:: C0SP2 semantic presentation

REAL is non empty V30() V185() V186() V187() V191() non bounded_below non bounded_above V268() set
NAT is non empty epsilon-transitive epsilon-connected ordinal V185() V186() V187() V188() V189() V190() V191() left_end bounded_below Element of bool REAL
bool REAL is non empty set
omega is non empty epsilon-transitive epsilon-connected ordinal V185() V186() V187() V188() V189() V190() V191() left_end bounded_below set
bool omega is non empty set
bool NAT is non empty set
[:NAT,REAL:] is non empty V152() V153() V154() set
bool [:NAT,REAL:] is non empty set
the_set_of_RealSequences is non empty set
[:the_set_of_RealSequences,the_set_of_RealSequences:] is non empty set
[:[:the_set_of_RealSequences,the_set_of_RealSequences:],the_set_of_RealSequences:] is non empty set
bool [:[:the_set_of_RealSequences,the_set_of_RealSequences:],the_set_of_RealSequences:] is non empty set
[:REAL,the_set_of_RealSequences:] is non empty set
[:[:REAL,the_set_of_RealSequences:],the_set_of_RealSequences:] is non empty set
bool [:[:REAL,the_set_of_RealSequences:],the_set_of_RealSequences:] is non empty set
Linear_Space_of_RealSequences is RLSStruct
the carrier of Linear_Space_of_RealSequences is set
bool the carrier of Linear_Space_of_RealSequences is non empty set
the_set_of_l2RealSequences is Element of bool the carrier of Linear_Space_of_RealSequences
[:the_set_of_l2RealSequences,the_set_of_l2RealSequences:] is set
[:[:the_set_of_l2RealSequences,the_set_of_l2RealSequences:],REAL:] is V152() V153() V154() set
bool [:[:the_set_of_l2RealSequences,the_set_of_l2RealSequences:],REAL:] is non empty set
the_set_of_l1RealSequences is Element of bool the carrier of Linear_Space_of_RealSequences
[:the_set_of_l1RealSequences,REAL:] is V152() V153() V154() set
bool [:the_set_of_l1RealSequences,REAL:] is non empty set
COMPLEX is non empty V30() V185() V191() set
ExtREAL is non empty V186() V268() set
RAT is non empty V30() V185() V186() V187() V188() V191() set
INT is non empty V30() V185() V186() V187() V188() V189() V191() set
2 is non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V117() V185() V186() V187() V188() V189() V190() V197() left_end bounded_below Element of NAT
K597() is V261() TopStruct
the carrier of K597() is V185() V186() V187() set
1 is non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V117() V185() V186() V187() V188() V189() V190() V197() left_end bounded_below Element of NAT
[:1,1:] is non empty RAT -valued INT -valued V152() V153() V154() V155() set
bool [:1,1:] is non empty set
[:[:1,1:],1:] is non empty RAT -valued INT -valued V152() V153() V154() V155() set
bool [:[:1,1:],1:] is non empty set
[:[:1,1:],REAL:] is non empty V152() V153() V154() set
bool [:[:1,1:],REAL:] is non empty set
[:REAL,REAL:] is non empty V152() V153() V154() set
[:[:REAL,REAL:],REAL:] is non empty V152() V153() V154() set
bool [:[:REAL,REAL:],REAL:] is non empty set
[:2,2:] is non empty RAT -valued INT -valued V152() V153() V154() V155() set
[:[:2,2:],REAL:] is non empty V152() V153() V154() set
bool [:[:2,2:],REAL:] is non empty set
K620() is V247() V261() L21()
R^1 is non empty strict TopSpace-like V261() TopStruct
[:COMPLEX,COMPLEX:] is non empty V152() set
bool [:COMPLEX,COMPLEX:] is non empty set
[:COMPLEX,REAL:] is non empty V152() V153() V154() set
bool [:COMPLEX,REAL:] is non empty set
bool [:REAL,REAL:] is non empty set
K689(2) is V281() L22()
the carrier of K689(2) is set
[: the carrier of K689(2),REAL:] is V152() V153() V154() set
bool [: the carrier of K689(2),REAL:] is non empty set
bool the carrier of K689(2) is non empty set
{} is empty epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural V11() real ext-real non positive non negative V185() V186() V187() V188() V189() V190() V191() bounded_below V268() set
0 is empty epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural V11() real ext-real non positive non negative V117() V185() V186() V187() V188() V189() V190() V191() V197() bounded_below V268() Element of NAT
|.0.| is V11() real ext-real V197() Element of REAL
3 is non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V117() V185() V186() V187() V188() V189() V190() V197() left_end bounded_below Element of NAT
- 1 is V11() real ext-real non positive Element of REAL
the carrier of R^1 is non empty V185() V186() V187() set
X is 1-sorted
the carrier of X is set
X is V11() real ext-real set
the carrier of X --> X is Relation-like the carrier of X -defined {X} -valued Function-like total quasi_total V152() V153() V154() Element of bool [: the carrier of X,{X}:]
{X} is non empty V185() V186() V187() set
[: the carrier of X,{X}:] is V152() V153() V154() set
bool [: the carrier of X,{X}:] is non empty set
[: the carrier of X,REAL:] is V152() V153() V154() set
bool [: the carrier of X,REAL:] is non empty set
X is TopSpace-like TopStruct
X is V11() real ext-real set
(X,X) is Relation-like the carrier of X -defined REAL -valued Function-like total quasi_total V152() V153() V154() Element of bool [: the carrier of X,REAL:]
the carrier of X is set
[: the carrier of X,REAL:] is V152() V153() V154() set
bool [: the carrier of X,REAL:] is non empty set
the carrier of X --> X is Relation-like the carrier of X -defined {X} -valued Function-like total quasi_total V152() V153() V154() Element of bool [: the carrier of X,{X}:]
{X} is non empty V185() V186() V187() set
[: the carrier of X,{X}:] is V152() V153() V154() set
bool [: the carrier of X,{X}:] is non empty set
[#] X is non proper closed Element of bool the carrier of X
bool the carrier of X is non empty set
FB is V185() V186() V187() Element of bool REAL
(X,X) " FB is Element of bool the carrier of X
{} X is empty epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural V11() real ext-real non positive non negative V185() V186() V187() V188() V189() V190() V191() closed bounded_below V268() compact 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,REAL:] is non empty V152() V153() V154() set
bool [: the carrier of X,REAL:] is non empty set
bool the carrier of X is non empty set
X is non empty Relation-like the carrier of X -defined REAL -valued Function-like total quasi_total V152() V153() V154() Element of bool [: the carrier of X,REAL:]
F is Element of the carrier of X
X . F is V11() real ext-real Element of REAL
G is V185() V186() V187() Element of bool REAL
GB is V11() real ext-real set
(X . F) - GB is V11() real ext-real Element of REAL
- GB is V11() real ext-real set
(X . F) + (- GB) is V11() real ext-real set
(X . F) + GB is V11() real ext-real Element of REAL
].((X . F) - GB),((X . F) + GB).[ is open V185() V186() V187() non left_end non right_end V268() Element of bool REAL
{ b₁ where b₁ is V11() real ext-real Element of REAL : ( not b₁ <= (X . F) - GB & not (X . F) + GB <= b₁ ) } is set
X " ].((X . F) - GB),((X . F) + GB).[ is Element of bool the carrier of X
(X . F) - (X . F) is V11() real ext-real Element of REAL
- (X . F) is V11() real ext-real set
(X . F) + (- (X . F)) is V11() real ext-real set
abs ((X . F) - (X . F)) is V11() real ext-real Element of REAL
aF is Element of bool the carrier of X
X .: (X " ].((X . F) - GB),((X . F) + GB).[) is V185() V186() V187() Element of bool REAL
X .: aF is V185() V186() V187() Element of bool REAL
F is Element of the carrier of X
X . F is V11() real ext-real Element of REAL
G is V185() V186() V187() Element of bool REAL
F is V185() V186() V187() Element of bool REAL
F ` is V185() V186() V187() Element of bool REAL
(F `) ` is V185() V186() V187() Element of bool REAL
X " F is Element of bool the carrier of X
(X " F) ` is Element of bool the carrier of X
G is Element of the carrier of X
X " (F `) is Element of bool the carrier of X
X . G is V11() real ext-real Element of REAL
FB is V185() V186() V187() Element of bool REAL
GB is Element of bool the carrier of X
X .: GB is V185() V186() V187() Element of bool REAL
X " (X .: GB) is Element of bool the carrier of 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,REAL:] is non empty V152() V153() V154() set
bool [: the carrier of X,REAL:] is non empty set
{ b₁ where b₁ is non empty Relation-like the carrier of X -defined REAL -valued Function-like total quasi_total V152() V153() V154() continuous Element of bool [: the carrier of X,REAL:] : verum } is set
RAlgebra the carrier of X is non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative unital associative commutative right-distributive right_unital well-unital left_unital strict vector-associative AlgebraStr
Funcs ( the carrier of X,REAL) is non empty functional FUNCTION_DOMAIN of the carrier of X, REAL
RealFuncMult the carrier of X is non empty Relation-like [:(Funcs ( the carrier of X,REAL)),(Funcs ( the carrier of X,REAL)):] -defined Funcs ( the carrier of X,REAL) -valued Function-like total quasi_total Element of bool [:[:(Funcs ( the carrier of X,REAL)),(Funcs ( the carrier of X,REAL)):],(Funcs ( the carrier of X,REAL)):]
[:(Funcs ( the carrier of X,REAL)),(Funcs ( the carrier of X,REAL)):] is non empty set
[:[:(Funcs ( the carrier of X,REAL)),(Funcs ( the carrier of X,REAL)):],(Funcs ( the carrier of X,REAL)):] is non empty set
bool [:[:(Funcs ( the carrier of X,REAL)),(Funcs ( the carrier of X,REAL)):],(Funcs ( the carrier of X,REAL)):] is non empty set
RealFuncAdd the carrier of X is non empty Relation-like [:(Funcs ( the carrier of X,REAL)),(Funcs ( the carrier of X,REAL)):] -defined Funcs ( the carrier of X,REAL) -valued Function-like total quasi_total Element of bool [:[:(Funcs ( the carrier of X,REAL)),(Funcs ( the carrier of X,REAL)):],(Funcs ( the carrier of X,REAL)):]
RealFuncExtMult the carrier of X is non empty Relation-like [:REAL,(Funcs ( the carrier of X,REAL)):] -defined Funcs ( the carrier of X,REAL) -valued Function-like total quasi_total Element of bool [:[:REAL,(Funcs ( the carrier of X,REAL)):],(Funcs ( the carrier of X,REAL)):]
[:REAL,(Funcs ( the carrier of X,REAL)):] is non empty set
[:[:REAL,(Funcs ( the carrier of X,REAL)):],(Funcs ( the carrier of X,REAL)):] is non empty set
bool [:[:REAL,(Funcs ( the carrier of X,REAL)):],(Funcs ( the carrier of X,REAL)):] is non empty set
RealFuncUnit the carrier of X is Relation-like the carrier of X -defined REAL -valued Function-like total quasi_total V152() V153() V154() Element of Funcs ( the carrier of X,REAL)
the carrier of X --> 1 is non empty Relation-like the carrier of X -defined NAT -valued RAT -valued INT -valued Function-like total quasi_total V152() V153() V154() V155() Element of bool [: the carrier of X,NAT:]
[: the carrier of X,NAT:] is non empty RAT -valued INT -valued V152() V153() V154() V155() set
bool [: the carrier of X,NAT:] is non empty set
RealFuncZero the carrier of X is Relation-like the carrier of X -defined REAL -valued Function-like total quasi_total V152() V153() V154() Element of Funcs ( the carrier of X,REAL)
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 V152() V153() V154() V155() Element of bool [: the carrier of X,NAT:]
AlgebraStr(# (Funcs ( the carrier of X,REAL)),(RealFuncMult the carrier of X),(RealFuncAdd the carrier of X),(RealFuncExtMult the carrier of X),(RealFuncUnit the carrier of X),(RealFuncZero the carrier of X) #) is strict AlgebraStr
the carrier of (RAlgebra the carrier of X) is non empty set
bool the carrier of (RAlgebra the carrier of X) is non empty set
F is set
G is non empty Relation-like the carrier of X -defined REAL -valued Function-like total quasi_total V152() V153() V154() continuous Element of bool [: the carrier of X,REAL:]
X is non empty TopSpace-like TopStruct
(X) is Element of bool the carrier of (RAlgebra the carrier of X)
the carrier of X is non empty set
RAlgebra the carrier of X is non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative unital associative commutative right-distributive right_unital well-unital left_unital strict vector-associative AlgebraStr
Funcs ( the carrier of X,REAL) is non empty functional FUNCTION_DOMAIN of the carrier of X, REAL
RealFuncMult the carrier of X is non empty Relation-like [:(Funcs ( the carrier of X,REAL)),(Funcs ( the carrier of X,REAL)):] -defined Funcs ( the carrier of X,REAL) -valued Function-like total quasi_total Element of bool [:[:(Funcs ( the carrier of X,REAL)),(Funcs ( the carrier of X,REAL)):],(Funcs ( the carrier of X,REAL)):]
[:(Funcs ( the carrier of X,REAL)),(Funcs ( the carrier of X,REAL)):] is non empty set
[:[:(Funcs ( the carrier of X,REAL)),(Funcs ( the carrier of X,REAL)):],(Funcs ( the carrier of X,REAL)):] is non empty set
bool [:[:(Funcs ( the carrier of X,REAL)),(Funcs ( the carrier of X,REAL)):],(Funcs ( the carrier of X,REAL)):] is non empty set
RealFuncAdd the carrier of X is non empty Relation-like [:(Funcs ( the carrier of X,REAL)),(Funcs ( the carrier of X,REAL)):] -defined Funcs ( the carrier of X,REAL) -valued Function-like total quasi_total Element of bool [:[:(Funcs ( the carrier of X,REAL)),(Funcs ( the carrier of X,REAL)):],(Funcs ( the carrier of X,REAL)):]
RealFuncExtMult the carrier of X is non empty Relation-like [:REAL,(Funcs ( the carrier of X,REAL)):] -defined Funcs ( the carrier of X,REAL) -valued Function-like total quasi_total Element of bool [:[:REAL,(Funcs ( the carrier of X,REAL)):],(Funcs ( the carrier of X,REAL)):]
[:REAL,(Funcs ( the carrier of X,REAL)):] is non empty set
[:[:REAL,(Funcs ( the carrier of X,REAL)):],(Funcs ( the carrier of X,REAL)):] is non empty set
bool [:[:REAL,(Funcs ( the carrier of X,REAL)):],(Funcs ( the carrier of X,REAL)):] is non empty set
RealFuncUnit the carrier of X is Relation-like the carrier of X -defined REAL -valued Function-like total quasi_total V152() V153() V154() Element of Funcs ( the carrier of X,REAL)
the carrier of X --> 1 is non empty Relation-like the carrier of X -defined NAT -valued RAT -valued INT -valued Function-like total quasi_total V152() V153() V154() V155() Element of bool [: the carrier of X,NAT:]
[: the carrier of X,NAT:] is non empty RAT -valued INT -valued V152() V153() V154() V155() set
bool [: the carrier of X,NAT:] is non empty set
RealFuncZero the carrier of X is Relation-like the carrier of X -defined REAL -valued Function-like total quasi_total V152() V153() V154() Element of Funcs ( the carrier of X,REAL)
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 V152() V153() V154() V155() Element of bool [: the carrier of X,NAT:]
AlgebraStr(# (Funcs ( the carrier of X,REAL)),(RealFuncMult the carrier of X),(RealFuncAdd the carrier of X),(RealFuncExtMult the carrier of X),(RealFuncUnit the carrier of X),(RealFuncZero the carrier of X) #) is strict AlgebraStr
the carrier of (RAlgebra the carrier of X) is non empty set
bool the carrier of (RAlgebra the carrier of X) is non empty set
[: the carrier of X,REAL:] is non empty V152() V153() V154() set
bool [: the carrier of X,REAL:] is non empty set
{ b₁ where b₁ is non empty Relation-like the carrier of X -defined REAL -valued Function-like total quasi_total V152() V153() V154() continuous Element of bool [: the carrier of X,REAL:] : verum } is set
(X,0) is non empty Relation-like the carrier of X -defined REAL -valued Function-like total quasi_total V152() V153() V154() continuous Element of bool [: the carrier of X,REAL:]
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 V152() V153() V154() V155() Element of bool [: the carrier of X,{0}:]
{0} is non empty V185() V186() V187() V188() V189() V190() left_end bounded_below set
[: the carrier of X,{0}:] is non empty RAT -valued INT -valued V152() V153() V154() V155() set
bool [: the carrier of X,{0}:] is non empty set
X is non empty TopSpace-like TopStruct
the carrier of X is non empty set
RAlgebra the carrier of X is non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative unital associative commutative right-distributive right_unital well-unital left_unital strict vector-associative AlgebraStr
Funcs ( the carrier of X,REAL) is non empty functional FUNCTION_DOMAIN of the carrier of X, REAL
RealFuncMult the carrier of X is non empty Relation-like [:(Funcs ( the carrier of X,REAL)),(Funcs ( the carrier of X,REAL)):] -defined Funcs ( the carrier of X,REAL) -valued Function-like total quasi_total Element of bool [:[:(Funcs ( the carrier of X,REAL)),(Funcs ( the carrier of X,REAL)):],(Funcs ( the carrier of X,REAL)):]
[:(Funcs ( the carrier of X,REAL)),(Funcs ( the carrier of X,REAL)):] is non empty set
[:[:(Funcs ( the carrier of X,REAL)),(Funcs ( the carrier of X,REAL)):],(Funcs ( the carrier of X,REAL)):] is non empty set
bool [:[:(Funcs ( the carrier of X,REAL)),(Funcs ( the carrier of X,REAL)):],(Funcs ( the carrier of X,REAL)):] is non empty set
RealFuncAdd the carrier of X is non empty Relation-like [:(Funcs ( the carrier of X,REAL)),(Funcs ( the carrier of X,REAL)):] -defined Funcs ( the carrier of X,REAL) -valued Function-like total quasi_total Element of bool [:[:(Funcs ( the carrier of X,REAL)),(Funcs ( the carrier of X,REAL)):],(Funcs ( the carrier of X,REAL)):]
RealFuncExtMult the carrier of X is non empty Relation-like [:REAL,(Funcs ( the carrier of X,REAL)):] -defined Funcs ( the carrier of X,REAL) -valued Function-like total quasi_total Element of bool [:[:REAL,(Funcs ( the carrier of X,REAL)):],(Funcs ( the carrier of X,REAL)):]
[:REAL,(Funcs ( the carrier of X,REAL)):] is non empty set
[:[:REAL,(Funcs ( the carrier of X,REAL)):],(Funcs ( the carrier of X,REAL)):] is non empty set
bool [:[:REAL,(Funcs ( the carrier of X,REAL)):],(Funcs ( the carrier of X,REAL)):] is non empty set
RealFuncUnit the carrier of X is Relation-like the carrier of X -defined REAL -valued Function-like total quasi_total V152() V153() V154() Element of Funcs ( the carrier of X,REAL)
the carrier of X --> 1 is non empty Relation-like the carrier of X -defined NAT -valued RAT -valued INT -valued Function-like total quasi_total V152() V153() V154() V155() Element of bool [: the carrier of X,NAT:]
[: the carrier of X,NAT:] is non empty RAT -valued INT -valued V152() V153() V154() V155() set
bool [: the carrier of X,NAT:] is non empty set
RealFuncZero the carrier of X is Relation-like the carrier of X -defined REAL -valued Function-like total quasi_total V152() V153() V154() Element of Funcs ( the carrier of X,REAL)
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 V152() V153() V154() V155() Element of bool [: the carrier of X,NAT:]
AlgebraStr(# (Funcs ( the carrier of X,REAL)),(RealFuncMult the carrier of X),(RealFuncAdd the carrier of X),(RealFuncExtMult the carrier of X),(RealFuncUnit the carrier of X),(RealFuncZero the carrier of X) #) is strict AlgebraStr
(X) is non empty Element of bool the carrier of (RAlgebra the carrier of X)
the carrier of (RAlgebra the carrier of X) is non empty set
bool the carrier of (RAlgebra the carrier of X) is non empty set
[: the carrier of X,REAL:] is non empty V152() V153() V154() set
bool [: the carrier of X,REAL:] is non empty set
{ b₁ where b₁ is non empty Relation-like the carrier of X -defined REAL -valued Function-like total quasi_total V152() V153() V154() continuous Element of bool [: the carrier of X,REAL:] : verum } is set
G is right_complementable Element of the carrier of (RAlgebra the carrier of X)
FB is right_complementable Element of the carrier of (RAlgebra the carrier of X)
G + FB is right_complementable Element of the carrier of (RAlgebra the carrier of X)
the addF of (RAlgebra the carrier of X) is non empty Relation-like [: the carrier of (RAlgebra the carrier of X), the carrier of (RAlgebra the carrier of X):] -defined the carrier of (RAlgebra the carrier of X) -valued Function-like total quasi_total Element of bool [:[: the carrier of (RAlgebra the carrier of X), the carrier of (RAlgebra the carrier of X):], the carrier of (RAlgebra the carrier of X):]
[: the carrier of (RAlgebra the carrier of X), the carrier of (RAlgebra the carrier of X):] is non empty set
[:[: the carrier of (RAlgebra the carrier of X), the carrier of (RAlgebra the carrier of X):], the carrier of (RAlgebra the carrier of X):] is non empty set
bool [:[: the carrier of (RAlgebra the carrier of X), the carrier of (RAlgebra the carrier of X):], the carrier of (RAlgebra the carrier of X):] is non empty set
the addF of (RAlgebra the carrier of X) . (G,FB) is right_complementable Element of the carrier of (RAlgebra 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 (RAlgebra the carrier of X) . [G,FB] is set
GB is non empty Relation-like the carrier of X -defined REAL -valued Function-like total quasi_total V152() V153() V154() continuous Element of bool [: the carrier of X,REAL:]
aFB is non empty Relation-like the carrier of X -defined REAL -valued Function-like total quasi_total V152() V153() V154() continuous Element of bool [: the carrier of X,REAL:]
GB + aFB is non empty Relation-like the carrier of X -defined REAL -valued Function-like total quasi_total V152() V153() V154() Element of bool [: the carrier of X,REAL:]
aF is Relation-like the carrier of X -defined REAL -valued Function-like total quasi_total V152() V153() V154() Element of Funcs ( the carrier of X,REAL)
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 V11() real ext-real set
GB . aG is V11() real ext-real set
aFB . aG is V11() real ext-real set
(GB . aG) + (aFB . aG) is V11() real ext-real set
aG is Relation-like Function-like set
dom aG is set
rng aG is set
G is right_complementable Element of the carrier of (RAlgebra the carrier of X)
- G is right_complementable Element of the carrier of (RAlgebra the carrier of X)
FB is non empty Relation-like the carrier of X -defined REAL -valued Function-like total quasi_total V152() V153() V154() continuous Element of bool [: the carrier of X,REAL:]
- FB is non empty Relation-like the carrier of X -defined REAL -valued Function-like total quasi_total V152() V153() V154() Element of bool [: the carrier of X,REAL:]
GB is Relation-like the carrier of X -defined REAL -valued Function-like total quasi_total V152() V153() V154() Element of Funcs ( the carrier of X,REAL)
(- 1) * G is right_complementable Element of the carrier of (RAlgebra the carrier of X)
the Mult of (RAlgebra the carrier of X) is non empty Relation-like [:REAL, the carrier of (RAlgebra the carrier of X):] -defined the carrier of (RAlgebra the carrier of X) -valued Function-like total quasi_total Element of bool [:[:REAL, the carrier of (RAlgebra the carrier of X):], the carrier of (RAlgebra the carrier of X):]
[:REAL, the carrier of (RAlgebra the carrier of X):] is non empty set
[:[:REAL, the carrier of (RAlgebra the carrier of X):], the carrier of (RAlgebra the carrier of X):] is non empty set
bool [:[:REAL, the carrier of (RAlgebra the carrier of X):], the carrier of (RAlgebra the carrier of X):] is non empty set
the Mult of (RAlgebra the carrier of X) . ((- 1),G) is set
[(- 1),G] is set
{(- 1),G} is non empty set
{(- 1)} is non empty V185() V186() V187() set
{{(- 1),G},{(- 1)}} is non empty set
the Mult of (RAlgebra the carrier of X) . [(- 1),G] is set
dom FB is Element of bool the carrier of X
bool the carrier of X is non empty set
aF is set
dom GB is Element of bool the carrier of X
GB . aF is V11() real ext-real set
aFB is Relation-like the carrier of X -defined REAL -valued Function-like total quasi_total V152() V153() V154() Element of Funcs ( the carrier of X,REAL)
aG is Element of the carrier of X
aFB . aG is V11() real ext-real Element of REAL
(- 1) * (aFB . aG) is V11() real ext-real Element of REAL
FB . aF is V11() real ext-real set
- (FB . aF) is V11() real ext-real set
aF is Relation-like Function-like set
dom aF is set
rng aF is set
G is V11() real ext-real Element of REAL
FB is right_complementable Element of the carrier of (RAlgebra the carrier of X)
G * FB is right_complementable Element of the carrier of (RAlgebra the carrier of X)
the Mult of (RAlgebra the carrier of X) is non empty Relation-like [:REAL, the carrier of (RAlgebra the carrier of X):] -defined the carrier of (RAlgebra the carrier of X) -valued Function-like total quasi_total Element of bool [:[:REAL, the carrier of (RAlgebra the carrier of X):], the carrier of (RAlgebra the carrier of X):]
[:REAL, the carrier of (RAlgebra the carrier of X):] is non empty set
[:[:REAL, the carrier of (RAlgebra the carrier of X):], the carrier of (RAlgebra the carrier of X):] is non empty set
bool [:[:REAL, the carrier of (RAlgebra the carrier of X):], the carrier of (RAlgebra the carrier of X):] is non empty set
the Mult of (RAlgebra the carrier of X) . (G,FB) is set
[G,FB] is set
{G,FB} is non empty set
{G} is non empty V185() V186() V187() set
{{G,FB},{G}} is non empty set
the Mult of (RAlgebra the carrier of X) . [G,FB] is set
GB is non empty Relation-like the carrier of X -defined REAL -valued Function-like total quasi_total V152() V153() V154() continuous Element of bool [: the carrier of X,REAL:]
G (#) GB is non empty Relation-like the carrier of X -defined REAL -valued Function-like total quasi_total V152() V153() V154() Element of bool [: the carrier of X,REAL:]
[: the carrier of X, the carrier of R^1:] is non empty V152() V153() V154() set
bool [: the carrier of X, the carrier of R^1:] is non empty set
aFB is non empty Relation-like the carrier of X -defined the carrier of R^1 -valued Function-like total quasi_total V152() V153() V154() continuous Element of bool [: the carrier of X, the carrier of R^1:]
aG is non empty Relation-like the carrier of X -defined the carrier of R^1 -valued Function-like total quasi_total V152() V153() V154() Element of bool [: the carrier of X, the carrier of R^1:]
r is non empty Relation-like the carrier of X -defined REAL -valued Function-like total quasi_total V152() V153() V154() Element of bool [: the carrier of X,REAL:]
dom r is Element of bool the carrier of X
bool the carrier of X is non empty set
dom GB is Element of bool the carrier of X
s1 is set
r . s1 is V11() real ext-real set
GB . s1 is V11() real ext-real set
G * (GB . s1) is V11() real ext-real Element of REAL
aFB is Relation-like the carrier of X -defined REAL -valued Function-like total quasi_total V152() V153() V154() Element of Funcs ( the carrier of X,REAL)
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 V11() real ext-real set
GB . aF is V11() real ext-real set
G * (GB . aF) is V11() real ext-real Element of REAL
aF is Relation-like Function-like set
dom aF is set
rng aF is set
G is right_complementable Element of the carrier of (RAlgebra the carrier of X)
FB is right_complementable Element of the carrier of (RAlgebra the carrier of X)
G * FB is right_complementable Element of the carrier of (RAlgebra the carrier of X)
the multF of (RAlgebra the carrier of X) is non empty Relation-like [: the carrier of (RAlgebra the carrier of X), the carrier of (RAlgebra the carrier of X):] -defined the carrier of (RAlgebra the carrier of X) -valued Function-like total quasi_total Element of bool [:[: the carrier of (RAlgebra the carrier of X), the carrier of (RAlgebra the carrier of X):], the carrier of (RAlgebra the carrier of X):]
[: the carrier of (RAlgebra the carrier of X), the carrier of (RAlgebra the carrier of X):] is non empty set
[:[: the carrier of (RAlgebra the carrier of X), the carrier of (RAlgebra the carrier of X):], the carrier of (RAlgebra the carrier of X):] is non empty set
bool [:[: the carrier of (RAlgebra the carrier of X), the carrier of (RAlgebra the carrier of X):], the carrier of (RAlgebra the carrier of X):] is non empty set
the multF of (RAlgebra the carrier of X) . (G,FB) is right_complementable Element of the carrier of (RAlgebra 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 (RAlgebra the carrier of X) . [G,FB] is set
GB is non empty Relation-like the carrier of X -defined REAL -valued Function-like total quasi_total V152() V153() V154() continuous Element of bool [: the carrier of X,REAL:]
aFB is non empty Relation-like the carrier of X -defined REAL -valued Function-like total quasi_total V152() V153() V154() continuous Element of bool [: the carrier of X,REAL:]
GB (#) aFB is non empty Relation-like the carrier of X -defined REAL -valued Function-like total quasi_total V152() V153() V154() Element of bool [: the carrier of X,REAL:]
[: the carrier of X, the carrier of R^1:] is non empty V152() V153() V154() set
bool [: the carrier of X, the carrier of R^1:] is non empty set
aF is non empty Relation-like the carrier of X -defined the carrier of R^1 -valued Function-like total quasi_total V152() V153() V154() continuous Element of bool [: the carrier of X, the carrier of R^1:]
aG is non empty Relation-like the carrier of X -defined the carrier of R^1 -valued Function-like total quasi_total V152() V153() V154() continuous Element of bool [: the carrier of X, the carrier of R^1:]
r is non empty Relation-like the carrier of X -defined the carrier of R^1 -valued Function-like total quasi_total V152() V153() V154() Element of bool [: the carrier of X, the carrier of R^1:]
s1 is non empty Relation-like the carrier of X -defined REAL -valued Function-like total quasi_total V152() V153() V154() Element of bool [: the carrier of X,REAL:]
dom s1 is Element of bool the carrier of X
bool the carrier of X is non empty set
the carrier of X /\ the carrier of X is set
dom aFB is Element of bool the carrier of X
the carrier of X /\ (dom aFB) is Element of bool the carrier of X
dom GB is Element of bool the carrier of X
(dom GB) /\ (dom aFB) is Element of bool the carrier of X
uu1 is set
s1 . uu1 is V11() real ext-real set
GB . uu1 is V11() real ext-real set
aFB . uu1 is V11() real ext-real set
(GB . uu1) * (aFB . uu1) is V11() real ext-real set
aF is Relation-like the carrier of X -defined REAL -valued Function-like total quasi_total V152() V153() V154() Element of Funcs ( the carrier of X,REAL)
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 V11() real ext-real set
GB . aG is V11() real ext-real set
aFB . aG is V11() real ext-real set
(GB . aG) * (aFB . aG) is V11() real ext-real set
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 REAL -valued Function-like total quasi_total V152() V153() V154() Element of bool [: the carrier of X,REAL:]
(X,1) is non empty Relation-like the carrier of X -defined REAL -valued Function-like total quasi_total V152() V153() V154() continuous Element of bool [: the carrier of X,REAL:]
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 V152() V153() V154() V155() Element of bool [: the carrier of X,{1}:]
{1} is non empty V185() V186() V187() V188() V189() V190() left_end bounded_below set
[: the carrier of X,{1}:] is non empty RAT -valued INT -valued V152() V153() V154() V155() set
bool [: the carrier of X,{1}:] is non empty set
1. (RAlgebra the carrier of X) is right_complementable Element of the carrier of (RAlgebra the carrier of X)
the OneF of (RAlgebra the carrier of X) is right_complementable Element of the carrier of (RAlgebra the carrier of X)
X is non empty TopSpace-like TopStruct
(X) is non empty add-closed having-inverse additively-closed multiplicatively-closed additively-linearly-closed Element of bool the carrier of (RAlgebra the carrier of X)
the carrier of X is non empty set
RAlgebra the carrier of X is non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative unital associative commutative right-distributive right_unital well-unital left_unital strict vector-associative AlgebraStr
Funcs ( the carrier of X,REAL) is non empty functional FUNCTION_DOMAIN of the carrier of X, REAL
RealFuncMult the carrier of X is non empty Relation-like [:(Funcs ( the carrier of X,REAL)),(Funcs ( the carrier of X,REAL)):] -defined Funcs ( the carrier of X,REAL) -valued Function-like total quasi_total Element of bool [:[:(Funcs ( the carrier of X,REAL)),(Funcs ( the carrier of X,REAL)):],(Funcs ( the carrier of X,REAL)):]
[:(Funcs ( the carrier of X,REAL)),(Funcs ( the carrier of X,REAL)):] is non empty set
[:[:(Funcs ( the carrier of X,REAL)),(Funcs ( the carrier of X,REAL)):],(Funcs ( the carrier of X,REAL)):] is non empty set
bool [:[:(Funcs ( the carrier of X,REAL)),(Funcs ( the carrier of X,REAL)):],(Funcs ( the carrier of X,REAL)):] is non empty set
RealFuncAdd the carrier of X is non empty Relation-like [:(Funcs ( the carrier of X,REAL)),(Funcs ( the carrier of X,REAL)):] -defined Funcs ( the carrier of X,REAL) -valued Function-like total quasi_total Element of bool [:[:(Funcs ( the carrier of X,REAL)),(Funcs ( the carrier of X,REAL)):],(Funcs ( the carrier of X,REAL)):]
RealFuncExtMult the carrier of X is non empty Relation-like [:REAL,(Funcs ( the carrier of X,REAL)):] -defined Funcs ( the carrier of X,REAL) -valued Function-like total quasi_total Element of bool [:[:REAL,(Funcs ( the carrier of X,REAL)):],(Funcs ( the carrier of X,REAL)):]
[:REAL,(Funcs ( the carrier of X,REAL)):] is non empty set
[:[:REAL,(Funcs ( the carrier of X,REAL)):],(Funcs ( the carrier of X,REAL)):] is non empty set
bool [:[:REAL,(Funcs ( the carrier of X,REAL)):],(Funcs ( the carrier of X,REAL)):] is non empty set
RealFuncUnit the carrier of X is Relation-like the carrier of X -defined REAL -valued Function-like total quasi_total V152() V153() V154() Element of Funcs ( the carrier of X,REAL)
the carrier of X --> 1 is non empty Relation-like the carrier of X -defined NAT -valued RAT -valued INT -valued Function-like total quasi_total V152() V153() V154() V155() Element of bool [: the carrier of X,NAT:]
[: the carrier of X,NAT:] is non empty RAT -valued INT -valued V152() V153() V154() V155() set
bool [: the carrier of X,NAT:] is non empty set
RealFuncZero the carrier of X is Relation-like the carrier of X -defined REAL -valued Function-like total quasi_total V152() V153() V154() Element of Funcs ( the carrier of X,REAL)
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 V152() V153() V154() V155() Element of bool [: the carrier of X,NAT:]
AlgebraStr(# (Funcs ( the carrier of X,REAL)),(RealFuncMult the carrier of X),(RealFuncAdd the carrier of X),(RealFuncExtMult the carrier of X),(RealFuncUnit the carrier of X),(RealFuncZero the carrier of X) #) is strict AlgebraStr
the carrier of (RAlgebra the carrier of X) is non empty set
bool the carrier of (RAlgebra the carrier of X) is non empty set
[: the carrier of X,REAL:] is non empty V152() V153() V154() set
bool [: the carrier of X,REAL:] is non empty set
{ b₁ where b₁ is non empty Relation-like the carrier of X -defined REAL -valued Function-like total quasi_total V152() V153() V154() continuous Element of bool [: the carrier of X,REAL:] : verum } is set
mult_ ((X),(RAlgebra 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),(RAlgebra 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),(RAlgebra the carrier of X)) is non empty Relation-like [:REAL,(X):] -defined (X) -valued Function-like total quasi_total Element of bool [:[:REAL,(X):],(X):]
[:REAL,(X):] is non empty set
[:[:REAL,(X):],(X):] is non empty set
bool [:[:REAL,(X):],(X):] is non empty set
One_ ((X),(RAlgebra the carrier of X)) is Element of (X)
Zero_ ((X),(RAlgebra the carrier of X)) is Element of (X)
AlgebraStr(# (X),(mult_ ((X),(RAlgebra the carrier of X))),(Add_ ((X),(RAlgebra the carrier of X))),(Mult_ ((X),(RAlgebra the carrier of X))),(One_ ((X),(RAlgebra the carrier of X))),(Zero_ ((X),(RAlgebra the carrier of X))) #) is strict AlgebraStr
X is non empty TopSpace-like TopStruct
(X) is AlgebraStr
(X) is non empty add-closed having-inverse additively-closed multiplicatively-closed additively-linearly-closed Element of bool the carrier of (RAlgebra the carrier of X)
the carrier of X is non empty set
RAlgebra the carrier of X is non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative unital associative commutative right-distributive right_unital well-unital left_unital strict vector-associative AlgebraStr
Funcs ( the carrier of X,REAL) is non empty functional FUNCTION_DOMAIN of the carrier of X, REAL
RealFuncMult the carrier of X is non empty Relation-like [:(Funcs ( the carrier of X,REAL)),(Funcs ( the carrier of X,REAL)):] -defined Funcs ( the carrier of X,REAL) -valued Function-like total quasi_total Element of bool [:[:(Funcs ( the carrier of X,REAL)),(Funcs ( the carrier of X,REAL)):],(Funcs ( the carrier of X,REAL)):]
[:(Funcs ( the carrier of X,REAL)),(Funcs ( the carrier of X,REAL)):] is non empty set
[:[:(Funcs ( the carrier of X,REAL)),(Funcs ( the carrier of X,REAL)):],(Funcs ( the carrier of X,REAL)):] is non empty set
bool [:[:(Funcs ( the carrier of X,REAL)),(Funcs ( the carrier of X,REAL)):],(Funcs ( the carrier of X,REAL)):] is non empty set
RealFuncAdd the carrier of X is non empty Relation-like [:(Funcs ( the carrier of X,REAL)),(Funcs ( the carrier of X,REAL)):] -defined Funcs ( the carrier of X,REAL) -valued Function-like total quasi_total Element of bool [:[:(Funcs ( the carrier of X,REAL)),(Funcs ( the carrier of X,REAL)):],(Funcs ( the carrier of X,REAL)):]
RealFuncExtMult the carrier of X is non empty Relation-like [:REAL,(Funcs ( the carrier of X,REAL)):] -defined Funcs ( the carrier of X,REAL) -valued Function-like total quasi_total Element of bool [:[:REAL,(Funcs ( the carrier of X,REAL)):],(Funcs ( the carrier of X,REAL)):]
[:REAL,(Funcs ( the carrier of X,REAL)):] is non empty set
[:[:REAL,(Funcs ( the carrier of X,REAL)):],(Funcs ( the carrier of X,REAL)):] is non empty set
bool [:[:REAL,(Funcs ( the carrier of X,REAL)):],(Funcs ( the carrier of X,REAL)):] is non empty set
RealFuncUnit the carrier of X is Relation-like the carrier of X -defined REAL -valued Function-like total quasi_total V152() V153() V154() Element of Funcs ( the carrier of X,REAL)
the carrier of X --> 1 is non empty Relation-like the carrier of X -defined NAT -valued RAT -valued INT -valued Function-like total quasi_total V152() V153() V154() V155() Element of bool [: the carrier of X,NAT:]
[: the carrier of X,NAT:] is non empty RAT -valued INT -valued V152() V153() V154() V155() set
bool [: the carrier of X,NAT:] is non empty set
RealFuncZero the carrier of X is Relation-like the carrier of X -defined REAL -valued Function-like total quasi_total V152() V153() V154() Element of Funcs ( the carrier of X,REAL)
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 V152() V153() V154() V155() Element of bool [: the carrier of X,NAT:]
AlgebraStr(# (Funcs ( the carrier of X,REAL)),(RealFuncMult the carrier of X),(RealFuncAdd the carrier of X),(RealFuncExtMult the carrier of X),(RealFuncUnit the carrier of X),(RealFuncZero the carrier of X) #) is strict AlgebraStr
the carrier of (RAlgebra the carrier of X) is non empty set
bool the carrier of (RAlgebra the carrier of X) is non empty set
[: the carrier of X,REAL:] is non empty V152() V153() V154() set
bool [: the carrier of X,REAL:] is non empty set
{ b₁ where b₁ is non empty Relation-like the carrier of X -defined REAL -valued Function-like total quasi_total V152() V153() V154() continuous Element of bool [: the carrier of X,REAL:] : verum } is set
mult_ ((X),(RAlgebra 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),(RAlgebra 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),(RAlgebra the carrier of X)) is non empty Relation-like [:REAL,(X):] -defined (X) -valued Function-like total quasi_total Element of bool [:[:REAL,(X):],(X):]
[:REAL,(X):] is non empty set
[:[:REAL,(X):],(X):] is non empty set
bool [:[:REAL,(X):],(X):] is non empty set
One_ ((X),(RAlgebra the carrier of X)) is Element of (X)
Zero_ ((X),(RAlgebra the carrier of X)) is Element of (X)
AlgebraStr(# (X),(mult_ ((X),(RAlgebra the carrier of X))),(Add_ ((X),(RAlgebra the carrier of X))),(Mult_ ((X),(RAlgebra the carrier of X))),(One_ ((X),(RAlgebra the carrier of X))),(Zero_ ((X),(RAlgebra the carrier of X))) #) is strict AlgebraStr
X is non empty TopSpace-like TopStruct
(X) is AlgebraStr
(X) is non empty add-closed having-inverse additively-closed multiplicatively-closed additively-linearly-closed Element of bool the carrier of (RAlgebra the carrier of X)
the carrier of X is non empty set
RAlgebra the carrier of X is non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative unital associative commutative right-distributive right_unital well-unital left_unital strict vector-associative AlgebraStr
Funcs ( the carrier of X,REAL) is non empty functional FUNCTION_DOMAIN of the carrier of X, REAL
RealFuncMult the carrier of X is non empty Relation-like [:(Funcs ( the carrier of X,REAL)),(Funcs ( the carrier of X,REAL)):] -defined Funcs ( the carrier of X,REAL) -valued Function-like total quasi_total Element of bool [:[:(Funcs ( the carrier of X,REAL)),(Funcs ( the carrier of X,REAL)):],(Funcs ( the carrier of X,REAL)):]
[:(Funcs ( the carrier of X,REAL)),(Funcs ( the carrier of X,REAL)):] is non empty set
[:[:(Funcs ( the carrier of X,REAL)),(Funcs ( the carrier of X,REAL)):],(Funcs ( the carrier of X,REAL)):] is non empty set
bool [:[:(Funcs ( the carrier of X,REAL)),(Funcs ( the carrier of X,REAL)):],(Funcs ( the carrier of X,REAL)):] is non empty set
RealFuncAdd the carrier of X is non empty Relation-like [:(Funcs ( the carrier of X,REAL)),(Funcs ( the carrier of X,REAL)):] -defined Funcs ( the carrier of X,REAL) -valued Function-like total quasi_total Element of bool [:[:(Funcs ( the carrier of X,REAL)),(Funcs ( the carrier of X,REAL)):],(Funcs ( the carrier of X,REAL)):]
RealFuncExtMult the carrier of X is non empty Relation-like [:REAL,(Funcs ( the carrier of X,REAL)):] -defined Funcs ( the carrier of X,REAL) -valued Function-like total quasi_total Element of bool [:[:REAL,(Funcs ( the carrier of X,REAL)):],(Funcs ( the carrier of X,REAL)):]
[:REAL,(Funcs ( the carrier of X,REAL)):] is non empty set
[:[:REAL,(Funcs ( the carrier of X,REAL)):],(Funcs ( the carrier of X,REAL)):] is non empty set
bool [:[:REAL,(Funcs ( the carrier of X,REAL)):],(Funcs ( the carrier of X,REAL)):] is non empty set
RealFuncUnit the carrier of X is Relation-like the carrier of X -defined REAL -valued Function-like total quasi_total V152() V153() V154() Element of Funcs ( the carrier of X,REAL)
the carrier of X --> 1 is non empty Relation-like the carrier of X -defined NAT -valued RAT -valued INT -valued Function-like total quasi_total V152() V153() V154() V155() Element of bool [: the carrier of X,NAT:]
[: the carrier of X,NAT:] is non empty RAT -valued INT -valued V152() V153() V154() V155() set
bool [: the carrier of X,NAT:] is non empty set
RealFuncZero the carrier of X is Relation-like the carrier of X -defined REAL -valued Function-like total quasi_total V152() V153() V154() Element of Funcs ( the carrier of X,REAL)
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 V152() V153() V154() V155() Element of bool [: the carrier of X,NAT:]
AlgebraStr(# (Funcs ( the carrier of X,REAL)),(RealFuncMult the carrier of X),(RealFuncAdd the carrier of X),(RealFuncExtMult the carrier of X),(RealFuncUnit the carrier of X),(RealFuncZero the carrier of X) #) is strict AlgebraStr
the carrier of (RAlgebra the carrier of X) is non empty set
bool the carrier of (RAlgebra the carrier of X) is non empty set
[: the carrier of X,REAL:] is non empty V152() V153() V154() set
bool [: the carrier of X,REAL:] is non empty set
{ b₁ where b₁ is non empty Relation-like the carrier of X -defined REAL -valued Function-like total quasi_total V152() V153() V154() continuous Element of bool [: the carrier of X,REAL:] : verum } is set
mult_ ((X),(RAlgebra 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),(RAlgebra 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),(RAlgebra the carrier of X)) is non empty Relation-like [:REAL,(X):] -defined (X) -valued Function-like total quasi_total Element of bool [:[:REAL,(X):],(X):]
[:REAL,(X):] is non empty set
[:[:REAL,(X):],(X):] is non empty set
bool [:[:REAL,(X):],(X):] is non empty set
One_ ((X),(RAlgebra the carrier of X)) is Element of (X)
Zero_ ((X),(RAlgebra the carrier of X)) is Element of (X)
AlgebraStr(# (X),(mult_ ((X),(RAlgebra the carrier of X))),(Add_ ((X),(RAlgebra the carrier of X))),(Mult_ ((X),(RAlgebra the carrier of X))),(One_ ((X),(RAlgebra the carrier of X))),(Zero_ ((X),(RAlgebra the carrier of X))) #) is strict AlgebraStr
X is non empty TopSpace-like TopStruct
(X) is non empty strict AlgebraStr
(X) is non empty add-closed having-inverse additively-closed multiplicatively-closed additively-linearly-closed Element of bool the carrier of (RAlgebra the carrier of X)
the carrier of X is non empty set
RAlgebra the carrier of X is non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative unital associative commutative right-distributive right_unital well-unital left_unital strict vector-associative AlgebraStr
Funcs ( the carrier of X,REAL) is non empty functional FUNCTION_DOMAIN of the carrier of X, REAL
RealFuncMult the carrier of X is non empty Relation-like [:(Funcs ( the carrier of X,REAL)),(Funcs ( the carrier of X,REAL)):] -defined Funcs ( the carrier of X,REAL) -valued Function-like total quasi_total Element of bool [:[:(Funcs ( the carrier of X,REAL)),(Funcs ( the carrier of X,REAL)):],(Funcs ( the carrier of X,REAL)):]
[:(Funcs ( the carrier of X,REAL)),(Funcs ( the carrier of X,REAL)):] is non empty set
[:[:(Funcs ( the carrier of X,REAL)),(Funcs ( the carrier of X,REAL)):],(Funcs ( the carrier of X,REAL)):] is non empty set
bool [:[:(Funcs ( the carrier of X,REAL)),(Funcs ( the carrier of X,REAL)):],(Funcs ( the carrier of X,REAL)):] is non empty set
RealFuncAdd the carrier of X is non empty Relation-like [:(Funcs ( the carrier of X,REAL)),(Funcs ( the carrier of X,REAL)):] -defined Funcs ( the carrier of X,REAL) -valued Function-like total quasi_total Element of bool [:[:(Funcs ( the carrier of X,REAL)),(Funcs ( the carrier of X,REAL)):],(Funcs ( the carrier of X,REAL)):]
RealFuncExtMult the carrier of X is non empty Relation-like [:REAL,(Funcs ( the carrier of X,REAL)):] -defined Funcs ( the carrier of X,REAL) -valued Function-like total quasi_total Element of bool [:[:REAL,(Funcs ( the carrier of X,REAL)):],(Funcs ( the carrier of X,REAL)):]
[:REAL,(Funcs ( the carrier of X,REAL)):] is non empty set
[:[:REAL,(Funcs ( the carrier of X,REAL)):],(Funcs ( the carrier of X,REAL)):] is non empty set
bool [:[:REAL,(Funcs ( the carrier of X,REAL)):],(Funcs ( the carrier of X,REAL)):] is non empty set
RealFuncUnit the carrier of X is Relation-like the carrier of X -defined REAL -valued Function-like total quasi_total V152() V153() V154() Element of Funcs ( the carrier of X,REAL)
the carrier of X --> 1 is non empty Relation-like the carrier of X -defined NAT -valued RAT -valued INT -valued Function-like total quasi_total V152() V153() V154() V155() Element of bool [: the carrier of X,NAT:]
[: the carrier of X,NAT:] is non empty RAT -valued INT -valued V152() V153() V154() V155() set
bool [: the carrier of X,NAT:] is non empty set
RealFuncZero the carrier of X is Relation-like the carrier of X -defined REAL -valued Function-like total quasi_total V152() V153() V154() Element of Funcs ( the carrier of X,REAL)
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 V152() V153() V154() V155() Element of bool [: the carrier of X,NAT:]
AlgebraStr(# (Funcs ( the carrier of X,REAL)),(RealFuncMult the carrier of X),(RealFuncAdd the carrier of X),(RealFuncExtMult the carrier of X),(RealFuncUnit the carrier of X),(RealFuncZero the carrier of X) #) is strict AlgebraStr
the carrier of (RAlgebra the carrier of X) is non empty set
bool the carrier of (RAlgebra the carrier of X) is non empty set
[: the carrier of X,REAL:] is non empty V152() V153() V154() set
bool [: the carrier of X,REAL:] is non empty set
{ b₁ where b₁ is non empty Relation-like the carrier of X -defined REAL -valued Function-like total quasi_total V152() V153() V154() continuous Element of bool [: the carrier of X,REAL:] : verum } is set
mult_ ((X),(RAlgebra 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),(RAlgebra 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),(RAlgebra the carrier of X)) is non empty Relation-like [:REAL,(X):] -defined (X) -valued Function-like total quasi_total Element of bool [:[:REAL,(X):],(X):]
[:REAL,(X):] is non empty set
[:[:REAL,(X):],(X):] is non empty set
bool [:[:REAL,(X):],(X):] is non empty set
One_ ((X),(RAlgebra the carrier of X)) is Element of (X)
Zero_ ((X),(RAlgebra the carrier of X)) is Element of (X)
AlgebraStr(# (X),(mult_ ((X),(RAlgebra the carrier of X))),(Add_ ((X),(RAlgebra the carrier of X))),(Mult_ ((X),(RAlgebra the carrier of X))),(One_ ((X),(RAlgebra the carrier of X))),(Zero_ ((X),(RAlgebra the carrier of X))) #) is strict AlgebraStr
the carrier of (X) is non empty set
X is Element of the carrier of (X)
1 * X is Element of the carrier of (X)
the Mult of (X) is non empty Relation-like [:REAL, the carrier of (X):] -defined the carrier of (X) -valued Function-like total quasi_total Element of bool [:[:REAL, the carrier of (X):], the carrier of (X):]
[:REAL, the carrier of (X):] is non empty set
[:[:REAL, the carrier of (X):], the carrier of (X):] is non empty set
bool [:[:REAL, the carrier of (X):], the carrier of (X):] is non empty set
the Mult of (X) . (1,X) is set
[1,X] is set
{1,X} is non empty set
{1} is non empty V185() V186() V187() V188() V189() V190() left_end bounded_below set
{{1,X},{1}} is non empty set
the Mult of (X) . [1,X] is set
F is right_complementable Element of the carrier of (RAlgebra the carrier of X)
1 * F is right_complementable Element of the carrier of (RAlgebra the carrier of X)
the Mult of (RAlgebra the carrier of X) is non empty Relation-like [:REAL, the carrier of (RAlgebra the carrier of X):] -defined the carrier of (RAlgebra the carrier of X) -valued Function-like total quasi_total Element of bool [:[:REAL, the carrier of (RAlgebra the carrier of X):], the carrier of (RAlgebra the carrier of X):]
[:REAL, the carrier of (RAlgebra the carrier of X):] is non empty set
[:[:REAL, the carrier of (RAlgebra the carrier of X):], the carrier of (RAlgebra the carrier of X):] is non empty set
bool [:[:REAL, the carrier of (RAlgebra the carrier of X):], the carrier of (RAlgebra the carrier of X):] is non empty set
the Mult of (RAlgebra the carrier of X) . (1,F) is set
[1,F] is set
{1,F} is non empty set
{{1,F},{1}} is non empty set
the Mult of (RAlgebra the carrier of X) . [1,F] is 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 unital associative commutative right-distributive right_unital well-unital left_unital strict vector-associative AlgebraStr
(X) is non empty add-closed having-inverse additively-closed multiplicatively-closed additively-linearly-closed Element of bool the carrier of (RAlgebra the carrier of X)
the carrier of X is non empty set
RAlgebra the carrier of X is non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative unital associative commutative right-distributive right_unital well-unital left_unital strict vector-associative AlgebraStr
Funcs ( the carrier of X,REAL) is non empty functional FUNCTION_DOMAIN of the carrier of X, REAL
RealFuncMult the carrier of X is non empty Relation-like [:(Funcs ( the carrier of X,REAL)),(Funcs ( the carrier of X,REAL)):] -defined Funcs ( the carrier of X,REAL) -valued Function-like total quasi_total Element of bool [:[:(Funcs ( the carrier of X,REAL)),(Funcs ( the carrier of X,REAL)):],(Funcs ( the carrier of X,REAL)):]
[:(Funcs ( the carrier of X,REAL)),(Funcs ( the carrier of X,REAL)):] is non empty set
[:[:(Funcs ( the carrier of X,REAL)),(Funcs ( the carrier of X,REAL)):],(Funcs ( the carrier of X,REAL)):] is non empty set
