X, the carrier of Y)):],(Funcs (X, the carrier of Y)):] is non empty set
FuncExtMult (X,Y) is non empty Relation-like [:COMPLEX,(Funcs (X, the carrier of Y)):] -defined Funcs (X, the carrier of Y) -valued Function-like V23([:COMPLEX,(Funcs (X, the carrier of Y)):]) quasi_total Function-yielding V135() Element of bool [:[:COMPLEX,(Funcs (X, the carrier of Y)):],(Funcs (X, the carrier of Y)):]
[:COMPLEX,(Funcs (X, the carrier of Y)):] is non empty Relation-like set
[:[:COMPLEX,(Funcs (X, the carrier of Y)):],(Funcs (X, the carrier of Y)):] is non empty Relation-like set
bool [:[:COMPLEX,(Funcs (X, the carrier of Y)):],(Funcs (X, the carrier of Y)):] is non empty set
CLSStruct(# (Funcs (X, the carrier of Y)),K340(X,Y),K338(X,Y),(FuncExtMult (X,Y)) #) is non empty strict CLSStruct
the carrier of (ComplexVectSpace (X,Y)) is non empty set
bool the carrier of (ComplexVectSpace (X,Y)) is non empty set
Zero_ ((X,Y),(ComplexVectSpace (X,Y))) is Element of (X,Y)
Add_ ((X,Y),(ComplexVectSpace (X,Y))) is non empty Relation-like [:(X,Y),(X,Y):] -defined (X,Y) -valued Function-like V23([:(X,Y),(X,Y):]) quasi_total Element of bool [:[:(X,Y),(X,Y):],(X,Y):]
[:(X,Y),(X,Y):] is non empty Relation-like set
[:[:(X,Y),(X,Y):],(X,Y):] is non empty Relation-like set
bool [:[:(X,Y),(X,Y):],(X,Y):] is non empty set
Mult_ ((X,Y),(ComplexVectSpace (X,Y))) is non empty Relation-like [:COMPLEX,(X,Y):] -defined (X,Y) -valued Function-like V23([:COMPLEX,(X,Y):]) quasi_total Element of bool [:[:COMPLEX,(X,Y):],(X,Y):]
[:COMPLEX,(X,Y):] is non empty Relation-like set
[:[:COMPLEX,(X,Y):],(X,Y):] is non empty Relation-like set
bool [:[:COMPLEX,(X,Y):],(X,Y):] is non empty set
CLSStruct(# (X,Y),(Zero_ ((X,Y),(ComplexVectSpace (X,Y)))),(Add_ ((X,Y),(ComplexVectSpace (X,Y)))),(Mult_ ((X,Y),(ComplexVectSpace (X,Y)))) #) is non empty right_complementable Abelian add-associative right_zeroed strict vector-distributive scalar-distributive scalar-associative scalar-unital CLSStruct
X is non empty set
Y is non empty right_complementable Abelian add-associative right_zeroed discerning reflexive vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like CNORMSTR
(X,Y) is non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital CLSStruct
(X,Y) is non empty Element of bool the carrier of (ComplexVectSpace (X,Y))
ComplexVectSpace (X,Y) is non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital CLSStruct
the carrier of Y is non empty set
Funcs (X, the carrier of Y) is non empty functional FUNCTION_DOMAIN of X, the carrier of Y
K340(X,Y) is Relation-like X -defined the carrier of Y -valued Function-like V23(X) quasi_total Element of Funcs (X, the carrier of Y)
K338(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 V23([:(Funcs (X, the carrier of Y)),(Funcs (X, the carrier of Y)):]) quasi_total Function-yielding V135() 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 Relation-like set
[:[:(Funcs (X, the carrier of Y)),(Funcs (X, the carrier of Y)):],(Funcs (X, the carrier of Y)):] is non empty Relation-like set
bool [:[:(Funcs (X, the carrier of Y)),(Funcs (X, the carrier of Y)):],(Funcs (X, the carrier of Y)):] is non empty set
FuncExtMult (X,Y) is non empty Relation-like [:COMPLEX,(Funcs (X, the carrier of Y)):] -defined Funcs (X, the carrier of Y) -valued Function-like V23([:COMPLEX,(Funcs (X, the carrier of Y)):]) quasi_total Function-yielding V135() Element of bool [:[:COMPLEX,(Funcs (X, the carrier of Y)):],(Funcs (X, the carrier of Y)):]
[:COMPLEX,(Funcs (X, the carrier of Y)):] is non empty Relation-like set
[:[:COMPLEX,(Funcs (X, the carrier of Y)):],(Funcs (X, the carrier of Y)):] is non empty Relation-like set
bool [:[:COMPLEX,(Funcs (X, the carrier of Y)):],(Funcs (X, the carrier of Y)):] is non empty set
CLSStruct(# (Funcs (X, the carrier of Y)),K340(X,Y),K338(X,Y),(FuncExtMult (X,Y)) #) is non empty strict CLSStruct
the carrier of (ComplexVectSpace (X,Y)) is non empty set
bool the carrier of (ComplexVectSpace (X,Y)) is non empty set
Zero_ ((X,Y),(ComplexVectSpace (X,Y))) is Element of (X,Y)
Add_ ((X,Y),(ComplexVectSpace (X,Y))) is non empty Relation-like [:(X,Y),(X,Y):] -defined (X,Y) -valued Function-like V23([:(X,Y),(X,Y):]) quasi_total Element of bool [:[:(X,Y),(X,Y):],(X,Y):]
[:(X,Y),(X,Y):] is non empty Relation-like set
[:[:(X,Y),(X,Y):],(X,Y):] is non empty Relation-like set
bool [:[:(X,Y),(X,Y):],(X,Y):] is non empty set
Mult_ ((X,Y),(ComplexVectSpace (X,Y))) is non empty Relation-like [:COMPLEX,(X,Y):] -defined (X,Y) -valued Function-like V23([:COMPLEX,(X,Y):]) quasi_total Element of bool [:[:COMPLEX,(X,Y):],(X,Y):]
[:COMPLEX,(X,Y):] is non empty Relation-like set
[:[:COMPLEX,(X,Y):],(X,Y):] is non empty Relation-like set
bool [:[:COMPLEX,(X,Y):],(X,Y):] is non empty set
CLSStruct(# (X,Y),(Zero_ ((X,Y),(ComplexVectSpace (X,Y)))),(Add_ ((X,Y),(ComplexVectSpace (X,Y)))),(Mult_ ((X,Y),(ComplexVectSpace (X,Y)))) #) is non empty right_complementable Abelian add-associative right_zeroed strict vector-distributive scalar-distributive scalar-associative scalar-unital CLSStruct
X is non empty set
Y is non empty right_complementable Abelian add-associative right_zeroed discerning reflexive vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like CNORMSTR
(X,Y) is non empty right_complementable Abelian add-associative right_zeroed strict vector-distributive scalar-distributive scalar-associative scalar-unital CLSStruct
(X,Y) is non empty Element of bool the carrier of (ComplexVectSpace (X,Y))
ComplexVectSpace (X,Y) is non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital CLSStruct
the carrier of Y is non empty set
Funcs (X, the carrier of Y) is non empty functional FUNCTION_DOMAIN of X, the carrier of Y
K340(X,Y) is Relation-like X -defined the carrier of Y -valued Function-like V23(X) quasi_total Element of Funcs (X, the carrier of Y)
K338(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 V23([:(Funcs (X, the carrier of Y)),(Funcs (X, the carrier of Y)):]) quasi_total Function-yielding V135() 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 Relation-like set
[:[:(Funcs (X, the carrier of Y)),(Funcs (X, the carrier of Y)):],(Funcs (X, the carrier of Y)):] is non empty Relation-like set
bool [:[:(Funcs (X, the carrier of Y)),(Funcs (X, the carrier of Y)):],(Funcs (X, the carrier of Y)):] is non empty set
FuncExtMult (X,Y) is non empty Relation-like [:COMPLEX,(Funcs (X, the carrier of Y)):] -defined Funcs (X, the carrier of Y) -valued Function-like V23([:COMPLEX,(Funcs (X, the carrier of Y)):]) quasi_total Function-yielding V135() Element of bool [:[:COMPLEX,(Funcs (X, the carrier of Y)):],(Funcs (X, the carrier of Y)):]
[:COMPLEX,(Funcs (X, the carrier of Y)):] is non empty Relation-like set
[:[:COMPLEX,(Funcs (X, the carrier of Y)):],(Funcs (X, the carrier of Y)):] is non empty Relation-like set
bool [:[:COMPLEX,(Funcs (X, the carrier of Y)):],(Funcs (X, the carrier of Y)):] is non empty set
CLSStruct(# (Funcs (X, the carrier of Y)),K340(X,Y),K338(X,Y),(FuncExtMult (X,Y)) #) is non empty strict CLSStruct
the carrier of (ComplexVectSpace (X,Y)) is non empty set
bool the carrier of (ComplexVectSpace (X,Y)) is non empty set
Zero_ ((X,Y),(ComplexVectSpace (X,Y))) is Element of (X,Y)
Add_ ((X,Y),(ComplexVectSpace (X,Y))) is non empty Relation-like [:(X,Y),(X,Y):] -defined (X,Y) -valued Function-like V23([:(X,Y),(X,Y):]) quasi_total Element of bool [:[:(X,Y),(X,Y):],(X,Y):]
[:(X,Y),(X,Y):] is non empty Relation-like set
[:[:(X,Y),(X,Y):],(X,Y):] is non empty Relation-like set
bool [:[:(X,Y),(X,Y):],(X,Y):] is non empty set
Mult_ ((X,Y),(ComplexVectSpace (X,Y))) is non empty Relation-like [:COMPLEX,(X,Y):] -defined (X,Y) -valued Function-like V23([:COMPLEX,(X,Y):]) quasi_total Element of bool [:[:COMPLEX,(X,Y):],(X,Y):]
[:COMPLEX,(X,Y):] is non empty Relation-like set
[:[:COMPLEX,(X,Y):],(X,Y):] is non empty Relation-like set
bool [:[:COMPLEX,(X,Y):],(X,Y):] is non empty set
CLSStruct(# (X,Y),(Zero_ ((X,Y),(ComplexVectSpace (X,Y)))),(Add_ ((X,Y),(ComplexVectSpace (X,Y)))),(Mult_ ((X,Y),(ComplexVectSpace (X,Y)))) #) is non empty right_complementable Abelian add-associative right_zeroed strict vector-distributive scalar-distributive scalar-associative scalar-unital CLSStruct
the carrier of (X,Y) is non empty set
[:X, the carrier of Y:] is non empty Relation-like set
bool [:X, the carrier of Y:] is non empty set
vseq is Element of the carrier of (X,Y)
f is Element of the carrier of (X,Y)
tseq is Element of the carrier of (X,Y)
vseq + f is 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 V23([: the carrier of (X,Y), the carrier of (X,Y):]) 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 Relation-like set
[:[: the carrier of (X,Y), the carrier of (X,Y):], the carrier of (X,Y):] is non empty Relation-like set
bool [:[: the carrier of (X,Y), the carrier of (X,Y):], the carrier of (X,Y):] is non empty set
K602( the carrier of (X,Y), the addF of (X,Y),vseq,f) is Element of the carrier of (X,Y)
[vseq,f] is set
{vseq,f} is set
{vseq} is set
{{vseq,f},{vseq}} is set
the addF of (X,Y) . [vseq,f] is set
m is non empty Relation-like X -defined the carrier of Y -valued Function-like V23(X) quasi_total (X,Y) Element of bool [:X, the carrier of Y:]
n is non empty Relation-like X -defined the carrier of Y -valued Function-like V23(X) quasi_total (X,Y) Element of bool [:X, the carrier of Y:]
n is non empty Relation-like X -defined the carrier of Y -valued Function-like V23(X) quasi_total (X,Y) Element of bool [:X, the carrier of Y:]
tv is Element of the carrier of (ComplexVectSpace (X,Y))
tseq is Element of the carrier of (ComplexVectSpace (X,Y))
e is Element of the carrier of (ComplexVectSpace (X,Y))
tseq + e is Element of the carrier of (ComplexVectSpace (X,Y))
the addF of (ComplexVectSpace (X,Y)) is non empty Relation-like [: the carrier of (ComplexVectSpace (X,Y)), the carrier of (ComplexVectSpace (X,Y)):] -defined the carrier of (ComplexVectSpace (X,Y)) -valued Function-like V23([: the carrier of (ComplexVectSpace (X,Y)), the carrier of (ComplexVectSpace (X,Y)):]) quasi_total Element of bool [:[: the carrier of (ComplexVectSpace (X,Y)), the carrier of (ComplexVectSpace (X,Y)):], the carrier of (ComplexVectSpace (X,Y)):]
[: the carrier of (ComplexVectSpace (X,Y)), the carrier of (ComplexVectSpace (X,Y)):] is non empty Relation-like set
[:[: the carrier of (ComplexVectSpace (X,Y)), the carrier of (ComplexVectSpace (X,Y)):], the carrier of (ComplexVectSpace (X,Y)):] is non empty Relation-like set
bool [:[: the carrier of (ComplexVectSpace (X,Y)), the carrier of (ComplexVectSpace (X,Y)):], the carrier of (ComplexVectSpace (X,Y)):] is non empty set
K602( the carrier of (ComplexVectSpace (X,Y)), the addF of (ComplexVectSpace (X,Y)),tseq,e) is Element of the carrier of (ComplexVectSpace (X,Y))
[tseq,e] is set
{tseq,e} is set
{tseq} is set
{{tseq,e},{tseq}} is set
the addF of (ComplexVectSpace (X,Y)) . [tseq,e] is set
h1 is Element of X
n . h1 is Element of the carrier of Y
m . h1 is Element of the carrier of Y
n . h1 is Element of the carrier of Y
(m . h1) + (n . h1) 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 V23([: the carrier of Y, the carrier of Y:]) 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 Relation-like set
[:[: the carrier of Y, the carrier of Y:], the carrier of Y:] is non empty Relation-like set
bool [:[: the carrier of Y, the carrier of Y:], the carrier of Y:] is non empty set
K602( the carrier of Y, the addF of Y,(m . h1),(n . h1)) is Element of the carrier of Y
[(m . h1),(n . h1)] is set
{(m . h1),(n . h1)} is set
{(m . h1)} is set
{{(m . h1),(n . h1)},{(m . h1)}} is set
the addF of Y . [(m . h1),(n . h1)] is set
h1 is Element of X
n . h1 is Element of the carrier of Y
m . h1 is Element of the carrier of Y
n . h1 is Element of the carrier of Y
(m . h1) + (n . h1) is Element of the carrier of Y
K602( the carrier of Y, the addF of Y,(m . h1),(n . h1)) is Element of the carrier of Y
[(m . h1),(n . h1)] is set
{(m . h1),(n . h1)} is set
{(m . h1)} is set
{{(m . h1),(n . h1)},{(m . h1)}} is set
the addF of Y . [(m . h1),(n . h1)] is set
f1 is Element of X
n . f1 is Element of the carrier of Y
m . f1 is Element of the carrier of Y
n . f1 is Element of the carrier of Y
(m . f1) + (n . f1) is Element of the carrier of Y
K602( the carrier of Y, the addF of Y,(m . f1),(n . f1)) is Element of the carrier of Y
[(m . f1),(n . f1)] is set
{(m . f1),(n . f1)} is set
{(m . f1)} is set
{{(m . f1),(n . f1)},{(m . f1)}} is set
the addF of Y . [(m . f1),(n . f1)] is set
X is non empty set
Y is non empty right_complementable Abelian add-associative right_zeroed discerning reflexive vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like CNORMSTR
(X,Y) is non empty right_complementable Abelian add-associative right_zeroed strict vector-distributive scalar-distributive scalar-associative scalar-unital CLSStruct
(X,Y) is non empty Element of bool the carrier of (ComplexVectSpace (X,Y))
ComplexVectSpace (X,Y) is non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital CLSStruct
the carrier of Y is non empty set
Funcs (X, the carrier of Y) is non empty functional FUNCTION_DOMAIN of X, the carrier of Y
K340(X,Y) is Relation-like X -defined the carrier of Y -valued Function-like V23(X) quasi_total Element of Funcs (X, the carrier of Y)
K338(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 V23([:(Funcs (X, the carrier of Y)),(Funcs (X, the carrier of Y)):]) quasi_total Function-yielding V135() 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 Relation-like set
[:[:(Funcs (X, the carrier of Y)),(Funcs (X, the carrier of Y)):],(Funcs (X, the carrier of Y)):] is non empty Relation-like set
bool [:[:(Funcs (X, the carrier of Y)),(Funcs (X, the carrier of Y)):],(Funcs (X, the carrier of Y)):] is non empty set
FuncExtMult (X,Y) is non empty Relation-like [:COMPLEX,(Funcs (X, the carrier of Y)):] -defined Funcs (X, the carrier of Y) -valued Function-like V23([:COMPLEX,(Funcs (X, the carrier of Y)):]) quasi_total Function-yielding V135() Element of bool [:[:COMPLEX,(Funcs (X, the carrier of Y)):],(Funcs (X, the carrier of Y)):]
[:COMPLEX,(Funcs (X, the carrier of Y)):] is non empty Relation-like set
[:[:COMPLEX,(Funcs (X, the carrier of Y)):],(Funcs (X, the carrier of Y)):] is non empty Relation-like set
bool [:[:COMPLEX,(Funcs (X, the carrier of Y)):],(Funcs (X, the carrier of Y)):] is non empty set
CLSStruct(# (Funcs (X, the carrier of Y)),K340(X,Y),K338(X,Y),(FuncExtMult (X,Y)) #) is non empty strict CLSStruct
the carrier of (ComplexVectSpace (X,Y)) is non empty set
bool the carrier of (ComplexVectSpace (X,Y)) is non empty set
Zero_ ((X,Y),(ComplexVectSpace (X,Y))) is Element of (X,Y)
Add_ ((X,Y),(ComplexVectSpace (X,Y))) is non empty Relation-like [:(X,Y),(X,Y):] -defined (X,Y) -valued Function-like V23([:(X,Y),(X,Y):]) quasi_total Element of bool [:[:(X,Y),(X,Y):],(X,Y):]
[:(X,Y),(X,Y):] is non empty Relation-like set
[:[:(X,Y),(X,Y):],(X,Y):] is non empty Relation-like set
bool [:[:(X,Y),(X,Y):],(X,Y):] is non empty set
Mult_ ((X,Y),(ComplexVectSpace (X,Y))) is non empty Relation-like [:COMPLEX,(X,Y):] -defined (X,Y) -valued Function-like V23([:COMPLEX,(X,Y):]) quasi_total Element of bool [:[:COMPLEX,(X,Y):],(X,Y):]
[:COMPLEX,(X,Y):] is non empty Relation-like set
[:[:COMPLEX,(X,Y):],(X,Y):] is non empty Relation-like set
bool [:[:COMPLEX,(X,Y):],(X,Y):] is non empty set
CLSStruct(# (X,Y),(Zero_ ((X,Y),(ComplexVectSpace (X,Y)))),(Add_ ((X,Y),(ComplexVectSpace (X,Y)))),(Mult_ ((X,Y),(ComplexVectSpace (X,Y)))) #) is non empty right_complementable Abelian add-associative right_zeroed strict vector-distributive scalar-distributive scalar-associative scalar-unital CLSStruct
the carrier of (X,Y) is non empty set
[:X, the carrier of Y:] is non empty Relation-like set
bool [:X, the carrier of Y:] is non empty set
vseq is Element of the carrier of (X,Y)
f is Element of the carrier of (X,Y)
tseq is non empty Relation-like X -defined the carrier of Y -valued Function-like V23(X) quasi_total (X,Y) Element of bool [:X, the carrier of Y:]
tseq is non empty Relation-like X -defined the carrier of Y -valued Function-like V23(X) quasi_total (X,Y) Element of bool [:X, the carrier of Y:]
tv is V11() set
tv * vseq is Element of the carrier of (X,Y)
m is Element of the carrier of (ComplexVectSpace (X,Y))
e is Element of the carrier of (ComplexVectSpace (X,Y))
tv * e is Element of the carrier of (ComplexVectSpace (X,Y))
n is Element of X
tseq . n is Element of the carrier of Y
tseq . n is Element of the carrier of Y
tv * (tseq . n) is Element of the carrier of Y
n is Element of X
tseq . n is Element of the carrier of Y
tseq . n is Element of the carrier of Y
tv * (tseq . n) is Element of the carrier of Y
n is Element of X
tseq . n is Element of the carrier of Y
tseq . n is Element of the carrier of Y
tv * (tseq . n) is Element of the carrier of Y
X is non empty set
Y is non empty right_complementable Abelian add-associative right_zeroed discerning reflexive vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like CNORMSTR
(X,Y) is non empty right_complementable Abelian add-associative right_zeroed strict vector-distributive scalar-distributive scalar-associative scalar-unital CLSStruct
(X,Y) is non empty Element of bool the carrier of (ComplexVectSpace (X,Y))
ComplexVectSpace (X,Y) is non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital CLSStruct
the carrier of Y is non empty set
Funcs (X, the carrier of Y) is non empty functional FUNCTION_DOMAIN of X, the carrier of Y
K340(X,Y) is Relation-like X -defined the carrier of Y -valued Function-like V23(X) quasi_total Element of Funcs (X, the carrier of Y)
K338(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 V23([:(Funcs (X, the carrier of Y)),(Funcs (X, the carrier of Y)):]) quasi_total Function-yielding V135() 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 Relation-like set
[:[:(Funcs (X, the carrier of Y)),(Funcs (X, the carrier of Y)):],(Funcs (X, the carrier of Y)):] is non empty Relation-like set
bool [:[:(Funcs (X, the carrier of Y)),(Funcs (X, the carrier of Y)):],(Funcs (X, the carrier of Y)):] is non empty set
FuncExtMult (X,Y) is non empty Relation-like [:COMPLEX,(Funcs (X, the carrier of Y)):] -defined Funcs (X, the carrier of Y) -valued Function-like V23([:COMPLEX,(Funcs (X, the carrier of Y)):]) quasi_total Function-yielding V135() Element of bool [:[:COMPLEX,(Funcs (X, the carrier of Y)):],(Funcs (X, the carrier of Y)):]
[:COMPLEX,(Funcs (X, the carrier of Y)):] is non empty Relation-like set
[:[:COMPLEX,(Funcs (X, the carrier of Y)):],(Funcs (X, the carrier of Y)):] is non empty Relation-like set
bool [:[:COMPLEX,(Funcs (X, the carrier of Y)):],(Funcs (X, the carrier of Y)):] is non empty set
CLSStruct(# (Funcs (X, the carrier of Y)),K340(X,Y),K338(X,Y),(FuncExtMult (X,Y)) #) is non empty strict CLSStruct
the carrier of (ComplexVectSpace (X,Y)) is non empty set
bool the carrier of (ComplexVectSpace (X,Y)) is non empty set
Zero_ ((X,Y),(ComplexVectSpace (X,Y))) is Element of (X,Y)
Add_ ((X,Y),(ComplexVectSpace (X,Y))) is non empty Relation-like [:(X,Y),(X,Y):] -defined (X,Y) -valued Function-like V23([:(X,Y),(X,Y):]) quasi_total Element of bool [:[:(X,Y),(X,Y):],(X,Y):]
