:: CC0SP2 semantic presentation

REAL is non empty V30() V176() V177() V178() V182() non bounded_below non bounded_above V197() set
NAT is non empty epsilon-transitive epsilon-connected ordinal V176() V177() V178() V179() V180() V181() V182() left_end bounded_below Element of bool REAL
bool REAL is non empty set
COMPLEX is non empty V30() V176() V182() set
bool COMPLEX is non empty set
omega is non empty epsilon-transitive epsilon-connected ordinal V176() V177() V178() V179() V180() V181() V182() left_end bounded_below set
bool omega is non empty set
bool NAT is non empty set
[:NAT,REAL:] is non empty V139() V140() V141() set
bool [:NAT,REAL:] is non empty set
{} is empty epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural complex real ext-real non positive non negative V176() V177() V178() V179() V180() V181() V182() bounded_below V197() set
1 is non empty epsilon-transitive epsilon-connected ordinal natural complex real V101() ext-real positive non negative V176() V177() V178() V179() V180() V181() V187() left_end bounded_below Element of NAT
{{},1} is non empty V176() V177() V178() V179() V180() V181() left_end bounded_below set
RAT is non empty V30() V176() V177() V178() V179() V182() set
INT is non empty V30() V176() V177() V178() V179() V180() V182() set
[:COMPLEX,COMPLEX:] is non empty V139() set
bool [:COMPLEX,COMPLEX:] is non empty set
[:[:COMPLEX,COMPLEX:],COMPLEX:] is non empty V139() set
bool [:[:COMPLEX,COMPLEX:],COMPLEX:] is non empty set
[:REAL,REAL:] is non empty V139() V140() V141() set
bool [:REAL,REAL:] is non empty set
[:[:REAL,REAL:],REAL:] is non empty V139() V140() V141() set
bool [:[:REAL,REAL:],REAL:] is non empty set
[:RAT,RAT:] is non empty RAT -valued V139() V140() V141() set
bool [:RAT,RAT:] is non empty set
[:[:RAT,RAT:],RAT:] is non empty RAT -valued V139() V140() V141() set
bool [:[:RAT,RAT:],RAT:] is non empty set
[:INT,INT:] is non empty RAT -valued INT -valued V139() V140() V141() set
bool [:INT,INT:] is non empty set
[:[:INT,INT:],INT:] is non empty RAT -valued INT -valued V139() V140() V141() set
bool [:[:INT,INT:],INT:] is non empty set
[:NAT,NAT:] is non empty RAT -valued INT -valued V139() V140() V141() V142() set
[:[:NAT,NAT:],NAT:] is non empty RAT -valued INT -valued V139() V140() V141() V142() set
bool [:[:NAT,NAT:],NAT:] is non empty set
[:COMPLEX,REAL:] is non empty V139() V140() V141() set
bool [:COMPLEX,REAL:] is non empty set
ExtREAL is non empty V177() V197() set
[:NAT,COMPLEX:] is non empty V139() set
bool [:NAT,COMPLEX:] is non empty set
K568() is non empty set
[:K568(),K568():] is non empty set
[:[:K568(),K568():],K568():] is non empty set
bool [:[:K568(),K568():],K568():] is non empty set
[:COMPLEX,K568():] is non empty set
[:[:COMPLEX,K568():],K568():] is non empty set
bool [:[:COMPLEX,K568():],K568():] is non empty set
K574() is non empty strict CLSStruct
the carrier of K574() is non empty set
bool the carrier of K574() is non empty set
K578() is Element of bool the carrier of K574()
[:K578(),K578():] is set
[:[:K578(),K578():],COMPLEX:] is V139() set
bool [:[:K578(),K578():],COMPLEX:] is non empty set
K586() is Element of bool the carrier of K574()
[:K586(),REAL:] is V139() V140() V141() set
bool [:K586(),REAL:] is non empty set
[:1,1:] is non empty RAT -valued INT -valued V139() V140() V141() V142() set
bool [:1,1:] is non empty set
[:[:1,1:],1:] is non empty RAT -valued INT -valued V139() V140() V141() V142() set
bool [:[:1,1:],1:] is non empty set
[:[:1,1:],REAL:] is non empty V139() V140() V141() set
bool [:[:1,1:],REAL:] is non empty set
2 is non empty epsilon-transitive epsilon-connected ordinal natural complex real V101() ext-real positive non negative V176() V177() V178() V179() V180() V181() V187() left_end bounded_below Element of NAT
[:2,2:] is non empty RAT -valued INT -valued V139() V140() V141() V142() set
[:[:2,2:],REAL:] is non empty V139() V140() V141() set
bool [:[:2,2:],REAL:] is non empty set
K760() is TopStruct
the carrier of K760() is set
K714() is V258() L25()
K765() is TopSpace-like TopStruct
the carrier of K765() is set
K800(2) is V294() L26()
the carrier of K800(2) is set
[: the carrier of K800(2),REAL:] is V139() V140() V141() set
bool [: the carrier of K800(2),REAL:] is non empty set
bool the carrier of K800(2) is non empty set
0 is empty epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural complex real V101() ext-real non positive non negative V176() V177() V178() V179() V180() V181() V182() V187() bounded_below V197() Element of NAT
1r is complex Element of COMPLEX
- 1r is complex Element of COMPLEX
X is TopStruct
the carrier of X is set
[: the carrier of X,COMPLEX:] is V139() set
bool [: the carrier of X,COMPLEX:] is non empty set
X is 1-sorted
the carrier of X is set
X is complex set
the carrier of X --> X is Relation-like the carrier of X -defined {X} -valued Function-like total quasi_total V139() Element of bool [: the carrier of X,{X}:]
{X} is non empty V176() set
[: the carrier of X,{X}:] is V139() set
bool [: the carrier of X,{X}:] is non empty set
[: the carrier of X,COMPLEX:] is V139() set
bool [: the carrier of X,COMPLEX:] is non empty set
X is non empty TopSpace-like TopStruct
the carrier of X is non empty set
[: the carrier of X,COMPLEX:] is non empty V139() set
bool [: the carrier of X,COMPLEX:] is non empty set
X is complex set
(X,X) is non empty Relation-like the carrier of X -defined COMPLEX -valued Function-like total quasi_total V139() Element of bool [: the carrier of X,COMPLEX:]
the carrier of X --> X is non empty Relation-like the carrier of X -defined {X} -valued Function-like total quasi_total V139() Element of bool [: the carrier of X,{X}:]
{X} is non empty V176() set
[: the carrier of X,{X}:] is non empty V139() set
bool [: the carrier of X,{X}:] is non empty set
F is non empty Relation-like the carrier of X -defined COMPLEX -valued Function-like total quasi_total V139() Element of bool [: the carrier of X,COMPLEX:]
[#] X is non empty non proper closed Element of bool the carrier of X
bool the carrier of X is non empty set
GB is V176() Element of bool COMPLEX
F " GB is Element of bool the carrier of X
{} X is empty proper epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural complex real ext-real non positive non negative V176() V177() V178() V179() V180() V181() V182() bounded_below V197() closed compact Element of bool the carrier of X
X is non empty TopSpace-like TopStruct
X is complex set
(X,X) is non empty Relation-like the carrier of X -defined COMPLEX -valued Function-like total quasi_total V139() Element of bool [: the carrier of X,COMPLEX:]
the carrier of X is non empty set
[: the carrier of X,COMPLEX:] is non empty V139() set
bool [: the carrier of X,COMPLEX:] is non empty set
the carrier of X --> X is non empty Relation-like the carrier of X -defined {X} -valued Function-like total quasi_total V139() Element of bool [: the carrier of X,{X}:]
{X} is non empty V176() set
[: the carrier of X,{X}:] is non empty V139() set
bool [: the carrier of X,{X}:] is non empty set
X is non empty TopSpace-like TopStruct
the carrier of X is non empty set
[: the carrier of X,COMPLEX:] is non empty V139() set
bool [: the carrier of X,COMPLEX:] is non empty set
0c is empty epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural complex real ext-real non positive non negative V176() V177() V178() V179() V180() V181() V182() bounded_below V197() Element of COMPLEX
(X,0c) is non empty Relation-like the carrier of X -defined COMPLEX -valued Function-like total quasi_total V139() (X) Element of bool [: the carrier of X,COMPLEX:]
the carrier of X --> 0c is non empty Relation-like the carrier of X -defined {0c} -valued Function-like total quasi_total V139() V140() V141() V142() Element of bool [: the carrier of X,{0c}:]
{0c} is non empty V176() V177() V178() V179() V180() V181() left_end bounded_below set
[: the carrier of X,{0c}:] is non empty RAT -valued INT -valued V139() V140() V141() V142() set
bool [: the carrier of X,{0c}:] is non empty set
X is non empty Relation-like the carrier of X -defined COMPLEX -valued Function-like total quasi_total V139() (X) Element of bool [: the carrier of X,COMPLEX:]
X is non empty TopSpace-like TopStruct
the carrier of X is non empty set
[: the carrier of X,COMPLEX:] is non empty V139() set
bool [: the carrier of X,COMPLEX:] is non empty set
X is non empty Relation-like the carrier of X -defined COMPLEX -valued Function-like total quasi_total V139() Element of bool [: the carrier of X,COMPLEX:]
F is V176() Element of bool COMPLEX
F ` is V176() Element of bool COMPLEX
COMPLEX \ F is V176() set
X " (F `) is Element of bool the carrier of X
bool the carrier of X is non empty set
X " COMPLEX is Element of bool the carrier of X
X " F is Element of bool the carrier of X
(X " COMPLEX) \ (X " F) is Element of bool the carrier of X
[#] X is non empty non proper closed Element of bool the carrier of X
([#] X) \ (X " F) is Element of bool the carrier of X
([#] X) \ (([#] X) \ (X " F)) is Element of bool the carrier of X
F is V176() Element of bool COMPLEX
X " F is Element of bool the carrier of X
F ` is V176() Element of bool COMPLEX
COMPLEX \ F is V176() set
(F `) ` is V176() Element of bool COMPLEX
COMPLEX \ (F `) is V176() set
X " (F `) is Element of bool the carrier of X
(X " COMPLEX) \ (X " F) is Element of bool the carrier of X
([#] X) \ (X " F) is Element of bool the carrier of X
X is non empty TopSpace-like TopStruct
the carrier of X is non empty set
[: the carrier of X,COMPLEX:] is non empty V139() set
bool [: the carrier of X,COMPLEX:] is non empty set
bool the carrier of X is non empty set
X is non empty Relation-like the carrier of X -defined COMPLEX -valued Function-like total quasi_total V139() Element of bool [: the carrier of X,COMPLEX:]
F is Element of the carrier of X
X . F is complex Element of COMPLEX
G is V176() Element of bool COMPLEX
GB is complex set
aFB is V176() Neighbourhood of GB
aF is complex real ext-real Element of REAL
{ b₁ where b₁ is complex set : not aF <= |.(b₁ - GB).| } is set
r is set
r is complex set
r - GB is complex set
- GB is complex set
r + (- GB) is complex set
|.(r - GB).| is complex real ext-real non negative Element of REAL
X " { b₁ where b₁ is complex set : not aF <= |.(b₁ - GB).| } is Element of bool the carrier of X
r is V176() Element of bool COMPLEX
r is Element of bool the carrier of X
X .: (X " { b₁ where b₁ is complex set : not aF <= |.(b₁ - GB).| } ) is V176() Element of bool COMPLEX
X .: r is V176() Element of bool COMPLEX
F is Element of the carrier of X
X . F is complex Element of COMPLEX
G is V176() Element of bool COMPLEX
F is V176() Element of bool COMPLEX
F ` is V176() Element of bool COMPLEX
COMPLEX \ F is V176() set
(F `) ` is V176() Element of bool COMPLEX
COMPLEX \ (F `) is V176() set
X " F is Element of bool the carrier of X
(X " F) ` is Element of bool the carrier of X
the carrier of X \ (X " F) is set
G is Element of the carrier of X
X " (F `) is Element of bool the carrier of X
X . G is complex Element of COMPLEX
FB is V176() Element of bool COMPLEX
GB is Element of bool the carrier of X
X .: GB is V176() Element of bool COMPLEX
X " (X .: GB) is Element of bool the carrier of X
((X " F) `) ` is Element of bool the carrier of X
the carrier of X \ ((X " F) `) is set
X is non empty TopSpace-like TopStruct
the carrier of X is non empty set
[: the carrier of X,COMPLEX:] is non empty V139() set
bool [: the carrier of X,COMPLEX:] is non empty set
X is non empty Relation-like the carrier of X -defined COMPLEX -valued Function-like total quasi_total V139() (X) Element of bool [: the carrier of X,COMPLEX:]
F is non empty Relation-like the carrier of X -defined COMPLEX -valued Function-like total quasi_total V139() (X) Element of bool [: the carrier of X,COMPLEX:]
X + F is Relation-like the carrier of X -defined Function-like total V139() set
rng (X + F) is V176() set
dom (X + F) is Element of bool the carrier of X
bool the carrier of X is non empty set
dom X is Element of bool the carrier of X
dom F is Element of bool the carrier of X
(dom X) /\ (dom F) is Element of bool the carrier of X
the carrier of X /\ (dom F) is Element of bool the carrier of X
the carrier of X /\ the carrier of X is set
FB is non empty Relation-like the carrier of X -defined COMPLEX -valued Function-like total quasi_total V139() Element of bool [: the carrier of X,COMPLEX:]
GB is Element of the carrier of X
FB . GB is complex Element of COMPLEX
X . GB is complex Element of COMPLEX
F . GB is complex Element of COMPLEX
(X . GB) + (F . GB) is complex Element of COMPLEX
GB is Element of the carrier of X
FB . GB is complex Element of COMPLEX
aFB is V176() Element of bool COMPLEX
aF is complex set
aG is V176() Neighbourhood of aF
r is complex real ext-real Element of REAL
{ b₁ where b₁ is complex set : not r <= |.(b₁ - aF).| } is set
r / 2 is complex real ext-real Element of COMPLEX
2 " is non empty complex real ext-real positive non negative set
r * (2 ") is complex real ext-real set
X . GB is complex Element of COMPLEX
s1 is complex set
{ b₁ where b₁ is complex set : not r / 2 <= |.(b₁ - s1).| } is set
A2 is set
A3 is complex set
A3 - s1 is complex set
- s1 is complex set
A3 + (- s1) is complex set
|.(A3 - s1).| is complex real ext-real non negative Element of REAL
A2 is V176() Element of bool COMPLEX
s1 - s1 is complex set
- s1 is complex set
s1 + (- s1) is complex set
|.(s1 - s1).| is complex real ext-real non negative Element of REAL
A3 is Element of bool the carrier of X
X .: A3 is V176() Element of bool COMPLEX
F . GB is complex Element of COMPLEX
vv1 is complex set
{ b₁ where b₁ is complex set : not r / 2 <= |.(b₁ - vv1).| } is set
fvu1 is set
zz0 is complex set
zz0 - vv1 is complex set
- vv1 is complex set
zz0 + (- vv1) is complex set
|.(zz0 - vv1).| is complex real ext-real non negative Element of REAL
fvu1 is V176() Element of bool COMPLEX
vv1 - vv1 is complex set
- vv1 is complex set
vv1 + (- vv1) is complex set
|.(vv1 - vv1).| is complex real ext-real non negative Element of REAL
zz0 is Element of bool the carrier of X
F .: zz0 is V176() Element of bool COMPLEX
A3 /\ zz0 is Element of bool the carrier of X
FB .: (A3 /\ zz0) is V176() Element of bool COMPLEX
hx0 is set
dom FB is Element of bool the carrier of X
W3 is set
FB . W3 is complex set
W is Element of the carrier of X
X . W is complex Element of COMPLEX
F . W is complex Element of COMPLEX
x3 is complex set
x3 - vv1 is complex set
x3 + (- vv1) is complex set
|.(x3 - vv1).| is complex real ext-real non negative Element of REAL
px is complex set
px - vv1 is complex set
px + (- vv1) is complex set
|.(px - vv1).| is complex real ext-real non negative Element of REAL
(FB . W3) - aF is complex set
- aF is complex set
(FB . W3) + (- aF) is complex set
|.((FB . W3) - aF).| is complex real ext-real non negative Element of REAL
(X . W) + (F . W) is complex Element of COMPLEX
((X . W) + (F . W)) - aF is complex set
((X . W) + (F . W)) + (- aF) is complex set
|.(((X . W) + (F . W)) - aF).| is complex real ext-real non negative Element of REAL
(X . GB) + (F . GB) is complex Element of COMPLEX
((X . W) + (F . W)) - ((X . GB) + (F . GB)) is complex Element of COMPLEX
- ((X . GB) + (F . GB)) is complex set
((X . W) + (F . W)) + (- ((X . GB) + (F . GB))) is complex set
|.(((X . W) + (F . W)) - ((X . GB) + (F . GB))).| is complex real ext-real non negative Element of REAL
z3 is complex set
z3 - s1 is complex set
z3 + (- s1) is complex set
(z3 - s1) + (x3 - vv1) is complex set
|.((z3 - s1) + (x3 - vv1)).| is complex real ext-real non negative Element of REAL
FB . W is complex Element of COMPLEX
(FB . W) - aF is complex set
(FB . W) + (- aF) is complex set
|.((FB . W) - aF).| is complex real ext-real non negative Element of REAL
|.(z3 - s1).| is complex real ext-real non negative Element of REAL
|.(z3 - s1).| + |.(x3 - vv1).| is complex real ext-real non negative Element of REAL
(r / 2) + |.(x3 - vv1).| is complex real ext-real Element of REAL
px is complex set
px - s1 is complex set
px + (- s1) is complex set
|.(px - s1).| is complex real ext-real non negative Element of REAL
(r / 2) + (r / 2) is complex real ext-real Element of COMPLEX
X is non empty TopSpace-like TopStruct
the carrier of X is non empty set
[: the carrier of X,COMPLEX:] is non empty V139() set
bool [: the carrier of X,COMPLEX:] is non empty set
X is complex set
F is non empty Relation-like the carrier of X -defined COMPLEX -valued Function-like total quasi_total V139() (X) Element of bool [: the carrier of X,COMPLEX:]
X (#) F is non empty Relation-like the carrier of X -defined COMPLEX -valued Function-like total quasi_total V139() Element of bool [: the carrier of X,COMPLEX:]
FB is Element of the carrier of X
(X (#) F) . FB is complex Element of COMPLEX
F . FB is complex Element of COMPLEX
X * (F . FB) is complex set
bool the carrier of X is non empty set
FB is Element of the carrier of X
(X (#) F) . FB is complex Element of COMPLEX
GB is V176() Element of bool COMPLEX
aFB is complex set
aF is V176() Neighbourhood of aFB
aG is complex real ext-real Element of REAL
{ b₁ where b₁ is complex set : not aG <= |.(b₁ - aFB).| } is set
|.X.| is complex real ext-real non negative Element of REAL
aG / |.X.| is complex real ext-real Element of COMPLEX
|.X.| " is complex real ext-real non negative set
aG * (|.X.| ") is complex real ext-real set
F . FB is complex Element of COMPLEX
r is complex set
{ b₁ where b₁ is complex set : not aG / |.X.| <= |.(b₁ - r).| } is set
fau1 is set
A2 is complex set
A2 - r is complex set
- r is complex set
A2 + (- r) is complex set
|.(A2 - r).| is complex real ext-real non negative Element of REAL
fau1 is V176() Element of bool COMPLEX
r - r is complex set
- r is complex set
r + (- r) is complex set
|.(r - r).| is complex real ext-real non negative Element of REAL
A2 is Element of bool the carrier of X
F .: A2 is V176() Element of bool COMPLEX
(X (#) F) .: A2 is V176() Element of bool COMPLEX
vv1 is set
dom (X (#) F) is Element of bool the carrier of X
uu1 is set
(X (#) F) . uu1 is complex set
fvu1 is Element of the carrier of X
dom F is Element of bool the carrier of X
F . fvu1 is complex Element of COMPLEX
zz0 is complex set
zz0 - r is complex set
zz0 + (- r) is complex set
|.(zz0 - r).| is complex real ext-real non negative Element of REAL
xx0 is complex set
xx0 - r is complex set
xx0 + (- r) is complex set
|.(xx0 - r).| is complex real ext-real non negative Element of REAL
((X (#) F) . uu1) - aFB is complex set
- aFB is complex set
((X (#) F) . uu1) + (- aFB) is complex set
|.(((X (#) F) . uu1) - aFB).| is complex real ext-real non negative Element of REAL
X * (F . fvu1) is complex set
(X * (F . fvu1)) - aFB is complex set
(X * (F . fvu1)) + (- aFB) is complex set
|.((X * (F . fvu1)) - aFB).| is complex real ext-real non negative Element of REAL
X * (F . FB) is complex set
(X * (F . fvu1)) - (X * (F . FB)) is complex set
- (X * (F . FB)) is complex set
(X * (F . fvu1)) + (- (X * (F . FB))) is complex set
|.((X * (F . fvu1)) - (X * (F . FB))).| is complex real ext-real non negative Element of REAL
(F . fvu1) - (F . FB) is complex Element of COMPLEX
- (F . FB) is complex set
(F . fvu1) + (- (F . FB)) is complex set
X * ((F . fvu1) - (F . FB)) is complex set
|.(X * ((F . fvu1) - (F . FB))).| is complex real ext-real non negative Element of REAL
|.X.| * |.(zz0 - r).| is complex real ext-real non negative Element of REAL
|.X.| * (aG / |.X.|) is complex real ext-real Element of REAL
|.X.| / |.X.| is complex real ext-real non negative Element of COMPLEX
|.X.| * (|.X.| ") is complex real ext-real non negative set
aG * (|.X.| / |.X.|) is complex real ext-real Element of REAL
aG * 1 is complex real ext-real Element of REAL
(X (#) F) . fvu1 is complex Element of COMPLEX
((X (#) F) . fvu1) - aFB is complex set
((X (#) F) . fvu1) + (- aFB) is complex set
|.(((X (#) F) . fvu1) - aFB).| is complex real ext-real non negative Element of REAL
0c is empty epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural complex real ext-real non positive non negative V176() V177() V178() V179() V180() V181() V182() bounded_below V197() Element of COMPLEX
(X,0c) is non empty Relation-like the carrier of X -defined COMPLEX -valued Function-like total quasi_total V139() (X) Element of bool [: the carrier of X,COMPLEX:]
the carrier of X --> 0c is non empty Relation-like the carrier of X -defined {0c} -valued Function-like total quasi_total V139() V140() V141() V142() Element of bool [: the carrier of X,{0c}:]
{0c} is non empty V176() V177() V178() V179() V180() V181() left_end bounded_below set
[: the carrier of X,{0c}:] is non empty RAT -valued INT -valued V139() V140() V141() V142() set
bool [: the carrier of X,{0c}:] is non empty set
dom (X,0c) is Element of bool the carrier of X
bool the carrier of X is non empty set
dom (X (#) F) is Element of bool the carrier of X
aFB is set
(X,0c) . aFB is complex set
(X (#) F) . aFB is complex set
F . aFB is complex set
0 * (F . aFB) is complex set
X is non empty TopSpace-like TopStruct
the carrier of X is non empty set
[: the carrier of X,COMPLEX:] is non empty V139() set
bool [: the carrier of X,COMPLEX:] is non empty set
X is non empty Relation-like the carrier of X -defined COMPLEX -valued Function-like total quasi_total V139() (X) Element of bool [: the carrier of X,COMPLEX:]
F is non empty Relation-like the carrier of X -defined COMPLEX -valued Function-like total quasi_total V139() (X) Element of bool [: the carrier of X,COMPLEX:]
X - F is non empty Relation-like the carrier of X -defined COMPLEX -valued Function-like total quasi_total V139() Element of bool [: the carrier of X,COMPLEX:]
- F is Relation-like the carrier of X -defined Function-like total V139() set
- 1 is non empty complex real ext-real non positive negative set
(- 1) (#) F is Relation-like the carrier of X -defined Function-like total V139() set
X + (- F) is Relation-like the carrier of X -defined Function-like total V139() set
- 1 is non empty complex real ext-real non positive negative Element of REAL
(- 1) (#) F is non empty Relation-like the carrier of X -defined COMPLEX -valued Function-like total quasi_total V139() Element of bool [: the carrier of X,COMPLEX:]
X is non empty TopSpace-like TopStruct
the carrier of X is non empty set
[: the carrier of X,COMPLEX:] is non empty V139() set
bool [: the carrier of X,COMPLEX:] is non empty set
X is non empty Relation-like the carrier of X -defined COMPLEX -valued Function-like total quasi_total V139() (X) Element of bool [: the carrier of X,COMPLEX:]
F is non empty Relation-like the carrier of X -defined COMPLEX -valued Function-like total quasi_total V139() (X) Element of bool [: the carrier of X,COMPLEX:]
X (#) F is non empty Relation-like the carrier of X -defined COMPLEX -valued Function-like total quasi_total V139() Element of bool [: the carrier of X,COMPLEX:]
FB is Element of the carrier of X
(X (#) F) . FB is complex Element of COMPLEX
X . FB is complex Element of COMPLEX
F . FB is complex Element of COMPLEX
(X . FB) * (F . FB) is complex Element of COMPLEX
bool the carrier of X is non empty set
FB is Element of the carrier of X
(X (#) F) . FB is complex Element of COMPLEX
GB is V176() Element of bool COMPLEX
aFB is complex set
aF is V176() Neighbourhood of aFB
aG is complex real ext-real Element of REAL
{ b₁ where b₁ is complex set : not aG <= |.(b₁ - aFB).| } is set
X . FB is complex Element of COMPLEX
F . FB is complex Element of COMPLEX
r is complex set
|.r.| is complex real ext-real non negative Element of REAL
s1 is complex set
|.s1.| is complex real ext-real non negative Element of REAL
|.r.| + |.s1.| is complex real ext-real non negative Element of REAL
(|.r.| + |.s1.|) + 1 is non empty complex real ext-real positive non negative Element of REAL
aG / ((|.r.| + |.s1.|) + 1) is complex real ext-real Element of COMPLEX
((|.r.| + |.s1.|) + 1) " is non empty complex real ext-real positive non negative set
aG * (((|.r.| + |.s1.|) + 1) ") is complex real ext-real set
{ b₁ where b₁ is complex set : not aG / ((|.r.| + |.s1.|) + 1) <= |.(b₁ - r).| } is set
A3 is set
vv1 is complex set
vv1 - r is complex set
- r is complex set
vv1 + (- r) is complex set
|.(vv1 - r).| is complex real ext-real non negative Element of REAL
A3 is V176() Element of bool COMPLEX
r - r is complex set
- r is complex set
r + (- r) is complex set
|.(r - r).| is complex real ext-real non negative Element of REAL
vv1 is Element of bool the carrier of X
X .: vv1 is V176() Element of bool COMPLEX
{ b₁ where b₁ is complex set : not aG / ((|.r.| + |.s1.|) + 1) <= |.(b₁ - s1).| } is set
fvu1 is set
zz0 is complex set
zz0 - s1 is complex set
- s1 is complex set
zz0 + (- s1) is complex set
|.(zz0 - s1).| is complex real ext-real non negative Element of REAL
fvu1 is V176() Element of bool COMPLEX
s1 - s1 is complex set
- s1 is complex set
s1 + (- s1) is complex set
|.(s1 - s1).| is complex real ext-real non negative Element of REAL
zz0 is Element of bool the carrier of X
F .: zz0 is V176() Element of bool COMPLEX
{ b₁ where b₁ is complex set : not 1 <= |.(b₁ - r).| } is set
hx0 is set
W3 is complex set
W3 - r is complex set
W3 + (- r) is complex set
|.(W3 - r).| is complex real ext-real non negative Element of REAL
hx0 is V176() Element of bool COMPLEX
W3 is Element of bool the carrier of X
X .: W3 is V176() Element of bool COMPLEX
vv1 /\ zz0 is Element of bool the carrier of X
(vv1 /\ zz0) /\ W3 is Element of bool the carrier of X
(X (#) F) .: ((vv1 /\ zz0) /\ W3) is V176() Element of bool COMPLEX
z3 is set
dom (X (#) F) is Element of bool the carrier of X
x3 is set
(X (#) F) . x3 is complex set
px is Element of the carrier of X
dom X is Element of bool the carrier of X
X . px is complex Element of COMPLEX
a1 is complex set
a1 - r is complex set
a1 + (- r) is complex set
|.(a1 - r).| is complex real ext-real non negative Element of REAL
a2 is complex set
a2 - r is complex set
a2 + (- r) is complex set
|.(a2 - r).| is complex real ext-real non negative Element of REAL
dom F is Element of bool the carrier of X
F . px is complex Element of COMPLEX
a2 is complex set
a2 - s1 is complex set
a2 + (- s1) is complex set
|.(a2 - s1).| is complex real ext-real non negative Element of REAL
a3 is complex set
a3 - s1 is complex set
a3 + (- s1) is complex set
|.(a3 - s1).| is complex real ext-real non negative Element of REAL
a3 is complex set
a3 - r is complex set
a3 + (- r) is complex set
|.(a3 - r).| is complex real ext-real non negative Element of REAL
c₃₀ is complex set
c₃₀ - r is complex set
c₃₀ + (- r) is complex set
|.(c₃₀ - r).| is complex real ext-real non negative Element of REAL
(a1 - r) + r is complex set
|.((a1 - r) + r).| is complex real ext-real non negative Element of REAL
|.(a1 - r).| + |.r.| is complex real ext-real non negative Element of REAL
|.a1.| is complex real ext-real non negative Element of REAL
|.a1.| - |.r.| is complex real ext-real Element of REAL
- |.r.| is complex real ext-real non positive set
|.a1.| + (- |.r.|) is complex real ext-real set
(|.(a1 - r).| + |.r.|) - |.r.| is complex real ext-real Element of REAL
(|.(a1 - r).| + |.r.|) + (- |.r.|) is complex real ext-real set
1 + |.r.| is non empty complex real ext-real positive non negative Element of REAL
(|.a1.| - |.r.|) + |.r.| is complex real ext-real Element of REAL
((X (#) F) . x3) - aFB is complex set
- aFB is complex set
((X (#) F) . x3) + (- aFB) is complex set
|.(((X (#) F) . x3) - aFB).| is complex real ext-real non negative Element of REAL
(X . px) * (F . px) is complex Element of COMPLEX
((X . px) * (F . px)) - aFB is complex set
((X . px) * (F . px)) + (- aFB) is complex set
|.(((X . px) * (F . px)) - aFB).| is complex real ext-real non negative Element of REAL
a1 * a2 is complex set
r * s1 is complex set
(a1 * a2) - (r * s1) is complex set
- (r * s1) is complex set
(a1 * a2) + (- (r * s1)) is complex set
|.((a1 * a2) - (r * s1)).| is complex real ext-real non negative Element of REAL
a1 * s1 is complex set
(a1 * a2) - (a1 * s1) is complex set
- (a1 * s1) is complex set
(a1 * a2) + (- (a1 * s1)) is complex set
(a1 * s1) - (r * s1) is complex set
(a1 * s1) + (- (r * s1)) is complex set
((a1 * a2) - (a1 * s1)) + ((a1 * s1) - (r * s1)) is complex set
|.(((a1 * a2) - (a1 * s1)) + ((a1 * s1) - (r * s1))).| is complex real ext-real non negative Element of REAL
|.((a1 * a2) - (a1 * s1)).| is complex real ext-real non negative Element of REAL
|.((a1 * s1) - (r * s1)).| is complex real ext-real non negative Element of REAL
|.((a1 * a2) - (a1 * s1)).| + |.((a1 * s1) - (r * s1)).| is complex real ext-real non negative Element of REAL
a1 * (a2 - s1) is complex set
|.(a1 * (a2 - s1)).| is complex real ext-real non negative Element of REAL
s1 * (a1 - r) is complex set
|.(s1 * (a1 - r)).| is complex real ext-real non negative Element of REAL
|.(a1 * (a2 - s1)).| + |.(s1 * (a1 - r)).| is complex real ext-real non negative Element of REAL
|.a1.| * |.(a2 - s1).| is complex real ext-real non negative Element of REAL
(|.a1.| * |.(a2 - s1).|) + |.(s1 * (a1 - r)).| is complex real ext-real non negative Element of REAL
|.s1.| * |.(a1 - r).| is complex real ext-real non negative Element of REAL
(|.a1.| * |.(a2 - s1).|) + (|.s1.| * |.(a1 - r).|) is complex real ext-real non negative Element of REAL
|.a1.| * (aG / ((|.r.| + |.s1.|) + 1)) is complex real ext-real Element of REAL
(1 + |.r.|) * (aG / ((|.r.| + |.s1.|) + 1)) is complex real ext-real Element of REAL
|.s1.| * (aG / ((|.r.| + |.s1.|) + 1)) is complex real ext-real Element of REAL
((1 + |.r.|) * (aG / ((|.r.| + |.s1.|) + 1))) + (|.s1.| * |.(a1 - r).|) is complex real ext-real Element of REAL
((1 + |.r.|) * (aG / ((|.r.| + |.s1.|) + 1))) + (|.s1.| * (aG / ((|.r.| + |.s1.|) + 1))) is complex real ext-real Element of REAL
((|.r.| + |.s1.|) + 1) / ((|.r.| + |.s1.|) + 1) is non empty complex real ext-real positive non negative Element of COMPLEX
((|.r.| + |.s1.|) + 1) * (((|.r.| + |.s1.|) + 1) ") is non empty complex real ext-real positive non negative set
aG * (((|.r.| + |.s1.|) + 1) / ((|.r.| + |.s1.|) + 1)) is complex real ext-real Element of REAL
(X (#) F) . px is complex Element of COMPLEX
((X (#) F) . px) - aFB is complex set
((X (#) F) . px) + (- aFB) is complex set
|.(((X (#) F) . px) - aFB).| is complex real ext-real non negative Element of REAL
X is non empty TopSpace-like TopStruct
the carrier of X is non empty set
[: the carrier of X,COMPLEX:] is non empty V139() set
bool [: the carrier of X,COMPLEX:] is non empty set
[: the carrier of X,REAL:] is non empty V139() V140() V141() set
bool [: the carrier of X,REAL:] is non empty set
X is non empty Relation-like the carrier of X -defined COMPLEX -valued Function-like total quasi_total V139() (X) Element of bool [: the carrier of X,COMPLEX:]
|.X.| is non empty Relation-like the carrier of X -defined REAL -valued Function-like total quasi_total V139() V140() V141() Element of bool [: the carrier of X,REAL:]
F is non empty Relation-like the carrier of X -defined REAL -valued Function-like total quasi_total V139() V140() V141() Element of bool [: the carrier of X,REAL:]
bool the carrier of X is non empty set
G is Element of the carrier of X
F . G is complex real ext-real Element of REAL
FB is V176() V177() V178() Element of bool REAL
GB is complex real ext-real Element of REAL
aFB is complex real ext-real set
GB - aFB is complex real ext-real Element of REAL
- aFB is complex real ext-real set
GB + (- aFB) is complex real ext-real set
GB + aFB is complex real ext-real Element of REAL
].(GB - aFB),(GB + aFB).[ is V176() V177() V178() non left_end non right_end V197() Element of bool REAL
X . G is complex Element of COMPLEX
aG is complex set
{ b₁ where b₁ is complex set : not aFB <= |.(b₁ - aG).| } is set
r is set
s1 is complex set
s1 - aG is complex set
- aG is complex set
s1 + (- aG) is complex set
|.(s1 - aG).| is complex real ext-real non negative Element of REAL
r is V176() Element of bool COMPLEX
aG - aG is complex set
- aG is complex set
aG + (- aG) is complex set
|.(aG - aG).| is complex real ext-real non negative Element of REAL
s1 is Element of bool the carrier of X
X .: s1 is V176() Element of bool COMPLEX
F .: s1 is V176() V177() V178() Element of bool REAL
A2 is set
dom F is Element of bool the carrier of X
A3 is set
F . A3 is complex real ext-real set
vv1 is Element of the carrier of X
dom X is Element of bool the carrier of X
X . vv1 is complex Element of COMPLEX
uu1 is complex set
uu1 - aG is complex set
uu1 + (- aG) is complex set
|.(uu1 - aG).| is complex real ext-real non negative Element of REAL
fvu1 is complex set
fvu1 - aG is complex set
fvu1 + (- aG) is complex set
|.(fvu1 - aG).| is complex real ext-real non negative Element of REAL
(F . A3) - GB is complex real ext-real Element of REAL
- GB is complex real ext-real set
(F . A3) + (- GB) is complex real ext-real set
|.((F . A3) - GB).| is complex real ext-real non negative Element of REAL
|.(X . vv1).| is complex real ext-real non negative Element of REAL
|.X.| . G is complex real ext-real Element of REAL
|.(X . vv1).| - (|.X.| . G) is complex real ext-real Element of REAL
- (|.X.| . G) is complex real ext-real set
|.(X . vv1).| + (- (|.X.| . G)) is complex real ext-real set
|.(|.(X . vv1).| - (|.X.| . G)).| is complex real ext-real non negative Element of REAL
|.(X . G).| is complex real ext-real non negative Element of REAL
|.(X . vv1).| - |.(X . G).| is complex real ext-real Element of REAL
- |.(X . G).| is complex real ext-real non positive set
|.(X . vv1).| + (- |.(X . G).|) is complex real ext-real set
|.(|.(X . vv1).| - |.(X . G).|).| is complex real ext-real non negative Element of REAL
(X . vv1) - (X . G) is complex Element of COMPLEX
- (X . G) is complex set
(X . vv1) + (- (X . G)) is complex set
|.((X . vv1) - (X . G)).| is complex real ext-real non negative Element of REAL
X is non empty TopSpace-like TopStruct
the carrier of X is non empty set
[: the carrier of X,COMPLEX:] is non empty V139() set
bool [: the carrier of X,COMPLEX:] is non empty set
{ b₁ where b₁ is non empty Relation-like the carrier of X -defined COMPLEX -valued Function-like total quasi_total V139() (X) Element of bool [: the carrier of X,COMPLEX:] : verum } is set
CAlgebra the carrier of X is non empty right_complementable Abelian add-associative right_zeroed unital associative commutative right-distributive right_unital well-unital left_unital vector-distributive scalar-distributive scalar-associative scalar-unital strict vector-associative ComplexAlgebraStr
Funcs ( the carrier of X,COMPLEX) is non empty functional FUNCTION_DOMAIN of the carrier of X, COMPLEX
ComplexFuncMult the carrier of X is non empty Relation-like [:(Funcs ( the carrier of X,COMPLEX)),(Funcs ( the carrier of X,COMPLEX)):] -defined Funcs ( the carrier of X,COMPLEX) -valued Function-like total quasi_total Element of bool [:[:(Funcs ( the carrier of X,COMPLEX)),(Funcs ( the carrier of X,COMPLEX)):],(Funcs ( the carrier of X,COMPLEX)):]
[:(Funcs ( the carrier of X,COMPLEX)),(Funcs ( the carrier of X,COMPLEX)):] is non empty set
[:[:(Funcs ( the carrier of X,COMPLEX)),(Funcs ( the carrier of X,COMPLEX)):],(Funcs ( the carrier of X,COMPLEX)):] is non empty set
bool [:[:(Funcs ( the carrier of X,COMPLEX)),(Funcs ( the carrier of X,COMPLEX)):],(Funcs ( the carrier of X,COMPLEX)):] is non empty set
ComplexFuncAdd the carrier of X is non empty Relation-like [:(Funcs ( the carrier of X,COMPLEX)),(Funcs ( the carrier of X,COMPLEX)):] -defined Funcs ( the carrier of X,COMPLEX) -valued Function-like total quasi_total Element of bool [:[:(Funcs ( the carrier of X,COMPLEX)),(Funcs ( the carrier of X,COMPLEX)):],(Funcs ( the carrier of X,COMPLEX)):]
ComplexFuncExtMult the carrier of X is non empty Relation-like [:COMPLEX,(Funcs ( the carrier of X,COMPLEX)):] -defined Funcs ( the carrier of X,COMPLEX) -valued Function-like total quasi_total Element of bool [:[:COMPLEX,(Funcs ( the carrier of X,COMPLEX)):],(Funcs ( the carrier of X,COMPLEX)):]
[:COMPLEX,(Funcs ( the carrier of X,COMPLEX)):] is non empty set
[:[:COMPLEX,(Funcs ( the carrier of X,COMPLEX)):],(Funcs ( the carrier of X,COMPLEX)):] is non empty set
bool [:[:COMPLEX,(Funcs ( the carrier of X,COMPLEX)):],(Funcs ( the carrier of X,COMPLEX)):] is non empty set
ComplexFuncUnit the carrier of X is Relation-like the carrier of X -defined COMPLEX -valued Function-like total quasi_total V139() Element of Funcs ( the carrier of X,COMPLEX)
the carrier of X --> 1r is non empty Relation-like the carrier of X -defined COMPLEX -valued Function-like total quasi_total V139() Element of bool [: the carrier of X,COMPLEX:]
ComplexFuncZero the carrier of X is Relation-like the carrier of X -defined COMPLEX -valued Function-like total quasi_total V139() Element of Funcs ( the carrier of X,COMPLEX)
the carrier of X --> 0 is non empty Relation-like the carrier of X -defined REAL -valued RAT -valued INT -valued Function-like total quasi_total V139() V140() V141() V142() Element of bool [: the carrier of X,REAL:]
[: the carrier of X,REAL:] is non empty V139() V140() V141() set
bool [: the carrier of X,REAL:] is non empty set
ComplexAlgebraStr(# (Funcs ( the carrier of X,COMPLEX)),(ComplexFuncMult the carrier of X),(ComplexFuncAdd the carrier of X),(ComplexFuncExtMult the carrier of X),(ComplexFuncUnit the carrier of X),(ComplexFuncZero the carrier of X) #) is strict ComplexAlgebraStr
the carrier of (CAlgebra the carrier of X) is non empty set
bool the carrier of (CAlgebra the carrier of X) is non empty set
F is set
G is non empty Relation-like the carrier of X -defined COMPLEX -valued Function-like total quasi_total V139() (X) Element of bool [: the carrier of X,COMPLEX:]
X is non empty TopSpace-like TopStruct
(X) is Element of bool the carrier of (CAlgebra the carrier of X)
the carrier of X is non empty set
CAlgebra the carrier of X is non empty right_complementable Abelian add-associative right_zeroed unital associative commutative right-distributive right_unital well-unital left_unital vector-distributive scalar-distributive scalar-associative scalar-unital strict vector-associative ComplexAlgebraStr
Funcs ( the carrier of X,COMPLEX) is non empty functional FUNCTION_DOMAIN of the carrier of X, COMPLEX
ComplexFuncMult the carrier of X is non empty Relation-like [:(Funcs ( the carrier of X,COMPLEX)),(Funcs ( the carrier of X,COMPLEX)):] -defined Funcs ( the carrier of X,COMPLEX) -valued Function-like total quasi_total Element of bool [:[:(Funcs ( the carrier of X,COMPLEX)),(Funcs ( the carrier of X,COMPLEX)):],(Funcs ( the carrier of X,COMPLEX)):]
[:(Funcs ( the carrier of X,COMPLEX)),(Funcs ( the carrier of X,COMPLEX)):] is non empty set
[:[:(Funcs ( the carrier of X,COMPLEX)),(Funcs ( the carrier of X,COMPLEX)):],(Funcs ( the carrier of X,COMPLEX)):] is non empty set
bool [:[:(Funcs ( the carrier of X,COMPLEX)),(Funcs ( the carrier of X,COMPLEX)):],(Funcs ( the carrier of X,COMPLEX)):] is non empty set
ComplexFuncAdd the carrier of X is non empty Relation-like [:(Funcs ( the carrier of X,COMPLEX)),(Funcs ( the carrier of X,COMPLEX)):] -defined Funcs ( the carrier of X,COMPLEX) -valued Function-like total quasi_total Element of bool [:[:(Funcs ( the carrier of X,COMPLEX)),(Funcs ( the carrier of X,COMPLEX)):],(Funcs ( the carrier of X,COMPLEX)):]
ComplexFuncExtMult the carrier of X is non empty Relation-like [:COMPLEX,(Funcs ( the carrier of X,COMPLEX)):] -defined Funcs ( the carrier of X,COMPLEX) -valued Function-like total quasi_total Element of bool [:[:COMPLEX,(Funcs ( the carrier of X,COMPLEX)):],(Funcs ( the carrier of X,COMPLEX)):]
[:COMPLEX,(Funcs ( the carrier of X,COMPLEX)):] is non empty set
[:[:COMPLEX,(Funcs ( the carrier of X,COMPLEX)):],(Funcs ( the carrier of X,COMPLEX)):] is non empty set
bool [:[:COMPLEX,(Funcs ( the carrier of X,COMPLEX)):],(Funcs ( the carrier of X,COMPLEX)):] is non empty set
ComplexFuncUnit the carrier of X is Relation-like the carrier of X -defined COMPLEX -valued Function-like total quasi_total V139() Element of Funcs ( the carrier of X,COMPLEX)
the carrier of X --> 1r is non empty Relation-like the carrier of X -defined COMPLEX -valued Function-like total quasi_total V139() Element of bool [: the carrier of X,COMPLEX:]
[: the carrier of X,COMPLEX:] is non empty V139() set
bool [: the carrier of X,COMPLEX:] is non empty set
ComplexFuncZero the carrier of X is Relation-like the carrier of X -defined COMPLEX -valued Function-like total quasi_total V139() Element of Funcs ( the carrier of X,COMPLEX)
the carrier of X --> 0 is non empty Relation-like the carrier of X -defined REAL -valued RAT -valued INT -valued Function-like total quasi_total V139() V140() V141() V142() Element of bool [: the carrier of X,REAL:]
[: the carrier of X,REAL:] is non empty V139() V140() V141() set
bool [: the carrier of X,REAL:] is non empty set
ComplexAlgebraStr(# (Funcs ( the carrier of X,COMPLEX)),(ComplexFuncMult the carrier of X),(ComplexFuncAdd the carrier of X),(ComplexFuncExtMult the carrier of X),(ComplexFuncUnit the carrier of X),(ComplexFuncZero the carrier of X) #) is strict ComplexAlgebraStr
the carrier of (CAlgebra the carrier of X) is non empty set
bool the carrier of (CAlgebra the carrier of X) is non empty set
{ b₁ where b₁ is non empty Relation-like the carrier of X -defined COMPLEX -valued Function-like total quasi_total V139() (X) Element of bool [: the carrier of X,COMPLEX:] : verum } is set
0c is empty epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural complex real ext-real non positive non negative V176() V177() V178() V179() V180() V181() V182() bounded_below V197() Element of COMPLEX
(X,0c) is non empty Relation-like the carrier of X -defined COMPLEX -valued Function-like total quasi_total V139() (X) Element of bool [: the carrier of X,COMPLEX:]
the carrier of X --> 0c is non empty Relation-like the carrier of X -defined {0c} -valued Function-like total quasi_total V139() V140() V141() V142() Element of bool [: the carrier of X,{0c}:]
{0c} is non empty V176() V177() V178() V179() V180() V181() left_end bounded_below set
[: the carrier of X,{0c}:] is non empty RAT -valued INT -valued V139() V140() V141() V142() set
bool [: the carrier of X,{0c}:] is non empty set
X is non empty TopSpace-like TopStruct
the carrier of X is non empty set
CAlgebra the carrier of X is non empty right_complementable Abelian add-associative right_zeroed unital associative commutative right-distributive right_unital well-unital left_unital vector-distributive scalar-distributive scalar-associative scalar-unital strict vector-associative ComplexAlgebraStr
Funcs ( the carrier of X,COMPLEX) is non empty functional FUNCTION_DOMAIN of the carrier of X, COMPLEX
ComplexFuncMult the carrier of X is non empty Relation-like [:(Funcs ( the carrier of X,COMPLEX)),(Funcs ( the carrier of X,COMPLEX)):] -defined Funcs ( the carrier of X,COMPLEX) -valued Function-like total quasi_total Element of bool [:[:(Funcs ( the carrier of X,COMPLEX)),(Funcs ( the carrier of X,COMPLEX)):],(Funcs ( the carrier of X,COMPLEX)):]
[:(Funcs ( the carrier of X,COMPLEX)),(Funcs ( the carrier of X,COMPLEX)):] is non empty set
[:[:(Funcs ( the carrier of X,COMPLEX)),(Funcs ( the carrier of X,COMPLEX)):],(Funcs ( the carrier of X,COMPLEX)):] is non empty set
bool [:[:(Funcs ( the carrier of X,COMPLEX)),(Funcs ( the carrier of X,COMPLEX)):],(Funcs ( the carrier of X,COMPLEX)):] is non empty set
ComplexFuncAdd the carrier of X is non empty Relation-like [:(Funcs ( the carrier of X,COMPLEX)),(Funcs ( the carrier of X,COMPLEX)):] -defined Funcs ( the carrier of X,COMPLEX) -valued Function-like total quasi_total Element of bool [:[:(Funcs ( the carrier of X,COMPLEX)),(Funcs ( the carrier of X,COMPLEX)):],(Funcs ( the carrier of X,COMPLEX)):]
ComplexFuncExtMult the carrier of X is non empty Relation-like [:COMPLEX,(Funcs ( the carrier of X,COMPLEX)):] -defined Funcs ( the carrier of X,COMPLEX) -valued Function-like total quasi_total Element of bool [:[:COMPLEX,(Funcs ( the carrier of X,COMPLEX)):],(Funcs ( the carrier of X,COMPLEX)):]
[:COMPLEX,(Funcs ( the carrier of X,COMPLEX)):] is non empty set
[:[:COMPLEX,(Funcs ( the carrier of X,COMPLEX)):],(Funcs ( the carrier of X,COMPLEX)):] is non empty set
bool [:[:COMPLEX,(Funcs ( the carrier of X,COMPLEX)):],(Funcs ( the carrier of X,COMPLEX)):] is non empty set
ComplexFuncUnit the carrier of X is Relation-like the carrier of X -defined COMPLEX -valued Function-like total quasi_total V139() Element of Funcs ( the carrier of X,COMPLEX)
the carrier of X --> 1r is non empty Relation-like the carrier of X -defined COMPLEX -valued Function-like total quasi_total V139() Element of bool [: the carrier of X,COMPLEX:]
[: the carrier of X,COMPLEX:] is non empty V139() set
bool [: the carrier of X,COMPLEX:] is non empty set
ComplexFuncZero the carrier of X is Relation-like the carrier of X -defined COMPLEX -valued Function-like total quasi_total V139() Element of Funcs ( the carrier of X,COMPLEX)
the carrier of X --> 0 is non empty Relation-like the carrier of X -defined REAL -valued RAT -valued INT -valued Function-like total quasi_total V139() V140() V141() V142() Element of bool [: the carrier of X,REAL:]
[: the carrier of X,REAL:] is non empty V139() V140() V141() set
bool [: the carrier of X,REAL:] is non empty set
ComplexAlgebraStr(# (Funcs ( the carrier of X,COMPLEX)),(ComplexFuncMult the carrier of X),(ComplexFuncAdd the carrier of X),(ComplexFuncExtMult the carrier of X),(ComplexFuncUnit the carrier of X),(ComplexFuncZero the carrier of X) #) is strict ComplexAlgebraStr
(X) is non empty Element of bool the carrier of (CAlgebra the carrier of X)
the carrier of (CAlgebra the carrier of X) is non empty set
bool the carrier of (CAlgebra the carrier of X) is non empty set
{ b₁ where b₁ is non empty Relation-like the carrier of X -defined COMPLEX -valued Function-like total quasi_total V139() (X) Element of bool [: the carrier of X,COMPLEX:] : verum } is set
G is Element of the carrier of (CAlgebra the carrier of X)
FB is Element of the carrier of (CAlgebra the carrier of X)
G + FB is Element of the carrier of (CAlgebra the carrier of X)
the addF of (CAlgebra the carrier of X) is non empty Relation-like [: the carrier of (CAlgebra the carrier of X), the carrier of (CAlgebra the carrier of X):] -defined the carrier of (CAlgebra the carrier of X) -valued Function-like total quasi_total Element of bool [:[: the carrier of (CAlgebra the carrier of X), the carrier of (CAlgebra the carrier of X):], the carrier of (CAlgebra the carrier of X):]
[: the carrier of (CAlgebra the carrier of X), the carrier of (CAlgebra the carrier of X):] is non empty set
[:[: the carrier of (CAlgebra the carrier of X), the carrier of (CAlgebra the carrier of X):], the carrier of (CAlgebra the carrier of X):] is non empty set
bool [:[: the carrier of (CAlgebra the carrier of X), the carrier of (CAlgebra the carrier of X):], the carrier of (CAlgebra the carrier of X):] is non empty set
the addF of (CAlgebra the carrier of X) . (G,FB) is Element of the carrier of (CAlgebra the carrier of X)
[G,FB] is set
{G,FB} is non empty set
{G} is non empty set
{{G,FB},{G}} is non empty set
the addF of (CAlgebra the carrier of X) . [G,FB] is set
GB is non empty Relation-like the carrier of X -defined COMPLEX -valued Function-like total quasi_total V139() (X) Element of bool [: the carrier of X,COMPLEX:]
aFB is non empty Relation-like the carrier of X -defined COMPLEX -valued Function-like total quasi_total V139() (X) Element of bool [: the carrier of X,COMPLEX:]
aF is Relation-like the carrier of X -defined COMPLEX -valued Function-like total quasi_total V139() Element of Funcs ( the carrier of X,COMPLEX)
dom GB is Element of bool the carrier of X
bool the carrier of X is non empty set
dom aFB is Element of bool the carrier of X
(dom GB) /\ (dom aFB) is Element of bool the carrier of X
the carrier of X /\ (dom aFB) is Element of bool the carrier of X
the carrier of X /\ the carrier of X is set
dom aF is Element of bool the carrier of X
aG is set
aF . aG is complex set
GB . aG is complex set
aFB . aG is complex set
(GB . aG) + (aFB . aG) is complex set
GB + aFB is Relation-like the carrier of X -defined Function-like total V139() set
aG is Relation-like Function-like set
dom aG is set
rng aG is set
G is Element of the carrier of (CAlgebra the carrier of X)
- G is Element of the carrier of (CAlgebra the carrier of X)
FB is non empty Relation-like the carrier of X -defined COMPLEX -valued Function-like total quasi_total V139() (X) Element of bool [: the carrier of X,COMPLEX:]
GB is Relation-like the carrier of X -defined COMPLEX -valued Function-like total quasi_total V139() Element of Funcs ( the carrier of X,COMPLEX)
aF is complex set
aF * G is Element of the carrier of (CAlgebra the carrier of X)
the Mult of (CAlgebra the carrier of X) is non empty Relation-like [:COMPLEX, the carrier of (CAlgebra the carrier of X):] -defined the carrier of (CAlgebra the carrier of X) -valued Function-like total quasi_total Element of bool [:[:COMPLEX, the carrier of (CAlgebra the carrier of X):], the carrier of (CAlgebra the carrier of X):]
[:COMPLEX, the carrier of (CAlgebra the carrier of X):] is non empty set
[:[:COMPLEX, the carrier of (CAlgebra the carrier of X):], the carrier of (CAlgebra the carrier of X):] is non empty set
bool [:[:COMPLEX, the carrier of (CAlgebra the carrier of X):], the carrier of (CAlgebra the carrier of X):] is non empty set
[aF,G] is set
{aF,G} is non empty set
{aF} is non empty V176() set
{{aF,G},{aF}} is non empty set
the Mult of (CAlgebra the carrier of X) . [aF,G] is set
dom FB is Element of bool the carrier of X
bool the carrier of X is non empty set
aG is set
dom GB is Element of bool the carrier of X
GB . aG is complex set
- 1 is non empty complex real ext-real non positive negative Element of REAL
aFB is Relation-like the carrier of X -defined COMPLEX -valued Function-like total quasi_total V139() Element of Funcs ( the carrier of X,COMPLEX)
r is Element of the carrier of X
aFB . r is complex Element of COMPLEX
(- 1) * (aFB . r) is complex set
FB . aG is complex set
- (FB . aG) is complex set
- FB is non empty Relation-like the carrier of X -defined COMPLEX -valued Function-like total quasi_total V139() Element of bool [: the carrier of X,COMPLEX:]
- 1 is non empty complex real ext-real non positive negative set
(- 1) (#) FB is Relation-like the carrier of X -defined Function-like total V139() set
aG is Relation-like Function-like set
dom aG is set
rng aG is set
aF (#) FB is non empty Relation-like the carrier of X -defined COMPLEX -valued Function-like total quasi_total V139() Element of bool [: the carrier of X,COMPLEX:]
G is complex set
FB is Element of the carrier of (CAlgebra the carrier of X)
G * FB is Element of the carrier of (CAlgebra the carrier of X)
the Mult of (CAlgebra the carrier of X) is non empty Relation-like [:COMPLEX, the carrier of (CAlgebra the carrier of X):] -defined the carrier of (CAlgebra the carrier of X) -valued Function-like total quasi_total Element of bool [:[:COMPLEX, the carrier of (CAlgebra the carrier of X):], the carrier of (CAlgebra the carrier of X):]
[:COMPLEX, the carrier of (CAlgebra the carrier of X):] is non empty set
[:[:COMPLEX, the carrier of (CAlgebra the carrier of X):], the carrier of (CAlgebra the carrier of X):] is non empty set
bool [:[:COMPLEX, the carrier of (CAlgebra the carrier of X):], the carrier of (CAlgebra the carrier of X):] is non empty set
[G,FB] is set
{G,FB} is non empty set
{G} is non empty V176() set
{{G,FB},{G}} is non empty set
the Mult of (CAlgebra the carrier of X) . [G,FB] is set
GB is non empty Relation-like the carrier of X -defined COMPLEX -valued Function-like total quasi_total V139() (X) Element of bool [: the carrier of X,COMPLEX:]
aFB is Relation-like the carrier of X -defined COMPLEX -valued Function-like total quasi_total V139() Element of Funcs ( the carrier of X,COMPLEX)
dom GB is Element of bool the carrier of X
bool the carrier of X is non empty set
dom aFB is Element of bool the carrier of X
aF is set
aFB . aF is complex set
GB . aF is complex set
G * (GB . aF) is complex set
G (#) GB is non empty Relation-like the carrier of X -defined COMPLEX -valued Function-like total quasi_total V139() Element of bool [: the carrier of X,COMPLEX:]
aF is Relation-like Function-like set
dom aF is set
rng aF is set
G is Element of the carrier of (CAlgebra the carrier of X)
FB is Element of the carrier of (CAlgebra the carrier of X)
G * FB is Element of the carrier of (CAlgebra the carrier of X)
the multF of (CAlgebra the carrier of X) is non empty Relation-like [: the carrier of (CAlgebra the carrier of X), the carrier of (CAlgebra the carrier of X):] -defined the carrier of (CAlgebra the carrier of X) -valued Function-like total quasi_total Element of bool [:[: the carrier of (CAlgebra the carrier of X), the carrier of (CAlgebra the carrier of X):], the carrier of (CAlgebra the carrier of X):]
[: the carrier of (CAlgebra the carrier of X), the carrier of (CAlgebra the carrier of X):] is non empty set
[:[: the carrier of (CAlgebra the carrier of X), the carrier of (CAlgebra the carrier of X):], the carrier of (CAlgebra the carrier of X):] is non empty set
bool [:[: the carrier of (CAlgebra the carrier of X), the carrier of (CAlgebra the carrier of X):], the carrier of (CAlgebra the carrier of X):] is non empty set
the multF of (CAlgebra the carrier of X) . (G,FB) is Element of the carrier of (CAlgebra the carrier of X)
[G,FB] is set
{G,FB} is non empty set
{G} is non empty set
{{G,FB},{G}} is non empty set
the multF of (CAlgebra the carrier of X) . [G,FB] is set
GB is non empty Relation-like the carrier of X -defined COMPLEX -valued Function-like total quasi_total V139() (X) Element of bool [: the carrier of X,COMPLEX:]
aFB is non empty Relation-like the carrier of X -defined COMPLEX -valued Function-like total quasi_total V139() (X) Element of bool [: the carrier of X,COMPLEX:]
aF is Relation-like the carrier of X -defined COMPLEX -valued Function-like total quasi_total V139() Element of Funcs ( the carrier of X,COMPLEX)
dom GB is Element of bool the carrier of X
bool the carrier of X is non empty set
dom aFB is Element of bool the carrier of X
(dom GB) /\ (dom aFB) is Element of bool the carrier of X
the carrier of X /\ (dom aFB) is Element of bool the carrier of X
the carrier of X /\ the carrier of X is set
dom aF is Element of bool the carrier of X
aG is set
aF . aG is complex set
GB . aG is complex set
aFB . aG is complex set
(GB . aG) * (aFB . aG) is complex set
GB (#) aFB is non empty Relation-like the carrier of X -defined COMPLEX -valued Function-like total quasi_total V139() Element of bool [: the carrier of X,COMPLEX:]
aG is Relation-like Function-like set
dom aG is set
rng aG is set
G is non empty Relation-like the carrier of X -defined COMPLEX -valued Function-like total quasi_total V139() Element of bool [: the carrier of X,COMPLEX:]
(X,1r) is non empty Relation-like the carrier of X -defined COMPLEX -valued Function-like total quasi_total V139() (X) Element of bool [: the carrier of X,COMPLEX:]
the carrier of X --> 1r is non empty Relation-like the carrier of X -defined {1r} -valued Function-like total quasi_total V139() Element of bool [: the carrier of X,{1r}:]
{1r} is non empty V176() set
[: the carrier of X,{1r}:] is non empty V139() set
bool [: the carrier of X,{1r}:] is non empty set
1. (CAlgebra the carrier of X) is Element of the carrier of (CAlgebra the carrier of X)
the OneF of (CAlgebra the carrier of X) is Element of the carrier of (CAlgebra the carrier of X)
X is non empty TopSpace-like TopStruct
(X) is non empty multiplicatively-closed Cadditively-linearly-closed Element of bool the carrier of (CAlgebra the carrier of X)
the carrier of X is non empty set
CAlgebra the carrier of X is non empty right_complementable Abelian add-associative right_zeroed unital associative commutative right-distributive right_unital well-unital left_unital vector-distributive scalar-distributive scalar-associative scalar-unital strict vector-associative ComplexAlgebraStr
Funcs ( the carrier of X,COMPLEX) is non empty functional FUNCTION_DOMAIN of the carrier of X, COMPLEX
ComplexFuncMult the carrier of X is non empty Relation-like [:(Funcs ( the carrier of X,COMPLEX)),(Funcs ( the carrier of X,COMPLEX)):] -defined Funcs ( the carrier of X,COMPLEX) -valued Function-like total quasi_total Element of bool [:[:(Funcs ( the carrier of X,COMPLEX)),(Funcs ( the carrier of X,COMPLEX)):],(Funcs ( the carrier of X,COMPLEX)):]
[:(Funcs ( the carrier of X,COMPLEX)),(Funcs ( the carrier of X,COMPLEX)):] is non empty set
[:[:(Funcs ( the carrier of X,COMPLEX)),(Funcs ( the carrier of X,COMPLEX)):],(Funcs ( the carrier of X,COMPLEX)):] is non empty set
bool [:[:(Funcs ( the carrier of X,COMPLEX)),(Funcs ( the carrier of X,COMPLEX)):],(Funcs ( the carrier of X,COMPLEX)):] is non empty set
ComplexFuncAdd the carrier of X is non empty Relation-like [:(Funcs ( the carrier of X,COMPLEX)),(Funcs ( the carrier of X,COMPLEX)):] -defined Funcs ( the carrier of X,COMPLEX) -valued Function-like total quasi_total Element of bool [:[:(Funcs ( the carrier of X,COMPLEX)),(Funcs ( the carrier of X,COMPLEX)):],(Funcs ( the carrier of X,COMPLEX)):]
ComplexFuncExtMult the carrier of X is non empty Relation-like [:COMPLEX,(Funcs ( the carrier of X,COMPLEX)):] -defined Funcs ( the carrier of X,COMPLEX) -valued Function-like total quasi_total Element of bool [:[:COMPLEX,(Funcs ( the carrier of X,COMPLEX)):],(Funcs ( the carrier of X,COMPLEX)):]
[:COMPLEX,(Funcs ( the carrier of X,COMPLEX)):] is non empty set
[:[:COMPLEX,(Funcs ( the carrier of X,COMPLEX)):],(Funcs ( the carrier of X,COMPLEX)):] is non empty set
bool [:[:COMPLEX,(Funcs ( the carrier of X,COMPLEX)):],(Funcs ( the carrier of X,COMPLEX)):] is non empty set
ComplexFuncUnit the carrier of X is Relation-like the carrier of X -defined COMPLEX -valued Function-like total quasi_total V139() Element of Funcs ( the carrier of X,COMPLEX)
the carrier of X --> 1r is non empty Relation-like the carrier of X -defined COMPLEX -valued Function-like total quasi_total V139() Element of bool [: the carrier of X,COMPLEX:]
[: the carrier of X,COMPLEX:] is non empty V139() set
bool [: the carrier of X,COMPLEX:] is non empty set
ComplexFuncZero the carrier of X is Relation-like the carrier of X -defined COMPLEX -valued Function-like total quasi_total V139() Element of Funcs ( the carrier of X,COMPLEX)
the carrier of X --> 0 is non empty Relation-like the carrier of X -defined REAL -valued RAT -valued INT -valued Function-like total quasi_total V139() V140() V141() V142() Element of bool [: the carrier of X,REAL:]
[: the carrier of X,REAL:] is non empty V139() V140() V141() set
bool [: the carrier of X,REAL:] is non empty set
ComplexAlgebraStr(# (Funcs ( the carrier of X,COMPLEX)),(ComplexFuncMult the carrier of X),(ComplexFuncAdd the carrier of X),(ComplexFuncExtMult the carrier of X),(ComplexFuncUnit the carrier of X),(ComplexFuncZero the carrier of X) #) is strict ComplexAlgebraStr
the carrier of (CAlgebra the carrier of X) is non empty set
bool the carrier of (CAlgebra the carrier of X) is non empty set
{ b₁ where b₁ is non empty Relation-like the carrier of X -defined COMPLEX -valued Function-like total quasi_total V139() (X) Element of bool [: the carrier of X,COMPLEX:] : verum } is set
mult_ ((X),(CAlgebra the carrier of X)) is non empty Relation-like [:(X),(X):] -defined (X) -valued Function-like total quasi_total Element of bool [:[:(X),(X):],(X):]
[:(X),(X):] is non empty set
[:[:(X),(X):],(X):] is non empty set
bool [:[:(X),(X):],(X):] is non empty set
Add_ ((X),(CAlgebra the carrier of X)) is non empty Relation-like [:(X),(X):] -defined (X) -valued Function-like total quasi_total Element of bool [:[:(X),(X):],(X):]
Mult_ ((X),(CAlgebra the carrier of X)) is non empty Relation-like [:COMPLEX,(X):] -defined (X) -valued Function-like total quasi_total Element of bool [:[:COMPLEX,(X):],(X):]
[:COMPLEX,(X):] is non empty set
[:[:COMPLEX,(X):],(X):] is non empty set
bool [:[:COMPLEX,(X):],(X):] is non empty set
One_ ((X),(CAlgebra the carrier of X)) is Element of (X)
Zero_ ((X),(CAlgebra the carrier of X)) is Element of (X)
ComplexAlgebraStr(# (X),(mult_ ((X),(CAlgebra the carrier of X))),(Add_ ((X),(CAlgebra the carrier of X))),(Mult_ ((X),(CAlgebra the carrier of X))),(One_ ((X),(CAlgebra the carrier of X))),(Zero_ ((X),(CAlgebra the carrier of X))) #) is strict ComplexAlgebraStr
X is non empty TopSpace-like TopStruct
(X) is non empty right_complementable Abelian add-associative right_zeroed unital associative commutative right-distributive right_unital well-unital left_unital vector-distributive scalar-distributive scalar-associative vector-associative ComplexAlgebraStr
(X) is non empty multiplicatively-closed Cadditively-linearly-closed Element of bool the carrier of (CAlgebra the carrier of X)
the carrier of X is non empty set
CAlgebra the carrier of X is non empty right_complementable Abelian add-associative right_zeroed unital associative commutative right-distributive right_unital well-unital left_unital vector-distributive scalar-distributive scalar-associative scalar-unital strict vector-associative ComplexAlgebraStr
Funcs ( the carrier of X,COMPLEX) is non empty functional FUNCTION_DOMAIN of the carrier of X, COMPLEX
ComplexFuncMult the carrier of X is non empty Relation-like [:(Funcs ( the carrier of X,COMPLEX)),(Funcs ( the carrier of X,COMPLEX)):] -defined Funcs ( the carrier of X,COMPLEX) -valued Function-like total quasi_total Element of bool [:[:(Funcs ( the carrier of X,COMPLEX)),(Funcs ( the carrier of X,COMPLEX)):],(Funcs ( the carrier of X,COMPLEX)):]
[:(Funcs ( the carrier of X,COMPLEX)),(Funcs ( the carrier of X,COMPLEX)):] is non empty set
[:[:(Funcs ( the carrier of X,COMPLEX)),(Funcs ( the carrier of X,COMPLEX)):],(Funcs ( the carrier of X,COMPLEX)):] is non empty set
bool [:[:(Funcs ( the carrier of X,COMPLEX)),(Funcs ( the carrier of X,COMPLEX)):],(Funcs ( the carrier of X,COMPLEX)):] is non empty set
ComplexFuncAdd the carrier of X is non empty Relation-like [:(Funcs ( the carrier of X,COMPLEX)),(Funcs ( the carrier of X,COMPLEX)):] -defined Funcs ( the carrier of X,COMPLEX) -valued Function-like total quasi_total Element of bool [:[:(Funcs ( the carrier of X,COMPLEX)),(Funcs ( the carrier of X,COMPLEX)):],(Funcs ( the carrier of X,COMPLEX)):]
ComplexFuncExtMult the carrier of X is non empty Relation-like [:COMPLEX,(Funcs ( the carrier of X,COMPLEX)):] -defined Funcs ( the carrier of X,COMPLEX) -valued Function-like total quasi_total Element of bool [:[:COMPLEX,(Funcs ( the carrier of X,COMPLEX)):],(Funcs ( the carrier of X,COMPLEX)):]
[:COMPLEX,(Funcs ( the carrier of X,COMPLEX)):] is non empty set
[:[:COMPLEX,(Funcs ( the carrier of X,COMPLEX)):],(Funcs ( the carrier of X,COMPLEX)):] is non empty set
bool [:[:COMPLEX,(Funcs ( the carrier of X,COMPLEX)):],(Funcs ( the carrier of X,COMPLEX)):] is non empty set
ComplexFuncUnit the carrier of X is Relation-like the carrier of X -defined COMPLEX -valued Function-like total quasi_total V139() Element of Funcs ( the carrier of X,COMPLEX)
the carrier of X --> 1r is non empty Relation-like the carrier of X -defined COMPLEX -valued Function-like total quasi_total V139() Element of bool [: the carrier of X,COMPLEX:]
[: the carrier of X,COMPLEX:] is non empty V139() set
bool [: the carrier of X,COMPLEX:] is non empty set
ComplexFuncZero the carrier of X is Relation-like the carrier of X -defined COMPLEX -valued Function-like total quasi_total V139() Element of Funcs ( the carrier of X,COMPLEX)
the carrier of X --> 0 is non empty Relation-like the carrier of X -defined REAL -valued RAT -valued INT -valued Function-like total quasi_total V139() V140() V141() V142() Element of bool [: the carrier of X,REAL:]
[: the carrier of X,REAL:] is non empty V139() V140() V141() set
bool [: the carrier of X,REAL:] is non empty set
ComplexAlgebraStr(# (Funcs ( the carrier of X,COMPLEX)),(ComplexFuncMult the carrier of X),(ComplexFuncAdd the carrier of X),(ComplexFuncExtMult the carrier of X),(ComplexFuncUnit the carrier of X),(ComplexFuncZero the carrier of X) #) is strict ComplexAlgebraStr
the carrier of (CAlgebra the carrier of X) is non empty set
bool the carrier of (CAlgebra the carrier of X) is non empty set
{ b₁ where b₁ is non empty Relation-like the carrier of X -defined COMPLEX -valued Function-like total quasi_total V139() (X) Element of bool [: the carrier of X,COMPLEX:] : verum } is set
mult_ ((X),(CAlgebra the carrier of X)) is non empty Relation-like [:(X),(X):] -defined (X) -valued Function-like total quasi_total Element of bool [:[:(X),(X):],(X):]
[:(X),(X):] is non empty set
[:[:(X),(X):],(X):] is non empty set
bool [:[:(X),(X):],(X):] is non empty set
Add_ ((X),(CAlgebra the carrier of X)) is non empty Relation-like [:(X),(X):] -defined (X) -valued Function-like total quasi_total Element of bool [:[:(X),(X):],(X):]
Mult_ ((X),(CAlgebra the carrier of X)) is non empty Relation-like [:COMPLEX,(X):] -defined (X) -valued Function-like total quasi_total Element of bool [:[:COMPLEX,(X):],(X):]
[:COMPLEX,(X):] is non empty set
[:[:COMPLEX,(X):],(X):] is non empty set
bool [:[:COMPLEX,(X):],(X):] is non empty set
One_ ((X),(CAlgebra the carrier of X)) is Element of (X)
Zero_ ((X),(CAlgebra the carrier of X)) is Element of (X)
ComplexAlgebraStr(# (X),(mult_ ((X),(CAlgebra the carrier of X))),(Add_ ((X),(CAlgebra the carrier of X))),(Mult_ ((X),(CAlgebra the carrier of X))),(One_ ((X),(CAlgebra the carrier of X))),(Zero_ ((X),(CAlgebra the carrier of X))) #) is strict ComplexAlgebraStr
X is non empty TopSpace-like TopStruct
(X) is non empty right_complementable Abelian add-associative right_zeroed unital associative commutative right-distributive right_unital well-unital left_unital vector-distributive scalar-distributive scalar-associative vector-associative ComplexAlgebraStr
(X) is non empty multiplicatively-closed Cadditively-linearly-closed Element of bool the carrier of (CAlgebra the carrier of X)
the carrier of X is non empty set
CAlgebra the carrier of X is non empty right_complementable Abelian add-associative right_zeroed unital associative commutative right-distributive right_unital well-unital left_unital vector-distributive scalar-distributive scalar-associative scalar-unital strict vector-associative ComplexAlgebraStr
Funcs ( the carrier of X,COMPLEX) is non empty functional FUNCTION_DOMAIN of the carrier of X, COMPLEX
ComplexFuncMult the carrier of X is non empty Relation-like [:(Funcs ( the carrier of X,COMPLEX)),(Funcs ( the carrier of X,COMPLEX)):] -defined Funcs ( the carrier of X,COMPLEX) -valued Function-like total quasi_total Element of bool [:[:(Funcs ( the carrier of X,COMPLEX)),(Funcs ( the carrier of X,COMPLEX)):],(Funcs ( the carrier of X,COMPLEX)):]
[:(Funcs ( the carrier of X,COMPLEX)),(Funcs ( the carrier of X,COMPLEX)):] is non empty set
[:[:(Funcs ( the carrier of X,COMPLEX)),(Funcs ( the carrier of X,COMPLEX)):],(Funcs ( the carrier of X,COMPLEX)):] is non empty set
bool [:[:(Funcs ( the carrier of X,COMPLEX)),(Funcs ( the carrier of X,COMPLEX)):],(Funcs ( the carrier of X,COMPLEX)):] is non empty set
ComplexFuncAdd the carrier of X is non empty Relation-like [:(Funcs ( the carrier of X,COMPLEX)),(Funcs ( the carrier of X,COMPLEX)):] -defined Funcs ( the carrier of X,COMPLEX) -valued Function-like total quasi_total Element of bool [:[:(Funcs ( the carrier of X,COMPLEX)),(Funcs ( the carrier of X,COMPLEX)):],(Funcs ( the carrier of X,COMPLEX)):]
ComplexFuncExtMult the carrier of X is non empty Relation-like [:COMPLEX,(Funcs ( the carrier of X,COMPLEX)):] -defined Funcs ( the carrier of X,COMPLEX) -valued Function-like total quasi_total Element of bool [:[:COMPLEX,(Funcs ( the carrier of X,COMPLEX)):],(Funcs ( the carrier of X,COMPLEX)):]
[:COMPLEX,(Funcs ( the carrier of X,COMPLEX)):] is non empty set
[:[:COMPLEX,(Funcs ( the carrier of X,COMPLEX)):],(Funcs ( the carrier of X,COMPLEX)):] is non empty set
bool [:[:COMPLEX,(Funcs ( the carrier of X,COMPLEX)):],(Funcs ( the carrier of X,COMPLEX)):] is non empty set
ComplexFuncUnit the carrier of X is Relation-like the carrier of X -defined COMPLEX -valued Function-like total quasi_total V139() Element of Funcs ( the carrier of X,COMPLEX)
the carrier of X --> 1r is non empty Relation-like the carrier of X -defined COMPLEX -valued Function-like total quasi_total V139() Element of bool [: the carrier of X,COMPLEX:]
[: the carrier of X,COMPLEX:] is non empty V139() set
bool [: the carrier of X,COMPLEX:] is non empty set
ComplexFuncZero the carrier of X is Relation-like the carrier of X -defined COMPLEX -valued Function-like total quasi_total V139() Element of Funcs ( the carrier of X,COMPLEX)
the carrier of X --> 0 is non empty Relation-like the carrier of X -defined REAL -valued RAT -valued INT -valued Function-like total quasi_total V139() V140() V141() V142() Element of bool [: the carrier of X,REAL:]
[: the carrier of X,REAL:] is non empty V139() V140() V141() set
bool [: the carrier of X,REAL:] is non empty set
ComplexAlgebraStr(# (Funcs ( the carrier of X,COMPLEX)),(ComplexFuncMult the carrier of X),(ComplexFuncAdd the carrier of X),(ComplexFuncExtMult the carrier of X),(ComplexFuncUnit the carrier of X),(ComplexFuncZero the carrier of X) #) is strict ComplexAlgebraStr
the carrier of (CAlgebra the carrier of X) is non empty set
bool the carrier of (CAlgebra the carrier of X) is non empty set
{ b₁ where b₁ is non empty Relation-like the carrier of X -defined COMPLEX -valued Function-like total quasi_total V139() (X) Element of bool [: the carrier of X,COMPLEX:] : verum } is set
mult_ ((X),(CAlgebra the carrier of X)) is non empty Relation-like [:(X),(X):] -defined (X) -valued Function-like total quasi_total Element of bool [:[:(X),(X):],(X):]
[:(X),(X):] is non empty set
[:[:(X),(X):],(X):] is non empty set
bool [:[:(X),(X):],(X):] is non empty set
Add_ ((X),(CAlgebra the carrier of X)) is non empty Relation-like [:(X),(X):] -defined (X) -valued Function-like total quasi_total Element of bool [:[:(X),(X):],(X):]
Mult_ ((X),(CAlgebra the carrier of X)) is non empty Relation-like [:COMPLEX,(X):] -defined (X) -valued Function-like total quasi_total Element of bool [:[:COMPLEX,(X):],(X):]
[:COMPLEX,(X):] is non empty set
[:[:COMPLEX,(X):],(X):] is non empty set
bool [:[:COMPLEX,(X):],(X):] is non empty set
One_ ((X),(CAlgebra the carrier of X)) is Element of (X)
Zero_ ((X),(CAlgebra the carrier of X)) is Element of (X)
ComplexAlgebraStr(# (X),(mult_ ((X),(CAlgebra the carrier of X))),(Add_ ((X),(CAlgebra the carrier of X))),(Mult_ ((X),(CAlgebra the carrier of X))),(One_ ((X),(CAlgebra the carrier of X))),(Zero_ ((X),(CAlgebra the carrier of X))) #) is strict ComplexAlgebraStr
X is non empty TopSpace-like TopStruct
(X) is non empty right_complementable Abelian add-associative right_zeroed unital associative commutative right-distributive right_unital well-unital left_unital vector-distributive scalar-distributive scalar-associative strict vector-associative ComplexAlgebraStr
(X) is non empty multiplicatively-closed Cadditively-linearly-closed Element of bool the carrier of (CAlgebra the carrier of X)
the carrier of X is non empty set
CAlgebra the carrier of X is non empty right_complementable Abelian add-associative right_zeroed unital associative commutative right-distributive right_unital well-unital left_unital vector-distributive scalar-distributive scalar-associative scalar-unital strict vector-associative ComplexAlgebraStr
Funcs ( the carrier of X,COMPLEX) is non empty functional FUNCTION_DOMAIN of the carrier of X, COMPLEX
ComplexFuncMult the carrier of X is non empty Relation-like [:(Funcs ( the carrier of X,COMPLEX)),(Funcs ( the carrier of X,COMPLEX)):] -defined Funcs ( the carrier of X,COMPLEX) -valued Function-like total quasi_total Element of bool [:[:(Funcs ( the carrier of X,COMPLEX)),(Funcs ( the carrier of X,COMPLEX)):],(Funcs ( the carrier of X,COMPLEX)):]
[:(Funcs ( the carrier of X,COMPLEX)),(Funcs ( the carrier of X,COMPLEX)):] is non empty set
[:[:(Funcs ( the carrier of X,COMPLEX)),(Funcs ( the carrier of X,COMPLEX)):],(Funcs ( the carrier of X,COMPLEX)):] is non empty set
bool [:[:(Funcs ( the carrier of X,COMPLEX)),(Funcs ( the carrier of X,COMPLEX)):],(Funcs ( the carrier of X,COMPLEX)):] is non empty set
ComplexFuncAdd the carrier of X is non empty Relation-like [:(Funcs ( the carrier of X,COMPLEX)),(Funcs ( the carrier of X,COMPLEX)):] -defined Funcs ( the carrier of X,COMPLEX) -valued Function-like total quasi_total Element of bool [:[:(Funcs ( the carrier of X,COMPLEX)),(Funcs ( the carrier of X,COMPLEX)):],(Funcs ( the carrier of X,COMPLEX)):]
ComplexFuncExtMult the carrier of X is non empty Relation-like [:COMPLEX,(Funcs ( the carrier of X,COMPLEX)):] -defined Funcs ( the carrier of X,COMPLEX) -valued Function-like total quasi_total Element of bool [:[:COMPLEX,(Funcs ( the carrier of X,COMPLEX)):],(Funcs ( the carrier of X,COMPLEX)):]
[:COMPLEX,(Funcs ( the carrier of X,COMPLEX)):] is non empty set
[:[:COMPLEX,(Funcs ( the carrier of X,COMPLEX)):],(Funcs ( the carrier of X,COMPLEX)):] is non empty set
bool [:[:COMPLEX,(Funcs ( the carrier of X,COMPLEX)):],(Funcs ( the carrier of X,COMPLEX)):] is non empty set
ComplexFuncUnit the carrier of X is Relation-like the carrier of X -defined COMPLEX -valued Function-like total quasi_total V139() Element of Funcs ( the carrier of X,COMPLEX)
the carrier of X --> 1r is non empty Relation-like the carrier of X -defined COMPLEX -valued Function-like total quasi_total V139() Element of bool [: the carrier of X,COMPLEX:]
[: the carrier of X,COMPLEX:] is non empty V139() set
bool [: the carrier of X,COMPLEX:] is non empty set
ComplexFuncZero the carrier of X is Relation-like the carrier of X -defined COMPLEX -valued Function-like total quasi_total V139() Element of Funcs ( the carrier of X,COMPLEX)
the carrier of X --> 0 is non empty Relation-like the carrier of X -defined REAL -valued RAT -valued INT -valued Function-like total quasi_total V139() V140() V141() V142() Element of bool [: the carrier of X,REAL:]
[: the carrier of X,REAL:] is non empty V139() V140() V141() set
bool [: the carrier of X,REAL:] is non empty set
ComplexAlgebraStr(# (Funcs ( the carrier of X,COMPLEX)),(ComplexFuncMult the carrier of X),(ComplexFuncAdd the carrier of X),(ComplexFuncExtMult the carrier of X),(ComplexFuncUnit the carrier of X),(ComplexFuncZero the carrier of X) #) is strict ComplexAlgebraStr
the carrier of (CAlgebra the carrier of X) is non empty set
bool the carrier of (CAlgebra the carrier of X) is non empty set
{ b₁ where b₁ is non empty Relation-like the carrier of X -defined COMPLEX -valued Function-like total quasi_total V139() (X) Element of bool [: the carrier of X,COMPLEX:] : verum } is set
mult_ ((X),(CAlgebra the carrier of X)) is non empty Relation-like [:(X),(X):] -defined (X) -valued Function-like total quasi_total Element of bool [:[:(X),(X):],(X):]
[:(X),(X):] is non empty set
[:[:(X),(X):],(X):] is non empty set
bool [:[:(X),(X):],(X):] is non empty set
Add_ ((X),(CAlgebra the carrier of X)) is non empty Relation-like [:(X),(X):] -defined (X) -valued Function-like total quasi_total Element of bool [:[:(X),(X):],(X):]
Mult_ ((X),(CAlgebra the carrier of X)) is non empty Relation-like [:COMPLEX,(X):] -defined (X) -valued Function-like total quasi_total Element of bool [:[:COMPLEX,(X):],(X):]
[:COMPLEX,(X):] is non empty set
[:[:COMPLEX,(X):],(X):] is non empty set
bool [:[:COMPLEX,(X):],(X):] is non empty set
One_ ((X),(CAlgebra the carrier of X)) is Element of (X)
Zero_ ((X),(CAlgebra the carrier of X)) is Element of (X)
ComplexAlgebraStr(# (X),(mult_ ((X),(CAlgebra the carrier of X))),(Add_ ((X),(CAlgebra the carrier of X))),(Mult_ ((X),(CAlgebra the carrier of X))),(One_ ((X),(CAlgebra the carrier of X))),(Zero_ ((X),(CAlgebra the carrier of X))) #) is strict ComplexAlgebraStr
the carrier of (X) is non empty set
X is Element of the carrier of (X)
1r * X is Element of the carrier of (X)
the Mult of (X) is non empty Relation-like [:COMPLEX, the carrier of (X):] -defined the carrier of (X) -valued Function-like total quasi_total Element of bool [:[:COMPLEX, the carrier of (X):], the carrier of (X):]
[:COMPLEX, the carrier of (X):] is non empty set
[:[:COMPLEX, the carrier of (X):], the carrier of (X):] is non empty set
bool [:[:COMPLEX, the carrier of (X):], the carrier of (X):] is non empty set
[1r,X] is set
{1r,X} is non empty set
{1r} is non empty V176() set
{{1r,X},{1r}} is non empty set
the Mult of (X) . [1r,X] is set
F is Element of the carrier of (CAlgebra the carrier of X)
1r * F is Element of the carrier of (CAlgebra the carrier of X)
the Mult of (CAlgebra the carrier of X) is non empty Relation-like [:COMPLEX, the carrier of (CAlgebra the carrier of X):] -defined the carrier of (CAlgebra the carrier of X) -valued Function-like total quasi_total Element of bool [:[:COMPLEX, the carrier of (CAlgebra the carrier of X):], the carrier of (CAlgebra the carrier of X):]
[:COMPLEX, the carrier of (CAlgebra the carrier of X):] is non empty set
[:[:COMPLEX, the carrier of (CAlgebra the carrier of X):], the carrier of (CAlgebra the carrier of X):] is non empty set
bool [:[:COMPLEX, the carrier of (CAlgebra the carrier of X):], the carrier of (CAlgebra the carrier of X):] is non empty set
[1r,F] is set
{1r,F} is non empty set
{{1r,F},{1r}} is non empty set
the Mult of (CAlgebra the carrier of X) . [1r,F] is set
X is non empty TopSpace-like TopStruct
(X) is non empty right_complementable Abelian add-associative right_zeroed unital associative commutative right-distributive right_unital well-unital left_unital vector-distributive scalar-distributive scalar-associative scalar-unital strict vector-associative ComplexAlgebraStr
(X) is non empty multiplicatively-closed Cadditively-linearly-closed Element of bool the carrier of (CAlgebra the carrier of X)
the carrier of X is non empty set
CAlgebra the carrier of X is non empty right_complementable Abelian add-associative right_zeroed unital associative commutative right-distributive right_unital well-unital left_unital vector-distributive scalar-distributive scalar-associative scalar-unital strict vector-associative ComplexAlgebraStr
Funcs ( the carrier of X,COMPLEX) is non empty functional FUNCTION_DOMAIN of the carrier of X, COMPLEX
ComplexFuncMult the carrier of X is non empty Relation-like [:(Funcs ( the carrier of X,COMPLEX)),(Funcs ( the carrier of X,COMPLEX)):] -defined Funcs ( the carrier of X,COMPLEX) -valued Function-like total quasi_total Element of bool [:[:(Funcs ( the carrier of X,COMPLEX)),(Funcs ( the carrier of X,COMPLEX)):],(Funcs ( the carrier of X,COMPLEX)):]
[:(Funcs ( the carrier of X,COMPLEX)),(Funcs ( the carrier of X,COMPLEX)):] is non empty set
[:[:(Funcs ( the carrier of X,COMPLEX)),(Funcs ( the carrier of X,COMPLEX)):],(Funcs ( the carrier of X,COMPLEX)):] is non empty set
bool [:[:(Funcs ( the carrier of X,COMPLEX)),(Funcs ( the carrier of X,COMPLEX)):],(Funcs ( the carrier of X,COMPLEX)):] is non empty set
ComplexFuncAdd the carrier of X is non empty Relation-like [:(Funcs ( the carrier of X,COMPLEX)),(Funcs ( the carrier of X,COMPLEX)):] -defined Funcs ( the carrier of X,COMPLEX) -valued Function-like total quasi_total Element of bool [:[:(Funcs ( the carrier of X,COMPLEX)),(Funcs ( the carrier of X,COMPLEX)):],(Funcs ( the carrier of X,COMPLEX)):]
ComplexFuncExtMult the carrier of X is non empty Relation-like [:COMPLEX,(Funcs ( the carrier of X,COMPLEX)):] -defined Funcs ( the carrier of X,COMPLEX) -valued Function-like total quasi_total Element of bool [:[:COMPLEX,(Funcs ( the carrier of X,COMPLEX)):],(Funcs ( the carrier of X,COMPLEX)):]
[:COMPLEX,(Funcs ( the carrier of X,COMPLEX)):] is non empty set
[:[:COMPLEX,(Funcs ( the carrier of X,COMPLEX)):],(Funcs ( the carrier of X,COMPLEX)):] is non empty set
bool [:[:COMPLEX,(Funcs ( the carrier of X,COMPLEX)):],(Funcs ( the carrier of X,COMPLEX)):] is non empty set
ComplexFuncUnit the carrier of X is Relation-like the carrier of X -defined COMPLEX -valued Function-like total quasi_total V139() Element of Funcs ( the carrier of X,COMPLEX)
the carrier of X --> 1r is non empty Relation-like the carrier of X -defined COMPLEX -valued Function-like total quasi_total V139() Element of bool [: the carrier of X,COMPLEX:]
[: the carrier of X,COMPLEX:] is non empty V139() set
bool [: the carrier of X,COMPLEX:] is non empty set
ComplexFuncZero the carrier of X is Relation-like the carrier of X -defined COMPLEX -valued Function-like total quasi_total V139() Element of Funcs ( the carrier of X,COMPLEX)
the carrier of X --> 0 is non empty Relation-like the carrier of X -defined REAL -valued RAT -valued INT -valued Function-like total quasi_total V139() V140() V141() V142() Element of bool [: the carrier of X,REAL:]
[: the carrier of X,REAL:] is non empty V139() V140() V141() set
bool [: the carrier of X,REAL:] is non empty set
ComplexAlgebraStr(# (Funcs ( the carrier of X,COMPLEX)),(ComplexFuncMult the carrier of X),(ComplexFuncAdd the carrier of X),(ComplexFuncExtMult the carrier of X),(ComplexFuncUnit the carrier of X),(ComplexFuncZero the carrier of X) #) is strict ComplexAlgebraStr
the carrier of (CAlgebra the carrier of X) is non empty set
bool the carrier of (CAlgebra the carrier of X) is non empty set
{ b₁ where b₁ is non empty Relation-like the carrier of X -defined COMPLEX -valued Function-like total quasi_total V139() (X) Element of bool [: the carrier of X,COMPLEX:] : verum } is set
mult_ ((X),(CAlgebra the carrier of X)) is non empty Relation-like [:(X),(X):] -defined (X) -valued Function-like total quasi_total Element of bool [:[:(X),(X):],(X):]
[:(X),(X):] is non empty set
[:[:(X),(X):],(X):] is non empty set
bool [:[:(X),(X):],(X):] is non empty set
Add_ ((X),(CAlgebra the carrier of X)) is non empty Relation-like [:(X),(X):] -defined (X) -valued Function-like total quasi_total Element of bool [:[:(X),(X):],(X):]
Mult_ ((X),(CAlgebra the carrier of X)) is non empty Relation-like [:COMPLEX,(X):] -defined (X) -valued Function-like total quasi_total Element of bool [:[:COMPLEX,(X):],(X):]
[:COMPLEX,(X):] is non empty set
[:[:COMPLEX,(X):],(X):] is non empty set
bool [:[:COMPLEX,(X):],(X):] is non empty set
One_ ((X),(CAlgebra the carrier of X)) is Element of (X)
Zero_ ((X),(CAlgebra the carrier of X)) is Element of (X)
ComplexAlgebraStr(# (X),(mult_ ((X),(CAlgebra the carrier of X))),(Add_ ((X),(CAlgebra the carrier of X))),(Mult_ ((X),(CAlgebra the carrier of X))),(One_ ((X),(CAlgebra the carrier of X))),(Zero_ ((X),(CAlgebra the carrier of X))) #) is strict ComplexAlgebraStr
the carrier of (X) is non empty set
X is Element of the carrier of (X)
F is Element of the carrier of (X)
G is Element of the carrier of (X)
X + F is Element of the carrier of (X)
the addF of (X) is non empty Relation-like [: the carrier of (X), the carrier of (X):] -defined the carrier of (X) -valued Function-like total quasi_total Element of bool [:[: the carrier of (X), the carrier of (X):], the carrier of (X):]
[: the carrier of (X), the carrier of (X):] is non empty set
[:[: the carrier of (X), the carrier of (X):], the carrier of (X):] is non empty set
bool [:[: the carrier of (X), the carrier of (X):], the carrier of (X):] is non empty set
the addF of (X) . (X,F) is Element of the carrier of (X)
[X,F] is set
{X,F} is non empty set
{X} is non empty set
{{X,F},{X}} is non empty set
the addF of (X) . [X,F] is set
FB is non empty Relation-like the carrier of X -defined COMPLEX -valued Function-like total quasi_total V139() Element of bool [: the carrier of X,COMPLEX:]
GB is non empty Relation-like the carrier of X -defined COMPLEX -valued Function-like total quasi_total V139() Element of bool [: the carrier of X,COMPLEX:]
aFB is non empty Relation-like the carrier of X -defined COMPLEX -valued Function-like total quasi_total V139() Element of bool [: the carrier of X,COMPLEX:]
r is Element of the carrier of (CAlgebra the carrier of X)
aF is Element of the carrier of (CAlgebra the carrier of X)
aG is Element of the carrier of (CAlgebra the carrier of X)
aF + aG is Element of the carrier of (CAlgebra the carrier of X)
the addF of (CAlgebra the carrier of X) is non empty Relation-like [: the carrier of (CAlgebra the carrier of X), the carrier of (CAlgebra the carrier of X):] -defined the carrier of (CAlgebra the carrier of X) -valued Function-like total quasi_total Element of bool [:[: the carrier of (CAlgebra the carrier of X), the carrier of (CAlgebra the carrier of X):], the carrier of (CAlgebra the carrier of X):]
[: the carrier of (CAlgebra the carrier of X), the carrier of (CAlgebra the carrier of X):] is non empty set
[:[: the carrier of (CAlgebra the carrier of X), the carrier of (CAlgebra the carrier of X):], the carrier of (CAlgebra the carrier of X):] is non empty set
bool [:[: the carrier of (CAlgebra the carrier of X), the carrier of (CAlgebra the carrier of X):], the carrier of (CAlgebra the carrier of X):] is non empty set
the addF of (CAlgebra the carrier of X) . (aF,aG) is Element of the carrier of (CAlgebra the carrier of X)
[aF,aG] is set
{aF,aG} is non empty set