bool [:[:(Funcs ( the carrier of X,REAL)),(Funcs ( the carrier of X,REAL)):],(Funcs ( the carrier of X,REAL)):] is non empty set
RealFuncAdd the carrier of X is non empty Relation-like [:(Funcs ( the carrier of X,REAL)),(Funcs ( the carrier of X,REAL)):] -defined Funcs ( the carrier of X,REAL) -valued Function-like total quasi_total Element of bool [:[:(Funcs ( the carrier of X,REAL)),(Funcs ( the carrier of X,REAL)):],(Funcs ( the carrier of X,REAL)):]
RealFuncExtMult the carrier of X is non empty Relation-like [:REAL,(Funcs ( the carrier of X,REAL)):] -defined Funcs ( the carrier of X,REAL) -valued Function-like total quasi_total Element of bool [:[:REAL,(Funcs ( the carrier of X,REAL)):],(Funcs ( the carrier of X,REAL)):]
[:REAL,(Funcs ( the carrier of X,REAL)):] is non empty set
[:[:REAL,(Funcs ( the carrier of X,REAL)):],(Funcs ( the carrier of X,REAL)):] is non empty set
bool [:[:REAL,(Funcs ( the carrier of X,REAL)):],(Funcs ( the carrier of X,REAL)):] is non empty set
RealFuncUnit the carrier of X is Relation-like the carrier of X -defined REAL -valued Function-like total quasi_total V152() V153() V154() Element of Funcs ( the carrier of X,REAL)
the carrier of X --> 1 is non empty Relation-like the carrier of X -defined NAT -valued RAT -valued INT -valued Function-like total quasi_total V152() V153() V154() V155() Element of bool [: the carrier of X,NAT:]
[: the carrier of X,NAT:] is non empty RAT -valued INT -valued V152() V153() V154() V155() set
bool [: the carrier of X,NAT:] is non empty set
RealFuncZero the carrier of X is Relation-like the carrier of X -defined REAL -valued Function-like total quasi_total V152() V153() V154() Element of Funcs ( the carrier of X,REAL)
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 V152() V153() V154() V155() Element of bool [: the carrier of X,NAT:]
AlgebraStr(# (Funcs ( the carrier of X,REAL)),(RealFuncMult the carrier of X),(RealFuncAdd the carrier of X),(RealFuncExtMult the carrier of X),(RealFuncUnit the carrier of X),(RealFuncZero the carrier of X) #) is strict AlgebraStr
the carrier of (RAlgebra the carrier of X) is non empty set
bool the carrier of (RAlgebra the carrier of X) is non empty set
[: the carrier of X,REAL:] is non empty V152() V153() V154() set
bool [: the carrier of X,REAL:] is non empty set
{ b₁ where b₁ is non empty Relation-like the carrier of X -defined REAL -valued Function-like total quasi_total V152() V153() V154() continuous Element of bool [: the carrier of X,REAL:] : verum } is set
mult_ ((X),(RAlgebra 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),(RAlgebra 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),(RAlgebra the carrier of X)) is non empty Relation-like [:REAL,(X):] -defined (X) -valued Function-like total quasi_total Element of bool [:[:REAL,(X):],(X):]
[:REAL,(X):] is non empty set
[:[:REAL,(X):],(X):] is non empty set
bool [:[:REAL,(X):],(X):] is non empty set
One_ ((X),(RAlgebra the carrier of X)) is Element of (X)
Zero_ ((X),(RAlgebra the carrier of X)) is Element of (X)
AlgebraStr(# (X),(mult_ ((X),(RAlgebra the carrier of X))),(Add_ ((X),(RAlgebra the carrier of X))),(Mult_ ((X),(RAlgebra the carrier of X))),(One_ ((X),(RAlgebra the carrier of X))),(Zero_ ((X),(RAlgebra the carrier of X))) #) is strict AlgebraStr
the carrier of (X) is non empty set
X is right_complementable Element of the carrier of (X)
F is right_complementable Element of the carrier of (X)
G is right_complementable Element of the carrier of (X)
X + F is right_complementable 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 right_complementable 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 REAL -valued Function-like total quasi_total V152() V153() V154() Element of bool [: the carrier of X,REAL:]
GB is non empty Relation-like the carrier of X -defined REAL -valued Function-like total quasi_total V152() V153() V154() Element of bool [: the carrier of X,REAL:]
aFB is non empty Relation-like the carrier of X -defined REAL -valued Function-like total quasi_total V152() V153() V154() Element of bool [: the carrier of X,REAL:]
r is right_complementable Element of the carrier of (RAlgebra the carrier of X)
aF is right_complementable Element of the carrier of (RAlgebra the carrier of X)
aG is right_complementable Element of the carrier of (RAlgebra the carrier of X)
aF + aG is right_complementable Element of the carrier of (RAlgebra the carrier of X)
the addF of (RAlgebra the carrier of X) is non empty Relation-like [: the carrier of (RAlgebra the carrier of X), the carrier of (RAlgebra the carrier of X):] -defined the carrier of (RAlgebra the carrier of X) -valued Function-like total quasi_total Element of bool [:[: the carrier of (RAlgebra the carrier of X), the carrier of (RAlgebra the carrier of X):], the carrier of (RAlgebra the carrier of X):]
[: the carrier of (RAlgebra the carrier of X), the carrier of (RAlgebra the carrier of X):] is non empty set
[:[: the carrier of (RAlgebra the carrier of X), the carrier of (RAlgebra the carrier of X):], the carrier of (RAlgebra the carrier of X):] is non empty set
bool [:[: the carrier of (RAlgebra the carrier of X), the carrier of (RAlgebra the carrier of X):], the carrier of (RAlgebra the carrier of X):] is non empty set
the addF of (RAlgebra the carrier of X) . (aF,aG) is right_complementable Element of the carrier of (RAlgebra 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 (RAlgebra the carrier of X) . [aF,aG] is set
s1 is Element of the carrier of X
aFB . s1 is V11() real ext-real Element of REAL
FB . s1 is V11() real ext-real Element of REAL
GB . s1 is V11() real ext-real Element of REAL
(FB . s1) + (GB . s1) is V11() real ext-real Element of REAL
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 unital associative commutative right-distributive right_unital well-unital left_unital strict vector-associative AlgebraStr
(X) is non empty add-closed having-inverse additively-closed multiplicatively-closed additively-linearly-closed Element of bool the carrier of (RAlgebra the carrier of X)
the carrier of X is non empty set
RAlgebra the carrier of X is non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative unital associative commutative right-distributive right_unital well-unital left_unital strict vector-associative AlgebraStr
Funcs ( the carrier of X,REAL) is non empty functional FUNCTION_DOMAIN of the carrier of X, REAL
RealFuncMult the carrier of X is non empty Relation-like [:(Funcs ( the carrier of X,REAL)),(Funcs ( the carrier of X,REAL)):] -defined Funcs ( the carrier of X,REAL) -valued Function-like total quasi_total Element of bool [:[:(Funcs ( the carrier of X,REAL)),(Funcs ( the carrier of X,REAL)):],(Funcs ( the carrier of X,REAL)):]
[:(Funcs ( the carrier of X,REAL)),(Funcs ( the carrier of X,REAL)):] is non empty set
[:[:(Funcs ( the carrier of X,REAL)),(Funcs ( the carrier of X,REAL)):],(Funcs ( the carrier of X,REAL)):] is non empty set
bool [:[:(Funcs ( the carrier of X,REAL)),(Funcs ( the carrier of X,REAL)):],(Funcs ( the carrier of X,REAL)):] is non empty set
RealFuncAdd the carrier of X is non empty Relation-like [:(Funcs ( the carrier of X,REAL)),(Funcs ( the carrier of X,REAL)):] -defined Funcs ( the carrier of X,REAL) -valued Function-like total quasi_total Element of bool [:[:(Funcs ( the carrier of X,REAL)),(Funcs ( the carrier of X,REAL)):],(Funcs ( the carrier of X,REAL)):]
RealFuncExtMult the carrier of X is non empty Relation-like [:REAL,(Funcs ( the carrier of X,REAL)):] -defined Funcs ( the carrier of X,REAL) -valued Function-like total quasi_total Element of bool [:[:REAL,(Funcs ( the carrier of X,REAL)):],(Funcs ( the carrier of X,REAL)):]
[:REAL,(Funcs ( the carrier of X,REAL)):] is non empty set
[:[:REAL,(Funcs ( the carrier of X,REAL)):],(Funcs ( the carrier of X,REAL)):] is non empty set
bool [:[:REAL,(Funcs ( the carrier of X,REAL)):],(Funcs ( the carrier of X,REAL)):] is non empty set
RealFuncUnit the carrier of X is Relation-like the carrier of X -defined REAL -valued Function-like total quasi_total V152() V153() V154() Element of Funcs ( the carrier of X,REAL)
the carrier of X --> 1 is non empty Relation-like the carrier of X -defined NAT -valued RAT -valued INT -valued Function-like total quasi_total V152() V153() V154() V155() Element of bool [: the carrier of X,NAT:]
[: the carrier of X,NAT:] is non empty RAT -valued INT -valued V152() V153() V154() V155() set
bool [: the carrier of X,NAT:] is non empty set
RealFuncZero the carrier of X is Relation-like the carrier of X -defined REAL -valued Function-like total quasi_total V152() V153() V154() Element of Funcs ( the carrier of X,REAL)
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 V152() V153() V154() V155() Element of bool [: the carrier of X,NAT:]
AlgebraStr(# (Funcs ( the carrier of X,REAL)),(RealFuncMult the carrier of X),(RealFuncAdd the carrier of X),(RealFuncExtMult the carrier of X),(RealFuncUnit the carrier of X),(RealFuncZero the carrier of X) #) is strict AlgebraStr
the carrier of (RAlgebra the carrier of X) is non empty set
bool the carrier of (RAlgebra the carrier of X) is non empty set
[: the carrier of X,REAL:] is non empty V152() V153() V154() set
bool [: the carrier of X,REAL:] is non empty set
{ b₁ where b₁ is non empty Relation-like the carrier of X -defined REAL -valued Function-like total quasi_total V152() V153() V154() continuous Element of bool [: the carrier of X,REAL:] : verum } is set
mult_ ((X),(RAlgebra 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),(RAlgebra 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),(RAlgebra the carrier of X)) is non empty Relation-like [:REAL,(X):] -defined (X) -valued Function-like total quasi_total Element of bool [:[:REAL,(X):],(X):]
[:REAL,(X):] is non empty set
[:[:REAL,(X):],(X):] is non empty set
bool [:[:REAL,(X):],(X):] is non empty set
One_ ((X),(RAlgebra the carrier of X)) is Element of (X)
Zero_ ((X),(RAlgebra the carrier of X)) is Element of (X)
AlgebraStr(# (X),(mult_ ((X),(RAlgebra the carrier of X))),(Add_ ((X),(RAlgebra the carrier of X))),(Mult_ ((X),(RAlgebra the carrier of X))),(One_ ((X),(RAlgebra the carrier of X))),(Zero_ ((X),(RAlgebra the carrier of X))) #) is strict AlgebraStr
the carrier of (X) is non empty set
X is right_complementable Element of the carrier of (X)
F is right_complementable Element of the carrier of (X)
FB is non empty Relation-like the carrier of X -defined REAL -valued Function-like total quasi_total V152() V153() V154() Element of bool [: the carrier of X,REAL:]
GB is non empty Relation-like the carrier of X -defined REAL -valued Function-like total quasi_total V152() V153() V154() Element of bool [: the carrier of X,REAL:]
aF is V11() real ext-real Element of REAL
aF * X is right_complementable Element of the carrier of (X)
the Mult of (X) is non empty Relation-like [:REAL, the carrier of (X):] -defined the carrier of (X) -valued Function-like total quasi_total Element of bool [:[:REAL, the carrier of (X):], the carrier of (X):]
[:REAL, the carrier of (X):] is non empty set
[:[:REAL, the carrier of (X):], the carrier of (X):] is non empty set
bool [:[:REAL, the carrier of (X):], the carrier of (X):] is non empty set
the Mult of (X) . (aF,X) is set
[aF,X] is set
{aF,X} is non empty set
{aF} is non empty V185() V186() V187() set
{{aF,X},{aF}} is non empty set
the Mult of (X) . [aF,X] is set
r is right_complementable Element of the carrier of (RAlgebra the carrier of X)
aG is right_complementable Element of the carrier of (RAlgebra the carrier of X)
aF * aG is right_complementable Element of the carrier of (RAlgebra the carrier of X)
the Mult of (RAlgebra the carrier of X) is non empty Relation-like [:REAL, the carrier of (RAlgebra the carrier of X):] -defined the carrier of (RAlgebra the carrier of X) -valued Function-like total quasi_total Element of bool [:[:REAL, the carrier of (RAlgebra the carrier of X):], the carrier of (RAlgebra the carrier of X):]
[:REAL, the carrier of (RAlgebra the carrier of X):] is non empty set
[:[:REAL, the carrier of (RAlgebra the carrier of X):], the carrier of (RAlgebra the carrier of X):] is non empty set
bool [:[:REAL, the carrier of (RAlgebra the carrier of X):], the carrier of (RAlgebra the carrier of X):] is non empty set
the Mult of (RAlgebra the carrier of X) . (aF,aG) is set
[aF,aG] is set
{aF,aG} is non empty set
{{aF,aG},{aF}} is non empty set
the Mult of (RAlgebra the carrier of X) . [aF,aG] is set
s1 is Element of the carrier of X
GB . s1 is V11() real ext-real Element of REAL
FB . s1 is V11() real ext-real Element of REAL
aF * (FB . s1) is V11() real ext-real Element of REAL
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 unital associative commutative right-distributive right_unital well-unital left_unital strict vector-associative AlgebraStr
(X) is non empty add-closed having-inverse additively-closed multiplicatively-closed additively-linearly-closed Element of bool the carrier of (RAlgebra the carrier of X)
the carrier of X is non empty set
RAlgebra the carrier of X is non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative unital associative commutative right-distributive right_unital well-unital left_unital strict vector-associative AlgebraStr
Funcs ( the carrier of X,REAL) is non empty functional FUNCTION_DOMAIN of the carrier of X, REAL
RealFuncMult the carrier of X is non empty Relation-like [:(Funcs ( the carrier of X,REAL)),(Funcs ( the carrier of X,REAL)):] -defined Funcs ( the carrier of X,REAL) -valued Function-like total quasi_total Element of bool [:[:(Funcs ( the carrier of X,REAL)),(Funcs ( the carrier of X,REAL)):],(Funcs ( the carrier of X,REAL)):]
[:(Funcs ( the carrier of X,REAL)),(Funcs ( the carrier of X,REAL)):] is non empty set
[:[:(Funcs ( the carrier of X,REAL)),(Funcs ( the carrier of X,REAL)):],(Funcs ( the carrier of X,REAL)):] is non empty set
bool [:[:(Funcs ( the carrier of X,REAL)),(Funcs ( the carrier of X,REAL)):],(Funcs ( the carrier of X,REAL)):] is non empty set
RealFuncAdd the carrier of X is non empty Relation-like [:(Funcs ( the carrier of X,REAL)),(Funcs ( the carrier of X,REAL)):] -defined Funcs ( the carrier of X,REAL) -valued Function-like total quasi_total Element of bool [:[:(Funcs ( the carrier of X,REAL)),(Funcs ( the carrier of X,REAL)):],(Funcs ( the carrier of X,REAL)):]
RealFuncExtMult the carrier of X is non empty Relation-like [:REAL,(Funcs ( the carrier of X,REAL)):] -defined Funcs ( the carrier of X,REAL) -valued Function-like total quasi_total Element of bool [:[:REAL,(Funcs ( the carrier of X,REAL)):],(Funcs ( the carrier of X,REAL)):]
[:REAL,(Funcs ( the carrier of X,REAL)):] is non empty set
[:[:REAL,(Funcs ( the carrier of X,REAL)):],(Funcs ( the carrier of X,REAL)):] is non empty set
bool [:[:REAL,(Funcs ( the carrier of X,REAL)):],(Funcs ( the carrier of X,REAL)):] is non empty set
RealFuncUnit the carrier of X is Relation-like the carrier of X -defined REAL -valued Function-like total quasi_total V152() V153() V154() Element of Funcs ( the carrier of X,REAL)
the carrier of X --> 1 is non empty Relation-like the carrier of X -defined NAT -valued RAT -valued INT -valued Function-like total quasi_total V152() V153() V154() V155() Element of bool [: the carrier of X,NAT:]
[: the carrier of X,NAT:] is non empty RAT -valued INT -valued V152() V153() V154() V155() set
bool [: the carrier of X,NAT:] is non empty set
RealFuncZero the carrier of X is Relation-like the carrier of X -defined REAL -valued Function-like total quasi_total V152() V153() V154() Element of Funcs ( the carrier of X,REAL)
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 V152() V153() V154() V155() Element of bool [: the carrier of X,NAT:]
AlgebraStr(# (Funcs ( the carrier of X,REAL)),(RealFuncMult the carrier of X),(RealFuncAdd the carrier of X),(RealFuncExtMult the carrier of X),(RealFuncUnit the carrier of X),(RealFuncZero the carrier of X) #) is strict AlgebraStr
the carrier of (RAlgebra the carrier of X) is non empty set
bool the carrier of (RAlgebra the carrier of X) is non empty set
[: the carrier of X,REAL:] is non empty V152() V153() V154() set
bool [: the carrier of X,REAL:] is non empty set
{ b₁ where b₁ is non empty Relation-like the carrier of X -defined REAL -valued Function-like total quasi_total V152() V153() V154() continuous Element of bool [: the carrier of X,REAL:] : verum } is set
mult_ ((X),(RAlgebra 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),(RAlgebra 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),(RAlgebra the carrier of X)) is non empty Relation-like [:REAL,(X):] -defined (X) -valued Function-like total quasi_total Element of bool [:[:REAL,(X):],(X):]
[:REAL,(X):] is non empty set
[:[:REAL,(X):],(X):] is non empty set
bool [:[:REAL,(X):],(X):] is non empty set
One_ ((X),(RAlgebra the carrier of X)) is Element of (X)
Zero_ ((X),(RAlgebra the carrier of X)) is Element of (X)
AlgebraStr(# (X),(mult_ ((X),(RAlgebra the carrier of X))),(Add_ ((X),(RAlgebra the carrier of X))),(Mult_ ((X),(RAlgebra the carrier of X))),(One_ ((X),(RAlgebra the carrier of X))),(Zero_ ((X),(RAlgebra the carrier of X))) #) is strict AlgebraStr
the carrier of (X) is non empty set
X is right_complementable Element of the carrier of (X)
F is right_complementable Element of the carrier of (X)
G is right_complementable Element of the carrier of (X)
X * F is right_complementable 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 right_complementable 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 REAL -valued Function-like total quasi_total V152() V153() V154() Element of bool [: the carrier of X,REAL:]
GB is non empty Relation-like the carrier of X -defined REAL -valued Function-like total quasi_total V152() V153() V154() Element of bool [: the carrier of X,REAL:]
aFB is non empty Relation-like the carrier of X -defined REAL -valued Function-like total quasi_total V152() V153() V154() Element of bool [: the carrier of X,REAL:]
r is right_complementable Element of the carrier of (RAlgebra the carrier of X)
aF is right_complementable Element of the carrier of (RAlgebra the carrier of X)
aG is right_complementable Element of the carrier of (RAlgebra the carrier of X)
aF * aG is right_complementable Element of the carrier of (RAlgebra the carrier of X)
the multF of (RAlgebra the carrier of X) is non empty Relation-like [: the carrier of (RAlgebra the carrier of X), the carrier of (RAlgebra the carrier of X):] -defined the carrier of (RAlgebra the carrier of X) -valued Function-like total quasi_total Element of bool [:[: the carrier of (RAlgebra the carrier of X), the carrier of (RAlgebra the carrier of X):], the carrier of (RAlgebra the carrier of X):]
[: the carrier of (RAlgebra the carrier of X), the carrier of (RAlgebra the carrier of X):] is non empty set
[:[: the carrier of (RAlgebra the carrier of X), the carrier of (RAlgebra the carrier of X):], the carrier of (RAlgebra the carrier of X):] is non empty set
bool [:[: the carrier of (RAlgebra the carrier of X), the carrier of (RAlgebra the carrier of X):], the carrier of (RAlgebra the carrier of X):] is non empty set
the multF of (RAlgebra the carrier of X) . (aF,aG) is right_complementable Element of the carrier of (RAlgebra 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 (RAlgebra the carrier of X) . [aF,aG] is set
s1 is Element of the carrier of X
aFB . s1 is V11() real ext-real Element of REAL
FB . s1 is V11() real ext-real Element of REAL
GB . s1 is V11() real ext-real Element of REAL
(FB . s1) * (GB . s1) is V11() real ext-real Element of REAL
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 unital associative commutative right-distributive right_unital well-unital left_unital strict vector-associative AlgebraStr
(X) is non empty add-closed having-inverse additively-closed multiplicatively-closed additively-linearly-closed Element of bool the carrier of (RAlgebra the carrier of X)
the carrier of X is non empty set
RAlgebra the carrier of X is non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative unital associative commutative right-distributive right_unital well-unital left_unital strict vector-associative AlgebraStr
Funcs ( the carrier of X,REAL) is non empty functional FUNCTION_DOMAIN of the carrier of X, REAL
RealFuncMult the carrier of X is non empty Relation-like [:(Funcs ( the carrier of X,REAL)),(Funcs ( the carrier of X,REAL)):] -defined Funcs ( the carrier of X,REAL) -valued Function-like total quasi_total Element of bool [:[:(Funcs ( the carrier of X,REAL)),(Funcs ( the carrier of X,REAL)):],(Funcs ( the carrier of X,REAL)):]
[:(Funcs ( the carrier of X,REAL)),(Funcs ( the carrier of X,REAL)):] is non empty set
[:[:(Funcs ( the carrier of X,REAL)),(Funcs ( the carrier of X,REAL)):],(Funcs ( the carrier of X,REAL)):] is non empty set
bool [:[:(Funcs ( the carrier of X,REAL)),(Funcs ( the carrier of X,REAL)):],(Funcs ( the carrier of X,REAL)):] is non empty set
RealFuncAdd the carrier of X is non empty Relation-like [:(Funcs ( the carrier of X,REAL)),(Funcs ( the carrier of X,REAL)):] -defined Funcs ( the carrier of X,REAL) -valued Function-like total quasi_total Element of bool [:[:(Funcs ( the carrier of X,REAL)),(Funcs ( the carrier of X,REAL)):],(Funcs ( the carrier of X,REAL)):]
RealFuncExtMult the carrier of X is non empty Relation-like [:REAL,(Funcs ( the carrier of X,REAL)):] -defined Funcs ( the carrier of X,REAL) -valued Function-like total quasi_total Element of bool [:[:REAL,(Funcs ( the carrier of X,REAL)):],(Funcs ( the carrier of X,REAL)):]
[:REAL,(Funcs ( the carrier of X,REAL)):] is non empty set
[:[:REAL,(Funcs ( the carrier of X,REAL)):],(Funcs ( the carrier of X,REAL)):] is non empty set
bool [:[:REAL,(Funcs ( the carrier of X,REAL)):],(Funcs ( the carrier of X,REAL)):] is non empty set
RealFuncUnit the carrier of X is Relation-like the carrier of X -defined REAL -valued Function-like total quasi_total V152() V153() V154() Element of Funcs ( the carrier of X,REAL)
the carrier of X --> 1 is non empty Relation-like the carrier of X -defined NAT -valued RAT -valued INT -valued Function-like total quasi_total V152() V153() V154() V155() Element of bool [: the carrier of X,NAT:]
[: the carrier of X,NAT:] is non empty RAT -valued INT -valued V152() V153() V154() V155() set
bool [: the carrier of X,NAT:] is non empty set
RealFuncZero the carrier of X is Relation-like the carrier of X -defined REAL -valued Function-like total quasi_total V152() V153() V154() Element of Funcs ( the carrier of X,REAL)
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 V152() V153() V154() V155() Element of bool [: the carrier of X,NAT:]
AlgebraStr(# (Funcs ( the carrier of X,REAL)),(RealFuncMult the carrier of X),(RealFuncAdd the carrier of X),(RealFuncExtMult the carrier of X),(RealFuncUnit the carrier of X),(RealFuncZero the carrier of X) #) is strict AlgebraStr
the carrier of (RAlgebra the carrier of X) is non empty set
bool the carrier of (RAlgebra the carrier of X) is non empty set
[: the carrier of X,REAL:] is non empty V152() V153() V154() set
bool [: the carrier of X,REAL:] is non empty set
{ b₁ where b₁ is non empty Relation-like the carrier of X -defined REAL -valued Function-like total quasi_total V152() V153() V154() continuous Element of bool [: the carrier of X,REAL:] : verum } is set
mult_ ((X),(RAlgebra 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),(RAlgebra 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),(RAlgebra the carrier of X)) is non empty Relation-like [:REAL,(X):] -defined (X) -valued Function-like total quasi_total Element of bool [:[:REAL,(X):],(X):]
[:REAL,(X):] is non empty set
[:[:REAL,(X):],(X):] is non empty set
bool [:[:REAL,(X):],(X):] is non empty set
One_ ((X),(RAlgebra the carrier of X)) is Element of (X)
Zero_ ((X),(RAlgebra the carrier of X)) is Element of (X)
AlgebraStr(# (X),(mult_ ((X),(RAlgebra the carrier of X))),(Add_ ((X),(RAlgebra the carrier of X))),(Mult_ ((X),(RAlgebra the carrier of X))),(One_ ((X),(RAlgebra the carrier of X))),(Zero_ ((X),(RAlgebra the carrier of X))) #) is strict AlgebraStr
0. (X) is V49((X)) right_complementable Element of the carrier of (X)
the carrier of (X) is non empty set
the ZeroF of (X) is right_complementable Element of the carrier of (X)
(X,0) is non empty Relation-like the carrier of X -defined REAL -valued Function-like total quasi_total V152() V153() V154() continuous Element of bool [: the carrier of X,REAL:]
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 V152() V153() V154() V155() Element of bool [: the carrier of X,{0}:]
{0} is non empty V185() V186() V187() V188() V189() V190() left_end bounded_below set
[: the carrier of X,{0}:] is non empty RAT -valued INT -valued V152() V153() V154() V155() set
bool [: the carrier of X,{0}:] is non empty set
0. (RAlgebra the carrier of X) is V49( RAlgebra the carrier of X) right_complementable Element of the carrier of (RAlgebra the carrier of X)
the ZeroF of (RAlgebra the carrier of X) is right_complementable Element of the carrier of (RAlgebra the carrier of X)
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 unital associative commutative right-distributive right_unital well-unital left_unital strict vector-associative AlgebraStr
(X) is non empty add-closed having-inverse additively-closed multiplicatively-closed additively-linearly-closed Element of bool the carrier of (RAlgebra the carrier of X)
the carrier of X is non empty set
RAlgebra the carrier of X is non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative unital associative commutative right-distributive right_unital well-unital left_unital strict vector-associative AlgebraStr
Funcs ( the carrier of X,REAL) is non empty functional FUNCTION_DOMAIN of the carrier of X, REAL
RealFuncMult the carrier of X is non empty Relation-like [:(Funcs ( the carrier of X,REAL)),(Funcs ( the carrier of X,REAL)):] -defined Funcs ( the carrier of X,REAL) -valued Function-like total quasi_total Element of bool [:[:(Funcs ( the carrier of X,REAL)),(Funcs ( the carrier of X,REAL)):],(Funcs ( the carrier of X,REAL)):]
[:(Funcs ( the carrier of X,REAL)),(Funcs ( the carrier of X,REAL)):] is non empty set
[:[:(Funcs ( the carrier of X,REAL)),(Funcs ( the carrier of X,REAL)):],(Funcs ( the carrier of X,REAL)):] is non empty set
bool [:[:(Funcs ( the carrier of X,REAL)),(Funcs ( the carrier of X,REAL)):],(Funcs ( the carrier of X,REAL)):] is non empty set
RealFuncAdd the carrier of X is non empty Relation-like [:(Funcs ( the carrier of X,REAL)),(Funcs ( the carrier of X,REAL)):] -defined Funcs ( the carrier of X,REAL) -valued Function-like total quasi_total Element of bool [:[:(Funcs ( the carrier of X,REAL)),(Funcs ( the carrier of X,REAL)):],(Funcs ( the carrier of X,REAL)):]
RealFuncExtMult the carrier of X is non empty Relation-like [:REAL,(Funcs ( the carrier of X,REAL)):] -defined Funcs ( the carrier of X,REAL) -valued Function-like total quasi_total Element of bool [:[:REAL,(Funcs ( the carrier of X,REAL)):],(Funcs ( the carrier of X,REAL)):]
[:REAL,(Funcs ( the carrier of X,REAL)):] is non empty set
[:[:REAL,(Funcs ( the carrier of X,REAL)):],(Funcs ( the carrier of X,REAL)):] is non empty set
bool [:[:REAL,(Funcs ( the carrier of X,REAL)):],(Funcs ( the carrier of X,REAL)):] is non empty set
RealFuncUnit the carrier of X is Relation-like the carrier of X -defined REAL -valued Function-like total quasi_total V152() V153() V154() Element of Funcs ( the carrier of X,REAL)
the carrier of X --> 1 is non empty Relation-like the carrier of X -defined NAT -valued RAT -valued INT -valued Function-like total quasi_total V152() V153() V154() V155() Element of bool [: the carrier of X,NAT:]
[: the carrier of X,NAT:] is non empty RAT -valued INT -valued V152() V153() V154() V155() set
bool [: the carrier of X,NAT:] is non empty set
RealFuncZero the carrier of X is Relation-like the carrier of X -defined REAL -valued Function-like total quasi_total V152() V153() V154() Element of Funcs ( the carrier of X,REAL)
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 V152() V153() V154() V155() Element of bool [: the carrier of X,NAT:]
AlgebraStr(# (Funcs ( the carrier of X,REAL)),(RealFuncMult the carrier of X),(RealFuncAdd the carrier of X),(RealFuncExtMult the carrier of X),(RealFuncUnit the carrier of X),(RealFuncZero the carrier of X) #) is strict AlgebraStr
the carrier of (RAlgebra the carrier of X) is non empty set
bool the carrier of (RAlgebra the carrier of X) is non empty set
[: the carrier of X,REAL:] is non empty V152() V153() V154() set
bool [: the carrier of X,REAL:] is non empty set
{ b₁ where b₁ is non empty Relation-like the carrier of X -defined REAL -valued Function-like total quasi_total V152() V153() V154() continuous Element of bool [: the carrier of X,REAL:] : verum } is set
mult_ ((X),(RAlgebra 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),(RAlgebra 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),(RAlgebra the carrier of X)) is non empty Relation-like [:REAL,(X):] -defined (X) -valued Function-like total quasi_total Element of bool [:[:REAL,(X):],(X):]
[:REAL,(X):] is non empty set
[:[:REAL,(X):],(X):] is non empty set
bool [:[:REAL,(X):],(X):] is non empty set
One_ ((X),(RAlgebra the carrier of X)) is Element of (X)
Zero_ ((X),(RAlgebra the carrier of X)) is Element of (X)
AlgebraStr(# (X),(mult_ ((X),(RAlgebra the carrier of X))),(Add_ ((X),(RAlgebra the carrier of X))),(Mult_ ((X),(RAlgebra the carrier of X))),(One_ ((X),(RAlgebra the carrier of X))),(Zero_ ((X),(RAlgebra the carrier of X))) #) is strict AlgebraStr
1_ (X) is right_complementable Element of the carrier of (X)
the carrier of (X) is non empty set
1. (X) is right_complementable Element of the carrier of (X)
the OneF of (X) is right_complementable Element of the carrier of (X)
(X,1) is non empty Relation-like the carrier of X -defined REAL -valued Function-like total quasi_total V152() V153() V154() continuous Element of bool [: the carrier of X,REAL:]
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 V152() V153() V154() V155() Element of bool [: the carrier of X,{1}:]
{1} is non empty V185() V186() V187() V188() V189() V190() left_end bounded_below set
[: the carrier of X,{1}:] is non empty RAT -valued INT -valued V152() V153() V154() V155() set
bool [: the carrier of X,{1}:] is non empty set
1_ (RAlgebra the carrier of X) is right_complementable Element of the carrier of (RAlgebra the carrier of X)
1. (RAlgebra the carrier of X) is right_complementable Element of the carrier of (RAlgebra the carrier of X)
the OneF of (RAlgebra the carrier of X) is right_complementable Element of the carrier of (RAlgebra the carrier of X)
X is non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative unital associative commutative right-distributive right_unital well-unital left_unital vector-associative AlgebraStr
X is non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative unital associative commutative right-distributive right_unital well-unital left_unital vector-associative Subalgebra of X
the carrier of X is non empty set
F is non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative unital associative commutative right-distributive right_unital well-unital left_unital vector-associative Subalgebra 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 [:REAL, the carrier of X:] -defined the carrier of X -valued Function-like total quasi_total Element of bool [:[:REAL, the carrier of X:], the carrier of X:]
[:REAL, the carrier of X:] is non empty set
[:[:REAL, the carrier of X:], the carrier of X:] is non empty set
bool [:[:REAL, the carrier of X:], the carrier of X:] is non empty set
0. X is V49(X) right_complementable Element of the carrier of X
the ZeroF of X is right_complementable Element of the carrier of X
0. X is V49(X) right_complementable Element of the carrier of X
the ZeroF of X is right_complementable Element of the carrier of X
0. F is V49(F) right_complementable Element of the carrier of F
the ZeroF of F is right_complementable Element of the carrier of F
1. X is right_complementable Element of the carrier of X
the OneF of X is right_complementable Element of the carrier of X
1. X is right_complementable Element of the carrier of X
the OneF of X is right_complementable Element of the carrier of X
1. F is right_complementable Element of the carrier of F
the OneF of F is right_complementable 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 [:REAL, the carrier of X:] -defined the carrier of X -valued Function-like total quasi_total Element of bool [:[:REAL, the carrier of X:], the carrier of X:]
[:REAL, the carrier of X:] is non empty set
[:[:REAL, the carrier of X:], the carrier of X:] is non empty set
bool [:[:REAL, the carrier of X:], the carrier of X:] is non empty set
[:REAL, the carrier of F:] is non empty set
the Mult of F is non empty Relation-like [:REAL, the carrier of F:] -defined the carrier of F -valued Function-like total quasi_total Element of bool [:[:REAL, the carrier of F:], the carrier of F:]
[:[:REAL, the carrier of F:], the carrier of F:] is non empty set
bool [:[:REAL, the carrier of F:], the carrier of F:] is non empty set
the Mult of F | [:REAL, the carrier of X:] is Relation-like [:REAL, the carrier of F:] -defined the carrier of F -valued Function-like Element of bool [:[:REAL, the carrier of F:], the carrier of F:]
the Mult of X | [:REAL, the carrier of X:] is Relation-like [:REAL, the carrier of X:] -defined the carrier of X -valued Function-like Element of bool [:[:REAL, the carrier of X:], the carrier of X:]
the Mult of X | [:REAL, the carrier of F:] is Relation-like [:REAL, the carrier of X:] -defined the carrier of X -valued Function-like Element of bool [:[:REAL, the carrier of X:], the carrier of X:]
X is non empty TopSpace-like pseudocompact compact TopStruct
(X) is non empty add-closed having-inverse additively-closed multiplicatively-closed additively-linearly-closed Element of bool the carrier of (RAlgebra the carrier of X)
the carrier of X is non empty set
RAlgebra the carrier of X is non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative unital associative commutative right-distributive right_unital well-unital left_unital strict vector-associative AlgebraStr
Funcs ( the carrier of X,REAL) is non empty functional FUNCTION_DOMAIN of the carrier of X, REAL
RealFuncMult the carrier of X is non empty Relation-like [:(Funcs ( the carrier of X,REAL)),(Funcs ( the carrier of X,REAL)):] -defined Funcs ( the carrier of X,REAL) -valued Function-like total quasi_total Element of bool [:[:(Funcs ( the carrier of X,REAL)),(Funcs ( the carrier of X,REAL)):],(Funcs ( the carrier of X,REAL)):]
[:(Funcs ( the carrier of X,REAL)),(Funcs ( the carrier of X,REAL)):] is non empty set
[:[:(Funcs ( the carrier of X,REAL)),(Funcs ( the carrier of X,REAL)):],(Funcs ( the carrier of X,REAL)):] is non empty set
bool [:[:(Funcs ( the carrier of X,REAL)),(Funcs ( the carrier of X,REAL)):],(Funcs ( the carrier of X,REAL)):] is non empty set
RealFuncAdd the carrier of X is non empty Relation-like [:(Funcs ( the carrier of X,REAL)),(Funcs ( the carrier of X,REAL)):] -defined Funcs ( the carrier of X,REAL) -valued Function-like total quasi_total Element of bool [:[:(Funcs ( the carrier of X,REAL)),(Funcs ( the carrier of X,REAL)):],(Funcs ( the carrier of X,REAL)):]
RealFuncExtMult the carrier of X is non empty Relation-like [:REAL,(Funcs ( the carrier of X,REAL)):] -defined Funcs ( the carrier of X,REAL) -valued Function-like total quasi_total Element of bool [:[:REAL,(Funcs ( the carrier of X,REAL)):],(Funcs ( the carrier of X,REAL)):]
[:REAL,(Funcs ( the carrier of X,REAL)):] is non empty set
[:[:REAL,(Funcs ( the carrier of X,REAL)):],(Funcs ( the carrier of X,REAL)):] is non empty set
bool [:[:REAL,(Funcs ( the carrier of X,REAL)):],(Funcs ( the carrier of X,REAL)):] is non empty set
RealFuncUnit the carrier of X is Relation-like the carrier of X -defined REAL -valued Function-like total quasi_total V152() V153() V154() Element of Funcs ( the carrier of X,REAL)
the carrier of X --> 1 is non empty Relation-like the carrier of X -defined NAT -valued RAT -valued INT -valued Function-like total quasi_total V152() V153() V154() V155() Element of bool [: the carrier of X,NAT:]
[: the carrier of X,NAT:] is non empty RAT -valued INT -valued V152() V153() V154() V155() set
bool [: the carrier of X,NAT:] is non empty set
RealFuncZero the carrier of X is Relation-like the carrier of X -defined REAL -valued Function-like total quasi_total V152() V153() V154() Element of Funcs ( the carrier of X,REAL)
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 V152() V153() V154() V155() Element of bool [: the carrier of X,NAT:]
AlgebraStr(# (Funcs ( the carrier of X,REAL)),(RealFuncMult the carrier of X),(RealFuncAdd the carrier of X),(RealFuncExtMult the carrier of X),(RealFuncUnit the carrier of X),(RealFuncZero the carrier of X) #) is strict AlgebraStr
the carrier of (RAlgebra the carrier of X) is non empty set
bool the carrier of (RAlgebra the carrier of X) is non empty set
[: the carrier of X,REAL:] is non empty V152() V153() V154() set
bool [: the carrier of X,REAL:] is non empty set
{ b₁ where b₁ is non empty Relation-like the carrier of X -defined REAL -valued Function-like total quasi_total V152() V153() V154() continuous Element of bool [: the carrier of X,REAL:] : verum } is set
BoundedFunctions the carrier of X is non empty add-closed having-inverse additively-closed multiplicatively-closed additively-linearly-closed Element of bool the carrier of (RAlgebra the carrier of X)
{ b₁ where b₁ is non empty Relation-like the carrier of X -defined REAL -valued Function-like total quasi_total V152() V153() V154() Element of bool [: the carrier of X,REAL:] : b₁ | the carrier of X is bounded } is set
X is set
F is non empty Relation-like the carrier of X -defined REAL -valued Function-like total quasi_total V152() V153() V154() V206( the carrier of X) V207( the carrier of X) continuous bounded Element of bool [: the carrier of X,REAL:]
dom F is Element of bool the carrier of X
bool the carrier of X is non empty set
G is V11() real ext-real set
the carrier of X /\ (dom F) is Element of bool the carrier of X
FB is set
F . FB is V11() real ext-real set
F | the carrier of X is Relation-like the carrier of X -defined REAL -valued Function-like V152() V153() V154() Element of bool [: the carrier of X,REAL:]
FB is V11() real ext-real set
GB is set
F . GB is V11() real ext-real set
the carrier of X /\ the carrier of X is set
X is non empty TopSpace-like pseudocompact compact TopStruct
(X) is non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital unital associative commutative right-distributive right_unital well-unital left_unital strict vector-associative AlgebraStr
(X) is non empty add-closed having-inverse additively-closed multiplicatively-closed additively-linearly-closed Element of bool the carrier of (RAlgebra the carrier of X)
the carrier of X is non empty set
RAlgebra the carrier of X is non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative unital associative commutative right-distributive right_unital well-unital left_unital strict vector-associative AlgebraStr
Funcs ( the carrier of X,REAL) is non empty functional FUNCTION_DOMAIN of the carrier of X, REAL
RealFuncMult the carrier of X is non empty Relation-like [:(Funcs ( the carrier of X,REAL)),(Funcs ( the carrier of X,REAL)):] -defined Funcs ( the carrier of X,REAL) -valued Function-like total quasi_total Element of bool [:[:(Funcs ( the carrier of X,REAL)),(Funcs ( the carrier of X,REAL)):],(Funcs ( the carrier of X,REAL)):]
[:(Funcs ( the carrier of X,REAL)),(Funcs ( the carrier of X,REAL)):] is non empty set
[:[:(Funcs ( the carrier of X,REAL)),(Funcs ( the carrier of X,REAL)):],(Funcs ( the carrier of X,REAL)):] is non empty set
bool [:[:(Funcs ( the carrier of X,REAL)),(Funcs ( the carrier of X,REAL)):],(Funcs ( the carrier of X,REAL)):] is non empty set
RealFuncAdd the carrier of X is non empty Relation-like [:(Funcs ( the carrier of X,REAL)),(Funcs ( the carrier of X,REAL)):] -defined Funcs ( the carrier of X,REAL) -valued Function-like total quasi_total Element of bool [:[:(Funcs ( the carrier of X,REAL)),(Funcs ( the carrier of X,REAL)):],(Funcs ( the carrier of X,REAL)):]
RealFuncExtMult the carrier of X is non empty Relation-like [:REAL,(Funcs ( the carrier of X,REAL)):] -defined Funcs ( the carrier of X,REAL) -valued Function-like total quasi_total Element of bool [:[:REAL,(Funcs ( the carrier of X,REAL)):],(Funcs ( the carrier of X,REAL)):]
[:REAL,(Funcs ( the carrier of X,REAL)):] is non empty set
[:[:REAL,(Funcs ( the carrier of X,REAL)):],(Funcs ( the carrier of X,REAL)):] is non empty set
bool [:[:REAL,(Funcs ( the carrier of X,REAL)):],(Funcs ( the carrier of X,REAL)):] is non empty set
RealFuncUnit the carrier of X is Relation-like the carrier of X -defined REAL -valued Function-like total quasi_total V152() V153() V154() Element of Funcs ( the carrier of X,REAL)
the carrier of X --> 1 is non empty Relation-like the carrier of X -defined NAT -valued RAT -valued INT -valued Function-like total quasi_total V152() V153() V154() V155() Element of bool [: the carrier of X,NAT:]
[: the carrier of X,NAT:] is non empty RAT -valued INT -valued V152() V153() V154() V155() set
bool [: the carrier of X,NAT:] is non empty set
RealFuncZero the carrier of X is Relation-like the carrier of X -defined REAL -valued Function-like total quasi_total V152() V153() V154() Element of Funcs ( the carrier of X,REAL)
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 V152() V153() V154() V155() Element of bool [: the carrier of X,NAT:]
AlgebraStr(# (Funcs ( the carrier of X,REAL)),(RealFuncMult the carrier of X),(RealFuncAdd the carrier of X),(RealFuncExtMult the carrier of X),(RealFuncUnit the carrier of X),(RealFuncZero the carrier of X) #) is strict AlgebraStr
the carrier of (RAlgebra the carrier of X) is non empty set
bool the carrier of (RAlgebra the carrier of X) is non empty set
[: the carrier of X,REAL:] is non empty V152() V153() V154() set
bool [: the carrier of X,REAL:] is non empty set
{ b₁ where b₁ is non empty Relation-like the carrier of X -defined REAL -valued Function-like total quasi_total V152() V153() V154() continuous Element of bool [: the carrier of X,REAL:] : verum } is set
mult_ ((X),(RAlgebra 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),(RAlgebra 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),(RAlgebra the carrier of X)) is non empty Relation-like [:REAL,(X):] -defined (X) -valued Function-like total quasi_total Element of bool [:[:REAL,(X):],(X):]
[:REAL,(X):] is non empty set
[:[:REAL,(X):],(X):] is non empty set
bool [:[:REAL,(X):],(X):] is non empty set
One_ ((X),(RAlgebra the carrier of X)) is Element of (X)
Zero_ ((X),(RAlgebra the carrier of X)) is Element of (X)
AlgebraStr(# (X),(mult_ ((X),(RAlgebra the carrier of X))),(Add_ ((X),(RAlgebra the carrier of X))),(Mult_ ((X),(RAlgebra the carrier of X))),(One_ ((X),(RAlgebra the carrier of X))),(Zero_ ((X),(RAlgebra the carrier of X))) #) is strict AlgebraStr
R_Algebra_of_BoundedFunctions the carrier of X is non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative unital associative commutative right-distributive right_unital well-unital left_unital vector-associative AlgebraStr
BoundedFunctions the carrier of X is non empty add-closed having-inverse additively-closed multiplicatively-closed additively-linearly-closed Element of bool the carrier of (RAlgebra the carrier of X)
{ b₁ where b₁ is non empty Relation-like the carrier of X -defined REAL -valued Function-like total quasi_total V152() V153() V154() Element of bool [: the carrier of X,REAL:] : b₁ | the carrier of X is bounded } is set
mult_ ((BoundedFunctions the carrier of X),(RAlgebra the carrier of X)) is non empty Relation-like [:(BoundedFunctions the carrier of X),(BoundedFunctions the carrier of X):] -defined BoundedFunctions the carrier of X -valued Function-like total quasi_total Element of bool [:[:(BoundedFunctions the carrier of X),(BoundedFunctions the carrier of X):],(BoundedFunctions the carrier of X):]
[:(BoundedFunctions the carrier of X),(BoundedFunctions the carrier of X):] is non empty set
[:[:(BoundedFunctions the carrier of X),(BoundedFunctions the carrier of X):],(BoundedFunctions the carrier of X):] is non empty set
bool [:[:(BoundedFunctions the carrier of X),(BoundedFunctions the carrier of X):],(BoundedFunctions the carrier of X):] is non empty set
Add_ ((BoundedFunctions the carrier of X),(RAlgebra the carrier of X)) is non empty Relation-like [:(BoundedFunctions the carrier of X),(BoundedFunctions the carrier of X):] -defined BoundedFunctions the carrier of X -valued Function-like total quasi_total Element of bool [:[:(BoundedFunctions the carrier of X),(BoundedFunctions the carrier of X):],(BoundedFunctions the carrier of X):]
Mult_ ((BoundedFunctions the carrier of X),(RAlgebra the carrier of X)) is non empty Relation-like [:REAL,(BoundedFunctions the carrier of X):] -defined BoundedFunctions the carrier of X -valued Function-like total quasi_total Element of bool [:[:REAL,(BoundedFunctions the carrier of X):],(BoundedFunctions the carrier of X):]
[:REAL,(BoundedFunctions the carrier of X):] is non empty set
[:[:REAL,(BoundedFunctions the carrier of X):],(BoundedFunctions the carrier of X):] is non empty set
bool [:[:REAL,(BoundedFunctions the carrier of X):],(BoundedFunctions the carrier of X):] is non empty set
One_ ((BoundedFunctions the carrier of X),(RAlgebra the carrier of X)) is Element of BoundedFunctions the carrier of X
Zero_ ((BoundedFunctions the carrier of X),(RAlgebra the carrier of X)) is Element of BoundedFunctions the carrier of X
AlgebraStr(# (BoundedFunctions the carrier of X),(mult_ ((BoundedFunctions the carrier of X),(RAlgebra the carrier of X))),(Add_ ((BoundedFunctions the carrier of X),(RAlgebra the carrier of X))),(Mult_ ((BoundedFunctions the carrier of X),(RAlgebra the carrier of X))),(One_ ((BoundedFunctions the carrier of X),(RAlgebra the carrier of X))),(Zero_ ((BoundedFunctions the carrier of X),(RAlgebra the carrier of X))) #) is strict AlgebraStr
the carrier of (X) is non empty set
the carrier of (R_Algebra_of_BoundedFunctions the carrier of X) is non empty set
X is set
X is non empty TopSpace-like pseudocompact compact TopStruct
the carrier of X is non empty set
BoundedFunctions the carrier of X is non empty add-closed having-inverse additively-closed multiplicatively-closed additively-linearly-closed Element of bool the carrier of (RAlgebra the carrier of X)
RAlgebra the carrier of X is non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative unital associative commutative right-distributive right_unital well-unital left_unital strict vector-associative AlgebraStr
Funcs ( the carrier of X,REAL) is non empty functional FUNCTION_DOMAIN of the carrier of X, REAL
RealFuncMult the carrier of X is non empty Relation-like [:(Funcs ( the carrier of X,REAL)),(Funcs ( the carrier of X,REAL)):] -defined Funcs ( the carrier of X,REAL) -valued Function-like total quasi_total Element of bool [:[:(Funcs ( the carrier of X,REAL)),(Funcs ( the carrier of X,REAL)):],(Funcs ( the carrier of X,REAL)):]
[:(Funcs ( the carrier of X,REAL)),(Funcs ( the carrier of X,REAL)):] is non empty set
[:[:(Funcs ( the carrier of X,REAL)),(Funcs ( the carrier of X,REAL)):],(Funcs ( the carrier of X,REAL)):] is non empty set
bool [:[:(Funcs ( the carrier of X,REAL)),(Funcs ( the carrier of X,REAL)):],(Funcs ( the carrier of X,REAL)):] is non empty set
RealFuncAdd the carrier of X is non empty Relation-like [:(Funcs ( the carrier of X,REAL)),(Funcs ( the carrier of X,REAL)):] -defined Funcs ( the carrier of X,REAL) -valued Function-like total quasi_total Element of bool [:[:(Funcs ( the carrier of X,REAL)),(Funcs ( the carrier of X,REAL)):],(Funcs ( the carrier of X,REAL)):]
RealFuncExtMult the carrier of X is non empty Relation-like [:REAL,(Funcs ( the carrier of X,REAL)):] -defined Funcs ( the carrier of X,REAL) -valued Function-like total quasi_total Element of bool [:[:REAL,(Funcs ( the carrier of X,REAL)):],(Funcs ( the carrier of X,REAL)):]
[:REAL,(Funcs ( the carrier of X,REAL)):] is non empty set
[:[:REAL,(Funcs ( the carrier of X,REAL)):],(Funcs ( the carrier of X,REAL)):] is non empty set
bool [:[:REAL,(Funcs ( the carrier of X,REAL)):],(Funcs ( the carrier of X,REAL)):] is non empty set
RealFuncUnit the carrier of X is Relation-like the carrier of X -defined REAL -valued Function-like total quasi_total V152() V153() V154() Element of Funcs ( the carrier of X,REAL)
the carrier of X --> 1 is non empty Relation-like the carrier of X -defined NAT -valued RAT -valued INT -valued Function-like total quasi_total V152() V153() V154() V155() Element of bool [: the carrier of X,NAT:]
[: the carrier of X,NAT:] is non empty RAT -valued INT -valued V152() V153() V154() V155() set
bool [: the carrier of X,NAT:] is non empty set
RealFuncZero the carrier of X is Relation-like the carrier of X -defined REAL -valued Function-like total quasi_total V152() V153() V154() Element of Funcs ( the carrier of X,REAL)
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 V152() V153() V154() V155() Element of bool [: the carrier of X,NAT:]
AlgebraStr(# (Funcs ( the carrier of X,REAL)),(RealFuncMult the carrier of X),(RealFuncAdd the carrier of X),(RealFuncExtMult the carrier of X),(RealFuncUnit the carrier of X),(RealFuncZero the carrier of X) #) is strict AlgebraStr
the carrier of (RAlgebra the carrier of X) is non empty set
bool the carrier of (RAlgebra the carrier of X) is non empty set
[: the carrier of X,REAL:] is non empty V152() V153() V154() set
bool [: the carrier of X,REAL:] is non empty set
{ b₁ where b₁ is non empty Relation-like the carrier of X -defined REAL -valued Function-like total quasi_total V152() V153() V154() Element of bool [: the carrier of X,REAL:] : b₁ | the carrier of X is bounded } is set
BoundedFunctionsNorm the carrier of X is non empty Relation-like BoundedFunctions the carrier of X -defined REAL -valued Function-like total quasi_total V152() V153() V154() Element of bool [:(BoundedFunctions the carrier of X),REAL:]
[:(BoundedFunctions the carrier of X),REAL:] is non empty V152() V153() V154() set
bool [:(BoundedFunctions the carrier of X),REAL:] is non empty set
(X) is non empty add-closed having-inverse additively-closed multiplicatively-closed additively-linearly-closed Element of bool the carrier of (RAlgebra the carrier of X)
{ b₁ where b₁ is non empty Relation-like the carrier of X -defined REAL -valued Function-like total quasi_total V152() V153() V154() continuous Element of bool [: the carrier of X,REAL:] : verum } is set
(BoundedFunctionsNorm the carrier of X) | (X) is Relation-like BoundedFunctions the carrier of X -defined REAL -valued Function-like V152() V153() V154() Element of bool [:(BoundedFunctions the carrier of X),REAL:]
[:(X),REAL:] is non empty V152() V153() V154() set
bool [:(X),REAL:] is non empty set
X is set
X is non empty TopSpace-like pseudocompact compact TopStruct
(X) is non empty add-closed having-inverse additively-closed multiplicatively-closed additively-linearly-closed Element of bool the carrier of (RAlgebra the carrier of X)