[:(X,Y),(X,Y):] is non empty Relation-like set
[:[:(X,Y),(X,Y):],(X,Y):] is non empty Relation-like set
bool [:[:(X,Y),(X,Y):],(X,Y):] is non empty set
Mult_ ((X,Y),(ComplexVectSpace (X,Y))) is non empty Relation-like [:COMPLEX,(X,Y):] -defined (X,Y) -valued Function-like V23([:COMPLEX,(X,Y):]) quasi_total Element of bool [:[:COMPLEX,(X,Y):],(X,Y):]
[:COMPLEX,(X,Y):] is non empty Relation-like set
[:[:COMPLEX,(X,Y):],(X,Y):] is non empty Relation-like set
bool [:[:COMPLEX,(X,Y):],(X,Y):] is non empty set
CLSStruct(# (X,Y),(Zero_ ((X,Y),(ComplexVectSpace (X,Y)))),(Add_ ((X,Y),(ComplexVectSpace (X,Y)))),(Mult_ ((X,Y),(ComplexVectSpace (X,Y)))) #) is non empty right_complementable Abelian add-associative right_zeroed strict vector-distributive scalar-distributive scalar-associative scalar-unital CLSStruct
0. (X,Y) is V82((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)
0. Y is V82(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 V23(X) quasi_total Element of bool [:X, the carrier of Y:]
[:X, the carrier of Y:] is non empty Relation-like set
bool [:X, the carrier of Y:] is non empty set
0. (ComplexVectSpace (X,Y)) is V82( ComplexVectSpace (X,Y)) Element of the carrier of (ComplexVectSpace (X,Y))
the ZeroF of (ComplexVectSpace (X,Y)) is Element of the carrier of (ComplexVectSpace (X,Y))
vseq is set
X is non empty set
Y is non empty right_complementable Abelian add-associative right_zeroed discerning reflexive vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like CNORMSTR
(X,Y) is non empty Element of bool the carrier of (ComplexVectSpace (X,Y))
ComplexVectSpace (X,Y) is non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital CLSStruct
the carrier of Y is non empty set
Funcs (X, the carrier of Y) is non empty functional FUNCTION_DOMAIN of X, the carrier of Y
K340(X,Y) is Relation-like X -defined the carrier of Y -valued Function-like V23(X) quasi_total Element of Funcs (X, the carrier of Y)
K338(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 V23([:(Funcs (X, the carrier of Y)),(Funcs (X, the carrier of Y)):]) quasi_total Function-yielding V135() 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 Relation-like set
[:[:(Funcs (X, the carrier of Y)),(Funcs (X, the carrier of Y)):],(Funcs (X, the carrier of Y)):] is non empty Relation-like set
bool [:[:(Funcs (X, the carrier of Y)),(Funcs (X, the carrier of Y)):],(Funcs (X, the carrier of Y)):] is non empty set
FuncExtMult (X,Y) is non empty Relation-like [:COMPLEX,(Funcs (X, the carrier of Y)):] -defined Funcs (X, the carrier of Y) -valued Function-like V23([:COMPLEX,(Funcs (X, the carrier of Y)):]) quasi_total Function-yielding V135() Element of bool [:[:COMPLEX,(Funcs (X, the carrier of Y)):],(Funcs (X, the carrier of Y)):]
[:COMPLEX,(Funcs (X, the carrier of Y)):] is non empty Relation-like set
[:[:COMPLEX,(Funcs (X, the carrier of Y)):],(Funcs (X, the carrier of Y)):] is non empty Relation-like set
bool [:[:COMPLEX,(Funcs (X, the carrier of Y)):],(Funcs (X, the carrier of Y)):] is non empty set
CLSStruct(# (Funcs (X, the carrier of Y)),K340(X,Y),K338(X,Y),(FuncExtMult (X,Y)) #) is non empty strict CLSStruct
the carrier of (ComplexVectSpace (X,Y)) is non empty set
bool the carrier of (ComplexVectSpace (X,Y)) is non empty set
[:X, the carrier of Y:] is non empty Relation-like set
bool [:X, the carrier of Y:] is non empty set
X is non empty set
Y is non empty right_complementable Abelian add-associative right_zeroed discerning reflexive vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like CNORMSTR
the carrier of Y is non empty set
[:X, the carrier of Y:] is non empty Relation-like set
bool [:X, the carrier of Y:] is non empty set
vseq is non empty Relation-like X -defined the carrier of Y -valued Function-like V23(X) quasi_total Element of bool [:X, the carrier of Y:]
{ ||.(vseq . b₁).|| where b₁ is Element of X : verum } is set
the Element of X is Element of X
tseq is set
tseq is Element of X
vseq . tseq is Element of the carrier of Y
||.(vseq . tseq).|| is V11() real ext-real Element of REAL
the normF of Y is non empty Relation-like the carrier of Y -defined REAL -valued Function-like V23( the carrier of Y) quasi_total V33() V34() V35() Element of bool [: the carrier of Y,REAL:]
[: the carrier of Y,REAL:] is non empty Relation-like V33() V34() V35() set
bool [: the carrier of Y,REAL:] is non empty set
the normF of Y . (vseq . tseq) is V11() real ext-real Element of REAL
vseq . the Element of X is Element of the carrier of Y
||.(vseq . the Element of X).|| is V11() real ext-real Element of REAL
the normF of Y is non empty Relation-like the carrier of Y -defined REAL -valued Function-like V23( the carrier of Y) quasi_total V33() V34() V35() Element of bool [: the carrier of Y,REAL:]
[: the carrier of Y,REAL:] is non empty Relation-like V33() V34() V35() set
bool [: the carrier of Y,REAL:] is non empty set
the normF of Y . (vseq . the Element of X) is V11() real ext-real Element of REAL
X is non empty set
Y is non empty right_complementable Abelian add-associative right_zeroed discerning reflexive vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like CNORMSTR
the carrier of Y is non empty set
[:X, the carrier of Y:] is non empty Relation-like set
bool [:X, the carrier of Y:] is non empty set
vseq is non empty Relation-like X -defined the carrier of Y -valued Function-like V23(X) quasi_total (X,Y) Element of bool [:X, the carrier of Y:]
(X,Y,vseq) is non empty V45() V46() V47() Element of bool REAL
{ ||.(vseq . b₁).|| where b₁ is Element of X : verum } is set
f is V11() real ext-real Element of REAL
tseq is ext-real set
tseq is Element of X
vseq . tseq is Element of the carrier of Y
||.(vseq . tseq).|| is V11() real ext-real Element of REAL
the normF of Y is non empty Relation-like the carrier of Y -defined REAL -valued Function-like V23( the carrier of Y) quasi_total V33() V34() V35() Element of bool [: the carrier of Y,REAL:]
[: the carrier of Y,REAL:] is non empty Relation-like V33() V34() V35() set
bool [: the carrier of Y,REAL:] is non empty set
the normF of Y . (vseq . tseq) is V11() real ext-real Element of REAL
X is non empty set
Y is non empty right_complementable Abelian add-associative right_zeroed discerning reflexive vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like CNORMSTR
the carrier of Y is non empty set
[:X, the carrier of Y:] is non empty Relation-like set
bool [:X, the carrier of Y:] is non empty set
vseq is non empty Relation-like X -defined the carrier of Y -valued Function-like V23(X) quasi_total Element of bool [:X, the carrier of Y:]
(X,Y,vseq) is non empty V45() V46() V47() Element of bool REAL
{ ||.(vseq . b₁).|| where b₁ is Element of X : verum } is set
upper_bound (X,Y,vseq) is V11() real ext-real Element of REAL
f is V11() real ext-real Element of REAL
tseq is set
tv is V11() real ext-real Element of REAL
e is Element of X
vseq . e is Element of the carrier of Y
||.(vseq . e).|| is V11() real ext-real Element of REAL
the normF of Y is non empty Relation-like the carrier of Y -defined REAL -valued Function-like V23( the carrier of Y) quasi_total V33() V34() V35() Element of bool [: the carrier of Y,REAL:]
[: the carrier of Y,REAL:] is non empty Relation-like V33() V34() V35() set
bool [: the carrier of Y,REAL:] is non empty set
the normF of Y . (vseq . e) is V11() real ext-real Element of REAL
tseq is V11() real ext-real Element of REAL
tseq is Element of X
vseq . tseq is Element of the carrier of Y
||.(vseq . tseq).|| is V11() real ext-real Element of REAL
the normF of Y is non empty Relation-like the carrier of Y -defined REAL -valued Function-like V23( the carrier of Y) quasi_total V33() V34() V35() Element of bool [: the carrier of Y,REAL:]
[: the carrier of Y,REAL:] is non empty Relation-like V33() V34() V35() set
bool [: the carrier of Y,REAL:] is non empty set
the normF of Y . (vseq . tseq) is V11() real ext-real Element of REAL
tseq is V11() real ext-real Element of REAL
X is non empty set
Y is non empty right_complementable Abelian add-associative right_zeroed discerning reflexive vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like CNORMSTR
(X,Y) is non empty Element of bool the carrier of (ComplexVectSpace (X,Y))
ComplexVectSpace (X,Y) is non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital CLSStruct
the carrier of Y is non empty set
Funcs (X, the carrier of Y) is non empty functional FUNCTION_DOMAIN of X, the carrier of Y
K340(X,Y) is Relation-like X -defined the carrier of Y -valued Function-like V23(X) quasi_total Element of Funcs (X, the carrier of Y)
K338(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 V23([:(Funcs (X, the carrier of Y)),(Funcs (X, the carrier of Y)):]) quasi_total Function-yielding V135() 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 Relation-like set
[:[:(Funcs (X, the carrier of Y)),(Funcs (X, the carrier of Y)):],(Funcs (X, the carrier of Y)):] is non empty Relation-like set
bool [:[:(Funcs (X, the carrier of Y)),(Funcs (X, the carrier of Y)):],(Funcs (X, the carrier of Y)):] is non empty set
FuncExtMult (X,Y) is non empty Relation-like [:COMPLEX,(Funcs (X, the carrier of Y)):] -defined Funcs (X, the carrier of Y) -valued Function-like V23([:COMPLEX,(Funcs (X, the carrier of Y)):]) quasi_total Function-yielding V135() Element of bool [:[:COMPLEX,(Funcs (X, the carrier of Y)):],(Funcs (X, the carrier of Y)):]
[:COMPLEX,(Funcs (X, the carrier of Y)):] is non empty Relation-like set
[:[:COMPLEX,(Funcs (X, the carrier of Y)):],(Funcs (X, the carrier of Y)):] is non empty Relation-like set
bool [:[:COMPLEX,(Funcs (X, the carrier of Y)):],(Funcs (X, the carrier of Y)):] is non empty set
CLSStruct(# (Funcs (X, the carrier of Y)),K340(X,Y),K338(X,Y),(FuncExtMult (X,Y)) #) is non empty strict CLSStruct
the carrier of (ComplexVectSpace (X,Y)) is non empty set
bool the carrier of (ComplexVectSpace (X,Y)) is non empty set
[:(X,Y),REAL:] is non empty Relation-like V33() V34() V35() set
bool [:(X,Y),REAL:] is non empty set
vseq is set
(X,Y,vseq) is non empty Relation-like X -defined the carrier of Y -valued Function-like V23(X) quasi_total (X,Y) Element of bool [:X, the carrier of Y:]
[:X, the carrier of Y:] is non empty Relation-like set
bool [:X, the carrier of Y:] is non empty set
(X,Y,(X,Y,vseq)) is non empty V45() V46() V47() Element of bool REAL
{ ||.((X,Y,vseq) . b₁).|| where b₁ is Element of X : verum } is set
upper_bound (X,Y,(X,Y,vseq)) is V11() real ext-real Element of REAL
vseq is non empty Relation-like (X,Y) -defined REAL -valued Function-like V23((X,Y)) quasi_total V33() V34() V35() Element of bool [:(X,Y),REAL:]
vseq is non empty Relation-like (X,Y) -defined REAL -valued Function-like V23((X,Y)) quasi_total V33() V34() V35() Element of bool [:(X,Y),REAL:]
f is non empty Relation-like (X,Y) -defined REAL -valued Function-like V23((X,Y)) quasi_total V33() V34() V35() Element of bool [:(X,Y),REAL:]
tseq is set
vseq . tseq is V11() real ext-real Element of REAL
f . tseq is V11() real ext-real Element of REAL
(X,Y,tseq) is non empty Relation-like X -defined the carrier of Y -valued Function-like V23(X) quasi_total (X,Y) Element of bool [:X, the carrier of Y:]
[:X, the carrier of Y:] is non empty Relation-like set
bool [:X, the carrier of Y:] is non empty set
(X,Y,(X,Y,tseq)) is non empty V45() V46() V47() Element of bool REAL
{ ||.((X,Y,tseq) . b₁).|| where b₁ is Element of X : verum } is set
upper_bound (X,Y,(X,Y,tseq)) is V11() real ext-real Element of REAL
dom vseq is non empty Element of bool (X,Y)
bool (X,Y) is non empty set
dom f is non empty Element of bool (X,Y)
X is non empty set
Y is non empty right_complementable Abelian add-associative right_zeroed discerning reflexive vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like CNORMSTR
the carrier of Y is non empty set
[:X, the carrier of Y:] is non empty Relation-like set
bool [:X, the carrier of Y:] is non empty set
vseq is non empty Relation-like X -defined the carrier of Y -valued Function-like V23(X) quasi_total (X,Y) Element of bool [:X, the carrier of Y:]
(X,Y,vseq) is non empty Relation-like X -defined the carrier of Y -valued Function-like V23(X) quasi_total (X,Y) Element of bool [:X, the carrier of Y:]
(X,Y) is non empty Element of bool the carrier of (ComplexVectSpace (X,Y))
ComplexVectSpace (X,Y) is non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital CLSStruct
Funcs (X, the carrier of Y) is non empty functional FUNCTION_DOMAIN of X, the carrier of Y
K340(X,Y) is Relation-like X -defined the carrier of Y -valued Function-like V23(X) quasi_total Element of Funcs (X, the carrier of Y)
K338(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 V23([:(Funcs (X, the carrier of Y)),(Funcs (X, the carrier of Y)):]) quasi_total Function-yielding V135() 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 Relation-like set
[:[:(Funcs (X, the carrier of Y)),(Funcs (X, the carrier of Y)):],(Funcs (X, the carrier of Y)):] is non empty Relation-like set
bool [:[:(Funcs (X, the carrier of Y)),(Funcs (X, the carrier of Y)):],(Funcs (X, the carrier of Y)):] is non empty set
FuncExtMult (X,Y) is non empty Relation-like [:COMPLEX,(Funcs (X, the carrier of Y)):] -defined Funcs (X, the carrier of Y) -valued Function-like V23([:COMPLEX,(Funcs (X, the carrier of Y)):]) quasi_total Function-yielding V135() Element of bool [:[:COMPLEX,(Funcs (X, the carrier of Y)):],(Funcs (X, the carrier of Y)):]
[:COMPLEX,(Funcs (X, the carrier of Y)):] is non empty Relation-like set
[:[:COMPLEX,(Funcs (X, the carrier of Y)):],(Funcs (X, the carrier of Y)):] is non empty Relation-like set
bool [:[:COMPLEX,(Funcs (X, the carrier of Y)):],(Funcs (X, the carrier of Y)):] is non empty set
CLSStruct(# (Funcs (X, the carrier of Y)),K340(X,Y),K338(X,Y),(FuncExtMult (X,Y)) #) is non empty strict CLSStruct
the carrier of (ComplexVectSpace (X,Y)) is non empty set
bool the carrier of (ComplexVectSpace (X,Y)) is non empty set
X is non empty set
Y is non empty right_complementable Abelian add-associative right_zeroed discerning reflexive vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like CNORMSTR
the carrier of Y is non empty set
[:X, the carrier of Y:] is non empty Relation-like set
bool [:X, the carrier of Y:] is non empty set
(X,Y) is non empty Relation-like (X,Y) -defined REAL -valued Function-like V23((X,Y)) quasi_total V33() V34() V35() Element of bool [:(X,Y),REAL:]
(X,Y) is non empty Element of bool the carrier of (ComplexVectSpace (X,Y))
ComplexVectSpace (X,Y) is non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital CLSStruct
Funcs (X, the carrier of Y) is non empty functional FUNCTION_DOMAIN of X, the carrier of Y
K340(X,Y) is Relation-like X -defined the carrier of Y -valued Function-like V23(X) quasi_total Element of Funcs (X, the carrier of Y)
K338(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 V23([:(Funcs (X, the carrier of Y)),(Funcs (X, the carrier of Y)):]) quasi_total Function-yielding V135() 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 Relation-like set
[:[:(Funcs (X, the carrier of Y)),(Funcs (X, the carrier of Y)):],(Funcs (X, the carrier of Y)):] is non empty Relation-like set
bool [:[:(Funcs (X, the carrier of Y)),(Funcs (X, the carrier of Y)):],(Funcs (X, the carrier of Y)):] is non empty set
FuncExtMult (X,Y) is non empty Relation-like [:COMPLEX,(Funcs (X, the carrier of Y)):] -defined Funcs (X, the carrier of Y) -valued Function-like V23([:COMPLEX,(Funcs (X, the carrier of Y)):]) quasi_total Function-yielding V135() Element of bool [:[:COMPLEX,(Funcs (X, the carrier of Y)):],(Funcs (X, the carrier of Y)):]
[:COMPLEX,(Funcs (X, the carrier of Y)):] is non empty Relation-like set
[:[:COMPLEX,(Funcs (X, the carrier of Y)):],(Funcs (X, the carrier of Y)):] is non empty Relation-like set
bool [:[:COMPLEX,(Funcs (X, the carrier of Y)):],(Funcs (X, the carrier of Y)):] is non empty set
CLSStruct(# (Funcs (X, the carrier of Y)),K340(X,Y),K338(X,Y),(FuncExtMult (X,Y)) #) is non empty strict CLSStruct
the carrier of (ComplexVectSpace (X,Y)) is non empty set
bool the carrier of (ComplexVectSpace (X,Y)) is non empty set
[:(X,Y),REAL:] is non empty Relation-like V33() V34() V35() set
bool [:(X,Y),REAL:] is non empty set
vseq is non empty Relation-like X -defined the carrier of Y -valued Function-like V23(X) quasi_total (X,Y) Element of bool [:X, the carrier of Y:]
(X,Y) . vseq is V11() real ext-real Element of REAL
(X,Y,vseq) is non empty V45() V46() V47() Element of bool REAL
{ ||.(vseq . b₁).|| where b₁ is Element of X : verum } is set
upper_bound (X,Y,vseq) is V11() real ext-real Element of REAL
f is set
(X,Y,f) is non empty Relation-like X -defined the carrier of Y -valued Function-like V23(X) quasi_total (X,Y) Element of bool [:X, the carrier of Y:]
(X,Y,(X,Y,f)) is non empty V45() V46() V47() Element of bool REAL
{ ||.((X,Y,f) . b₁).|| where b₁ is Element of X : verum } is set
upper_bound (X,Y,(X,Y,f)) is V11() real ext-real Element of REAL
X is non empty set
Y is non empty right_complementable Abelian add-associative right_zeroed discerning reflexive vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like CNORMSTR
(X,Y) is non empty Element of bool the carrier of (ComplexVectSpace (X,Y))
ComplexVectSpace (X,Y) is non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital CLSStruct
the carrier of Y is non empty set
Funcs (X, the carrier of Y) is non empty functional FUNCTION_DOMAIN of X, the carrier of Y
K340(X,Y) is Relation-like X -defined the carrier of Y -valued Function-like V23(X) quasi_total Element of Funcs (X, the carrier of Y)
K338(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 V23([:(Funcs (X, the carrier of Y)),(Funcs (X, the carrier of Y)):]) quasi_total Function-yielding V135() 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 Relation-like set
[:[:(Funcs (X, the carrier of Y)),(Funcs (X, the carrier of Y)):],(Funcs (X, the carrier of Y)):] is non empty Relation-like set
bool [:[:(Funcs (X, the carrier of Y)),(Funcs (X, the carrier of Y)):],(Funcs (X, the carrier of Y)):] is non empty set
FuncExtMult (X,Y) is non empty Relation-like [:COMPLEX,(Funcs (X, the carrier of Y)):] -defined Funcs (X, the carrier of Y) -valued Function-like V23([:COMPLEX,(Funcs (X, the carrier of Y)):]) quasi_total Function-yielding V135() Element of bool [:[:COMPLEX,(Funcs (X, the carrier of Y)):],(Funcs (X, the carrier of Y)):]
[:COMPLEX,(Funcs (X, the carrier of Y)):] is non empty Relation-like set
[:[:COMPLEX,(Funcs (X, the carrier of Y)):],(Funcs (X, the carrier of Y)):] is non empty Relation-like set
bool [:[:COMPLEX,(Funcs (X, the carrier of Y)):],(Funcs (X, the carrier of Y)):] is non empty set
CLSStruct(# (Funcs (X, the carrier of Y)),K340(X,Y),K338(X,Y),(FuncExtMult (X,Y)) #) is non empty strict CLSStruct
the carrier of (ComplexVectSpace (X,Y)) is non empty set
bool the carrier of (ComplexVectSpace (X,Y)) is non empty set
Zero_ ((X,Y),(ComplexVectSpace (X,Y))) is Element of (X,Y)
Add_ ((X,Y),(ComplexVectSpace (X,Y))) is non empty Relation-like [:(X,Y),(X,Y):] -defined (X,Y) -valued Function-like V23([:(X,Y),(X,Y):]) quasi_total Element of bool [:[:(X,Y),(X,Y):],(X,Y):]
[:(X,Y),(X,Y):] is non empty Relation-like set