{aF} is non empty set
{{aF,aG},{aF}} is non empty set
the addF of (CAlgebra the carrier of X) . [aF,aG] is set
r is Element of the carrier of X
aFB . r is complex Element of COMPLEX
FB . r is complex Element of COMPLEX
GB . r is complex Element of COMPLEX
(FB . r) + (GB . r) is complex Element of COMPLEX
X is non empty TopSpace-like TopStruct
(X) is non empty right_complementable Abelian add-associative right_zeroed unital associative commutative right-distributive right_unital well-unital left_unital vector-distributive scalar-distributive scalar-associative scalar-unital strict vector-associative ComplexAlgebraStr
(X) is non empty multiplicatively-closed Cadditively-linearly-closed Element of bool the carrier of (CAlgebra the carrier of X)
the carrier of X is non empty set
CAlgebra the carrier of X is non empty right_complementable Abelian add-associative right_zeroed unital associative commutative right-distributive right_unital well-unital left_unital vector-distributive scalar-distributive scalar-associative scalar-unital strict vector-associative ComplexAlgebraStr
Funcs ( the carrier of X,COMPLEX) is non empty functional FUNCTION_DOMAIN of the carrier of X, COMPLEX
ComplexFuncMult the carrier of X is non empty Relation-like [:(Funcs ( the carrier of X,COMPLEX)),(Funcs ( the carrier of X,COMPLEX)):] -defined Funcs ( the carrier of X,COMPLEX) -valued Function-like total quasi_total Element of bool [:[:(Funcs ( the carrier of X,COMPLEX)),(Funcs ( the carrier of X,COMPLEX)):],(Funcs ( the carrier of X,COMPLEX)):]
[:(Funcs ( the carrier of X,COMPLEX)),(Funcs ( the carrier of X,COMPLEX)):] is non empty set
[:[:(Funcs ( the carrier of X,COMPLEX)),(Funcs ( the carrier of X,COMPLEX)):],(Funcs ( the carrier of X,COMPLEX)):] is non empty set
bool [:[:(Funcs ( the carrier of X,COMPLEX)),(Funcs ( the carrier of X,COMPLEX)):],(Funcs ( the carrier of X,COMPLEX)):] is non empty set
ComplexFuncAdd the carrier of X is non empty Relation-like [:(Funcs ( the carrier of X,COMPLEX)),(Funcs ( the carrier of X,COMPLEX)):] -defined Funcs ( the carrier of X,COMPLEX) -valued Function-like total quasi_total Element of bool [:[:(Funcs ( the carrier of X,COMPLEX)),(Funcs ( the carrier of X,COMPLEX)):],(Funcs ( the carrier of X,COMPLEX)):]
ComplexFuncExtMult the carrier of X is non empty Relation-like [:COMPLEX,(Funcs ( the carrier of X,COMPLEX)):] -defined Funcs ( the carrier of X,COMPLEX) -valued Function-like total quasi_total Element of bool [:[:COMPLEX,(Funcs ( the carrier of X,COMPLEX)):],(Funcs ( the carrier of X,COMPLEX)):]
[:COMPLEX,(Funcs ( the carrier of X,COMPLEX)):] is non empty set
[:[:COMPLEX,(Funcs ( the carrier of X,COMPLEX)):],(Funcs ( the carrier of X,COMPLEX)):] is non empty set
bool [:[:COMPLEX,(Funcs ( the carrier of X,COMPLEX)):],(Funcs ( the carrier of X,COMPLEX)):] is non empty set
ComplexFuncUnit the carrier of X is Relation-like the carrier of X -defined COMPLEX -valued Function-like total quasi_total V139() Element of Funcs ( the carrier of X,COMPLEX)
the carrier of X --> 1r is non empty Relation-like the carrier of X -defined COMPLEX -valued Function-like total quasi_total V139() Element of bool [: the carrier of X,COMPLEX:]
[: the carrier of X,COMPLEX:] is non empty V139() set
bool [: the carrier of X,COMPLEX:] is non empty set
ComplexFuncZero the carrier of X is Relation-like the carrier of X -defined COMPLEX -valued Function-like total quasi_total V139() Element of Funcs ( the carrier of X,COMPLEX)
the carrier of X --> 0 is non empty Relation-like the carrier of X -defined REAL -valued RAT -valued INT -valued Function-like total quasi_total V139() V140() V141() V142() Element of bool [: the carrier of X,REAL:]
[: the carrier of X,REAL:] is non empty V139() V140() V141() set
bool [: the carrier of X,REAL:] is non empty set
ComplexAlgebraStr(# (Funcs ( the carrier of X,COMPLEX)),(ComplexFuncMult the carrier of X),(ComplexFuncAdd the carrier of X),(ComplexFuncExtMult the carrier of X),(ComplexFuncUnit the carrier of X),(ComplexFuncZero the carrier of X) #) is strict ComplexAlgebraStr
the carrier of (CAlgebra the carrier of X) is non empty set
bool the carrier of (CAlgebra the carrier of X) is non empty set
{ b₁ where b₁ is non empty Relation-like the carrier of X -defined COMPLEX -valued Function-like total quasi_total V139() (X) Element of bool [: the carrier of X,COMPLEX:] : verum } is set
mult_ ((X),(CAlgebra the carrier of X)) is non empty Relation-like [:(X),(X):] -defined (X) -valued Function-like total quasi_total Element of bool [:[:(X),(X):],(X):]
[:(X),(X):] is non empty set
[:[:(X),(X):],(X):] is non empty set
bool [:[:(X),(X):],(X):] is non empty set
Add_ ((X),(CAlgebra the carrier of X)) is non empty Relation-like [:(X),(X):] -defined (X) -valued Function-like total quasi_total Element of bool [:[:(X),(X):],(X):]
Mult_ ((X),(CAlgebra the carrier of X)) is non empty Relation-like [:COMPLEX,(X):] -defined (X) -valued Function-like total quasi_total Element of bool [:[:COMPLEX,(X):],(X):]
[:COMPLEX,(X):] is non empty set
[:[:COMPLEX,(X):],(X):] is non empty set
bool [:[:COMPLEX,(X):],(X):] is non empty set
One_ ((X),(CAlgebra the carrier of X)) is Element of (X)
Zero_ ((X),(CAlgebra the carrier of X)) is Element of (X)
ComplexAlgebraStr(# (X),(mult_ ((X),(CAlgebra the carrier of X))),(Add_ ((X),(CAlgebra the carrier of X))),(Mult_ ((X),(CAlgebra the carrier of X))),(One_ ((X),(CAlgebra the carrier of X))),(Zero_ ((X),(CAlgebra the carrier of X))) #) is strict ComplexAlgebraStr
the carrier of (X) is non empty set
X is Element of the carrier of (X)
F is Element of the carrier of (X)
G is non empty Relation-like the carrier of X -defined COMPLEX -valued Function-like total quasi_total V139() Element of bool [: the carrier of X,COMPLEX:]
FB is non empty Relation-like the carrier of X -defined COMPLEX -valued Function-like total quasi_total V139() Element of bool [: the carrier of X,COMPLEX:]
GB is complex set
GB * X is Element of the carrier of (X)
the Mult of (X) is non empty Relation-like [:COMPLEX, the carrier of (X):] -defined the carrier of (X) -valued Function-like total quasi_total Element of bool [:[:COMPLEX, the carrier of (X):], the carrier of (X):]
[:COMPLEX, the carrier of (X):] is non empty set
[:[:COMPLEX, the carrier of (X):], the carrier of (X):] is non empty set
bool [:[:COMPLEX, the carrier of (X):], the carrier of (X):] is non empty set
[GB,X] is set
{GB,X} is non empty set
{GB} is non empty V176() set
{{GB,X},{GB}} is non empty set
the Mult of (X) . [GB,X] is set
aF is Element of the carrier of (CAlgebra the carrier of X)
aFB is Element of the carrier of (CAlgebra the carrier of X)
GB * aFB is Element of the carrier of (CAlgebra the carrier of X)
the Mult of (CAlgebra the carrier of X) is non empty Relation-like [:COMPLEX, the carrier of (CAlgebra the carrier of X):] -defined the carrier of (CAlgebra the carrier of X) -valued Function-like total quasi_total Element of bool [:[:COMPLEX, the carrier of (CAlgebra the carrier of X):], the carrier of (CAlgebra the carrier of X):]
[:COMPLEX, the carrier of (CAlgebra the carrier of X):] is non empty set
[:[:COMPLEX, the carrier of (CAlgebra the carrier of X):], the carrier of (CAlgebra the carrier of X):] is non empty set
bool [:[:COMPLEX, the carrier of (CAlgebra the carrier of X):], the carrier of (CAlgebra the carrier of X):] is non empty set
[GB,aFB] is set
{GB,aFB} is non empty set
{{GB,aFB},{GB}} is non empty set
the Mult of (CAlgebra the carrier of X) . [GB,aFB] is set
aG is Element of the carrier of X
FB . aG is complex Element of COMPLEX
G . aG is complex Element of COMPLEX
GB * (G . aG) is complex set
X is non empty TopSpace-like TopStruct
(X) is non empty right_complementable Abelian add-associative right_zeroed unital associative commutative right-distributive right_unital well-unital left_unital vector-distributive scalar-distributive scalar-associative scalar-unital strict vector-associative ComplexAlgebraStr
(X) is non empty multiplicatively-closed Cadditively-linearly-closed Element of bool the carrier of (CAlgebra the carrier of X)
the carrier of X is non empty set
CAlgebra the carrier of X is non empty right_complementable Abelian add-associative right_zeroed unital associative commutative right-distributive right_unital well-unital left_unital vector-distributive scalar-distributive scalar-associative scalar-unital strict vector-associative ComplexAlgebraStr
Funcs ( the carrier of X,COMPLEX) is non empty functional FUNCTION_DOMAIN of the carrier of X, COMPLEX
ComplexFuncMult the carrier of X is non empty Relation-like [:(Funcs ( the carrier of X,COMPLEX)),(Funcs ( the carrier of X,COMPLEX)):] -defined Funcs ( the carrier of X,COMPLEX) -valued Function-like total quasi_total Element of bool [:[:(Funcs ( the carrier of X,COMPLEX)),(Funcs ( the carrier of X,COMPLEX)):],(Funcs ( the carrier of X,COMPLEX)):]
[:(Funcs ( the carrier of X,COMPLEX)),(Funcs ( the carrier of X,COMPLEX)):] is non empty set
[:[:(Funcs ( the carrier of X,COMPLEX)),(Funcs ( the carrier of X,COMPLEX)):],(Funcs ( the carrier of X,COMPLEX)):] is non empty set
bool [:[:(Funcs ( the carrier of X,COMPLEX)),(Funcs ( the carrier of X,COMPLEX)):],(Funcs ( the carrier of X,COMPLEX)):] is non empty set
ComplexFuncAdd the carrier of X is non empty Relation-like [:(Funcs ( the carrier of X,COMPLEX)),(Funcs ( the carrier of X,COMPLEX)):] -defined Funcs ( the carrier of X,COMPLEX) -valued Function-like total quasi_total Element of bool [:[:(Funcs ( the carrier of X,COMPLEX)),(Funcs ( the carrier of X,COMPLEX)):],(Funcs ( the carrier of X,COMPLEX)):]
ComplexFuncExtMult the carrier of X is non empty Relation-like [:COMPLEX,(Funcs ( the carrier of X,COMPLEX)):] -defined Funcs ( the carrier of X,COMPLEX) -valued Function-like total quasi_total Element of bool [:[:COMPLEX,(Funcs ( the carrier of X,COMPLEX)):],(Funcs ( the carrier of X,COMPLEX)):]
[:COMPLEX,(Funcs ( the carrier of X,COMPLEX)):] is non empty set
[:[:COMPLEX,(Funcs ( the carrier of X,COMPLEX)):],(Funcs ( the carrier of X,COMPLEX)):] is non empty set
bool [:[:COMPLEX,(Funcs ( the carrier of X,COMPLEX)):],(Funcs ( the carrier of X,COMPLEX)):] is non empty set
ComplexFuncUnit the carrier of X is Relation-like the carrier of X -defined COMPLEX -valued Function-like total quasi_total V139() Element of Funcs ( the carrier of X,COMPLEX)
the carrier of X --> 1r is non empty Relation-like the carrier of X -defined COMPLEX -valued Function-like total quasi_total V139() Element of bool [: the carrier of X,COMPLEX:]
[: the carrier of X,COMPLEX:] is non empty V139() set
bool [: the carrier of X,COMPLEX:] is non empty set
ComplexFuncZero the carrier of X is Relation-like the carrier of X -defined COMPLEX -valued Function-like total quasi_total V139() Element of Funcs ( the carrier of X,COMPLEX)
the carrier of X --> 0 is non empty Relation-like the carrier of X -defined REAL -valued RAT -valued INT -valued Function-like total quasi_total V139() V140() V141() V142() Element of bool [: the carrier of X,REAL:]
[: the carrier of X,REAL:] is non empty V139() V140() V141() set
bool [: the carrier of X,REAL:] is non empty set
ComplexAlgebraStr(# (Funcs ( the carrier of X,COMPLEX)),(ComplexFuncMult the carrier of X),(ComplexFuncAdd the carrier of X),(ComplexFuncExtMult the carrier of X),(ComplexFuncUnit the carrier of X),(ComplexFuncZero the carrier of X) #) is strict ComplexAlgebraStr
the carrier of (CAlgebra the carrier of X) is non empty set
bool the carrier of (CAlgebra the carrier of X) is non empty set
{ b₁ where b₁ is non empty Relation-like the carrier of X -defined COMPLEX -valued Function-like total quasi_total V139() (X) Element of bool [: the carrier of X,COMPLEX:] : verum } is set
mult_ ((X),(CAlgebra the carrier of X)) is non empty Relation-like [:(X),(X):] -defined (X) -valued Function-like total quasi_total Element of bool [:[:(X),(X):],(X):]
[:(X),(X):] is non empty set
[:[:(X),(X):],(X):] is non empty set
bool [:[:(X),(X):],(X):] is non empty set
Add_ ((X),(CAlgebra the carrier of X)) is non empty Relation-like [:(X),(X):] -defined (X) -valued Function-like total quasi_total Element of bool [:[:(X),(X):],(X):]
Mult_ ((X),(CAlgebra the carrier of X)) is non empty Relation-like [:COMPLEX,(X):] -defined (X) -valued Function-like total quasi_total Element of bool [:[:COMPLEX,(X):],(X):]
[:COMPLEX,(X):] is non empty set
[:[:COMPLEX,(X):],(X):] is non empty set
bool [:[:COMPLEX,(X):],(X):] is non empty set
One_ ((X),(CAlgebra the carrier of X)) is Element of (X)
Zero_ ((X),(CAlgebra the carrier of X)) is Element of (X)
ComplexAlgebraStr(# (X),(mult_ ((X),(CAlgebra the carrier of X))),(Add_ ((X),(CAlgebra the carrier of X))),(Mult_ ((X),(CAlgebra the carrier of X))),(One_ ((X),(CAlgebra the carrier of X))),(Zero_ ((X),(CAlgebra the carrier of X))) #) is strict ComplexAlgebraStr
the carrier of (X) is non empty set
X is Element of the carrier of (X)
F is Element of the carrier of (X)
G is Element of the carrier of (X)
X * F is Element of the carrier of (X)
the multF of (X) is non empty Relation-like [: the carrier of (X), the carrier of (X):] -defined the carrier of (X) -valued Function-like total quasi_total Element of bool [:[: the carrier of (X), the carrier of (X):], the carrier of (X):]
[: the carrier of (X), the carrier of (X):] is non empty set
[:[: the carrier of (X), the carrier of (X):], the carrier of (X):] is non empty set
bool [:[: the carrier of (X), the carrier of (X):], the carrier of (X):] is non empty set
the multF of (X) . (X,F) is Element of the carrier of (X)
[X,F] is set
{X,F} is non empty set
{X} is non empty set
{{X,F},{X}} is non empty set
the multF of (X) . [X,F] is set
FB is non empty Relation-like the carrier of X -defined COMPLEX -valued Function-like total quasi_total V139() Element of bool [: the carrier of X,COMPLEX:]
GB is non empty Relation-like the carrier of X -defined COMPLEX -valued Function-like total quasi_total V139() Element of bool [: the carrier of X,COMPLEX:]
aFB is non empty Relation-like the carrier of X -defined COMPLEX -valued Function-like total quasi_total V139() Element of bool [: the carrier of X,COMPLEX:]
r is Element of the carrier of (CAlgebra the carrier of X)
aF is Element of the carrier of (CAlgebra the carrier of X)
aG is Element of the carrier of (CAlgebra the carrier of X)
aF * aG is Element of the carrier of (CAlgebra the carrier of X)
the multF of (CAlgebra the carrier of X) is non empty Relation-like [: the carrier of (CAlgebra the carrier of X), the carrier of (CAlgebra the carrier of X):] -defined the carrier of (CAlgebra the carrier of X) -valued Function-like total quasi_total Element of bool [:[: the carrier of (CAlgebra the carrier of X), the carrier of (CAlgebra the carrier of X):], the carrier of (CAlgebra the carrier of X):]
[: the carrier of (CAlgebra the carrier of X), the carrier of (CAlgebra the carrier of X):] is non empty set
[:[: the carrier of (CAlgebra the carrier of X), the carrier of (CAlgebra the carrier of X):], the carrier of (CAlgebra the carrier of X):] is non empty set
bool [:[: the carrier of (CAlgebra the carrier of X), the carrier of (CAlgebra the carrier of X):], the carrier of (CAlgebra the carrier of X):] is non empty set
the multF of (CAlgebra the carrier of X) . (aF,aG) is Element of the carrier of (CAlgebra the carrier of X)
[aF,aG] is set
{aF,aG} is non empty set
{aF} is non empty set
{{aF,aG},{aF}} is non empty set
the multF of (CAlgebra the carrier of X) . [aF,aG] is set
r is Element of the carrier of X
aFB . r is complex Element of COMPLEX
FB . r is complex Element of COMPLEX
GB . r is complex Element of COMPLEX
(FB . r) * (GB . r) is complex Element of COMPLEX
0c is empty epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural complex real ext-real non positive non negative V176() V177() V178() V179() V180() V181() V182() bounded_below V197() Element of COMPLEX
X is non empty TopSpace-like TopStruct
(X) is non empty right_complementable Abelian add-associative right_zeroed unital associative commutative right-distributive right_unital well-unital left_unital vector-distributive scalar-distributive scalar-associative scalar-unital strict vector-associative ComplexAlgebraStr
(X) is non empty multiplicatively-closed Cadditively-linearly-closed Element of bool the carrier of (CAlgebra the carrier of X)
the carrier of X is non empty set
CAlgebra the carrier of X is non empty right_complementable Abelian add-associative right_zeroed unital associative commutative right-distributive right_unital well-unital left_unital vector-distributive scalar-distributive scalar-associative scalar-unital strict vector-associative ComplexAlgebraStr
Funcs ( the carrier of X,COMPLEX) is non empty functional FUNCTION_DOMAIN of the carrier of X, COMPLEX
ComplexFuncMult the carrier of X is non empty Relation-like [:(Funcs ( the carrier of X,COMPLEX)),(Funcs ( the carrier of X,COMPLEX)):] -defined Funcs ( the carrier of X,COMPLEX) -valued Function-like total quasi_total Element of bool [:[:(Funcs ( the carrier of X,COMPLEX)),(Funcs ( the carrier of X,COMPLEX)):],(Funcs ( the carrier of X,COMPLEX)):]
[:(Funcs ( the carrier of X,COMPLEX)),(Funcs ( the carrier of X,COMPLEX)):] is non empty set
[:[:(Funcs ( the carrier of X,COMPLEX)),(Funcs ( the carrier of X,COMPLEX)):],(Funcs ( the carrier of X,COMPLEX)):] is non empty set
bool [:[:(Funcs ( the carrier of X,COMPLEX)),(Funcs ( the carrier of X,COMPLEX)):],(Funcs ( the carrier of X,COMPLEX)):] is non empty set
ComplexFuncAdd the carrier of X is non empty Relation-like [:(Funcs ( the carrier of X,COMPLEX)),(Funcs ( the carrier of X,COMPLEX)):] -defined Funcs ( the carrier of X,COMPLEX) -valued Function-like total quasi_total Element of bool [:[:(Funcs ( the carrier of X,COMPLEX)),(Funcs ( the carrier of X,COMPLEX)):],(Funcs ( the carrier of X,COMPLEX)):]
ComplexFuncExtMult the carrier of X is non empty Relation-like [:COMPLEX,(Funcs ( the carrier of X,COMPLEX)):] -defined Funcs ( the carrier of X,COMPLEX) -valued Function-like total quasi_total Element of bool [:[:COMPLEX,(Funcs ( the carrier of X,COMPLEX)):],(Funcs ( the carrier of X,COMPLEX)):]
[:COMPLEX,(Funcs ( the carrier of X,COMPLEX)):] is non empty set
[:[:COMPLEX,(Funcs ( the carrier of X,COMPLEX)):],(Funcs ( the carrier of X,COMPLEX)):] is non empty set
bool [:[:COMPLEX,(Funcs ( the carrier of X,COMPLEX)):],(Funcs ( the carrier of X,COMPLEX)):] is non empty set
ComplexFuncUnit the carrier of X is Relation-like the carrier of X -defined COMPLEX -valued Function-like total quasi_total V139() Element of Funcs ( the carrier of X,COMPLEX)
the carrier of X --> 1r is non empty Relation-like the carrier of X -defined COMPLEX -valued Function-like total quasi_total V139() Element of bool [: the carrier of X,COMPLEX:]
[: the carrier of X,COMPLEX:] is non empty V139() set
bool [: the carrier of X,COMPLEX:] is non empty set
ComplexFuncZero the carrier of X is Relation-like the carrier of X -defined COMPLEX -valued Function-like total quasi_total V139() Element of Funcs ( the carrier of X,COMPLEX)
the carrier of X --> 0 is non empty Relation-like the carrier of X -defined REAL -valued RAT -valued INT -valued Function-like total quasi_total V139() V140() V141() V142() Element of bool [: the carrier of X,REAL:]
[: the carrier of X,REAL:] is non empty V139() V140() V141() set
bool [: the carrier of X,REAL:] is non empty set
ComplexAlgebraStr(# (Funcs ( the carrier of X,COMPLEX)),(ComplexFuncMult the carrier of X),(ComplexFuncAdd the carrier of X),(ComplexFuncExtMult the carrier of X),(ComplexFuncUnit the carrier of X),(ComplexFuncZero the carrier of X) #) is strict ComplexAlgebraStr
the carrier of (CAlgebra the carrier of X) is non empty set
bool the carrier of (CAlgebra the carrier of X) is non empty set
{ b₁ where b₁ is non empty Relation-like the carrier of X -defined COMPLEX -valued Function-like total quasi_total V139() (X) Element of bool [: the carrier of X,COMPLEX:] : verum } is set
mult_ ((X),(CAlgebra the carrier of X)) is non empty Relation-like [:(X),(X):] -defined (X) -valued Function-like total quasi_total Element of bool [:[:(X),(X):],(X):]
[:(X),(X):] is non empty set
[:[:(X),(X):],(X):] is non empty set
bool [:[:(X),(X):],(X):] is non empty set
Add_ ((X),(CAlgebra the carrier of X)) is non empty Relation-like [:(X),(X):] -defined (X) -valued Function-like total quasi_total Element of bool [:[:(X),(X):],(X):]
Mult_ ((X),(CAlgebra the carrier of X)) is non empty Relation-like [:COMPLEX,(X):] -defined (X) -valued Function-like total quasi_total Element of bool [:[:COMPLEX,(X):],(X):]
[:COMPLEX,(X):] is non empty set
[:[:COMPLEX,(X):],(X):] is non empty set
bool [:[:COMPLEX,(X):],(X):] is non empty set
One_ ((X),(CAlgebra the carrier of X)) is Element of (X)
Zero_ ((X),(CAlgebra the carrier of X)) is Element of (X)
ComplexAlgebraStr(# (X),(mult_ ((X),(CAlgebra the carrier of X))),(Add_ ((X),(CAlgebra the carrier of X))),(Mult_ ((X),(CAlgebra the carrier of X))),(One_ ((X),(CAlgebra the carrier of X))),(Zero_ ((X),(CAlgebra the carrier of X))) #) is strict ComplexAlgebraStr
0. (X) is zero Element of the carrier of (X)
the carrier of (X) is non empty set
the ZeroF of (X) is Element of the carrier of (X)
(X,0c) is non empty Relation-like the carrier of X -defined COMPLEX -valued Function-like total quasi_total V139() (X) Element of bool [: the carrier of X,COMPLEX:]
the carrier of X --> 0c is non empty Relation-like the carrier of X -defined {0c} -valued Function-like total quasi_total V139() V140() V141() V142() Element of bool [: the carrier of X,{0c}:]
{0c} is non empty V176() V177() V178() V179() V180() V181() left_end bounded_below set
[: the carrier of X,{0c}:] is non empty RAT -valued INT -valued V139() V140() V141() V142() set
bool [: the carrier of X,{0c}:] is non empty set
0. (CAlgebra the carrier of X) is zero Element of the carrier of (CAlgebra the carrier of X)
the ZeroF of (CAlgebra the carrier of X) is Element of the carrier of (CAlgebra the carrier of X)
(X,0) is non empty Relation-like the carrier of X -defined COMPLEX -valued Function-like total quasi_total V139() (X) Element of bool [: the carrier of X,COMPLEX:]
the carrier of X --> 0 is non empty Relation-like the carrier of X -defined RAT -valued INT -valued {0} -valued Function-like total quasi_total V139() V140() V141() V142() Element of bool [: the carrier of X,{0}:]
{0} is non empty V176() V177() V178() V179() V180() V181() left_end bounded_below set
[: the carrier of X,{0}:] is non empty RAT -valued INT -valued V139() V140() V141() V142() set
bool [: the carrier of X,{0}:] is non empty set
X is non empty TopSpace-like TopStruct
(X) is non empty right_complementable Abelian add-associative right_zeroed unital associative commutative right-distributive right_unital well-unital left_unital vector-distributive scalar-distributive scalar-associative scalar-unital strict vector-associative ComplexAlgebraStr
(X) is non empty multiplicatively-closed Cadditively-linearly-closed Element of bool the carrier of (CAlgebra the carrier of X)
the carrier of X is non empty set
CAlgebra the carrier of X is non empty right_complementable Abelian add-associative right_zeroed unital associative commutative right-distributive right_unital well-unital left_unital vector-distributive scalar-distributive scalar-associative scalar-unital strict vector-associative ComplexAlgebraStr
Funcs ( the carrier of X,COMPLEX) is non empty functional FUNCTION_DOMAIN of the carrier of X, COMPLEX
ComplexFuncMult the carrier of X is non empty Relation-like [:(Funcs ( the carrier of X,COMPLEX)),(Funcs ( the carrier of X,COMPLEX)):] -defined Funcs ( the carrier of X,COMPLEX) -valued Function-like total quasi_total Element of bool [:[:(Funcs ( the carrier of X,COMPLEX)),(Funcs ( the carrier of X,COMPLEX)):],(Funcs ( the carrier of X,COMPLEX)):]
[:(Funcs ( the carrier of X,COMPLEX)),(Funcs ( the carrier of X,COMPLEX)):] is non empty set
[:[:(Funcs ( the carrier of X,COMPLEX)),(Funcs ( the carrier of X,COMPLEX)):],(Funcs ( the carrier of X,COMPLEX)):] is non empty set
bool [:[:(Funcs ( the carrier of X,COMPLEX)),(Funcs ( the carrier of X,COMPLEX)):],(Funcs ( the carrier of X,COMPLEX)):] is non empty set
ComplexFuncAdd the carrier of X is non empty Relation-like [:(Funcs ( the carrier of X,COMPLEX)),(Funcs ( the carrier of X,COMPLEX)):] -defined Funcs ( the carrier of X,COMPLEX) -valued Function-like total quasi_total Element of bool [:[:(Funcs ( the carrier of X,COMPLEX)),(Funcs ( the carrier of X,COMPLEX)):],(Funcs ( the carrier of X,COMPLEX)):]
ComplexFuncExtMult the carrier of X is non empty Relation-like [:COMPLEX,(Funcs ( the carrier of X,COMPLEX)):] -defined Funcs ( the carrier of X,COMPLEX) -valued Function-like total quasi_total Element of bool [:[:COMPLEX,(Funcs ( the carrier of X,COMPLEX)):],(Funcs ( the carrier of X,COMPLEX)):]
[:COMPLEX,(Funcs ( the carrier of X,COMPLEX)):] is non empty set
[:[:COMPLEX,(Funcs ( the carrier of X,COMPLEX)):],(Funcs ( the carrier of X,COMPLEX)):] is non empty set
bool [:[:COMPLEX,(Funcs ( the carrier of X,COMPLEX)):],(Funcs ( the carrier of X,COMPLEX)):] is non empty set
ComplexFuncUnit the carrier of X is Relation-like the carrier of X -defined COMPLEX -valued Function-like total quasi_total V139() Element of Funcs ( the carrier of X,COMPLEX)
the carrier of X --> 1r is non empty Relation-like the carrier of X -defined COMPLEX -valued Function-like total quasi_total V139() Element of bool [: the carrier of X,COMPLEX:]
[: the carrier of X,COMPLEX:] is non empty V139() set
bool [: the carrier of X,COMPLEX:] is non empty set
ComplexFuncZero the carrier of X is Relation-like the carrier of X -defined COMPLEX -valued Function-like total quasi_total V139() Element of Funcs ( the carrier of X,COMPLEX)
the carrier of X --> 0 is non empty Relation-like the carrier of X -defined REAL -valued RAT -valued INT -valued Function-like total quasi_total V139() V140() V141() V142() Element of bool [: the carrier of X,REAL:]
[: the carrier of X,REAL:] is non empty V139() V140() V141() set
bool [: the carrier of X,REAL:] is non empty set
ComplexAlgebraStr(# (Funcs ( the carrier of X,COMPLEX)),(ComplexFuncMult the carrier of X),(ComplexFuncAdd the carrier of X),(ComplexFuncExtMult the carrier of X),(ComplexFuncUnit the carrier of X),(ComplexFuncZero the carrier of X) #) is strict ComplexAlgebraStr
the carrier of (CAlgebra the carrier of X) is non empty set
bool the carrier of (CAlgebra the carrier of X) is non empty set
{ b₁ where b₁ is non empty Relation-like the carrier of X -defined COMPLEX -valued Function-like total quasi_total V139() (X) Element of bool [: the carrier of X,COMPLEX:] : verum } is set
mult_ ((X),(CAlgebra the carrier of X)) is non empty Relation-like [:(X),(X):] -defined (X) -valued Function-like total quasi_total Element of bool [:[:(X),(X):],(X):]
[:(X),(X):] is non empty set
[:[:(X),(X):],(X):] is non empty set
bool [:[:(X),(X):],(X):] is non empty set
Add_ ((X),(CAlgebra the carrier of X)) is non empty Relation-like [:(X),(X):] -defined (X) -valued Function-like total quasi_total Element of bool [:[:(X),(X):],(X):]
Mult_ ((X),(CAlgebra the carrier of X)) is non empty Relation-like [:COMPLEX,(X):] -defined (X) -valued Function-like total quasi_total Element of bool [:[:COMPLEX,(X):],(X):]
[:COMPLEX,(X):] is non empty set
[:[:COMPLEX,(X):],(X):] is non empty set
bool [:[:COMPLEX,(X):],(X):] is non empty set
One_ ((X),(CAlgebra the carrier of X)) is Element of (X)
Zero_ ((X),(CAlgebra the carrier of X)) is Element of (X)
ComplexAlgebraStr(# (X),(mult_ ((X),(CAlgebra the carrier of X))),(Add_ ((X),(CAlgebra the carrier of X))),(Mult_ ((X),(CAlgebra the carrier of X))),(One_ ((X),(CAlgebra the carrier of X))),(Zero_ ((X),(CAlgebra the carrier of X))) #) is strict ComplexAlgebraStr
1_ (X) is Element of the carrier of (X)
the carrier of (X) is non empty set
1. (X) is Element of the carrier of (X)
the OneF of (X) is Element of the carrier of (X)
(X,1r) is non empty Relation-like the carrier of X -defined COMPLEX -valued Function-like total quasi_total V139() (X) Element of bool [: the carrier of X,COMPLEX:]
the carrier of X --> 1r is non empty Relation-like the carrier of X -defined {1r} -valued Function-like total quasi_total V139() Element of bool [: the carrier of X,{1r}:]
{1r} is non empty V176() set
[: the carrier of X,{1r}:] is non empty V139() set
bool [: the carrier of X,{1r}:] is non empty set
1_ (CAlgebra the carrier of X) is Element of the carrier of (CAlgebra the carrier of X)
1. (CAlgebra the carrier of X) is Element of the carrier of (CAlgebra the carrier of X)
the OneF of (CAlgebra the carrier of X) is Element of the carrier of (CAlgebra the carrier of X)
X is non empty right_complementable Abelian add-associative right_zeroed unital associative commutative right-distributive right_unital well-unital left_unital vector-distributive scalar-distributive scalar-associative vector-associative ComplexAlgebraStr
X is non empty right_complementable Abelian add-associative right_zeroed unital associative commutative right-distributive right_unital well-unital left_unital vector-distributive scalar-distributive scalar-associative vector-associative ComplexSubAlgebra of X
the carrier of X is non empty set
F is non empty right_complementable Abelian add-associative right_zeroed unital associative commutative right-distributive right_unital well-unital left_unital vector-distributive scalar-distributive scalar-associative vector-associative ComplexSubAlgebra of X
the carrier of F is non empty set
the addF of X is non empty Relation-like [: the carrier of X, the carrier of X:] -defined the carrier of X -valued Function-like total quasi_total Element of bool [:[: the carrier of X, the carrier of X:], the carrier of X:]
the carrier of X is non empty set
[: the carrier of X, the carrier of X:] is non empty set
[:[: the carrier of X, the carrier of X:], the carrier of X:] is non empty set
bool [:[: the carrier of X, the carrier of X:], the carrier of X:] is non empty set
the multF of X is non empty Relation-like [: the carrier of X, the carrier of X:] -defined the carrier of X -valued Function-like total quasi_total Element of bool [:[: the carrier of X, the carrier of X:], the carrier of X:]
the Mult of X is non empty Relation-like [:COMPLEX, the carrier of X:] -defined the carrier of X -valued Function-like total quasi_total Element of bool [:[:COMPLEX, the carrier of X:], the carrier of X:]
[:COMPLEX, the carrier of X:] is non empty set
[:[:COMPLEX, the carrier of X:], the carrier of X:] is non empty set
bool [:[:COMPLEX, the carrier of X:], the carrier of X:] is non empty set
0. X is zero Element of the carrier of X
the ZeroF of X is Element of the carrier of X
0. X is zero Element of the carrier of X
the ZeroF of X is Element of the carrier of X
0. F is zero Element of the carrier of F
the ZeroF of F is Element of the carrier of F
1. X is Element of the carrier of X
the OneF of X is Element of the carrier of X
1. X is Element of the carrier of X
the OneF of X is Element of the carrier of X
1. F is Element of the carrier of F
the OneF of F is Element of the carrier of F
the addF of X is non empty Relation-like [: the carrier of X, the carrier of X:] -defined the carrier of X -valued Function-like total quasi_total Element of bool [:[: the carrier of X, the carrier of X:], the carrier of X:]
[: the carrier of X, the carrier of X:] is non empty set
[:[: the carrier of X, the carrier of X:], the carrier of X:] is non empty set
bool [:[: the carrier of X, the carrier of X:], the carrier of X:] is non empty set
the addF of F is non empty Relation-like [: the carrier of F, the carrier of F:] -defined the carrier of F -valued Function-like total quasi_total Element of bool [:[: the carrier of F, the carrier of F:], the carrier of F:]
[: the carrier of F, the carrier of F:] is non empty set
[:[: the carrier of F, the carrier of F:], the carrier of F:] is non empty set
bool [:[: the carrier of F, the carrier of F:], the carrier of F:] is non empty set
the addF of F || the carrier of X is Relation-like Function-like set
the addF of F | [: the carrier of X, the carrier of X:] is Relation-like set
the addF of X || the carrier of X is Relation-like Function-like set
the addF of X | [: the carrier of X, the carrier of X:] is Relation-like set
the addF of X || the carrier of F is Relation-like Function-like set
the addF of X | [: the carrier of F, the carrier of F:] is Relation-like set
the multF of X is non empty Relation-like [: the carrier of X, the carrier of X:] -defined the carrier of X -valued Function-like total quasi_total Element of bool [:[: the carrier of X, the carrier of X:], the carrier of X:]
the multF of F is non empty Relation-like [: the carrier of F, the carrier of F:] -defined the carrier of F -valued Function-like total quasi_total Element of bool [:[: the carrier of F, the carrier of F:], the carrier of F:]
the multF of F || the carrier of X is Relation-like Function-like set
the multF of F | [: the carrier of X, the carrier of X:] is Relation-like set
the multF of X || the carrier of X is Relation-like Function-like set
the multF of X | [: the carrier of X, the carrier of X:] is Relation-like set
the multF of X || the carrier of F is Relation-like Function-like set
the multF of X | [: the carrier of F, the carrier of F:] is Relation-like set
the Mult of X is non empty Relation-like [:COMPLEX, the carrier of X:] -defined the carrier of X -valued Function-like total quasi_total Element of bool [:[:COMPLEX, the carrier of X:], the carrier of X:]
[:COMPLEX, the carrier of X:] is non empty set
[:[:COMPLEX, the carrier of X:], the carrier of X:] is non empty set
bool [:[:COMPLEX, the carrier of X:], the carrier of X:] is non empty set
[:COMPLEX, the carrier of F:] is non empty set
the Mult of F is non empty Relation-like [:COMPLEX, the carrier of F:] -defined the carrier of F -valued Function-like total quasi_total Element of bool [:[:COMPLEX, the carrier of F:], the carrier of F:]
[:[:COMPLEX, the carrier of F:], the carrier of F:] is non empty set
bool [:[:COMPLEX, the carrier of F:], the carrier of F:] is non empty set
the Mult of F | [:COMPLEX, the carrier of X:] is Relation-like [:COMPLEX, the carrier of F:] -defined the carrier of F -valued Function-like Element of bool [:[:COMPLEX, the carrier of F:], the carrier of F:]
the Mult of X | [:COMPLEX, the carrier of X:] is Relation-like [:COMPLEX, the carrier of X:] -defined the carrier of X -valued Function-like Element of bool [:[:COMPLEX, the carrier of X:], the carrier of X:]
the Mult of X | [:COMPLEX, the carrier of F:] is Relation-like [:COMPLEX, the carrier of X:] -defined the carrier of X -valued Function-like Element of bool [:[:COMPLEX, the carrier of X:], the carrier of X:]
X is non empty TopSpace-like V300() compact TopStruct
(X) is non empty multiplicatively-closed Cadditively-linearly-closed Element of bool the carrier of (CAlgebra the carrier of X)
the carrier of X is non empty set
CAlgebra the carrier of X is non empty right_complementable Abelian add-associative right_zeroed unital associative commutative right-distributive right_unital well-unital left_unital vector-distributive scalar-distributive scalar-associative scalar-unital strict vector-associative ComplexAlgebraStr
Funcs ( the carrier of X,COMPLEX) is non empty functional FUNCTION_DOMAIN of the carrier of X, COMPLEX
ComplexFuncMult the carrier of X is non empty Relation-like [:(Funcs ( the carrier of X,COMPLEX)),(Funcs ( the carrier of X,COMPLEX)):] -defined Funcs ( the carrier of X,COMPLEX) -valued Function-like total quasi_total Element of bool [:[:(Funcs ( the carrier of X,COMPLEX)),(Funcs ( the carrier of X,COMPLEX)):],(Funcs ( the carrier of X,COMPLEX)):]
[:(Funcs ( the carrier of X,COMPLEX)),(Funcs ( the carrier of X,COMPLEX)):] is non empty set
[:[:(Funcs ( the carrier of X,COMPLEX)),(Funcs ( the carrier of X,COMPLEX)):],(Funcs ( the carrier of X,COMPLEX)):] is non empty set
bool [:[:(Funcs ( the carrier of X,COMPLEX)),(Funcs ( the carrier of X,COMPLEX)):],(Funcs ( the carrier of X,COMPLEX)):] is non empty set
ComplexFuncAdd the carrier of X is non empty Relation-like [:(Funcs ( the carrier of X,COMPLEX)),(Funcs ( the carrier of X,COMPLEX)):] -defined Funcs ( the carrier of X,COMPLEX) -valued Function-like total quasi_total Element of bool [:[:(Funcs ( the carrier of X,COMPLEX)),(Funcs ( the carrier of X,COMPLEX)):],(Funcs ( the carrier of X,COMPLEX)):]
ComplexFuncExtMult the carrier of X is non empty Relation-like [:COMPLEX,(Funcs ( the carrier of X,COMPLEX)):] -defined Funcs ( the carrier of X,COMPLEX) -valued Function-like total quasi_total Element of bool [:[:COMPLEX,(Funcs ( the carrier of X,COMPLEX)):],(Funcs ( the carrier of X,COMPLEX)):]
[:COMPLEX,(Funcs ( the carrier of X,COMPLEX)):] is non empty set
[:[:COMPLEX,(Funcs ( the carrier of X,COMPLEX)):],(Funcs ( the carrier of X,COMPLEX)):] is non empty set
bool [:[:COMPLEX,(Funcs ( the carrier of X,COMPLEX)):],(Funcs ( the carrier of X,COMPLEX)):] is non empty set
ComplexFuncUnit the carrier of X is Relation-like the carrier of X -defined COMPLEX -valued Function-like total quasi_total V139() Element of Funcs ( the carrier of X,COMPLEX)
the carrier of X --> 1r is non empty Relation-like the carrier of X -defined COMPLEX -valued Function-like total quasi_total V139() Element of bool [: the carrier of X,COMPLEX:]
[: the carrier of X,COMPLEX:] is non empty V139() set
bool [: the carrier of X,COMPLEX:] is non empty set
ComplexFuncZero the carrier of X is Relation-like the carrier of X -defined COMPLEX -valued Function-like total quasi_total V139() Element of Funcs ( the carrier of X,COMPLEX)
the carrier of X --> 0 is non empty Relation-like the carrier of X -defined REAL -valued RAT -valued INT -valued Function-like total quasi_total V139() V140() V141() V142() Element of bool [: the carrier of X,REAL:]
[: the carrier of X,REAL:] is non empty V139() V140() V141() set
bool [: the carrier of X,REAL:] is non empty set
ComplexAlgebraStr(# (Funcs ( the carrier of X,COMPLEX)),(ComplexFuncMult the carrier of X),(ComplexFuncAdd the carrier of X),(ComplexFuncExtMult the carrier of X),(ComplexFuncUnit the carrier of X),(ComplexFuncZero the carrier of X) #) is strict ComplexAlgebraStr
the carrier of (CAlgebra the carrier of X) is non empty set
bool the carrier of (CAlgebra the carrier of X) is non empty set
{ b₁ where b₁ is non empty Relation-like the carrier of X -defined COMPLEX -valued Function-like total quasi_total V139() (X) Element of bool [: the carrier of X,COMPLEX:] : verum } is set
ComplexBoundedFunctions the carrier of X is non empty multiplicatively-closed Cadditively-linearly-closed Element of bool the carrier of (CAlgebra the carrier of X)
{ b₁ where b₁ is non empty Relation-like the carrier of X -defined COMPLEX -valued Function-like total quasi_total V139() Element of bool [: the carrier of X,COMPLEX:] : b₁ | the carrier of X is bounded } is set
X is set
F is non empty Relation-like the carrier of X -defined COMPLEX -valued Function-like total quasi_total V139() (X) Element of bool [: the carrier of X,COMPLEX:]
|.F.| is non empty Relation-like the carrier of X -defined REAL -valued Function-like total quasi_total V139() V140() V141() Element of bool [: the carrier of X,REAL:]
G is non empty Relation-like the carrier of X -defined REAL -valued Function-like total quasi_total V139() V140() V141() Element of bool [: the carrier of X,REAL:]
dom G is Element of bool the carrier of X
bool the carrier of X is non empty set
FB is complex real ext-real set
dom F is Element of bool the carrier of X
GB is set
F . GB is complex set
abs (F . GB) is complex real ext-real non negative Element of REAL
G . GB is complex real ext-real set
|.(F . GB).| is complex real ext-real non negative Element of REAL
F | the carrier of X is Relation-like the carrier of X -defined COMPLEX -valued Function-like V139() Element of bool [: the carrier of X,COMPLEX:]
X is non empty TopSpace-like V300() compact TopStruct
(X) is non empty right_complementable Abelian add-associative right_zeroed unital associative commutative right-distributive right_unital well-unital left_unital vector-distributive scalar-distributive scalar-associative scalar-unital strict vector-associative ComplexAlgebraStr
(X) is non empty multiplicatively-closed Cadditively-linearly-closed Element of bool the carrier of (CAlgebra the carrier of X)
the carrier of X is non empty set
CAlgebra the carrier of X is non empty right_complementable Abelian add-associative right_zeroed unital associative commutative right-distributive right_unital well-unital left_unital vector-distributive scalar-distributive scalar-associative scalar-unital strict vector-associative ComplexAlgebraStr
Funcs ( the carrier of X,COMPLEX) is non empty functional FUNCTION_DOMAIN of the carrier of X, COMPLEX
ComplexFuncMult the carrier of X is non empty Relation-like [:(Funcs ( the carrier of X,COMPLEX)),(Funcs ( the carrier of X,COMPLEX)):] -defined Funcs ( the carrier of X,COMPLEX) -valued Function-like total quasi_total Element of bool [:[:(Funcs ( the carrier of X,COMPLEX)),(Funcs ( the carrier of X,COMPLEX)):],(Funcs ( the carrier of X,COMPLEX)):]
[:(Funcs ( the carrier of X,COMPLEX)),(Funcs ( the carrier of X,COMPLEX)):] is non empty set
[:[:(Funcs ( the carrier of X,COMPLEX)),(Funcs ( the carrier of X,COMPLEX)):],(Funcs ( the carrier of X,COMPLEX)):] is non empty set
bool [:[:(Funcs ( the carrier of X,COMPLEX)),(Funcs ( the carrier of X,COMPLEX)):],(Funcs ( the carrier of X,COMPLEX)):] is non empty set
ComplexFuncAdd the carrier of X is non empty Relation-like [:(Funcs ( the carrier of X,COMPLEX)),(Funcs ( the carrier of X,COMPLEX)):] -defined Funcs ( the carrier of X,COMPLEX) -valued Function-like total quasi_total Element of bool [:[:(Funcs ( the carrier of X,COMPLEX)),(Funcs ( the carrier of X,COMPLEX)):],(Funcs ( the carrier of X,COMPLEX)):]
ComplexFuncExtMult the carrier of X is non empty Relation-like [:COMPLEX,(Funcs ( the carrier of X,COMPLEX)):] -defined Funcs ( the carrier of X,COMPLEX) -valued Function-like total quasi_total Element of bool [:[:COMPLEX,(Funcs ( the carrier of X,COMPLEX)):],(Funcs ( the carrier of X,COMPLEX)):]
[:COMPLEX,(Funcs ( the carrier of X,COMPLEX)):] is non empty set
[:[:COMPLEX,(Funcs ( the carrier of X,COMPLEX)):],(Funcs ( the carrier of X,COMPLEX)):] is non empty set
bool [:[:COMPLEX,(Funcs ( the carrier of X,COMPLEX)):],(Funcs ( the carrier of X,COMPLEX)):] is non empty set
ComplexFuncUnit the carrier of X is Relation-like the carrier of X -defined COMPLEX -valued Function-like total quasi_total V139() Element of Funcs ( the carrier of X,COMPLEX)
the carrier of X --> 1r is non empty Relation-like the carrier of X -defined COMPLEX -valued Function-like total quasi_total V139() Element of bool [: the carrier of X,COMPLEX:]
[: the carrier of X,COMPLEX:] is non empty V139() set
bool [: the carrier of X,COMPLEX:] is non empty set
ComplexFuncZero the carrier of X is Relation-like the carrier of X -defined COMPLEX -valued Function-like total quasi_total V139() Element of Funcs ( the carrier of X,COMPLEX)
the carrier of X --> 0 is non empty Relation-like the carrier of X -defined REAL -valued RAT -valued INT -valued Function-like total quasi_total V139() V140() V141() V142() Element of bool [: the carrier of X,REAL:]
[: the carrier of X,REAL:] is non empty V139() V140() V141() set
bool [: the carrier of X,REAL:] is non empty set
ComplexAlgebraStr(# (Funcs ( the carrier of X,COMPLEX)),(ComplexFuncMult the carrier of X),(ComplexFuncAdd the carrier of X),(ComplexFuncExtMult the carrier of X),(ComplexFuncUnit the carrier of X),(ComplexFuncZero the carrier of X) #) is strict ComplexAlgebraStr
the carrier of (CAlgebra the carrier of X) is non empty set
bool the carrier of (CAlgebra the carrier of X) is non empty set
{ b₁ where b₁ is non empty Relation-like the carrier of X -defined COMPLEX -valued Function-like total quasi_total V139() (X) Element of bool [: the carrier of X,COMPLEX:] : verum } is set
mult_ ((X),(CAlgebra the carrier of X)) is non empty Relation-like [:(X),(X):] -defined (X) -valued Function-like total quasi_total Element of bool [:[:(X),(X):],(X):]
[:(X),(X):] is non empty set
[:[:(X),(X):],(X):] is non empty set
bool [:[:(X),(X):],(X):] is non empty set
Add_ ((X),(CAlgebra the carrier of X)) is non empty Relation-like [:(X),(X):] -defined (X) -valued Function-like total quasi_total Element of bool [:[:(X),(X):],(X):]
Mult_ ((X),(CAlgebra the carrier of X)) is non empty Relation-like [:COMPLEX,(X):] -defined (X) -valued Function-like total quasi_total Element of bool [:[:COMPLEX,(X):],(X):]
[:COMPLEX,(X):] is non empty set
[:[:COMPLEX,(X):],(X):] is non empty set
bool [:[:COMPLEX,(X):],(X):] is non empty set
One_ ((X),(CAlgebra the carrier of X)) is Element of (X)
Zero_ ((X),(CAlgebra the carrier of X)) is Element of (X)
ComplexAlgebraStr(# (X),(mult_ ((X),(CAlgebra the carrier of X))),(Add_ ((X),(CAlgebra the carrier of X))),(Mult_ ((X),(CAlgebra the carrier of X))),(One_ ((X),(CAlgebra the carrier of X))),(Zero_ ((X),(CAlgebra the carrier of X))) #) is strict ComplexAlgebraStr
C_Algebra_of_BoundedFunctions the carrier of X is non empty right_complementable Abelian add-associative right_zeroed unital associative commutative right-distributive right_unital well-unital left_unital vector-distributive scalar-distributive scalar-associative scalar-unital vector-associative ComplexAlgebraStr
ComplexBoundedFunctions the carrier of X is non empty multiplicatively-closed Cadditively-linearly-closed Element of bool the carrier of (CAlgebra the carrier of X)
{ b₁ where b₁ is non empty Relation-like the carrier of X -defined COMPLEX -valued Function-like total quasi_total V139() Element of bool [: the carrier of X,COMPLEX:] : b₁ | the carrier of X is bounded } is set
mult_ ((ComplexBoundedFunctions the carrier of X),(CAlgebra the carrier of X)) is non empty Relation-like [:(ComplexBoundedFunctions the carrier of X),(ComplexBoundedFunctions the carrier of X):] -defined ComplexBoundedFunctions the carrier of X -valued Function-like total quasi_total Element of bool [:[:(ComplexBoundedFunctions the carrier of X),(ComplexBoundedFunctions the carrier of X):],(ComplexBoundedFunctions the carrier of X):]
[:(ComplexBoundedFunctions the carrier of X),(ComplexBoundedFunctions the carrier of X):] is non empty set
[:[:(ComplexBoundedFunctions the carrier of X),(ComplexBoundedFunctions the carrier of X):],(ComplexBoundedFunctions the carrier of X):] is non empty set
bool [:[:(ComplexBoundedFunctions the carrier of X),(ComplexBoundedFunctions the carrier of X):],(ComplexBoundedFunctions the carrier of X):] is non empty set
Add_ ((ComplexBoundedFunctions the carrier of X),(CAlgebra the carrier of X)) is non empty Relation-like [:(ComplexBoundedFunctions the carrier of X),(ComplexBoundedFunctions the carrier of X):] -defined ComplexBoundedFunctions the carrier of X -valued Function-like total quasi_total Element of bool [:[:(ComplexBoundedFunctions the carrier of X),(ComplexBoundedFunctions the carrier of X):],(ComplexBoundedFunctions the carrier of X):]
Mult_ ((ComplexBoundedFunctions the carrier of X),(CAlgebra the carrier of X)) is non empty Relation-like [:COMPLEX,(ComplexBoundedFunctions the carrier of X):] -defined ComplexBoundedFunctions the carrier of X -valued Function-like total quasi_total Element of bool [:[:COMPLEX,(ComplexBoundedFunctions the carrier of X):],(ComplexBoundedFunctions the carrier of X):]
[:COMPLEX,(ComplexBoundedFunctions the carrier of X):] is non empty set
[:[:COMPLEX,(ComplexBoundedFunctions the carrier of X):],(ComplexBoundedFunctions the carrier of X):] is non empty set
bool [:[:COMPLEX,(ComplexBoundedFunctions the carrier of X):],(ComplexBoundedFunctions the carrier of X):] is non empty set
One_ ((ComplexBoundedFunctions the carrier of X),(CAlgebra the carrier of X)) is Element of ComplexBoundedFunctions the carrier of X
Zero_ ((ComplexBoundedFunctions the carrier of X),(CAlgebra the carrier of X)) is Element of ComplexBoundedFunctions the carrier of X
ComplexAlgebraStr(# (ComplexBoundedFunctions the carrier of X),(mult_ ((ComplexBoundedFunctions the carrier of X),(CAlgebra the carrier of X))),(Add_ ((ComplexBoundedFunctions the carrier of X),(CAlgebra the carrier of X))),(Mult_ ((ComplexBoundedFunctions the carrier of X),(CAlgebra the carrier of X))),(One_ ((ComplexBoundedFunctions the carrier of X),(CAlgebra the carrier of X))),(Zero_ ((ComplexBoundedFunctions the carrier of X),(CAlgebra the carrier of X))) #) is strict ComplexAlgebraStr
the carrier of (X) is non empty set
the carrier of (C_Algebra_of_BoundedFunctions the carrier of X) is non empty set
X is set
X is non empty TopSpace-like V300() compact TopStruct
the carrier of X is non empty set
ComplexBoundedFunctions the carrier of X is non empty multiplicatively-closed Cadditively-linearly-closed Element of bool the carrier of (CAlgebra the carrier of X)
CAlgebra the carrier of X is non empty right_complementable Abelian add-associative right_zeroed unital associative commutative right-distributive right_unital well-unital left_unital vector-distributive scalar-distributive scalar-associative scalar-unital strict vector-associative ComplexAlgebraStr
Funcs ( the carrier of X,COMPLEX) is non empty functional FUNCTION_DOMAIN of the carrier of X, COMPLEX
ComplexFuncMult the carrier of X is non empty Relation-like [:(Funcs ( the carrier of X,COMPLEX)),(Funcs ( the carrier of X,COMPLEX)):] -defined Funcs ( the carrier of X,COMPLEX) -valued Function-like total quasi_total Element of bool [:[:(Funcs ( the carrier of X,COMPLEX)),(Funcs ( the carrier of X,COMPLEX)):],(Funcs ( the carrier of X,COMPLEX)):]
[:(Funcs ( the carrier of X,COMPLEX)),(Funcs ( the carrier of X,COMPLEX)):] is non empty set
[:[:(Funcs ( the carrier of X,COMPLEX)),(Funcs ( the carrier of X,COMPLEX)):],(Funcs ( the carrier of X,COMPLEX)):] is non empty set
bool [:[:(Funcs ( the carrier of X,COMPLEX)),(Funcs ( the carrier of X,COMPLEX)):],(Funcs ( the carrier of X,COMPLEX)):] is non empty set
ComplexFuncAdd the carrier of X is non empty Relation-like [:(Funcs ( the carrier of X,COMPLEX)),(Funcs ( the carrier of X,COMPLEX)):] -defined Funcs ( the carrier of X,COMPLEX) -valued Function-like total quasi_total Element of bool [:[:(Funcs ( the carrier of X,COMPLEX)),(Funcs ( the carrier of X,COMPLEX)):],(Funcs ( the carrier of X,COMPLEX)):]
ComplexFuncExtMult the carrier of X is non empty Relation-like [:COMPLEX,(Funcs ( the carrier of X,COMPLEX)):] -defined Funcs ( the carrier of X,COMPLEX) -valued Function-like total quasi_total Element of bool [:[:COMPLEX,(Funcs ( the carrier of X,COMPLEX)):],(Funcs ( the carrier of X,COMPLEX)):]
[:COMPLEX,(Funcs ( the carrier of X,COMPLEX)):] is non empty set
[:[:COMPLEX,(Funcs ( the carrier of X,COMPLEX)):],(Funcs ( the carrier of X,COMPLEX)):] is non empty set
bool [:[:COMPLEX,(Funcs ( the carrier of X,COMPLEX)):],(Funcs ( the carrier of X,COMPLEX)):] is non empty set
ComplexFuncUnit the carrier of X is Relation-like the carrier of X -defined COMPLEX -valued Function-like total quasi_total V139() Element of Funcs ( the carrier of X,COMPLEX)
the carrier of X --> 1r is non empty Relation-like the carrier of X -defined COMPLEX -valued Function-like total quasi_total V139() Element of bool [: the carrier of X,COMPLEX:]
[: the carrier of X,COMPLEX:] is non empty V139() set
bool [: the carrier of X,COMPLEX:] is non empty set
ComplexFuncZero the carrier of X is Relation-like the carrier of X -defined COMPLEX -valued Function-like total quasi_total V139() Element of Funcs ( the carrier of X,COMPLEX)
the carrier of X --> 0 is non empty Relation-like the carrier of X -defined REAL -valued RAT -valued INT -valued Function-like total quasi_total V139() V140() V141() V142() Element of bool [: the carrier of X,REAL:]
[: the carrier of X,REAL:] is non empty V139() V140() V141() set
bool [: the carrier of X,REAL:] is non empty set
ComplexAlgebraStr(# (Funcs ( the carrier of X,COMPLEX)),(ComplexFuncMult the carrier of X),(ComplexFuncAdd the carrier of X),(ComplexFuncExtMult the carrier of X),(ComplexFuncUnit the carrier of X),(ComplexFuncZero the carrier of X) #) is strict ComplexAlgebraStr
the carrier of (CAlgebra the carrier of X) is non empty set
bool the carrier of (CAlgebra the carrier of X) is non empty set
{ b₁ where b₁ is non empty Relation-like the carrier of X -defined COMPLEX -valued Function-like total quasi_total V139() Element of bool [: the carrier of X,COMPLEX:] : b₁ | the carrier of X is bounded } is set
ComplexBoundedFunctionsNorm the carrier of X is non empty Relation-like ComplexBoundedFunctions the carrier of X -defined REAL -valued Function-like total quasi_total V139() V140() V141() Element of bool [:(ComplexBoundedFunctions the carrier of X),REAL:]
[:(ComplexBoundedFunctions the carrier of X),REAL:] is non empty V139() V140() V141() set
bool [:(ComplexBoundedFunctions the carrier of X),REAL:] is non empty set
(X) is non empty multiplicatively-closed Cadditively-linearly-closed Element of bool the carrier of (CAlgebra the carrier of X)
{ b₁ where b₁ is non empty Relation-like the carrier of X -defined COMPLEX -valued Function-like total quasi_total V139() (X) Element of bool [: the carrier of X,COMPLEX:] : verum } is set
(ComplexBoundedFunctionsNorm the carrier of X) | (X) is Relation-like ComplexBoundedFunctions the carrier of X -defined REAL -valued Function-like V139() V140() V141() Element of bool [:(ComplexBoundedFunctions the carrier of X),REAL:]
[:(X),REAL:] is non empty V139() V140() V141() set
bool [:(X),REAL:] is non empty set
X is set
X is non empty TopSpace-like V300() compact TopStruct
(X) is non empty multiplicatively-closed Cadditively-linearly-closed Element of bool the carrier of (CAlgebra the carrier of X)
the carrier of X is non empty set
CAlgebra the carrier of X is non empty right_complementable Abelian add-associative right_zeroed unital associative commutative right-distributive right_unital well-unital left_unital vector-distributive scalar-distributive scalar-associative scalar-unital strict vector-associative ComplexAlgebraStr
Funcs ( the carrier of X,COMPLEX) is non empty functional FUNCTION_DOMAIN of the carrier of X, COMPLEX
ComplexFuncMult the carrier of X is non empty Relation-like [:(Funcs ( the carrier of X,COMPLEX)),(Funcs ( the carrier of X,COMPLEX)):] -defined Funcs ( the carrier of X,COMPLEX) -valued Function-like total quasi_total Element of bool [:[:(Funcs ( the carrier of X,COMPLEX)),(Funcs ( the carrier of X,COMPLEX)):],(Funcs ( the carrier of X,COMPLEX)):]
[:(Funcs ( the carrier of X,COMPLEX)),(Funcs ( the carrier of X,COMPLEX)):] is non empty set
[:[:(Funcs ( the carrier of X,COMPLEX)),(Funcs ( the carrier of X,COMPLEX)):],(Funcs ( the carrier of X,COMPLEX)):] is non empty set
bool [:[:(Funcs ( the carrier of X,COMPLEX)),(Funcs ( the carrier of X,COMPLEX)):],(Funcs ( the carrier of X,COMPLEX)):] is non empty set
ComplexFuncAdd the carrier of X is non empty Relation-like [:(Funcs ( the carrier of X,COMPLEX)),(Funcs ( the carrier of X,COMPLEX)):] -defined Funcs ( the carrier of X,COMPLEX) -valued Function-like total quasi_total Element of bool [:[:(Funcs ( the carrier of X,COMPLEX)),(Funcs ( the carrier of X,COMPLEX)):],(Funcs ( the carrier of X,COMPLEX)):]
ComplexFuncExtMult the carrier of X is non empty Relation-like [:COMPLEX,(Funcs ( the carrier of X,COMPLEX)):] -defined Funcs ( the carrier of X,COMPLEX) -valued Function-like total quasi_total Element of bool [:[:COMPLEX,(Funcs ( the carrier of X,COMPLEX)):],(Funcs ( the carrier of X,COMPLEX)):]
[:COMPLEX,(Funcs ( the carrier of X,COMPLEX)):] is non empty set
[:[:COMPLEX,(Funcs ( the carrier of X,COMPLEX)):],(Funcs ( the carrier of X,COMPLEX)):] is non empty set
bool [:[:COMPLEX,(Funcs ( the carrier of X,COMPLEX)):],(Funcs ( the carrier of X,COMPLEX)):] is non empty set
ComplexFuncUnit the carrier of X is Relation-like the carrier of X -defined COMPLEX -valued Function-like total quasi_total V139() Element of Funcs ( the carrier of X,COMPLEX)
the carrier of X --> 1r is non empty Relation-like the carrier of X -defined COMPLEX -valued Function-like total quasi_total V139() Element of bool [: the carrier of X,COMPLEX:]
[: the carrier of X,COMPLEX:] is non empty V139() set
bool [: the carrier of X,COMPLEX:] is non empty set
ComplexFuncZero the carrier of X is Relation-like the carrier of X -defined COMPLEX -valued Function-like total quasi_total V139() Element of Funcs ( the carrier of X,COMPLEX)
the carrier of X --> 0 is non empty Relation-like the carrier of X -defined REAL -valued RAT -valued INT -valued Function-like total quasi_total V139() V140() V141() V142() Element of bool [: the carrier of X,REAL:]
[: the carrier of X,REAL:] is non empty V139() V140() V141() set
bool [: the carrier of X,REAL:] is non empty set
ComplexAlgebraStr(# (Funcs ( the carrier of X,COMPLEX)),(ComplexFuncMult the carrier of X),(ComplexFuncAdd the carrier of X),(ComplexFuncExtMult the carrier of X),(ComplexFuncUnit the carrier of X),(ComplexFuncZero the carrier of X) #) is strict ComplexAlgebraStr
the carrier of (CAlgebra the carrier of X) is non empty set
bool the carrier of (CAlgebra the carrier of X) is non empty set
{ b₁ where b₁ is non empty Relation-like the carrier of X -defined COMPLEX -valued Function-like total quasi_total V139() (X) Element of bool [: the carrier of X,COMPLEX:] : verum } is set
mult_ ((X),(CAlgebra the carrier of X)) is non empty Relation-like [:(X),(X):] -defined (X) -valued Function-like total quasi_total Element of bool [:[:(X),(X):],(X):]
[:(X),(X):] is non empty set
[:[:(X),(X):],(X):] is non empty set
bool [:[:(X),(X):],(X):] is non empty set
Add_ ((X),(CAlgebra the carrier of X)) is non empty Relation-like [:(X),(X):] -defined (X) -valued Function-like total quasi_total Element of bool [:[:(X),(X):],(X):]
Mult_ ((X),(CAlgebra the carrier of X)) is non empty Relation-like [:COMPLEX,(X):] -defined (X) -valued Function-like total quasi_total Element of bool [:[:COMPLEX,(X):],(X):]
[:COMPLEX,(X):] is non empty set
[:[:COMPLEX,(X):],(X):] is non empty set
bool [:[:COMPLEX,(X):],(X):] is non empty set
One_ ((X),(CAlgebra the carrier of X)) is Element of (X)
Zero_ ((X),(CAlgebra the carrier of X)) is Element of (X)
(X) is non empty Relation-like (X) -defined REAL -valued Function-like total quasi_total V139() V140() V141() Element of bool [:(X),REAL:]
[:(X),REAL:] is non empty V139() V140() V141() set
bool [:(X),REAL:] is non empty set
ComplexBoundedFunctions the carrier of X is non empty multiplicatively-closed Cadditively-linearly-closed Element of bool the carrier of (CAlgebra the carrier of X)
{ b₁ where b₁ is non empty Relation-like the carrier of X -defined COMPLEX -valued Function-like total quasi_total V139() Element of bool [: the carrier of X,COMPLEX:] : b₁ | the carrier of X is bounded } is set
ComplexBoundedFunctionsNorm the carrier of X is non empty Relation-like ComplexBoundedFunctions the carrier of X -defined REAL -valued Function-like total quasi_total V139() V140() V141() Element of bool [:(ComplexBoundedFunctions the carrier of X),REAL:]
[:(ComplexBoundedFunctions the carrier of X),REAL:] is non empty V139() V140() V141() set
bool [:(ComplexBoundedFunctions the carrier of X),REAL:] is non empty set
(ComplexBoundedFunctionsNorm the carrier of X) | (X) is Relation-like ComplexBoundedFunctions the carrier of X -defined REAL -valued Function-like V139() V140() V141() Element of bool [:(ComplexBoundedFunctions the carrier of X),REAL:]
Normed_Complex_AlgebraStr(# (X),(mult_ ((X),(CAlgebra the carrier of X))),(Add_ ((X),(CAlgebra the carrier of X))),(Mult_ ((X),(CAlgebra the carrier of X))),(One_ ((X),(CAlgebra the carrier of X))),(Zero_ ((X),(CAlgebra the carrier of X))),(X) #) is strict Normed_Complex_AlgebraStr
X is non empty TopSpace-like V300() compact TopStruct
(X) is Normed_Complex_AlgebraStr
(X) is non empty multiplicatively-closed Cadditively-linearly-closed Element of bool the carrier of (CAlgebra the carrier of X)
the carrier of X is non empty set
CAlgebra the carrier of X is non empty right_complementable Abelian add-associative right_zeroed unital associative commutative right-distributive right_unital well-unital left_unital vector-distributive scalar-distributive scalar-associative scalar-unital strict vector-associative ComplexAlgebraStr
Funcs ( the carrier of X,COMPLEX) is non empty functional FUNCTION_DOMAIN of the carrier of X, COMPLEX
ComplexFuncMult the carrier of X is non empty Relation-like [:(Funcs ( the carrier of X,COMPLEX)),(Funcs ( the carrier of X,COMPLEX)):] -defined Funcs ( the carrier of X,COMPLEX) -valued Function-like total quasi_total Element of bool [:[:(Funcs ( the carrier of X,COMPLEX)),(Funcs ( the carrier of X,COMPLEX)):],(Funcs ( the carrier of X,COMPLEX)):]
[:(Funcs ( the carrier of X,COMPLEX)),(Funcs ( the carrier of X,COMPLEX)):] is non empty set
[:[:(Funcs ( the carrier of X,COMPLEX)),(Funcs ( the carrier of X,COMPLEX)):],(Funcs ( the carrier of X,COMPLEX)):] is non empty set
bool [:[:(Funcs ( the carrier of X,COMPLEX)),(Funcs ( the carrier of X,COMPLEX)):],(Funcs ( the carrier of X,COMPLEX)):] is non empty set
ComplexFuncAdd the carrier of X is non empty Relation-like [:(Funcs ( the carrier of X,COMPLEX)),(Funcs ( the carrier of X,COMPLEX)):] -defined Funcs ( the carrier of X,COMPLEX) -valued Function-like total quasi_total Element of bool [:[:(Funcs ( the carrier of X,COMPLEX)),(Funcs ( the carrier of X,COMPLEX)):],(Funcs ( the carrier of X,COMPLEX)):]
ComplexFuncExtMult the carrier of X is non empty Relation-like [:COMPLEX,(Funcs ( the carrier of X,COMPLEX)):] -defined Funcs ( the carrier of X,COMPLEX) -valued Function-like total quasi_total Element of bool [:[:COMPLEX,(Funcs ( the carrier of X,COMPLEX)):],(Funcs ( the carrier of X,COMPLEX)):]
[:COMPLEX,(Funcs ( the carrier of X,COMPLEX)):] is non empty set
[:[:COMPLEX,(Funcs ( the carrier of X,COMPLEX)):],(Funcs ( the carrier of X,COMPLEX)):] is non empty set
bool [:[:COMPLEX,(Funcs ( the carrier of X,COMPLEX)):],(Funcs ( the carrier of X,COMPLEX)):] is non empty set
ComplexFuncUnit the carrier of X is Relation-like the carrier of X -defined COMPLEX -valued Function-like total quasi_total V139() Element of Funcs ( the carrier of X,COMPLEX)
the carrier of X --> 1r is non empty Relation-like the carrier of X -defined COMPLEX -valued Function-like total quasi_total V139() Element of bool [: the carrier of X,COMPLEX:]
[: the carrier of X,COMPLEX:] is non empty V139() set
bool [: the carrier of X,COMPLEX:] is non empty set
ComplexFuncZero the carrier of X is Relation-like the carrier of X -defined COMPLEX -valued Function-like total quasi_total V139() Element of Funcs ( the carrier of X,COMPLEX)
the carrier of X --> 0 is non empty Relation-like the carrier of X -defined REAL -valued RAT -valued INT -valued Function-like total quasi_total V139() V140() V141() V142() Element of bool [: the carrier of X,REAL:]
[: the carrier of X,REAL:] is non empty V139() V140() V141() set
bool [: the carrier of X,REAL:] is non empty set
ComplexAlgebraStr(# (Funcs ( the carrier of X,COMPLEX)),(ComplexFuncMult the carrier of X),(ComplexFuncAdd the carrier of X),(ComplexFuncExtMult the carrier of X),(ComplexFuncUnit the carrier of X),(ComplexFuncZero the carrier of X) #) is strict ComplexAlgebraStr
the carrier of (CAlgebra the carrier of X) is non empty set
bool the carrier of (CAlgebra the carrier of X) is non empty set
{ b₁ where b₁ is non empty Relation-like the carrier of X -defined COMPLEX -valued Function-like total quasi_total V139() (X) Element of bool [: the carrier of X,COMPLEX:] : verum } is set
mult_ ((X),(CAlgebra the carrier of X)) is non empty Relation-like [:(X),(X):] -defined (X) -valued Function-like total quasi_total Element of bool [:[:(X),(X):],(X):]
[:(X),(X):] is non empty set
[:[:(X),(X):],(X):] is non empty set
bool [:[:(X),(X):],(X):] is non empty set
Add_ ((X),(CAlgebra the carrier of X)) is non empty Relation-like [:(X),(X):] -defined (X) -valued Function-like total quasi_total Element of bool [:[:(X),(X):],(X):]
Mult_ ((X),(CAlgebra the carrier of X)) is non empty Relation-like [:COMPLEX,(X):] -defined (X) -valued Function-like total quasi_total Element of bool [:[:COMPLEX,(X):],(X):]
[:COMPLEX,(X):] is non empty set
[:[:COMPLEX,(X):],(X):] is non empty set
bool [:[:COMPLEX,(X):],(X):] is non empty set
One_ ((X),(CAlgebra the carrier of X)) is Element of (X)
Zero_ ((X),(CAlgebra the carrier of X)) is Element of (X)
(X) is non empty Relation-like (X) -defined REAL -valued Function-like total quasi_total V139() V140() V141() Element of bool [:(X),REAL:]
[:(X),REAL:] is non empty V139() V140() V141() set
bool [:(X),REAL:] is non empty set
ComplexBoundedFunctions the carrier of X is non empty multiplicatively-closed Cadditively-linearly-closed Element of bool the carrier of (CAlgebra the carrier of X)
{ b₁ where b₁ is non empty Relation-like the carrier of X -defined COMPLEX -valued Function-like total quasi_total V139() Element of bool [: the carrier of X,COMPLEX:] : b₁ | the carrier of X is bounded } is set
ComplexBoundedFunctionsNorm the carrier of X is non empty Relation-like ComplexBoundedFunctions the carrier of X -defined REAL -valued Function-like total quasi_total V139() V140() V141() Element of bool [:(ComplexBoundedFunctions the carrier of X),REAL:]
[:(ComplexBoundedFunctions the carrier of X),REAL:] is non empty V139() V140() V141() set
bool [:(ComplexBoundedFunctions the carrier of X),REAL:] is non empty set
(ComplexBoundedFunctionsNorm the carrier of X) | (X) is Relation-like ComplexBoundedFunctions the carrier of X -defined REAL -valued Function-like V139() V140() V141() Element of bool [:(ComplexBoundedFunctions the carrier of X),REAL:]
Normed_Complex_AlgebraStr(# (X),(mult_ ((X),(CAlgebra the carrier of X))),(Add_ ((X),(CAlgebra the carrier of X))),(Mult_ ((X),(CAlgebra the carrier of X))),(One_ ((X),(CAlgebra the carrier of X))),(Zero_ ((X),(CAlgebra the carrier of X))),(X) #) is strict Normed_Complex_AlgebraStr
X is non empty TopSpace-like V300() compact TopStruct
(X) is non empty strict Normed_Complex_AlgebraStr
(X) is non empty multiplicatively-closed Cadditively-linearly-closed Element of bool the carrier of (CAlgebra the carrier of X)
the carrier of X is non empty set
CAlgebra the carrier of X is non empty right_complementable Abelian add-associative right_zeroed unital associative commutative right-distributive right_unital well-unital left_unital vector-distributive scalar-distributive scalar-associative scalar-unital strict vector-associative ComplexAlgebraStr
Funcs ( the carrier of X,COMPLEX) is non empty functional FUNCTION_DOMAIN of the carrier of X, COMPLEX
ComplexFuncMult the carrier of X is non empty Relation-like [:(Funcs ( the carrier of X,COMPLEX)),(Funcs ( the carrier of X,COMPLEX)):] -defined Funcs ( the carrier of X,COMPLEX) -valued Function-like total quasi_total Element of bool [:[:(Funcs ( the carrier of X,COMPLEX)),(Funcs ( the carrier of X,COMPLEX)):],(Funcs ( the carrier of X,COMPLEX)):]
[:(Funcs ( the carrier of X,COMPLEX)),(Funcs ( the carrier of X,COMPLEX)):] is non empty set
[:[:(Funcs ( the carrier of X,COMPLEX)),(Funcs ( the carrier of X,COMPLEX)):],(Funcs ( the carrier of X,COMPLEX)):] is non empty set
bool [:[:(Funcs ( the carrier of X,COMPLEX)),(Funcs ( the carrier of X,COMPLEX)):],(Funcs ( the carrier of X,COMPLEX)):] is non empty set
ComplexFuncAdd the carrier of X is non empty Relation-like [:(Funcs ( the carrier of X,COMPLEX)),(Funcs ( the carrier of X,COMPLEX)):] -defined Funcs ( the carrier of X,COMPLEX) -valued Function-like total quasi_total Element of bool [:[:(Funcs ( the carrier of X,COMPLEX)),(Funcs ( the carrier of X,COMPLEX)):],(Funcs ( the carrier of X,COMPLEX)):]
ComplexFuncExtMult the carrier of X is non empty Relation-like [:COMPLEX,(Funcs ( the carrier of X,COMPLEX)):] -defined Funcs ( the carrier of X,COMPLEX) -valued Function-like total quasi_total Element of bool [:[:COMPLEX,(Funcs ( the carrier of X,COMPLEX)):],(Funcs ( the carrier of X,COMPLEX)):]
[:COMPLEX,(Funcs ( the carrier of X,COMPLEX)):] is non empty set
[:[:COMPLEX,(Funcs ( the carrier of X,COMPLEX)):],(Funcs ( the carrier of X,COMPLEX)):] is non empty set
bool [:[:COMPLEX,(Funcs ( the carrier of X,COMPLEX)):],(Funcs ( the carrier of X,COMPLEX)):] is non empty set
ComplexFuncUnit the carrier of X is Relation-like the carrier of X -defined COMPLEX -valued Function-like total quasi_total V139() Element of Funcs ( the carrier of X,COMPLEX)
the carrier of X --> 1r is non empty Relation-like the carrier of X -defined COMPLEX -valued Function-like total quasi_total V139() Element of bool [: the carrier of X,COMPLEX:]
[: the carrier of X,COMPLEX:] is non empty V139() set
bool [: the carrier of X,COMPLEX:] is non empty set
ComplexFuncZero the carrier of X is Relation-like the carrier of X -defined COMPLEX -valued Function-like total quasi_total V139() Element of Funcs ( the carrier of X,COMPLEX)
the carrier of X --> 0 is non empty Relation-like the carrier of X -defined REAL -valued RAT -valued INT -valued Function-like total quasi_total V139() V140() V141() V142() Element of bool [: the carrier of X,REAL:]
[: the carrier of X,REAL:] is non empty V139() V140() V141() set
bool [: the carrier of X,REAL:] is non empty set
ComplexAlgebraStr(# (Funcs ( the carrier of X,COMPLEX)),(ComplexFuncMult the carrier of X),(ComplexFuncAdd the carrier of X),(ComplexFuncExtMult the carrier of X),(ComplexFuncUnit the carrier of X),(ComplexFuncZero the carrier of X) #) is strict ComplexAlgebraStr
the carrier of (CAlgebra the carrier of X) is non empty set
bool the carrier of (CAlgebra the carrier of X) is non empty set
{ b₁ where b₁ is non empty Relation-like the carrier of X -defined COMPLEX -valued Function-like total quasi_total V139() (X) Element of bool [: the carrier of X,COMPLEX:] : verum } is set
mult_ ((X),(CAlgebra the carrier of X)) is non empty Relation-like [:(X),(X):] -defined (X) -valued Function-like total quasi_total Element of bool [:[:(X),(X):],(X):]
[:(X),(X):] is non empty set
[:[:(X),(X):],(X):] is non empty set
bool [:[:(X),(X):],(X):] is non empty set
Add_ ((X),(CAlgebra the carrier of X)) is non empty Relation-like [:(X),(X):] -defined (X) -valued Function-like total quasi_total Element of bool [:[:(X),(X):],(X):]
Mult_ ((X),(CAlgebra the carrier of X)) is non empty Relation-like [:COMPLEX,(X):] -defined (X) -valued Function-like total quasi_total Element of bool [:[:COMPLEX,(X):],(X):]
[:COMPLEX,(X):] is non empty set
[:[:COMPLEX,(X):],(X):] is non empty set
bool [:[:COMPLEX,(X):],(X):] is non empty set
One_ ((X),(CAlgebra the carrier of X)) is Element of (X)
Zero_ ((X),(CAlgebra the carrier of X)) is Element of (X)
(X) is non empty Relation-like (X) -defined REAL -valued Function-like total quasi_total V139() V140() V141() Element of bool [:(X),REAL:]
[:(X),REAL:] is non empty V139() V140() V141() set
bool [:(X),REAL:] is non empty set
ComplexBoundedFunctions the carrier of X is non empty multiplicatively-closed Cadditively-linearly-closed Element of bool the carrier of (CAlgebra the carrier of X)
{ b₁ where b₁ is non empty Relation-like the carrier of X -defined COMPLEX -valued Function-like total quasi_total V139() Element of bool [: the carrier of X,COMPLEX:] : b₁ | the carrier of X is bounded } is set
ComplexBoundedFunctionsNorm the carrier of X is non empty Relation-like ComplexBoundedFunctions the carrier of X -defined REAL -valued Function-like total quasi_total V139() V140() V141() Element of bool [:(ComplexBoundedFunctions the carrier of X),REAL:]
[:(ComplexBoundedFunctions the carrier of X),REAL:] is non empty V139() V140() V141() set
bool [:(ComplexBoundedFunctions the carrier of X),REAL:] is non empty set
(ComplexBoundedFunctionsNorm the carrier of X) | (X) is Relation-like ComplexBoundedFunctions the carrier of X -defined REAL -valued Function-like V139() V140() V141() Element of bool [:(ComplexBoundedFunctions the carrier of X),REAL:]
Normed_Complex_AlgebraStr(# (X),(mult_ ((X),(CAlgebra the carrier of X))),(Add_ ((X),(CAlgebra the carrier of X))),(Mult_ ((X),(CAlgebra the carrier of X))),(One_ ((X),(CAlgebra the carrier of X))),(Zero_ ((X),(CAlgebra the carrier of X))),(X) #) is strict Normed_Complex_AlgebraStr
the carrier of (X) is non empty set
G is Element of the carrier of (X)
F is Element of the carrier of (X)
1_ (CAlgebra the carrier of X) is Element of the carrier of (CAlgebra the carrier of X)
1. (CAlgebra the carrier of X) is Element of the carrier of (CAlgebra the carrier of X)
the OneF of (CAlgebra the carrier of X) is Element of the carrier of (CAlgebra the carrier of X)
(X,1) is non empty Relation-like the carrier of X -defined COMPLEX -valued Function-like total quasi_total V139() (X) Element of bool [: the carrier of X,COMPLEX:]
the carrier of X --> 1 is non empty Relation-like the carrier of X -defined RAT -valued INT -valued {1} -valued Function-like total quasi_total V139() V140() V141() V142() Element of bool [: the carrier of X,{1}:]
{1} is non empty V176() V177() V178() V179() V180() V181() left_end bounded_below set
[: the carrier of X,{1}:] is non empty RAT -valued INT -valued V139() V140() V141() V142() set
bool [: the carrier of X,{1}:] is non empty set
(X) is non empty right_complementable Abelian add-associative right_zeroed unital associative commutative right-distributive right_unital well-unital left_unital vector-distributive scalar-distributive scalar-associative scalar-unital strict vector-associative ComplexAlgebraStr
ComplexAlgebraStr(# (X),(mult_ ((X),(CAlgebra the carrier of X))),(Add_ ((X),(CAlgebra the carrier of X))),(Mult_ ((X),(CAlgebra the carrier of X))),(One_ ((X),(CAlgebra the carrier of X))),(Zero_ ((X),(CAlgebra the carrier of X))) #) is strict ComplexAlgebraStr
1_ (X) is Element of the carrier of (X)
the carrier of (X) is non empty set
1. (X) is Element of the carrier of (X)
the OneF of (X) is Element of the carrier of (X)
GB is Element of (X)
[GB,(1_ (CAlgebra the carrier of X))] is Element of [:(X), the carrier of (CAlgebra the carrier of X):]
[:(X), the carrier of (CAlgebra the carrier of X):] is non empty set
{GB,(1_ (CAlgebra the carrier of X))} is non empty set
{GB} is non empty set
{{GB,(1_ (CAlgebra the carrier of X))},{GB}} is non empty set
[(1_ (CAlgebra the carrier of X)),GB] is Element of [: the carrier of (CAlgebra the carrier of X),(X):]
[: the carrier of (CAlgebra the carrier of X),(X):] is non empty set
{(1_ (CAlgebra the carrier of X)),GB} is non empty set
{(1_ (CAlgebra the carrier of X))} is non empty set
{{(1_ (CAlgebra the carrier of X)),GB},{(1_ (CAlgebra the carrier of X))}} is non empty set
F * G is Element of the carrier of (X)
the multF of (X) is non empty Relation-like [: the carrier of (X), the carrier of (X):] -defined the carrier of (X) -valued Function-like total quasi_total Element of bool [:[: the carrier of (X), the carrier of (X):], the carrier of (X):]
[: the carrier of (X), the carrier of (X):] is non empty set
[:[: the carrier of (X), the carrier of (X):], the carrier of (X):] is non empty set
bool [:[: the carrier of (X), the carrier of (X):], the carrier of (X):] is non empty set
the multF of (X) . (F,G) is Element of the carrier of (X)
[F,G] is set
{F,G} is non empty set
{F} is non empty set
{{F,G},{F}} is non empty set
the multF of (X) . [F,G] is set
(mult_ ((X),(CAlgebra the carrier of X))) . (GB,(1_ (CAlgebra the carrier of X))) is set
[GB,(1_ (CAlgebra the carrier of X))] is set
(mult_ ((X),(CAlgebra the carrier of X))) . [GB,(1_ (CAlgebra the carrier of X))] is set
the multF of (CAlgebra the carrier of X) is non empty Relation-like [: the carrier of (CAlgebra the carrier of X), the carrier of (CAlgebra the carrier of X):] -defined the carrier of (CAlgebra the carrier of X) -valued Function-like total quasi_total Element of bool [:[: the carrier of (CAlgebra the carrier of X), the carrier of (CAlgebra the carrier of X):], the carrier of (CAlgebra the carrier of X):]
[: the carrier of (CAlgebra the carrier of X), the carrier of (CAlgebra the carrier of X):] is non empty set
[:[: the carrier of (CAlgebra the carrier of X), the carrier of (CAlgebra the carrier of X):], the carrier of (CAlgebra the carrier of X):] is non empty set
bool [:[: the carrier of (CAlgebra the carrier of X), the carrier of (CAlgebra the carrier of X):], the carrier of (CAlgebra the carrier of X):] is non empty set
the multF of (CAlgebra the carrier of X) || (X) is Relation-like Function-like set
the multF of (CAlgebra the carrier of X) | [:(X),(X):] is Relation-like set
( the multF of (CAlgebra the carrier of X) || (X)) . (GB,(1_ (CAlgebra the carrier of X))) is set
( the multF of (CAlgebra the carrier of X) || (X)) . [GB,(1_ (CAlgebra the carrier of X))] is set
GB * (1_ (CAlgebra the carrier of X)) is Element of the carrier of (CAlgebra the carrier of X)
the multF of (CAlgebra the carrier of X) . (GB,(1_ (CAlgebra the carrier of X))) is Element of the carrier of (CAlgebra the carrier of X)
the multF of (CAlgebra the carrier of X) . [GB,(1_ (CAlgebra the carrier of X))] is set
G * F is Element of the carrier of (X)
the multF of (X) . (G,F) is Element of the carrier of (X)
[G,F] is set
{G,F} is non empty set
{G} is non empty set
{{G,F},{G}} is non empty set
the multF of (X) . [G,F] is set
(mult_ ((X),(CAlgebra the carrier of X))) . ((1_ (CAlgebra the carrier of X)),GB) is set
[(1_ (CAlgebra the carrier of X)),GB] is set
(mult_ ((X),(CAlgebra the carrier of X))) . [(1_ (CAlgebra the carrier of X)),GB] is set
( the multF of (CAlgebra the carrier of X) || (X)) . ((1_ (CAlgebra the carrier of X)),GB) is set
( the multF of (CAlgebra the carrier of X) || (X)) . [(1_ (CAlgebra the carrier of X)),GB] is set
(1_ (CAlgebra the carrier of X)) * GB is Element of the carrier of (CAlgebra the carrier of X)
the multF of (CAlgebra the carrier of X) . ((1_ (CAlgebra the carrier of X)),GB) is Element of the carrier of (CAlgebra the carrier of X)
the multF of (CAlgebra the carrier of X) . [(1_ (CAlgebra the carrier of X)),GB] is set
X is non empty TopSpace-like V300() compact TopStruct
(X) is non empty strict Normed_Complex_AlgebraStr
(X) is non empty multiplicatively-closed Cadditively-linearly-closed Element of bool the carrier of (CAlgebra the carrier of X)
the carrier of X is non empty set
CAlgebra the carrier of X is non empty right_complementable Abelian add-associative right_zeroed unital associative commutative right-distributive right_unital well-unital left_unital vector-distributive scalar-distributive scalar-associative scalar-unital strict vector-associative ComplexAlgebraStr
Funcs ( the carrier of X,COMPLEX) is non empty functional FUNCTION_DOMAIN of the carrier of X, COMPLEX
ComplexFuncMult the carrier of X is non empty Relation-like [:(Funcs ( the carrier of X,COMPLEX)),(Funcs ( the carrier of X,COMPLEX)):] -defined Funcs ( the carrier of X,COMPLEX) -valued Function-like total quasi_total Element of bool [:[:(Funcs ( the carrier of X,COMPLEX)),(Funcs ( the carrier of X,COMPLEX)):],(Funcs ( the carrier of X,COMPLEX)):]
[:(Funcs ( the carrier of X,COMPLEX)),(Funcs ( the carrier of X,COMPLEX)):] is non empty set
[:[:(Funcs ( the carrier of X,COMPLEX)),(Funcs ( the carrier of X,COMPLEX)):],(Funcs ( the carrier of X,COMPLEX)):] is non empty set
bool [:[:(Funcs ( the carrier of X,COMPLEX)),(Funcs ( the carrier of X,COMPLEX)):],(Funcs ( the carrier of X,COMPLEX)):] is non empty set
ComplexFuncAdd the carrier of X is non empty Relation-like [:(Funcs ( the carrier of X,COMPLEX)),(Funcs ( the carrier of X,COMPLEX)):] -defined Funcs ( the carrier of X,COMPLEX) -valued Function-like total quasi_total Element of bool [:[:(Funcs ( the carrier of X,COMPLEX)),(Funcs ( the carrier of X,COMPLEX)):],(Funcs ( the carrier of X,COMPLEX)):]
ComplexFuncExtMult the carrier of X is non empty Relation-like [:COMPLEX,(Funcs ( the carrier of X,COMPLEX)):] -defined Funcs ( the carrier of X,COMPLEX) -valued Function-like total quasi_total Element of bool [:[:COMPLEX,(Funcs ( the carrier of X,COMPLEX)):],(Funcs ( the carrier of X,COMPLEX)):]
[:COMPLEX,(Funcs ( the carrier of X,COMPLEX)):] is non empty set
[:[:COMPLEX,(Funcs ( the carrier of X,COMPLEX)):],(Funcs ( the carrier of X,COMPLEX)):] is non empty set
bool [:[:COMPLEX,(Funcs ( the carrier of X,COMPLEX)):],(Funcs ( the carrier of X,COMPLEX)):] is non empty set
ComplexFuncUnit the carrier of X is Relation-like the carrier of X -defined COMPLEX -valued Function-like total quasi_total V139() Element of Funcs ( the carrier of X,COMPLEX)
the carrier of X --> 1r is non empty Relation-like the carrier of X -defined COMPLEX -valued Function-like total quasi_total V139() Element of bool [: the carrier of X,COMPLEX:]
[: the carrier of X,COMPLEX:] is non empty V139() set
bool [: the carrier of X,COMPLEX:] is non empty set
ComplexFuncZero the carrier of X is Relation-like the carrier of X -defined COMPLEX -valued Function-like total quasi_total V139() Element of Funcs ( the carrier of X,COMPLEX)
the carrier of X --> 0 is non empty Relation-like the carrier of X -defined REAL -valued RAT -valued INT -valued Function-like total quasi_total V139() V140() V141() V142() Element of bool [: the carrier of X,REAL:]
[: the carrier of X,REAL:] is non empty V139() V140() V141() set
bool [: the carrier of X,REAL:] is non empty set
ComplexAlgebraStr(# (Funcs ( the carrier of X,COMPLEX)),(ComplexFuncMult the carrier of X),(ComplexFuncAdd the carrier of X),(ComplexFuncExtMult the carrier of X),(ComplexFuncUnit the carrier of X),(ComplexFuncZero the carrier of X) #) is strict ComplexAlgebraStr
the carrier of (CAlgebra the carrier of X) is non empty set
bool the carrier of (CAlgebra the carrier of X) is non empty set
{ b₁ where b₁ is non empty Relation-like the carrier of X -defined COMPLEX -valued Function-like total quasi_total V139() (X) Element of bool [: the carrier of X,COMPLEX:] : verum } is set
mult_ ((X),(CAlgebra the carrier of X)) is non empty Relation-like [:(X),(X):] -defined (X) -valued Function-like total quasi_total Element of bool [:[:(X),(X):],(X):]
[:(X),(X):] is non empty set
[:[:(X),(X):],(X):] is non empty set
bool [:[:(X),(X):],(X):] is non empty set
Add_ ((X),(CAlgebra the carrier of X)) is non empty Relation-like [:(X),(X):] -defined (X) -valued Function-like total quasi_total Element of bool [:[:(X),(X):],(X):]
Mult_ ((X),(CAlgebra the carrier of X)) is non empty Relation-like [:COMPLEX,(X):] -defined (X) -valued Function-like total quasi_total Element of bool [:[:COMPLEX,(X):],(X):]
[:COMPLEX,(X):] is non empty set
[:[:COMPLEX,(X):],(X):] is non empty set
bool [:[:COMPLEX,(X):],(X):] is non empty set
One_ ((X),(CAlgebra the carrier of X)) is Element of (X)
Zero_ ((X),(CAlgebra the carrier of X)) is Element of (X)
(X) is non empty Relation-like (X) -defined REAL -valued Function-like total quasi_total V139() V140() V141() Element of bool [:(X),REAL:]
[:(X),REAL:] is non empty V139() V140() V141() set
bool [:(X),REAL:] is non empty set
ComplexBoundedFunctions the carrier of X is non empty multiplicatively-closed Cadditively-linearly-closed Element of bool the carrier of (CAlgebra the carrier of X)
{ b₁ where b₁ is non empty Relation-like the carrier of X -defined COMPLEX -valued Function-like total quasi_total V139() Element of bool [: the carrier of X,COMPLEX:] : b₁ | the carrier of X is bounded } is set
ComplexBoundedFunctionsNorm the carrier of X is non empty Relation-like ComplexBoundedFunctions the carrier of X -defined REAL -valued Function-like total quasi_total V139() V140() V141() Element of bool [:(ComplexBoundedFunctions the carrier of X),REAL:]
[:(ComplexBoundedFunctions the carrier of X),REAL:] is non empty V139() V140() V141() set
bool [:(ComplexBoundedFunctions the carrier of X),REAL:] is non empty set
(ComplexBoundedFunctionsNorm the carrier of X) | (X) is Relation-like ComplexBoundedFunctions the carrier of X -defined REAL -valued Function-like V139() V140() V141() Element of bool [:(ComplexBoundedFunctions the carrier of X),REAL:]
Normed_Complex_AlgebraStr(# (X),(mult_ ((X),(CAlgebra the carrier of X))),(Add_ ((X),(CAlgebra the carrier of X))),(Mult_ ((X),(CAlgebra the carrier of X))),(One_ ((X),(CAlgebra the carrier of X))),(Zero_ ((X),(CAlgebra the carrier of X))),(X) #) is strict Normed_Complex_AlgebraStr
the carrier of (X) is non empty set
X is Element of the carrier of (X)
F is Element of the carrier of (X)
F * X is Element of the carrier of (X)
the multF of (X) is non empty Relation-like [: the carrier of (X), the carrier of (X):] -defined the carrier of (X) -valued Function-like total quasi_total Element of bool [:[: the carrier of (X), the carrier of (X):], the carrier of (X):]
[: the carrier of (X), the carrier of (X):] is non empty set
[:[: the carrier of (X), the carrier of (X):], the carrier of (X):] is non empty set
bool [:[: the carrier of (X), the carrier of (X):], the carrier of (X):] is non empty set
the multF of (X) . (F,X) is Element of the carrier of (X)
[F,X] is set
{F,X} is non empty set
{F} is non empty set
{{F,X},{F}} is non empty set
the multF of (X) . [F,X] is set
G is Element of the carrier of (X)
X * G is Element of the carrier of (X)
the multF of (X) . (X,G) is Element of the carrier of (X)