the carrier of X is non empty set
RAlgebra the carrier of X is non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative unital associative commutative right-distributive right_unital well-unital left_unital strict vector-associative AlgebraStr
Funcs ( the carrier of X,REAL) is non empty functional FUNCTION_DOMAIN of the carrier of X, REAL
RealFuncMult the carrier of X is non empty Relation-like [:(Funcs ( the carrier of X,REAL)),(Funcs ( the carrier of X,REAL)):] -defined Funcs ( the carrier of X,REAL) -valued Function-like total quasi_total Element of bool [:[:(Funcs ( the carrier of X,REAL)),(Funcs ( the carrier of X,REAL)):],(Funcs ( the carrier of X,REAL)):]
[:(Funcs ( the carrier of X,REAL)),(Funcs ( the carrier of X,REAL)):] is non empty set
[:[:(Funcs ( the carrier of X,REAL)),(Funcs ( the carrier of X,REAL)):],(Funcs ( the carrier of X,REAL)):] is non empty set
bool [:[:(Funcs ( the carrier of X,REAL)),(Funcs ( the carrier of X,REAL)):],(Funcs ( the carrier of X,REAL)):] is non empty set
RealFuncAdd the carrier of X is non empty Relation-like [:(Funcs ( the carrier of X,REAL)),(Funcs ( the carrier of X,REAL)):] -defined Funcs ( the carrier of X,REAL) -valued Function-like total quasi_total Element of bool [:[:(Funcs ( the carrier of X,REAL)),(Funcs ( the carrier of X,REAL)):],(Funcs ( the carrier of X,REAL)):]
RealFuncExtMult the carrier of X is non empty Relation-like [:REAL,(Funcs ( the carrier of X,REAL)):] -defined Funcs ( the carrier of X,REAL) -valued Function-like total quasi_total Element of bool [:[:REAL,(Funcs ( the carrier of X,REAL)):],(Funcs ( the carrier of X,REAL)):]
[:REAL,(Funcs ( the carrier of X,REAL)):] is non empty set
[:[:REAL,(Funcs ( the carrier of X,REAL)):],(Funcs ( the carrier of X,REAL)):] is non empty set
bool [:[:REAL,(Funcs ( the carrier of X,REAL)):],(Funcs ( the carrier of X,REAL)):] is non empty set
RealFuncUnit the carrier of X is Relation-like the carrier of X -defined REAL -valued Function-like total quasi_total V152() V153() V154() Element of Funcs ( the carrier of X,REAL)
the carrier of X --> 1 is non empty Relation-like the carrier of X -defined NAT -valued RAT -valued INT -valued Function-like total quasi_total V152() V153() V154() V155() Element of bool [: the carrier of X,NAT:]
[: the carrier of X,NAT:] is non empty RAT -valued INT -valued V152() V153() V154() V155() set
bool [: the carrier of X,NAT:] is non empty set
RealFuncZero the carrier of X is Relation-like the carrier of X -defined REAL -valued Function-like total quasi_total V152() V153() V154() Element of Funcs ( the carrier of X,REAL)
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 V152() V153() V154() V155() Element of bool [: the carrier of X,NAT:]
AlgebraStr(# (Funcs ( the carrier of X,REAL)),(RealFuncMult the carrier of X),(RealFuncAdd the carrier of X),(RealFuncExtMult the carrier of X),(RealFuncUnit the carrier of X),(RealFuncZero the carrier of X) #) is strict AlgebraStr
the carrier of (RAlgebra the carrier of X) is non empty set
bool the carrier of (RAlgebra the carrier of X) is non empty set
[: the carrier of X,REAL:] is non empty V152() V153() V154() set
bool [: the carrier of X,REAL:] is non empty set
{ b₁ where b₁ is non empty Relation-like the carrier of X -defined REAL -valued Function-like total quasi_total V152() V153() V154() continuous Element of bool [: the carrier of X,REAL:] : verum } is set
mult_ ((X),(RAlgebra 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),(RAlgebra 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),(RAlgebra the carrier of X)) is non empty Relation-like [:REAL,(X):] -defined (X) -valued Function-like total quasi_total Element of bool [:[:REAL,(X):],(X):]
[:REAL,(X):] is non empty set
[:[:REAL,(X):],(X):] is non empty set
bool [:[:REAL,(X):],(X):] is non empty set
One_ ((X),(RAlgebra the carrier of X)) is Element of (X)
Zero_ ((X),(RAlgebra the carrier of X)) is Element of (X)
(X) is non empty Relation-like (X) -defined REAL -valued Function-like total quasi_total V152() V153() V154() Element of bool [:(X),REAL:]
[:(X),REAL:] is non empty V152() V153() V154() set
bool [:(X),REAL:] is non empty set
BoundedFunctions the carrier of X is non empty add-closed having-inverse additively-closed multiplicatively-closed additively-linearly-closed Element of bool the carrier of (RAlgebra the carrier of X)
{ b₁ where b₁ is non empty Relation-like the carrier of X -defined REAL -valued Function-like total quasi_total V152() V153() V154() Element of bool [: the carrier of X,REAL:] : b₁ | the carrier of X is bounded } is set
BoundedFunctionsNorm the carrier of X is non empty Relation-like BoundedFunctions the carrier of X -defined REAL -valued Function-like total quasi_total V152() V153() V154() Element of bool [:(BoundedFunctions the carrier of X),REAL:]
[:(BoundedFunctions the carrier of X),REAL:] is non empty V152() V153() V154() set
bool [:(BoundedFunctions the carrier of X),REAL:] is non empty set
(BoundedFunctionsNorm the carrier of X) | (X) is Relation-like BoundedFunctions the carrier of X -defined REAL -valued Function-like V152() V153() V154() Element of bool [:(BoundedFunctions the carrier of X),REAL:]
Normed_AlgebraStr(# (X),(mult_ ((X),(RAlgebra the carrier of X))),(Add_ ((X),(RAlgebra the carrier of X))),(Mult_ ((X),(RAlgebra the carrier of X))),(One_ ((X),(RAlgebra the carrier of X))),(Zero_ ((X),(RAlgebra the carrier of X))),(X) #) is strict Normed_AlgebraStr
X is non empty TopSpace-like pseudocompact compact TopStruct
(X) is Normed_AlgebraStr
(X) is non empty add-closed having-inverse additively-closed multiplicatively-closed additively-linearly-closed Element of bool the carrier of (RAlgebra the carrier of X)
the carrier of X is non empty set
RAlgebra the carrier of X is non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative unital associative commutative right-distributive right_unital well-unital left_unital strict vector-associative AlgebraStr
Funcs ( the carrier of X,REAL) is non empty functional FUNCTION_DOMAIN of the carrier of X, REAL
RealFuncMult the carrier of X is non empty Relation-like [:(Funcs ( the carrier of X,REAL)),(Funcs ( the carrier of X,REAL)):] -defined Funcs ( the carrier of X,REAL) -valued Function-like total quasi_total Element of bool [:[:(Funcs ( the carrier of X,REAL)),(Funcs ( the carrier of X,REAL)):],(Funcs ( the carrier of X,REAL)):]
[:(Funcs ( the carrier of X,REAL)),(Funcs ( the carrier of X,REAL)):] is non empty set
[:[:(Funcs ( the carrier of X,REAL)),(Funcs ( the carrier of X,REAL)):],(Funcs ( the carrier of X,REAL)):] is non empty set
bool [:[:(Funcs ( the carrier of X,REAL)),(Funcs ( the carrier of X,REAL)):],(Funcs ( the carrier of X,REAL)):] is non empty set
RealFuncAdd the carrier of X is non empty Relation-like [:(Funcs ( the carrier of X,REAL)),(Funcs ( the carrier of X,REAL)):] -defined Funcs ( the carrier of X,REAL) -valued Function-like total quasi_total Element of bool [:[:(Funcs ( the carrier of X,REAL)),(Funcs ( the carrier of X,REAL)):],(Funcs ( the carrier of X,REAL)):]
RealFuncExtMult the carrier of X is non empty Relation-like [:REAL,(Funcs ( the carrier of X,REAL)):] -defined Funcs ( the carrier of X,REAL) -valued Function-like total quasi_total Element of bool [:[:REAL,(Funcs ( the carrier of X,REAL)):],(Funcs ( the carrier of X,REAL)):]
[:REAL,(Funcs ( the carrier of X,REAL)):] is non empty set
[:[:REAL,(Funcs ( the carrier of X,REAL)):],(Funcs ( the carrier of X,REAL)):] is non empty set
bool [:[:REAL,(Funcs ( the carrier of X,REAL)):],(Funcs ( the carrier of X,REAL)):] is non empty set
RealFuncUnit the carrier of X is Relation-like the carrier of X -defined REAL -valued Function-like total quasi_total V152() V153() V154() Element of Funcs ( the carrier of X,REAL)
the carrier of X --> 1 is non empty Relation-like the carrier of X -defined NAT -valued RAT -valued INT -valued Function-like total quasi_total V152() V153() V154() V155() Element of bool [: the carrier of X,NAT:]
[: the carrier of X,NAT:] is non empty RAT -valued INT -valued V152() V153() V154() V155() set
bool [: the carrier of X,NAT:] is non empty set
RealFuncZero the carrier of X is Relation-like the carrier of X -defined REAL -valued Function-like total quasi_total V152() V153() V154() Element of Funcs ( the carrier of X,REAL)
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 V152() V153() V154() V155() Element of bool [: the carrier of X,NAT:]
AlgebraStr(# (Funcs ( the carrier of X,REAL)),(RealFuncMult the carrier of X),(RealFuncAdd the carrier of X),(RealFuncExtMult the carrier of X),(RealFuncUnit the carrier of X),(RealFuncZero the carrier of X) #) is strict AlgebraStr
the carrier of (RAlgebra the carrier of X) is non empty set
bool the carrier of (RAlgebra the carrier of X) is non empty set
[: the carrier of X,REAL:] is non empty V152() V153() V154() set
bool [: the carrier of X,REAL:] is non empty set
{ b₁ where b₁ is non empty Relation-like the carrier of X -defined REAL -valued Function-like total quasi_total V152() V153() V154() continuous Element of bool [: the carrier of X,REAL:] : verum } is set
mult_ ((X),(RAlgebra 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),(RAlgebra 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),(RAlgebra the carrier of X)) is non empty Relation-like [:REAL,(X):] -defined (X) -valued Function-like total quasi_total Element of bool [:[:REAL,(X):],(X):]
[:REAL,(X):] is non empty set
[:[:REAL,(X):],(X):] is non empty set
bool [:[:REAL,(X):],(X):] is non empty set
One_ ((X),(RAlgebra the carrier of X)) is Element of (X)
Zero_ ((X),(RAlgebra the carrier of X)) is Element of (X)
(X) is non empty Relation-like (X) -defined REAL -valued Function-like total quasi_total V152() V153() V154() Element of bool [:(X),REAL:]
[:(X),REAL:] is non empty V152() V153() V154() set
bool [:(X),REAL:] is non empty set
BoundedFunctions the carrier of X is non empty add-closed having-inverse additively-closed multiplicatively-closed additively-linearly-closed Element of bool the carrier of (RAlgebra the carrier of X)
{ b₁ where b₁ is non empty Relation-like the carrier of X -defined REAL -valued Function-like total quasi_total V152() V153() V154() Element of bool [: the carrier of X,REAL:] : b₁ | the carrier of X is bounded } is set
BoundedFunctionsNorm the carrier of X is non empty Relation-like BoundedFunctions the carrier of X -defined REAL -valued Function-like total quasi_total V152() V153() V154() Element of bool [:(BoundedFunctions the carrier of X),REAL:]
[:(BoundedFunctions the carrier of X),REAL:] is non empty V152() V153() V154() set
bool [:(BoundedFunctions the carrier of X),REAL:] is non empty set
(BoundedFunctionsNorm the carrier of X) | (X) is Relation-like BoundedFunctions the carrier of X -defined REAL -valued Function-like V152() V153() V154() Element of bool [:(BoundedFunctions the carrier of X),REAL:]
Normed_AlgebraStr(# (X),(mult_ ((X),(RAlgebra the carrier of X))),(Add_ ((X),(RAlgebra the carrier of X))),(Mult_ ((X),(RAlgebra the carrier of X))),(One_ ((X),(RAlgebra the carrier of X))),(Zero_ ((X),(RAlgebra the carrier of X))),(X) #) is strict Normed_AlgebraStr
X is non empty TopSpace-like pseudocompact compact TopStruct
(X) is non empty strict Normed_AlgebraStr
(X) is non empty add-closed having-inverse additively-closed multiplicatively-closed additively-linearly-closed Element of bool the carrier of (RAlgebra the carrier of X)
the carrier of X is non empty set
RAlgebra the carrier of X is non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative unital associative commutative right-distributive right_unital well-unital left_unital strict vector-associative AlgebraStr
Funcs ( the carrier of X,REAL) is non empty functional FUNCTION_DOMAIN of the carrier of X, REAL
RealFuncMult the carrier of X is non empty Relation-like [:(Funcs ( the carrier of X,REAL)),(Funcs ( the carrier of X,REAL)):] -defined Funcs ( the carrier of X,REAL) -valued Function-like total quasi_total Element of bool [:[:(Funcs ( the carrier of X,REAL)),(Funcs ( the carrier of X,REAL)):],(Funcs ( the carrier of X,REAL)):]
[:(Funcs ( the carrier of X,REAL)),(Funcs ( the carrier of X,REAL)):] is non empty set
[:[:(Funcs ( the carrier of X,REAL)),(Funcs ( the carrier of X,REAL)):],(Funcs ( the carrier of X,REAL)):] is non empty set
bool [:[:(Funcs ( the carrier of X,REAL)),(Funcs ( the carrier of X,REAL)):],(Funcs ( the carrier of X,REAL)):] is non empty set
RealFuncAdd the carrier of X is non empty Relation-like [:(Funcs ( the carrier of X,REAL)),(Funcs ( the carrier of X,REAL)):] -defined Funcs ( the carrier of X,REAL) -valued Function-like total quasi_total Element of bool [:[:(Funcs ( the carrier of X,REAL)),(Funcs ( the carrier of X,REAL)):],(Funcs ( the carrier of X,REAL)):]
RealFuncExtMult the carrier of X is non empty Relation-like [:REAL,(Funcs ( the carrier of X,REAL)):] -defined Funcs ( the carrier of X,REAL) -valued Function-like total quasi_total Element of bool [:[:REAL,(Funcs ( the carrier of X,REAL)):],(Funcs ( the carrier of X,REAL)):]
[:REAL,(Funcs ( the carrier of X,REAL)):] is non empty set
[:[:REAL,(Funcs ( the carrier of X,REAL)):],(Funcs ( the carrier of X,REAL)):] is non empty set
bool [:[:REAL,(Funcs ( the carrier of X,REAL)):],(Funcs ( the carrier of X,REAL)):] is non empty set
RealFuncUnit the carrier of X is Relation-like the carrier of X -defined REAL -valued Function-like total quasi_total V152() V153() V154() Element of Funcs ( the carrier of X,REAL)
the carrier of X --> 1 is non empty Relation-like the carrier of X -defined NAT -valued RAT -valued INT -valued Function-like total quasi_total V152() V153() V154() V155() Element of bool [: the carrier of X,NAT:]
[: the carrier of X,NAT:] is non empty RAT -valued INT -valued V152() V153() V154() V155() set
bool [: the carrier of X,NAT:] is non empty set
RealFuncZero the carrier of X is Relation-like the carrier of X -defined REAL -valued Function-like total quasi_total V152() V153() V154() Element of Funcs ( the carrier of X,REAL)
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 V152() V153() V154() V155() Element of bool [: the carrier of X,NAT:]
AlgebraStr(# (Funcs ( the carrier of X,REAL)),(RealFuncMult the carrier of X),(RealFuncAdd the carrier of X),(RealFuncExtMult the carrier of X),(RealFuncUnit the carrier of X),(RealFuncZero the carrier of X) #) is strict AlgebraStr
the carrier of (RAlgebra the carrier of X) is non empty set
bool the carrier of (RAlgebra the carrier of X) is non empty set
[: the carrier of X,REAL:] is non empty V152() V153() V154() set
bool [: the carrier of X,REAL:] is non empty set
{ b₁ where b₁ is non empty Relation-like the carrier of X -defined REAL -valued Function-like total quasi_total V152() V153() V154() continuous Element of bool [: the carrier of X,REAL:] : verum } is set
mult_ ((X),(RAlgebra 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),(RAlgebra 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),(RAlgebra the carrier of X)) is non empty Relation-like [:REAL,(X):] -defined (X) -valued Function-like total quasi_total Element of bool [:[:REAL,(X):],(X):]
[:REAL,(X):] is non empty set
[:[:REAL,(X):],(X):] is non empty set
bool [:[:REAL,(X):],(X):] is non empty set
One_ ((X),(RAlgebra the carrier of X)) is Element of (X)
Zero_ ((X),(RAlgebra the carrier of X)) is Element of (X)
(X) is non empty Relation-like (X) -defined REAL -valued Function-like total quasi_total V152() V153() V154() Element of bool [:(X),REAL:]
[:(X),REAL:] is non empty V152() V153() V154() set
bool [:(X),REAL:] is non empty set
BoundedFunctions the carrier of X is non empty add-closed having-inverse additively-closed multiplicatively-closed additively-linearly-closed Element of bool the carrier of (RAlgebra the carrier of X)
{ b₁ where b₁ is non empty Relation-like the carrier of X -defined REAL -valued Function-like total quasi_total V152() V153() V154() Element of bool [: the carrier of X,REAL:] : b₁ | the carrier of X is bounded } is set
BoundedFunctionsNorm the carrier of X is non empty Relation-like BoundedFunctions the carrier of X -defined REAL -valued Function-like total quasi_total V152() V153() V154() Element of bool [:(BoundedFunctions the carrier of X),REAL:]
[:(BoundedFunctions the carrier of X),REAL:] is non empty V152() V153() V154() set
bool [:(BoundedFunctions the carrier of X),REAL:] is non empty set
(BoundedFunctionsNorm the carrier of X) | (X) is Relation-like BoundedFunctions the carrier of X -defined REAL -valued Function-like V152() V153() V154() Element of bool [:(BoundedFunctions the carrier of X),REAL:]
Normed_AlgebraStr(# (X),(mult_ ((X),(RAlgebra the carrier of X))),(Add_ ((X),(RAlgebra the carrier of X))),(Mult_ ((X),(RAlgebra the carrier of X))),(One_ ((X),(RAlgebra the carrier of X))),(Zero_ ((X),(RAlgebra the carrier of X))),(X) #) is strict Normed_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_ (RAlgebra the carrier of X) is right_complementable Element of the carrier of (RAlgebra the carrier of X)
1. (RAlgebra the carrier of X) is right_complementable Element of the carrier of (RAlgebra the carrier of X)
the OneF of (RAlgebra the carrier of X) is right_complementable Element of the carrier of (RAlgebra the carrier of X)
(X,1) is non empty Relation-like the carrier of X -defined REAL -valued Function-like total quasi_total V152() V153() V154() V206( the carrier of X) V207( the carrier of X) continuous bounded Element of bool [: the carrier of X,REAL:]
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 V152() V153() V154() V155() Element of bool [: the carrier of X,{1}:]
{1} is non empty V185() V186() V187() V188() V189() V190() left_end bounded_below set
[: the carrier of X,{1}:] is non empty RAT -valued INT -valued V152() V153() V154() V155() set
bool [: the carrier of X,{1}:] is non empty set
(X) is non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital unital associative commutative right-distributive right_unital well-unital left_unital strict vector-associative AlgebraStr
AlgebraStr(# (X),(mult_ ((X),(RAlgebra the carrier of X))),(Add_ ((X),(RAlgebra the carrier of X))),(Mult_ ((X),(RAlgebra the carrier of X))),(One_ ((X),(RAlgebra the carrier of X))),(Zero_ ((X),(RAlgebra the carrier of X))) #) is strict AlgebraStr
1_ (X) is right_complementable Element of the carrier of (X)
the carrier of (X) is non empty set
1. (X) is right_complementable Element of the carrier of (X)
the OneF of (X) is right_complementable Element of the carrier of (X)
GB is Element of (X)
[GB,(1_ (RAlgebra the carrier of X))] is Element of [:(X), the carrier of (RAlgebra the carrier of X):]
[:(X), the carrier of (RAlgebra the carrier of X):] is non empty set
{GB,(1_ (RAlgebra the carrier of X))} is non empty set
{GB} is non empty set
{{GB,(1_ (RAlgebra the carrier of X))},{GB}} is non empty set
[(1_ (RAlgebra the carrier of X)),GB] is Element of [: the carrier of (RAlgebra the carrier of X),(X):]
[: the carrier of (RAlgebra the carrier of X),(X):] is non empty set
{(1_ (RAlgebra the carrier of X)),GB} is non empty set
{(1_ (RAlgebra the carrier of X))} is non empty set
{{(1_ (RAlgebra the carrier of X)),GB},{(1_ (RAlgebra 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),(RAlgebra the carrier of X))) . (GB,(1_ (RAlgebra the carrier of X))) is set
[GB,(1_ (RAlgebra the carrier of X))] is set
(mult_ ((X),(RAlgebra the carrier of X))) . [GB,(1_ (RAlgebra the carrier of X))] is set
the multF of (RAlgebra the carrier of X) is non empty Relation-like [: the carrier of (RAlgebra the carrier of X), the carrier of (RAlgebra the carrier of X):] -defined the carrier of (RAlgebra the carrier of X) -valued Function-like total quasi_total Element of bool [:[: the carrier of (RAlgebra the carrier of X), the carrier of (RAlgebra the carrier of X):], the carrier of (RAlgebra the carrier of X):]
[: the carrier of (RAlgebra the carrier of X), the carrier of (RAlgebra the carrier of X):] is non empty set
[:[: the carrier of (RAlgebra the carrier of X), the carrier of (RAlgebra the carrier of X):], the carrier of (RAlgebra the carrier of X):] is non empty set
bool [:[: the carrier of (RAlgebra the carrier of X), the carrier of (RAlgebra the carrier of X):], the carrier of (RAlgebra the carrier of X):] is non empty set
the multF of (RAlgebra the carrier of X) || (X) is Relation-like Function-like set
the multF of (RAlgebra the carrier of X) | [:(X),(X):] is Relation-like set
( the multF of (RAlgebra the carrier of X) || (X)) . (GB,(1_ (RAlgebra the carrier of X))) is set
( the multF of (RAlgebra the carrier of X) || (X)) . [GB,(1_ (RAlgebra the carrier of X))] is set
GB * (1_ (RAlgebra the carrier of X)) is right_complementable Element of the carrier of (RAlgebra the carrier of X)
the multF of (RAlgebra the carrier of X) . (GB,(1_ (RAlgebra the carrier of X))) is right_complementable Element of the carrier of (RAlgebra the carrier of X)
the multF of (RAlgebra the carrier of X) . [GB,(1_ (RAlgebra 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),(RAlgebra the carrier of X))) . ((1_ (RAlgebra the carrier of X)),GB) is set
[(1_ (RAlgebra the carrier of X)),GB] is set
(mult_ ((X),(RAlgebra the carrier of X))) . [(1_ (RAlgebra the carrier of X)),GB] is set
( the multF of (RAlgebra the carrier of X) || (X)) . ((1_ (RAlgebra the carrier of X)),GB) is set
( the multF of (RAlgebra the carrier of X) || (X)) . [(1_ (RAlgebra the carrier of X)),GB] is set
(1_ (RAlgebra the carrier of X)) * GB is right_complementable Element of the carrier of (RAlgebra the carrier of X)
the multF of (RAlgebra the carrier of X) . ((1_ (RAlgebra the carrier of X)),GB) is right_complementable Element of the carrier of (RAlgebra the carrier of X)
the multF of (RAlgebra the carrier of X) . [(1_ (RAlgebra the carrier of X)),GB] is set
X is non empty TopSpace-like pseudocompact compact TopStruct
(X) is non empty strict Normed_AlgebraStr
(X) is non empty add-closed having-inverse additively-closed multiplicatively-closed additively-linearly-closed Element of bool the carrier of (RAlgebra the carrier of X)
the carrier of X is non empty set
RAlgebra the carrier of X is non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative unital associative commutative right-distributive right_unital well-unital left_unital strict vector-associative AlgebraStr
Funcs ( the carrier of X,REAL) is non empty functional FUNCTION_DOMAIN of the carrier of X, REAL
RealFuncMult the carrier of X is non empty Relation-like [:(Funcs ( the carrier of X,REAL)),(Funcs ( the carrier of X,REAL)):] -defined Funcs ( the carrier of X,REAL) -valued Function-like total quasi_total Element of bool [:[:(Funcs ( the carrier of X,REAL)),(Funcs ( the carrier of X,REAL)):],(Funcs ( the carrier of X,REAL)):]
[:(Funcs ( the carrier of X,REAL)),(Funcs ( the carrier of X,REAL)):] is non empty set
[:[:(Funcs ( the carrier of X,REAL)),(Funcs ( the carrier of X,REAL)):],(Funcs ( the carrier of X,REAL)):] is non empty set
bool [:[:(Funcs ( the carrier of X,REAL)),(Funcs ( the carrier of X,REAL)):],(Funcs ( the carrier of X,REAL)):] is non empty set
RealFuncAdd the carrier of X is non empty Relation-like [:(Funcs ( the carrier of X,REAL)),(Funcs ( the carrier of X,REAL)):] -defined Funcs ( the carrier of X,REAL) -valued Function-like total quasi_total Element of bool [:[:(Funcs ( the carrier of X,REAL)),(Funcs ( the carrier of X,REAL)):],(Funcs ( the carrier of X,REAL)):]
RealFuncExtMult the carrier of X is non empty Relation-like [:REAL,(Funcs ( the carrier of X,REAL)):] -defined Funcs ( the carrier of X,REAL) -valued Function-like total quasi_total Element of bool [:[:REAL,(Funcs ( the carrier of X,REAL)):],(Funcs ( the carrier of X,REAL)):]
[:REAL,(Funcs ( the carrier of X,REAL)):] is non empty set
[:[:REAL,(Funcs ( the carrier of X,REAL)):],(Funcs ( the carrier of X,REAL)):] is non empty set
bool [:[:REAL,(Funcs ( the carrier of X,REAL)):],(Funcs ( the carrier of X,REAL)):] is non empty set
RealFuncUnit the carrier of X is Relation-like the carrier of X -defined REAL -valued Function-like total quasi_total V152() V153() V154() Element of Funcs ( the carrier of X,REAL)
the carrier of X --> 1 is non empty Relation-like the carrier of X -defined NAT -valued RAT -valued INT -valued Function-like total quasi_total V152() V153() V154() V155() Element of bool [: the carrier of X,NAT:]
[: the carrier of X,NAT:] is non empty RAT -valued INT -valued V152() V153() V154() V155() set
bool [: the carrier of X,NAT:] is non empty set
RealFuncZero the carrier of X is Relation-like the carrier of X -defined REAL -valued Function-like total quasi_total V152() V153() V154() Element of Funcs ( the carrier of X,REAL)
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 V152() V153() V154() V155() Element of bool [: the carrier of X,NAT:]
AlgebraStr(# (Funcs ( the carrier of X,REAL)),(RealFuncMult the carrier of X),(RealFuncAdd the carrier of X),(RealFuncExtMult the carrier of X),(RealFuncUnit the carrier of X),(RealFuncZero the carrier of X) #) is strict AlgebraStr
the carrier of (RAlgebra the carrier of X) is non empty set
bool the carrier of (RAlgebra the carrier of X) is non empty set
[: the carrier of X,REAL:] is non empty V152() V153() V154() set
bool [: the carrier of X,REAL:] is non empty set
{ b₁ where b₁ is non empty Relation-like the carrier of X -defined REAL -valued Function-like total quasi_total V152() V153() V154() continuous Element of bool [: the carrier of X,REAL:] : verum } is set
mult_ ((X),(RAlgebra 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),(RAlgebra 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),(RAlgebra the carrier of X)) is non empty Relation-like [:REAL,(X):] -defined (X) -valued Function-like total quasi_total Element of bool [:[:REAL,(X):],(X):]
[:REAL,(X):] is non empty set
[:[:REAL,(X):],(X):] is non empty set
bool [:[:REAL,(X):],(X):] is non empty set
One_ ((X),(RAlgebra the carrier of X)) is Element of (X)
Zero_ ((X),(RAlgebra the carrier of X)) is Element of (X)
(X) is non empty Relation-like (X) -defined REAL -valued Function-like total quasi_total V152() V153() V154() Element of bool [:(X),REAL:]
[:(X),REAL:] is non empty V152() V153() V154() set
bool [:(X),REAL:] is non empty set
BoundedFunctions the carrier of X is non empty add-closed having-inverse additively-closed multiplicatively-closed additively-linearly-closed Element of bool the carrier of (RAlgebra the carrier of X)
{ b₁ where b₁ is non empty Relation-like the carrier of X -defined REAL -valued Function-like total quasi_total V152() V153() V154() Element of bool [: the carrier of X,REAL:] : b₁ | the carrier of X is bounded } is set
BoundedFunctionsNorm the carrier of X is non empty Relation-like BoundedFunctions the carrier of X -defined REAL -valued Function-like total quasi_total V152() V153() V154() Element of bool [:(BoundedFunctions the carrier of X),REAL:]
[:(BoundedFunctions the carrier of X),REAL:] is non empty V152() V153() V154() set
bool [:(BoundedFunctions the carrier of X),REAL:] is non empty set
(BoundedFunctionsNorm the carrier of X) | (X) is Relation-like BoundedFunctions the carrier of X -defined REAL -valued Function-like V152() V153() V154() Element of bool [:(BoundedFunctions the carrier of X),REAL:]
Normed_AlgebraStr(# (X),(mult_ ((X),(RAlgebra the carrier of X))),(Add_ ((X),(RAlgebra the carrier of X))),(Mult_ ((X),(RAlgebra the carrier of X))),(One_ ((X),(RAlgebra the carrier of X))),(Zero_ ((X),(RAlgebra the carrier of X))),(X) #) is strict Normed_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 Normed_AlgebraStr
the carrier of X is set
the multF of X is Relation-like [: the carrier of X, the carrier of X:] -defined the carrier of X -valued Function-like 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 set
[:[: the carrier of X, the carrier of X:], the carrier of X:] is set
bool [:[: the carrier of X, the carrier of X:], the carrier of X:] is non empty set
the addF of X is Relation-like [: the carrier of X, the carrier of X:] -defined the carrier of X -valued Function-like quasi_total Element of bool [:[: the carrier of X, the carrier of X:], the carrier of X:]
the Mult of X is Relation-like [:REAL, the carrier of X:] -defined the carrier of X -valued Function-like quasi_total Element of bool [:[:REAL, the carrier of X:], the carrier of X:]
[:REAL, the carrier of X:] is set
[:[:REAL, the carrier of X:], the carrier of X:] is set
bool [:[:REAL, the carrier of X:], the carrier of X:] is non empty set
the OneF of X is Element of the carrier of X
the ZeroF of X is Element of the carrier of X
AlgebraStr(# the carrier of X, the multF of X, the addF of X, the Mult of X, the OneF of X, the ZeroF of X #) is strict AlgebraStr
X is non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative unital associative commutative right-distributive right_unital well-unital left_unital vector-associative AlgebraStr
F is non empty AlgebraStr
the carrier of F is non empty set
G is Element of the carrier of F
FB is Element of the carrier of F
G + FB is Element of the carrier of F
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 . (G,FB) is Element of the carrier of F
[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 F . [G,FB] is set
FB + G is Element of the carrier of F
the addF of F . (FB,G) is Element of the carrier of F
[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 F . [FB,G] is set
the carrier of X is non empty set
GB is right_complementable Element of the carrier of X
aFB is right_complementable Element of the carrier of X
GB + aFB is right_complementable 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 . (GB,aFB) is right_complementable Element of 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 X . [GB,aFB] is set
aFB + GB is right_complementable Element of the carrier of X
the addF of X . (aFB,GB) is right_complementable 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
G is Element of the carrier of F
FB is Element of the carrier of F
G + FB is Element of the carrier of F
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 . (G,FB) is Element of the carrier of F
[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 F . [G,FB] is set
GB is Element of the carrier of F
(G + FB) + GB is Element of the carrier of F
the addF of F . ((G + FB),GB) is Element of the carrier of F
[(G + FB),GB] is set
{(G + FB),GB} is non empty set
{(G + FB)} is non empty set
{{(G + FB),GB},{(G + FB)}} is non empty set
the addF of F . [(G + FB),GB] is set
FB + GB is Element of the carrier of F
the addF of F . (FB,GB) is Element of the carrier of F
[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 F . [FB,GB] is set
G + (FB + GB) is Element of the carrier of F
the addF of F . (G,(FB + GB)) is Element of the carrier of F
[G,(FB + GB)] is set
{G,(FB + GB)} is non empty set
{{G,(FB + GB)},{G}} is non empty set
the addF of F . [G,(FB + GB)] is set
the carrier of X is non empty set
aFB is right_complementable Element of the carrier of X
aF is right_complementable Element of the carrier of X
aFB + aF is right_complementable 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 right_complementable 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
aG is right_complementable Element of the carrier of X
(aFB + aF) + aG is right_complementable Element of the carrier of X
the addF of X . ((aFB + aF),aG) is right_complementable Element of the carrier of X
[(aFB + aF),aG] is set
{(aFB + aF),aG} is non empty set
{(aFB + aF)} is non empty set
{{(aFB + aF),aG},{(aFB + aF)}} is non empty set
the addF of X . [(aFB + aF),aG] is set
aF + aG is right_complementable Element of the carrier of X
the addF of X . (aF,aG) is right_complementable 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
aFB + (aF + aG) is right_complementable Element of the carrier of X
the addF of X . (aFB,(aF + aG)) is right_complementable Element of the carrier of X
[aFB,(aF + aG)] is set
{aFB,(aF + aG)} is non empty set
{{aFB,(aF + aG)},{aFB}} is non empty set
the addF of X . [aFB,(aF + aG)] is set
0. F is V49(F) Element of the carrier of F
the ZeroF of F is Element of the carrier of F
G is Element of the carrier of F
G + (0. F) is Element of the carrier of F
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 . (G,(0. F)) is Element of the carrier of F
[G,(0. F)] is set
{G,(0. F)} is non empty set
{G} is non empty set
{{G,(0. F)},{G}} is non empty set
the addF of F . [G,(0. F)] is set
the carrier of X is non empty set
FB is right_complementable Element of the carrier of X
0. X is V49(X) right_complementable Element of the carrier of X
the ZeroF of X is right_complementable Element of the carrier of X
FB + (0. X) is right_complementable 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,(0. X)) is right_complementable Element of the carrier of X
[FB,(0. X)] is set
{FB,(0. X)} is non empty set
{FB} is non empty set
{{FB,(0. X)},{FB}} is non empty set
the addF of X . [FB,(0. X)] is set
G is Element of the carrier of F
the carrier of X is non empty set
FB is right_complementable Element of the carrier of X
0. X is V49(X) right_complementable Element of the carrier of X
the ZeroF of X is right_complementable Element of the carrier of X
GB is right_complementable Element of the carrier of X
FB + GB is right_complementable 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 right_complementable 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 is Element of the carrier of F
G + aFB is Element of the carrier of F
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 . (G,aFB) is Element of the carrier of F
[G,aFB] is set
{G,aFB} is non empty set
{G} is non empty set
{{G,aFB},{G}} is non empty set
the addF of F . [G,aFB] is set
G is Element of the carrier of F
FB is Element of the carrier of F
G * FB is Element of the carrier of F
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 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 multF of F . (G,FB) is Element of the carrier of F
[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 F . [G,FB] is set
FB * G is Element of the carrier of F
the multF of F . (FB,G) is Element of the carrier of F
[FB,G] is set
{FB,G} is non empty set
{FB} is non empty set
{{FB,G},{FB}} is non empty set
the multF of F . [FB,G] is set
the carrier of X is non empty set
GB is right_complementable Element of the carrier of X
aFB is right_complementable Element of the carrier of X
GB * aFB is right_complementable 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 . (GB,aFB) is right_complementable Element of 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 X . [GB,aFB] is set
aFB * GB is right_complementable Element of the carrier of X
the multF of X . (aFB,GB) is right_complementable 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 multF of X . [aFB,GB] is set
G is Element of the carrier of F
FB is Element of the carrier of F
G * FB is Element of the carrier of F
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 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 multF of F . (G,FB) is Element of the carrier of F
[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 F . [G,FB] is set
GB is Element of the carrier of F
(G * FB) * GB is Element of the carrier of F
the multF of F . ((G * FB),GB) is Element of the carrier of F
[(G * FB),GB] is set
{(G * FB),GB} is non empty set
{(G * FB)} is non empty set
{{(G * FB),GB},{(G * FB)}} is non empty set
the multF of F . [(G * FB),GB] is set
FB * GB is Element of the carrier of F
the multF of F . (FB,GB) is Element of the carrier of F
[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 F . [FB,GB] is set
G * (FB * GB) is Element of the carrier of F
the multF of F . (G,(FB * GB)) is Element of the carrier of F
[G,(FB * GB)] is set
{G,(FB * GB)} is non empty set
{{G,(FB * GB)},{G}} is non empty set
the multF of F . [G,(FB * GB)] is set
the carrier of X is non empty set
aF is right_complementable Element of the carrier of X
aG is right_complementable Element of the carrier of X
aF * aG is right_complementable 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 right_complementable 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
aFB is right_complementable Element of the carrier of X
(aF * aG) * aFB is right_complementable Element of the carrier of X
the multF of X . ((aF * aG),aFB) is right_complementable Element of the carrier of X
[(aF * aG),aFB] is set
{(aF * aG),aFB} is non empty set
{(aF * aG)} is non empty set
{{(aF * aG),aFB},{(aF * aG)}} is non empty set
the multF of X . [(aF * aG),aFB] is set
aG * aFB is right_complementable Element of the carrier of X
the multF of X . (aG,aFB) is right_complementable Element of the carrier of X
[aG,aFB] is set
{aG,aFB} is non empty set
{aG} is non empty set
{{aG,aFB},{aG}} is non empty set
the multF of X . [aG,aFB] is set
aF * (aG * aFB) is right_complementable Element of the carrier of X
the multF of X . (aF,(aG * aFB)) is right_complementable Element of the carrier of X
[aF,(aG * aFB)] is set
{aF,(aG * aFB)} is non empty set
{{aF,(aG * aFB)},{aF}} is non empty set
the multF of X . [aF,(aG * aFB)] is set
G is Element of the carrier of F
1. F is Element of the carrier of F
the OneF of F is Element of the carrier of F
G * (1. F) is Element of the carrier of F
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 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 multF of F . (G,(1. F)) is Element of the carrier of F
[G,(1. F)] is set
{G,(1. F)} is non empty set
{G} is non empty set
{{G,(1. F)},{G}} is non empty set
the multF of F . [G,(1. F)] is set
the carrier of X is non empty set
FB is right_complementable Element of the carrier of X
1. X is right_complementable Element of the carrier of X
the OneF of X is right_complementable Element of the carrier of X
FB * (1. X) is right_complementable 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,(1. X)) is right_complementable Element of the carrier of X
[FB,(1. X)] is set
{FB,(1. X)} is non empty set
{FB} is non empty set
{{FB,(1. X)},{FB}} is non empty set
the multF of X . [FB,(1. X)] is set
G is Element of the carrier of F
FB is Element of the carrier of F
GB is Element of the carrier of F
FB + GB is Element of the carrier of F
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 . (FB,GB) is Element of the carrier of F
[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 F . [FB,GB] is set
G * (FB + GB) is Element of the carrier of F
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 . (G,(FB + GB)) is Element of the carrier of F
[G,(FB + GB)] is set
{G,(FB + GB)} is non empty set
{G} is non empty set
{{G,(FB + GB)},{G}} is non empty set
the multF of F . [G,(FB + GB)] is set
G * FB is Element of the carrier of F
the multF of F . (G,FB) is Element of the carrier of F
[G,FB] is set
{G,FB} is non empty set
{{G,FB},{G}} is non empty set
the multF of F . [G,FB] is set
G * GB is Element of the carrier of F
the multF of F . (G,GB) is Element of the carrier of F
[G,GB] is set
{G,GB} is non empty set
{{G,GB},{G}} is non empty set
the multF of F . [G,GB] is set
(G * FB) + (G * GB) is Element of the carrier of F
the addF of F . ((G * FB),(G * GB)) is Element of the carrier of F
[(G * FB),(G * GB)] is set
{(G * FB),(G * GB)} is non empty set
{(G * FB)} is non empty set
{{(G * FB),(G * GB)},{(G * FB)}} is non empty set
the addF of F . [(G * FB),(G * GB)] is set
the carrier of X is non empty set
aFB is right_complementable Element of the carrier of X
aF is right_complementable Element of the carrier of X
aG is right_complementable Element of the carrier of X
aF + aG is right_complementable 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 right_complementable 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
aFB * (aF + aG) is right_complementable 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 multF of X . (aFB,(aF + aG)) is right_complementable Element of the carrier of X
[aFB,(aF + aG)] is set
{aFB,(aF + aG)} is non empty set
{aFB} is non empty set
{{aFB,(aF + aG)},{aFB}} is non empty set
the multF of X . [aFB,(aF + aG)] is set
aFB * aF is right_complementable Element of the carrier of X
the multF of X . (aFB,aF) is right_complementable Element of the carrier of X
[aFB,aF] is set
{aFB,aF} is non empty set
{{aFB,aF},{aFB}} is non empty set
the multF of X . [aFB,aF] is set
aFB * aG is right_complementable Element of the carrier of X
the multF of X . (aFB,aG) is right_complementable Element of the carrier of X
[aFB,aG] is set
{aFB,aG} is non empty set
{{aFB,aG},{aFB}} is non empty set
the multF of X . [aFB,aG] is set
(aFB * aF) + (aFB * aG) is right_complementable Element of the carrier of X
the addF of X . ((aFB * aF),(aFB * aG)) is right_complementable Element of the carrier of X
[(aFB * aF),(aFB * aG)] is set
{(aFB * aF),(aFB * aG)} is non empty set
{(aFB * aF)} is non empty set
{{(aFB * aF),(aFB * aG)},{(aFB * aF)}} is non empty set
the addF of X . [(aFB * aF),(aFB * aG)] is set
G is V11() real ext-real set
FB is Element of the carrier of F
GB is Element of the carrier of F
FB + GB is Element of the carrier of F
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 . (FB,GB) is Element of the carrier of F
[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 F . [FB,GB] is set
G * (FB + GB) is Element of the carrier of F
the Mult of F is non empty Relation-like [:REAL, the carrier of F:] -defined the carrier of F -valued Function-like total quasi_total Element of bool [:[:REAL, the carrier of F:], the carrier of F:]
[:REAL, the carrier of F:] is non empty set
[:[:REAL, the carrier of F:], the carrier of F:] is non empty set
bool [:[:REAL, the carrier of F:], the carrier of F:] is non empty set
the Mult of F . (G,(FB + GB)) is set
[G,(FB + GB)] is set
{G,(FB + GB)} is non empty set
{G} is non empty V185() V186() V187() set
{{G,(FB + GB)},{G}} is non empty set
the Mult of F . [G,(FB + GB)] is set
G * FB is Element of the carrier of F
the Mult of F . (G,FB) is set
[G,FB] is set
{G,FB} is non empty set
{{G,FB},{G}} is non empty set
the Mult of F . [G,FB] is set
G * GB is Element of the carrier of F
the Mult of F . (G,GB) is set
[G,GB] is set
{G,GB} is non empty set
{{G,GB},{G}} is non empty set
the Mult of F . [G,GB] is set
(G * FB) + (G * GB) is Element of the carrier of F
the addF of F . ((G * FB),(G * GB)) is Element of the carrier of F
[(G * FB),(G * GB)] is set
{(G * FB),(G * GB)} is non empty set
{(G * FB)} is non empty set
{{(G * FB),(G * GB)},{(G * FB)}} is non empty set
the addF of F . [(G * FB),(G * GB)] is set
the carrier of X is non empty set
aFB is right_complementable Element of the carrier of X
aF is right_complementable Element of the carrier of X
aFB + aF is right_complementable 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 right_complementable 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
G * (aFB + aF) is right_complementable Element of the carrier of X
the Mult of X is non empty Relation-like [:REAL, the carrier of X:] -defined the carrier of X -valued Function-like total quasi_total Element of bool [:[:REAL, the carrier of X:], the carrier of X:]
[:REAL, the carrier of X:] is non empty set
[:[:REAL, the carrier of X:], the carrier of X:] is non empty set
bool [:[:REAL, the carrier of X:], the carrier of X:] is non empty set
the Mult of X . (G,(aFB + aF)) is set
[G,(aFB + aF)] is set
{G,(aFB + aF)} is non empty set
{{G,(aFB + aF)},{G}} is non empty set
the Mult of X . [G,(aFB + aF)] is set
G * aFB is right_complementable Element of the carrier of X
the Mult of X . (G,aFB) is set
[G,aFB] is set
{G,aFB} is non empty set
{{G,aFB},{G}} is non empty set
the Mult of X . [G,aFB] is set
G * aF is right_complementable Element of the carrier of X
the Mult of X . (G,aF) is set
[G,aF] is set
{G,aF} is non empty set
{{G,aF},{G}} is non empty set
the Mult of X . [G,aF] is set
(G * aFB) + (G * aF) is right_complementable Element of the carrier of X
the addF of X . ((G * aFB),(G * aF)) is right_complementable Element of the carrier of X
[(G * aFB),(G * aF)] is set
{(G * aFB),(G * aF)} is non empty set
{(G * aFB)} is non empty set
{{(G * aFB),(G * aF)},{(G * aFB)}} is non empty set
the addF of X . [(G * aFB),(G * aF)] is set
G is V11() real ext-real set
FB is V11() real ext-real set
G + FB is V11() real ext-real set
GB is Element of the carrier of F
(G + FB) * GB is Element of the carrier of F
the Mult of F is non empty Relation-like [:REAL, the carrier of F:] -defined the carrier of F -valued Function-like total quasi_total Element of bool [:[:REAL, the carrier of F:], the carrier of F:]
[:REAL, the carrier of F:] is non empty set
[:[:REAL, the carrier of F:], the carrier of F:] is non empty set
bool [:[:REAL, the carrier of F:], the carrier of F:] is non empty set
the Mult of F . ((G + FB),GB) is set
[(G + FB),GB] is set
{(G + FB),GB} is non empty set
{(G + FB)} is non empty V185() V186() V187() set
{{(G + FB),GB},{(G + FB)}} is non empty set
the Mult of F . [(G + FB),GB] is set
G * GB is Element of the carrier of F
the Mult of F . (G,GB) is set
[G,GB] is set
{G,GB} is non empty set
{G} is non empty V185() V186() V187() set
{{G,GB},{G}} is non empty set
the Mult of F . [G,GB] is set
FB * GB is Element of the carrier of F
the Mult of F . (FB,GB) is set
[FB,GB] is set
{FB,GB} is non empty set
{FB} is non empty V185() V186() V187() set
{{FB,GB},{FB}} is non empty set
the Mult of F . [FB,GB] is set
(G * GB) + (FB * GB) is Element of the carrier of F
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 . ((G * GB),(FB * GB)) is Element of the carrier of F
[(G * GB),(FB * GB)] is set
{(G * GB),(FB * GB)} is non empty set
{(G * GB)} is non empty set
{{(G * GB),(FB * GB)},{(G * GB)}} is non empty set
the addF of F . [(G * GB),(FB * GB)] is set
the carrier of X is non empty set
aFB is right_complementable Element of the carrier of X
(G + FB) * aFB is right_complementable Element of the carrier of X
the Mult of X is non empty Relation-like [:REAL, the carrier of X:] -defined the carrier of X -valued Function-like total quasi_total Element of bool [:[:REAL, the carrier of X:], the carrier of X:]
[:REAL, the carrier of X:] is non empty set
[:[:REAL, the carrier of X:], the carrier of X:] is non empty set
bool [:[:REAL, the carrier of X:], the carrier of X:] is non empty set
the Mult of X . ((G + FB),aFB) is set
[(G + FB),aFB] is set
{(G + FB),aFB} is non empty set
{{(G + FB),aFB},{(G + FB)}} is non empty set
the Mult of X . [(G + FB),aFB] is set
G * aFB is right_complementable Element of the carrier of X
the Mult of X . (G,aFB) is set
[G,aFB] is set
{G,aFB} is non empty set
{{G,aFB},{G}} is non empty set
the Mult of X . [G,aFB] is set
FB * aFB is right_complementable Element of the carrier of X
the Mult of X . (FB,aFB) is set
[FB,aFB] is set
{FB,aFB} is non empty set
{{FB,aFB},{FB}} is non empty set
the Mult of X . [FB,aFB] is set
(G * aFB) + (FB * aFB) is right_complementable 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 * aFB),(FB * aFB)) is right_complementable Element of the carrier of X
[(G * aFB),(FB * aFB)] is set
{(G * aFB),(FB * aFB)} is non empty set
{(G * aFB)} is non empty set
{{(G * aFB),(FB * aFB)},{(G * aFB)}} is non empty set
the addF of X . [(G * aFB),(FB * aFB)] is set
G is V11() real ext-real set
FB is V11() real ext-real set
G * FB is V11() real ext-real set
GB is Element of the carrier of F
(G * FB) * GB is Element of the carrier of F
the Mult of F is non empty Relation-like [:REAL, the carrier of F:] -defined the carrier of F -valued Function-like total quasi_total Element of bool [:[:REAL, the carrier of F:], the carrier of F:]
[:REAL, the carrier of F:] is non empty set
[:[:REAL, the carrier of F:], the carrier of F:] is non empty set
bool [:[:REAL, the carrier of F:], the carrier of F:] is non empty set
the Mult of F . ((G * FB),GB) is set
[(G * FB),GB] is set
{(G * FB),GB} is non empty set
{(G * FB)} is non empty V185() V186() V187() set
{{(G * FB),GB},{(G * FB)}} is non empty set
the Mult of F . [(G * FB),GB] is set
FB * GB is Element of the carrier of F
the Mult of F . (FB,GB) is set
[FB,GB] is set
{FB,GB} is non empty set
{FB} is non empty V185() V186() V187() set
{{FB,GB},{FB}} is non empty set
the Mult of F . [FB,GB] is set
G * (FB * GB) is Element of the carrier of F
the Mult of F . (G,(FB * GB)) is set
[G,(FB * GB)] is set
{G,(FB * GB)} is non empty set
{G} is non empty V185() V186() V187() set
{{G,(FB * GB)},{G}} is non empty set
the Mult of F . [G,(FB * GB)] is set
the carrier of X is non empty set
aFB is right_complementable Element of the carrier of X
(G * FB) * aFB is right_complementable Element of the carrier of X
the Mult of X is non empty Relation-like [:REAL, the carrier of X:] -defined the carrier of X -valued Function-like total quasi_total Element of bool [:[:REAL, the carrier of X:], the carrier of X:]
[:REAL, the carrier of X:] is non empty set
[:[:REAL, the carrier of X:], the carrier of X:] is non empty set
bool [:[:REAL, the carrier of X:], the carrier of X:] is non empty set
the Mult of X . ((G * FB),aFB) is set
[(G * FB),aFB] is set
{(G * FB),aFB} is non empty set
{{(G * FB),aFB},{(G * FB)}} is non empty set
the Mult of X . [(G * FB),aFB] is set
FB * aFB is right_complementable Element of the carrier of X
the Mult of X . (FB,aFB) is set
[FB,aFB] is set
{FB,aFB} is non empty set
{{FB,aFB},{FB}} is non empty set
the Mult of X . [FB,aFB] is set
G * (FB * aFB) is right_complementable Element of the carrier of X
the Mult of X . (G,(FB * aFB)) is set
[G,(FB * aFB)] is set
{G,(FB * aFB)} is non empty set
{{G,(FB * aFB)},{G}} is non empty set
the Mult of X . [G,(FB * aFB)] is set
G is Element of the carrier of F
FB is Element of the carrier of F
G * FB is Element of the carrier of F
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 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 multF of F . (G,FB) is Element of the carrier of F
[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 F . [G,FB] is set
GB is V11() real ext-real Element of REAL
GB * (G * FB) is Element of the carrier of F
the Mult of F is non empty Relation-like [:REAL, the carrier of F:] -defined the carrier of F -valued Function-like total quasi_total Element of bool [:[:REAL, the carrier of F:], the carrier of F:]
[:REAL, the carrier of F:] is non empty set
[:[:REAL, the carrier of F:], the carrier of F:] is non empty set
bool [:[:REAL, the carrier of F:], the carrier of F:] is non empty set
the Mult of F . (GB,(G * FB)) is set
[GB,(G * FB)] is set
{GB,(G * FB)} is non empty set
{GB} is non empty V185() V186() V187() set
{{GB,(G * FB)},{GB}} is non empty set
the Mult of F . [GB,(G * FB)] is set
GB * G is Element of the carrier of F
the Mult of F . (GB,G) is set
[GB,G] is set
{GB,G} is non empty set
{{GB,G},{GB}} is non empty set
the Mult of F . [GB,G] is set
(GB * G) * FB is Element of the carrier of F
the multF of F . ((GB * G),FB) is Element of the carrier of F
[(GB * G),FB] is set
{(GB * G),FB} is non empty set
{(GB * G)} is non empty set
{{(GB * G),FB},{(GB * G)}} is non empty set
the multF of F . [(GB * G),FB] is set
the carrier of X is non empty set
aFB is right_complementable Element of the carrier of X
aF is right_complementable Element of the carrier of X
aFB * aF is right_complementable 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 . (aFB,aF) is right_complementable 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 multF of X . [aFB,aF] is set