[:[:(X,Y),(X,Y):],(X,Y):] is non empty Relation-like set
bool [:[:(X,Y),(X,Y):],(X,Y):] is non empty set
Mult_ ((X,Y),(ComplexVectSpace (X,Y))) is non empty Relation-like [:COMPLEX,(X,Y):] -defined (X,Y) -valued Function-like V23([:COMPLEX,(X,Y):]) quasi_total Element of bool [:[:COMPLEX,(X,Y):],(X,Y):]
[:COMPLEX,(X,Y):] is non empty Relation-like set
[:[:COMPLEX,(X,Y):],(X,Y):] is non empty Relation-like set
bool [:[:COMPLEX,(X,Y):],(X,Y):] is non empty set
(X,Y) is non empty Relation-like (X,Y) -defined REAL -valued Function-like V23((X,Y)) quasi_total V33() V34() V35() Element of bool [:(X,Y),REAL:]
[:(X,Y),REAL:] is non empty Relation-like V33() V34() V35() set
bool [:(X,Y),REAL:] is non empty set
CNORMSTR(# (X,Y),(Zero_ ((X,Y),(ComplexVectSpace (X,Y)))),(Add_ ((X,Y),(ComplexVectSpace (X,Y)))),(Mult_ ((X,Y),(ComplexVectSpace (X,Y)))),(X,Y) #) is non empty strict CNORMSTR
X is non empty set
Y is non empty right_complementable Abelian add-associative right_zeroed discerning reflexive vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like CNORMSTR
the carrier of Y is non empty set
0. Y is V82(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 V23(X) quasi_total Element of bool [:X, the carrier of Y:]
[:X, the carrier of Y:] is non empty Relation-like set
bool [:X, the carrier of Y:] is non empty set
(X,Y) is non empty CNORMSTR
(X,Y) is non empty Element of bool the carrier of (ComplexVectSpace (X,Y))
ComplexVectSpace (X,Y) is non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital CLSStruct
Funcs (X, the carrier of Y) is non empty functional FUNCTION_DOMAIN of X, the carrier of Y
K340(X,Y) is Relation-like X -defined the carrier of Y -valued Function-like V23(X) quasi_total Element of Funcs (X, the carrier of Y)
K338(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 V23([:(Funcs (X, the carrier of Y)),(Funcs (X, the carrier of Y)):]) quasi_total Function-yielding V135() 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 Relation-like set
[:[:(Funcs (X, the carrier of Y)),(Funcs (X, the carrier of Y)):],(Funcs (X, the carrier of Y)):] is non empty Relation-like set
bool [:[:(Funcs (X, the carrier of Y)),(Funcs (X, the carrier of Y)):],(Funcs (X, the carrier of Y)):] is non empty set
FuncExtMult (X,Y) is non empty Relation-like [:COMPLEX,(Funcs (X, the carrier of Y)):] -defined Funcs (X, the carrier of Y) -valued Function-like V23([:COMPLEX,(Funcs (X, the carrier of Y)):]) quasi_total Function-yielding V135() Element of bool [:[:COMPLEX,(Funcs (X, the carrier of Y)):],(Funcs (X, the carrier of Y)):]
[:COMPLEX,(Funcs (X, the carrier of Y)):] is non empty Relation-like set
[:[:COMPLEX,(Funcs (X, the carrier of Y)):],(Funcs (X, the carrier of Y)):] is non empty Relation-like set
bool [:[:COMPLEX,(Funcs (X, the carrier of Y)):],(Funcs (X, the carrier of Y)):] is non empty set
CLSStruct(# (Funcs (X, the carrier of Y)),K340(X,Y),K338(X,Y),(FuncExtMult (X,Y)) #) is non empty strict CLSStruct
the carrier of (ComplexVectSpace (X,Y)) is non empty set
bool the carrier of (ComplexVectSpace (X,Y)) is non empty set
Zero_ ((X,Y),(ComplexVectSpace (X,Y))) is Element of (X,Y)
Add_ ((X,Y),(ComplexVectSpace (X,Y))) is non empty Relation-like [:(X,Y),(X,Y):] -defined (X,Y) -valued Function-like V23([:(X,Y),(X,Y):]) quasi_total Element of bool [:[:(X,Y),(X,Y):],(X,Y):]
[:(X,Y),(X,Y):] is non empty Relation-like set
[:[:(X,Y),(X,Y):],(X,Y):] is non empty Relation-like set
bool [:[:(X,Y),(X,Y):],(X,Y):] is non empty set
Mult_ ((X,Y),(ComplexVectSpace (X,Y))) is non empty Relation-like [:COMPLEX,(X,Y):] -defined (X,Y) -valued Function-like V23([:COMPLEX,(X,Y):]) quasi_total Element of bool [:[:COMPLEX,(X,Y):],(X,Y):]
[:COMPLEX,(X,Y):] is non empty Relation-like set
[:[:COMPLEX,(X,Y):],(X,Y):] is non empty Relation-like set
bool [:[:COMPLEX,(X,Y):],(X,Y):] is non empty set
(X,Y) is non empty Relation-like (X,Y) -defined REAL -valued Function-like V23((X,Y)) quasi_total V33() V34() V35() Element of bool [:(X,Y),REAL:]
[:(X,Y),REAL:] is non empty Relation-like V33() V34() V35() set
bool [:(X,Y),REAL:] is non empty set
CNORMSTR(# (X,Y),(Zero_ ((X,Y),(ComplexVectSpace (X,Y)))),(Add_ ((X,Y),(ComplexVectSpace (X,Y)))),(Mult_ ((X,Y),(ComplexVectSpace (X,Y)))),(X,Y) #) is non empty strict CNORMSTR
0. (X,Y) is V82((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 right_complementable Abelian add-associative right_zeroed strict vector-distributive scalar-distributive scalar-associative scalar-unital CLSStruct
CLSStruct(# (X,Y),(Zero_ ((X,Y),(ComplexVectSpace (X,Y)))),(Add_ ((X,Y),(ComplexVectSpace (X,Y)))),(Mult_ ((X,Y),(ComplexVectSpace (X,Y)))) #) is non empty right_complementable Abelian add-associative right_zeroed strict vector-distributive scalar-distributive scalar-associative scalar-unital CLSStruct
0. (X,Y) is V82((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 is non empty set
Y is non empty right_complementable Abelian add-associative right_zeroed discerning reflexive vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like CNORMSTR
(X,Y) is non empty CNORMSTR
(X,Y) is non empty Element of bool the carrier of (ComplexVectSpace (X,Y))
ComplexVectSpace (X,Y) is non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital CLSStruct
the carrier of Y is non empty set
Funcs (X, the carrier of Y) is non empty functional FUNCTION_DOMAIN of X, the carrier of Y
K340(X,Y) is Relation-like X -defined the carrier of Y -valued Function-like V23(X) quasi_total Element of Funcs (X, the carrier of Y)
K338(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 V23([:(Funcs (X, the carrier of Y)),(Funcs (X, the carrier of Y)):]) quasi_total Function-yielding V135() 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 Relation-like set
[:[:(Funcs (X, the carrier of Y)),(Funcs (X, the carrier of Y)):],(Funcs (X, the carrier of Y)):] is non empty Relation-like set
bool [:[:(Funcs (X, the carrier of Y)),(Funcs (X, the carrier of Y)):],(Funcs (X, the carrier of Y)):] is non empty set
FuncExtMult (X,Y) is non empty Relation-like [:COMPLEX,(Funcs (X, the carrier of Y)):] -defined Funcs (X, the carrier of Y) -valued Function-like V23([:COMPLEX,(Funcs (X, the carrier of Y)):]) quasi_total Function-yielding V135() Element of bool [:[:COMPLEX,(Funcs (X, the carrier of Y)):],(Funcs (X, the carrier of Y)):]
[:COMPLEX,(Funcs (X, the carrier of Y)):] is non empty Relation-like set
[:[:COMPLEX,(Funcs (X, the carrier of Y)):],(Funcs (X, the carrier of Y)):] is non empty Relation-like set
bool [:[:COMPLEX,(Funcs (X, the carrier of Y)):],(Funcs (X, the carrier of Y)):] is non empty set
CLSStruct(# (Funcs (X, the carrier of Y)),K340(X,Y),K338(X,Y),(FuncExtMult (X,Y)) #) is non empty strict CLSStruct
the carrier of (ComplexVectSpace (X,Y)) is non empty set
bool the carrier of (ComplexVectSpace (X,Y)) is non empty set
Zero_ ((X,Y),(ComplexVectSpace (X,Y))) is Element of (X,Y)
Add_ ((X,Y),(ComplexVectSpace (X,Y))) is non empty Relation-like [:(X,Y),(X,Y):] -defined (X,Y) -valued Function-like V23([:(X,Y),(X,Y):]) quasi_total Element of bool [:[:(X,Y),(X,Y):],(X,Y):]
[:(X,Y),(X,Y):] is non empty Relation-like set
[:[:(X,Y),(X,Y):],(X,Y):] is non empty Relation-like set
bool [:[:(X,Y),(X,Y):],(X,Y):] is non empty set
Mult_ ((X,Y),(ComplexVectSpace (X,Y))) is non empty Relation-like [:COMPLEX,(X,Y):] -defined (X,Y) -valued Function-like V23([:COMPLEX,(X,Y):]) quasi_total Element of bool [:[:COMPLEX,(X,Y):],(X,Y):]
[:COMPLEX,(X,Y):] is non empty Relation-like set
[:[:COMPLEX,(X,Y):],(X,Y):] is non empty Relation-like set
bool [:[:COMPLEX,(X,Y):],(X,Y):] is non empty set
(X,Y) is non empty Relation-like (X,Y) -defined REAL -valued Function-like V23((X,Y)) quasi_total V33() V34() V35() Element of bool [:(X,Y),REAL:]
[:(X,Y),REAL:] is non empty Relation-like V33() V34() V35() set
bool [:(X,Y),REAL:] is non empty set
CNORMSTR(# (X,Y),(Zero_ ((X,Y),(ComplexVectSpace (X,Y)))),(Add_ ((X,Y),(ComplexVectSpace (X,Y)))),(Mult_ ((X,Y),(ComplexVectSpace (X,Y)))),(X,Y) #) is non empty strict CNORMSTR
the carrier of (X,Y) is non empty set
[:X, the carrier of Y:] is non empty Relation-like set
bool [:X, the carrier of Y:] is non empty set
vseq is Element of the carrier of (X,Y)
||.vseq.|| is V11() real ext-real Element of REAL
the normF of (X,Y) is non empty Relation-like the carrier of (X,Y) -defined REAL -valued Function-like V23( the carrier of (X,Y)) quasi_total V33() V34() V35() Element of bool [: the carrier of (X,Y),REAL:]
[: the carrier of (X,Y),REAL:] is non empty Relation-like V33() V34() V35() set
bool [: the carrier of (X,Y),REAL:] is non empty set
the normF of (X,Y) . vseq is V11() real ext-real Element of REAL
f is non empty Relation-like X -defined the carrier of Y -valued Function-like V23(X) quasi_total (X,Y) Element of bool [:X, the carrier of Y:]
(X,Y,f) is non empty V45() V46() V47() Element of bool REAL
{ ||.(f . b₁).|| where b₁ is Element of X : verum } is set
tseq is Element of X
f . tseq is Element of the carrier of Y
||.(f . tseq).|| is V11() real ext-real Element of REAL
the normF of Y is non empty Relation-like the carrier of Y -defined REAL -valued Function-like V23( the carrier of Y) quasi_total V33() V34() V35() Element of bool [: the carrier of Y,REAL:]
[: the carrier of Y,REAL:] is non empty Relation-like V33() V34() V35() set
bool [: the carrier of Y,REAL:] is non empty set
the normF of Y . (f . tseq) is V11() real ext-real Element of REAL
upper_bound (X,Y,f) is V11() real ext-real Element of REAL
(X,Y) . f is V11() real ext-real Element of REAL
tseq is Element of X
f . tseq is Element of the carrier of Y
||.(f . tseq).|| is V11() real ext-real Element of REAL
the normF of Y . (f . tseq) is V11() real ext-real Element of REAL
X is non empty set
Y is non empty right_complementable Abelian add-associative right_zeroed discerning reflexive vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like CNORMSTR
(X,Y) is non empty CNORMSTR
(X,Y) is non empty Element of bool the carrier of (ComplexVectSpace (X,Y))
ComplexVectSpace (X,Y) is non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital CLSStruct
the carrier of Y is non empty set
Funcs (X, the carrier of Y) is non empty functional FUNCTION_DOMAIN of X, the carrier of Y
K340(X,Y) is Relation-like X -defined the carrier of Y -valued Function-like V23(X) quasi_total Element of Funcs (X, the carrier of Y)
K338(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 V23([:(Funcs (X, the carrier of Y)),(Funcs (X, the carrier of Y)):]) quasi_total Function-yielding V135() 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 Relation-like set
[:[:(Funcs (X, the carrier of Y)),(Funcs (X, the carrier of Y)):],(Funcs (X, the carrier of Y)):] is non empty Relation-like set
bool [:[:(Funcs (X, the carrier of Y)),(Funcs (X, the carrier of Y)):],(Funcs (X, the carrier of Y)):] is non empty set
FuncExtMult (X,Y) is non empty Relation-like [:COMPLEX,(Funcs (X, the carrier of Y)):] -defined Funcs (X, the carrier of Y) -valued Function-like V23([:COMPLEX,(Funcs (X, the carrier of Y)):]) quasi_total Function-yielding V135() Element of bool [:[:COMPLEX,(Funcs (X, the carrier of Y)):],(Funcs (X, the carrier of Y)):]
[:COMPLEX,(Funcs (X, the carrier of Y)):] is non empty Relation-like set
[:[:COMPLEX,(Funcs (X, the carrier of Y)):],(Funcs (X, the carrier of Y)):] is non empty Relation-like set
bool [:[:COMPLEX,(Funcs (X, the carrier of Y)):],(Funcs (X, the carrier of Y)):] is non empty set
CLSStruct(# (Funcs (X, the carrier of Y)),K340(X,Y),K338(X,Y),(FuncExtMult (X,Y)) #) is non empty strict CLSStruct
the carrier of (ComplexVectSpace (X,Y)) is non empty set
bool the carrier of (ComplexVectSpace (X,Y)) is non empty set
Zero_ ((X,Y),(ComplexVectSpace (X,Y))) is Element of (X,Y)
Add_ ((X,Y),(ComplexVectSpace (X,Y))) is non empty Relation-like [:(X,Y),(X,Y):] -defined (X,Y) -valued Function-like V23([:(X,Y),(X,Y):]) quasi_total Element of bool [:[:(X,Y),(X,Y):],(X,Y):]
[:(X,Y),(X,Y):] is non empty Relation-like set
[:[:(X,Y),(X,Y):],(X,Y):] is non empty Relation-like set
bool [:[:(X,Y),(X,Y):],(X,Y):] is non empty set
Mult_ ((X,Y),(ComplexVectSpace (X,Y))) is non empty Relation-like [:COMPLEX,(X,Y):] -defined (X,Y) -valued Function-like V23([:COMPLEX,(X,Y):]) quasi_total Element of bool [:[:COMPLEX,(X,Y):],(X,Y):]
[:COMPLEX,(X,Y):] is non empty Relation-like set
[:[:COMPLEX,(X,Y):],(X,Y):] is non empty Relation-like set
bool [:[:COMPLEX,(X,Y):],(X,Y):] is non empty set
(X,Y) is non empty Relation-like (X,Y) -defined REAL -valued Function-like V23((X,Y)) quasi_total V33() V34() V35() Element of bool [:(X,Y),REAL:]
[:(X,Y),REAL:] is non empty Relation-like V33() V34() V35() set
bool [:(X,Y),REAL:] is non empty set
CNORMSTR(# (X,Y),(Zero_ ((X,Y),(ComplexVectSpace (X,Y)))),(Add_ ((X,Y),(ComplexVectSpace (X,Y)))),(Mult_ ((X,Y),(ComplexVectSpace (X,Y)))),(X,Y) #) is non empty strict CNORMSTR
the carrier of (X,Y) is non empty set
vseq is Element of the carrier of (X,Y)
||.vseq.|| is V11() real ext-real Element of REAL
the normF of (X,Y) is non empty Relation-like the carrier of (X,Y) -defined REAL -valued Function-like V23( the carrier of (X,Y)) quasi_total V33() V34() V35() Element of bool [: the carrier of (X,Y),REAL:]
[: the carrier of (X,Y),REAL:] is non empty Relation-like V33() V34() V35() set
bool [: the carrier of (X,Y),REAL:] is non empty set
the normF of (X,Y) . vseq is V11() real ext-real Element of REAL
[:X, the carrier of Y:] is non empty Relation-like set
bool [:X, the carrier of Y:] is non empty set
f is non empty Relation-like X -defined the carrier of Y -valued Function-like V23(X) quasi_total (X,Y) Element of bool [:X, the carrier of Y:]
(X,Y,f) is non empty V45() V46() V47() Element of bool REAL
{ ||.(f . b₁).|| where b₁ is Element of X : verum } is set
tseq is set
(X,Y) . vseq is V11() real ext-real Element of REAL
upper_bound (X,Y,f) is V11() real ext-real Element of REAL
tv is V11() real ext-real Element of REAL
e is Element of X
f . e is Element of the carrier of Y
||.(f . e).|| is V11() real ext-real Element of REAL
the normF of Y is non empty Relation-like the carrier of Y -defined REAL -valued Function-like V23( the carrier of Y) quasi_total V33() V34() V35() Element of bool [: the carrier of Y,REAL:]
[: the carrier of Y,REAL:] is non empty Relation-like V33() V34() V35() set
bool [: the carrier of Y,REAL:] is non empty set
the normF of Y . (f . e) is V11() real ext-real Element of REAL
tseq is V11() real ext-real Element of REAL
X is non empty set
Y is non empty right_complementable Abelian add-associative right_zeroed discerning reflexive vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like CNORMSTR
(X,Y) is non empty CNORMSTR
(X,Y) is non empty Element of bool the carrier of (ComplexVectSpace (X,Y))
ComplexVectSpace (X,Y) is non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital CLSStruct
the carrier of Y is non empty set
Funcs (X, the carrier of Y) is non empty functional FUNCTION_DOMAIN of X, the carrier of Y
K340(X,Y) is Relation-like X -defined the carrier of Y -valued Function-like V23(X) quasi_total Element of Funcs (X, the carrier of Y)
K338(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 V23([:(Funcs (X, the carrier of Y)),(Funcs (X, the carrier of Y)):]) quasi_total Function-yielding V135() 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 Relation-like set
[:[:(Funcs (X, the carrier of Y)),(Funcs (X, the carrier of Y)):],(Funcs (X, the carrier of Y)):] is non empty Relation-like set
bool [:[:(Funcs (X, the carrier of Y)),(Funcs (X, the carrier of Y)):],(Funcs (X, the carrier of Y)):] is non empty set
FuncExtMult (X,Y) is non empty Relation-like [:COMPLEX,(Funcs (X, the carrier of Y)):] -defined Funcs (X, the carrier of Y) -valued Function-like V23([:COMPLEX,(Funcs (X, the carrier of Y)):]) quasi_total Function-yielding V135() Element of bool [:[:COMPLEX,(Funcs (X, the carrier of Y)):],(Funcs (X, the carrier of Y)):]
[:COMPLEX,(Funcs (X, the carrier of Y)):] is non empty Relation-like set
[:[:COMPLEX,(Funcs (X, the carrier of Y)):],(Funcs (X, the carrier of Y)):] is non empty Relation-like set
bool [:[:COMPLEX,(Funcs (X, the carrier of Y)):],(Funcs (X, the carrier of Y)):] is non empty set
CLSStruct(# (Funcs (X, the carrier of Y)),K340(X,Y),K338(X,Y),(FuncExtMult (X,Y)) #) is non empty strict CLSStruct
the carrier of (ComplexVectSpace (X,Y)) is non empty set
bool the carrier of (ComplexVectSpace (X,Y)) is non empty set
Zero_ ((X,Y),(ComplexVectSpace (X,Y))) is Element of (X,Y)
Add_ ((X,Y),(ComplexVectSpace (X,Y))) is non empty Relation-like [:(X,Y),(X,Y):] -defined (X,Y) -valued Function-like V23([:(X,Y),(X,Y):]) quasi_total Element of bool [:[:(X,Y),(X,Y):],(X,Y):]
[:(X,Y),(X,Y):] is non empty Relation-like set
[:[:(X,Y),(X,Y):],(X,Y):] is non empty Relation-like set
bool [:[:(X,Y),(X,Y):],(X,Y):] is non empty set
Mult_ ((X,Y),(ComplexVectSpace (X,Y))) is non empty Relation-like [:COMPLEX,(X,Y):] -defined (X,Y) -valued Function-like V23([:COMPLEX,(X,Y):]) quasi_total Element of bool [:[:COMPLEX,(X,Y):],(X,Y):]
[:COMPLEX,(X,Y):] is non empty Relation-like set
[:[:COMPLEX,(X,Y):],(X,Y):] is non empty Relation-like set
bool [:[:COMPLEX,(X,Y):],(X,Y):] is non empty set
(X,Y) is non empty Relation-like (X,Y) -defined REAL -valued Function-like V23((X,Y)) quasi_total V33() V34() V35() Element of bool [:(X,Y),REAL:]
[:(X,Y),REAL:] is non empty Relation-like V33() V34() V35() set
bool [:(X,Y),REAL:] is non empty set
CNORMSTR(# (X,Y),(Zero_ ((X,Y),(ComplexVectSpace (X,Y)))),(Add_ ((X,Y),(ComplexVectSpace (X,Y)))),(Mult_ ((X,Y),(ComplexVectSpace (X,Y)))),(X,Y) #) is non empty strict CNORMSTR
the carrier of (X,Y) is non empty set
0. (X,Y) is V82((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 V11() real ext-real Element of REAL
the normF of (X,Y) is non empty Relation-like the carrier of (X,Y) -defined REAL -valued Function-like V23( the carrier of (X,Y)) quasi_total V33() V34() V35() Element of bool [: the carrier of (X,Y),REAL:]
[: the carrier of (X,Y),REAL:] is non empty Relation-like V33() V34() V35() set
bool [: the carrier of (X,Y),REAL:] is non empty set
the normF of (X,Y) . vseq is V11() real ext-real Element of REAL
[:X, the carrier of Y:] is non empty Relation-like set
bool [:X, the carrier of Y:] is non empty set
0. Y is V82(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 V23(X) quasi_total Element of bool [:X, the carrier of Y:]
f is non empty Relation-like X -defined the carrier of Y -valued Function-like V23(X) quasi_total (X,Y) Element of bool [:X, the carrier of Y:]
(X,Y,f) is non empty V45() V46() V47() Element of bool REAL
{ ||.(f . b₁).|| where b₁ is Element of X : verum } is set
tv is set
upper_bound (X,Y,f) is V11() real ext-real Element of REAL
tseq is non empty Relation-like X -defined the carrier of Y -valued Function-like V23(X) quasi_total Element of bool [:X, the carrier of Y:]
m is V11() real ext-real Element of REAL
n is Element of X
f . n is Element of the carrier of Y
||.(f . n).|| is V11() real ext-real Element of REAL
the normF of Y is non empty Relation-like the carrier of Y -defined REAL -valued Function-like V23( the carrier of Y) quasi_total V33() V34() V35() Element of bool [: the carrier of Y,REAL:]