[X,G] is set
{X,G} is non empty set
{X} is non empty set
{{X,G},{X}} is non empty set
the multF of (X) . [X,G] is set
X is non empty TopSpace-like V300() compact TopStruct
(X) is non empty unital strict Normed_Complex_AlgebraStr
(X) is non empty multiplicatively-closed Cadditively-linearly-closed Element of bool the carrier of (CAlgebra the carrier of X)
the carrier of X is non empty set
CAlgebra the carrier of X is non empty right_complementable Abelian add-associative right_zeroed unital associative commutative right-distributive right_unital well-unital left_unital vector-distributive scalar-distributive scalar-associative scalar-unital strict vector-associative ComplexAlgebraStr
Funcs ( the carrier of X,COMPLEX) is non empty functional FUNCTION_DOMAIN of the carrier of X, COMPLEX
ComplexFuncMult the carrier of X is non empty Relation-like [:(Funcs ( the carrier of X,COMPLEX)),(Funcs ( the carrier of X,COMPLEX)):] -defined Funcs ( the carrier of X,COMPLEX) -valued Function-like total quasi_total Element of bool [:[:(Funcs ( the carrier of X,COMPLEX)),(Funcs ( the carrier of X,COMPLEX)):],(Funcs ( the carrier of X,COMPLEX)):]
[:(Funcs ( the carrier of X,COMPLEX)),(Funcs ( the carrier of X,COMPLEX)):] is non empty set
[:[:(Funcs ( the carrier of X,COMPLEX)),(Funcs ( the carrier of X,COMPLEX)):],(Funcs ( the carrier of X,COMPLEX)):] is non empty set
bool [:[:(Funcs ( the carrier of X,COMPLEX)),(Funcs ( the carrier of X,COMPLEX)):],(Funcs ( the carrier of X,COMPLEX)):] is non empty set
ComplexFuncAdd the carrier of X is non empty Relation-like [:(Funcs ( the carrier of X,COMPLEX)),(Funcs ( the carrier of X,COMPLEX)):] -defined Funcs ( the carrier of X,COMPLEX) -valued Function-like total quasi_total Element of bool [:[:(Funcs ( the carrier of X,COMPLEX)),(Funcs ( the carrier of X,COMPLEX)):],(Funcs ( the carrier of X,COMPLEX)):]
ComplexFuncExtMult the carrier of X is non empty Relation-like [:COMPLEX,(Funcs ( the carrier of X,COMPLEX)):] -defined Funcs ( the carrier of X,COMPLEX) -valued Function-like total quasi_total Element of bool [:[:COMPLEX,(Funcs ( the carrier of X,COMPLEX)):],(Funcs ( the carrier of X,COMPLEX)):]
[:COMPLEX,(Funcs ( the carrier of X,COMPLEX)):] is non empty set
[:[:COMPLEX,(Funcs ( the carrier of X,COMPLEX)):],(Funcs ( the carrier of X,COMPLEX)):] is non empty set
bool [:[:COMPLEX,(Funcs ( the carrier of X,COMPLEX)):],(Funcs ( the carrier of X,COMPLEX)):] is non empty set
ComplexFuncUnit the carrier of X is Relation-like the carrier of X -defined COMPLEX -valued Function-like total quasi_total V139() Element of Funcs ( the carrier of X,COMPLEX)
the carrier of X --> 1r is non empty Relation-like the carrier of X -defined COMPLEX -valued Function-like total quasi_total V139() Element of bool [: the carrier of X,COMPLEX:]
[: the carrier of X,COMPLEX:] is non empty V139() set
bool [: the carrier of X,COMPLEX:] is non empty set
ComplexFuncZero the carrier of X is Relation-like the carrier of X -defined COMPLEX -valued Function-like total quasi_total V139() Element of Funcs ( the carrier of X,COMPLEX)
the carrier of X --> 0 is non empty Relation-like the carrier of X -defined REAL -valued RAT -valued INT -valued Function-like total quasi_total V139() V140() V141() V142() Element of bool [: the carrier of X,REAL:]
[: the carrier of X,REAL:] is non empty V139() V140() V141() set
bool [: the carrier of X,REAL:] is non empty set
ComplexAlgebraStr(# (Funcs ( the carrier of X,COMPLEX)),(ComplexFuncMult the carrier of X),(ComplexFuncAdd the carrier of X),(ComplexFuncExtMult the carrier of X),(ComplexFuncUnit the carrier of X),(ComplexFuncZero the carrier of X) #) is strict ComplexAlgebraStr
the carrier of (CAlgebra the carrier of X) is non empty set
bool the carrier of (CAlgebra the carrier of X) is non empty set
{ b₁ where b₁ is non empty Relation-like the carrier of X -defined COMPLEX -valued Function-like total quasi_total V139() (X) Element of bool [: the carrier of X,COMPLEX:] : verum } is set
mult_ ((X),(CAlgebra the carrier of X)) is non empty Relation-like [:(X),(X):] -defined (X) -valued Function-like total quasi_total Element of bool [:[:(X),(X):],(X):]
[:(X),(X):] is non empty set
[:[:(X),(X):],(X):] is non empty set
bool [:[:(X),(X):],(X):] is non empty set
Add_ ((X),(CAlgebra the carrier of X)) is non empty Relation-like [:(X),(X):] -defined (X) -valued Function-like total quasi_total Element of bool [:[:(X),(X):],(X):]
Mult_ ((X),(CAlgebra the carrier of X)) is non empty Relation-like [:COMPLEX,(X):] -defined (X) -valued Function-like total quasi_total Element of bool [:[:COMPLEX,(X):],(X):]
[:COMPLEX,(X):] is non empty set
[:[:COMPLEX,(X):],(X):] is non empty set
bool [:[:COMPLEX,(X):],(X):] is non empty set
One_ ((X),(CAlgebra the carrier of X)) is Element of (X)
Zero_ ((X),(CAlgebra the carrier of X)) is Element of (X)
(X) is non empty Relation-like (X) -defined REAL -valued Function-like total quasi_total V139() V140() V141() Element of bool [:(X),REAL:]
[:(X),REAL:] is non empty V139() V140() V141() set
bool [:(X),REAL:] is non empty set
ComplexBoundedFunctions the carrier of X is non empty multiplicatively-closed Cadditively-linearly-closed Element of bool the carrier of (CAlgebra the carrier of X)
{ b₁ where b₁ is non empty Relation-like the carrier of X -defined COMPLEX -valued Function-like total quasi_total V139() Element of bool [: the carrier of X,COMPLEX:] : b₁ | the carrier of X is bounded } is set
ComplexBoundedFunctionsNorm the carrier of X is non empty Relation-like ComplexBoundedFunctions the carrier of X -defined REAL -valued Function-like total quasi_total V139() V140() V141() Element of bool [:(ComplexBoundedFunctions the carrier of X),REAL:]
[:(ComplexBoundedFunctions the carrier of X),REAL:] is non empty V139() V140() V141() set
bool [:(ComplexBoundedFunctions the carrier of X),REAL:] is non empty set
(ComplexBoundedFunctionsNorm the carrier of X) | (X) is Relation-like ComplexBoundedFunctions the carrier of X -defined REAL -valued Function-like V139() V140() V141() Element of bool [:(ComplexBoundedFunctions the carrier of X),REAL:]
Normed_Complex_AlgebraStr(# (X),(mult_ ((X),(CAlgebra the carrier of X))),(Add_ ((X),(CAlgebra the carrier of X))),(Mult_ ((X),(CAlgebra the carrier of X))),(One_ ((X),(CAlgebra the carrier of X))),(Zero_ ((X),(CAlgebra the carrier of X))),(X) #) is strict Normed_Complex_AlgebraStr
the carrier of (X) is non empty set
C_Normed_Algebra_of_BoundedFunctions the carrier of X is non empty right_complementable Abelian add-associative right_zeroed unital discerning reflexive vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like complete Normed_Complex_AlgebraStr
mult_ ((ComplexBoundedFunctions the carrier of X),(CAlgebra the carrier of X)) is non empty Relation-like [:(ComplexBoundedFunctions the carrier of X),(ComplexBoundedFunctions the carrier of X):] -defined ComplexBoundedFunctions the carrier of X -valued Function-like total quasi_total Element of bool [:[:(ComplexBoundedFunctions the carrier of X),(ComplexBoundedFunctions the carrier of X):],(ComplexBoundedFunctions the carrier of X):]
[:(ComplexBoundedFunctions the carrier of X),(ComplexBoundedFunctions the carrier of X):] is non empty set
[:[:(ComplexBoundedFunctions the carrier of X),(ComplexBoundedFunctions the carrier of X):],(ComplexBoundedFunctions the carrier of X):] is non empty set
bool [:[:(ComplexBoundedFunctions the carrier of X),(ComplexBoundedFunctions the carrier of X):],(ComplexBoundedFunctions the carrier of X):] is non empty set
Add_ ((ComplexBoundedFunctions the carrier of X),(CAlgebra the carrier of X)) is non empty Relation-like [:(ComplexBoundedFunctions the carrier of X),(ComplexBoundedFunctions the carrier of X):] -defined ComplexBoundedFunctions the carrier of X -valued Function-like total quasi_total Element of bool [:[:(ComplexBoundedFunctions the carrier of X),(ComplexBoundedFunctions the carrier of X):],(ComplexBoundedFunctions the carrier of X):]
Mult_ ((ComplexBoundedFunctions the carrier of X),(CAlgebra the carrier of X)) is non empty Relation-like [:COMPLEX,(ComplexBoundedFunctions the carrier of X):] -defined ComplexBoundedFunctions the carrier of X -valued Function-like total quasi_total Element of bool [:[:COMPLEX,(ComplexBoundedFunctions the carrier of X):],(ComplexBoundedFunctions the carrier of X):]
[:COMPLEX,(ComplexBoundedFunctions the carrier of X):] is non empty set
[:[:COMPLEX,(ComplexBoundedFunctions the carrier of X):],(ComplexBoundedFunctions the carrier of X):] is non empty set
bool [:[:COMPLEX,(ComplexBoundedFunctions the carrier of X):],(ComplexBoundedFunctions the carrier of X):] is non empty set
One_ ((ComplexBoundedFunctions the carrier of X),(CAlgebra the carrier of X)) is Element of ComplexBoundedFunctions the carrier of X
Zero_ ((ComplexBoundedFunctions the carrier of X),(CAlgebra the carrier of X)) is Element of ComplexBoundedFunctions the carrier of X
Normed_Complex_AlgebraStr(# (ComplexBoundedFunctions the carrier of X),(mult_ ((ComplexBoundedFunctions the carrier of X),(CAlgebra the carrier of X))),(Add_ ((ComplexBoundedFunctions the carrier of X),(CAlgebra the carrier of X))),(Mult_ ((ComplexBoundedFunctions the carrier of X),(CAlgebra the carrier of X))),(One_ ((ComplexBoundedFunctions the carrier of X),(CAlgebra the carrier of X))),(Zero_ ((ComplexBoundedFunctions the carrier of X),(CAlgebra the carrier of X))),(ComplexBoundedFunctionsNorm the carrier of X) #) is strict Normed_Complex_AlgebraStr
the carrier of (C_Normed_Algebra_of_BoundedFunctions the carrier of X) is non empty set
X is Element of the carrier of (X)
F is Element of the carrier of (C_Normed_Algebra_of_BoundedFunctions the carrier of X)
||.X.|| is complex real ext-real Element of REAL
the U9 of (X) is non empty Relation-like the carrier of (X) -defined REAL -valued Function-like total quasi_total V139() V140() V141() Element of bool [: the carrier of (X),REAL:]
[: the carrier of (X),REAL:] is non empty V139() V140() V141() set
bool [: the carrier of (X),REAL:] is non empty set
the U9 of (X) . X is complex real ext-real Element of REAL
||.F.|| is complex real ext-real Element of REAL
the U9 of (C_Normed_Algebra_of_BoundedFunctions the carrier of X) is non empty Relation-like the carrier of (C_Normed_Algebra_of_BoundedFunctions the carrier of X) -defined REAL -valued Function-like total quasi_total V139() V140() V141() Element of bool [: the carrier of (C_Normed_Algebra_of_BoundedFunctions the carrier of X),REAL:]
[: the carrier of (C_Normed_Algebra_of_BoundedFunctions the carrier of X),REAL:] is non empty V139() V140() V141() set
bool [: the carrier of (C_Normed_Algebra_of_BoundedFunctions the carrier of X),REAL:] is non empty set
the U9 of (C_Normed_Algebra_of_BoundedFunctions the carrier of X) . F is complex real ext-real Element of REAL
X is non empty TopSpace-like V300() compact TopStruct
(X) is non empty unital strict Normed_Complex_AlgebraStr
(X) is non empty multiplicatively-closed Cadditively-linearly-closed Element of bool the carrier of (CAlgebra the carrier of X)
the carrier of X is non empty set
CAlgebra the carrier of X is non empty right_complementable Abelian add-associative right_zeroed unital associative commutative right-distributive right_unital well-unital left_unital vector-distributive scalar-distributive scalar-associative scalar-unital strict vector-associative ComplexAlgebraStr
Funcs ( the carrier of X,COMPLEX) is non empty functional FUNCTION_DOMAIN of the carrier of X, COMPLEX
ComplexFuncMult the carrier of X is non empty Relation-like [:(Funcs ( the carrier of X,COMPLEX)),(Funcs ( the carrier of X,COMPLEX)):] -defined Funcs ( the carrier of X,COMPLEX) -valued Function-like total quasi_total Element of bool [:[:(Funcs ( the carrier of X,COMPLEX)),(Funcs ( the carrier of X,COMPLEX)):],(Funcs ( the carrier of X,COMPLEX)):]
[:(Funcs ( the carrier of X,COMPLEX)),(Funcs ( the carrier of X,COMPLEX)):] is non empty set
[:[:(Funcs ( the carrier of X,COMPLEX)),(Funcs ( the carrier of X,COMPLEX)):],(Funcs ( the carrier of X,COMPLEX)):] is non empty set
bool [:[:(Funcs ( the carrier of X,COMPLEX)),(Funcs ( the carrier of X,COMPLEX)):],(Funcs ( the carrier of X,COMPLEX)):] is non empty set
ComplexFuncAdd the carrier of X is non empty Relation-like [:(Funcs ( the carrier of X,COMPLEX)),(Funcs ( the carrier of X,COMPLEX)):] -defined Funcs ( the carrier of X,COMPLEX) -valued Function-like total quasi_total Element of bool [:[:(Funcs ( the carrier of X,COMPLEX)),(Funcs ( the carrier of X,COMPLEX)):],(Funcs ( the carrier of X,COMPLEX)):]
ComplexFuncExtMult the carrier of X is non empty Relation-like [:COMPLEX,(Funcs ( the carrier of X,COMPLEX)):] -defined Funcs ( the carrier of X,COMPLEX) -valued Function-like total quasi_total Element of bool [:[:COMPLEX,(Funcs ( the carrier of X,COMPLEX)):],(Funcs ( the carrier of X,COMPLEX)):]
[:COMPLEX,(Funcs ( the carrier of X,COMPLEX)):] is non empty set
[:[:COMPLEX,(Funcs ( the carrier of X,COMPLEX)):],(Funcs ( the carrier of X,COMPLEX)):] is non empty set
bool [:[:COMPLEX,(Funcs ( the carrier of X,COMPLEX)):],(Funcs ( the carrier of X,COMPLEX)):] is non empty set
ComplexFuncUnit the carrier of X is Relation-like the carrier of X -defined COMPLEX -valued Function-like total quasi_total V139() Element of Funcs ( the carrier of X,COMPLEX)
the carrier of X --> 1r is non empty Relation-like the carrier of X -defined COMPLEX -valued Function-like total quasi_total V139() Element of bool [: the carrier of X,COMPLEX:]
[: the carrier of X,COMPLEX:] is non empty V139() set
bool [: the carrier of X,COMPLEX:] is non empty set
ComplexFuncZero the carrier of X is Relation-like the carrier of X -defined COMPLEX -valued Function-like total quasi_total V139() Element of Funcs ( the carrier of X,COMPLEX)
the carrier of X --> 0 is non empty Relation-like the carrier of X -defined REAL -valued RAT -valued INT -valued Function-like total quasi_total V139() V140() V141() V142() Element of bool [: the carrier of X,REAL:]
[: the carrier of X,REAL:] is non empty V139() V140() V141() set
bool [: the carrier of X,REAL:] is non empty set
ComplexAlgebraStr(# (Funcs ( the carrier of X,COMPLEX)),(ComplexFuncMult the carrier of X),(ComplexFuncAdd the carrier of X),(ComplexFuncExtMult the carrier of X),(ComplexFuncUnit the carrier of X),(ComplexFuncZero the carrier of X) #) is strict ComplexAlgebraStr
the carrier of (CAlgebra the carrier of X) is non empty set
bool the carrier of (CAlgebra the carrier of X) is non empty set
{ b₁ where b₁ is non empty Relation-like the carrier of X -defined COMPLEX -valued Function-like total quasi_total V139() (X) Element of bool [: the carrier of X,COMPLEX:] : verum } is set
mult_ ((X),(CAlgebra the carrier of X)) is non empty Relation-like [:(X),(X):] -defined (X) -valued Function-like total quasi_total Element of bool [:[:(X),(X):],(X):]
[:(X),(X):] is non empty set
[:[:(X),(X):],(X):] is non empty set
bool [:[:(X),(X):],(X):] is non empty set
Add_ ((X),(CAlgebra the carrier of X)) is non empty Relation-like [:(X),(X):] -defined (X) -valued Function-like total quasi_total Element of bool [:[:(X),(X):],(X):]
Mult_ ((X),(CAlgebra the carrier of X)) is non empty Relation-like [:COMPLEX,(X):] -defined (X) -valued Function-like total quasi_total Element of bool [:[:COMPLEX,(X):],(X):]
[:COMPLEX,(X):] is non empty set
[:[:COMPLEX,(X):],(X):] is non empty set
bool [:[:COMPLEX,(X):],(X):] is non empty set
One_ ((X),(CAlgebra the carrier of X)) is Element of (X)
Zero_ ((X),(CAlgebra the carrier of X)) is Element of (X)
(X) is non empty Relation-like (X) -defined REAL -valued Function-like total quasi_total V139() V140() V141() Element of bool [:(X),REAL:]
[:(X),REAL:] is non empty V139() V140() V141() set
bool [:(X),REAL:] is non empty set
ComplexBoundedFunctions the carrier of X is non empty multiplicatively-closed Cadditively-linearly-closed Element of bool the carrier of (CAlgebra the carrier of X)
{ b₁ where b₁ is non empty Relation-like the carrier of X -defined COMPLEX -valued Function-like total quasi_total V139() Element of bool [: the carrier of X,COMPLEX:] : b₁ | the carrier of X is bounded } is set
ComplexBoundedFunctionsNorm the carrier of X is non empty Relation-like ComplexBoundedFunctions the carrier of X -defined REAL -valued Function-like total quasi_total V139() V140() V141() Element of bool [:(ComplexBoundedFunctions the carrier of X),REAL:]
[:(ComplexBoundedFunctions the carrier of X),REAL:] is non empty V139() V140() V141() set
bool [:(ComplexBoundedFunctions the carrier of X),REAL:] is non empty set
(ComplexBoundedFunctionsNorm the carrier of X) | (X) is Relation-like ComplexBoundedFunctions the carrier of X -defined REAL -valued Function-like V139() V140() V141() Element of bool [:(ComplexBoundedFunctions the carrier of X),REAL:]
Normed_Complex_AlgebraStr(# (X),(mult_ ((X),(CAlgebra the carrier of X))),(Add_ ((X),(CAlgebra the carrier of X))),(Mult_ ((X),(CAlgebra the carrier of X))),(One_ ((X),(CAlgebra the carrier of X))),(Zero_ ((X),(CAlgebra the carrier of X))),(X) #) is strict Normed_Complex_AlgebraStr
the carrier of (X) is non empty set
C_Normed_Algebra_of_BoundedFunctions the carrier of X is non empty right_complementable Abelian add-associative right_zeroed unital discerning reflexive vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like complete Normed_Complex_AlgebraStr
mult_ ((ComplexBoundedFunctions the carrier of X),(CAlgebra the carrier of X)) is non empty Relation-like [:(ComplexBoundedFunctions the carrier of X),(ComplexBoundedFunctions the carrier of X):] -defined ComplexBoundedFunctions the carrier of X -valued Function-like total quasi_total Element of bool [:[:(ComplexBoundedFunctions the carrier of X),(ComplexBoundedFunctions the carrier of X):],(ComplexBoundedFunctions the carrier of X):]
[:(ComplexBoundedFunctions the carrier of X),(ComplexBoundedFunctions the carrier of X):] is non empty set
[:[:(ComplexBoundedFunctions the carrier of X),(ComplexBoundedFunctions the carrier of X):],(ComplexBoundedFunctions the carrier of X):] is non empty set
bool [:[:(ComplexBoundedFunctions the carrier of X),(ComplexBoundedFunctions the carrier of X):],(ComplexBoundedFunctions the carrier of X):] is non empty set
Add_ ((ComplexBoundedFunctions the carrier of X),(CAlgebra the carrier of X)) is non empty Relation-like [:(ComplexBoundedFunctions the carrier of X),(ComplexBoundedFunctions the carrier of X):] -defined ComplexBoundedFunctions the carrier of X -valued Function-like total quasi_total Element of bool [:[:(ComplexBoundedFunctions the carrier of X),(ComplexBoundedFunctions the carrier of X):],(ComplexBoundedFunctions the carrier of X):]
Mult_ ((ComplexBoundedFunctions the carrier of X),(CAlgebra the carrier of X)) is non empty Relation-like [:COMPLEX,(ComplexBoundedFunctions the carrier of X):] -defined ComplexBoundedFunctions the carrier of X -valued Function-like total quasi_total Element of bool [:[:COMPLEX,(ComplexBoundedFunctions the carrier of X):],(ComplexBoundedFunctions the carrier of X):]
[:COMPLEX,(ComplexBoundedFunctions the carrier of X):] is non empty set
[:[:COMPLEX,(ComplexBoundedFunctions the carrier of X):],(ComplexBoundedFunctions the carrier of X):] is non empty set
bool [:[:COMPLEX,(ComplexBoundedFunctions the carrier of X):],(ComplexBoundedFunctions the carrier of X):] is non empty set
One_ ((ComplexBoundedFunctions the carrier of X),(CAlgebra the carrier of X)) is Element of ComplexBoundedFunctions the carrier of X
Zero_ ((ComplexBoundedFunctions the carrier of X),(CAlgebra the carrier of X)) is Element of ComplexBoundedFunctions the carrier of X
Normed_Complex_AlgebraStr(# (ComplexBoundedFunctions the carrier of X),(mult_ ((ComplexBoundedFunctions the carrier of X),(CAlgebra the carrier of X))),(Add_ ((ComplexBoundedFunctions the carrier of X),(CAlgebra the carrier of X))),(Mult_ ((ComplexBoundedFunctions the carrier of X),(CAlgebra the carrier of X))),(One_ ((ComplexBoundedFunctions the carrier of X),(CAlgebra the carrier of X))),(Zero_ ((ComplexBoundedFunctions the carrier of X),(CAlgebra the carrier of X))),(ComplexBoundedFunctionsNorm the carrier of X) #) is strict Normed_Complex_AlgebraStr
the carrier of (C_Normed_Algebra_of_BoundedFunctions the carrier of X) is non empty set
X is Element of the carrier of (X)
F is Element of the carrier of (X)
X + F is Element of the carrier of (X)
the addF of (X) is non empty Relation-like [: the carrier of (X), the carrier of (X):] -defined the carrier of (X) -valued Function-like total quasi_total Element of bool [:[: the carrier of (X), the carrier of (X):], the carrier of (X):]
[: the carrier of (X), the carrier of (X):] is non empty set
[:[: the carrier of (X), the carrier of (X):], the carrier of (X):] is non empty set
bool [:[: the carrier of (X), the carrier of (X):], the carrier of (X):] is non empty set
the addF of (X) . (X,F) is Element of the carrier of (X)
[X,F] is set
{X,F} is non empty set
{X} is non empty set
{{X,F},{X}} is non empty set
the addF of (X) . [X,F] is set
G is Element of the carrier of (C_Normed_Algebra_of_BoundedFunctions the carrier of X)
FB is Element of the carrier of (C_Normed_Algebra_of_BoundedFunctions the carrier of X)
G + FB is Element of the carrier of (C_Normed_Algebra_of_BoundedFunctions the carrier of X)
the addF of (C_Normed_Algebra_of_BoundedFunctions the carrier of X) is non empty Relation-like [: the carrier of (C_Normed_Algebra_of_BoundedFunctions the carrier of X), the carrier of (C_Normed_Algebra_of_BoundedFunctions the carrier of X):] -defined the carrier of (C_Normed_Algebra_of_BoundedFunctions the carrier of X) -valued Function-like total quasi_total Element of bool [:[: the carrier of (C_Normed_Algebra_of_BoundedFunctions the carrier of X), the carrier of (C_Normed_Algebra_of_BoundedFunctions the carrier of X):], the carrier of (C_Normed_Algebra_of_BoundedFunctions the carrier of X):]
[: the carrier of (C_Normed_Algebra_of_BoundedFunctions the carrier of X), the carrier of (C_Normed_Algebra_of_BoundedFunctions the carrier of X):] is non empty set
[:[: the carrier of (C_Normed_Algebra_of_BoundedFunctions the carrier of X), the carrier of (C_Normed_Algebra_of_BoundedFunctions the carrier of X):], the carrier of (C_Normed_Algebra_of_BoundedFunctions the carrier of X):] is non empty set
bool [:[: the carrier of (C_Normed_Algebra_of_BoundedFunctions the carrier of X), the carrier of (C_Normed_Algebra_of_BoundedFunctions the carrier of X):], the carrier of (C_Normed_Algebra_of_BoundedFunctions the carrier of X):] is non empty set
the addF of (C_Normed_Algebra_of_BoundedFunctions the carrier of X) . (G,FB) is Element of the carrier of (C_Normed_Algebra_of_BoundedFunctions the carrier of X)
[G,FB] is set
{G,FB} is non empty set
{G} is non empty set
{{G,FB},{G}} is non empty set
the addF of (C_Normed_Algebra_of_BoundedFunctions the carrier of X) . [G,FB] is set
the addF of (CAlgebra the carrier of X) is non empty Relation-like [: the carrier of (CAlgebra the carrier of X), the carrier of (CAlgebra the carrier of X):] -defined the carrier of (CAlgebra the carrier of X) -valued Function-like total quasi_total Element of bool [:[: the carrier of (CAlgebra the carrier of X), the carrier of (CAlgebra the carrier of X):], the carrier of (CAlgebra the carrier of X):]
[: the carrier of (CAlgebra the carrier of X), the carrier of (CAlgebra the carrier of X):] is non empty set
[:[: the carrier of (CAlgebra the carrier of X), the carrier of (CAlgebra the carrier of X):], the carrier of (CAlgebra the carrier of X):] is non empty set
bool [:[: the carrier of (CAlgebra the carrier of X), the carrier of (CAlgebra the carrier of X):], the carrier of (CAlgebra the carrier of X):] is non empty set
the addF of (CAlgebra the carrier of X) || (X) is Relation-like Function-like set
the addF of (CAlgebra the carrier of X) | [:(X),(X):] is Relation-like set
[X,F] is Element of [: the carrier of (X), the carrier of (X):]
( the addF of (CAlgebra the carrier of X) || (X)) . [X,F] is set
the addF of (CAlgebra the carrier of X) . [X,F] is set
the addF of (CAlgebra the carrier of X) || (ComplexBoundedFunctions the carrier of X) is Relation-like Function-like set
the addF of (CAlgebra the carrier of X) | [:(ComplexBoundedFunctions the carrier of X),(ComplexBoundedFunctions the carrier of X):] is Relation-like set
[G,FB] is Element of [: the carrier of (C_Normed_Algebra_of_BoundedFunctions the carrier of X), the carrier of (C_Normed_Algebra_of_BoundedFunctions the carrier of X):]
( the addF of (CAlgebra the carrier of X) || (ComplexBoundedFunctions the carrier of X)) . [G,FB] is set
X is non empty TopSpace-like V300() compact TopStruct
(X) is non empty unital strict Normed_Complex_AlgebraStr
(X) is non empty multiplicatively-closed Cadditively-linearly-closed Element of bool the carrier of (CAlgebra the carrier of X)
the carrier of X is non empty set
CAlgebra the carrier of X is non empty right_complementable Abelian add-associative right_zeroed unital associative commutative right-distributive right_unital well-unital left_unital vector-distributive scalar-distributive scalar-associative scalar-unital strict vector-associative ComplexAlgebraStr
Funcs ( the carrier of X,COMPLEX) is non empty functional FUNCTION_DOMAIN of the carrier of X, COMPLEX
ComplexFuncMult the carrier of X is non empty Relation-like [:(Funcs ( the carrier of X,COMPLEX)),(Funcs ( the carrier of X,COMPLEX)):] -defined Funcs ( the carrier of X,COMPLEX) -valued Function-like total quasi_total Element of bool [:[:(Funcs ( the carrier of X,COMPLEX)),(Funcs ( the carrier of X,COMPLEX)):],(Funcs ( the carrier of X,COMPLEX)):]
[:(Funcs ( the carrier of X,COMPLEX)),(Funcs ( the carrier of X,COMPLEX)):] is non empty set
[:[:(Funcs ( the carrier of X,COMPLEX)),(Funcs ( the carrier of X,COMPLEX)):],(Funcs ( the carrier of X,COMPLEX)):] is non empty set
bool [:[:(Funcs ( the carrier of X,COMPLEX)),(Funcs ( the carrier of X,COMPLEX)):],(Funcs ( the carrier of X,COMPLEX)):] is non empty set
ComplexFuncAdd the carrier of X is non empty Relation-like [:(Funcs ( the carrier of X,COMPLEX)),(Funcs ( the carrier of X,COMPLEX)):] -defined Funcs ( the carrier of X,COMPLEX) -valued Function-like total quasi_total Element of bool [:[:(Funcs ( the carrier of X,COMPLEX)),(Funcs ( the carrier of X,COMPLEX)):],(Funcs ( the carrier of X,COMPLEX)):]
ComplexFuncExtMult the carrier of X is non empty Relation-like [:COMPLEX,(Funcs ( the carrier of X,COMPLEX)):] -defined Funcs ( the carrier of X,COMPLEX) -valued Function-like total quasi_total Element of bool [:[:COMPLEX,(Funcs ( the carrier of X,COMPLEX)):],(Funcs ( the carrier of X,COMPLEX)):]
[:COMPLEX,(Funcs ( the carrier of X,COMPLEX)):] is non empty set
[:[:COMPLEX,(Funcs ( the carrier of X,COMPLEX)):],(Funcs ( the carrier of X,COMPLEX)):] is non empty set
bool [:[:COMPLEX,(Funcs ( the carrier of X,COMPLEX)):],(Funcs ( the carrier of X,COMPLEX)):] is non empty set
ComplexFuncUnit the carrier of X is Relation-like the carrier of X -defined COMPLEX -valued Function-like total quasi_total V139() Element of Funcs ( the carrier of X,COMPLEX)
the carrier of X --> 1r is non empty Relation-like the carrier of X -defined COMPLEX -valued Function-like total quasi_total V139() Element of bool [: the carrier of X,COMPLEX:]
[: the carrier of X,COMPLEX:] is non empty V139() set
bool [: the carrier of X,COMPLEX:] is non empty set
ComplexFuncZero the carrier of X is Relation-like the carrier of X -defined COMPLEX -valued Function-like total quasi_total V139() Element of Funcs ( the carrier of X,COMPLEX)
the carrier of X --> 0 is non empty Relation-like the carrier of X -defined REAL -valued RAT -valued INT -valued Function-like total quasi_total V139() V140() V141() V142() Element of bool [: the carrier of X,REAL:]
[: the carrier of X,REAL:] is non empty V139() V140() V141() set
bool [: the carrier of X,REAL:] is non empty set
ComplexAlgebraStr(# (Funcs ( the carrier of X,COMPLEX)),(ComplexFuncMult the carrier of X),(ComplexFuncAdd the carrier of X),(ComplexFuncExtMult the carrier of X),(ComplexFuncUnit the carrier of X),(ComplexFuncZero the carrier of X) #) is strict ComplexAlgebraStr
the carrier of (CAlgebra the carrier of X) is non empty set
bool the carrier of (CAlgebra the carrier of X) is non empty set
{ b₁ where b₁ is non empty Relation-like the carrier of X -defined COMPLEX -valued Function-like total quasi_total V139() (X) Element of bool [: the carrier of X,COMPLEX:] : verum } is set
mult_ ((X),(CAlgebra the carrier of X)) is non empty Relation-like [:(X),(X):] -defined (X) -valued Function-like total quasi_total Element of bool [:[:(X),(X):],(X):]
[:(X),(X):] is non empty set
[:[:(X),(X):],(X):] is non empty set
bool [:[:(X),(X):],(X):] is non empty set
Add_ ((X),(CAlgebra the carrier of X)) is non empty Relation-like [:(X),(X):] -defined (X) -valued Function-like total quasi_total Element of bool [:[:(X),(X):],(X):]
Mult_ ((X),(CAlgebra the carrier of X)) is non empty Relation-like [:COMPLEX,(X):] -defined (X) -valued Function-like total quasi_total Element of bool [:[:COMPLEX,(X):],(X):]
[:COMPLEX,(X):] is non empty set
[:[:COMPLEX,(X):],(X):] is non empty set
bool [:[:COMPLEX,(X):],(X):] is non empty set
One_ ((X),(CAlgebra the carrier of X)) is Element of (X)
Zero_ ((X),(CAlgebra the carrier of X)) is Element of (X)
(X) is non empty Relation-like (X) -defined REAL -valued Function-like total quasi_total V139() V140() V141() Element of bool [:(X),REAL:]
[:(X),REAL:] is non empty V139() V140() V141() set
bool [:(X),REAL:] is non empty set
ComplexBoundedFunctions the carrier of X is non empty multiplicatively-closed Cadditively-linearly-closed Element of bool the carrier of (CAlgebra the carrier of X)
{ b₁ where b₁ is non empty Relation-like the carrier of X -defined COMPLEX -valued Function-like total quasi_total V139() Element of bool [: the carrier of X,COMPLEX:] : b₁ | the carrier of X is bounded } is set
ComplexBoundedFunctionsNorm the carrier of X is non empty Relation-like ComplexBoundedFunctions the carrier of X -defined REAL -valued Function-like total quasi_total V139() V140() V141() Element of bool [:(ComplexBoundedFunctions the carrier of X),REAL:]
[:(ComplexBoundedFunctions the carrier of X),REAL:] is non empty V139() V140() V141() set
bool [:(ComplexBoundedFunctions the carrier of X),REAL:] is non empty set
(ComplexBoundedFunctionsNorm the carrier of X) | (X) is Relation-like ComplexBoundedFunctions the carrier of X -defined REAL -valued Function-like V139() V140() V141() Element of bool [:(ComplexBoundedFunctions the carrier of X),REAL:]
Normed_Complex_AlgebraStr(# (X),(mult_ ((X),(CAlgebra the carrier of X))),(Add_ ((X),(CAlgebra the carrier of X))),(Mult_ ((X),(CAlgebra the carrier of X))),(One_ ((X),(CAlgebra the carrier of X))),(Zero_ ((X),(CAlgebra the carrier of X))),(X) #) is strict Normed_Complex_AlgebraStr
the carrier of (X) is non empty set
C_Normed_Algebra_of_BoundedFunctions the carrier of X is non empty right_complementable Abelian add-associative right_zeroed unital discerning reflexive vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like complete Normed_Complex_AlgebraStr
mult_ ((ComplexBoundedFunctions the carrier of X),(CAlgebra the carrier of X)) is non empty Relation-like [:(ComplexBoundedFunctions the carrier of X),(ComplexBoundedFunctions the carrier of X):] -defined ComplexBoundedFunctions the carrier of X -valued Function-like total quasi_total Element of bool [:[:(ComplexBoundedFunctions the carrier of X),(ComplexBoundedFunctions the carrier of X):],(ComplexBoundedFunctions the carrier of X):]
[:(ComplexBoundedFunctions the carrier of X),(ComplexBoundedFunctions the carrier of X):] is non empty set
[:[:(ComplexBoundedFunctions the carrier of X),(ComplexBoundedFunctions the carrier of X):],(ComplexBoundedFunctions the carrier of X):] is non empty set
bool [:[:(ComplexBoundedFunctions the carrier of X),(ComplexBoundedFunctions the carrier of X):],(ComplexBoundedFunctions the carrier of X):] is non empty set
Add_ ((ComplexBoundedFunctions the carrier of X),(CAlgebra the carrier of X)) is non empty Relation-like [:(ComplexBoundedFunctions the carrier of X),(ComplexBoundedFunctions the carrier of X):] -defined ComplexBoundedFunctions the carrier of X -valued Function-like total quasi_total Element of bool [:[:(ComplexBoundedFunctions the carrier of X),(ComplexBoundedFunctions the carrier of X):],(ComplexBoundedFunctions the carrier of X):]
Mult_ ((ComplexBoundedFunctions the carrier of X),(CAlgebra the carrier of X)) is non empty Relation-like [:COMPLEX,(ComplexBoundedFunctions the carrier of X):] -defined ComplexBoundedFunctions the carrier of X -valued Function-like total quasi_total Element of bool [:[:COMPLEX,(ComplexBoundedFunctions the carrier of X):],(ComplexBoundedFunctions the carrier of X):]
[:COMPLEX,(ComplexBoundedFunctions the carrier of X):] is non empty set
[:[:COMPLEX,(ComplexBoundedFunctions the carrier of X):],(ComplexBoundedFunctions the carrier of X):] is non empty set
bool [:[:COMPLEX,(ComplexBoundedFunctions the carrier of X):],(ComplexBoundedFunctions the carrier of X):] is non empty set
One_ ((ComplexBoundedFunctions the carrier of X),(CAlgebra the carrier of X)) is Element of ComplexBoundedFunctions the carrier of X
Zero_ ((ComplexBoundedFunctions the carrier of X),(CAlgebra the carrier of X)) is Element of ComplexBoundedFunctions the carrier of X
Normed_Complex_AlgebraStr(# (ComplexBoundedFunctions the carrier of X),(mult_ ((ComplexBoundedFunctions the carrier of X),(CAlgebra the carrier of X))),(Add_ ((ComplexBoundedFunctions the carrier of X),(CAlgebra the carrier of X))),(Mult_ ((ComplexBoundedFunctions the carrier of X),(CAlgebra the carrier of X))),(One_ ((ComplexBoundedFunctions the carrier of X),(CAlgebra the carrier of X))),(Zero_ ((ComplexBoundedFunctions the carrier of X),(CAlgebra the carrier of X))),(ComplexBoundedFunctionsNorm the carrier of X) #) is strict Normed_Complex_AlgebraStr
the carrier of (C_Normed_Algebra_of_BoundedFunctions the carrier of X) is non empty set
X is complex set
F is Element of the carrier of (X)
X * F is Element of the carrier of (X)
the Mult of (X) is non empty Relation-like [:COMPLEX, the carrier of (X):] -defined the carrier of (X) -valued Function-like total quasi_total Element of bool [:[:COMPLEX, the carrier of (X):], the carrier of (X):]
[:COMPLEX, the carrier of (X):] is non empty set
[:[:COMPLEX, the carrier of (X):], the carrier of (X):] is non empty set
bool [:[:COMPLEX, the carrier of (X):], the carrier of (X):] is non empty set
[X,F] is set
{X,F} is non empty set
{X} is non empty V176() set
{{X,F},{X}} is non empty set
the Mult of (X) . [X,F] is set
G is Element of the carrier of (C_Normed_Algebra_of_BoundedFunctions the carrier of X)
X * G is Element of the carrier of (C_Normed_Algebra_of_BoundedFunctions the carrier of X)
the Mult of (C_Normed_Algebra_of_BoundedFunctions the carrier of X) is non empty Relation-like [:COMPLEX, the carrier of (C_Normed_Algebra_of_BoundedFunctions the carrier of X):] -defined the carrier of (C_Normed_Algebra_of_BoundedFunctions the carrier of X) -valued Function-like total quasi_total Element of bool [:[:COMPLEX, the carrier of (C_Normed_Algebra_of_BoundedFunctions the carrier of X):], the carrier of (C_Normed_Algebra_of_BoundedFunctions the carrier of X):]
[:COMPLEX, the carrier of (C_Normed_Algebra_of_BoundedFunctions the carrier of X):] is non empty set
[:[:COMPLEX, the carrier of (C_Normed_Algebra_of_BoundedFunctions the carrier of X):], the carrier of (C_Normed_Algebra_of_BoundedFunctions the carrier of X):] is non empty set
bool [:[:COMPLEX, the carrier of (C_Normed_Algebra_of_BoundedFunctions the carrier of X):], the carrier of (C_Normed_Algebra_of_BoundedFunctions the carrier of X):] is non empty set
[X,G] is set
{X,G} is non empty set
{{X,G},{X}} is non empty set
the Mult of (C_Normed_Algebra_of_BoundedFunctions the carrier of X) . [X,G] is set
[:COMPLEX, the carrier of (CAlgebra the carrier of X):] is non empty set
the Mult of (CAlgebra the carrier of X) is non empty Relation-like [:COMPLEX, the carrier of (CAlgebra the carrier of X):] -defined the carrier of (CAlgebra the carrier of X) -valued Function-like total quasi_total Element of bool [:[:COMPLEX, the carrier of (CAlgebra the carrier of X):], the carrier of (CAlgebra the carrier of X):]
[:[:COMPLEX, the carrier of (CAlgebra the carrier of X):], the carrier of (CAlgebra the carrier of X):] is non empty set
bool [:[:COMPLEX, the carrier of (CAlgebra the carrier of X):], the carrier of (CAlgebra the carrier of X):] is non empty set
the Mult of (CAlgebra the carrier of X) | [:COMPLEX,(X):] is Relation-like [:COMPLEX, the carrier of (CAlgebra the carrier of X):] -defined the carrier of (CAlgebra the carrier of X) -valued Function-like Element of bool [:[:COMPLEX, the carrier of (CAlgebra the carrier of X):], the carrier of (CAlgebra the carrier of X):]
FB is complex Element of COMPLEX
[FB,F] is Element of [:COMPLEX, the carrier of (X):]
{FB,F} is non empty set
{FB} is non empty V176() set
{{FB,F},{FB}} is non empty set
( the Mult of (CAlgebra the carrier of X) | [:COMPLEX,(X):]) . [FB,F] is set
the Mult of (CAlgebra the carrier of X) . [FB,F] is set
the Mult of (CAlgebra the carrier of X) | [:COMPLEX,(ComplexBoundedFunctions the carrier of X):] is Relation-like [:COMPLEX, the carrier of (CAlgebra the carrier of X):] -defined the carrier of (CAlgebra the carrier of X) -valued Function-like Element of bool [:[:COMPLEX, the carrier of (CAlgebra the carrier of X):], the carrier of (CAlgebra the carrier of X):]
[FB,G] is Element of [:COMPLEX, the carrier of (C_Normed_Algebra_of_BoundedFunctions the carrier of X):]
{FB,G} is non empty set
{{FB,G},{FB}} is non empty set
( the Mult of (CAlgebra the carrier of X) | [:COMPLEX,(ComplexBoundedFunctions the carrier of X):]) . [FB,G] is set
X is non empty TopSpace-like V300() compact TopStruct
(X) is non empty unital strict Normed_Complex_AlgebraStr
(X) is non empty multiplicatively-closed Cadditively-linearly-closed Element of bool the carrier of (CAlgebra the carrier of X)
the carrier of X is non empty set
CAlgebra the carrier of X is non empty right_complementable Abelian add-associative right_zeroed unital associative commutative right-distributive right_unital well-unital left_unital vector-distributive scalar-distributive scalar-associative scalar-unital strict vector-associative ComplexAlgebraStr
Funcs ( the carrier of X,COMPLEX) is non empty functional FUNCTION_DOMAIN of the carrier of X, COMPLEX
ComplexFuncMult the carrier of X is non empty Relation-like [:(Funcs ( the carrier of X,COMPLEX)),(Funcs ( the carrier of X,COMPLEX)):] -defined Funcs ( the carrier of X,COMPLEX) -valued Function-like total quasi_total Element of bool [:[:(Funcs ( the carrier of X,COMPLEX)),(Funcs ( the carrier of X,COMPLEX)):],(Funcs ( the carrier of X,COMPLEX)):]
[:(Funcs ( the carrier of X,COMPLEX)),(Funcs ( the carrier of X,COMPLEX)):] is non empty set
[:[:(Funcs ( the carrier of X,COMPLEX)),(Funcs ( the carrier of X,COMPLEX)):],(Funcs ( the carrier of X,COMPLEX)):] is non empty set
bool [:[:(Funcs ( the carrier of X,COMPLEX)),(Funcs ( the carrier of X,COMPLEX)):],(Funcs ( the carrier of X,COMPLEX)):] is non empty set
ComplexFuncAdd the carrier of X is non empty Relation-like [:(Funcs ( the carrier of X,COMPLEX)),(Funcs ( the carrier of X,COMPLEX)):] -defined Funcs ( the carrier of X,COMPLEX) -valued Function-like total quasi_total Element of bool [:[:(Funcs ( the carrier of X,COMPLEX)),(Funcs ( the carrier of X,COMPLEX)):],(Funcs ( the carrier of X,COMPLEX)):]
ComplexFuncExtMult the carrier of X is non empty Relation-like [:COMPLEX,(Funcs ( the carrier of X,COMPLEX)):] -defined Funcs ( the carrier of X,COMPLEX) -valued Function-like total quasi_total Element of bool [:[:COMPLEX,(Funcs ( the carrier of X,COMPLEX)):],(Funcs ( the carrier of X,COMPLEX)):]
[:COMPLEX,(Funcs ( the carrier of X,COMPLEX)):] is non empty set
[:[:COMPLEX,(Funcs ( the carrier of X,COMPLEX)):],(Funcs ( the carrier of X,COMPLEX)):] is non empty set
bool [:[:COMPLEX,(Funcs ( the carrier of X,COMPLEX)):],(Funcs ( the carrier of X,COMPLEX)):] is non empty set
ComplexFuncUnit the carrier of X is Relation-like the carrier of X -defined COMPLEX -valued Function-like total quasi_total V139() Element of Funcs ( the carrier of X,COMPLEX)
the carrier of X --> 1r is non empty Relation-like the carrier of X -defined COMPLEX -valued Function-like total quasi_total V139() Element of bool [: the carrier of X,COMPLEX:]
[: the carrier of X,COMPLEX:] is non empty V139() set
bool [: the carrier of X,COMPLEX:] is non empty set
ComplexFuncZero the carrier of X is Relation-like the carrier of X -defined COMPLEX -valued Function-like total quasi_total V139() Element of Funcs ( the carrier of X,COMPLEX)
the carrier of X --> 0 is non empty Relation-like the carrier of X -defined REAL -valued RAT -valued INT -valued Function-like total quasi_total V139() V140() V141() V142() Element of bool [: the carrier of X,REAL:]
[: the carrier of X,REAL:] is non empty V139() V140() V141() set
bool [: the carrier of X,REAL:] is non empty set
ComplexAlgebraStr(# (Funcs ( the carrier of X,COMPLEX)),(ComplexFuncMult the carrier of X),(ComplexFuncAdd the carrier of X),(ComplexFuncExtMult the carrier of X),(ComplexFuncUnit the carrier of X),(ComplexFuncZero the carrier of X) #) is strict ComplexAlgebraStr
the carrier of (CAlgebra the carrier of X) is non empty set
bool the carrier of (CAlgebra the carrier of X) is non empty set
{ b₁ where b₁ is non empty Relation-like the carrier of X -defined COMPLEX -valued Function-like total quasi_total V139() (X) Element of bool [: the carrier of X,COMPLEX:] : verum } is set
mult_ ((X),(CAlgebra the carrier of X)) is non empty Relation-like [:(X),(X):] -defined (X) -valued Function-like total quasi_total Element of bool [:[:(X),(X):],(X):]
[:(X),(X):] is non empty set
[:[:(X),(X):],(X):] is non empty set
bool [:[:(X),(X):],(X):] is non empty set
Add_ ((X),(CAlgebra the carrier of X)) is non empty Relation-like [:(X),(X):] -defined (X) -valued Function-like total quasi_total Element of bool [:[:(X),(X):],(X):]
Mult_ ((X),(CAlgebra the carrier of X)) is non empty Relation-like [:COMPLEX,(X):] -defined (X) -valued Function-like total quasi_total Element of bool [:[:COMPLEX,(X):],(X):]
[:COMPLEX,(X):] is non empty set
[:[:COMPLEX,(X):],(X):] is non empty set
bool [:[:COMPLEX,(X):],(X):] is non empty set
One_ ((X),(CAlgebra the carrier of X)) is Element of (X)
Zero_ ((X),(CAlgebra the carrier of X)) is Element of (X)
(X) is non empty Relation-like (X) -defined REAL -valued Function-like total quasi_total V139() V140() V141() Element of bool [:(X),REAL:]
[:(X),REAL:] is non empty V139() V140() V141() set
bool [:(X),REAL:] is non empty set
ComplexBoundedFunctions the carrier of X is non empty multiplicatively-closed Cadditively-linearly-closed Element of bool the carrier of (CAlgebra the carrier of X)
{ b₁ where b₁ is non empty Relation-like the carrier of X -defined COMPLEX -valued Function-like total quasi_total V139() Element of bool [: the carrier of X,COMPLEX:] : b₁ | the carrier of X is bounded } is set
ComplexBoundedFunctionsNorm the carrier of X is non empty Relation-like ComplexBoundedFunctions the carrier of X -defined REAL -valued Function-like total quasi_total V139() V140() V141() Element of bool [:(ComplexBoundedFunctions the carrier of X),REAL:]
[:(ComplexBoundedFunctions the carrier of X),REAL:] is non empty V139() V140() V141() set
bool [:(ComplexBoundedFunctions the carrier of X),REAL:] is non empty set
(ComplexBoundedFunctionsNorm the carrier of X) | (X) is Relation-like ComplexBoundedFunctions the carrier of X -defined REAL -valued Function-like V139() V140() V141() Element of bool [:(ComplexBoundedFunctions the carrier of X),REAL:]
Normed_Complex_AlgebraStr(# (X),(mult_ ((X),(CAlgebra the carrier of X))),(Add_ ((X),(CAlgebra the carrier of X))),(Mult_ ((X),(CAlgebra the carrier of X))),(One_ ((X),(CAlgebra the carrier of X))),(Zero_ ((X),(CAlgebra the carrier of X))),(X) #) is strict Normed_Complex_AlgebraStr
the carrier of (X) is non empty set
C_Normed_Algebra_of_BoundedFunctions the carrier of X is non empty right_complementable Abelian add-associative right_zeroed unital discerning reflexive vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like complete Normed_Complex_AlgebraStr
mult_ ((ComplexBoundedFunctions the carrier of X),(CAlgebra the carrier of X)) is non empty Relation-like [:(ComplexBoundedFunctions the carrier of X),(ComplexBoundedFunctions the carrier of X):] -defined ComplexBoundedFunctions the carrier of X -valued Function-like total quasi_total Element of bool [:[:(ComplexBoundedFunctions the carrier of X),(ComplexBoundedFunctions the carrier of X):],(ComplexBoundedFunctions the carrier of X):]
[:(ComplexBoundedFunctions the carrier of X),(ComplexBoundedFunctions the carrier of X):] is non empty set
[:[:(ComplexBoundedFunctions the carrier of X),(ComplexBoundedFunctions the carrier of X):],(ComplexBoundedFunctions the carrier of X):] is non empty set
bool [:[:(ComplexBoundedFunctions the carrier of X),(ComplexBoundedFunctions the carrier of X):],(ComplexBoundedFunctions the carrier of X):] is non empty set
Add_ ((ComplexBoundedFunctions the carrier of X),(CAlgebra the carrier of X)) is non empty Relation-like [:(ComplexBoundedFunctions the carrier of X),(ComplexBoundedFunctions the carrier of X):] -defined ComplexBoundedFunctions the carrier of X -valued Function-like total quasi_total Element of bool [:[:(ComplexBoundedFunctions the carrier of X),(ComplexBoundedFunctions the carrier of X):],(ComplexBoundedFunctions the carrier of X):]
Mult_ ((ComplexBoundedFunctions the carrier of X),(CAlgebra the carrier of X)) is non empty Relation-like [:COMPLEX,(ComplexBoundedFunctions the carrier of X):] -defined ComplexBoundedFunctions the carrier of X -valued Function-like total quasi_total Element of bool [:[:COMPLEX,(ComplexBoundedFunctions the carrier of X):],(ComplexBoundedFunctions the carrier of X):]
[:COMPLEX,(ComplexBoundedFunctions the carrier of X):] is non empty set
[:[:COMPLEX,(ComplexBoundedFunctions the carrier of X):],(ComplexBoundedFunctions the carrier of X):] is non empty set
bool [:[:COMPLEX,(ComplexBoundedFunctions the carrier of X):],(ComplexBoundedFunctions the carrier of X):] is non empty set
One_ ((ComplexBoundedFunctions the carrier of X),(CAlgebra the carrier of X)) is Element of ComplexBoundedFunctions the carrier of X
Zero_ ((ComplexBoundedFunctions the carrier of X),(CAlgebra the carrier of X)) is Element of ComplexBoundedFunctions the carrier of X
Normed_Complex_AlgebraStr(# (ComplexBoundedFunctions the carrier of X),(mult_ ((ComplexBoundedFunctions the carrier of X),(CAlgebra the carrier of X))),(Add_ ((ComplexBoundedFunctions the carrier of X),(CAlgebra the carrier of X))),(Mult_ ((ComplexBoundedFunctions the carrier of X),(CAlgebra the carrier of X))),(One_ ((ComplexBoundedFunctions the carrier of X),(CAlgebra the carrier of X))),(Zero_ ((ComplexBoundedFunctions the carrier of X),(CAlgebra the carrier of X))),(ComplexBoundedFunctionsNorm the carrier of X) #) is strict Normed_Complex_AlgebraStr
the carrier of (C_Normed_Algebra_of_BoundedFunctions the carrier of X) is non empty set
X is Element of the carrier of (X)
F is Element of the carrier of (X)
X * F is Element of the carrier of (X)
the multF of (X) is non empty Relation-like [: the carrier of (X), the carrier of (X):] -defined the carrier of (X) -valued Function-like total quasi_total Element of bool [:[: the carrier of (X), the carrier of (X):], the carrier of (X):]
[: the carrier of (X), the carrier of (X):] is non empty set
[:[: the carrier of (X), the carrier of (X):], the carrier of (X):] is non empty set
bool [:[: the carrier of (X), the carrier of (X):], the carrier of (X):] is non empty set
the multF of (X) . (X,F) is Element of the carrier of (X)
[X,F] is set
{X,F} is non empty set
{X} is non empty set
{{X,F},{X}} is non empty set
the multF of (X) . [X,F] is set
G is Element of the carrier of (C_Normed_Algebra_of_BoundedFunctions the carrier of X)
FB is Element of the carrier of (C_Normed_Algebra_of_BoundedFunctions the carrier of X)
G * FB is Element of the carrier of (C_Normed_Algebra_of_BoundedFunctions the carrier of X)
the multF of (C_Normed_Algebra_of_BoundedFunctions the carrier of X) is non empty Relation-like [: the carrier of (C_Normed_Algebra_of_BoundedFunctions the carrier of X), the carrier of (C_Normed_Algebra_of_BoundedFunctions the carrier of X):] -defined the carrier of (C_Normed_Algebra_of_BoundedFunctions the carrier of X) -valued Function-like total quasi_total Element of bool [:[: the carrier of (C_Normed_Algebra_of_BoundedFunctions the carrier of X), the carrier of (C_Normed_Algebra_of_BoundedFunctions the carrier of X):], the carrier of (C_Normed_Algebra_of_BoundedFunctions the carrier of X):]
[: the carrier of (C_Normed_Algebra_of_BoundedFunctions the carrier of X), the carrier of (C_Normed_Algebra_of_BoundedFunctions the carrier of X):] is non empty set
[:[: the carrier of (C_Normed_Algebra_of_BoundedFunctions the carrier of X), the carrier of (C_Normed_Algebra_of_BoundedFunctions the carrier of X):], the carrier of (C_Normed_Algebra_of_BoundedFunctions the carrier of X):] is non empty set
bool [:[: the carrier of (C_Normed_Algebra_of_BoundedFunctions the carrier of X), the carrier of (C_Normed_Algebra_of_BoundedFunctions the carrier of X):], the carrier of (C_Normed_Algebra_of_BoundedFunctions the carrier of X):] is non empty set
the multF of (C_Normed_Algebra_of_BoundedFunctions the carrier of X) . (G,FB) is Element of the carrier of (C_Normed_Algebra_of_BoundedFunctions the carrier of X)
[G,FB] is set
{G,FB} is non empty set
{G} is non empty set
{{G,FB},{G}} is non empty set
the multF of (C_Normed_Algebra_of_BoundedFunctions the carrier of X) . [G,FB] is set
the multF of (CAlgebra the carrier of X) is non empty Relation-like [: the carrier of (CAlgebra the carrier of X), the carrier of (CAlgebra the carrier of X):] -defined the carrier of (CAlgebra the carrier of X) -valued Function-like total quasi_total Element of bool [:[: the carrier of (CAlgebra the carrier of X), the carrier of (CAlgebra the carrier of X):], the carrier of (CAlgebra the carrier of X):]
[: the carrier of (CAlgebra the carrier of X), the carrier of (CAlgebra the carrier of X):] is non empty set
[:[: the carrier of (CAlgebra the carrier of X), the carrier of (CAlgebra the carrier of X):], the carrier of (CAlgebra the carrier of X):] is non empty set
bool [:[: the carrier of (CAlgebra the carrier of X), the carrier of (CAlgebra the carrier of X):], the carrier of (CAlgebra the carrier of X):] is non empty set
the multF of (CAlgebra the carrier of X) || (X) is Relation-like Function-like set
the multF of (CAlgebra the carrier of X) | [:(X),(X):] is Relation-like set
[X,F] is Element of [: the carrier of (X), the carrier of (X):]
( the multF of (CAlgebra the carrier of X) || (X)) . [X,F] is set
the multF of (CAlgebra the carrier of X) . [X,F] is set
the multF of (CAlgebra the carrier of X) || (ComplexBoundedFunctions the carrier of X) is Relation-like Function-like set
the multF of (CAlgebra the carrier of X) | [:(ComplexBoundedFunctions the carrier of X),(ComplexBoundedFunctions the carrier of X):] is Relation-like set
[G,FB] is Element of [: the carrier of (C_Normed_Algebra_of_BoundedFunctions the carrier of X), the carrier of (C_Normed_Algebra_of_BoundedFunctions the carrier of X):]
( the multF of (CAlgebra the carrier of X) || (ComplexBoundedFunctions the carrier of X)) . [G,FB] is set
X is non empty TopSpace-like V300() compact TopStruct
(X) is non empty unital strict Normed_Complex_AlgebraStr
(X) is non empty multiplicatively-closed Cadditively-linearly-closed Element of bool the carrier of (CAlgebra the carrier of X)
the carrier of X is non empty set
CAlgebra the carrier of X is non empty right_complementable Abelian add-associative right_zeroed unital associative commutative right-distributive right_unital well-unital left_unital vector-distributive scalar-distributive scalar-associative scalar-unital strict vector-associative ComplexAlgebraStr
Funcs ( the carrier of X,COMPLEX) is non empty functional FUNCTION_DOMAIN of the carrier of X, COMPLEX
ComplexFuncMult the carrier of X is non empty Relation-like [:(Funcs ( the carrier of X,COMPLEX)),(Funcs ( the carrier of X,COMPLEX)):] -defined Funcs ( the carrier of X,COMPLEX) -valued Function-like total quasi_total Element of bool [:[:(Funcs ( the carrier of X,COMPLEX)),(Funcs ( the carrier of X,COMPLEX)):],(Funcs ( the carrier of X,COMPLEX)):]
[:(Funcs ( the carrier of X,COMPLEX)),(Funcs ( the carrier of X,COMPLEX)):] is non empty set
[:[:(Funcs ( the carrier of X,COMPLEX)),(Funcs ( the carrier of X,COMPLEX)):],(Funcs ( the carrier of X,COMPLEX)):] is non empty set
bool [:[:(Funcs ( the carrier of X,COMPLEX)),(Funcs ( the carrier of X,COMPLEX)):],(Funcs ( the carrier of X,COMPLEX)):] is non empty set
ComplexFuncAdd the carrier of X is non empty Relation-like [:(Funcs ( the carrier of X,COMPLEX)),(Funcs ( the carrier of X,COMPLEX)):] -defined Funcs ( the carrier of X,COMPLEX) -valued Function-like total quasi_total Element of bool [:[:(Funcs ( the carrier of X,COMPLEX)),(Funcs ( the carrier of X,COMPLEX)):],(Funcs ( the carrier of X,COMPLEX)):]
ComplexFuncExtMult the carrier of X is non empty Relation-like [:COMPLEX,(Funcs ( the carrier of X,COMPLEX)):] -defined Funcs ( the carrier of X,COMPLEX) -valued Function-like total quasi_total Element of bool [:[:COMPLEX,(Funcs ( the carrier of X,COMPLEX)):],(Funcs ( the carrier of X,COMPLEX)):]
[:COMPLEX,(Funcs ( the carrier of X,COMPLEX)):] is non empty set
[:[:COMPLEX,(Funcs ( the carrier of X,COMPLEX)):],(Funcs ( the carrier of X,COMPLEX)):] is non empty set
bool [:[:COMPLEX,(Funcs ( the carrier of X,COMPLEX)):],(Funcs ( the carrier of X,COMPLEX)):] is non empty set
ComplexFuncUnit the carrier of X is Relation-like the carrier of X -defined COMPLEX -valued Function-like total quasi_total V139() Element of Funcs ( the carrier of X,COMPLEX)
the carrier of X --> 1r is non empty Relation-like the carrier of X -defined COMPLEX -valued Function-like total quasi_total V139() Element of bool [: the carrier of X,COMPLEX:]
[: the carrier of X,COMPLEX:] is non empty V139() set
bool [: the carrier of X,COMPLEX:] is non empty set
ComplexFuncZero the carrier of X is Relation-like the carrier of X -defined COMPLEX -valued Function-like total quasi_total V139() Element of Funcs ( the carrier of X,COMPLEX)
the carrier of X --> 0 is non empty Relation-like the carrier of X -defined REAL -valued RAT -valued INT -valued Function-like total quasi_total V139() V140() V141() V142() Element of bool [: the carrier of X,REAL:]
[: the carrier of X,REAL:] is non empty V139() V140() V141() set
bool [: the carrier of X,REAL:] is non empty set
ComplexAlgebraStr(# (Funcs ( the carrier of X,COMPLEX)),(ComplexFuncMult the carrier of X),(ComplexFuncAdd the carrier of X),(ComplexFuncExtMult the carrier of X),(ComplexFuncUnit the carrier of X),(ComplexFuncZero the carrier of X) #) is strict ComplexAlgebraStr
the carrier of (CAlgebra the carrier of X) is non empty set
bool the carrier of (CAlgebra the carrier of X) is non empty set
{ b₁ where b₁ is non empty Relation-like the carrier of X -defined COMPLEX -valued Function-like total quasi_total V139() (X) Element of bool [: the carrier of X,COMPLEX:] : verum } is set
mult_ ((X),(CAlgebra the carrier of X)) is non empty Relation-like [:(X),(X):] -defined (X) -valued Function-like total quasi_total Element of bool [:[:(X),(X):],(X):]
[:(X),(X):] is non empty set
[:[:(X),(X):],(X):] is non empty set
bool [:[:(X),(X):],(X):] is non empty set
Add_ ((X),(CAlgebra the carrier of X)) is non empty Relation-like [:(X),(X):] -defined (X) -valued Function-like total quasi_total Element of bool [:[:(X),(X):],(X):]
Mult_ ((X),(CAlgebra the carrier of X)) is non empty Relation-like [:COMPLEX,(X):] -defined (X) -valued Function-like total quasi_total Element of bool [:[:COMPLEX,(X):],(X):]
[:COMPLEX,(X):] is non empty set
[:[:COMPLEX,(X):],(X):] is non empty set
bool [:[:COMPLEX,(X):],(X):] is non empty set
One_ ((X),(CAlgebra the carrier of X)) is Element of (X)
Zero_ ((X),(CAlgebra the carrier of X)) is Element of (X)
(X) is non empty Relation-like (X) -defined REAL -valued Function-like total quasi_total V139() V140() V141() Element of bool [:(X),REAL:]
[:(X),REAL:] is non empty V139() V140() V141() set
bool [:(X),REAL:] is non empty set
ComplexBoundedFunctions the carrier of X is non empty multiplicatively-closed Cadditively-linearly-closed Element of bool the carrier of (CAlgebra the carrier of X)
{ b₁ where b₁ is non empty Relation-like the carrier of X -defined COMPLEX -valued Function-like total quasi_total V139() Element of bool [: the carrier of X,COMPLEX:] : b₁ | the carrier of X is bounded } is set
ComplexBoundedFunctionsNorm the carrier of X is non empty Relation-like ComplexBoundedFunctions the carrier of X -defined REAL -valued Function-like total quasi_total V139() V140() V141() Element of bool [:(ComplexBoundedFunctions the carrier of X),REAL:]
[:(ComplexBoundedFunctions the carrier of X),REAL:] is non empty V139() V140() V141() set
bool [:(ComplexBoundedFunctions the carrier of X),REAL:] is non empty set
(ComplexBoundedFunctionsNorm the carrier of X) | (X) is Relation-like ComplexBoundedFunctions the carrier of X -defined REAL -valued Function-like V139() V140() V141() Element of bool [:(ComplexBoundedFunctions the carrier of X),REAL:]
Normed_Complex_AlgebraStr(# (X),(mult_ ((X),(CAlgebra the carrier of X))),(Add_ ((X),(CAlgebra the carrier of X))),(Mult_ ((X),(CAlgebra the carrier of X))),(One_ ((X),(CAlgebra the carrier of X))),(Zero_ ((X),(CAlgebra the carrier of X))),(X) #) is strict Normed_Complex_AlgebraStr
1. (X) is Element of the carrier of (X)
the carrier of (X) is non empty set
the OneF of (X) is Element of the carrier of (X)
(X) is non empty right_complementable Abelian add-associative right_zeroed unital associative commutative right-distributive right_unital well-unital left_unital vector-distributive scalar-distributive scalar-associative scalar-unital strict vector-associative ComplexAlgebraStr
ComplexAlgebraStr(# (X),(mult_ ((X),(CAlgebra the carrier of X))),(Add_ ((X),(CAlgebra the carrier of X))),(Mult_ ((X),(CAlgebra the carrier of X))),(One_ ((X),(CAlgebra the carrier of X))),(Zero_ ((X),(CAlgebra the carrier of X))) #) is strict ComplexAlgebraStr
1_ (X) is Element of the carrier of (X)
the carrier of (X) is non empty set
1. (X) is Element of the carrier of (X)
the OneF of (X) is Element of the carrier of (X)
X is non empty TopSpace-like V300() compact TopStruct
(X) is non empty unital strict Normed_Complex_AlgebraStr
(X) is non empty multiplicatively-closed Cadditively-linearly-closed Element of bool the carrier of (CAlgebra the carrier of X)
the carrier of X is non empty set
CAlgebra the carrier of X is non empty right_complementable Abelian add-associative right_zeroed unital associative commutative right-distributive right_unital well-unital left_unital vector-distributive scalar-distributive scalar-associative scalar-unital strict vector-associative ComplexAlgebraStr
Funcs ( the carrier of X,COMPLEX) is non empty functional FUNCTION_DOMAIN of the carrier of X, COMPLEX
ComplexFuncMult the carrier of X is non empty Relation-like [:(Funcs ( the carrier of X,COMPLEX)),(Funcs ( the carrier of X,COMPLEX)):] -defined Funcs ( the carrier of X,COMPLEX) -valued Function-like total quasi_total Element of bool [:[:(Funcs ( the carrier of X,COMPLEX)),(Funcs ( the carrier of X,COMPLEX)):],(Funcs ( the carrier of X,COMPLEX)):]
[:(Funcs ( the carrier of X,COMPLEX)),(Funcs ( the carrier of X,COMPLEX)):] is non empty set
[:[:(Funcs ( the carrier of X,COMPLEX)),(Funcs ( the carrier of X,COMPLEX)):],(Funcs ( the carrier of X,COMPLEX)):] is non empty set
bool [:[:(Funcs ( the carrier of X,COMPLEX)),(Funcs ( the carrier of X,COMPLEX)):],(Funcs ( the carrier of X,COMPLEX)):] is non empty set
ComplexFuncAdd the carrier of X is non empty Relation-like [:(Funcs ( the carrier of X,COMPLEX)),(Funcs ( the carrier of X,COMPLEX)):] -defined Funcs ( the carrier of X,COMPLEX) -valued Function-like total quasi_total Element of bool [:[:(Funcs ( the carrier of X,COMPLEX)),(Funcs ( the carrier of X,COMPLEX)):],(Funcs ( the carrier of X,COMPLEX)):]
ComplexFuncExtMult the carrier of X is non empty Relation-like [:COMPLEX,(Funcs ( the carrier of X,COMPLEX)):] -defined Funcs ( the carrier of X,COMPLEX) -valued Function-like total quasi_total Element of bool [:[:COMPLEX,(Funcs ( the carrier of X,COMPLEX)):],(Funcs ( the carrier of X,COMPLEX)):]
[:COMPLEX,(Funcs ( the carrier of X,COMPLEX)):] is non empty set
[:[:COMPLEX,(Funcs ( the carrier of X,COMPLEX)):],(Funcs ( the carrier of X,COMPLEX)):] is non empty set
bool [:[:COMPLEX,(Funcs ( the carrier of X,COMPLEX)):],(Funcs ( the carrier of X,COMPLEX)):] is non empty set
ComplexFuncUnit the carrier of X is Relation-like the carrier of X -defined COMPLEX -valued Function-like total quasi_total V139() Element of Funcs ( the carrier of X,COMPLEX)
the carrier of X --> 1r is non empty Relation-like the carrier of X -defined COMPLEX -valued Function-like total quasi_total V139() Element of bool [: the carrier of X,COMPLEX:]
[: the carrier of X,COMPLEX:] is non empty V139() set
bool [: the carrier of X,COMPLEX:] is non empty set
ComplexFuncZero the carrier of X is Relation-like the carrier of X -defined COMPLEX -valued Function-like total quasi_total V139() Element of Funcs ( the carrier of X,COMPLEX)
the carrier of X --> 0 is non empty Relation-like the carrier of X -defined REAL -valued RAT -valued INT -valued Function-like total quasi_total V139() V140() V141() V142() Element of bool [: the carrier of X,REAL:]
[: the carrier of X,REAL:] is non empty V139() V140() V141() set
bool [: the carrier of X,REAL:] is non empty set
ComplexAlgebraStr(# (Funcs ( the carrier of X,COMPLEX)),(ComplexFuncMult the carrier of X),(ComplexFuncAdd the carrier of X),(ComplexFuncExtMult the carrier of X),(ComplexFuncUnit the carrier of X),(ComplexFuncZero the carrier of X) #) is strict ComplexAlgebraStr
the carrier of (CAlgebra the carrier of X) is non empty set
bool the carrier of (CAlgebra the carrier of X) is non empty set
{ b₁ where b₁ is non empty Relation-like the carrier of X -defined COMPLEX -valued Function-like total quasi_total V139() (X) Element of bool [: the carrier of X,COMPLEX:] : verum } is set
mult_ ((X),(CAlgebra the carrier of X)) is non empty Relation-like [:(X),(X):] -defined (X) -valued Function-like total quasi_total Element of bool [:[:(X),(X):],(X):]
[:(X),(X):] is non empty set
[:[:(X),(X):],(X):] is non empty set
bool [:[:(X),(X):],(X):] is non empty set
Add_ ((X),(CAlgebra the carrier of X)) is non empty Relation-like [:(X),(X):] -defined (X) -valued Function-like total quasi_total Element of bool [:[:(X),(X):],(X):]
Mult_ ((X),(CAlgebra the carrier of X)) is non empty Relation-like [:COMPLEX,(X):] -defined (X) -valued Function-like total quasi_total Element of bool [:[:COMPLEX,(X):],(X):]
[:COMPLEX,(X):] is non empty set
[:[:COMPLEX,(X):],(X):] is non empty set
bool [:[:COMPLEX,(X):],(X):] is non empty set
One_ ((X),(CAlgebra the carrier of X)) is Element of (X)
Zero_ ((X),(CAlgebra the carrier of X)) is Element of (X)
(X) is non empty Relation-like (X) -defined REAL -valued Function-like total quasi_total V139() V140() V141() Element of bool [:(X),REAL:]
[:(X),REAL:] is non empty V139() V140() V141() set
bool [:(X),REAL:] is non empty set
ComplexBoundedFunctions the carrier of X is non empty multiplicatively-closed Cadditively-linearly-closed Element of bool the carrier of (CAlgebra the carrier of X)
{ b₁ where b₁ is non empty Relation-like the carrier of X -defined COMPLEX -valued Function-like total quasi_total V139() Element of bool [: the carrier of X,COMPLEX:] : b₁ | the carrier of X is bounded } is set
ComplexBoundedFunctionsNorm the carrier of X is non empty Relation-like ComplexBoundedFunctions the carrier of X -defined REAL -valued Function-like total quasi_total V139() V140() V141() Element of bool [:(ComplexBoundedFunctions the carrier of X),REAL:]
[:(ComplexBoundedFunctions the carrier of X),REAL:] is non empty V139() V140() V141() set
bool [:(ComplexBoundedFunctions the carrier of X),REAL:] is non empty set
(ComplexBoundedFunctionsNorm the carrier of X) | (X) is Relation-like ComplexBoundedFunctions the carrier of X -defined REAL -valued Function-like V139() V140() V141() Element of bool [:(ComplexBoundedFunctions the carrier of X),REAL:]
Normed_Complex_AlgebraStr(# (X),(mult_ ((X),(CAlgebra the carrier of X))),(Add_ ((X),(CAlgebra the carrier of X))),(Mult_ ((X),(CAlgebra the carrier of X))),(One_ ((X),(CAlgebra the carrier of X))),(Zero_ ((X),(CAlgebra the carrier of X))),(X) #) is strict Normed_Complex_AlgebraStr
X is non empty TopSpace-like V300() compact TopStruct
(X) is non empty right_complementable Abelian add-associative right_zeroed unital associative commutative right-distributive right_unital well-unital left_unital vector-distributive scalar-distributive scalar-associative vector-associative strict Normed_Complex_AlgebraStr
(X) is non empty multiplicatively-closed Cadditively-linearly-closed Element of bool the carrier of (CAlgebra the carrier of X)
the carrier of X is non empty set
CAlgebra the carrier of X is non empty right_complementable Abelian add-associative right_zeroed unital associative commutative right-distributive right_unital well-unital left_unital vector-distributive scalar-distributive scalar-associative scalar-unital strict vector-associative ComplexAlgebraStr
Funcs ( the carrier of X,COMPLEX) is non empty functional FUNCTION_DOMAIN of the carrier of X, COMPLEX
ComplexFuncMult the carrier of X is non empty Relation-like [:(Funcs ( the carrier of X,COMPLEX)),(Funcs ( the carrier of X,COMPLEX)):] -defined Funcs ( the carrier of X,COMPLEX) -valued Function-like total quasi_total Element of bool [:[:(Funcs ( the carrier of X,COMPLEX)),(Funcs ( the carrier of X,COMPLEX)):],(Funcs ( the carrier of X,COMPLEX)):]
[:(Funcs ( the carrier of X,COMPLEX)),(Funcs ( the carrier of X,COMPLEX)):] is non empty set
[:[:(Funcs ( the carrier of X,COMPLEX)),(Funcs ( the carrier of X,COMPLEX)):],(Funcs ( the carrier of X,COMPLEX)):] is non empty set
bool [:[:(Funcs ( the carrier of X,COMPLEX)),(Funcs ( the carrier of X,COMPLEX)):],(Funcs ( the carrier of X,COMPLEX)):] is non empty set
ComplexFuncAdd the carrier of X is non empty Relation-like [:(Funcs ( the carrier of X,COMPLEX)),(Funcs ( the carrier of X,COMPLEX)):] -defined Funcs ( the carrier of X,COMPLEX) -valued Function-like total quasi_total Element of bool [:[:(Funcs ( the carrier of X,COMPLEX)),(Funcs ( the carrier of X,COMPLEX)):],(Funcs ( the carrier of X,COMPLEX)):]
ComplexFuncExtMult the carrier of X is non empty Relation-like [:COMPLEX,(Funcs ( the carrier of X,COMPLEX)):] -defined Funcs ( the carrier of X,COMPLEX) -valued Function-like total quasi_total Element of bool [:[:COMPLEX,(Funcs ( the carrier of X,COMPLEX)):],(Funcs ( the carrier of X,COMPLEX)):]
[:COMPLEX,(Funcs ( the carrier of X,COMPLEX)):] is non empty set
[:[:COMPLEX,(Funcs ( the carrier of X,COMPLEX)):],(Funcs ( the carrier of X,COMPLEX)):] is non empty set
bool [:[:COMPLEX,(Funcs ( the carrier of X,COMPLEX)):],(Funcs ( the carrier of X,COMPLEX)):] is non empty set
ComplexFuncUnit the carrier of X is Relation-like the carrier of X -defined COMPLEX -valued Function-like total quasi_total V139() Element of Funcs ( the carrier of X,COMPLEX)
the carrier of X --> 1r is non empty Relation-like the carrier of X -defined COMPLEX -valued Function-like total quasi_total V139() Element of bool [: the carrier of X,COMPLEX:]
[: the carrier of X,COMPLEX:] is non empty V139() set
bool [: the carrier of X,COMPLEX:] is non empty set
ComplexFuncZero the carrier of X is Relation-like the carrier of X -defined COMPLEX -valued Function-like total quasi_total V139() Element of Funcs ( the carrier of X,COMPLEX)
the carrier of X --> 0 is non empty Relation-like the carrier of X -defined REAL -valued RAT -valued INT -valued Function-like total quasi_total V139() V140() V141() V142() Element of bool [: the carrier of X,REAL:]
[: the carrier of X,REAL:] is non empty V139() V140() V141() set
bool [: the carrier of X,REAL:] is non empty set
ComplexAlgebraStr(# (Funcs ( the carrier of X,COMPLEX)),(ComplexFuncMult the carrier of X),(ComplexFuncAdd the carrier of X),(ComplexFuncExtMult the carrier of X),(ComplexFuncUnit the carrier of X),(ComplexFuncZero the carrier of X) #) is strict ComplexAlgebraStr
the carrier of (CAlgebra the carrier of X) is non empty set
bool the carrier of (CAlgebra the carrier of X) is non empty set
{ b₁ where b₁ is non empty Relation-like the carrier of X -defined COMPLEX -valued Function-like total quasi_total V139() (X) Element of bool [: the carrier of X,COMPLEX:] : verum } is set
mult_ ((X),(CAlgebra the carrier of X)) is non empty Relation-like [:(X),(X):] -defined (X) -valued Function-like total quasi_total Element of bool [:[:(X),(X):],(X):]
[:(X),(X):] is non empty set
[:[:(X),(X):],(X):] is non empty set
bool [:[:(X),(X):],(X):] is non empty set
Add_ ((X),(CAlgebra the carrier of X)) is non empty Relation-like [:(X),(X):] -defined (X) -valued Function-like total quasi_total Element of bool [:[:(X),(X):],(X):]
Mult_ ((X),(CAlgebra the carrier of X)) is non empty Relation-like [:COMPLEX,(X):] -defined (X) -valued Function-like total quasi_total Element of bool [:[:COMPLEX,(X):],(X):]
[:COMPLEX,(X):] is non empty set
[:[:COMPLEX,(X):],(X):] is non empty set
bool [:[:COMPLEX,(X):],(X):] is non empty set
One_ ((X),(CAlgebra the carrier of X)) is Element of (X)
Zero_ ((X),(CAlgebra the carrier of X)) is Element of (X)
(X) is non empty Relation-like (X) -defined REAL -valued Function-like total quasi_total V139() V140() V141() Element of bool [:(X),REAL:]
[:(X),REAL:] is non empty V139() V140() V141() set
bool [:(X),REAL:] is non empty set
ComplexBoundedFunctions the carrier of X is non empty multiplicatively-closed Cadditively-linearly-closed Element of bool the carrier of (CAlgebra the carrier of X)
{ b₁ where b₁ is non empty Relation-like the carrier of X -defined COMPLEX -valued Function-like total quasi_total V139() Element of bool [: the carrier of X,COMPLEX:] : b₁ | the carrier of X is bounded } is set
ComplexBoundedFunctionsNorm the carrier of X is non empty Relation-like ComplexBoundedFunctions the carrier of X -defined REAL -valued Function-like total quasi_total V139() V140() V141() Element of bool [:(ComplexBoundedFunctions the carrier of X),REAL:]
[:(ComplexBoundedFunctions the carrier of X),REAL:] is non empty V139() V140() V141() set
bool [:(ComplexBoundedFunctions the carrier of X),REAL:] is non empty set
(ComplexBoundedFunctionsNorm the carrier of X) | (X) is Relation-like ComplexBoundedFunctions the carrier of X -defined REAL -valued Function-like V139() V140() V141() Element of bool [:(ComplexBoundedFunctions the carrier of X),REAL:]
Normed_Complex_AlgebraStr(# (X),(mult_ ((X),(CAlgebra the carrier of X))),(Add_ ((X),(CAlgebra the carrier of X))),(Mult_ ((X),(CAlgebra the carrier of X))),(One_ ((X),(CAlgebra the carrier of X))),(Zero_ ((X),(CAlgebra the carrier of X))),(X) #) is strict Normed_Complex_AlgebraStr
the carrier of (X) is non empty set
X is Element of the carrier of (X)
(Mult_ ((X),(CAlgebra the carrier of X))) . (1r,X) is set
[1r,X] is set
{1r,X} is non empty set
{1r} is non empty V176() set
{{1r,X},{1r}} is non empty set
(Mult_ ((X),(CAlgebra the carrier of X))) . [1r,X] is set
G is Element of (X)
[1,G] is Element of [:NAT,(X):]
[:NAT,(X):] is non empty set
{1,G} is non empty set
{{1,G},{1}} is non empty set
(Mult_ ((X),(CAlgebra the carrier of X))) . (1,X) is set
[1,X] is set
{1,X} is non empty set
{{1,X},{1}} is non empty set
(Mult_ ((X),(CAlgebra the carrier of X))) . [1,X] is set
[:COMPLEX, the carrier of (CAlgebra the carrier of X):] is non empty set
the Mult of (CAlgebra the carrier of X) is non empty Relation-like [:COMPLEX, the carrier of (CAlgebra the carrier of X):] -defined the carrier of (CAlgebra the carrier of X) -valued Function-like total quasi_total Element of bool [:[:COMPLEX, the carrier of (CAlgebra the carrier of X):], the carrier of (CAlgebra the carrier of X):]
[:[:COMPLEX, the carrier of (CAlgebra the carrier of X):], the carrier of (CAlgebra the carrier of X):] is non empty set
bool [:[:COMPLEX, the carrier of (CAlgebra the carrier of X):], the carrier of (CAlgebra the carrier of X):] is non empty set
the Mult of (CAlgebra the carrier of X) | [:COMPLEX,(X):] is Relation-like [:COMPLEX, the carrier of (CAlgebra the carrier of X):] -defined the carrier of (CAlgebra the carrier of X) -valued Function-like Element of bool [:[:COMPLEX, the carrier of (CAlgebra the carrier of X):], the carrier of (CAlgebra the carrier of X):]
( the Mult of (CAlgebra the carrier of X) | [:COMPLEX,(X):]) . (1,G) is set
[1,G] is set
( the Mult of (CAlgebra the carrier of X) | [:COMPLEX,(X):]) . [1,G] is set
the Mult of (CAlgebra the carrier of X) . (1,G) is set
the Mult of (CAlgebra the carrier of X) . [1,G] is set
X is non empty TopSpace-like V300() compact TopStruct
(X) is non empty right_complementable Abelian add-associative right_zeroed unital associative commutative right-distributive right_unital well-unital left_unital vector-distributive scalar-distributive scalar-associative vector-associative strict Normed_Complex_AlgebraStr
(X) is non empty multiplicatively-closed Cadditively-linearly-closed Element of bool the carrier of (CAlgebra the carrier of X)
the carrier of X is non empty set
CAlgebra the carrier of X is non empty right_complementable Abelian add-associative right_zeroed unital associative commutative right-distributive right_unital well-unital left_unital vector-distributive scalar-distributive scalar-associative scalar-unital strict vector-associative ComplexAlgebraStr
Funcs ( the carrier of X,COMPLEX) is non empty functional FUNCTION_DOMAIN of the carrier of X, COMPLEX
ComplexFuncMult the carrier of X is non empty Relation-like [:(Funcs ( the carrier of X,COMPLEX)),(Funcs ( the carrier of X,COMPLEX)):] -defined Funcs ( the carrier of X,COMPLEX) -valued Function-like total quasi_total Element of bool [:[:(Funcs ( the carrier of X,COMPLEX)),(Funcs ( the carrier of X,COMPLEX)):],(Funcs ( the carrier of X,COMPLEX)):]
[:(Funcs ( the carrier of X,COMPLEX)),(Funcs ( the carrier of X,COMPLEX)):] is non empty set
[:[:(Funcs ( the carrier of X,COMPLEX)),(Funcs ( the carrier of X,COMPLEX)):],(Funcs ( the carrier of X,COMPLEX)):] is non empty set
bool [:[:(Funcs ( the carrier of X,COMPLEX)),(Funcs ( the carrier of X,COMPLEX)):],(Funcs ( the carrier of X,COMPLEX)):] is non empty set
ComplexFuncAdd the carrier of X is non empty Relation-like [:(Funcs ( the carrier of X,COMPLEX)),(Funcs ( the carrier of X,COMPLEX)):] -defined Funcs ( the carrier of X,COMPLEX) -valued Function-like total quasi_total Element of bool [:[:(Funcs ( the carrier of X,COMPLEX)),(Funcs ( the carrier of X,COMPLEX)):],(Funcs ( the carrier of X,COMPLEX)):]
ComplexFuncExtMult the carrier of X is non empty Relation-like [:COMPLEX,(Funcs ( the carrier of X,COMPLEX)):] -defined Funcs ( the carrier of X,COMPLEX) -valued Function-like total quasi_total Element of bool [:[:COMPLEX,(Funcs ( the carrier of X,COMPLEX)):],(Funcs ( the carrier of X,COMPLEX)):]
[:COMPLEX,(Funcs ( the carrier of X,COMPLEX)):] is non empty set
[:[:COMPLEX,(Funcs ( the carrier of X,COMPLEX)):],(Funcs ( the carrier of X,COMPLEX)):] is non empty set
bool [:[:COMPLEX,(Funcs ( the carrier of X,COMPLEX)):],(Funcs ( the carrier of X,COMPLEX)):] is non empty set
ComplexFuncUnit the carrier of X is Relation-like the carrier of X -defined COMPLEX -valued Function-like total quasi_total V139() Element of Funcs ( the carrier of X,COMPLEX)
the carrier of X --> 1r is non empty Relation-like the carrier of X -defined COMPLEX -valued Function-like total quasi_total V139() Element of bool [: the carrier of X,COMPLEX:]
[: the carrier of X,COMPLEX:] is non empty V139() set
bool [: the carrier of X,COMPLEX:] is non empty set
ComplexFuncZero the carrier of X is Relation-like the carrier of X -defined COMPLEX -valued Function-like total quasi_total V139() Element of Funcs ( the carrier of X,COMPLEX)
the carrier of X --> 0 is non empty Relation-like the carrier of X -defined REAL -valued RAT -valued INT -valued Function-like total quasi_total V139() V140() V141() V142() Element of bool [: the carrier of X,REAL:]
[: the carrier of X,REAL:] is non empty V139() V140() V141() set
bool [: the carrier of X,REAL:] is non empty set
ComplexAlgebraStr(# (Funcs ( the carrier of X,COMPLEX)),(ComplexFuncMult the carrier of X),(ComplexFuncAdd the carrier of X),(ComplexFuncExtMult the carrier of X),(ComplexFuncUnit the carrier of X),(ComplexFuncZero the carrier of X) #) is strict ComplexAlgebraStr
the carrier of (CAlgebra the carrier of X) is non empty set
bool the carrier of (CAlgebra the carrier of X) is non empty set
{ b₁ where b₁ is non empty Relation-like the carrier of X -defined COMPLEX -valued Function-like total quasi_total V139() (X) Element of bool [: the carrier of X,COMPLEX:] : verum } is set
mult_ ((X),(CAlgebra the carrier of X)) is non empty Relation-like [:(X),(X):] -defined (X) -valued Function-like total quasi_total Element of bool [:[:(X),(X):],(X):]
[:(X),(X):] is non empty set
[:[:(X),(X):],(X):] is non empty set
bool [:[:(X),(X):],(X):] is non empty set
Add_ ((X),(CAlgebra the carrier of X)) is non empty Relation-like [:(X),(X):] -defined (X) -valued Function-like total quasi_total Element of bool [:[:(X),(X):],(X):]
Mult_ ((X),(CAlgebra the carrier of X)) is non empty Relation-like [:COMPLEX,(X):] -defined (X) -valued Function-like total quasi_total Element of bool [:[:COMPLEX,(X):],(X):]
[:COMPLEX,(X):] is non empty set
[:[:COMPLEX,(X):],(X):] is non empty set
bool [:[:COMPLEX,(X):],(X):] is non empty set
One_ ((X),(CAlgebra the carrier of X)) is Element of (X)
Zero_ ((X),(CAlgebra the carrier of X)) is Element of (X)
(X) is non empty Relation-like (X) -defined REAL -valued Function-like total quasi_total V139() V140() V141() Element of bool [:(X),REAL:]
[:(X),REAL:] is non empty V139() V140() V141() set
bool [:(X),REAL:] is non empty set
ComplexBoundedFunctions the carrier of X is non empty multiplicatively-closed Cadditively-linearly-closed Element of bool the carrier of (CAlgebra the carrier of X)
{ b₁ where b₁ is non empty Relation-like the carrier of X -defined COMPLEX -valued Function-like total quasi_total V139() Element of bool [: the carrier of X,COMPLEX:] : b₁ | the carrier of X is bounded } is set
ComplexBoundedFunctionsNorm the carrier of X is non empty Relation-like ComplexBoundedFunctions the carrier of X -defined REAL -valued Function-like total quasi_total V139() V140() V141() Element of bool [:(ComplexBoundedFunctions the carrier of X),REAL:]
[:(ComplexBoundedFunctions the carrier of X),REAL:] is non empty V139() V140() V141() set
bool [:(ComplexBoundedFunctions the carrier of X),REAL:] is non empty set
(ComplexBoundedFunctionsNorm the carrier of X) | (X) is Relation-like ComplexBoundedFunctions the carrier of X -defined REAL -valued Function-like V139() V140() V141() Element of bool [:(ComplexBoundedFunctions the carrier of X),REAL:]
Normed_Complex_AlgebraStr(# (X),(mult_ ((X),(CAlgebra the carrier of X))),(Add_ ((X),(CAlgebra the carrier of X))),(Mult_ ((X),(CAlgebra the carrier of X))),(One_ ((X),(CAlgebra the carrier of X))),(Zero_ ((X),(CAlgebra the carrier of X))),(X) #) is strict Normed_Complex_AlgebraStr
the carrier of (X) is non empty set
X is Element of the carrier of (X)
1r * X is Element of the carrier of (X)
the Mult of (X) is non empty Relation-like [:COMPLEX, the carrier of (X):] -defined the carrier of (X) -valued Function-like total quasi_total Element of bool [:[:COMPLEX, the carrier of (X):], the carrier of (X):]
[:COMPLEX, the carrier of (X):] is non empty set
[:[:COMPLEX, the carrier of (X):], the carrier of (X):] is non empty set
bool [:[:COMPLEX, the carrier of (X):], the carrier of (X):] is non empty set
[1r,X] is set
{1r,X} is non empty set
{1r} is non empty V176() set
{{1r,X},{1r}} is non empty set
the Mult of (X) . [1r,X] is set
[:COMPLEX, the carrier of (CAlgebra the carrier of X):] is non empty set
the Mult of (CAlgebra the carrier of X) is non empty Relation-like [:COMPLEX, the carrier of (CAlgebra the carrier of X):] -defined the carrier of (CAlgebra the carrier of X) -valued Function-like total quasi_total Element of bool [:[:COMPLEX, the carrier of (CAlgebra the carrier of X):], the carrier of (CAlgebra the carrier of X):]
[:[:COMPLEX, the carrier of (CAlgebra the carrier of X):], the carrier of (CAlgebra the carrier of X):] is non empty set
bool [:[:COMPLEX, the carrier of (CAlgebra the carrier of X):], the carrier of (CAlgebra the carrier of X):] is non empty set
the Mult of (CAlgebra the carrier of X) | [:COMPLEX,(X):] is Relation-like [:COMPLEX, the carrier of (CAlgebra the carrier of X):] -defined the carrier of (CAlgebra the carrier of X) -valued Function-like Element of bool [:[:COMPLEX, the carrier of (CAlgebra the carrier of X):], the carrier of (CAlgebra the carrier of X):]
F is Element of (X)
[1r,F] is Element of [:COMPLEX,(X):]
{1r,F} is non empty set
{{1r,F},{1r}} is non empty set
( the Mult of (CAlgebra the carrier of X) | [:COMPLEX,(X):]) . [1r,F] is set
the Mult of (CAlgebra the carrier of X) . (1r,F) is Element of the carrier of (CAlgebra the carrier of X)
[1r,F] is set
the Mult of (CAlgebra the carrier of X) . [1r,F] is set
X is non empty TopSpace-like V300() compact TopStruct
(X) is non empty right_complementable Abelian add-associative right_zeroed unital associative commutative right-distributive right_unital well-unital left_unital vector-distributive scalar-distributive scalar-associative scalar-unital vector-associative strict Normed_Complex_AlgebraStr
(X) is non empty multiplicatively-closed Cadditively-linearly-closed Element of bool the carrier of (CAlgebra the carrier of X)
the carrier of X is non empty set
CAlgebra the carrier of X is non empty right_complementable Abelian add-associative right_zeroed unital associative commutative right-distributive right_unital well-unital left_unital vector-distributive scalar-distributive scalar-associative scalar-unital strict vector-associative ComplexAlgebraStr
Funcs ( the carrier of X,COMPLEX) is non empty functional FUNCTION_DOMAIN of the carrier of X, COMPLEX
ComplexFuncMult the carrier of X is non empty Relation-like [:(Funcs ( the carrier of X,COMPLEX)),(Funcs ( the carrier of X,COMPLEX)):] -defined Funcs ( the carrier of X,COMPLEX) -valued Function-like total quasi_total Element of bool [:[:(Funcs ( the carrier of X,COMPLEX)),(Funcs ( the carrier of X,COMPLEX)):],(Funcs ( the carrier of X,COMPLEX)):]
[:(Funcs ( the carrier of X,COMPLEX)),(Funcs ( the carrier of X,COMPLEX)):] is non empty set
[:[:(Funcs ( the carrier of X,COMPLEX)),(Funcs ( the carrier of X,COMPLEX)):],(Funcs ( the carrier of X,COMPLEX)):] is non empty set
bool [:[:(Funcs ( the carrier of X,COMPLEX)),(Funcs ( the carrier of X,COMPLEX)):],(Funcs ( the carrier of X,COMPLEX)):] is non empty set
ComplexFuncAdd the carrier of X is non empty Relation-like [:(Funcs ( the carrier of X,COMPLEX)),(Funcs ( the carrier of X,COMPLEX)):] -defined Funcs ( the carrier of X,COMPLEX) -valued Function-like total quasi_total Element of bool [:[:(Funcs ( the carrier of X,COMPLEX)),(Funcs ( the carrier of X,COMPLEX)):],(Funcs ( the carrier of X,COMPLEX)):]
ComplexFuncExtMult the carrier of X is non empty Relation-like [:COMPLEX,(Funcs ( the carrier of X,COMPLEX)):] -defined Funcs ( the carrier of X,COMPLEX) -valued Function-like total quasi_total Element of bool [:[:COMPLEX,(Funcs ( the carrier of X,COMPLEX)):],(Funcs ( the carrier of X,COMPLEX)):]
[:COMPLEX,(Funcs ( the carrier of X,COMPLEX)):] is non empty set
[:[:COMPLEX,(Funcs ( the carrier of X,COMPLEX)):],(Funcs ( the carrier of X,COMPLEX)):] is non empty set
bool [:[:COMPLEX,(Funcs ( the carrier of X,COMPLEX)):],(Funcs ( the carrier of X,COMPLEX)):] is non empty set
ComplexFuncUnit the carrier of X is Relation-like the carrier of X -defined COMPLEX -valued Function-like total quasi_total V139() Element of Funcs ( the carrier of X,COMPLEX)
the carrier of X --> 1r is non empty Relation-like the carrier of X -defined COMPLEX -valued Function-like total quasi_total V139() Element of bool [: the carrier of X,COMPLEX:]
[: the carrier of X,COMPLEX:] is non empty V139() set
bool [: the carrier of X,COMPLEX:] is non empty set
ComplexFuncZero the carrier of X is Relation-like the carrier of X -defined COMPLEX -valued Function-like total quasi_total V139() Element of Funcs ( the carrier of X,COMPLEX)
the carrier of X --> 0 is non empty Relation-like the carrier of X -defined REAL -valued RAT -valued INT -valued Function-like total quasi_total V139() V140() V141() V142() Element of bool [: the carrier of X,REAL:]
[: the carrier of X,REAL:] is non empty V139() V140() V141() set