GB * (aFB * aF) is right_complementable Element of the carrier of X
the Mult of X is non empty Relation-like [:REAL, the carrier of X:] -defined the carrier of X -valued Function-like total quasi_total Element of bool [:[:REAL, the carrier of X:], the carrier of X:]
[:REAL, the carrier of X:] is non empty set
[:[:REAL, the carrier of X:], the carrier of X:] is non empty set
bool [:[:REAL, the carrier of X:], the carrier of X:] is non empty set
the Mult of X . (GB,(aFB * aF)) is set
[GB,(aFB * aF)] is set
{GB,(aFB * aF)} is non empty set
{{GB,(aFB * aF)},{GB}} is non empty set
the Mult of X . [GB,(aFB * aF)] is set
GB * aFB is right_complementable Element of the carrier of X
the Mult of X . (GB,aFB) is set
[GB,aFB] is set
{GB,aFB} is non empty set
{{GB,aFB},{GB}} is non empty set
the Mult of X . [GB,aFB] is set
(GB * aFB) * aF is right_complementable Element of the carrier of X
the multF of X . ((GB * aFB),aF) is right_complementable Element of the carrier of X
[(GB * aFB),aF] is set
{(GB * aFB),aF} is non empty set
{(GB * aFB)} is non empty set
{{(GB * aFB),aF},{(GB * aFB)}} is non empty set
the multF of X . [(GB * aFB),aF] is set
X is non empty TopSpace-like pseudocompact compact TopStruct
(X) is non empty unital strict Normed_AlgebraStr
(X) is non empty add-closed having-inverse additively-closed multiplicatively-closed additively-linearly-closed Element of bool the carrier of (RAlgebra the carrier of X)
the carrier of X is non empty set
RAlgebra the carrier of X is non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative unital associative commutative right-distributive right_unital well-unital left_unital strict vector-associative AlgebraStr
Funcs ( the carrier of X,REAL) is non empty functional FUNCTION_DOMAIN of the carrier of X, REAL
RealFuncMult the carrier of X is non empty Relation-like [:(Funcs ( the carrier of X,REAL)),(Funcs ( the carrier of X,REAL)):] -defined Funcs ( the carrier of X,REAL) -valued Function-like total quasi_total Element of bool [:[:(Funcs ( the carrier of X,REAL)),(Funcs ( the carrier of X,REAL)):],(Funcs ( the carrier of X,REAL)):]
[:(Funcs ( the carrier of X,REAL)),(Funcs ( the carrier of X,REAL)):] is non empty set
[:[:(Funcs ( the carrier of X,REAL)),(Funcs ( the carrier of X,REAL)):],(Funcs ( the carrier of X,REAL)):] is non empty set
bool [:[:(Funcs ( the carrier of X,REAL)),(Funcs ( the carrier of X,REAL)):],(Funcs ( the carrier of X,REAL)):] is non empty set
RealFuncAdd the carrier of X is non empty Relation-like [:(Funcs ( the carrier of X,REAL)),(Funcs ( the carrier of X,REAL)):] -defined Funcs ( the carrier of X,REAL) -valued Function-like total quasi_total Element of bool [:[:(Funcs ( the carrier of X,REAL)),(Funcs ( the carrier of X,REAL)):],(Funcs ( the carrier of X,REAL)):]
RealFuncExtMult the carrier of X is non empty Relation-like [:REAL,(Funcs ( the carrier of X,REAL)):] -defined Funcs ( the carrier of X,REAL) -valued Function-like total quasi_total Element of bool [:[:REAL,(Funcs ( the carrier of X,REAL)):],(Funcs ( the carrier of X,REAL)):]
[:REAL,(Funcs ( the carrier of X,REAL)):] is non empty set
[:[:REAL,(Funcs ( the carrier of X,REAL)):],(Funcs ( the carrier of X,REAL)):] is non empty set
bool [:[:REAL,(Funcs ( the carrier of X,REAL)):],(Funcs ( the carrier of X,REAL)):] is non empty set
RealFuncUnit the carrier of X is Relation-like the carrier of X -defined REAL -valued Function-like total quasi_total V152() V153() V154() Element of Funcs ( the carrier of X,REAL)
the carrier of X --> 1 is non empty Relation-like the carrier of X -defined NAT -valued RAT -valued INT -valued Function-like total quasi_total V152() V153() V154() V155() Element of bool [: the carrier of X,NAT:]
[: the carrier of X,NAT:] is non empty RAT -valued INT -valued V152() V153() V154() V155() set
bool [: the carrier of X,NAT:] is non empty set
RealFuncZero the carrier of X is Relation-like the carrier of X -defined REAL -valued Function-like total quasi_total V152() V153() V154() Element of Funcs ( the carrier of X,REAL)
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 V152() V153() V154() V155() Element of bool [: the carrier of X,NAT:]
AlgebraStr(# (Funcs ( the carrier of X,REAL)),(RealFuncMult the carrier of X),(RealFuncAdd the carrier of X),(RealFuncExtMult the carrier of X),(RealFuncUnit the carrier of X),(RealFuncZero the carrier of X) #) is strict AlgebraStr
the carrier of (RAlgebra the carrier of X) is non empty set
bool the carrier of (RAlgebra the carrier of X) is non empty set
[: the carrier of X,REAL:] is non empty V152() V153() V154() set
bool [: the carrier of X,REAL:] is non empty set
{ b₁ where b₁ is non empty Relation-like the carrier of X -defined REAL -valued Function-like total quasi_total V152() V153() V154() continuous Element of bool [: the carrier of X,REAL:] : verum } is set
mult_ ((X),(RAlgebra 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),(RAlgebra 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),(RAlgebra the carrier of X)) is non empty Relation-like [:REAL,(X):] -defined (X) -valued Function-like total quasi_total Element of bool [:[:REAL,(X):],(X):]
[:REAL,(X):] is non empty set
[:[:REAL,(X):],(X):] is non empty set
bool [:[:REAL,(X):],(X):] is non empty set
One_ ((X),(RAlgebra the carrier of X)) is Element of (X)
Zero_ ((X),(RAlgebra the carrier of X)) is Element of (X)
(X) is non empty Relation-like (X) -defined REAL -valued Function-like total quasi_total V152() V153() V154() Element of bool [:(X),REAL:]
[:(X),REAL:] is non empty V152() V153() V154() set
bool [:(X),REAL:] is non empty set
BoundedFunctions the carrier of X is non empty add-closed having-inverse additively-closed multiplicatively-closed additively-linearly-closed Element of bool the carrier of (RAlgebra the carrier of X)
{ b₁ where b₁ is non empty Relation-like the carrier of X -defined REAL -valued Function-like total quasi_total V152() V153() V154() Element of bool [: the carrier of X,REAL:] : b₁ | the carrier of X is bounded } is set
BoundedFunctionsNorm the carrier of X is non empty Relation-like BoundedFunctions the carrier of X -defined REAL -valued Function-like total quasi_total V152() V153() V154() Element of bool [:(BoundedFunctions the carrier of X),REAL:]
[:(BoundedFunctions the carrier of X),REAL:] is non empty V152() V153() V154() set
bool [:(BoundedFunctions the carrier of X),REAL:] is non empty set
(BoundedFunctionsNorm the carrier of X) | (X) is Relation-like BoundedFunctions the carrier of X -defined REAL -valued Function-like V152() V153() V154() Element of bool [:(BoundedFunctions the carrier of X),REAL:]
Normed_AlgebraStr(# (X),(mult_ ((X),(RAlgebra the carrier of X))),(Add_ ((X),(RAlgebra the carrier of X))),(Mult_ ((X),(RAlgebra the carrier of X))),(One_ ((X),(RAlgebra the carrier of X))),(Zero_ ((X),(RAlgebra the carrier of X))),(X) #) is strict Normed_AlgebraStr
the carrier of (X) is non empty set
R_Normed_Algebra_of_BoundedFunctions the carrier of X is non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital discerning reflexive RealNormSpace-like unital complete Normed_AlgebraStr
mult_ ((BoundedFunctions the carrier of X),(RAlgebra the carrier of X)) is non empty Relation-like [:(BoundedFunctions the carrier of X),(BoundedFunctions the carrier of X):] -defined BoundedFunctions the carrier of X -valued Function-like total quasi_total Element of bool [:[:(BoundedFunctions the carrier of X),(BoundedFunctions the carrier of X):],(BoundedFunctions the carrier of X):]
[:(BoundedFunctions the carrier of X),(BoundedFunctions the carrier of X):] is non empty set
[:[:(BoundedFunctions the carrier of X),(BoundedFunctions the carrier of X):],(BoundedFunctions the carrier of X):] is non empty set
bool [:[:(BoundedFunctions the carrier of X),(BoundedFunctions the carrier of X):],(BoundedFunctions the carrier of X):] is non empty set
Add_ ((BoundedFunctions the carrier of X),(RAlgebra the carrier of X)) is non empty Relation-like [:(BoundedFunctions the carrier of X),(BoundedFunctions the carrier of X):] -defined BoundedFunctions the carrier of X -valued Function-like total quasi_total Element of bool [:[:(BoundedFunctions the carrier of X),(BoundedFunctions the carrier of X):],(BoundedFunctions the carrier of X):]
Mult_ ((BoundedFunctions the carrier of X),(RAlgebra the carrier of X)) is non empty Relation-like [:REAL,(BoundedFunctions the carrier of X):] -defined BoundedFunctions the carrier of X -valued Function-like total quasi_total Element of bool [:[:REAL,(BoundedFunctions the carrier of X):],(BoundedFunctions the carrier of X):]
[:REAL,(BoundedFunctions the carrier of X):] is non empty set
[:[:REAL,(BoundedFunctions the carrier of X):],(BoundedFunctions the carrier of X):] is non empty set
bool [:[:REAL,(BoundedFunctions the carrier of X):],(BoundedFunctions the carrier of X):] is non empty set
One_ ((BoundedFunctions the carrier of X),(RAlgebra the carrier of X)) is Element of BoundedFunctions the carrier of X
Zero_ ((BoundedFunctions the carrier of X),(RAlgebra the carrier of X)) is Element of BoundedFunctions the carrier of X
Normed_AlgebraStr(# (BoundedFunctions the carrier of X),(mult_ ((BoundedFunctions the carrier of X),(RAlgebra the carrier of X))),(Add_ ((BoundedFunctions the carrier of X),(RAlgebra the carrier of X))),(Mult_ ((BoundedFunctions the carrier of X),(RAlgebra the carrier of X))),(One_ ((BoundedFunctions the carrier of X),(RAlgebra the carrier of X))),(Zero_ ((BoundedFunctions the carrier of X),(RAlgebra the carrier of X))),(BoundedFunctionsNorm the carrier of X) #) is strict Normed_AlgebraStr
the carrier of (R_Normed_Algebra_of_BoundedFunctions the carrier of X) is non empty set
X is Element of the carrier of (X)
F is right_complementable Element of the carrier of (R_Normed_Algebra_of_BoundedFunctions the carrier of X)
||.X.|| is V11() real ext-real Element of REAL
the U8 of (X) is non empty Relation-like the carrier of (X) -defined REAL -valued Function-like total quasi_total V152() V153() V154() Element of bool [: the carrier of (X),REAL:]
[: the carrier of (X),REAL:] is non empty V152() V153() V154() set
bool [: the carrier of (X),REAL:] is non empty set
the U8 of (X) . X is V11() real ext-real Element of REAL
||.F.|| is V11() real ext-real Element of REAL
the U8 of (R_Normed_Algebra_of_BoundedFunctions the carrier of X) is non empty Relation-like the carrier of (R_Normed_Algebra_of_BoundedFunctions the carrier of X) -defined REAL -valued Function-like total quasi_total V152() V153() V154() Element of bool [: the carrier of (R_Normed_Algebra_of_BoundedFunctions the carrier of X),REAL:]
[: the carrier of (R_Normed_Algebra_of_BoundedFunctions the carrier of X),REAL:] is non empty V152() V153() V154() set
bool [: the carrier of (R_Normed_Algebra_of_BoundedFunctions the carrier of X),REAL:] is non empty set
the U8 of (R_Normed_Algebra_of_BoundedFunctions the carrier of X) . F is V11() real ext-real Element of REAL
X is non empty TopSpace-like pseudocompact compact TopStruct
(X) is non empty unital strict Normed_AlgebraStr
(X) is non empty add-closed having-inverse additively-closed multiplicatively-closed additively-linearly-closed Element of bool the carrier of (RAlgebra the carrier of X)
the carrier of X is non empty set
RAlgebra the carrier of X is non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative unital associative commutative right-distributive right_unital well-unital left_unital strict vector-associative AlgebraStr
Funcs ( the carrier of X,REAL) is non empty functional FUNCTION_DOMAIN of the carrier of X, REAL
RealFuncMult the carrier of X is non empty Relation-like [:(Funcs ( the carrier of X,REAL)),(Funcs ( the carrier of X,REAL)):] -defined Funcs ( the carrier of X,REAL) -valued Function-like total quasi_total Element of bool [:[:(Funcs ( the carrier of X,REAL)),(Funcs ( the carrier of X,REAL)):],(Funcs ( the carrier of X,REAL)):]
[:(Funcs ( the carrier of X,REAL)),(Funcs ( the carrier of X,REAL)):] is non empty set
[:[:(Funcs ( the carrier of X,REAL)),(Funcs ( the carrier of X,REAL)):],(Funcs ( the carrier of X,REAL)):] is non empty set
bool [:[:(Funcs ( the carrier of X,REAL)),(Funcs ( the carrier of X,REAL)):],(Funcs ( the carrier of X,REAL)):] is non empty set
RealFuncAdd the carrier of X is non empty Relation-like [:(Funcs ( the carrier of X,REAL)),(Funcs ( the carrier of X,REAL)):] -defined Funcs ( the carrier of X,REAL) -valued Function-like total quasi_total Element of bool [:[:(Funcs ( the carrier of X,REAL)),(Funcs ( the carrier of X,REAL)):],(Funcs ( the carrier of X,REAL)):]
RealFuncExtMult the carrier of X is non empty Relation-like [:REAL,(Funcs ( the carrier of X,REAL)):] -defined Funcs ( the carrier of X,REAL) -valued Function-like total quasi_total Element of bool [:[:REAL,(Funcs ( the carrier of X,REAL)):],(Funcs ( the carrier of X,REAL)):]
[:REAL,(Funcs ( the carrier of X,REAL)):] is non empty set
[:[:REAL,(Funcs ( the carrier of X,REAL)):],(Funcs ( the carrier of X,REAL)):] is non empty set
bool [:[:REAL,(Funcs ( the carrier of X,REAL)):],(Funcs ( the carrier of X,REAL)):] is non empty set
RealFuncUnit the carrier of X is Relation-like the carrier of X -defined REAL -valued Function-like total quasi_total V152() V153() V154() Element of Funcs ( the carrier of X,REAL)
the carrier of X --> 1 is non empty Relation-like the carrier of X -defined NAT -valued RAT -valued INT -valued Function-like total quasi_total V152() V153() V154() V155() Element of bool [: the carrier of X,NAT:]
[: the carrier of X,NAT:] is non empty RAT -valued INT -valued V152() V153() V154() V155() set
bool [: the carrier of X,NAT:] is non empty set
RealFuncZero the carrier of X is Relation-like the carrier of X -defined REAL -valued Function-like total quasi_total V152() V153() V154() Element of Funcs ( the carrier of X,REAL)
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 V152() V153() V154() V155() Element of bool [: the carrier of X,NAT:]
AlgebraStr(# (Funcs ( the carrier of X,REAL)),(RealFuncMult the carrier of X),(RealFuncAdd the carrier of X),(RealFuncExtMult the carrier of X),(RealFuncUnit the carrier of X),(RealFuncZero the carrier of X) #) is strict AlgebraStr
the carrier of (RAlgebra the carrier of X) is non empty set
bool the carrier of (RAlgebra the carrier of X) is non empty set
[: the carrier of X,REAL:] is non empty V152() V153() V154() set
bool [: the carrier of X,REAL:] is non empty set
{ b₁ where b₁ is non empty Relation-like the carrier of X -defined REAL -valued Function-like total quasi_total V152() V153() V154() continuous Element of bool [: the carrier of X,REAL:] : verum } is set
mult_ ((X),(RAlgebra 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),(RAlgebra 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),(RAlgebra the carrier of X)) is non empty Relation-like [:REAL,(X):] -defined (X) -valued Function-like total quasi_total Element of bool [:[:REAL,(X):],(X):]
[:REAL,(X):] is non empty set
[:[:REAL,(X):],(X):] is non empty set
bool [:[:REAL,(X):],(X):] is non empty set
One_ ((X),(RAlgebra the carrier of X)) is Element of (X)
Zero_ ((X),(RAlgebra the carrier of X)) is Element of (X)
(X) is non empty Relation-like (X) -defined REAL -valued Function-like total quasi_total V152() V153() V154() Element of bool [:(X),REAL:]
[:(X),REAL:] is non empty V152() V153() V154() set
bool [:(X),REAL:] is non empty set
BoundedFunctions the carrier of X is non empty add-closed having-inverse additively-closed multiplicatively-closed additively-linearly-closed Element of bool the carrier of (RAlgebra the carrier of X)
{ b₁ where b₁ is non empty Relation-like the carrier of X -defined REAL -valued Function-like total quasi_total V152() V153() V154() Element of bool [: the carrier of X,REAL:] : b₁ | the carrier of X is bounded } is set
BoundedFunctionsNorm the carrier of X is non empty Relation-like BoundedFunctions the carrier of X -defined REAL -valued Function-like total quasi_total V152() V153() V154() Element of bool [:(BoundedFunctions the carrier of X),REAL:]
[:(BoundedFunctions the carrier of X),REAL:] is non empty V152() V153() V154() set
bool [:(BoundedFunctions the carrier of X),REAL:] is non empty set
(BoundedFunctionsNorm the carrier of X) | (X) is Relation-like BoundedFunctions the carrier of X -defined REAL -valued Function-like V152() V153() V154() Element of bool [:(BoundedFunctions the carrier of X),REAL:]
Normed_AlgebraStr(# (X),(mult_ ((X),(RAlgebra the carrier of X))),(Add_ ((X),(RAlgebra the carrier of X))),(Mult_ ((X),(RAlgebra the carrier of X))),(One_ ((X),(RAlgebra the carrier of X))),(Zero_ ((X),(RAlgebra the carrier of X))),(X) #) is strict Normed_AlgebraStr
the carrier of (X) is non empty set
R_Normed_Algebra_of_BoundedFunctions the carrier of X is non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital discerning reflexive RealNormSpace-like unital complete Normed_AlgebraStr
mult_ ((BoundedFunctions the carrier of X),(RAlgebra the carrier of X)) is non empty Relation-like [:(BoundedFunctions the carrier of X),(BoundedFunctions the carrier of X):] -defined BoundedFunctions the carrier of X -valued Function-like total quasi_total Element of bool [:[:(BoundedFunctions the carrier of X),(BoundedFunctions the carrier of X):],(BoundedFunctions the carrier of X):]
[:(BoundedFunctions the carrier of X),(BoundedFunctions the carrier of X):] is non empty set
[:[:(BoundedFunctions the carrier of X),(BoundedFunctions the carrier of X):],(BoundedFunctions the carrier of X):] is non empty set
bool [:[:(BoundedFunctions the carrier of X),(BoundedFunctions the carrier of X):],(BoundedFunctions the carrier of X):] is non empty set
Add_ ((BoundedFunctions the carrier of X),(RAlgebra the carrier of X)) is non empty Relation-like [:(BoundedFunctions the carrier of X),(BoundedFunctions the carrier of X):] -defined BoundedFunctions the carrier of X -valued Function-like total quasi_total Element of bool [:[:(BoundedFunctions the carrier of X),(BoundedFunctions the carrier of X):],(BoundedFunctions the carrier of X):]
Mult_ ((BoundedFunctions the carrier of X),(RAlgebra the carrier of X)) is non empty Relation-like [:REAL,(BoundedFunctions the carrier of X):] -defined BoundedFunctions the carrier of X -valued Function-like total quasi_total Element of bool [:[:REAL,(BoundedFunctions the carrier of X):],(BoundedFunctions the carrier of X):]
[:REAL,(BoundedFunctions the carrier of X):] is non empty set
[:[:REAL,(BoundedFunctions the carrier of X):],(BoundedFunctions the carrier of X):] is non empty set
bool [:[:REAL,(BoundedFunctions the carrier of X):],(BoundedFunctions the carrier of X):] is non empty set
One_ ((BoundedFunctions the carrier of X),(RAlgebra the carrier of X)) is Element of BoundedFunctions the carrier of X
Zero_ ((BoundedFunctions the carrier of X),(RAlgebra the carrier of X)) is Element of BoundedFunctions the carrier of X
Normed_AlgebraStr(# (BoundedFunctions the carrier of X),(mult_ ((BoundedFunctions the carrier of X),(RAlgebra the carrier of X))),(Add_ ((BoundedFunctions the carrier of X),(RAlgebra the carrier of X))),(Mult_ ((BoundedFunctions the carrier of X),(RAlgebra the carrier of X))),(One_ ((BoundedFunctions the carrier of X),(RAlgebra the carrier of X))),(Zero_ ((BoundedFunctions the carrier of X),(RAlgebra the carrier of X))),(BoundedFunctionsNorm the carrier of X) #) is strict Normed_AlgebraStr
the carrier of (R_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 right_complementable Element of the carrier of (R_Normed_Algebra_of_BoundedFunctions the carrier of X)
FB is right_complementable Element of the carrier of (R_Normed_Algebra_of_BoundedFunctions the carrier of X)
G + FB is right_complementable Element of the carrier of (R_Normed_Algebra_of_BoundedFunctions the carrier of X)
the addF of (R_Normed_Algebra_of_BoundedFunctions the carrier of X) is non empty Relation-like [: the carrier of (R_Normed_Algebra_of_BoundedFunctions the carrier of X), the carrier of (R_Normed_Algebra_of_BoundedFunctions the carrier of X):] -defined the carrier of (R_Normed_Algebra_of_BoundedFunctions the carrier of X) -valued Function-like total quasi_total Element of bool [:[: the carrier of (R_Normed_Algebra_of_BoundedFunctions the carrier of X), the carrier of (R_Normed_Algebra_of_BoundedFunctions the carrier of X):], the carrier of (R_Normed_Algebra_of_BoundedFunctions the carrier of X):]
[: the carrier of (R_Normed_Algebra_of_BoundedFunctions the carrier of X), the carrier of (R_Normed_Algebra_of_BoundedFunctions the carrier of X):] is non empty set
[:[: the carrier of (R_Normed_Algebra_of_BoundedFunctions the carrier of X), the carrier of (R_Normed_Algebra_of_BoundedFunctions the carrier of X):], the carrier of (R_Normed_Algebra_of_BoundedFunctions the carrier of X):] is non empty set
bool [:[: the carrier of (R_Normed_Algebra_of_BoundedFunctions the carrier of X), the carrier of (R_Normed_Algebra_of_BoundedFunctions the carrier of X):], the carrier of (R_Normed_Algebra_of_BoundedFunctions the carrier of X):] is non empty set
the addF of (R_Normed_Algebra_of_BoundedFunctions the carrier of X) . (G,FB) is right_complementable Element of the carrier of (R_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 (R_Normed_Algebra_of_BoundedFunctions the carrier of X) . [G,FB] is set
the addF of (RAlgebra the carrier of X) is non empty Relation-like [: the carrier of (RAlgebra the carrier of X), the carrier of (RAlgebra the carrier of X):] -defined the carrier of (RAlgebra the carrier of X) -valued Function-like total quasi_total Element of bool [:[: the carrier of (RAlgebra the carrier of X), the carrier of (RAlgebra the carrier of X):], the carrier of (RAlgebra the carrier of X):]
[: the carrier of (RAlgebra the carrier of X), the carrier of (RAlgebra the carrier of X):] is non empty set
[:[: the carrier of (RAlgebra the carrier of X), the carrier of (RAlgebra the carrier of X):], the carrier of (RAlgebra the carrier of X):] is non empty set
bool [:[: the carrier of (RAlgebra the carrier of X), the carrier of (RAlgebra the carrier of X):], the carrier of (RAlgebra the carrier of X):] is non empty set
the addF of (RAlgebra the carrier of X) || (X) is Relation-like Function-like set
the addF of (RAlgebra 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 (RAlgebra the carrier of X) || (X)) . [X,F] is set
the addF of (RAlgebra the carrier of X) . [X,F] is set
the addF of (RAlgebra the carrier of X) || (BoundedFunctions the carrier of X) is Relation-like Function-like set
the addF of (RAlgebra the carrier of X) | [:(BoundedFunctions the carrier of X),(BoundedFunctions the carrier of X):] is Relation-like set
[G,FB] is Element of [: the carrier of (R_Normed_Algebra_of_BoundedFunctions the carrier of X), the carrier of (R_Normed_Algebra_of_BoundedFunctions the carrier of X):]
( the addF of (RAlgebra the carrier of X) || (BoundedFunctions the carrier of X)) . [G,FB] is set
X is non empty TopSpace-like pseudocompact compact TopStruct
(X) is non empty unital strict Normed_AlgebraStr
(X) is non empty add-closed having-inverse additively-closed multiplicatively-closed additively-linearly-closed Element of bool the carrier of (RAlgebra the carrier of X)
the carrier of X is non empty set
RAlgebra the carrier of X is non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative unital associative commutative right-distributive right_unital well-unital left_unital strict vector-associative AlgebraStr
Funcs ( the carrier of X,REAL) is non empty functional FUNCTION_DOMAIN of the carrier of X, REAL
RealFuncMult the carrier of X is non empty Relation-like [:(Funcs ( the carrier of X,REAL)),(Funcs ( the carrier of X,REAL)):] -defined Funcs ( the carrier of X,REAL) -valued Function-like total quasi_total Element of bool [:[:(Funcs ( the carrier of X,REAL)),(Funcs ( the carrier of X,REAL)):],(Funcs ( the carrier of X,REAL)):]
[:(Funcs ( the carrier of X,REAL)),(Funcs ( the carrier of X,REAL)):] is non empty set
[:[:(Funcs ( the carrier of X,REAL)),(Funcs ( the carrier of X,REAL)):],(Funcs ( the carrier of X,REAL)):] is non empty set
bool [:[:(Funcs ( the carrier of X,REAL)),(Funcs ( the carrier of X,REAL)):],(Funcs ( the carrier of X,REAL)):] is non empty set
RealFuncAdd the carrier of X is non empty Relation-like [:(Funcs ( the carrier of X,REAL)),(Funcs ( the carrier of X,REAL)):] -defined Funcs ( the carrier of X,REAL) -valued Function-like total quasi_total Element of bool [:[:(Funcs ( the carrier of X,REAL)),(Funcs ( the carrier of X,REAL)):],(Funcs ( the carrier of X,REAL)):]
RealFuncExtMult the carrier of X is non empty Relation-like [:REAL,(Funcs ( the carrier of X,REAL)):] -defined Funcs ( the carrier of X,REAL) -valued Function-like total quasi_total Element of bool [:[:REAL,(Funcs ( the carrier of X,REAL)):],(Funcs ( the carrier of X,REAL)):]
[:REAL,(Funcs ( the carrier of X,REAL)):] is non empty set
[:[:REAL,(Funcs ( the carrier of X,REAL)):],(Funcs ( the carrier of X,REAL)):] is non empty set
bool [:[:REAL,(Funcs ( the carrier of X,REAL)):],(Funcs ( the carrier of X,REAL)):] is non empty set
RealFuncUnit the carrier of X is Relation-like the carrier of X -defined REAL -valued Function-like total quasi_total V152() V153() V154() Element of Funcs ( the carrier of X,REAL)
the carrier of X --> 1 is non empty Relation-like the carrier of X -defined NAT -valued RAT -valued INT -valued Function-like total quasi_total V152() V153() V154() V155() Element of bool [: the carrier of X,NAT:]
[: the carrier of X,NAT:] is non empty RAT -valued INT -valued V152() V153() V154() V155() set
bool [: the carrier of X,NAT:] is non empty set
RealFuncZero the carrier of X is Relation-like the carrier of X -defined REAL -valued Function-like total quasi_total V152() V153() V154() Element of Funcs ( the carrier of X,REAL)
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 V152() V153() V154() V155() Element of bool [: the carrier of X,NAT:]
AlgebraStr(# (Funcs ( the carrier of X,REAL)),(RealFuncMult the carrier of X),(RealFuncAdd the carrier of X),(RealFuncExtMult the carrier of X),(RealFuncUnit the carrier of X),(RealFuncZero the carrier of X) #) is strict AlgebraStr
the carrier of (RAlgebra the carrier of X) is non empty set
bool the carrier of (RAlgebra the carrier of X) is non empty set
[: the carrier of X,REAL:] is non empty V152() V153() V154() set
bool [: the carrier of X,REAL:] is non empty set
{ b₁ where b₁ is non empty Relation-like the carrier of X -defined REAL -valued Function-like total quasi_total V152() V153() V154() continuous Element of bool [: the carrier of X,REAL:] : verum } is set
mult_ ((X),(RAlgebra 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),(RAlgebra 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),(RAlgebra the carrier of X)) is non empty Relation-like [:REAL,(X):] -defined (X) -valued Function-like total quasi_total Element of bool [:[:REAL,(X):],(X):]
[:REAL,(X):] is non empty set
[:[:REAL,(X):],(X):] is non empty set
bool [:[:REAL,(X):],(X):] is non empty set
One_ ((X),(RAlgebra the carrier of X)) is Element of (X)
Zero_ ((X),(RAlgebra the carrier of X)) is Element of (X)
(X) is non empty Relation-like (X) -defined REAL -valued Function-like total quasi_total V152() V153() V154() Element of bool [:(X),REAL:]
[:(X),REAL:] is non empty V152() V153() V154() set
bool [:(X),REAL:] is non empty set
BoundedFunctions the carrier of X is non empty add-closed having-inverse additively-closed multiplicatively-closed additively-linearly-closed Element of bool the carrier of (RAlgebra the carrier of X)
{ b₁ where b₁ is non empty Relation-like the carrier of X -defined REAL -valued Function-like total quasi_total V152() V153() V154() Element of bool [: the carrier of X,REAL:] : b₁ | the carrier of X is bounded } is set
BoundedFunctionsNorm the carrier of X is non empty Relation-like BoundedFunctions the carrier of X -defined REAL -valued Function-like total quasi_total V152() V153() V154() Element of bool [:(BoundedFunctions the carrier of X),REAL:]
[:(BoundedFunctions the carrier of X),REAL:] is non empty V152() V153() V154() set
bool [:(BoundedFunctions the carrier of X),REAL:] is non empty set
(BoundedFunctionsNorm the carrier of X) | (X) is Relation-like BoundedFunctions the carrier of X -defined REAL -valued Function-like V152() V153() V154() Element of bool [:(BoundedFunctions the carrier of X),REAL:]
Normed_AlgebraStr(# (X),(mult_ ((X),(RAlgebra the carrier of X))),(Add_ ((X),(RAlgebra the carrier of X))),(Mult_ ((X),(RAlgebra the carrier of X))),(One_ ((X),(RAlgebra the carrier of X))),(Zero_ ((X),(RAlgebra the carrier of X))),(X) #) is strict Normed_AlgebraStr
the carrier of (X) is non empty set
R_Normed_Algebra_of_BoundedFunctions the carrier of X is non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital discerning reflexive RealNormSpace-like unital complete Normed_AlgebraStr
mult_ ((BoundedFunctions the carrier of X),(RAlgebra the carrier of X)) is non empty Relation-like [:(BoundedFunctions the carrier of X),(BoundedFunctions the carrier of X):] -defined BoundedFunctions the carrier of X -valued Function-like total quasi_total Element of bool [:[:(BoundedFunctions the carrier of X),(BoundedFunctions the carrier of X):],(BoundedFunctions the carrier of X):]
[:(BoundedFunctions the carrier of X),(BoundedFunctions the carrier of X):] is non empty set
[:[:(BoundedFunctions the carrier of X),(BoundedFunctions the carrier of X):],(BoundedFunctions the carrier of X):] is non empty set
bool [:[:(BoundedFunctions the carrier of X),(BoundedFunctions the carrier of X):],(BoundedFunctions the carrier of X):] is non empty set
Add_ ((BoundedFunctions the carrier of X),(RAlgebra the carrier of X)) is non empty Relation-like [:(BoundedFunctions the carrier of X),(BoundedFunctions the carrier of X):] -defined BoundedFunctions the carrier of X -valued Function-like total quasi_total Element of bool [:[:(BoundedFunctions the carrier of X),(BoundedFunctions the carrier of X):],(BoundedFunctions the carrier of X):]
Mult_ ((BoundedFunctions the carrier of X),(RAlgebra the carrier of X)) is non empty Relation-like [:REAL,(BoundedFunctions the carrier of X):] -defined BoundedFunctions the carrier of X -valued Function-like total quasi_total Element of bool [:[:REAL,(BoundedFunctions the carrier of X):],(BoundedFunctions the carrier of X):]
[:REAL,(BoundedFunctions the carrier of X):] is non empty set
[:[:REAL,(BoundedFunctions the carrier of X):],(BoundedFunctions the carrier of X):] is non empty set
bool [:[:REAL,(BoundedFunctions the carrier of X):],(BoundedFunctions the carrier of X):] is non empty set
One_ ((BoundedFunctions the carrier of X),(RAlgebra the carrier of X)) is Element of BoundedFunctions the carrier of X
Zero_ ((BoundedFunctions the carrier of X),(RAlgebra the carrier of X)) is Element of BoundedFunctions the carrier of X
Normed_AlgebraStr(# (BoundedFunctions the carrier of X),(mult_ ((BoundedFunctions the carrier of X),(RAlgebra the carrier of X))),(Add_ ((BoundedFunctions the carrier of X),(RAlgebra the carrier of X))),(Mult_ ((BoundedFunctions the carrier of X),(RAlgebra the carrier of X))),(One_ ((BoundedFunctions the carrier of X),(RAlgebra the carrier of X))),(Zero_ ((BoundedFunctions the carrier of X),(RAlgebra the carrier of X))),(BoundedFunctionsNorm the carrier of X) #) is strict Normed_AlgebraStr
the carrier of (R_Normed_Algebra_of_BoundedFunctions the carrier of X) is non empty set
X is V11() real ext-real Element of REAL
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 [:REAL, the carrier of (X):] -defined the carrier of (X) -valued Function-like total quasi_total Element of bool [:[:REAL, the carrier of (X):], the carrier of (X):]
[:REAL, the carrier of (X):] is non empty set
[:[:REAL, the carrier of (X):], the carrier of (X):] is non empty set
bool [:[:REAL, the carrier of (X):], the carrier of (X):] is non empty set
the Mult of (X) . (X,F) is set
[X,F] is set
{X,F} is non empty set
{X} is non empty V185() V186() V187() set
{{X,F},{X}} is non empty set
the Mult of (X) . [X,F] is set
G is right_complementable Element of the carrier of (R_Normed_Algebra_of_BoundedFunctions the carrier of X)
X * G is right_complementable Element of the carrier of (R_Normed_Algebra_of_BoundedFunctions the carrier of X)
the Mult of (R_Normed_Algebra_of_BoundedFunctions the carrier of X) is non empty Relation-like [:REAL, the carrier of (R_Normed_Algebra_of_BoundedFunctions the carrier of X):] -defined the carrier of (R_Normed_Algebra_of_BoundedFunctions the carrier of X) -valued Function-like total quasi_total Element of bool [:[:REAL, the carrier of (R_Normed_Algebra_of_BoundedFunctions the carrier of X):], the carrier of (R_Normed_Algebra_of_BoundedFunctions the carrier of X):]
[:REAL, the carrier of (R_Normed_Algebra_of_BoundedFunctions the carrier of X):] is non empty set
[:[:REAL, the carrier of (R_Normed_Algebra_of_BoundedFunctions the carrier of X):], the carrier of (R_Normed_Algebra_of_BoundedFunctions the carrier of X):] is non empty set
bool [:[:REAL, the carrier of (R_Normed_Algebra_of_BoundedFunctions the carrier of X):], the carrier of (R_Normed_Algebra_of_BoundedFunctions the carrier of X):] is non empty set
the Mult of (R_Normed_Algebra_of_BoundedFunctions the carrier of X) . (X,G) is set
[X,G] is set
{X,G} is non empty set
{{X,G},{X}} is non empty set
the Mult of (R_Normed_Algebra_of_BoundedFunctions the carrier of X) . [X,G] is set
[:REAL, the carrier of (RAlgebra the carrier of X):] is non empty set
the Mult of (RAlgebra the carrier of X) is non empty Relation-like [:REAL, the carrier of (RAlgebra the carrier of X):] -defined the carrier of (RAlgebra the carrier of X) -valued Function-like total quasi_total Element of bool [:[:REAL, the carrier of (RAlgebra the carrier of X):], the carrier of (RAlgebra the carrier of X):]
[:[:REAL, the carrier of (RAlgebra the carrier of X):], the carrier of (RAlgebra the carrier of X):] is non empty set
bool [:[:REAL, the carrier of (RAlgebra the carrier of X):], the carrier of (RAlgebra the carrier of X):] is non empty set
the Mult of (RAlgebra the carrier of X) | [:REAL,(X):] is Relation-like [:REAL, the carrier of (RAlgebra the carrier of X):] -defined the carrier of (RAlgebra the carrier of X) -valued Function-like Element of bool [:[:REAL, the carrier of (RAlgebra the carrier of X):], the carrier of (RAlgebra the carrier of X):]
[X,F] is Element of [:REAL, the carrier of (X):]
( the Mult of (RAlgebra the carrier of X) | [:REAL,(X):]) . [X,F] is set
the Mult of (RAlgebra the carrier of X) . [X,F] is set
the Mult of (RAlgebra the carrier of X) | [:REAL,(BoundedFunctions the carrier of X):] is Relation-like [:REAL, the carrier of (RAlgebra the carrier of X):] -defined the carrier of (RAlgebra the carrier of X) -valued Function-like Element of bool [:[:REAL, the carrier of (RAlgebra the carrier of X):], the carrier of (RAlgebra the carrier of X):]
[X,G] is Element of [:REAL, the carrier of (R_Normed_Algebra_of_BoundedFunctions the carrier of X):]
( the Mult of (RAlgebra the carrier of X) | [:REAL,(BoundedFunctions the carrier of X):]) . [X,G] is set
X is non empty TopSpace-like pseudocompact compact TopStruct
(X) is non empty unital strict Normed_AlgebraStr
(X) is non empty add-closed having-inverse additively-closed multiplicatively-closed additively-linearly-closed Element of bool the carrier of (RAlgebra the carrier of X)
the carrier of X is non empty set
RAlgebra the carrier of X is non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative unital associative commutative right-distributive right_unital well-unital left_unital strict vector-associative AlgebraStr
Funcs ( the carrier of X,REAL) is non empty functional FUNCTION_DOMAIN of the carrier of X, REAL
RealFuncMult the carrier of X is non empty Relation-like [:(Funcs ( the carrier of X,REAL)),(Funcs ( the carrier of X,REAL)):] -defined Funcs ( the carrier of X,REAL) -valued Function-like total quasi_total Element of bool [:[:(Funcs ( the carrier of X,REAL)),(Funcs ( the carrier of X,REAL)):],(Funcs ( the carrier of X,REAL)):]
[:(Funcs ( the carrier of X,REAL)),(Funcs ( the carrier of X,REAL)):] is non empty set
[:[:(Funcs ( the carrier of X,REAL)),(Funcs ( the carrier of X,REAL)):],(Funcs ( the carrier of X,REAL)):] is non empty set
bool [:[:(Funcs ( the carrier of X,REAL)),(Funcs ( the carrier of X,REAL)):],(Funcs ( the carrier of X,REAL)):] is non empty set
RealFuncAdd the carrier of X is non empty Relation-like [:(Funcs ( the carrier of X,REAL)),(Funcs ( the carrier of X,REAL)):] -defined Funcs ( the carrier of X,REAL) -valued Function-like total quasi_total Element of bool [:[:(Funcs ( the carrier of X,REAL)),(Funcs ( the carrier of X,REAL)):],(Funcs ( the carrier of X,REAL)):]
RealFuncExtMult the carrier of X is non empty Relation-like [:REAL,(Funcs ( the carrier of X,REAL)):] -defined Funcs ( the carrier of X,REAL) -valued Function-like total quasi_total Element of bool [:[:REAL,(Funcs ( the carrier of X,REAL)):],(Funcs ( the carrier of X,REAL)):]
[:REAL,(Funcs ( the carrier of X,REAL)):] is non empty set
[:[:REAL,(Funcs ( the carrier of X,REAL)):],(Funcs ( the carrier of X,REAL)):] is non empty set
bool [:[:REAL,(Funcs ( the carrier of X,REAL)):],(Funcs ( the carrier of X,REAL)):] is non empty set
RealFuncUnit the carrier of X is Relation-like the carrier of X -defined REAL -valued Function-like total quasi_total V152() V153() V154() Element of Funcs ( the carrier of X,REAL)
the carrier of X --> 1 is non empty Relation-like the carrier of X -defined NAT -valued RAT -valued INT -valued Function-like total quasi_total V152() V153() V154() V155() Element of bool [: the carrier of X,NAT:]
[: the carrier of X,NAT:] is non empty RAT -valued INT -valued V152() V153() V154() V155() set
bool [: the carrier of X,NAT:] is non empty set
RealFuncZero the carrier of X is Relation-like the carrier of X -defined REAL -valued Function-like total quasi_total V152() V153() V154() Element of Funcs ( the carrier of X,REAL)
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 V152() V153() V154() V155() Element of bool [: the carrier of X,NAT:]
AlgebraStr(# (Funcs ( the carrier of X,REAL)),(RealFuncMult the carrier of X),(RealFuncAdd the carrier of X),(RealFuncExtMult the carrier of X),(RealFuncUnit the carrier of X),(RealFuncZero the carrier of X) #) is strict AlgebraStr
the carrier of (RAlgebra the carrier of X) is non empty set
bool the carrier of (RAlgebra the carrier of X) is non empty set
[: the carrier of X,REAL:] is non empty V152() V153() V154() set
bool [: the carrier of X,REAL:] is non empty set
{ b₁ where b₁ is non empty Relation-like the carrier of X -defined REAL -valued Function-like total quasi_total V152() V153() V154() continuous Element of bool [: the carrier of X,REAL:] : verum } is set
mult_ ((X),(RAlgebra 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),(RAlgebra 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),(RAlgebra the carrier of X)) is non empty Relation-like [:REAL,(X):] -defined (X) -valued Function-like total quasi_total Element of bool [:[:REAL,(X):],(X):]
[:REAL,(X):] is non empty set
[:[:REAL,(X):],(X):] is non empty set
bool [:[:REAL,(X):],(X):] is non empty set
One_ ((X),(RAlgebra the carrier of X)) is Element of (X)
Zero_ ((X),(RAlgebra the carrier of X)) is Element of (X)
(X) is non empty Relation-like (X) -defined REAL -valued Function-like total quasi_total V152() V153() V154() Element of bool [:(X),REAL:]
[:(X),REAL:] is non empty V152() V153() V154() set
bool [:(X),REAL:] is non empty set
BoundedFunctions the carrier of X is non empty add-closed having-inverse additively-closed multiplicatively-closed additively-linearly-closed Element of bool the carrier of (RAlgebra the carrier of X)
{ b₁ where b₁ is non empty Relation-like the carrier of X -defined REAL -valued Function-like total quasi_total V152() V153() V154() Element of bool [: the carrier of X,REAL:] : b₁ | the carrier of X is bounded } is set
BoundedFunctionsNorm the carrier of X is non empty Relation-like BoundedFunctions the carrier of X -defined REAL -valued Function-like total quasi_total V152() V153() V154() Element of bool [:(BoundedFunctions the carrier of X),REAL:]
[:(BoundedFunctions the carrier of X),REAL:] is non empty V152() V153() V154() set
bool [:(BoundedFunctions the carrier of X),REAL:] is non empty set
(BoundedFunctionsNorm the carrier of X) | (X) is Relation-like BoundedFunctions the carrier of X -defined REAL -valued Function-like V152() V153() V154() Element of bool [:(BoundedFunctions the carrier of X),REAL:]
Normed_AlgebraStr(# (X),(mult_ ((X),(RAlgebra the carrier of X))),(Add_ ((X),(RAlgebra the carrier of X))),(Mult_ ((X),(RAlgebra the carrier of X))),(One_ ((X),(RAlgebra the carrier of X))),(Zero_ ((X),(RAlgebra the carrier of X))),(X) #) is strict Normed_AlgebraStr
the carrier of (X) is non empty set
R_Normed_Algebra_of_BoundedFunctions the carrier of X is non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital discerning reflexive RealNormSpace-like unital complete Normed_AlgebraStr
mult_ ((BoundedFunctions the carrier of X),(RAlgebra the carrier of X)) is non empty Relation-like [:(BoundedFunctions the carrier of X),(BoundedFunctions the carrier of X):] -defined BoundedFunctions the carrier of X -valued Function-like total quasi_total Element of bool [:[:(BoundedFunctions the carrier of X),(BoundedFunctions the carrier of X):],(BoundedFunctions the carrier of X):]
[:(BoundedFunctions the carrier of X),(BoundedFunctions the carrier of X):] is non empty set
[:[:(BoundedFunctions the carrier of X),(BoundedFunctions the carrier of X):],(BoundedFunctions the carrier of X):] is non empty set
bool [:[:(BoundedFunctions the carrier of X),(BoundedFunctions the carrier of X):],(BoundedFunctions the carrier of X):] is non empty set
Add_ ((BoundedFunctions the carrier of X),(RAlgebra the carrier of X)) is non empty Relation-like [:(BoundedFunctions the carrier of X),(BoundedFunctions the carrier of X):] -defined BoundedFunctions the carrier of X -valued Function-like total quasi_total Element of bool [:[:(BoundedFunctions the carrier of X),(BoundedFunctions the carrier of X):],(BoundedFunctions the carrier of X):]
Mult_ ((BoundedFunctions the carrier of X),(RAlgebra the carrier of X)) is non empty Relation-like [:REAL,(BoundedFunctions the carrier of X):] -defined BoundedFunctions the carrier of X -valued Function-like total quasi_total Element of bool [:[:REAL,(BoundedFunctions the carrier of X):],(BoundedFunctions the carrier of X):]
[:REAL,(BoundedFunctions the carrier of X):] is non empty set
[:[:REAL,(BoundedFunctions the carrier of X):],(BoundedFunctions the carrier of X):] is non empty set
bool [:[:REAL,(BoundedFunctions the carrier of X):],(BoundedFunctions the carrier of X):] is non empty set
One_ ((BoundedFunctions the carrier of X),(RAlgebra the carrier of X)) is Element of BoundedFunctions the carrier of X
Zero_ ((BoundedFunctions the carrier of X),(RAlgebra the carrier of X)) is Element of BoundedFunctions the carrier of X
Normed_AlgebraStr(# (BoundedFunctions the carrier of X),(mult_ ((BoundedFunctions the carrier of X),(RAlgebra the carrier of X))),(Add_ ((BoundedFunctions the carrier of X),(RAlgebra the carrier of X))),(Mult_ ((BoundedFunctions the carrier of X),(RAlgebra the carrier of X))),(One_ ((BoundedFunctions the carrier of X),(RAlgebra the carrier of X))),(Zero_ ((BoundedFunctions the carrier of X),(RAlgebra the carrier of X))),(BoundedFunctionsNorm the carrier of X) #) is strict Normed_AlgebraStr
the carrier of (R_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 right_complementable Element of the carrier of (R_Normed_Algebra_of_BoundedFunctions the carrier of X)
FB is right_complementable Element of the carrier of (R_Normed_Algebra_of_BoundedFunctions the carrier of X)
G * FB is right_complementable Element of the carrier of (R_Normed_Algebra_of_BoundedFunctions the carrier of X)
the multF of (R_Normed_Algebra_of_BoundedFunctions the carrier of X) is non empty Relation-like [: the carrier of (R_Normed_Algebra_of_BoundedFunctions the carrier of X), the carrier of (R_Normed_Algebra_of_BoundedFunctions the carrier of X):] -defined the carrier of (R_Normed_Algebra_of_BoundedFunctions the carrier of X) -valued Function-like total quasi_total Element of bool [:[: the carrier of (R_Normed_Algebra_of_BoundedFunctions the carrier of X), the carrier of (R_Normed_Algebra_of_BoundedFunctions the carrier of X):], the carrier of (R_Normed_Algebra_of_BoundedFunctions the carrier of X):]
[: the carrier of (R_Normed_Algebra_of_BoundedFunctions the carrier of X), the carrier of (R_Normed_Algebra_of_BoundedFunctions the carrier of X):] is non empty set
[:[: the carrier of (R_Normed_Algebra_of_BoundedFunctions the carrier of X), the carrier of (R_Normed_Algebra_of_BoundedFunctions the carrier of X):], the carrier of (R_Normed_Algebra_of_BoundedFunctions the carrier of X):] is non empty set
bool [:[: the carrier of (R_Normed_Algebra_of_BoundedFunctions the carrier of X), the carrier of (R_Normed_Algebra_of_BoundedFunctions the carrier of X):], the carrier of (R_Normed_Algebra_of_BoundedFunctions the carrier of X):] is non empty set
the multF of (R_Normed_Algebra_of_BoundedFunctions the carrier of X) . (G,FB) is right_complementable Element of the carrier of (R_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 (R_Normed_Algebra_of_BoundedFunctions the carrier of X) . [G,FB] is set
the multF of (RAlgebra the carrier of X) is non empty Relation-like [: the carrier of (RAlgebra the carrier of X), the carrier of (RAlgebra the carrier of X):] -defined the carrier of (RAlgebra the carrier of X) -valued Function-like total quasi_total Element of bool [:[: the carrier of (RAlgebra the carrier of X), the carrier of (RAlgebra the carrier of X):], the carrier of (RAlgebra the carrier of X):]
[: the carrier of (RAlgebra the carrier of X), the carrier of (RAlgebra the carrier of X):] is non empty set
[:[: the carrier of (RAlgebra the carrier of X), the carrier of (RAlgebra the carrier of X):], the carrier of (RAlgebra the carrier of X):] is non empty set
bool [:[: the carrier of (RAlgebra the carrier of X), the carrier of (RAlgebra the carrier of X):], the carrier of (RAlgebra the carrier of X):] is non empty set
the multF of (RAlgebra the carrier of X) || (X) is Relation-like Function-like set
the multF of (RAlgebra 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 (RAlgebra the carrier of X) || (X)) . [X,F] is set
the multF of (RAlgebra the carrier of X) . [X,F] is set
the multF of (RAlgebra the carrier of X) || (BoundedFunctions the carrier of X) is Relation-like Function-like set
the multF of (RAlgebra the carrier of X) | [:(BoundedFunctions the carrier of X),(BoundedFunctions the carrier of X):] is Relation-like set
[G,FB] is Element of [: the carrier of (R_Normed_Algebra_of_BoundedFunctions the carrier of X), the carrier of (R_Normed_Algebra_of_BoundedFunctions the carrier of X):]
( the multF of (RAlgebra the carrier of X) || (BoundedFunctions the carrier of X)) . [G,FB] is set
X is non empty TopSpace-like pseudocompact compact TopStruct
(X) is non empty unital strict Normed_AlgebraStr
(X) is non empty add-closed having-inverse additively-closed multiplicatively-closed additively-linearly-closed Element of bool the carrier of (RAlgebra the carrier of X)
the carrier of X is non empty set
RAlgebra the carrier of X is non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative unital associative commutative right-distributive right_unital well-unital left_unital strict vector-associative AlgebraStr
Funcs ( the carrier of X,REAL) is non empty functional FUNCTION_DOMAIN of the carrier of X, REAL
RealFuncMult the carrier of X is non empty Relation-like [:(Funcs ( the carrier of X,REAL)),(Funcs ( the carrier of X,REAL)):] -defined Funcs ( the carrier of X,REAL) -valued Function-like total quasi_total Element of bool [:[:(Funcs ( the carrier of X,REAL)),(Funcs ( the carrier of X,REAL)):],(Funcs ( the carrier of X,REAL)):]
[:(Funcs ( the carrier of X,REAL)),(Funcs ( the carrier of X,REAL)):] is non empty set
[:[:(Funcs ( the carrier of X,REAL)),(Funcs ( the carrier of X,REAL)):],(Funcs ( the carrier of X,REAL)):] is non empty set
bool [:[:(Funcs ( the carrier of X,REAL)),(Funcs ( the carrier of X,REAL)):],(Funcs ( the carrier of X,REAL)):] is non empty set
RealFuncAdd the carrier of X is non empty Relation-like [:(Funcs ( the carrier of X,REAL)),(Funcs ( the carrier of X,REAL)):] -defined Funcs ( the carrier of X,REAL) -valued Function-like total quasi_total Element of bool [:[:(Funcs ( the carrier of X,REAL)),(Funcs ( the carrier of X,REAL)):],(Funcs ( the carrier of X,REAL)):]
RealFuncExtMult the carrier of X is non empty Relation-like [:REAL,(Funcs ( the carrier of X,REAL)):] -defined Funcs ( the carrier of X,REAL) -valued Function-like total quasi_total Element of bool [:[:REAL,(Funcs ( the carrier of X,REAL)):],(Funcs ( the carrier of X,REAL)):]
[:REAL,(Funcs ( the carrier of X,REAL)):] is non empty set
[:[:REAL,(Funcs ( the carrier of X,REAL)):],(Funcs ( the carrier of X,REAL)):] is non empty set
bool [:[:REAL,(Funcs ( the carrier of X,REAL)):],(Funcs ( the carrier of X,REAL)):] is non empty set
RealFuncUnit the carrier of X is Relation-like the carrier of X -defined REAL -valued Function-like total quasi_total V152() V153() V154() Element of Funcs ( the carrier of X,REAL)
the carrier of X --> 1 is non empty Relation-like the carrier of X -defined NAT -valued RAT -valued INT -valued Function-like total quasi_total V152() V153() V154() V155() Element of bool [: the carrier of X,NAT:]
[: the carrier of X,NAT:] is non empty RAT -valued INT -valued V152() V153() V154() V155() set
bool [: the carrier of X,NAT:] is non empty set
RealFuncZero the carrier of X is Relation-like the carrier of X -defined REAL -valued Function-like total quasi_total V152() V153() V154() Element of Funcs ( the carrier of X,REAL)
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 V152() V153() V154() V155() Element of bool [: the carrier of X,NAT:]
AlgebraStr(# (Funcs ( the carrier of X,REAL)),(RealFuncMult the carrier of X),(RealFuncAdd the carrier of X),(RealFuncExtMult the carrier of X),(RealFuncUnit the carrier of X),(RealFuncZero the carrier of X) #) is strict AlgebraStr
the carrier of (RAlgebra the carrier of X) is non empty set
bool the carrier of (RAlgebra the carrier of X) is non empty set
[: the carrier of X,REAL:] is non empty V152() V153() V154() set
bool [: the carrier of X,REAL:] is non empty set
{ b₁ where b₁ is non empty Relation-like the carrier of X -defined REAL -valued Function-like total quasi_total V152() V153() V154() continuous Element of bool [: the carrier of X,REAL:] : verum } is set
mult_ ((X),(RAlgebra 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),(RAlgebra 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),(RAlgebra the carrier of X)) is non empty Relation-like [:REAL,(X):] -defined (X) -valued Function-like total quasi_total Element of bool [:[:REAL,(X):],(X):]
[:REAL,(X):] is non empty set
[:[:REAL,(X):],(X):] is non empty set
bool [:[:REAL,(X):],(X):] is non empty set
One_ ((X),(RAlgebra the carrier of X)) is Element of (X)
Zero_ ((X),(RAlgebra the carrier of X)) is Element of (X)
(X) is non empty Relation-like (X) -defined REAL -valued Function-like total quasi_total V152() V153() V154() Element of bool [:(X),REAL:]