[: the carrier of Y,REAL:] is non empty Relation-like V33() V34() V35() set
bool [: the carrier of Y,REAL:] is non empty set
the normF of Y . (f . n) is V11() real ext-real Element of REAL
||.(0. Y).|| is empty epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural V11() real Relation-like non-empty empty-yielding RAT -valued ext-real non positive non negative V33() V34() V35() V36() V45() V46() V47() V48() V49() V50() V51() Element of REAL
the normF of Y . (0. Y) is V11() real ext-real Element of REAL
e is V11() real ext-real Element of REAL
m is V11() real ext-real set
(X,Y) . vseq is V11() real ext-real Element of REAL
X is non empty set
Y is non empty right_complementable Abelian add-associative right_zeroed discerning reflexive vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like CNORMSTR
(X,Y) is non empty CNORMSTR
(X,Y) is non empty Element of bool the carrier of (ComplexVectSpace (X,Y))
ComplexVectSpace (X,Y) is non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital CLSStruct
the carrier of Y is non empty set
Funcs (X, the carrier of Y) is non empty functional FUNCTION_DOMAIN of X, the carrier of Y
K340(X,Y) is Relation-like X -defined the carrier of Y -valued Function-like V23(X) quasi_total Element of Funcs (X, the carrier of Y)
K338(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 V23([:(Funcs (X, the carrier of Y)),(Funcs (X, the carrier of Y)):]) quasi_total Function-yielding V135() 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 Relation-like set
[:[:(Funcs (X, the carrier of Y)),(Funcs (X, the carrier of Y)):],(Funcs (X, the carrier of Y)):] is non empty Relation-like set
bool [:[:(Funcs (X, the carrier of Y)),(Funcs (X, the carrier of Y)):],(Funcs (X, the carrier of Y)):] is non empty set
FuncExtMult (X,Y) is non empty Relation-like [:COMPLEX,(Funcs (X, the carrier of Y)):] -defined Funcs (X, the carrier of Y) -valued Function-like V23([:COMPLEX,(Funcs (X, the carrier of Y)):]) quasi_total Function-yielding V135() Element of bool [:[:COMPLEX,(Funcs (X, the carrier of Y)):],(Funcs (X, the carrier of Y)):]
[:COMPLEX,(Funcs (X, the carrier of Y)):] is non empty Relation-like set
[:[:COMPLEX,(Funcs (X, the carrier of Y)):],(Funcs (X, the carrier of Y)):] is non empty Relation-like set
bool [:[:COMPLEX,(Funcs (X, the carrier of Y)):],(Funcs (X, the carrier of Y)):] is non empty set
CLSStruct(# (Funcs (X, the carrier of Y)),K340(X,Y),K338(X,Y),(FuncExtMult (X,Y)) #) is non empty strict CLSStruct
the carrier of (ComplexVectSpace (X,Y)) is non empty set
bool the carrier of (ComplexVectSpace (X,Y)) is non empty set
Zero_ ((X,Y),(ComplexVectSpace (X,Y))) is Element of (X,Y)
Add_ ((X,Y),(ComplexVectSpace (X,Y))) is non empty Relation-like [:(X,Y),(X,Y):] -defined (X,Y) -valued Function-like V23([:(X,Y),(X,Y):]) quasi_total Element of bool [:[:(X,Y),(X,Y):],(X,Y):]
[:(X,Y),(X,Y):] is non empty Relation-like set
[:[:(X,Y),(X,Y):],(X,Y):] is non empty Relation-like set
bool [:[:(X,Y),(X,Y):],(X,Y):] is non empty set
Mult_ ((X,Y),(ComplexVectSpace (X,Y))) is non empty Relation-like [:COMPLEX,(X,Y):] -defined (X,Y) -valued Function-like V23([:COMPLEX,(X,Y):]) quasi_total Element of bool [:[:COMPLEX,(X,Y):],(X,Y):]
[:COMPLEX,(X,Y):] is non empty Relation-like set
[:[:COMPLEX,(X,Y):],(X,Y):] is non empty Relation-like set
bool [:[:COMPLEX,(X,Y):],(X,Y):] is non empty set
(X,Y) is non empty Relation-like (X,Y) -defined REAL -valued Function-like V23((X,Y)) quasi_total V33() V34() V35() Element of bool [:(X,Y),REAL:]
[:(X,Y),REAL:] is non empty Relation-like V33() V34() V35() set
bool [:(X,Y),REAL:] is non empty set
CNORMSTR(# (X,Y),(Zero_ ((X,Y),(ComplexVectSpace (X,Y)))),(Add_ ((X,Y),(ComplexVectSpace (X,Y)))),(Mult_ ((X,Y),(ComplexVectSpace (X,Y)))),(X,Y) #) is non empty strict CNORMSTR
the carrier of (X,Y) is non empty set
[:X, the carrier of Y:] is non empty Relation-like set
bool [:X, the carrier of Y:] is non empty set
vseq is Element of the carrier of (X,Y)
f is Element of the carrier of (X,Y)
tseq is Element of the carrier of (X,Y)
vseq + f is 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 V23([: the carrier of (X,Y), the carrier of (X,Y):]) 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 Relation-like set
[:[: the carrier of (X,Y), the carrier of (X,Y):], the carrier of (X,Y):] is non empty Relation-like set
bool [:[: the carrier of (X,Y), the carrier of (X,Y):], the carrier of (X,Y):] is non empty set
K602( the carrier of (X,Y), the addF of (X,Y),vseq,f) is Element of the carrier of (X,Y)
[vseq,f] is set
{vseq,f} is set
{vseq} is set
{{vseq,f},{vseq}} is set
the addF of (X,Y) . [vseq,f] is set
(X,Y) is non empty right_complementable Abelian add-associative right_zeroed strict vector-distributive scalar-distributive scalar-associative scalar-unital CLSStruct
CLSStruct(# (X,Y),(Zero_ ((X,Y),(ComplexVectSpace (X,Y)))),(Add_ ((X,Y),(ComplexVectSpace (X,Y)))),(Mult_ ((X,Y),(ComplexVectSpace (X,Y)))) #) is non empty right_complementable Abelian add-associative right_zeroed strict vector-distributive scalar-distributive scalar-associative scalar-unital CLSStruct
the carrier of (X,Y) is non empty set
e is Element of the carrier of (X,Y)
tseq is Element of the carrier of (X,Y)
tv is Element of the carrier of (X,Y)
tseq + tv is 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 V23([: the carrier of (X,Y), the carrier of (X,Y):]) 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 Relation-like set
[:[: the carrier of (X,Y), the carrier of (X,Y):], the carrier of (X,Y):] is non empty Relation-like set
bool [:[: the carrier of (X,Y), the carrier of (X,Y):], the carrier of (X,Y):] is non empty set
K602( the carrier of (X,Y), the addF of (X,Y),tseq,tv) is Element of the carrier of (X,Y)
[tseq,tv] is set
{tseq,tv} is set
{tseq} is set
{{tseq,tv},{tseq}} is set
the addF of (X,Y) . [tseq,tv] is set
m is non empty Relation-like X -defined the carrier of Y -valued Function-like V23(X) quasi_total (X,Y) Element of bool [:X, the carrier of Y:]
n is non empty Relation-like X -defined the carrier of Y -valued Function-like V23(X) quasi_total (X,Y) Element of bool [:X, the carrier of Y:]
n is non empty Relation-like X -defined the carrier of Y -valued Function-like V23(X) quasi_total (X,Y) Element of bool [:X, the carrier of Y:]
h1 is Element of X
n . h1 is Element of the carrier of Y
m . h1 is Element of the carrier of Y
n . h1 is Element of the carrier of Y
(m . h1) + (n . h1) 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 V23([: the carrier of Y, the carrier of Y:]) 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 Relation-like set
[:[: the carrier of Y, the carrier of Y:], the carrier of Y:] is non empty Relation-like set
bool [:[: the carrier of Y, the carrier of Y:], the carrier of Y:] is non empty set
K602( the carrier of Y, the addF of Y,(m . h1),(n . h1)) is Element of the carrier of Y
[(m . h1),(n . h1)] is set
{(m . h1),(n . h1)} is set
{(m . h1)} is set
{{(m . h1),(n . h1)},{(m . h1)}} is set
the addF of Y . [(m . h1),(n . h1)] is set
X is non empty set
Y is non empty right_complementable Abelian add-associative right_zeroed discerning reflexive vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like CNORMSTR
(X,Y) is non empty CNORMSTR
(X,Y) is non empty Element of bool the carrier of (ComplexVectSpace (X,Y))
ComplexVectSpace (X,Y) is non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital CLSStruct
the carrier of Y is non empty set
Funcs (X, the carrier of Y) is non empty functional FUNCTION_DOMAIN of X, the carrier of Y
K340(X,Y) is Relation-like X -defined the carrier of Y -valued Function-like V23(X) quasi_total Element of Funcs (X, the carrier of Y)
K338(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 V23([:(Funcs (X, the carrier of Y)),(Funcs (X, the carrier of Y)):]) quasi_total Function-yielding V135() 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 Relation-like set
[:[:(Funcs (X, the carrier of Y)),(Funcs (X, the carrier of Y)):],(Funcs (X, the carrier of Y)):] is non empty Relation-like set
bool [:[:(Funcs (X, the carrier of Y)),(Funcs (X, the carrier of Y)):],(Funcs (X, the carrier of Y)):] is non empty set
FuncExtMult (X,Y) is non empty Relation-like [:COMPLEX,(Funcs (X, the carrier of Y)):] -defined Funcs (X, the carrier of Y) -valued Function-like V23([:COMPLEX,(Funcs (X, the carrier of Y)):]) quasi_total Function-yielding V135() Element of bool [:[:COMPLEX,(Funcs (X, the carrier of Y)):],(Funcs (X, the carrier of Y)):]
[:COMPLEX,(Funcs (X, the carrier of Y)):] is non empty Relation-like set
[:[:COMPLEX,(Funcs (X, the carrier of Y)):],(Funcs (X, the carrier of Y)):] is non empty Relation-like set
bool [:[:COMPLEX,(Funcs (X, the carrier of Y)):],(Funcs (X, the carrier of Y)):] is non empty set
CLSStruct(# (Funcs (X, the carrier of Y)),K340(X,Y),K338(X,Y),(FuncExtMult (X,Y)) #) is non empty strict CLSStruct
the carrier of (ComplexVectSpace (X,Y)) is non empty set
bool the carrier of (ComplexVectSpace (X,Y)) is non empty set
Zero_ ((X,Y),(ComplexVectSpace (X,Y))) is Element of (X,Y)
Add_ ((X,Y),(ComplexVectSpace (X,Y))) is non empty Relation-like [:(X,Y),(X,Y):] -defined (X,Y) -valued Function-like V23([:(X,Y),(X,Y):]) quasi_total Element of bool [:[:(X,Y),(X,Y):],(X,Y):]
[:(X,Y),(X,Y):] is non empty Relation-like set
[:[:(X,Y),(X,Y):],(X,Y):] is non empty Relation-like set
bool [:[:(X,Y),(X,Y):],(X,Y):] is non empty set
Mult_ ((X,Y),(ComplexVectSpace (X,Y))) is non empty Relation-like [:COMPLEX,(X,Y):] -defined (X,Y) -valued Function-like V23([:COMPLEX,(X,Y):]) quasi_total Element of bool [:[:COMPLEX,(X,Y):],(X,Y):]
[:COMPLEX,(X,Y):] is non empty Relation-like set
[:[:COMPLEX,(X,Y):],(X,Y):] is non empty Relation-like set
bool [:[:COMPLEX,(X,Y):],(X,Y):] is non empty set
(X,Y) is non empty Relation-like (X,Y) -defined REAL -valued Function-like V23((X,Y)) quasi_total V33() V34() V35() Element of bool [:(X,Y),REAL:]
[:(X,Y),REAL:] is non empty Relation-like V33() V34() V35() set
bool [:(X,Y),REAL:] is non empty set
CNORMSTR(# (X,Y),(Zero_ ((X,Y),(ComplexVectSpace (X,Y)))),(Add_ ((X,Y),(ComplexVectSpace (X,Y)))),(Mult_ ((X,Y),(ComplexVectSpace (X,Y)))),(X,Y) #) is non empty strict CNORMSTR
the carrier of (X,Y) is non empty set
[:X, the carrier of Y:] is non empty Relation-like set
bool [:X, the carrier of Y:] is non empty set
vseq is Element of the carrier of (X,Y)
f is Element of the carrier of (X,Y)
tseq is non empty Relation-like X -defined the carrier of Y -valued Function-like V23(X) quasi_total (X,Y) Element of bool [:X, the carrier of Y:]
tseq is non empty Relation-like X -defined the carrier of Y -valued Function-like V23(X) quasi_total (X,Y) Element of bool [:X, the carrier of Y:]
(X,Y) is non empty right_complementable Abelian add-associative right_zeroed strict vector-distributive scalar-distributive scalar-associative scalar-unital CLSStruct
CLSStruct(# (X,Y),(Zero_ ((X,Y),(ComplexVectSpace (X,Y)))),(Add_ ((X,Y),(ComplexVectSpace (X,Y)))),(Mult_ ((X,Y),(ComplexVectSpace (X,Y)))) #) is non empty right_complementable Abelian add-associative right_zeroed strict vector-distributive scalar-distributive scalar-associative scalar-unital CLSStruct
the carrier of (X,Y) is non empty set
m is V11() set
m * vseq is Element of the carrier of (X,Y)
tv is Element of the carrier of (X,Y)
e is Element of the carrier of (X,Y)
m * e is Element of the carrier of (X,Y)
[m,e] is set
{m,e} is set
{m} is V45() set
{{m,e},{m}} is set
(Mult_ ((X,Y),(ComplexVectSpace (X,Y)))) . [m,e] is set
[m,vseq] is set
{m,vseq} is set
{{m,vseq},{m}} is set
(Mult_ ((X,Y),(ComplexVectSpace (X,Y)))) . [m,vseq] is set
n is Element of X
tseq . n is Element of the carrier of Y
tseq . n is Element of the carrier of Y
m * (tseq . n) is Element of the carrier of Y
X is non empty set
Y is non empty right_complementable Abelian add-associative right_zeroed discerning reflexive vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like CNORMSTR
(X,Y) is non empty CNORMSTR
(X,Y) is non empty Element of bool the carrier of (ComplexVectSpace (X,Y))
ComplexVectSpace (X,Y) is non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital CLSStruct
the carrier of Y is non empty set
Funcs (X, the carrier of Y) is non empty functional FUNCTION_DOMAIN of X, the carrier of Y
K340(X,Y) is Relation-like X -defined the carrier of Y -valued Function-like V23(X) quasi_total Element of Funcs (X, the carrier of Y)
K338(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 V23([:(Funcs (X, the carrier of Y)),(Funcs (X, the carrier of Y)):]) quasi_total Function-yielding V135() 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 Relation-like set
[:[:(Funcs (X, the carrier of Y)),(Funcs (X, the carrier of Y)):],(Funcs (X, the carrier of Y)):] is non empty Relation-like set
bool [:[:(Funcs (X, the carrier of Y)),(Funcs (X, the carrier of Y)):],(Funcs (X, the carrier of Y)):] is non empty set
FuncExtMult (X,Y) is non empty Relation-like [:COMPLEX,(Funcs (X, the carrier of Y)):] -defined Funcs (X, the carrier of Y) -valued Function-like V23([:COMPLEX,(Funcs (X, the carrier of Y)):]) quasi_total Function-yielding V135() Element of bool [:[:COMPLEX,(Funcs (X, the carrier of Y)):],(Funcs (X, the carrier of Y)):]
[:COMPLEX,(Funcs (X, the carrier of Y)):] is non empty Relation-like set
[:[:COMPLEX,(Funcs (X, the carrier of Y)):],(Funcs (X, the carrier of Y)):] is non empty Relation-like set
bool [:[:COMPLEX,(Funcs (X, the carrier of Y)):],(Funcs (X, the carrier of Y)):] is non empty set
CLSStruct(# (Funcs (X, the carrier of Y)),K340(X,Y),K338(X,Y),(FuncExtMult (X,Y)) #) is non empty strict CLSStruct
the carrier of (ComplexVectSpace (X,Y)) is non empty set
bool the carrier of (ComplexVectSpace (X,Y)) is non empty set
Zero_ ((X,Y),(ComplexVectSpace (X,Y))) is Element of (X,Y)
Add_ ((X,Y),(ComplexVectSpace (X,Y))) is non empty Relation-like [:(X,Y),(X,Y):] -defined (X,Y) -valued Function-like V23([:(X,Y),(X,Y):]) quasi_total Element of bool [:[:(X,Y),(X,Y):],(X,Y):]
[:(X,Y),(X,Y):] is non empty Relation-like set
[:[:(X,Y),(X,Y):],(X,Y):] is non empty Relation-like set
bool [:[:(X,Y),(X,Y):],(X,Y):] is non empty set
Mult_ ((X,Y),(ComplexVectSpace (X,Y))) is non empty Relation-like [:COMPLEX,(X,Y):] -defined (X,Y) -valued Function-like V23([:COMPLEX,(X,Y):]) quasi_total Element of bool [:[:COMPLEX,(X,Y):],(X,Y):]
[:COMPLEX,(X,Y):] is non empty Relation-like set
[:[:COMPLEX,(X,Y):],(X,Y):] is non empty Relation-like set
bool [:[:COMPLEX,(X,Y):],(X,Y):] is non empty set
(X,Y) is non empty Relation-like (X,Y) -defined REAL -valued Function-like V23((X,Y)) quasi_total V33() V34() V35() Element of bool [:(X,Y),REAL:]
[:(X,Y),REAL:] is non empty Relation-like V33() V34() V35() set
bool [:(X,Y),REAL:] is non empty set
CNORMSTR(# (X,Y),(Zero_ ((X,Y),(ComplexVectSpace (X,Y)))),(Add_ ((X,Y),(ComplexVectSpace (X,Y)))),(Mult_ ((X,Y),(ComplexVectSpace (X,Y)))),(X,Y) #) is non empty strict CNORMSTR
the carrier of (X,Y) is non empty set
0. (X,Y) is V82((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 V11() real ext-real Element of REAL
the normF of (X,Y) is non empty Relation-like the carrier of (X,Y) -defined REAL -valued Function-like V23( the carrier of (X,Y)) quasi_total V33() V34() V35() Element of bool [: the carrier of (X,Y),REAL:]
[: the carrier of (X,Y),REAL:] is non empty Relation-like V33() V34() V35() set
bool [: the carrier of (X,Y),REAL:] is non empty set
the normF of (X,Y) . vseq is V11() real ext-real Element of REAL
f is Element of the carrier of (X,Y)
vseq + f is 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 V23([: the carrier of (X,Y), the carrier of (X,Y):]) 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 Relation-like set
[:[: the carrier of (X,Y), the carrier of (X,Y):], the carrier of (X,Y):] is non empty Relation-like set
bool [:[: the carrier of (X,Y), the carrier of (X,Y):], the carrier of (X,Y):] is non empty set
K602( the carrier of (X,Y), the addF of (X,Y),vseq,f) is Element of the carrier of (X,Y)
[vseq,f] is set
{vseq,f} is set
{vseq} is set
{{vseq,f},{vseq}} is set
the addF of (X,Y) . [vseq,f] is set
||.(vseq + f).|| is V11() real ext-real Element of REAL
the normF of (X,Y) . (vseq + f) is V11() real ext-real Element of REAL
||.f.|| is V11() real ext-real Element of REAL
the normF of (X,Y) . f is V11() real ext-real Element of REAL
||.vseq.|| + ||.f.|| is V11() real ext-real Element of REAL
tseq is V11() set
tseq * vseq is Element of the carrier of (X,Y)
||.(tseq * vseq).|| is V11() real ext-real Element of REAL
the normF of (X,Y) . (tseq * vseq) is V11() real ext-real Element of REAL
|.tseq.| is V11() real ext-real Element of REAL
|.tseq.| * ||.vseq.|| is V11() real ext-real Element of REAL
[:X, the carrier of Y:] is non empty Relation-like set
bool [:X, the carrier of Y:] is non empty set
0. Y is V82(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 V23(X) quasi_total Element of bool [:X, the carrier of Y:]
tseq is non empty Relation-like X -defined the carrier of Y -valued Function-like V23(X) quasi_total (X,Y) Element of bool [:X, the carrier of Y:]
(X,Y,tseq) is non empty V45() V46() V47() Element of bool REAL
{ ||.(tseq . b₁).|| where b₁ is Element of X : verum } is set
m is set
upper_bound (X,Y,tseq) is V11() real ext-real Element of REAL
e is non empty Relation-like X -defined the carrier of Y -valued Function-like V23(X) quasi_total Element of bool [:X, the carrier of Y:]
n is V11() real ext-real Element of REAL
h1 is Element of X
tseq . h1 is Element of the carrier of Y
||.(tseq . h1).|| is V11() real ext-real Element of REAL
the normF of Y is non empty Relation-like the carrier of Y -defined REAL -valued Function-like V23( the carrier of Y) quasi_total V33() V34() V35() Element of bool [: the carrier of Y,REAL:]
[: the carrier of Y,REAL:] is non empty Relation-like V33() V34() V35() set
bool [: the carrier of Y,REAL:] is non empty set
the normF of Y . (tseq . h1) is V11() real ext-real Element of REAL
||.(0. Y).|| is empty epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural V11() real Relation-like non-empty empty-yielding RAT -valued ext-real non positive non negative V33() V34() V35() V36() V45() V46() V47() V48() V49() V50() V51() Element of REAL
the normF of Y . (0. Y) is V11() real ext-real Element of REAL
n is V11() real ext-real Element of REAL
n is V11() real ext-real set
(X,Y) . vseq is V11() real ext-real Element of REAL
[:X, the carrier of Y:] is non empty Relation-like set
bool [:X, the carrier of Y:] is non empty set
e is non empty Relation-like X -defined the carrier of Y -valued Function-like V23(X) quasi_total (X,Y) Element of bool [:X, the carrier of Y:]
(X,Y,e) is non empty V45() V46() V47() Element of bool REAL
{ ||.(e . b₁).|| where b₁ is Element of X : verum } is set
upper_bound (X,Y,e) is V11() real ext-real Element of REAL
tseq is non empty Relation-like X -defined the carrier of Y -valued Function-like V23(X) quasi_total (X,Y) Element of bool [:X, the carrier of Y:]
m is Element of X
tseq . m is Element of the carrier of Y
||.(tseq . m).|| is V11() real ext-real Element of REAL
the normF of Y is non empty Relation-like the carrier of Y -defined REAL -valued Function-like V23( the carrier of Y) quasi_total V33() V34() V35() Element of bool [: the carrier of Y,REAL:]