bool [: the carrier of X,REAL:] is non empty set
ComplexAlgebraStr(# (Funcs ( the carrier of X,COMPLEX)),(ComplexFuncMult the carrier of X),(ComplexFuncAdd the carrier of X),(ComplexFuncExtMult the carrier of X),(ComplexFuncUnit the carrier of X),(ComplexFuncZero the carrier of X) #) is strict ComplexAlgebraStr
the carrier of (CAlgebra the carrier of X) is non empty set
bool the carrier of (CAlgebra the carrier of X) is non empty set
{ b₁ where b₁ is non empty Relation-like the carrier of X -defined COMPLEX -valued Function-like total quasi_total V139() (X) Element of bool [: the carrier of X,COMPLEX:] : verum } is set
mult_ ((X),(CAlgebra the carrier of X)) is non empty Relation-like [:(X),(X):] -defined (X) -valued Function-like total quasi_total Element of bool [:[:(X),(X):],(X):]
[:(X),(X):] is non empty set
[:[:(X),(X):],(X):] is non empty set
bool [:[:(X),(X):],(X):] is non empty set
Add_ ((X),(CAlgebra the carrier of X)) is non empty Relation-like [:(X),(X):] -defined (X) -valued Function-like total quasi_total Element of bool [:[:(X),(X):],(X):]
Mult_ ((X),(CAlgebra the carrier of X)) is non empty Relation-like [:COMPLEX,(X):] -defined (X) -valued Function-like total quasi_total Element of bool [:[:COMPLEX,(X):],(X):]
[:COMPLEX,(X):] is non empty set
[:[:COMPLEX,(X):],(X):] is non empty set
bool [:[:COMPLEX,(X):],(X):] is non empty set
One_ ((X),(CAlgebra the carrier of X)) is Element of (X)
Zero_ ((X),(CAlgebra the carrier of X)) is Element of (X)
(X) is non empty Relation-like (X) -defined REAL -valued Function-like total quasi_total V139() V140() V141() Element of bool [:(X),REAL:]
[:(X),REAL:] is non empty V139() V140() V141() set
bool [:(X),REAL:] is non empty set
ComplexBoundedFunctions the carrier of X is non empty multiplicatively-closed Cadditively-linearly-closed Element of bool the carrier of (CAlgebra the carrier of X)
{ b₁ where b₁ is non empty Relation-like the carrier of X -defined COMPLEX -valued Function-like total quasi_total V139() Element of bool [: the carrier of X,COMPLEX:] : b₁ | the carrier of X is bounded } is set
ComplexBoundedFunctionsNorm the carrier of X is non empty Relation-like ComplexBoundedFunctions the carrier of X -defined REAL -valued Function-like total quasi_total V139() V140() V141() Element of bool [:(ComplexBoundedFunctions the carrier of X),REAL:]
[:(ComplexBoundedFunctions the carrier of X),REAL:] is non empty V139() V140() V141() set
bool [:(ComplexBoundedFunctions the carrier of X),REAL:] is non empty set
(ComplexBoundedFunctionsNorm the carrier of X) | (X) is Relation-like ComplexBoundedFunctions the carrier of X -defined REAL -valued Function-like V139() V140() V141() Element of bool [:(ComplexBoundedFunctions the carrier of X),REAL:]
Normed_Complex_AlgebraStr(# (X),(mult_ ((X),(CAlgebra the carrier of X))),(Add_ ((X),(CAlgebra the carrier of X))),(Mult_ ((X),(CAlgebra the carrier of X))),(One_ ((X),(CAlgebra the carrier of X))),(Zero_ ((X),(CAlgebra the carrier of X))),(X) #) is strict Normed_Complex_AlgebraStr
X is non empty TopSpace-like V300() compact TopStruct
(X,0) is non empty Relation-like the carrier of X -defined COMPLEX -valued Function-like total quasi_total V139() (X) Element of bool [: the carrier of X,COMPLEX:]
the carrier of X is non empty set
[: the carrier of X,COMPLEX:] is non empty V139() set
bool [: the carrier of X,COMPLEX:] is non empty set
the carrier of X --> 0 is non empty Relation-like the carrier of X -defined RAT -valued INT -valued {0} -valued Function-like total quasi_total V139() V140() V141() V142() Element of bool [: the carrier of X,{0}:]
{0} is non empty V176() V177() V178() V179() V180() V181() left_end bounded_below set
[: the carrier of X,{0}:] is non empty RAT -valued INT -valued V139() V140() V141() V142() set
bool [: the carrier of X,{0}:] is non empty set
(X) is non empty right_complementable Abelian add-associative right_zeroed unital associative commutative right-distributive right_unital well-unital left_unital vector-distributive scalar-distributive scalar-associative scalar-unital vector-associative strict Normed_Complex_AlgebraStr
(X) is non empty multiplicatively-closed Cadditively-linearly-closed Element of bool the carrier of (CAlgebra the carrier of X)
CAlgebra the carrier of X is non empty right_complementable Abelian add-associative right_zeroed unital associative commutative right-distributive right_unital well-unital left_unital vector-distributive scalar-distributive scalar-associative scalar-unital strict vector-associative ComplexAlgebraStr
Funcs ( the carrier of X,COMPLEX) is non empty functional FUNCTION_DOMAIN of the carrier of X, COMPLEX
ComplexFuncMult the carrier of X is non empty Relation-like [:(Funcs ( the carrier of X,COMPLEX)),(Funcs ( the carrier of X,COMPLEX)):] -defined Funcs ( the carrier of X,COMPLEX) -valued Function-like total quasi_total Element of bool [:[:(Funcs ( the carrier of X,COMPLEX)),(Funcs ( the carrier of X,COMPLEX)):],(Funcs ( the carrier of X,COMPLEX)):]
[:(Funcs ( the carrier of X,COMPLEX)),(Funcs ( the carrier of X,COMPLEX)):] is non empty set
[:[:(Funcs ( the carrier of X,COMPLEX)),(Funcs ( the carrier of X,COMPLEX)):],(Funcs ( the carrier of X,COMPLEX)):] is non empty set
bool [:[:(Funcs ( the carrier of X,COMPLEX)),(Funcs ( the carrier of X,COMPLEX)):],(Funcs ( the carrier of X,COMPLEX)):] is non empty set
ComplexFuncAdd the carrier of X is non empty Relation-like [:(Funcs ( the carrier of X,COMPLEX)),(Funcs ( the carrier of X,COMPLEX)):] -defined Funcs ( the carrier of X,COMPLEX) -valued Function-like total quasi_total Element of bool [:[:(Funcs ( the carrier of X,COMPLEX)),(Funcs ( the carrier of X,COMPLEX)):],(Funcs ( the carrier of X,COMPLEX)):]
ComplexFuncExtMult the carrier of X is non empty Relation-like [:COMPLEX,(Funcs ( the carrier of X,COMPLEX)):] -defined Funcs ( the carrier of X,COMPLEX) -valued Function-like total quasi_total Element of bool [:[:COMPLEX,(Funcs ( the carrier of X,COMPLEX)):],(Funcs ( the carrier of X,COMPLEX)):]
[:COMPLEX,(Funcs ( the carrier of X,COMPLEX)):] is non empty set
[:[:COMPLEX,(Funcs ( the carrier of X,COMPLEX)):],(Funcs ( the carrier of X,COMPLEX)):] is non empty set
bool [:[:COMPLEX,(Funcs ( the carrier of X,COMPLEX)):],(Funcs ( the carrier of X,COMPLEX)):] is non empty set
ComplexFuncUnit the carrier of X is Relation-like the carrier of X -defined COMPLEX -valued Function-like total quasi_total V139() Element of Funcs ( the carrier of X,COMPLEX)
the carrier of X --> 1r is non empty Relation-like the carrier of X -defined COMPLEX -valued Function-like total quasi_total V139() Element of bool [: the carrier of X,COMPLEX:]
ComplexFuncZero the carrier of X is Relation-like the carrier of X -defined COMPLEX -valued Function-like total quasi_total V139() Element of Funcs ( the carrier of X,COMPLEX)
the carrier of X --> 0 is non empty Relation-like the carrier of X -defined REAL -valued RAT -valued INT -valued Function-like total quasi_total V139() V140() V141() V142() Element of bool [: the carrier of X,REAL:]
[: the carrier of X,REAL:] is non empty V139() V140() V141() set
bool [: the carrier of X,REAL:] is non empty set
ComplexAlgebraStr(# (Funcs ( the carrier of X,COMPLEX)),(ComplexFuncMult the carrier of X),(ComplexFuncAdd the carrier of X),(ComplexFuncExtMult the carrier of X),(ComplexFuncUnit the carrier of X),(ComplexFuncZero the carrier of X) #) is strict ComplexAlgebraStr
the carrier of (CAlgebra the carrier of X) is non empty set
bool the carrier of (CAlgebra the carrier of X) is non empty set
{ b₁ where b₁ is non empty Relation-like the carrier of X -defined COMPLEX -valued Function-like total quasi_total V139() (X) Element of bool [: the carrier of X,COMPLEX:] : verum } is set
mult_ ((X),(CAlgebra the carrier of X)) is non empty Relation-like [:(X),(X):] -defined (X) -valued Function-like total quasi_total Element of bool [:[:(X),(X):],(X):]
[:(X),(X):] is non empty set
[:[:(X),(X):],(X):] is non empty set
bool [:[:(X),(X):],(X):] is non empty set
Add_ ((X),(CAlgebra the carrier of X)) is non empty Relation-like [:(X),(X):] -defined (X) -valued Function-like total quasi_total Element of bool [:[:(X),(X):],(X):]
Mult_ ((X),(CAlgebra the carrier of X)) is non empty Relation-like [:COMPLEX,(X):] -defined (X) -valued Function-like total quasi_total Element of bool [:[:COMPLEX,(X):],(X):]
[:COMPLEX,(X):] is non empty set
[:[:COMPLEX,(X):],(X):] is non empty set
bool [:[:COMPLEX,(X):],(X):] is non empty set
One_ ((X),(CAlgebra the carrier of X)) is Element of (X)
Zero_ ((X),(CAlgebra the carrier of X)) is Element of (X)
(X) is non empty Relation-like (X) -defined REAL -valued Function-like total quasi_total V139() V140() V141() Element of bool [:(X),REAL:]
[:(X),REAL:] is non empty V139() V140() V141() set
bool [:(X),REAL:] is non empty set
ComplexBoundedFunctions the carrier of X is non empty multiplicatively-closed Cadditively-linearly-closed Element of bool the carrier of (CAlgebra the carrier of X)
{ b₁ where b₁ is non empty Relation-like the carrier of X -defined COMPLEX -valued Function-like total quasi_total V139() Element of bool [: the carrier of X,COMPLEX:] : b₁ | the carrier of X is bounded } is set
ComplexBoundedFunctionsNorm the carrier of X is non empty Relation-like ComplexBoundedFunctions the carrier of X -defined REAL -valued Function-like total quasi_total V139() V140() V141() Element of bool [:(ComplexBoundedFunctions the carrier of X),REAL:]
[:(ComplexBoundedFunctions the carrier of X),REAL:] is non empty V139() V140() V141() set
bool [:(ComplexBoundedFunctions the carrier of X),REAL:] is non empty set
(ComplexBoundedFunctionsNorm the carrier of X) | (X) is Relation-like ComplexBoundedFunctions the carrier of X -defined REAL -valued Function-like V139() V140() V141() Element of bool [:(ComplexBoundedFunctions the carrier of X),REAL:]
Normed_Complex_AlgebraStr(# (X),(mult_ ((X),(CAlgebra the carrier of X))),(Add_ ((X),(CAlgebra the carrier of X))),(Mult_ ((X),(CAlgebra the carrier of X))),(One_ ((X),(CAlgebra the carrier of X))),(Zero_ ((X),(CAlgebra the carrier of X))),(X) #) is strict Normed_Complex_AlgebraStr
0. (X) is zero Element of the carrier of (X)
the carrier of (X) is non empty set
the ZeroF of (X) is Element of the carrier of (X)
(X) is non empty right_complementable Abelian add-associative right_zeroed unital associative commutative right-distributive right_unital well-unital left_unital vector-distributive scalar-distributive scalar-associative scalar-unital strict vector-associative ComplexAlgebraStr
ComplexAlgebraStr(# (X),(mult_ ((X),(CAlgebra the carrier of X))),(Add_ ((X),(CAlgebra the carrier of X))),(Mult_ ((X),(CAlgebra the carrier of X))),(One_ ((X),(CAlgebra the carrier of X))),(Zero_ ((X),(CAlgebra the carrier of X))) #) is strict ComplexAlgebraStr
0. (X) is zero Element of the carrier of (X)
the carrier of (X) is non empty set
the ZeroF of (X) is Element of the carrier of (X)
X is non empty TopSpace-like V300() compact TopStruct
(X) is non empty right_complementable Abelian add-associative right_zeroed unital associative commutative right-distributive right_unital well-unital left_unital vector-distributive scalar-distributive scalar-associative scalar-unital vector-associative strict Normed_Complex_AlgebraStr
(X) is non empty multiplicatively-closed Cadditively-linearly-closed Element of bool the carrier of (CAlgebra the carrier of X)
the carrier of X is non empty set
CAlgebra the carrier of X is non empty right_complementable Abelian add-associative right_zeroed unital associative commutative right-distributive right_unital well-unital left_unital vector-distributive scalar-distributive scalar-associative scalar-unital strict vector-associative ComplexAlgebraStr
Funcs ( the carrier of X,COMPLEX) is non empty functional FUNCTION_DOMAIN of the carrier of X, COMPLEX
ComplexFuncMult the carrier of X is non empty Relation-like [:(Funcs ( the carrier of X,COMPLEX)),(Funcs ( the carrier of X,COMPLEX)):] -defined Funcs ( the carrier of X,COMPLEX) -valued Function-like total quasi_total Element of bool [:[:(Funcs ( the carrier of X,COMPLEX)),(Funcs ( the carrier of X,COMPLEX)):],(Funcs ( the carrier of X,COMPLEX)):]
[:(Funcs ( the carrier of X,COMPLEX)),(Funcs ( the carrier of X,COMPLEX)):] is non empty set
[:[:(Funcs ( the carrier of X,COMPLEX)),(Funcs ( the carrier of X,COMPLEX)):],(Funcs ( the carrier of X,COMPLEX)):] is non empty set
bool [:[:(Funcs ( the carrier of X,COMPLEX)),(Funcs ( the carrier of X,COMPLEX)):],(Funcs ( the carrier of X,COMPLEX)):] is non empty set
ComplexFuncAdd the carrier of X is non empty Relation-like [:(Funcs ( the carrier of X,COMPLEX)),(Funcs ( the carrier of X,COMPLEX)):] -defined Funcs ( the carrier of X,COMPLEX) -valued Function-like total quasi_total Element of bool [:[:(Funcs ( the carrier of X,COMPLEX)),(Funcs ( the carrier of X,COMPLEX)):],(Funcs ( the carrier of X,COMPLEX)):]
ComplexFuncExtMult the carrier of X is non empty Relation-like [:COMPLEX,(Funcs ( the carrier of X,COMPLEX)):] -defined Funcs ( the carrier of X,COMPLEX) -valued Function-like total quasi_total Element of bool [:[:COMPLEX,(Funcs ( the carrier of X,COMPLEX)):],(Funcs ( the carrier of X,COMPLEX)):]
[:COMPLEX,(Funcs ( the carrier of X,COMPLEX)):] is non empty set
[:[:COMPLEX,(Funcs ( the carrier of X,COMPLEX)):],(Funcs ( the carrier of X,COMPLEX)):] is non empty set
bool [:[:COMPLEX,(Funcs ( the carrier of X,COMPLEX)):],(Funcs ( the carrier of X,COMPLEX)):] is non empty set
ComplexFuncUnit the carrier of X is Relation-like the carrier of X -defined COMPLEX -valued Function-like total quasi_total V139() Element of Funcs ( the carrier of X,COMPLEX)
the carrier of X --> 1r is non empty Relation-like the carrier of X -defined COMPLEX -valued Function-like total quasi_total V139() Element of bool [: the carrier of X,COMPLEX:]
[: the carrier of X,COMPLEX:] is non empty V139() set
bool [: the carrier of X,COMPLEX:] is non empty set
ComplexFuncZero the carrier of X is Relation-like the carrier of X -defined COMPLEX -valued Function-like total quasi_total V139() Element of Funcs ( the carrier of X,COMPLEX)
the carrier of X --> 0 is non empty Relation-like the carrier of X -defined REAL -valued RAT -valued INT -valued Function-like total quasi_total V139() V140() V141() V142() Element of bool [: the carrier of X,REAL:]
[: the carrier of X,REAL:] is non empty V139() V140() V141() set
bool [: the carrier of X,REAL:] is non empty set
ComplexAlgebraStr(# (Funcs ( the carrier of X,COMPLEX)),(ComplexFuncMult the carrier of X),(ComplexFuncAdd the carrier of X),(ComplexFuncExtMult the carrier of X),(ComplexFuncUnit the carrier of X),(ComplexFuncZero the carrier of X) #) is strict ComplexAlgebraStr
the carrier of (CAlgebra the carrier of X) is non empty set
bool the carrier of (CAlgebra the carrier of X) is non empty set
{ b₁ where b₁ is non empty Relation-like the carrier of X -defined COMPLEX -valued Function-like total quasi_total V139() (X) Element of bool [: the carrier of X,COMPLEX:] : verum } is set
mult_ ((X),(CAlgebra the carrier of X)) is non empty Relation-like [:(X),(X):] -defined (X) -valued Function-like total quasi_total Element of bool [:[:(X),(X):],(X):]
[:(X),(X):] is non empty set
[:[:(X),(X):],(X):] is non empty set
bool [:[:(X),(X):],(X):] is non empty set
Add_ ((X),(CAlgebra the carrier of X)) is non empty Relation-like [:(X),(X):] -defined (X) -valued Function-like total quasi_total Element of bool [:[:(X),(X):],(X):]
Mult_ ((X),(CAlgebra the carrier of X)) is non empty Relation-like [:COMPLEX,(X):] -defined (X) -valued Function-like total quasi_total Element of bool [:[:COMPLEX,(X):],(X):]
[:COMPLEX,(X):] is non empty set
[:[:COMPLEX,(X):],(X):] is non empty set
bool [:[:COMPLEX,(X):],(X):] is non empty set
One_ ((X),(CAlgebra the carrier of X)) is Element of (X)
Zero_ ((X),(CAlgebra the carrier of X)) is Element of (X)
(X) is non empty Relation-like (X) -defined REAL -valued Function-like total quasi_total V139() V140() V141() Element of bool [:(X),REAL:]
[:(X),REAL:] is non empty V139() V140() V141() set
bool [:(X),REAL:] is non empty set
ComplexBoundedFunctions the carrier of X is non empty multiplicatively-closed Cadditively-linearly-closed Element of bool the carrier of (CAlgebra the carrier of X)
{ b₁ where b₁ is non empty Relation-like the carrier of X -defined COMPLEX -valued Function-like total quasi_total V139() Element of bool [: the carrier of X,COMPLEX:] : b₁ | the carrier of X is bounded } is set
ComplexBoundedFunctionsNorm the carrier of X is non empty Relation-like ComplexBoundedFunctions the carrier of X -defined REAL -valued Function-like total quasi_total V139() V140() V141() Element of bool [:(ComplexBoundedFunctions the carrier of X),REAL:]
[:(ComplexBoundedFunctions the carrier of X),REAL:] is non empty V139() V140() V141() set
bool [:(ComplexBoundedFunctions the carrier of X),REAL:] is non empty set
(ComplexBoundedFunctionsNorm the carrier of X) | (X) is Relation-like ComplexBoundedFunctions the carrier of X -defined REAL -valued Function-like V139() V140() V141() Element of bool [:(ComplexBoundedFunctions the carrier of X),REAL:]
Normed_Complex_AlgebraStr(# (X),(mult_ ((X),(CAlgebra the carrier of X))),(Add_ ((X),(CAlgebra the carrier of X))),(Mult_ ((X),(CAlgebra the carrier of X))),(One_ ((X),(CAlgebra the carrier of X))),(Zero_ ((X),(CAlgebra the carrier of X))),(X) #) is strict Normed_Complex_AlgebraStr
0. (X) is zero Element of the carrier of (X)
the carrier of (X) is non empty set
the ZeroF of (X) is Element of the carrier of (X)
C_Normed_Algebra_of_BoundedFunctions the carrier of X is non empty right_complementable Abelian add-associative right_zeroed unital discerning reflexive vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like complete Normed_Complex_AlgebraStr
mult_ ((ComplexBoundedFunctions the carrier of X),(CAlgebra the carrier of X)) is non empty Relation-like [:(ComplexBoundedFunctions the carrier of X),(ComplexBoundedFunctions the carrier of X):] -defined ComplexBoundedFunctions the carrier of X -valued Function-like total quasi_total Element of bool [:[:(ComplexBoundedFunctions the carrier of X),(ComplexBoundedFunctions the carrier of X):],(ComplexBoundedFunctions the carrier of X):]
[:(ComplexBoundedFunctions the carrier of X),(ComplexBoundedFunctions the carrier of X):] is non empty set
[:[:(ComplexBoundedFunctions the carrier of X),(ComplexBoundedFunctions the carrier of X):],(ComplexBoundedFunctions the carrier of X):] is non empty set
bool [:[:(ComplexBoundedFunctions the carrier of X),(ComplexBoundedFunctions the carrier of X):],(ComplexBoundedFunctions the carrier of X):] is non empty set
Add_ ((ComplexBoundedFunctions the carrier of X),(CAlgebra the carrier of X)) is non empty Relation-like [:(ComplexBoundedFunctions the carrier of X),(ComplexBoundedFunctions the carrier of X):] -defined ComplexBoundedFunctions the carrier of X -valued Function-like total quasi_total Element of bool [:[:(ComplexBoundedFunctions the carrier of X),(ComplexBoundedFunctions the carrier of X):],(ComplexBoundedFunctions the carrier of X):]
Mult_ ((ComplexBoundedFunctions the carrier of X),(CAlgebra the carrier of X)) is non empty Relation-like [:COMPLEX,(ComplexBoundedFunctions the carrier of X):] -defined ComplexBoundedFunctions the carrier of X -valued Function-like total quasi_total Element of bool [:[:COMPLEX,(ComplexBoundedFunctions the carrier of X):],(ComplexBoundedFunctions the carrier of X):]
[:COMPLEX,(ComplexBoundedFunctions the carrier of X):] is non empty set
[:[:COMPLEX,(ComplexBoundedFunctions the carrier of X):],(ComplexBoundedFunctions the carrier of X):] is non empty set
bool [:[:COMPLEX,(ComplexBoundedFunctions the carrier of X):],(ComplexBoundedFunctions the carrier of X):] is non empty set
One_ ((ComplexBoundedFunctions the carrier of X),(CAlgebra the carrier of X)) is Element of ComplexBoundedFunctions the carrier of X
Zero_ ((ComplexBoundedFunctions the carrier of X),(CAlgebra the carrier of X)) is Element of ComplexBoundedFunctions the carrier of X
Normed_Complex_AlgebraStr(# (ComplexBoundedFunctions the carrier of X),(mult_ ((ComplexBoundedFunctions the carrier of X),(CAlgebra the carrier of X))),(Add_ ((ComplexBoundedFunctions the carrier of X),(CAlgebra the carrier of X))),(Mult_ ((ComplexBoundedFunctions the carrier of X),(CAlgebra the carrier of X))),(One_ ((ComplexBoundedFunctions the carrier of X),(CAlgebra the carrier of X))),(Zero_ ((ComplexBoundedFunctions the carrier of X),(CAlgebra the carrier of X))),(ComplexBoundedFunctionsNorm the carrier of X) #) is strict Normed_Complex_AlgebraStr
0. (C_Normed_Algebra_of_BoundedFunctions the carrier of X) is zero Element of the carrier of (C_Normed_Algebra_of_BoundedFunctions the carrier of X)
the carrier of (C_Normed_Algebra_of_BoundedFunctions the carrier of X) is non empty set
the ZeroF of (C_Normed_Algebra_of_BoundedFunctions the carrier of X) is Element of the carrier of (C_Normed_Algebra_of_BoundedFunctions the carrier of X)
(X,0) is non empty Relation-like the carrier of X -defined COMPLEX -valued Function-like total quasi_total V139() (X) Element of bool [: the carrier of X,COMPLEX:]
the carrier of X --> 0 is non empty Relation-like the carrier of X -defined RAT -valued INT -valued {0} -valued Function-like total quasi_total V139() V140() V141() V142() Element of bool [: the carrier of X,{0}:]
{0} is non empty V176() V177() V178() V179() V180() V181() left_end bounded_below set
[: the carrier of X,{0}:] is non empty RAT -valued INT -valued V139() V140() V141() V142() set
bool [: the carrier of X,{0}:] is non empty set
X is non empty TopSpace-like V300() compact TopStruct
(X) is non empty right_complementable Abelian add-associative right_zeroed unital associative commutative right-distributive right_unital well-unital left_unital vector-distributive scalar-distributive scalar-associative scalar-unital vector-associative strict Normed_Complex_AlgebraStr
(X) is non empty multiplicatively-closed Cadditively-linearly-closed Element of bool the carrier of (CAlgebra the carrier of X)
the carrier of X is non empty set
CAlgebra the carrier of X is non empty right_complementable Abelian add-associative right_zeroed unital associative commutative right-distributive right_unital well-unital left_unital vector-distributive scalar-distributive scalar-associative scalar-unital strict vector-associative ComplexAlgebraStr
Funcs ( the carrier of X,COMPLEX) is non empty functional FUNCTION_DOMAIN of the carrier of X, COMPLEX
ComplexFuncMult the carrier of X is non empty Relation-like [:(Funcs ( the carrier of X,COMPLEX)),(Funcs ( the carrier of X,COMPLEX)):] -defined Funcs ( the carrier of X,COMPLEX) -valued Function-like total quasi_total Element of bool [:[:(Funcs ( the carrier of X,COMPLEX)),(Funcs ( the carrier of X,COMPLEX)):],(Funcs ( the carrier of X,COMPLEX)):]
[:(Funcs ( the carrier of X,COMPLEX)),(Funcs ( the carrier of X,COMPLEX)):] is non empty set
[:[:(Funcs ( the carrier of X,COMPLEX)),(Funcs ( the carrier of X,COMPLEX)):],(Funcs ( the carrier of X,COMPLEX)):] is non empty set
bool [:[:(Funcs ( the carrier of X,COMPLEX)),(Funcs ( the carrier of X,COMPLEX)):],(Funcs ( the carrier of X,COMPLEX)):] is non empty set
ComplexFuncAdd the carrier of X is non empty Relation-like [:(Funcs ( the carrier of X,COMPLEX)),(Funcs ( the carrier of X,COMPLEX)):] -defined Funcs ( the carrier of X,COMPLEX) -valued Function-like total quasi_total Element of bool [:[:(Funcs ( the carrier of X,COMPLEX)),(Funcs ( the carrier of X,COMPLEX)):],(Funcs ( the carrier of X,COMPLEX)):]
ComplexFuncExtMult the carrier of X is non empty Relation-like [:COMPLEX,(Funcs ( the carrier of X,COMPLEX)):] -defined Funcs ( the carrier of X,COMPLEX) -valued Function-like total quasi_total Element of bool [:[:COMPLEX,(Funcs ( the carrier of X,COMPLEX)):],(Funcs ( the carrier of X,COMPLEX)):]
[:COMPLEX,(Funcs ( the carrier of X,COMPLEX)):] is non empty set
[:[:COMPLEX,(Funcs ( the carrier of X,COMPLEX)):],(Funcs ( the carrier of X,COMPLEX)):] is non empty set
bool [:[:COMPLEX,(Funcs ( the carrier of X,COMPLEX)):],(Funcs ( the carrier of X,COMPLEX)):] is non empty set
ComplexFuncUnit the carrier of X is Relation-like the carrier of X -defined COMPLEX -valued Function-like total quasi_total V139() Element of Funcs ( the carrier of X,COMPLEX)
the carrier of X --> 1r is non empty Relation-like the carrier of X -defined COMPLEX -valued Function-like total quasi_total V139() Element of bool [: the carrier of X,COMPLEX:]
[: the carrier of X,COMPLEX:] is non empty V139() set
bool [: the carrier of X,COMPLEX:] is non empty set
ComplexFuncZero the carrier of X is Relation-like the carrier of X -defined COMPLEX -valued Function-like total quasi_total V139() Element of Funcs ( the carrier of X,COMPLEX)
the carrier of X --> 0 is non empty Relation-like the carrier of X -defined REAL -valued RAT -valued INT -valued Function-like total quasi_total V139() V140() V141() V142() Element of bool [: the carrier of X,REAL:]
[: the carrier of X,REAL:] is non empty V139() V140() V141() set
bool [: the carrier of X,REAL:] is non empty set
ComplexAlgebraStr(# (Funcs ( the carrier of X,COMPLEX)),(ComplexFuncMult the carrier of X),(ComplexFuncAdd the carrier of X),(ComplexFuncExtMult the carrier of X),(ComplexFuncUnit the carrier of X),(ComplexFuncZero the carrier of X) #) is strict ComplexAlgebraStr
the carrier of (CAlgebra the carrier of X) is non empty set
bool the carrier of (CAlgebra the carrier of X) is non empty set
{ b₁ where b₁ is non empty Relation-like the carrier of X -defined COMPLEX -valued Function-like total quasi_total V139() (X) Element of bool [: the carrier of X,COMPLEX:] : verum } is set
mult_ ((X),(CAlgebra the carrier of X)) is non empty Relation-like [:(X),(X):] -defined (X) -valued Function-like total quasi_total Element of bool [:[:(X),(X):],(X):]
[:(X),(X):] is non empty set
[:[:(X),(X):],(X):] is non empty set
bool [:[:(X),(X):],(X):] is non empty set
Add_ ((X),(CAlgebra the carrier of X)) is non empty Relation-like [:(X),(X):] -defined (X) -valued Function-like total quasi_total Element of bool [:[:(X),(X):],(X):]
Mult_ ((X),(CAlgebra the carrier of X)) is non empty Relation-like [:COMPLEX,(X):] -defined (X) -valued Function-like total quasi_total Element of bool [:[:COMPLEX,(X):],(X):]
[:COMPLEX,(X):] is non empty set
[:[:COMPLEX,(X):],(X):] is non empty set
bool [:[:COMPLEX,(X):],(X):] is non empty set
One_ ((X),(CAlgebra the carrier of X)) is Element of (X)
Zero_ ((X),(CAlgebra the carrier of X)) is Element of (X)
(X) is non empty Relation-like (X) -defined REAL -valued Function-like total quasi_total V139() V140() V141() Element of bool [:(X),REAL:]
[:(X),REAL:] is non empty V139() V140() V141() set
bool [:(X),REAL:] is non empty set
ComplexBoundedFunctions the carrier of X is non empty multiplicatively-closed Cadditively-linearly-closed Element of bool the carrier of (CAlgebra the carrier of X)
{ b₁ where b₁ is non empty Relation-like the carrier of X -defined COMPLEX -valued Function-like total quasi_total V139() Element of bool [: the carrier of X,COMPLEX:] : b₁ | the carrier of X is bounded } is set
ComplexBoundedFunctionsNorm the carrier of X is non empty Relation-like ComplexBoundedFunctions the carrier of X -defined REAL -valued Function-like total quasi_total V139() V140() V141() Element of bool [:(ComplexBoundedFunctions the carrier of X),REAL:]
[:(ComplexBoundedFunctions the carrier of X),REAL:] is non empty V139() V140() V141() set
bool [:(ComplexBoundedFunctions the carrier of X),REAL:] is non empty set
(ComplexBoundedFunctionsNorm the carrier of X) | (X) is Relation-like ComplexBoundedFunctions the carrier of X -defined REAL -valued Function-like V139() V140() V141() Element of bool [:(ComplexBoundedFunctions the carrier of X),REAL:]
Normed_Complex_AlgebraStr(# (X),(mult_ ((X),(CAlgebra the carrier of X))),(Add_ ((X),(CAlgebra the carrier of X))),(Mult_ ((X),(CAlgebra the carrier of X))),(One_ ((X),(CAlgebra the carrier of X))),(Zero_ ((X),(CAlgebra the carrier of X))),(X) #) is strict Normed_Complex_AlgebraStr
1. (X) is Element of the carrier of (X)
the carrier of (X) is non empty set
the OneF of (X) is Element of the carrier of (X)
C_Normed_Algebra_of_BoundedFunctions the carrier of X is non empty right_complementable Abelian add-associative right_zeroed unital discerning reflexive vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like complete Normed_Complex_AlgebraStr
mult_ ((ComplexBoundedFunctions the carrier of X),(CAlgebra the carrier of X)) is non empty Relation-like [:(ComplexBoundedFunctions the carrier of X),(ComplexBoundedFunctions the carrier of X):] -defined ComplexBoundedFunctions the carrier of X -valued Function-like total quasi_total Element of bool [:[:(ComplexBoundedFunctions the carrier of X),(ComplexBoundedFunctions the carrier of X):],(ComplexBoundedFunctions the carrier of X):]
[:(ComplexBoundedFunctions the carrier of X),(ComplexBoundedFunctions the carrier of X):] is non empty set
[:[:(ComplexBoundedFunctions the carrier of X),(ComplexBoundedFunctions the carrier of X):],(ComplexBoundedFunctions the carrier of X):] is non empty set
bool [:[:(ComplexBoundedFunctions the carrier of X),(ComplexBoundedFunctions the carrier of X):],(ComplexBoundedFunctions the carrier of X):] is non empty set
Add_ ((ComplexBoundedFunctions the carrier of X),(CAlgebra the carrier of X)) is non empty Relation-like [:(ComplexBoundedFunctions the carrier of X),(ComplexBoundedFunctions the carrier of X):] -defined ComplexBoundedFunctions the carrier of X -valued Function-like total quasi_total Element of bool [:[:(ComplexBoundedFunctions the carrier of X),(ComplexBoundedFunctions the carrier of X):],(ComplexBoundedFunctions the carrier of X):]
Mult_ ((ComplexBoundedFunctions the carrier of X),(CAlgebra the carrier of X)) is non empty Relation-like [:COMPLEX,(ComplexBoundedFunctions the carrier of X):] -defined ComplexBoundedFunctions the carrier of X -valued Function-like total quasi_total Element of bool [:[:COMPLEX,(ComplexBoundedFunctions the carrier of X):],(ComplexBoundedFunctions the carrier of X):]
[:COMPLEX,(ComplexBoundedFunctions the carrier of X):] is non empty set
[:[:COMPLEX,(ComplexBoundedFunctions the carrier of X):],(ComplexBoundedFunctions the carrier of X):] is non empty set
bool [:[:COMPLEX,(ComplexBoundedFunctions the carrier of X):],(ComplexBoundedFunctions the carrier of X):] is non empty set
One_ ((ComplexBoundedFunctions the carrier of X),(CAlgebra the carrier of X)) is Element of ComplexBoundedFunctions the carrier of X
Zero_ ((ComplexBoundedFunctions the carrier of X),(CAlgebra the carrier of X)) is Element of ComplexBoundedFunctions the carrier of X
Normed_Complex_AlgebraStr(# (ComplexBoundedFunctions the carrier of X),(mult_ ((ComplexBoundedFunctions the carrier of X),(CAlgebra the carrier of X))),(Add_ ((ComplexBoundedFunctions the carrier of X),(CAlgebra the carrier of X))),(Mult_ ((ComplexBoundedFunctions the carrier of X),(CAlgebra the carrier of X))),(One_ ((ComplexBoundedFunctions the carrier of X),(CAlgebra the carrier of X))),(Zero_ ((ComplexBoundedFunctions the carrier of X),(CAlgebra the carrier of X))),(ComplexBoundedFunctionsNorm the carrier of X) #) is strict Normed_Complex_AlgebraStr
1. (C_Normed_Algebra_of_BoundedFunctions the carrier of X) is Element of the carrier of (C_Normed_Algebra_of_BoundedFunctions the carrier of X)
the carrier of (C_Normed_Algebra_of_BoundedFunctions the carrier of X) is non empty set
the OneF of (C_Normed_Algebra_of_BoundedFunctions the carrier of X) is Element of the carrier of (C_Normed_Algebra_of_BoundedFunctions the carrier of X)
(X) is non empty right_complementable Abelian add-associative right_zeroed unital associative commutative right-distributive right_unital well-unital left_unital vector-distributive scalar-distributive scalar-associative scalar-unital strict vector-associative ComplexAlgebraStr
ComplexAlgebraStr(# (X),(mult_ ((X),(CAlgebra the carrier of X))),(Add_ ((X),(CAlgebra the carrier of X))),(Mult_ ((X),(CAlgebra the carrier of X))),(One_ ((X),(CAlgebra the carrier of X))),(Zero_ ((X),(CAlgebra the carrier of X))) #) is strict ComplexAlgebraStr
1_ (X) is Element of the carrier of (X)
the carrier of (X) is non empty set
1. (X) is Element of the carrier of (X)
the OneF of (X) is Element of the carrier of (X)
(X,1r) is non empty Relation-like the carrier of X -defined COMPLEX -valued Function-like total quasi_total V139() (X) Element of bool [: the carrier of X,COMPLEX:]
the carrier of X --> 1r is non empty Relation-like the carrier of X -defined {1r} -valued Function-like total quasi_total V139() Element of bool [: the carrier of X,{1r}:]
{1r} is non empty V176() set
[: the carrier of X,{1r}:] is non empty V139() set
bool [: the carrier of X,{1r}:] is non empty set
C_Algebra_of_BoundedFunctions the carrier of X is non empty right_complementable Abelian add-associative right_zeroed unital associative commutative right-distributive right_unital well-unital left_unital vector-distributive scalar-distributive scalar-associative scalar-unital vector-associative ComplexAlgebraStr
ComplexAlgebraStr(# (ComplexBoundedFunctions the carrier of X),(mult_ ((ComplexBoundedFunctions the carrier of X),(CAlgebra the carrier of X))),(Add_ ((ComplexBoundedFunctions the carrier of X),(CAlgebra the carrier of X))),(Mult_ ((ComplexBoundedFunctions the carrier of X),(CAlgebra the carrier of X))),(One_ ((ComplexBoundedFunctions the carrier of X),(CAlgebra the carrier of X))),(Zero_ ((ComplexBoundedFunctions the carrier of X),(CAlgebra the carrier of X))) #) is strict ComplexAlgebraStr
1_ (C_Algebra_of_BoundedFunctions the carrier of X) is Element of the carrier of (C_Algebra_of_BoundedFunctions the carrier of X)
the carrier of (C_Algebra_of_BoundedFunctions the carrier of X) is non empty set
1. (C_Algebra_of_BoundedFunctions the carrier of X) is Element of the carrier of (C_Algebra_of_BoundedFunctions the carrier of X)
the OneF of (C_Algebra_of_BoundedFunctions the carrier of X) is Element of the carrier of (C_Algebra_of_BoundedFunctions the carrier of X)
X is non empty TopSpace-like V300() compact TopStruct
(X) is non empty right_complementable Abelian add-associative right_zeroed unital associative commutative right-distributive right_unital well-unital left_unital vector-distributive scalar-distributive scalar-associative scalar-unital vector-associative strict Normed_Complex_AlgebraStr
(X) is non empty multiplicatively-closed Cadditively-linearly-closed Element of bool the carrier of (CAlgebra the carrier of X)
the carrier of X is non empty set
CAlgebra the carrier of X is non empty right_complementable Abelian add-associative right_zeroed unital associative commutative right-distributive right_unital well-unital left_unital vector-distributive scalar-distributive scalar-associative scalar-unital strict vector-associative ComplexAlgebraStr
Funcs ( the carrier of X,COMPLEX) is non empty functional FUNCTION_DOMAIN of the carrier of X, COMPLEX
ComplexFuncMult the carrier of X is non empty Relation-like [:(Funcs ( the carrier of X,COMPLEX)),(Funcs ( the carrier of X,COMPLEX)):] -defined Funcs ( the carrier of X,COMPLEX) -valued Function-like total quasi_total Element of bool [:[:(Funcs ( the carrier of X,COMPLEX)),(Funcs ( the carrier of X,COMPLEX)):],(Funcs ( the carrier of X,COMPLEX)):]
[:(Funcs ( the carrier of X,COMPLEX)),(Funcs ( the carrier of X,COMPLEX)):] is non empty set
[:[:(Funcs ( the carrier of X,COMPLEX)),(Funcs ( the carrier of X,COMPLEX)):],(Funcs ( the carrier of X,COMPLEX)):] is non empty set
bool [:[:(Funcs ( the carrier of X,COMPLEX)),(Funcs ( the carrier of X,COMPLEX)):],(Funcs ( the carrier of X,COMPLEX)):] is non empty set
ComplexFuncAdd the carrier of X is non empty Relation-like [:(Funcs ( the carrier of X,COMPLEX)),(Funcs ( the carrier of X,COMPLEX)):] -defined Funcs ( the carrier of X,COMPLEX) -valued Function-like total quasi_total Element of bool [:[:(Funcs ( the carrier of X,COMPLEX)),(Funcs ( the carrier of X,COMPLEX)):],(Funcs ( the carrier of X,COMPLEX)):]
ComplexFuncExtMult the carrier of X is non empty Relation-like [:COMPLEX,(Funcs ( the carrier of X,COMPLEX)):] -defined Funcs ( the carrier of X,COMPLEX) -valued Function-like total quasi_total Element of bool [:[:COMPLEX,(Funcs ( the carrier of X,COMPLEX)):],(Funcs ( the carrier of X,COMPLEX)):]
[:COMPLEX,(Funcs ( the carrier of X,COMPLEX)):] is non empty set
[:[:COMPLEX,(Funcs ( the carrier of X,COMPLEX)):],(Funcs ( the carrier of X,COMPLEX)):] is non empty set
bool [:[:COMPLEX,(Funcs ( the carrier of X,COMPLEX)):],(Funcs ( the carrier of X,COMPLEX)):] is non empty set
ComplexFuncUnit the carrier of X is Relation-like the carrier of X -defined COMPLEX -valued Function-like total quasi_total V139() Element of Funcs ( the carrier of X,COMPLEX)
the carrier of X --> 1r is non empty Relation-like the carrier of X -defined COMPLEX -valued Function-like total quasi_total V139() Element of bool [: the carrier of X,COMPLEX:]
[: the carrier of X,COMPLEX:] is non empty V139() set
bool [: the carrier of X,COMPLEX:] is non empty set
ComplexFuncZero the carrier of X is Relation-like the carrier of X -defined COMPLEX -valued Function-like total quasi_total V139() Element of Funcs ( the carrier of X,COMPLEX)
the carrier of X --> 0 is non empty Relation-like the carrier of X -defined REAL -valued RAT -valued INT -valued Function-like total quasi_total V139() V140() V141() V142() Element of bool [: the carrier of X,REAL:]
[: the carrier of X,REAL:] is non empty V139() V140() V141() set
bool [: the carrier of X,REAL:] is non empty set
ComplexAlgebraStr(# (Funcs ( the carrier of X,COMPLEX)),(ComplexFuncMult the carrier of X),(ComplexFuncAdd the carrier of X),(ComplexFuncExtMult the carrier of X),(ComplexFuncUnit the carrier of X),(ComplexFuncZero the carrier of X) #) is strict ComplexAlgebraStr
the carrier of (CAlgebra the carrier of X) is non empty set
bool the carrier of (CAlgebra the carrier of X) is non empty set
{ b₁ where b₁ is non empty Relation-like the carrier of X -defined COMPLEX -valued Function-like total quasi_total V139() (X) Element of bool [: the carrier of X,COMPLEX:] : verum } is set
mult_ ((X),(CAlgebra the carrier of X)) is non empty Relation-like [:(X),(X):] -defined (X) -valued Function-like total quasi_total Element of bool [:[:(X),(X):],(X):]
[:(X),(X):] is non empty set
[:[:(X),(X):],(X):] is non empty set
bool [:[:(X),(X):],(X):] is non empty set
Add_ ((X),(CAlgebra the carrier of X)) is non empty Relation-like [:(X),(X):] -defined (X) -valued Function-like total quasi_total Element of bool [:[:(X),(X):],(X):]
Mult_ ((X),(CAlgebra the carrier of X)) is non empty Relation-like [:COMPLEX,(X):] -defined (X) -valued Function-like total quasi_total Element of bool [:[:COMPLEX,(X):],(X):]
[:COMPLEX,(X):] is non empty set
[:[:COMPLEX,(X):],(X):] is non empty set
bool [:[:COMPLEX,(X):],(X):] is non empty set
One_ ((X),(CAlgebra the carrier of X)) is Element of (X)
Zero_ ((X),(CAlgebra the carrier of X)) is Element of (X)
(X) is non empty Relation-like (X) -defined REAL -valued Function-like total quasi_total V139() V140() V141() Element of bool [:(X),REAL:]
[:(X),REAL:] is non empty V139() V140() V141() set
bool [:(X),REAL:] is non empty set
ComplexBoundedFunctions the carrier of X is non empty multiplicatively-closed Cadditively-linearly-closed Element of bool the carrier of (CAlgebra the carrier of X)
{ b₁ where b₁ is non empty Relation-like the carrier of X -defined COMPLEX -valued Function-like total quasi_total V139() Element of bool [: the carrier of X,COMPLEX:] : b₁ | the carrier of X is bounded } is set
ComplexBoundedFunctionsNorm the carrier of X is non empty Relation-like ComplexBoundedFunctions the carrier of X -defined REAL -valued Function-like total quasi_total V139() V140() V141() Element of bool [:(ComplexBoundedFunctions the carrier of X),REAL:]
[:(ComplexBoundedFunctions the carrier of X),REAL:] is non empty V139() V140() V141() set
bool [:(ComplexBoundedFunctions the carrier of X),REAL:] is non empty set
(ComplexBoundedFunctionsNorm the carrier of X) | (X) is Relation-like ComplexBoundedFunctions the carrier of X -defined REAL -valued Function-like V139() V140() V141() Element of bool [:(ComplexBoundedFunctions the carrier of X),REAL:]
Normed_Complex_AlgebraStr(# (X),(mult_ ((X),(CAlgebra the carrier of X))),(Add_ ((X),(CAlgebra the carrier of X))),(Mult_ ((X),(CAlgebra the carrier of X))),(One_ ((X),(CAlgebra the carrier of X))),(Zero_ ((X),(CAlgebra the carrier of X))),(X) #) is strict Normed_Complex_AlgebraStr
the carrier of (X) is non empty set
X is Element of the carrier of (X)
||.X.|| is complex real ext-real Element of REAL
the U9 of (X) is non empty Relation-like the carrier of (X) -defined REAL -valued Function-like total quasi_total V139() V140() V141() Element of bool [: the carrier of (X),REAL:]
[: the carrier of (X),REAL:] is non empty V139() V140() V141() set
bool [: the carrier of (X),REAL:] is non empty set
the U9 of (X) . X is complex real ext-real Element of REAL
C_Normed_Algebra_of_BoundedFunctions the carrier of X is non empty right_complementable Abelian add-associative right_zeroed unital discerning reflexive vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like complete Normed_Complex_AlgebraStr
mult_ ((ComplexBoundedFunctions the carrier of X),(CAlgebra the carrier of X)) is non empty Relation-like [:(ComplexBoundedFunctions the carrier of X),(ComplexBoundedFunctions the carrier of X):] -defined ComplexBoundedFunctions the carrier of X -valued Function-like total quasi_total Element of bool [:[:(ComplexBoundedFunctions the carrier of X),(ComplexBoundedFunctions the carrier of X):],(ComplexBoundedFunctions the carrier of X):]
[:(ComplexBoundedFunctions the carrier of X),(ComplexBoundedFunctions the carrier of X):] is non empty set
[:[:(ComplexBoundedFunctions the carrier of X),(ComplexBoundedFunctions the carrier of X):],(ComplexBoundedFunctions the carrier of X):] is non empty set
bool [:[:(ComplexBoundedFunctions the carrier of X),(ComplexBoundedFunctions the carrier of X):],(ComplexBoundedFunctions the carrier of X):] is non empty set
Add_ ((ComplexBoundedFunctions the carrier of X),(CAlgebra the carrier of X)) is non empty Relation-like [:(ComplexBoundedFunctions the carrier of X),(ComplexBoundedFunctions the carrier of X):] -defined ComplexBoundedFunctions the carrier of X -valued Function-like total quasi_total Element of bool [:[:(ComplexBoundedFunctions the carrier of X),(ComplexBoundedFunctions the carrier of X):],(ComplexBoundedFunctions the carrier of X):]
Mult_ ((ComplexBoundedFunctions the carrier of X),(CAlgebra the carrier of X)) is non empty Relation-like [:COMPLEX,(ComplexBoundedFunctions the carrier of X):] -defined ComplexBoundedFunctions the carrier of X -valued Function-like total quasi_total Element of bool [:[:COMPLEX,(ComplexBoundedFunctions the carrier of X):],(ComplexBoundedFunctions the carrier of X):]
[:COMPLEX,(ComplexBoundedFunctions the carrier of X):] is non empty set
[:[:COMPLEX,(ComplexBoundedFunctions the carrier of X):],(ComplexBoundedFunctions the carrier of X):] is non empty set
bool [:[:COMPLEX,(ComplexBoundedFunctions the carrier of X):],(ComplexBoundedFunctions the carrier of X):] is non empty set
One_ ((ComplexBoundedFunctions the carrier of X),(CAlgebra the carrier of X)) is Element of ComplexBoundedFunctions the carrier of X
Zero_ ((ComplexBoundedFunctions the carrier of X),(CAlgebra the carrier of X)) is Element of ComplexBoundedFunctions the carrier of X
Normed_Complex_AlgebraStr(# (ComplexBoundedFunctions the carrier of X),(mult_ ((ComplexBoundedFunctions the carrier of X),(CAlgebra the carrier of X))),(Add_ ((ComplexBoundedFunctions the carrier of X),(CAlgebra the carrier of X))),(Mult_ ((ComplexBoundedFunctions the carrier of X),(CAlgebra the carrier of X))),(One_ ((ComplexBoundedFunctions the carrier of X),(CAlgebra the carrier of X))),(Zero_ ((ComplexBoundedFunctions the carrier of X),(CAlgebra the carrier of X))),(ComplexBoundedFunctionsNorm the carrier of X) #) is strict Normed_Complex_AlgebraStr
the carrier of (C_Normed_Algebra_of_BoundedFunctions the carrier of X) is non empty set
F is Element of the carrier of (C_Normed_Algebra_of_BoundedFunctions the carrier of X)
||.F.|| is complex real ext-real Element of REAL
the U9 of (C_Normed_Algebra_of_BoundedFunctions the carrier of X) is non empty Relation-like the carrier of (C_Normed_Algebra_of_BoundedFunctions the carrier of X) -defined REAL -valued Function-like total quasi_total V139() V140() V141() Element of bool [: the carrier of (C_Normed_Algebra_of_BoundedFunctions the carrier of X),REAL:]
[: the carrier of (C_Normed_Algebra_of_BoundedFunctions the carrier of X),REAL:] is non empty V139() V140() V141() set
bool [: the carrier of (C_Normed_Algebra_of_BoundedFunctions the carrier of X),REAL:] is non empty set
the U9 of (C_Normed_Algebra_of_BoundedFunctions the carrier of X) . F is complex real ext-real Element of REAL
X is non empty TopSpace-like V300() compact TopStruct
the carrier of X is non empty set
[: the carrier of X,COMPLEX:] is non empty V139() set
bool [: the carrier of X,COMPLEX:] is non empty set
(X) is non empty right_complementable Abelian add-associative right_zeroed unital associative commutative right-distributive right_unital well-unital left_unital vector-distributive scalar-distributive scalar-associative scalar-unital vector-associative strict Normed_Complex_AlgebraStr
(X) is non empty multiplicatively-closed Cadditively-linearly-closed Element of bool the carrier of (CAlgebra the carrier of X)
CAlgebra the carrier of X is non empty right_complementable Abelian add-associative right_zeroed unital associative commutative right-distributive right_unital well-unital left_unital vector-distributive scalar-distributive scalar-associative scalar-unital strict vector-associative ComplexAlgebraStr
Funcs ( the carrier of X,COMPLEX) is non empty functional FUNCTION_DOMAIN of the carrier of X, COMPLEX
ComplexFuncMult the carrier of X is non empty Relation-like [:(Funcs ( the carrier of X,COMPLEX)),(Funcs ( the carrier of X,COMPLEX)):] -defined Funcs ( the carrier of X,COMPLEX) -valued Function-like total quasi_total Element of bool [:[:(Funcs ( the carrier of X,COMPLEX)),(Funcs ( the carrier of X,COMPLEX)):],(Funcs ( the carrier of X,COMPLEX)):]
[:(Funcs ( the carrier of X,COMPLEX)),(Funcs ( the carrier of X,COMPLEX)):] is non empty set
[:[:(Funcs ( the carrier of X,COMPLEX)),(Funcs ( the carrier of X,COMPLEX)):],(Funcs ( the carrier of X,COMPLEX)):] is non empty set
bool [:[:(Funcs ( the carrier of X,COMPLEX)),(Funcs ( the carrier of X,COMPLEX)):],(Funcs ( the carrier of X,COMPLEX)):] is non empty set
ComplexFuncAdd the carrier of X is non empty Relation-like [:(Funcs ( the carrier of X,COMPLEX)),(Funcs ( the carrier of X,COMPLEX)):] -defined Funcs ( the carrier of X,COMPLEX) -valued Function-like total quasi_total Element of bool [:[:(Funcs ( the carrier of X,COMPLEX)),(Funcs ( the carrier of X,COMPLEX)):],(Funcs ( the carrier of X,COMPLEX)):]
ComplexFuncExtMult the carrier of X is non empty Relation-like [:COMPLEX,(Funcs ( the carrier of X,COMPLEX)):] -defined Funcs ( the carrier of X,COMPLEX) -valued Function-like total quasi_total Element of bool [:[:COMPLEX,(Funcs ( the carrier of X,COMPLEX)):],(Funcs ( the carrier of X,COMPLEX)):]
[:COMPLEX,(Funcs ( the carrier of X,COMPLEX)):] is non empty set
[:[:COMPLEX,(Funcs ( the carrier of X,COMPLEX)):],(Funcs ( the carrier of X,COMPLEX)):] is non empty set
bool [:[:COMPLEX,(Funcs ( the carrier of X,COMPLEX)):],(Funcs ( the carrier of X,COMPLEX)):] is non empty set
ComplexFuncUnit the carrier of X is Relation-like the carrier of X -defined COMPLEX -valued Function-like total quasi_total V139() Element of Funcs ( the carrier of X,COMPLEX)
the carrier of X --> 1r is non empty Relation-like the carrier of X -defined COMPLEX -valued Function-like total quasi_total V139() Element of bool [: the carrier of X,COMPLEX:]
ComplexFuncZero the carrier of X is Relation-like the carrier of X -defined COMPLEX -valued Function-like total quasi_total V139() Element of Funcs ( the carrier of X,COMPLEX)
the carrier of X --> 0 is non empty Relation-like the carrier of X -defined REAL -valued RAT -valued INT -valued Function-like total quasi_total V139() V140() V141() V142() Element of bool [: the carrier of X,REAL:]
[: the carrier of X,REAL:] is non empty V139() V140() V141() set
bool [: the carrier of X,REAL:] is non empty set
ComplexAlgebraStr(# (Funcs ( the carrier of X,COMPLEX)),(ComplexFuncMult the carrier of X),(ComplexFuncAdd the carrier of X),(ComplexFuncExtMult the carrier of X),(ComplexFuncUnit the carrier of X),(ComplexFuncZero the carrier of X) #) is strict ComplexAlgebraStr
the carrier of (CAlgebra the carrier of X) is non empty set
bool the carrier of (CAlgebra the carrier of X) is non empty set
{ b₁ where b₁ is non empty Relation-like the carrier of X -defined COMPLEX -valued Function-like total quasi_total V139() (X) Element of bool [: the carrier of X,COMPLEX:] : verum } is set
mult_ ((X),(CAlgebra the carrier of X)) is non empty Relation-like [:(X),(X):] -defined (X) -valued Function-like total quasi_total Element of bool [:[:(X),(X):],(X):]
[:(X),(X):] is non empty set
[:[:(X),(X):],(X):] is non empty set
bool [:[:(X),(X):],(X):] is non empty set
Add_ ((X),(CAlgebra the carrier of X)) is non empty Relation-like [:(X),(X):] -defined (X) -valued Function-like total quasi_total Element of bool [:[:(X),(X):],(X):]
Mult_ ((X),(CAlgebra the carrier of X)) is non empty Relation-like [:COMPLEX,(X):] -defined (X) -valued Function-like total quasi_total Element of bool [:[:COMPLEX,(X):],(X):]
[:COMPLEX,(X):] is non empty set
[:[:COMPLEX,(X):],(X):] is non empty set
bool [:[:COMPLEX,(X):],(X):] is non empty set
One_ ((X),(CAlgebra the carrier of X)) is Element of (X)
Zero_ ((X),(CAlgebra the carrier of X)) is Element of (X)
(X) is non empty Relation-like (X) -defined REAL -valued Function-like total quasi_total V139() V140() V141() Element of bool [:(X),REAL:]
[:(X),REAL:] is non empty V139() V140() V141() set
bool [:(X),REAL:] is non empty set
ComplexBoundedFunctions the carrier of X is non empty multiplicatively-closed Cadditively-linearly-closed Element of bool the carrier of (CAlgebra the carrier of X)
{ b₁ where b₁ is non empty Relation-like the carrier of X -defined COMPLEX -valued Function-like total quasi_total V139() Element of bool [: the carrier of X,COMPLEX:] : b₁ | the carrier of X is bounded } is set
ComplexBoundedFunctionsNorm the carrier of X is non empty Relation-like ComplexBoundedFunctions the carrier of X -defined REAL -valued Function-like total quasi_total V139() V140() V141() Element of bool [:(ComplexBoundedFunctions the carrier of X),REAL:]
[:(ComplexBoundedFunctions the carrier of X),REAL:] is non empty V139() V140() V141() set
bool [:(ComplexBoundedFunctions the carrier of X),REAL:] is non empty set
(ComplexBoundedFunctionsNorm the carrier of X) | (X) is Relation-like ComplexBoundedFunctions the carrier of X -defined REAL -valued Function-like V139() V140() V141() Element of bool [:(ComplexBoundedFunctions the carrier of X),REAL:]
Normed_Complex_AlgebraStr(# (X),(mult_ ((X),(CAlgebra the carrier of X))),(Add_ ((X),(CAlgebra the carrier of X))),(Mult_ ((X),(CAlgebra the carrier of X))),(One_ ((X),(CAlgebra the carrier of X))),(Zero_ ((X),(CAlgebra the carrier of X))),(X) #) is strict Normed_Complex_AlgebraStr
the carrier of (X) is non empty set
X is non empty Relation-like the carrier of X -defined COMPLEX -valued Function-like total quasi_total V139() Element of bool [: the carrier of X,COMPLEX:]
F is non empty Relation-like the carrier of X -defined COMPLEX -valued Function-like total quasi_total V139() Element of bool [: the carrier of X,COMPLEX:]
G is non empty Relation-like the carrier of X -defined COMPLEX -valued Function-like total quasi_total V139() Element of bool [: the carrier of X,COMPLEX:]
FB is Element of the carrier of (X)
GB is Element of the carrier of (X)
aFB is Element of the carrier of (X)
FB + GB is Element of the carrier of (X)
the addF of (X) is non empty Relation-like [: the carrier of (X), the carrier of (X):] -defined the carrier of (X) -valued Function-like total quasi_total Element of bool [:[: the carrier of (X), the carrier of (X):], the carrier of (X):]
[: the carrier of (X), the carrier of (X):] is non empty set
[:[: the carrier of (X), the carrier of (X):], the carrier of (X):] is non empty set
bool [:[: the carrier of (X), the carrier of (X):], the carrier of (X):] is non empty set
the addF of (X) . (FB,GB) is Element of the carrier of (X)
[FB,GB] is set
{FB,GB} is non empty set
{FB} is non empty set
{{FB,GB},{FB}} is non empty set
the addF of (X) . [FB,GB] is set
(X) is non empty right_complementable Abelian add-associative right_zeroed unital associative commutative right-distributive right_unital well-unital left_unital vector-distributive scalar-distributive scalar-associative scalar-unital strict vector-associative ComplexAlgebraStr
ComplexAlgebraStr(# (X),(mult_ ((X),(CAlgebra the carrier of X))),(Add_ ((X),(CAlgebra the carrier of X))),(Mult_ ((X),(CAlgebra the carrier of X))),(One_ ((X),(CAlgebra the carrier of X))),(Zero_ ((X),(CAlgebra the carrier of X))) #) is strict ComplexAlgebraStr
the carrier of (X) is non empty set
r is Element of the carrier of (X)
aF is Element of the carrier of (X)
aG is Element of the carrier of (X)
aF + aG is Element of the carrier of (X)
the addF of (X) is non empty Relation-like [: the carrier of (X), the carrier of (X):] -defined the carrier of (X) -valued Function-like total quasi_total Element of bool [:[: the carrier of (X), the carrier of (X):], the carrier of (X):]
[: the carrier of (X), the carrier of (X):] is non empty set
[:[: the carrier of (X), the carrier of (X):], the carrier of (X):] is non empty set
bool [:[: the carrier of (X), the carrier of (X):], the carrier of (X):] is non empty set
the addF of (X) . (aF,aG) is Element of the carrier of (X)
[aF,aG] is set
{aF,aG} is non empty set
{aF} is non empty set
{{aF,aG},{aF}} is non empty set
the addF of (X) . [aF,aG] is set
r is Element of the carrier of X
G . r is complex Element of COMPLEX
X . r is complex Element of COMPLEX
F . r is complex Element of COMPLEX
(X . r) + (F . r) is complex Element of COMPLEX
X is complex set
X is non empty TopSpace-like V300() compact TopStruct
the carrier of X is non empty set
[: the carrier of X,COMPLEX:] is non empty V139() set
bool [: the carrier of X,COMPLEX:] is non empty set
(X) is non empty right_complementable Abelian add-associative right_zeroed unital associative commutative right-distributive right_unital well-unital left_unital vector-distributive scalar-distributive scalar-associative scalar-unital vector-associative strict Normed_Complex_AlgebraStr
(X) is non empty multiplicatively-closed Cadditively-linearly-closed Element of bool the carrier of (CAlgebra the carrier of X)
CAlgebra the carrier of X is non empty right_complementable Abelian add-associative right_zeroed unital associative commutative right-distributive right_unital well-unital left_unital vector-distributive scalar-distributive scalar-associative scalar-unital strict vector-associative ComplexAlgebraStr
Funcs ( the carrier of X,COMPLEX) is non empty functional FUNCTION_DOMAIN of the carrier of X, COMPLEX
ComplexFuncMult the carrier of X is non empty Relation-like [:(Funcs ( the carrier of X,COMPLEX)),(Funcs ( the carrier of X,COMPLEX)):] -defined Funcs ( the carrier of X,COMPLEX) -valued Function-like total quasi_total Element of bool [:[:(Funcs ( the carrier of X,COMPLEX)),(Funcs ( the carrier of X,COMPLEX)):],(Funcs ( the carrier of X,COMPLEX)):]
[:(Funcs ( the carrier of X,COMPLEX)),(Funcs ( the carrier of X,COMPLEX)):] is non empty set
[:[:(Funcs ( the carrier of X,COMPLEX)),(Funcs ( the carrier of X,COMPLEX)):],(Funcs ( the carrier of X,COMPLEX)):] is non empty set
bool [:[:(Funcs ( the carrier of X,COMPLEX)),(Funcs ( the carrier of X,COMPLEX)):],(Funcs ( the carrier of X,COMPLEX)):] is non empty set
ComplexFuncAdd the carrier of X is non empty Relation-like [:(Funcs ( the carrier of X,COMPLEX)),(Funcs ( the carrier of X,COMPLEX)):] -defined Funcs ( the carrier of X,COMPLEX) -valued Function-like total quasi_total Element of bool [:[:(Funcs ( the carrier of X,COMPLEX)),(Funcs ( the carrier of X,COMPLEX)):],(Funcs ( the carrier of X,COMPLEX)):]
ComplexFuncExtMult the carrier of X is non empty Relation-like [:COMPLEX,(Funcs ( the carrier of X,COMPLEX)):] -defined Funcs ( the carrier of X,COMPLEX) -valued Function-like total quasi_total Element of bool [:[:COMPLEX,(Funcs ( the carrier of X,COMPLEX)):],(Funcs ( the carrier of X,COMPLEX)):]
[:COMPLEX,(Funcs ( the carrier of X,COMPLEX)):] is non empty set
[:[:COMPLEX,(Funcs ( the carrier of X,COMPLEX)):],(Funcs ( the carrier of X,COMPLEX)):] is non empty set
bool [:[:COMPLEX,(Funcs ( the carrier of X,COMPLEX)):],(Funcs ( the carrier of X,COMPLEX)):] is non empty set
ComplexFuncUnit the carrier of X is Relation-like the carrier of X -defined COMPLEX -valued Function-like total quasi_total V139() Element of Funcs ( the carrier of X,COMPLEX)
the carrier of X --> 1r is non empty Relation-like the carrier of X -defined COMPLEX -valued Function-like total quasi_total V139() Element of bool [: the carrier of X,COMPLEX:]
ComplexFuncZero the carrier of X is Relation-like the carrier of X -defined COMPLEX -valued Function-like total quasi_total V139() Element of Funcs ( the carrier of X,COMPLEX)
the carrier of X --> 0 is non empty Relation-like the carrier of X -defined REAL -valued RAT -valued INT -valued Function-like total quasi_total V139() V140() V141() V142() Element of bool [: the carrier of X,REAL:]
[: the carrier of X,REAL:] is non empty V139() V140() V141() set
bool [: the carrier of X,REAL:] is non empty set
ComplexAlgebraStr(# (Funcs ( the carrier of X,COMPLEX)),(ComplexFuncMult the carrier of X),(ComplexFuncAdd the carrier of X),(ComplexFuncExtMult the carrier of X),(ComplexFuncUnit the carrier of X),(ComplexFuncZero the carrier of X) #) is strict ComplexAlgebraStr
the carrier of (CAlgebra the carrier of X) is non empty set
bool the carrier of (CAlgebra the carrier of X) is non empty set
{ b₁ where b₁ is non empty Relation-like the carrier of X -defined COMPLEX -valued Function-like total quasi_total V139() (X) Element of bool [: the carrier of X,COMPLEX:] : verum } is set
mult_ ((X),(CAlgebra the carrier of X)) is non empty Relation-like [:(X),(X):] -defined (X) -valued Function-like total quasi_total Element of bool [:[:(X),(X):],(X):]
[:(X),(X):] is non empty set
[:[:(X),(X):],(X):] is non empty set
bool [:[:(X),(X):],(X):] is non empty set
Add_ ((X),(CAlgebra the carrier of X)) is non empty Relation-like [:(X),(X):] -defined (X) -valued Function-like total quasi_total Element of bool [:[:(X),(X):],(X):]
Mult_ ((X),(CAlgebra the carrier of X)) is non empty Relation-like [:COMPLEX,(X):] -defined (X) -valued Function-like total quasi_total Element of bool [:[:COMPLEX,(X):],(X):]
[:COMPLEX,(X):] is non empty set
[:[:COMPLEX,(X):],(X):] is non empty set
bool [:[:COMPLEX,(X):],(X):] is non empty set
One_ ((X),(CAlgebra the carrier of X)) is Element of (X)
Zero_ ((X),(CAlgebra the carrier of X)) is Element of (X)
(X) is non empty Relation-like (X) -defined REAL -valued Function-like total quasi_total V139() V140() V141() Element of bool [:(X),REAL:]
[:(X),REAL:] is non empty V139() V140() V141() set
bool [:(X),REAL:] is non empty set
ComplexBoundedFunctions the carrier of X is non empty multiplicatively-closed Cadditively-linearly-closed Element of bool the carrier of (CAlgebra the carrier of X)
{ b₁ where b₁ is non empty Relation-like the carrier of X -defined COMPLEX -valued Function-like total quasi_total V139() Element of bool [: the carrier of X,COMPLEX:] : b₁ | the carrier of X is bounded } is set
ComplexBoundedFunctionsNorm the carrier of X is non empty Relation-like ComplexBoundedFunctions the carrier of X -defined REAL -valued Function-like total quasi_total V139() V140() V141() Element of bool [:(ComplexBoundedFunctions the carrier of X),REAL:]
[:(ComplexBoundedFunctions the carrier of X),REAL:] is non empty V139() V140() V141() set
bool [:(ComplexBoundedFunctions the carrier of X),REAL:] is non empty set
(ComplexBoundedFunctionsNorm the carrier of X) | (X) is Relation-like ComplexBoundedFunctions the carrier of X -defined REAL -valued Function-like V139() V140() V141() Element of bool [:(ComplexBoundedFunctions the carrier of X),REAL:]
Normed_Complex_AlgebraStr(# (X),(mult_ ((X),(CAlgebra the carrier of X))),(Add_ ((X),(CAlgebra the carrier of X))),(Mult_ ((X),(CAlgebra the carrier of X))),(One_ ((X),(CAlgebra the carrier of X))),(Zero_ ((X),(CAlgebra the carrier of X))),(X) #) is strict Normed_Complex_AlgebraStr
the carrier of (X) is non empty set
F is non empty Relation-like the carrier of X -defined COMPLEX -valued Function-like total quasi_total V139() Element of bool [: the carrier of X,COMPLEX:]
G is non empty Relation-like the carrier of X -defined COMPLEX -valued Function-like total quasi_total V139() Element of bool [: the carrier of X,COMPLEX:]
FB is Element of the carrier of (X)
GB is Element of the carrier of (X)
X * FB is Element of the carrier of (X)
the Mult of (X) is non empty Relation-like [:COMPLEX, the carrier of (X):] -defined the carrier of (X) -valued Function-like total quasi_total Element of bool [:[:COMPLEX, the carrier of (X):], the carrier of (X):]
[:COMPLEX, the carrier of (X):] is non empty set
[:[:COMPLEX, the carrier of (X):], the carrier of (X):] is non empty set
bool [:[:COMPLEX, the carrier of (X):], the carrier of (X):] is non empty set
[X,FB] is set
{X,FB} is non empty set
{X} is non empty V176() set
{{X,FB},{X}} is non empty set
the Mult of (X) . [X,FB] is set
(X) is non empty right_complementable Abelian add-associative right_zeroed unital associative commutative right-distributive right_unital well-unital left_unital vector-distributive scalar-distributive scalar-associative scalar-unital strict vector-associative ComplexAlgebraStr
ComplexAlgebraStr(# (X),(mult_ ((X),(CAlgebra the carrier of X))),(Add_ ((X),(CAlgebra the carrier of X))),(Mult_ ((X),(CAlgebra the carrier of X))),(One_ ((X),(CAlgebra the carrier of X))),(Zero_ ((X),(CAlgebra the carrier of X))) #) is strict ComplexAlgebraStr
the carrier of (X) is non empty set
aF is Element of the carrier of (X)
aFB is Element of the carrier of (X)
X * aFB is Element of the carrier of (X)
the Mult of (X) is non empty Relation-like [:COMPLEX, the carrier of (X):] -defined the carrier of (X) -valued Function-like total quasi_total Element of bool [:[:COMPLEX, the carrier of (X):], the carrier of (X):]
[:COMPLEX, the carrier of (X):] is non empty set
[:[:COMPLEX, the carrier of (X):], the carrier of (X):] is non empty set
bool [:[:COMPLEX, the carrier of (X):], the carrier of (X):] is non empty set
[X,aFB] is set
{X,aFB} is non empty set
{{X,aFB},{X}} is non empty set
the Mult of (X) . [X,aFB] is set
aG is Element of the carrier of X
G . aG is complex Element of COMPLEX
F . aG is complex Element of COMPLEX
X * (F . aG) is complex set
X is non empty TopSpace-like V300() compact TopStruct
the carrier of X is non empty set
[: the carrier of X,COMPLEX:] is non empty V139() set
bool [: the carrier of X,COMPLEX:] is non empty set
(X) is non empty right_complementable Abelian add-associative right_zeroed unital associative commutative right-distributive right_unital well-unital left_unital vector-distributive scalar-distributive scalar-associative scalar-unital vector-associative strict Normed_Complex_AlgebraStr
(X) is non empty multiplicatively-closed Cadditively-linearly-closed Element of bool the carrier of (CAlgebra the carrier of X)
CAlgebra the carrier of X is non empty right_complementable Abelian add-associative right_zeroed unital associative commutative right-distributive right_unital well-unital left_unital vector-distributive scalar-distributive scalar-associative scalar-unital strict vector-associative ComplexAlgebraStr
Funcs ( the carrier of X,COMPLEX) is non empty functional FUNCTION_DOMAIN of the carrier of X, COMPLEX
ComplexFuncMult the carrier of X is non empty Relation-like [:(Funcs ( the carrier of X,COMPLEX)),(Funcs ( the carrier of X,COMPLEX)):] -defined Funcs ( the carrier of X,COMPLEX) -valued Function-like total quasi_total Element of bool [:[:(Funcs ( the carrier of X,COMPLEX)),(Funcs ( the carrier of X,COMPLEX)):],(Funcs ( the carrier of X,COMPLEX)):]
[:(Funcs ( the carrier of X,COMPLEX)),(Funcs ( the carrier of X,COMPLEX)):] is non empty set
[:[:(Funcs ( the carrier of X,COMPLEX)),(Funcs ( the carrier of X,COMPLEX)):],(Funcs ( the carrier of X,COMPLEX)):] is non empty set
bool [:[:(Funcs ( the carrier of X,COMPLEX)),(Funcs ( the carrier of X,COMPLEX)):],(Funcs ( the carrier of X,COMPLEX)):] is non empty set
ComplexFuncAdd the carrier of X is non empty Relation-like [:(Funcs ( the carrier of X,COMPLEX)),(Funcs ( the carrier of X,COMPLEX)):] -defined Funcs ( the carrier of X,COMPLEX) -valued Function-like total quasi_total Element of bool [:[:(Funcs ( the carrier of X,COMPLEX)),(Funcs ( the carrier of X,COMPLEX)):],(Funcs ( the carrier of X,COMPLEX)):]
ComplexFuncExtMult the carrier of X is non empty Relation-like [:COMPLEX,(Funcs ( the carrier of X,COMPLEX)):] -defined Funcs ( the carrier of X,COMPLEX) -valued Function-like total quasi_total Element of bool [:[:COMPLEX,(Funcs ( the carrier of X,COMPLEX)):],(Funcs ( the carrier of X,COMPLEX)):]
[:COMPLEX,(Funcs ( the carrier of X,COMPLEX)):] is non empty set
[:[:COMPLEX,(Funcs ( the carrier of X,COMPLEX)):],(Funcs ( the carrier of X,COMPLEX)):] is non empty set
bool [:[:COMPLEX,(Funcs ( the carrier of X,COMPLEX)):],(Funcs ( the carrier of X,COMPLEX)):] is non empty set
ComplexFuncUnit the carrier of X is Relation-like the carrier of X -defined COMPLEX -valued Function-like total quasi_total V139() Element of Funcs ( the carrier of X,COMPLEX)
the carrier of X --> 1r is non empty Relation-like the carrier of X -defined COMPLEX -valued Function-like total quasi_total V139() Element of bool [: the carrier of X,COMPLEX:]
ComplexFuncZero the carrier of X is Relation-like the carrier of X -defined COMPLEX -valued Function-like total quasi_total V139() Element of Funcs ( the carrier of X,COMPLEX)
the carrier of X --> 0 is non empty Relation-like the carrier of X -defined REAL -valued RAT -valued INT -valued Function-like total quasi_total V139() V140() V141() V142() Element of bool [: the carrier of X,REAL:]
[: the carrier of X,REAL:] is non empty V139() V140() V141() set
bool [: the carrier of X,REAL:] is non empty set
ComplexAlgebraStr(# (Funcs ( the carrier of X,COMPLEX)),(ComplexFuncMult the carrier of X),(ComplexFuncAdd the carrier of X),(ComplexFuncExtMult the carrier of X),(ComplexFuncUnit the carrier of X),(ComplexFuncZero the carrier of X) #) is strict ComplexAlgebraStr
the carrier of (CAlgebra the carrier of X) is non empty set
bool the carrier of (CAlgebra the carrier of X) is non empty set
{ b₁ where b₁ is non empty Relation-like the carrier of X -defined COMPLEX -valued Function-like total quasi_total V139() (X) Element of bool [: the carrier of X,COMPLEX:] : verum } is set
mult_ ((X),(CAlgebra the carrier of X)) is non empty Relation-like [:(X),(X):] -defined (X) -valued Function-like total quasi_total Element of bool [:[:(X),(X):],(X):]
[:(X),(X):] is non empty set
[:[:(X),(X):],(X):] is non empty set
bool [:[:(X),(X):],(X):] is non empty set
Add_ ((X),(CAlgebra the carrier of X)) is non empty Relation-like [:(X),(X):] -defined (X) -valued Function-like total quasi_total Element of bool [:[:(X),(X):],(X):]
Mult_ ((X),(CAlgebra the carrier of X)) is non empty Relation-like [:COMPLEX,(X):] -defined (X) -valued Function-like total quasi_total Element of bool [:[:COMPLEX,(X):],(X):]
[:COMPLEX,(X):] is non empty set
[:[:COMPLEX,(X):],(X):] is non empty set
bool [:[:COMPLEX,(X):],(X):] is non empty set
One_ ((X),(CAlgebra the carrier of X)) is Element of (X)
Zero_ ((X),(CAlgebra the carrier of X)) is Element of (X)
(X) is non empty Relation-like (X) -defined REAL -valued Function-like total quasi_total V139() V140() V141() Element of bool [:(X),REAL:]
[:(X),REAL:] is non empty V139() V140() V141() set
bool [:(X),REAL:] is non empty set
ComplexBoundedFunctions the carrier of X is non empty multiplicatively-closed Cadditively-linearly-closed Element of bool the carrier of (CAlgebra the carrier of X)
{ b₁ where b₁ is non empty Relation-like the carrier of X -defined COMPLEX -valued Function-like total quasi_total V139() Element of bool [: the carrier of X,COMPLEX:] : b₁ | the carrier of X is bounded } is set
ComplexBoundedFunctionsNorm the carrier of X is non empty Relation-like ComplexBoundedFunctions the carrier of X -defined REAL -valued Function-like total quasi_total V139() V140() V141() Element of bool [:(ComplexBoundedFunctions the carrier of X),REAL:]
[:(ComplexBoundedFunctions the carrier of X),REAL:] is non empty V139() V140() V141() set
bool [:(ComplexBoundedFunctions the carrier of X),REAL:] is non empty set
(ComplexBoundedFunctionsNorm the carrier of X) | (X) is Relation-like ComplexBoundedFunctions the carrier of X -defined REAL -valued Function-like V139() V140() V141() Element of bool [:(ComplexBoundedFunctions the carrier of X),REAL:]
Normed_Complex_AlgebraStr(# (X),(mult_ ((X),(CAlgebra the carrier of X))),(Add_ ((X),(CAlgebra the carrier of X))),(Mult_ ((X),(CAlgebra the carrier of X))),(One_ ((X),(CAlgebra the carrier of X))),(Zero_ ((X),(CAlgebra the carrier of X))),(X) #) is strict Normed_Complex_AlgebraStr
the carrier of (X) is non empty set
X is non empty Relation-like the carrier of X -defined COMPLEX -valued Function-like total quasi_total V139() Element of bool [: the carrier of X,COMPLEX:]
F is non empty Relation-like the carrier of X -defined COMPLEX -valued Function-like total quasi_total V139() Element of bool [: the carrier of X,COMPLEX:]
G is non empty Relation-like the carrier of X -defined COMPLEX -valued Function-like total quasi_total V139() Element of bool [: the carrier of X,COMPLEX:]
FB is Element of the carrier of (X)
GB is Element of the carrier of (X)
aFB is Element of the carrier of (X)
FB * GB is Element of the carrier of (X)
the multF of (X) is non empty Relation-like [: the carrier of (X), the carrier of (X):] -defined the carrier of (X) -valued Function-like total quasi_total Element of bool [:[: the carrier of (X), the carrier of (X):], the carrier of (X):]
[: the carrier of (X), the carrier of (X):] is non empty set
[:[: the carrier of (X), the carrier of (X):], the carrier of (X):] is non empty set
bool [:[: the carrier of (X), the carrier of (X):], the carrier of (X):] is non empty set
the multF of (X) . (FB,GB) is Element of the carrier of (X)
[FB,GB] is set
{FB,GB} is non empty set
{FB} is non empty set
{{FB,GB},{FB}} is non empty set
the multF of (X) . [FB,GB] is set
(X) is non empty right_complementable Abelian add-associative right_zeroed unital associative commutative right-distributive right_unital well-unital left_unital vector-distributive scalar-distributive scalar-associative scalar-unital strict vector-associative ComplexAlgebraStr
ComplexAlgebraStr(# (X),(mult_ ((X),(CAlgebra the carrier of X))),(Add_ ((X),(CAlgebra the carrier of X))),(Mult_ ((X),(CAlgebra the carrier of X))),(One_ ((X),(CAlgebra the carrier of X))),(Zero_ ((X),(CAlgebra the carrier of X))) #) is strict ComplexAlgebraStr
the carrier of (X) is non empty set
r is Element of the carrier of (X)
aF is Element of the carrier of (X)
aG is Element of the carrier of (X)
aF * aG is Element of the carrier of (X)
the multF of (X) is non empty Relation-like [: the carrier of (X), the carrier of (X):] -defined the carrier of (X) -valued Function-like total quasi_total Element of bool [:[: the carrier of (X), the carrier of (X):], the carrier of (X):]
[: the carrier of (X), the carrier of (X):] is non empty set
[:[: the carrier of (X), the carrier of (X):], the carrier of (X):] is non empty set
bool [:[: the carrier of (X), the carrier of (X):], the carrier of (X):] is non empty set
the multF of (X) . (aF,aG) is Element of the carrier of (X)
[aF,aG] is set
{aF,aG} is non empty set
{aF} is non empty set
{{aF,aG},{aF}} is non empty set
the multF of (X) . [aF,aG] is set
r is Element of the carrier of X
G . r is complex Element of COMPLEX
X . r is complex Element of COMPLEX
F . r is complex Element of COMPLEX
(X . r) * (F . r) is complex Element of COMPLEX
X is non empty TopSpace-like V300() compact TopStruct
(X) is non empty right_complementable Abelian add-associative right_zeroed unital associative commutative right-distributive right_unital well-unital left_unital vector-distributive scalar-distributive scalar-associative scalar-unital vector-associative strict Normed_Complex_AlgebraStr
(X) is non empty multiplicatively-closed Cadditively-linearly-closed Element of bool the carrier of (CAlgebra the carrier of X)
the carrier of X is non empty set
CAlgebra the carrier of X is non empty right_complementable Abelian add-associative right_zeroed unital associative commutative right-distributive right_unital well-unital left_unital vector-distributive scalar-distributive scalar-associative scalar-unital strict vector-associative ComplexAlgebraStr
Funcs ( the carrier of X,COMPLEX) is non empty functional FUNCTION_DOMAIN of the carrier of X, COMPLEX
ComplexFuncMult the carrier of X is non empty Relation-like [:(Funcs ( the carrier of X,COMPLEX)),(Funcs ( the carrier of X,COMPLEX)):] -defined Funcs ( the carrier of X,COMPLEX) -valued Function-like total quasi_total Element of bool [:[:(Funcs ( the carrier of X,COMPLEX)),(Funcs ( the carrier of X,COMPLEX)):],(Funcs ( the carrier of X,COMPLEX)):]
[:(Funcs ( the carrier of X,COMPLEX)),(Funcs ( the carrier of X,COMPLEX)):] is non empty set
[:[:(Funcs ( the carrier of X,COMPLEX)),(Funcs ( the carrier of X,COMPLEX)):],(Funcs ( the carrier of X,COMPLEX)):] is non empty set
bool [:[:(Funcs ( the carrier of X,COMPLEX)),(Funcs ( the carrier of X,COMPLEX)):],(Funcs ( the carrier of X,COMPLEX)):] is non empty set
ComplexFuncAdd the carrier of X is non empty Relation-like [:(Funcs ( the carrier of X,COMPLEX)),(Funcs ( the carrier of X,COMPLEX)):] -defined Funcs ( the carrier of X,COMPLEX) -valued Function-like total quasi_total Element of bool [:[:(Funcs ( the carrier of X,COMPLEX)),(Funcs ( the carrier of X,COMPLEX)):],(Funcs ( the carrier of X,COMPLEX)):]
ComplexFuncExtMult the carrier of X is non empty Relation-like [:COMPLEX,(Funcs ( the carrier of X,COMPLEX)):] -defined Funcs ( the carrier of X,COMPLEX) -valued Function-like total quasi_total Element of bool [:[:COMPLEX,(Funcs ( the carrier of X,COMPLEX)):],(Funcs ( the carrier of X,COMPLEX)):]
[:COMPLEX,(Funcs ( the carrier of X,COMPLEX)):] is non empty set
[:[:COMPLEX,(Funcs ( the carrier of X,COMPLEX)):],(Funcs ( the carrier of X,COMPLEX)):] is non empty set
bool [:[:COMPLEX,(Funcs ( the carrier of X,COMPLEX)):],(Funcs ( the carrier of X,COMPLEX)):] is non empty set
ComplexFuncUnit the carrier of X is Relation-like the carrier of X -defined COMPLEX -valued Function-like total quasi_total V139() Element of Funcs ( the carrier of X,COMPLEX)
the carrier of X --> 1r is non empty Relation-like the carrier of X -defined COMPLEX -valued Function-like total quasi_total V139() Element of bool [: the carrier of X,COMPLEX:]
[: the carrier of X,COMPLEX:] is non empty V139() set
bool [: the carrier of X,COMPLEX:] is non empty set
ComplexFuncZero the carrier of X is Relation-like the carrier of X -defined COMPLEX -valued Function-like total quasi_total V139() Element of Funcs ( the carrier of X,COMPLEX)
the carrier of X --> 0 is non empty Relation-like the carrier of X -defined REAL -valued RAT -valued INT -valued Function-like total quasi_total V139() V140() V141() V142() Element of bool [: the carrier of X,REAL:]
[: the carrier of X,REAL:] is non empty V139() V140() V141() set
bool [: the carrier of X,REAL:] is non empty set
ComplexAlgebraStr(# (Funcs ( the carrier of X,COMPLEX)),(ComplexFuncMult the carrier of X),(ComplexFuncAdd the carrier of X),(ComplexFuncExtMult the carrier of X),(ComplexFuncUnit the carrier of X),(ComplexFuncZero the carrier of X) #) is strict ComplexAlgebraStr
the carrier of (CAlgebra the carrier of X) is non empty set
bool the carrier of (CAlgebra the carrier of X) is non empty set
{ b₁ where b₁ is non empty Relation-like the carrier of X -defined COMPLEX -valued Function-like total quasi_total V139() (X) Element of bool [: the carrier of X,COMPLEX:] : verum } is set
mult_ ((X),(CAlgebra the carrier of X)) is non empty Relation-like [:(X),(X):] -defined (X) -valued Function-like total quasi_total Element of bool [:[:(X),(X):],(X):]
[:(X),(X):] is non empty set
[:[:(X),(X):],(X):] is non empty set
bool [:[:(X),(X):],(X):] is non empty set
Add_ ((X),(CAlgebra the carrier of X)) is non empty Relation-like [:(X),(X):] -defined (X) -valued Function-like total quasi_total Element of bool [:[:(X),(X):],(X):]
Mult_ ((X),(CAlgebra the carrier of X)) is non empty Relation-like [:COMPLEX,(X):] -defined (X) -valued Function-like total quasi_total Element of bool [:[:COMPLEX,(X):],(X):]
[:COMPLEX,(X):] is non empty set
[:[:COMPLEX,(X):],(X):] is non empty set
bool [:[:COMPLEX,(X):],(X):] is non empty set
One_ ((X),(CAlgebra the carrier of X)) is Element of (X)
Zero_ ((X),(CAlgebra the carrier of X)) is Element of (X)
(X) is non empty Relation-like (X) -defined REAL -valued Function-like total quasi_total V139() V140() V141() Element of bool [:(X),REAL:]
[:(X),REAL:] is non empty V139() V140() V141() set
bool [:(X),REAL:] is non empty set
ComplexBoundedFunctions the carrier of X is non empty multiplicatively-closed Cadditively-linearly-closed Element of bool the carrier of (CAlgebra the carrier of X)
{ b₁ where b₁ is non empty Relation-like the carrier of X -defined COMPLEX -valued Function-like total quasi_total V139() Element of bool [: the carrier of X,COMPLEX:] : b₁ | the carrier of X is bounded } is set
ComplexBoundedFunctionsNorm the carrier of X is non empty Relation-like ComplexBoundedFunctions the carrier of X -defined REAL -valued Function-like total quasi_total V139() V140() V141() Element of bool [:(ComplexBoundedFunctions the carrier of X),REAL:]
[:(ComplexBoundedFunctions the carrier of X),REAL:] is non empty V139() V140() V141() set
bool [:(ComplexBoundedFunctions the carrier of X),REAL:] is non empty set
(ComplexBoundedFunctionsNorm the carrier of X) | (X) is Relation-like ComplexBoundedFunctions the carrier of X -defined REAL -valued Function-like V139() V140() V141() Element of bool [:(ComplexBoundedFunctions the carrier of X),REAL:]
Normed_Complex_AlgebraStr(# (X),(mult_ ((X),(CAlgebra the carrier of X))),(Add_ ((X),(CAlgebra the carrier of X))),(Mult_ ((X),(CAlgebra the carrier of X))),(One_ ((X),(CAlgebra the carrier of X))),(Zero_ ((X),(CAlgebra the carrier of X))),(X) #) is strict Normed_Complex_AlgebraStr
0. (X) is zero Element of the carrier of (X)
the carrier of (X) is non empty set
the ZeroF of (X) is Element of the carrier of (X)
||.(0. (X)).|| is complex real ext-real Element of REAL
the U9 of (X) is non empty Relation-like the carrier of (X) -defined REAL -valued Function-like total quasi_total V139() V140() V141() Element of bool [: the carrier of (X),REAL:]
[: the carrier of (X),REAL:] is non empty V139() V140() V141() set
bool [: the carrier of (X),REAL:] is non empty set
the U9 of (X) . (0. (X)) is complex real ext-real Element of REAL
C_Normed_Algebra_of_BoundedFunctions the carrier of X is non empty right_complementable Abelian add-associative right_zeroed unital discerning reflexive vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like complete Normed_Complex_AlgebraStr
mult_ ((ComplexBoundedFunctions the carrier of X),(CAlgebra the carrier of X)) is non empty Relation-like [:(ComplexBoundedFunctions the carrier of X),(ComplexBoundedFunctions the carrier of X):] -defined ComplexBoundedFunctions the carrier of X -valued Function-like total quasi_total Element of bool [:[:(ComplexBoundedFunctions the carrier of X),(ComplexBoundedFunctions the carrier of X):],(ComplexBoundedFunctions the carrier of X):]
[:(ComplexBoundedFunctions the carrier of X),(ComplexBoundedFunctions the carrier of X):] is non empty set
[:[:(ComplexBoundedFunctions the carrier of X),(ComplexBoundedFunctions the carrier of X):],(ComplexBoundedFunctions the carrier of X):] is non empty set
bool [:[:(ComplexBoundedFunctions the carrier of X),(ComplexBoundedFunctions the carrier of X):],(ComplexBoundedFunctions the carrier of X):] is non empty set
Add_ ((ComplexBoundedFunctions the carrier of X),(CAlgebra the carrier of X)) is non empty Relation-like [:(ComplexBoundedFunctions the carrier of X),(ComplexBoundedFunctions the carrier of X):] -defined ComplexBoundedFunctions the carrier of X -valued Function-like total quasi_total Element of bool [:[:(ComplexBoundedFunctions the carrier of X),(ComplexBoundedFunctions the carrier of X):],(ComplexBoundedFunctions the carrier of X):]
Mult_ ((ComplexBoundedFunctions the carrier of X),(CAlgebra the carrier of X)) is non empty Relation-like [:COMPLEX,(ComplexBoundedFunctions the carrier of X):] -defined ComplexBoundedFunctions the carrier of X -valued Function-like total quasi_total Element of bool [:[:COMPLEX,(ComplexBoundedFunctions the carrier of X):],(ComplexBoundedFunctions the carrier of X):]
[:COMPLEX,(ComplexBoundedFunctions the carrier of X):] is non empty set
[:[:COMPLEX,(ComplexBoundedFunctions the carrier of X):],(ComplexBoundedFunctions the carrier of X):] is non empty set
bool [:[:COMPLEX,(ComplexBoundedFunctions the carrier of X):],(ComplexBoundedFunctions the carrier of X):] is non empty set
One_ ((ComplexBoundedFunctions the carrier of X),(CAlgebra the carrier of X)) is Element of ComplexBoundedFunctions the carrier of X
Zero_ ((ComplexBoundedFunctions the carrier of X),(CAlgebra the carrier of X)) is Element of ComplexBoundedFunctions the carrier of X
Normed_Complex_AlgebraStr(# (ComplexBoundedFunctions the carrier of X),(mult_ ((ComplexBoundedFunctions the carrier of X),(CAlgebra the carrier of X))),(Add_ ((ComplexBoundedFunctions the carrier of X),(CAlgebra the carrier of X))),(Mult_ ((ComplexBoundedFunctions the carrier of X),(CAlgebra the carrier of X))),(One_ ((ComplexBoundedFunctions the carrier of X),(CAlgebra the carrier of X))),(Zero_ ((ComplexBoundedFunctions the carrier of X),(CAlgebra the carrier of X))),(ComplexBoundedFunctionsNorm the carrier of X) #) is strict Normed_Complex_AlgebraStr
the carrier of (C_Normed_Algebra_of_BoundedFunctions the carrier of X) is non empty set
F is Element of the carrier of (C_Normed_Algebra_of_BoundedFunctions the carrier of X)
||.F.|| is complex real ext-real Element of REAL
the U9 of (C_Normed_Algebra_of_BoundedFunctions the carrier of X) is non empty Relation-like the carrier of (C_Normed_Algebra_of_BoundedFunctions the carrier of X) -defined REAL -valued Function-like total quasi_total V139() V140() V141() Element of bool [: the carrier of (C_Normed_Algebra_of_BoundedFunctions the carrier of X),REAL:]
[: the carrier of (C_Normed_Algebra_of_BoundedFunctions the carrier of X),REAL:] is non empty V139() V140() V141() set
bool [: the carrier of (C_Normed_Algebra_of_BoundedFunctions the carrier of X),REAL:] is non empty set
the U9 of (C_Normed_Algebra_of_BoundedFunctions the carrier of X) . F is complex real ext-real Element of REAL
0. (C_Normed_Algebra_of_BoundedFunctions the carrier of X) is zero Element of the carrier of (C_Normed_Algebra_of_BoundedFunctions the carrier of X)
the ZeroF of (C_Normed_Algebra_of_BoundedFunctions the carrier of X) is Element of the carrier of (C_Normed_Algebra_of_BoundedFunctions the carrier of X)
X is non empty TopSpace-like V300() compact TopStruct
(X) is non empty right_complementable Abelian add-associative right_zeroed unital associative commutative right-distributive right_unital well-unital left_unital vector-distributive scalar-distributive scalar-associative scalar-unital vector-associative strict Normed_Complex_AlgebraStr
(X) is non empty multiplicatively-closed Cadditively-linearly-closed Element of bool the carrier of (CAlgebra the carrier of X)
the carrier of X is non empty set
CAlgebra the carrier of X is non empty right_complementable Abelian add-associative right_zeroed unital associative commutative right-distributive right_unital well-unital left_unital vector-distributive scalar-distributive scalar-associative scalar-unital strict vector-associative ComplexAlgebraStr
Funcs ( the carrier of X,COMPLEX) is non empty functional FUNCTION_DOMAIN of the carrier of X, COMPLEX
ComplexFuncMult the carrier of X is non empty Relation-like [:(Funcs ( the carrier of X,COMPLEX)),(Funcs ( the carrier of X,COMPLEX)):] -defined Funcs ( the carrier of X,COMPLEX) -valued Function-like total quasi_total Element of bool [:[:(Funcs ( the carrier of X,COMPLEX)),(Funcs ( the carrier of X,COMPLEX)):],(Funcs ( the carrier of X,COMPLEX)):]
[:(Funcs ( the carrier of X,COMPLEX)),(Funcs ( the carrier of X,COMPLEX)):] is non empty set