[:(X),REAL:] is non empty V152() V153() V154() set
bool [:(X),REAL:] is non empty set
BoundedFunctions the carrier of X is non empty add-closed having-inverse additively-closed multiplicatively-closed additively-linearly-closed Element of bool the carrier of (RAlgebra the carrier of X)
{ b₁ where b₁ is non empty Relation-like the carrier of X -defined REAL -valued Function-like total quasi_total V152() V153() V154() Element of bool [: the carrier of X,REAL:] : b₁ | the carrier of X is bounded } is set
BoundedFunctionsNorm the carrier of X is non empty Relation-like BoundedFunctions the carrier of X -defined REAL -valued Function-like total quasi_total V152() V153() V154() Element of bool [:(BoundedFunctions the carrier of X),REAL:]
[:(BoundedFunctions the carrier of X),REAL:] is non empty V152() V153() V154() set
bool [:(BoundedFunctions the carrier of X),REAL:] is non empty set
(BoundedFunctionsNorm the carrier of X) | (X) is Relation-like BoundedFunctions the carrier of X -defined REAL -valued Function-like V152() V153() V154() Element of bool [:(BoundedFunctions the carrier of X),REAL:]
Normed_AlgebraStr(# (X),(mult_ ((X),(RAlgebra the carrier of X))),(Add_ ((X),(RAlgebra the carrier of X))),(Mult_ ((X),(RAlgebra the carrier of X))),(One_ ((X),(RAlgebra the carrier of X))),(Zero_ ((X),(RAlgebra the carrier of X))),(X) #) is strict Normed_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 vector-distributive scalar-distributive scalar-associative scalar-unital unital associative commutative right-distributive right_unital well-unital left_unital strict vector-associative AlgebraStr
AlgebraStr(# (X),(mult_ ((X),(RAlgebra the carrier of X))),(Add_ ((X),(RAlgebra the carrier of X))),(Mult_ ((X),(RAlgebra the carrier of X))),(One_ ((X),(RAlgebra the carrier of X))),(Zero_ ((X),(RAlgebra the carrier of X))) #) is strict AlgebraStr
1_ (X) is right_complementable Element of the carrier of (X)
the carrier of (X) is non empty set
1. (X) is right_complementable Element of the carrier of (X)
the OneF of (X) is right_complementable Element of the carrier of (X)
X is non empty TopSpace-like pseudocompact compact TopStruct
(X) is non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative unital associative commutative right-distributive right_unital well-unital left_unital vector-associative strict Normed_AlgebraStr
(X) is non empty add-closed having-inverse additively-closed multiplicatively-closed additively-linearly-closed Element of bool the carrier of (RAlgebra the carrier of X)
the carrier of X is non empty set
RAlgebra the carrier of X is non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative unital associative commutative right-distributive right_unital well-unital left_unital strict vector-associative AlgebraStr
Funcs ( the carrier of X,REAL) is non empty functional FUNCTION_DOMAIN of the carrier of X, REAL
RealFuncMult the carrier of X is non empty Relation-like [:(Funcs ( the carrier of X,REAL)),(Funcs ( the carrier of X,REAL)):] -defined Funcs ( the carrier of X,REAL) -valued Function-like total quasi_total Element of bool [:[:(Funcs ( the carrier of X,REAL)),(Funcs ( the carrier of X,REAL)):],(Funcs ( the carrier of X,REAL)):]
[:(Funcs ( the carrier of X,REAL)),(Funcs ( the carrier of X,REAL)):] is non empty set
[:[:(Funcs ( the carrier of X,REAL)),(Funcs ( the carrier of X,REAL)):],(Funcs ( the carrier of X,REAL)):] is non empty set
bool [:[:(Funcs ( the carrier of X,REAL)),(Funcs ( the carrier of X,REAL)):],(Funcs ( the carrier of X,REAL)):] is non empty set
RealFuncAdd the carrier of X is non empty Relation-like [:(Funcs ( the carrier of X,REAL)),(Funcs ( the carrier of X,REAL)):] -defined Funcs ( the carrier of X,REAL) -valued Function-like total quasi_total Element of bool [:[:(Funcs ( the carrier of X,REAL)),(Funcs ( the carrier of X,REAL)):],(Funcs ( the carrier of X,REAL)):]
RealFuncExtMult the carrier of X is non empty Relation-like [:REAL,(Funcs ( the carrier of X,REAL)):] -defined Funcs ( the carrier of X,REAL) -valued Function-like total quasi_total Element of bool [:[:REAL,(Funcs ( the carrier of X,REAL)):],(Funcs ( the carrier of X,REAL)):]
[:REAL,(Funcs ( the carrier of X,REAL)):] is non empty set
[:[:REAL,(Funcs ( the carrier of X,REAL)):],(Funcs ( the carrier of X,REAL)):] is non empty set
bool [:[:REAL,(Funcs ( the carrier of X,REAL)):],(Funcs ( the carrier of X,REAL)):] is non empty set
RealFuncUnit the carrier of X is Relation-like the carrier of X -defined REAL -valued Function-like total quasi_total V152() V153() V154() Element of Funcs ( the carrier of X,REAL)
the carrier of X --> 1 is non empty Relation-like the carrier of X -defined NAT -valued RAT -valued INT -valued Function-like total quasi_total V152() V153() V154() V155() Element of bool [: the carrier of X,NAT:]
[: the carrier of X,NAT:] is non empty RAT -valued INT -valued V152() V153() V154() V155() set
bool [: the carrier of X,NAT:] is non empty set
RealFuncZero the carrier of X is Relation-like the carrier of X -defined REAL -valued Function-like total quasi_total V152() V153() V154() Element of Funcs ( the carrier of X,REAL)
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 V152() V153() V154() V155() Element of bool [: the carrier of X,NAT:]
AlgebraStr(# (Funcs ( the carrier of X,REAL)),(RealFuncMult the carrier of X),(RealFuncAdd the carrier of X),(RealFuncExtMult the carrier of X),(RealFuncUnit the carrier of X),(RealFuncZero the carrier of X) #) is strict AlgebraStr
the carrier of (RAlgebra the carrier of X) is non empty set
bool the carrier of (RAlgebra the carrier of X) is non empty set
[: the carrier of X,REAL:] is non empty V152() V153() V154() set
bool [: the carrier of X,REAL:] is non empty set
{ b₁ where b₁ is non empty Relation-like the carrier of X -defined REAL -valued Function-like total quasi_total V152() V153() V154() continuous Element of bool [: the carrier of X,REAL:] : verum } is set
mult_ ((X),(RAlgebra 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),(RAlgebra 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),(RAlgebra the carrier of X)) is non empty Relation-like [:REAL,(X):] -defined (X) -valued Function-like total quasi_total Element of bool [:[:REAL,(X):],(X):]
[:REAL,(X):] is non empty set
[:[:REAL,(X):],(X):] is non empty set
bool [:[:REAL,(X):],(X):] is non empty set
One_ ((X),(RAlgebra the carrier of X)) is Element of (X)
Zero_ ((X),(RAlgebra the carrier of X)) is Element of (X)
(X) is non empty Relation-like (X) -defined REAL -valued Function-like total quasi_total V152() V153() V154() Element of bool [:(X),REAL:]
[:(X),REAL:] is non empty V152() V153() V154() set
bool [:(X),REAL:] is non empty set
BoundedFunctions the carrier of X is non empty add-closed having-inverse additively-closed multiplicatively-closed additively-linearly-closed Element of bool the carrier of (RAlgebra the carrier of X)
{ b₁ where b₁ is non empty Relation-like the carrier of X -defined REAL -valued Function-like total quasi_total V152() V153() V154() Element of bool [: the carrier of X,REAL:] : b₁ | the carrier of X is bounded } is set
BoundedFunctionsNorm the carrier of X is non empty Relation-like BoundedFunctions the carrier of X -defined REAL -valued Function-like total quasi_total V152() V153() V154() Element of bool [:(BoundedFunctions the carrier of X),REAL:]
[:(BoundedFunctions the carrier of X),REAL:] is non empty V152() V153() V154() set
bool [:(BoundedFunctions the carrier of X),REAL:] is non empty set
(BoundedFunctionsNorm the carrier of X) | (X) is Relation-like BoundedFunctions the carrier of X -defined REAL -valued Function-like V152() V153() V154() Element of bool [:(BoundedFunctions the carrier of X),REAL:]
Normed_AlgebraStr(# (X),(mult_ ((X),(RAlgebra the carrier of X))),(Add_ ((X),(RAlgebra the carrier of X))),(Mult_ ((X),(RAlgebra the carrier of X))),(One_ ((X),(RAlgebra the carrier of X))),(Zero_ ((X),(RAlgebra the carrier of X))),(X) #) is strict Normed_AlgebraStr
the carrier of (X) is non empty set
X is right_complementable Element of the carrier of (X)
(Mult_ ((X),(RAlgebra 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),(RAlgebra the carrier of X))) . [1,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
[:REAL, the carrier of (RAlgebra the carrier of X):] is non empty set
the Mult of (RAlgebra the carrier of X) is non empty Relation-like [:REAL, the carrier of (RAlgebra the carrier of X):] -defined the carrier of (RAlgebra the carrier of X) -valued Function-like total quasi_total Element of bool [:[:REAL, the carrier of (RAlgebra the carrier of X):], the carrier of (RAlgebra the carrier of X):]
[:[:REAL, the carrier of (RAlgebra the carrier of X):], the carrier of (RAlgebra the carrier of X):] is non empty set
bool [:[:REAL, the carrier of (RAlgebra the carrier of X):], the carrier of (RAlgebra the carrier of X):] is non empty set
the Mult of (RAlgebra the carrier of X) | [:REAL,(X):] is Relation-like [:REAL, the carrier of (RAlgebra the carrier of X):] -defined the carrier of (RAlgebra the carrier of X) -valued Function-like Element of bool [:[:REAL, the carrier of (RAlgebra the carrier of X):], the carrier of (RAlgebra the carrier of X):]
( the Mult of (RAlgebra the carrier of X) | [:REAL,(X):]) . (1,G) is set
[1,G] is set
( the Mult of (RAlgebra the carrier of X) | [:REAL,(X):]) . [1,G] is set
the Mult of (RAlgebra the carrier of X) . (1,G) is right_complementable Element of the carrier of (RAlgebra the carrier of X)
the Mult of (RAlgebra the carrier of X) . [1,G] is set
X is non empty TopSpace-like pseudocompact compact TopStruct
(X) is non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative unital associative commutative right-distributive right_unital well-unital left_unital vector-associative strict Normed_AlgebraStr
(X) is non empty add-closed having-inverse additively-closed multiplicatively-closed additively-linearly-closed Element of bool the carrier of (RAlgebra the carrier of X)
the carrier of X is non empty set
RAlgebra the carrier of X is non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative unital associative commutative right-distributive right_unital well-unital left_unital strict vector-associative AlgebraStr
Funcs ( the carrier of X,REAL) is non empty functional FUNCTION_DOMAIN of the carrier of X, REAL
RealFuncMult the carrier of X is non empty Relation-like [:(Funcs ( the carrier of X,REAL)),(Funcs ( the carrier of X,REAL)):] -defined Funcs ( the carrier of X,REAL) -valued Function-like total quasi_total Element of bool [:[:(Funcs ( the carrier of X,REAL)),(Funcs ( the carrier of X,REAL)):],(Funcs ( the carrier of X,REAL)):]
[:(Funcs ( the carrier of X,REAL)),(Funcs ( the carrier of X,REAL)):] is non empty set
[:[:(Funcs ( the carrier of X,REAL)),(Funcs ( the carrier of X,REAL)):],(Funcs ( the carrier of X,REAL)):] is non empty set
bool [:[:(Funcs ( the carrier of X,REAL)),(Funcs ( the carrier of X,REAL)):],(Funcs ( the carrier of X,REAL)):] is non empty set
RealFuncAdd the carrier of X is non empty Relation-like [:(Funcs ( the carrier of X,REAL)),(Funcs ( the carrier of X,REAL)):] -defined Funcs ( the carrier of X,REAL) -valued Function-like total quasi_total Element of bool [:[:(Funcs ( the carrier of X,REAL)),(Funcs ( the carrier of X,REAL)):],(Funcs ( the carrier of X,REAL)):]
RealFuncExtMult the carrier of X is non empty Relation-like [:REAL,(Funcs ( the carrier of X,REAL)):] -defined Funcs ( the carrier of X,REAL) -valued Function-like total quasi_total Element of bool [:[:REAL,(Funcs ( the carrier of X,REAL)):],(Funcs ( the carrier of X,REAL)):]
[:REAL,(Funcs ( the carrier of X,REAL)):] is non empty set
[:[:REAL,(Funcs ( the carrier of X,REAL)):],(Funcs ( the carrier of X,REAL)):] is non empty set
bool [:[:REAL,(Funcs ( the carrier of X,REAL)):],(Funcs ( the carrier of X,REAL)):] is non empty set
RealFuncUnit the carrier of X is Relation-like the carrier of X -defined REAL -valued Function-like total quasi_total V152() V153() V154() Element of Funcs ( the carrier of X,REAL)
the carrier of X --> 1 is non empty Relation-like the carrier of X -defined NAT -valued RAT -valued INT -valued Function-like total quasi_total V152() V153() V154() V155() Element of bool [: the carrier of X,NAT:]
[: the carrier of X,NAT:] is non empty RAT -valued INT -valued V152() V153() V154() V155() set
bool [: the carrier of X,NAT:] is non empty set
RealFuncZero the carrier of X is Relation-like the carrier of X -defined REAL -valued Function-like total quasi_total V152() V153() V154() Element of Funcs ( the carrier of X,REAL)
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 V152() V153() V154() V155() Element of bool [: the carrier of X,NAT:]
AlgebraStr(# (Funcs ( the carrier of X,REAL)),(RealFuncMult the carrier of X),(RealFuncAdd the carrier of X),(RealFuncExtMult the carrier of X),(RealFuncUnit the carrier of X),(RealFuncZero the carrier of X) #) is strict AlgebraStr
the carrier of (RAlgebra the carrier of X) is non empty set
bool the carrier of (RAlgebra the carrier of X) is non empty set
[: the carrier of X,REAL:] is non empty V152() V153() V154() set
bool [: the carrier of X,REAL:] is non empty set
{ b₁ where b₁ is non empty Relation-like the carrier of X -defined REAL -valued Function-like total quasi_total V152() V153() V154() continuous Element of bool [: the carrier of X,REAL:] : verum } is set
mult_ ((X),(RAlgebra 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),(RAlgebra 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),(RAlgebra the carrier of X)) is non empty Relation-like [:REAL,(X):] -defined (X) -valued Function-like total quasi_total Element of bool [:[:REAL,(X):],(X):]
[:REAL,(X):] is non empty set
[:[:REAL,(X):],(X):] is non empty set
bool [:[:REAL,(X):],(X):] is non empty set
One_ ((X),(RAlgebra the carrier of X)) is Element of (X)
Zero_ ((X),(RAlgebra the carrier of X)) is Element of (X)
(X) is non empty Relation-like (X) -defined REAL -valued Function-like total quasi_total V152() V153() V154() Element of bool [:(X),REAL:]
[:(X),REAL:] is non empty V152() V153() V154() set
bool [:(X),REAL:] is non empty set
BoundedFunctions the carrier of X is non empty add-closed having-inverse additively-closed multiplicatively-closed additively-linearly-closed Element of bool the carrier of (RAlgebra the carrier of X)
{ b₁ where b₁ is non empty Relation-like the carrier of X -defined REAL -valued Function-like total quasi_total V152() V153() V154() Element of bool [: the carrier of X,REAL:] : b₁ | the carrier of X is bounded } is set
BoundedFunctionsNorm the carrier of X is non empty Relation-like BoundedFunctions the carrier of X -defined REAL -valued Function-like total quasi_total V152() V153() V154() Element of bool [:(BoundedFunctions the carrier of X),REAL:]
[:(BoundedFunctions the carrier of X),REAL:] is non empty V152() V153() V154() set
bool [:(BoundedFunctions the carrier of X),REAL:] is non empty set
(BoundedFunctionsNorm the carrier of X) | (X) is Relation-like BoundedFunctions the carrier of X -defined REAL -valued Function-like V152() V153() V154() Element of bool [:(BoundedFunctions the carrier of X),REAL:]
Normed_AlgebraStr(# (X),(mult_ ((X),(RAlgebra the carrier of X))),(Add_ ((X),(RAlgebra the carrier of X))),(Mult_ ((X),(RAlgebra the carrier of X))),(One_ ((X),(RAlgebra the carrier of X))),(Zero_ ((X),(RAlgebra the carrier of X))),(X) #) is strict Normed_AlgebraStr
the carrier of (X) is non empty set
X is right_complementable Element of the carrier of (X)
1 * X is right_complementable Element of the carrier of (X)
the Mult of (X) is non empty Relation-like [:REAL, the carrier of (X):] -defined the carrier of (X) -valued Function-like total quasi_total Element of bool [:[:REAL, the carrier of (X):], the carrier of (X):]
[:REAL, the carrier of (X):] is non empty set
[:[:REAL, the carrier of (X):], the carrier of (X):] is non empty set
bool [:[:REAL, the carrier of (X):], the carrier of (X):] is non empty set
the Mult of (X) . (1,X) is set
[1,X] is set
{1,X} is non empty set
{{1,X},{1}} is non empty set
the Mult of (X) . [1,X] is set
X is non empty TopSpace-like pseudocompact compact TopStruct
(X,0) is non empty Relation-like the carrier of X -defined REAL -valued Function-like total quasi_total V152() V153() V154() V206( the carrier of X) V207( the carrier of X) continuous bounded Element of bool [: the carrier of X,REAL:]
the carrier of X is non empty set
[: the carrier of X,REAL:] is non empty V152() V153() V154() set
bool [: the carrier of X,REAL:] 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 V152() V153() V154() V155() Element of bool [: the carrier of X,{0}:]
{0} is non empty V185() V186() V187() V188() V189() V190() left_end bounded_below set
[: the carrier of X,{0}:] is non empty RAT -valued INT -valued V152() V153() V154() V155() 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 unital associative commutative right-distributive right_unital well-unital left_unital vector-associative strict Normed_AlgebraStr
(X) is non empty add-closed having-inverse additively-closed multiplicatively-closed additively-linearly-closed Element of bool the carrier of (RAlgebra the carrier of X)
RAlgebra the carrier of X is non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative unital associative commutative right-distributive right_unital well-unital left_unital strict vector-associative AlgebraStr
Funcs ( the carrier of X,REAL) is non empty functional FUNCTION_DOMAIN of the carrier of X, REAL
RealFuncMult the carrier of X is non empty Relation-like [:(Funcs ( the carrier of X,REAL)),(Funcs ( the carrier of X,REAL)):] -defined Funcs ( the carrier of X,REAL) -valued Function-like total quasi_total Element of bool [:[:(Funcs ( the carrier of X,REAL)),(Funcs ( the carrier of X,REAL)):],(Funcs ( the carrier of X,REAL)):]
[:(Funcs ( the carrier of X,REAL)),(Funcs ( the carrier of X,REAL)):] is non empty set
[:[:(Funcs ( the carrier of X,REAL)),(Funcs ( the carrier of X,REAL)):],(Funcs ( the carrier of X,REAL)):] is non empty set
bool [:[:(Funcs ( the carrier of X,REAL)),(Funcs ( the carrier of X,REAL)):],(Funcs ( the carrier of X,REAL)):] is non empty set
RealFuncAdd the carrier of X is non empty Relation-like [:(Funcs ( the carrier of X,REAL)),(Funcs ( the carrier of X,REAL)):] -defined Funcs ( the carrier of X,REAL) -valued Function-like total quasi_total Element of bool [:[:(Funcs ( the carrier of X,REAL)),(Funcs ( the carrier of X,REAL)):],(Funcs ( the carrier of X,REAL)):]
RealFuncExtMult the carrier of X is non empty Relation-like [:REAL,(Funcs ( the carrier of X,REAL)):] -defined Funcs ( the carrier of X,REAL) -valued Function-like total quasi_total Element of bool [:[:REAL,(Funcs ( the carrier of X,REAL)):],(Funcs ( the carrier of X,REAL)):]
[:REAL,(Funcs ( the carrier of X,REAL)):] is non empty set
[:[:REAL,(Funcs ( the carrier of X,REAL)):],(Funcs ( the carrier of X,REAL)):] is non empty set
bool [:[:REAL,(Funcs ( the carrier of X,REAL)):],(Funcs ( the carrier of X,REAL)):] is non empty set
RealFuncUnit the carrier of X is Relation-like the carrier of X -defined REAL -valued Function-like total quasi_total V152() V153() V154() Element of Funcs ( the carrier of X,REAL)
the carrier of X --> 1 is non empty Relation-like the carrier of X -defined NAT -valued RAT -valued INT -valued Function-like total quasi_total V152() V153() V154() V155() Element of bool [: the carrier of X,NAT:]
[: the carrier of X,NAT:] is non empty RAT -valued INT -valued V152() V153() V154() V155() set
bool [: the carrier of X,NAT:] is non empty set
RealFuncZero the carrier of X is Relation-like the carrier of X -defined REAL -valued Function-like total quasi_total V152() V153() V154() Element of Funcs ( the carrier of X,REAL)
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 V152() V153() V154() V155() Element of bool [: the carrier of X,NAT:]
AlgebraStr(# (Funcs ( the carrier of X,REAL)),(RealFuncMult the carrier of X),(RealFuncAdd the carrier of X),(RealFuncExtMult the carrier of X),(RealFuncUnit the carrier of X),(RealFuncZero the carrier of X) #) is strict AlgebraStr
the carrier of (RAlgebra the carrier of X) is non empty set
bool the carrier of (RAlgebra the carrier of X) is non empty set
{ b₁ where b₁ is non empty Relation-like the carrier of X -defined REAL -valued Function-like total quasi_total V152() V153() V154() continuous Element of bool [: the carrier of X,REAL:] : verum } is set
mult_ ((X),(RAlgebra 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),(RAlgebra 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),(RAlgebra the carrier of X)) is non empty Relation-like [:REAL,(X):] -defined (X) -valued Function-like total quasi_total Element of bool [:[:REAL,(X):],(X):]
[:REAL,(X):] is non empty set
[:[:REAL,(X):],(X):] is non empty set
bool [:[:REAL,(X):],(X):] is non empty set
One_ ((X),(RAlgebra the carrier of X)) is Element of (X)
Zero_ ((X),(RAlgebra the carrier of X)) is Element of (X)
(X) is non empty Relation-like (X) -defined REAL -valued Function-like total quasi_total V152() V153() V154() Element of bool [:(X),REAL:]
[:(X),REAL:] is non empty V152() V153() V154() set
bool [:(X),REAL:] is non empty set
BoundedFunctions the carrier of X is non empty add-closed having-inverse additively-closed multiplicatively-closed additively-linearly-closed Element of bool the carrier of (RAlgebra the carrier of X)
{ b₁ where b₁ is non empty Relation-like the carrier of X -defined REAL -valued Function-like total quasi_total V152() V153() V154() Element of bool [: the carrier of X,REAL:] : b₁ | the carrier of X is bounded } is set
BoundedFunctionsNorm the carrier of X is non empty Relation-like BoundedFunctions the carrier of X -defined REAL -valued Function-like total quasi_total V152() V153() V154() Element of bool [:(BoundedFunctions the carrier of X),REAL:]
[:(BoundedFunctions the carrier of X),REAL:] is non empty V152() V153() V154() set
bool [:(BoundedFunctions the carrier of X),REAL:] is non empty set
(BoundedFunctionsNorm the carrier of X) | (X) is Relation-like BoundedFunctions the carrier of X -defined REAL -valued Function-like V152() V153() V154() Element of bool [:(BoundedFunctions the carrier of X),REAL:]
Normed_AlgebraStr(# (X),(mult_ ((X),(RAlgebra the carrier of X))),(Add_ ((X),(RAlgebra the carrier of X))),(Mult_ ((X),(RAlgebra the carrier of X))),(One_ ((X),(RAlgebra the carrier of X))),(Zero_ ((X),(RAlgebra the carrier of X))),(X) #) is strict Normed_AlgebraStr
0. (X) is V49((X)) right_complementable Element of the carrier of (X)
the carrier of (X) is non empty set
the ZeroF of (X) is right_complementable Element of the carrier of (X)
(X) is non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital unital associative commutative right-distributive right_unital well-unital left_unital strict vector-associative AlgebraStr
AlgebraStr(# (X),(mult_ ((X),(RAlgebra the carrier of X))),(Add_ ((X),(RAlgebra the carrier of X))),(Mult_ ((X),(RAlgebra the carrier of X))),(One_ ((X),(RAlgebra the carrier of X))),(Zero_ ((X),(RAlgebra the carrier of X))) #) is strict AlgebraStr
0. (X) is V49((X)) right_complementable Element of the carrier of (X)
the carrier of (X) is non empty set
the ZeroF of (X) is right_complementable Element of the carrier of (X)
X is non empty TopSpace-like pseudocompact compact TopStruct
(X) is non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital unital associative commutative right-distributive right_unital well-unital left_unital vector-associative strict Normed_AlgebraStr
(X) is non empty add-closed having-inverse additively-closed multiplicatively-closed additively-linearly-closed Element of bool the carrier of (RAlgebra the carrier of X)
the carrier of X is non empty set
RAlgebra the carrier of X is non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative unital associative commutative right-distributive right_unital well-unital left_unital strict vector-associative AlgebraStr
Funcs ( the carrier of X,REAL) is non empty functional FUNCTION_DOMAIN of the carrier of X, REAL
RealFuncMult the carrier of X is non empty Relation-like [:(Funcs ( the carrier of X,REAL)),(Funcs ( the carrier of X,REAL)):] -defined Funcs ( the carrier of X,REAL) -valued Function-like total quasi_total Element of bool [:[:(Funcs ( the carrier of X,REAL)),(Funcs ( the carrier of X,REAL)):],(Funcs ( the carrier of X,REAL)):]
[:(Funcs ( the carrier of X,REAL)),(Funcs ( the carrier of X,REAL)):] is non empty set
[:[:(Funcs ( the carrier of X,REAL)),(Funcs ( the carrier of X,REAL)):],(Funcs ( the carrier of X,REAL)):] is non empty set
bool [:[:(Funcs ( the carrier of X,REAL)),(Funcs ( the carrier of X,REAL)):],(Funcs ( the carrier of X,REAL)):] is non empty set
RealFuncAdd the carrier of X is non empty Relation-like [:(Funcs ( the carrier of X,REAL)),(Funcs ( the carrier of X,REAL)):] -defined Funcs ( the carrier of X,REAL) -valued Function-like total quasi_total Element of bool [:[:(Funcs ( the carrier of X,REAL)),(Funcs ( the carrier of X,REAL)):],(Funcs ( the carrier of X,REAL)):]
RealFuncExtMult the carrier of X is non empty Relation-like [:REAL,(Funcs ( the carrier of X,REAL)):] -defined Funcs ( the carrier of X,REAL) -valued Function-like total quasi_total Element of bool [:[:REAL,(Funcs ( the carrier of X,REAL)):],(Funcs ( the carrier of X,REAL)):]
[:REAL,(Funcs ( the carrier of X,REAL)):] is non empty set
[:[:REAL,(Funcs ( the carrier of X,REAL)):],(Funcs ( the carrier of X,REAL)):] is non empty set
bool [:[:REAL,(Funcs ( the carrier of X,REAL)):],(Funcs ( the carrier of X,REAL)):] is non empty set
RealFuncUnit the carrier of X is Relation-like the carrier of X -defined REAL -valued Function-like total quasi_total V152() V153() V154() Element of Funcs ( the carrier of X,REAL)
the carrier of X --> 1 is non empty Relation-like the carrier of X -defined NAT -valued RAT -valued INT -valued Function-like total quasi_total V152() V153() V154() V155() Element of bool [: the carrier of X,NAT:]
[: the carrier of X,NAT:] is non empty RAT -valued INT -valued V152() V153() V154() V155() set
bool [: the carrier of X,NAT:] is non empty set
RealFuncZero the carrier of X is Relation-like the carrier of X -defined REAL -valued Function-like total quasi_total V152() V153() V154() Element of Funcs ( the carrier of X,REAL)
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 V152() V153() V154() V155() Element of bool [: the carrier of X,NAT:]
AlgebraStr(# (Funcs ( the carrier of X,REAL)),(RealFuncMult the carrier of X),(RealFuncAdd the carrier of X),(RealFuncExtMult the carrier of X),(RealFuncUnit the carrier of X),(RealFuncZero the carrier of X) #) is strict AlgebraStr
the carrier of (RAlgebra the carrier of X) is non empty set
bool the carrier of (RAlgebra the carrier of X) is non empty set
[: the carrier of X,REAL:] is non empty V152() V153() V154() set
bool [: the carrier of X,REAL:] is non empty set
{ b₁ where b₁ is non empty Relation-like the carrier of X -defined REAL -valued Function-like total quasi_total V152() V153() V154() continuous Element of bool [: the carrier of X,REAL:] : verum } is set
mult_ ((X),(RAlgebra 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),(RAlgebra 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),(RAlgebra the carrier of X)) is non empty Relation-like [:REAL,(X):] -defined (X) -valued Function-like total quasi_total Element of bool [:[:REAL,(X):],(X):]
[:REAL,(X):] is non empty set
[:[:REAL,(X):],(X):] is non empty set
bool [:[:REAL,(X):],(X):] is non empty set
One_ ((X),(RAlgebra the carrier of X)) is Element of (X)
Zero_ ((X),(RAlgebra the carrier of X)) is Element of (X)
(X) is non empty Relation-like (X) -defined REAL -valued Function-like total quasi_total V152() V153() V154() Element of bool [:(X),REAL:]
[:(X),REAL:] is non empty V152() V153() V154() set
bool [:(X),REAL:] is non empty set
BoundedFunctions the carrier of X is non empty add-closed having-inverse additively-closed multiplicatively-closed additively-linearly-closed Element of bool the carrier of (RAlgebra the carrier of X)
{ b₁ where b₁ is non empty Relation-like the carrier of X -defined REAL -valued Function-like total quasi_total V152() V153() V154() Element of bool [: the carrier of X,REAL:] : b₁ | the carrier of X is bounded } is set
BoundedFunctionsNorm the carrier of X is non empty Relation-like BoundedFunctions the carrier of X -defined REAL -valued Function-like total quasi_total V152() V153() V154() Element of bool [:(BoundedFunctions the carrier of X),REAL:]
[:(BoundedFunctions the carrier of X),REAL:] is non empty V152() V153() V154() set
bool [:(BoundedFunctions the carrier of X),REAL:] is non empty set
(BoundedFunctionsNorm the carrier of X) | (X) is Relation-like BoundedFunctions the carrier of X -defined REAL -valued Function-like V152() V153() V154() Element of bool [:(BoundedFunctions the carrier of X),REAL:]
Normed_AlgebraStr(# (X),(mult_ ((X),(RAlgebra the carrier of X))),(Add_ ((X),(RAlgebra the carrier of X))),(Mult_ ((X),(RAlgebra the carrier of X))),(One_ ((X),(RAlgebra the carrier of X))),(Zero_ ((X),(RAlgebra the carrier of X))),(X) #) is strict Normed_AlgebraStr
0. (X) is V49((X)) right_complementable Element of the carrier of (X)
the carrier of (X) is non empty set
the ZeroF of (X) is right_complementable Element of the carrier of (X)
R_Normed_Algebra_of_BoundedFunctions the carrier of X is non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital discerning reflexive RealNormSpace-like unital complete Normed_AlgebraStr
mult_ ((BoundedFunctions the carrier of X),(RAlgebra the carrier of X)) is non empty Relation-like [:(BoundedFunctions the carrier of X),(BoundedFunctions the carrier of X):] -defined BoundedFunctions the carrier of X -valued Function-like total quasi_total Element of bool [:[:(BoundedFunctions the carrier of X),(BoundedFunctions the carrier of X):],(BoundedFunctions the carrier of X):]
[:(BoundedFunctions the carrier of X),(BoundedFunctions the carrier of X):] is non empty set
[:[:(BoundedFunctions the carrier of X),(BoundedFunctions the carrier of X):],(BoundedFunctions the carrier of X):] is non empty set
bool [:[:(BoundedFunctions the carrier of X),(BoundedFunctions the carrier of X):],(BoundedFunctions the carrier of X):] is non empty set
Add_ ((BoundedFunctions the carrier of X),(RAlgebra the carrier of X)) is non empty Relation-like [:(BoundedFunctions the carrier of X),(BoundedFunctions the carrier of X):] -defined BoundedFunctions the carrier of X -valued Function-like total quasi_total Element of bool [:[:(BoundedFunctions the carrier of X),(BoundedFunctions the carrier of X):],(BoundedFunctions the carrier of X):]
Mult_ ((BoundedFunctions the carrier of X),(RAlgebra the carrier of X)) is non empty Relation-like [:REAL,(BoundedFunctions the carrier of X):] -defined BoundedFunctions the carrier of X -valued Function-like total quasi_total Element of bool [:[:REAL,(BoundedFunctions the carrier of X):],(BoundedFunctions the carrier of X):]
[:REAL,(BoundedFunctions the carrier of X):] is non empty set
[:[:REAL,(BoundedFunctions the carrier of X):],(BoundedFunctions the carrier of X):] is non empty set
bool [:[:REAL,(BoundedFunctions the carrier of X):],(BoundedFunctions the carrier of X):] is non empty set
One_ ((BoundedFunctions the carrier of X),(RAlgebra the carrier of X)) is Element of BoundedFunctions the carrier of X
Zero_ ((BoundedFunctions the carrier of X),(RAlgebra the carrier of X)) is Element of BoundedFunctions the carrier of X
Normed_AlgebraStr(# (BoundedFunctions the carrier of X),(mult_ ((BoundedFunctions the carrier of X),(RAlgebra the carrier of X))),(Add_ ((BoundedFunctions the carrier of X),(RAlgebra the carrier of X))),(Mult_ ((BoundedFunctions the carrier of X),(RAlgebra the carrier of X))),(One_ ((BoundedFunctions the carrier of X),(RAlgebra the carrier of X))),(Zero_ ((BoundedFunctions the carrier of X),(RAlgebra the carrier of X))),(BoundedFunctionsNorm the carrier of X) #) is strict Normed_AlgebraStr
0. (R_Normed_Algebra_of_BoundedFunctions the carrier of X) is V49( R_Normed_Algebra_of_BoundedFunctions the carrier of X) right_complementable Element of the carrier of (R_Normed_Algebra_of_BoundedFunctions the carrier of X)
the carrier of (R_Normed_Algebra_of_BoundedFunctions the carrier of X) is non empty set
the ZeroF of (R_Normed_Algebra_of_BoundedFunctions the carrier of X) is right_complementable Element of the carrier of (R_Normed_Algebra_of_BoundedFunctions the carrier of X)
(X,0) is non empty Relation-like the carrier of X -defined REAL -valued Function-like total quasi_total V152() V153() V154() V206( the carrier of X) V207( the carrier of X) continuous bounded Element of bool [: the carrier of X,REAL:]
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 V152() V153() V154() V155() Element of bool [: the carrier of X,{0}:]
{0} is non empty V185() V186() V187() V188() V189() V190() left_end bounded_below set
[: the carrier of X,{0}:] is non empty RAT -valued INT -valued V152() V153() V154() V155() set
bool [: the carrier of X,{0}:] is non empty set
X is non empty TopSpace-like pseudocompact compact TopStruct
(X) is non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital unital associative commutative right-distributive right_unital well-unital left_unital vector-associative strict Normed_AlgebraStr
(X) is non empty add-closed having-inverse additively-closed multiplicatively-closed additively-linearly-closed Element of bool the carrier of (RAlgebra the carrier of X)
the carrier of X is non empty set
RAlgebra the carrier of X is non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative unital associative commutative right-distributive right_unital well-unital left_unital strict vector-associative AlgebraStr
Funcs ( the carrier of X,REAL) is non empty functional FUNCTION_DOMAIN of the carrier of X, REAL
RealFuncMult the carrier of X is non empty Relation-like [:(Funcs ( the carrier of X,REAL)),(Funcs ( the carrier of X,REAL)):] -defined Funcs ( the carrier of X,REAL) -valued Function-like total quasi_total Element of bool [:[:(Funcs ( the carrier of X,REAL)),(Funcs ( the carrier of X,REAL)):],(Funcs ( the carrier of X,REAL)):]
[:(Funcs ( the carrier of X,REAL)),(Funcs ( the carrier of X,REAL)):] is non empty set
[:[:(Funcs ( the carrier of X,REAL)),(Funcs ( the carrier of X,REAL)):],(Funcs ( the carrier of X,REAL)):] is non empty set
bool [:[:(Funcs ( the carrier of X,REAL)),(Funcs ( the carrier of X,REAL)):],(Funcs ( the carrier of X,REAL)):] is non empty set
RealFuncAdd the carrier of X is non empty Relation-like [:(Funcs ( the carrier of X,REAL)),(Funcs ( the carrier of X,REAL)):] -defined Funcs ( the carrier of X,REAL) -valued Function-like total quasi_total Element of bool [:[:(Funcs ( the carrier of X,REAL)),(Funcs ( the carrier of X,REAL)):],(Funcs ( the carrier of X,REAL)):]
RealFuncExtMult the carrier of X is non empty Relation-like [:REAL,(Funcs ( the carrier of X,REAL)):] -defined Funcs ( the carrier of X,REAL) -valued Function-like total quasi_total Element of bool [:[:REAL,(Funcs ( the carrier of X,REAL)):],(Funcs ( the carrier of X,REAL)):]
[:REAL,(Funcs ( the carrier of X,REAL)):] is non empty set
[:[:REAL,(Funcs ( the carrier of X,REAL)):],(Funcs ( the carrier of X,REAL)):] is non empty set
bool [:[:REAL,(Funcs ( the carrier of X,REAL)):],(Funcs ( the carrier of X,REAL)):] is non empty set
RealFuncUnit the carrier of X is Relation-like the carrier of X -defined REAL -valued Function-like total quasi_total V152() V153() V154() Element of Funcs ( the carrier of X,REAL)
the carrier of X --> 1 is non empty Relation-like the carrier of X -defined NAT -valued RAT -valued INT -valued Function-like total quasi_total V152() V153() V154() V155() Element of bool [: the carrier of X,NAT:]
[: the carrier of X,NAT:] is non empty RAT -valued INT -valued V152() V153() V154() V155() set
bool [: the carrier of X,NAT:] is non empty set
RealFuncZero the carrier of X is Relation-like the carrier of X -defined REAL -valued Function-like total quasi_total V152() V153() V154() Element of Funcs ( the carrier of X,REAL)
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 V152() V153() V154() V155() Element of bool [: the carrier of X,NAT:]
AlgebraStr(# (Funcs ( the carrier of X,REAL)),(RealFuncMult the carrier of X),(RealFuncAdd the carrier of X),(RealFuncExtMult the carrier of X),(RealFuncUnit the carrier of X),(RealFuncZero the carrier of X) #) is strict AlgebraStr
the carrier of (RAlgebra the carrier of X) is non empty set
bool the carrier of (RAlgebra the carrier of X) is non empty set
[: the carrier of X,REAL:] is non empty V152() V153() V154() set
bool [: the carrier of X,REAL:] is non empty set
{ b₁ where b₁ is non empty Relation-like the carrier of X -defined REAL -valued Function-like total quasi_total V152() V153() V154() continuous Element of bool [: the carrier of X,REAL:] : verum } is set
mult_ ((X),(RAlgebra 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),(RAlgebra 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),(RAlgebra the carrier of X)) is non empty Relation-like [:REAL,(X):] -defined (X) -valued Function-like total quasi_total Element of bool [:[:REAL,(X):],(X):]
[:REAL,(X):] is non empty set
[:[:REAL,(X):],(X):] is non empty set
bool [:[:REAL,(X):],(X):] is non empty set
One_ ((X),(RAlgebra the carrier of X)) is Element of (X)
Zero_ ((X),(RAlgebra the carrier of X)) is Element of (X)
(X) is non empty Relation-like (X) -defined REAL -valued Function-like total quasi_total V152() V153() V154() Element of bool [:(X),REAL:]
[:(X),REAL:] is non empty V152() V153() V154() set
bool [:(X),REAL:] is non empty set
BoundedFunctions the carrier of X is non empty add-closed having-inverse additively-closed multiplicatively-closed additively-linearly-closed Element of bool the carrier of (RAlgebra the carrier of X)
{ b₁ where b₁ is non empty Relation-like the carrier of X -defined REAL -valued Function-like total quasi_total V152() V153() V154() Element of bool [: the carrier of X,REAL:] : b₁ | the carrier of X is bounded } is set
BoundedFunctionsNorm the carrier of X is non empty Relation-like BoundedFunctions the carrier of X -defined REAL -valued Function-like total quasi_total V152() V153() V154() Element of bool [:(BoundedFunctions the carrier of X),REAL:]
[:(BoundedFunctions the carrier of X),REAL:] is non empty V152() V153() V154() set
bool [:(BoundedFunctions the carrier of X),REAL:] is non empty set
(BoundedFunctionsNorm the carrier of X) | (X) is Relation-like BoundedFunctions the carrier of X -defined REAL -valued Function-like V152() V153() V154() Element of bool [:(BoundedFunctions the carrier of X),REAL:]
Normed_AlgebraStr(# (X),(mult_ ((X),(RAlgebra the carrier of X))),(Add_ ((X),(RAlgebra the carrier of X))),(Mult_ ((X),(RAlgebra the carrier of X))),(One_ ((X),(RAlgebra the carrier of X))),(Zero_ ((X),(RAlgebra the carrier of X))),(X) #) is strict Normed_AlgebraStr
1. (X) is right_complementable Element of the carrier of (X)
the carrier of (X) is non empty set
the OneF of (X) is right_complementable Element of the carrier of (X)
R_Normed_Algebra_of_BoundedFunctions the carrier of X is non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital discerning reflexive RealNormSpace-like unital complete Normed_AlgebraStr
mult_ ((BoundedFunctions the carrier of X),(RAlgebra the carrier of X)) is non empty Relation-like [:(BoundedFunctions the carrier of X),(BoundedFunctions the carrier of X):] -defined BoundedFunctions the carrier of X -valued Function-like total quasi_total Element of bool [:[:(BoundedFunctions the carrier of X),(BoundedFunctions the carrier of X):],(BoundedFunctions the carrier of X):]
[:(BoundedFunctions the carrier of X),(BoundedFunctions the carrier of X):] is non empty set
[:[:(BoundedFunctions the carrier of X),(BoundedFunctions the carrier of X):],(BoundedFunctions the carrier of X):] is non empty set
bool [:[:(BoundedFunctions the carrier of X),(BoundedFunctions the carrier of X):],(BoundedFunctions the carrier of X):] is non empty set
Add_ ((BoundedFunctions the carrier of X),(RAlgebra the carrier of X)) is non empty Relation-like [:(BoundedFunctions the carrier of X),(BoundedFunctions the carrier of X):] -defined BoundedFunctions the carrier of X -valued Function-like total quasi_total Element of bool [:[:(BoundedFunctions the carrier of X),(BoundedFunctions the carrier of X):],(BoundedFunctions the carrier of X):]
Mult_ ((BoundedFunctions the carrier of X),(RAlgebra the carrier of X)) is non empty Relation-like [:REAL,(BoundedFunctions the carrier of X):] -defined BoundedFunctions the carrier of X -valued Function-like total quasi_total Element of bool [:[:REAL,(BoundedFunctions the carrier of X):],(BoundedFunctions the carrier of X):]
[:REAL,(BoundedFunctions the carrier of X):] is non empty set
[:[:REAL,(BoundedFunctions the carrier of X):],(BoundedFunctions the carrier of X):] is non empty set
bool [:[:REAL,(BoundedFunctions the carrier of X):],(BoundedFunctions the carrier of X):] is non empty set
One_ ((BoundedFunctions the carrier of X),(RAlgebra the carrier of X)) is Element of BoundedFunctions the carrier of X
Zero_ ((BoundedFunctions the carrier of X),(RAlgebra the carrier of X)) is Element of BoundedFunctions the carrier of X
Normed_AlgebraStr(# (BoundedFunctions the carrier of X),(mult_ ((BoundedFunctions the carrier of X),(RAlgebra the carrier of X))),(Add_ ((BoundedFunctions the carrier of X),(RAlgebra the carrier of X))),(Mult_ ((BoundedFunctions the carrier of X),(RAlgebra the carrier of X))),(One_ ((BoundedFunctions the carrier of X),(RAlgebra the carrier of X))),(Zero_ ((BoundedFunctions the carrier of X),(RAlgebra the carrier of X))),(BoundedFunctionsNorm the carrier of X) #) is strict Normed_AlgebraStr
1. (R_Normed_Algebra_of_BoundedFunctions the carrier of X) is right_complementable Element of the carrier of (R_Normed_Algebra_of_BoundedFunctions the carrier of X)
the carrier of (R_Normed_Algebra_of_BoundedFunctions the carrier of X) is non empty set
the OneF of (R_Normed_Algebra_of_BoundedFunctions the carrier of X) is right_complementable Element of the carrier of (R_Normed_Algebra_of_BoundedFunctions the carrier of X)
(X) is non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital unital associative commutative right-distributive right_unital well-unital left_unital strict vector-associative AlgebraStr
AlgebraStr(# (X),(mult_ ((X),(RAlgebra the carrier of X))),(Add_ ((X),(RAlgebra the carrier of X))),(Mult_ ((X),(RAlgebra the carrier of X))),(One_ ((X),(RAlgebra the carrier of X))),(Zero_ ((X),(RAlgebra the carrier of X))) #) is strict AlgebraStr
1_ (X) is right_complementable Element of the carrier of (X)
the carrier of (X) is non empty set
1. (X) is right_complementable Element of the carrier of (X)
the OneF of (X) is right_complementable Element of the carrier of (X)
(X,1) is non empty Relation-like the carrier of X -defined REAL -valued Function-like total quasi_total V152() V153() V154() V206( the carrier of X) V207( the carrier of X) continuous bounded Element of bool [: the carrier of X,REAL:]
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 V152() V153() V154() V155() Element of bool [: the carrier of X,{1}:]
[: the carrier of X,{1}:] is non empty RAT -valued INT -valued V152() V153() V154() V155() set
bool [: the carrier of X,{1}:] is non empty set
R_Algebra_of_BoundedFunctions the carrier of X is non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative unital associative commutative right-distributive right_unital well-unital left_unital vector-associative AlgebraStr
AlgebraStr(# (BoundedFunctions the carrier of X),(mult_ ((BoundedFunctions the carrier of X),(RAlgebra the carrier of X))),(Add_ ((BoundedFunctions the carrier of X),(RAlgebra the carrier of X))),(Mult_ ((BoundedFunctions the carrier of X),(RAlgebra the carrier of X))),(One_ ((BoundedFunctions the carrier of X),(RAlgebra the carrier of X))),(Zero_ ((BoundedFunctions the carrier of X),(RAlgebra the carrier of X))) #) is strict AlgebraStr
1_ (R_Algebra_of_BoundedFunctions the carrier of X) is right_complementable Element of the carrier of (R_Algebra_of_BoundedFunctions the carrier of X)
the carrier of (R_Algebra_of_BoundedFunctions the carrier of X) is non empty set
1. (R_Algebra_of_BoundedFunctions the carrier of X) is right_complementable Element of the carrier of (R_Algebra_of_BoundedFunctions the carrier of X)
the OneF of (R_Algebra_of_BoundedFunctions the carrier of X) is right_complementable Element of the carrier of (R_Algebra_of_BoundedFunctions the carrier of X)
X is non empty TopSpace-like pseudocompact compact TopStruct
(X) is non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital unital associative commutative right-distributive right_unital well-unital left_unital vector-associative strict Normed_AlgebraStr
(X) is non empty add-closed having-inverse additively-closed multiplicatively-closed additively-linearly-closed Element of bool the carrier of (RAlgebra the carrier of X)
the carrier of X is non empty set
RAlgebra the carrier of X is non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative unital associative commutative right-distributive right_unital well-unital left_unital strict vector-associative AlgebraStr
Funcs ( the carrier of X,REAL) is non empty functional FUNCTION_DOMAIN of the carrier of X, REAL
RealFuncMult the carrier of X is non empty Relation-like [:(Funcs ( the carrier of X,REAL)),(Funcs ( the carrier of X,REAL)):] -defined Funcs ( the carrier of X,REAL) -valued Function-like total quasi_total Element of bool [:[:(Funcs ( the carrier of X,REAL)),(Funcs ( the carrier of X,REAL)):],(Funcs ( the carrier of X,REAL)):]
[:(Funcs ( the carrier of X,REAL)),(Funcs ( the carrier of X,REAL)):] is non empty set
[:[:(Funcs ( the carrier of X,REAL)),(Funcs ( the carrier of X,REAL)):],(Funcs ( the carrier of X,REAL)):] is non empty set
bool [:[:(Funcs ( the carrier of X,REAL)),(Funcs ( the carrier of X,REAL)):],(Funcs ( the carrier of X,REAL)):] is non empty set
RealFuncAdd the carrier of X is non empty Relation-like [:(Funcs ( the carrier of X,REAL)),(Funcs ( the carrier of X,REAL)):] -defined Funcs ( the carrier of X,REAL) -valued Function-like total quasi_total Element of bool [:[:(Funcs ( the carrier of X,REAL)),(Funcs ( the carrier of X,REAL)):],(Funcs ( the carrier of X,REAL)):]
RealFuncExtMult the carrier of X is non empty Relation-like [:REAL,(Funcs ( the carrier of X,REAL)):] -defined Funcs ( the carrier of X,REAL) -valued Function-like total quasi_total Element of bool [:[:REAL,(Funcs ( the carrier of X,REAL)):],(Funcs ( the carrier of X,REAL)):]
[:REAL,(Funcs ( the carrier of X,REAL)):] is non empty set
[:[:REAL,(Funcs ( the carrier of X,REAL)):],(Funcs ( the carrier of X,REAL)):] is non empty set
bool [:[:REAL,(Funcs ( the carrier of X,REAL)):],(Funcs ( the carrier of X,REAL)):] is non empty set
RealFuncUnit the carrier of X is Relation-like the carrier of X -defined REAL -valued Function-like total quasi_total V152() V153() V154() Element of Funcs ( the carrier of X,REAL)
the carrier of X --> 1 is non empty Relation-like the carrier of X -defined NAT -valued RAT -valued INT -valued Function-like total quasi_total V152() V153() V154() V155() Element of bool [: the carrier of X,NAT:]
[: the carrier of X,NAT:] is non empty RAT -valued INT -valued V152() V153() V154() V155() set
bool [: the carrier of X,NAT:] is non empty set
RealFuncZero the carrier of X is Relation-like the carrier of X -defined REAL -valued Function-like total quasi_total V152() V153() V154() Element of Funcs ( the carrier of X,REAL)
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 V152() V153() V154() V155() Element of bool [: the carrier of X,NAT:]
AlgebraStr(# (Funcs ( the carrier of X,REAL)),(RealFuncMult the carrier of X),(RealFuncAdd the carrier of X),(RealFuncExtMult the carrier of X),(RealFuncUnit the carrier of X),(RealFuncZero the carrier of X) #) is strict AlgebraStr
the carrier of (RAlgebra the carrier of X) is non empty set
bool the carrier of (RAlgebra the carrier of X) is non empty set
[: the carrier of X,REAL:] is non empty V152() V153() V154() set
bool [: the carrier of X,REAL:] is non empty set
{ b₁ where b₁ is non empty Relation-like the carrier of X -defined REAL -valued Function-like total quasi_total V152() V153() V154() continuous Element of bool [: the carrier of X,REAL:] : verum } is set
mult_ ((X),(RAlgebra 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),(RAlgebra 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),(RAlgebra the carrier of X)) is non empty Relation-like [:REAL,(X):] -defined (X) -valued Function-like total quasi_total Element of bool [:[:REAL,(X):],(X):]
[:REAL,(X):] is non empty set
[:[:REAL,(X):],(X):] is non empty set
bool [:[:REAL,(X):],(X):] is non empty set
One_ ((X),(RAlgebra the carrier of X)) is Element of (X)
Zero_ ((X),(RAlgebra the carrier of X)) is Element of (X)
(X) is non empty Relation-like (X) -defined REAL -valued Function-like total quasi_total V152() V153() V154() Element of bool [:(X),REAL:]
[:(X),REAL:] is non empty V152() V153() V154() set
bool [:(X),REAL:] is non empty set
BoundedFunctions the carrier of X is non empty add-closed having-inverse additively-closed multiplicatively-closed additively-linearly-closed Element of bool the carrier of (RAlgebra the carrier of X)
{ b₁ where b₁ is non empty Relation-like the carrier of X -defined REAL -valued Function-like total quasi_total V152() V153() V154() Element of bool [: the carrier of X,REAL:] : b₁ | the carrier of X is bounded } is set
BoundedFunctionsNorm the carrier of X is non empty Relation-like BoundedFunctions the carrier of X -defined REAL -valued Function-like total quasi_total V152() V153() V154() Element of bool [:(BoundedFunctions the carrier of X),REAL:]
[:(BoundedFunctions the carrier of X),REAL:] is non empty V152() V153() V154() set
bool [:(BoundedFunctions the carrier of X),REAL:] is non empty set
(BoundedFunctionsNorm the carrier of X) | (X) is Relation-like BoundedFunctions the carrier of X -defined REAL -valued Function-like V152() V153() V154() Element of bool [:(BoundedFunctions the carrier of X),REAL:]
Normed_AlgebraStr(# (X),(mult_ ((X),(RAlgebra the carrier of X))),(Add_ ((X),(RAlgebra the carrier of X))),(Mult_ ((X),(RAlgebra the carrier of X))),(One_ ((X),(RAlgebra the carrier of X))),(Zero_ ((X),(RAlgebra the carrier of X))),(X) #) is strict Normed_AlgebraStr
the carrier of (X) is non empty set
X is right_complementable Element of the carrier of (X)
||.X.|| is V11() real ext-real Element of REAL
the U8 of (X) is non empty Relation-like the carrier of (X) -defined REAL -valued Function-like total quasi_total V152() V153() V154() Element of bool [: the carrier of (X),REAL:]
[: the carrier of (X),REAL:] is non empty V152() V153() V154() set
bool [: the carrier of (X),REAL:] is non empty set
the U8 of (X) . X is V11() real ext-real Element of REAL
R_Normed_Algebra_of_BoundedFunctions the carrier of X is non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital discerning reflexive RealNormSpace-like unital complete Normed_AlgebraStr