[: the carrier of Y,REAL:] is non empty Relation-like V33() V34() V35() set
bool [: the carrier of Y,REAL:] is non empty set
the normF of Y . (tseq . m) is V11() real ext-real Element of REAL
tv is non empty Relation-like X -defined the carrier of Y -valued Function-like V23(X) quasi_total (X,Y) Element of bool [:X, the carrier of Y:]
tv . m is Element of the carrier of Y
||.(tv . m).|| is V11() real ext-real Element of REAL
the normF of Y . (tv . m) is V11() real ext-real Element of REAL
||.(tseq . m).|| + ||.(tv . m).|| is V11() real ext-real Element of REAL
e . m is Element of the carrier of Y
||.(e . m).|| is V11() real ext-real Element of REAL
the normF of Y . (e . m) is V11() real ext-real Element of REAL
(tseq . m) + (tv . 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 V23([: the carrier of Y, the carrier of Y:]) 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 Relation-like set
[:[: the carrier of Y, the carrier of Y:], the carrier of Y:] is non empty Relation-like set
bool [:[: the carrier of Y, the carrier of Y:], the carrier of Y:] is non empty set
K602( the carrier of Y, the addF of Y,(tseq . m),(tv . m)) is Element of the carrier of Y
[(tseq . m),(tv . m)] is set
{(tseq . m),(tv . m)} is set
{(tseq . m)} is set
{{(tseq . m),(tv . m)},{(tseq . m)}} is set
the addF of Y . [(tseq . m),(tv . m)] is set
||.((tseq . m) + (tv . m)).|| is V11() real ext-real Element of REAL
the normF of Y . ((tseq . m) + (tv . m)) is V11() real ext-real Element of REAL
m is V11() real ext-real Element of REAL
n is Element of X
e . n is Element of the carrier of Y
||.(e . n).|| is V11() real ext-real Element of REAL
the normF of Y . (e . n) is V11() real ext-real Element of REAL
(X,Y) . (vseq + f) is V11() real ext-real Element of REAL
m is V11() real ext-real set
[:X, the carrier of Y:] is non empty Relation-like set
bool [:X, the carrier of Y:] is non empty set
tv is non empty Relation-like X -defined the carrier of Y -valued Function-like V23(X) quasi_total (X,Y) Element of bool [:X, the carrier of Y:]
(X,Y,tv) is non empty V45() V46() V47() Element of bool REAL
{ ||.(tv . b₁).|| where b₁ is Element of X : verum } is set
upper_bound (X,Y,tv) is V11() real ext-real Element of REAL
e is Element of X
tv . e is Element of the carrier of Y
||.(tv . e).|| is V11() real ext-real Element of REAL
the normF of Y is non empty Relation-like the carrier of Y -defined REAL -valued Function-like V23( the carrier of Y) quasi_total V33() V34() V35() Element of bool [: the carrier of Y,REAL:]
[: the carrier of Y,REAL:] is non empty Relation-like V33() V34() V35() set
bool [: the carrier of Y,REAL:] is non empty set
the normF of Y . (tv . e) is V11() real ext-real Element of REAL
tseq is non empty Relation-like X -defined the carrier of Y -valued Function-like V23(X) quasi_total (X,Y) Element of bool [:X, the carrier of Y:]
tseq . e is Element of the carrier of Y
tseq * (tseq . e) is Element of the carrier of Y
||.(tseq * (tseq . e)).|| is V11() real ext-real Element of REAL
the normF of Y . (tseq * (tseq . e)) is V11() real ext-real Element of REAL
||.(tseq . e).|| is V11() real ext-real Element of REAL
the normF of Y . (tseq . e) is V11() real ext-real Element of REAL
|.tseq.| * ||.(tseq . e).|| is V11() real ext-real Element of REAL
e is V11() real ext-real Element of REAL
m is Element of X
tv . m is Element of the carrier of Y
||.(tv . m).|| is V11() real ext-real Element of REAL
the normF of Y . (tv . m) is V11() real ext-real Element of REAL
tseq " is V11() set
|.(tseq ").| is V11() real ext-real Element of REAL
1r / tseq is V11() Element of COMPLEX
|.(1r / tseq).| is V11() real ext-real Element of REAL
1 / |.tseq.| is V11() real ext-real Element of REAL
|.tseq.| " is V11() real ext-real Element of REAL
1 * (|.tseq.| ") is V11() real ext-real Element of REAL
e is Element of X
tv . e is Element of the carrier of Y
tseq . e is Element of the carrier of Y
tseq * (tseq . e) is Element of the carrier of Y
(tseq ") * (tv . e) is Element of the carrier of Y
(tseq ") * tseq is V11() set
((tseq ") * tseq) * (tseq . e) is Element of the carrier of Y
1r * (tseq . e) is Element of the carrier of Y
||.((tseq ") * (tv . e)).|| is V11() real ext-real Element of REAL
the normF of Y . ((tseq ") * (tv . e)) is V11() real ext-real Element of REAL
||.(tv . e).|| is V11() real ext-real Element of REAL
the normF of Y . (tv . e) is V11() real ext-real Element of REAL
|.(tseq ").| * ||.(tv . e).|| is V11() real ext-real Element of REAL
||.(tseq . e).|| is V11() real ext-real Element of REAL
the normF of Y . (tseq . e) is V11() real ext-real Element of REAL
(|.tseq.| ") * ||.(tseq * vseq).|| is V11() real ext-real Element of REAL
e is V11() real ext-real Element of REAL
(X,Y,tseq) is non empty V45() V46() V47() Element of bool REAL
{ ||.(tseq . b₁).|| where b₁ is Element of X : verum } is set
m is Element of X
tseq . m is Element of the carrier of Y
||.(tseq . m).|| is V11() real ext-real Element of REAL
the normF of Y . (tseq . m) is V11() real ext-real Element of REAL
upper_bound (X,Y,tseq) is V11() real ext-real Element of REAL
(X,Y) . vseq is V11() real ext-real Element of REAL
e is V11() real ext-real set
|.tseq.| * ((|.tseq.| ") * ||.(tseq * vseq).||) is V11() real ext-real Element of REAL
|.tseq.| * (|.tseq.| ") is V11() real ext-real Element of REAL
(|.tseq.| * (|.tseq.| ")) * ||.(tseq * vseq).|| is V11() real ext-real Element of REAL
1 * ||.(tseq * vseq).|| is V11() real ext-real Element of REAL
(X,Y) is non empty right_complementable Abelian add-associative right_zeroed strict vector-distributive scalar-distributive scalar-associative scalar-unital CLSStruct
CLSStruct(# (X,Y),(Zero_ ((X,Y),(ComplexVectSpace (X,Y)))),(Add_ ((X,Y),(ComplexVectSpace (X,Y)))),(Mult_ ((X,Y),(ComplexVectSpace (X,Y)))) #) is non empty right_complementable Abelian add-associative right_zeroed strict vector-distributive scalar-distributive scalar-associative scalar-unital CLSStruct
the carrier of (X,Y) is non empty set
[tseq,vseq] is set
{tseq,vseq} is set
{tseq} is V45() set
{{tseq,vseq},{tseq}} is set
(Mult_ ((X,Y),(ComplexVectSpace (X,Y)))) . [tseq,vseq] is set
e is Element of the carrier of (X,Y)
tseq * e is Element of the carrier of (X,Y)
0. (X,Y) is V82((X,Y)) Element of the carrier of (X,Y)
the ZeroF of (X,Y) is Element of the carrier of (X,Y)
(X,Y) . (tseq * vseq) is V11() real ext-real Element of REAL
e is V11() real ext-real set
[:X, the carrier of Y:] is non empty Relation-like set
bool [:X, the carrier of Y:] is non empty set
0. Y is V82(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 V23(X) quasi_total Element of bool [:X, the carrier of Y:]
tseq is non empty Relation-like X -defined the carrier of Y -valued Function-like V23(X) quasi_total (X,Y) Element of bool [:X, the carrier of Y:]
m is Element of X
tseq . m is Element of the carrier of Y
||.(tseq . m).|| is V11() real ext-real Element of REAL
the normF of Y is non empty Relation-like the carrier of Y -defined REAL -valued Function-like V23( the carrier of Y) quasi_total V33() V34() V35() Element of bool [: the carrier of Y,REAL:]
[: the carrier of Y,REAL:] is non empty Relation-like V33() V34() V35() set
bool [: the carrier of Y,REAL:] is non empty set
the normF of Y . (tseq . m) is V11() real ext-real Element of REAL
e is non empty Relation-like X -defined the carrier of Y -valued Function-like V23(X) quasi_total Element of bool [:X, the carrier of Y:]
e . m is Element of the carrier of Y
X is non empty set
Y is non empty right_complementable Abelian add-associative right_zeroed discerning reflexive vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like CNORMSTR
(X,Y) is non empty CNORMSTR
(X,Y) is non empty Element of bool the carrier of (ComplexVectSpace (X,Y))
ComplexVectSpace (X,Y) is non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital CLSStruct
the carrier of Y is non empty set
Funcs (X, the carrier of Y) is non empty functional FUNCTION_DOMAIN of X, the carrier of Y
K340(X,Y) is Relation-like X -defined the carrier of Y -valued Function-like V23(X) quasi_total Element of Funcs (X, the carrier of Y)
K338(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 V23([:(Funcs (X, the carrier of Y)),(Funcs (X, the carrier of Y)):]) quasi_total Function-yielding V135() 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 Relation-like set
[:[:(Funcs (X, the carrier of Y)),(Funcs (X, the carrier of Y)):],(Funcs (X, the carrier of Y)):] is non empty Relation-like set
bool [:[:(Funcs (X, the carrier of Y)),(Funcs (X, the carrier of Y)):],(Funcs (X, the carrier of Y)):] is non empty set
FuncExtMult (X,Y) is non empty Relation-like [:COMPLEX,(Funcs (X, the carrier of Y)):] -defined Funcs (X, the carrier of Y) -valued Function-like V23([:COMPLEX,(Funcs (X, the carrier of Y)):]) quasi_total Function-yielding V135() Element of bool [:[:COMPLEX,(Funcs (X, the carrier of Y)):],(Funcs (X, the carrier of Y)):]
[:COMPLEX,(Funcs (X, the carrier of Y)):] is non empty Relation-like set
[:[:COMPLEX,(Funcs (X, the carrier of Y)):],(Funcs (X, the carrier of Y)):] is non empty Relation-like set
bool [:[:COMPLEX,(Funcs (X, the carrier of Y)):],(Funcs (X, the carrier of Y)):] is non empty set
CLSStruct(# (Funcs (X, the carrier of Y)),K340(X,Y),K338(X,Y),(FuncExtMult (X,Y)) #) is non empty strict CLSStruct
the carrier of (ComplexVectSpace (X,Y)) is non empty set
bool the carrier of (ComplexVectSpace (X,Y)) is non empty set
Zero_ ((X,Y),(ComplexVectSpace (X,Y))) is Element of (X,Y)
Add_ ((X,Y),(ComplexVectSpace (X,Y))) is non empty Relation-like [:(X,Y),(X,Y):] -defined (X,Y) -valued Function-like V23([:(X,Y),(X,Y):]) quasi_total Element of bool [:[:(X,Y),(X,Y):],(X,Y):]
[:(X,Y),(X,Y):] is non empty Relation-like set
[:[:(X,Y),(X,Y):],(X,Y):] is non empty Relation-like set
bool [:[:(X,Y),(X,Y):],(X,Y):] is non empty set
Mult_ ((X,Y),(ComplexVectSpace (X,Y))) is non empty Relation-like [:COMPLEX,(X,Y):] -defined (X,Y) -valued Function-like V23([:COMPLEX,(X,Y):]) quasi_total Element of bool [:[:COMPLEX,(X,Y):],(X,Y):]
[:COMPLEX,(X,Y):] is non empty Relation-like set
[:[:COMPLEX,(X,Y):],(X,Y):] is non empty Relation-like set
bool [:[:COMPLEX,(X,Y):],(X,Y):] is non empty set
(X,Y) is non empty Relation-like (X,Y) -defined REAL -valued Function-like V23((X,Y)) quasi_total V33() V34() V35() Element of bool [:(X,Y),REAL:]
[:(X,Y),REAL:] is non empty Relation-like V33() V34() V35() set
bool [:(X,Y),REAL:] is non empty set
CNORMSTR(# (X,Y),(Zero_ ((X,Y),(ComplexVectSpace (X,Y)))),(Add_ ((X,Y),(ComplexVectSpace (X,Y)))),(Mult_ ((X,Y),(ComplexVectSpace (X,Y)))),(X,Y) #) is non empty strict CNORMSTR
0. (X,Y) is V82((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)
||.(0. (X,Y)).|| is V11() real ext-real Element of REAL
the normF of (X,Y) is non empty Relation-like the carrier of (X,Y) -defined REAL -valued Function-like V23( the carrier of (X,Y)) quasi_total V33() V34() V35() Element of bool [: the carrier of (X,Y),REAL:]
[: the carrier of (X,Y),REAL:] is non empty Relation-like V33() V34() V35() set
bool [: the carrier of (X,Y),REAL:] is non empty set
the normF of (X,Y) . (0. (X,Y)) is V11() real ext-real Element of REAL
vseq is Element of the carrier of (X,Y)
||.vseq.|| is V11() real ext-real Element of REAL
the normF of (X,Y) . vseq is V11() real ext-real Element of REAL
f is Element of the carrier of (X,Y)
||.f.|| is V11() real ext-real Element of REAL
the normF of (X,Y) . f is V11() real ext-real Element of REAL
tseq is Element of the carrier of (X,Y)
tseq is V11() set
tseq * tseq is Element of the carrier of (X,Y)
||.(tseq * tseq).|| is V11() real ext-real Element of REAL
the normF of (X,Y) . (tseq * tseq) is V11() real ext-real Element of REAL
|.tseq.| is V11() real ext-real Element of REAL
||.tseq.|| is V11() real ext-real Element of REAL
the normF of (X,Y) . tseq is V11() real ext-real Element of REAL
|.tseq.| * ||.tseq.|| is V11() real ext-real Element of REAL
tv is Element of the carrier of (X,Y)
e is Element of the carrier of (X,Y)
tv + e is 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 V23([: the carrier of (X,Y), the carrier of (X,Y):]) 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 Relation-like set
[:[: the carrier of (X,Y), the carrier of (X,Y):], the carrier of (X,Y):] is non empty Relation-like set
bool [:[: the carrier of (X,Y), the carrier of (X,Y):], the carrier of (X,Y):] is non empty set
K602( the carrier of (X,Y), the addF of (X,Y),tv,e) is Element of the carrier of (X,Y)
[tv,e] is set
{tv,e} is set
{tv} is set
{{tv,e},{tv}} is set
the addF of (X,Y) . [tv,e] is set
||.(tv + e).|| is V11() real ext-real Element of REAL
the normF of (X,Y) . (tv + e) is V11() real ext-real Element of REAL
||.tv.|| is V11() real ext-real Element of REAL
the normF of (X,Y) . tv is V11() real ext-real Element of REAL
||.e.|| is V11() real ext-real Element of REAL
the normF of (X,Y) . e is V11() real ext-real Element of REAL
||.tv.|| + ||.e.|| is V11() real ext-real Element of REAL
X is non empty set
Y is non empty right_complementable Abelian add-associative right_zeroed discerning reflexive vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like CNORMSTR
(X,Y) is non empty CNORMSTR
(X,Y) is non empty Element of bool the carrier of (ComplexVectSpace (X,Y))
ComplexVectSpace (X,Y) is non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital CLSStruct
the carrier of Y is non empty set
Funcs (X, the carrier of Y) is non empty functional FUNCTION_DOMAIN of X, the carrier of Y
K340(X,Y) is Relation-like X -defined the carrier of Y -valued Function-like V23(X) quasi_total Element of Funcs (X, the carrier of Y)
K338(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 V23([:(Funcs (X, the carrier of Y)),(Funcs (X, the carrier of Y)):]) quasi_total Function-yielding V135() 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 Relation-like set
[:[:(Funcs (X, the carrier of Y)),(Funcs (X, the carrier of Y)):],(Funcs (X, the carrier of Y)):] is non empty Relation-like set
bool [:[:(Funcs (X, the carrier of Y)),(Funcs (X, the carrier of Y)):],(Funcs (X, the carrier of Y)):] is non empty set
FuncExtMult (X,Y) is non empty Relation-like [:COMPLEX,(Funcs (X, the carrier of Y)):] -defined Funcs (X, the carrier of Y) -valued Function-like V23([:COMPLEX,(Funcs (X, the carrier of Y)):]) quasi_total Function-yielding V135() Element of bool [:[:COMPLEX,(Funcs (X, the carrier of Y)):],(Funcs (X, the carrier of Y)):]
[:COMPLEX,(Funcs (X, the carrier of Y)):] is non empty Relation-like set
[:[:COMPLEX,(Funcs (X, the carrier of Y)):],(Funcs (X, the carrier of Y)):] is non empty Relation-like set
bool [:[:COMPLEX,(Funcs (X, the carrier of Y)):],(Funcs (X, the carrier of Y)):] is non empty set
CLSStruct(# (Funcs (X, the carrier of Y)),K340(X,Y),K338(X,Y),(FuncExtMult (X,Y)) #) is non empty strict CLSStruct
the carrier of (ComplexVectSpace (X,Y)) is non empty set
bool the carrier of (ComplexVectSpace (X,Y)) is non empty set
Zero_ ((X,Y),(ComplexVectSpace (X,Y))) is Element of (X,Y)
Add_ ((X,Y),(ComplexVectSpace (X,Y))) is non empty Relation-like [:(X,Y),(X,Y):] -defined (X,Y) -valued Function-like V23([:(X,Y),(X,Y):]) quasi_total Element of bool [:[:(X,Y),(X,Y):],(X,Y):]
[:(X,Y),(X,Y):] is non empty Relation-like set
[:[:(X,Y),(X,Y):],(X,Y):] is non empty Relation-like set
bool [:[:(X,Y),(X,Y):],(X,Y):] is non empty set
Mult_ ((X,Y),(ComplexVectSpace (X,Y))) is non empty Relation-like [:COMPLEX,(X,Y):] -defined (X,Y) -valued Function-like V23([:COMPLEX,(X,Y):]) quasi_total Element of bool [:[:COMPLEX,(X,Y):],(X,Y):]
[:COMPLEX,(X,Y):] is non empty Relation-like set
[:[:COMPLEX,(X,Y):],(X,Y):] is non empty Relation-like set
bool [:[:COMPLEX,(X,Y):],(X,Y):] is non empty set
(X,Y) is non empty Relation-like (X,Y) -defined REAL -valued Function-like V23((X,Y)) quasi_total V33() V34() V35() Element of bool [:(X,Y),REAL:]
[:(X,Y),REAL:] is non empty Relation-like V33() V34() V35() set
bool [:(X,Y),REAL:] is non empty set
CNORMSTR(# (X,Y),(Zero_ ((X,Y),(ComplexVectSpace (X,Y)))),(Add_ ((X,Y),(ComplexVectSpace (X,Y)))),(Mult_ ((X,Y),(ComplexVectSpace (X,Y)))),(X,Y) #) is non empty strict CNORMSTR
CLSStruct(# (X,Y),(Zero_ ((X,Y),(ComplexVectSpace (X,Y)))),(Add_ ((X,Y),(ComplexVectSpace (X,Y)))),(Mult_ ((X,Y),(ComplexVectSpace (X,Y)))) #) is non empty right_complementable Abelian add-associative right_zeroed strict vector-distributive scalar-distributive scalar-associative scalar-unital CLSStruct
X is non empty set
Y is non empty right_complementable Abelian add-associative right_zeroed discerning reflexive vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like CNORMSTR
(X,Y) is non empty CNORMSTR
(X,Y) is non empty Element of bool the carrier of (ComplexVectSpace (X,Y))
ComplexVectSpace (X,Y) is non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital CLSStruct
the carrier of Y is non empty set
Funcs (X, the carrier of Y) is non empty functional FUNCTION_DOMAIN of X, the carrier of Y
K340(X,Y) is Relation-like X -defined the carrier of Y -valued Function-like V23(X) quasi_total Element of Funcs (X, the carrier of Y)
K338(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 V23([:(Funcs (X, the carrier of Y)),(Funcs (X, the carrier of Y)):]) quasi_total Function-yielding V135() 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 Relation-like set
[:[:(Funcs (X, the carrier of Y)),(Funcs (X, the carrier of Y)):],(Funcs (X, the carrier of Y)):] is non empty Relation-like set
bool [:[:(Funcs (X, the carrier of Y)),(Funcs (X, the carrier of Y)):],(Funcs (X, the carrier of Y)):] is non empty set
FuncExtMult (X,Y) is non empty Relation-like [:COMPLEX,(Funcs (X, the carrier of Y)):] -defined Funcs (X, the carrier of Y) -valued Function-like V23([:COMPLEX,(Funcs (X, the carrier of Y)):]) quasi_total Function-yielding V135() Element of bool [:[:COMPLEX,(Funcs (X, the carrier of Y)):],(Funcs (X, the carrier of Y)):]
[:COMPLEX,(Funcs (X, the carrier of Y)):] is non empty Relation-like set
[:[:COMPLEX,(Funcs (X, the carrier of Y)):],(Funcs (X, the carrier of Y)):] is non empty Relation-like set
bool [:[:COMPLEX,(Funcs (X, the carrier of Y)):],(Funcs (X, the carrier of Y)):] is non empty set
CLSStruct(# (Funcs (X, the carrier of Y)),K340(X,Y),K338(X,Y),(FuncExtMult (X,Y)) #) is non empty strict CLSStruct
the carrier of (ComplexVectSpace (X,Y)) is non empty set
bool the carrier of (ComplexVectSpace (X,Y)) is non empty set
Zero_ ((X,Y),(ComplexVectSpace (X,Y))) is Element of (X,Y)
Add_ ((X,Y),(ComplexVectSpace (X,Y))) is non empty Relation-like [:(X,Y),(X,Y):] -defined (X,Y) -valued Function-like V23([:(X,Y),(X,Y):]) quasi_total Element of bool [:[:(X,Y),(X,Y):],(X,Y):]
[:(X,Y),(X,Y):] is non empty Relation-like set
[:[:(X,Y),(X,Y):],(X,Y):] is non empty Relation-like set
bool [:[:(X,Y),(X,Y):],(X,Y):] is non empty set
Mult_ ((X,Y),(ComplexVectSpace (X,Y))) is non empty Relation-like [:COMPLEX,(X,Y):] -defined (X,Y) -valued Function-like V23([:COMPLEX,(X,Y):]) quasi_total Element of bool [:[:COMPLEX,(X,Y):],(X,Y):]
[:COMPLEX,(X,Y):] is non empty Relation-like set
[:[:COMPLEX,(X,Y):],(X,Y):] is non empty Relation-like set
bool [:[:COMPLEX,(X,Y):],(X,Y):] is non empty set