[:[:(Funcs ( the carrier of X,COMPLEX)),(Funcs ( the carrier of X,COMPLEX)):],(Funcs ( the carrier of X,COMPLEX)):] is non empty set
bool [:[:(Funcs ( the carrier of X,COMPLEX)),(Funcs ( the carrier of X,COMPLEX)):],(Funcs ( the carrier of X,COMPLEX)):] is non empty set
ComplexFuncAdd the carrier of X is non empty Relation-like [:(Funcs ( the carrier of X,COMPLEX)),(Funcs ( the carrier of X,COMPLEX)):] -defined Funcs ( the carrier of X,COMPLEX) -valued Function-like total quasi_total Element of bool [:[:(Funcs ( the carrier of X,COMPLEX)),(Funcs ( the carrier of X,COMPLEX)):],(Funcs ( the carrier of X,COMPLEX)):]
ComplexFuncExtMult the carrier of X is non empty Relation-like [:COMPLEX,(Funcs ( the carrier of X,COMPLEX)):] -defined Funcs ( the carrier of X,COMPLEX) -valued Function-like total quasi_total Element of bool [:[:COMPLEX,(Funcs ( the carrier of X,COMPLEX)):],(Funcs ( the carrier of X,COMPLEX)):]
[:COMPLEX,(Funcs ( the carrier of X,COMPLEX)):] is non empty set
[:[:COMPLEX,(Funcs ( the carrier of X,COMPLEX)):],(Funcs ( the carrier of X,COMPLEX)):] is non empty set
bool [:[:COMPLEX,(Funcs ( the carrier of X,COMPLEX)):],(Funcs ( the carrier of X,COMPLEX)):] is non empty set
ComplexFuncUnit the carrier of X is Relation-like the carrier of X -defined COMPLEX -valued Function-like total quasi_total V139() Element of Funcs ( the carrier of X,COMPLEX)
the carrier of X --> 1r is non empty Relation-like the carrier of X -defined COMPLEX -valued Function-like total quasi_total V139() Element of bool [: the carrier of X,COMPLEX:]
[: the carrier of X,COMPLEX:] is non empty V139() set
bool [: the carrier of X,COMPLEX:] is non empty set
ComplexFuncZero the carrier of X is Relation-like the carrier of X -defined COMPLEX -valued Function-like total quasi_total V139() Element of Funcs ( the carrier of X,COMPLEX)
the carrier of X --> 0 is non empty Relation-like the carrier of X -defined REAL -valued RAT -valued INT -valued Function-like total quasi_total V139() V140() V141() V142() Element of bool [: the carrier of X,REAL:]
[: the carrier of X,REAL:] is non empty V139() V140() V141() set
bool [: the carrier of X,REAL:] is non empty set
ComplexAlgebraStr(# (Funcs ( the carrier of X,COMPLEX)),(ComplexFuncMult the carrier of X),(ComplexFuncAdd the carrier of X),(ComplexFuncExtMult the carrier of X),(ComplexFuncUnit the carrier of X),(ComplexFuncZero the carrier of X) #) is strict ComplexAlgebraStr
the carrier of (CAlgebra the carrier of X) is non empty set
bool the carrier of (CAlgebra the carrier of X) is non empty set
{ b₁ where b₁ is non empty Relation-like the carrier of X -defined COMPLEX -valued Function-like total quasi_total V139() (X) Element of bool [: the carrier of X,COMPLEX:] : verum } is set
mult_ ((X),(CAlgebra the carrier of X)) is non empty Relation-like [:(X),(X):] -defined (X) -valued Function-like total quasi_total Element of bool [:[:(X),(X):],(X):]
[:(X),(X):] is non empty set
[:[:(X),(X):],(X):] is non empty set
bool [:[:(X),(X):],(X):] is non empty set
Add_ ((X),(CAlgebra the carrier of X)) is non empty Relation-like [:(X),(X):] -defined (X) -valued Function-like total quasi_total Element of bool [:[:(X),(X):],(X):]
Mult_ ((X),(CAlgebra the carrier of X)) is non empty Relation-like [:COMPLEX,(X):] -defined (X) -valued Function-like total quasi_total Element of bool [:[:COMPLEX,(X):],(X):]
[:COMPLEX,(X):] is non empty set
[:[:COMPLEX,(X):],(X):] is non empty set
bool [:[:COMPLEX,(X):],(X):] is non empty set
One_ ((X),(CAlgebra the carrier of X)) is Element of (X)
Zero_ ((X),(CAlgebra the carrier of X)) is Element of (X)
(X) is non empty Relation-like (X) -defined REAL -valued Function-like total quasi_total V139() V140() V141() Element of bool [:(X),REAL:]
[:(X),REAL:] is non empty V139() V140() V141() set
bool [:(X),REAL:] is non empty set
ComplexBoundedFunctions the carrier of X is non empty multiplicatively-closed Cadditively-linearly-closed Element of bool the carrier of (CAlgebra the carrier of X)
{ b₁ where b₁ is non empty Relation-like the carrier of X -defined COMPLEX -valued Function-like total quasi_total V139() Element of bool [: the carrier of X,COMPLEX:] : b₁ | the carrier of X is bounded } is set
ComplexBoundedFunctionsNorm the carrier of X is non empty Relation-like ComplexBoundedFunctions the carrier of X -defined REAL -valued Function-like total quasi_total V139() V140() V141() Element of bool [:(ComplexBoundedFunctions the carrier of X),REAL:]
[:(ComplexBoundedFunctions the carrier of X),REAL:] is non empty V139() V140() V141() set
bool [:(ComplexBoundedFunctions the carrier of X),REAL:] is non empty set
(ComplexBoundedFunctionsNorm the carrier of X) | (X) is Relation-like ComplexBoundedFunctions the carrier of X -defined REAL -valued Function-like V139() V140() V141() Element of bool [:(ComplexBoundedFunctions the carrier of X),REAL:]
Normed_Complex_AlgebraStr(# (X),(mult_ ((X),(CAlgebra the carrier of X))),(Add_ ((X),(CAlgebra the carrier of X))),(Mult_ ((X),(CAlgebra the carrier of X))),(One_ ((X),(CAlgebra the carrier of X))),(Zero_ ((X),(CAlgebra the carrier of X))),(X) #) is strict Normed_Complex_AlgebraStr
the carrier of (X) is non empty set
0. (X) is zero Element of the carrier of (X)
the ZeroF of (X) is Element of the carrier of (X)
X is Element of the carrier of (X)
||.X.|| is complex real ext-real Element of REAL
the U9 of (X) is non empty Relation-like the carrier of (X) -defined REAL -valued Function-like total quasi_total V139() V140() V141() Element of bool [: the carrier of (X),REAL:]
[: the carrier of (X),REAL:] is non empty V139() V140() V141() set
bool [: the carrier of (X),REAL:] is non empty set
the U9 of (X) . X is complex real ext-real Element of REAL
C_Normed_Algebra_of_BoundedFunctions the carrier of X is non empty right_complementable Abelian add-associative right_zeroed unital discerning reflexive vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like complete Normed_Complex_AlgebraStr
mult_ ((ComplexBoundedFunctions the carrier of X),(CAlgebra the carrier of X)) is non empty Relation-like [:(ComplexBoundedFunctions the carrier of X),(ComplexBoundedFunctions the carrier of X):] -defined ComplexBoundedFunctions the carrier of X -valued Function-like total quasi_total Element of bool [:[:(ComplexBoundedFunctions the carrier of X),(ComplexBoundedFunctions the carrier of X):],(ComplexBoundedFunctions the carrier of X):]
[:(ComplexBoundedFunctions the carrier of X),(ComplexBoundedFunctions the carrier of X):] is non empty set
[:[:(ComplexBoundedFunctions the carrier of X),(ComplexBoundedFunctions the carrier of X):],(ComplexBoundedFunctions the carrier of X):] is non empty set
bool [:[:(ComplexBoundedFunctions the carrier of X),(ComplexBoundedFunctions the carrier of X):],(ComplexBoundedFunctions the carrier of X):] is non empty set
Add_ ((ComplexBoundedFunctions the carrier of X),(CAlgebra the carrier of X)) is non empty Relation-like [:(ComplexBoundedFunctions the carrier of X),(ComplexBoundedFunctions the carrier of X):] -defined ComplexBoundedFunctions the carrier of X -valued Function-like total quasi_total Element of bool [:[:(ComplexBoundedFunctions the carrier of X),(ComplexBoundedFunctions the carrier of X):],(ComplexBoundedFunctions the carrier of X):]
Mult_ ((ComplexBoundedFunctions the carrier of X),(CAlgebra the carrier of X)) is non empty Relation-like [:COMPLEX,(ComplexBoundedFunctions the carrier of X):] -defined ComplexBoundedFunctions the carrier of X -valued Function-like total quasi_total Element of bool [:[:COMPLEX,(ComplexBoundedFunctions the carrier of X):],(ComplexBoundedFunctions the carrier of X):]
[:COMPLEX,(ComplexBoundedFunctions the carrier of X):] is non empty set
[:[:COMPLEX,(ComplexBoundedFunctions the carrier of X):],(ComplexBoundedFunctions the carrier of X):] is non empty set
bool [:[:COMPLEX,(ComplexBoundedFunctions the carrier of X):],(ComplexBoundedFunctions the carrier of X):] is non empty set
One_ ((ComplexBoundedFunctions the carrier of X),(CAlgebra the carrier of X)) is Element of ComplexBoundedFunctions the carrier of X
Zero_ ((ComplexBoundedFunctions the carrier of X),(CAlgebra the carrier of X)) is Element of ComplexBoundedFunctions the carrier of X
Normed_Complex_AlgebraStr(# (ComplexBoundedFunctions the carrier of X),(mult_ ((ComplexBoundedFunctions the carrier of X),(CAlgebra the carrier of X))),(Add_ ((ComplexBoundedFunctions the carrier of X),(CAlgebra the carrier of X))),(Mult_ ((ComplexBoundedFunctions the carrier of X),(CAlgebra the carrier of X))),(One_ ((ComplexBoundedFunctions the carrier of X),(CAlgebra the carrier of X))),(Zero_ ((ComplexBoundedFunctions the carrier of X),(CAlgebra the carrier of X))),(ComplexBoundedFunctionsNorm the carrier of X) #) is strict Normed_Complex_AlgebraStr
the carrier of (C_Normed_Algebra_of_BoundedFunctions the carrier of X) is non empty set
F is Element of the carrier of (C_Normed_Algebra_of_BoundedFunctions the carrier of X)
||.F.|| is complex real ext-real Element of REAL
the U9 of (C_Normed_Algebra_of_BoundedFunctions the carrier of X) is non empty Relation-like the carrier of (C_Normed_Algebra_of_BoundedFunctions the carrier of X) -defined REAL -valued Function-like total quasi_total V139() V140() V141() Element of bool [: the carrier of (C_Normed_Algebra_of_BoundedFunctions the carrier of X),REAL:]
[: the carrier of (C_Normed_Algebra_of_BoundedFunctions the carrier of X),REAL:] is non empty V139() V140() V141() set
bool [: the carrier of (C_Normed_Algebra_of_BoundedFunctions the carrier of X),REAL:] is non empty set
the U9 of (C_Normed_Algebra_of_BoundedFunctions the carrier of X) . F is complex real ext-real Element of REAL
0. (C_Normed_Algebra_of_BoundedFunctions the carrier of X) is zero Element of the carrier of (C_Normed_Algebra_of_BoundedFunctions the carrier of X)
the ZeroF of (C_Normed_Algebra_of_BoundedFunctions the carrier of X) is Element of the carrier of (C_Normed_Algebra_of_BoundedFunctions the carrier of X)
X is complex set
abs X is complex real ext-real non negative Element of REAL
X is non empty TopSpace-like V300() compact TopStruct
(X) is non empty right_complementable Abelian add-associative right_zeroed unital associative commutative right-distributive right_unital well-unital left_unital vector-distributive scalar-distributive scalar-associative scalar-unital vector-associative strict Normed_Complex_AlgebraStr
(X) is non empty multiplicatively-closed Cadditively-linearly-closed Element of bool the carrier of (CAlgebra the carrier of X)
the carrier of X is non empty set
CAlgebra the carrier of X is non empty right_complementable Abelian add-associative right_zeroed unital associative commutative right-distributive right_unital well-unital left_unital vector-distributive scalar-distributive scalar-associative scalar-unital strict vector-associative ComplexAlgebraStr
Funcs ( the carrier of X,COMPLEX) is non empty functional FUNCTION_DOMAIN of the carrier of X, COMPLEX
ComplexFuncMult the carrier of X is non empty Relation-like [:(Funcs ( the carrier of X,COMPLEX)),(Funcs ( the carrier of X,COMPLEX)):] -defined Funcs ( the carrier of X,COMPLEX) -valued Function-like total quasi_total Element of bool [:[:(Funcs ( the carrier of X,COMPLEX)),(Funcs ( the carrier of X,COMPLEX)):],(Funcs ( the carrier of X,COMPLEX)):]
[:(Funcs ( the carrier of X,COMPLEX)),(Funcs ( the carrier of X,COMPLEX)):] is non empty set
[:[:(Funcs ( the carrier of X,COMPLEX)),(Funcs ( the carrier of X,COMPLEX)):],(Funcs ( the carrier of X,COMPLEX)):] is non empty set
bool [:[:(Funcs ( the carrier of X,COMPLEX)),(Funcs ( the carrier of X,COMPLEX)):],(Funcs ( the carrier of X,COMPLEX)):] is non empty set
ComplexFuncAdd the carrier of X is non empty Relation-like [:(Funcs ( the carrier of X,COMPLEX)),(Funcs ( the carrier of X,COMPLEX)):] -defined Funcs ( the carrier of X,COMPLEX) -valued Function-like total quasi_total Element of bool [:[:(Funcs ( the carrier of X,COMPLEX)),(Funcs ( the carrier of X,COMPLEX)):],(Funcs ( the carrier of X,COMPLEX)):]
ComplexFuncExtMult the carrier of X is non empty Relation-like [:COMPLEX,(Funcs ( the carrier of X,COMPLEX)):] -defined Funcs ( the carrier of X,COMPLEX) -valued Function-like total quasi_total Element of bool [:[:COMPLEX,(Funcs ( the carrier of X,COMPLEX)):],(Funcs ( the carrier of X,COMPLEX)):]
[:COMPLEX,(Funcs ( the carrier of X,COMPLEX)):] is non empty set
[:[:COMPLEX,(Funcs ( the carrier of X,COMPLEX)):],(Funcs ( the carrier of X,COMPLEX)):] is non empty set
bool [:[:COMPLEX,(Funcs ( the carrier of X,COMPLEX)):],(Funcs ( the carrier of X,COMPLEX)):] is non empty set
ComplexFuncUnit the carrier of X is Relation-like the carrier of X -defined COMPLEX -valued Function-like total quasi_total V139() Element of Funcs ( the carrier of X,COMPLEX)
the carrier of X --> 1r is non empty Relation-like the carrier of X -defined COMPLEX -valued Function-like total quasi_total V139() Element of bool [: the carrier of X,COMPLEX:]
[: the carrier of X,COMPLEX:] is non empty V139() set
bool [: the carrier of X,COMPLEX:] is non empty set
ComplexFuncZero the carrier of X is Relation-like the carrier of X -defined COMPLEX -valued Function-like total quasi_total V139() Element of Funcs ( the carrier of X,COMPLEX)
the carrier of X --> 0 is non empty Relation-like the carrier of X -defined REAL -valued RAT -valued INT -valued Function-like total quasi_total V139() V140() V141() V142() Element of bool [: the carrier of X,REAL:]
[: the carrier of X,REAL:] is non empty V139() V140() V141() set
bool [: the carrier of X,REAL:] is non empty set
ComplexAlgebraStr(# (Funcs ( the carrier of X,COMPLEX)),(ComplexFuncMult the carrier of X),(ComplexFuncAdd the carrier of X),(ComplexFuncExtMult the carrier of X),(ComplexFuncUnit the carrier of X),(ComplexFuncZero the carrier of X) #) is strict ComplexAlgebraStr
the carrier of (CAlgebra the carrier of X) is non empty set
bool the carrier of (CAlgebra the carrier of X) is non empty set
{ b₁ where b₁ is non empty Relation-like the carrier of X -defined COMPLEX -valued Function-like total quasi_total V139() (X) Element of bool [: the carrier of X,COMPLEX:] : verum } is set
mult_ ((X),(CAlgebra the carrier of X)) is non empty Relation-like [:(X),(X):] -defined (X) -valued Function-like total quasi_total Element of bool [:[:(X),(X):],(X):]
[:(X),(X):] is non empty set
[:[:(X),(X):],(X):] is non empty set
bool [:[:(X),(X):],(X):] is non empty set
Add_ ((X),(CAlgebra the carrier of X)) is non empty Relation-like [:(X),(X):] -defined (X) -valued Function-like total quasi_total Element of bool [:[:(X),(X):],(X):]
Mult_ ((X),(CAlgebra the carrier of X)) is non empty Relation-like [:COMPLEX,(X):] -defined (X) -valued Function-like total quasi_total Element of bool [:[:COMPLEX,(X):],(X):]
[:COMPLEX,(X):] is non empty set
[:[:COMPLEX,(X):],(X):] is non empty set
bool [:[:COMPLEX,(X):],(X):] is non empty set
One_ ((X),(CAlgebra the carrier of X)) is Element of (X)
Zero_ ((X),(CAlgebra the carrier of X)) is Element of (X)
(X) is non empty Relation-like (X) -defined REAL -valued Function-like total quasi_total V139() V140() V141() Element of bool [:(X),REAL:]
[:(X),REAL:] is non empty V139() V140() V141() set
bool [:(X),REAL:] is non empty set
ComplexBoundedFunctions the carrier of X is non empty multiplicatively-closed Cadditively-linearly-closed Element of bool the carrier of (CAlgebra the carrier of X)
{ b₁ where b₁ is non empty Relation-like the carrier of X -defined COMPLEX -valued Function-like total quasi_total V139() Element of bool [: the carrier of X,COMPLEX:] : b₁ | the carrier of X is bounded } is set
ComplexBoundedFunctionsNorm the carrier of X is non empty Relation-like ComplexBoundedFunctions the carrier of X -defined REAL -valued Function-like total quasi_total V139() V140() V141() Element of bool [:(ComplexBoundedFunctions the carrier of X),REAL:]
[:(ComplexBoundedFunctions the carrier of X),REAL:] is non empty V139() V140() V141() set
bool [:(ComplexBoundedFunctions the carrier of X),REAL:] is non empty set
(ComplexBoundedFunctionsNorm the carrier of X) | (X) is Relation-like ComplexBoundedFunctions the carrier of X -defined REAL -valued Function-like V139() V140() V141() Element of bool [:(ComplexBoundedFunctions the carrier of X),REAL:]
Normed_Complex_AlgebraStr(# (X),(mult_ ((X),(CAlgebra the carrier of X))),(Add_ ((X),(CAlgebra the carrier of X))),(Mult_ ((X),(CAlgebra the carrier of X))),(One_ ((X),(CAlgebra the carrier of X))),(Zero_ ((X),(CAlgebra the carrier of X))),(X) #) is strict Normed_Complex_AlgebraStr
the carrier of (X) is non empty set
F is Element of the carrier of (X)
X * F is Element of the carrier of (X)
the Mult of (X) is non empty Relation-like [:COMPLEX, the carrier of (X):] -defined the carrier of (X) -valued Function-like total quasi_total Element of bool [:[:COMPLEX, the carrier of (X):], the carrier of (X):]
[:COMPLEX, the carrier of (X):] is non empty set
[:[:COMPLEX, the carrier of (X):], the carrier of (X):] is non empty set
bool [:[:COMPLEX, the carrier of (X):], the carrier of (X):] is non empty set
[X,F] is set
{X,F} is non empty set
{X} is non empty V176() set
{{X,F},{X}} is non empty set
the Mult of (X) . [X,F] is set
||.(X * F).|| is complex real ext-real Element of REAL
the U9 of (X) is non empty Relation-like the carrier of (X) -defined REAL -valued Function-like total quasi_total V139() V140() V141() Element of bool [: the carrier of (X),REAL:]
[: the carrier of (X),REAL:] is non empty V139() V140() V141() set
bool [: the carrier of (X),REAL:] is non empty set
the U9 of (X) . (X * F) is complex real ext-real Element of REAL
||.F.|| is complex real ext-real Element of REAL
the U9 of (X) . F is complex real ext-real Element of REAL
(abs X) * ||.F.|| is complex real ext-real Element of REAL
C_Normed_Algebra_of_BoundedFunctions the carrier of X is non empty right_complementable Abelian add-associative right_zeroed unital discerning reflexive vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like complete Normed_Complex_AlgebraStr
mult_ ((ComplexBoundedFunctions the carrier of X),(CAlgebra the carrier of X)) is non empty Relation-like [:(ComplexBoundedFunctions the carrier of X),(ComplexBoundedFunctions the carrier of X):] -defined ComplexBoundedFunctions the carrier of X -valued Function-like total quasi_total Element of bool [:[:(ComplexBoundedFunctions the carrier of X),(ComplexBoundedFunctions the carrier of X):],(ComplexBoundedFunctions the carrier of X):]
[:(ComplexBoundedFunctions the carrier of X),(ComplexBoundedFunctions the carrier of X):] is non empty set
[:[:(ComplexBoundedFunctions the carrier of X),(ComplexBoundedFunctions the carrier of X):],(ComplexBoundedFunctions the carrier of X):] is non empty set
bool [:[:(ComplexBoundedFunctions the carrier of X),(ComplexBoundedFunctions the carrier of X):],(ComplexBoundedFunctions the carrier of X):] is non empty set
Add_ ((ComplexBoundedFunctions the carrier of X),(CAlgebra the carrier of X)) is non empty Relation-like [:(ComplexBoundedFunctions the carrier of X),(ComplexBoundedFunctions the carrier of X):] -defined ComplexBoundedFunctions the carrier of X -valued Function-like total quasi_total Element of bool [:[:(ComplexBoundedFunctions the carrier of X),(ComplexBoundedFunctions the carrier of X):],(ComplexBoundedFunctions the carrier of X):]
Mult_ ((ComplexBoundedFunctions the carrier of X),(CAlgebra the carrier of X)) is non empty Relation-like [:COMPLEX,(ComplexBoundedFunctions the carrier of X):] -defined ComplexBoundedFunctions the carrier of X -valued Function-like total quasi_total Element of bool [:[:COMPLEX,(ComplexBoundedFunctions the carrier of X):],(ComplexBoundedFunctions the carrier of X):]
[:COMPLEX,(ComplexBoundedFunctions the carrier of X):] is non empty set
[:[:COMPLEX,(ComplexBoundedFunctions the carrier of X):],(ComplexBoundedFunctions the carrier of X):] is non empty set
bool [:[:COMPLEX,(ComplexBoundedFunctions the carrier of X):],(ComplexBoundedFunctions the carrier of X):] is non empty set
One_ ((ComplexBoundedFunctions the carrier of X),(CAlgebra the carrier of X)) is Element of ComplexBoundedFunctions the carrier of X
Zero_ ((ComplexBoundedFunctions the carrier of X),(CAlgebra the carrier of X)) is Element of ComplexBoundedFunctions the carrier of X
Normed_Complex_AlgebraStr(# (ComplexBoundedFunctions the carrier of X),(mult_ ((ComplexBoundedFunctions the carrier of X),(CAlgebra the carrier of X))),(Add_ ((ComplexBoundedFunctions the carrier of X),(CAlgebra the carrier of X))),(Mult_ ((ComplexBoundedFunctions the carrier of X),(CAlgebra the carrier of X))),(One_ ((ComplexBoundedFunctions the carrier of X),(CAlgebra the carrier of X))),(Zero_ ((ComplexBoundedFunctions the carrier of X),(CAlgebra the carrier of X))),(ComplexBoundedFunctionsNorm the carrier of X) #) is strict Normed_Complex_AlgebraStr
the carrier of (C_Normed_Algebra_of_BoundedFunctions the carrier of X) is non empty set
G is Element of the carrier of (C_Normed_Algebra_of_BoundedFunctions the carrier of X)
X * G is Element of the carrier of (C_Normed_Algebra_of_BoundedFunctions the carrier of X)
the Mult of (C_Normed_Algebra_of_BoundedFunctions the carrier of X) is non empty Relation-like [:COMPLEX, the carrier of (C_Normed_Algebra_of_BoundedFunctions the carrier of X):] -defined the carrier of (C_Normed_Algebra_of_BoundedFunctions the carrier of X) -valued Function-like total quasi_total Element of bool [:[:COMPLEX, the carrier of (C_Normed_Algebra_of_BoundedFunctions the carrier of X):], the carrier of (C_Normed_Algebra_of_BoundedFunctions the carrier of X):]
[:COMPLEX, the carrier of (C_Normed_Algebra_of_BoundedFunctions the carrier of X):] is non empty set
[:[:COMPLEX, the carrier of (C_Normed_Algebra_of_BoundedFunctions the carrier of X):], the carrier of (C_Normed_Algebra_of_BoundedFunctions the carrier of X):] is non empty set
bool [:[:COMPLEX, the carrier of (C_Normed_Algebra_of_BoundedFunctions the carrier of X):], the carrier of (C_Normed_Algebra_of_BoundedFunctions the carrier of X):] is non empty set
[X,G] is set
{X,G} is non empty set
{{X,G},{X}} is non empty set
the Mult of (C_Normed_Algebra_of_BoundedFunctions the carrier of X) . [X,G] is set
||.(X * G).|| is complex real ext-real Element of REAL
the U9 of (C_Normed_Algebra_of_BoundedFunctions the carrier of X) is non empty Relation-like the carrier of (C_Normed_Algebra_of_BoundedFunctions the carrier of X) -defined REAL -valued Function-like total quasi_total V139() V140() V141() Element of bool [: the carrier of (C_Normed_Algebra_of_BoundedFunctions the carrier of X),REAL:]
[: the carrier of (C_Normed_Algebra_of_BoundedFunctions the carrier of X),REAL:] is non empty V139() V140() V141() set
bool [: the carrier of (C_Normed_Algebra_of_BoundedFunctions the carrier of X),REAL:] is non empty set
the U9 of (C_Normed_Algebra_of_BoundedFunctions the carrier of X) . (X * G) is complex real ext-real Element of REAL
||.G.|| is complex real ext-real Element of REAL
the U9 of (C_Normed_Algebra_of_BoundedFunctions the carrier of X) . G is complex real ext-real Element of REAL
(abs X) * ||.G.|| is complex real ext-real Element of REAL
X is non empty TopSpace-like V300() compact TopStruct
(X) is non empty right_complementable Abelian add-associative right_zeroed unital associative commutative right-distributive right_unital well-unital left_unital vector-distributive scalar-distributive scalar-associative scalar-unital vector-associative strict Normed_Complex_AlgebraStr
(X) is non empty multiplicatively-closed Cadditively-linearly-closed Element of bool the carrier of (CAlgebra the carrier of X)
the carrier of X is non empty set
CAlgebra the carrier of X is non empty right_complementable Abelian add-associative right_zeroed unital associative commutative right-distributive right_unital well-unital left_unital vector-distributive scalar-distributive scalar-associative scalar-unital strict vector-associative ComplexAlgebraStr
Funcs ( the carrier of X,COMPLEX) is non empty functional FUNCTION_DOMAIN of the carrier of X, COMPLEX
ComplexFuncMult the carrier of X is non empty Relation-like [:(Funcs ( the carrier of X,COMPLEX)),(Funcs ( the carrier of X,COMPLEX)):] -defined Funcs ( the carrier of X,COMPLEX) -valued Function-like total quasi_total Element of bool [:[:(Funcs ( the carrier of X,COMPLEX)),(Funcs ( the carrier of X,COMPLEX)):],(Funcs ( the carrier of X,COMPLEX)):]
[:(Funcs ( the carrier of X,COMPLEX)),(Funcs ( the carrier of X,COMPLEX)):] is non empty set
[:[:(Funcs ( the carrier of X,COMPLEX)),(Funcs ( the carrier of X,COMPLEX)):],(Funcs ( the carrier of X,COMPLEX)):] is non empty set
bool [:[:(Funcs ( the carrier of X,COMPLEX)),(Funcs ( the carrier of X,COMPLEX)):],(Funcs ( the carrier of X,COMPLEX)):] is non empty set
ComplexFuncAdd the carrier of X is non empty Relation-like [:(Funcs ( the carrier of X,COMPLEX)),(Funcs ( the carrier of X,COMPLEX)):] -defined Funcs ( the carrier of X,COMPLEX) -valued Function-like total quasi_total Element of bool [:[:(Funcs ( the carrier of X,COMPLEX)),(Funcs ( the carrier of X,COMPLEX)):],(Funcs ( the carrier of X,COMPLEX)):]
ComplexFuncExtMult the carrier of X is non empty Relation-like [:COMPLEX,(Funcs ( the carrier of X,COMPLEX)):] -defined Funcs ( the carrier of X,COMPLEX) -valued Function-like total quasi_total Element of bool [:[:COMPLEX,(Funcs ( the carrier of X,COMPLEX)):],(Funcs ( the carrier of X,COMPLEX)):]
[:COMPLEX,(Funcs ( the carrier of X,COMPLEX)):] is non empty set
[:[:COMPLEX,(Funcs ( the carrier of X,COMPLEX)):],(Funcs ( the carrier of X,COMPLEX)):] is non empty set
bool [:[:COMPLEX,(Funcs ( the carrier of X,COMPLEX)):],(Funcs ( the carrier of X,COMPLEX)):] is non empty set
ComplexFuncUnit the carrier of X is Relation-like the carrier of X -defined COMPLEX -valued Function-like total quasi_total V139() Element of Funcs ( the carrier of X,COMPLEX)
the carrier of X --> 1r is non empty Relation-like the carrier of X -defined COMPLEX -valued Function-like total quasi_total V139() Element of bool [: the carrier of X,COMPLEX:]
[: the carrier of X,COMPLEX:] is non empty V139() set
bool [: the carrier of X,COMPLEX:] is non empty set
ComplexFuncZero the carrier of X is Relation-like the carrier of X -defined COMPLEX -valued Function-like total quasi_total V139() Element of Funcs ( the carrier of X,COMPLEX)
the carrier of X --> 0 is non empty Relation-like the carrier of X -defined REAL -valued RAT -valued INT -valued Function-like total quasi_total V139() V140() V141() V142() Element of bool [: the carrier of X,REAL:]
[: the carrier of X,REAL:] is non empty V139() V140() V141() set
bool [: the carrier of X,REAL:] is non empty set
ComplexAlgebraStr(# (Funcs ( the carrier of X,COMPLEX)),(ComplexFuncMult the carrier of X),(ComplexFuncAdd the carrier of X),(ComplexFuncExtMult the carrier of X),(ComplexFuncUnit the carrier of X),(ComplexFuncZero the carrier of X) #) is strict ComplexAlgebraStr
the carrier of (CAlgebra the carrier of X) is non empty set
bool the carrier of (CAlgebra the carrier of X) is non empty set
{ b₁ where b₁ is non empty Relation-like the carrier of X -defined COMPLEX -valued Function-like total quasi_total V139() (X) Element of bool [: the carrier of X,COMPLEX:] : verum } is set
mult_ ((X),(CAlgebra the carrier of X)) is non empty Relation-like [:(X),(X):] -defined (X) -valued Function-like total quasi_total Element of bool [:[:(X),(X):],(X):]
[:(X),(X):] is non empty set
[:[:(X),(X):],(X):] is non empty set
bool [:[:(X),(X):],(X):] is non empty set
Add_ ((X),(CAlgebra the carrier of X)) is non empty Relation-like [:(X),(X):] -defined (X) -valued Function-like total quasi_total Element of bool [:[:(X),(X):],(X):]
Mult_ ((X),(CAlgebra the carrier of X)) is non empty Relation-like [:COMPLEX,(X):] -defined (X) -valued Function-like total quasi_total Element of bool [:[:COMPLEX,(X):],(X):]
[:COMPLEX,(X):] is non empty set
[:[:COMPLEX,(X):],(X):] is non empty set
bool [:[:COMPLEX,(X):],(X):] is non empty set
One_ ((X),(CAlgebra the carrier of X)) is Element of (X)
Zero_ ((X),(CAlgebra the carrier of X)) is Element of (X)
(X) is non empty Relation-like (X) -defined REAL -valued Function-like total quasi_total V139() V140() V141() Element of bool [:(X),REAL:]
[:(X),REAL:] is non empty V139() V140() V141() set
bool [:(X),REAL:] is non empty set
ComplexBoundedFunctions the carrier of X is non empty multiplicatively-closed Cadditively-linearly-closed Element of bool the carrier of (CAlgebra the carrier of X)
{ b₁ where b₁ is non empty Relation-like the carrier of X -defined COMPLEX -valued Function-like total quasi_total V139() Element of bool [: the carrier of X,COMPLEX:] : b₁ | the carrier of X is bounded } is set
ComplexBoundedFunctionsNorm the carrier of X is non empty Relation-like ComplexBoundedFunctions the carrier of X -defined REAL -valued Function-like total quasi_total V139() V140() V141() Element of bool [:(ComplexBoundedFunctions the carrier of X),REAL:]
[:(ComplexBoundedFunctions the carrier of X),REAL:] is non empty V139() V140() V141() set
bool [:(ComplexBoundedFunctions the carrier of X),REAL:] is non empty set
(ComplexBoundedFunctionsNorm the carrier of X) | (X) is Relation-like ComplexBoundedFunctions the carrier of X -defined REAL -valued Function-like V139() V140() V141() Element of bool [:(ComplexBoundedFunctions the carrier of X),REAL:]
Normed_Complex_AlgebraStr(# (X),(mult_ ((X),(CAlgebra the carrier of X))),(Add_ ((X),(CAlgebra the carrier of X))),(Mult_ ((X),(CAlgebra the carrier of X))),(One_ ((X),(CAlgebra the carrier of X))),(Zero_ ((X),(CAlgebra the carrier of X))),(X) #) is strict Normed_Complex_AlgebraStr
the carrier of (X) is non empty set
X is Element of the carrier of (X)
F is Element of the carrier of (X)
X + F is Element of the carrier of (X)
the addF of (X) is non empty Relation-like [: the carrier of (X), the carrier of (X):] -defined the carrier of (X) -valued Function-like total quasi_total Element of bool [:[: the carrier of (X), the carrier of (X):], the carrier of (X):]
[: the carrier of (X), the carrier of (X):] is non empty set
[:[: the carrier of (X), the carrier of (X):], the carrier of (X):] is non empty set
bool [:[: the carrier of (X), the carrier of (X):], the carrier of (X):] is non empty set
the addF of (X) . (X,F) is Element of the carrier of (X)
[X,F] is set
{X,F} is non empty set
{X} is non empty set
{{X,F},{X}} is non empty set
the addF of (X) . [X,F] is set
||.(X + F).|| is complex real ext-real Element of REAL
the U9 of (X) is non empty Relation-like the carrier of (X) -defined REAL -valued Function-like total quasi_total V139() V140() V141() Element of bool [: the carrier of (X),REAL:]
[: the carrier of (X),REAL:] is non empty V139() V140() V141() set
bool [: the carrier of (X),REAL:] is non empty set
the U9 of (X) . (X + F) is complex real ext-real Element of REAL
||.X.|| is complex real ext-real Element of REAL
the U9 of (X) . X is complex real ext-real Element of REAL
||.F.|| is complex real ext-real Element of REAL
the U9 of (X) . F is complex real ext-real Element of REAL
||.X.|| + ||.F.|| is complex real ext-real Element of REAL
C_Normed_Algebra_of_BoundedFunctions the carrier of X is non empty right_complementable Abelian add-associative right_zeroed unital discerning reflexive vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like complete Normed_Complex_AlgebraStr
mult_ ((ComplexBoundedFunctions the carrier of X),(CAlgebra the carrier of X)) is non empty Relation-like [:(ComplexBoundedFunctions the carrier of X),(ComplexBoundedFunctions the carrier of X):] -defined ComplexBoundedFunctions the carrier of X -valued Function-like total quasi_total Element of bool [:[:(ComplexBoundedFunctions the carrier of X),(ComplexBoundedFunctions the carrier of X):],(ComplexBoundedFunctions the carrier of X):]
[:(ComplexBoundedFunctions the carrier of X),(ComplexBoundedFunctions the carrier of X):] is non empty set
[:[:(ComplexBoundedFunctions the carrier of X),(ComplexBoundedFunctions the carrier of X):],(ComplexBoundedFunctions the carrier of X):] is non empty set
bool [:[:(ComplexBoundedFunctions the carrier of X),(ComplexBoundedFunctions the carrier of X):],(ComplexBoundedFunctions the carrier of X):] is non empty set
Add_ ((ComplexBoundedFunctions the carrier of X),(CAlgebra the carrier of X)) is non empty Relation-like [:(ComplexBoundedFunctions the carrier of X),(ComplexBoundedFunctions the carrier of X):] -defined ComplexBoundedFunctions the carrier of X -valued Function-like total quasi_total Element of bool [:[:(ComplexBoundedFunctions the carrier of X),(ComplexBoundedFunctions the carrier of X):],(ComplexBoundedFunctions the carrier of X):]
Mult_ ((ComplexBoundedFunctions the carrier of X),(CAlgebra the carrier of X)) is non empty Relation-like [:COMPLEX,(ComplexBoundedFunctions the carrier of X):] -defined ComplexBoundedFunctions the carrier of X -valued Function-like total quasi_total Element of bool [:[:COMPLEX,(ComplexBoundedFunctions the carrier of X):],(ComplexBoundedFunctions the carrier of X):]
[:COMPLEX,(ComplexBoundedFunctions the carrier of X):] is non empty set
[:[:COMPLEX,(ComplexBoundedFunctions the carrier of X):],(ComplexBoundedFunctions the carrier of X):] is non empty set
bool [:[:COMPLEX,(ComplexBoundedFunctions the carrier of X):],(ComplexBoundedFunctions the carrier of X):] is non empty set
One_ ((ComplexBoundedFunctions the carrier of X),(CAlgebra the carrier of X)) is Element of ComplexBoundedFunctions the carrier of X
Zero_ ((ComplexBoundedFunctions the carrier of X),(CAlgebra the carrier of X)) is Element of ComplexBoundedFunctions the carrier of X
Normed_Complex_AlgebraStr(# (ComplexBoundedFunctions the carrier of X),(mult_ ((ComplexBoundedFunctions the carrier of X),(CAlgebra the carrier of X))),(Add_ ((ComplexBoundedFunctions the carrier of X),(CAlgebra the carrier of X))),(Mult_ ((ComplexBoundedFunctions the carrier of X),(CAlgebra the carrier of X))),(One_ ((ComplexBoundedFunctions the carrier of X),(CAlgebra the carrier of X))),(Zero_ ((ComplexBoundedFunctions the carrier of X),(CAlgebra the carrier of X))),(ComplexBoundedFunctionsNorm the carrier of X) #) is strict Normed_Complex_AlgebraStr
the carrier of (C_Normed_Algebra_of_BoundedFunctions the carrier of X) is non empty set
G is Element of the carrier of (C_Normed_Algebra_of_BoundedFunctions the carrier of X)
||.G.|| is complex real ext-real Element of REAL
the U9 of (C_Normed_Algebra_of_BoundedFunctions the carrier of X) is non empty Relation-like the carrier of (C_Normed_Algebra_of_BoundedFunctions the carrier of X) -defined REAL -valued Function-like total quasi_total V139() V140() V141() Element of bool [: the carrier of (C_Normed_Algebra_of_BoundedFunctions the carrier of X),REAL:]
[: the carrier of (C_Normed_Algebra_of_BoundedFunctions the carrier of X),REAL:] is non empty V139() V140() V141() set
bool [: the carrier of (C_Normed_Algebra_of_BoundedFunctions the carrier of X),REAL:] is non empty set
the U9 of (C_Normed_Algebra_of_BoundedFunctions the carrier of X) . G is complex real ext-real Element of REAL
FB is Element of the carrier of (C_Normed_Algebra_of_BoundedFunctions the carrier of X)
||.FB.|| is complex real ext-real Element of REAL
the U9 of (C_Normed_Algebra_of_BoundedFunctions the carrier of X) . FB is complex real ext-real Element of REAL
G + FB is Element of the carrier of (C_Normed_Algebra_of_BoundedFunctions the carrier of X)
the addF of (C_Normed_Algebra_of_BoundedFunctions the carrier of X) is non empty Relation-like [: the carrier of (C_Normed_Algebra_of_BoundedFunctions the carrier of X), the carrier of (C_Normed_Algebra_of_BoundedFunctions the carrier of X):] -defined the carrier of (C_Normed_Algebra_of_BoundedFunctions the carrier of X) -valued Function-like total quasi_total Element of bool [:[: the carrier of (C_Normed_Algebra_of_BoundedFunctions the carrier of X), the carrier of (C_Normed_Algebra_of_BoundedFunctions the carrier of X):], the carrier of (C_Normed_Algebra_of_BoundedFunctions the carrier of X):]
[: the carrier of (C_Normed_Algebra_of_BoundedFunctions the carrier of X), the carrier of (C_Normed_Algebra_of_BoundedFunctions the carrier of X):] is non empty set
[:[: the carrier of (C_Normed_Algebra_of_BoundedFunctions the carrier of X), the carrier of (C_Normed_Algebra_of_BoundedFunctions the carrier of X):], the carrier of (C_Normed_Algebra_of_BoundedFunctions the carrier of X):] is non empty set
bool [:[: the carrier of (C_Normed_Algebra_of_BoundedFunctions the carrier of X), the carrier of (C_Normed_Algebra_of_BoundedFunctions the carrier of X):], the carrier of (C_Normed_Algebra_of_BoundedFunctions the carrier of X):] is non empty set
the addF of (C_Normed_Algebra_of_BoundedFunctions the carrier of X) . (G,FB) is Element of the carrier of (C_Normed_Algebra_of_BoundedFunctions the carrier of X)
[G,FB] is set
{G,FB} is non empty set
{G} is non empty set
{{G,FB},{G}} is non empty set
the addF of (C_Normed_Algebra_of_BoundedFunctions the carrier of X) . [G,FB] is set
||.(G + FB).|| is complex real ext-real Element of REAL
the U9 of (C_Normed_Algebra_of_BoundedFunctions the carrier of X) . (G + FB) is complex real ext-real Element of REAL
X is non empty TopSpace-like V300() compact TopStruct
(X) is non empty right_complementable Abelian add-associative right_zeroed unital associative commutative right-distributive right_unital well-unital left_unital vector-distributive scalar-distributive scalar-associative scalar-unital vector-associative strict Normed_Complex_AlgebraStr
(X) is non empty multiplicatively-closed Cadditively-linearly-closed Element of bool the carrier of (CAlgebra the carrier of X)
the carrier of X is non empty set
CAlgebra the carrier of X is non empty right_complementable Abelian add-associative right_zeroed unital associative commutative right-distributive right_unital well-unital left_unital vector-distributive scalar-distributive scalar-associative scalar-unital strict vector-associative ComplexAlgebraStr
Funcs ( the carrier of X,COMPLEX) is non empty functional FUNCTION_DOMAIN of the carrier of X, COMPLEX
ComplexFuncMult the carrier of X is non empty Relation-like [:(Funcs ( the carrier of X,COMPLEX)),(Funcs ( the carrier of X,COMPLEX)):] -defined Funcs ( the carrier of X,COMPLEX) -valued Function-like total quasi_total Element of bool [:[:(Funcs ( the carrier of X,COMPLEX)),(Funcs ( the carrier of X,COMPLEX)):],(Funcs ( the carrier of X,COMPLEX)):]
[:(Funcs ( the carrier of X,COMPLEX)),(Funcs ( the carrier of X,COMPLEX)):] is non empty set
[:[:(Funcs ( the carrier of X,COMPLEX)),(Funcs ( the carrier of X,COMPLEX)):],(Funcs ( the carrier of X,COMPLEX)):] is non empty set
bool [:[:(Funcs ( the carrier of X,COMPLEX)),(Funcs ( the carrier of X,COMPLEX)):],(Funcs ( the carrier of X,COMPLEX)):] is non empty set
ComplexFuncAdd the carrier of X is non empty Relation-like [:(Funcs ( the carrier of X,COMPLEX)),(Funcs ( the carrier of X,COMPLEX)):] -defined Funcs ( the carrier of X,COMPLEX) -valued Function-like total quasi_total Element of bool [:[:(Funcs ( the carrier of X,COMPLEX)),(Funcs ( the carrier of X,COMPLEX)):],(Funcs ( the carrier of X,COMPLEX)):]
ComplexFuncExtMult the carrier of X is non empty Relation-like [:COMPLEX,(Funcs ( the carrier of X,COMPLEX)):] -defined Funcs ( the carrier of X,COMPLEX) -valued Function-like total quasi_total Element of bool [:[:COMPLEX,(Funcs ( the carrier of X,COMPLEX)):],(Funcs ( the carrier of X,COMPLEX)):]
[:COMPLEX,(Funcs ( the carrier of X,COMPLEX)):] is non empty set
[:[:COMPLEX,(Funcs ( the carrier of X,COMPLEX)):],(Funcs ( the carrier of X,COMPLEX)):] is non empty set
bool [:[:COMPLEX,(Funcs ( the carrier of X,COMPLEX)):],(Funcs ( the carrier of X,COMPLEX)):] is non empty set
ComplexFuncUnit the carrier of X is Relation-like the carrier of X -defined COMPLEX -valued Function-like total quasi_total V139() Element of Funcs ( the carrier of X,COMPLEX)
the carrier of X --> 1r is non empty Relation-like the carrier of X -defined COMPLEX -valued Function-like total quasi_total V139() Element of bool [: the carrier of X,COMPLEX:]
[: the carrier of X,COMPLEX:] is non empty V139() set
bool [: the carrier of X,COMPLEX:] is non empty set
ComplexFuncZero the carrier of X is Relation-like the carrier of X -defined COMPLEX -valued Function-like total quasi_total V139() Element of Funcs ( the carrier of X,COMPLEX)
the carrier of X --> 0 is non empty Relation-like the carrier of X -defined REAL -valued RAT -valued INT -valued Function-like total quasi_total V139() V140() V141() V142() Element of bool [: the carrier of X,REAL:]
[: the carrier of X,REAL:] is non empty V139() V140() V141() set
bool [: the carrier of X,REAL:] is non empty set
ComplexAlgebraStr(# (Funcs ( the carrier of X,COMPLEX)),(ComplexFuncMult the carrier of X),(ComplexFuncAdd the carrier of X),(ComplexFuncExtMult the carrier of X),(ComplexFuncUnit the carrier of X),(ComplexFuncZero the carrier of X) #) is strict ComplexAlgebraStr
the carrier of (CAlgebra the carrier of X) is non empty set
bool the carrier of (CAlgebra the carrier of X) is non empty set
{ b₁ where b₁ is non empty Relation-like the carrier of X -defined COMPLEX -valued Function-like total quasi_total V139() (X) Element of bool [: the carrier of X,COMPLEX:] : verum } is set
mult_ ((X),(CAlgebra the carrier of X)) is non empty Relation-like [:(X),(X):] -defined (X) -valued Function-like total quasi_total Element of bool [:[:(X),(X):],(X):]
[:(X),(X):] is non empty set
[:[:(X),(X):],(X):] is non empty set
bool [:[:(X),(X):],(X):] is non empty set
Add_ ((X),(CAlgebra the carrier of X)) is non empty Relation-like [:(X),(X):] -defined (X) -valued Function-like total quasi_total Element of bool [:[:(X),(X):],(X):]
Mult_ ((X),(CAlgebra the carrier of X)) is non empty Relation-like [:COMPLEX,(X):] -defined (X) -valued Function-like total quasi_total Element of bool [:[:COMPLEX,(X):],(X):]
[:COMPLEX,(X):] is non empty set
[:[:COMPLEX,(X):],(X):] is non empty set
bool [:[:COMPLEX,(X):],(X):] is non empty set
One_ ((X),(CAlgebra the carrier of X)) is Element of (X)
Zero_ ((X),(CAlgebra the carrier of X)) is Element of (X)
(X) is non empty Relation-like (X) -defined REAL -valued Function-like total quasi_total V139() V140() V141() Element of bool [:(X),REAL:]
[:(X),REAL:] is non empty V139() V140() V141() set
bool [:(X),REAL:] is non empty set
ComplexBoundedFunctions the carrier of X is non empty multiplicatively-closed Cadditively-linearly-closed Element of bool the carrier of (CAlgebra the carrier of X)
{ b₁ where b₁ is non empty Relation-like the carrier of X -defined COMPLEX -valued Function-like total quasi_total V139() Element of bool [: the carrier of X,COMPLEX:] : b₁ | the carrier of X is bounded } is set
ComplexBoundedFunctionsNorm the carrier of X is non empty Relation-like ComplexBoundedFunctions the carrier of X -defined REAL -valued Function-like total quasi_total V139() V140() V141() Element of bool [:(ComplexBoundedFunctions the carrier of X),REAL:]
[:(ComplexBoundedFunctions the carrier of X),REAL:] is non empty V139() V140() V141() set
bool [:(ComplexBoundedFunctions the carrier of X),REAL:] is non empty set
(ComplexBoundedFunctionsNorm the carrier of X) | (X) is Relation-like ComplexBoundedFunctions the carrier of X -defined REAL -valued Function-like V139() V140() V141() Element of bool [:(ComplexBoundedFunctions the carrier of X),REAL:]
Normed_Complex_AlgebraStr(# (X),(mult_ ((X),(CAlgebra the carrier of X))),(Add_ ((X),(CAlgebra the carrier of X))),(Mult_ ((X),(CAlgebra the carrier of X))),(One_ ((X),(CAlgebra the carrier of X))),(Zero_ ((X),(CAlgebra the carrier of X))),(X) #) is strict Normed_Complex_AlgebraStr
0. (X) is zero Element of the carrier of (X)
the carrier of (X) is non empty set
the ZeroF of (X) is Element of the carrier of (X)
||.(0. (X)).|| is complex real ext-real Element of REAL
the U9 of (X) is non empty Relation-like the carrier of (X) -defined REAL -valued Function-like total quasi_total V139() V140() V141() Element of bool [: the carrier of (X),REAL:]
[: the carrier of (X),REAL:] is non empty V139() V140() V141() set
bool [: the carrier of (X),REAL:] is non empty set
the U9 of (X) . (0. (X)) is complex real ext-real Element of REAL
F is Element of the carrier of (X)
||.F.|| is complex real ext-real Element of REAL
the U9 of (X) . F is complex real ext-real Element of REAL
G is Element of the carrier of (X)
||.G.|| is complex real ext-real Element of REAL
the U9 of (X) . G is complex real ext-real Element of REAL
FB is Element of the carrier of (X)
GB is complex set
GB * FB is Element of the carrier of (X)
the Mult of (X) is non empty Relation-like [:COMPLEX, the carrier of (X):] -defined the carrier of (X) -valued Function-like total quasi_total Element of bool [:[:COMPLEX, the carrier of (X):], the carrier of (X):]
[:COMPLEX, the carrier of (X):] is non empty set
[:[:COMPLEX, the carrier of (X):], the carrier of (X):] is non empty set
bool [:[:COMPLEX, the carrier of (X):], the carrier of (X):] is non empty set
[GB,FB] is set
{GB,FB} is non empty set
{GB} is non empty V176() set
{{GB,FB},{GB}} is non empty set
the Mult of (X) . [GB,FB] is set
||.(GB * FB).|| is complex real ext-real Element of REAL
the U9 of (X) . (GB * FB) is complex real ext-real Element of REAL
abs GB is complex real ext-real non negative Element of REAL
||.FB.|| is complex real ext-real Element of REAL
the U9 of (X) . FB is complex real ext-real Element of REAL
(abs GB) * ||.FB.|| is complex real ext-real Element of REAL
aFB is Element of the carrier of (X)
aF is Element of the carrier of (X)
aFB + aF is Element of the carrier of (X)
the addF of (X) is non empty Relation-like [: the carrier of (X), the carrier of (X):] -defined the carrier of (X) -valued Function-like total quasi_total Element of bool [:[: the carrier of (X), the carrier of (X):], the carrier of (X):]
[: the carrier of (X), the carrier of (X):] is non empty set
[:[: the carrier of (X), the carrier of (X):], the carrier of (X):] is non empty set
bool [:[: the carrier of (X), the carrier of (X):], the carrier of (X):] is non empty set
the addF of (X) . (aFB,aF) is Element of the carrier of (X)
[aFB,aF] is set
{aFB,aF} is non empty set
{aFB} is non empty set
{{aFB,aF},{aFB}} is non empty set
the addF of (X) . [aFB,aF] is set
||.(aFB + aF).|| is complex real ext-real Element of REAL
the U9 of (X) . (aFB + aF) is complex real ext-real Element of REAL
||.aFB.|| is complex real ext-real Element of REAL
the U9 of (X) . aFB is complex real ext-real Element of REAL
||.aF.|| is complex real ext-real Element of REAL
the U9 of (X) . aF is complex real ext-real Element of REAL
||.aFB.|| + ||.aF.|| is complex real ext-real Element of REAL
X is non empty TopSpace-like V300() compact TopStruct
(X) is non empty right_complementable Abelian add-associative right_zeroed unital associative commutative right-distributive right_unital well-unital left_unital discerning reflexive vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like vector-associative strict Normed_Complex_AlgebraStr
(X) is non empty multiplicatively-closed Cadditively-linearly-closed Element of bool the carrier of (CAlgebra the carrier of X)
the carrier of X is non empty set
CAlgebra the carrier of X is non empty right_complementable Abelian add-associative right_zeroed unital associative commutative right-distributive right_unital well-unital left_unital vector-distributive scalar-distributive scalar-associative scalar-unital strict vector-associative ComplexAlgebraStr
Funcs ( the carrier of X,COMPLEX) is non empty functional FUNCTION_DOMAIN of the carrier of X, COMPLEX
ComplexFuncMult the carrier of X is non empty Relation-like [:(Funcs ( the carrier of X,COMPLEX)),(Funcs ( the carrier of X,COMPLEX)):] -defined Funcs ( the carrier of X,COMPLEX) -valued Function-like total quasi_total Element of bool [:[:(Funcs ( the carrier of X,COMPLEX)),(Funcs ( the carrier of X,COMPLEX)):],(Funcs ( the carrier of X,COMPLEX)):]
[:(Funcs ( the carrier of X,COMPLEX)),(Funcs ( the carrier of X,COMPLEX)):] is non empty set
[:[:(Funcs ( the carrier of X,COMPLEX)),(Funcs ( the carrier of X,COMPLEX)):],(Funcs ( the carrier of X,COMPLEX)):] is non empty set
bool [:[:(Funcs ( the carrier of X,COMPLEX)),(Funcs ( the carrier of X,COMPLEX)):],(Funcs ( the carrier of X,COMPLEX)):] is non empty set
ComplexFuncAdd the carrier of X is non empty Relation-like [:(Funcs ( the carrier of X,COMPLEX)),(Funcs ( the carrier of X,COMPLEX)):] -defined Funcs ( the carrier of X,COMPLEX) -valued Function-like total quasi_total Element of bool [:[:(Funcs ( the carrier of X,COMPLEX)),(Funcs ( the carrier of X,COMPLEX)):],(Funcs ( the carrier of X,COMPLEX)):]
ComplexFuncExtMult the carrier of X is non empty Relation-like [:COMPLEX,(Funcs ( the carrier of X,COMPLEX)):] -defined Funcs ( the carrier of X,COMPLEX) -valued Function-like total quasi_total Element of bool [:[:COMPLEX,(Funcs ( the carrier of X,COMPLEX)):],(Funcs ( the carrier of X,COMPLEX)):]
[:COMPLEX,(Funcs ( the carrier of X,COMPLEX)):] is non empty set
[:[:COMPLEX,(Funcs ( the carrier of X,COMPLEX)):],(Funcs ( the carrier of X,COMPLEX)):] is non empty set
bool [:[:COMPLEX,(Funcs ( the carrier of X,COMPLEX)):],(Funcs ( the carrier of X,COMPLEX)):] is non empty set
ComplexFuncUnit the carrier of X is Relation-like the carrier of X -defined COMPLEX -valued Function-like total quasi_total V139() Element of Funcs ( the carrier of X,COMPLEX)
the carrier of X --> 1r is non empty Relation-like the carrier of X -defined COMPLEX -valued Function-like total quasi_total V139() Element of bool [: the carrier of X,COMPLEX:]
[: the carrier of X,COMPLEX:] is non empty V139() set
bool [: the carrier of X,COMPLEX:] is non empty set
ComplexFuncZero the carrier of X is Relation-like the carrier of X -defined COMPLEX -valued Function-like total quasi_total V139() Element of Funcs ( the carrier of X,COMPLEX)
the carrier of X --> 0 is non empty Relation-like the carrier of X -defined REAL -valued RAT -valued INT -valued Function-like total quasi_total V139() V140() V141() V142() Element of bool [: the carrier of X,REAL:]
[: the carrier of X,REAL:] is non empty V139() V140() V141() set
bool [: the carrier of X,REAL:] is non empty set
ComplexAlgebraStr(# (Funcs ( the carrier of X,COMPLEX)),(ComplexFuncMult the carrier of X),(ComplexFuncAdd the carrier of X),(ComplexFuncExtMult the carrier of X),(ComplexFuncUnit the carrier of X),(ComplexFuncZero the carrier of X) #) is strict ComplexAlgebraStr
the carrier of (CAlgebra the carrier of X) is non empty set
bool the carrier of (CAlgebra the carrier of X) is non empty set
{ b₁ where b₁ is non empty Relation-like the carrier of X -defined COMPLEX -valued Function-like total quasi_total V139() (X) Element of bool [: the carrier of X,COMPLEX:] : verum } is set
mult_ ((X),(CAlgebra the carrier of X)) is non empty Relation-like [:(X),(X):] -defined (X) -valued Function-like total quasi_total Element of bool [:[:(X),(X):],(X):]
[:(X),(X):] is non empty set
[:[:(X),(X):],(X):] is non empty set
bool [:[:(X),(X):],(X):] is non empty set
Add_ ((X),(CAlgebra the carrier of X)) is non empty Relation-like [:(X),(X):] -defined (X) -valued Function-like total quasi_total Element of bool [:[:(X),(X):],(X):]
Mult_ ((X),(CAlgebra the carrier of X)) is non empty Relation-like [:COMPLEX,(X):] -defined (X) -valued Function-like total quasi_total Element of bool [:[:COMPLEX,(X):],(X):]
[:COMPLEX,(X):] is non empty set
[:[:COMPLEX,(X):],(X):] is non empty set
bool [:[:COMPLEX,(X):],(X):] is non empty set
One_ ((X),(CAlgebra the carrier of X)) is Element of (X)
Zero_ ((X),(CAlgebra the carrier of X)) is Element of (X)
(X) is non empty Relation-like (X) -defined REAL -valued Function-like total quasi_total V139() V140() V141() Element of bool [:(X),REAL:]
[:(X),REAL:] is non empty V139() V140() V141() set
bool [:(X),REAL:] is non empty set
ComplexBoundedFunctions the carrier of X is non empty multiplicatively-closed Cadditively-linearly-closed Element of bool the carrier of (CAlgebra the carrier of X)
{ b₁ where b₁ is non empty Relation-like the carrier of X -defined COMPLEX -valued Function-like total quasi_total V139() Element of bool [: the carrier of X,COMPLEX:] : b₁ | the carrier of X is bounded } is set
ComplexBoundedFunctionsNorm the carrier of X is non empty Relation-like ComplexBoundedFunctions the carrier of X -defined REAL -valued Function-like total quasi_total V139() V140() V141() Element of bool [:(ComplexBoundedFunctions the carrier of X),REAL:]
[:(ComplexBoundedFunctions the carrier of X),REAL:] is non empty V139() V140() V141() set
bool [:(ComplexBoundedFunctions the carrier of X),REAL:] is non empty set
(ComplexBoundedFunctionsNorm the carrier of X) | (X) is Relation-like ComplexBoundedFunctions the carrier of X -defined REAL -valued Function-like V139() V140() V141() Element of bool [:(ComplexBoundedFunctions the carrier of X),REAL:]
Normed_Complex_AlgebraStr(# (X),(mult_ ((X),(CAlgebra the carrier of X))),(Add_ ((X),(CAlgebra the carrier of X))),(Mult_ ((X),(CAlgebra the carrier of X))),(One_ ((X),(CAlgebra the carrier of X))),(Zero_ ((X),(CAlgebra the carrier of X))),(X) #) is strict Normed_Complex_AlgebraStr
the carrier of (X) is non empty set
C_Normed_Algebra_of_BoundedFunctions the carrier of X is non empty right_complementable Abelian add-associative right_zeroed unital discerning reflexive vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like complete Normed_Complex_AlgebraStr
mult_ ((ComplexBoundedFunctions the carrier of X),(CAlgebra the carrier of X)) is non empty Relation-like [:(ComplexBoundedFunctions the carrier of X),(ComplexBoundedFunctions the carrier of X):] -defined ComplexBoundedFunctions the carrier of X -valued Function-like total quasi_total Element of bool [:[:(ComplexBoundedFunctions the carrier of X),(ComplexBoundedFunctions the carrier of X):],(ComplexBoundedFunctions the carrier of X):]
[:(ComplexBoundedFunctions the carrier of X),(ComplexBoundedFunctions the carrier of X):] is non empty set
[:[:(ComplexBoundedFunctions the carrier of X),(ComplexBoundedFunctions the carrier of X):],(ComplexBoundedFunctions the carrier of X):] is non empty set
bool [:[:(ComplexBoundedFunctions the carrier of X),(ComplexBoundedFunctions the carrier of X):],(ComplexBoundedFunctions the carrier of X):] is non empty set
Add_ ((ComplexBoundedFunctions the carrier of X),(CAlgebra the carrier of X)) is non empty Relation-like [:(ComplexBoundedFunctions the carrier of X),(ComplexBoundedFunctions the carrier of X):] -defined ComplexBoundedFunctions the carrier of X -valued Function-like total quasi_total Element of bool [:[:(ComplexBoundedFunctions the carrier of X),(ComplexBoundedFunctions the carrier of X):],(ComplexBoundedFunctions the carrier of X):]
Mult_ ((ComplexBoundedFunctions the carrier of X),(CAlgebra the carrier of X)) is non empty Relation-like [:COMPLEX,(ComplexBoundedFunctions the carrier of X):] -defined ComplexBoundedFunctions the carrier of X -valued Function-like total quasi_total Element of bool [:[:COMPLEX,(ComplexBoundedFunctions the carrier of X):],(ComplexBoundedFunctions the carrier of X):]
[:COMPLEX,(ComplexBoundedFunctions the carrier of X):] is non empty set
[:[:COMPLEX,(ComplexBoundedFunctions the carrier of X):],(ComplexBoundedFunctions the carrier of X):] is non empty set
bool [:[:COMPLEX,(ComplexBoundedFunctions the carrier of X):],(ComplexBoundedFunctions the carrier of X):] is non empty set
One_ ((ComplexBoundedFunctions the carrier of X),(CAlgebra the carrier of X)) is Element of ComplexBoundedFunctions the carrier of X
Zero_ ((ComplexBoundedFunctions the carrier of X),(CAlgebra the carrier of X)) is Element of ComplexBoundedFunctions the carrier of X
Normed_Complex_AlgebraStr(# (ComplexBoundedFunctions the carrier of X),(mult_ ((ComplexBoundedFunctions the carrier of X),(CAlgebra the carrier of X))),(Add_ ((ComplexBoundedFunctions the carrier of X),(CAlgebra the carrier of X))),(Mult_ ((ComplexBoundedFunctions the carrier of X),(CAlgebra the carrier of X))),(One_ ((ComplexBoundedFunctions the carrier of X),(CAlgebra the carrier of X))),(Zero_ ((ComplexBoundedFunctions the carrier of X),(CAlgebra the carrier of X))),(ComplexBoundedFunctionsNorm the carrier of X) #) is strict Normed_Complex_AlgebraStr
the carrier of (C_Normed_Algebra_of_BoundedFunctions the carrier of X) is non empty set
X is Element of the carrier of (X)
F is Element of the carrier of (X)
X - F is Element of the carrier of (X)
- F is Element of the carrier of (X)
X + (- F) is Element of the carrier of (X)
the addF of (X) is non empty Relation-like [: the carrier of (X), the carrier of (X):] -defined the carrier of (X) -valued Function-like total quasi_total Element of bool [:[: the carrier of (X), the carrier of (X):], the carrier of (X):]
[: the carrier of (X), the carrier of (X):] is non empty set
[:[: the carrier of (X), the carrier of (X):], the carrier of (X):] is non empty set
bool [:[: the carrier of (X), the carrier of (X):], the carrier of (X):] is non empty set
the addF of (X) . (X,(- F)) is Element of the carrier of (X)
[X,(- F)] is set
{X,(- F)} is non empty set
{X} is non empty set
{{X,(- F)},{X}} is non empty set
the addF of (X) . [X,(- F)] is set
G is Element of the carrier of (C_Normed_Algebra_of_BoundedFunctions the carrier of X)
FB is Element of the carrier of (C_Normed_Algebra_of_BoundedFunctions the carrier of X)
G - FB is Element of the carrier of (C_Normed_Algebra_of_BoundedFunctions the carrier of X)
- FB is Element of the carrier of (C_Normed_Algebra_of_BoundedFunctions the carrier of X)
G + (- FB) is Element of the carrier of (C_Normed_Algebra_of_BoundedFunctions the carrier of X)
the addF of (C_Normed_Algebra_of_BoundedFunctions the carrier of X) is non empty Relation-like [: the carrier of (C_Normed_Algebra_of_BoundedFunctions the carrier of X), the carrier of (C_Normed_Algebra_of_BoundedFunctions the carrier of X):] -defined the carrier of (C_Normed_Algebra_of_BoundedFunctions the carrier of X) -valued Function-like total quasi_total Element of bool [:[: the carrier of (C_Normed_Algebra_of_BoundedFunctions the carrier of X), the carrier of (C_Normed_Algebra_of_BoundedFunctions the carrier of X):], the carrier of (C_Normed_Algebra_of_BoundedFunctions the carrier of X):]
[: the carrier of (C_Normed_Algebra_of_BoundedFunctions the carrier of X), the carrier of (C_Normed_Algebra_of_BoundedFunctions the carrier of X):] is non empty set
[:[: the carrier of (C_Normed_Algebra_of_BoundedFunctions the carrier of X), the carrier of (C_Normed_Algebra_of_BoundedFunctions the carrier of X):], the carrier of (C_Normed_Algebra_of_BoundedFunctions the carrier of X):] is non empty set
bool [:[: the carrier of (C_Normed_Algebra_of_BoundedFunctions the carrier of X), the carrier of (C_Normed_Algebra_of_BoundedFunctions the carrier of X):], the carrier of (C_Normed_Algebra_of_BoundedFunctions the carrier of X):] is non empty set
the addF of (C_Normed_Algebra_of_BoundedFunctions the carrier of X) . (G,(- FB)) is Element of the carrier of (C_Normed_Algebra_of_BoundedFunctions the carrier of X)
[G,(- FB)] is set
{G,(- FB)} is non empty set
{G} is non empty set
{{G,(- FB)},{G}} is non empty set
the addF of (C_Normed_Algebra_of_BoundedFunctions the carrier of X) . [G,(- FB)] is set
aFB is complex set
aFB * F is Element of the carrier of (X)
the Mult of (X) is non empty Relation-like [:COMPLEX, the carrier of (X):] -defined the carrier of (X) -valued Function-like total quasi_total Element of bool [:[:COMPLEX, the carrier of (X):], the carrier of (X):]
[:COMPLEX, the carrier of (X):] is non empty set
[:[:COMPLEX, the carrier of (X):], the carrier of (X):] is non empty set
bool [:[:COMPLEX, the carrier of (X):], the carrier of (X):] is non empty set
[aFB,F] is set
{aFB,F} is non empty set
{aFB} is non empty V176() set
{{aFB,F},{aFB}} is non empty set
the Mult of (X) . [aFB,F] is set
aFB * FB is Element of the carrier of (C_Normed_Algebra_of_BoundedFunctions the carrier of X)
the Mult of (C_Normed_Algebra_of_BoundedFunctions the carrier of X) is non empty Relation-like [:COMPLEX, the carrier of (C_Normed_Algebra_of_BoundedFunctions the carrier of X):] -defined the carrier of (C_Normed_Algebra_of_BoundedFunctions the carrier of X) -valued Function-like total quasi_total Element of bool [:[:COMPLEX, the carrier of (C_Normed_Algebra_of_BoundedFunctions the carrier of X):], the carrier of (C_Normed_Algebra_of_BoundedFunctions the carrier of X):]
[:COMPLEX, the carrier of (C_Normed_Algebra_of_BoundedFunctions the carrier of X):] is non empty set
[:[:COMPLEX, the carrier of (C_Normed_Algebra_of_BoundedFunctions the carrier of X):], the carrier of (C_Normed_Algebra_of_BoundedFunctions the carrier of X):] is non empty set
bool [:[:COMPLEX, the carrier of (C_Normed_Algebra_of_BoundedFunctions the carrier of X):], the carrier of (C_Normed_Algebra_of_BoundedFunctions the carrier of X):] is non empty set
[aFB,FB] is set
{aFB,FB} is non empty set
{{aFB,FB},{aFB}} is non empty set
the Mult of (C_Normed_Algebra_of_BoundedFunctions the carrier of X) . [aFB,FB] is set
X is non empty TopSpace-like V300() compact TopStruct
the carrier of X is non empty set
[: the carrier of X,COMPLEX:] is non empty V139() set
bool [: the carrier of X,COMPLEX:] is non empty set
(X) is non empty right_complementable Abelian add-associative right_zeroed unital associative commutative right-distributive right_unital well-unital left_unital discerning reflexive vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like vector-associative strict Normed_Complex_AlgebraStr
(X) is non empty multiplicatively-closed Cadditively-linearly-closed Element of bool the carrier of (CAlgebra the carrier of X)
CAlgebra the carrier of X is non empty right_complementable Abelian add-associative right_zeroed unital associative commutative right-distributive right_unital well-unital left_unital vector-distributive scalar-distributive scalar-associative scalar-unital strict vector-associative ComplexAlgebraStr
Funcs ( the carrier of X,COMPLEX) is non empty functional FUNCTION_DOMAIN of the carrier of X, COMPLEX
ComplexFuncMult the carrier of X is non empty Relation-like [:(Funcs ( the carrier of X,COMPLEX)),(Funcs ( the carrier of X,COMPLEX)):] -defined Funcs ( the carrier of X,COMPLEX) -valued Function-like total quasi_total Element of bool [:[:(Funcs ( the carrier of X,COMPLEX)),(Funcs ( the carrier of X,COMPLEX)):],(Funcs ( the carrier of X,COMPLEX)):]
[:(Funcs ( the carrier of X,COMPLEX)),(Funcs ( the carrier of X,COMPLEX)):] is non empty set
[:[:(Funcs ( the carrier of X,COMPLEX)),(Funcs ( the carrier of X,COMPLEX)):],(Funcs ( the carrier of X,COMPLEX)):] is non empty set
bool [:[:(Funcs ( the carrier of X,COMPLEX)),(Funcs ( the carrier of X,COMPLEX)):],(Funcs ( the carrier of X,COMPLEX)):] is non empty set
ComplexFuncAdd the carrier of X is non empty Relation-like [:(Funcs ( the carrier of X,COMPLEX)),(Funcs ( the carrier of X,COMPLEX)):] -defined Funcs ( the carrier of X,COMPLEX) -valued Function-like total quasi_total Element of bool [:[:(Funcs ( the carrier of X,COMPLEX)),(Funcs ( the carrier of X,COMPLEX)):],(Funcs ( the carrier of X,COMPLEX)):]
ComplexFuncExtMult the carrier of X is non empty Relation-like [:COMPLEX,(Funcs ( the carrier of X,COMPLEX)):] -defined Funcs ( the carrier of X,COMPLEX) -valued Function-like total quasi_total Element of bool [:[:COMPLEX,(Funcs ( the carrier of X,COMPLEX)):],(Funcs ( the carrier of X,COMPLEX)):]
[:COMPLEX,(Funcs ( the carrier of X,COMPLEX)):] is non empty set
[:[:COMPLEX,(Funcs ( the carrier of X,COMPLEX)):],(Funcs ( the carrier of X,COMPLEX)):] is non empty set
bool [:[:COMPLEX,(Funcs ( the carrier of X,COMPLEX)):],(Funcs ( the carrier of X,COMPLEX)):] is non empty set
ComplexFuncUnit the carrier of X is Relation-like the carrier of X -defined COMPLEX -valued Function-like total quasi_total V139() Element of Funcs ( the carrier of X,COMPLEX)
the carrier of X --> 1r is non empty Relation-like the carrier of X -defined COMPLEX -valued Function-like total quasi_total V139() Element of bool [: the carrier of X,COMPLEX:]
ComplexFuncZero the carrier of X is Relation-like the carrier of X -defined COMPLEX -valued Function-like total quasi_total V139() Element of Funcs ( the carrier of X,COMPLEX)
the carrier of X --> 0 is non empty Relation-like the carrier of X -defined REAL -valued RAT -valued INT -valued Function-like total quasi_total V139() V140() V141() V142() Element of bool [: the carrier of X,REAL:]
[: the carrier of X,REAL:] is non empty V139() V140() V141() set
bool [: the carrier of X,REAL:] is non empty set
ComplexAlgebraStr(# (Funcs ( the carrier of X,COMPLEX)),(ComplexFuncMult the carrier of X),(ComplexFuncAdd the carrier of X),(ComplexFuncExtMult the carrier of X),(ComplexFuncUnit the carrier of X),(ComplexFuncZero the carrier of X) #) is strict ComplexAlgebraStr
the carrier of (CAlgebra the carrier of X) is non empty set
bool the carrier of (CAlgebra the carrier of X) is non empty set
{ b₁ where b₁ is non empty Relation-like the carrier of X -defined COMPLEX -valued Function-like total quasi_total V139() (X) Element of bool [: the carrier of X,COMPLEX:] : verum } is set
mult_ ((X),(CAlgebra the carrier of X)) is non empty Relation-like [:(X),(X):] -defined (X) -valued Function-like total quasi_total Element of bool [:[:(X),(X):],(X):]
[:(X),(X):] is non empty set
[:[:(X),(X):],(X):] is non empty set
bool [:[:(X),(X):],(X):] is non empty set
Add_ ((X),(CAlgebra the carrier of X)) is non empty Relation-like [:(X),(X):] -defined (X) -valued Function-like total quasi_total Element of bool [:[:(X),(X):],(X):]
Mult_ ((X),(CAlgebra the carrier of X)) is non empty Relation-like [:COMPLEX,(X):] -defined (X) -valued Function-like total quasi_total Element of bool [:[:COMPLEX,(X):],(X):]
[:COMPLEX,(X):] is non empty set
[:[:COMPLEX,(X):],(X):] is non empty set
bool [:[:COMPLEX,(X):],(X):] is non empty set
One_ ((X),(CAlgebra the carrier of X)) is Element of (X)
Zero_ ((X),(CAlgebra the carrier of X)) is Element of (X)
(X) is non empty Relation-like (X) -defined REAL -valued Function-like total quasi_total V139() V140() V141() Element of bool [:(X),REAL:]
[:(X),REAL:] is non empty V139() V140() V141() set
bool [:(X),REAL:] is non empty set
ComplexBoundedFunctions the carrier of X is non empty multiplicatively-closed Cadditively-linearly-closed Element of bool the carrier of (CAlgebra the carrier of X)
{ b₁ where b₁ is non empty Relation-like the carrier of X -defined COMPLEX -valued Function-like total quasi_total V139() Element of bool [: the carrier of X,COMPLEX:] : b₁ | the carrier of X is bounded } is set
ComplexBoundedFunctionsNorm the carrier of X is non empty Relation-like ComplexBoundedFunctions the carrier of X -defined REAL -valued Function-like total quasi_total V139() V140() V141() Element of bool [:(ComplexBoundedFunctions the carrier of X),REAL:]
[:(ComplexBoundedFunctions the carrier of X),REAL:] is non empty V139() V140() V141() set
bool [:(ComplexBoundedFunctions the carrier of X),REAL:] is non empty set
(ComplexBoundedFunctionsNorm the carrier of X) | (X) is Relation-like ComplexBoundedFunctions the carrier of X -defined REAL -valued Function-like V139() V140() V141() Element of bool [:(ComplexBoundedFunctions the carrier of X),REAL:]
Normed_Complex_AlgebraStr(# (X),(mult_ ((X),(CAlgebra the carrier of X))),(Add_ ((X),(CAlgebra the carrier of X))),(Mult_ ((X),(CAlgebra the carrier of X))),(One_ ((X),(CAlgebra the carrier of X))),(Zero_ ((X),(CAlgebra the carrier of X))),(X) #) is strict Normed_Complex_AlgebraStr
the carrier of (X) is non empty set
X is non empty Relation-like the carrier of X -defined COMPLEX -valued Function-like total quasi_total V139() Element of bool [: the carrier of X,COMPLEX:]
F is non empty Relation-like the carrier of X -defined COMPLEX -valued Function-like total quasi_total V139() Element of bool [: the carrier of X,COMPLEX:]
G is non empty Relation-like the carrier of X -defined COMPLEX -valued Function-like total quasi_total V139() Element of bool [: the carrier of X,COMPLEX:]
FB is Element of the carrier of (X)
GB is Element of the carrier of (X)
aFB is Element of the carrier of (X)
FB - GB is Element of the carrier of (X)
- GB is Element of the carrier of (X)
FB + (- GB) is Element of the carrier of (X)
the addF of (X) is non empty Relation-like [: the carrier of (X), the carrier of (X):] -defined the carrier of (X) -valued Function-like total quasi_total Element of bool [:[: the carrier of (X), the carrier of (X):], the carrier of (X):]
[: the carrier of (X), the carrier of (X):] is non empty set
[:[: the carrier of (X), the carrier of (X):], the carrier of (X):] is non empty set
bool [:[: the carrier of (X), the carrier of (X):], the carrier of (X):] is non empty set
the addF of (X) . (FB,(- GB)) is Element of the carrier of (X)
[FB,(- GB)] is set
{FB,(- GB)} is non empty set
{FB} is non empty set
{{FB,(- GB)},{FB}} is non empty set
the addF of (X) . [FB,(- GB)] is set
aFB + GB is Element of the carrier of (X)
the addF of (X) . (aFB,GB) is Element of the carrier of (X)
[aFB,GB] is set
{aFB,GB} is non empty set
{aFB} is non empty set
{{aFB,GB},{aFB}} is non empty set
the addF of (X) . [aFB,GB] is set
GB - GB is Element of the carrier of (X)
GB + (- GB) is Element of the carrier of (X)
the addF of (X) . (GB,(- GB)) is Element of the carrier of (X)
[GB,(- GB)] is set
{GB,(- GB)} is non empty set
{GB} is non empty set
{{GB,(- GB)},{GB}} is non empty set
the addF of (X) . [GB,(- GB)] is set
FB - (GB - GB) is Element of the carrier of (X)
- (GB - GB) is Element of the carrier of (X)
FB + (- (GB - GB)) is Element of the carrier of (X)
the addF of (X) . (FB,(- (GB - GB))) is Element of the carrier of (X)
[FB,(- (GB - GB))] is set
{FB,(- (GB - GB))} is non empty set
{{FB,(- (GB - GB))},{FB}} is non empty set
the addF of (X) . [FB,(- (GB - GB))] is set
0. (X) is zero Element of the carrier of (X)
the ZeroF of (X) is Element of the carrier of (X)
FB - (0. (X)) is Element of the carrier of (X)
- (0. (X)) is Element of the carrier of (X)
FB + (- (0. (X))) is Element of the carrier of (X)
the addF of (X) . (FB,(- (0. (X)))) is Element of the carrier of (X)
[FB,(- (0. (X)))] is set
{FB,(- (0. (X)))} is non empty set
{{FB,(- (0. (X)))},{FB}} is non empty set
the addF of (X) . [FB,(- (0. (X)))] is set
aF is Element of the carrier of X
X . aF is complex Element of COMPLEX
G . aF is complex Element of COMPLEX
F . aF is complex Element of COMPLEX
(G . aF) + (F . aF) is complex Element of COMPLEX
(X . aF) - (F . aF) is complex Element of COMPLEX
- (F . aF) is complex set
(X . aF) + (- (F . aF)) is complex set
aF is Element of the carrier of X
G . aF is complex Element of COMPLEX
X . aF is complex Element of COMPLEX
F . aF is complex Element of COMPLEX
(X . aF) - (F . aF) is complex Element of COMPLEX
- (F . aF) is complex set
(X . aF) + (- (F . aF)) is complex set
aF is Element of the carrier of X
G . aF is complex Element of COMPLEX
X . aF is complex Element of COMPLEX
F . aF is complex Element of COMPLEX
(X . aF) - (F . aF) is complex Element of COMPLEX
- (F . aF) is complex set
(X . aF) + (- (F . aF)) is complex set
(G . aF) + (F . aF) is complex Element of COMPLEX
aFB + (GB - GB) is Element of the carrier of (X)
the addF of (X) . (aFB,(GB - GB)) is Element of the carrier of (X)
[aFB,(GB - GB)] is set
{aFB,(GB - GB)} is non empty set
{{aFB,(GB - GB)},{aFB}} is non empty set
the addF of (X) . [aFB,(GB - GB)] is set
aFB + (0. (X)) is Element of the carrier of (X)
the addF of (X) . (aFB,(0. (X))) is Element of the carrier of (X)
[aFB,(0. (X))] is set
{aFB,(0. (X))} is non empty set
{{aFB,(0. (X))},{aFB}} is non empty set
the addF of (X) . [aFB,(0. (X))] is set
aF is Element of the carrier of X
G . aF is complex Element of COMPLEX
X . aF is complex Element of COMPLEX
F . aF is complex Element of COMPLEX
(X . aF) - (F . aF) is complex Element of COMPLEX
- (F . aF) is complex set
(X . aF) + (- (F . aF)) is complex set
aG is Element of the carrier of X
G . aG is complex Element of COMPLEX
X . aG is complex Element of COMPLEX
F . aG is complex Element of COMPLEX
(X . aG) - (F . aG) is complex Element of COMPLEX
- (F . aG) is complex set
(X . aG) + (- (F . aG)) is complex set
X is non empty right_complementable Abelian add-associative right_zeroed discerning reflexive vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like complete CNORMSTR
the carrier of X is non empty set
bool the carrier of X is non empty set
[:NAT, the carrier of X:] is non empty set
bool [:NAT, the carrier of X:] is non empty set
X is Element of bool the carrier of X
F is non empty Relation-like NAT -defined the carrier of X -valued Function-like total quasi_total Element of bool [:NAT, the carrier of X:]
rng F is Element of bool the carrier of X
lim F is Element of the carrier of X
X is non empty TopSpace-like V300() compact TopStruct
the carrier of X is non empty set
C_Normed_Algebra_of_BoundedFunctions the carrier of X is non empty right_complementable Abelian add-associative right_zeroed unital discerning reflexive vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like complete Normed_Complex_AlgebraStr
ComplexBoundedFunctions the carrier of X is non empty multiplicatively-closed Cadditively-linearly-closed Element of bool the carrier of (CAlgebra the carrier of X)
CAlgebra the carrier of X is non empty right_complementable Abelian add-associative right_zeroed unital associative commutative right-distributive right_unital well-unital left_unital vector-distributive scalar-distributive scalar-associative scalar-unital strict vector-associative ComplexAlgebraStr
Funcs ( the carrier of X,COMPLEX) is non empty functional FUNCTION_DOMAIN of the carrier of X, COMPLEX
ComplexFuncMult the carrier of X is non empty Relation-like [:(Funcs ( the carrier of X,COMPLEX)),(Funcs ( the carrier of X,COMPLEX)):] -defined Funcs ( the carrier of X,COMPLEX) -valued Function-like total quasi_total Element of bool [:[:(Funcs ( the carrier of X,COMPLEX)),(Funcs ( the carrier of X,COMPLEX)):],(Funcs ( the carrier of X,COMPLEX)):]
[:(Funcs ( the carrier of X,COMPLEX)),(Funcs ( the carrier of X,COMPLEX)):] is non empty set
[:[:(Funcs ( the carrier of X,COMPLEX)),(Funcs ( the carrier of X,COMPLEX)):],(Funcs ( the carrier of X,COMPLEX)):] is non empty set
bool [:[:(Funcs ( the carrier of X,COMPLEX)),(Funcs ( the carrier of X,COMPLEX)):],(Funcs ( the carrier of X,COMPLEX)):] is non empty set
ComplexFuncAdd the carrier of X is non empty Relation-like [:(Funcs ( the carrier of X,COMPLEX)),(Funcs ( the carrier of X,COMPLEX)):] -defined Funcs ( the carrier of X,COMPLEX) -valued Function-like total quasi_total Element of bool [:[:(Funcs ( the carrier of X,COMPLEX)),(Funcs ( the carrier of X,COMPLEX)):],(Funcs ( the carrier of X,COMPLEX)):]
ComplexFuncExtMult the carrier of X is non empty Relation-like [:COMPLEX,(Funcs ( the carrier of X,COMPLEX)):] -defined Funcs ( the carrier of X,COMPLEX) -valued Function-like total quasi_total Element of bool [:[:COMPLEX,(Funcs ( the carrier of X,COMPLEX)):],(Funcs ( the carrier of X,COMPLEX)):]
[:COMPLEX,(Funcs ( the carrier of X,COMPLEX)):] is non empty set
[:[:COMPLEX,(Funcs ( the carrier of X,COMPLEX)):],(Funcs ( the carrier of X,COMPLEX)):] is non empty set
bool [:[:COMPLEX,(Funcs ( the carrier of X,COMPLEX)):],(Funcs ( the carrier of X,COMPLEX)):] is non empty set
ComplexFuncUnit the carrier of X is Relation-like the carrier of X -defined COMPLEX -valued Function-like total quasi_total V139() Element of Funcs ( the carrier of X,COMPLEX)
the carrier of X --> 1r is non empty Relation-like the carrier of X -defined COMPLEX -valued Function-like total quasi_total V139() Element of bool [: the carrier of X,COMPLEX:]
[: the carrier of X,COMPLEX:] is non empty V139() set
bool [: the carrier of X,COMPLEX:] is non empty set
ComplexFuncZero the carrier of X is Relation-like the carrier of X -defined COMPLEX -valued Function-like total quasi_total V139() Element of Funcs ( the carrier of X,COMPLEX)
the carrier of X --> 0 is non empty Relation-like the carrier of X -defined REAL -valued RAT -valued INT -valued Function-like total quasi_total V139() V140() V141() V142() Element of bool [: the carrier of X,REAL:]
[: the carrier of X,REAL:] is non empty V139() V140() V141() set
bool [: the carrier of X,REAL:] is non empty set
ComplexAlgebraStr(# (Funcs ( the carrier of X,COMPLEX)),(ComplexFuncMult the carrier of X),(ComplexFuncAdd the carrier of X),(ComplexFuncExtMult the carrier of X),(ComplexFuncUnit the carrier of X),(ComplexFuncZero the carrier of X) #) is strict ComplexAlgebraStr
the carrier of (CAlgebra the carrier of X) is non empty set
bool the carrier of (CAlgebra the carrier of X) is non empty set
{ b₁ where b₁ is non empty Relation-like the carrier of X -defined COMPLEX -valued Function-like total quasi_total V139() Element of bool [: the carrier of X,COMPLEX:] : b₁ | the carrier of X is bounded } is set
mult_ ((ComplexBoundedFunctions the carrier of X),(CAlgebra the carrier of X)) is non empty Relation-like [:(ComplexBoundedFunctions the carrier of X),(ComplexBoundedFunctions the carrier of X):] -defined ComplexBoundedFunctions the carrier of X -valued Function-like total quasi_total Element of bool [:[:(ComplexBoundedFunctions the carrier of X),(ComplexBoundedFunctions the carrier of X):],(ComplexBoundedFunctions the carrier of X):]
[:(ComplexBoundedFunctions the carrier of X),(ComplexBoundedFunctions the carrier of X):] is non empty set
[:[:(ComplexBoundedFunctions the carrier of X),(ComplexBoundedFunctions the carrier of X):],(ComplexBoundedFunctions the carrier of X):] is non empty set
bool [:[:(ComplexBoundedFunctions the carrier of X),(ComplexBoundedFunctions the carrier of X):],(ComplexBoundedFunctions the carrier of X):] is non empty set
Add_ ((ComplexBoundedFunctions the carrier of X),(CAlgebra the carrier of X)) is non empty Relation-like [:(ComplexBoundedFunctions the carrier of X),(ComplexBoundedFunctions the carrier of X):] -defined ComplexBoundedFunctions the carrier of X -valued Function-like total quasi_total Element of bool [:[:(ComplexBoundedFunctions the carrier of X),(ComplexBoundedFunctions the carrier of X):],(ComplexBoundedFunctions the carrier of X):]
Mult_ ((ComplexBoundedFunctions the carrier of X),(CAlgebra the carrier of X)) is non empty Relation-like [:COMPLEX,(ComplexBoundedFunctions the carrier of X):] -defined ComplexBoundedFunctions the carrier of X -valued Function-like total quasi_total Element of bool [:[:COMPLEX,(ComplexBoundedFunctions the carrier of X):],(ComplexBoundedFunctions the carrier of X):]
[:COMPLEX,(ComplexBoundedFunctions the carrier of X):] is non empty set
[:[:COMPLEX,(ComplexBoundedFunctions the carrier of X):],(ComplexBoundedFunctions the carrier of X):] is non empty set
bool [:[:COMPLEX,(ComplexBoundedFunctions the carrier of X):],(ComplexBoundedFunctions the carrier of X):] is non empty set
One_ ((ComplexBoundedFunctions the carrier of X),(CAlgebra the carrier of X)) is Element of ComplexBoundedFunctions the carrier of X
Zero_ ((ComplexBoundedFunctions the carrier of X),(CAlgebra the carrier of X)) is Element of ComplexBoundedFunctions the carrier of X
ComplexBoundedFunctionsNorm the carrier of X is non empty Relation-like ComplexBoundedFunctions the carrier of X -defined REAL -valued Function-like total quasi_total V139() V140() V141() Element of bool [:(ComplexBoundedFunctions the carrier of X),REAL:]
[:(ComplexBoundedFunctions the carrier of X),REAL:] is non empty V139() V140() V141() set
bool [:(ComplexBoundedFunctions the carrier of X),REAL:] is non empty set
Normed_Complex_AlgebraStr(# (ComplexBoundedFunctions the carrier of X),(mult_ ((ComplexBoundedFunctions the carrier of X),(CAlgebra the carrier of X))),(Add_ ((ComplexBoundedFunctions the carrier of X),(CAlgebra the carrier of X))),(Mult_ ((ComplexBoundedFunctions the carrier of X),(CAlgebra the carrier of X))),(One_ ((ComplexBoundedFunctions the carrier of X),(CAlgebra the carrier of X))),(Zero_ ((ComplexBoundedFunctions the carrier of X),(CAlgebra the carrier of X))),(ComplexBoundedFunctionsNorm the carrier of X) #) is strict Normed_Complex_AlgebraStr
the carrier of (C_Normed_Algebra_of_BoundedFunctions the carrier of X) is non empty set
bool the carrier of (C_Normed_Algebra_of_BoundedFunctions the carrier of X) is non empty set
(X) is non empty multiplicatively-closed Cadditively-linearly-closed Element of bool the carrier of (CAlgebra the carrier of X)
{ b₁ where b₁ is non empty Relation-like the carrier of X -defined COMPLEX -valued Function-like total quasi_total V139() (X) Element of bool [: the carrier of X,COMPLEX:] : verum } is set
X is Element of bool the carrier of (C_Normed_Algebra_of_BoundedFunctions the carrier of X)
[:NAT, the carrier of (C_Normed_Algebra_of_BoundedFunctions the carrier of X):] is non empty set
bool [:NAT, the carrier of (C_Normed_Algebra_of_BoundedFunctions the carrier of X):] is non empty set
F is non empty Relation-like NAT -defined the carrier of (C_Normed_Algebra_of_BoundedFunctions the carrier of X) -valued Function-like total quasi_total Element of bool [:NAT, the carrier of (C_Normed_Algebra_of_BoundedFunctions the carrier of X):]
rng F is Element of bool the carrier of (C_Normed_Algebra_of_BoundedFunctions the carrier of X)
lim F is Element of the carrier of (C_Normed_Algebra_of_BoundedFunctions the carrier of X)
G is non empty Relation-like the carrier of X -defined COMPLEX -valued Function-like total quasi_total V139() Element of bool [: the carrier of X,COMPLEX:]
G | the carrier of X is Relation-like the carrier of X -defined COMPLEX -valued Function-like V139() Element of bool [: the carrier of X,COMPLEX:]
FB is set
PFuncs ( the carrier of X,COMPLEX) is set
GB is non empty Relation-like the carrier of X -defined COMPLEX -valued Function-like total quasi_total V139() Element of bool [: the carrier of X,COMPLEX:]
GB | the carrier of X is Relation-like the carrier of X -defined COMPLEX -valued Function-like V139() Element of bool [: the carrier of X,COMPLEX:]
[:NAT,(PFuncs ( the carrier of X,COMPLEX)):] is set
bool [:NAT,(PFuncs ( the carrier of X,COMPLEX)):] is non empty set
FB is Relation-like NAT -defined PFuncs ( the carrier of X,COMPLEX) -valued Function-like quasi_total Element of bool [:NAT,(PFuncs ( the carrier of X,COMPLEX)):]
GB is complex real ext-real Element of REAL
aFB is epsilon-transitive epsilon-connected ordinal natural complex real V101() ext-real V176() V177() V178() V179() V180() V181() V187() bounded_below Element of NAT
aF is epsilon-transitive epsilon-connected ordinal natural complex real V101() ext-real V176() V177() V178() V179() V180() V181() V187() bounded_below Element of NAT
aG is set
F . aF is Element of the carrier of (C_Normed_Algebra_of_BoundedFunctions the carrier of X)
(F . aF) - (lim F) is Element of the carrier of (C_Normed_Algebra_of_BoundedFunctions the carrier of X)
- (lim F) is Element of the carrier of (C_Normed_Algebra_of_BoundedFunctions the carrier of X)
(F . aF) + (- (lim F)) is Element of the carrier of (C_Normed_Algebra_of_BoundedFunctions the carrier of X)
the addF of (C_Normed_Algebra_of_BoundedFunctions the carrier of X) is non empty Relation-like [: the carrier of (C_Normed_Algebra_of_BoundedFunctions the carrier of X), the carrier of (C_Normed_Algebra_of_BoundedFunctions the carrier of X):] -defined the carrier of (C_Normed_Algebra_of_BoundedFunctions the carrier of X) -valued Function-like total quasi_total Element of bool [:[: the carrier of (C_Normed_Algebra_of_BoundedFunctions the carrier of X), the carrier of (C_Normed_Algebra_of_BoundedFunctions the carrier of X):], the carrier of (C_Normed_Algebra_of_BoundedFunctions the carrier of X):]
[: the carrier of (C_Normed_Algebra_of_BoundedFunctions the carrier of X), the carrier of (C_Normed_Algebra_of_BoundedFunctions the carrier of X):] is non empty set
[:[: the carrier of (C_Normed_Algebra_of_BoundedFunctions the carrier of X), the carrier of (C_Normed_Algebra_of_BoundedFunctions the carrier of X):], the carrier of (C_Normed_Algebra_of_BoundedFunctions the carrier of X):] is non empty set
bool [:[: the carrier of (C_Normed_Algebra_of_BoundedFunctions the carrier of X), the carrier of (C_Normed_Algebra_of_BoundedFunctions the carrier of X):], the carrier of (C_Normed_Algebra_of_BoundedFunctions the carrier of X):] is non empty set
the addF of (C_Normed_Algebra_of_BoundedFunctions the carrier of X) . ((F . aF),(- (lim F))) is Element of the carrier of (C_Normed_Algebra_of_BoundedFunctions the carrier of X)
[(F . aF),(- (lim F))] is set
{(F . aF),(- (lim F))} is non empty set
{(F . aF)} is non empty set
{{(F . aF),(- (lim F))},{(F . aF)}} is non empty set
the addF of (C_Normed_Algebra_of_BoundedFunctions the carrier of X) . [(F . aF),(- (lim F))] is set
||.((F . aF) - (lim F)).|| is complex real ext-real Element of REAL
the U9 of (C_Normed_Algebra_of_BoundedFunctions the carrier of X) is non empty Relation-like the carrier of (C_Normed_Algebra_of_BoundedFunctions the carrier of X) -defined REAL -valued Function-like total quasi_total V139() V140() V141() Element of bool [: the carrier of (C_Normed_Algebra_of_BoundedFunctions the carrier of X),REAL:]