mult_ ((BoundedFunctions the carrier of X),(RAlgebra the carrier of X)) is non empty Relation-like [:(BoundedFunctions the carrier of X),(BoundedFunctions the carrier of X):] -defined BoundedFunctions the carrier of X -valued Function-like total quasi_total Element of bool [:[:(BoundedFunctions the carrier of X),(BoundedFunctions the carrier of X):],(BoundedFunctions the carrier of X):]
[:(BoundedFunctions the carrier of X),(BoundedFunctions the carrier of X):] is non empty set
[:[:(BoundedFunctions the carrier of X),(BoundedFunctions the carrier of X):],(BoundedFunctions the carrier of X):] is non empty set
bool [:[:(BoundedFunctions the carrier of X),(BoundedFunctions the carrier of X):],(BoundedFunctions the carrier of X):] is non empty set
Add_ ((BoundedFunctions the carrier of X),(RAlgebra the carrier of X)) is non empty Relation-like [:(BoundedFunctions the carrier of X),(BoundedFunctions the carrier of X):] -defined BoundedFunctions the carrier of X -valued Function-like total quasi_total Element of bool [:[:(BoundedFunctions the carrier of X),(BoundedFunctions the carrier of X):],(BoundedFunctions the carrier of X):]
Mult_ ((BoundedFunctions the carrier of X),(RAlgebra the carrier of X)) is non empty Relation-like [:REAL,(BoundedFunctions the carrier of X):] -defined BoundedFunctions the carrier of X -valued Function-like total quasi_total Element of bool [:[:REAL,(BoundedFunctions the carrier of X):],(BoundedFunctions the carrier of X):]
[:REAL,(BoundedFunctions the carrier of X):] is non empty set
[:[:REAL,(BoundedFunctions the carrier of X):],(BoundedFunctions the carrier of X):] is non empty set
bool [:[:REAL,(BoundedFunctions the carrier of X):],(BoundedFunctions the carrier of X):] is non empty set
One_ ((BoundedFunctions the carrier of X),(RAlgebra the carrier of X)) is Element of BoundedFunctions the carrier of X
Zero_ ((BoundedFunctions the carrier of X),(RAlgebra the carrier of X)) is Element of BoundedFunctions the carrier of X
Normed_AlgebraStr(# (BoundedFunctions the carrier of X),(mult_ ((BoundedFunctions the carrier of X),(RAlgebra the carrier of X))),(Add_ ((BoundedFunctions the carrier of X),(RAlgebra the carrier of X))),(Mult_ ((BoundedFunctions the carrier of X),(RAlgebra the carrier of X))),(One_ ((BoundedFunctions the carrier of X),(RAlgebra the carrier of X))),(Zero_ ((BoundedFunctions the carrier of X),(RAlgebra the carrier of X))),(BoundedFunctionsNorm the carrier of X) #) is strict Normed_AlgebraStr
the carrier of (R_Normed_Algebra_of_BoundedFunctions the carrier of X) is non empty set
F is right_complementable Element of the carrier of (R_Normed_Algebra_of_BoundedFunctions the carrier of X)
||.F.|| is V11() real ext-real Element of REAL
the U8 of (R_Normed_Algebra_of_BoundedFunctions the carrier of X) is non empty Relation-like the carrier of (R_Normed_Algebra_of_BoundedFunctions the carrier of X) -defined REAL -valued Function-like total quasi_total V152() V153() V154() Element of bool [: the carrier of (R_Normed_Algebra_of_BoundedFunctions the carrier of X),REAL:]
[: the carrier of (R_Normed_Algebra_of_BoundedFunctions the carrier of X),REAL:] is non empty V152() V153() V154() set
bool [: the carrier of (R_Normed_Algebra_of_BoundedFunctions the carrier of X),REAL:] is non empty set
the U8 of (R_Normed_Algebra_of_BoundedFunctions the carrier of X) . F is V11() real ext-real Element of REAL
X is non empty TopSpace-like pseudocompact compact TopStruct
(X) is non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital unital associative commutative right-distributive right_unital well-unital left_unital vector-associative strict Normed_AlgebraStr
(X) is non empty add-closed having-inverse additively-closed multiplicatively-closed additively-linearly-closed Element of bool the carrier of (RAlgebra the carrier of X)
the carrier of X is non empty set
RAlgebra the carrier of X is non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative unital associative commutative right-distributive right_unital well-unital left_unital strict vector-associative AlgebraStr
Funcs ( the carrier of X,REAL) is non empty functional FUNCTION_DOMAIN of the carrier of X, REAL
RealFuncMult the carrier of X is non empty Relation-like [:(Funcs ( the carrier of X,REAL)),(Funcs ( the carrier of X,REAL)):] -defined Funcs ( the carrier of X,REAL) -valued Function-like total quasi_total Element of bool [:[:(Funcs ( the carrier of X,REAL)),(Funcs ( the carrier of X,REAL)):],(Funcs ( the carrier of X,REAL)):]
[:(Funcs ( the carrier of X,REAL)),(Funcs ( the carrier of X,REAL)):] is non empty set
[:[:(Funcs ( the carrier of X,REAL)),(Funcs ( the carrier of X,REAL)):],(Funcs ( the carrier of X,REAL)):] is non empty set
bool [:[:(Funcs ( the carrier of X,REAL)),(Funcs ( the carrier of X,REAL)):],(Funcs ( the carrier of X,REAL)):] is non empty set
RealFuncAdd the carrier of X is non empty Relation-like [:(Funcs ( the carrier of X,REAL)),(Funcs ( the carrier of X,REAL)):] -defined Funcs ( the carrier of X,REAL) -valued Function-like total quasi_total Element of bool [:[:(Funcs ( the carrier of X,REAL)),(Funcs ( the carrier of X,REAL)):],(Funcs ( the carrier of X,REAL)):]
RealFuncExtMult the carrier of X is non empty Relation-like [:REAL,(Funcs ( the carrier of X,REAL)):] -defined Funcs ( the carrier of X,REAL) -valued Function-like total quasi_total Element of bool [:[:REAL,(Funcs ( the carrier of X,REAL)):],(Funcs ( the carrier of X,REAL)):]
[:REAL,(Funcs ( the carrier of X,REAL)):] is non empty set
[:[:REAL,(Funcs ( the carrier of X,REAL)):],(Funcs ( the carrier of X,REAL)):] is non empty set
bool [:[:REAL,(Funcs ( the carrier of X,REAL)):],(Funcs ( the carrier of X,REAL)):] is non empty set
RealFuncUnit the carrier of X is Relation-like the carrier of X -defined REAL -valued Function-like total quasi_total V152() V153() V154() Element of Funcs ( the carrier of X,REAL)
the carrier of X --> 1 is non empty Relation-like the carrier of X -defined NAT -valued RAT -valued INT -valued Function-like total quasi_total V152() V153() V154() V155() Element of bool [: the carrier of X,NAT:]
[: the carrier of X,NAT:] is non empty RAT -valued INT -valued V152() V153() V154() V155() set
bool [: the carrier of X,NAT:] is non empty set
RealFuncZero the carrier of X is Relation-like the carrier of X -defined REAL -valued Function-like total quasi_total V152() V153() V154() Element of Funcs ( the carrier of X,REAL)
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 V152() V153() V154() V155() Element of bool [: the carrier of X,NAT:]
AlgebraStr(# (Funcs ( the carrier of X,REAL)),(RealFuncMult the carrier of X),(RealFuncAdd the carrier of X),(RealFuncExtMult the carrier of X),(RealFuncUnit the carrier of X),(RealFuncZero the carrier of X) #) is strict AlgebraStr
the carrier of (RAlgebra the carrier of X) is non empty set
bool the carrier of (RAlgebra the carrier of X) is non empty set
[: the carrier of X,REAL:] is non empty V152() V153() V154() set
bool [: the carrier of X,REAL:] is non empty set
{ b₁ where b₁ is non empty Relation-like the carrier of X -defined REAL -valued Function-like total quasi_total V152() V153() V154() continuous Element of bool [: the carrier of X,REAL:] : verum } is set
mult_ ((X),(RAlgebra 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),(RAlgebra 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),(RAlgebra the carrier of X)) is non empty Relation-like [:REAL,(X):] -defined (X) -valued Function-like total quasi_total Element of bool [:[:REAL,(X):],(X):]
[:REAL,(X):] is non empty set
[:[:REAL,(X):],(X):] is non empty set
bool [:[:REAL,(X):],(X):] is non empty set
One_ ((X),(RAlgebra the carrier of X)) is Element of (X)
Zero_ ((X),(RAlgebra the carrier of X)) is Element of (X)
(X) is non empty Relation-like (X) -defined REAL -valued Function-like total quasi_total V152() V153() V154() Element of bool [:(X),REAL:]
[:(X),REAL:] is non empty V152() V153() V154() set
bool [:(X),REAL:] is non empty set
BoundedFunctions the carrier of X is non empty add-closed having-inverse additively-closed multiplicatively-closed additively-linearly-closed Element of bool the carrier of (RAlgebra the carrier of X)
{ b₁ where b₁ is non empty Relation-like the carrier of X -defined REAL -valued Function-like total quasi_total V152() V153() V154() Element of bool [: the carrier of X,REAL:] : b₁ | the carrier of X is bounded } is set
BoundedFunctionsNorm the carrier of X is non empty Relation-like BoundedFunctions the carrier of X -defined REAL -valued Function-like total quasi_total V152() V153() V154() Element of bool [:(BoundedFunctions the carrier of X),REAL:]
[:(BoundedFunctions the carrier of X),REAL:] is non empty V152() V153() V154() set
bool [:(BoundedFunctions the carrier of X),REAL:] is non empty set
(BoundedFunctionsNorm the carrier of X) | (X) is Relation-like BoundedFunctions the carrier of X -defined REAL -valued Function-like V152() V153() V154() Element of bool [:(BoundedFunctions the carrier of X),REAL:]
Normed_AlgebraStr(# (X),(mult_ ((X),(RAlgebra the carrier of X))),(Add_ ((X),(RAlgebra the carrier of X))),(Mult_ ((X),(RAlgebra the carrier of X))),(One_ ((X),(RAlgebra the carrier of X))),(Zero_ ((X),(RAlgebra the carrier of X))),(X) #) is strict Normed_AlgebraStr
the carrier of (X) is non empty set
0. (X) is V49((X)) right_complementable Element of the carrier of (X)
the ZeroF of (X) is right_complementable Element of the carrier of (X)
X is right_complementable Element of the carrier of (X)
||.X.|| is V11() real ext-real Element of REAL
the U8 of (X) is non empty Relation-like the carrier of (X) -defined REAL -valued Function-like total quasi_total V152() V153() V154() Element of bool [: the carrier of (X),REAL:]
[: the carrier of (X),REAL:] is non empty V152() V153() V154() set
bool [: the carrier of (X),REAL:] is non empty set
the U8 of (X) . X is V11() real ext-real Element of REAL
R_Normed_Algebra_of_BoundedFunctions the carrier of X is non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital discerning reflexive RealNormSpace-like unital complete Normed_AlgebraStr
mult_ ((BoundedFunctions the carrier of X),(RAlgebra the carrier of X)) is non empty Relation-like [:(BoundedFunctions the carrier of X),(BoundedFunctions the carrier of X):] -defined BoundedFunctions the carrier of X -valued Function-like total quasi_total Element of bool [:[:(BoundedFunctions the carrier of X),(BoundedFunctions the carrier of X):],(BoundedFunctions the carrier of X):]
[:(BoundedFunctions the carrier of X),(BoundedFunctions the carrier of X):] is non empty set
[:[:(BoundedFunctions the carrier of X),(BoundedFunctions the carrier of X):],(BoundedFunctions the carrier of X):] is non empty set
bool [:[:(BoundedFunctions the carrier of X),(BoundedFunctions the carrier of X):],(BoundedFunctions the carrier of X):] is non empty set
Add_ ((BoundedFunctions the carrier of X),(RAlgebra the carrier of X)) is non empty Relation-like [:(BoundedFunctions the carrier of X),(BoundedFunctions the carrier of X):] -defined BoundedFunctions the carrier of X -valued Function-like total quasi_total Element of bool [:[:(BoundedFunctions the carrier of X),(BoundedFunctions the carrier of X):],(BoundedFunctions the carrier of X):]
Mult_ ((BoundedFunctions the carrier of X),(RAlgebra the carrier of X)) is non empty Relation-like [:REAL,(BoundedFunctions the carrier of X):] -defined BoundedFunctions the carrier of X -valued Function-like total quasi_total Element of bool [:[:REAL,(BoundedFunctions the carrier of X):],(BoundedFunctions the carrier of X):]
[:REAL,(BoundedFunctions the carrier of X):] is non empty set
[:[:REAL,(BoundedFunctions the carrier of X):],(BoundedFunctions the carrier of X):] is non empty set
bool [:[:REAL,(BoundedFunctions the carrier of X):],(BoundedFunctions the carrier of X):] is non empty set
One_ ((BoundedFunctions the carrier of X),(RAlgebra the carrier of X)) is Element of BoundedFunctions the carrier of X
Zero_ ((BoundedFunctions the carrier of X),(RAlgebra the carrier of X)) is Element of BoundedFunctions the carrier of X
Normed_AlgebraStr(# (BoundedFunctions the carrier of X),(mult_ ((BoundedFunctions the carrier of X),(RAlgebra the carrier of X))),(Add_ ((BoundedFunctions the carrier of X),(RAlgebra the carrier of X))),(Mult_ ((BoundedFunctions the carrier of X),(RAlgebra the carrier of X))),(One_ ((BoundedFunctions the carrier of X),(RAlgebra the carrier of X))),(Zero_ ((BoundedFunctions the carrier of X),(RAlgebra the carrier of X))),(BoundedFunctionsNorm the carrier of X) #) is strict Normed_AlgebraStr
the carrier of (R_Normed_Algebra_of_BoundedFunctions the carrier of X) is non empty set
F is right_complementable Element of the carrier of (R_Normed_Algebra_of_BoundedFunctions the carrier of X)
||.F.|| is V11() real ext-real Element of REAL
the U8 of (R_Normed_Algebra_of_BoundedFunctions the carrier of X) is non empty Relation-like the carrier of (R_Normed_Algebra_of_BoundedFunctions the carrier of X) -defined REAL -valued Function-like total quasi_total V152() V153() V154() Element of bool [: the carrier of (R_Normed_Algebra_of_BoundedFunctions the carrier of X),REAL:]
[: the carrier of (R_Normed_Algebra_of_BoundedFunctions the carrier of X),REAL:] is non empty V152() V153() V154() set
bool [: the carrier of (R_Normed_Algebra_of_BoundedFunctions the carrier of X),REAL:] is non empty set
the U8 of (R_Normed_Algebra_of_BoundedFunctions the carrier of X) . F is V11() real ext-real Element of REAL
(X,0) is non empty Relation-like the carrier of X -defined REAL -valued Function-like total quasi_total V152() V153() V154() V206( the carrier of X) V207( the carrier of X) continuous bounded Element of bool [: the carrier of X,REAL:]
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 V152() V153() V154() V155() Element of bool [: the carrier of X,{0}:]
{0} is non empty V185() V186() V187() V188() V189() V190() left_end bounded_below set
[: the carrier of X,{0}:] is non empty RAT -valued INT -valued V152() V153() V154() V155() set
bool [: the carrier of X,{0}:] is non empty set
0. (R_Normed_Algebra_of_BoundedFunctions the carrier of X) is V49( R_Normed_Algebra_of_BoundedFunctions the carrier of X) right_complementable Element of the carrier of (R_Normed_Algebra_of_BoundedFunctions the carrier of X)
the ZeroF of (R_Normed_Algebra_of_BoundedFunctions the carrier of X) is right_complementable Element of the carrier of (R_Normed_Algebra_of_BoundedFunctions the carrier of X)
X is non empty TopSpace-like pseudocompact compact TopStruct
(X) is non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital unital associative commutative right-distributive right_unital well-unital left_unital vector-associative strict Normed_AlgebraStr
(X) is non empty add-closed having-inverse additively-closed multiplicatively-closed additively-linearly-closed Element of bool the carrier of (RAlgebra the carrier of X)
the carrier of X is non empty set
RAlgebra the carrier of X is non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative unital associative commutative right-distributive right_unital well-unital left_unital strict vector-associative AlgebraStr
Funcs ( the carrier of X,REAL) is non empty functional FUNCTION_DOMAIN of the carrier of X, REAL
RealFuncMult the carrier of X is non empty Relation-like [:(Funcs ( the carrier of X,REAL)),(Funcs ( the carrier of X,REAL)):] -defined Funcs ( the carrier of X,REAL) -valued Function-like total quasi_total Element of bool [:[:(Funcs ( the carrier of X,REAL)),(Funcs ( the carrier of X,REAL)):],(Funcs ( the carrier of X,REAL)):]
[:(Funcs ( the carrier of X,REAL)),(Funcs ( the carrier of X,REAL)):] is non empty set
[:[:(Funcs ( the carrier of X,REAL)),(Funcs ( the carrier of X,REAL)):],(Funcs ( the carrier of X,REAL)):] is non empty set
bool [:[:(Funcs ( the carrier of X,REAL)),(Funcs ( the carrier of X,REAL)):],(Funcs ( the carrier of X,REAL)):] is non empty set
RealFuncAdd the carrier of X is non empty Relation-like [:(Funcs ( the carrier of X,REAL)),(Funcs ( the carrier of X,REAL)):] -defined Funcs ( the carrier of X,REAL) -valued Function-like total quasi_total Element of bool [:[:(Funcs ( the carrier of X,REAL)),(Funcs ( the carrier of X,REAL)):],(Funcs ( the carrier of X,REAL)):]
RealFuncExtMult the carrier of X is non empty Relation-like [:REAL,(Funcs ( the carrier of X,REAL)):] -defined Funcs ( the carrier of X,REAL) -valued Function-like total quasi_total Element of bool [:[:REAL,(Funcs ( the carrier of X,REAL)):],(Funcs ( the carrier of X,REAL)):]
[:REAL,(Funcs ( the carrier of X,REAL)):] is non empty set
[:[:REAL,(Funcs ( the carrier of X,REAL)):],(Funcs ( the carrier of X,REAL)):] is non empty set
bool [:[:REAL,(Funcs ( the carrier of X,REAL)):],(Funcs ( the carrier of X,REAL)):] is non empty set
RealFuncUnit the carrier of X is Relation-like the carrier of X -defined REAL -valued Function-like total quasi_total V152() V153() V154() Element of Funcs ( the carrier of X,REAL)
the carrier of X --> 1 is non empty Relation-like the carrier of X -defined NAT -valued RAT -valued INT -valued Function-like total quasi_total V152() V153() V154() V155() Element of bool [: the carrier of X,NAT:]
[: the carrier of X,NAT:] is non empty RAT -valued INT -valued V152() V153() V154() V155() set
bool [: the carrier of X,NAT:] is non empty set
RealFuncZero the carrier of X is Relation-like the carrier of X -defined REAL -valued Function-like total quasi_total V152() V153() V154() Element of Funcs ( the carrier of X,REAL)
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 V152() V153() V154() V155() Element of bool [: the carrier of X,NAT:]
AlgebraStr(# (Funcs ( the carrier of X,REAL)),(RealFuncMult the carrier of X),(RealFuncAdd the carrier of X),(RealFuncExtMult the carrier of X),(RealFuncUnit the carrier of X),(RealFuncZero the carrier of X) #) is strict AlgebraStr
the carrier of (RAlgebra the carrier of X) is non empty set
bool the carrier of (RAlgebra the carrier of X) is non empty set
[: the carrier of X,REAL:] is non empty V152() V153() V154() set
bool [: the carrier of X,REAL:] is non empty set
{ b₁ where b₁ is non empty Relation-like the carrier of X -defined REAL -valued Function-like total quasi_total V152() V153() V154() continuous Element of bool [: the carrier of X,REAL:] : verum } is set
mult_ ((X),(RAlgebra 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),(RAlgebra 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),(RAlgebra the carrier of X)) is non empty Relation-like [:REAL,(X):] -defined (X) -valued Function-like total quasi_total Element of bool [:[:REAL,(X):],(X):]
[:REAL,(X):] is non empty set
[:[:REAL,(X):],(X):] is non empty set
bool [:[:REAL,(X):],(X):] is non empty set
One_ ((X),(RAlgebra the carrier of X)) is Element of (X)
Zero_ ((X),(RAlgebra the carrier of X)) is Element of (X)
(X) is non empty Relation-like (X) -defined REAL -valued Function-like total quasi_total V152() V153() V154() Element of bool [:(X),REAL:]
[:(X),REAL:] is non empty V152() V153() V154() set
bool [:(X),REAL:] is non empty set
BoundedFunctions the carrier of X is non empty add-closed having-inverse additively-closed multiplicatively-closed additively-linearly-closed Element of bool the carrier of (RAlgebra the carrier of X)
{ b₁ where b₁ is non empty Relation-like the carrier of X -defined REAL -valued Function-like total quasi_total V152() V153() V154() Element of bool [: the carrier of X,REAL:] : b₁ | the carrier of X is bounded } is set
BoundedFunctionsNorm the carrier of X is non empty Relation-like BoundedFunctions the carrier of X -defined REAL -valued Function-like total quasi_total V152() V153() V154() Element of bool [:(BoundedFunctions the carrier of X),REAL:]
[:(BoundedFunctions the carrier of X),REAL:] is non empty V152() V153() V154() set
bool [:(BoundedFunctions the carrier of X),REAL:] is non empty set
(BoundedFunctionsNorm the carrier of X) | (X) is Relation-like BoundedFunctions the carrier of X -defined REAL -valued Function-like V152() V153() V154() Element of bool [:(BoundedFunctions the carrier of X),REAL:]
Normed_AlgebraStr(# (X),(mult_ ((X),(RAlgebra the carrier of X))),(Add_ ((X),(RAlgebra the carrier of X))),(Mult_ ((X),(RAlgebra the carrier of X))),(One_ ((X),(RAlgebra the carrier of X))),(Zero_ ((X),(RAlgebra the carrier of X))),(X) #) is strict Normed_AlgebraStr
the carrier of (X) is non empty set
X is right_complementable Element of the carrier of (X)
F is right_complementable Element of the carrier of (X)
G is right_complementable Element of the carrier of (X)
X + F is right_complementable 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 right_complementable 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 REAL -valued Function-like total quasi_total V152() V153() V154() Element of bool [: the carrier of X,REAL:]
GB is non empty Relation-like the carrier of X -defined REAL -valued Function-like total quasi_total V152() V153() V154() Element of bool [: the carrier of X,REAL:]
aFB is non empty Relation-like the carrier of X -defined REAL -valued Function-like total quasi_total V152() V153() V154() Element of bool [: the carrier of X,REAL:]
(X) is non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital unital associative commutative right-distributive right_unital well-unital left_unital strict vector-associative AlgebraStr
AlgebraStr(# (X),(mult_ ((X),(RAlgebra the carrier of X))),(Add_ ((X),(RAlgebra the carrier of X))),(Mult_ ((X),(RAlgebra the carrier of X))),(One_ ((X),(RAlgebra the carrier of X))),(Zero_ ((X),(RAlgebra the carrier of X))) #) is strict AlgebraStr
the carrier of (X) is non empty set
r is right_complementable Element of the carrier of (X)
aF is right_complementable Element of the carrier of (X)
aG is right_complementable Element of the carrier of (X)
aF + aG is right_complementable 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 right_complementable 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
s1 is Element of the carrier of X
aFB . s1 is V11() real ext-real Element of REAL
FB . s1 is V11() real ext-real Element of REAL
GB . s1 is V11() real ext-real Element of REAL
(FB . s1) + (GB . s1) is V11() real ext-real Element of REAL
X is V11() real ext-real Element of REAL
X is non empty TopSpace-like pseudocompact compact TopStruct
(X) is non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital unital associative commutative right-distributive right_unital well-unital left_unital vector-associative strict Normed_AlgebraStr
(X) is non empty add-closed having-inverse additively-closed multiplicatively-closed additively-linearly-closed Element of bool the carrier of (RAlgebra the carrier of X)
the carrier of X is non empty set
RAlgebra the carrier of X is non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative unital associative commutative right-distributive right_unital well-unital left_unital strict vector-associative AlgebraStr
Funcs ( the carrier of X,REAL) is non empty functional FUNCTION_DOMAIN of the carrier of X, REAL
RealFuncMult the carrier of X is non empty Relation-like [:(Funcs ( the carrier of X,REAL)),(Funcs ( the carrier of X,REAL)):] -defined Funcs ( the carrier of X,REAL) -valued Function-like total quasi_total Element of bool [:[:(Funcs ( the carrier of X,REAL)),(Funcs ( the carrier of X,REAL)):],(Funcs ( the carrier of X,REAL)):]
[:(Funcs ( the carrier of X,REAL)),(Funcs ( the carrier of X,REAL)):] is non empty set
[:[:(Funcs ( the carrier of X,REAL)),(Funcs ( the carrier of X,REAL)):],(Funcs ( the carrier of X,REAL)):] is non empty set
bool [:[:(Funcs ( the carrier of X,REAL)),(Funcs ( the carrier of X,REAL)):],(Funcs ( the carrier of X,REAL)):] is non empty set
RealFuncAdd the carrier of X is non empty Relation-like [:(Funcs ( the carrier of X,REAL)),(Funcs ( the carrier of X,REAL)):] -defined Funcs ( the carrier of X,REAL) -valued Function-like total quasi_total Element of bool [:[:(Funcs ( the carrier of X,REAL)),(Funcs ( the carrier of X,REAL)):],(Funcs ( the carrier of X,REAL)):]
RealFuncExtMult the carrier of X is non empty Relation-like [:REAL,(Funcs ( the carrier of X,REAL)):] -defined Funcs ( the carrier of X,REAL) -valued Function-like total quasi_total Element of bool [:[:REAL,(Funcs ( the carrier of X,REAL)):],(Funcs ( the carrier of X,REAL)):]
[:REAL,(Funcs ( the carrier of X,REAL)):] is non empty set
[:[:REAL,(Funcs ( the carrier of X,REAL)):],(Funcs ( the carrier of X,REAL)):] is non empty set
bool [:[:REAL,(Funcs ( the carrier of X,REAL)):],(Funcs ( the carrier of X,REAL)):] is non empty set
RealFuncUnit the carrier of X is Relation-like the carrier of X -defined REAL -valued Function-like total quasi_total V152() V153() V154() Element of Funcs ( the carrier of X,REAL)
the carrier of X --> 1 is non empty Relation-like the carrier of X -defined NAT -valued RAT -valued INT -valued Function-like total quasi_total V152() V153() V154() V155() Element of bool [: the carrier of X,NAT:]
[: the carrier of X,NAT:] is non empty RAT -valued INT -valued V152() V153() V154() V155() set
bool [: the carrier of X,NAT:] is non empty set
RealFuncZero the carrier of X is Relation-like the carrier of X -defined REAL -valued Function-like total quasi_total V152() V153() V154() Element of Funcs ( the carrier of X,REAL)
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 V152() V153() V154() V155() Element of bool [: the carrier of X,NAT:]
AlgebraStr(# (Funcs ( the carrier of X,REAL)),(RealFuncMult the carrier of X),(RealFuncAdd the carrier of X),(RealFuncExtMult the carrier of X),(RealFuncUnit the carrier of X),(RealFuncZero the carrier of X) #) is strict AlgebraStr
the carrier of (RAlgebra the carrier of X) is non empty set
bool the carrier of (RAlgebra the carrier of X) is non empty set
[: the carrier of X,REAL:] is non empty V152() V153() V154() set
bool [: the carrier of X,REAL:] is non empty set
{ b₁ where b₁ is non empty Relation-like the carrier of X -defined REAL -valued Function-like total quasi_total V152() V153() V154() continuous Element of bool [: the carrier of X,REAL:] : verum } is set
mult_ ((X),(RAlgebra 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),(RAlgebra 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),(RAlgebra the carrier of X)) is non empty Relation-like [:REAL,(X):] -defined (X) -valued Function-like total quasi_total Element of bool [:[:REAL,(X):],(X):]
[:REAL,(X):] is non empty set
[:[:REAL,(X):],(X):] is non empty set
bool [:[:REAL,(X):],(X):] is non empty set
One_ ((X),(RAlgebra the carrier of X)) is Element of (X)
Zero_ ((X),(RAlgebra the carrier of X)) is Element of (X)
(X) is non empty Relation-like (X) -defined REAL -valued Function-like total quasi_total V152() V153() V154() Element of bool [:(X),REAL:]
[:(X),REAL:] is non empty V152() V153() V154() set
bool [:(X),REAL:] is non empty set
BoundedFunctions the carrier of X is non empty add-closed having-inverse additively-closed multiplicatively-closed additively-linearly-closed Element of bool the carrier of (RAlgebra the carrier of X)
{ b₁ where b₁ is non empty Relation-like the carrier of X -defined REAL -valued Function-like total quasi_total V152() V153() V154() Element of bool [: the carrier of X,REAL:] : b₁ | the carrier of X is bounded } is set
BoundedFunctionsNorm the carrier of X is non empty Relation-like BoundedFunctions the carrier of X -defined REAL -valued Function-like total quasi_total V152() V153() V154() Element of bool [:(BoundedFunctions the carrier of X),REAL:]
[:(BoundedFunctions the carrier of X),REAL:] is non empty V152() V153() V154() set
bool [:(BoundedFunctions the carrier of X),REAL:] is non empty set
(BoundedFunctionsNorm the carrier of X) | (X) is Relation-like BoundedFunctions the carrier of X -defined REAL -valued Function-like V152() V153() V154() Element of bool [:(BoundedFunctions the carrier of X),REAL:]
Normed_AlgebraStr(# (X),(mult_ ((X),(RAlgebra the carrier of X))),(Add_ ((X),(RAlgebra the carrier of X))),(Mult_ ((X),(RAlgebra the carrier of X))),(One_ ((X),(RAlgebra the carrier of X))),(Zero_ ((X),(RAlgebra the carrier of X))),(X) #) is strict Normed_AlgebraStr
the carrier of (X) is non empty set
F is right_complementable Element of the carrier of (X)
G is right_complementable Element of the carrier of (X)
X * F is right_complementable Element of the carrier of (X)
the Mult of (X) is non empty Relation-like [:REAL, the carrier of (X):] -defined the carrier of (X) -valued Function-like total quasi_total Element of bool [:[:REAL, the carrier of (X):], the carrier of (X):]
[:REAL, the carrier of (X):] is non empty set
[:[:REAL, the carrier of (X):], the carrier of (X):] is non empty set
bool [:[:REAL, the carrier of (X):], the carrier of (X):] is non empty set
the Mult of (X) . (X,F) is set
[X,F] is set
{X,F} is non empty set
{X} is non empty V185() V186() V187() set
{{X,F},{X}} is non empty set
the Mult of (X) . [X,F] is set
FB is non empty Relation-like the carrier of X -defined REAL -valued Function-like total quasi_total V152() V153() V154() Element of bool [: the carrier of X,REAL:]
GB is non empty Relation-like the carrier of X -defined REAL -valued Function-like total quasi_total V152() V153() V154() Element of bool [: the carrier of X,REAL:]
(X) is non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital unital associative commutative right-distributive right_unital well-unital left_unital strict vector-associative AlgebraStr
AlgebraStr(# (X),(mult_ ((X),(RAlgebra the carrier of X))),(Add_ ((X),(RAlgebra the carrier of X))),(Mult_ ((X),(RAlgebra the carrier of X))),(One_ ((X),(RAlgebra the carrier of X))),(Zero_ ((X),(RAlgebra the carrier of X))) #) is strict AlgebraStr
the carrier of (X) is non empty set
aF is right_complementable Element of the carrier of (X)
aFB is right_complementable Element of the carrier of (X)
X * aFB is right_complementable Element of the carrier of (X)
the Mult of (X) is non empty Relation-like [:REAL, the carrier of (X):] -defined the carrier of (X) -valued Function-like total quasi_total Element of bool [:[:REAL, the carrier of (X):], the carrier of (X):]
[:REAL, the carrier of (X):] is non empty set
[:[:REAL, the carrier of (X):], the carrier of (X):] is non empty set
bool [:[:REAL, the carrier of (X):], the carrier of (X):] is non empty set
the Mult of (X) . (X,aFB) is 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
GB . aG is V11() real ext-real Element of REAL
FB . aG is V11() real ext-real Element of REAL
X * (FB . aG) is V11() real ext-real Element of REAL
X is non empty TopSpace-like pseudocompact compact TopStruct
(X) is non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital unital associative commutative right-distributive right_unital well-unital left_unital vector-associative strict Normed_AlgebraStr
(X) is non empty add-closed having-inverse additively-closed multiplicatively-closed additively-linearly-closed Element of bool the carrier of (RAlgebra the carrier of X)
the carrier of X is non empty set
RAlgebra the carrier of X is non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative unital associative commutative right-distributive right_unital well-unital left_unital strict vector-associative AlgebraStr
Funcs ( the carrier of X,REAL) is non empty functional FUNCTION_DOMAIN of the carrier of X, REAL
RealFuncMult the carrier of X is non empty Relation-like [:(Funcs ( the carrier of X,REAL)),(Funcs ( the carrier of X,REAL)):] -defined Funcs ( the carrier of X,REAL) -valued Function-like total quasi_total Element of bool [:[:(Funcs ( the carrier of X,REAL)),(Funcs ( the carrier of X,REAL)):],(Funcs ( the carrier of X,REAL)):]
[:(Funcs ( the carrier of X,REAL)),(Funcs ( the carrier of X,REAL)):] is non empty set
[:[:(Funcs ( the carrier of X,REAL)),(Funcs ( the carrier of X,REAL)):],(Funcs ( the carrier of X,REAL)):] is non empty set
bool [:[:(Funcs ( the carrier of X,REAL)),(Funcs ( the carrier of X,REAL)):],(Funcs ( the carrier of X,REAL)):] is non empty set
RealFuncAdd the carrier of X is non empty Relation-like [:(Funcs ( the carrier of X,REAL)),(Funcs ( the carrier of X,REAL)):] -defined Funcs ( the carrier of X,REAL) -valued Function-like total quasi_total Element of bool [:[:(Funcs ( the carrier of X,REAL)),(Funcs ( the carrier of X,REAL)):],(Funcs ( the carrier of X,REAL)):]
RealFuncExtMult the carrier of X is non empty Relation-like [:REAL,(Funcs ( the carrier of X,REAL)):] -defined Funcs ( the carrier of X,REAL) -valued Function-like total quasi_total Element of bool [:[:REAL,(Funcs ( the carrier of X,REAL)):],(Funcs ( the carrier of X,REAL)):]
[:REAL,(Funcs ( the carrier of X,REAL)):] is non empty set
[:[:REAL,(Funcs ( the carrier of X,REAL)):],(Funcs ( the carrier of X,REAL)):] is non empty set
bool [:[:REAL,(Funcs ( the carrier of X,REAL)):],(Funcs ( the carrier of X,REAL)):] is non empty set
RealFuncUnit the carrier of X is Relation-like the carrier of X -defined REAL -valued Function-like total quasi_total V152() V153() V154() Element of Funcs ( the carrier of X,REAL)
the carrier of X --> 1 is non empty Relation-like the carrier of X -defined NAT -valued RAT -valued INT -valued Function-like total quasi_total V152() V153() V154() V155() Element of bool [: the carrier of X,NAT:]
[: the carrier of X,NAT:] is non empty RAT -valued INT -valued V152() V153() V154() V155() set
bool [: the carrier of X,NAT:] is non empty set
RealFuncZero the carrier of X is Relation-like the carrier of X -defined REAL -valued Function-like total quasi_total V152() V153() V154() Element of Funcs ( the carrier of X,REAL)
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 V152() V153() V154() V155() Element of bool [: the carrier of X,NAT:]
AlgebraStr(# (Funcs ( the carrier of X,REAL)),(RealFuncMult the carrier of X),(RealFuncAdd the carrier of X),(RealFuncExtMult the carrier of X),(RealFuncUnit the carrier of X),(RealFuncZero the carrier of X) #) is strict AlgebraStr
the carrier of (RAlgebra the carrier of X) is non empty set
bool the carrier of (RAlgebra the carrier of X) is non empty set
[: the carrier of X,REAL:] is non empty V152() V153() V154() set
bool [: the carrier of X,REAL:] is non empty set
{ b₁ where b₁ is non empty Relation-like the carrier of X -defined REAL -valued Function-like total quasi_total V152() V153() V154() continuous Element of bool [: the carrier of X,REAL:] : verum } is set
mult_ ((X),(RAlgebra 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),(RAlgebra 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),(RAlgebra the carrier of X)) is non empty Relation-like [:REAL,(X):] -defined (X) -valued Function-like total quasi_total Element of bool [:[:REAL,(X):],(X):]
[:REAL,(X):] is non empty set
[:[:REAL,(X):],(X):] is non empty set
bool [:[:REAL,(X):],(X):] is non empty set
One_ ((X),(RAlgebra the carrier of X)) is Element of (X)
Zero_ ((X),(RAlgebra the carrier of X)) is Element of (X)
(X) is non empty Relation-like (X) -defined REAL -valued Function-like total quasi_total V152() V153() V154() Element of bool [:(X),REAL:]
[:(X),REAL:] is non empty V152() V153() V154() set
bool [:(X),REAL:] is non empty set
BoundedFunctions the carrier of X is non empty add-closed having-inverse additively-closed multiplicatively-closed additively-linearly-closed Element of bool the carrier of (RAlgebra the carrier of X)
{ b₁ where b₁ is non empty Relation-like the carrier of X -defined REAL -valued Function-like total quasi_total V152() V153() V154() Element of bool [: the carrier of X,REAL:] : b₁ | the carrier of X is bounded } is set
BoundedFunctionsNorm the carrier of X is non empty Relation-like BoundedFunctions the carrier of X -defined REAL -valued Function-like total quasi_total V152() V153() V154() Element of bool [:(BoundedFunctions the carrier of X),REAL:]
[:(BoundedFunctions the carrier of X),REAL:] is non empty V152() V153() V154() set
bool [:(BoundedFunctions the carrier of X),REAL:] is non empty set
(BoundedFunctionsNorm the carrier of X) | (X) is Relation-like BoundedFunctions the carrier of X -defined REAL -valued Function-like V152() V153() V154() Element of bool [:(BoundedFunctions the carrier of X),REAL:]
Normed_AlgebraStr(# (X),(mult_ ((X),(RAlgebra the carrier of X))),(Add_ ((X),(RAlgebra the carrier of X))),(Mult_ ((X),(RAlgebra the carrier of X))),(One_ ((X),(RAlgebra the carrier of X))),(Zero_ ((X),(RAlgebra the carrier of X))),(X) #) is strict Normed_AlgebraStr
the carrier of (X) is non empty set
X is right_complementable Element of the carrier of (X)
F is right_complementable Element of the carrier of (X)
G is right_complementable Element of the carrier of (X)
X * F is right_complementable 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 right_complementable 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 REAL -valued Function-like total quasi_total V152() V153() V154() Element of bool [: the carrier of X,REAL:]
GB is non empty Relation-like the carrier of X -defined REAL -valued Function-like total quasi_total V152() V153() V154() Element of bool [: the carrier of X,REAL:]
aFB is non empty Relation-like the carrier of X -defined REAL -valued Function-like total quasi_total V152() V153() V154() Element of bool [: the carrier of X,REAL:]
(X) is non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital unital associative commutative right-distributive right_unital well-unital left_unital strict vector-associative AlgebraStr
AlgebraStr(# (X),(mult_ ((X),(RAlgebra the carrier of X))),(Add_ ((X),(RAlgebra the carrier of X))),(Mult_ ((X),(RAlgebra the carrier of X))),(One_ ((X),(RAlgebra the carrier of X))),(Zero_ ((X),(RAlgebra the carrier of X))) #) is strict AlgebraStr
the carrier of (X) is non empty set
r is right_complementable Element of the carrier of (X)
aF is right_complementable Element of the carrier of (X)
aG is right_complementable Element of the carrier of (X)
aF * aG is right_complementable 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 right_complementable 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
s1 is Element of the carrier of X
aFB . s1 is V11() real ext-real Element of REAL
FB . s1 is V11() real ext-real Element of REAL
GB . s1 is V11() real ext-real Element of REAL
(FB . s1) * (GB . s1) is V11() real ext-real Element of REAL
X is V11() real ext-real Element of REAL
abs X is V11() real ext-real Element of REAL
X is non empty TopSpace-like pseudocompact compact TopStruct
(X) is non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital unital associative commutative right-distributive right_unital well-unital left_unital vector-associative strict Normed_AlgebraStr
(X) is non empty add-closed having-inverse additively-closed multiplicatively-closed additively-linearly-closed Element of bool the carrier of (RAlgebra the carrier of X)
the carrier of X is non empty set
RAlgebra the carrier of X is non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative unital associative commutative right-distributive right_unital well-unital left_unital strict vector-associative AlgebraStr
Funcs ( the carrier of X,REAL) is non empty functional FUNCTION_DOMAIN of the carrier of X, REAL
RealFuncMult the carrier of X is non empty Relation-like [:(Funcs ( the carrier of X,REAL)),(Funcs ( the carrier of X,REAL)):] -defined Funcs ( the carrier of X,REAL) -valued Function-like total quasi_total Element of bool [:[:(Funcs ( the carrier of X,REAL)),(Funcs ( the carrier of X,REAL)):],(Funcs ( the carrier of X,REAL)):]
[:(Funcs ( the carrier of X,REAL)),(Funcs ( the carrier of X,REAL)):] is non empty set
[:[:(Funcs ( the carrier of X,REAL)),(Funcs ( the carrier of X,REAL)):],(Funcs ( the carrier of X,REAL)):] is non empty set
bool [:[:(Funcs ( the carrier of X,REAL)),(Funcs ( the carrier of X,REAL)):],(Funcs ( the carrier of X,REAL)):] is non empty set
RealFuncAdd the carrier of X is non empty Relation-like [:(Funcs ( the carrier of X,REAL)),(Funcs ( the carrier of X,REAL)):] -defined Funcs ( the carrier of X,REAL) -valued Function-like total quasi_total Element of bool [:[:(Funcs ( the carrier of X,REAL)),(Funcs ( the carrier of X,REAL)):],(Funcs ( the carrier of X,REAL)):]
RealFuncExtMult the carrier of X is non empty Relation-like [:REAL,(Funcs ( the carrier of X,REAL)):] -defined Funcs ( the carrier of X,REAL) -valued Function-like total quasi_total Element of bool [:[:REAL,(Funcs ( the carrier of X,REAL)):],(Funcs ( the carrier of X,REAL)):]
[:REAL,(Funcs ( the carrier of X,REAL)):] is non empty set
[:[:REAL,(Funcs ( the carrier of X,REAL)):],(Funcs ( the carrier of X,REAL)):] is non empty set
bool [:[:REAL,(Funcs ( the carrier of X,REAL)):],(Funcs ( the carrier of X,REAL)):] is non empty set
RealFuncUnit the carrier of X is Relation-like the carrier of X -defined REAL -valued Function-like total quasi_total V152() V153() V154() Element of Funcs ( the carrier of X,REAL)
the carrier of X --> 1 is non empty Relation-like the carrier of X -defined NAT -valued RAT -valued INT -valued Function-like total quasi_total V152() V153() V154() V155() Element of bool [: the carrier of X,NAT:]
[: the carrier of X,NAT:] is non empty RAT -valued INT -valued V152() V153() V154() V155() set
bool [: the carrier of X,NAT:] is non empty set
RealFuncZero the carrier of X is Relation-like the carrier of X -defined REAL -valued Function-like total quasi_total V152() V153() V154() Element of Funcs ( the carrier of X,REAL)
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 V152() V153() V154() V155() Element of bool [: the carrier of X,NAT:]
AlgebraStr(# (Funcs ( the carrier of X,REAL)),(RealFuncMult the carrier of X),(RealFuncAdd the carrier of X),(RealFuncExtMult the carrier of X),(RealFuncUnit the carrier of X),(RealFuncZero the carrier of X) #) is strict AlgebraStr
the carrier of (RAlgebra the carrier of X) is non empty set
bool the carrier of (RAlgebra the carrier of X) is non empty set
[: the carrier of X,REAL:] is non empty V152() V153() V154() set
bool [: the carrier of X,REAL:] is non empty set
{ b₁ where b₁ is non empty Relation-like the carrier of X -defined REAL -valued Function-like total quasi_total V152() V153() V154() continuous Element of bool [: the carrier of X,REAL:] : verum } is set
mult_ ((X),(RAlgebra 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),(RAlgebra 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),(RAlgebra the carrier of X)) is non empty Relation-like [:REAL,(X):] -defined (X) -valued Function-like total quasi_total Element of bool [:[:REAL,(X):],(X):]
[:REAL,(X):] is non empty set
[:[:REAL,(X):],(X):] is non empty set
bool [:[:REAL,(X):],(X):] is non empty set
One_ ((X),(RAlgebra the carrier of X)) is Element of (X)
Zero_ ((X),(RAlgebra the carrier of X)) is Element of (X)
(X) is non empty Relation-like (X) -defined REAL -valued Function-like total quasi_total V152() V153() V154() Element of bool [:(X),REAL:]
[:(X),REAL:] is non empty V152() V153() V154() set
bool [:(X),REAL:] is non empty set
BoundedFunctions the carrier of X is non empty add-closed having-inverse additively-closed multiplicatively-closed additively-linearly-closed Element of bool the carrier of (RAlgebra the carrier of X)
{ b₁ where b₁ is non empty Relation-like the carrier of X -defined REAL -valued Function-like total quasi_total V152() V153() V154() Element of bool [: the carrier of X,REAL:] : b₁ | the carrier of X is bounded } is set
BoundedFunctionsNorm the carrier of X is non empty Relation-like BoundedFunctions the carrier of X -defined REAL -valued Function-like total quasi_total V152() V153() V154() Element of bool [:(BoundedFunctions the carrier of X),REAL:]
[:(BoundedFunctions the carrier of X),REAL:] is non empty V152() V153() V154() set
bool [:(BoundedFunctions the carrier of X),REAL:] is non empty set
(BoundedFunctionsNorm the carrier of X) | (X) is Relation-like BoundedFunctions the carrier of X -defined REAL -valued Function-like V152() V153() V154() Element of bool [:(BoundedFunctions the carrier of X),REAL:]
Normed_AlgebraStr(# (X),(mult_ ((X),(RAlgebra the carrier of X))),(Add_ ((X),(RAlgebra the carrier of X))),(Mult_ ((X),(RAlgebra the carrier of X))),(One_ ((X),(RAlgebra the carrier of X))),(Zero_ ((X),(RAlgebra the carrier of X))),(X) #) is strict Normed_AlgebraStr
the carrier of (X) is non empty set
0. (X) is V49((X)) right_complementable Element of the carrier of (X)
the ZeroF of (X) is right_complementable Element of the carrier of (X)
F is right_complementable Element of the carrier of (X)
||.F.|| is V11() real ext-real Element of REAL
the U8 of (X) is non empty Relation-like the carrier of (X) -defined REAL -valued Function-like total quasi_total V152() V153() V154() Element of bool [: the carrier of (X),REAL:]
[: the carrier of (X),REAL:] is non empty V152() V153() V154() set
bool [: the carrier of (X),REAL:] is non empty set
the U8 of (X) . F is V11() real ext-real Element of REAL
X * F is right_complementable Element of the carrier of (X)
the Mult of (X) is non empty Relation-like [:REAL, the carrier of (X):] -defined the carrier of (X) -valued Function-like total quasi_total Element of bool [:[:REAL, the carrier of (X):], the carrier of (X):]
[:REAL, the carrier of (X):] is non empty set
[:[:REAL, the carrier of (X):], the carrier of (X):] is non empty set
bool [:[:REAL, the carrier of (X):], the carrier of (X):] is non empty set
the Mult of (X) . (X,F) is set
[X,F] is set
{X,F} is non empty set
{X} is non empty V185() V186() V187() set
{{X,F},{X}} is non empty set
the Mult of (X) . [X,F] is set
||.(X * F).|| is V11() real ext-real Element of REAL
the U8 of (X) . (X * F) is V11() real ext-real Element of REAL
(abs X) * ||.F.|| is V11() real ext-real Element of REAL
G is right_complementable Element of the carrier of (X)
F + G is right_complementable 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 right_complementable 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 V11() real ext-real Element of REAL
the U8 of (X) . (F + G) is V11() real ext-real Element of REAL
||.G.|| is V11() real ext-real Element of REAL
the U8 of (X) . G is V11() real ext-real Element of REAL
||.F.|| + ||.G.|| is V11() real ext-real Element of REAL
R_Normed_Algebra_of_BoundedFunctions the carrier of X is non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital discerning reflexive RealNormSpace-like unital complete Normed_AlgebraStr
mult_ ((BoundedFunctions the carrier of X),(RAlgebra the carrier of X)) is non empty Relation-like [:(BoundedFunctions the carrier of X),(BoundedFunctions the carrier of X):] -defined BoundedFunctions the carrier of X -valued Function-like total quasi_total Element of bool [:[:(BoundedFunctions the carrier of X),(BoundedFunctions the carrier of X):],(BoundedFunctions the carrier of X):]
[:(BoundedFunctions the carrier of X),(BoundedFunctions the carrier of X):] is non empty set
[:[:(BoundedFunctions the carrier of X),(BoundedFunctions the carrier of X):],(BoundedFunctions the carrier of X):] is non empty set
bool [:[:(BoundedFunctions the carrier of X),(BoundedFunctions the carrier of X):],(BoundedFunctions the carrier of X):] is non empty set
Add_ ((BoundedFunctions the carrier of X),(RAlgebra the carrier of X)) is non empty Relation-like [:(BoundedFunctions the carrier of X),(BoundedFunctions the carrier of X):] -defined BoundedFunctions the carrier of X -valued Function-like total quasi_total Element of bool [:[:(BoundedFunctions the carrier of X),(BoundedFunctions the carrier of X):],(BoundedFunctions the carrier of X):]
Mult_ ((BoundedFunctions the carrier of X),(RAlgebra the carrier of X)) is non empty Relation-like [:REAL,(BoundedFunctions the carrier of X):] -defined BoundedFunctions the carrier of X -valued Function-like total quasi_total Element of bool [:[:REAL,(BoundedFunctions the carrier of X):],(BoundedFunctions the carrier of X):]
[:REAL,(BoundedFunctions the carrier of X):] is non empty set
[:[:REAL,(BoundedFunctions the carrier of X):],(BoundedFunctions the carrier of X):] is non empty set
bool [:[:REAL,(BoundedFunctions the carrier of X):],(BoundedFunctions the carrier of X):] is non empty set
One_ ((BoundedFunctions the carrier of X),(RAlgebra the carrier of X)) is Element of BoundedFunctions the carrier of X
Zero_ ((BoundedFunctions the carrier of X),(RAlgebra the carrier of X)) is Element of BoundedFunctions the carrier of X
Normed_AlgebraStr(# (BoundedFunctions the carrier of X),(mult_ ((BoundedFunctions the carrier of X),(RAlgebra the carrier of X))),(Add_ ((BoundedFunctions the carrier of X),(RAlgebra the carrier of X))),(Mult_ ((BoundedFunctions the carrier of X),(RAlgebra the carrier of X))),(One_ ((BoundedFunctions the carrier of X),(RAlgebra the carrier of X))),(Zero_ ((BoundedFunctions the carrier of X),(RAlgebra the carrier of X))),(BoundedFunctionsNorm the carrier of X) #) is strict Normed_AlgebraStr
the carrier of (R_Normed_Algebra_of_BoundedFunctions the carrier of X) is non empty set
FB is right_complementable Element of the carrier of (R_Normed_Algebra_of_BoundedFunctions the carrier of X)
||.FB.|| is V11() real ext-real Element of REAL
the U8 of (R_Normed_Algebra_of_BoundedFunctions the carrier of X) is non empty Relation-like the carrier of (R_Normed_Algebra_of_BoundedFunctions the carrier of X) -defined REAL -valued Function-like total quasi_total V152() V153() V154() Element of bool [: the carrier of (R_Normed_Algebra_of_BoundedFunctions the carrier of X),REAL:]
[: the carrier of (R_Normed_Algebra_of_BoundedFunctions the carrier of X),REAL:] is non empty V152() V153() V154() set
bool [: the carrier of (R_Normed_Algebra_of_BoundedFunctions the carrier of X),REAL:] is non empty set
the U8 of (R_Normed_Algebra_of_BoundedFunctions the carrier of X) . FB is V11() real ext-real Element of REAL
GB is right_complementable Element of the carrier of (R_Normed_Algebra_of_BoundedFunctions the carrier of X)
||.GB.|| is V11() real ext-real Element of REAL
the U8 of (R_Normed_Algebra_of_BoundedFunctions the carrier of X) . GB is V11() real ext-real Element of REAL
FB + GB is right_complementable Element of the carrier of (R_Normed_Algebra_of_BoundedFunctions the carrier of X)
the addF of (R_Normed_Algebra_of_BoundedFunctions the carrier of X) is non empty Relation-like [: the carrier of (R_Normed_Algebra_of_BoundedFunctions the carrier of X), the carrier of (R_Normed_Algebra_of_BoundedFunctions the carrier of X):] -defined the carrier of (R_Normed_Algebra_of_BoundedFunctions the carrier of X) -valued Function-like total quasi_total Element of bool [:[: the carrier of (R_Normed_Algebra_of_BoundedFunctions the carrier of X), the carrier of (R_Normed_Algebra_of_BoundedFunctions the carrier of X):], the carrier of (R_Normed_Algebra_of_BoundedFunctions the carrier of X):]
[: the carrier of (R_Normed_Algebra_of_BoundedFunctions the carrier of X), the carrier of (R_Normed_Algebra_of_BoundedFunctions the carrier of X):] is non empty set
[:[: the carrier of (R_Normed_Algebra_of_BoundedFunctions the carrier of X), the carrier of (R_Normed_Algebra_of_BoundedFunctions the carrier of X):], the carrier of (R_Normed_Algebra_of_BoundedFunctions the carrier of X):] is non empty set
bool [:[: the carrier of (R_Normed_Algebra_of_BoundedFunctions the carrier of X), the carrier of (R_Normed_Algebra_of_BoundedFunctions the carrier of X):], the carrier of (R_Normed_Algebra_of_BoundedFunctions the carrier of X):] is non empty set
the addF of (R_Normed_Algebra_of_BoundedFunctions the carrier of X) . (FB,GB) is right_complementable Element of the carrier of (R_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 (R_Normed_Algebra_of_BoundedFunctions the carrier of X) . [FB,GB] is set