(X,Y) is non empty Relation-like (X,Y) -defined REAL -valued Function-like V23((X,Y)) quasi_total V33() V34() V35() Element of bool [:(X,Y),REAL:]
[:(X,Y),REAL:] is non empty Relation-like V33() V34() V35() set
bool [:(X,Y),REAL:] is non empty set
CNORMSTR(# (X,Y),(Zero_ ((X,Y),(ComplexVectSpace (X,Y)))),(Add_ ((X,Y),(ComplexVectSpace (X,Y)))),(Mult_ ((X,Y),(ComplexVectSpace (X,Y)))),(X,Y) #) is non empty strict CNORMSTR
X is non empty set
Y is non empty right_complementable Abelian add-associative right_zeroed discerning reflexive vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like CNORMSTR
(X,Y) is non empty right_complementable Abelian add-associative right_zeroed discerning reflexive vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like CNORMSTR
(X,Y) is non empty Element of bool the carrier of (ComplexVectSpace (X,Y))
ComplexVectSpace (X,Y) is non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital CLSStruct
the carrier of Y is non empty set
Funcs (X, the carrier of Y) is non empty functional FUNCTION_DOMAIN of X, the carrier of Y
K340(X,Y) is Relation-like X -defined the carrier of Y -valued Function-like V23(X) quasi_total Element of Funcs (X, the carrier of Y)
K338(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 V23([:(Funcs (X, the carrier of Y)),(Funcs (X, the carrier of Y)):]) quasi_total Function-yielding V135() 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 Relation-like set
[:[:(Funcs (X, the carrier of Y)),(Funcs (X, the carrier of Y)):],(Funcs (X, the carrier of Y)):] is non empty Relation-like set
bool [:[:(Funcs (X, the carrier of Y)),(Funcs (X, the carrier of Y)):],(Funcs (X, the carrier of Y)):] is non empty set
FuncExtMult (X,Y) is non empty Relation-like [:COMPLEX,(Funcs (X, the carrier of Y)):] -defined Funcs (X, the carrier of Y) -valued Function-like V23([:COMPLEX,(Funcs (X, the carrier of Y)):]) quasi_total Function-yielding V135() Element of bool [:[:COMPLEX,(Funcs (X, the carrier of Y)):],(Funcs (X, the carrier of Y)):]
[:COMPLEX,(Funcs (X, the carrier of Y)):] is non empty Relation-like set
[:[:COMPLEX,(Funcs (X, the carrier of Y)):],(Funcs (X, the carrier of Y)):] is non empty Relation-like set
bool [:[:COMPLEX,(Funcs (X, the carrier of Y)):],(Funcs (X, the carrier of Y)):] is non empty set
CLSStruct(# (Funcs (X, the carrier of Y)),K340(X,Y),K338(X,Y),(FuncExtMult (X,Y)) #) is non empty strict CLSStruct
the carrier of (ComplexVectSpace (X,Y)) is non empty set
bool the carrier of (ComplexVectSpace (X,Y)) is non empty set
Zero_ ((X,Y),(ComplexVectSpace (X,Y))) is Element of (X,Y)
Add_ ((X,Y),(ComplexVectSpace (X,Y))) is non empty Relation-like [:(X,Y),(X,Y):] -defined (X,Y) -valued Function-like V23([:(X,Y),(X,Y):]) quasi_total Element of bool [:[:(X,Y),(X,Y):],(X,Y):]
[:(X,Y),(X,Y):] is non empty Relation-like set
[:[:(X,Y),(X,Y):],(X,Y):] is non empty Relation-like set
bool [:[:(X,Y),(X,Y):],(X,Y):] is non empty set
Mult_ ((X,Y),(ComplexVectSpace (X,Y))) is non empty Relation-like [:COMPLEX,(X,Y):] -defined (X,Y) -valued Function-like V23([:COMPLEX,(X,Y):]) quasi_total Element of bool [:[:COMPLEX,(X,Y):],(X,Y):]
[:COMPLEX,(X,Y):] is non empty Relation-like set
[:[:COMPLEX,(X,Y):],(X,Y):] is non empty Relation-like set
bool [:[:COMPLEX,(X,Y):],(X,Y):] is non empty set
(X,Y) is non empty Relation-like (X,Y) -defined REAL -valued Function-like V23((X,Y)) quasi_total V33() V34() V35() Element of bool [:(X,Y),REAL:]
[:(X,Y),REAL:] is non empty Relation-like V33() V34() V35() set
bool [:(X,Y),REAL:] is non empty set
CNORMSTR(# (X,Y),(Zero_ ((X,Y),(ComplexVectSpace (X,Y)))),(Add_ ((X,Y),(ComplexVectSpace (X,Y)))),(Mult_ ((X,Y),(ComplexVectSpace (X,Y)))),(X,Y) #) is non empty strict CNORMSTR
the carrier of (X,Y) is non empty set
[:X, the carrier of Y:] is non empty Relation-like set
bool [:X, the carrier of Y:] is non empty set
vseq is Element of the carrier of (X,Y)
f is Element of the carrier of (X,Y)
tseq is Element of the carrier of (X,Y)
vseq - f is Element of the carrier of (X,Y)
- f is Element of the carrier of (X,Y)
vseq + (- f) is 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 V23([: the carrier of (X,Y), the carrier of (X,Y):]) 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 Relation-like set
[:[: the carrier of (X,Y), the carrier of (X,Y):], the carrier of (X,Y):] is non empty Relation-like set
bool [:[: the carrier of (X,Y), the carrier of (X,Y):], the carrier of (X,Y):] is non empty set
K602( the carrier of (X,Y), the addF of (X,Y),vseq,(- f)) is Element of the carrier of (X,Y)
[vseq,(- f)] is set
{vseq,(- f)} is set
{vseq} is set
{{vseq,(- f)},{vseq}} is set
the addF of (X,Y) . [vseq,(- f)] is set
tseq is non empty Relation-like X -defined the carrier of Y -valued Function-like V23(X) quasi_total (X,Y) Element of bool [:X, the carrier of Y:]
tv is non empty Relation-like X -defined the carrier of Y -valued Function-like V23(X) quasi_total (X,Y) Element of bool [:X, the carrier of Y:]
e is non empty Relation-like X -defined the carrier of Y -valued Function-like V23(X) quasi_total (X,Y) Element of bool [:X, the carrier of Y:]
m is Element of X
e . m is Element of the carrier of Y
tseq . m is Element of the carrier of Y
tv . m is Element of the carrier of Y
(tseq . m) - (tv . m) is Element of the carrier of Y
- (tv . m) is Element of the carrier of Y
(tseq . m) + (- (tv . 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 V23([: the carrier of Y, the carrier of Y:]) 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 Relation-like set
[:[: the carrier of Y, the carrier of Y:], the carrier of Y:] is non empty Relation-like set
bool [:[: the carrier of Y, the carrier of Y:], the carrier of Y:] is non empty set
K602( the carrier of Y, the addF of Y,(tseq . m),(- (tv . m))) is Element of the carrier of Y
[(tseq . m),(- (tv . m))] is set
{(tseq . m),(- (tv . m))} is set
{(tseq . m)} is set
{{(tseq . m),(- (tv . m))},{(tseq . m)}} is set
the addF of Y . [(tseq . m),(- (tv . m))] is set
(e . m) + (tv . m) is Element of the carrier of Y
K602( the carrier of Y, the addF of Y,(e . m),(tv . m)) is Element of the carrier of Y
[(e . m),(tv . m)] is set
{(e . m),(tv . m)} is set
{(e . m)} is set
{{(e . m),(tv . m)},{(e . m)}} is set
the addF of Y . [(e . m),(tv . m)] is set
(tv . m) - (tv . m) is Element of the carrier of Y
(tv . m) + (- (tv . m)) is Element of the carrier of Y
K602( the carrier of Y, the addF of Y,(tv . m),(- (tv . m))) is Element of the carrier of Y
[(tv . m),(- (tv . m))] is set
{(tv . m),(- (tv . m))} is set
{(tv . m)} is set
{{(tv . m),(- (tv . m))},{(tv . m)}} is set
the addF of Y . [(tv . m),(- (tv . m))] is set
(tseq . m) - ((tv . m) - (tv . m)) is Element of the carrier of Y
- ((tv . m) - (tv . m)) is Element of the carrier of Y
(tseq . m) + (- ((tv . m) - (tv . m))) is Element of the carrier of Y
K602( the carrier of Y, the addF of Y,(tseq . m),(- ((tv . m) - (tv . m)))) is Element of the carrier of Y
[(tseq . m),(- ((tv . m) - (tv . m)))] is set
{(tseq . m),(- ((tv . m) - (tv . m)))} is set
{{(tseq . m),(- ((tv . m) - (tv . m)))},{(tseq . m)}} is set
the addF of Y . [(tseq . m),(- ((tv . m) - (tv . m)))] is set
0. Y is V82(Y) Element of the carrier of Y
the ZeroF of Y is Element of the carrier of Y
(tseq . m) - (0. Y) is Element of the carrier of Y
- (0. Y) is Element of the carrier of Y
(tseq . m) + (- (0. Y)) is Element of the carrier of Y
K602( the carrier of Y, the addF of Y,(tseq . m),(- (0. Y))) is Element of the carrier of Y
[(tseq . m),(- (0. Y))] is set
{(tseq . m),(- (0. Y))} is set
{{(tseq . m),(- (0. Y))},{(tseq . m)}} is set
the addF of Y . [(tseq . m),(- (0. Y))] is set
tseq + f is Element of the carrier of (X,Y)
K602( the carrier of (X,Y), the addF of (X,Y),tseq,f) is Element of the carrier of (X,Y)
[tseq,f] is set
{tseq,f} is set
{tseq} is set
{{tseq,f},{tseq}} is set
the addF of (X,Y) . [tseq,f] is set
f - f is Element of the carrier of (X,Y)
f + (- f) is Element of the carrier of (X,Y)
K602( the carrier of (X,Y), the addF of (X,Y),f,(- f)) is Element of the carrier of (X,Y)
[f,(- f)] is set
{f,(- f)} is set
{f} is set
{{f,(- f)},{f}} is set
the addF of (X,Y) . [f,(- f)] is set
tseq + (f - f) is Element of the carrier of (X,Y)
K602( the carrier of (X,Y), the addF of (X,Y),tseq,(f - f)) is Element of the carrier of (X,Y)
[tseq,(f - f)] is set
{tseq,(f - f)} is set
{{tseq,(f - f)},{tseq}} is set
the addF of (X,Y) . [tseq,(f - f)] is set
0. (X,Y) is V82((X,Y)) Element of the carrier of (X,Y)
the ZeroF of (X,Y) is Element of the carrier of (X,Y)
tseq + (0. (X,Y)) is Element of the carrier of (X,Y)
K602( the carrier of (X,Y), the addF of (X,Y),tseq,(0. (X,Y))) is Element of the carrier of (X,Y)
[tseq,(0. (X,Y))] is set
{tseq,(0. (X,Y))} is set
{{tseq,(0. (X,Y))},{tseq}} is set
the addF of (X,Y) . [tseq,(0. (X,Y))] is set
vseq - (f - f) is Element of the carrier of (X,Y)
- (f - f) is Element of the carrier of (X,Y)
vseq + (- (f - f)) is Element of the carrier of (X,Y)
K602( the carrier of (X,Y), the addF of (X,Y),vseq,(- (f - f))) is Element of the carrier of (X,Y)
[vseq,(- (f - f))] is set
{vseq,(- (f - f))} is set
{{vseq,(- (f - f))},{vseq}} is set
the addF of (X,Y) . [vseq,(- (f - f))] is set
vseq - (0. (X,Y)) is Element of the carrier of (X,Y)
- (0. (X,Y)) is Element of the carrier of (X,Y)
vseq + (- (0. (X,Y))) is Element of the carrier of (X,Y)
K602( the carrier of (X,Y), the addF of (X,Y),vseq,(- (0. (X,Y)))) is Element of the carrier of (X,Y)
[vseq,(- (0. (X,Y)))] is set
{vseq,(- (0. (X,Y)))} is set
{{vseq,(- (0. (X,Y)))},{vseq}} is set
the addF of (X,Y) . [vseq,(- (0. (X,Y)))] is set
m is Element of X
tseq . m is Element of the carrier of Y
e . m is Element of the carrier of Y
tv . m is Element of the carrier of Y
(e . m) + (tv . m) is Element of the carrier of Y
K602( the carrier of Y, the addF of Y,(e . m),(tv . m)) is Element of the carrier of Y
[(e . m),(tv . m)] is set
{(e . m),(tv . m)} is set
{(e . m)} is set
{{(e . m),(tv . m)},{(e . m)}} is set
the addF of Y . [(e . m),(tv . m)] is set
(tseq . m) - (tv . m) is Element of the carrier of Y
- (tv . m) is Element of the carrier of Y
(tseq . m) + (- (tv . m)) is Element of the carrier of Y
K602( the carrier of Y, the addF of Y,(tseq . m),(- (tv . m))) is Element of the carrier of Y
[(tseq . m),(- (tv . m))] is set
{(tseq . m),(- (tv . m))} is set
{(tseq . m)} is set
{{(tseq . m),(- (tv . m))},{(tseq . m)}} is set
the addF of Y . [(tseq . m),(- (tv . m))] is set
(tv . m) - (tv . m) is Element of the carrier of Y
(tv . m) + (- (tv . m)) is Element of the carrier of Y
K602( the carrier of Y, the addF of Y,(tv . m),(- (tv . m))) is Element of the carrier of Y
[(tv . m),(- (tv . m))] is set
{(tv . m),(- (tv . m))} is set
{(tv . m)} is set
{{(tv . m),(- (tv . m))},{(tv . m)}} is set
the addF of Y . [(tv . m),(- (tv . m))] is set
(e . m) + ((tv . m) - (tv . m)) is Element of the carrier of Y
K602( the carrier of Y, the addF of Y,(e . m),((tv . m) - (tv . m))) is Element of the carrier of Y
[(e . m),((tv . m) - (tv . m))] is set
{(e . m),((tv . m) - (tv . m))} is set
{{(e . m),((tv . m) - (tv . m))},{(e . m)}} is set
the addF of Y . [(e . m),((tv . m) - (tv . m))] is set
(e . m) + (0. Y) is Element of the carrier of Y
K602( the carrier of Y, the addF of Y,(e . m),(0. Y)) is Element of the carrier of Y
[(e . m),(0. Y)] is set
{(e . m),(0. Y)} is set
{{(e . m),(0. Y)},{(e . m)}} is set
the addF of Y . [(e . m),(0. Y)] is set
m is Element of X
e . m is Element of the carrier of Y
tseq . m is Element of the carrier of Y
tv . m is Element of the carrier of Y
(tseq . m) - (tv . m) is Element of the carrier of Y
- (tv . m) is Element of the carrier of Y
(tseq . m) + (- (tv . m)) is Element of the carrier of Y
K602( the carrier of Y, the addF of Y,(tseq . m),(- (tv . m))) is Element of the carrier of Y
[(tseq . m),(- (tv . m))] is set
{(tseq . m),(- (tv . m))} is set
{(tseq . m)} is set
{{(tseq . m),(- (tv . m))},{(tseq . m)}} is set
the addF of Y . [(tseq . m),(- (tv . m))] is set
m is Element of X
e . m is Element of the carrier of Y
tseq . m is Element of the carrier of Y
tv . m is Element of the carrier of Y
(tseq . m) - (tv . m) is Element of the carrier of Y
- (tv . m) is Element of the carrier of Y
(tseq . m) + (- (tv . m)) is Element of the carrier of Y
K602( the carrier of Y, the addF of Y,(tseq . m),(- (tv . m))) is Element of the carrier of Y
[(tseq . m),(- (tv . m))] is set
{(tseq . m),(- (tv . m))} is set
{(tseq . m)} is set
{{(tseq . m),(- (tv . m))},{(tseq . m)}} is set
the addF of Y . [(tseq . m),(- (tv . m))] is set
n is Element of X
e . n is Element of the carrier of Y
tseq . n is Element of the carrier of Y
tv . n is Element of the carrier of Y
(tseq . n) - (tv . n) is Element of the carrier of Y
- (tv . n) is Element of the carrier of Y
(tseq . n) + (- (tv . n)) is Element of the carrier of Y
K602( the carrier of Y, the addF of Y,(tseq . n),(- (tv . n))) is Element of the carrier of Y
[(tseq . n),(- (tv . n))] is set
{(tseq . n),(- (tv . n))} is set
{(tseq . n)} is set
{{(tseq . n),(- (tv . n))},{(tseq . n)}} is set
the addF of Y . [(tseq . n),(- (tv . n))] is set
X is V11() real ext-real Element of REAL
Y is non empty Relation-like NAT -defined REAL -valued Function-like V23( NAT ) quasi_total V33() V34() V35() Element of bool [:NAT,REAL:]
lim Y is V11() real ext-real Element of REAL
vseq is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real V45() V46() V47() V48() V49() V50() V136() V181() Element of NAT
vseq is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real V45() V46() V47() V48() V49() V50() V136() V181() Element of NAT
NAT --> X is non empty Relation-like NAT -defined REAL -valued Function-like constant V23( NAT ) quasi_total V33() V34() V35() bounded convergent bounded_above bounded_below Element of bool [:NAT,REAL:]
f is non empty Relation-like NAT -defined REAL -valued Function-like V23( NAT ) quasi_total V33() V34() V35() Element of bool [:NAT,REAL:]
lim f is V11() real ext-real Element of REAL
f . 0 is V11() real ext-real Element of REAL
Y ^\ vseq is non empty Relation-like NAT -defined REAL -valued Function-like V23( NAT ) quasi_total V33() V34() V35() Element of bool [:NAT,REAL:]
tseq is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real V45() V46() V47() V48() V49() V50() V136() V181() Element of NAT
(Y ^\ vseq) . tseq is V11() real ext-real Element of REAL
vseq + tseq is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real V45() V46() V47() V48() V49() V50() V136() V181() Element of NAT
Y . (vseq + tseq) is V11() real ext-real Element of REAL
f . tseq is V11() real ext-real Element of REAL
lim (Y ^\ vseq) is V11() real ext-real Element of REAL
X is non empty set
Y is non empty right_complementable Abelian add-associative right_zeroed discerning reflexive vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like CNORMSTR
(X,Y) is non empty right_complementable Abelian add-associative right_zeroed discerning reflexive vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like CNORMSTR
(X,Y) is non empty Element of bool the carrier of (ComplexVectSpace (X,Y))
ComplexVectSpace (X,Y) is non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital CLSStruct
the carrier of Y is non empty set
Funcs (X, the carrier of Y) is non empty functional FUNCTION_DOMAIN of X, the carrier of Y
K340(X,Y) is Relation-like X -defined the carrier of Y -valued Function-like V23(X) quasi_total Element of Funcs (X, the carrier of Y)
K338(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 V23([:(Funcs (X, the carrier of Y)),(Funcs (X, the carrier of Y)):]) quasi_total Function-yielding V135() 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 Relation-like set
[:[:(Funcs (X, the carrier of Y)),(Funcs (X, the carrier of Y)):],(Funcs (X, the carrier of Y)):] is non empty Relation-like set
bool [:[:(Funcs (X, the carrier of Y)),(Funcs (X, the carrier of Y)):],(Funcs (X, the carrier of Y)):] is non empty set
FuncExtMult (X,Y) is non empty Relation-like [:COMPLEX,(Funcs (X, the carrier of Y)):] -defined Funcs (X, the carrier of Y) -valued Function-like V23([:COMPLEX,(Funcs (X, the carrier of Y)):]) quasi_total Function-yielding V135() Element of bool [:[:COMPLEX,(Funcs (X, the carrier of Y)):],(Funcs (X, the carrier of Y)):]
[:COMPLEX,(Funcs (X, the carrier of Y)):] is non empty Relation-like set
[:[:COMPLEX,(Funcs (X, the carrier of Y)):],(Funcs (X, the carrier of Y)):] is non empty Relation-like set
bool [:[:COMPLEX,(Funcs (X, the carrier of Y)):],(Funcs (X, the carrier of Y)):] is non empty set
CLSStruct(# (Funcs (X, the carrier of Y)),K340(X,Y),K338(X,Y),(FuncExtMult (X,Y)) #) is non empty strict CLSStruct
the carrier of (ComplexVectSpace (X,Y)) is non empty set
bool the carrier of (ComplexVectSpace (X,Y)) is non empty set
Zero_ ((X,Y),(ComplexVectSpace (X,Y))) is Element of (X,Y)
Add_ ((X,Y),(ComplexVectSpace (X,Y))) is non empty Relation-like [:(X,Y),(X,Y):] -defined (X,Y) -valued Function-like V23([:(X,Y),(X,Y):]) quasi_total Element of bool [:[:(X,Y),(X,Y):],(X,Y):]
[:(X,Y),(X,Y):] is non empty Relation-like set
[:[:(X,Y),(X,Y):],(X,Y):] is non empty Relation-like set
bool [:[:(X,Y),(X,Y):],(X,Y):] is non empty set
Mult_ ((X,Y),(ComplexVectSpace (X,Y))) is non empty Relation-like [:COMPLEX,(X,Y):] -defined (X,Y) -valued Function-like V23([:COMPLEX,(X,Y):]) quasi_total Element of bool [:[:COMPLEX,(X,Y):],(X,Y):]
[:COMPLEX,(X,Y):] is non empty Relation-like set
[:[:COMPLEX,(X,Y):],(X,Y):] is non empty Relation-like set
bool [:[:COMPLEX,(X,Y):],(X,Y):] is non empty set
(X,Y) is non empty Relation-like (X,Y) -defined REAL -valued Function-like V23((X,Y)) quasi_total V33() V34() V35() Element of bool [:(X,Y),REAL:]
[:(X,Y),REAL:] is non empty Relation-like V33() V34() V35() set
bool [:(X,Y),REAL:] is non empty set
CNORMSTR(# (X,Y),(Zero_ ((X,Y),(ComplexVectSpace (X,Y)))),(Add_ ((X,Y),(ComplexVectSpace (X,Y)))),(Mult_ ((X,Y),(ComplexVectSpace (X,Y)))),(X,Y) #) is non empty strict CNORMSTR
the carrier of (X,Y) is non empty set
[:NAT, the carrier of (X,Y):] is non empty Relation-like 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 V23( NAT ) quasi_total Element of bool [:NAT, the carrier of (X,Y):]
[:NAT, the carrier of Y:] is non empty Relation-like set
bool [:NAT, the carrier of Y:] is non empty set
f is Element of X
tseq is non empty Relation-like NAT -defined the carrier of Y -valued Function-like V23( NAT ) quasi_total Element of bool [:NAT, the carrier of Y:]
lim tseq is Element of the carrier of Y
tseq is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real V45() V46() V47() V48() V49() V50() V136() V181() Element of NAT
tseq . tseq is Element of the carrier of Y
tv is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real V45() V46() V47() V48() V49() V50() V136() V181() Element of NAT
tseq . tv is Element of the carrier of Y
(tseq . tseq) - (tseq . tv) is Element of the carrier of Y
- (tseq . tv) is Element of the carrier of Y
(tseq . tseq) + (- (tseq . tv)) is Element of the carrier of Y
the addF of Y is non empty Relation-like [: the carrier of Y, the carrier of Y:] -defined the carrier of Y -valued Function-like V23([: the carrier of Y, the carrier of Y:]) 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 Relation-like set