[: the carrier of (C_Normed_Algebra_of_BoundedFunctions the carrier of X),REAL:] is non empty V139() V140() V141() set
bool [: the carrier of (C_Normed_Algebra_of_BoundedFunctions the carrier of X),REAL:] is non empty set
the U9 of (C_Normed_Algebra_of_BoundedFunctions the carrier of X) . ((F . aF) - (lim F)) is complex real ext-real Element of REAL
r is non empty Relation-like the carrier of X -defined COMPLEX -valued Function-like total quasi_total V139() Element of bool [: the carrier of X,COMPLEX:]
r | the carrier of X is Relation-like the carrier of X -defined COMPLEX -valued Function-like V139() Element of bool [: the carrier of X,COMPLEX:]
r is non empty Relation-like the carrier of X -defined COMPLEX -valued Function-like total quasi_total V139() Element of bool [: the carrier of X,COMPLEX:]
r | the carrier of X is Relation-like the carrier of X -defined COMPLEX -valued Function-like V139() Element of bool [: the carrier of X,COMPLEX:]
s1 is Element of the carrier of X
r . s1 is complex Element of COMPLEX
r . s1 is complex Element of COMPLEX
G . s1 is complex Element of COMPLEX
(r . s1) - (G . s1) is complex Element of COMPLEX
- (G . s1) is complex set
(r . s1) + (- (G . s1)) is complex set
abs (r . s1) is complex real ext-real non negative Element of REAL
FB . aF is Relation-like the carrier of X -defined COMPLEX -valued Function-like V139() Element of bool [: the carrier of X,COMPLEX:]
(FB . aF) . aG is complex set
G . aG is complex set
((FB . aF) . aG) - (G . aG) is complex set
- (G . aG) is complex set
((FB . aF) . aG) + (- (G . aG)) is complex set
abs (((FB . aF) . aG) - (G . aG)) is complex real ext-real non negative Element of REAL
the epsilon-transitive epsilon-connected ordinal natural complex real V101() ext-real V176() V177() V178() V179() V180() V181() V187() bounded_below Element of NAT is epsilon-transitive epsilon-connected ordinal natural complex real V101() ext-real V176() V177() V178() V179() V180() V181() V187() bounded_below Element of NAT
bool the carrier of X is non empty set
aFB is Element of the carrier of X
G . aFB is complex Element of COMPLEX
aF is V176() Element of bool COMPLEX
aG is complex set
r is V176() Neighbourhood of aG
r is complex real ext-real Element of REAL
{ b₁ where b₁ is complex set : not r <= |.(b₁ - aG).| } is set
3 is non empty epsilon-transitive epsilon-connected ordinal natural complex real V101() ext-real positive non negative V176() V177() V178() V179() V180() V181() V187() left_end bounded_below Element of NAT
r / 3 is complex real ext-real Element of COMPLEX
3 " is non empty complex real ext-real positive non negative set
r * (3 ") is complex real ext-real set
fau1 is epsilon-transitive epsilon-connected ordinal natural complex real V101() ext-real V176() V177() V178() V179() V180() V181() V187() bounded_below Element of NAT
dom F is V176() V177() V178() V179() V180() V181() bounded_below Element of bool NAT
FB . fau1 is Relation-like the carrier of X -defined COMPLEX -valued Function-like V139() Element of bool [: the carrier of X,COMPLEX:]
A2 is non empty Relation-like the carrier of X -defined COMPLEX -valued Function-like total quasi_total V139() (X) Element of bool [: the carrier of X,COMPLEX:]
A2 . aFB is complex Element of COMPLEX
{ b₁ where b₁ is complex set : not r / 3 <= |.(b₁ - (A2 . aFB)).| } is set
uu1 is set
fvu1 is complex set
fvu1 - (A2 . aFB) is complex set
- (A2 . aFB) is complex set
fvu1 + (- (A2 . aFB)) is complex set
|.(fvu1 - (A2 . aFB)).| is complex real ext-real non negative Element of REAL
uu1 is V176() Element of bool COMPLEX
(A2 . aFB) - (A2 . aFB) is complex Element of COMPLEX
- (A2 . aFB) is complex set
(A2 . aFB) + (- (A2 . aFB)) is complex set
|.((A2 . aFB) - (A2 . aFB)).| is complex real ext-real non negative Element of REAL
fvu1 is Element of bool the carrier of X
A2 .: fvu1 is V176() Element of bool COMPLEX
zz0 is set
G .: fvu1 is V176() Element of bool COMPLEX
dom G is Element of bool the carrier of X
xx0 is set
G . xx0 is complex set
A2 . xx0 is complex set
hx0 is complex set
hx0 - (A2 . aFB) is complex set
hx0 + (- (A2 . aFB)) is complex set
|.(hx0 - (A2 . aFB)).| is complex real ext-real non negative Element of REAL
(A2 . xx0) - (G . xx0) is complex set
- (G . xx0) is complex set
(A2 . xx0) + (- (G . xx0)) is complex set
|.((A2 . xx0) - (G . xx0)).| is complex real ext-real non negative Element of REAL
- ((A2 . xx0) - (G . xx0)) is complex set
|.(- ((A2 . xx0) - (G . xx0))).| is complex real ext-real non negative Element of REAL
(G . xx0) - (A2 . xx0) is complex set
- (A2 . xx0) is complex set
(G . xx0) + (- (A2 . xx0)) is complex set
|.((G . xx0) - (A2 . xx0)).| is complex real ext-real non negative Element of REAL
(A2 . aFB) - (G . aFB) is complex Element of COMPLEX
- (G . aFB) is complex set
(A2 . aFB) + (- (G . aFB)) is complex set
|.((A2 . aFB) - (G . aFB)).| is complex real ext-real non negative Element of REAL
(r / 3) + (r / 3) is complex real ext-real Element of COMPLEX
(A2 . xx0) - (A2 . aFB) is complex set
(A2 . xx0) + (- (A2 . aFB)) is complex set
|.((A2 . xx0) - (A2 . aFB)).| is complex real ext-real non negative Element of REAL
|.((G . xx0) - (A2 . xx0)).| + |.((A2 . xx0) - (A2 . aFB)).| is complex real ext-real non negative Element of REAL
((r / 3) + (r / 3)) + (r / 3) is complex real ext-real Element of COMPLEX
(|.((G . xx0) - (A2 . xx0)).| + |.((A2 . xx0) - (A2 . aFB)).|) + |.((A2 . aFB) - (G . aFB)).| is complex real ext-real non negative Element of REAL
(G . xx0) - (G . aFB) is complex set
(G . xx0) + (- (G . aFB)) is complex set
|.((G . xx0) - (G . aFB)).| is complex real ext-real non negative Element of REAL
((G . xx0) - (A2 . xx0)) + ((A2 . xx0) - (A2 . aFB)) is complex set
(((G . xx0) - (A2 . xx0)) + ((A2 . xx0) - (A2 . aFB))) + ((A2 . aFB) - (G . aFB)) is complex set
|.((((G . xx0) - (A2 . xx0)) + ((A2 . xx0) - (A2 . aFB))) + ((A2 . aFB) - (G . aFB))).| is complex real ext-real non negative Element of REAL
|.(((G . xx0) - (A2 . xx0)) + ((A2 . xx0) - (A2 . aFB))).| is complex real ext-real non negative Element of REAL
|.(((G . xx0) - (A2 . xx0)) + ((A2 . xx0) - (A2 . aFB))).| + |.((A2 . aFB) - (G . aFB)).| is complex real ext-real non negative Element of REAL
X is non empty TopSpace-like V300() compact TopStruct
(X) is non empty right_complementable Abelian add-associative right_zeroed unital associative commutative right-distributive right_unital well-unital left_unital discerning reflexive vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like vector-associative strict Normed_Complex_AlgebraStr
(X) is non empty multiplicatively-closed Cadditively-linearly-closed Element of bool the carrier of (CAlgebra the carrier of X)
the carrier of X is non empty set
CAlgebra the carrier of X is non empty right_complementable Abelian add-associative right_zeroed unital associative commutative right-distributive right_unital well-unital left_unital vector-distributive scalar-distributive scalar-associative scalar-unital strict vector-associative ComplexAlgebraStr
Funcs ( the carrier of X,COMPLEX) is non empty functional FUNCTION_DOMAIN of the carrier of X, COMPLEX
ComplexFuncMult the carrier of X is non empty Relation-like [:(Funcs ( the carrier of X,COMPLEX)),(Funcs ( the carrier of X,COMPLEX)):] -defined Funcs ( the carrier of X,COMPLEX) -valued Function-like total quasi_total Element of bool [:[:(Funcs ( the carrier of X,COMPLEX)),(Funcs ( the carrier of X,COMPLEX)):],(Funcs ( the carrier of X,COMPLEX)):]
[:(Funcs ( the carrier of X,COMPLEX)),(Funcs ( the carrier of X,COMPLEX)):] is non empty set
[:[:(Funcs ( the carrier of X,COMPLEX)),(Funcs ( the carrier of X,COMPLEX)):],(Funcs ( the carrier of X,COMPLEX)):] is non empty set
bool [:[:(Funcs ( the carrier of X,COMPLEX)),(Funcs ( the carrier of X,COMPLEX)):],(Funcs ( the carrier of X,COMPLEX)):] is non empty set
ComplexFuncAdd the carrier of X is non empty Relation-like [:(Funcs ( the carrier of X,COMPLEX)),(Funcs ( the carrier of X,COMPLEX)):] -defined Funcs ( the carrier of X,COMPLEX) -valued Function-like total quasi_total Element of bool [:[:(Funcs ( the carrier of X,COMPLEX)),(Funcs ( the carrier of X,COMPLEX)):],(Funcs ( the carrier of X,COMPLEX)):]
ComplexFuncExtMult the carrier of X is non empty Relation-like [:COMPLEX,(Funcs ( the carrier of X,COMPLEX)):] -defined Funcs ( the carrier of X,COMPLEX) -valued Function-like total quasi_total Element of bool [:[:COMPLEX,(Funcs ( the carrier of X,COMPLEX)):],(Funcs ( the carrier of X,COMPLEX)):]
[:COMPLEX,(Funcs ( the carrier of X,COMPLEX)):] is non empty set
[:[:COMPLEX,(Funcs ( the carrier of X,COMPLEX)):],(Funcs ( the carrier of X,COMPLEX)):] is non empty set
bool [:[:COMPLEX,(Funcs ( the carrier of X,COMPLEX)):],(Funcs ( the carrier of X,COMPLEX)):] is non empty set
ComplexFuncUnit the carrier of X is Relation-like the carrier of X -defined COMPLEX -valued Function-like total quasi_total V139() Element of Funcs ( the carrier of X,COMPLEX)
the carrier of X --> 1r is non empty Relation-like the carrier of X -defined COMPLEX -valued Function-like total quasi_total V139() Element of bool [: the carrier of X,COMPLEX:]
[: the carrier of X,COMPLEX:] is non empty V139() set
bool [: the carrier of X,COMPLEX:] is non empty set
ComplexFuncZero the carrier of X is Relation-like the carrier of X -defined COMPLEX -valued Function-like total quasi_total V139() Element of Funcs ( the carrier of X,COMPLEX)
the carrier of X --> 0 is non empty Relation-like the carrier of X -defined REAL -valued RAT -valued INT -valued Function-like total quasi_total V139() V140() V141() V142() Element of bool [: the carrier of X,REAL:]
[: the carrier of X,REAL:] is non empty V139() V140() V141() set
bool [: the carrier of X,REAL:] is non empty set
ComplexAlgebraStr(# (Funcs ( the carrier of X,COMPLEX)),(ComplexFuncMult the carrier of X),(ComplexFuncAdd the carrier of X),(ComplexFuncExtMult the carrier of X),(ComplexFuncUnit the carrier of X),(ComplexFuncZero the carrier of X) #) is strict ComplexAlgebraStr
the carrier of (CAlgebra the carrier of X) is non empty set
bool the carrier of (CAlgebra the carrier of X) is non empty set
{ b₁ where b₁ is non empty Relation-like the carrier of X -defined COMPLEX -valued Function-like total quasi_total V139() (X) Element of bool [: the carrier of X,COMPLEX:] : verum } is set
mult_ ((X),(CAlgebra the carrier of X)) is non empty Relation-like [:(X),(X):] -defined (X) -valued Function-like total quasi_total Element of bool [:[:(X),(X):],(X):]
[:(X),(X):] is non empty set
[:[:(X),(X):],(X):] is non empty set
bool [:[:(X),(X):],(X):] is non empty set
Add_ ((X),(CAlgebra the carrier of X)) is non empty Relation-like [:(X),(X):] -defined (X) -valued Function-like total quasi_total Element of bool [:[:(X),(X):],(X):]
Mult_ ((X),(CAlgebra the carrier of X)) is non empty Relation-like [:COMPLEX,(X):] -defined (X) -valued Function-like total quasi_total Element of bool [:[:COMPLEX,(X):],(X):]
[:COMPLEX,(X):] is non empty set
[:[:COMPLEX,(X):],(X):] is non empty set
bool [:[:COMPLEX,(X):],(X):] is non empty set
One_ ((X),(CAlgebra the carrier of X)) is Element of (X)
Zero_ ((X),(CAlgebra the carrier of X)) is Element of (X)
(X) is non empty Relation-like (X) -defined REAL -valued Function-like total quasi_total V139() V140() V141() Element of bool [:(X),REAL:]
[:(X),REAL:] is non empty V139() V140() V141() set
bool [:(X),REAL:] is non empty set
ComplexBoundedFunctions the carrier of X is non empty multiplicatively-closed Cadditively-linearly-closed Element of bool the carrier of (CAlgebra the carrier of X)
{ b₁ where b₁ is non empty Relation-like the carrier of X -defined COMPLEX -valued Function-like total quasi_total V139() Element of bool [: the carrier of X,COMPLEX:] : b₁ | the carrier of X is bounded } is set
ComplexBoundedFunctionsNorm the carrier of X is non empty Relation-like ComplexBoundedFunctions the carrier of X -defined REAL -valued Function-like total quasi_total V139() V140() V141() Element of bool [:(ComplexBoundedFunctions the carrier of X),REAL:]
[:(ComplexBoundedFunctions the carrier of X),REAL:] is non empty V139() V140() V141() set
bool [:(ComplexBoundedFunctions the carrier of X),REAL:] is non empty set
(ComplexBoundedFunctionsNorm the carrier of X) | (X) is Relation-like ComplexBoundedFunctions the carrier of X -defined REAL -valued Function-like V139() V140() V141() Element of bool [:(ComplexBoundedFunctions the carrier of X),REAL:]
Normed_Complex_AlgebraStr(# (X),(mult_ ((X),(CAlgebra the carrier of X))),(Add_ ((X),(CAlgebra the carrier of X))),(Mult_ ((X),(CAlgebra the carrier of X))),(One_ ((X),(CAlgebra the carrier of X))),(Zero_ ((X),(CAlgebra the carrier of X))),(X) #) is strict Normed_Complex_AlgebraStr
the carrier of (X) is non empty set
[:NAT, the carrier of (X):] is non empty set
bool [:NAT, the carrier of (X):] is non empty set
X is non empty Relation-like NAT -defined the carrier of (X) -valued Function-like total quasi_total Element of bool [:NAT, the carrier of (X):]
F is set
rng X is Element of bool the carrier of (X)
bool the carrier of (X) is non empty set
C_Normed_Algebra_of_BoundedFunctions the carrier of X is non empty right_complementable Abelian add-associative right_zeroed unital discerning reflexive vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like complete Normed_Complex_AlgebraStr
mult_ ((ComplexBoundedFunctions the carrier of X),(CAlgebra the carrier of X)) is non empty Relation-like [:(ComplexBoundedFunctions the carrier of X),(ComplexBoundedFunctions the carrier of X):] -defined ComplexBoundedFunctions the carrier of X -valued Function-like total quasi_total Element of bool [:[:(ComplexBoundedFunctions the carrier of X),(ComplexBoundedFunctions the carrier of X):],(ComplexBoundedFunctions the carrier of X):]
[:(ComplexBoundedFunctions the carrier of X),(ComplexBoundedFunctions the carrier of X):] is non empty set
[:[:(ComplexBoundedFunctions the carrier of X),(ComplexBoundedFunctions the carrier of X):],(ComplexBoundedFunctions the carrier of X):] is non empty set
bool [:[:(ComplexBoundedFunctions the carrier of X),(ComplexBoundedFunctions the carrier of X):],(ComplexBoundedFunctions the carrier of X):] is non empty set
Add_ ((ComplexBoundedFunctions the carrier of X),(CAlgebra the carrier of X)) is non empty Relation-like [:(ComplexBoundedFunctions the carrier of X),(ComplexBoundedFunctions the carrier of X):] -defined ComplexBoundedFunctions the carrier of X -valued Function-like total quasi_total Element of bool [:[:(ComplexBoundedFunctions the carrier of X),(ComplexBoundedFunctions the carrier of X):],(ComplexBoundedFunctions the carrier of X):]
Mult_ ((ComplexBoundedFunctions the carrier of X),(CAlgebra the carrier of X)) is non empty Relation-like [:COMPLEX,(ComplexBoundedFunctions the carrier of X):] -defined ComplexBoundedFunctions the carrier of X -valued Function-like total quasi_total Element of bool [:[:COMPLEX,(ComplexBoundedFunctions the carrier of X):],(ComplexBoundedFunctions the carrier of X):]
[:COMPLEX,(ComplexBoundedFunctions the carrier of X):] is non empty set
[:[:COMPLEX,(ComplexBoundedFunctions the carrier of X):],(ComplexBoundedFunctions the carrier of X):] is non empty set
bool [:[:COMPLEX,(ComplexBoundedFunctions the carrier of X):],(ComplexBoundedFunctions the carrier of X):] is non empty set
One_ ((ComplexBoundedFunctions the carrier of X),(CAlgebra the carrier of X)) is Element of ComplexBoundedFunctions the carrier of X
Zero_ ((ComplexBoundedFunctions the carrier of X),(CAlgebra the carrier of X)) is Element of ComplexBoundedFunctions the carrier of X
Normed_Complex_AlgebraStr(# (ComplexBoundedFunctions the carrier of X),(mult_ ((ComplexBoundedFunctions the carrier of X),(CAlgebra the carrier of X))),(Add_ ((ComplexBoundedFunctions the carrier of X),(CAlgebra the carrier of X))),(Mult_ ((ComplexBoundedFunctions the carrier of X),(CAlgebra the carrier of X))),(One_ ((ComplexBoundedFunctions the carrier of X),(CAlgebra the carrier of X))),(Zero_ ((ComplexBoundedFunctions the carrier of X),(CAlgebra the carrier of X))),(ComplexBoundedFunctionsNorm the carrier of X) #) is strict Normed_Complex_AlgebraStr
the carrier of (C_Normed_Algebra_of_BoundedFunctions the carrier of X) is non empty set
[:NAT, the carrier of (C_Normed_Algebra_of_BoundedFunctions the carrier of X):] is non empty set
bool [:NAT, the carrier of (C_Normed_Algebra_of_BoundedFunctions the carrier of X):] is non empty set
G is complex real ext-real Element of REAL
FB is epsilon-transitive epsilon-connected ordinal natural complex real V101() ext-real V176() V177() V178() V179() V180() V181() V187() bounded_below Element of NAT
GB is epsilon-transitive epsilon-connected ordinal natural complex real V101() ext-real V176() V177() V178() V179() V180() V181() V187() bounded_below Element of NAT
aFB is epsilon-transitive epsilon-connected ordinal natural complex real V101() ext-real V176() V177() V178() V179() V180() V181() V187() bounded_below Element of NAT
aF is epsilon-transitive epsilon-connected ordinal natural complex real V101() ext-real V176() V177() V178() V179() V180() V181() V187() bounded_below Element of NAT
X . aFB is Element of the carrier of (X)
X . aF is Element of the carrier of (X)
(X . aFB) - (X . aF) is Element of the carrier of (X)
- (X . aF) is Element of the carrier of (X)
(X . aFB) + (- (X . aF)) is Element of the carrier of (X)
the addF of (X) is non empty Relation-like [: the carrier of (X), the carrier of (X):] -defined the carrier of (X) -valued Function-like total quasi_total Element of bool [:[: the carrier of (X), the carrier of (X):], the carrier of (X):]
[: the carrier of (X), the carrier of (X):] is non empty set
[:[: the carrier of (X), the carrier of (X):], the carrier of (X):] is non empty set
bool [:[: the carrier of (X), the carrier of (X):], the carrier of (X):] is non empty set
the addF of (X) . ((X . aFB),(- (X . aF))) is Element of the carrier of (X)
[(X . aFB),(- (X . aF))] is set
{(X . aFB),(- (X . aF))} is non empty set
{(X . aFB)} is non empty set
{{(X . aFB),(- (X . aF))},{(X . aFB)}} is non empty set
the addF of (X) . [(X . aFB),(- (X . aF))] is set
||.((X . aFB) - (X . aF)).|| is complex real ext-real Element of REAL
the U9 of (X) is non empty Relation-like the carrier of (X) -defined REAL -valued Function-like total quasi_total V139() V140() V141() Element of bool [: the carrier of (X),REAL:]
[: the carrier of (X),REAL:] is non empty V139() V140() V141() set
bool [: the carrier of (X),REAL:] is non empty set
the U9 of (X) . ((X . aFB) - (X . aF)) is complex real ext-real Element of REAL
F is non empty Relation-like NAT -defined the carrier of (C_Normed_Algebra_of_BoundedFunctions the carrier of X) -valued Function-like total quasi_total Element of bool [:NAT, the carrier of (C_Normed_Algebra_of_BoundedFunctions the carrier of X):]
F . aFB is Element of the carrier of (C_Normed_Algebra_of_BoundedFunctions the carrier of X)
F . aF is Element of the carrier of (C_Normed_Algebra_of_BoundedFunctions the carrier of X)
(F . aFB) - (F . aF) is Element of the carrier of (C_Normed_Algebra_of_BoundedFunctions the carrier of X)
- (F . aF) is Element of the carrier of (C_Normed_Algebra_of_BoundedFunctions the carrier of X)
(F . aFB) + (- (F . aF)) is Element of the carrier of (C_Normed_Algebra_of_BoundedFunctions the carrier of X)
the addF of (C_Normed_Algebra_of_BoundedFunctions the carrier of X) is non empty Relation-like [: the carrier of (C_Normed_Algebra_of_BoundedFunctions the carrier of X), the carrier of (C_Normed_Algebra_of_BoundedFunctions the carrier of X):] -defined the carrier of (C_Normed_Algebra_of_BoundedFunctions the carrier of X) -valued Function-like total quasi_total Element of bool [:[: the carrier of (C_Normed_Algebra_of_BoundedFunctions the carrier of X), the carrier of (C_Normed_Algebra_of_BoundedFunctions the carrier of X):], the carrier of (C_Normed_Algebra_of_BoundedFunctions the carrier of X):]
[: the carrier of (C_Normed_Algebra_of_BoundedFunctions the carrier of X), the carrier of (C_Normed_Algebra_of_BoundedFunctions the carrier of X):] is non empty set
[:[: the carrier of (C_Normed_Algebra_of_BoundedFunctions the carrier of X), the carrier of (C_Normed_Algebra_of_BoundedFunctions the carrier of X):], the carrier of (C_Normed_Algebra_of_BoundedFunctions the carrier of X):] is non empty set
bool [:[: the carrier of (C_Normed_Algebra_of_BoundedFunctions the carrier of X), the carrier of (C_Normed_Algebra_of_BoundedFunctions the carrier of X):], the carrier of (C_Normed_Algebra_of_BoundedFunctions the carrier of X):] is non empty set
the addF of (C_Normed_Algebra_of_BoundedFunctions the carrier of X) . ((F . aFB),(- (F . aF))) is Element of the carrier of (C_Normed_Algebra_of_BoundedFunctions the carrier of X)
[(F . aFB),(- (F . aF))] is set
{(F . aFB),(- (F . aF))} is non empty set
{(F . aFB)} is non empty set
{{(F . aFB),(- (F . aF))},{(F . aFB)}} is non empty set
the addF of (C_Normed_Algebra_of_BoundedFunctions the carrier of X) . [(F . aFB),(- (F . aF))] is set
||.((F . aFB) - (F . aF)).|| is complex real ext-real Element of REAL
the U9 of (C_Normed_Algebra_of_BoundedFunctions the carrier of X) is non empty Relation-like the carrier of (C_Normed_Algebra_of_BoundedFunctions the carrier of X) -defined REAL -valued Function-like total quasi_total V139() V140() V141() Element of bool [: the carrier of (C_Normed_Algebra_of_BoundedFunctions the carrier of X),REAL:]
[: the carrier of (C_Normed_Algebra_of_BoundedFunctions the carrier of X),REAL:] is non empty V139() V140() V141() set
bool [: the carrier of (C_Normed_Algebra_of_BoundedFunctions the carrier of X),REAL:] is non empty set
the U9 of (C_Normed_Algebra_of_BoundedFunctions the carrier of X) . ((F . aFB) - (F . aF)) is complex real ext-real Element of REAL
aFB is epsilon-transitive epsilon-connected ordinal natural complex real V101() ext-real V176() V177() V178() V179() V180() V181() V187() bounded_below Element of NAT
aF is epsilon-transitive epsilon-connected ordinal natural complex real V101() ext-real V176() V177() V178() V179() V180() V181() V187() bounded_below Element of NAT
F . aFB is Element of the carrier of (C_Normed_Algebra_of_BoundedFunctions the carrier of X)
F . aF is Element of the carrier of (C_Normed_Algebra_of_BoundedFunctions the carrier of X)
(F . aFB) - (F . aF) is Element of the carrier of (C_Normed_Algebra_of_BoundedFunctions the carrier of X)
- (F . aF) is Element of the carrier of (C_Normed_Algebra_of_BoundedFunctions the carrier of X)
(F . aFB) + (- (F . aF)) is Element of the carrier of (C_Normed_Algebra_of_BoundedFunctions the carrier of X)
the addF of (C_Normed_Algebra_of_BoundedFunctions the carrier of X) . ((F . aFB),(- (F . aF))) is Element of the carrier of (C_Normed_Algebra_of_BoundedFunctions the carrier of X)
[(F . aFB),(- (F . aF))] is set
{(F . aFB),(- (F . aF))} is non empty set
{(F . aFB)} is non empty set
{{(F . aFB),(- (F . aF))},{(F . aFB)}} is non empty set
the addF of (C_Normed_Algebra_of_BoundedFunctions the carrier of X) . [(F . aFB),(- (F . aF))] is set
||.((F . aFB) - (F . aF)).|| is complex real ext-real Element of REAL
the U9 of (C_Normed_Algebra_of_BoundedFunctions the carrier of X) . ((F . aFB) - (F . aF)) is complex real ext-real Element of REAL
bool the carrier of (C_Normed_Algebra_of_BoundedFunctions the carrier of X) is non empty set
G is Element of bool the carrier of (C_Normed_Algebra_of_BoundedFunctions the carrier of X)
lim F is Element of the carrier of (C_Normed_Algebra_of_BoundedFunctions the carrier of X)
FB is Element of the carrier of (X)
GB is complex real ext-real Element of REAL
aFB is epsilon-transitive epsilon-connected ordinal natural complex real V101() ext-real V176() V177() V178() V179() V180() V181() V187() bounded_below Element of NAT
aF is epsilon-transitive epsilon-connected ordinal natural complex real V101() ext-real V176() V177() V178() V179() V180() V181() V187() bounded_below Element of NAT
F . aF is Element of the carrier of (C_Normed_Algebra_of_BoundedFunctions the carrier of X)
(F . aF) - (lim F) is Element of the carrier of (C_Normed_Algebra_of_BoundedFunctions the carrier of X)
- (lim F) is Element of the carrier of (C_Normed_Algebra_of_BoundedFunctions the carrier of X)
(F . aF) + (- (lim F)) is Element of the carrier of (C_Normed_Algebra_of_BoundedFunctions the carrier of X)
the addF of (C_Normed_Algebra_of_BoundedFunctions the carrier of X) . ((F . aF),(- (lim F))) is Element of the carrier of (C_Normed_Algebra_of_BoundedFunctions the carrier of X)
[(F . aF),(- (lim F))] is set
{(F . aF),(- (lim F))} is non empty set
{(F . aF)} is non empty set
{{(F . aF),(- (lim F))},{(F . aF)}} is non empty set
the addF of (C_Normed_Algebra_of_BoundedFunctions the carrier of X) . [(F . aF),(- (lim F))] is set
||.((F . aF) - (lim F)).|| is complex real ext-real Element of REAL
the U9 of (C_Normed_Algebra_of_BoundedFunctions the carrier of X) . ((F . aF) - (lim F)) is complex real ext-real Element of REAL
X . aF is Element of the carrier of (X)
(X . aF) - FB is Element of the carrier of (X)
- FB is Element of the carrier of (X)
(X . aF) + (- FB) is Element of the carrier of (X)
the addF of (X) . ((X . aF),(- FB)) is Element of the carrier of (X)
[(X . aF),(- FB)] is set
{(X . aF),(- FB)} is non empty set
{(X . aF)} is non empty set
{{(X . aF),(- FB)},{(X . aF)}} is non empty set
the addF of (X) . [(X . aF),(- FB)] is set
||.((X . aF) - FB).|| is complex real ext-real Element of REAL
the U9 of (X) . ((X . aF) - FB) is complex real ext-real Element of REAL
aF is epsilon-transitive epsilon-connected ordinal natural complex real V101() ext-real V176() V177() V178() V179() V180() V181() V187() bounded_below Element of NAT
X . aF is Element of the carrier of (X)
(X . aF) - FB is Element of the carrier of (X)
(X . aF) + (- FB) is Element of the carrier of (X)
the addF of (X) . ((X . aF),(- FB)) is Element of the carrier of (X)
[(X . aF),(- FB)] is set
{(X . aF),(- FB)} is non empty set
{(X . aF)} is non empty set
{{(X . aF),(- FB)},{(X . aF)}} is non empty set
the addF of (X) . [(X . aF),(- FB)] is set
||.((X . aF) - FB).|| is complex real ext-real Element of REAL
the U9 of (X) . ((X . aF) - FB) is complex real ext-real Element of REAL
X is non empty TopSpace-like V300() compact TopStruct
(X) is non empty right_complementable Abelian add-associative right_zeroed unital associative commutative right-distributive right_unital well-unital left_unital discerning reflexive vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like vector-associative strict Normed_Complex_AlgebraStr
(X) is non empty multiplicatively-closed Cadditively-linearly-closed Element of bool the carrier of (CAlgebra the carrier of X)
the carrier of X is non empty set
CAlgebra the carrier of X is non empty right_complementable Abelian add-associative right_zeroed unital associative commutative right-distributive right_unital well-unital left_unital vector-distributive scalar-distributive scalar-associative scalar-unital strict vector-associative ComplexAlgebraStr
Funcs ( the carrier of X,COMPLEX) is non empty functional FUNCTION_DOMAIN of the carrier of X, COMPLEX
ComplexFuncMult the carrier of X is non empty Relation-like [:(Funcs ( the carrier of X,COMPLEX)),(Funcs ( the carrier of X,COMPLEX)):] -defined Funcs ( the carrier of X,COMPLEX) -valued Function-like total quasi_total Element of bool [:[:(Funcs ( the carrier of X,COMPLEX)),(Funcs ( the carrier of X,COMPLEX)):],(Funcs ( the carrier of X,COMPLEX)):]
[:(Funcs ( the carrier of X,COMPLEX)),(Funcs ( the carrier of X,COMPLEX)):] is non empty set
[:[:(Funcs ( the carrier of X,COMPLEX)),(Funcs ( the carrier of X,COMPLEX)):],(Funcs ( the carrier of X,COMPLEX)):] is non empty set
bool [:[:(Funcs ( the carrier of X,COMPLEX)),(Funcs ( the carrier of X,COMPLEX)):],(Funcs ( the carrier of X,COMPLEX)):] is non empty set
ComplexFuncAdd the carrier of X is non empty Relation-like [:(Funcs ( the carrier of X,COMPLEX)),(Funcs ( the carrier of X,COMPLEX)):] -defined Funcs ( the carrier of X,COMPLEX) -valued Function-like total quasi_total Element of bool [:[:(Funcs ( the carrier of X,COMPLEX)),(Funcs ( the carrier of X,COMPLEX)):],(Funcs ( the carrier of X,COMPLEX)):]
ComplexFuncExtMult the carrier of X is non empty Relation-like [:COMPLEX,(Funcs ( the carrier of X,COMPLEX)):] -defined Funcs ( the carrier of X,COMPLEX) -valued Function-like total quasi_total Element of bool [:[:COMPLEX,(Funcs ( the carrier of X,COMPLEX)):],(Funcs ( the carrier of X,COMPLEX)):]
[:COMPLEX,(Funcs ( the carrier of X,COMPLEX)):] is non empty set
[:[:COMPLEX,(Funcs ( the carrier of X,COMPLEX)):],(Funcs ( the carrier of X,COMPLEX)):] is non empty set
bool [:[:COMPLEX,(Funcs ( the carrier of X,COMPLEX)):],(Funcs ( the carrier of X,COMPLEX)):] is non empty set
ComplexFuncUnit the carrier of X is Relation-like the carrier of X -defined COMPLEX -valued Function-like total quasi_total V139() Element of Funcs ( the carrier of X,COMPLEX)
the carrier of X --> 1r is non empty Relation-like the carrier of X -defined COMPLEX -valued Function-like total quasi_total V139() Element of bool [: the carrier of X,COMPLEX:]
[: the carrier of X,COMPLEX:] is non empty V139() set
bool [: the carrier of X,COMPLEX:] is non empty set
ComplexFuncZero the carrier of X is Relation-like the carrier of X -defined COMPLEX -valued Function-like total quasi_total V139() Element of Funcs ( the carrier of X,COMPLEX)
the carrier of X --> 0 is non empty Relation-like the carrier of X -defined REAL -valued RAT -valued INT -valued Function-like total quasi_total V139() V140() V141() V142() Element of bool [: the carrier of X,REAL:]
[: the carrier of X,REAL:] is non empty V139() V140() V141() set
bool [: the carrier of X,REAL:] is non empty set
ComplexAlgebraStr(# (Funcs ( the carrier of X,COMPLEX)),(ComplexFuncMult the carrier of X),(ComplexFuncAdd the carrier of X),(ComplexFuncExtMult the carrier of X),(ComplexFuncUnit the carrier of X),(ComplexFuncZero the carrier of X) #) is strict ComplexAlgebraStr
the carrier of (CAlgebra the carrier of X) is non empty set
bool the carrier of (CAlgebra the carrier of X) is non empty set
{ b₁ where b₁ is non empty Relation-like the carrier of X -defined COMPLEX -valued Function-like total quasi_total V139() (X) Element of bool [: the carrier of X,COMPLEX:] : verum } is set
mult_ ((X),(CAlgebra the carrier of X)) is non empty Relation-like [:(X),(X):] -defined (X) -valued Function-like total quasi_total Element of bool [:[:(X),(X):],(X):]
[:(X),(X):] is non empty set
[:[:(X),(X):],(X):] is non empty set
bool [:[:(X),(X):],(X):] is non empty set
Add_ ((X),(CAlgebra the carrier of X)) is non empty Relation-like [:(X),(X):] -defined (X) -valued Function-like total quasi_total Element of bool [:[:(X),(X):],(X):]
Mult_ ((X),(CAlgebra the carrier of X)) is non empty Relation-like [:COMPLEX,(X):] -defined (X) -valued Function-like total quasi_total Element of bool [:[:COMPLEX,(X):],(X):]
[:COMPLEX,(X):] is non empty set
[:[:COMPLEX,(X):],(X):] is non empty set
bool [:[:COMPLEX,(X):],(X):] is non empty set
One_ ((X),(CAlgebra the carrier of X)) is Element of (X)
Zero_ ((X),(CAlgebra the carrier of X)) is Element of (X)
(X) is non empty Relation-like (X) -defined REAL -valued Function-like total quasi_total V139() V140() V141() Element of bool [:(X),REAL:]
[:(X),REAL:] is non empty V139() V140() V141() set
bool [:(X),REAL:] is non empty set
ComplexBoundedFunctions the carrier of X is non empty multiplicatively-closed Cadditively-linearly-closed Element of bool the carrier of (CAlgebra the carrier of X)
{ b₁ where b₁ is non empty Relation-like the carrier of X -defined COMPLEX -valued Function-like total quasi_total V139() Element of bool [: the carrier of X,COMPLEX:] : b₁ | the carrier of X is bounded } is set
ComplexBoundedFunctionsNorm the carrier of X is non empty Relation-like ComplexBoundedFunctions the carrier of X -defined REAL -valued Function-like total quasi_total V139() V140() V141() Element of bool [:(ComplexBoundedFunctions the carrier of X),REAL:]
[:(ComplexBoundedFunctions the carrier of X),REAL:] is non empty V139() V140() V141() set
bool [:(ComplexBoundedFunctions the carrier of X),REAL:] is non empty set
(ComplexBoundedFunctionsNorm the carrier of X) | (X) is Relation-like ComplexBoundedFunctions the carrier of X -defined REAL -valued Function-like V139() V140() V141() Element of bool [:(ComplexBoundedFunctions the carrier of X),REAL:]
Normed_Complex_AlgebraStr(# (X),(mult_ ((X),(CAlgebra the carrier of X))),(Add_ ((X),(CAlgebra the carrier of X))),(Mult_ ((X),(CAlgebra the carrier of X))),(One_ ((X),(CAlgebra the carrier of X))),(Zero_ ((X),(CAlgebra the carrier of X))),(X) #) is strict Normed_Complex_AlgebraStr
the carrier of (X) is non empty set
[:NAT, the carrier of (X):] is non empty set
bool [:NAT, the carrier of (X):] is non empty set
X is non empty Relation-like NAT -defined the carrier of (X) -valued Function-like total quasi_total Element of bool [:NAT, the carrier of (X):]
X is non empty TopSpace-like V300() compact TopStruct
(X) is non empty right_complementable Abelian add-associative right_zeroed unital associative commutative right-distributive right_unital well-unital left_unital discerning reflexive vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like complete vector-associative strict Normed_Complex_AlgebraStr
(X) is non empty multiplicatively-closed Cadditively-linearly-closed Element of bool the carrier of (CAlgebra the carrier of X)
the carrier of X is non empty set
CAlgebra the carrier of X is non empty right_complementable Abelian add-associative right_zeroed unital associative commutative right-distributive right_unital well-unital left_unital vector-distributive scalar-distributive scalar-associative scalar-unital strict vector-associative ComplexAlgebraStr
Funcs ( the carrier of X,COMPLEX) is non empty functional FUNCTION_DOMAIN of the carrier of X, COMPLEX
ComplexFuncMult the carrier of X is non empty Relation-like [:(Funcs ( the carrier of X,COMPLEX)),(Funcs ( the carrier of X,COMPLEX)):] -defined Funcs ( the carrier of X,COMPLEX) -valued Function-like total quasi_total Element of bool [:[:(Funcs ( the carrier of X,COMPLEX)),(Funcs ( the carrier of X,COMPLEX)):],(Funcs ( the carrier of X,COMPLEX)):]
[:(Funcs ( the carrier of X,COMPLEX)),(Funcs ( the carrier of X,COMPLEX)):] is non empty set
[:[:(Funcs ( the carrier of X,COMPLEX)),(Funcs ( the carrier of X,COMPLEX)):],(Funcs ( the carrier of X,COMPLEX)):] is non empty set
bool [:[:(Funcs ( the carrier of X,COMPLEX)),(Funcs ( the carrier of X,COMPLEX)):],(Funcs ( the carrier of X,COMPLEX)):] is non empty set
ComplexFuncAdd the carrier of X is non empty Relation-like [:(Funcs ( the carrier of X,COMPLEX)),(Funcs ( the carrier of X,COMPLEX)):] -defined Funcs ( the carrier of X,COMPLEX) -valued Function-like total quasi_total Element of bool [:[:(Funcs ( the carrier of X,COMPLEX)),(Funcs ( the carrier of X,COMPLEX)):],(Funcs ( the carrier of X,COMPLEX)):]
ComplexFuncExtMult the carrier of X is non empty Relation-like [:COMPLEX,(Funcs ( the carrier of X,COMPLEX)):] -defined Funcs ( the carrier of X,COMPLEX) -valued Function-like total quasi_total Element of bool [:[:COMPLEX,(Funcs ( the carrier of X,COMPLEX)):],(Funcs ( the carrier of X,COMPLEX)):]
[:COMPLEX,(Funcs ( the carrier of X,COMPLEX)):] is non empty set
[:[:COMPLEX,(Funcs ( the carrier of X,COMPLEX)):],(Funcs ( the carrier of X,COMPLEX)):] is non empty set
bool [:[:COMPLEX,(Funcs ( the carrier of X,COMPLEX)):],(Funcs ( the carrier of X,COMPLEX)):] is non empty set
ComplexFuncUnit the carrier of X is Relation-like the carrier of X -defined COMPLEX -valued Function-like total quasi_total V139() Element of Funcs ( the carrier of X,COMPLEX)
the carrier of X --> 1r is non empty Relation-like the carrier of X -defined COMPLEX -valued Function-like total quasi_total V139() Element of bool [: the carrier of X,COMPLEX:]
[: the carrier of X,COMPLEX:] is non empty V139() set
bool [: the carrier of X,COMPLEX:] is non empty set
ComplexFuncZero the carrier of X is Relation-like the carrier of X -defined COMPLEX -valued Function-like total quasi_total V139() Element of Funcs ( the carrier of X,COMPLEX)
the carrier of X --> 0 is non empty Relation-like the carrier of X -defined REAL -valued RAT -valued INT -valued Function-like total quasi_total V139() V140() V141() V142() Element of bool [: the carrier of X,REAL:]
[: the carrier of X,REAL:] is non empty V139() V140() V141() set
bool [: the carrier of X,REAL:] is non empty set
ComplexAlgebraStr(# (Funcs ( the carrier of X,COMPLEX)),(ComplexFuncMult the carrier of X),(ComplexFuncAdd the carrier of X),(ComplexFuncExtMult the carrier of X),(ComplexFuncUnit the carrier of X),(ComplexFuncZero the carrier of X) #) is strict ComplexAlgebraStr
the carrier of (CAlgebra the carrier of X) is non empty set
bool the carrier of (CAlgebra the carrier of X) is non empty set
{ b₁ where b₁ is non empty Relation-like the carrier of X -defined COMPLEX -valued Function-like total quasi_total V139() (X) Element of bool [: the carrier of X,COMPLEX:] : verum } is set
mult_ ((X),(CAlgebra the carrier of X)) is non empty Relation-like [:(X),(X):] -defined (X) -valued Function-like total quasi_total Element of bool [:[:(X),(X):],(X):]
[:(X),(X):] is non empty set
[:[:(X),(X):],(X):] is non empty set
bool [:[:(X),(X):],(X):] is non empty set
Add_ ((X),(CAlgebra the carrier of X)) is non empty Relation-like [:(X),(X):] -defined (X) -valued Function-like total quasi_total Element of bool [:[:(X),(X):],(X):]
Mult_ ((X),(CAlgebra the carrier of X)) is non empty Relation-like [:COMPLEX,(X):] -defined (X) -valued Function-like total quasi_total Element of bool [:[:COMPLEX,(X):],(X):]
[:COMPLEX,(X):] is non empty set
[:[:COMPLEX,(X):],(X):] is non empty set
bool [:[:COMPLEX,(X):],(X):] is non empty set
One_ ((X),(CAlgebra the carrier of X)) is Element of (X)
Zero_ ((X),(CAlgebra the carrier of X)) is Element of (X)
(X) is non empty Relation-like (X) -defined REAL -valued Function-like total quasi_total V139() V140() V141() Element of bool [:(X),REAL:]
[:(X),REAL:] is non empty V139() V140() V141() set
bool [:(X),REAL:] is non empty set
ComplexBoundedFunctions the carrier of X is non empty multiplicatively-closed Cadditively-linearly-closed Element of bool the carrier of (CAlgebra the carrier of X)
{ b₁ where b₁ is non empty Relation-like the carrier of X -defined COMPLEX -valued Function-like total quasi_total V139() Element of bool [: the carrier of X,COMPLEX:] : b₁ | the carrier of X is bounded } is set
ComplexBoundedFunctionsNorm the carrier of X is non empty Relation-like ComplexBoundedFunctions the carrier of X -defined REAL -valued Function-like total quasi_total V139() V140() V141() Element of bool [:(ComplexBoundedFunctions the carrier of X),REAL:]
[:(ComplexBoundedFunctions the carrier of X),REAL:] is non empty V139() V140() V141() set
bool [:(ComplexBoundedFunctions the carrier of X),REAL:] is non empty set
(ComplexBoundedFunctionsNorm the carrier of X) | (X) is Relation-like ComplexBoundedFunctions the carrier of X -defined REAL -valued Function-like V139() V140() V141() Element of bool [:(ComplexBoundedFunctions the carrier of X),REAL:]
Normed_Complex_AlgebraStr(# (X),(mult_ ((X),(CAlgebra the carrier of X))),(Add_ ((X),(CAlgebra the carrier of X))),(Mult_ ((X),(CAlgebra the carrier of X))),(One_ ((X),(CAlgebra the carrier of X))),(Zero_ ((X),(CAlgebra the carrier of X))),(X) #) is strict Normed_Complex_AlgebraStr
the carrier of (X) is non empty set
C_Normed_Algebra_of_BoundedFunctions the carrier of X is non empty right_complementable Abelian add-associative right_zeroed unital discerning reflexive vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like complete Normed_Complex_AlgebraStr
mult_ ((ComplexBoundedFunctions the carrier of X),(CAlgebra the carrier of X)) is non empty Relation-like [:(ComplexBoundedFunctions the carrier of X),(ComplexBoundedFunctions the carrier of X):] -defined ComplexBoundedFunctions the carrier of X -valued Function-like total quasi_total Element of bool [:[:(ComplexBoundedFunctions the carrier of X),(ComplexBoundedFunctions the carrier of X):],(ComplexBoundedFunctions the carrier of X):]
[:(ComplexBoundedFunctions the carrier of X),(ComplexBoundedFunctions the carrier of X):] is non empty set
[:[:(ComplexBoundedFunctions the carrier of X),(ComplexBoundedFunctions the carrier of X):],(ComplexBoundedFunctions the carrier of X):] is non empty set
bool [:[:(ComplexBoundedFunctions the carrier of X),(ComplexBoundedFunctions the carrier of X):],(ComplexBoundedFunctions the carrier of X):] is non empty set
Add_ ((ComplexBoundedFunctions the carrier of X),(CAlgebra the carrier of X)) is non empty Relation-like [:(ComplexBoundedFunctions the carrier of X),(ComplexBoundedFunctions the carrier of X):] -defined ComplexBoundedFunctions the carrier of X -valued Function-like total quasi_total Element of bool [:[:(ComplexBoundedFunctions the carrier of X),(ComplexBoundedFunctions the carrier of X):],(ComplexBoundedFunctions the carrier of X):]
Mult_ ((ComplexBoundedFunctions the carrier of X),(CAlgebra the carrier of X)) is non empty Relation-like [:COMPLEX,(ComplexBoundedFunctions the carrier of X):] -defined ComplexBoundedFunctions the carrier of X -valued Function-like total quasi_total Element of bool [:[:COMPLEX,(ComplexBoundedFunctions the carrier of X):],(ComplexBoundedFunctions the carrier of X):]
[:COMPLEX,(ComplexBoundedFunctions the carrier of X):] is non empty set
[:[:COMPLEX,(ComplexBoundedFunctions the carrier of X):],(ComplexBoundedFunctions the carrier of X):] is non empty set
bool [:[:COMPLEX,(ComplexBoundedFunctions the carrier of X):],(ComplexBoundedFunctions the carrier of X):] is non empty set
One_ ((ComplexBoundedFunctions the carrier of X),(CAlgebra the carrier of X)) is Element of ComplexBoundedFunctions the carrier of X
Zero_ ((ComplexBoundedFunctions the carrier of X),(CAlgebra the carrier of X)) is Element of ComplexBoundedFunctions the carrier of X
Normed_Complex_AlgebraStr(# (ComplexBoundedFunctions the carrier of X),(mult_ ((ComplexBoundedFunctions the carrier of X),(CAlgebra the carrier of X))),(Add_ ((ComplexBoundedFunctions the carrier of X),(CAlgebra the carrier of X))),(Mult_ ((ComplexBoundedFunctions the carrier of X),(CAlgebra the carrier of X))),(One_ ((ComplexBoundedFunctions the carrier of X),(CAlgebra the carrier of X))),(Zero_ ((ComplexBoundedFunctions the carrier of X),(CAlgebra the carrier of X))),(ComplexBoundedFunctionsNorm the carrier of X) #) is strict Normed_Complex_AlgebraStr
the carrier of (C_Normed_Algebra_of_BoundedFunctions the carrier of X) is non empty set
F is Element of the carrier of (X)
G is Element of the carrier of (X)
||.F.|| is complex real ext-real Element of REAL
the U9 of (X) is non empty Relation-like the carrier of (X) -defined REAL -valued Function-like total quasi_total V139() V140() V141() Element of bool [: the carrier of (X),REAL:]
[: the carrier of (X),REAL:] is non empty V139() V140() V141() set
bool [: the carrier of (X),REAL:] is non empty set
the U9 of (X) . F is complex real ext-real Element of REAL
FB is Element of the carrier of (C_Normed_Algebra_of_BoundedFunctions the carrier of X)
||.FB.|| is complex real ext-real Element of REAL
the U9 of (C_Normed_Algebra_of_BoundedFunctions the carrier of X) is non empty Relation-like the carrier of (C_Normed_Algebra_of_BoundedFunctions the carrier of X) -defined REAL -valued Function-like total quasi_total V139() V140() V141() Element of bool [: the carrier of (C_Normed_Algebra_of_BoundedFunctions the carrier of X),REAL:]
[: the carrier of (C_Normed_Algebra_of_BoundedFunctions the carrier of X),REAL:] is non empty V139() V140() V141() set
bool [: the carrier of (C_Normed_Algebra_of_BoundedFunctions the carrier of X),REAL:] is non empty set
the U9 of (C_Normed_Algebra_of_BoundedFunctions the carrier of X) . FB is complex real ext-real Element of REAL
||.G.|| is complex real ext-real Element of REAL
the U9 of (X) . G is complex real ext-real Element of REAL
GB is Element of the carrier of (C_Normed_Algebra_of_BoundedFunctions the carrier of X)
||.GB.|| is complex real ext-real Element of REAL
the U9 of (C_Normed_Algebra_of_BoundedFunctions the carrier of X) . GB is complex real ext-real Element of REAL
F * G is Element of the carrier of (X)
the multF of (X) is non empty Relation-like [: the carrier of (X), the carrier of (X):] -defined the carrier of (X) -valued Function-like total quasi_total Element of bool [:[: the carrier of (X), the carrier of (X):], the carrier of (X):]
[: the carrier of (X), the carrier of (X):] is non empty set
[:[: the carrier of (X), the carrier of (X):], the carrier of (X):] is non empty set
bool [:[: the carrier of (X), the carrier of (X):], the carrier of (X):] is non empty set
the multF of (X) . (F,G) is Element of the carrier of (X)
[F,G] is set
{F,G} is non empty set
{F} is non empty set
{{F,G},{F}} is non empty set
the multF of (X) . [F,G] is set
||.(F * G).|| is complex real ext-real Element of REAL
the U9 of (X) . (F * G) is complex real ext-real Element of REAL
FB * GB is Element of the carrier of (C_Normed_Algebra_of_BoundedFunctions the carrier of X)
the multF of (C_Normed_Algebra_of_BoundedFunctions the carrier of X) is non empty Relation-like [: the carrier of (C_Normed_Algebra_of_BoundedFunctions the carrier of X), the carrier of (C_Normed_Algebra_of_BoundedFunctions the carrier of X):] -defined the carrier of (C_Normed_Algebra_of_BoundedFunctions the carrier of X) -valued Function-like total quasi_total Element of bool [:[: the carrier of (C_Normed_Algebra_of_BoundedFunctions the carrier of X), the carrier of (C_Normed_Algebra_of_BoundedFunctions the carrier of X):], the carrier of (C_Normed_Algebra_of_BoundedFunctions the carrier of X):]
[: the carrier of (C_Normed_Algebra_of_BoundedFunctions the carrier of X), the carrier of (C_Normed_Algebra_of_BoundedFunctions the carrier of X):] is non empty set
[:[: the carrier of (C_Normed_Algebra_of_BoundedFunctions the carrier of X), the carrier of (C_Normed_Algebra_of_BoundedFunctions the carrier of X):], the carrier of (C_Normed_Algebra_of_BoundedFunctions the carrier of X):] is non empty set
bool [:[: the carrier of (C_Normed_Algebra_of_BoundedFunctions the carrier of X), the carrier of (C_Normed_Algebra_of_BoundedFunctions the carrier of X):], the carrier of (C_Normed_Algebra_of_BoundedFunctions the carrier of X):] is non empty set
the multF of (C_Normed_Algebra_of_BoundedFunctions the carrier of X) . (FB,GB) is Element of the carrier of (C_Normed_Algebra_of_BoundedFunctions the carrier of X)
[FB,GB] is set
{FB,GB} is non empty set
{FB} is non empty set
{{FB,GB},{FB}} is non empty set
the multF of (C_Normed_Algebra_of_BoundedFunctions the carrier of X) . [FB,GB] is set
||.(FB * GB).|| is complex real ext-real Element of REAL
the U9 of (C_Normed_Algebra_of_BoundedFunctions the carrier of X) . (FB * GB) is complex real ext-real Element of REAL
||.F.|| * ||.G.|| is complex real ext-real Element of REAL
1. (X) is Element of the carrier of (X)
the OneF of (X) is Element of the carrier of (X)
||.(1. (X)).|| is complex real ext-real Element of REAL
the U9 of (X) . (1. (X)) is complex real ext-real Element of REAL
1. (C_Normed_Algebra_of_BoundedFunctions the carrier of X) is Element of the carrier of (C_Normed_Algebra_of_BoundedFunctions the carrier of X)
the OneF of (C_Normed_Algebra_of_BoundedFunctions the carrier of X) is Element of the carrier of (C_Normed_Algebra_of_BoundedFunctions the carrier of X)
||.(1. (C_Normed_Algebra_of_BoundedFunctions the carrier of X)).|| is complex real ext-real Element of REAL
the U9 of (C_Normed_Algebra_of_BoundedFunctions the carrier of X) . (1. (C_Normed_Algebra_of_BoundedFunctions the carrier of X)) is complex real ext-real Element of REAL
G is Element of the carrier of (X)
FB is Element of the carrier of (X)
aFB is Element of the carrier of (C_Normed_Algebra_of_BoundedFunctions the carrier of X)
F is complex set
F * aFB is Element of the carrier of (C_Normed_Algebra_of_BoundedFunctions the carrier of X)
the Mult of (C_Normed_Algebra_of_BoundedFunctions the carrier of X) is non empty Relation-like [:COMPLEX, the carrier of (C_Normed_Algebra_of_BoundedFunctions the carrier of X):] -defined the carrier of (C_Normed_Algebra_of_BoundedFunctions the carrier of X) -valued Function-like total quasi_total Element of bool [:[:COMPLEX, the carrier of (C_Normed_Algebra_of_BoundedFunctions the carrier of X):], the carrier of (C_Normed_Algebra_of_BoundedFunctions the carrier of X):]
[:COMPLEX, the carrier of (C_Normed_Algebra_of_BoundedFunctions the carrier of X):] is non empty set
[:[:COMPLEX, the carrier of (C_Normed_Algebra_of_BoundedFunctions the carrier of X):], the carrier of (C_Normed_Algebra_of_BoundedFunctions the carrier of X):] is non empty set
bool [:[:COMPLEX, the carrier of (C_Normed_Algebra_of_BoundedFunctions the carrier of X):], the carrier of (C_Normed_Algebra_of_BoundedFunctions the carrier of X):] is non empty set
[F,aFB] is set
{F,aFB} is non empty set
{F} is non empty V176() set
{{F,aFB},{F}} is non empty set
the Mult of (C_Normed_Algebra_of_BoundedFunctions the carrier of X) . [F,aFB] is set
F * FB is Element of the carrier of (X)
the Mult of (X) is non empty Relation-like [:COMPLEX, the carrier of (X):] -defined the carrier of (X) -valued Function-like total quasi_total Element of bool [:[:COMPLEX, the carrier of (X):], the carrier of (X):]
[:COMPLEX, the carrier of (X):] is non empty set
[:[:COMPLEX, the carrier of (X):], the carrier of (X):] is non empty set
bool [:[:COMPLEX, the carrier of (X):], the carrier of (X):] is non empty set
[F,FB] is set
{F,FB} is non empty set
{{F,FB},{F}} is non empty set
the Mult of (X) . [F,FB] is set
G * FB is Element of the carrier of (X)
the multF of (X) . (G,FB) is Element of the carrier of (X)
[G,FB] is set
{G,FB} is non empty set
{G} is non empty set
{{G,FB},{G}} is non empty set
the multF of (X) . [G,FB] is set
GB is Element of the carrier of (C_Normed_Algebra_of_BoundedFunctions the carrier of X)
GB * aFB is Element of the carrier of (C_Normed_Algebra_of_BoundedFunctions the carrier of X)
the multF of (C_Normed_Algebra_of_BoundedFunctions the carrier of X) . (GB,aFB) is Element of the carrier of (C_Normed_Algebra_of_BoundedFunctions the carrier of X)
[GB,aFB] is set
{GB,aFB} is non empty set
{GB} is non empty set
{{GB,aFB},{GB}} is non empty set
the multF of (C_Normed_Algebra_of_BoundedFunctions the carrier of X) . [GB,aFB] is set
F * (G * FB) is Element of the carrier of (X)
[F,(G * FB)] is set
{F,(G * FB)} is non empty set
{{F,(G * FB)},{F}} is non empty set
the Mult of (X) . [F,(G * FB)] is set
F * (GB * aFB) is Element of the carrier of (C_Normed_Algebra_of_BoundedFunctions the carrier of X)
[F,(GB * aFB)] is set
{F,(GB * aFB)} is non empty set
{{F,(GB * aFB)},{F}} is non empty set
the Mult of (C_Normed_Algebra_of_BoundedFunctions the carrier of X) . [F,(GB * aFB)] is set
GB * (F * aFB) is Element of the carrier of (C_Normed_Algebra_of_BoundedFunctions the carrier of X)
the multF of (C_Normed_Algebra_of_BoundedFunctions the carrier of X) . (GB,(F * aFB)) is Element of the carrier of (C_Normed_Algebra_of_BoundedFunctions the carrier of X)
[GB,(F * aFB)] is set
{GB,(F * aFB)} is non empty set
{{GB,(F * aFB)},{GB}} is non empty set
the multF of (C_Normed_Algebra_of_BoundedFunctions the carrier of X) . [GB,(F * aFB)] is set
G * (F * FB) is Element of the carrier of (X)
the multF of (X) . (G,(F * FB)) is Element of the carrier of (X)
[G,(F * FB)] is set
{G,(F * FB)} is non empty set
{{G,(F * FB)},{G}} is non empty set
the multF of (X) . [G,(F * FB)] is set
F is Element of the carrier of (X)
G is Element of the carrier of (X)
FB is Element of the carrier of (X)
G + FB is Element of the carrier of (X)
the addF of (X) is non empty Relation-like [: the carrier of (X), the carrier of (X):] -defined the carrier of (X) -valued Function-like total quasi_total Element of bool [:[: the carrier of (X), the carrier of (X):], the carrier of (X):]
the addF of (X) . (G,FB) is Element of the carrier of (X)
[G,FB] is set
{G,FB} is non empty set
{G} is non empty set
{{G,FB},{G}} is non empty set
the addF of (X) . [G,FB] is set
aFB is Element of the carrier of (C_Normed_Algebra_of_BoundedFunctions the carrier of X)
aF is Element of the carrier of (C_Normed_Algebra_of_BoundedFunctions the carrier of X)
aFB + aF is Element of the carrier of (C_Normed_Algebra_of_BoundedFunctions the carrier of X)
the addF of (C_Normed_Algebra_of_BoundedFunctions the carrier of X) is non empty Relation-like [: the carrier of (C_Normed_Algebra_of_BoundedFunctions the carrier of X), the carrier of (C_Normed_Algebra_of_BoundedFunctions the carrier of X):] -defined the carrier of (C_Normed_Algebra_of_BoundedFunctions the carrier of X) -valued Function-like total quasi_total Element of bool [:[: the carrier of (C_Normed_Algebra_of_BoundedFunctions the carrier of X), the carrier of (C_Normed_Algebra_of_BoundedFunctions the carrier of X):], the carrier of (C_Normed_Algebra_of_BoundedFunctions the carrier of X):]
the addF of (C_Normed_Algebra_of_BoundedFunctions the carrier of X) . (aFB,aF) is Element of the carrier of (C_Normed_Algebra_of_BoundedFunctions the carrier of X)
[aFB,aF] is set
{aFB,aF} is non empty set
{aFB} is non empty set
{{aFB,aF},{aFB}} is non empty set
the addF of (C_Normed_Algebra_of_BoundedFunctions the carrier of X) . [aFB,aF] is set
G * F is Element of the carrier of (X)
the multF of (X) . (G,F) is Element of the carrier of (X)
[G,F] is set
{G,F} is non empty set
{{G,F},{G}} is non empty set
the multF of (X) . [G,F] is set
GB is Element of the carrier of (C_Normed_Algebra_of_BoundedFunctions the carrier of X)
aFB * GB is Element of the carrier of (C_Normed_Algebra_of_BoundedFunctions the carrier of X)
the multF of (C_Normed_Algebra_of_BoundedFunctions the carrier of X) . (aFB,GB) is Element of the carrier of (C_Normed_Algebra_of_BoundedFunctions the carrier of X)
[aFB,GB] is set
{aFB,GB} is non empty set
{{aFB,GB},{aFB}} is non empty set
the multF of (C_Normed_Algebra_of_BoundedFunctions the carrier of X) . [aFB,GB] is set
FB * F is Element of the carrier of (X)
the multF of (X) . (FB,F) is Element of the carrier of (X)
[FB,F] is set
{FB,F} is non empty set
{FB} is non empty set
{{FB,F},{FB}} is non empty set
the multF of (X) . [FB,F] is set
aF * GB is Element of the carrier of (C_Normed_Algebra_of_BoundedFunctions the carrier of X)
the multF of (C_Normed_Algebra_of_BoundedFunctions the carrier of X) . (aF,GB) is Element of the carrier of (C_Normed_Algebra_of_BoundedFunctions the carrier of X)
[aF,GB] is set
{aF,GB} is non empty set
{aF} is non empty set
{{aF,GB},{aF}} is non empty set
the multF of (C_Normed_Algebra_of_BoundedFunctions the carrier of X) . [aF,GB] is set
(G + FB) * F is Element of the carrier of (X)
the multF of (X) . ((G + FB),F) is Element of the carrier of (X)
[(G + FB),F] is set
{(G + FB),F} is non empty set
{(G + FB)} is non empty set
{{(G + FB),F},{(G + FB)}} is non empty set
the multF of (X) . [(G + FB),F] is set
(aFB + aF) * GB is Element of the carrier of (C_Normed_Algebra_of_BoundedFunctions the carrier of X)
the multF of (C_Normed_Algebra_of_BoundedFunctions the carrier of X) . ((aFB + aF),GB) is Element of the carrier of (C_Normed_Algebra_of_BoundedFunctions the carrier of X)
[(aFB + aF),GB] is set
{(aFB + aF),GB} is non empty set
{(aFB + aF)} is non empty set
{{(aFB + aF),GB},{(aFB + aF)}} is non empty set
the multF of (C_Normed_Algebra_of_BoundedFunctions the carrier of X) . [(aFB + aF),GB] is set
(aFB * GB) + (aF * GB) is Element of the carrier of (C_Normed_Algebra_of_BoundedFunctions the carrier of X)
the addF of (C_Normed_Algebra_of_BoundedFunctions the carrier of X) . ((aFB * GB),(aF * GB)) is Element of the carrier of (C_Normed_Algebra_of_BoundedFunctions the carrier of X)
[(aFB * GB),(aF * GB)] is set
{(aFB * GB),(aF * GB)} is non empty set
{(aFB * GB)} is non empty set
{{(aFB * GB),(aF * GB)},{(aFB * GB)}} is non empty set
the addF of (C_Normed_Algebra_of_BoundedFunctions the carrier of X) . [(aFB * GB),(aF * GB)] is set
(G * F) + (FB * F) is Element of the carrier of (X)
the addF of (X) . ((G * F),(FB * F)) is Element of the carrier of (X)
[(G * F),(FB * F)] is set
{(G * F),(FB * F)} is non empty set
{(G * F)} is non empty set
{{(G * F),(FB * F)},{(G * F)}} is non empty set
the addF of (X) . [(G * F),(FB * F)] is set
X is non empty TopSpace-like TopStruct
the carrier of X is non empty set
[: the carrier of X,COMPLEX:] is non empty V139() set
bool [: the carrier of X,COMPLEX:] is non empty set
X is non empty Relation-like the carrier of X -defined COMPLEX -valued Function-like total quasi_total V139() Element of bool [: the carrier of X,COMPLEX:]
F is non empty Relation-like the carrier of X -defined COMPLEX -valued Function-like total quasi_total V139() Element of bool [: the carrier of X,COMPLEX:]
X + F is Relation-like the carrier of X -defined Function-like total V139() set
support (X + F) is set
support X is set
support F is set
(support X) \/ (support F) is set
rng (X + F) is V176() set
dom (X + F) is Element of bool the carrier of X
bool the carrier of X is non empty set
dom X is Element of bool the carrier of X
dom F is Element of bool the carrier of X
(dom X) /\ (dom F) is Element of bool the carrier of X
the carrier of X /\ (dom F) is Element of bool the carrier of X
the carrier of X /\ the carrier of X is set
GB is non empty Relation-like the carrier of X -defined COMPLEX -valued Function-like total quasi_total V139() Element of bool [: the carrier of X,COMPLEX:]
dom GB is Element of bool the carrier of X
support GB is set
aFB is set
the carrier of X \ (support X) is Element of bool the carrier of X
the carrier of X \ (support F) is Element of bool the carrier of X
( the carrier of X \ (support X)) /\ ( the carrier of X \ (support F)) is Element of bool the carrier of X
X . aFB is complex set
F . aFB is complex set
(X + F) . aFB is complex set
(X . aFB) + (F . aFB) is complex set
the carrier of X \ (support (X + F)) is Element of bool the carrier of X
the carrier of X \ ((support X) \/ (support F)) is Element of bool the carrier of X
the carrier of X \ ( the carrier of X \ (support (X + F))) is Element of bool the carrier of X
the carrier of X \ ( the carrier of X \ ((support X) \/ (support F))) is Element of bool the carrier of X
the carrier of X /\ (support (X + F)) is set
the carrier of X /\ ((support X) \/ (support F)) is set
the carrier of X /\ (support X) is set
the carrier of X /\ (support F) is set
( the carrier of X /\ (support X)) \/ ( the carrier of X /\ (support F)) is set
(support X) \/ ( the carrier of X /\ (support F)) is set
X is non empty TopSpace-like TopStruct
the carrier of X is non empty set
[: the carrier of X,COMPLEX:] is non empty V139() set
bool [: the carrier of X,COMPLEX:] is non empty set
X is complex set
F is non empty Relation-like the carrier of X -defined COMPLEX -valued Function-like total quasi_total V139() Element of bool [: the carrier of X,COMPLEX:]
X (#) F is non empty Relation-like the carrier of X -defined COMPLEX -valued Function-like total quasi_total V139() Element of bool [: the carrier of X,COMPLEX:]
support (X (#) F) is set
support F is set
GB is set
(X (#) F) . GB is complex set
F . GB is complex set
X * (F . GB) is complex set
X is non empty TopSpace-like TopStruct
the carrier of X is non empty set
[: the carrier of X,COMPLEX:] is non empty V139() set
bool [: the carrier of X,COMPLEX:] is non empty set
X is non empty Relation-like the carrier of X -defined COMPLEX -valued Function-like total quasi_total V139() Element of bool [: the carrier of X,COMPLEX:]
F is non empty Relation-like the carrier of X -defined COMPLEX -valued Function-like total quasi_total V139() Element of bool [: the carrier of X,COMPLEX:]
X (#) F is non empty Relation-like the carrier of X -defined COMPLEX -valued Function-like total quasi_total V139() Element of bool [: the carrier of X,COMPLEX:]
support (X (#) F) is set
support X is set
support F is set
(support X) \/ (support F) is set
dom X is Element of bool the carrier of X
bool the carrier of X is non empty set
dom F is Element of bool the carrier of X
FB is non empty Relation-like the carrier of X -defined COMPLEX -valued Function-like total quasi_total V139() Element of bool [: the carrier of X,COMPLEX:]
dom FB is Element of bool the carrier of X
support FB is set
GB is set
the carrier of X \ (support X) is Element of bool the carrier of X
the carrier of X \ (support F) is Element of bool the carrier of X
( the carrier of X \ (support X)) /\ ( the carrier of X \ (support F)) is Element of bool the carrier of X
X . GB is complex set
F . GB is complex set
(X (#) F) . GB is complex set
0 * 0 is empty epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural complex real ext-real non positive non negative V176() V177() V178() V179() V180() V181() V182() bounded_below V197() Element of REAL
the carrier of X \ (support (X (#) F)) is Element of bool the carrier of X
the carrier of X \ ((support X) \/ (support F)) is Element of bool the carrier of X
the carrier of X \ ( the carrier of X \ (support (X (#) F))) is Element of bool the carrier of X
the carrier of X \ ( the carrier of X \ ((support X) \/ (support F))) is Element of bool the carrier of X
the carrier of X /\ (support (X (#) F)) is set
the carrier of X /\ ((support X) \/ (support F)) is set
the carrier of X /\ (support X) is set
the carrier of X /\ (support F) is set
( the carrier of X /\ (support X)) \/ ( the carrier of X /\ (support F)) is set
(support X) \/ ( the carrier of X /\ (support F)) is set
X is non empty TopSpace-like TopStruct
the carrier of X is non empty set
[: the carrier of X,COMPLEX:] is non empty V139() set
bool [: the carrier of X,COMPLEX:] is non empty set
bool the carrier of X is non empty set
{ b₁ where b₁ is non empty Relation-like the carrier of X -defined COMPLEX -valued Function-like total quasi_total V139() Element of bool [: the carrier of X,COMPLEX:] : ( b₁ is (X) & ex b₂ being non empty Element of bool the carrier of X st
( b₂ is compact & ( for b₃ being Element of bool the carrier of X holds
( not b₃ = support b₁ or Cl b₃ is Element of bool b₂ ) ) ) ) } is set
ComplexVectSpace the carrier of X is non empty right_complementable Abelian add-associative right_zeroed strict vector-distributive scalar-distributive scalar-associative scalar-unital CLSStruct
Funcs ( the carrier of X,COMPLEX) is non empty functional FUNCTION_DOMAIN of the carrier of X, COMPLEX
ComplexFuncZero the carrier of X is Relation-like the carrier of X -defined COMPLEX -valued Function-like total quasi_total V139() Element of Funcs ( the carrier of X,COMPLEX)
the carrier of X --> 0 is non empty Relation-like the carrier of X -defined REAL -valued RAT -valued INT -valued Function-like total quasi_total V139() V140() V141() V142() Element of bool [: the carrier of X,REAL:]
[: the carrier of X,REAL:] is non empty V139() V140() V141() set
bool [: the carrier of X,REAL:] is non empty set
ComplexFuncAdd the carrier of X is non empty Relation-like [:(Funcs ( the carrier of X,COMPLEX)),(Funcs ( the carrier of X,COMPLEX)):] -defined Funcs ( the carrier of X,COMPLEX) -valued Function-like total quasi_total Element of bool [:[:(Funcs ( the carrier of X,COMPLEX)),(Funcs ( the carrier of X,COMPLEX)):],(Funcs ( the carrier of X,COMPLEX)):]
[:(Funcs ( the carrier of X,COMPLEX)),(Funcs ( the carrier of X,COMPLEX)):] is non empty set
[:[:(Funcs ( the carrier of X,COMPLEX)),(Funcs ( the carrier of X,COMPLEX)):],(Funcs ( the carrier of X,COMPLEX)):] is non empty set
bool [:[:(Funcs ( the carrier of X,COMPLEX)),(Funcs ( the carrier of X,COMPLEX)):],(Funcs ( the carrier of X,COMPLEX)):] is non empty set
ComplexFuncExtMult the carrier of X is non empty Relation-like [:COMPLEX,(Funcs ( the carrier of X,COMPLEX)):] -defined Funcs ( the carrier of X,COMPLEX) -valued Function-like total quasi_total Element of bool [:[:COMPLEX,(Funcs ( the carrier of X,COMPLEX)):],(Funcs ( the carrier of X,COMPLEX)):]
[:COMPLEX,(Funcs ( the carrier of X,COMPLEX)):] is non empty set
[:[:COMPLEX,(Funcs ( the carrier of X,COMPLEX)):],(Funcs ( the carrier of X,COMPLEX)):] is non empty set
bool [:[:COMPLEX,(Funcs ( the carrier of X,COMPLEX)):],(Funcs ( the carrier of X,COMPLEX)):] is non empty set
CLSStruct(# (Funcs ( the carrier of X,COMPLEX)),(ComplexFuncZero the carrier of X),(ComplexFuncAdd the carrier of X),(ComplexFuncExtMult the carrier of X) #) is strict CLSStruct
the carrier of (ComplexVectSpace the carrier of X) is non empty set
bool the carrier of (ComplexVectSpace the carrier of X) is non empty set
X is set
F is non empty Relation-like the carrier of X -defined COMPLEX -valued Function-like total quasi_total V139() Element of bool [: the carrier of X,COMPLEX:]
G is non empty Element of bool the carrier of X
support F is set
bool G is non empty set
the carrier of X --> 0c is non empty Relation-like the carrier of X -defined COMPLEX -valued Function-like total quasi_total V139() V140() V141() V142() Element of bool [: the carrier of X,COMPLEX:]
F is non empty Relation-like the carrier of X -defined COMPLEX -valued Function-like total quasi_total V139() Element of bool [: the carrier of X,COMPLEX:]
G is V176() Element of bool COMPLEX
F " G is Element of bool the carrier of X
the carrier of X --> 0 is non empty Relation-like the carrier of X -defined NAT -valued RAT -valued INT -valued Function-like total quasi_total V139() V140() V141() V142() Element of bool [: the carrier of X,NAT:]
[: the carrier of X,NAT:] is non empty RAT -valued INT -valued V139() V140() V141() V142() set
bool [: the carrier of X,NAT:] is non empty set
[#] X is non empty non proper closed Element of bool the carrier of X
( the carrier of X --> 0) " G is Element of bool the carrier of X
{} X is empty proper epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural complex real ext-real non positive non negative V176() V177() V178() V179() V180() V181() V182() bounded_below V197() closed compact Element of bool the carrier of X
support F is set
the non empty compact Element of bool the carrier of X is non empty compact Element of bool the carrier of X
bool the non empty compact Element of bool the carrier of X is non empty set
FB is Element of bool the carrier of X
Cl FB is Element of bool the carrier of X
the Element of support F is Element of support F
F . the Element of support F is complex set
{} X is empty proper epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural complex real ext-real non positive non negative V176() V177() V178() V179() V180() V181() V182() bounded_below V197() closed compact Element of bool the carrier of X
Cl ({} X) is Element of bool the carrier of X
G is non empty Element of bool the carrier of X
bool G is non empty set
X is non empty TopSpace-like TopStruct
(X) is non empty Element of bool the carrier of (ComplexVectSpace the carrier of X)
the carrier of X is non empty set
ComplexVectSpace the carrier of X is non empty right_complementable Abelian add-associative right_zeroed strict vector-distributive scalar-distributive scalar-associative scalar-unital CLSStruct
Funcs ( the carrier of X,COMPLEX) is non empty functional FUNCTION_DOMAIN of the carrier of X, COMPLEX
ComplexFuncZero the carrier of X is Relation-like the carrier of X -defined COMPLEX -valued Function-like total quasi_total V139() Element of Funcs ( the carrier of X,COMPLEX)
the carrier of X --> 0 is non empty Relation-like the carrier of X -defined REAL -valued RAT -valued INT -valued Function-like total quasi_total V139() V140() V141() V142() Element of bool [: the carrier of X,REAL:]
[: the carrier of X,REAL:] is non empty V139() V140() V141() set
bool [: the carrier of X,REAL:] is non empty set
ComplexFuncAdd the carrier of X is non empty Relation-like [:(Funcs ( the carrier of X,COMPLEX)),(Funcs ( the carrier of X,COMPLEX)):] -defined Funcs ( the carrier of X,COMPLEX) -valued Function-like total quasi_total Element of bool [:[:(Funcs ( the carrier of X,COMPLEX)),(Funcs ( the carrier of X,COMPLEX)):],(Funcs ( the carrier of X,COMPLEX)):]
[:(Funcs ( the carrier of X,COMPLEX)),(Funcs ( the carrier of X,COMPLEX)):] is non empty set
[:[:(Funcs ( the carrier of X,COMPLEX)),(Funcs ( the carrier of X,COMPLEX)):],(Funcs ( the carrier of X,COMPLEX)):] is non empty set
bool [:[:(Funcs ( the carrier of X,COMPLEX)),(Funcs ( the carrier of X,COMPLEX)):],(Funcs ( the carrier of X,COMPLEX)):] is non empty set
ComplexFuncExtMult the carrier of X is non empty Relation-like [:COMPLEX,(Funcs ( the carrier of X,COMPLEX)):] -defined Funcs ( the carrier of X,COMPLEX) -valued Function-like total quasi_total Element of bool [:[:COMPLEX,(Funcs ( the carrier of X,COMPLEX)):],(Funcs ( the carrier of X,COMPLEX)):]
[:COMPLEX,(Funcs ( the carrier of X,COMPLEX)):] is non empty set
[:[:COMPLEX,(Funcs ( the carrier of X,COMPLEX)):],(Funcs ( the carrier of X,COMPLEX)):] is non empty set
bool [:[:COMPLEX,(Funcs ( the carrier of X,COMPLEX)):],(Funcs ( the carrier of X,COMPLEX)):] is non empty set
CLSStruct(# (Funcs ( the carrier of X,COMPLEX)),(ComplexFuncZero the carrier of X),(ComplexFuncAdd the carrier of X),(ComplexFuncExtMult the carrier of X) #) is strict CLSStruct
the carrier of (ComplexVectSpace the carrier of X) is non empty set
bool the carrier of (ComplexVectSpace the carrier of X) is non empty set
[: the carrier of X,COMPLEX:] is non empty V139() set
bool [: the carrier of X,COMPLEX:] is non empty set
bool the carrier of X is non empty set
{ b₁ where b₁ is non empty Relation-like the carrier of X -defined COMPLEX -valued Function-like total quasi_total V139() Element of bool [: the carrier of X,COMPLEX:] : ( b₁ is (X) & ex b₂ being non empty Element of bool the carrier of X st
( b₂ is compact & ( for b₃ being Element of bool the carrier of X holds
( not b₃ = support b₁ or Cl b₃ is Element of bool b₂ ) ) ) ) } is set
CAlgebra the carrier of X is non empty right_complementable Abelian add-associative right_zeroed unital associative commutative right-distributive right_unital well-unital left_unital vector-distributive scalar-distributive scalar-associative scalar-unital strict vector-associative ComplexAlgebraStr
ComplexFuncMult the carrier of X is non empty Relation-like [:(Funcs ( the carrier of X,COMPLEX)),(Funcs ( the carrier of X,COMPLEX)):] -defined Funcs ( the carrier of X,COMPLEX) -valued Function-like total quasi_total Element of bool [:[:(Funcs ( the carrier of X,COMPLEX)),(Funcs ( the carrier of X,COMPLEX)):],(Funcs ( the carrier of X,COMPLEX)):]
ComplexFuncUnit the carrier of X is Relation-like the carrier of X -defined COMPLEX -valued Function-like total quasi_total V139() Element of Funcs ( the carrier of X,COMPLEX)
the carrier of X --> 1r is non empty Relation-like the carrier of X -defined COMPLEX -valued Function-like total quasi_total V139() Element of bool [: the carrier of X,COMPLEX:]
ComplexAlgebraStr(# (Funcs ( the carrier of X,COMPLEX)),(ComplexFuncMult the carrier of X),(ComplexFuncAdd the carrier of X),(ComplexFuncExtMult the carrier of X),(ComplexFuncUnit the carrier of X),(ComplexFuncZero the carrier of X) #) is strict ComplexAlgebraStr
the carrier of (CAlgebra the carrier of X) is non empty set
bool the carrier of (CAlgebra the carrier of X) is non empty set
X is non empty TopSpace-like TopStruct
the carrier of X is non empty set
CAlgebra the carrier of X is non empty right_complementable Abelian add-associative right_zeroed unital associative commutative right-distributive right_unital well-unital left_unital vector-distributive scalar-distributive scalar-associative scalar-unital strict vector-associative ComplexAlgebraStr
Funcs ( the carrier of X,COMPLEX) is non empty functional FUNCTION_DOMAIN of the carrier of X, COMPLEX
ComplexFuncMult the carrier of X is non empty Relation-like [:(Funcs ( the carrier of X,COMPLEX)),(Funcs ( the carrier of X,COMPLEX)):] -defined Funcs ( the carrier of X,COMPLEX) -valued Function-like total quasi_total Element of bool [:[:(Funcs ( the carrier of X,COMPLEX)),(Funcs ( the carrier of X,COMPLEX)):],(Funcs ( the carrier of X,COMPLEX)):]
[:(Funcs ( the carrier of X,COMPLEX)),(Funcs ( the carrier of X,COMPLEX)):] is non empty set
[:[:(Funcs ( the carrier of X,COMPLEX)),(Funcs ( the carrier of X,COMPLEX)):],(Funcs ( the carrier of X,COMPLEX)):] is non empty set
bool [:[:(Funcs ( the carrier of X,COMPLEX)),(Funcs ( the carrier of X,COMPLEX)):],(Funcs ( the carrier of X,COMPLEX)):] is non empty set
ComplexFuncAdd the carrier of X is non empty Relation-like [:(Funcs ( the carrier of X,COMPLEX)),(Funcs ( the carrier of X,COMPLEX)):] -defined Funcs ( the carrier of X,COMPLEX) -valued Function-like total quasi_total Element of bool [:[:(Funcs ( the carrier of X,COMPLEX)),(Funcs ( the carrier of X,COMPLEX)):],(Funcs ( the carrier of X,COMPLEX)):]
ComplexFuncExtMult the carrier of X is non empty Relation-like [:COMPLEX,(Funcs ( the carrier of X,COMPLEX)):] -defined Funcs ( the carrier of X,COMPLEX) -valued Function-like total quasi_total Element of bool [:[:COMPLEX,(Funcs ( the carrier of X,COMPLEX)):],(Funcs ( the carrier of X,COMPLEX)):]
[:COMPLEX,(Funcs ( the carrier of X,COMPLEX)):] is non empty set
[:[:COMPLEX,(Funcs ( the carrier of X,COMPLEX)):],(Funcs ( the carrier of X,COMPLEX)):] is non empty set
bool [:[:COMPLEX,(Funcs ( the carrier of X,COMPLEX)):],(Funcs ( the carrier of X,COMPLEX)):] is non empty set
ComplexFuncUnit the carrier of X is Relation-like the carrier of X -defined COMPLEX -valued Function-like total quasi_total V139() Element of Funcs ( the carrier of X,COMPLEX)
the carrier of X --> 1r is non empty Relation-like the carrier of X -defined COMPLEX -valued Function-like total quasi_total V139() Element of bool [: the carrier of X,COMPLEX:]
[: the carrier of X,COMPLEX:] is non empty V139() set
bool [: the carrier of X,COMPLEX:] is non empty set
ComplexFuncZero the carrier of X is Relation-like the carrier of X -defined COMPLEX -valued Function-like total quasi_total V139() Element of Funcs ( the carrier of X,COMPLEX)
the carrier of X --> 0 is non empty Relation-like the carrier of X -defined REAL -valued RAT -valued INT -valued Function-like total quasi_total V139() V140() V141() V142() Element of bool [: the carrier of X,REAL:]
[: the carrier of X,REAL:] is non empty V139() V140() V141() set
bool [: the carrier of X,REAL:] is non empty set
ComplexAlgebraStr(# (Funcs ( the carrier of X,COMPLEX)),(ComplexFuncMult the carrier of X),(ComplexFuncAdd the carrier of X),(ComplexFuncExtMult the carrier of X),(ComplexFuncUnit the carrier of X),(ComplexFuncZero the carrier of X) #) is strict ComplexAlgebraStr
the carrier of (CAlgebra the carrier of X) is non empty set
(X) is non empty Element of bool the carrier of (ComplexVectSpace the carrier of X)
ComplexVectSpace the carrier of X is non empty right_complementable Abelian add-associative right_zeroed strict vector-distributive scalar-distributive scalar-associative scalar-unital CLSStruct
CLSStruct(# (Funcs ( the carrier of X,COMPLEX)),(ComplexFuncZero the carrier of X),(ComplexFuncAdd the carrier of X),(ComplexFuncExtMult the carrier of X) #) is strict CLSStruct
the carrier of (ComplexVectSpace the carrier of X) is non empty set
bool the carrier of (ComplexVectSpace the carrier of X) is non empty set
bool the carrier of X is non empty set
{ b₁ where b₁ is non empty Relation-like the carrier of X -defined COMPLEX -valued Function-like total quasi_total V139() Element of bool [: the carrier of X,COMPLEX:] : ( b₁ is (X) & ex b₂ being non empty Element of bool the carrier of X st
( b₂ is compact & ( for b₃ being Element of bool the carrier of X holds
( not b₃ = support b₁ or Cl b₃ is Element of bool b₂ ) ) ) ) } is set
G is Element of the carrier of (CAlgebra the carrier of X)
FB is Element of the carrier of (CAlgebra the carrier of X)
G + FB is Element of the carrier of (CAlgebra the carrier of X)
the addF of (CAlgebra the carrier of X) is non empty Relation-like [: the carrier of (CAlgebra the carrier of X), the carrier of (CAlgebra the carrier of X):] -defined the carrier of (CAlgebra the carrier of X) -valued Function-like total quasi_total Element of bool [:[: the carrier of (CAlgebra the carrier of X), the carrier of (CAlgebra the carrier of X):], the carrier of (CAlgebra the carrier of X):]
[: the carrier of (CAlgebra the carrier of X), the carrier of (CAlgebra the carrier of X):] is non empty set
[:[: the carrier of (CAlgebra the carrier of X), the carrier of (CAlgebra the carrier of X):], the carrier of (CAlgebra the carrier of X):] is non empty set
bool [:[: the carrier of (CAlgebra the carrier of X), the carrier of (CAlgebra the carrier of X):], the carrier of (CAlgebra the carrier of X):] is non empty set
the addF of (CAlgebra the carrier of X) . (G,FB) is Element of the carrier of (CAlgebra the carrier of X)
[G,FB] is set
{G,FB} is non empty set
{G} is non empty set
{{G,FB},{G}} is non empty set
the addF of (CAlgebra the carrier of X) . [G,FB] is set
GB is non empty Relation-like the carrier of X -defined COMPLEX -valued Function-like total quasi_total V139() Element of bool [: the carrier of X,COMPLEX:]
support GB is set
aFB is non empty Element of bool the carrier of X
bool aFB is non empty set
aFB is non empty Element of bool the carrier of X
bool aFB is non empty set
aF is non empty Relation-like the carrier of X -defined COMPLEX -valued Function-like total quasi_total V139() Element of bool [: the carrier of X,COMPLEX:]
support aF is set
aG is non empty Element of bool the carrier of X
bool aG is non empty set
aG is non empty Element of bool the carrier of X
bool aG is non empty set
(X) is non empty multiplicatively-closed Cadditively-linearly-closed Element of bool the carrier of (CAlgebra the carrier of X)
bool the carrier of (CAlgebra the carrier of X) is non empty set
{ b₁ where b₁ is non empty Relation-like the carrier of X -defined COMPLEX -valued Function-like total quasi_total V139() (X) Element of bool [: the carrier of X,COMPLEX:] : verum } is set
r is non empty Element of bool the carrier of X
bool r is non empty set
r is non empty Element of bool the carrier of X
bool r is non empty set
FB + G is Element of the carrier of (CAlgebra the carrier of X)
the addF of (CAlgebra the carrier of X) . (FB,G) is Element of the carrier of (CAlgebra the carrier of X)
[FB,G] is set
{FB,G} is non empty set
{FB} is non empty set
{{FB,G},{FB}} is non empty set
the addF of (CAlgebra the carrier of X) . [FB,G] is set
r is non empty Relation-like the carrier of X -defined COMPLEX -valued Function-like total quasi_total V139() (X) Element of bool [: the carrier of X,COMPLEX:]
aFB \/ aG is non empty Element of bool the carrier of X
dom GB is Element of bool the carrier of X
dom aF is Element of bool the carrier of X
(dom aF) /\ (dom GB) is Element of bool the carrier of X
the carrier of X /\ (dom GB) is Element of bool the carrier of X
the carrier of X /\ the carrier of X is set
aF + GB is Relation-like the carrier of X -defined Function-like total V139() set
dom (aF + GB) is Element of bool the carrier of X
support (aF + GB) is set
A3 is Element of bool the carrier of X
Cl A3 is Element of bool the carrier of X
A2 is Element of bool the carrier of X
fau1 is Element of bool the carrier of X
A2 \/ fau1 is Element of bool the carrier of X
Cl (A2 \/ fau1) is Element of bool the carrier of X
Cl A2 is Element of bool the carrier of X
Cl fau1 is Element of bool the carrier of X
(Cl A2) \/ (Cl fau1) is Element of bool the carrier of X
aG \/ aFB is non empty Element of bool the carrier of X
bool (aG \/ aFB) is non empty set
vv1 is Element of bool the carrier of X
Cl vv1 is Element of bool the carrier of X
fvu1 is Relation-like the carrier of X -defined COMPLEX -valued Function-like total quasi_total V139() Element of Funcs ( the carrier of X,COMPLEX)
vv1 is Relation-like the carrier of X -defined COMPLEX -valued Function-like total quasi_total V139() Element of Funcs ( the carrier of X,COMPLEX)
uu1 is Relation-like the carrier of X -defined COMPLEX -valued Function-like total quasi_total V139() Element of Funcs ( the carrier of X,COMPLEX)
zz0 is Element of the carrier of X
fvu1 . zz0 is complex Element of COMPLEX
vv1 . zz0 is complex Element of COMPLEX
uu1 . zz0 is complex Element of COMPLEX
(vv1 . zz0) + (uu1 . zz0) is complex Element of COMPLEX
dom fvu1 is Element of bool the carrier of X
zz0 is set
fvu1 . zz0 is complex set
aF . zz0 is complex set
GB . zz0 is complex set
(aF . zz0) + (GB . zz0) is complex set
xx0 is non empty Element of bool the carrier of X
bool xx0 is non empty set
hx0 is non empty Element of bool the carrier of X
bool hx0 is non empty set
X is non empty TopSpace-like TopStruct
the carrier of X is non empty set
CAlgebra the carrier of X is non empty right_complementable Abelian add-associative right_zeroed unital associative commutative right-distributive right_unital well-unital left_unital vector-distributive scalar-distributive scalar-associative scalar-unital strict vector-associative ComplexAlgebraStr
Funcs ( the carrier of X,COMPLEX) is non empty functional FUNCTION_DOMAIN of the carrier of X, COMPLEX
ComplexFuncMult the carrier of X is non empty Relation-like [:(Funcs ( the carrier of X,COMPLEX)),(Funcs ( the carrier of X,COMPLEX)):] -defined Funcs ( the carrier of X,COMPLEX) -valued Function-like total quasi_total Element of bool [:[:(Funcs ( the carrier of X,COMPLEX)),(Funcs ( the carrier of X,COMPLEX)):],(Funcs ( the carrier of X,COMPLEX)):]
[:(Funcs ( the carrier of X,COMPLEX)),(Funcs ( the carrier of X,COMPLEX)):] is non empty set
[:[:(Funcs ( the carrier of X,COMPLEX)),(Funcs ( the carrier of X,COMPLEX)):],(Funcs ( the carrier of X,COMPLEX)):] is non empty set
bool [:[:(Funcs ( the carrier of X,COMPLEX)),(Funcs ( the carrier of X,COMPLEX)):],(Funcs ( the carrier of X,COMPLEX)):] is non empty set
ComplexFuncAdd the carrier of X is non empty Relation-like [:(Funcs ( the carrier of X,COMPLEX)),(Funcs ( the carrier of X,COMPLEX)):] -defined Funcs ( the carrier of X,COMPLEX) -valued Function-like total quasi_total Element of bool [:[:(Funcs ( the carrier of X,COMPLEX)),(Funcs ( the carrier of X,COMPLEX)):],(Funcs ( the carrier of X,COMPLEX)):]
ComplexFuncExtMult the carrier of X is non empty Relation-like [:COMPLEX,(Funcs ( the carrier of X,COMPLEX)):] -defined Funcs ( the carrier of X,COMPLEX) -valued Function-like total quasi_total Element of bool [:[:COMPLEX,(Funcs ( the carrier of X,COMPLEX)):],(Funcs ( the carrier of X,COMPLEX)):]
[:COMPLEX,(Funcs ( the carrier of X,COMPLEX)):] is non empty set
[:[:COMPLEX,(Funcs ( the carrier of X,COMPLEX)):],(Funcs ( the carrier of X,COMPLEX)):] is non empty set
bool [:[:COMPLEX,(Funcs ( the carrier of X,COMPLEX)):],(Funcs ( the carrier of X,COMPLEX)):] is non empty set
ComplexFuncUnit the carrier of X is Relation-like the carrier of X -defined COMPLEX -valued Function-like total quasi_total V139() Element of Funcs ( the carrier of X,COMPLEX)
the carrier of X --> 1r is non empty Relation-like the carrier of X -defined COMPLEX -valued Function-like total quasi_total V139() Element of bool [: the carrier of X,COMPLEX:]
[: the carrier of X,COMPLEX:] is non empty V139() set
bool [: the carrier of X,COMPLEX:] is non empty set
ComplexFuncZero the carrier of X is Relation-like the carrier of X -defined COMPLEX -valued Function-like total quasi_total V139() Element of Funcs ( the carrier of X,COMPLEX)
the carrier of X --> 0 is non empty Relation-like the carrier of X -defined REAL -valued RAT -valued INT -valued Function-like total quasi_total V139() V140() V141() V142() Element of bool [: the carrier of X,REAL:]
[: the carrier of X,REAL:] is non empty V139() V140() V141() set
bool [: the carrier of X,REAL:] is non empty set
ComplexAlgebraStr(# (Funcs ( the carrier of X,COMPLEX)),(ComplexFuncMult the carrier of X),(ComplexFuncAdd the carrier of X),(ComplexFuncExtMult the carrier of X),(ComplexFuncUnit the carrier of X),(ComplexFuncZero the carrier of X) #) is strict ComplexAlgebraStr
the carrier of (CAlgebra the carrier of X) is non empty set
(X) is non empty Element of bool the carrier of (ComplexVectSpace the carrier of X)
ComplexVectSpace the carrier of X is non empty right_complementable Abelian add-associative right_zeroed strict vector-distributive scalar-distributive scalar-associative scalar-unital CLSStruct
CLSStruct(# (Funcs ( the carrier of X,COMPLEX)),(ComplexFuncZero the carrier of X),(ComplexFuncAdd the carrier of X),(ComplexFuncExtMult the carrier of X) #) is strict CLSStruct
the carrier of (ComplexVectSpace the carrier of X) is non empty set
bool the carrier of (ComplexVectSpace the carrier of X) is non empty set
bool the carrier of X is non empty set
{ b₁ where b₁ is non empty Relation-like the carrier of X -defined COMPLEX -valued Function-like total quasi_total V139() Element of bool [: the carrier of X,COMPLEX:] : ( b₁ is (X) & ex b₂ being non empty Element of bool the carrier of X st
( b₂ is compact & ( for b₃ being Element of bool the carrier of X holds
( not b₃ = support b₁ or Cl b₃ is Element of bool b₂ ) ) ) ) } is set
G is complex set
FB is Element of the carrier of (CAlgebra the carrier of X)
G * FB is Element of the carrier of (CAlgebra the carrier of X)