||.(FB + GB).|| is V11() real ext-real Element of REAL
the U8 of (R_Normed_Algebra_of_BoundedFunctions the carrier of X) . (FB + GB) is V11() real ext-real Element of REAL
0. (R_Normed_Algebra_of_BoundedFunctions the carrier of X) is V49( R_Normed_Algebra_of_BoundedFunctions the carrier of X) right_complementable Element of the carrier of (R_Normed_Algebra_of_BoundedFunctions the carrier of X)
the ZeroF of (R_Normed_Algebra_of_BoundedFunctions the carrier of X) is right_complementable Element of the carrier of (R_Normed_Algebra_of_BoundedFunctions the carrier of X)
X * FB is right_complementable Element of the carrier of (R_Normed_Algebra_of_BoundedFunctions the carrier of X)
the Mult of (R_Normed_Algebra_of_BoundedFunctions the carrier of X) is non empty Relation-like [:REAL, the carrier of (R_Normed_Algebra_of_BoundedFunctions the carrier of X):] -defined the carrier of (R_Normed_Algebra_of_BoundedFunctions the carrier of X) -valued Function-like total quasi_total Element of bool [:[:REAL, the carrier of (R_Normed_Algebra_of_BoundedFunctions the carrier of X):], the carrier of (R_Normed_Algebra_of_BoundedFunctions the carrier of X):]
[:REAL, the carrier of (R_Normed_Algebra_of_BoundedFunctions the carrier of X):] is non empty set
[:[:REAL, the carrier of (R_Normed_Algebra_of_BoundedFunctions the carrier of X):], the carrier of (R_Normed_Algebra_of_BoundedFunctions the carrier of X):] is non empty set
bool [:[:REAL, the carrier of (R_Normed_Algebra_of_BoundedFunctions the carrier of X):], the carrier of (R_Normed_Algebra_of_BoundedFunctions the carrier of X):] is non empty set
the Mult of (R_Normed_Algebra_of_BoundedFunctions the carrier of X) . (X,FB) is set
[X,FB] is set
{X,FB} is non empty set
{{X,FB},{X}} is non empty set
the Mult of (R_Normed_Algebra_of_BoundedFunctions the carrier of X) . [X,FB] is set
||.(X * FB).|| is V11() real ext-real Element of REAL
the U8 of (R_Normed_Algebra_of_BoundedFunctions the carrier of X) . (X * FB) is V11() real ext-real Element of REAL
(abs X) * ||.FB.|| is V11() real ext-real Element of REAL
X is non empty TopSpace-like pseudocompact compact TopStruct
(X) is non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital unital associative commutative right-distributive right_unital well-unital left_unital vector-associative strict Normed_AlgebraStr
(X) is non empty add-closed having-inverse additively-closed multiplicatively-closed additively-linearly-closed Element of bool the carrier of (RAlgebra the carrier of X)
the carrier of X is non empty set
RAlgebra the carrier of X is non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative unital associative commutative right-distributive right_unital well-unital left_unital strict vector-associative AlgebraStr
Funcs ( the carrier of X,REAL) is non empty functional FUNCTION_DOMAIN of the carrier of X, REAL
RealFuncMult the carrier of X is non empty Relation-like [:(Funcs ( the carrier of X,REAL)),(Funcs ( the carrier of X,REAL)):] -defined Funcs ( the carrier of X,REAL) -valued Function-like total quasi_total Element of bool [:[:(Funcs ( the carrier of X,REAL)),(Funcs ( the carrier of X,REAL)):],(Funcs ( the carrier of X,REAL)):]
[:(Funcs ( the carrier of X,REAL)),(Funcs ( the carrier of X,REAL)):] is non empty set
[:[:(Funcs ( the carrier of X,REAL)),(Funcs ( the carrier of X,REAL)):],(Funcs ( the carrier of X,REAL)):] is non empty set
bool [:[:(Funcs ( the carrier of X,REAL)),(Funcs ( the carrier of X,REAL)):],(Funcs ( the carrier of X,REAL)):] is non empty set
RealFuncAdd the carrier of X is non empty Relation-like [:(Funcs ( the carrier of X,REAL)),(Funcs ( the carrier of X,REAL)):] -defined Funcs ( the carrier of X,REAL) -valued Function-like total quasi_total Element of bool [:[:(Funcs ( the carrier of X,REAL)),(Funcs ( the carrier of X,REAL)):],(Funcs ( the carrier of X,REAL)):]
RealFuncExtMult the carrier of X is non empty Relation-like [:REAL,(Funcs ( the carrier of X,REAL)):] -defined Funcs ( the carrier of X,REAL) -valued Function-like total quasi_total Element of bool [:[:REAL,(Funcs ( the carrier of X,REAL)):],(Funcs ( the carrier of X,REAL)):]
[:REAL,(Funcs ( the carrier of X,REAL)):] is non empty set
[:[:REAL,(Funcs ( the carrier of X,REAL)):],(Funcs ( the carrier of X,REAL)):] is non empty set
bool [:[:REAL,(Funcs ( the carrier of X,REAL)):],(Funcs ( the carrier of X,REAL)):] is non empty set
RealFuncUnit the carrier of X is Relation-like the carrier of X -defined REAL -valued Function-like total quasi_total V152() V153() V154() Element of Funcs ( the carrier of X,REAL)
the carrier of X --> 1 is non empty Relation-like the carrier of X -defined NAT -valued RAT -valued INT -valued Function-like total quasi_total V152() V153() V154() V155() Element of bool [: the carrier of X,NAT:]
[: the carrier of X,NAT:] is non empty RAT -valued INT -valued V152() V153() V154() V155() set
bool [: the carrier of X,NAT:] is non empty set
RealFuncZero the carrier of X is Relation-like the carrier of X -defined REAL -valued Function-like total quasi_total V152() V153() V154() Element of Funcs ( the carrier of X,REAL)
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 V152() V153() V154() V155() Element of bool [: the carrier of X,NAT:]
AlgebraStr(# (Funcs ( the carrier of X,REAL)),(RealFuncMult the carrier of X),(RealFuncAdd the carrier of X),(RealFuncExtMult the carrier of X),(RealFuncUnit the carrier of X),(RealFuncZero the carrier of X) #) is strict AlgebraStr
the carrier of (RAlgebra the carrier of X) is non empty set
bool the carrier of (RAlgebra the carrier of X) is non empty set
[: the carrier of X,REAL:] is non empty V152() V153() V154() set
bool [: the carrier of X,REAL:] is non empty set
{ b₁ where b₁ is non empty Relation-like the carrier of X -defined REAL -valued Function-like total quasi_total V152() V153() V154() continuous Element of bool [: the carrier of X,REAL:] : verum } is set
mult_ ((X),(RAlgebra 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),(RAlgebra 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),(RAlgebra the carrier of X)) is non empty Relation-like [:REAL,(X):] -defined (X) -valued Function-like total quasi_total Element of bool [:[:REAL,(X):],(X):]
[:REAL,(X):] is non empty set
[:[:REAL,(X):],(X):] is non empty set
bool [:[:REAL,(X):],(X):] is non empty set
One_ ((X),(RAlgebra the carrier of X)) is Element of (X)
Zero_ ((X),(RAlgebra the carrier of X)) is Element of (X)
(X) is non empty Relation-like (X) -defined REAL -valued Function-like total quasi_total V152() V153() V154() Element of bool [:(X),REAL:]
[:(X),REAL:] is non empty V152() V153() V154() set
bool [:(X),REAL:] is non empty set
BoundedFunctions the carrier of X is non empty add-closed having-inverse additively-closed multiplicatively-closed additively-linearly-closed Element of bool the carrier of (RAlgebra the carrier of X)
{ b₁ where b₁ is non empty Relation-like the carrier of X -defined REAL -valued Function-like total quasi_total V152() V153() V154() Element of bool [: the carrier of X,REAL:] : b₁ | the carrier of X is bounded } is set
BoundedFunctionsNorm the carrier of X is non empty Relation-like BoundedFunctions the carrier of X -defined REAL -valued Function-like total quasi_total V152() V153() V154() Element of bool [:(BoundedFunctions the carrier of X),REAL:]
[:(BoundedFunctions the carrier of X),REAL:] is non empty V152() V153() V154() set
bool [:(BoundedFunctions the carrier of X),REAL:] is non empty set
(BoundedFunctionsNorm the carrier of X) | (X) is Relation-like BoundedFunctions the carrier of X -defined REAL -valued Function-like V152() V153() V154() Element of bool [:(BoundedFunctions the carrier of X),REAL:]
Normed_AlgebraStr(# (X),(mult_ ((X),(RAlgebra the carrier of X))),(Add_ ((X),(RAlgebra the carrier of X))),(Mult_ ((X),(RAlgebra the carrier of X))),(One_ ((X),(RAlgebra the carrier of X))),(Zero_ ((X),(RAlgebra the carrier of X))),(X) #) is strict Normed_AlgebraStr
0. (X) is V49((X)) right_complementable Element of the carrier of (X)
the carrier of (X) is non empty set
the ZeroF of (X) is right_complementable Element of the carrier of (X)
||.(0. (X)).|| is V11() real ext-real Element of REAL
the U8 of (X) is non empty Relation-like the carrier of (X) -defined REAL -valued Function-like total quasi_total V152() V153() V154() Element of bool [: the carrier of (X),REAL:]
[: the carrier of (X),REAL:] is non empty V152() V153() V154() set
bool [: the carrier of (X),REAL:] is non empty set
the U8 of (X) . (0. (X)) is V11() real ext-real Element of REAL
F is right_complementable Element of the carrier of (X)
||.F.|| is V11() real ext-real Element of REAL
the U8 of (X) . F is V11() real ext-real Element of REAL
G is right_complementable Element of the carrier of (X)
||.G.|| is V11() real ext-real Element of REAL
the U8 of (X) . G is V11() real ext-real Element of REAL
FB is right_complementable Element of the carrier of (X)
GB is V11() real ext-real Element of REAL
GB * FB is right_complementable Element of the carrier of (X)
the Mult of (X) is non empty Relation-like [:REAL, the carrier of (X):] -defined the carrier of (X) -valued Function-like total quasi_total Element of bool [:[:REAL, the carrier of (X):], the carrier of (X):]
[:REAL, the carrier of (X):] is non empty set
[:[:REAL, the carrier of (X):], the carrier of (X):] is non empty set
bool [:[:REAL, the carrier of (X):], the carrier of (X):] is non empty set
the Mult of (X) . (GB,FB) is set
[GB,FB] is set
{GB,FB} is non empty set
{GB} is non empty V185() V186() V187() set
{{GB,FB},{GB}} is non empty set
the Mult of (X) . [GB,FB] is set
||.(GB * FB).|| is V11() real ext-real Element of REAL
the U8 of (X) . (GB * FB) is V11() real ext-real Element of REAL
abs GB is V11() real ext-real Element of REAL
||.FB.|| is V11() real ext-real Element of REAL
the U8 of (X) . FB is V11() real ext-real Element of REAL
(abs GB) * ||.FB.|| is V11() real ext-real Element of REAL
aFB is right_complementable Element of the carrier of (X)
aF is right_complementable Element of the carrier of (X)
aFB + aF is right_complementable 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 right_complementable 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 V11() real ext-real Element of REAL
the U8 of (X) . (aFB + aF) is V11() real ext-real Element of REAL
||.aFB.|| is V11() real ext-real Element of REAL
the U8 of (X) . aFB is V11() real ext-real Element of REAL
||.aF.|| is V11() real ext-real Element of REAL
the U8 of (X) . aF is V11() real ext-real Element of REAL
||.aFB.|| + ||.aF.|| is V11() real ext-real Element of REAL
X is non empty TopSpace-like pseudocompact compact TopStruct
(X) is non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital discerning reflexive RealNormSpace-like unital associative commutative right-distributive right_unital well-unital left_unital vector-associative strict Normed_AlgebraStr
(X) is non empty add-closed having-inverse additively-closed multiplicatively-closed additively-linearly-closed Element of bool the carrier of (RAlgebra the carrier of X)
the carrier of X is non empty set
RAlgebra the carrier of X is non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative unital associative commutative right-distributive right_unital well-unital left_unital strict vector-associative AlgebraStr
Funcs ( the carrier of X,REAL) is non empty functional FUNCTION_DOMAIN of the carrier of X, REAL
RealFuncMult the carrier of X is non empty Relation-like [:(Funcs ( the carrier of X,REAL)),(Funcs ( the carrier of X,REAL)):] -defined Funcs ( the carrier of X,REAL) -valued Function-like total quasi_total Element of bool [:[:(Funcs ( the carrier of X,REAL)),(Funcs ( the carrier of X,REAL)):],(Funcs ( the carrier of X,REAL)):]
[:(Funcs ( the carrier of X,REAL)),(Funcs ( the carrier of X,REAL)):] is non empty set
[:[:(Funcs ( the carrier of X,REAL)),(Funcs ( the carrier of X,REAL)):],(Funcs ( the carrier of X,REAL)):] is non empty set
bool [:[:(Funcs ( the carrier of X,REAL)),(Funcs ( the carrier of X,REAL)):],(Funcs ( the carrier of X,REAL)):] is non empty set
RealFuncAdd the carrier of X is non empty Relation-like [:(Funcs ( the carrier of X,REAL)),(Funcs ( the carrier of X,REAL)):] -defined Funcs ( the carrier of X,REAL) -valued Function-like total quasi_total Element of bool [:[:(Funcs ( the carrier of X,REAL)),(Funcs ( the carrier of X,REAL)):],(Funcs ( the carrier of X,REAL)):]
RealFuncExtMult the carrier of X is non empty Relation-like [:REAL,(Funcs ( the carrier of X,REAL)):] -defined Funcs ( the carrier of X,REAL) -valued Function-like total quasi_total Element of bool [:[:REAL,(Funcs ( the carrier of X,REAL)):],(Funcs ( the carrier of X,REAL)):]
[:REAL,(Funcs ( the carrier of X,REAL)):] is non empty set
[:[:REAL,(Funcs ( the carrier of X,REAL)):],(Funcs ( the carrier of X,REAL)):] is non empty set
bool [:[:REAL,(Funcs ( the carrier of X,REAL)):],(Funcs ( the carrier of X,REAL)):] is non empty set
RealFuncUnit the carrier of X is Relation-like the carrier of X -defined REAL -valued Function-like total quasi_total V152() V153() V154() Element of Funcs ( the carrier of X,REAL)
the carrier of X --> 1 is non empty Relation-like the carrier of X -defined NAT -valued RAT -valued INT -valued Function-like total quasi_total V152() V153() V154() V155() Element of bool [: the carrier of X,NAT:]
[: the carrier of X,NAT:] is non empty RAT -valued INT -valued V152() V153() V154() V155() set
bool [: the carrier of X,NAT:] is non empty set
RealFuncZero the carrier of X is Relation-like the carrier of X -defined REAL -valued Function-like total quasi_total V152() V153() V154() Element of Funcs ( the carrier of X,REAL)
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 V152() V153() V154() V155() Element of bool [: the carrier of X,NAT:]
AlgebraStr(# (Funcs ( the carrier of X,REAL)),(RealFuncMult the carrier of X),(RealFuncAdd the carrier of X),(RealFuncExtMult the carrier of X),(RealFuncUnit the carrier of X),(RealFuncZero the carrier of X) #) is strict AlgebraStr
the carrier of (RAlgebra the carrier of X) is non empty set
bool the carrier of (RAlgebra the carrier of X) is non empty set
[: the carrier of X,REAL:] is non empty V152() V153() V154() set
bool [: the carrier of X,REAL:] is non empty set
{ b₁ where b₁ is non empty Relation-like the carrier of X -defined REAL -valued Function-like total quasi_total V152() V153() V154() continuous Element of bool [: the carrier of X,REAL:] : verum } is set
mult_ ((X),(RAlgebra 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),(RAlgebra 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),(RAlgebra the carrier of X)) is non empty Relation-like [:REAL,(X):] -defined (X) -valued Function-like total quasi_total Element of bool [:[:REAL,(X):],(X):]
[:REAL,(X):] is non empty set
[:[:REAL,(X):],(X):] is non empty set
bool [:[:REAL,(X):],(X):] is non empty set
One_ ((X),(RAlgebra the carrier of X)) is Element of (X)
Zero_ ((X),(RAlgebra the carrier of X)) is Element of (X)
(X) is non empty Relation-like (X) -defined REAL -valued Function-like total quasi_total V152() V153() V154() Element of bool [:(X),REAL:]
[:(X),REAL:] is non empty V152() V153() V154() set
bool [:(X),REAL:] is non empty set
BoundedFunctions the carrier of X is non empty add-closed having-inverse additively-closed multiplicatively-closed additively-linearly-closed Element of bool the carrier of (RAlgebra the carrier of X)
{ b₁ where b₁ is non empty Relation-like the carrier of X -defined REAL -valued Function-like total quasi_total V152() V153() V154() Element of bool [: the carrier of X,REAL:] : b₁ | the carrier of X is bounded } is set
BoundedFunctionsNorm the carrier of X is non empty Relation-like BoundedFunctions the carrier of X -defined REAL -valued Function-like total quasi_total V152() V153() V154() Element of bool [:(BoundedFunctions the carrier of X),REAL:]
[:(BoundedFunctions the carrier of X),REAL:] is non empty V152() V153() V154() set
bool [:(BoundedFunctions the carrier of X),REAL:] is non empty set
(BoundedFunctionsNorm the carrier of X) | (X) is Relation-like BoundedFunctions the carrier of X -defined REAL -valued Function-like V152() V153() V154() Element of bool [:(BoundedFunctions the carrier of X),REAL:]
Normed_AlgebraStr(# (X),(mult_ ((X),(RAlgebra the carrier of X))),(Add_ ((X),(RAlgebra the carrier of X))),(Mult_ ((X),(RAlgebra the carrier of X))),(One_ ((X),(RAlgebra the carrier of X))),(Zero_ ((X),(RAlgebra the carrier of X))),(X) #) is strict Normed_AlgebraStr
the carrier of (X) is non empty set
R_Normed_Algebra_of_BoundedFunctions the carrier of X is non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital discerning reflexive RealNormSpace-like unital complete Normed_AlgebraStr
mult_ ((BoundedFunctions the carrier of X),(RAlgebra the carrier of X)) is non empty Relation-like [:(BoundedFunctions the carrier of X),(BoundedFunctions the carrier of X):] -defined BoundedFunctions the carrier of X -valued Function-like total quasi_total Element of bool [:[:(BoundedFunctions the carrier of X),(BoundedFunctions the carrier of X):],(BoundedFunctions the carrier of X):]
[:(BoundedFunctions the carrier of X),(BoundedFunctions the carrier of X):] is non empty set
[:[:(BoundedFunctions the carrier of X),(BoundedFunctions the carrier of X):],(BoundedFunctions the carrier of X):] is non empty set
bool [:[:(BoundedFunctions the carrier of X),(BoundedFunctions the carrier of X):],(BoundedFunctions the carrier of X):] is non empty set
Add_ ((BoundedFunctions the carrier of X),(RAlgebra the carrier of X)) is non empty Relation-like [:(BoundedFunctions the carrier of X),(BoundedFunctions the carrier of X):] -defined BoundedFunctions the carrier of X -valued Function-like total quasi_total Element of bool [:[:(BoundedFunctions the carrier of X),(BoundedFunctions the carrier of X):],(BoundedFunctions the carrier of X):]
Mult_ ((BoundedFunctions the carrier of X),(RAlgebra the carrier of X)) is non empty Relation-like [:REAL,(BoundedFunctions the carrier of X):] -defined BoundedFunctions the carrier of X -valued Function-like total quasi_total Element of bool [:[:REAL,(BoundedFunctions the carrier of X):],(BoundedFunctions the carrier of X):]
[:REAL,(BoundedFunctions the carrier of X):] is non empty set
[:[:REAL,(BoundedFunctions the carrier of X):],(BoundedFunctions the carrier of X):] is non empty set
bool [:[:REAL,(BoundedFunctions the carrier of X):],(BoundedFunctions the carrier of X):] is non empty set
One_ ((BoundedFunctions the carrier of X),(RAlgebra the carrier of X)) is Element of BoundedFunctions the carrier of X
Zero_ ((BoundedFunctions the carrier of X),(RAlgebra the carrier of X)) is Element of BoundedFunctions the carrier of X
Normed_AlgebraStr(# (BoundedFunctions the carrier of X),(mult_ ((BoundedFunctions the carrier of X),(RAlgebra the carrier of X))),(Add_ ((BoundedFunctions the carrier of X),(RAlgebra the carrier of X))),(Mult_ ((BoundedFunctions the carrier of X),(RAlgebra the carrier of X))),(One_ ((BoundedFunctions the carrier of X),(RAlgebra the carrier of X))),(Zero_ ((BoundedFunctions the carrier of X),(RAlgebra the carrier of X))),(BoundedFunctionsNorm the carrier of X) #) is strict Normed_AlgebraStr
the carrier of (R_Normed_Algebra_of_BoundedFunctions the carrier of X) is non empty set
X is right_complementable Element of the carrier of (X)
F is right_complementable Element of the carrier of (X)
X - F is right_complementable Element of the carrier of (X)
- F is right_complementable Element of the carrier of (X)
X + (- F) is right_complementable 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 right_complementable 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 right_complementable Element of the carrier of (R_Normed_Algebra_of_BoundedFunctions the carrier of X)
FB is right_complementable Element of the carrier of (R_Normed_Algebra_of_BoundedFunctions the carrier of X)
G - FB is right_complementable Element of the carrier of (R_Normed_Algebra_of_BoundedFunctions the carrier of X)
- FB is right_complementable Element of the carrier of (R_Normed_Algebra_of_BoundedFunctions the carrier of X)
G + (- FB) is right_complementable Element of the carrier of (R_Normed_Algebra_of_BoundedFunctions the carrier of X)
the addF of (R_Normed_Algebra_of_BoundedFunctions the carrier of X) is non empty Relation-like [: the carrier of (R_Normed_Algebra_of_BoundedFunctions the carrier of X), the carrier of (R_Normed_Algebra_of_BoundedFunctions the carrier of X):] -defined the carrier of (R_Normed_Algebra_of_BoundedFunctions the carrier of X) -valued Function-like total quasi_total Element of bool [:[: the carrier of (R_Normed_Algebra_of_BoundedFunctions the carrier of X), the carrier of (R_Normed_Algebra_of_BoundedFunctions the carrier of X):], the carrier of (R_Normed_Algebra_of_BoundedFunctions the carrier of X):]
[: the carrier of (R_Normed_Algebra_of_BoundedFunctions the carrier of X), the carrier of (R_Normed_Algebra_of_BoundedFunctions the carrier of X):] is non empty set
[:[: the carrier of (R_Normed_Algebra_of_BoundedFunctions the carrier of X), the carrier of (R_Normed_Algebra_of_BoundedFunctions the carrier of X):], the carrier of (R_Normed_Algebra_of_BoundedFunctions the carrier of X):] is non empty set
bool [:[: the carrier of (R_Normed_Algebra_of_BoundedFunctions the carrier of X), the carrier of (R_Normed_Algebra_of_BoundedFunctions the carrier of X):], the carrier of (R_Normed_Algebra_of_BoundedFunctions the carrier of X):] is non empty set
the addF of (R_Normed_Algebra_of_BoundedFunctions the carrier of X) . (G,(- FB)) is right_complementable Element of the carrier of (R_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 (R_Normed_Algebra_of_BoundedFunctions the carrier of X) . [G,(- FB)] is set
(- 1) * F is right_complementable Element of the carrier of (X)
the Mult of (X) is non empty Relation-like [:REAL, the carrier of (X):] -defined the carrier of (X) -valued Function-like total quasi_total Element of bool [:[:REAL, the carrier of (X):], the carrier of (X):]
[:REAL, the carrier of (X):] is non empty set
[:[:REAL, the carrier of (X):], the carrier of (X):] is non empty set
bool [:[:REAL, the carrier of (X):], the carrier of (X):] is non empty set
the Mult of (X) . ((- 1),F) is set
[(- 1),F] is set
{(- 1),F} is non empty set
{(- 1)} is non empty V185() V186() V187() set
{{(- 1),F},{(- 1)}} is non empty set
the Mult of (X) . [(- 1),F] is set
(- 1) * FB is right_complementable Element of the carrier of (R_Normed_Algebra_of_BoundedFunctions the carrier of X)
the Mult of (R_Normed_Algebra_of_BoundedFunctions the carrier of X) is non empty Relation-like [:REAL, the carrier of (R_Normed_Algebra_of_BoundedFunctions the carrier of X):] -defined the carrier of (R_Normed_Algebra_of_BoundedFunctions the carrier of X) -valued Function-like total quasi_total Element of bool [:[:REAL, the carrier of (R_Normed_Algebra_of_BoundedFunctions the carrier of X):], the carrier of (R_Normed_Algebra_of_BoundedFunctions the carrier of X):]
[:REAL, the carrier of (R_Normed_Algebra_of_BoundedFunctions the carrier of X):] is non empty set
[:[:REAL, the carrier of (R_Normed_Algebra_of_BoundedFunctions the carrier of X):], the carrier of (R_Normed_Algebra_of_BoundedFunctions the carrier of X):] is non empty set
bool [:[:REAL, the carrier of (R_Normed_Algebra_of_BoundedFunctions the carrier of X):], the carrier of (R_Normed_Algebra_of_BoundedFunctions the carrier of X):] is non empty set
the Mult of (R_Normed_Algebra_of_BoundedFunctions the carrier of X) . ((- 1),FB) is set
[(- 1),FB] is set
{(- 1),FB} is non empty set
{{(- 1),FB},{(- 1)}} is non empty set
the Mult of (R_Normed_Algebra_of_BoundedFunctions the carrier of X) . [(- 1),FB] is set
X is non empty TopSpace-like pseudocompact compact TopStruct
(X) is non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital discerning reflexive RealNormSpace-like unital associative commutative right-distributive right_unital well-unital left_unital vector-associative strict Normed_AlgebraStr
(X) is non empty add-closed having-inverse additively-closed multiplicatively-closed additively-linearly-closed Element of bool the carrier of (RAlgebra the carrier of X)
the carrier of X is non empty set
RAlgebra the carrier of X is non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative unital associative commutative right-distributive right_unital well-unital left_unital strict vector-associative AlgebraStr
Funcs ( the carrier of X,REAL) is non empty functional FUNCTION_DOMAIN of the carrier of X, REAL
RealFuncMult the carrier of X is non empty Relation-like [:(Funcs ( the carrier of X,REAL)),(Funcs ( the carrier of X,REAL)):] -defined Funcs ( the carrier of X,REAL) -valued Function-like total quasi_total Element of bool [:[:(Funcs ( the carrier of X,REAL)),(Funcs ( the carrier of X,REAL)):],(Funcs ( the carrier of X,REAL)):]
[:(Funcs ( the carrier of X,REAL)),(Funcs ( the carrier of X,REAL)):] is non empty set
[:[:(Funcs ( the carrier of X,REAL)),(Funcs ( the carrier of X,REAL)):],(Funcs ( the carrier of X,REAL)):] is non empty set
bool [:[:(Funcs ( the carrier of X,REAL)),(Funcs ( the carrier of X,REAL)):],(Funcs ( the carrier of X,REAL)):] is non empty set
RealFuncAdd the carrier of X is non empty Relation-like [:(Funcs ( the carrier of X,REAL)),(Funcs ( the carrier of X,REAL)):] -defined Funcs ( the carrier of X,REAL) -valued Function-like total quasi_total Element of bool [:[:(Funcs ( the carrier of X,REAL)),(Funcs ( the carrier of X,REAL)):],(Funcs ( the carrier of X,REAL)):]
RealFuncExtMult the carrier of X is non empty Relation-like [:REAL,(Funcs ( the carrier of X,REAL)):] -defined Funcs ( the carrier of X,REAL) -valued Function-like total quasi_total Element of bool [:[:REAL,(Funcs ( the carrier of X,REAL)):],(Funcs ( the carrier of X,REAL)):]
[:REAL,(Funcs ( the carrier of X,REAL)):] is non empty set
[:[:REAL,(Funcs ( the carrier of X,REAL)):],(Funcs ( the carrier of X,REAL)):] is non empty set
bool [:[:REAL,(Funcs ( the carrier of X,REAL)):],(Funcs ( the carrier of X,REAL)):] is non empty set
RealFuncUnit the carrier of X is Relation-like the carrier of X -defined REAL -valued Function-like total quasi_total V152() V153() V154() Element of Funcs ( the carrier of X,REAL)
the carrier of X --> 1 is non empty Relation-like the carrier of X -defined NAT -valued RAT -valued INT -valued Function-like total quasi_total V152() V153() V154() V155() Element of bool [: the carrier of X,NAT:]
[: the carrier of X,NAT:] is non empty RAT -valued INT -valued V152() V153() V154() V155() set
bool [: the carrier of X,NAT:] is non empty set
RealFuncZero the carrier of X is Relation-like the carrier of X -defined REAL -valued Function-like total quasi_total V152() V153() V154() Element of Funcs ( the carrier of X,REAL)
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 V152() V153() V154() V155() Element of bool [: the carrier of X,NAT:]
AlgebraStr(# (Funcs ( the carrier of X,REAL)),(RealFuncMult the carrier of X),(RealFuncAdd the carrier of X),(RealFuncExtMult the carrier of X),(RealFuncUnit the carrier of X),(RealFuncZero the carrier of X) #) is strict AlgebraStr
the carrier of (RAlgebra the carrier of X) is non empty set
bool the carrier of (RAlgebra the carrier of X) is non empty set
[: the carrier of X,REAL:] is non empty V152() V153() V154() set
bool [: the carrier of X,REAL:] is non empty set
{ b₁ where b₁ is non empty Relation-like the carrier of X -defined REAL -valued Function-like total quasi_total V152() V153() V154() continuous Element of bool [: the carrier of X,REAL:] : verum } is set
mult_ ((X),(RAlgebra 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),(RAlgebra 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),(RAlgebra the carrier of X)) is non empty Relation-like [:REAL,(X):] -defined (X) -valued Function-like total quasi_total Element of bool [:[:REAL,(X):],(X):]
[:REAL,(X):] is non empty set
[:[:REAL,(X):],(X):] is non empty set
bool [:[:REAL,(X):],(X):] is non empty set
One_ ((X),(RAlgebra the carrier of X)) is Element of (X)
Zero_ ((X),(RAlgebra the carrier of X)) is Element of (X)
(X) is non empty Relation-like (X) -defined REAL -valued Function-like total quasi_total V152() V153() V154() Element of bool [:(X),REAL:]
[:(X),REAL:] is non empty V152() V153() V154() set
bool [:(X),REAL:] is non empty set
BoundedFunctions the carrier of X is non empty add-closed having-inverse additively-closed multiplicatively-closed additively-linearly-closed Element of bool the carrier of (RAlgebra the carrier of X)
{ b₁ where b₁ is non empty Relation-like the carrier of X -defined REAL -valued Function-like total quasi_total V152() V153() V154() Element of bool [: the carrier of X,REAL:] : b₁ | the carrier of X is bounded } is set
BoundedFunctionsNorm the carrier of X is non empty Relation-like BoundedFunctions the carrier of X -defined REAL -valued Function-like total quasi_total V152() V153() V154() Element of bool [:(BoundedFunctions the carrier of X),REAL:]
[:(BoundedFunctions the carrier of X),REAL:] is non empty V152() V153() V154() set
bool [:(BoundedFunctions the carrier of X),REAL:] is non empty set
(BoundedFunctionsNorm the carrier of X) | (X) is Relation-like BoundedFunctions the carrier of X -defined REAL -valued Function-like V152() V153() V154() Element of bool [:(BoundedFunctions the carrier of X),REAL:]
Normed_AlgebraStr(# (X),(mult_ ((X),(RAlgebra the carrier of X))),(Add_ ((X),(RAlgebra the carrier of X))),(Mult_ ((X),(RAlgebra the carrier of X))),(One_ ((X),(RAlgebra the carrier of X))),(Zero_ ((X),(RAlgebra the carrier of X))),(X) #) is strict Normed_AlgebraStr
the carrier of (X) is non empty set
X is right_complementable Element of the carrier of (X)
F is right_complementable Element of the carrier of (X)
G is right_complementable Element of the carrier of (X)
X - F is right_complementable Element of the carrier of (X)
- F is right_complementable Element of the carrier of (X)
X + (- F) is right_complementable 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 right_complementable 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 REAL -valued Function-like total quasi_total V152() V153() V154() Element of bool [: the carrier of X,REAL:]
GB is non empty Relation-like the carrier of X -defined REAL -valued Function-like total quasi_total V152() V153() V154() Element of bool [: the carrier of X,REAL:]
aFB is non empty Relation-like the carrier of X -defined REAL -valued Function-like total quasi_total V152() V153() V154() Element of bool [: the carrier of X,REAL:]
G + F is right_complementable Element of the carrier of (X)
the addF of (X) . (G,F) is right_complementable 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 addF of (X) . [G,F] is set
F - F is right_complementable Element of the carrier of (X)
F + (- F) is right_complementable Element of the carrier of (X)
the addF of (X) . (F,(- F)) is right_complementable Element of the carrier of (X)
[F,(- F)] is set
{F,(- F)} is non empty set
{F} is non empty set
{{F,(- F)},{F}} is non empty set
the addF of (X) . [F,(- F)] is set
X - (F - F) is right_complementable Element of the carrier of (X)
- (F - F) is right_complementable Element of the carrier of (X)
X + (- (F - F)) is right_complementable Element of the carrier of (X)
the addF of (X) . (X,(- (F - F))) is right_complementable Element of the carrier of (X)
[X,(- (F - F))] is set
{X,(- (F - F))} is non empty set
{{X,(- (F - F))},{X}} is non empty set
the addF of (X) . [X,(- (F - F))] is set
0. (X) is V49((X)) right_complementable Element of the carrier of (X)
the ZeroF of (X) is right_complementable Element of the carrier of (X)
X - (0. (X)) is right_complementable Element of the carrier of (X)
- (0. (X)) is right_complementable Element of the carrier of (X)
X + (- (0. (X))) is right_complementable Element of the carrier of (X)
the addF of (X) . (X,(- (0. (X)))) is right_complementable Element of the carrier of (X)
[X,(- (0. (X)))] is set
{X,(- (0. (X)))} is non empty set
{{X,(- (0. (X)))},{X}} is non empty set
the addF of (X) . [X,(- (0. (X)))] is set
aF is Element of the carrier of X
FB . aF is V11() real ext-real Element of REAL
aFB . aF is V11() real ext-real Element of REAL
GB . aF is V11() real ext-real Element of REAL
(aFB . aF) + (GB . aF) is V11() real ext-real Element of REAL
(FB . aF) - (GB . aF) is V11() real ext-real Element of REAL
- (GB . aF) is V11() real ext-real set
(FB . aF) + (- (GB . aF)) is V11() real ext-real set
aF is Element of the carrier of X
aFB . aF is V11() real ext-real Element of REAL
FB . aF is V11() real ext-real Element of REAL
GB . aF is V11() real ext-real Element of REAL
(FB . aF) - (GB . aF) is V11() real ext-real Element of REAL
- (GB . aF) is V11() real ext-real set
(FB . aF) + (- (GB . aF)) is V11() real ext-real set
aF is Element of the carrier of X
aFB . aF is V11() real ext-real Element of REAL
FB . aF is V11() real ext-real Element of REAL
GB . aF is V11() real ext-real Element of REAL
(FB . aF) - (GB . aF) is V11() real ext-real Element of REAL
- (GB . aF) is V11() real ext-real set
(FB . aF) + (- (GB . aF)) is V11() real ext-real set
(aFB . aF) + (GB . aF) is V11() real ext-real Element of REAL
G + (F - F) is right_complementable Element of the carrier of (X)
the addF of (X) . (G,(F - F)) is right_complementable Element of the carrier of (X)
[G,(F - F)] is set
{G,(F - F)} is non empty set
{{G,(F - F)},{G}} is non empty set
the addF of (X) . [G,(F - F)] is set
G + (0. (X)) is right_complementable Element of the carrier of (X)
the addF of (X) . (G,(0. (X))) is right_complementable Element of the carrier of (X)
[G,(0. (X))] is set
{G,(0. (X))} is non empty set
{{G,(0. (X))},{G}} is non empty set
the addF of (X) . [G,(0. (X))] is set
aF is Element of the carrier of X
aFB . aF is V11() real ext-real Element of REAL
FB . aF is V11() real ext-real Element of REAL
GB . aF is V11() real ext-real Element of REAL
(FB . aF) - (GB . aF) is V11() real ext-real Element of REAL
- (GB . aF) is V11() real ext-real set
(FB . aF) + (- (GB . aF)) is V11() real ext-real set
aG is Element of the carrier of X
aFB . aG is V11() real ext-real Element of REAL
FB . aG is V11() real ext-real Element of REAL
GB . aG is V11() real ext-real Element of REAL
(FB . aG) - (GB . aG) is V11() real ext-real Element of REAL
- (GB . aG) is V11() real ext-real set
(FB . aG) + (- (GB . aG)) is V11() real ext-real set
X is non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital discerning reflexive RealNormSpace-like complete NORMSTR
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 right_complementable Element of the carrier of X
X is non empty TopSpace-like pseudocompact compact TopStruct
the carrier of X is non empty set
R_Normed_Algebra_of_BoundedFunctions the carrier of X is non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital discerning reflexive RealNormSpace-like unital complete Normed_AlgebraStr
BoundedFunctions the carrier of X is non empty add-closed having-inverse additively-closed multiplicatively-closed additively-linearly-closed Element of bool the carrier of (RAlgebra the carrier of X)
RAlgebra the carrier of X is non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative unital associative commutative right-distributive right_unital well-unital left_unital strict vector-associative AlgebraStr
Funcs ( the carrier of X,REAL) is non empty functional FUNCTION_DOMAIN of the carrier of X, REAL
RealFuncMult the carrier of X is non empty Relation-like [:(Funcs ( the carrier of X,REAL)),(Funcs ( the carrier of X,REAL)):] -defined Funcs ( the carrier of X,REAL) -valued Function-like total quasi_total Element of bool [:[:(Funcs ( the carrier of X,REAL)),(Funcs ( the carrier of X,REAL)):],(Funcs ( the carrier of X,REAL)):]
[:(Funcs ( the carrier of X,REAL)),(Funcs ( the carrier of X,REAL)):] is non empty set
[:[:(Funcs ( the carrier of X,REAL)),(Funcs ( the carrier of X,REAL)):],(Funcs ( the carrier of X,REAL)):] is non empty set
bool [:[:(Funcs ( the carrier of X,REAL)),(Funcs ( the carrier of X,REAL)):],(Funcs ( the carrier of X,REAL)):] is non empty set
RealFuncAdd the carrier of X is non empty Relation-like [:(Funcs ( the carrier of X,REAL)),(Funcs ( the carrier of X,REAL)):] -defined Funcs ( the carrier of X,REAL) -valued Function-like total quasi_total Element of bool [:[:(Funcs ( the carrier of X,REAL)),(Funcs ( the carrier of X,REAL)):],(Funcs ( the carrier of X,REAL)):]
RealFuncExtMult the carrier of X is non empty Relation-like [:REAL,(Funcs ( the carrier of X,REAL)):] -defined Funcs ( the carrier of X,REAL) -valued Function-like total quasi_total Element of bool [:[:REAL,(Funcs ( the carrier of X,REAL)):],(Funcs ( the carrier of X,REAL)):]
[:REAL,(Funcs ( the carrier of X,REAL)):] is non empty set
[:[:REAL,(Funcs ( the carrier of X,REAL)):],(Funcs ( the carrier of X,REAL)):] is non empty set
bool [:[:REAL,(Funcs ( the carrier of X,REAL)):],(Funcs ( the carrier of X,REAL)):] is non empty set
RealFuncUnit the carrier of X is Relation-like the carrier of X -defined REAL -valued Function-like total quasi_total V152() V153() V154() Element of Funcs ( the carrier of X,REAL)
the carrier of X --> 1 is non empty Relation-like the carrier of X -defined NAT -valued RAT -valued INT -valued Function-like total quasi_total V152() V153() V154() V155() Element of bool [: the carrier of X,NAT:]
[: the carrier of X,NAT:] is non empty RAT -valued INT -valued V152() V153() V154() V155() set
bool [: the carrier of X,NAT:] is non empty set
RealFuncZero the carrier of X is Relation-like the carrier of X -defined REAL -valued Function-like total quasi_total V152() V153() V154() Element of Funcs ( the carrier of X,REAL)
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 V152() V153() V154() V155() Element of bool [: the carrier of X,NAT:]
AlgebraStr(# (Funcs ( the carrier of X,REAL)),(RealFuncMult the carrier of X),(RealFuncAdd the carrier of X),(RealFuncExtMult the carrier of X),(RealFuncUnit the carrier of X),(RealFuncZero the carrier of X) #) is strict AlgebraStr
the carrier of (RAlgebra the carrier of X) is non empty set
bool the carrier of (RAlgebra the carrier of X) is non empty set
[: the carrier of X,REAL:] is non empty V152() V153() V154() set
bool [: the carrier of X,REAL:] is non empty set
{ b₁ where b₁ is non empty Relation-like the carrier of X -defined REAL -valued Function-like total quasi_total V152() V153() V154() Element of bool [: the carrier of X,REAL:] : b₁ | the carrier of X is bounded } is set
mult_ ((BoundedFunctions the carrier of X),(RAlgebra the carrier of X)) is non empty Relation-like [:(BoundedFunctions the carrier of X),(BoundedFunctions the carrier of X):] -defined BoundedFunctions the carrier of X -valued Function-like total quasi_total Element of bool [:[:(BoundedFunctions the carrier of X),(BoundedFunctions the carrier of X):],(BoundedFunctions the carrier of X):]
[:(BoundedFunctions the carrier of X),(BoundedFunctions the carrier of X):] is non empty set
[:[:(BoundedFunctions the carrier of X),(BoundedFunctions the carrier of X):],(BoundedFunctions the carrier of X):] is non empty set
bool [:[:(BoundedFunctions the carrier of X),(BoundedFunctions the carrier of X):],(BoundedFunctions the carrier of X):] is non empty set
Add_ ((BoundedFunctions the carrier of X),(RAlgebra the carrier of X)) is non empty Relation-like [:(BoundedFunctions the carrier of X),(BoundedFunctions the carrier of X):] -defined BoundedFunctions the carrier of X -valued Function-like total quasi_total Element of bool [:[:(BoundedFunctions the carrier of X),(BoundedFunctions the carrier of X):],(BoundedFunctions the carrier of X):]
Mult_ ((BoundedFunctions the carrier of X),(RAlgebra the carrier of X)) is non empty Relation-like [:REAL,(BoundedFunctions the carrier of X):] -defined BoundedFunctions the carrier of X -valued Function-like total quasi_total Element of bool [:[:REAL,(BoundedFunctions the carrier of X):],(BoundedFunctions the carrier of X):]
[:REAL,(BoundedFunctions the carrier of X):] is non empty set
[:[:REAL,(BoundedFunctions the carrier of X):],(BoundedFunctions the carrier of X):] is non empty set
bool [:[:REAL,(BoundedFunctions the carrier of X):],(BoundedFunctions the carrier of X):] is non empty set
One_ ((BoundedFunctions the carrier of X),(RAlgebra the carrier of X)) is Element of BoundedFunctions the carrier of X
Zero_ ((BoundedFunctions the carrier of X),(RAlgebra the carrier of X)) is Element of BoundedFunctions the carrier of X
BoundedFunctionsNorm the carrier of X is non empty Relation-like BoundedFunctions the carrier of X -defined REAL -valued Function-like total quasi_total V152() V153() V154() Element of bool [:(BoundedFunctions the carrier of X),REAL:]
[:(BoundedFunctions the carrier of X),REAL:] is non empty V152() V153() V154() set
bool [:(BoundedFunctions the carrier of X),REAL:] is non empty set
Normed_AlgebraStr(# (BoundedFunctions the carrier of X),(mult_ ((BoundedFunctions the carrier of X),(RAlgebra the carrier of X))),(Add_ ((BoundedFunctions the carrier of X),(RAlgebra the carrier of X))),(Mult_ ((BoundedFunctions the carrier of X),(RAlgebra the carrier of X))),(One_ ((BoundedFunctions the carrier of X),(RAlgebra the carrier of X))),(Zero_ ((BoundedFunctions the carrier of X),(RAlgebra the carrier of X))),(BoundedFunctionsNorm the carrier of X) #) is strict Normed_AlgebraStr
the carrier of (R_Normed_Algebra_of_BoundedFunctions the carrier of X) is non empty set
bool the carrier of (R_Normed_Algebra_of_BoundedFunctions the carrier of X) is non empty set
(X) is non empty add-closed having-inverse additively-closed multiplicatively-closed additively-linearly-closed Element of bool the carrier of (RAlgebra the carrier of X)
{ b₁ where b₁ is non empty Relation-like the carrier of X -defined REAL -valued Function-like total quasi_total V152() V153() V154() continuous Element of bool [: the carrier of X,REAL:] : verum } is set
X is Element of bool the carrier of (R_Normed_Algebra_of_BoundedFunctions the carrier of X)
[:NAT, the carrier of (R_Normed_Algebra_of_BoundedFunctions the carrier of X):] is non empty set
bool [:NAT, the carrier of (R_Normed_Algebra_of_BoundedFunctions the carrier of X):] is non empty set
F is non empty Relation-like NAT -defined the carrier of (R_Normed_Algebra_of_BoundedFunctions the carrier of X) -valued Function-like total quasi_total Element of bool [:NAT, the carrier of (R_Normed_Algebra_of_BoundedFunctions the carrier of X):]
rng F is Element of bool the carrier of (R_Normed_Algebra_of_BoundedFunctions the carrier of X)
lim F is right_complementable Element of the carrier of (R_Normed_Algebra_of_BoundedFunctions the carrier of X)
G is non empty Relation-like the carrier of X -defined REAL -valued Function-like total quasi_total V152() V153() V154() Element of bool [: the carrier of X,REAL:]
G | the carrier of X is Relation-like the carrier of X -defined REAL -valued Function-like V152() V153() V154() Element of bool [: the carrier of X,REAL:]
FB is set
PFuncs ( the carrier of X,REAL) is set
GB is non empty Relation-like the carrier of X -defined REAL -valued Function-like total quasi_total V152() V153() V154() Element of bool [: the carrier of X,REAL:]
GB | the carrier of X is Relation-like the carrier of X -defined REAL -valued Function-like V152() V153() V154() Element of bool [: the carrier of X,REAL:]
[:NAT,(PFuncs ( the carrier of X,REAL)):] is set
bool [:NAT,(PFuncs ( the carrier of X,REAL)):] is non empty set
FB is Relation-like NAT -defined PFuncs ( the carrier of X,REAL) -valued Function-like quasi_total Element of bool [:NAT,(PFuncs ( the carrier of X,REAL)):]
GB is V11() real ext-real Element of REAL
aFB is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real V117() V185() V186() V187() V188() V189() V190() V197() bounded_below Element of NAT
aF is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real V117() V185() V186() V187() V188() V189() V190() V197() bounded_below Element of NAT
aG is set
F . aF is right_complementable Element of the carrier of (R_Normed_Algebra_of_BoundedFunctions the carrier of X)
(F . aF) - (lim F) is right_complementable Element of the carrier of (R_Normed_Algebra_of_BoundedFunctions the carrier of X)
- (lim F) is right_complementable Element of the carrier of (R_Normed_Algebra_of_BoundedFunctions the carrier of X)
(F . aF) + (- (lim F)) is right_complementable Element of the carrier of (R_Normed_Algebra_of_BoundedFunctions the carrier of X)
the addF of (R_Normed_Algebra_of_BoundedFunctions the carrier of X) is non empty Relation-like [: the carrier of (R_Normed_Algebra_of_BoundedFunctions the carrier of X), the carrier of (R_Normed_Algebra_of_BoundedFunctions the carrier of X):] -defined the carrier of (R_Normed_Algebra_of_BoundedFunctions the carrier of X) -valued Function-like total quasi_total Element of bool [:[: the carrier of (R_Normed_Algebra_of_BoundedFunctions the carrier of X), the carrier of (R_Normed_Algebra_of_BoundedFunctions the carrier of X):], the carrier of (R_Normed_Algebra_of_BoundedFunctions the carrier of X):]
[: the carrier of (R_Normed_Algebra_of_BoundedFunctions the carrier of X), the carrier of (R_Normed_Algebra_of_BoundedFunctions the carrier of X):] is non empty set
[:[: the carrier of (R_Normed_Algebra_of_BoundedFunctions the carrier of X), the carrier of (R_Normed_Algebra_of_BoundedFunctions the carrier of X):], the carrier of (R_Normed_Algebra_of_BoundedFunctions the carrier of X):] is non empty set
bool [:[: the carrier of (R_Normed_Algebra_of_BoundedFunctions the carrier of X), the carrier of (R_Normed_Algebra_of_BoundedFunctions the carrier of X):], the carrier of (R_Normed_Algebra_of_BoundedFunctions the carrier of X):] is non empty set
the addF of (R_Normed_Algebra_of_BoundedFunctions the carrier of X) . ((F . aF),(- (lim F))) is right_complementable Element of the carrier of (R_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 (R_Normed_Algebra_of_BoundedFunctions the carrier of X) . [(F . aF),(- (lim F))] is set
||.((F . aF) - (lim F)).|| is V11() real ext-real Element of REAL
the U8 of (R_Normed_Algebra_of_BoundedFunctions the carrier of X) is non empty Relation-like the carrier of (R_Normed_Algebra_of_BoundedFunctions the carrier of X) -defined REAL -valued Function-like total quasi_total V152() V153() V154() Element of bool [: the carrier of (R_Normed_Algebra_of_BoundedFunctions the carrier of X),REAL:]
[: the carrier of (R_Normed_Algebra_of_BoundedFunctions the carrier of X),REAL:] is non empty V152() V153() V154() set
bool [: the carrier of (R_Normed_Algebra_of_BoundedFunctions the carrier of X),REAL:] is non empty set
the U8 of (R_Normed_Algebra_of_BoundedFunctions the carrier of X) . ((F . aF) - (lim F)) is V11() real ext-real Element of REAL
r is non empty Relation-like the carrier of X -defined REAL -valued Function-like total quasi_total V152() V153() V154() Element of bool [: the carrier of X,REAL:]
r | the carrier of X is Relation-like the carrier of X -defined REAL -valued Function-like V152() V153() V154() Element of bool [: the carrier of X,REAL:]
s1 is non empty Relation-like the carrier of X -defined REAL -valued Function-like total quasi_total V152() V153() V154() Element of bool [: the carrier of X,REAL:]
s1 | the carrier of X is Relation-like the carrier of X -defined REAL -valued Function-like V152() V153() V154() Element of bool [: the carrier of X,REAL:]
uu1 is Element of the carrier of X
r . uu1 is V11() real ext-real Element of REAL
s1 . uu1 is V11() real ext-real Element of REAL
G . uu1 is V11() real ext-real Element of REAL
(s1 . uu1) - (G . uu1) is V11() real ext-real Element of REAL
- (G . uu1) is V11() real ext-real set
(s1 . uu1) + (- (G . uu1)) is V11() real ext-real set
abs (r . uu1) is V11() real ext-real Element of REAL
FB . aF is Relation-like the carrier of X -defined REAL -valued Function-like V152() V153() V154() Element of bool [: the carrier of X,REAL:]
(FB . aF) . aG is V11() real ext-real set
G . aG is V11() real ext-real set
((FB . aF) . aG) - (G . aG) is V11() real ext-real set
- (G . aG) is V11() real ext-real set
((FB . aF) . aG) + (- (G . aG)) is V11() real ext-real set
abs (((FB . aF) . aG) - (G . aG)) is V11() real ext-real Element of REAL
bool the carrier of X is non empty set
GB is Element of the carrier of X
G . GB is V11() real ext-real Element of REAL
aFB is V185() V186() V187() Element of bool REAL
aG is V11() real ext-real set
(G . GB) - aG is V11() real ext-real Element of REAL
- aG is V11() real ext-real set
(G . GB) + (- aG) is V11() real ext-real set
(G . GB) + aG is V11() real ext-real Element of REAL
].((G . GB) - aG),((G . GB) + aG).[ is open V185() V186() V187() non left_end non right_end V268() Element of bool REAL
{ b₁ where b₁ is V11() real ext-real Element of REAL : ( not b₁ <= (G . GB) - aG & not (G . GB) + aG <= b₁ ) } is set
aG / 3 is V11() real ext-real Element of REAL
3 " is non empty V11() real ext-real positive non negative set
aG * (3 ") is V11() real ext-real set
r is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real V117() V185() V186() V187() V188() V189() V190() V197() bounded_below Element of NAT
FB . r is Relation-like the carrier of X -defined REAL -valued Function-like V152() V153() V154() Element of bool [: the carrier of X,REAL:]
(FB . r) . GB is V11() real ext-real set
((FB . r) . GB) - (G . GB) is V11() real ext-real Element of REAL
- (G . GB) is V11() real ext-real set
((FB . r) . GB) + (- (G . GB)) is V11() real ext-real set
abs (((FB . r) . GB) - (G . GB)) is V11() real ext-real Element of REAL
dom F is V185() V186() V187() V188() V189() V190() bounded_below Element of bool NAT
s1 is non empty Relation-like the carrier of X -defined REAL -valued Function-like total quasi_total V152() V153() V154() V206( the carrier of X) V207( the carrier of X) continuous bounded Element of bool [: the carrier of X,REAL:]
s1 . GB is V11() real ext-real Element of REAL
(s1 . GB) - (aG / 3) is V11() real ext-real Element of REAL
- (aG / 3) is V11() real ext-real set
(s1 . GB) + (- (aG / 3)) is V11() real ext-real set
(s1 . GB) + (aG / 3) is V11() real ext-real Element of REAL
].((s1 . GB) - (aG / 3)),((s1 . GB) + (aG / 3)).[ is open V185() V186() V187() non left_end non right_end V268() Element of bool REAL
{ b₁ where b₁ is V11() real ext-real Element of REAL : ( not b₁ <= (s1 . GB) - (aG / 3) & not (s1 . GB) + (aG / 3) <= b₁ ) } is set
A2 is Element of bool the carrier of X
s1 .: A2 is V185() V186() V187() Element of bool REAL
A3 is set
G .: A2 is V185() V186() V187() Element of bool REAL
dom G is Element of bool the carrier of X
vv1 is set
G . vv1 is V11() real ext-real set
s1 . vv1 is V11() real ext-real set
(s1 . vv1) - (s1 . GB) is V11() real ext-real Element of REAL
- (s1 . GB) is V11() real ext-real set
(s1 . vv1) + (- (s1 . GB)) is V11() real ext-real set
((s1 . GB) - (aG / 3)) - (s1 . GB) is V11() real ext-real Element of REAL
((s1 . GB) - (aG / 3)) + (- (s1 . GB)) is V11() real ext-real set
((s1 . GB) + (aG / 3)) - (s1 . GB) is V11() real ext-real Element of REAL
((s1 . GB) + (aG / 3)) + (- (s1 . GB)) is V11() real ext-real set
abs ((s1 . vv1) - (s1 . GB)) is V11() real ext-real Element of REAL
(s1 . GB) - (G . GB) is V11() real ext-real Element of REAL
(s1 . GB) + (- (G . GB)) is V11() real ext-real set