[:[: the carrier of Y, the carrier of Y:], the carrier of Y:] is non empty Relation-like set
bool [:[: the carrier of Y, the carrier of Y:], the carrier of Y:] is non empty set
K602( the carrier of Y, the addF of Y,(tseq . tseq),(- (tseq . tv))) is Element of the carrier of Y
[(tseq . tseq),(- (tseq . tv))] is set
{(tseq . tseq),(- (tseq . tv))} is set
{(tseq . tseq)} is set
{{(tseq . tseq),(- (tseq . tv))},{(tseq . tseq)}} is set
the addF of Y . [(tseq . tseq),(- (tseq . tv))] is set
||.((tseq . tseq) - (tseq . tv)).|| is V11() real ext-real Element of REAL
the normF of Y is non empty Relation-like the carrier of Y -defined REAL -valued Function-like V23( the carrier of Y) quasi_total V33() V34() V35() Element of bool [: the carrier of Y,REAL:]
[: the carrier of Y,REAL:] is non empty Relation-like V33() V34() V35() set
bool [: the carrier of Y,REAL:] is non empty set
the normF of Y . ((tseq . tseq) - (tseq . tv)) is V11() real ext-real Element of REAL
vseq . tseq is Element of the carrier of (X,Y)
vseq . tv is Element of the carrier of (X,Y)
(vseq . tseq) - (vseq . tv) is Element of the carrier of (X,Y)
- (vseq . tv) is Element of the carrier of (X,Y)
(vseq . tseq) + (- (vseq . tv)) is 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 V23([: the carrier of (X,Y), the carrier of (X,Y):]) 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 Relation-like set
[:[: the carrier of (X,Y), the carrier of (X,Y):], the carrier of (X,Y):] is non empty Relation-like set
bool [:[: the carrier of (X,Y), the carrier of (X,Y):], the carrier of (X,Y):] is non empty set
K602( the carrier of (X,Y), the addF of (X,Y),(vseq . tseq),(- (vseq . tv))) is Element of the carrier of (X,Y)
[(vseq . tseq),(- (vseq . tv))] is set
{(vseq . tseq),(- (vseq . tv))} is set
{(vseq . tseq)} is set
{{(vseq . tseq),(- (vseq . tv))},{(vseq . tseq)}} is set
the addF of (X,Y) . [(vseq . tseq),(- (vseq . tv))] is set
||.((vseq . tseq) - (vseq . tv)).|| is V11() real ext-real Element of REAL
the normF of (X,Y) is non empty Relation-like the carrier of (X,Y) -defined REAL -valued Function-like V23( the carrier of (X,Y)) quasi_total V33() V34() V35() Element of bool [: the carrier of (X,Y),REAL:]
[: the carrier of (X,Y),REAL:] is non empty Relation-like V33() V34() V35() set
bool [: the carrier of (X,Y),REAL:] is non empty set
the normF of (X,Y) . ((vseq . tseq) - (vseq . tv)) is V11() real ext-real Element of REAL
[:X, the carrier of Y:] is non empty Relation-like set
bool [:X, the carrier of Y:] is non empty set
(X,Y,(vseq . tv)) is non empty Relation-like X -defined the carrier of Y -valued Function-like V23(X) quasi_total (X,Y) Element of bool [:X, the carrier of Y:]
(X,Y,(vseq . tseq)) is non empty Relation-like X -defined the carrier of Y -valued Function-like V23(X) quasi_total (X,Y) Element of bool [:X, the carrier of Y:]
(X,Y,(vseq . tseq)) . f is Element of the carrier of Y
(X,Y,(vseq . tv)) . f is Element of the carrier of Y
e is non empty Relation-like X -defined the carrier of Y -valued Function-like V23(X) quasi_total (X,Y) Element of bool [:X, the carrier of Y:]
e . f is Element of the carrier of Y
tseq is V11() real ext-real Element of REAL
tv is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real V45() V46() V47() V48() V49() V50() V136() V181() Element of NAT
e is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real V45() V46() V47() V48() V49() V50() V136() V181() Element of NAT
m is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real V45() V46() V47() V48() V49() V50() V136() V181() 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 V23([: the carrier of Y, the carrier of Y:]) 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 Relation-like set
[:[: the carrier of Y, the carrier of Y:], the carrier of Y:] is non empty Relation-like set
bool [:[: the carrier of Y, the carrier of Y:], the carrier of Y:] is non empty set
K602( 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
{(tseq . e),(- (tseq . m))} is set
{(tseq . e)} is set
{{(tseq . e),(- (tseq . m))},{(tseq . e)}} is set
the addF of Y . [(tseq . e),(- (tseq . m))] is set
||.((tseq . e) - (tseq . m)).|| is V11() real ext-real Element of REAL
the normF of Y is non empty Relation-like the carrier of Y -defined REAL -valued Function-like V23( the carrier of Y) quasi_total V33() V34() V35() Element of bool [: the carrier of Y,REAL:]
[: the carrier of Y,REAL:] is non empty Relation-like V33() V34() V35() set
bool [: the carrier of Y,REAL:] is non empty set
the normF of Y . ((tseq . e) - (tseq . m)) is V11() real ext-real Element of REAL
vseq . e is Element of the carrier of (X,Y)
vseq . m is Element of the carrier of (X,Y)
(vseq . e) - (vseq . m) is Element of the carrier of (X,Y)
- (vseq . m) is Element of the carrier of (X,Y)
(vseq . e) + (- (vseq . m)) is 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 V23([: the carrier of (X,Y), the carrier of (X,Y):]) 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 Relation-like set
[:[: the carrier of (X,Y), the carrier of (X,Y):], the carrier of (X,Y):] is non empty Relation-like set
bool [:[: the carrier of (X,Y), the carrier of (X,Y):], the carrier of (X,Y):] is non empty set
K602( the carrier of (X,Y), the addF of (X,Y),(vseq . e),(- (vseq . m))) is Element of the carrier of (X,Y)
[(vseq . e),(- (vseq . m))] is set
{(vseq . e),(- (vseq . m))} is set
{(vseq . e)} is set
{{(vseq . e),(- (vseq . m))},{(vseq . e)}} is set
the addF of (X,Y) . [(vseq . e),(- (vseq . m))] is set
||.((vseq . e) - (vseq . m)).|| is V11() real ext-real Element of REAL
the normF of (X,Y) is non empty Relation-like the carrier of (X,Y) -defined REAL -valued Function-like V23( the carrier of (X,Y)) quasi_total V33() V34() V35() Element of bool [: the carrier of (X,Y),REAL:]
[: the carrier of (X,Y),REAL:] is non empty Relation-like V33() V34() V35() set
bool [: the carrier of (X,Y),REAL:] is non empty set
the normF of (X,Y) . ((vseq . e) - (vseq . m)) is V11() real ext-real Element of REAL
[:X, the carrier of Y:] is non empty Relation-like set
bool [:X, the carrier of Y:] is non empty set
f is non empty Relation-like X -defined the carrier of Y -valued Function-like V23(X) quasi_total Element of bool [:X, the carrier of Y:]
tseq is V11() real ext-real set
tv is V11() real ext-real Element of REAL
e is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real V45() V46() V47() V48() V49() V50() V136() V181() Element of NAT
m is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real V45() V46() V47() V48() V49() V50() V136() V181() Element of NAT
n is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real V45() V46() V47() V48() V49() V50() V136() V181() Element of NAT
vseq . n is Element of the carrier of (X,Y)
||.(vseq . n).|| is V11() real ext-real Element of REAL
the normF of (X,Y) is non empty Relation-like the carrier of (X,Y) -defined REAL -valued Function-like V23( the carrier of (X,Y)) quasi_total V33() V34() V35() Element of bool [: the carrier of (X,Y),REAL:]
[: the carrier of (X,Y),REAL:] is non empty Relation-like V33() V34() V35() set
bool [: the carrier of (X,Y),REAL:] is non empty set
the normF of (X,Y) . (vseq . n) is V11() real ext-real Element of REAL
||.vseq.|| is non empty Relation-like NAT -defined REAL -valued Function-like V23( NAT ) quasi_total V33() V34() V35() Element of bool [:NAT,REAL:]
||.vseq.|| . n is V11() real ext-real Element of REAL
vseq . m is Element of the carrier of (X,Y)
(vseq . n) - (vseq . m) is Element of the carrier of (X,Y)
- (vseq . m) is Element of the carrier of (X,Y)
(vseq . n) + (- (vseq . m)) is 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 V23([: the carrier of (X,Y), the carrier of (X,Y):]) 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 Relation-like set
[:[: the carrier of (X,Y), the carrier of (X,Y):], the carrier of (X,Y):] is non empty Relation-like set
bool [:[: the carrier of (X,Y), the carrier of (X,Y):], the carrier of (X,Y):] is non empty set
K602( the carrier of (X,Y), the addF of (X,Y),(vseq . n),(- (vseq . m))) is Element of the carrier of (X,Y)
[(vseq . n),(- (vseq . m))] is set
{(vseq . n),(- (vseq . m))} is set
{(vseq . n)} is set
{{(vseq . n),(- (vseq . m))},{(vseq . n)}} is set
the addF of (X,Y) . [(vseq . n),(- (vseq . m))] is set
||.((vseq . n) - (vseq . m)).|| is V11() real ext-real Element of REAL
the normF of (X,Y) . ((vseq . n) - (vseq . m)) is V11() real ext-real Element of REAL
||.(vseq . m).|| is V11() real ext-real Element of REAL
the normF of (X,Y) . (vseq . m) is V11() real ext-real Element of REAL
||.(vseq . n).|| - ||.(vseq . m).|| is V11() real ext-real Element of REAL
abs (||.(vseq . n).|| - ||.(vseq . m).||) is V11() real ext-real Element of REAL
||.vseq.|| . m is V11() real ext-real Element of REAL
(||.vseq.|| . n) - (||.vseq.|| . m) is V11() real ext-real Element of REAL
abs ((||.vseq.|| . n) - (||.vseq.|| . m)) is V11() real ext-real Element of REAL
n is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real V45() V46() V47() V48() V49() V50() V136() V181() Element of NAT
||.vseq.|| . n is V11() real ext-real Element of REAL
(||.vseq.|| . n) - (||.vseq.|| . m) is V11() real ext-real Element of REAL
abs ((||.vseq.|| . n) - (||.vseq.|| . m)) is V11() real ext-real Element of REAL
tseq is non empty Relation-like X -defined the carrier of Y -valued Function-like V23(X) quasi_total Element of bool [:X, the carrier of Y:]
lim ||.vseq.|| is V11() real ext-real Element of REAL
tseq is Element 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 V23( NAT ) quasi_total Element of bool [:NAT, the carrier of Y:]
lim tv is Element of the carrier of Y
e is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real V45() V46() V47() V48() V49() V50() V136() V181() Element of NAT
tv . e is Element of the carrier of Y
||.(tv . e).|| is V11() real ext-real Element of REAL
the normF of Y is non empty Relation-like the carrier of Y -defined REAL -valued Function-like V23( the carrier of Y) quasi_total V33() V34() V35() Element of bool [: the carrier of Y,REAL:]
[: the carrier of Y,REAL:] is non empty Relation-like V33() V34() V35() set
bool [: the carrier of Y,REAL:] is non empty set
the normF of Y . (tv . e) is V11() real ext-real Element of REAL
vseq . e is Element of the carrier of (X,Y)
||.(vseq . e).|| is V11() real ext-real Element of REAL
the normF of (X,Y) . (vseq . e) is V11() real ext-real Element of REAL
(X,Y,(vseq . e)) is non empty Relation-like X -defined the carrier of Y -valued Function-like V23(X) quasi_total (X,Y) Element of bool [:X, the carrier of Y:]
(X,Y,(vseq . e)) . tseq is Element of the carrier of Y
||.tv.|| is non empty Relation-like NAT -defined REAL -valued Function-like V23( NAT ) quasi_total V33() V34() V35() Element of bool [:NAT,REAL:]
e is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real V45() V46() V47() V48() V49() V50() V136() V181() Element of NAT
||.tv.|| . e is V11() real ext-real Element of REAL
||.vseq.|| . e is V11() real ext-real Element of REAL
tv . e is Element of the carrier of Y
||.(tv . e).|| is V11() real ext-real Element of REAL
the normF of Y is non empty Relation-like the carrier of Y -defined REAL -valued Function-like V23( the carrier of Y) quasi_total V33() V34() V35() Element of bool [: the carrier of Y,REAL:]
[: the carrier of Y,REAL:] is non empty Relation-like V33() V34() V35() set
bool [: the carrier of Y,REAL:] is non empty set
the normF of Y . (tv . e) is V11() real ext-real Element of REAL
vseq . e is Element of the carrier of (X,Y)
||.(vseq . e).|| is V11() real ext-real Element of REAL
the normF of (X,Y) . (vseq . e) is V11() real ext-real Element of REAL
||.(tseq . tseq).|| is V11() real ext-real Element of REAL
the normF of Y is non empty Relation-like the carrier of Y -defined REAL -valued Function-like V23( the carrier of Y) quasi_total V33() V34() V35() Element of bool [: the carrier of Y,REAL:]
[: the carrier of Y,REAL:] is non empty Relation-like V33() V34() V35() set
bool [: the carrier of Y,REAL:] is non empty set
the normF of Y . (tseq . tseq) is V11() real ext-real Element of REAL
lim ||.tv.|| is V11() real ext-real Element of REAL
tseq is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real V45() V46() V47() V48() V49() V50() V136() V181() Element of NAT
vseq . tseq is Element of the carrier of (X,Y)
||.(vseq . tseq).|| is V11() real ext-real Element of REAL
the normF of (X,Y) . (vseq . tseq) is V11() real ext-real Element of REAL
||.vseq.|| . tseq is V11() real ext-real Element of REAL
tseq is Element of X
tseq . tseq is Element of the carrier of Y
||.(tseq . tseq).|| is V11() real ext-real Element of REAL
the normF of Y . (tseq . tseq) is V11() real ext-real Element of REAL
tseq is V11() real ext-real Element of REAL
tv is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real V45() V46() V47() V48() V49() V50() V136() V181() Element of NAT
e is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real V45() V46() V47() V48() V49() V50() V136() V181() Element of NAT
m is Element 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 V23( NAT ) quasi_total Element of bool [:NAT, the carrier of Y:]
lim n is Element of the carrier of Y
n is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real V45() V46() V47() V48() V49() V50() V136() V181() Element of NAT
n . n is Element of the carrier of Y
h1 is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real V45() V46() V47() V48() V49() V50() V136() V181() 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 is non empty Relation-like [: the carrier of Y, the carrier of Y:] -defined the carrier of Y -valued Function-like V23([: the carrier of Y, the carrier of Y:]) 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 Relation-like set
[:[: the carrier of Y, the carrier of Y:], the carrier of Y:] is non empty Relation-like set
bool [:[: the carrier of Y, the carrier of Y:], the carrier of Y:] is non empty set
K602( 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
{(n . n),(- (n . h1))} is set
{(n . n)} is set
{{(n . n),(- (n . h1))},{(n . n)}} is set
the addF of Y . [(n . n),(- (n . h1))] is set
||.((n . n) - (n . h1)).|| is V11() real ext-real Element of REAL
the normF of Y is non empty Relation-like the carrier of Y -defined REAL -valued Function-like V23( the carrier of Y) quasi_total V33() V34() V35() Element of bool [: the carrier of Y,REAL:]
[: the carrier of Y,REAL:] is non empty Relation-like V33() V34() V35() set
bool [: the carrier of Y,REAL:] is non empty set
the normF of Y . ((n . n) - (n . h1)) is V11() real ext-real Element of REAL
vseq . n is Element of the carrier of (X,Y)
vseq . h1 is Element of the carrier of (X,Y)
(vseq . n) - (vseq . h1) is Element of the carrier of (X,Y)
- (vseq . h1) is Element of the carrier of (X,Y)
(vseq . n) + (- (vseq . h1)) is Element of the carrier of (X,Y)
K602( the carrier of (X,Y), the addF of (X,Y),(vseq . n),(- (vseq . h1))) is Element of the carrier of (X,Y)
[(vseq . n),(- (vseq . h1))] is set
{(vseq . n),(- (vseq . h1))} is set
{(vseq . n)} is set
{{(vseq . n),(- (vseq . h1))},{(vseq . n)}} is set
the addF of (X,Y) . [(vseq . n),(- (vseq . h1))] is set
||.((vseq . n) - (vseq . h1)).|| is V11() real ext-real Element of REAL
the normF of (X,Y) . ((vseq . n) - (vseq . h1)) is V11() real ext-real Element of REAL
(X,Y,(vseq . h1)) is non empty Relation-like X -defined the carrier of Y -valued Function-like V23(X) quasi_total (X,Y) Element of bool [:X, the carrier of Y:]
(X,Y,(vseq . n)) is non empty Relation-like X -defined the carrier of Y -valued Function-like V23(X) quasi_total (X,Y) Element of bool [:X, the carrier of Y:]
(X,Y,(vseq . n)) . m is Element of the carrier of Y
(X,Y,(vseq . h1)) . m is Element of the carrier of Y
f1 is non empty Relation-like X -defined the carrier of Y -valued Function-like V23(X) quasi_total (X,Y) Element of bool [:X, the carrier of Y:]
f1 . m is Element of the carrier of Y
n . e is Element of the carrier of Y
n is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real V45() V46() V47() V48() V49() V50() V136() V181() 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 is non empty Relation-like [: the carrier of Y, the carrier of Y:] -defined the carrier of Y -valued Function-like V23([: the carrier of Y, the carrier of Y:]) 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 Relation-like set
[:[: the carrier of Y, the carrier of Y:], the carrier of Y:] is non empty Relation-like set
bool [:[: the carrier of Y, the carrier of Y:], the carrier of Y:] is non empty set
K602( 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
{(n . e),(- (n . n))} is set
{(n . e)} is set
{{(n . e),(- (n . n))},{(n . e)}} is set
the addF of Y . [(n . e),(- (n . n))] is set
||.((n . e) - (n . n)).|| is V11() real ext-real Element of REAL
the normF of Y is non empty Relation-like the carrier of Y -defined REAL -valued Function-like V23( the carrier of Y) quasi_total V33() V34() V35() Element of bool [: the carrier of Y,REAL:]
[: the carrier of Y,REAL:] is non empty Relation-like V33() V34() V35() set
bool [: the carrier of Y,REAL:] is non empty set
the normF of Y . ((n . e) - (n . n)) is V11() real ext-real Element of REAL
vseq . e is Element of the carrier of (X,Y)
vseq . n is Element of the carrier of (X,Y)
(vseq . e) - (vseq . n) is Element of the carrier of (X,Y)
- (vseq . n) is Element of the carrier of (X,Y)
(vseq . e) + (- (vseq . n)) is Element of the carrier of (X,Y)
K602( the carrier of (X,Y), the addF of (X,Y),(vseq . e),(- (vseq . n))) is Element of the carrier of (X,Y)
[(vseq . e),(- (vseq . n))] is set
{(vseq . e),(- (vseq . n))} is set
{(vseq . e)} is set
{{(vseq . e),(- (vseq . n))},{(vseq . e)}} is set
the addF of (X,Y) . [(vseq . e),(- (vseq . n))] is set
||.((vseq . e) - (vseq . n)).|| is V11() real ext-real Element of REAL
the normF of (X,Y) . ((vseq . e) - (vseq . n)) is V11() real ext-real Element of REAL
(n . e) - (tseq . m) is Element of the carrier of Y
- (tseq . m) is Element of the carrier of Y
(n . e) + (- (tseq . m)) is Element of the carrier of Y
the addF of Y is non empty Relation-like [: the carrier of Y, the carrier of Y:] -defined the carrier of Y -valued Function-like V23([: the carrier of Y, the carrier of Y:]) 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 Relation-like set
[:[: the carrier of Y, the carrier of Y:], the carrier of Y:] is non empty Relation-like set
bool [:[: the carrier of Y, the carrier of Y:], the carrier of Y:] is non empty set
K602( 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
{(n . e),(- (tseq . m))} is set
{(n . e)} is set
{{(n . e),(- (tseq . m))},{(n . e)}} is set
the addF of Y . [(n . e),(- (tseq . m))] is set
||.((n . e) - (tseq . m)).|| is V11() real ext-real Element of REAL
the normF of Y is non empty Relation-like the carrier of Y -defined REAL -valued Function-like V23( the carrier of Y) quasi_total V33() V34() V35() Element of bool [: the carrier of Y,REAL:]
[: the carrier of Y,REAL:] is non empty Relation-like V33() V34() V35() set
bool [: the carrier of Y,REAL:] is non empty set
the normF of Y . ((n . e) - (tseq . m)) is V11() real ext-real Element of REAL
n is non empty Relation-like NAT -defined REAL -valued Function-like V23( NAT ) quasi_total V33() V34() V35() Element of bool [:NAT,REAL:]