the Mult of (CAlgebra the carrier of X) is non empty Relation-like [:COMPLEX, the carrier of (CAlgebra the carrier of X):] -defined the carrier of (CAlgebra the carrier of X) -valued Function-like total quasi_total Element of bool [:[:COMPLEX, the carrier of (CAlgebra the carrier of X):], the carrier of (CAlgebra the carrier of X):]
[:COMPLEX, the carrier of (CAlgebra the carrier of X):] is non empty set
[:[:COMPLEX, the carrier of (CAlgebra the carrier of X):], the carrier of (CAlgebra the carrier of X):] is non empty set
bool [:[:COMPLEX, the carrier of (CAlgebra the carrier of X):], the carrier of (CAlgebra the carrier of X):] is non empty set
[G,FB] is set
{G,FB} is non empty set
{G} is non empty V176() set
{{G,FB},{G}} is non empty set
the Mult of (CAlgebra the carrier of X) . [G,FB] is set
GB is non empty Relation-like the carrier of X -defined COMPLEX -valued Function-like total quasi_total V139() Element of bool [: the carrier of X,COMPLEX:]
support GB is set
aFB is non empty Element of bool the carrier of X
bool aFB is non empty set
aFB is non empty Element of bool the carrier of X
bool aFB is non empty set
(X) is non empty multiplicatively-closed Cadditively-linearly-closed Element of bool the carrier of (CAlgebra the carrier of X)
bool the carrier of (CAlgebra the carrier of X) is non empty set
{ b₁ where b₁ is non empty Relation-like the carrier of X -defined COMPLEX -valued Function-like total quasi_total V139() (X) Element of bool [: the carrier of X,COMPLEX:] : verum } is set
aF is non empty Element of bool the carrier of X
bool aF is non empty set
aF is non empty Relation-like the carrier of X -defined COMPLEX -valued Function-like total quasi_total V139() (X) Element of bool [: the carrier of X,COMPLEX:]
dom GB is Element of bool the carrier of X
G (#) GB is non empty Relation-like the carrier of X -defined COMPLEX -valued Function-like total quasi_total V139() Element of bool [: the carrier of X,COMPLEX:]
dom (G (#) GB) is Element of bool the carrier of X
support (G (#) GB) is set
r is Element of bool the carrier of X
Cl r is Element of bool the carrier of X
r is Element of bool the carrier of X
Cl r is Element of bool the carrier of X
s1 is Element of bool the carrier of X
Cl s1 is Element of bool the carrier of X
fau1 is Relation-like the carrier of X -defined COMPLEX -valued Function-like total quasi_total V139() Element of Funcs ( the carrier of X,COMPLEX)
s1 is Relation-like the carrier of X -defined COMPLEX -valued Function-like total quasi_total V139() Element of Funcs ( the carrier of X,COMPLEX)
A2 is Element of the carrier of X
fau1 . A2 is complex Element of COMPLEX
s1 . A2 is complex Element of COMPLEX
G * (s1 . A2) is complex set
dom fau1 is Element of bool the carrier of X
A2 is set
fau1 . A2 is complex set
GB . A2 is complex set
G * (GB . A2) is complex set
A3 is non empty Element of bool the carrier of X
bool A3 is non empty set
X is non empty TopSpace-like TopStruct
the carrier of X is non empty set
CAlgebra the carrier of X is non empty right_complementable Abelian add-associative right_zeroed unital associative commutative right-distributive right_unital well-unital left_unital vector-distributive scalar-distributive scalar-associative scalar-unital strict vector-associative ComplexAlgebraStr
Funcs ( the carrier of X,COMPLEX) is non empty functional FUNCTION_DOMAIN of the carrier of X, COMPLEX
ComplexFuncMult the carrier of X is non empty Relation-like [:(Funcs ( the carrier of X,COMPLEX)),(Funcs ( the carrier of X,COMPLEX)):] -defined Funcs ( the carrier of X,COMPLEX) -valued Function-like total quasi_total Element of bool [:[:(Funcs ( the carrier of X,COMPLEX)),(Funcs ( the carrier of X,COMPLEX)):],(Funcs ( the carrier of X,COMPLEX)):]
[:(Funcs ( the carrier of X,COMPLEX)),(Funcs ( the carrier of X,COMPLEX)):] is non empty set
[:[:(Funcs ( the carrier of X,COMPLEX)),(Funcs ( the carrier of X,COMPLEX)):],(Funcs ( the carrier of X,COMPLEX)):] is non empty set
bool [:[:(Funcs ( the carrier of X,COMPLEX)),(Funcs ( the carrier of X,COMPLEX)):],(Funcs ( the carrier of X,COMPLEX)):] is non empty set
ComplexFuncAdd the carrier of X is non empty Relation-like [:(Funcs ( the carrier of X,COMPLEX)),(Funcs ( the carrier of X,COMPLEX)):] -defined Funcs ( the carrier of X,COMPLEX) -valued Function-like total quasi_total Element of bool [:[:(Funcs ( the carrier of X,COMPLEX)),(Funcs ( the carrier of X,COMPLEX)):],(Funcs ( the carrier of X,COMPLEX)):]
ComplexFuncExtMult the carrier of X is non empty Relation-like [:COMPLEX,(Funcs ( the carrier of X,COMPLEX)):] -defined Funcs ( the carrier of X,COMPLEX) -valued Function-like total quasi_total Element of bool [:[:COMPLEX,(Funcs ( the carrier of X,COMPLEX)):],(Funcs ( the carrier of X,COMPLEX)):]
[:COMPLEX,(Funcs ( the carrier of X,COMPLEX)):] is non empty set
[:[:COMPLEX,(Funcs ( the carrier of X,COMPLEX)):],(Funcs ( the carrier of X,COMPLEX)):] is non empty set
bool [:[:COMPLEX,(Funcs ( the carrier of X,COMPLEX)):],(Funcs ( the carrier of X,COMPLEX)):] is non empty set
ComplexFuncUnit the carrier of X is Relation-like the carrier of X -defined COMPLEX -valued Function-like total quasi_total V139() Element of Funcs ( the carrier of X,COMPLEX)
the carrier of X --> 1r is non empty Relation-like the carrier of X -defined COMPLEX -valued Function-like total quasi_total V139() Element of bool [: the carrier of X,COMPLEX:]
[: the carrier of X,COMPLEX:] is non empty V139() set
bool [: the carrier of X,COMPLEX:] is non empty set
ComplexFuncZero the carrier of X is Relation-like the carrier of X -defined COMPLEX -valued Function-like total quasi_total V139() Element of Funcs ( the carrier of X,COMPLEX)
the carrier of X --> 0 is non empty Relation-like the carrier of X -defined REAL -valued RAT -valued INT -valued Function-like total quasi_total V139() V140() V141() V142() Element of bool [: the carrier of X,REAL:]
[: the carrier of X,REAL:] is non empty V139() V140() V141() set
bool [: the carrier of X,REAL:] is non empty set
ComplexAlgebraStr(# (Funcs ( the carrier of X,COMPLEX)),(ComplexFuncMult the carrier of X),(ComplexFuncAdd the carrier of X),(ComplexFuncExtMult the carrier of X),(ComplexFuncUnit the carrier of X),(ComplexFuncZero the carrier of X) #) is strict ComplexAlgebraStr
the carrier of (CAlgebra the carrier of X) is non empty set
(X) is non empty Element of bool the carrier of (ComplexVectSpace the carrier of X)
ComplexVectSpace the carrier of X is non empty right_complementable Abelian add-associative right_zeroed strict vector-distributive scalar-distributive scalar-associative scalar-unital CLSStruct
CLSStruct(# (Funcs ( the carrier of X,COMPLEX)),(ComplexFuncZero the carrier of X),(ComplexFuncAdd the carrier of X),(ComplexFuncExtMult the carrier of X) #) is strict CLSStruct
the carrier of (ComplexVectSpace the carrier of X) is non empty set
bool the carrier of (ComplexVectSpace the carrier of X) is non empty set
bool the carrier of X is non empty set
{ b₁ where b₁ is non empty Relation-like the carrier of X -defined COMPLEX -valued Function-like total quasi_total V139() Element of bool [: the carrier of X,COMPLEX:] : ( b₁ is (X) & ex b₂ being non empty Element of bool the carrier of X st
( b₂ is compact & ( for b₃ being Element of bool the carrier of X holds
( not b₃ = support b₁ or Cl b₃ is Element of bool b₂ ) ) ) ) } is set
G is Element of the carrier of (CAlgebra the carrier of X)
- G is Element of the carrier of (CAlgebra the carrier of X)
(- 1r) * G is Element of the carrier of (CAlgebra the carrier of X)
the Mult of (CAlgebra the carrier of X) is non empty Relation-like [:COMPLEX, the carrier of (CAlgebra the carrier of X):] -defined the carrier of (CAlgebra the carrier of X) -valued Function-like total quasi_total Element of bool [:[:COMPLEX, the carrier of (CAlgebra the carrier of X):], the carrier of (CAlgebra the carrier of X):]
[:COMPLEX, the carrier of (CAlgebra the carrier of X):] is non empty set
[:[:COMPLEX, the carrier of (CAlgebra the carrier of X):], the carrier of (CAlgebra the carrier of X):] is non empty set
bool [:[:COMPLEX, the carrier of (CAlgebra the carrier of X):], the carrier of (CAlgebra the carrier of X):] is non empty set
[(- 1r),G] is set
{(- 1r),G} is non empty set
{(- 1r)} is non empty V176() set
{{(- 1r),G},{(- 1r)}} is non empty set
the Mult of (CAlgebra the carrier of X) . [(- 1r),G] is set
X is non empty TopSpace-like TopStruct
the carrier of X is non empty set
CAlgebra the carrier of X is non empty right_complementable Abelian add-associative right_zeroed unital associative commutative right-distributive right_unital well-unital left_unital vector-distributive scalar-distributive scalar-associative scalar-unital strict vector-associative ComplexAlgebraStr
Funcs ( the carrier of X,COMPLEX) is non empty functional FUNCTION_DOMAIN of the carrier of X, COMPLEX
ComplexFuncMult the carrier of X is non empty Relation-like [:(Funcs ( the carrier of X,COMPLEX)),(Funcs ( the carrier of X,COMPLEX)):] -defined Funcs ( the carrier of X,COMPLEX) -valued Function-like total quasi_total Element of bool [:[:(Funcs ( the carrier of X,COMPLEX)),(Funcs ( the carrier of X,COMPLEX)):],(Funcs ( the carrier of X,COMPLEX)):]
[:(Funcs ( the carrier of X,COMPLEX)),(Funcs ( the carrier of X,COMPLEX)):] is non empty set
[:[:(Funcs ( the carrier of X,COMPLEX)),(Funcs ( the carrier of X,COMPLEX)):],(Funcs ( the carrier of X,COMPLEX)):] is non empty set
bool [:[:(Funcs ( the carrier of X,COMPLEX)),(Funcs ( the carrier of X,COMPLEX)):],(Funcs ( the carrier of X,COMPLEX)):] is non empty set
ComplexFuncAdd the carrier of X is non empty Relation-like [:(Funcs ( the carrier of X,COMPLEX)),(Funcs ( the carrier of X,COMPLEX)):] -defined Funcs ( the carrier of X,COMPLEX) -valued Function-like total quasi_total Element of bool [:[:(Funcs ( the carrier of X,COMPLEX)),(Funcs ( the carrier of X,COMPLEX)):],(Funcs ( the carrier of X,COMPLEX)):]
ComplexFuncExtMult the carrier of X is non empty Relation-like [:COMPLEX,(Funcs ( the carrier of X,COMPLEX)):] -defined Funcs ( the carrier of X,COMPLEX) -valued Function-like total quasi_total Element of bool [:[:COMPLEX,(Funcs ( the carrier of X,COMPLEX)):],(Funcs ( the carrier of X,COMPLEX)):]
[:COMPLEX,(Funcs ( the carrier of X,COMPLEX)):] is non empty set
[:[:COMPLEX,(Funcs ( the carrier of X,COMPLEX)):],(Funcs ( the carrier of X,COMPLEX)):] is non empty set
bool [:[:COMPLEX,(Funcs ( the carrier of X,COMPLEX)):],(Funcs ( the carrier of X,COMPLEX)):] is non empty set
ComplexFuncUnit the carrier of X is Relation-like the carrier of X -defined COMPLEX -valued Function-like total quasi_total V139() Element of Funcs ( the carrier of X,COMPLEX)
the carrier of X --> 1r is non empty Relation-like the carrier of X -defined COMPLEX -valued Function-like total quasi_total V139() Element of bool [: the carrier of X,COMPLEX:]
[: the carrier of X,COMPLEX:] is non empty V139() set
bool [: the carrier of X,COMPLEX:] is non empty set
ComplexFuncZero the carrier of X is Relation-like the carrier of X -defined COMPLEX -valued Function-like total quasi_total V139() Element of Funcs ( the carrier of X,COMPLEX)
the carrier of X --> 0 is non empty Relation-like the carrier of X -defined REAL -valued RAT -valued INT -valued Function-like total quasi_total V139() V140() V141() V142() Element of bool [: the carrier of X,REAL:]
[: the carrier of X,REAL:] is non empty V139() V140() V141() set
bool [: the carrier of X,REAL:] is non empty set
ComplexAlgebraStr(# (Funcs ( the carrier of X,COMPLEX)),(ComplexFuncMult the carrier of X),(ComplexFuncAdd the carrier of X),(ComplexFuncExtMult the carrier of X),(ComplexFuncUnit the carrier of X),(ComplexFuncZero the carrier of X) #) is strict ComplexAlgebraStr
the carrier of (CAlgebra the carrier of X) is non empty set
bool the carrier of (CAlgebra the carrier of X) is non empty set
(X) is non empty Element of bool the carrier of (ComplexVectSpace the carrier of X)
ComplexVectSpace the carrier of X is non empty right_complementable Abelian add-associative right_zeroed strict vector-distributive scalar-distributive scalar-associative scalar-unital CLSStruct
CLSStruct(# (Funcs ( the carrier of X,COMPLEX)),(ComplexFuncZero the carrier of X),(ComplexFuncAdd the carrier of X),(ComplexFuncExtMult the carrier of X) #) is strict CLSStruct
the carrier of (ComplexVectSpace the carrier of X) is non empty set
bool the carrier of (ComplexVectSpace the carrier of X) is non empty set
bool the carrier of X is non empty set
{ b₁ where b₁ is non empty Relation-like the carrier of X -defined COMPLEX -valued Function-like total quasi_total V139() Element of bool [: the carrier of X,COMPLEX:] : ( b₁ is (X) & ex b₂ being non empty Element of bool the carrier of X st
( b₂ is compact & ( for b₃ being Element of bool the carrier of X holds
( not b₃ = support b₁ or Cl b₃ is Element of bool b₂ ) ) ) ) } is set
X is non empty Element of bool the carrier of (CAlgebra the carrier of X)
G is Element of the carrier of (CAlgebra the carrier of X)
FB is Element of the carrier of (CAlgebra the carrier of X)
G + FB is Element of the carrier of (CAlgebra the carrier of X)
the addF of (CAlgebra the carrier of X) is non empty Relation-like [: the carrier of (CAlgebra the carrier of X), the carrier of (CAlgebra the carrier of X):] -defined the carrier of (CAlgebra the carrier of X) -valued Function-like total quasi_total Element of bool [:[: the carrier of (CAlgebra the carrier of X), the carrier of (CAlgebra the carrier of X):], the carrier of (CAlgebra the carrier of X):]
[: the carrier of (CAlgebra the carrier of X), the carrier of (CAlgebra the carrier of X):] is non empty set
[:[: the carrier of (CAlgebra the carrier of X), the carrier of (CAlgebra the carrier of X):], the carrier of (CAlgebra the carrier of X):] is non empty set
bool [:[: the carrier of (CAlgebra the carrier of X), the carrier of (CAlgebra the carrier of X):], the carrier of (CAlgebra the carrier of X):] is non empty set
the addF of (CAlgebra the carrier of X) . (G,FB) is Element of the carrier of (CAlgebra the carrier of X)
[G,FB] is set
{G,FB} is non empty set
{G} is non empty set
{{G,FB},{G}} is non empty set
the addF of (CAlgebra the carrier of X) . [G,FB] is set
G is Element of the carrier of (CAlgebra the carrier of X)
- G is Element of the carrier of (CAlgebra the carrier of X)
FB is Element of the carrier of (CAlgebra the carrier of X)
G is complex set
G * FB is Element of the carrier of (CAlgebra the carrier of X)
the Mult of (CAlgebra the carrier of X) is non empty Relation-like [:COMPLEX, the carrier of (CAlgebra the carrier of X):] -defined the carrier of (CAlgebra the carrier of X) -valued Function-like total quasi_total Element of bool [:[:COMPLEX, the carrier of (CAlgebra the carrier of X):], the carrier of (CAlgebra the carrier of X):]
[:COMPLEX, the carrier of (CAlgebra the carrier of X):] is non empty set
[:[:COMPLEX, the carrier of (CAlgebra the carrier of X):], the carrier of (CAlgebra the carrier of X):] is non empty set
bool [:[:COMPLEX, the carrier of (CAlgebra the carrier of X):], the carrier of (CAlgebra the carrier of X):] is non empty set
[G,FB] is set
{G,FB} is non empty set
{G} is non empty V176() set
{{G,FB},{G}} is non empty set
the Mult of (CAlgebra the carrier of X) . [G,FB] is set
X is non empty TopSpace-like TopStruct
the carrier of X is non empty set
ComplexVectSpace the carrier of X is non empty right_complementable Abelian add-associative right_zeroed strict vector-distributive scalar-distributive scalar-associative scalar-unital CLSStruct
Funcs ( the carrier of X,COMPLEX) is non empty functional FUNCTION_DOMAIN of the carrier of X, COMPLEX
ComplexFuncZero the carrier of X is Relation-like the carrier of X -defined COMPLEX -valued Function-like total quasi_total V139() Element of Funcs ( the carrier of X,COMPLEX)
the carrier of X --> 0 is non empty Relation-like the carrier of X -defined REAL -valued RAT -valued INT -valued Function-like total quasi_total V139() V140() V141() V142() Element of bool [: the carrier of X,REAL:]
[: the carrier of X,REAL:] is non empty V139() V140() V141() set
bool [: the carrier of X,REAL:] is non empty set
ComplexFuncAdd the carrier of X is non empty Relation-like [:(Funcs ( the carrier of X,COMPLEX)),(Funcs ( the carrier of X,COMPLEX)):] -defined Funcs ( the carrier of X,COMPLEX) -valued Function-like total quasi_total Element of bool [:[:(Funcs ( the carrier of X,COMPLEX)),(Funcs ( the carrier of X,COMPLEX)):],(Funcs ( the carrier of X,COMPLEX)):]
[:(Funcs ( the carrier of X,COMPLEX)),(Funcs ( the carrier of X,COMPLEX)):] is non empty set
[:[:(Funcs ( the carrier of X,COMPLEX)),(Funcs ( the carrier of X,COMPLEX)):],(Funcs ( the carrier of X,COMPLEX)):] is non empty set
bool [:[:(Funcs ( the carrier of X,COMPLEX)),(Funcs ( the carrier of X,COMPLEX)):],(Funcs ( the carrier of X,COMPLEX)):] is non empty set
ComplexFuncExtMult the carrier of X is non empty Relation-like [:COMPLEX,(Funcs ( the carrier of X,COMPLEX)):] -defined Funcs ( the carrier of X,COMPLEX) -valued Function-like total quasi_total Element of bool [:[:COMPLEX,(Funcs ( the carrier of X,COMPLEX)):],(Funcs ( the carrier of X,COMPLEX)):]
[:COMPLEX,(Funcs ( the carrier of X,COMPLEX)):] is non empty set
[:[:COMPLEX,(Funcs ( the carrier of X,COMPLEX)):],(Funcs ( the carrier of X,COMPLEX)):] is non empty set
bool [:[:COMPLEX,(Funcs ( the carrier of X,COMPLEX)):],(Funcs ( the carrier of X,COMPLEX)):] is non empty set
CLSStruct(# (Funcs ( the carrier of X,COMPLEX)),(ComplexFuncZero the carrier of X),(ComplexFuncAdd the carrier of X),(ComplexFuncExtMult the carrier of X) #) is strict CLSStruct
(X) is non empty Element of bool the carrier of (ComplexVectSpace the carrier of X)
the carrier of (ComplexVectSpace the carrier of X) is non empty set
bool the carrier of (ComplexVectSpace the carrier of X) is non empty set
[: the carrier of X,COMPLEX:] is non empty V139() set
bool [: the carrier of X,COMPLEX:] is non empty set
bool the carrier of X is non empty set
{ b₁ where b₁ is non empty Relation-like the carrier of X -defined COMPLEX -valued Function-like total quasi_total V139() Element of bool [: the carrier of X,COMPLEX:] : ( b₁ is (X) & ex b₂ being non empty Element of bool the carrier of X st
( b₂ is compact & ( for b₃ being Element of bool the carrier of X holds
( not b₃ = support b₁ or Cl b₃ is Element of bool b₂ ) ) ) ) } is set
G is Element of the carrier of (ComplexVectSpace the carrier of X)
FB is Element of the carrier of (ComplexVectSpace the carrier of X)
G + FB is Element of the carrier of (ComplexVectSpace the carrier of X)
the addF of (ComplexVectSpace the carrier of X) is non empty Relation-like [: the carrier of (ComplexVectSpace the carrier of X), the carrier of (ComplexVectSpace the carrier of X):] -defined the carrier of (ComplexVectSpace the carrier of X) -valued Function-like total quasi_total Element of bool [:[: the carrier of (ComplexVectSpace the carrier of X), the carrier of (ComplexVectSpace the carrier of X):], the carrier of (ComplexVectSpace the carrier of X):]
[: the carrier of (ComplexVectSpace the carrier of X), the carrier of (ComplexVectSpace the carrier of X):] is non empty set
[:[: the carrier of (ComplexVectSpace the carrier of X), the carrier of (ComplexVectSpace the carrier of X):], the carrier of (ComplexVectSpace the carrier of X):] is non empty set
bool [:[: the carrier of (ComplexVectSpace the carrier of X), the carrier of (ComplexVectSpace the carrier of X):], the carrier of (ComplexVectSpace the carrier of X):] is non empty set
the addF of (ComplexVectSpace the carrier of X) . (G,FB) is Element of the carrier of (ComplexVectSpace the carrier of X)
[G,FB] is set
{G,FB} is non empty set
{G} is non empty set
{{G,FB},{G}} is non empty set
the addF of (ComplexVectSpace the carrier of X) . [G,FB] is set
CAlgebra the carrier of X is non empty right_complementable Abelian add-associative right_zeroed unital associative commutative right-distributive right_unital well-unital left_unital vector-distributive scalar-distributive scalar-associative scalar-unital strict vector-associative ComplexAlgebraStr
ComplexFuncMult the carrier of X is non empty Relation-like [:(Funcs ( the carrier of X,COMPLEX)),(Funcs ( the carrier of X,COMPLEX)):] -defined Funcs ( the carrier of X,COMPLEX) -valued Function-like total quasi_total Element of bool [:[:(Funcs ( the carrier of X,COMPLEX)),(Funcs ( the carrier of X,COMPLEX)):],(Funcs ( the carrier of X,COMPLEX)):]
ComplexFuncUnit the carrier of X is Relation-like the carrier of X -defined COMPLEX -valued Function-like total quasi_total V139() Element of Funcs ( the carrier of X,COMPLEX)
the carrier of X --> 1r is non empty Relation-like the carrier of X -defined COMPLEX -valued Function-like total quasi_total V139() Element of bool [: the carrier of X,COMPLEX:]
ComplexAlgebraStr(# (Funcs ( the carrier of X,COMPLEX)),(ComplexFuncMult the carrier of X),(ComplexFuncAdd the carrier of X),(ComplexFuncExtMult the carrier of X),(ComplexFuncUnit the carrier of X),(ComplexFuncZero the carrier of X) #) is strict ComplexAlgebraStr
the carrier of (CAlgebra the carrier of X) is non empty set
GB is Element of the carrier of (CAlgebra the carrier of X)
aFB is Element of the carrier of (CAlgebra the carrier of X)
GB + aFB is Element of the carrier of (CAlgebra the carrier of X)
the addF of (CAlgebra the carrier of X) is non empty Relation-like [: the carrier of (CAlgebra the carrier of X), the carrier of (CAlgebra the carrier of X):] -defined the carrier of (CAlgebra the carrier of X) -valued Function-like total quasi_total Element of bool [:[: the carrier of (CAlgebra the carrier of X), the carrier of (CAlgebra the carrier of X):], the carrier of (CAlgebra the carrier of X):]
[: the carrier of (CAlgebra the carrier of X), the carrier of (CAlgebra the carrier of X):] is non empty set
[:[: the carrier of (CAlgebra the carrier of X), the carrier of (CAlgebra the carrier of X):], the carrier of (CAlgebra the carrier of X):] is non empty set
bool [:[: the carrier of (CAlgebra the carrier of X), the carrier of (CAlgebra the carrier of X):], the carrier of (CAlgebra the carrier of X):] is non empty set
the addF of (CAlgebra the carrier of X) . (GB,aFB) is Element of the carrier of (CAlgebra the carrier of X)
[GB,aFB] is set
{GB,aFB} is non empty set
{GB} is non empty set
{{GB,aFB},{GB}} is non empty set
the addF of (CAlgebra the carrier of X) . [GB,aFB] is set
X is non empty TopSpace-like TopStruct
the carrier of X is non empty set
ComplexVectSpace the carrier of X is non empty right_complementable Abelian add-associative right_zeroed strict vector-distributive scalar-distributive scalar-associative scalar-unital CLSStruct
Funcs ( the carrier of X,COMPLEX) is non empty functional FUNCTION_DOMAIN of the carrier of X, COMPLEX
ComplexFuncZero the carrier of X is Relation-like the carrier of X -defined COMPLEX -valued Function-like total quasi_total V139() Element of Funcs ( the carrier of X,COMPLEX)
the carrier of X --> 0 is non empty Relation-like the carrier of X -defined REAL -valued RAT -valued INT -valued Function-like total quasi_total V139() V140() V141() V142() Element of bool [: the carrier of X,REAL:]
[: the carrier of X,REAL:] is non empty V139() V140() V141() set
bool [: the carrier of X,REAL:] is non empty set
ComplexFuncAdd the carrier of X is non empty Relation-like [:(Funcs ( the carrier of X,COMPLEX)),(Funcs ( the carrier of X,COMPLEX)):] -defined Funcs ( the carrier of X,COMPLEX) -valued Function-like total quasi_total Element of bool [:[:(Funcs ( the carrier of X,COMPLEX)),(Funcs ( the carrier of X,COMPLEX)):],(Funcs ( the carrier of X,COMPLEX)):]
[:(Funcs ( the carrier of X,COMPLEX)),(Funcs ( the carrier of X,COMPLEX)):] is non empty set
[:[:(Funcs ( the carrier of X,COMPLEX)),(Funcs ( the carrier of X,COMPLEX)):],(Funcs ( the carrier of X,COMPLEX)):] is non empty set
bool [:[:(Funcs ( the carrier of X,COMPLEX)),(Funcs ( the carrier of X,COMPLEX)):],(Funcs ( the carrier of X,COMPLEX)):] is non empty set
ComplexFuncExtMult the carrier of X is non empty Relation-like [:COMPLEX,(Funcs ( the carrier of X,COMPLEX)):] -defined Funcs ( the carrier of X,COMPLEX) -valued Function-like total quasi_total Element of bool [:[:COMPLEX,(Funcs ( the carrier of X,COMPLEX)):],(Funcs ( the carrier of X,COMPLEX)):]
[:COMPLEX,(Funcs ( the carrier of X,COMPLEX)):] is non empty set
[:[:COMPLEX,(Funcs ( the carrier of X,COMPLEX)):],(Funcs ( the carrier of X,COMPLEX)):] is non empty set
bool [:[:COMPLEX,(Funcs ( the carrier of X,COMPLEX)):],(Funcs ( the carrier of X,COMPLEX)):] is non empty set
CLSStruct(# (Funcs ( the carrier of X,COMPLEX)),(ComplexFuncZero the carrier of X),(ComplexFuncAdd the carrier of X),(ComplexFuncExtMult the carrier of X) #) is strict CLSStruct
(X) is non empty Element of bool the carrier of (ComplexVectSpace the carrier of X)
the carrier of (ComplexVectSpace the carrier of X) is non empty set
bool the carrier of (ComplexVectSpace the carrier of X) is non empty set
[: the carrier of X,COMPLEX:] is non empty V139() set
bool [: the carrier of X,COMPLEX:] is non empty set
bool the carrier of X is non empty set
{ b₁ where b₁ is non empty Relation-like the carrier of X -defined COMPLEX -valued Function-like total quasi_total V139() Element of bool [: the carrier of X,COMPLEX:] : ( b₁ is (X) & ex b₂ being non empty Element of bool the carrier of X st
( b₂ is compact & ( for b₃ being Element of bool the carrier of X holds
( not b₃ = support b₁ or Cl b₃ is Element of bool b₂ ) ) ) ) } is set
G is Element of the carrier of (ComplexVectSpace the carrier of X)
FB is Element of the carrier of (ComplexVectSpace the carrier of X)
G + FB is Element of the carrier of (ComplexVectSpace the carrier of X)
the addF of (ComplexVectSpace the carrier of X) is non empty Relation-like [: the carrier of (ComplexVectSpace the carrier of X), the carrier of (ComplexVectSpace the carrier of X):] -defined the carrier of (ComplexVectSpace the carrier of X) -valued Function-like total quasi_total Element of bool [:[: the carrier of (ComplexVectSpace the carrier of X), the carrier of (ComplexVectSpace the carrier of X):], the carrier of (ComplexVectSpace the carrier of X):]
[: the carrier of (ComplexVectSpace the carrier of X), the carrier of (ComplexVectSpace the carrier of X):] is non empty set
[:[: the carrier of (ComplexVectSpace the carrier of X), the carrier of (ComplexVectSpace the carrier of X):], the carrier of (ComplexVectSpace the carrier of X):] is non empty set
bool [:[: the carrier of (ComplexVectSpace the carrier of X), the carrier of (ComplexVectSpace the carrier of X):], the carrier of (ComplexVectSpace the carrier of X):] is non empty set
the addF of (ComplexVectSpace the carrier of X) . (G,FB) is Element of the carrier of (ComplexVectSpace the carrier of X)
[G,FB] is set
{G,FB} is non empty set
{G} is non empty set
{{G,FB},{G}} is non empty set
the addF of (ComplexVectSpace the carrier of X) . [G,FB] is set
CAlgebra the carrier of X is non empty right_complementable Abelian add-associative right_zeroed unital associative commutative right-distributive right_unital well-unital left_unital vector-distributive scalar-distributive scalar-associative scalar-unital strict vector-associative ComplexAlgebraStr
ComplexFuncMult the carrier of X is non empty Relation-like [:(Funcs ( the carrier of X,COMPLEX)),(Funcs ( the carrier of X,COMPLEX)):] -defined Funcs ( the carrier of X,COMPLEX) -valued Function-like total quasi_total Element of bool [:[:(Funcs ( the carrier of X,COMPLEX)),(Funcs ( the carrier of X,COMPLEX)):],(Funcs ( the carrier of X,COMPLEX)):]
ComplexFuncUnit the carrier of X is Relation-like the carrier of X -defined COMPLEX -valued Function-like total quasi_total V139() Element of Funcs ( the carrier of X,COMPLEX)
the carrier of X --> 1r is non empty Relation-like the carrier of X -defined COMPLEX -valued Function-like total quasi_total V139() Element of bool [: the carrier of X,COMPLEX:]
ComplexAlgebraStr(# (Funcs ( the carrier of X,COMPLEX)),(ComplexFuncMult the carrier of X),(ComplexFuncAdd the carrier of X),(ComplexFuncExtMult the carrier of X),(ComplexFuncUnit the carrier of X),(ComplexFuncZero the carrier of X) #) is strict ComplexAlgebraStr
the carrier of (CAlgebra the carrier of X) is non empty set
aF is Element of the carrier of (CAlgebra the carrier of X)
aG is Element of the carrier of (CAlgebra the carrier of X)
aF + aG is Element of the carrier of (CAlgebra the carrier of X)
the addF of (CAlgebra the carrier of X) is non empty Relation-like [: the carrier of (CAlgebra the carrier of X), the carrier of (CAlgebra the carrier of X):] -defined the carrier of (CAlgebra the carrier of X) -valued Function-like total quasi_total Element of bool [:[: the carrier of (CAlgebra the carrier of X), the carrier of (CAlgebra the carrier of X):], the carrier of (CAlgebra the carrier of X):]
[: the carrier of (CAlgebra the carrier of X), the carrier of (CAlgebra the carrier of X):] is non empty set
[:[: the carrier of (CAlgebra the carrier of X), the carrier of (CAlgebra the carrier of X):], the carrier of (CAlgebra the carrier of X):] is non empty set
bool [:[: the carrier of (CAlgebra the carrier of X), the carrier of (CAlgebra the carrier of X):], the carrier of (CAlgebra the carrier of X):] is non empty set
the addF of (CAlgebra the carrier of X) . (aF,aG) is Element of the carrier of (CAlgebra the carrier of X)
[aF,aG] is set
{aF,aG} is non empty set
{aF} is non empty set
{{aF,aG},{aF}} is non empty set
the addF of (CAlgebra the carrier of X) . [aF,aG] is set
G is complex set
FB is Element of the carrier of (ComplexVectSpace the carrier of X)
G * FB is Element of the carrier of (ComplexVectSpace the carrier of X)
the Mult of (ComplexVectSpace the carrier of X) is non empty Relation-like [:COMPLEX, the carrier of (ComplexVectSpace the carrier of X):] -defined the carrier of (ComplexVectSpace the carrier of X) -valued Function-like total quasi_total Element of bool [:[:COMPLEX, the carrier of (ComplexVectSpace the carrier of X):], the carrier of (ComplexVectSpace the carrier of X):]
[:COMPLEX, the carrier of (ComplexVectSpace the carrier of X):] is non empty set
[:[:COMPLEX, the carrier of (ComplexVectSpace the carrier of X):], the carrier of (ComplexVectSpace the carrier of X):] is non empty set
bool [:[:COMPLEX, the carrier of (ComplexVectSpace the carrier of X):], the carrier of (ComplexVectSpace the carrier of X):] is non empty set
[G,FB] is set
{G,FB} is non empty set
{G} is non empty V176() set
{{G,FB},{G}} is non empty set
the Mult of (ComplexVectSpace the carrier of X) . [G,FB] is set
CAlgebra the carrier of X is non empty right_complementable Abelian add-associative right_zeroed unital associative commutative right-distributive right_unital well-unital left_unital vector-distributive scalar-distributive scalar-associative scalar-unital strict vector-associative ComplexAlgebraStr
ComplexFuncMult the carrier of X is non empty Relation-like [:(Funcs ( the carrier of X,COMPLEX)),(Funcs ( the carrier of X,COMPLEX)):] -defined Funcs ( the carrier of X,COMPLEX) -valued Function-like total quasi_total Element of bool [:[:(Funcs ( the carrier of X,COMPLEX)),(Funcs ( the carrier of X,COMPLEX)):],(Funcs ( the carrier of X,COMPLEX)):]
ComplexFuncUnit the carrier of X is Relation-like the carrier of X -defined COMPLEX -valued Function-like total quasi_total V139() Element of Funcs ( the carrier of X,COMPLEX)
the carrier of X --> 1r is non empty Relation-like the carrier of X -defined COMPLEX -valued Function-like total quasi_total V139() Element of bool [: the carrier of X,COMPLEX:]
ComplexAlgebraStr(# (Funcs ( the carrier of X,COMPLEX)),(ComplexFuncMult the carrier of X),(ComplexFuncAdd the carrier of X),(ComplexFuncExtMult the carrier of X),(ComplexFuncUnit the carrier of X),(ComplexFuncZero the carrier of X) #) is strict ComplexAlgebraStr
the carrier of (CAlgebra the carrier of X) is non empty set
aFB is Element of the carrier of (CAlgebra the carrier of X)
G * aFB is Element of the carrier of (CAlgebra the carrier of X)
the Mult of (CAlgebra the carrier of X) is non empty Relation-like [:COMPLEX, the carrier of (CAlgebra the carrier of X):] -defined the carrier of (CAlgebra the carrier of X) -valued Function-like total quasi_total Element of bool [:[:COMPLEX, the carrier of (CAlgebra the carrier of X):], the carrier of (CAlgebra the carrier of X):]
[:COMPLEX, the carrier of (CAlgebra the carrier of X):] is non empty set
[:[:COMPLEX, the carrier of (CAlgebra the carrier of X):], the carrier of (CAlgebra the carrier of X):] is non empty set
bool [:[:COMPLEX, the carrier of (CAlgebra the carrier of X):], the carrier of (CAlgebra the carrier of X):] is non empty set
[G,aFB] is set
{G,aFB} is non empty set
{{G,aFB},{G}} is non empty set
the Mult of (CAlgebra the carrier of X) . [G,aFB] is set
X is non empty TopSpace-like TopStruct
(X) is non empty Element of bool the carrier of (ComplexVectSpace the carrier of X)
the carrier of X is non empty set
ComplexVectSpace the carrier of X is non empty right_complementable Abelian add-associative right_zeroed strict vector-distributive scalar-distributive scalar-associative scalar-unital CLSStruct
Funcs ( the carrier of X,COMPLEX) is non empty functional FUNCTION_DOMAIN of the carrier of X, COMPLEX
ComplexFuncZero the carrier of X is Relation-like the carrier of X -defined COMPLEX -valued Function-like total quasi_total V139() Element of Funcs ( the carrier of X,COMPLEX)
the carrier of X --> 0 is non empty Relation-like the carrier of X -defined REAL -valued RAT -valued INT -valued Function-like total quasi_total V139() V140() V141() V142() Element of bool [: the carrier of X,REAL:]
[: the carrier of X,REAL:] is non empty V139() V140() V141() set
bool [: the carrier of X,REAL:] is non empty set
ComplexFuncAdd the carrier of X is non empty Relation-like [:(Funcs ( the carrier of X,COMPLEX)),(Funcs ( the carrier of X,COMPLEX)):] -defined Funcs ( the carrier of X,COMPLEX) -valued Function-like total quasi_total Element of bool [:[:(Funcs ( the carrier of X,COMPLEX)),(Funcs ( the carrier of X,COMPLEX)):],(Funcs ( the carrier of X,COMPLEX)):]
[:(Funcs ( the carrier of X,COMPLEX)),(Funcs ( the carrier of X,COMPLEX)):] is non empty set
[:[:(Funcs ( the carrier of X,COMPLEX)),(Funcs ( the carrier of X,COMPLEX)):],(Funcs ( the carrier of X,COMPLEX)):] is non empty set
bool [:[:(Funcs ( the carrier of X,COMPLEX)),(Funcs ( the carrier of X,COMPLEX)):],(Funcs ( the carrier of X,COMPLEX)):] is non empty set
ComplexFuncExtMult the carrier of X is non empty Relation-like [:COMPLEX,(Funcs ( the carrier of X,COMPLEX)):] -defined Funcs ( the carrier of X,COMPLEX) -valued Function-like total quasi_total Element of bool [:[:COMPLEX,(Funcs ( the carrier of X,COMPLEX)):],(Funcs ( the carrier of X,COMPLEX)):]
[:COMPLEX,(Funcs ( the carrier of X,COMPLEX)):] is non empty set
[:[:COMPLEX,(Funcs ( the carrier of X,COMPLEX)):],(Funcs ( the carrier of X,COMPLEX)):] is non empty set
bool [:[:COMPLEX,(Funcs ( the carrier of X,COMPLEX)):],(Funcs ( the carrier of X,COMPLEX)):] is non empty set
CLSStruct(# (Funcs ( the carrier of X,COMPLEX)),(ComplexFuncZero the carrier of X),(ComplexFuncAdd the carrier of X),(ComplexFuncExtMult the carrier of X) #) is strict CLSStruct
the carrier of (ComplexVectSpace the carrier of X) is non empty set
bool the carrier of (ComplexVectSpace the carrier of X) is non empty set
[: the carrier of X,COMPLEX:] is non empty V139() set
bool [: the carrier of X,COMPLEX:] is non empty set
bool the carrier of X is non empty set
{ b₁ where b₁ is non empty Relation-like the carrier of X -defined COMPLEX -valued Function-like total quasi_total V139() Element of bool [: the carrier of X,COMPLEX:] : ( b₁ is (X) & ex b₂ being non empty Element of bool the carrier of X st
( b₂ is compact & ( for b₃ being Element of bool the carrier of X holds
( not b₃ = support b₁ or Cl b₃ is Element of bool b₂ ) ) ) ) } is set
X is non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital CLSStruct
the carrier of X is non empty set
bool the carrier of X is non empty set
X is Element of bool the carrier of X
Zero_ (X,X) is Element of X
Add_ (X,X) is Relation-like [:X,X:] -defined X -valued Function-like quasi_total Element of bool [:[:X,X:],X:]
[:X,X:] is set
[:[:X,X:],X:] is set
bool [:[:X,X:],X:] is non empty set
Mult_ (X,X) is Relation-like [:COMPLEX,X:] -defined X -valued Function-like quasi_total Element of bool [:[:COMPLEX,X:],X:]
[:COMPLEX,X:] is set
[:[:COMPLEX,X:],X:] is set
bool [:[:COMPLEX,X:],X:] is non empty set
CLSStruct(# X,(Zero_ (X,X)),(Add_ (X,X)),(Mult_ (X,X)) #) is strict CLSStruct
F is Element of the carrier of X
G is Element of the carrier of X
F + G is Element of the carrier of X
the addF of X is non empty Relation-like [: the carrier of X, the carrier of X:] -defined the carrier of X -valued Function-like total quasi_total Element of bool [:[: the carrier of X, the carrier of X:], the carrier of X:]
[: the carrier of X, the carrier of X:] is non empty set
[:[: the carrier of X, the carrier of X:], the carrier of X:] is non empty set
bool [:[: the carrier of X, the carrier of X:], the carrier of X:] is non empty set
the addF of X . (F,G) is Element of the carrier of X
[F,G] is set
{F,G} is non empty set
{F} is non empty set
{{F,G},{F}} is non empty set
the addF of X . [F,G] is set
the addF of X is non empty Relation-like [: the carrier of X, the carrier of X:] -defined the carrier of X -valued Function-like total quasi_total Element of bool [:[: the carrier of X, the carrier of X:], the carrier of X:]
[: the carrier of X, the carrier of X:] is non empty set
[:[: the carrier of X, the carrier of X:], the carrier of X:] is non empty set
bool [:[: the carrier of X, the carrier of X:], the carrier of X:] is non empty set
the addF of X || X is Relation-like Function-like set
the addF of X | [:X,X:] is Relation-like set
[:COMPLEX, the carrier of X:] is non empty set
the Mult of X is non empty Relation-like [:COMPLEX, the carrier of X:] -defined the carrier of X -valued Function-like total quasi_total Element of bool [:[:COMPLEX, the carrier of X:], the carrier of X:]
[:[:COMPLEX, the carrier of X:], the carrier of X:] is non empty set
bool [:[:COMPLEX, the carrier of X:], the carrier of X:] is non empty set
the Mult of X | [:COMPLEX,X:] is Relation-like [:COMPLEX, the carrier of X:] -defined the carrier of X -valued Function-like Element of bool [:[:COMPLEX, the carrier of X:], the carrier of X:]
0. X is zero Element of the carrier of X
the ZeroF of X is Element of the carrier of X
X is non empty TopSpace-like TopStruct
(X) is non empty linearly-closed Element of bool the carrier of (ComplexVectSpace the carrier of X)
the carrier of X is non empty set
ComplexVectSpace the carrier of X is non empty right_complementable Abelian add-associative right_zeroed strict vector-distributive scalar-distributive scalar-associative scalar-unital CLSStruct
Funcs ( the carrier of X,COMPLEX) is non empty functional FUNCTION_DOMAIN of the carrier of X, COMPLEX
ComplexFuncZero the carrier of X is Relation-like the carrier of X -defined COMPLEX -valued Function-like total quasi_total V139() Element of Funcs ( the carrier of X,COMPLEX)
the carrier of X --> 0 is non empty Relation-like the carrier of X -defined REAL -valued RAT -valued INT -valued Function-like total quasi_total V139() V140() V141() V142() Element of bool [: the carrier of X,REAL:]
[: the carrier of X,REAL:] is non empty V139() V140() V141() set
bool [: the carrier of X,REAL:] is non empty set
ComplexFuncAdd the carrier of X is non empty Relation-like [:(Funcs ( the carrier of X,COMPLEX)),(Funcs ( the carrier of X,COMPLEX)):] -defined Funcs ( the carrier of X,COMPLEX) -valued Function-like total quasi_total Element of bool [:[:(Funcs ( the carrier of X,COMPLEX)),(Funcs ( the carrier of X,COMPLEX)):],(Funcs ( the carrier of X,COMPLEX)):]
[:(Funcs ( the carrier of X,COMPLEX)),(Funcs ( the carrier of X,COMPLEX)):] is non empty set
[:[:(Funcs ( the carrier of X,COMPLEX)),(Funcs ( the carrier of X,COMPLEX)):],(Funcs ( the carrier of X,COMPLEX)):] is non empty set
bool [:[:(Funcs ( the carrier of X,COMPLEX)),(Funcs ( the carrier of X,COMPLEX)):],(Funcs ( the carrier of X,COMPLEX)):] is non empty set
ComplexFuncExtMult the carrier of X is non empty Relation-like [:COMPLEX,(Funcs ( the carrier of X,COMPLEX)):] -defined Funcs ( the carrier of X,COMPLEX) -valued Function-like total quasi_total Element of bool [:[:COMPLEX,(Funcs ( the carrier of X,COMPLEX)):],(Funcs ( the carrier of X,COMPLEX)):]
[:COMPLEX,(Funcs ( the carrier of X,COMPLEX)):] is non empty set
[:[:COMPLEX,(Funcs ( the carrier of X,COMPLEX)):],(Funcs ( the carrier of X,COMPLEX)):] is non empty set
bool [:[:COMPLEX,(Funcs ( the carrier of X,COMPLEX)):],(Funcs ( the carrier of X,COMPLEX)):] is non empty set
CLSStruct(# (Funcs ( the carrier of X,COMPLEX)),(ComplexFuncZero the carrier of X),(ComplexFuncAdd the carrier of X),(ComplexFuncExtMult the carrier of X) #) is strict CLSStruct
the carrier of (ComplexVectSpace the carrier of X) is non empty set
bool the carrier of (ComplexVectSpace the carrier of X) is non empty set
[: the carrier of X,COMPLEX:] is non empty V139() set
bool [: the carrier of X,COMPLEX:] is non empty set
bool the carrier of X is non empty set
{ b₁ where b₁ is non empty Relation-like the carrier of X -defined COMPLEX -valued Function-like total quasi_total V139() Element of bool [: the carrier of X,COMPLEX:] : ( b₁ is (X) & ex b₂ being non empty Element of bool the carrier of X st
( b₂ is compact & ( for b₃ being Element of bool the carrier of X holds
( not b₃ = support b₁ or Cl b₃ is Element of bool b₂ ) ) ) ) } is set
Zero_ ((X),(ComplexVectSpace the carrier of X)) is Element of (X)
Add_ ((X),(ComplexVectSpace the carrier of X)) is non empty Relation-like [:(X),(X):] -defined (X) -valued Function-like total quasi_total Element of bool [:[:(X),(X):],(X):]
[:(X),(X):] is non empty set
[:[:(X),(X):],(X):] is non empty set
bool [:[:(X),(X):],(X):] is non empty set
Mult_ ((X),(ComplexVectSpace the carrier of X)) is non empty Relation-like [:COMPLEX,(X):] -defined (X) -valued Function-like total quasi_total Element of bool [:[:COMPLEX,(X):],(X):]
[:COMPLEX,(X):] is non empty set
[:[:COMPLEX,(X):],(X):] is non empty set
bool [:[:COMPLEX,(X):],(X):] is non empty set
CLSStruct(# (X),(Zero_ ((X),(ComplexVectSpace the carrier of X))),(Add_ ((X),(ComplexVectSpace the carrier of X))),(Mult_ ((X),(ComplexVectSpace the carrier of X))) #) is strict CLSStruct
X is non empty TopSpace-like TopStruct
(X) is non empty linearly-closed Element of bool the carrier of (ComplexVectSpace the carrier of X)
the carrier of X is non empty set
ComplexVectSpace the carrier of X is non empty right_complementable Abelian add-associative right_zeroed strict vector-distributive scalar-distributive scalar-associative scalar-unital CLSStruct
Funcs ( the carrier of X,COMPLEX) is non empty functional FUNCTION_DOMAIN of the carrier of X, COMPLEX
ComplexFuncZero the carrier of X is Relation-like the carrier of X -defined COMPLEX -valued Function-like total quasi_total V139() Element of Funcs ( the carrier of X,COMPLEX)
the carrier of X --> 0 is non empty Relation-like the carrier of X -defined REAL -valued RAT -valued INT -valued Function-like total quasi_total V139() V140() V141() V142() Element of bool [: the carrier of X,REAL:]
[: the carrier of X,REAL:] is non empty V139() V140() V141() set
bool [: the carrier of X,REAL:] is non empty set
ComplexFuncAdd the carrier of X is non empty Relation-like [:(Funcs ( the carrier of X,COMPLEX)),(Funcs ( the carrier of X,COMPLEX)):] -defined Funcs ( the carrier of X,COMPLEX) -valued Function-like total quasi_total Element of bool [:[:(Funcs ( the carrier of X,COMPLEX)),(Funcs ( the carrier of X,COMPLEX)):],(Funcs ( the carrier of X,COMPLEX)):]
[:(Funcs ( the carrier of X,COMPLEX)),(Funcs ( the carrier of X,COMPLEX)):] is non empty set
[:[:(Funcs ( the carrier of X,COMPLEX)),(Funcs ( the carrier of X,COMPLEX)):],(Funcs ( the carrier of X,COMPLEX)):] is non empty set
bool [:[:(Funcs ( the carrier of X,COMPLEX)),(Funcs ( the carrier of X,COMPLEX)):],(Funcs ( the carrier of X,COMPLEX)):] is non empty set
ComplexFuncExtMult the carrier of X is non empty Relation-like [:COMPLEX,(Funcs ( the carrier of X,COMPLEX)):] -defined Funcs ( the carrier of X,COMPLEX) -valued Function-like total quasi_total Element of bool [:[:COMPLEX,(Funcs ( the carrier of X,COMPLEX)):],(Funcs ( the carrier of X,COMPLEX)):]
[:COMPLEX,(Funcs ( the carrier of X,COMPLEX)):] is non empty set
[:[:COMPLEX,(Funcs ( the carrier of X,COMPLEX)):],(Funcs ( the carrier of X,COMPLEX)):] is non empty set
bool [:[:COMPLEX,(Funcs ( the carrier of X,COMPLEX)):],(Funcs ( the carrier of X,COMPLEX)):] is non empty set
CLSStruct(# (Funcs ( the carrier of X,COMPLEX)),(ComplexFuncZero the carrier of X),(ComplexFuncAdd the carrier of X),(ComplexFuncExtMult the carrier of X) #) is strict CLSStruct
the carrier of (ComplexVectSpace the carrier of X) is non empty set
bool the carrier of (ComplexVectSpace the carrier of X) is non empty set
[: the carrier of X,COMPLEX:] is non empty V139() set
bool [: the carrier of X,COMPLEX:] is non empty set
bool the carrier of X is non empty set
{ b₁ where b₁ is non empty Relation-like the carrier of X -defined COMPLEX -valued Function-like total quasi_total V139() Element of bool [: the carrier of X,COMPLEX:] : ( b₁ is (X) & ex b₂ being non empty Element of bool the carrier of X st
( b₂ is compact & ( for b₃ being Element of bool the carrier of X holds
( not b₃ = support b₁ or Cl b₃ is Element of bool b₂ ) ) ) ) } is set
Zero_ ((X),(ComplexVectSpace the carrier of X)) is Element of (X)
Add_ ((X),(ComplexVectSpace the carrier of X)) is non empty Relation-like [:(X),(X):] -defined (X) -valued Function-like total quasi_total Element of bool [:[:(X),(X):],(X):]
[:(X),(X):] is non empty set
[:[:(X),(X):],(X):] is non empty set
bool [:[:(X),(X):],(X):] is non empty set
Mult_ ((X),(ComplexVectSpace the carrier of X)) is non empty Relation-like [:COMPLEX,(X):] -defined (X) -valued Function-like total quasi_total Element of bool [:[:COMPLEX,(X):],(X):]
[:COMPLEX,(X):] is non empty set
[:[:COMPLEX,(X):],(X):] is non empty set
bool [:[:COMPLEX,(X):],(X):] is non empty set
CLSStruct(# (X),(Zero_ ((X),(ComplexVectSpace the carrier of X))),(Add_ ((X),(ComplexVectSpace the carrier of X))),(Mult_ ((X),(ComplexVectSpace the carrier of X))) #) is strict CLSStruct
X is non empty TopSpace-like TopStruct
(X) is non empty linearly-closed Element of bool the carrier of (ComplexVectSpace the carrier of X)
the carrier of X is non empty set
ComplexVectSpace the carrier of X is non empty right_complementable Abelian add-associative right_zeroed strict vector-distributive scalar-distributive scalar-associative scalar-unital CLSStruct
Funcs ( the carrier of X,COMPLEX) is non empty functional FUNCTION_DOMAIN of the carrier of X, COMPLEX
ComplexFuncZero the carrier of X is Relation-like the carrier of X -defined COMPLEX -valued Function-like total quasi_total V139() Element of Funcs ( the carrier of X,COMPLEX)
the carrier of X --> 0 is non empty Relation-like the carrier of X -defined REAL -valued RAT -valued INT -valued Function-like total quasi_total V139() V140() V141() V142() Element of bool [: the carrier of X,REAL:]
[: the carrier of X,REAL:] is non empty V139() V140() V141() set
bool [: the carrier of X,REAL:] is non empty set
ComplexFuncAdd the carrier of X is non empty Relation-like [:(Funcs ( the carrier of X,COMPLEX)),(Funcs ( the carrier of X,COMPLEX)):] -defined Funcs ( the carrier of X,COMPLEX) -valued Function-like total quasi_total Element of bool [:[:(Funcs ( the carrier of X,COMPLEX)),(Funcs ( the carrier of X,COMPLEX)):],(Funcs ( the carrier of X,COMPLEX)):]
[:(Funcs ( the carrier of X,COMPLEX)),(Funcs ( the carrier of X,COMPLEX)):] is non empty set
[:[:(Funcs ( the carrier of X,COMPLEX)),(Funcs ( the carrier of X,COMPLEX)):],(Funcs ( the carrier of X,COMPLEX)):] is non empty set
bool [:[:(Funcs ( the carrier of X,COMPLEX)),(Funcs ( the carrier of X,COMPLEX)):],(Funcs ( the carrier of X,COMPLEX)):] is non empty set
ComplexFuncExtMult the carrier of X is non empty Relation-like [:COMPLEX,(Funcs ( the carrier of X,COMPLEX)):] -defined Funcs ( the carrier of X,COMPLEX) -valued Function-like total quasi_total Element of bool [:[:COMPLEX,(Funcs ( the carrier of X,COMPLEX)):],(Funcs ( the carrier of X,COMPLEX)):]
[:COMPLEX,(Funcs ( the carrier of X,COMPLEX)):] is non empty set
[:[:COMPLEX,(Funcs ( the carrier of X,COMPLEX)):],(Funcs ( the carrier of X,COMPLEX)):] is non empty set
bool [:[:COMPLEX,(Funcs ( the carrier of X,COMPLEX)):],(Funcs ( the carrier of X,COMPLEX)):] is non empty set
CLSStruct(# (Funcs ( the carrier of X,COMPLEX)),(ComplexFuncZero the carrier of X),(ComplexFuncAdd the carrier of X),(ComplexFuncExtMult the carrier of X) #) is strict CLSStruct
the carrier of (ComplexVectSpace the carrier of X) is non empty set
bool the carrier of (ComplexVectSpace the carrier of X) is non empty set
[: the carrier of X,COMPLEX:] is non empty V139() set
bool [: the carrier of X,COMPLEX:] is non empty set
bool the carrier of X is non empty set
{ b₁ where b₁ is non empty Relation-like the carrier of X -defined COMPLEX -valued Function-like total quasi_total V139() Element of bool [: the carrier of X,COMPLEX:] : ( b₁ is (X) & ex b₂ being non empty Element of bool the carrier of X st
( b₂ is compact & ( for b₃ being Element of bool the carrier of X holds
( not b₃ = support b₁ or Cl b₃ is Element of bool b₂ ) ) ) ) } is set
ComplexBoundedFunctions the carrier of X is non empty multiplicatively-closed Cadditively-linearly-closed Element of bool the carrier of (CAlgebra the carrier of X)
CAlgebra the carrier of X is non empty right_complementable Abelian add-associative right_zeroed unital associative commutative right-distributive right_unital well-unital left_unital vector-distributive scalar-distributive scalar-associative scalar-unital strict vector-associative ComplexAlgebraStr
ComplexFuncMult the carrier of X is non empty Relation-like [:(Funcs ( the carrier of X,COMPLEX)),(Funcs ( the carrier of X,COMPLEX)):] -defined Funcs ( the carrier of X,COMPLEX) -valued Function-like total quasi_total Element of bool [:[:(Funcs ( the carrier of X,COMPLEX)),(Funcs ( the carrier of X,COMPLEX)):],(Funcs ( the carrier of X,COMPLEX)):]
ComplexFuncUnit the carrier of X is Relation-like the carrier of X -defined COMPLEX -valued Function-like total quasi_total V139() Element of Funcs ( the carrier of X,COMPLEX)
the carrier of X --> 1r is non empty Relation-like the carrier of X -defined COMPLEX -valued Function-like total quasi_total V139() Element of bool [: the carrier of X,COMPLEX:]
ComplexAlgebraStr(# (Funcs ( the carrier of X,COMPLEX)),(ComplexFuncMult the carrier of X),(ComplexFuncAdd the carrier of X),(ComplexFuncExtMult the carrier of X),(ComplexFuncUnit the carrier of X),(ComplexFuncZero the carrier of X) #) is strict ComplexAlgebraStr
the carrier of (CAlgebra the carrier of X) is non empty set
bool the carrier of (CAlgebra the carrier of X) is non empty set
{ b₁ where b₁ is non empty Relation-like the carrier of X -defined COMPLEX -valued Function-like total quasi_total V139() Element of bool [: the carrier of X,COMPLEX:] : b₁ | the carrier of X is bounded } is set
X is set
F is non empty Relation-like the carrier of X -defined COMPLEX -valued Function-like total quasi_total V139() Element of bool [: the carrier of X,COMPLEX:]
support F is set
G is non empty Element of bool the carrier of X
bool G is non empty set
G is non empty Element of bool the carrier of X
bool G is non empty set
dom F is Element of bool the carrier of X
|.F.| is non empty Relation-like the carrier of X -defined REAL -valued Function-like total quasi_total V139() V140() V141() Element of bool [: the carrier of X,REAL:]
GB is non empty Element of bool the carrier of X
bool GB is non empty set
GB is non empty Relation-like the carrier of X -defined REAL -valued Function-like total quasi_total V139() V140() V141() Element of bool [: the carrier of X,REAL:]
GB .: G is non empty V176() V177() V178() Element of bool REAL
aFB is non empty V176() V177() V178() bounded_below bounded_above real-bounded Element of bool REAL
inf aFB is complex real ext-real set
sup aFB is complex real ext-real set
[.(inf aFB),(sup aFB).] is V176() V177() V178() V197() compact Element of bool REAL
aF is complex real ext-real set
abs aF is complex real ext-real non negative Element of REAL
aG is complex real ext-real set
abs aG is complex real ext-real non negative Element of REAL
(abs aF) + (abs aG) is complex real ext-real non negative Element of REAL
((abs aF) + (abs aG)) + 1 is non empty complex real ext-real positive non negative Element of REAL
r is complex real ext-real set
- r is complex real ext-real set
r is Element of G
GB . r is complex real ext-real Element of REAL
[.aF,aG.] is V176() V177() V178() V197() compact Element of bool REAL
{ b₁ where b₁ is complex real ext-real Element of REAL : ( aF <= b₁ & b₁ <= aG ) } is set
s1 is complex real ext-real Element of REAL
- (abs aF) is complex real ext-real non positive Element of REAL
(- (abs aF)) - (abs aG) is complex real ext-real non positive Element of REAL
- (abs aG) is complex real ext-real non positive set
(- (abs aF)) + (- (abs aG)) is complex real ext-real non positive set
aF - 0 is complex real ext-real Element of REAL
- 0 is empty epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural complex real ext-real non positive non negative V176() V177() V178() V179() V180() V181() V182() bounded_below V197() set
aF + (- 0) is complex real ext-real set
((- (abs aF)) - (abs aG)) - 1 is non empty complex real ext-real non positive negative Element of REAL
- 1 is non empty complex real ext-real non positive negative set
((- (abs aF)) - (abs aG)) + (- 1) is non empty complex real ext-real non positive negative set
aG + 0 is complex real ext-real Element of REAL
(abs aG) + (abs aF) is complex real ext-real non negative Element of REAL
r is complex real ext-real set
- r is complex real ext-real set
s1 is Element of the carrier of X
GB . s1 is complex real ext-real Element of REAL
the carrier of X \ G is Element of bool the carrier of X
FB is Element of bool the carrier of X
Cl FB is Element of bool the carrier of X
F . s1 is complex Element of COMPLEX
|.(F . s1).| is complex real ext-real non negative Element of REAL
s1 is complex real ext-real set
- s1 is complex real ext-real set
dom GB is Element of bool the carrier of X
the carrier of X /\ (dom GB) is Element of bool the carrier of X
fau1 is set
GB . fau1 is complex real ext-real set
GB | the carrier of X is Relation-like the carrier of X -defined REAL -valued Function-like V139() V140() V141() Element of bool [: the carrier of X,REAL:]
fau1 is set
GB . fau1 is complex real ext-real set
the carrier of X /\ the carrier of X is set
F | the carrier of X is Relation-like the carrier of X -defined COMPLEX -valued Function-like V139() Element of bool [: the carrier of X,COMPLEX:]
|.(F | the carrier of X).| is Relation-like the carrier of X -defined REAL -valued Function-like V139() V140() V141() Element of bool [: the carrier of X,REAL:]
fau1 is non empty Element of bool the carrier of X
bool fau1 is non empty set
X is non empty TopSpace-like TopStruct
the carrier of X is non empty set
ComplexBoundedFunctions the carrier of X is non empty multiplicatively-closed Cadditively-linearly-closed Element of bool the carrier of (CAlgebra the carrier of X)
CAlgebra the carrier of X is non empty right_complementable Abelian add-associative right_zeroed unital associative commutative right-distributive right_unital well-unital left_unital vector-distributive scalar-distributive scalar-associative scalar-unital strict vector-associative ComplexAlgebraStr
Funcs ( the carrier of X,COMPLEX) is non empty functional FUNCTION_DOMAIN of the carrier of X, COMPLEX
ComplexFuncMult the carrier of X is non empty Relation-like [:(Funcs ( the carrier of X,COMPLEX)),(Funcs ( the carrier of X,COMPLEX)):] -defined Funcs ( the carrier of X,COMPLEX) -valued Function-like total quasi_total Element of bool [:[:(Funcs ( the carrier of X,COMPLEX)),(Funcs ( the carrier of X,COMPLEX)):],(Funcs ( the carrier of X,COMPLEX)):]
[:(Funcs ( the carrier of X,COMPLEX)),(Funcs ( the carrier of X,COMPLEX)):] is non empty set
[:[:(Funcs ( the carrier of X,COMPLEX)),(Funcs ( the carrier of X,COMPLEX)):],(Funcs ( the carrier of X,COMPLEX)):] is non empty set
bool [:[:(Funcs ( the carrier of X,COMPLEX)),(Funcs ( the carrier of X,COMPLEX)):],(Funcs ( the carrier of X,COMPLEX)):] is non empty set
ComplexFuncAdd the carrier of X is non empty Relation-like [:(Funcs ( the carrier of X,COMPLEX)),(Funcs ( the carrier of X,COMPLEX)):] -defined Funcs ( the carrier of X,COMPLEX) -valued Function-like total quasi_total Element of bool [:[:(Funcs ( the carrier of X,COMPLEX)),(Funcs ( the carrier of X,COMPLEX)):],(Funcs ( the carrier of X,COMPLEX)):]
ComplexFuncExtMult the carrier of X is non empty Relation-like [:COMPLEX,(Funcs ( the carrier of X,COMPLEX)):] -defined Funcs ( the carrier of X,COMPLEX) -valued Function-like total quasi_total Element of bool [:[:COMPLEX,(Funcs ( the carrier of X,COMPLEX)):],(Funcs ( the carrier of X,COMPLEX)):]
[:COMPLEX,(Funcs ( the carrier of X,COMPLEX)):] is non empty set
[:[:COMPLEX,(Funcs ( the carrier of X,COMPLEX)):],(Funcs ( the carrier of X,COMPLEX)):] is non empty set
bool [:[:COMPLEX,(Funcs ( the carrier of X,COMPLEX)):],(Funcs ( the carrier of X,COMPLEX)):] is non empty set
ComplexFuncUnit the carrier of X is Relation-like the carrier of X -defined COMPLEX -valued Function-like total quasi_total V139() Element of Funcs ( the carrier of X,COMPLEX)
the carrier of X --> 1r is non empty Relation-like the carrier of X -defined COMPLEX -valued Function-like total quasi_total V139() Element of bool [: the carrier of X,COMPLEX:]
[: the carrier of X,COMPLEX:] is non empty V139() set
bool [: the carrier of X,COMPLEX:] is non empty set
ComplexFuncZero the carrier of X is Relation-like the carrier of X -defined COMPLEX -valued Function-like total quasi_total V139() Element of Funcs ( the carrier of X,COMPLEX)
the carrier of X --> 0 is non empty Relation-like the carrier of X -defined REAL -valued RAT -valued INT -valued Function-like total quasi_total V139() V140() V141() V142() Element of bool [: the carrier of X,REAL:]
[: the carrier of X,REAL:] is non empty V139() V140() V141() set
bool [: the carrier of X,REAL:] is non empty set
ComplexAlgebraStr(# (Funcs ( the carrier of X,COMPLEX)),(ComplexFuncMult the carrier of X),(ComplexFuncAdd the carrier of X),(ComplexFuncExtMult the carrier of X),(ComplexFuncUnit the carrier of X),(ComplexFuncZero the carrier of X) #) is strict ComplexAlgebraStr
the carrier of (CAlgebra the carrier of X) is non empty set
bool the carrier of (CAlgebra the carrier of X) is non empty set
{ b₁ where b₁ is non empty Relation-like the carrier of X -defined COMPLEX -valued Function-like total quasi_total V139() Element of bool [: the carrier of X,COMPLEX:] : b₁ | the carrier of X is bounded } is set
ComplexBoundedFunctionsNorm the carrier of X is non empty Relation-like ComplexBoundedFunctions the carrier of X -defined REAL -valued Function-like total quasi_total V139() V140() V141() Element of bool [:(ComplexBoundedFunctions the carrier of X),REAL:]
[:(ComplexBoundedFunctions the carrier of X),REAL:] is non empty V139() V140() V141() set
bool [:(ComplexBoundedFunctions the carrier of X),REAL:] is non empty set
(X) is non empty linearly-closed Element of bool the carrier of (ComplexVectSpace the carrier of X)
ComplexVectSpace the carrier of X is non empty right_complementable Abelian add-associative right_zeroed strict vector-distributive scalar-distributive scalar-associative scalar-unital CLSStruct
CLSStruct(# (Funcs ( the carrier of X,COMPLEX)),(ComplexFuncZero the carrier of X),(ComplexFuncAdd the carrier of X),(ComplexFuncExtMult the carrier of X) #) is strict CLSStruct
the carrier of (ComplexVectSpace the carrier of X) is non empty set
bool the carrier of (ComplexVectSpace the carrier of X) is non empty set
bool the carrier of X is non empty set
{ b₁ where b₁ is non empty Relation-like the carrier of X -defined COMPLEX -valued Function-like total quasi_total V139() Element of bool [: the carrier of X,COMPLEX:] : ( b₁ is (X) & ex b₂ being non empty Element of bool the carrier of X st
( b₂ is compact & ( for b₃ being Element of bool the carrier of X holds
( not b₃ = support b₁ or Cl b₃ is Element of bool b₂ ) ) ) ) } is set
(ComplexBoundedFunctionsNorm the carrier of X) | (X) is Relation-like ComplexBoundedFunctions the carrier of X -defined REAL -valued Function-like V139() V140() V141() Element of bool [:(ComplexBoundedFunctions the carrier of X),REAL:]
[:(X),REAL:] is non empty V139() V140() V141() set
bool [:(X),REAL:] is non empty set
X is set
X is non empty TopSpace-like TopStruct
(X) is non empty linearly-closed Element of bool the carrier of (ComplexVectSpace the carrier of X)
the carrier of X is non empty set
ComplexVectSpace the carrier of X is non empty right_complementable Abelian add-associative right_zeroed strict vector-distributive scalar-distributive scalar-associative scalar-unital CLSStruct
Funcs ( the carrier of X,COMPLEX) is non empty functional FUNCTION_DOMAIN of the carrier of X, COMPLEX
ComplexFuncZero the carrier of X is Relation-like the carrier of X -defined COMPLEX -valued Function-like total quasi_total V139() Element of Funcs ( the carrier of X,COMPLEX)
the carrier of X --> 0 is non empty Relation-like the carrier of X -defined REAL -valued RAT -valued INT -valued Function-like total quasi_total V139() V140() V141() V142() Element of bool [: the carrier of X,REAL:]
[: the carrier of X,REAL:] is non empty V139() V140() V141() set
bool [: the carrier of X,REAL:] is non empty set
ComplexFuncAdd the carrier of X is non empty Relation-like [:(Funcs ( the carrier of X,COMPLEX)),(Funcs ( the carrier of X,COMPLEX)):] -defined Funcs ( the carrier of X,COMPLEX) -valued Function-like total quasi_total Element of bool [:[:(Funcs ( the carrier of X,COMPLEX)),(Funcs ( the carrier of X,COMPLEX)):],(Funcs ( the carrier of X,COMPLEX)):]
[:(Funcs ( the carrier of X,COMPLEX)),(Funcs ( the carrier of X,COMPLEX)):] is non empty set
[:[:(Funcs ( the carrier of X,COMPLEX)),(Funcs ( the carrier of X,COMPLEX)):],(Funcs ( the carrier of X,COMPLEX)):] is non empty set
bool [:[:(Funcs ( the carrier of X,COMPLEX)),(Funcs ( the carrier of X,COMPLEX)):],(Funcs ( the carrier of X,COMPLEX)):] is non empty set
ComplexFuncExtMult the carrier of X is non empty Relation-like [:COMPLEX,(Funcs ( the carrier of X,COMPLEX)):] -defined Funcs ( the carrier of X,COMPLEX) -valued Function-like total quasi_total Element of bool [:[:COMPLEX,(Funcs ( the carrier of X,COMPLEX)):],(Funcs ( the carrier of X,COMPLEX)):]
[:COMPLEX,(Funcs ( the carrier of X,COMPLEX)):] is non empty set
[:[:COMPLEX,(Funcs ( the carrier of X,COMPLEX)):],(Funcs ( the carrier of X,COMPLEX)):] is non empty set
bool [:[:COMPLEX,(Funcs ( the carrier of X,COMPLEX)):],(Funcs ( the carrier of X,COMPLEX)):] is non empty set
CLSStruct(# (Funcs ( the carrier of X,COMPLEX)),(ComplexFuncZero the carrier of X),(ComplexFuncAdd the carrier of X),(ComplexFuncExtMult the carrier of X) #) is strict CLSStruct
the carrier of (ComplexVectSpace the carrier of X) is non empty set
bool the carrier of (ComplexVectSpace the carrier of X) is non empty set
[: the carrier of X,COMPLEX:] is non empty V139() set
bool [: the carrier of X,COMPLEX:] is non empty set
bool the carrier of X is non empty set
{ b₁ where b₁ is non empty Relation-like the carrier of X -defined COMPLEX -valued Function-like total quasi_total V139() Element of bool [: the carrier of X,COMPLEX:] : ( b₁ is (X) & ex b₂ being non empty Element of bool the carrier of X st
( b₂ is compact & ( for b₃ being Element of bool the carrier of X holds
( not b₃ = support b₁ or Cl b₃ is Element of bool b₂ ) ) ) ) } is set
Zero_ ((X),(ComplexVectSpace the carrier of X)) is Element of (X)
Add_ ((X),(ComplexVectSpace the carrier of X)) is non empty Relation-like [:(X),(X):] -defined (X) -valued Function-like total quasi_total Element of bool [:[:(X),(X):],(X):]
[:(X),(X):] is non empty set
[:[:(X),(X):],(X):] is non empty set
bool [:[:(X),(X):],(X):] is non empty set
Mult_ ((X),(ComplexVectSpace the carrier of X)) is non empty Relation-like [:COMPLEX,(X):] -defined (X) -valued Function-like total quasi_total Element of bool [:[:COMPLEX,(X):],(X):]
[:COMPLEX,(X):] is non empty set
[:[:COMPLEX,(X):],(X):] is non empty set
bool [:[:COMPLEX,(X):],(X):] is non empty set
(X) is non empty Relation-like (X) -defined REAL -valued Function-like total quasi_total V139() V140() V141() Element of bool [:(X),REAL:]
[:(X),REAL:] is non empty V139() V140() V141() set
bool [:(X),REAL:] is non empty set
ComplexBoundedFunctions the carrier of X is non empty multiplicatively-closed Cadditively-linearly-closed Element of bool the carrier of (CAlgebra the carrier of X)
CAlgebra the carrier of X is non empty right_complementable Abelian add-associative right_zeroed unital associative commutative right-distributive right_unital well-unital left_unital vector-distributive scalar-distributive scalar-associative scalar-unital strict vector-associative ComplexAlgebraStr
ComplexFuncMult the carrier of X is non empty Relation-like [:(Funcs ( the carrier of X,COMPLEX)),(Funcs ( the carrier of X,COMPLEX)):] -defined Funcs ( the carrier of X,COMPLEX) -valued Function-like total quasi_total Element of bool [:[:(Funcs ( the carrier of X,COMPLEX)),(Funcs ( the carrier of X,COMPLEX)):],(Funcs ( the carrier of X,COMPLEX)):]
ComplexFuncUnit the carrier of X is Relation-like the carrier of X -defined COMPLEX -valued Function-like total quasi_total V139() Element of Funcs ( the carrier of X,COMPLEX)
the carrier of X --> 1r is non empty Relation-like the carrier of X -defined COMPLEX -valued Function-like total quasi_total V139() Element of bool [: the carrier of X,COMPLEX:]
ComplexAlgebraStr(# (Funcs ( the carrier of X,COMPLEX)),(ComplexFuncMult the carrier of X),(ComplexFuncAdd the carrier of X),(ComplexFuncExtMult the carrier of X),(ComplexFuncUnit the carrier of X),(ComplexFuncZero the carrier of X) #) is strict ComplexAlgebraStr
the carrier of (CAlgebra the carrier of X) is non empty set
bool the carrier of (CAlgebra the carrier of X) is non empty set
{ b₁ where b₁ is non empty Relation-like the carrier of X -defined COMPLEX -valued Function-like total quasi_total V139() Element of bool [: the carrier of X,COMPLEX:] : b₁ | the carrier of X is bounded } is set
ComplexBoundedFunctionsNorm the carrier of X is non empty Relation-like ComplexBoundedFunctions the carrier of X -defined REAL -valued Function-like total quasi_total V139() V140() V141() Element of bool [:(ComplexBoundedFunctions the carrier of X),REAL:]
[:(ComplexBoundedFunctions the carrier of X),REAL:] is non empty V139() V140() V141() set
bool [:(ComplexBoundedFunctions the carrier of X),REAL:] is non empty set
(ComplexBoundedFunctionsNorm the carrier of X) | (X) is Relation-like ComplexBoundedFunctions the carrier of X -defined REAL -valued Function-like V139() V140() V141() Element of bool [:(ComplexBoundedFunctions the carrier of X),REAL:]
CNORMSTR(# (X),(Zero_ ((X),(ComplexVectSpace the carrier of X))),(Add_ ((X),(ComplexVectSpace the carrier of X))),(Mult_ ((X),(ComplexVectSpace the carrier of X))),(X) #) is strict CNORMSTR
X is non empty TopSpace-like TopStruct
(X) is CNORMSTR
(X) is non empty linearly-closed Element of bool the carrier of (ComplexVectSpace the carrier of X)
the carrier of X is non empty set
ComplexVectSpace the carrier of X is non empty right_complementable Abelian add-associative right_zeroed strict vector-distributive scalar-distributive scalar-associative scalar-unital CLSStruct
Funcs ( the carrier of X,COMPLEX) is non empty functional FUNCTION_DOMAIN of the carrier of X, COMPLEX
ComplexFuncZero the carrier of X is Relation-like the carrier of X -defined COMPLEX -valued Function-like total quasi_total V139() Element of Funcs ( the carrier of X,COMPLEX)
the carrier of X --> 0 is non empty Relation-like the carrier of X -defined REAL -valued RAT -valued INT -valued Function-like total quasi_total V139() V140() V141() V142() Element of bool [: the carrier of X,REAL:]
[: the carrier of X,REAL:] is non empty V139() V140() V141() set
bool [: the carrier of X,REAL:] is non empty set
ComplexFuncAdd the carrier of X is non empty Relation-like [:(Funcs ( the carrier of X,COMPLEX)),(Funcs ( the carrier of X,COMPLEX)):] -defined Funcs ( the carrier of X,COMPLEX) -valued Function-like total quasi_total Element of bool [:[:(Funcs ( the carrier of X,COMPLEX)),(Funcs ( the carrier of X,COMPLEX)):],(Funcs ( the carrier of X,COMPLEX)):]
[:(Funcs ( the carrier of X,COMPLEX)),(Funcs ( the carrier of X,COMPLEX)):] is non empty set
[:[:(Funcs ( the carrier of X,COMPLEX)),(Funcs ( the carrier of X,COMPLEX)):],(Funcs ( the carrier of X,COMPLEX)):] is non empty set
bool [:[:(Funcs ( the carrier of X,COMPLEX)),(Funcs ( the carrier of X,COMPLEX)):],(Funcs ( the carrier of X,COMPLEX)):] is non empty set
ComplexFuncExtMult the carrier of X is non empty Relation-like [:COMPLEX,(Funcs ( the carrier of X,COMPLEX)):] -defined Funcs ( the carrier of X,COMPLEX) -valued Function-like total quasi_total Element of bool [:[:COMPLEX,(Funcs ( the carrier of X,COMPLEX)):],(Funcs ( the carrier of X,COMPLEX)):]
[:COMPLEX,(Funcs ( the carrier of X,COMPLEX)):] is non empty set
[:[:COMPLEX,(Funcs ( the carrier of X,COMPLEX)):],(Funcs ( the carrier of X,COMPLEX)):] is non empty set
bool [:[:COMPLEX,(Funcs ( the carrier of X,COMPLEX)):],(Funcs ( the carrier of X,COMPLEX)):] is non empty set
CLSStruct(# (Funcs ( the carrier of X,COMPLEX)),(ComplexFuncZero the carrier of X),(ComplexFuncAdd the carrier of X),(ComplexFuncExtMult the carrier of X) #) is strict CLSStruct
the carrier of (ComplexVectSpace the carrier of X) is non empty set
bool the carrier of (ComplexVectSpace the carrier of X) is non empty set
[: the carrier of X,COMPLEX:] is non empty V139() set
bool [: the carrier of X,COMPLEX:] is non empty set
bool the carrier of X is non empty set
{ b₁ where b₁ is non empty Relation-like the carrier of X -defined COMPLEX -valued Function-like total quasi_total V139() Element of bool [: the carrier of X,COMPLEX:] : ( b₁ is (X) & ex b₂ being non empty Element of bool the carrier of X st
( b₂ is compact & ( for b₃ being Element of bool the carrier of X holds
( not b₃ = support b₁ or Cl b₃ is Element of bool b₂ ) ) ) ) } is set
Zero_ ((X),(ComplexVectSpace the carrier of X)) is Element of (X)
Add_ ((X),(ComplexVectSpace the carrier of X)) is non empty Relation-like [:(X),(X):] -defined (X) -valued Function-like total quasi_total Element of bool [:[:(X),(X):],(X):]
[:(X),(X):] is non empty set
[:[:(X),(X):],(X):] is non empty set
bool [:[:(X),(X):],(X):] is non empty set
Mult_ ((X),(ComplexVectSpace the carrier of X)) is non empty Relation-like [:COMPLEX,(X):] -defined (X) -valued Function-like total quasi_total Element of bool [:[:COMPLEX,(X):],(X):]
[:COMPLEX,(X):] is non empty set
[:[:COMPLEX,(X):],(X):] is non empty set
bool [:[:COMPLEX,(X):],(X):] is non empty set
(X) is non empty Relation-like (X) -defined REAL -valued Function-like total quasi_total V139() V140() V141() Element of bool [:(X),REAL:]
[:(X),REAL:] is non empty V139() V140() V141() set
bool [:(X),REAL:] is non empty set
ComplexBoundedFunctions the carrier of X is non empty multiplicatively-closed Cadditively-linearly-closed Element of bool the carrier of (CAlgebra the carrier of X)
CAlgebra the carrier of X is non empty right_complementable Abelian add-associative right_zeroed unital associative commutative right-distributive right_unital well-unital left_unital vector-distributive scalar-distributive scalar-associative scalar-unital strict vector-associative ComplexAlgebraStr
ComplexFuncMult the carrier of X is non empty Relation-like [:(Funcs ( the carrier of X,COMPLEX)),(Funcs ( the carrier of X,COMPLEX)):] -defined Funcs ( the carrier of X,COMPLEX) -valued Function-like total quasi_total Element of bool [:[:(Funcs ( the carrier of X,COMPLEX)),(Funcs ( the carrier of X,COMPLEX)):],(Funcs ( the carrier of X,COMPLEX)):]
ComplexFuncUnit the carrier of X is Relation-like the carrier of X -defined COMPLEX -valued Function-like total quasi_total V139() Element of Funcs ( the carrier of X,COMPLEX)
the carrier of X --> 1r is non empty Relation-like the carrier of X -defined COMPLEX -valued Function-like total quasi_total V139() Element of bool [: the carrier of X,COMPLEX:]
ComplexAlgebraStr(# (Funcs ( the carrier of X,COMPLEX)),(ComplexFuncMult the carrier of X),(ComplexFuncAdd the carrier of X),(ComplexFuncExtMult the carrier of X),(ComplexFuncUnit the carrier of X),(ComplexFuncZero the carrier of X) #) is strict ComplexAlgebraStr
the carrier of (CAlgebra the carrier of X) is non empty set
bool the carrier of (CAlgebra the carrier of X) is non empty set
{ b₁ where b₁ is non empty Relation-like the carrier of X -defined COMPLEX -valued Function-like total quasi_total V139() Element of bool [: the carrier of X,COMPLEX:] : b₁ | the carrier of X is bounded } is set
ComplexBoundedFunctionsNorm the carrier of X is non empty Relation-like ComplexBoundedFunctions the carrier of X -defined REAL -valued Function-like total quasi_total V139() V140() V141() Element of bool [:(ComplexBoundedFunctions the carrier of X),REAL:]
[:(ComplexBoundedFunctions the carrier of X),REAL:] is non empty V139() V140() V141() set
bool [:(ComplexBoundedFunctions the carrier of X),REAL:] is non empty set
(ComplexBoundedFunctionsNorm the carrier of X) | (X) is Relation-like ComplexBoundedFunctions the carrier of X -defined REAL -valued Function-like V139() V140() V141() Element of bool [:(ComplexBoundedFunctions the carrier of X),REAL:]
CNORMSTR(# (X),(Zero_ ((X),(ComplexVectSpace the carrier of X))),(Add_ ((X),(ComplexVectSpace the carrier of X))),(Mult_ ((X),(ComplexVectSpace the carrier of X))),(X) #) is strict CNORMSTR
X is non empty TopSpace-like TopStruct
(X) is non empty linearly-closed Element of bool the carrier of (ComplexVectSpace the carrier of X)
the carrier of X is non empty set
ComplexVectSpace the carrier of X is non empty right_complementable Abelian add-associative right_zeroed strict vector-distributive scalar-distributive scalar-associative scalar-unital CLSStruct
Funcs ( the carrier of X,COMPLEX) is non empty functional FUNCTION_DOMAIN of the carrier of X, COMPLEX
ComplexFuncZero the carrier of X is Relation-like the carrier of X -defined COMPLEX -valued Function-like total quasi_total V139() Element of Funcs ( the carrier of X,COMPLEX)
the carrier of X --> 0 is non empty Relation-like the carrier of X -defined REAL -valued RAT -valued INT -valued Function-like total quasi_total V139() V140() V141() V142() Element of bool [: the carrier of X,REAL:]
[: the carrier of X,REAL:] is non empty V139() V140() V141() set
bool [: the carrier of X,REAL:] is non empty set
ComplexFuncAdd the carrier of X is non empty Relation-like [:(Funcs ( the carrier of X,COMPLEX)),(Funcs ( the carrier of X,COMPLEX)):] -defined Funcs ( the carrier of X,COMPLEX) -valued Function-like total quasi_total Element of bool [:[:(Funcs ( the carrier of X,COMPLEX)),(Funcs ( the carrier of X,COMPLEX)):],(Funcs ( the carrier of X,COMPLEX)):]
[:(Funcs ( the carrier of X,COMPLEX)),(Funcs ( the carrier of X,COMPLEX)):] is non empty set
[:[:(Funcs ( the carrier of X,COMPLEX)),(Funcs ( the carrier of X,COMPLEX)):],(Funcs ( the carrier of X,COMPLEX)):] is non empty set
bool [:[:(Funcs ( the carrier of X,COMPLEX)),(Funcs ( the carrier of X,COMPLEX)):],(Funcs ( the carrier of X,COMPLEX)):] is non empty set
ComplexFuncExtMult the carrier of X is non empty Relation-like [:COMPLEX,(Funcs ( the carrier of X,COMPLEX)):] -defined Funcs ( the carrier of X,COMPLEX) -valued Function-like total quasi_total Element of bool [:[:COMPLEX,(Funcs ( the carrier of X,COMPLEX)):],(Funcs ( the carrier of X,COMPLEX)):]
[:COMPLEX,(Funcs ( the carrier of X,COMPLEX)):] is non empty set
[:[:COMPLEX,(Funcs ( the carrier of X,COMPLEX)):],(Funcs ( the carrier of X,COMPLEX)):] is non empty set
bool [:[:COMPLEX,(Funcs ( the carrier of X,COMPLEX)):],(Funcs ( the carrier of X,COMPLEX)):] is non empty set
CLSStruct(# (Funcs ( the carrier of X,COMPLEX)),(ComplexFuncZero the carrier of X),(ComplexFuncAdd the carrier of X),(ComplexFuncExtMult the carrier of X) #) is strict CLSStruct
the carrier of (ComplexVectSpace the carrier of X) is non empty set
bool the carrier of (ComplexVectSpace the carrier of X) is non empty set
[: the carrier of X,COMPLEX:] is non empty V139() set
bool [: the carrier of X,COMPLEX:] is non empty set
bool the carrier of X is non empty set
{ b₁ where b₁ is non empty Relation-like the carrier of X -defined COMPLEX -valued Function-like total quasi_total V139() Element of bool [: the carrier of X,COMPLEX:] : ( b₁ is (X) & ex b₂ being non empty Element of bool the carrier of X st
( b₂ is compact & ( for b₃ being Element of bool the carrier of X holds
( not b₃ = support b₁ or Cl b₃ is Element of bool b₂ ) ) ) ) } is set
(X) is non empty multiplicatively-closed Cadditively-linearly-closed Element of bool the carrier of (CAlgebra the carrier of X)
CAlgebra the carrier of X is non empty right_complementable Abelian add-associative right_zeroed unital associative commutative right-distributive right_unital well-unital left_unital vector-distributive scalar-distributive scalar-associative scalar-unital strict vector-associative ComplexAlgebraStr
ComplexFuncMult the carrier of X is non empty Relation-like [:(Funcs ( the carrier of X,COMPLEX)),(Funcs ( the carrier of X,COMPLEX)):] -defined Funcs ( the carrier of X,COMPLEX) -valued Function-like total quasi_total Element of bool [:[:(Funcs ( the carrier of X,COMPLEX)),(Funcs ( the carrier of X,COMPLEX)):],(Funcs ( the carrier of X,COMPLEX)):]
ComplexFuncUnit the carrier of X is Relation-like the carrier of X -defined COMPLEX -valued Function-like total quasi_total V139() Element of Funcs ( the carrier of X,COMPLEX)
the carrier of X --> 1r is non empty Relation-like the carrier of X -defined COMPLEX -valued Function-like total quasi_total V139() Element of bool [: the carrier of X,COMPLEX:]
ComplexAlgebraStr(# (Funcs ( the carrier of X,COMPLEX)),(ComplexFuncMult the carrier of X),(ComplexFuncAdd the carrier of X),(ComplexFuncExtMult the carrier of X),(ComplexFuncUnit the carrier of X),(ComplexFuncZero the carrier of X) #) is strict ComplexAlgebraStr
the carrier of (CAlgebra the carrier of X) is non empty set
bool the carrier of (CAlgebra the carrier of X) is non empty set
{ b₁ where b₁ is non empty Relation-like the carrier of X -defined COMPLEX -valued Function-like total quasi_total V139() (X) Element of bool [: the carrier of X,COMPLEX:] : verum } is set
X is set
F is non empty Relation-like the carrier of X -defined COMPLEX -valued Function-like total quasi_total V139() Element of bool [: the carrier of X,COMPLEX:]
support F is set
G is non empty Element of bool the carrier of X
bool G is non empty set
X is non empty TopSpace-like TopStruct
(X) is non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital CLSStruct
(X) is non empty linearly-closed Element of bool the carrier of (ComplexVectSpace the carrier of X)
the carrier of X is non empty set
ComplexVectSpace the carrier of X is non empty right_complementable Abelian add-associative right_zeroed strict vector-distributive scalar-distributive scalar-associative scalar-unital CLSStruct
Funcs ( the carrier of X,COMPLEX) is non empty functional FUNCTION_DOMAIN of the carrier of X, COMPLEX
ComplexFuncZero the carrier of X is Relation-like the carrier of X -defined COMPLEX -valued Function-like total quasi_total V139() Element of Funcs ( the carrier of X,COMPLEX)
the carrier of X --> 0 is non empty Relation-like the carrier of X -defined REAL -valued RAT -valued INT -valued Function-like total quasi_total V139() V140() V141() V142() Element of bool [: the carrier of X,REAL:]
[: the carrier of X,REAL:] is non empty V139() V140() V141() set
bool [: the carrier of X,REAL:] is non empty set
ComplexFuncAdd the carrier of X is non empty Relation-like [:(Funcs ( the carrier of X,COMPLEX)),(Funcs ( the carrier of X,COMPLEX)):] -defined Funcs ( the carrier of X,COMPLEX) -valued Function-like total quasi_total Element of bool [:[:(Funcs ( the carrier of X,COMPLEX)),(Funcs ( the carrier of X,COMPLEX)):],(Funcs ( the carrier of X,COMPLEX)):]
[:(Funcs ( the carrier of X,COMPLEX)),(Funcs ( the carrier of X,COMPLEX)):] is non empty set
[:[:(Funcs ( the carrier of X,COMPLEX)),(Funcs ( the carrier of X,COMPLEX)):],(Funcs ( the carrier of X,COMPLEX)):] is non empty set
bool [:[:(Funcs ( the carrier of X,COMPLEX)),(Funcs ( the carrier of X,COMPLEX)):],(Funcs ( the carrier of X,COMPLEX)):] is non empty set
ComplexFuncExtMult the carrier of X is non empty Relation-like [:COMPLEX,(Funcs ( the carrier of X,COMPLEX)):] -defined Funcs ( the carrier of X,COMPLEX) -valued Function-like total quasi_total Element of bool [:[:COMPLEX,(Funcs ( the carrier of X,COMPLEX)):],(Funcs ( the carrier of X,COMPLEX)):]
[:COMPLEX,(Funcs ( the carrier of X,COMPLEX)):] is non empty set
[:[:COMPLEX,(Funcs ( the carrier of X,COMPLEX)):],(Funcs ( the carrier of X,COMPLEX)):] is non empty set
bool [:[:COMPLEX,(Funcs ( the carrier of X,COMPLEX)):],(Funcs ( the carrier of X,COMPLEX)):] is non empty set
CLSStruct(# (Funcs ( the carrier of X,COMPLEX)),(ComplexFuncZero the carrier of X),(ComplexFuncAdd the carrier of X),(ComplexFuncExtMult the carrier of X) #) is strict CLSStruct
the carrier of (ComplexVectSpace the carrier of X) is non empty set
bool the carrier of (ComplexVectSpace the carrier of X) is non empty set
[: the carrier of X,COMPLEX:] is non empty V139() set
bool [: the carrier of X,COMPLEX:] is non empty set
bool the carrier of X is non empty set
{ b₁ where b₁ is non empty Relation-like the carrier of X -defined COMPLEX -valued Function-like total quasi_total V139() Element of bool [: the carrier of X,COMPLEX:] : ( b₁ is (X) & ex b₂ being non empty Element of bool the carrier of X st
( b₂ is compact & ( for b₃ being Element of bool the carrier of X holds
( not b₃ = support b₁ or Cl b₃ is Element of bool b₂ ) ) ) ) } is set
Zero_ ((X),(ComplexVectSpace the carrier of X)) is Element of (X)
Add_ ((X),(ComplexVectSpace the carrier of X)) is non empty Relation-like [:(X),(X):] -defined (X) -valued Function-like total quasi_total Element of bool [:[:(X),(X):],(X):]
[:(X),(X):] is non empty set
[:[:(X),(X):],(X):] is non empty set
bool [:[:(X),(X):],(X):] is non empty set
Mult_ ((X),(ComplexVectSpace the carrier of X)) is non empty Relation-like [:COMPLEX,(X):] -defined (X) -valued Function-like total quasi_total Element of bool [:[:COMPLEX,(X):],(X):]
[:COMPLEX,(X):] is non empty set
[:[:COMPLEX,(X):],(X):] is non empty set
bool [:[:COMPLEX,(X):],(X):] is non empty set
CLSStruct(# (X),(Zero_ ((X),(ComplexVectSpace the carrier of X))),(Add_ ((X),(ComplexVectSpace the carrier of X))),(Mult_ ((X),(ComplexVectSpace the carrier of X))) #) is strict CLSStruct
0. (X) is zero Element of the carrier of (X)
the carrier of (X) is non empty set
the ZeroF of (X) is Element of the carrier of (X)
(X,0) is non empty Relation-like the carrier of X -defined COMPLEX -valued Function-like total quasi_total V139() (X) Element of bool [: the carrier of X,COMPLEX:]
the carrier of X --> 0 is non empty Relation-like the carrier of X -defined RAT -valued INT -valued {0} -valued Function-like total quasi_total V139() V140() V141() V142() Element of bool [: the carrier of X,{0}:]
{0} is non empty V176() V177() V178() V179() V180() V181() left_end bounded_below set
[: the carrier of X,{0}:] is non empty RAT -valued INT -valued V139() V140() V141() V142() set
bool [: the carrier of X,{0}:] is non empty set
0. (ComplexVectSpace the carrier of X) is zero Element of the carrier of (ComplexVectSpace the carrier of X)
the ZeroF of (ComplexVectSpace the carrier of X) is Element of the carrier of (ComplexVectSpace the carrier of X)
X is non empty TopSpace-like TopStruct
(X) is non empty strict CNORMSTR
(X) is non empty linearly-closed Element of bool the carrier of (ComplexVectSpace the carrier of X)
the carrier of X is non empty set
ComplexVectSpace the carrier of X is non empty right_complementable Abelian add-associative right_zeroed strict vector-distributive scalar-distributive scalar-associative scalar-unital CLSStruct
Funcs ( the carrier of X,COMPLEX) is non empty functional FUNCTION_DOMAIN of the carrier of X, COMPLEX
ComplexFuncZero the carrier of X is Relation-like the carrier of X -defined COMPLEX -valued Function-like total quasi_total V139() Element of Funcs ( the carrier of X,COMPLEX)
the carrier of X --> 0 is non empty Relation-like the carrier of X -defined REAL -valued RAT -valued INT -valued Function-like total quasi_total V139() V140() V141() V142() Element of bool [: the carrier of X,REAL:]
[: the carrier of X,REAL:] is non empty V139() V140() V141() set