h1 is set
n . h1 is V11() real ext-real Element of REAL
f1 is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real V45() V46() V47() V48() V49() V50() V136() V181() 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
K602( 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
{(n . f1),(- (n . e))} is set
{(n . f1)} is set
{{(n . f1),(- (n . e))},{(n . f1)}} is set
the addF of Y . [(n . f1),(- (n . e))] is set
||.((n . f1) - (n . e)).|| is V11() real ext-real Element of REAL
the normF of Y . ((n . f1) - (n . e)) is V11() real ext-real Element of REAL
n - (n . e) is non empty Relation-like NAT -defined the carrier of Y -valued Function-like V23( NAT ) 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 V11() real ext-real Element of REAL
the normF of Y . ((n - (n . e)) . f1) is V11() real ext-real Element of REAL
||.(n - (n . e)).|| is non empty Relation-like NAT -defined REAL -valued Function-like V23( NAT ) quasi_total V33() V34() V35() Element of bool [:NAT,REAL:]
||.(n - (n . e)).|| . h1 is V11() real ext-real Element of REAL
lim (n - (n . e)) is Element of the carrier of Y
(tseq . m) - (n . e) is Element of the carrier of Y
(tseq . m) + (- (n . e)) is Element of the carrier of Y
K602( 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
{(tseq . m),(- (n . e))} is set
{(tseq . m)} is set
{{(tseq . m),(- (n . e))},{(tseq . m)}} is set
the addF of Y . [(tseq . m),(- (n . e))] is set
lim n is V11() real ext-real Element of REAL
||.((tseq . m) - (n . e)).|| is V11() real ext-real Element of REAL
the normF of Y . ((tseq . m) - (n . e)) is V11() real ext-real Element of REAL
h1 is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real V45() V46() V47() V48() V49() V50() V136() V181() Element of NAT
n . h1 is V11() real ext-real Element of REAL
n . h1 is Element of the carrier of Y
(n . h1) - (n . e) is Element of the carrier of Y
(n . h1) + (- (n . e)) is Element of the carrier of Y
K602( 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
{(n . h1),(- (n . e))} is set
{(n . h1)} is set
{{(n . h1),(- (n . e))},{(n . h1)}} is set
the addF of Y . [(n . h1),(- (n . e))] is set
||.((n . h1) - (n . e)).|| is V11() real ext-real Element of REAL
the normF of Y . ((n . h1) - (n . e)) is V11() real ext-real Element of REAL
(n . e) - (n . h1) is Element of the carrier of Y
- (n . h1) is Element of the carrier of Y
(n . e) + (- (n . h1)) is Element of the carrier of Y
K602( 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
{(n . e),(- (n . h1))} is set
{{(n . e),(- (n . h1))},{(n . e)}} is set
the addF of Y . [(n . e),(- (n . h1))] is set
||.((n . e) - (n . h1)).|| is V11() real ext-real Element of REAL
the normF of Y . ((n . e) - (n . h1)) is V11() real ext-real Element of REAL
vseq . e is Element of the carrier of (X,Y)
(X,Y,(vseq . e)) is non empty Relation-like X -defined the carrier of Y -valued Function-like V23(X) quasi_total (X,Y) Element of bool [: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
K602( 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
{((X,Y,(vseq . e)) . m),(- (tseq . m))} is set
{((X,Y,(vseq . e)) . m)} is set
{{((X,Y,(vseq . e)) . m),(- (tseq . m))},{((X,Y,(vseq . e)) . m)}} is set
the addF of Y . [((X,Y,(vseq . e)) . m),(- (tseq . m))] is set
||.(((X,Y,(vseq . e)) . m) - (tseq . m)).|| is V11() real ext-real Element of REAL
the normF of Y . (((X,Y,(vseq . e)) . m) - (tseq . m)) is V11() real ext-real Element of REAL
m is Element 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
K602( 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
{((X,Y,(vseq . e)) . m),(- (tseq . m))} is set
{((X,Y,(vseq . e)) . m)} is set
{{((X,Y,(vseq . e)) . m),(- (tseq . m))},{((X,Y,(vseq . e)) . m)}} is set
the addF of Y . [((X,Y,(vseq . e)) . m),(- (tseq . m))] is set
||.(((X,Y,(vseq . e)) . m) - (tseq . m)).|| is V11() real ext-real Element of REAL
the normF of Y . (((X,Y,(vseq . e)) . m) - (tseq . m)) is V11() real ext-real Element of REAL
e is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real V45() V46() V47() V48() V49() V50() V136() V181() Element of NAT
vseq . e is Element of the carrier of (X,Y)
(X,Y,(vseq . e)) is non empty Relation-like X -defined the carrier of Y -valued Function-like V23(X) quasi_total (X,Y) Element of bool [:X, the carrier of Y:]
m is Element 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
K602( 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
{((X,Y,(vseq . e)) . m),(- (tseq . m))} is set
{((X,Y,(vseq . e)) . m)} is set
{{((X,Y,(vseq . e)) . m),(- (tseq . m))},{((X,Y,(vseq . e)) . m)}} is set
the addF of Y . [((X,Y,(vseq . e)) . m),(- (tseq . m))] is set
||.(((X,Y,(vseq . e)) . m) - (tseq . m)).|| is V11() real ext-real Element of REAL
the normF of Y . (((X,Y,(vseq . e)) . m) - (tseq . m)) is V11() real ext-real Element of REAL
tseq is non empty Relation-like X -defined the carrier of Y -valued Function-like V23(X) quasi_total (X,Y) Element of bool [:X, the carrier of Y:]
tv is Element of the carrier of (X,Y)
e is V11() real ext-real Element of REAL
m is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real V45() V46() V47() V48() V49() V50() V136() V181() Element of NAT
n is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real V45() V46() V47() V48() V49() V50() V136() V181() Element of NAT
vseq . n is Element of the carrier of (X,Y)
(vseq . n) - tv is Element of the carrier of (X,Y)
- tv is Element of the carrier of (X,Y)
(vseq . n) + (- tv) is Element of the carrier of (X,Y)
K602( the carrier of (X,Y), the addF of (X,Y),(vseq . n),(- tv)) is Element of the carrier of (X,Y)
[(vseq . n),(- tv)] is set
{(vseq . n),(- tv)} is set
{(vseq . n)} is set
{{(vseq . n),(- tv)},{(vseq . n)}} is set
the addF of (X,Y) . [(vseq . n),(- tv)] is set
(X,Y,(vseq . n)) is non empty Relation-like X -defined the carrier of Y -valued Function-like V23(X) quasi_total (X,Y) Element of bool [:X, the carrier of Y:]
h1 is non empty Relation-like X -defined the carrier of Y -valued Function-like V23(X) quasi_total (X,Y) Element of bool [:X, the carrier of Y:]
r is Element of X
h1 . r is Element of the carrier of Y
||.(h1 . r).|| is V11() real ext-real Element of REAL
the normF of Y is non empty Relation-like the carrier of Y -defined REAL -valued Function-like V23( the carrier of Y) quasi_total V33() V34() V35() Element of bool [: the carrier of Y,REAL:]
[: the carrier of Y,REAL:] is non empty Relation-like V33() V34() V35() set
bool [: the carrier of Y,REAL:] is non empty set
the normF of Y . (h1 . r) is V11() real ext-real Element of REAL
(X,Y,(vseq . n)) . r is Element of the carrier of Y
tseq . r is Element of the carrier of Y
((X,Y,(vseq . n)) . r) - (tseq . r) is Element of the carrier of Y
- (tseq . r) is Element of the carrier of Y
((X,Y,(vseq . n)) . r) + (- (tseq . r)) is Element of the carrier of Y
the addF of Y is non empty Relation-like [: the carrier of Y, the carrier of Y:] -defined the carrier of Y -valued Function-like V23([: the carrier of Y, the carrier of Y:]) 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 Relation-like set
[:[: the carrier of Y, the carrier of Y:], the carrier of Y:] is non empty Relation-like set
bool [:[: the carrier of Y, the carrier of Y:], the carrier of Y:] is non empty set
K602( 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
{((X,Y,(vseq . n)) . r),(- (tseq . r))} is set
{((X,Y,(vseq . n)) . r)} is set
{{((X,Y,(vseq . n)) . r),(- (tseq . r))},{((X,Y,(vseq . n)) . r)}} is set
the addF of Y . [((X,Y,(vseq . n)) . r),(- (tseq . r))] is set
||.(((X,Y,(vseq . n)) . r) - (tseq . r)).|| is V11() real ext-real Element of REAL
the normF of Y . (((X,Y,(vseq . n)) . r) - (tseq . r)) is V11() real ext-real Element of REAL
r is V11() real ext-real Element of REAL
(X,Y,h1) is non empty V45() V46() V47() Element of bool REAL
{ ||.(h1 . b₁).|| where b₁ is Element of X : verum } is set
c₁₅ is Element of X
h1 . c₁₅ is Element of the carrier of Y
||.(h1 . c₁₅).|| is V11() real ext-real Element of REAL
the normF of Y . (h1 . c₁₅) is V11() real ext-real Element of REAL
upper_bound (X,Y,h1) is V11() real ext-real Element of REAL
(X,Y) . ((vseq . n) - tv) is V11() real ext-real Element of REAL
||.((vseq . n) - tv).|| is V11() real ext-real Element of REAL
the normF of (X,Y) . ((vseq . n) - tv) is V11() real ext-real Element of REAL
r is V11() real ext-real set
e is V11() real ext-real Element of REAL
e / 2 is V11() real ext-real Element of REAL
m is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real V45() V46() V47() V48() V49() V50() V136() V181() Element of NAT
n is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real V45() V46() V47() V48() V49() V50() V136() V181() Element of NAT
vseq . n is Element of the carrier of (X,Y)
(vseq . n) - tv is Element of the carrier of (X,Y)
- tv is Element of the carrier of (X,Y)
(vseq . n) + (- tv) is Element of the carrier of (X,Y)
K602( the carrier of (X,Y), the addF of (X,Y),(vseq . n),(- tv)) is Element of the carrier of (X,Y)
[(vseq . n),(- tv)] is set
{(vseq . n),(- tv)} is set
{(vseq . n)} is set
{{(vseq . n),(- tv)},{(vseq . n)}} is set
the addF of (X,Y) . [(vseq . n),(- tv)] is set
||.((vseq . n) - tv).|| is V11() real ext-real Element of REAL
the normF of (X,Y) . ((vseq . n) - tv) is V11() real ext-real Element of REAL
X is non empty set
Y is non empty right_complementable Abelian add-associative right_zeroed discerning reflexive vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like complete CNORMSTR
(X,Y) is non empty right_complementable Abelian add-associative right_zeroed discerning reflexive vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like CNORMSTR
(X,Y) is non empty Element of bool the carrier of (ComplexVectSpace (X,Y))
ComplexVectSpace (X,Y) is non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital CLSStruct
the carrier of Y is non empty set
Funcs (X, the carrier of Y) is non empty functional FUNCTION_DOMAIN of X, the carrier of Y
K340(X,Y) is Relation-like X -defined the carrier of Y -valued Function-like V23(X) quasi_total Element of Funcs (X, the carrier of Y)
K338(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 V23([:(Funcs (X, the carrier of Y)),(Funcs (X, the carrier of Y)):]) quasi_total Function-yielding V135() 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 Relation-like set
[:[:(Funcs (X, the carrier of Y)),(Funcs (X, the carrier of Y)):],(Funcs (X, the carrier of Y)):] is non empty Relation-like set
bool [:[:(Funcs (X, the carrier of Y)),(Funcs (X, the carrier of Y)):],(Funcs (X, the carrier of Y)):] is non empty set
FuncExtMult (X,Y) is non empty Relation-like [:COMPLEX,(Funcs (X, the carrier of Y)):] -defined Funcs (X, the carrier of Y) -valued Function-like V23([:COMPLEX,(Funcs (X, the carrier of Y)):]) quasi_total Function-yielding V135() Element of bool [:[:COMPLEX,(Funcs (X, the carrier of Y)):],(Funcs (X, the carrier of Y)):]
[:COMPLEX,(Funcs (X, the carrier of Y)):] is non empty Relation-like set
[:[:COMPLEX,(Funcs (X, the carrier of Y)):],(Funcs (X, the carrier of Y)):] is non empty Relation-like set
bool [:[:COMPLEX,(Funcs (X, the carrier of Y)):],(Funcs (X, the carrier of Y)):] is non empty set
CLSStruct(# (Funcs (X, the carrier of Y)),K340(X,Y),K338(X,Y),(FuncExtMult (X,Y)) #) is non empty strict CLSStruct
the carrier of (ComplexVectSpace (X,Y)) is non empty set
bool the carrier of (ComplexVectSpace (X,Y)) is non empty set
Zero_ ((X,Y),(ComplexVectSpace (X,Y))) is Element of (X,Y)
Add_ ((X,Y),(ComplexVectSpace (X,Y))) is non empty Relation-like [:(X,Y),(X,Y):] -defined (X,Y) -valued Function-like V23([:(X,Y),(X,Y):]) quasi_total Element of bool [:[:(X,Y),(X,Y):],(X,Y):]
[:(X,Y),(X,Y):] is non empty Relation-like set
[:[:(X,Y),(X,Y):],(X,Y):] is non empty Relation-like set
bool [:[:(X,Y),(X,Y):],(X,Y):] is non empty set
Mult_ ((X,Y),(ComplexVectSpace (X,Y))) is non empty Relation-like [:COMPLEX,(X,Y):] -defined (X,Y) -valued Function-like V23([:COMPLEX,(X,Y):]) quasi_total Element of bool [:[:COMPLEX,(X,Y):],(X,Y):]
[:COMPLEX,(X,Y):] is non empty Relation-like set
[:[:COMPLEX,(X,Y):],(X,Y):] is non empty Relation-like set
bool [:[:COMPLEX,(X,Y):],(X,Y):] is non empty set
(X,Y) is non empty Relation-like (X,Y) -defined REAL -valued Function-like V23((X,Y)) quasi_total V33() V34() V35() Element of bool [:(X,Y),REAL:]
[:(X,Y),REAL:] is non empty Relation-like V33() V34() V35() set
bool [:(X,Y),REAL:] is non empty set
CNORMSTR(# (X,Y),(Zero_ ((X,Y),(ComplexVectSpace (X,Y)))),(Add_ ((X,Y),(ComplexVectSpace (X,Y)))),(Mult_ ((X,Y),(ComplexVectSpace (X,Y)))),(X,Y) #) is non empty strict CNORMSTR
the carrier of (X,Y) is non empty set
[:NAT, the carrier of (X,Y):] is non empty Relation-like 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 V23( NAT ) quasi_total Element of bool [:NAT, the carrier of (X,Y):]
X is non empty set
Y is non empty right_complementable Abelian add-associative right_zeroed discerning reflexive vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like complete CNORMSTR
(X,Y) is non empty right_complementable Abelian add-associative right_zeroed discerning reflexive vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like CNORMSTR
(X,Y) is non empty Element of bool the carrier of (ComplexVectSpace (X,Y))
ComplexVectSpace (X,Y) is non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital CLSStruct
the carrier of Y is non empty set
Funcs (X, the carrier of Y) is non empty functional FUNCTION_DOMAIN of X, the carrier of Y
K340(X,Y) is Relation-like X -defined the carrier of Y -valued Function-like V23(X) quasi_total Element of Funcs (X, the carrier of Y)
K338(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 V23([:(Funcs (X, the carrier of Y)),(Funcs (X, the carrier of Y)):]) quasi_total Function-yielding V135() 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 Relation-like set
[:[:(Funcs (X, the carrier of Y)),(Funcs (X, the carrier of Y)):],(Funcs (X, the carrier of Y)):] is non empty Relation-like set
bool [:[:(Funcs (X, the carrier of Y)),(Funcs (X, the carrier of Y)):],(Funcs (X, the carrier of Y)):] is non empty set
FuncExtMult (X,Y) is non empty Relation-like [:COMPLEX,(Funcs (X, the carrier of Y)):] -defined Funcs (X, the carrier of Y) -valued Function-like V23([:COMPLEX,(Funcs (X, the carrier of Y)):]) quasi_total Function-yielding V135() Element of bool [:[:COMPLEX,(Funcs (X, the carrier of Y)):],(Funcs (X, the carrier of Y)):]
[:COMPLEX,(Funcs (X, the carrier of Y)):] is non empty Relation-like set
[:[:COMPLEX,(Funcs (X, the carrier of Y)):],(Funcs (X, the carrier of Y)):] is non empty Relation-like set
bool [:[:COMPLEX,(Funcs (X, the carrier of Y)):],(Funcs (X, the carrier of Y)):] is non empty set
CLSStruct(# (Funcs (X, the carrier of Y)),K340(X,Y),K338(X,Y),(FuncExtMult (X,Y)) #) is non empty strict CLSStruct
the carrier of (ComplexVectSpace (X,Y)) is non empty set
bool the carrier of (ComplexVectSpace (X,Y)) is non empty set
Zero_ ((X,Y),(ComplexVectSpace (X,Y))) is Element of (X,Y)
Add_ ((X,Y),(ComplexVectSpace (X,Y))) is non empty Relation-like [:(X,Y),(X,Y):] -defined (X,Y) -valued Function-like V23([:(X,Y),(X,Y):]) quasi_total Element of bool [:[:(X,Y),(X,Y):],(X,Y):]
[:(X,Y),(X,Y):] is non empty Relation-like set
[:[:(X,Y),(X,Y):],(X,Y):] is non empty Relation-like set
bool [:[:(X,Y),(X,Y):],(X,Y):] is non empty set
Mult_ ((X,Y),(ComplexVectSpace (X,Y))) is non empty Relation-like [:COMPLEX,(X,Y):] -defined (X,Y) -valued Function-like V23([:COMPLEX,(X,Y):]) quasi_total Element of bool [:[:COMPLEX,(X,Y):],(X,Y):]
[:COMPLEX,(X,Y):] is non empty Relation-like set
[:[:COMPLEX,(X,Y):],(X,Y):] is non empty Relation-like set
bool [:[:COMPLEX,(X,Y):],(X,Y):] is non empty set
(X,Y) is non empty Relation-like (X,Y) -defined REAL -valued Function-like V23((X,Y)) quasi_total V33() V34() V35() Element of bool [:(X,Y),REAL:]
[:(X,Y),REAL:] is non empty Relation-like V33() V34() V35() set
bool [:(X,Y),REAL:] is non empty set
CNORMSTR(# (X,Y),(Zero_ ((X,Y),(ComplexVectSpace (X,Y)))),(Add_ ((X,Y),(ComplexVectSpace (X,Y)))),(Mult_ ((X,Y),(ComplexVectSpace (X,Y)))),(X,Y) #) is non empty strict CNORMSTR
X is non empty Relation-like NAT -defined COMPLEX -valued Function-like V23( NAT ) quasi_total V33() Element of bool [:NAT,COMPLEX:]
Y is non empty Relation-like NAT -defined COMPLEX -valued Function-like V23( NAT ) quasi_total V33() Element of bool [:NAT,COMPLEX:]
X + Y is non empty Relation-like NAT -defined COMPLEX -valued Function-like V23( NAT ) quasi_total V33() Element of bool [:NAT,COMPLEX:]
Y is non empty Relation-like NAT -defined COMPLEX -valued Function-like V23( NAT ) quasi_total V33() Element of bool [:NAT,COMPLEX:]
X is V11() set
X (#) Y is non empty Relation-like NAT -defined COMPLEX -valued Function-like V23( NAT ) quasi_total V33() Element of bool [:NAT,COMPLEX:]
X is non empty Relation-like NAT -defined COMPLEX -valued Function-like V23( NAT ) quasi_total V33() Element of bool [:NAT,COMPLEX:]
|.X.| is non empty Relation-like NAT -defined REAL -valued Function-like V23( NAT ) quasi_total V33() V34() V35() Element of bool [:NAT,REAL:]
Y is non empty Relation-like NAT -defined COMPLEX -valued Function-like V23( NAT ) quasi_total V33() Element of bool [:NAT,COMPLEX:]
|.Y.| is non empty Relation-like NAT -defined REAL -valued Function-like V23( NAT ) quasi_total V33() V34() V35() Element of bool [:NAT,REAL:]
X is non empty Relation-like NAT -defined COMPLEX -valued Function-like V23( NAT ) quasi_total V33() Element of bool [:NAT,COMPLEX:]
Y is non empty Relation-like NAT -defined COMPLEX -valued Function-like V23( NAT ) quasi_total V33() Element of bool [:NAT,COMPLEX:]
vseq is non empty Relation-like NAT -defined COMPLEX -valued Function-like V23( NAT ) quasi_total V33() Element of bool [:NAT,COMPLEX:]
Y - vseq is non empty Relation-like NAT -defined COMPLEX -valued Function-like V23( NAT ) quasi_total V33() Element of bool [:NAT,COMPLEX:]
- vseq is Relation-like NAT -defined Function-like V23( NAT ) V33() set
- 1 is non empty V11() real ext-real non positive negative set
(- 1) (#) vseq is Relation-like NAT -defined Function-like V23( NAT ) V33() set
Y + (- vseq) is Relation-like NAT -defined Function-like V23( NAT ) V33() set
f is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real V45() V46() V47() V48() V49() V50() V136() V181() Element of NAT
X . f is V11() Element of COMPLEX
Y . f is V11() Element of COMPLEX
vseq . f is V11() Element of COMPLEX
(Y . f) - (vseq . f) is V11() Element of COMPLEX
tseq is non empty Relation-like NAT -defined COMPLEX -valued Function-like V23( NAT ) quasi_total V33() Element of bool [:NAT,COMPLEX:]
tseq is non empty Relation-like NAT -defined COMPLEX -valued Function-like V23( NAT ) quasi_total V33() Element of bool [:NAT,COMPLEX:]
tv is non empty Relation-like NAT -defined COMPLEX -valued Function-like V23( NAT ) quasi_total V33() Element of bool [:NAT,COMPLEX:]
tseq - tv is non empty Relation-like NAT -defined COMPLEX -valued Function-like V23( NAT ) quasi_total V33() Element of bool [:NAT,COMPLEX:]
- tv is Relation-like NAT -defined Function-like V23( NAT ) V33() set
(- 1) (#) tv is Relation-like NAT -defined Function-like V23( NAT ) V33() set
tseq + (- tv) is Relation-like NAT -defined Function-like V23( NAT ) V33() set