bool [: the carrier of X,REAL:] is non empty set
ComplexFuncAdd the carrier of X is non empty Relation-like [:(Funcs ( the carrier of X,COMPLEX)),(Funcs ( the carrier of X,COMPLEX)):] -defined Funcs ( the carrier of X,COMPLEX) -valued Function-like total quasi_total Element of bool [:[:(Funcs ( the carrier of X,COMPLEX)),(Funcs ( the carrier of X,COMPLEX)):],(Funcs ( the carrier of X,COMPLEX)):]
[:(Funcs ( the carrier of X,COMPLEX)),(Funcs ( the carrier of X,COMPLEX)):] is non empty set
[:[:(Funcs ( the carrier of X,COMPLEX)),(Funcs ( the carrier of X,COMPLEX)):],(Funcs ( the carrier of X,COMPLEX)):] is non empty set
bool [:[:(Funcs ( the carrier of X,COMPLEX)),(Funcs ( the carrier of X,COMPLEX)):],(Funcs ( the carrier of X,COMPLEX)):] is non empty set
ComplexFuncExtMult the carrier of X is non empty Relation-like [:COMPLEX,(Funcs ( the carrier of X,COMPLEX)):] -defined Funcs ( the carrier of X,COMPLEX) -valued Function-like total quasi_total Element of bool [:[:COMPLEX,(Funcs ( the carrier of X,COMPLEX)):],(Funcs ( the carrier of X,COMPLEX)):]
[:COMPLEX,(Funcs ( the carrier of X,COMPLEX)):] is non empty set
[:[:COMPLEX,(Funcs ( the carrier of X,COMPLEX)):],(Funcs ( the carrier of X,COMPLEX)):] is non empty set
bool [:[:COMPLEX,(Funcs ( the carrier of X,COMPLEX)):],(Funcs ( the carrier of X,COMPLEX)):] is non empty set
CLSStruct(# (Funcs ( the carrier of X,COMPLEX)),(ComplexFuncZero the carrier of X),(ComplexFuncAdd the carrier of X),(ComplexFuncExtMult the carrier of X) #) is strict CLSStruct
the carrier of (ComplexVectSpace the carrier of X) is non empty set
bool the carrier of (ComplexVectSpace the carrier of X) is non empty set
[: the carrier of X,COMPLEX:] is non empty V139() set
bool [: the carrier of X,COMPLEX:] is non empty set
bool the carrier of X is non empty set
{ b₁ where b₁ is non empty Relation-like the carrier of X -defined COMPLEX -valued Function-like total quasi_total V139() Element of bool [: the carrier of X,COMPLEX:] : ( b₁ is (X) & ex b₂ being non empty Element of bool the carrier of X st
( b₂ is compact & ( for b₃ being Element of bool the carrier of X holds
( not b₃ = support b₁ or Cl b₃ is Element of bool b₂ ) ) ) ) } is set
Zero_ ((X),(ComplexVectSpace the carrier of X)) is Element of (X)
Add_ ((X),(ComplexVectSpace the carrier of X)) is non empty Relation-like [:(X),(X):] -defined (X) -valued Function-like total quasi_total Element of bool [:[:(X),(X):],(X):]
[:(X),(X):] is non empty set
[:[:(X),(X):],(X):] is non empty set
bool [:[:(X),(X):],(X):] is non empty set
Mult_ ((X),(ComplexVectSpace the carrier of X)) is non empty Relation-like [:COMPLEX,(X):] -defined (X) -valued Function-like total quasi_total Element of bool [:[:COMPLEX,(X):],(X):]
[:COMPLEX,(X):] is non empty set
[:[:COMPLEX,(X):],(X):] is non empty set
bool [:[:COMPLEX,(X):],(X):] is non empty set
(X) is non empty Relation-like (X) -defined REAL -valued Function-like total quasi_total V139() V140() V141() Element of bool [:(X),REAL:]
[:(X),REAL:] is non empty V139() V140() V141() set
bool [:(X),REAL:] is non empty set
ComplexBoundedFunctions the carrier of X is non empty multiplicatively-closed Cadditively-linearly-closed Element of bool the carrier of (CAlgebra the carrier of X)
CAlgebra the carrier of X is non empty right_complementable Abelian add-associative right_zeroed unital associative commutative right-distributive right_unital well-unital left_unital vector-distributive scalar-distributive scalar-associative scalar-unital strict vector-associative ComplexAlgebraStr
ComplexFuncMult the carrier of X is non empty Relation-like [:(Funcs ( the carrier of X,COMPLEX)),(Funcs ( the carrier of X,COMPLEX)):] -defined Funcs ( the carrier of X,COMPLEX) -valued Function-like total quasi_total Element of bool [:[:(Funcs ( the carrier of X,COMPLEX)),(Funcs ( the carrier of X,COMPLEX)):],(Funcs ( the carrier of X,COMPLEX)):]
ComplexFuncUnit the carrier of X is Relation-like the carrier of X -defined COMPLEX -valued Function-like total quasi_total V139() Element of Funcs ( the carrier of X,COMPLEX)
the carrier of X --> 1r is non empty Relation-like the carrier of X -defined COMPLEX -valued Function-like total quasi_total V139() Element of bool [: the carrier of X,COMPLEX:]
ComplexAlgebraStr(# (Funcs ( the carrier of X,COMPLEX)),(ComplexFuncMult the carrier of X),(ComplexFuncAdd the carrier of X),(ComplexFuncExtMult the carrier of X),(ComplexFuncUnit the carrier of X),(ComplexFuncZero the carrier of X) #) is strict ComplexAlgebraStr
the carrier of (CAlgebra the carrier of X) is non empty set
bool the carrier of (CAlgebra the carrier of X) is non empty set
{ b₁ where b₁ is non empty Relation-like the carrier of X -defined COMPLEX -valued Function-like total quasi_total V139() Element of bool [: the carrier of X,COMPLEX:] : b₁ | the carrier of X is bounded } is set
ComplexBoundedFunctionsNorm the carrier of X is non empty Relation-like ComplexBoundedFunctions the carrier of X -defined REAL -valued Function-like total quasi_total V139() V140() V141() Element of bool [:(ComplexBoundedFunctions the carrier of X),REAL:]
[:(ComplexBoundedFunctions the carrier of X),REAL:] is non empty V139() V140() V141() set
bool [:(ComplexBoundedFunctions the carrier of X),REAL:] is non empty set
(ComplexBoundedFunctionsNorm the carrier of X) | (X) is Relation-like ComplexBoundedFunctions the carrier of X -defined REAL -valued Function-like V139() V140() V141() Element of bool [:(ComplexBoundedFunctions the carrier of X),REAL:]
CNORMSTR(# (X),(Zero_ ((X),(ComplexVectSpace the carrier of X))),(Add_ ((X),(ComplexVectSpace the carrier of X))),(Mult_ ((X),(ComplexVectSpace the carrier of X))),(X) #) is strict CNORMSTR
0. (X) is zero Element of the carrier of (X)
the carrier of (X) is non empty set
the ZeroF of (X) is Element of the carrier of (X)
(X,0) is non empty Relation-like the carrier of X -defined COMPLEX -valued Function-like total quasi_total V139() (X) Element of bool [: the carrier of X,COMPLEX:]
the carrier of X --> 0 is non empty Relation-like the carrier of X -defined RAT -valued INT -valued {0} -valued Function-like total quasi_total V139() V140() V141() V142() Element of bool [: the carrier of X,{0}:]
{0} is non empty V176() V177() V178() V179() V180() V181() left_end bounded_below set
[: the carrier of X,{0}:] is non empty RAT -valued INT -valued V139() V140() V141() V142() set
bool [: the carrier of X,{0}:] is non empty set
(X) is non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital CLSStruct
CLSStruct(# (X),(Zero_ ((X),(ComplexVectSpace the carrier of X))),(Add_ ((X),(ComplexVectSpace the carrier of X))),(Mult_ ((X),(ComplexVectSpace the carrier of X))) #) is strict CLSStruct
0. (X) is zero Element of the carrier of (X)
the carrier of (X) is non empty set
the ZeroF of (X) is Element of the carrier of (X)
X is non empty TopSpace-like TopStruct
(X) is non empty strict CNORMSTR
(X) is non empty linearly-closed Element of bool the carrier of (ComplexVectSpace the carrier of X)
the carrier of X is non empty set
ComplexVectSpace the carrier of X is non empty right_complementable Abelian add-associative right_zeroed strict vector-distributive scalar-distributive scalar-associative scalar-unital CLSStruct
Funcs ( the carrier of X,COMPLEX) is non empty functional FUNCTION_DOMAIN of the carrier of X, COMPLEX
ComplexFuncZero the carrier of X is Relation-like the carrier of X -defined COMPLEX -valued Function-like total quasi_total V139() Element of Funcs ( the carrier of X,COMPLEX)
the carrier of X --> 0 is non empty Relation-like the carrier of X -defined REAL -valued RAT -valued INT -valued Function-like total quasi_total V139() V140() V141() V142() Element of bool [: the carrier of X,REAL:]
[: the carrier of X,REAL:] is non empty V139() V140() V141() set
bool [: the carrier of X,REAL:] is non empty set
ComplexFuncAdd the carrier of X is non empty Relation-like [:(Funcs ( the carrier of X,COMPLEX)),(Funcs ( the carrier of X,COMPLEX)):] -defined Funcs ( the carrier of X,COMPLEX) -valued Function-like total quasi_total Element of bool [:[:(Funcs ( the carrier of X,COMPLEX)),(Funcs ( the carrier of X,COMPLEX)):],(Funcs ( the carrier of X,COMPLEX)):]
[:(Funcs ( the carrier of X,COMPLEX)),(Funcs ( the carrier of X,COMPLEX)):] is non empty set
[:[:(Funcs ( the carrier of X,COMPLEX)),(Funcs ( the carrier of X,COMPLEX)):],(Funcs ( the carrier of X,COMPLEX)):] is non empty set
bool [:[:(Funcs ( the carrier of X,COMPLEX)),(Funcs ( the carrier of X,COMPLEX)):],(Funcs ( the carrier of X,COMPLEX)):] is non empty set
ComplexFuncExtMult the carrier of X is non empty Relation-like [:COMPLEX,(Funcs ( the carrier of X,COMPLEX)):] -defined Funcs ( the carrier of X,COMPLEX) -valued Function-like total quasi_total Element of bool [:[:COMPLEX,(Funcs ( the carrier of X,COMPLEX)):],(Funcs ( the carrier of X,COMPLEX)):]
[:COMPLEX,(Funcs ( the carrier of X,COMPLEX)):] is non empty set
[:[:COMPLEX,(Funcs ( the carrier of X,COMPLEX)):],(Funcs ( the carrier of X,COMPLEX)):] is non empty set
bool [:[:COMPLEX,(Funcs ( the carrier of X,COMPLEX)):],(Funcs ( the carrier of X,COMPLEX)):] is non empty set
CLSStruct(# (Funcs ( the carrier of X,COMPLEX)),(ComplexFuncZero the carrier of X),(ComplexFuncAdd the carrier of X),(ComplexFuncExtMult the carrier of X) #) is strict CLSStruct
the carrier of (ComplexVectSpace the carrier of X) is non empty set
bool the carrier of (ComplexVectSpace the carrier of X) is non empty set
[: the carrier of X,COMPLEX:] is non empty V139() set
bool [: the carrier of X,COMPLEX:] is non empty set
bool the carrier of X is non empty set
{ b₁ where b₁ is non empty Relation-like the carrier of X -defined COMPLEX -valued Function-like total quasi_total V139() Element of bool [: the carrier of X,COMPLEX:] : ( b₁ is (X) & ex b₂ being non empty Element of bool the carrier of X st
( b₂ is compact & ( for b₃ being Element of bool the carrier of X holds
( not b₃ = support b₁ or Cl b₃ is Element of bool b₂ ) ) ) ) } is set
Zero_ ((X),(ComplexVectSpace the carrier of X)) is Element of (X)
Add_ ((X),(ComplexVectSpace the carrier of X)) is non empty Relation-like [:(X),(X):] -defined (X) -valued Function-like total quasi_total Element of bool [:[:(X),(X):],(X):]
[:(X),(X):] is non empty set
[:[:(X),(X):],(X):] is non empty set
bool [:[:(X),(X):],(X):] is non empty set
Mult_ ((X),(ComplexVectSpace the carrier of X)) is non empty Relation-like [:COMPLEX,(X):] -defined (X) -valued Function-like total quasi_total Element of bool [:[:COMPLEX,(X):],(X):]
[:COMPLEX,(X):] is non empty set
[:[:COMPLEX,(X):],(X):] is non empty set
bool [:[:COMPLEX,(X):],(X):] is non empty set
(X) is non empty Relation-like (X) -defined REAL -valued Function-like total quasi_total V139() V140() V141() Element of bool [:(X),REAL:]
[:(X),REAL:] is non empty V139() V140() V141() set
bool [:(X),REAL:] is non empty set
ComplexBoundedFunctions the carrier of X is non empty multiplicatively-closed Cadditively-linearly-closed Element of bool the carrier of (CAlgebra the carrier of X)
CAlgebra the carrier of X is non empty right_complementable Abelian add-associative right_zeroed unital associative commutative right-distributive right_unital well-unital left_unital vector-distributive scalar-distributive scalar-associative scalar-unital strict vector-associative ComplexAlgebraStr
ComplexFuncMult the carrier of X is non empty Relation-like [:(Funcs ( the carrier of X,COMPLEX)),(Funcs ( the carrier of X,COMPLEX)):] -defined Funcs ( the carrier of X,COMPLEX) -valued Function-like total quasi_total Element of bool [:[:(Funcs ( the carrier of X,COMPLEX)),(Funcs ( the carrier of X,COMPLEX)):],(Funcs ( the carrier of X,COMPLEX)):]
ComplexFuncUnit the carrier of X is Relation-like the carrier of X -defined COMPLEX -valued Function-like total quasi_total V139() Element of Funcs ( the carrier of X,COMPLEX)
the carrier of X --> 1r is non empty Relation-like the carrier of X -defined COMPLEX -valued Function-like total quasi_total V139() Element of bool [: the carrier of X,COMPLEX:]
ComplexAlgebraStr(# (Funcs ( the carrier of X,COMPLEX)),(ComplexFuncMult the carrier of X),(ComplexFuncAdd the carrier of X),(ComplexFuncExtMult the carrier of X),(ComplexFuncUnit the carrier of X),(ComplexFuncZero the carrier of X) #) is strict ComplexAlgebraStr
the carrier of (CAlgebra the carrier of X) is non empty set
bool the carrier of (CAlgebra the carrier of X) is non empty set
{ b₁ where b₁ is non empty Relation-like the carrier of X -defined COMPLEX -valued Function-like total quasi_total V139() Element of bool [: the carrier of X,COMPLEX:] : b₁ | the carrier of X is bounded } is set
ComplexBoundedFunctionsNorm the carrier of X is non empty Relation-like ComplexBoundedFunctions the carrier of X -defined REAL -valued Function-like total quasi_total V139() V140() V141() Element of bool [:(ComplexBoundedFunctions the carrier of X),REAL:]
[:(ComplexBoundedFunctions the carrier of X),REAL:] is non empty V139() V140() V141() set
bool [:(ComplexBoundedFunctions the carrier of X),REAL:] is non empty set
(ComplexBoundedFunctionsNorm the carrier of X) | (X) is Relation-like ComplexBoundedFunctions the carrier of X -defined REAL -valued Function-like V139() V140() V141() Element of bool [:(ComplexBoundedFunctions the carrier of X),REAL:]
CNORMSTR(# (X),(Zero_ ((X),(ComplexVectSpace the carrier of X))),(Add_ ((X),(ComplexVectSpace the carrier of X))),(Mult_ ((X),(ComplexVectSpace the carrier of X))),(X) #) is strict CNORMSTR
the carrier of (X) is non empty set
C_Normed_Algebra_of_BoundedFunctions the carrier of X is non empty right_complementable Abelian add-associative right_zeroed unital discerning reflexive vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like complete Normed_Complex_AlgebraStr
mult_ ((ComplexBoundedFunctions the carrier of X),(CAlgebra the carrier of X)) is non empty Relation-like [:(ComplexBoundedFunctions the carrier of X),(ComplexBoundedFunctions the carrier of X):] -defined ComplexBoundedFunctions the carrier of X -valued Function-like total quasi_total Element of bool [:[:(ComplexBoundedFunctions the carrier of X),(ComplexBoundedFunctions the carrier of X):],(ComplexBoundedFunctions the carrier of X):]
[:(ComplexBoundedFunctions the carrier of X),(ComplexBoundedFunctions the carrier of X):] is non empty set
[:[:(ComplexBoundedFunctions the carrier of X),(ComplexBoundedFunctions the carrier of X):],(ComplexBoundedFunctions the carrier of X):] is non empty set
bool [:[:(ComplexBoundedFunctions the carrier of X),(ComplexBoundedFunctions the carrier of X):],(ComplexBoundedFunctions the carrier of X):] is non empty set
Add_ ((ComplexBoundedFunctions the carrier of X),(CAlgebra the carrier of X)) is non empty Relation-like [:(ComplexBoundedFunctions the carrier of X),(ComplexBoundedFunctions the carrier of X):] -defined ComplexBoundedFunctions the carrier of X -valued Function-like total quasi_total Element of bool [:[:(ComplexBoundedFunctions the carrier of X),(ComplexBoundedFunctions the carrier of X):],(ComplexBoundedFunctions the carrier of X):]
Mult_ ((ComplexBoundedFunctions the carrier of X),(CAlgebra the carrier of X)) is non empty Relation-like [:COMPLEX,(ComplexBoundedFunctions the carrier of X):] -defined ComplexBoundedFunctions the carrier of X -valued Function-like total quasi_total Element of bool [:[:COMPLEX,(ComplexBoundedFunctions the carrier of X):],(ComplexBoundedFunctions the carrier of X):]
[:COMPLEX,(ComplexBoundedFunctions the carrier of X):] is non empty set
[:[:COMPLEX,(ComplexBoundedFunctions the carrier of X):],(ComplexBoundedFunctions the carrier of X):] is non empty set
bool [:[:COMPLEX,(ComplexBoundedFunctions the carrier of X):],(ComplexBoundedFunctions the carrier of X):] is non empty set
One_ ((ComplexBoundedFunctions the carrier of X),(CAlgebra the carrier of X)) is Element of ComplexBoundedFunctions the carrier of X
Zero_ ((ComplexBoundedFunctions the carrier of X),(CAlgebra the carrier of X)) is Element of ComplexBoundedFunctions the carrier of X
Normed_Complex_AlgebraStr(# (ComplexBoundedFunctions the carrier of X),(mult_ ((ComplexBoundedFunctions the carrier of X),(CAlgebra the carrier of X))),(Add_ ((ComplexBoundedFunctions the carrier of X),(CAlgebra the carrier of X))),(Mult_ ((ComplexBoundedFunctions the carrier of X),(CAlgebra the carrier of X))),(One_ ((ComplexBoundedFunctions the carrier of X),(CAlgebra the carrier of X))),(Zero_ ((ComplexBoundedFunctions the carrier of X),(CAlgebra the carrier of X))),(ComplexBoundedFunctionsNorm the carrier of X) #) is strict Normed_Complex_AlgebraStr
the carrier of (C_Normed_Algebra_of_BoundedFunctions the carrier of X) is non empty set
X is Element of the carrier of (X)
F is Element of the carrier of (X)
X + F is Element of the carrier of (X)
the addF of (X) is non empty Relation-like [: the carrier of (X), the carrier of (X):] -defined the carrier of (X) -valued Function-like total quasi_total Element of bool [:[: the carrier of (X), the carrier of (X):], the carrier of (X):]
[: the carrier of (X), the carrier of (X):] is non empty set
[:[: the carrier of (X), the carrier of (X):], the carrier of (X):] is non empty set
bool [:[: the carrier of (X), the carrier of (X):], the carrier of (X):] is non empty set
the addF of (X) . (X,F) is Element of the carrier of (X)
[X,F] is set
{X,F} is non empty set
{X} is non empty set
{{X,F},{X}} is non empty set
the addF of (X) . [X,F] is set
G is Element of the carrier of (C_Normed_Algebra_of_BoundedFunctions the carrier of X)
FB is Element of the carrier of (C_Normed_Algebra_of_BoundedFunctions the carrier of X)
G + FB is Element of the carrier of (C_Normed_Algebra_of_BoundedFunctions the carrier of X)
the addF of (C_Normed_Algebra_of_BoundedFunctions the carrier of X) is non empty Relation-like [: the carrier of (C_Normed_Algebra_of_BoundedFunctions the carrier of X), the carrier of (C_Normed_Algebra_of_BoundedFunctions the carrier of X):] -defined the carrier of (C_Normed_Algebra_of_BoundedFunctions the carrier of X) -valued Function-like total quasi_total Element of bool [:[: the carrier of (C_Normed_Algebra_of_BoundedFunctions the carrier of X), the carrier of (C_Normed_Algebra_of_BoundedFunctions the carrier of X):], the carrier of (C_Normed_Algebra_of_BoundedFunctions the carrier of X):]
[: the carrier of (C_Normed_Algebra_of_BoundedFunctions the carrier of X), the carrier of (C_Normed_Algebra_of_BoundedFunctions the carrier of X):] is non empty set
[:[: the carrier of (C_Normed_Algebra_of_BoundedFunctions the carrier of X), the carrier of (C_Normed_Algebra_of_BoundedFunctions the carrier of X):], the carrier of (C_Normed_Algebra_of_BoundedFunctions the carrier of X):] is non empty set
bool [:[: the carrier of (C_Normed_Algebra_of_BoundedFunctions the carrier of X), the carrier of (C_Normed_Algebra_of_BoundedFunctions the carrier of X):], the carrier of (C_Normed_Algebra_of_BoundedFunctions the carrier of X):] is non empty set
the addF of (C_Normed_Algebra_of_BoundedFunctions the carrier of X) . (G,FB) is Element of the carrier of (C_Normed_Algebra_of_BoundedFunctions the carrier of X)
[G,FB] is set
{G,FB} is non empty set
{G} is non empty set
{{G,FB},{G}} is non empty set
the addF of (C_Normed_Algebra_of_BoundedFunctions the carrier of X) . [G,FB] is set
the addF of (ComplexVectSpace the carrier of X) is non empty Relation-like [: the carrier of (ComplexVectSpace the carrier of X), the carrier of (ComplexVectSpace the carrier of X):] -defined the carrier of (ComplexVectSpace the carrier of X) -valued Function-like total quasi_total Element of bool [:[: the carrier of (ComplexVectSpace the carrier of X), the carrier of (ComplexVectSpace the carrier of X):], the carrier of (ComplexVectSpace the carrier of X):]
[: the carrier of (ComplexVectSpace the carrier of X), the carrier of (ComplexVectSpace the carrier of X):] is non empty set
[:[: the carrier of (ComplexVectSpace the carrier of X), the carrier of (ComplexVectSpace the carrier of X):], the carrier of (ComplexVectSpace the carrier of X):] is non empty set
bool [:[: the carrier of (ComplexVectSpace the carrier of X), the carrier of (ComplexVectSpace the carrier of X):], the carrier of (ComplexVectSpace the carrier of X):] is non empty set
the addF of (ComplexVectSpace the carrier of X) || (X) is Relation-like Function-like set
the addF of (ComplexVectSpace the carrier of X) | [:(X),(X):] is Relation-like set
[X,F] is Element of [: the carrier of (X), the carrier of (X):]
( the addF of (ComplexVectSpace the carrier of X) || (X)) . [X,F] is set
the addF of (CAlgebra the carrier of X) is non empty Relation-like [: the carrier of (CAlgebra the carrier of X), the carrier of (CAlgebra the carrier of X):] -defined the carrier of (CAlgebra the carrier of X) -valued Function-like total quasi_total Element of bool [:[: the carrier of (CAlgebra the carrier of X), the carrier of (CAlgebra the carrier of X):], the carrier of (CAlgebra the carrier of X):]
[: the carrier of (CAlgebra the carrier of X), the carrier of (CAlgebra the carrier of X):] is non empty set
[:[: the carrier of (CAlgebra the carrier of X), the carrier of (CAlgebra the carrier of X):], the carrier of (CAlgebra the carrier of X):] is non empty set
bool [:[: the carrier of (CAlgebra the carrier of X), the carrier of (CAlgebra the carrier of X):], the carrier of (CAlgebra the carrier of X):] is non empty set
the addF of (CAlgebra the carrier of X) . [X,F] is set
the addF of (CAlgebra the carrier of X) || (ComplexBoundedFunctions the carrier of X) is Relation-like Function-like set
the addF of (CAlgebra the carrier of X) | [:(ComplexBoundedFunctions the carrier of X),(ComplexBoundedFunctions the carrier of X):] is Relation-like set
[G,FB] is Element of [: the carrier of (C_Normed_Algebra_of_BoundedFunctions the carrier of X), the carrier of (C_Normed_Algebra_of_BoundedFunctions the carrier of X):]
( the addF of (CAlgebra the carrier of X) || (ComplexBoundedFunctions the carrier of X)) . [G,FB] is set
X is non empty TopSpace-like TopStruct
(X) is non empty strict CNORMSTR
(X) is non empty linearly-closed Element of bool the carrier of (ComplexVectSpace the carrier of X)
the carrier of X is non empty set
ComplexVectSpace the carrier of X is non empty right_complementable Abelian add-associative right_zeroed strict vector-distributive scalar-distributive scalar-associative scalar-unital CLSStruct
Funcs ( the carrier of X,COMPLEX) is non empty functional FUNCTION_DOMAIN of the carrier of X, COMPLEX
ComplexFuncZero the carrier of X is Relation-like the carrier of X -defined COMPLEX -valued Function-like total quasi_total V139() Element of Funcs ( the carrier of X,COMPLEX)
the carrier of X --> 0 is non empty Relation-like the carrier of X -defined REAL -valued RAT -valued INT -valued Function-like total quasi_total V139() V140() V141() V142() Element of bool [: the carrier of X,REAL:]
[: the carrier of X,REAL:] is non empty V139() V140() V141() set
bool [: the carrier of X,REAL:] is non empty set
ComplexFuncAdd the carrier of X is non empty Relation-like [:(Funcs ( the carrier of X,COMPLEX)),(Funcs ( the carrier of X,COMPLEX)):] -defined Funcs ( the carrier of X,COMPLEX) -valued Function-like total quasi_total Element of bool [:[:(Funcs ( the carrier of X,COMPLEX)),(Funcs ( the carrier of X,COMPLEX)):],(Funcs ( the carrier of X,COMPLEX)):]
[:(Funcs ( the carrier of X,COMPLEX)),(Funcs ( the carrier of X,COMPLEX)):] is non empty set
[:[:(Funcs ( the carrier of X,COMPLEX)),(Funcs ( the carrier of X,COMPLEX)):],(Funcs ( the carrier of X,COMPLEX)):] is non empty set
bool [:[:(Funcs ( the carrier of X,COMPLEX)),(Funcs ( the carrier of X,COMPLEX)):],(Funcs ( the carrier of X,COMPLEX)):] is non empty set
ComplexFuncExtMult the carrier of X is non empty Relation-like [:COMPLEX,(Funcs ( the carrier of X,COMPLEX)):] -defined Funcs ( the carrier of X,COMPLEX) -valued Function-like total quasi_total Element of bool [:[:COMPLEX,(Funcs ( the carrier of X,COMPLEX)):],(Funcs ( the carrier of X,COMPLEX)):]
[:COMPLEX,(Funcs ( the carrier of X,COMPLEX)):] is non empty set
[:[:COMPLEX,(Funcs ( the carrier of X,COMPLEX)):],(Funcs ( the carrier of X,COMPLEX)):] is non empty set
bool [:[:COMPLEX,(Funcs ( the carrier of X,COMPLEX)):],(Funcs ( the carrier of X,COMPLEX)):] is non empty set
CLSStruct(# (Funcs ( the carrier of X,COMPLEX)),(ComplexFuncZero the carrier of X),(ComplexFuncAdd the carrier of X),(ComplexFuncExtMult the carrier of X) #) is strict CLSStruct
the carrier of (ComplexVectSpace the carrier of X) is non empty set
bool the carrier of (ComplexVectSpace the carrier of X) is non empty set
[: the carrier of X,COMPLEX:] is non empty V139() set
bool [: the carrier of X,COMPLEX:] is non empty set
bool the carrier of X is non empty set
{ b₁ where b₁ is non empty Relation-like the carrier of X -defined COMPLEX -valued Function-like total quasi_total V139() Element of bool [: the carrier of X,COMPLEX:] : ( b₁ is (X) & ex b₂ being non empty Element of bool the carrier of X st
( b₂ is compact & ( for b₃ being Element of bool the carrier of X holds
( not b₃ = support b₁ or Cl b₃ is Element of bool b₂ ) ) ) ) } is set
Zero_ ((X),(ComplexVectSpace the carrier of X)) is Element of (X)
Add_ ((X),(ComplexVectSpace the carrier of X)) is non empty Relation-like [:(X),(X):] -defined (X) -valued Function-like total quasi_total Element of bool [:[:(X),(X):],(X):]
[:(X),(X):] is non empty set
[:[:(X),(X):],(X):] is non empty set
bool [:[:(X),(X):],(X):] is non empty set
Mult_ ((X),(ComplexVectSpace the carrier of X)) is non empty Relation-like [:COMPLEX,(X):] -defined (X) -valued Function-like total quasi_total Element of bool [:[:COMPLEX,(X):],(X):]
[:COMPLEX,(X):] is non empty set
[:[:COMPLEX,(X):],(X):] is non empty set
bool [:[:COMPLEX,(X):],(X):] is non empty set
(X) is non empty Relation-like (X) -defined REAL -valued Function-like total quasi_total V139() V140() V141() Element of bool [:(X),REAL:]
[:(X),REAL:] is non empty V139() V140() V141() set
bool [:(X),REAL:] is non empty set
ComplexBoundedFunctions the carrier of X is non empty multiplicatively-closed Cadditively-linearly-closed Element of bool the carrier of (CAlgebra the carrier of X)
CAlgebra the carrier of X is non empty right_complementable Abelian add-associative right_zeroed unital associative commutative right-distributive right_unital well-unital left_unital vector-distributive scalar-distributive scalar-associative scalar-unital strict vector-associative ComplexAlgebraStr
ComplexFuncMult the carrier of X is non empty Relation-like [:(Funcs ( the carrier of X,COMPLEX)),(Funcs ( the carrier of X,COMPLEX)):] -defined Funcs ( the carrier of X,COMPLEX) -valued Function-like total quasi_total Element of bool [:[:(Funcs ( the carrier of X,COMPLEX)),(Funcs ( the carrier of X,COMPLEX)):],(Funcs ( the carrier of X,COMPLEX)):]
ComplexFuncUnit the carrier of X is Relation-like the carrier of X -defined COMPLEX -valued Function-like total quasi_total V139() Element of Funcs ( the carrier of X,COMPLEX)
the carrier of X --> 1r is non empty Relation-like the carrier of X -defined COMPLEX -valued Function-like total quasi_total V139() Element of bool [: the carrier of X,COMPLEX:]
ComplexAlgebraStr(# (Funcs ( the carrier of X,COMPLEX)),(ComplexFuncMult the carrier of X),(ComplexFuncAdd the carrier of X),(ComplexFuncExtMult the carrier of X),(ComplexFuncUnit the carrier of X),(ComplexFuncZero the carrier of X) #) is strict ComplexAlgebraStr
the carrier of (CAlgebra the carrier of X) is non empty set
bool the carrier of (CAlgebra the carrier of X) is non empty set
{ b₁ where b₁ is non empty Relation-like the carrier of X -defined COMPLEX -valued Function-like total quasi_total V139() Element of bool [: the carrier of X,COMPLEX:] : b₁ | the carrier of X is bounded } is set
ComplexBoundedFunctionsNorm the carrier of X is non empty Relation-like ComplexBoundedFunctions the carrier of X -defined REAL -valued Function-like total quasi_total V139() V140() V141() Element of bool [:(ComplexBoundedFunctions the carrier of X),REAL:]
[:(ComplexBoundedFunctions the carrier of X),REAL:] is non empty V139() V140() V141() set
bool [:(ComplexBoundedFunctions the carrier of X),REAL:] is non empty set
(ComplexBoundedFunctionsNorm the carrier of X) | (X) is Relation-like ComplexBoundedFunctions the carrier of X -defined REAL -valued Function-like V139() V140() V141() Element of bool [:(ComplexBoundedFunctions the carrier of X),REAL:]
CNORMSTR(# (X),(Zero_ ((X),(ComplexVectSpace the carrier of X))),(Add_ ((X),(ComplexVectSpace the carrier of X))),(Mult_ ((X),(ComplexVectSpace the carrier of X))),(X) #) is strict CNORMSTR
the carrier of (X) is non empty set
C_Normed_Algebra_of_BoundedFunctions the carrier of X is non empty right_complementable Abelian add-associative right_zeroed unital discerning reflexive vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like complete Normed_Complex_AlgebraStr
mult_ ((ComplexBoundedFunctions the carrier of X),(CAlgebra the carrier of X)) is non empty Relation-like [:(ComplexBoundedFunctions the carrier of X),(ComplexBoundedFunctions the carrier of X):] -defined ComplexBoundedFunctions the carrier of X -valued Function-like total quasi_total Element of bool [:[:(ComplexBoundedFunctions the carrier of X),(ComplexBoundedFunctions the carrier of X):],(ComplexBoundedFunctions the carrier of X):]
[:(ComplexBoundedFunctions the carrier of X),(ComplexBoundedFunctions the carrier of X):] is non empty set
[:[:(ComplexBoundedFunctions the carrier of X),(ComplexBoundedFunctions the carrier of X):],(ComplexBoundedFunctions the carrier of X):] is non empty set
bool [:[:(ComplexBoundedFunctions the carrier of X),(ComplexBoundedFunctions the carrier of X):],(ComplexBoundedFunctions the carrier of X):] is non empty set
Add_ ((ComplexBoundedFunctions the carrier of X),(CAlgebra the carrier of X)) is non empty Relation-like [:(ComplexBoundedFunctions the carrier of X),(ComplexBoundedFunctions the carrier of X):] -defined ComplexBoundedFunctions the carrier of X -valued Function-like total quasi_total Element of bool [:[:(ComplexBoundedFunctions the carrier of X),(ComplexBoundedFunctions the carrier of X):],(ComplexBoundedFunctions the carrier of X):]
Mult_ ((ComplexBoundedFunctions the carrier of X),(CAlgebra the carrier of X)) is non empty Relation-like [:COMPLEX,(ComplexBoundedFunctions the carrier of X):] -defined ComplexBoundedFunctions the carrier of X -valued Function-like total quasi_total Element of bool [:[:COMPLEX,(ComplexBoundedFunctions the carrier of X):],(ComplexBoundedFunctions the carrier of X):]
[:COMPLEX,(ComplexBoundedFunctions the carrier of X):] is non empty set
[:[:COMPLEX,(ComplexBoundedFunctions the carrier of X):],(ComplexBoundedFunctions the carrier of X):] is non empty set
bool [:[:COMPLEX,(ComplexBoundedFunctions the carrier of X):],(ComplexBoundedFunctions the carrier of X):] is non empty set
One_ ((ComplexBoundedFunctions the carrier of X),(CAlgebra the carrier of X)) is Element of ComplexBoundedFunctions the carrier of X
Zero_ ((ComplexBoundedFunctions the carrier of X),(CAlgebra the carrier of X)) is Element of ComplexBoundedFunctions the carrier of X
Normed_Complex_AlgebraStr(# (ComplexBoundedFunctions the carrier of X),(mult_ ((ComplexBoundedFunctions the carrier of X),(CAlgebra the carrier of X))),(Add_ ((ComplexBoundedFunctions the carrier of X),(CAlgebra the carrier of X))),(Mult_ ((ComplexBoundedFunctions the carrier of X),(CAlgebra the carrier of X))),(One_ ((ComplexBoundedFunctions the carrier of X),(CAlgebra the carrier of X))),(Zero_ ((ComplexBoundedFunctions the carrier of X),(CAlgebra the carrier of X))),(ComplexBoundedFunctionsNorm the carrier of X) #) is strict Normed_Complex_AlgebraStr
the carrier of (C_Normed_Algebra_of_BoundedFunctions the carrier of X) is non empty set
X is complex set
F is Element of the carrier of (X)
X * F is Element of the carrier of (X)
the Mult of (X) is non empty Relation-like [:COMPLEX, the carrier of (X):] -defined the carrier of (X) -valued Function-like total quasi_total Element of bool [:[:COMPLEX, the carrier of (X):], the carrier of (X):]
[:COMPLEX, the carrier of (X):] is non empty set
[:[:COMPLEX, the carrier of (X):], the carrier of (X):] is non empty set
bool [:[:COMPLEX, the carrier of (X):], the carrier of (X):] is non empty set
[X,F] is set
{X,F} is non empty set
{X} is non empty V176() set
{{X,F},{X}} is non empty set
the Mult of (X) . [X,F] is set
G is Element of the carrier of (C_Normed_Algebra_of_BoundedFunctions the carrier of X)
X * G is Element of the carrier of (C_Normed_Algebra_of_BoundedFunctions the carrier of X)
the Mult of (C_Normed_Algebra_of_BoundedFunctions the carrier of X) is non empty Relation-like [:COMPLEX, the carrier of (C_Normed_Algebra_of_BoundedFunctions the carrier of X):] -defined the carrier of (C_Normed_Algebra_of_BoundedFunctions the carrier of X) -valued Function-like total quasi_total Element of bool [:[:COMPLEX, the carrier of (C_Normed_Algebra_of_BoundedFunctions the carrier of X):], the carrier of (C_Normed_Algebra_of_BoundedFunctions the carrier of X):]
[:COMPLEX, the carrier of (C_Normed_Algebra_of_BoundedFunctions the carrier of X):] is non empty set
[:[:COMPLEX, the carrier of (C_Normed_Algebra_of_BoundedFunctions the carrier of X):], the carrier of (C_Normed_Algebra_of_BoundedFunctions the carrier of X):] is non empty set
bool [:[:COMPLEX, the carrier of (C_Normed_Algebra_of_BoundedFunctions the carrier of X):], the carrier of (C_Normed_Algebra_of_BoundedFunctions the carrier of X):] is non empty set
[X,G] is set
{X,G} is non empty set
{{X,G},{X}} is non empty set
the Mult of (C_Normed_Algebra_of_BoundedFunctions the carrier of X) . [X,G] is set
[:COMPLEX, the carrier of (CAlgebra the carrier of X):] is non empty set
the Mult of (CAlgebra the carrier of X) is non empty Relation-like [:COMPLEX, the carrier of (CAlgebra the carrier of X):] -defined the carrier of (CAlgebra the carrier of X) -valued Function-like total quasi_total Element of bool [:[:COMPLEX, the carrier of (CAlgebra the carrier of X):], the carrier of (CAlgebra the carrier of X):]
[:[:COMPLEX, the carrier of (CAlgebra the carrier of X):], the carrier of (CAlgebra the carrier of X):] is non empty set
bool [:[:COMPLEX, the carrier of (CAlgebra the carrier of X):], the carrier of (CAlgebra the carrier of X):] is non empty set
the Mult of (CAlgebra the carrier of X) | [:COMPLEX,(X):] is Relation-like [:COMPLEX, the carrier of (CAlgebra the carrier of X):] -defined the carrier of (CAlgebra the carrier of X) -valued Function-like Element of bool [:[:COMPLEX, the carrier of (CAlgebra the carrier of X):], the carrier of (CAlgebra the carrier of X):]
FB is complex Element of COMPLEX
[FB,F] is Element of [:COMPLEX, the carrier of (X):]
{FB,F} is non empty set
{FB} is non empty V176() set
{{FB,F},{FB}} is non empty set
( the Mult of (CAlgebra the carrier of X) | [:COMPLEX,(X):]) . [FB,F] is set
the Mult of (CAlgebra the carrier of X) . [FB,F] is set
the Mult of (CAlgebra the carrier of X) | [:COMPLEX,(ComplexBoundedFunctions the carrier of X):] is Relation-like [:COMPLEX, the carrier of (CAlgebra the carrier of X):] -defined the carrier of (CAlgebra the carrier of X) -valued Function-like Element of bool [:[:COMPLEX, the carrier of (CAlgebra the carrier of X):], the carrier of (CAlgebra the carrier of X):]
[FB,G] is Element of [:COMPLEX, the carrier of (C_Normed_Algebra_of_BoundedFunctions the carrier of X):]
{FB,G} is non empty set
{{FB,G},{FB}} is non empty set
( the Mult of (CAlgebra the carrier of X) | [:COMPLEX,(ComplexBoundedFunctions the carrier of X):]) . [FB,G] is set
X is complex set
abs X is complex real ext-real non negative Element of REAL
X is non empty TopSpace-like TopStruct
(X) is non empty strict CNORMSTR
(X) is non empty linearly-closed Element of bool the carrier of (ComplexVectSpace the carrier of X)
the carrier of X is non empty set
ComplexVectSpace the carrier of X is non empty right_complementable Abelian add-associative right_zeroed strict vector-distributive scalar-distributive scalar-associative scalar-unital CLSStruct
Funcs ( the carrier of X,COMPLEX) is non empty functional FUNCTION_DOMAIN of the carrier of X, COMPLEX
ComplexFuncZero the carrier of X is Relation-like the carrier of X -defined COMPLEX -valued Function-like total quasi_total V139() Element of Funcs ( the carrier of X,COMPLEX)
the carrier of X --> 0 is non empty Relation-like the carrier of X -defined REAL -valued RAT -valued INT -valued Function-like total quasi_total V139() V140() V141() V142() Element of bool [: the carrier of X,REAL:]
[: the carrier of X,REAL:] is non empty V139() V140() V141() set
bool [: the carrier of X,REAL:] is non empty set
ComplexFuncAdd the carrier of X is non empty Relation-like [:(Funcs ( the carrier of X,COMPLEX)),(Funcs ( the carrier of X,COMPLEX)):] -defined Funcs ( the carrier of X,COMPLEX) -valued Function-like total quasi_total Element of bool [:[:(Funcs ( the carrier of X,COMPLEX)),(Funcs ( the carrier of X,COMPLEX)):],(Funcs ( the carrier of X,COMPLEX)):]
[:(Funcs ( the carrier of X,COMPLEX)),(Funcs ( the carrier of X,COMPLEX)):] is non empty set
[:[:(Funcs ( the carrier of X,COMPLEX)),(Funcs ( the carrier of X,COMPLEX)):],(Funcs ( the carrier of X,COMPLEX)):] is non empty set
bool [:[:(Funcs ( the carrier of X,COMPLEX)),(Funcs ( the carrier of X,COMPLEX)):],(Funcs ( the carrier of X,COMPLEX)):] is non empty set
ComplexFuncExtMult the carrier of X is non empty Relation-like [:COMPLEX,(Funcs ( the carrier of X,COMPLEX)):] -defined Funcs ( the carrier of X,COMPLEX) -valued Function-like total quasi_total Element of bool [:[:COMPLEX,(Funcs ( the carrier of X,COMPLEX)):],(Funcs ( the carrier of X,COMPLEX)):]
[:COMPLEX,(Funcs ( the carrier of X,COMPLEX)):] is non empty set
[:[:COMPLEX,(Funcs ( the carrier of X,COMPLEX)):],(Funcs ( the carrier of X,COMPLEX)):] is non empty set
bool [:[:COMPLEX,(Funcs ( the carrier of X,COMPLEX)):],(Funcs ( the carrier of X,COMPLEX)):] is non empty set
CLSStruct(# (Funcs ( the carrier of X,COMPLEX)),(ComplexFuncZero the carrier of X),(ComplexFuncAdd the carrier of X),(ComplexFuncExtMult the carrier of X) #) is strict CLSStruct
the carrier of (ComplexVectSpace the carrier of X) is non empty set
bool the carrier of (ComplexVectSpace the carrier of X) is non empty set
[: the carrier of X,COMPLEX:] is non empty V139() set
bool [: the carrier of X,COMPLEX:] is non empty set
bool the carrier of X is non empty set
{ b₁ where b₁ is non empty Relation-like the carrier of X -defined COMPLEX -valued Function-like total quasi_total V139() Element of bool [: the carrier of X,COMPLEX:] : ( b₁ is (X) & ex b₂ being non empty Element of bool the carrier of X st
( b₂ is compact & ( for b₃ being Element of bool the carrier of X holds
( not b₃ = support b₁ or Cl b₃ is Element of bool b₂ ) ) ) ) } is set
Zero_ ((X),(ComplexVectSpace the carrier of X)) is Element of (X)
Add_ ((X),(ComplexVectSpace the carrier of X)) is non empty Relation-like [:(X),(X):] -defined (X) -valued Function-like total quasi_total Element of bool [:[:(X),(X):],(X):]
[:(X),(X):] is non empty set
[:[:(X),(X):],(X):] is non empty set
bool [:[:(X),(X):],(X):] is non empty set
Mult_ ((X),(ComplexVectSpace the carrier of X)) is non empty Relation-like [:COMPLEX,(X):] -defined (X) -valued Function-like total quasi_total Element of bool [:[:COMPLEX,(X):],(X):]
[:COMPLEX,(X):] is non empty set
[:[:COMPLEX,(X):],(X):] is non empty set
bool [:[:COMPLEX,(X):],(X):] is non empty set
(X) is non empty Relation-like (X) -defined REAL -valued Function-like total quasi_total V139() V140() V141() Element of bool [:(X),REAL:]
[:(X),REAL:] is non empty V139() V140() V141() set
bool [:(X),REAL:] is non empty set
ComplexBoundedFunctions the carrier of X is non empty multiplicatively-closed Cadditively-linearly-closed Element of bool the carrier of (CAlgebra the carrier of X)
CAlgebra the carrier of X is non empty right_complementable Abelian add-associative right_zeroed unital associative commutative right-distributive right_unital well-unital left_unital vector-distributive scalar-distributive scalar-associative scalar-unital strict vector-associative ComplexAlgebraStr
ComplexFuncMult the carrier of X is non empty Relation-like [:(Funcs ( the carrier of X,COMPLEX)),(Funcs ( the carrier of X,COMPLEX)):] -defined Funcs ( the carrier of X,COMPLEX) -valued Function-like total quasi_total Element of bool [:[:(Funcs ( the carrier of X,COMPLEX)),(Funcs ( the carrier of X,COMPLEX)):],(Funcs ( the carrier of X,COMPLEX)):]
ComplexFuncUnit the carrier of X is Relation-like the carrier of X -defined COMPLEX -valued Function-like total quasi_total V139() Element of Funcs ( the carrier of X,COMPLEX)
the carrier of X --> 1r is non empty Relation-like the carrier of X -defined COMPLEX -valued Function-like total quasi_total V139() Element of bool [: the carrier of X,COMPLEX:]
ComplexAlgebraStr(# (Funcs ( the carrier of X,COMPLEX)),(ComplexFuncMult the carrier of X),(ComplexFuncAdd the carrier of X),(ComplexFuncExtMult the carrier of X),(ComplexFuncUnit the carrier of X),(ComplexFuncZero the carrier of X) #) is strict ComplexAlgebraStr
the carrier of (CAlgebra the carrier of X) is non empty set
bool the carrier of (CAlgebra the carrier of X) is non empty set
{ b₁ where b₁ is non empty Relation-like the carrier of X -defined COMPLEX -valued Function-like total quasi_total V139() Element of bool [: the carrier of X,COMPLEX:] : b₁ | the carrier of X is bounded } is set
ComplexBoundedFunctionsNorm the carrier of X is non empty Relation-like ComplexBoundedFunctions the carrier of X -defined REAL -valued Function-like total quasi_total V139() V140() V141() Element of bool [:(ComplexBoundedFunctions the carrier of X),REAL:]
[:(ComplexBoundedFunctions the carrier of X),REAL:] is non empty V139() V140() V141() set
bool [:(ComplexBoundedFunctions the carrier of X),REAL:] is non empty set
(ComplexBoundedFunctionsNorm the carrier of X) | (X) is Relation-like ComplexBoundedFunctions the carrier of X -defined REAL -valued Function-like V139() V140() V141() Element of bool [:(ComplexBoundedFunctions the carrier of X),REAL:]
CNORMSTR(# (X),(Zero_ ((X),(ComplexVectSpace the carrier of X))),(Add_ ((X),(ComplexVectSpace the carrier of X))),(Mult_ ((X),(ComplexVectSpace the carrier of X))),(X) #) is strict CNORMSTR
the carrier of (X) is non empty set
0. (X) is zero Element of the carrier of (X)
the ZeroF of (X) is Element of the carrier of (X)
F is Element of the carrier of (X)
||.F.|| is complex real ext-real Element of REAL
the U9 of (X) is non empty Relation-like the carrier of (X) -defined REAL -valued Function-like total quasi_total V139() V140() V141() Element of bool [: the carrier of (X),REAL:]
[: the carrier of (X),REAL:] is non empty V139() V140() V141() set
bool [: the carrier of (X),REAL:] is non empty set
the U9 of (X) . F is complex real ext-real Element of REAL
X * F is Element of the carrier of (X)
the Mult of (X) is non empty Relation-like [:COMPLEX, the carrier of (X):] -defined the carrier of (X) -valued Function-like total quasi_total Element of bool [:[:COMPLEX, the carrier of (X):], the carrier of (X):]
[:COMPLEX, the carrier of (X):] is non empty set
[:[:COMPLEX, the carrier of (X):], the carrier of (X):] is non empty set
bool [:[:COMPLEX, the carrier of (X):], the carrier of (X):] is non empty set
[X,F] is set
{X,F} is non empty set
{X} is non empty V176() set
{{X,F},{X}} is non empty set
the Mult of (X) . [X,F] is set
||.(X * F).|| is complex real ext-real Element of REAL
the U9 of (X) . (X * F) is complex real ext-real Element of REAL
(abs X) * ||.F.|| is complex real ext-real Element of REAL
G is Element of the carrier of (X)
F + G is Element of the carrier of (X)
the addF of (X) is non empty Relation-like [: the carrier of (X), the carrier of (X):] -defined the carrier of (X) -valued Function-like total quasi_total Element of bool [:[: the carrier of (X), the carrier of (X):], the carrier of (X):]
[: the carrier of (X), the carrier of (X):] is non empty set
[:[: the carrier of (X), the carrier of (X):], the carrier of (X):] is non empty set
bool [:[: the carrier of (X), the carrier of (X):], the carrier of (X):] is non empty set
the addF of (X) . (F,G) is Element of the carrier of (X)
[F,G] is set
{F,G} is non empty set
{F} is non empty set
{{F,G},{F}} is non empty set
the addF of (X) . [F,G] is set
||.(F + G).|| is complex real ext-real Element of REAL
the U9 of (X) . (F + G) is complex real ext-real Element of REAL
||.G.|| is complex real ext-real Element of REAL
the U9 of (X) . G is complex real ext-real Element of REAL
||.F.|| + ||.G.|| is complex real ext-real Element of REAL
C_Normed_Algebra_of_BoundedFunctions the carrier of X is non empty right_complementable Abelian add-associative right_zeroed unital discerning reflexive vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like complete Normed_Complex_AlgebraStr
mult_ ((ComplexBoundedFunctions the carrier of X),(CAlgebra the carrier of X)) is non empty Relation-like [:(ComplexBoundedFunctions the carrier of X),(ComplexBoundedFunctions the carrier of X):] -defined ComplexBoundedFunctions the carrier of X -valued Function-like total quasi_total Element of bool [:[:(ComplexBoundedFunctions the carrier of X),(ComplexBoundedFunctions the carrier of X):],(ComplexBoundedFunctions the carrier of X):]
[:(ComplexBoundedFunctions the carrier of X),(ComplexBoundedFunctions the carrier of X):] is non empty set
[:[:(ComplexBoundedFunctions the carrier of X),(ComplexBoundedFunctions the carrier of X):],(ComplexBoundedFunctions the carrier of X):] is non empty set
bool [:[:(ComplexBoundedFunctions the carrier of X),(ComplexBoundedFunctions the carrier of X):],(ComplexBoundedFunctions the carrier of X):] is non empty set
Add_ ((ComplexBoundedFunctions the carrier of X),(CAlgebra the carrier of X)) is non empty Relation-like [:(ComplexBoundedFunctions the carrier of X),(ComplexBoundedFunctions the carrier of X):] -defined ComplexBoundedFunctions the carrier of X -valued Function-like total quasi_total Element of bool [:[:(ComplexBoundedFunctions the carrier of X),(ComplexBoundedFunctions the carrier of X):],(ComplexBoundedFunctions the carrier of X):]
Mult_ ((ComplexBoundedFunctions the carrier of X),(CAlgebra the carrier of X)) is non empty Relation-like [:COMPLEX,(ComplexBoundedFunctions the carrier of X):] -defined ComplexBoundedFunctions the carrier of X -valued Function-like total quasi_total Element of bool [:[:COMPLEX,(ComplexBoundedFunctions the carrier of X):],(ComplexBoundedFunctions the carrier of X):]
[:COMPLEX,(ComplexBoundedFunctions the carrier of X):] is non empty set
[:[:COMPLEX,(ComplexBoundedFunctions the carrier of X):],(ComplexBoundedFunctions the carrier of X):] is non empty set
bool [:[:COMPLEX,(ComplexBoundedFunctions the carrier of X):],(ComplexBoundedFunctions the carrier of X):] is non empty set
One_ ((ComplexBoundedFunctions the carrier of X),(CAlgebra the carrier of X)) is Element of ComplexBoundedFunctions the carrier of X
Zero_ ((ComplexBoundedFunctions the carrier of X),(CAlgebra the carrier of X)) is Element of ComplexBoundedFunctions the carrier of X
Normed_Complex_AlgebraStr(# (ComplexBoundedFunctions the carrier of X),(mult_ ((ComplexBoundedFunctions the carrier of X),(CAlgebra the carrier of X))),(Add_ ((ComplexBoundedFunctions the carrier of X),(CAlgebra the carrier of X))),(Mult_ ((ComplexBoundedFunctions the carrier of X),(CAlgebra the carrier of X))),(One_ ((ComplexBoundedFunctions the carrier of X),(CAlgebra the carrier of X))),(Zero_ ((ComplexBoundedFunctions the carrier of X),(CAlgebra the carrier of X))),(ComplexBoundedFunctionsNorm the carrier of X) #) is strict Normed_Complex_AlgebraStr
the carrier of (C_Normed_Algebra_of_BoundedFunctions the carrier of X) is non empty set
FB is Element of the carrier of (C_Normed_Algebra_of_BoundedFunctions the carrier of X)
||.FB.|| is complex real ext-real Element of REAL
the U9 of (C_Normed_Algebra_of_BoundedFunctions the carrier of X) is non empty Relation-like the carrier of (C_Normed_Algebra_of_BoundedFunctions the carrier of X) -defined REAL -valued Function-like total quasi_total V139() V140() V141() Element of bool [: the carrier of (C_Normed_Algebra_of_BoundedFunctions the carrier of X),REAL:]
[: the carrier of (C_Normed_Algebra_of_BoundedFunctions the carrier of X),REAL:] is non empty V139() V140() V141() set
bool [: the carrier of (C_Normed_Algebra_of_BoundedFunctions the carrier of X),REAL:] is non empty set
the U9 of (C_Normed_Algebra_of_BoundedFunctions the carrier of X) . FB is complex real ext-real Element of REAL
0. (C_Normed_Algebra_of_BoundedFunctions the carrier of X) is zero Element of the carrier of (C_Normed_Algebra_of_BoundedFunctions the carrier of X)
the ZeroF of (C_Normed_Algebra_of_BoundedFunctions the carrier of X) is Element of the carrier of (C_Normed_Algebra_of_BoundedFunctions the carrier of X)
(X,0) is non empty Relation-like the carrier of X -defined COMPLEX -valued Function-like total quasi_total V139() (X) Element of bool [: the carrier of X,COMPLEX:]
the carrier of X --> 0 is non empty Relation-like the carrier of X -defined RAT -valued INT -valued {0} -valued Function-like total quasi_total V139() V140() V141() V142() Element of bool [: the carrier of X,{0}:]
{0} is non empty V176() V177() V178() V179() V180() V181() left_end bounded_below set
[: the carrier of X,{0}:] is non empty RAT -valued INT -valued V139() V140() V141() V142() set
bool [: the carrier of X,{0}:] is non empty set
C_Normed_Algebra_of_BoundedFunctions the carrier of X is non empty right_complementable Abelian add-associative right_zeroed unital discerning reflexive vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like complete Normed_Complex_AlgebraStr
mult_ ((ComplexBoundedFunctions the carrier of X),(CAlgebra the carrier of X)) is non empty Relation-like [:(ComplexBoundedFunctions the carrier of X),(ComplexBoundedFunctions the carrier of X):] -defined ComplexBoundedFunctions the carrier of X -valued Function-like total quasi_total Element of bool [:[:(ComplexBoundedFunctions the carrier of X),(ComplexBoundedFunctions the carrier of X):],(ComplexBoundedFunctions the carrier of X):]
[:(ComplexBoundedFunctions the carrier of X),(ComplexBoundedFunctions the carrier of X):] is non empty set
[:[:(ComplexBoundedFunctions the carrier of X),(ComplexBoundedFunctions the carrier of X):],(ComplexBoundedFunctions the carrier of X):] is non empty set
bool [:[:(ComplexBoundedFunctions the carrier of X),(ComplexBoundedFunctions the carrier of X):],(ComplexBoundedFunctions the carrier of X):] is non empty set
Add_ ((ComplexBoundedFunctions the carrier of X),(CAlgebra the carrier of X)) is non empty Relation-like [:(ComplexBoundedFunctions the carrier of X),(ComplexBoundedFunctions the carrier of X):] -defined ComplexBoundedFunctions the carrier of X -valued Function-like total quasi_total Element of bool [:[:(ComplexBoundedFunctions the carrier of X),(ComplexBoundedFunctions the carrier of X):],(ComplexBoundedFunctions the carrier of X):]
Mult_ ((ComplexBoundedFunctions the carrier of X),(CAlgebra the carrier of X)) is non empty Relation-like [:COMPLEX,(ComplexBoundedFunctions the carrier of X):] -defined ComplexBoundedFunctions the carrier of X -valued Function-like total quasi_total Element of bool [:[:COMPLEX,(ComplexBoundedFunctions the carrier of X):],(ComplexBoundedFunctions the carrier of X):]
[:COMPLEX,(ComplexBoundedFunctions the carrier of X):] is non empty set
[:[:COMPLEX,(ComplexBoundedFunctions the carrier of X):],(ComplexBoundedFunctions the carrier of X):] is non empty set
bool [:[:COMPLEX,(ComplexBoundedFunctions the carrier of X):],(ComplexBoundedFunctions the carrier of X):] is non empty set
One_ ((ComplexBoundedFunctions the carrier of X),(CAlgebra the carrier of X)) is Element of ComplexBoundedFunctions the carrier of X
Zero_ ((ComplexBoundedFunctions the carrier of X),(CAlgebra the carrier of X)) is Element of ComplexBoundedFunctions the carrier of X
Normed_Complex_AlgebraStr(# (ComplexBoundedFunctions the carrier of X),(mult_ ((ComplexBoundedFunctions the carrier of X),(CAlgebra the carrier of X))),(Add_ ((ComplexBoundedFunctions the carrier of X),(CAlgebra the carrier of X))),(Mult_ ((ComplexBoundedFunctions the carrier of X),(CAlgebra the carrier of X))),(One_ ((ComplexBoundedFunctions the carrier of X),(CAlgebra the carrier of X))),(Zero_ ((ComplexBoundedFunctions the carrier of X),(CAlgebra the carrier of X))),(ComplexBoundedFunctionsNorm the carrier of X) #) is strict Normed_Complex_AlgebraStr
the carrier of (C_Normed_Algebra_of_BoundedFunctions the carrier of X) is non empty set
FB is Element of the carrier of (C_Normed_Algebra_of_BoundedFunctions the carrier of X)
||.FB.|| is complex real ext-real Element of REAL
the U9 of (C_Normed_Algebra_of_BoundedFunctions the carrier of X) is non empty Relation-like the carrier of (C_Normed_Algebra_of_BoundedFunctions the carrier of X) -defined REAL -valued Function-like total quasi_total V139() V140() V141() Element of bool [: the carrier of (C_Normed_Algebra_of_BoundedFunctions the carrier of X),REAL:]
[: the carrier of (C_Normed_Algebra_of_BoundedFunctions the carrier of X),REAL:] is non empty V139() V140() V141() set
bool [: the carrier of (C_Normed_Algebra_of_BoundedFunctions the carrier of X),REAL:] is non empty set
the U9 of (C_Normed_Algebra_of_BoundedFunctions the carrier of X) . FB is complex real ext-real Element of REAL
X * FB is Element of the carrier of (C_Normed_Algebra_of_BoundedFunctions the carrier of X)
the Mult of (C_Normed_Algebra_of_BoundedFunctions the carrier of X) is non empty Relation-like [:COMPLEX, the carrier of (C_Normed_Algebra_of_BoundedFunctions the carrier of X):] -defined the carrier of (C_Normed_Algebra_of_BoundedFunctions the carrier of X) -valued Function-like total quasi_total Element of bool [:[:COMPLEX, the carrier of (C_Normed_Algebra_of_BoundedFunctions the carrier of X):], the carrier of (C_Normed_Algebra_of_BoundedFunctions the carrier of X):]
[:COMPLEX, the carrier of (C_Normed_Algebra_of_BoundedFunctions the carrier of X):] is non empty set
[:[:COMPLEX, the carrier of (C_Normed_Algebra_of_BoundedFunctions the carrier of X):], the carrier of (C_Normed_Algebra_of_BoundedFunctions the carrier of X):] is non empty set
bool [:[:COMPLEX, the carrier of (C_Normed_Algebra_of_BoundedFunctions the carrier of X):], the carrier of (C_Normed_Algebra_of_BoundedFunctions the carrier of X):] is non empty set
[X,FB] is set
{X,FB} is non empty set
{{X,FB},{X}} is non empty set
the Mult of (C_Normed_Algebra_of_BoundedFunctions the carrier of X) . [X,FB] is set
GB is Element of the carrier of (C_Normed_Algebra_of_BoundedFunctions the carrier of X)
||.GB.|| is complex real ext-real Element of REAL
the U9 of (C_Normed_Algebra_of_BoundedFunctions the carrier of X) . GB is complex real ext-real Element of REAL
aFB is Element of the carrier of (X)
||.aFB.|| is complex real ext-real Element of REAL
the U9 of (X) . aFB is complex real ext-real Element of REAL
||.(X * FB).|| is complex real ext-real Element of REAL
the U9 of (C_Normed_Algebra_of_BoundedFunctions the carrier of X) . (X * FB) is complex real ext-real Element of REAL
(abs X) * ||.FB.|| is complex real ext-real Element of REAL
C_Normed_Algebra_of_BoundedFunctions the carrier of X is non empty right_complementable Abelian add-associative right_zeroed unital discerning reflexive vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like complete Normed_Complex_AlgebraStr
mult_ ((ComplexBoundedFunctions the carrier of X),(CAlgebra the carrier of X)) is non empty Relation-like [:(ComplexBoundedFunctions the carrier of X),(ComplexBoundedFunctions the carrier of X):] -defined ComplexBoundedFunctions the carrier of X -valued Function-like total quasi_total Element of bool [:[:(ComplexBoundedFunctions the carrier of X),(ComplexBoundedFunctions the carrier of X):],(ComplexBoundedFunctions the carrier of X):]
[:(ComplexBoundedFunctions the carrier of X),(ComplexBoundedFunctions the carrier of X):] is non empty set
[:[:(ComplexBoundedFunctions the carrier of X),(ComplexBoundedFunctions the carrier of X):],(ComplexBoundedFunctions the carrier of X):] is non empty set
bool [:[:(ComplexBoundedFunctions the carrier of X),(ComplexBoundedFunctions the carrier of X):],(ComplexBoundedFunctions the carrier of X):] is non empty set
Add_ ((ComplexBoundedFunctions the carrier of X),(CAlgebra the carrier of X)) is non empty Relation-like [:(ComplexBoundedFunctions the carrier of X),(ComplexBoundedFunctions the carrier of X):] -defined ComplexBoundedFunctions the carrier of X -valued Function-like total quasi_total Element of bool [:[:(ComplexBoundedFunctions the carrier of X),(ComplexBoundedFunctions the carrier of X):],(ComplexBoundedFunctions the carrier of X):]
Mult_ ((ComplexBoundedFunctions the carrier of X),(CAlgebra the carrier of X)) is non empty Relation-like [:COMPLEX,(ComplexBoundedFunctions the carrier of X):] -defined ComplexBoundedFunctions the carrier of X -valued Function-like total quasi_total Element of bool [:[:COMPLEX,(ComplexBoundedFunctions the carrier of X):],(ComplexBoundedFunctions the carrier of X):]
[:COMPLEX,(ComplexBoundedFunctions the carrier of X):] is non empty set
[:[:COMPLEX,(ComplexBoundedFunctions the carrier of X):],(ComplexBoundedFunctions the carrier of X):] is non empty set
bool [:[:COMPLEX,(ComplexBoundedFunctions the carrier of X):],(ComplexBoundedFunctions the carrier of X):] is non empty set
One_ ((ComplexBoundedFunctions the carrier of X),(CAlgebra the carrier of X)) is Element of ComplexBoundedFunctions the carrier of X
Zero_ ((ComplexBoundedFunctions the carrier of X),(CAlgebra the carrier of X)) is Element of ComplexBoundedFunctions the carrier of X
Normed_Complex_AlgebraStr(# (ComplexBoundedFunctions the carrier of X),(mult_ ((ComplexBoundedFunctions the carrier of X),(CAlgebra the carrier of X))),(Add_ ((ComplexBoundedFunctions the carrier of X),(CAlgebra the carrier of X))),(Mult_ ((ComplexBoundedFunctions the carrier of X),(CAlgebra the carrier of X))),(One_ ((ComplexBoundedFunctions the carrier of X),(CAlgebra the carrier of X))),(Zero_ ((ComplexBoundedFunctions the carrier of X),(CAlgebra the carrier of X))),(ComplexBoundedFunctionsNorm the carrier of X) #) is strict Normed_Complex_AlgebraStr
the carrier of (C_Normed_Algebra_of_BoundedFunctions the carrier of X) is non empty set
FB is Element of the carrier of (C_Normed_Algebra_of_BoundedFunctions the carrier of X)
||.FB.|| is complex real ext-real Element of REAL
the U9 of (C_Normed_Algebra_of_BoundedFunctions the carrier of X) is non empty Relation-like the carrier of (C_Normed_Algebra_of_BoundedFunctions the carrier of X) -defined REAL -valued Function-like total quasi_total V139() V140() V141() Element of bool [: the carrier of (C_Normed_Algebra_of_BoundedFunctions the carrier of X),REAL:]
[: the carrier of (C_Normed_Algebra_of_BoundedFunctions the carrier of X),REAL:] is non empty V139() V140() V141() set
bool [: the carrier of (C_Normed_Algebra_of_BoundedFunctions the carrier of X),REAL:] is non empty set
the U9 of (C_Normed_Algebra_of_BoundedFunctions the carrier of X) . FB is complex real ext-real Element of REAL
GB is Element of the carrier of (C_Normed_Algebra_of_BoundedFunctions the carrier of X)
||.GB.|| is complex real ext-real Element of REAL
the U9 of (C_Normed_Algebra_of_BoundedFunctions the carrier of X) . GB is complex real ext-real Element of REAL
FB + GB is Element of the carrier of (C_Normed_Algebra_of_BoundedFunctions the carrier of X)
the addF of (C_Normed_Algebra_of_BoundedFunctions the carrier of X) is non empty Relation-like [: the carrier of (C_Normed_Algebra_of_BoundedFunctions the carrier of X), the carrier of (C_Normed_Algebra_of_BoundedFunctions the carrier of X):] -defined the carrier of (C_Normed_Algebra_of_BoundedFunctions the carrier of X) -valued Function-like total quasi_total Element of bool [:[: the carrier of (C_Normed_Algebra_of_BoundedFunctions the carrier of X), the carrier of (C_Normed_Algebra_of_BoundedFunctions the carrier of X):], the carrier of (C_Normed_Algebra_of_BoundedFunctions the carrier of X):]
[: the carrier of (C_Normed_Algebra_of_BoundedFunctions the carrier of X), the carrier of (C_Normed_Algebra_of_BoundedFunctions the carrier of X):] is non empty set
[:[: the carrier of (C_Normed_Algebra_of_BoundedFunctions the carrier of X), the carrier of (C_Normed_Algebra_of_BoundedFunctions the carrier of X):], the carrier of (C_Normed_Algebra_of_BoundedFunctions the carrier of X):] is non empty set
bool [:[: the carrier of (C_Normed_Algebra_of_BoundedFunctions the carrier of X), the carrier of (C_Normed_Algebra_of_BoundedFunctions the carrier of X):], the carrier of (C_Normed_Algebra_of_BoundedFunctions the carrier of X):] is non empty set
the addF of (C_Normed_Algebra_of_BoundedFunctions the carrier of X) . (FB,GB) is Element of the carrier of (C_Normed_Algebra_of_BoundedFunctions the carrier of X)
[FB,GB] is set
{FB,GB} is non empty set
{FB} is non empty set
{{FB,GB},{FB}} is non empty set
the addF of (C_Normed_Algebra_of_BoundedFunctions the carrier of X) . [FB,GB] is set
||.(FB + GB).|| is complex real ext-real Element of REAL
the U9 of (C_Normed_Algebra_of_BoundedFunctions the carrier of X) . (FB + GB) is complex real ext-real Element of REAL
||.FB.|| + ||.GB.|| is complex real ext-real Element of REAL
X is non empty TopSpace-like TopStruct
(X) is non empty strict CNORMSTR
(X) is non empty linearly-closed Element of bool the carrier of (ComplexVectSpace the carrier of X)
the carrier of X is non empty set
ComplexVectSpace the carrier of X is non empty right_complementable Abelian add-associative right_zeroed strict vector-distributive scalar-distributive scalar-associative scalar-unital CLSStruct
Funcs ( the carrier of X,COMPLEX) is non empty functional FUNCTION_DOMAIN of the carrier of X, COMPLEX
ComplexFuncZero the carrier of X is Relation-like the carrier of X -defined COMPLEX -valued Function-like total quasi_total V139() Element of Funcs ( the carrier of X,COMPLEX)
the carrier of X --> 0 is non empty Relation-like the carrier of X -defined REAL -valued RAT -valued INT -valued Function-like total quasi_total V139() V140() V141() V142() Element of bool [: the carrier of X,REAL:]
[: the carrier of X,REAL:] is non empty V139() V140() V141() set
bool [: the carrier of X,REAL:] is non empty set
ComplexFuncAdd the carrier of X is non empty Relation-like [:(Funcs ( the carrier of X,COMPLEX)),(Funcs ( the carrier of X,COMPLEX)):] -defined Funcs ( the carrier of X,COMPLEX) -valued Function-like total quasi_total Element of bool [:[:(Funcs ( the carrier of X,COMPLEX)),(Funcs ( the carrier of X,COMPLEX)):],(Funcs ( the carrier of X,COMPLEX)):]
[:(Funcs ( the carrier of X,COMPLEX)),(Funcs ( the carrier of X,COMPLEX)):] is non empty set
[:[:(Funcs ( the carrier of X,COMPLEX)),(Funcs ( the carrier of X,COMPLEX)):],(Funcs ( the carrier of X,COMPLEX)):] is non empty set
bool [:[:(Funcs ( the carrier of X,COMPLEX)),(Funcs ( the carrier of X,COMPLEX)):],(Funcs ( the carrier of X,COMPLEX)):] is non empty set
ComplexFuncExtMult the carrier of X is non empty Relation-like [:COMPLEX,(Funcs ( the carrier of X,COMPLEX)):] -defined Funcs ( the carrier of X,COMPLEX) -valued Function-like total quasi_total Element of bool [:[:COMPLEX,(Funcs ( the carrier of X,COMPLEX)):],(Funcs ( the carrier of X,COMPLEX)):]
[:COMPLEX,(Funcs ( the carrier of X,COMPLEX)):] is non empty set
[:[:COMPLEX,(Funcs ( the carrier of X,COMPLEX)):],(Funcs ( the carrier of X,COMPLEX)):] is non empty set
bool [:[:COMPLEX,(Funcs ( the carrier of X,COMPLEX)):],(Funcs ( the carrier of X,COMPLEX)):] is non empty set
CLSStruct(# (Funcs ( the carrier of X,COMPLEX)),(ComplexFuncZero the carrier of X),(ComplexFuncAdd the carrier of X),(ComplexFuncExtMult the carrier of X) #) is strict CLSStruct
the carrier of (ComplexVectSpace the carrier of X) is non empty set
bool the carrier of (ComplexVectSpace the carrier of X) is non empty set
[: the carrier of X,COMPLEX:] is non empty V139() set
bool [: the carrier of X,COMPLEX:] is non empty set
bool the carrier of X is non empty set
{ b₁ where b₁ is non empty Relation-like the carrier of X -defined COMPLEX -valued Function-like total quasi_total V139() Element of bool [: the carrier of X,COMPLEX:] : ( b₁ is (X) & ex b₂ being non empty Element of bool the carrier of X st
( b₂ is compact & ( for b₃ being Element of bool the carrier of X holds
( not b₃ = support b₁ or Cl b₃ is Element of bool b₂ ) ) ) ) } is set
Zero_ ((X),(ComplexVectSpace the carrier of X)) is Element of (X)
Add_ ((X),(ComplexVectSpace the carrier of X)) is non empty Relation-like [:(X),(X):] -defined (X) -valued Function-like total quasi_total Element of bool [:[:(X),(X):],(X):]
[:(X),(X):] is non empty set
[:[:(X),(X):],(X):] is non empty set
bool [:[:(X),(X):],(X):] is non empty set
Mult_ ((X),(ComplexVectSpace the carrier of X)) is non empty Relation-like [:COMPLEX,(X):] -defined (X) -valued Function-like total quasi_total Element of bool [:[:COMPLEX,(X):],(X):]
[:COMPLEX,(X):] is non empty set
[:[:COMPLEX,(X):],(X):] is non empty set
bool [:[:COMPLEX,(X):],(X):] is non empty set
(X) is non empty Relation-like (X) -defined REAL -valued Function-like total quasi_total V139() V140() V141() Element of bool [:(X),REAL:]
[:(X),REAL:] is non empty V139() V140() V141() set
bool [:(X),REAL:] is non empty set
ComplexBoundedFunctions the carrier of X is non empty multiplicatively-closed Cadditively-linearly-closed Element of bool the carrier of (CAlgebra the carrier of X)
CAlgebra the carrier of X is non empty right_complementable Abelian add-associative right_zeroed unital associative commutative right-distributive right_unital well-unital left_unital vector-distributive scalar-distributive scalar-associative scalar-unital strict vector-associative ComplexAlgebraStr
ComplexFuncMult the carrier of X is non empty Relation-like [:(Funcs ( the carrier of X,COMPLEX)),(Funcs ( the carrier of X,COMPLEX)):] -defined Funcs ( the carrier of X,COMPLEX) -valued Function-like total quasi_total Element of bool [:[:(Funcs ( the carrier of X,COMPLEX)),(Funcs ( the carrier of X,COMPLEX)):],(Funcs ( the carrier of X,COMPLEX)):]
ComplexFuncUnit the carrier of X is Relation-like the carrier of X -defined COMPLEX -valued Function-like total quasi_total V139() Element of Funcs ( the carrier of X,COMPLEX)
the carrier of X --> 1r is non empty Relation-like the carrier of X -defined COMPLEX -valued Function-like total quasi_total V139() Element of bool [: the carrier of X,COMPLEX:]
ComplexAlgebraStr(# (Funcs ( the carrier of X,COMPLEX)),(ComplexFuncMult the carrier of X),(ComplexFuncAdd the carrier of X),(ComplexFuncExtMult the carrier of X),(ComplexFuncUnit the carrier of X),(ComplexFuncZero the carrier of X) #) is strict ComplexAlgebraStr
the carrier of (CAlgebra the carrier of X) is non empty set
bool the carrier of (CAlgebra the carrier of X) is non empty set
{ b₁ where b₁ is non empty Relation-like the carrier of X -defined COMPLEX -valued Function-like total quasi_total V139() Element of bool [: the carrier of X,COMPLEX:] : b₁ | the carrier of X is bounded } is set
ComplexBoundedFunctionsNorm the carrier of X is non empty Relation-like ComplexBoundedFunctions the carrier of X -defined REAL -valued Function-like total quasi_total V139() V140() V141() Element of bool [:(ComplexBoundedFunctions the carrier of X),REAL:]
[:(ComplexBoundedFunctions the carrier of X),REAL:] is non empty V139() V140() V141() set
bool [:(ComplexBoundedFunctions the carrier of X),REAL:] is non empty set
(ComplexBoundedFunctionsNorm the carrier of X) | (X) is Relation-like ComplexBoundedFunctions the carrier of X -defined REAL -valued Function-like V139() V140() V141() Element of bool [:(ComplexBoundedFunctions the carrier of X),REAL:]
CNORMSTR(# (X),(Zero_ ((X),(ComplexVectSpace the carrier of X))),(Add_ ((X),(ComplexVectSpace the carrier of X))),(Mult_ ((X),(ComplexVectSpace the carrier of X))),(X) #) is strict CNORMSTR
the carrier of (X) is non empty set
X is Element of the carrier of (X)
F is complex set
F * X is Element of the carrier of (X)
the Mult of (X) is non empty Relation-like [:COMPLEX, the carrier of (X):] -defined the carrier of (X) -valued Function-like total quasi_total Element of bool [:[:COMPLEX, the carrier of (X):], the carrier of (X):]
[:COMPLEX, the carrier of (X):] is non empty set
[:[:COMPLEX, the carrier of (X):], the carrier of (X):] is non empty set
bool [:[:COMPLEX, the carrier of (X):], the carrier of (X):] is non empty set
[F,X] is set
{F,X} is non empty set
{F} is non empty V176() set
{{F,X},{F}} is non empty set
the Mult of (X) . [F,X] is set
||.(F * X).|| is complex real ext-real Element of REAL
the U9 of (X) is non empty Relation-like the carrier of (X) -defined REAL -valued Function-like total quasi_total V139() V140() V141() Element of bool [: the carrier of (X),REAL:]
[: the carrier of (X),REAL:] is non empty V139() V140() V141() set
bool [: the carrier of (X),REAL:] is non empty set
the U9 of (X) . (F * X) is complex real ext-real Element of REAL
abs F is complex real ext-real non negative Element of REAL
||.X.|| is complex real ext-real Element of REAL
the U9 of (X) . X is complex real ext-real Element of REAL
(abs F) * ||.X.|| is complex real ext-real Element of REAL
G is Element of the carrier of (X)
FB is Element of the carrier of (X)
G + FB is Element of the carrier of (X)
the addF of (X) is non empty Relation-like [: the carrier of (X), the carrier of (X):] -defined the carrier of (X) -valued Function-like total quasi_total Element of bool [:[: the carrier of (X), the carrier of (X):], the carrier of (X):]
[: the carrier of (X), the carrier of (X):] is non empty set
[:[: the carrier of (X), the carrier of (X):], the carrier of (X):] is non empty set
bool [:[: the carrier of (X), the carrier of (X):], the carrier of (X):] is non empty set
the addF of (X) . (G,FB) is Element of the carrier of (X)
[G,FB] is set
{G,FB} is non empty set
{G} is non empty set
{{G,FB},{G}} is non empty set
the addF of (X) . [G,FB] is set
||.(G + FB).|| is complex real ext-real Element of REAL
the U9 of (X) . (G + FB) is complex real ext-real Element of REAL
||.G.|| is complex real ext-real Element of REAL
the U9 of (X) . G is complex real ext-real Element of REAL
||.FB.|| is complex real ext-real Element of REAL
the U9 of (X) . FB is complex real ext-real Element of REAL
||.G.|| + ||.FB.|| is complex real ext-real Element of REAL
X is non empty TopSpace-like TopStruct
(X) is non empty strict CNORMSTR
(X) is non empty linearly-closed Element of bool the carrier of (ComplexVectSpace the carrier of X)
the carrier of X is non empty set
ComplexVectSpace the carrier of X is non empty right_complementable Abelian add-associative right_zeroed strict vector-distributive scalar-distributive scalar-associative scalar-unital CLSStruct
Funcs ( the carrier of X,COMPLEX) is non empty functional FUNCTION_DOMAIN of the carrier of X, COMPLEX
ComplexFuncZero the carrier of X is Relation-like the carrier of X -defined COMPLEX -valued Function-like total quasi_total V139() Element of Funcs ( the carrier of X,COMPLEX)
the carrier of X --> 0 is non empty Relation-like the carrier of X -defined REAL -valued RAT -valued INT -valued Function-like total quasi_total V139() V140() V141() V142() Element of bool [: the carrier of X,REAL:]
[: the carrier of X,REAL:] is non empty V139() V140() V141() set
bool [: the carrier of X,REAL:] is non empty set
ComplexFuncAdd the carrier of X is non empty Relation-like [:(Funcs ( the carrier of X,COMPLEX)),(Funcs ( the carrier of X,COMPLEX)):] -defined Funcs ( the carrier of X,COMPLEX) -valued Function-like total quasi_total Element of bool [:[:(Funcs ( the carrier of X,COMPLEX)),(Funcs ( the carrier of X,COMPLEX)):],(Funcs ( the carrier of X,COMPLEX)):]
[:(Funcs ( the carrier of X,COMPLEX)),(Funcs ( the carrier of X,COMPLEX)):] is non empty set
[:[:(Funcs ( the carrier of X,COMPLEX)),(Funcs ( the carrier of X,COMPLEX)):],(Funcs ( the carrier of X,COMPLEX)):] is non empty set
bool [:[:(Funcs ( the carrier of X,COMPLEX)),(Funcs ( the carrier of X,COMPLEX)):],(Funcs ( the carrier of X,COMPLEX)):] is non empty set
ComplexFuncExtMult the carrier of X is non empty Relation-like [:COMPLEX,(Funcs ( the carrier of X,COMPLEX)):] -defined Funcs ( the carrier of X,COMPLEX) -valued Function-like total quasi_total Element of bool [:[:COMPLEX,(Funcs ( the carrier of X,COMPLEX)):],(Funcs ( the carrier of X,COMPLEX)):]
[:COMPLEX,(Funcs ( the carrier of X,COMPLEX)):] is non empty set
[:[:COMPLEX,(Funcs ( the carrier of X,COMPLEX)):],(Funcs ( the carrier of X,COMPLEX)):] is non empty set
bool [:[:COMPLEX,(Funcs ( the carrier of X,COMPLEX)):],(Funcs ( the carrier of X,COMPLEX)):] is non empty set
CLSStruct(# (Funcs ( the carrier of X,COMPLEX)),(ComplexFuncZero the carrier of X),(ComplexFuncAdd the carrier of X),(ComplexFuncExtMult the carrier of X) #) is strict CLSStruct
the carrier of (ComplexVectSpace the carrier of X) is non empty set
bool the carrier of (ComplexVectSpace the carrier of X) is non empty set
[: the carrier of X,COMPLEX:] is non empty V139() set
bool [: the carrier of X,COMPLEX:] is non empty set
bool the carrier of X is non empty set
{ b₁ where b₁ is non empty Relation-like the carrier of X -defined COMPLEX -valued Function-like total quasi_total V139() Element of bool [: the carrier of X,COMPLEX:] : ( b₁ is (X) & ex b₂ being non empty Element of bool the carrier of X st
( b₂ is compact & ( for b₃ being Element of bool the carrier of X holds
( not b₃ = support b₁ or Cl b₃ is Element of bool b₂ ) ) ) ) } is set
Zero_ ((X),(ComplexVectSpace the carrier of X)) is Element of (X)
Add_ ((X),(ComplexVectSpace the carrier of X)) is non empty Relation-like [:(X),(X):] -defined (X) -valued Function-like total quasi_total Element of bool [:[:(X),(X):],(X):]
[:(X),(X):] is non empty set
[:[:(X),(X):],(X):] is non empty set
bool [:[:(X),(X):],(X):] is non empty set
Mult_ ((X),(ComplexVectSpace the carrier of X)) is non empty Relation-like [:COMPLEX,(X):] -defined (X) -valued Function-like total quasi_total Element of bool [:[:COMPLEX,(X):],(X):]
[:COMPLEX,(X):] is non empty set
[:[:COMPLEX,(X):],(X):] is non empty set
bool [:[:COMPLEX,(X):],(X):] is non empty set
(X) is non empty Relation-like (X) -defined REAL -valued Function-like total quasi_total V139() V140() V141() Element of bool [:(X),REAL:]
[:(X),REAL:] is non empty V139() V140() V141() set
bool [:(X),REAL:] is non empty set
ComplexBoundedFunctions the carrier of X is non empty multiplicatively-closed Cadditively-linearly-closed Element of bool the carrier of (CAlgebra the carrier of X)
CAlgebra the carrier of X is non empty right_complementable Abelian add-associative right_zeroed unital associative commutative right-distributive right_unital well-unital left_unital vector-distributive scalar-distributive scalar-associative scalar-unital strict vector-associative ComplexAlgebraStr
ComplexFuncMult the carrier of X is non empty Relation-like [:(Funcs ( the carrier of X,COMPLEX)),(Funcs ( the carrier of X,COMPLEX)):] -defined Funcs ( the carrier of X,COMPLEX) -valued Function-like total quasi_total Element of bool [:[:(Funcs ( the carrier of X,COMPLEX)),(Funcs ( the carrier of X,COMPLEX)):],(Funcs ( the carrier of X,COMPLEX)):]
ComplexFuncUnit the carrier of X is Relation-like the carrier of X -defined COMPLEX -valued Function-like total quasi_total V139() Element of Funcs ( the carrier of X,COMPLEX)
the carrier of X --> 1r is non empty Relation-like the carrier of X -defined COMPLEX -valued Function-like total quasi_total V139() Element of bool [: the carrier of X,COMPLEX:]
ComplexAlgebraStr(# (Funcs ( the carrier of X,COMPLEX)),(ComplexFuncMult the carrier of X),(ComplexFuncAdd the carrier of X),(ComplexFuncExtMult the carrier of X),(ComplexFuncUnit the carrier of X),(ComplexFuncZero the carrier of X) #) is strict ComplexAlgebraStr
the carrier of (CAlgebra the carrier of X) is non empty set
bool the carrier of (CAlgebra the carrier of X) is non empty set
{ b₁ where b₁ is non empty Relation-like the carrier of X -defined COMPLEX -valued Function-like total quasi_total V139() Element of bool [: the carrier of X,COMPLEX:] : b₁ | the carrier of X is bounded } is set
ComplexBoundedFunctionsNorm the carrier of X is non empty Relation-like ComplexBoundedFunctions the carrier of X -defined REAL -valued Function-like total quasi_total V139() V140() V141() Element of bool [:(ComplexBoundedFunctions the carrier of X),REAL:]
[:(ComplexBoundedFunctions the carrier of X),REAL:] is non empty V139() V140() V141() set
bool [:(ComplexBoundedFunctions the carrier of X),REAL:] is non empty set
(ComplexBoundedFunctionsNorm the carrier of X) | (X) is Relation-like ComplexBoundedFunctions the carrier of X -defined REAL -valued Function-like V139() V140() V141() Element of bool [:(ComplexBoundedFunctions the carrier of X),REAL:]
CNORMSTR(# (X),(Zero_ ((X),(ComplexVectSpace the carrier of X))),(Add_ ((X),(ComplexVectSpace the carrier of X))),(Mult_ ((X),(ComplexVectSpace the carrier of X))),(X) #) is strict CNORMSTR
(X) is non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital CLSStruct
CLSStruct(# (X),(Zero_ ((X),(ComplexVectSpace the carrier of X))),(Add_ ((X),(ComplexVectSpace the carrier of X))),(Mult_ ((X),(ComplexVectSpace the carrier of X))) #) is strict CLSStruct
the carrier of (X) is non empty set
0. (X) is zero Element of the carrier of (X)
the ZeroF of (X) is Element of the carrier of (X)
X is Element of the carrier of (X)
||.X.|| is complex real ext-real Element of REAL
the U9 of (X) is non empty Relation-like the carrier of (X) -defined REAL -valued Function-like total quasi_total V139() V140() V141() Element of bool [: the carrier of (X),REAL:]
[: the carrier of (X),REAL:] is non empty V139() V140() V141() set
bool [: the carrier of (X),REAL:] is non empty set
the U9 of (X) . X is complex real ext-real Element of REAL
||.(0. (X)).|| is complex real ext-real Element of REAL
the U9 of (X) is non empty Relation-like the carrier of (X) -defined REAL -valued Function-like total quasi_total V139() V140() V141() Element of bool [: the carrier of (X),REAL:]
[: the carrier of (X),REAL:] is non empty V139() V140() V141() set
bool [: the carrier of (X),REAL:] is non empty set
the U9 of (X) . (0. (X)) is complex real ext-real Element of REAL
X is non empty TopSpace-like TopStruct
(X) is non empty right_complementable Abelian add-associative right_zeroed discerning reflexive vector-distributive scalar-distributive scalar-associative scalar-unital strict ComplexNormSpace-like CNORMSTR
(X) is non empty linearly-closed Element of bool the carrier of (ComplexVectSpace the carrier of X)
the carrier of X is non empty set
ComplexVectSpace the carrier of X is non empty right_complementable Abelian add-associative right_zeroed strict vector-distributive scalar-distributive scalar-associative scalar-unital CLSStruct
Funcs ( the carrier of X,COMPLEX) is non empty functional FUNCTION_DOMAIN of the carrier of X, COMPLEX
ComplexFuncZero the carrier of X is Relation-like the carrier of X -defined COMPLEX -valued Function-like total quasi_total V139() Element of Funcs ( the carrier of X,COMPLEX)
the carrier of X --> 0 is non empty Relation-like the carrier of X -defined REAL -valued RAT -valued INT -valued Function-like total quasi_total V139() V140() V141() V142() Element of bool [: the carrier of X,REAL:]
[: the carrier of X,REAL:] is non empty V139() V140() V141() set
bool [: the carrier of X,REAL:] is non empty set
ComplexFuncAdd the carrier of X is non empty Relation-like [:(Funcs ( the carrier of X,COMPLEX)),(Funcs ( the carrier of X,COMPLEX)):] -defined Funcs ( the carrier of X,COMPLEX) -valued Function-like total quasi_total Element of bool [:[:(Funcs ( the carrier of X,COMPLEX)),(Funcs ( the carrier of X,COMPLEX)):],(Funcs ( the carrier of X,COMPLEX)):]
[:(Funcs ( the carrier of X,COMPLEX)),(Funcs ( the carrier of X,COMPLEX)):] is non empty set
[:[:(Funcs ( the carrier of X,COMPLEX)),(Funcs ( the carrier of X,COMPLEX)):],(Funcs ( the carrier of X,COMPLEX)):] is non empty set
bool [:[:(Funcs ( the carrier of X,COMPLEX)),(Funcs ( the carrier of X,COMPLEX)):],(Funcs ( the carrier of X,COMPLEX)):] is non empty set
ComplexFuncExtMult the carrier of X is non empty Relation-like [:COMPLEX,(Funcs ( the carrier of X,COMPLEX)):] -defined Funcs ( the carrier of X,COMPLEX) -valued Function-like total quasi_total Element of bool [:[:COMPLEX,(Funcs ( the carrier of X,COMPLEX)):],(Funcs ( the carrier of X,COMPLEX)):]
[:COMPLEX,(Funcs ( the carrier of X,COMPLEX)):] is non empty set
[:[:COMPLEX,(Funcs ( the carrier of X,COMPLEX)):],(Funcs ( the carrier of X,COMPLEX)):] is non empty set
bool [:[:COMPLEX,(Funcs ( the carrier of X,COMPLEX)):],(Funcs ( the carrier of X,COMPLEX)):] is non empty set
CLSStruct(# (Funcs ( the carrier of X,COMPLEX)),(ComplexFuncZero the carrier of X),(ComplexFuncAdd the carrier of X),(ComplexFuncExtMult the carrier of X) #) is strict CLSStruct
the carrier of (ComplexVectSpace the carrier of X) is non empty set
bool the carrier of (ComplexVectSpace the carrier of X) is non empty set
[: the carrier of X,COMPLEX:] is non empty V139() set
bool [: the carrier of X,COMPLEX:] is non empty set
bool the carrier of X is non empty set
{ b₁ where b₁ is non empty Relation-like the carrier of X -defined COMPLEX -valued Function-like total quasi_total V139() Element of bool [: the carrier of X,COMPLEX:] : ( b₁ is (X) & ex b₂ being non empty Element of bool the carrier of X st
( b₂ is compact & ( for b₃ being Element of bool the carrier of X holds
( not b₃ = support b₁ or Cl b₃ is Element of bool b₂ ) ) ) ) } is set
Zero_ ((X),(ComplexVectSpace the carrier of X)) is Element of (X)
Add_ ((X),(ComplexVectSpace the carrier of X)) is non empty Relation-like [:(X),(X):] -defined (X) -valued Function-like total quasi_total Element of bool [:[:(X),(X):],(X):]
[:(X),(X):] is non empty set
[:[:(X),(X):],(X):] is non empty set
bool [:[:(X),(X):],(X):] is non empty set
Mult_ ((X),(ComplexVectSpace the carrier of X)) is non empty Relation-like [:COMPLEX,(X):] -defined (X) -valued Function-like total quasi_total Element of bool [:[:COMPLEX,(X):],(X):]
[:COMPLEX,(X):] is non empty set
[:[:COMPLEX,(X):],(X):] is non empty set
bool [:[:COMPLEX,(X):],(X):] is non empty set
(X) is non empty Relation-like (X) -defined REAL -valued Function-like total quasi_total V139() V140() V141() Element of bool [:(X),REAL:]
[:(X),REAL:] is non empty V139() V140() V141() set
bool [:(X),REAL:] is non empty set
ComplexBoundedFunctions the carrier of X is non empty multiplicatively-closed Cadditively-linearly-closed Element of bool the carrier of (CAlgebra the carrier of X)
CAlgebra the carrier of X is non empty right_complementable Abelian add-associative right_zeroed unital associative commutative right-distributive right_unital well-unital left_unital vector-distributive scalar-distributive scalar-associative scalar-unital strict vector-associative ComplexAlgebraStr
ComplexFuncMult the carrier of X is non empty Relation-like [:(Funcs ( the carrier of X,COMPLEX)),(Funcs ( the carrier of X,COMPLEX)):] -defined Funcs ( the carrier of X,COMPLEX) -valued Function-like total quasi_total Element of bool [:[:(Funcs ( the carrier of X,COMPLEX)),(Funcs ( the carrier of X,COMPLEX)):],(Funcs ( the carrier of X,COMPLEX)):]
ComplexFuncUnit the carrier of X is Relation-like the carrier of X -defined COMPLEX -valued Function-like total quasi_total V139() Element of Funcs ( the carrier of X,COMPLEX)
the carrier of X --> 1r is non empty Relation-like the carrier of X -defined COMPLEX -valued Function-like total quasi_total V139() Element of bool [: the carrier of X,COMPLEX:]
ComplexAlgebraStr(# (Funcs ( the carrier of X,COMPLEX)),(ComplexFuncMult the carrier of X),(ComplexFuncAdd the carrier of X),(ComplexFuncExtMult the carrier of X),(ComplexFuncUnit the carrier of X),(ComplexFuncZero the carrier of X) #) is strict ComplexAlgebraStr
the carrier of (CAlgebra the carrier of X) is non empty set
bool the carrier of (CAlgebra the carrier of X) is non empty set
{ b₁ where b₁ is non empty Relation-like the carrier of X -defined COMPLEX -valued Function-like total quasi_total V139() Element of bool [: the carrier of X,COMPLEX:] : b₁ | the carrier of X is bounded } is set
ComplexBoundedFunctionsNorm the carrier of X is non empty Relation-like ComplexBoundedFunctions the carrier of X -defined REAL -valued Function-like total quasi_total V139() V140() V141() Element of bool [:(ComplexBoundedFunctions the carrier of X),REAL:]
[:(ComplexBoundedFunctions the carrier of X),REAL:] is non empty V139() V140() V141() set
bool [:(ComplexBoundedFunctions the carrier of X),REAL:] is non empty set
(ComplexBoundedFunctionsNorm the carrier of X) | (X) is Relation-like ComplexBoundedFunctions the carrier of X -defined REAL -valued Function-like V139() V140() V141() Element of bool [:(ComplexBoundedFunctions the carrier of X),REAL:]
CNORMSTR(# (X),(Zero_ ((X),(ComplexVectSpace the carrier of X))),(Add_ ((X),(ComplexVectSpace the carrier of X))),(Mult_ ((X),(ComplexVectSpace the carrier of X))),(X) #) is strict CNORMSTR