- ((s1 . GB) - (G . GB)) is V11() real ext-real Element of REAL
abs (- ((s1 . GB) - (G . GB))) is V11() real ext-real Element of REAL
(s1 . vv1) - (G . vv1) is V11() real ext-real set
- (G . vv1) is V11() real ext-real set
(s1 . vv1) + (- (G . vv1)) is V11() real ext-real set
abs ((s1 . vv1) - (G . vv1)) is V11() real ext-real Element of REAL
- ((s1 . vv1) - (G . vv1)) is V11() real ext-real set
abs (- ((s1 . vv1) - (G . vv1))) is V11() real ext-real Element of REAL
(aG / 3) + (aG / 3) is V11() real ext-real Element of REAL
(G . vv1) - (s1 . vv1) is V11() real ext-real set
- (s1 . vv1) is V11() real ext-real set
(G . vv1) + (- (s1 . vv1)) is V11() real ext-real set
abs ((G . vv1) - (s1 . vv1)) is V11() real ext-real Element of REAL
(G . GB) - (s1 . GB) is V11() real ext-real Element of REAL
(G . GB) + (- (s1 . GB)) is V11() real ext-real set
abs ((G . GB) - (s1 . GB)) is V11() real ext-real Element of REAL
(abs ((G . vv1) - (s1 . vv1))) + (abs ((G . GB) - (s1 . GB))) is V11() real ext-real Element of REAL
((aG / 3) + (aG / 3)) + (aG / 3) is V11() real ext-real Element of REAL
((abs ((G . vv1) - (s1 . vv1))) + (abs ((G . GB) - (s1 . GB)))) + (abs ((s1 . vv1) - (s1 . GB))) is V11() real ext-real Element of REAL
(G . vv1) - (G . GB) is V11() real ext-real Element of REAL
(G . vv1) + (- (G . GB)) is V11() real ext-real set
abs ((G . vv1) - (G . GB)) is V11() real ext-real Element of REAL
((G . vv1) - (s1 . vv1)) - ((G . GB) - (s1 . GB)) is V11() real ext-real Element of REAL
- ((G . GB) - (s1 . GB)) is V11() real ext-real set
((G . vv1) - (s1 . vv1)) + (- ((G . GB) - (s1 . GB))) is V11() real ext-real set
(((G . vv1) - (s1 . vv1)) - ((G . GB) - (s1 . GB))) + ((s1 . vv1) - (s1 . GB)) is V11() real ext-real Element of REAL
abs ((((G . vv1) - (s1 . vv1)) - ((G . GB) - (s1 . GB))) + ((s1 . vv1) - (s1 . GB))) is V11() real ext-real Element of REAL
abs (((G . vv1) - (s1 . vv1)) - ((G . GB) - (s1 . GB))) is V11() real ext-real Element of REAL
(abs (((G . vv1) - (s1 . vv1)) - ((G . GB) - (s1 . GB)))) + (abs ((s1 . vv1) - (s1 . GB))) is V11() real ext-real Element of REAL
((G . vv1) - (G . GB)) + (G . GB) is V11() real ext-real Element of REAL
(- aG) + (G . GB) is V11() real ext-real Element of REAL
aG + (G . GB) is V11() real ext-real Element of REAL
X is non empty TopSpace-like pseudocompact compact TopStruct
(X) is non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital discerning reflexive RealNormSpace-like unital associative commutative right-distributive right_unital well-unital left_unital vector-associative strict Normed_AlgebraStr
(X) is non empty add-closed having-inverse additively-closed multiplicatively-closed additively-linearly-closed Element of bool the carrier of (RAlgebra the carrier of X)
the carrier of X is non empty set
RAlgebra the carrier of X is non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative unital associative commutative right-distributive right_unital well-unital left_unital strict vector-associative AlgebraStr
Funcs ( the carrier of X,REAL) is non empty functional FUNCTION_DOMAIN of the carrier of X, REAL
RealFuncMult the carrier of X is non empty Relation-like [:(Funcs ( the carrier of X,REAL)),(Funcs ( the carrier of X,REAL)):] -defined Funcs ( the carrier of X,REAL) -valued Function-like total quasi_total Element of bool [:[:(Funcs ( the carrier of X,REAL)),(Funcs ( the carrier of X,REAL)):],(Funcs ( the carrier of X,REAL)):]
[:(Funcs ( the carrier of X,REAL)),(Funcs ( the carrier of X,REAL)):] is non empty set
[:[:(Funcs ( the carrier of X,REAL)),(Funcs ( the carrier of X,REAL)):],(Funcs ( the carrier of X,REAL)):] is non empty set
bool [:[:(Funcs ( the carrier of X,REAL)),(Funcs ( the carrier of X,REAL)):],(Funcs ( the carrier of X,REAL)):] is non empty set
RealFuncAdd the carrier of X is non empty Relation-like [:(Funcs ( the carrier of X,REAL)),(Funcs ( the carrier of X,REAL)):] -defined Funcs ( the carrier of X,REAL) -valued Function-like total quasi_total Element of bool [:[:(Funcs ( the carrier of X,REAL)),(Funcs ( the carrier of X,REAL)):],(Funcs ( the carrier of X,REAL)):]
RealFuncExtMult the carrier of X is non empty Relation-like [:REAL,(Funcs ( the carrier of X,REAL)):] -defined Funcs ( the carrier of X,REAL) -valued Function-like total quasi_total Element of bool [:[:REAL,(Funcs ( the carrier of X,REAL)):],(Funcs ( the carrier of X,REAL)):]
[:REAL,(Funcs ( the carrier of X,REAL)):] is non empty set
[:[:REAL,(Funcs ( the carrier of X,REAL)):],(Funcs ( the carrier of X,REAL)):] is non empty set
bool [:[:REAL,(Funcs ( the carrier of X,REAL)):],(Funcs ( the carrier of X,REAL)):] is non empty set
RealFuncUnit the carrier of X is Relation-like the carrier of X -defined REAL -valued Function-like total quasi_total V152() V153() V154() Element of Funcs ( the carrier of X,REAL)
the carrier of X --> 1 is non empty Relation-like the carrier of X -defined NAT -valued RAT -valued INT -valued Function-like total quasi_total V152() V153() V154() V155() Element of bool [: the carrier of X,NAT:]
[: the carrier of X,NAT:] is non empty RAT -valued INT -valued V152() V153() V154() V155() set
bool [: the carrier of X,NAT:] is non empty set
RealFuncZero the carrier of X is Relation-like the carrier of X -defined REAL -valued Function-like total quasi_total V152() V153() V154() Element of Funcs ( the carrier of X,REAL)
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 V152() V153() V154() V155() Element of bool [: the carrier of X,NAT:]
AlgebraStr(# (Funcs ( the carrier of X,REAL)),(RealFuncMult the carrier of X),(RealFuncAdd the carrier of X),(RealFuncExtMult the carrier of X),(RealFuncUnit the carrier of X),(RealFuncZero the carrier of X) #) is strict AlgebraStr
the carrier of (RAlgebra the carrier of X) is non empty set
bool the carrier of (RAlgebra the carrier of X) is non empty set
[: the carrier of X,REAL:] is non empty V152() V153() V154() set
bool [: the carrier of X,REAL:] is non empty set
{ b₁ where b₁ is non empty Relation-like the carrier of X -defined REAL -valued Function-like total quasi_total V152() V153() V154() continuous Element of bool [: the carrier of X,REAL:] : verum } is set
mult_ ((X),(RAlgebra 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),(RAlgebra 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),(RAlgebra the carrier of X)) is non empty Relation-like [:REAL,(X):] -defined (X) -valued Function-like total quasi_total Element of bool [:[:REAL,(X):],(X):]
[:REAL,(X):] is non empty set
[:[:REAL,(X):],(X):] is non empty set
bool [:[:REAL,(X):],(X):] is non empty set
One_ ((X),(RAlgebra the carrier of X)) is Element of (X)
Zero_ ((X),(RAlgebra the carrier of X)) is Element of (X)
(X) is non empty Relation-like (X) -defined REAL -valued Function-like total quasi_total V152() V153() V154() Element of bool [:(X),REAL:]
[:(X),REAL:] is non empty V152() V153() V154() set
bool [:(X),REAL:] is non empty set
BoundedFunctions the carrier of X is non empty add-closed having-inverse additively-closed multiplicatively-closed additively-linearly-closed Element of bool the carrier of (RAlgebra the carrier of X)
{ b₁ where b₁ is non empty Relation-like the carrier of X -defined REAL -valued Function-like total quasi_total V152() V153() V154() Element of bool [: the carrier of X,REAL:] : b₁ | the carrier of X is bounded } is set
BoundedFunctionsNorm the carrier of X is non empty Relation-like BoundedFunctions the carrier of X -defined REAL -valued Function-like total quasi_total V152() V153() V154() Element of bool [:(BoundedFunctions the carrier of X),REAL:]
[:(BoundedFunctions the carrier of X),REAL:] is non empty V152() V153() V154() set
bool [:(BoundedFunctions the carrier of X),REAL:] is non empty set
(BoundedFunctionsNorm the carrier of X) | (X) is Relation-like BoundedFunctions the carrier of X -defined REAL -valued Function-like V152() V153() V154() Element of bool [:(BoundedFunctions the carrier of X),REAL:]
Normed_AlgebraStr(# (X),(mult_ ((X),(RAlgebra the carrier of X))),(Add_ ((X),(RAlgebra the carrier of X))),(Mult_ ((X),(RAlgebra the carrier of X))),(One_ ((X),(RAlgebra the carrier of X))),(Zero_ ((X),(RAlgebra the carrier of X))),(X) #) is strict Normed_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
R_Normed_Algebra_of_BoundedFunctions the carrier of X is non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital discerning reflexive RealNormSpace-like unital complete Normed_AlgebraStr
mult_ ((BoundedFunctions the carrier of X),(RAlgebra the carrier of X)) is non empty Relation-like [:(BoundedFunctions the carrier of X),(BoundedFunctions the carrier of X):] -defined BoundedFunctions the carrier of X -valued Function-like total quasi_total Element of bool [:[:(BoundedFunctions the carrier of X),(BoundedFunctions the carrier of X):],(BoundedFunctions the carrier of X):]
[:(BoundedFunctions the carrier of X),(BoundedFunctions the carrier of X):] is non empty set
[:[:(BoundedFunctions the carrier of X),(BoundedFunctions the carrier of X):],(BoundedFunctions the carrier of X):] is non empty set
bool [:[:(BoundedFunctions the carrier of X),(BoundedFunctions the carrier of X):],(BoundedFunctions the carrier of X):] is non empty set
Add_ ((BoundedFunctions the carrier of X),(RAlgebra the carrier of X)) is non empty Relation-like [:(BoundedFunctions the carrier of X),(BoundedFunctions the carrier of X):] -defined BoundedFunctions the carrier of X -valued Function-like total quasi_total Element of bool [:[:(BoundedFunctions the carrier of X),(BoundedFunctions the carrier of X):],(BoundedFunctions the carrier of X):]
Mult_ ((BoundedFunctions the carrier of X),(RAlgebra the carrier of X)) is non empty Relation-like [:REAL,(BoundedFunctions the carrier of X):] -defined BoundedFunctions the carrier of X -valued Function-like total quasi_total Element of bool [:[:REAL,(BoundedFunctions the carrier of X):],(BoundedFunctions the carrier of X):]
[:REAL,(BoundedFunctions the carrier of X):] is non empty set
[:[:REAL,(BoundedFunctions the carrier of X):],(BoundedFunctions the carrier of X):] is non empty set
bool [:[:REAL,(BoundedFunctions the carrier of X):],(BoundedFunctions the carrier of X):] is non empty set
One_ ((BoundedFunctions the carrier of X),(RAlgebra the carrier of X)) is Element of BoundedFunctions the carrier of X
Zero_ ((BoundedFunctions the carrier of X),(RAlgebra the carrier of X)) is Element of BoundedFunctions the carrier of X
Normed_AlgebraStr(# (BoundedFunctions the carrier of X),(mult_ ((BoundedFunctions the carrier of X),(RAlgebra the carrier of X))),(Add_ ((BoundedFunctions the carrier of X),(RAlgebra the carrier of X))),(Mult_ ((BoundedFunctions the carrier of X),(RAlgebra the carrier of X))),(One_ ((BoundedFunctions the carrier of X),(RAlgebra the carrier of X))),(Zero_ ((BoundedFunctions the carrier of X),(RAlgebra the carrier of X))),(BoundedFunctionsNorm the carrier of X) #) is strict Normed_AlgebraStr
the carrier of (R_Normed_Algebra_of_BoundedFunctions the carrier of X) is non empty set
[:NAT, the carrier of (R_Normed_Algebra_of_BoundedFunctions the carrier of X):] is non empty set
bool [:NAT, the carrier of (R_Normed_Algebra_of_BoundedFunctions the carrier of X):] is non empty set
G is V11() real ext-real Element of REAL
FB is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real V117() V185() V186() V187() V188() V189() V190() V197() bounded_below Element of NAT
GB is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real V117() V185() V186() V187() V188() V189() V190() V197() bounded_below Element of NAT
aFB is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real V117() V185() V186() V187() V188() V189() V190() V197() bounded_below Element of NAT
aF is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real V117() V185() V186() V187() V188() V189() V190() V197() bounded_below Element of NAT
X . aFB is right_complementable Element of the carrier of (X)
X . aF is right_complementable Element of the carrier of (X)
(X . aFB) - (X . aF) is right_complementable Element of the carrier of (X)
- (X . aF) is right_complementable Element of the carrier of (X)
(X . aFB) + (- (X . aF)) is right_complementable 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 right_complementable 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 V11() real ext-real Element of REAL
the U8 of (X) is non empty Relation-like the carrier of (X) -defined REAL -valued Function-like total quasi_total V152() V153() V154() Element of bool [: the carrier of (X),REAL:]
[: the carrier of (X),REAL:] is non empty V152() V153() V154() set
bool [: the carrier of (X),REAL:] is non empty set
the U8 of (X) . ((X . aFB) - (X . aF)) is V11() real ext-real Element of REAL
F is non empty Relation-like NAT -defined the carrier of (R_Normed_Algebra_of_BoundedFunctions the carrier of X) -valued Function-like total quasi_total Element of bool [:NAT, the carrier of (R_Normed_Algebra_of_BoundedFunctions the carrier of X):]
F . aFB is right_complementable Element of the carrier of (R_Normed_Algebra_of_BoundedFunctions the carrier of X)
F . aF is right_complementable Element of the carrier of (R_Normed_Algebra_of_BoundedFunctions the carrier of X)
(F . aFB) - (F . aF) is right_complementable Element of the carrier of (R_Normed_Algebra_of_BoundedFunctions the carrier of X)
- (F . aF) is right_complementable Element of the carrier of (R_Normed_Algebra_of_BoundedFunctions the carrier of X)
(F . aFB) + (- (F . aF)) is right_complementable Element of the carrier of (R_Normed_Algebra_of_BoundedFunctions the carrier of X)
the addF of (R_Normed_Algebra_of_BoundedFunctions the carrier of X) is non empty Relation-like [: the carrier of (R_Normed_Algebra_of_BoundedFunctions the carrier of X), the carrier of (R_Normed_Algebra_of_BoundedFunctions the carrier of X):] -defined the carrier of (R_Normed_Algebra_of_BoundedFunctions the carrier of X) -valued Function-like total quasi_total Element of bool [:[: the carrier of (R_Normed_Algebra_of_BoundedFunctions the carrier of X), the carrier of (R_Normed_Algebra_of_BoundedFunctions the carrier of X):], the carrier of (R_Normed_Algebra_of_BoundedFunctions the carrier of X):]
[: the carrier of (R_Normed_Algebra_of_BoundedFunctions the carrier of X), the carrier of (R_Normed_Algebra_of_BoundedFunctions the carrier of X):] is non empty set
[:[: the carrier of (R_Normed_Algebra_of_BoundedFunctions the carrier of X), the carrier of (R_Normed_Algebra_of_BoundedFunctions the carrier of X):], the carrier of (R_Normed_Algebra_of_BoundedFunctions the carrier of X):] is non empty set
bool [:[: the carrier of (R_Normed_Algebra_of_BoundedFunctions the carrier of X), the carrier of (R_Normed_Algebra_of_BoundedFunctions the carrier of X):], the carrier of (R_Normed_Algebra_of_BoundedFunctions the carrier of X):] is non empty set
the addF of (R_Normed_Algebra_of_BoundedFunctions the carrier of X) . ((F . aFB),(- (F . aF))) is right_complementable Element of the carrier of (R_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 (R_Normed_Algebra_of_BoundedFunctions the carrier of X) . [(F . aFB),(- (F . aF))] is set
||.((F . aFB) - (F . aF)).|| is V11() real ext-real Element of REAL
the U8 of (R_Normed_Algebra_of_BoundedFunctions the carrier of X) is non empty Relation-like the carrier of (R_Normed_Algebra_of_BoundedFunctions the carrier of X) -defined REAL -valued Function-like total quasi_total V152() V153() V154() Element of bool [: the carrier of (R_Normed_Algebra_of_BoundedFunctions the carrier of X),REAL:]
[: the carrier of (R_Normed_Algebra_of_BoundedFunctions the carrier of X),REAL:] is non empty V152() V153() V154() set
bool [: the carrier of (R_Normed_Algebra_of_BoundedFunctions the carrier of X),REAL:] is non empty set
the U8 of (R_Normed_Algebra_of_BoundedFunctions the carrier of X) . ((F . aFB) - (F . aF)) is V11() real ext-real Element of REAL
aFB is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real V117() V185() V186() V187() V188() V189() V190() V197() bounded_below Element of NAT
aF is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real V117() V185() V186() V187() V188() V189() V190() V197() bounded_below Element of NAT
F . aFB is right_complementable Element of the carrier of (R_Normed_Algebra_of_BoundedFunctions the carrier of X)
F . aF is right_complementable Element of the carrier of (R_Normed_Algebra_of_BoundedFunctions the carrier of X)
(F . aFB) - (F . aF) is right_complementable Element of the carrier of (R_Normed_Algebra_of_BoundedFunctions the carrier of X)
- (F . aF) is right_complementable Element of the carrier of (R_Normed_Algebra_of_BoundedFunctions the carrier of X)
(F . aFB) + (- (F . aF)) is right_complementable Element of the carrier of (R_Normed_Algebra_of_BoundedFunctions the carrier of X)
the addF of (R_Normed_Algebra_of_BoundedFunctions the carrier of X) . ((F . aFB),(- (F . aF))) is right_complementable Element of the carrier of (R_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 (R_Normed_Algebra_of_BoundedFunctions the carrier of X) . [(F . aFB),(- (F . aF))] is set
||.((F . aFB) - (F . aF)).|| is V11() real ext-real Element of REAL
the U8 of (R_Normed_Algebra_of_BoundedFunctions the carrier of X) . ((F . aFB) - (F . aF)) is V11() real ext-real Element of REAL
bool the carrier of (R_Normed_Algebra_of_BoundedFunctions the carrier of X) is non empty set
G is Element of bool the carrier of (R_Normed_Algebra_of_BoundedFunctions the carrier of X)
lim F is right_complementable Element of the carrier of (R_Normed_Algebra_of_BoundedFunctions the carrier of X)
FB is right_complementable Element of the carrier of (X)
GB is V11() real ext-real Element of REAL
aFB is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real V117() V185() V186() V187() V188() V189() V190() V197() bounded_below Element of NAT
aF is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real V117() V185() V186() V187() V188() V189() V190() V197() bounded_below Element of NAT
F . aF is right_complementable Element of the carrier of (R_Normed_Algebra_of_BoundedFunctions the carrier of X)
(F . aF) - (lim F) is right_complementable Element of the carrier of (R_Normed_Algebra_of_BoundedFunctions the carrier of X)
- (lim F) is right_complementable Element of the carrier of (R_Normed_Algebra_of_BoundedFunctions the carrier of X)
(F . aF) + (- (lim F)) is right_complementable Element of the carrier of (R_Normed_Algebra_of_BoundedFunctions the carrier of X)
the addF of (R_Normed_Algebra_of_BoundedFunctions the carrier of X) . ((F . aF),(- (lim F))) is right_complementable Element of the carrier of (R_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 (R_Normed_Algebra_of_BoundedFunctions the carrier of X) . [(F . aF),(- (lim F))] is set
||.((F . aF) - (lim F)).|| is V11() real ext-real Element of REAL
the U8 of (R_Normed_Algebra_of_BoundedFunctions the carrier of X) . ((F . aF) - (lim F)) is V11() real ext-real Element of REAL
X . aF is right_complementable Element of the carrier of (X)
(X . aF) - FB is right_complementable Element of the carrier of (X)
- FB is right_complementable Element of the carrier of (X)
(X . aF) + (- FB) is right_complementable Element of the carrier of (X)
the addF of (X) . ((X . aF),(- FB)) is right_complementable 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 V11() real ext-real Element of REAL
the U8 of (X) . ((X . aF) - FB) is V11() real ext-real Element of REAL
aF is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real V117() V185() V186() V187() V188() V189() V190() V197() bounded_below Element of NAT
X . aF is right_complementable Element of the carrier of (X)
(X . aF) - FB is right_complementable Element of the carrier of (X)
(X . aF) + (- FB) is right_complementable Element of the carrier of (X)
the addF of (X) . ((X . aF),(- FB)) is right_complementable 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 V11() real ext-real Element of REAL
the U8 of (X) . ((X . aF) - FB) is V11() real ext-real Element of REAL
X is non empty TopSpace-like pseudocompact compact TopStruct
(X) is non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital discerning reflexive RealNormSpace-like unital associative commutative right-distributive right_unital well-unital left_unital vector-associative strict Normed_AlgebraStr
(X) is non empty add-closed having-inverse additively-closed multiplicatively-closed additively-linearly-closed Element of bool the carrier of (RAlgebra the carrier of X)
the carrier of X is non empty set
RAlgebra the carrier of X is non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative unital associative commutative right-distributive right_unital well-unital left_unital strict vector-associative AlgebraStr
Funcs ( the carrier of X,REAL) is non empty functional FUNCTION_DOMAIN of the carrier of X, REAL
RealFuncMult the carrier of X is non empty Relation-like [:(Funcs ( the carrier of X,REAL)),(Funcs ( the carrier of X,REAL)):] -defined Funcs ( the carrier of X,REAL) -valued Function-like total quasi_total Element of bool [:[:(Funcs ( the carrier of X,REAL)),(Funcs ( the carrier of X,REAL)):],(Funcs ( the carrier of X,REAL)):]
[:(Funcs ( the carrier of X,REAL)),(Funcs ( the carrier of X,REAL)):] is non empty set
[:[:(Funcs ( the carrier of X,REAL)),(Funcs ( the carrier of X,REAL)):],(Funcs ( the carrier of X,REAL)):] is non empty set
bool [:[:(Funcs ( the carrier of X,REAL)),(Funcs ( the carrier of X,REAL)):],(Funcs ( the carrier of X,REAL)):] is non empty set
RealFuncAdd the carrier of X is non empty Relation-like [:(Funcs ( the carrier of X,REAL)),(Funcs ( the carrier of X,REAL)):] -defined Funcs ( the carrier of X,REAL) -valued Function-like total quasi_total Element of bool [:[:(Funcs ( the carrier of X,REAL)),(Funcs ( the carrier of X,REAL)):],(Funcs ( the carrier of X,REAL)):]
RealFuncExtMult the carrier of X is non empty Relation-like [:REAL,(Funcs ( the carrier of X,REAL)):] -defined Funcs ( the carrier of X,REAL) -valued Function-like total quasi_total Element of bool [:[:REAL,(Funcs ( the carrier of X,REAL)):],(Funcs ( the carrier of X,REAL)):]
[:REAL,(Funcs ( the carrier of X,REAL)):] is non empty set
[:[:REAL,(Funcs ( the carrier of X,REAL)):],(Funcs ( the carrier of X,REAL)):] is non empty set
bool [:[:REAL,(Funcs ( the carrier of X,REAL)):],(Funcs ( the carrier of X,REAL)):] is non empty set
RealFuncUnit the carrier of X is Relation-like the carrier of X -defined REAL -valued Function-like total quasi_total V152() V153() V154() Element of Funcs ( the carrier of X,REAL)
the carrier of X --> 1 is non empty Relation-like the carrier of X -defined NAT -valued RAT -valued INT -valued Function-like total quasi_total V152() V153() V154() V155() Element of bool [: the carrier of X,NAT:]
[: the carrier of X,NAT:] is non empty RAT -valued INT -valued V152() V153() V154() V155() set
bool [: the carrier of X,NAT:] is non empty set
RealFuncZero the carrier of X is Relation-like the carrier of X -defined REAL -valued Function-like total quasi_total V152() V153() V154() Element of Funcs ( the carrier of X,REAL)
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 V152() V153() V154() V155() Element of bool [: the carrier of X,NAT:]
AlgebraStr(# (Funcs ( the carrier of X,REAL)),(RealFuncMult the carrier of X),(RealFuncAdd the carrier of X),(RealFuncExtMult the carrier of X),(RealFuncUnit the carrier of X),(RealFuncZero the carrier of X) #) is strict AlgebraStr
the carrier of (RAlgebra the carrier of X) is non empty set
bool the carrier of (RAlgebra the carrier of X) is non empty set
[: the carrier of X,REAL:] is non empty V152() V153() V154() set
bool [: the carrier of X,REAL:] is non empty set
{ b₁ where b₁ is non empty Relation-like the carrier of X -defined REAL -valued Function-like total quasi_total V152() V153() V154() continuous Element of bool [: the carrier of X,REAL:] : verum } is set
mult_ ((X),(RAlgebra 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),(RAlgebra 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),(RAlgebra the carrier of X)) is non empty Relation-like [:REAL,(X):] -defined (X) -valued Function-like total quasi_total Element of bool [:[:REAL,(X):],(X):]
[:REAL,(X):] is non empty set
[:[:REAL,(X):],(X):] is non empty set
bool [:[:REAL,(X):],(X):] is non empty set
One_ ((X),(RAlgebra the carrier of X)) is Element of (X)
Zero_ ((X),(RAlgebra the carrier of X)) is Element of (X)
(X) is non empty Relation-like (X) -defined REAL -valued Function-like total quasi_total V152() V153() V154() Element of bool [:(X),REAL:]
[:(X),REAL:] is non empty V152() V153() V154() set
bool [:(X),REAL:] is non empty set
BoundedFunctions the carrier of X is non empty add-closed having-inverse additively-closed multiplicatively-closed additively-linearly-closed Element of bool the carrier of (RAlgebra the carrier of X)
{ b₁ where b₁ is non empty Relation-like the carrier of X -defined REAL -valued Function-like total quasi_total V152() V153() V154() Element of bool [: the carrier of X,REAL:] : b₁ | the carrier of X is bounded } is set
BoundedFunctionsNorm the carrier of X is non empty Relation-like BoundedFunctions the carrier of X -defined REAL -valued Function-like total quasi_total V152() V153() V154() Element of bool [:(BoundedFunctions the carrier of X),REAL:]
[:(BoundedFunctions the carrier of X),REAL:] is non empty V152() V153() V154() set
bool [:(BoundedFunctions the carrier of X),REAL:] is non empty set
(BoundedFunctionsNorm the carrier of X) | (X) is Relation-like BoundedFunctions the carrier of X -defined REAL -valued Function-like V152() V153() V154() Element of bool [:(BoundedFunctions the carrier of X),REAL:]
Normed_AlgebraStr(# (X),(mult_ ((X),(RAlgebra the carrier of X))),(Add_ ((X),(RAlgebra the carrier of X))),(Mult_ ((X),(RAlgebra the carrier of X))),(One_ ((X),(RAlgebra the carrier of X))),(Zero_ ((X),(RAlgebra the carrier of X))),(X) #) is strict Normed_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 pseudocompact compact TopStruct
(X) is non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital discerning reflexive RealNormSpace-like unital associative commutative right-distributive right_unital well-unital left_unital vector-associative complete strict Normed_AlgebraStr
(X) is non empty add-closed having-inverse additively-closed multiplicatively-closed additively-linearly-closed Element of bool the carrier of (RAlgebra the carrier of X)
the carrier of X is non empty set
RAlgebra the carrier of X is non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative unital associative commutative right-distributive right_unital well-unital left_unital strict vector-associative AlgebraStr
Funcs ( the carrier of X,REAL) is non empty functional FUNCTION_DOMAIN of the carrier of X, REAL
RealFuncMult the carrier of X is non empty Relation-like [:(Funcs ( the carrier of X,REAL)),(Funcs ( the carrier of X,REAL)):] -defined Funcs ( the carrier of X,REAL) -valued Function-like total quasi_total Element of bool [:[:(Funcs ( the carrier of X,REAL)),(Funcs ( the carrier of X,REAL)):],(Funcs ( the carrier of X,REAL)):]
[:(Funcs ( the carrier of X,REAL)),(Funcs ( the carrier of X,REAL)):] is non empty set
[:[:(Funcs ( the carrier of X,REAL)),(Funcs ( the carrier of X,REAL)):],(Funcs ( the carrier of X,REAL)):] is non empty set
bool [:[:(Funcs ( the carrier of X,REAL)),(Funcs ( the carrier of X,REAL)):],(Funcs ( the carrier of X,REAL)):] is non empty set
RealFuncAdd the carrier of X is non empty Relation-like [:(Funcs ( the carrier of X,REAL)),(Funcs ( the carrier of X,REAL)):] -defined Funcs ( the carrier of X,REAL) -valued Function-like total quasi_total Element of bool [:[:(Funcs ( the carrier of X,REAL)),(Funcs ( the carrier of X,REAL)):],(Funcs ( the carrier of X,REAL)):]
RealFuncExtMult the carrier of X is non empty Relation-like [:REAL,(Funcs ( the carrier of X,REAL)):] -defined Funcs ( the carrier of X,REAL) -valued Function-like total quasi_total Element of bool [:[:REAL,(Funcs ( the carrier of X,REAL)):],(Funcs ( the carrier of X,REAL)):]
[:REAL,(Funcs ( the carrier of X,REAL)):] is non empty set
[:[:REAL,(Funcs ( the carrier of X,REAL)):],(Funcs ( the carrier of X,REAL)):] is non empty set
bool [:[:REAL,(Funcs ( the carrier of X,REAL)):],(Funcs ( the carrier of X,REAL)):] is non empty set
RealFuncUnit the carrier of X is Relation-like the carrier of X -defined REAL -valued Function-like total quasi_total V152() V153() V154() Element of Funcs ( the carrier of X,REAL)
the carrier of X --> 1 is non empty Relation-like the carrier of X -defined NAT -valued RAT -valued INT -valued Function-like total quasi_total V152() V153() V154() V155() Element of bool [: the carrier of X,NAT:]
[: the carrier of X,NAT:] is non empty RAT -valued INT -valued V152() V153() V154() V155() set
bool [: the carrier of X,NAT:] is non empty set
RealFuncZero the carrier of X is Relation-like the carrier of X -defined REAL -valued Function-like total quasi_total V152() V153() V154() Element of Funcs ( the carrier of X,REAL)
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 V152() V153() V154() V155() Element of bool [: the carrier of X,NAT:]
AlgebraStr(# (Funcs ( the carrier of X,REAL)),(RealFuncMult the carrier of X),(RealFuncAdd the carrier of X),(RealFuncExtMult the carrier of X),(RealFuncUnit the carrier of X),(RealFuncZero the carrier of X) #) is strict AlgebraStr
the carrier of (RAlgebra the carrier of X) is non empty set
bool the carrier of (RAlgebra the carrier of X) is non empty set
[: the carrier of X,REAL:] is non empty V152() V153() V154() set
bool [: the carrier of X,REAL:] is non empty set
{ b₁ where b₁ is non empty Relation-like the carrier of X -defined REAL -valued Function-like total quasi_total V152() V153() V154() continuous Element of bool [: the carrier of X,REAL:] : verum } is set
mult_ ((X),(RAlgebra 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),(RAlgebra 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),(RAlgebra the carrier of X)) is non empty Relation-like [:REAL,(X):] -defined (X) -valued Function-like total quasi_total Element of bool [:[:REAL,(X):],(X):]
[:REAL,(X):] is non empty set
[:[:REAL,(X):],(X):] is non empty set
bool [:[:REAL,(X):],(X):] is non empty set
One_ ((X),(RAlgebra the carrier of X)) is Element of (X)
Zero_ ((X),(RAlgebra the carrier of X)) is Element of (X)
(X) is non empty Relation-like (X) -defined REAL -valued Function-like total quasi_total V152() V153() V154() Element of bool [:(X),REAL:]
[:(X),REAL:] is non empty V152() V153() V154() set
bool [:(X),REAL:] is non empty set
BoundedFunctions the carrier of X is non empty add-closed having-inverse additively-closed multiplicatively-closed additively-linearly-closed Element of bool the carrier of (RAlgebra the carrier of X)
{ b₁ where b₁ is non empty Relation-like the carrier of X -defined REAL -valued Function-like total quasi_total V152() V153() V154() Element of bool [: the carrier of X,REAL:] : b₁ | the carrier of X is bounded } is set
BoundedFunctionsNorm the carrier of X is non empty Relation-like BoundedFunctions the carrier of X -defined REAL -valued Function-like total quasi_total V152() V153() V154() Element of bool [:(BoundedFunctions the carrier of X),REAL:]
[:(BoundedFunctions the carrier of X),REAL:] is non empty V152() V153() V154() set
bool [:(BoundedFunctions the carrier of X),REAL:] is non empty set
(BoundedFunctionsNorm the carrier of X) | (X) is Relation-like BoundedFunctions the carrier of X -defined REAL -valued Function-like V152() V153() V154() Element of bool [:(BoundedFunctions the carrier of X),REAL:]
Normed_AlgebraStr(# (X),(mult_ ((X),(RAlgebra the carrier of X))),(Add_ ((X),(RAlgebra the carrier of X))),(Mult_ ((X),(RAlgebra the carrier of X))),(One_ ((X),(RAlgebra the carrier of X))),(Zero_ ((X),(RAlgebra the carrier of X))),(X) #) is strict Normed_AlgebraStr
the carrier of (X) is non empty set
R_Normed_Algebra_of_BoundedFunctions the carrier of X is non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital discerning reflexive RealNormSpace-like unital complete Normed_AlgebraStr
mult_ ((BoundedFunctions the carrier of X),(RAlgebra the carrier of X)) is non empty Relation-like [:(BoundedFunctions the carrier of X),(BoundedFunctions the carrier of X):] -defined BoundedFunctions the carrier of X -valued Function-like total quasi_total Element of bool [:[:(BoundedFunctions the carrier of X),(BoundedFunctions the carrier of X):],(BoundedFunctions the carrier of X):]
[:(BoundedFunctions the carrier of X),(BoundedFunctions the carrier of X):] is non empty set
[:[:(BoundedFunctions the carrier of X),(BoundedFunctions the carrier of X):],(BoundedFunctions the carrier of X):] is non empty set
bool [:[:(BoundedFunctions the carrier of X),(BoundedFunctions the carrier of X):],(BoundedFunctions the carrier of X):] is non empty set
Add_ ((BoundedFunctions the carrier of X),(RAlgebra the carrier of X)) is non empty Relation-like [:(BoundedFunctions the carrier of X),(BoundedFunctions the carrier of X):] -defined BoundedFunctions the carrier of X -valued Function-like total quasi_total Element of bool [:[:(BoundedFunctions the carrier of X),(BoundedFunctions the carrier of X):],(BoundedFunctions the carrier of X):]
Mult_ ((BoundedFunctions the carrier of X),(RAlgebra the carrier of X)) is non empty Relation-like [:REAL,(BoundedFunctions the carrier of X):] -defined BoundedFunctions the carrier of X -valued Function-like total quasi_total Element of bool [:[:REAL,(BoundedFunctions the carrier of X):],(BoundedFunctions the carrier of X):]
[:REAL,(BoundedFunctions the carrier of X):] is non empty set
[:[:REAL,(BoundedFunctions the carrier of X):],(BoundedFunctions the carrier of X):] is non empty set
bool [:[:REAL,(BoundedFunctions the carrier of X):],(BoundedFunctions the carrier of X):] is non empty set
One_ ((BoundedFunctions the carrier of X),(RAlgebra the carrier of X)) is Element of BoundedFunctions the carrier of X
Zero_ ((BoundedFunctions the carrier of X),(RAlgebra the carrier of X)) is Element of BoundedFunctions the carrier of X
Normed_AlgebraStr(# (BoundedFunctions the carrier of X),(mult_ ((BoundedFunctions the carrier of X),(RAlgebra the carrier of X))),(Add_ ((BoundedFunctions the carrier of X),(RAlgebra the carrier of X))),(Mult_ ((BoundedFunctions the carrier of X),(RAlgebra the carrier of X))),(One_ ((BoundedFunctions the carrier of X),(RAlgebra the carrier of X))),(Zero_ ((BoundedFunctions the carrier of X),(RAlgebra the carrier of X))),(BoundedFunctionsNorm the carrier of X) #) is strict Normed_AlgebraStr
the carrier of (R_Normed_Algebra_of_BoundedFunctions the carrier of X) is non empty set
F is right_complementable Element of the carrier of (X)
G is right_complementable Element of the carrier of (X)
||.F.|| is V11() real ext-real Element of REAL
the U8 of (X) is non empty Relation-like the carrier of (X) -defined REAL -valued Function-like total quasi_total V152() V153() V154() Element of bool [: the carrier of (X),REAL:]
[: the carrier of (X),REAL:] is non empty V152() V153() V154() set
bool [: the carrier of (X),REAL:] is non empty set
the U8 of (X) . F is V11() real ext-real Element of REAL
FB is right_complementable Element of the carrier of (R_Normed_Algebra_of_BoundedFunctions the carrier of X)
||.FB.|| is V11() real ext-real Element of REAL
the U8 of (R_Normed_Algebra_of_BoundedFunctions the carrier of X) is non empty Relation-like the carrier of (R_Normed_Algebra_of_BoundedFunctions the carrier of X) -defined REAL -valued Function-like total quasi_total V152() V153() V154() Element of bool [: the carrier of (R_Normed_Algebra_of_BoundedFunctions the carrier of X),REAL:]
[: the carrier of (R_Normed_Algebra_of_BoundedFunctions the carrier of X),REAL:] is non empty V152() V153() V154() set
bool [: the carrier of (R_Normed_Algebra_of_BoundedFunctions the carrier of X),REAL:] is non empty set
the U8 of (R_Normed_Algebra_of_BoundedFunctions the carrier of X) . FB is V11() real ext-real Element of REAL
||.G.|| is V11() real ext-real Element of REAL
the U8 of (X) . G is V11() real ext-real Element of REAL
GB is right_complementable Element of the carrier of (R_Normed_Algebra_of_BoundedFunctions the carrier of X)
||.GB.|| is V11() real ext-real Element of REAL
the U8 of (R_Normed_Algebra_of_BoundedFunctions the carrier of X) . GB is V11() real ext-real Element of REAL
F * G is right_complementable 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 right_complementable 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 V11() real ext-real Element of REAL
the U8 of (X) . (F * G) is V11() real ext-real Element of REAL
FB * GB is right_complementable Element of the carrier of (R_Normed_Algebra_of_BoundedFunctions the carrier of X)
the multF of (R_Normed_Algebra_of_BoundedFunctions the carrier of X) is non empty Relation-like [: the carrier of (R_Normed_Algebra_of_BoundedFunctions the carrier of X), the carrier of (R_Normed_Algebra_of_BoundedFunctions the carrier of X):] -defined the carrier of (R_Normed_Algebra_of_BoundedFunctions the carrier of X) -valued Function-like total quasi_total Element of bool [:[: the carrier of (R_Normed_Algebra_of_BoundedFunctions the carrier of X), the carrier of (R_Normed_Algebra_of_BoundedFunctions the carrier of X):], the carrier of (R_Normed_Algebra_of_BoundedFunctions the carrier of X):]
[: the carrier of (R_Normed_Algebra_of_BoundedFunctions the carrier of X), the carrier of (R_Normed_Algebra_of_BoundedFunctions the carrier of X):] is non empty set
[:[: the carrier of (R_Normed_Algebra_of_BoundedFunctions the carrier of X), the carrier of (R_Normed_Algebra_of_BoundedFunctions the carrier of X):], the carrier of (R_Normed_Algebra_of_BoundedFunctions the carrier of X):] is non empty set
bool [:[: the carrier of (R_Normed_Algebra_of_BoundedFunctions the carrier of X), the carrier of (R_Normed_Algebra_of_BoundedFunctions the carrier of X):], the carrier of (R_Normed_Algebra_of_BoundedFunctions the carrier of X):] is non empty set
the multF of (R_Normed_Algebra_of_BoundedFunctions the carrier of X) . (FB,GB) is right_complementable Element of the carrier of (R_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 (R_Normed_Algebra_of_BoundedFunctions the carrier of X) . [FB,GB] is set
||.(FB * GB).|| is V11() real ext-real Element of REAL
the U8 of (R_Normed_Algebra_of_BoundedFunctions the carrier of X) . (FB * GB) is V11() real ext-real Element of REAL
||.F.|| * ||.G.|| is V11() real ext-real Element of REAL
1. (X) is right_complementable Element of the carrier of (X)
the OneF of (X) is right_complementable Element of the carrier of (X)
||.(1. (X)).|| is V11() real ext-real Element of REAL
the U8 of (X) . (1. (X)) is V11() real ext-real Element of REAL
1. (R_Normed_Algebra_of_BoundedFunctions the carrier of X) is right_complementable Element of the carrier of (R_Normed_Algebra_of_BoundedFunctions the carrier of X)
the OneF of (R_Normed_Algebra_of_BoundedFunctions the carrier of X) is right_complementable Element of the carrier of (R_Normed_Algebra_of_BoundedFunctions the carrier of X)
||.(1. (R_Normed_Algebra_of_BoundedFunctions the carrier of X)).|| is V11() real ext-real Element of REAL
the U8 of (R_Normed_Algebra_of_BoundedFunctions the carrier of X) . (1. (R_Normed_Algebra_of_BoundedFunctions the carrier of X)) is V11() real ext-real Element of REAL
G is right_complementable Element of the carrier of (X)
FB is right_complementable Element of the carrier of (X)
aFB is right_complementable Element of the carrier of (R_Normed_Algebra_of_BoundedFunctions the carrier of X)
F is V11() real ext-real Element of REAL
F * aFB is right_complementable Element of the carrier of (R_Normed_Algebra_of_BoundedFunctions the carrier of X)
the Mult of (R_Normed_Algebra_of_BoundedFunctions the carrier of X) is non empty Relation-like [:REAL, the carrier of (R_Normed_Algebra_of_BoundedFunctions the carrier of X):] -defined the carrier of (R_Normed_Algebra_of_BoundedFunctions the carrier of X) -valued Function-like total quasi_total Element of bool [:[:REAL, the carrier of (R_Normed_Algebra_of_BoundedFunctions the carrier of X):], the carrier of (R_Normed_Algebra_of_BoundedFunctions the carrier of X):]
[:REAL, the carrier of (R_Normed_Algebra_of_BoundedFunctions the carrier of X):] is non empty set
[:[:REAL, the carrier of (R_Normed_Algebra_of_BoundedFunctions the carrier of X):], the carrier of (R_Normed_Algebra_of_BoundedFunctions the carrier of X):] is non empty set
bool [:[:REAL, the carrier of (R_Normed_Algebra_of_BoundedFunctions the carrier of X):], the carrier of (R_Normed_Algebra_of_BoundedFunctions the carrier of X):] is non empty set
the Mult of (R_Normed_Algebra_of_BoundedFunctions the carrier of X) . (F,aFB) is set
[F,aFB] is set
{F,aFB} is non empty set
{F} is non empty V185() V186() V187() set
{{F,aFB},{F}} is non empty set
the Mult of (R_Normed_Algebra_of_BoundedFunctions the carrier of X) . [F,aFB] is set
F * FB is right_complementable Element of the carrier of (X)
the Mult of (X) is non empty Relation-like [:REAL, the carrier of (X):] -defined the carrier of (X) -valued Function-like total quasi_total Element of bool [:[:REAL, the carrier of (X):], the carrier of (X):]
[:REAL, the carrier of (X):] is non empty set
[:[:REAL, the carrier of (X):], the carrier of (X):] is non empty set
bool [:[:REAL, the carrier of (X):], the carrier of (X):] is non empty set
the Mult of (X) . (F,FB) is 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 right_complementable Element of the carrier of (X)
the multF of (X) . (G,FB) is right_complementable 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 right_complementable Element of the carrier of (R_Normed_Algebra_of_BoundedFunctions the carrier of X)
GB * aFB is right_complementable Element of the carrier of (R_Normed_Algebra_of_BoundedFunctions the carrier of X)
the multF of (R_Normed_Algebra_of_BoundedFunctions the carrier of X) . (GB,aFB) is right_complementable Element of the carrier of (R_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 (R_Normed_Algebra_of_BoundedFunctions the carrier of X) . [GB,aFB] is set
F * (G * FB) is right_complementable Element of the carrier of (X)
the Mult of (X) . (F,(G * FB)) is set
[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 right_complementable Element of the carrier of (R_Normed_Algebra_of_BoundedFunctions the carrier of X)
the Mult of (R_Normed_Algebra_of_BoundedFunctions the carrier of X) . (F,(GB * aFB)) is set
[F,(GB * aFB)] is set
{F,(GB * aFB)} is non empty set
{{F,(GB * aFB)},{F}} is non empty set
the Mult of (R_Normed_Algebra_of_BoundedFunctions the carrier of X) . [F,(GB * aFB)] is set
GB * (F * aFB) is right_complementable Element of the carrier of (R_Normed_Algebra_of_BoundedFunctions the carrier of X)
the multF of (R_Normed_Algebra_of_BoundedFunctions the carrier of X) . (GB,(F * aFB)) is right_complementable Element of the carrier of (R_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 (R_Normed_Algebra_of_BoundedFunctions the carrier of X) . [GB,(F * aFB)] is set
G * (F * FB) is right_complementable Element of the carrier of (X)
the multF of (X) . (G,(F * FB)) is right_complementable 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 right_complementable Element of the carrier of (X)
G is right_complementable Element of the carrier of (X)
FB is right_complementable Element of the carrier of (X)
G + FB is right_complementable 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 right_complementable 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 right_complementable Element of the carrier of (R_Normed_Algebra_of_BoundedFunctions the carrier of X)
aF is right_complementable Element of the carrier of (R_Normed_Algebra_of_BoundedFunctions the carrier of X)
aFB + aF is right_complementable Element of the carrier of (R_Normed_Algebra_of_BoundedFunctions the carrier of X)
the addF of (R_Normed_Algebra_of_BoundedFunctions the carrier of X) is non empty Relation-like [: the carrier of (R_Normed_Algebra_of_BoundedFunctions the carrier of X), the carrier of (R_Normed_Algebra_of_BoundedFunctions the carrier of X):] -defined the carrier of (R_Normed_Algebra_of_BoundedFunctions the carrier of X) -valued Function-like total quasi_total Element of bool [:[: the carrier of (R_Normed_Algebra_of_BoundedFunctions the carrier of X), the carrier of (R_Normed_Algebra_of_BoundedFunctions the carrier of X):], the carrier of (R_Normed_Algebra_of_BoundedFunctions the carrier of X):]
the addF of (R_Normed_Algebra_of_BoundedFunctions the carrier of X) . (aFB,aF) is right_complementable Element of the carrier of (R_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 (R_Normed_Algebra_of_BoundedFunctions the carrier of X) . [aFB,aF] is set
G * F is right_complementable Element of the carrier of (X)
the multF of (X) . (G,F) is right_complementable 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 right_complementable Element of the carrier of (R_Normed_Algebra_of_BoundedFunctions the carrier of X)
aFB * GB is right_complementable Element of the carrier of (R_Normed_Algebra_of_BoundedFunctions the carrier of X)
the multF of (R_Normed_Algebra_of_BoundedFunctions the carrier of X) . (aFB,GB) is right_complementable Element of the carrier of (R_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 (R_Normed_Algebra_of_BoundedFunctions the carrier of X) . [aFB,GB] is set
FB * F is right_complementable Element of the carrier of (X)
the multF of (X) . (FB,F) is right_complementable 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 right_complementable Element of the carrier of (R_Normed_Algebra_of_BoundedFunctions the carrier of X)
the multF of (R_Normed_Algebra_of_BoundedFunctions the carrier of X) . (aF,GB) is right_complementable Element of the carrier of (R_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 (R_Normed_Algebra_of_BoundedFunctions the carrier of X) . [aF,GB] is set
(G + FB) * F is right_complementable Element of the carrier of (X)
the multF of (X) . ((G + FB),F) is right_complementable 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 right_complementable Element of the carrier of (R_Normed_Algebra_of_BoundedFunctions the carrier of X)
the multF of (R_Normed_Algebra_of_BoundedFunctions the carrier of X) . ((aFB + aF),GB) is right_complementable Element of the carrier of (R_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 (R_Normed_Algebra_of_BoundedFunctions the carrier of X) . [(aFB + aF),GB] is set
(aFB * GB) + (aF * GB) is right_complementable Element of the carrier of (R_Normed_Algebra_of_BoundedFunctions the carrier of X)
the addF of (R_Normed_Algebra_of_BoundedFunctions the carrier of X) . ((aFB * GB),(aF * GB)) is right_complementable Element of the carrier of (R_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 (R_Normed_Algebra_of_BoundedFunctions the carrier of X) . [(aFB * GB),(aF * GB)] is set
(G * F) + (FB * F) is right_complementable Element of the carrier of (X)
the addF of (X) . ((G * F),(FB * F)) is right_complementable 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,REAL:] is non empty V152() V153() V154() set
bool [: the carrier of X,REAL:] is non empty set
X is non empty Relation-like the carrier of X -defined REAL -valued Function-like total quasi_total V152() V153() V154() 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 V152() V153() V154() Element of bool [: the carrier of X,REAL:]
X + F is non empty Relation-like the carrier of X -defined REAL -valued Function-like total quasi_total V152() V153() V154() Element of bool [: the carrier of X,REAL:]
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 REAL -valued Function-like total quasi_total V152() V153() V154() Element of bool [: the carrier of X,REAL:]
dom FB is Element of bool the carrier of X
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 V11() real ext-real set
F . GB is V11() real ext-real set
(X + F) . GB is V11() real ext-real set
0 + 0 is empty epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural V11() real ext-real non positive non negative V185() V186() V187() V188() V189() V190() V191() bounded_below V268() 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,REAL:] is non empty V152() V153() V154() set
bool [: the carrier of X,REAL:] is non empty set
X is V11() real ext-real Element of REAL
F is non empty Relation-like the carrier of X -defined REAL -valued Function-like total quasi_total V152() V153() V154() Element of bool [: the carrier of X,REAL:]
X (#) F is non empty Relation-like the carrier of X -defined REAL -valued Function-like total quasi_total V152() V153() V154() Element of bool [: the carrier of X,REAL:]
support (X (#) F) is set
support F is set
GB is set
(X (#) F) . GB is V11() real ext-real set
F . GB is V11() real ext-real set
X * (F . GB) is V11() real ext-real Element of REAL
X is non empty TopSpace-like TopStruct
the carrier of X is non empty set
[: the carrier of X,REAL:] is non empty V152() V153() V154() set
bool [: the carrier of X,REAL:] is non empty set
X is non empty Relation-like the carrier of X -defined REAL -valued Function-like total quasi_total V152() V153() V154() 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 V152() V153() V154() Element of bool [: the carrier of X,REAL:]
X (#) F is non empty Relation-like the carrier of X -defined REAL -valued Function-like total quasi_total V152() V153() V154() Element of bool [: the carrier of X,REAL:]
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 REAL -valued Function-like total quasi_total V152() V153() V154() Element of bool [: the carrier of X,REAL:]
dom FB is Element of bool the carrier of X
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 V11() real ext-real set
F . GB is V11() real ext-real set
(X (#) F) . GB is V11() real ext-real set
0 * 0 is empty epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural V11() real ext-real non positive non negative V185() V186() V187() V188() V189() V190() V191() bounded_below V268() 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,REAL:] is non empty V152() V153() V154() set
bool [: the carrier of X,REAL:] 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 REAL -valued Function-like total quasi_total V152() V153() V154() Element of bool [: the carrier of X,REAL:] : ( b₁ is continuous & 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
RealVectSpace the carrier of X is non empty right_complementable strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RLSStruct
Funcs ( the carrier of X,REAL) is non empty functional FUNCTION_DOMAIN of the carrier of X, REAL
RealFuncZero the carrier of X is Relation-like the carrier of X -defined REAL -valued Function-like total quasi_total V152() V153() V154() Element of Funcs ( the carrier of X,REAL)
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 V152() V153() V154() V155() Element of bool [: the carrier of X,NAT:]
[: the carrier of X,NAT:] is non empty RAT -valued INT -valued V152() V153() V154() V155() set
bool [: the carrier of X,NAT:] is non empty set
RealFuncAdd the carrier of X is non empty Relation-like [:(Funcs ( the carrier of X,REAL)),(Funcs ( the carrier of X,REAL)):] -defined Funcs ( the carrier of X,REAL) -valued Function-like total quasi_total Element of bool [:[:(Funcs ( the carrier of X,REAL)),(Funcs ( the carrier of X,REAL)):],(Funcs ( the carrier of X,REAL)):]
[:(Funcs ( the carrier of X,REAL)),(Funcs ( the carrier of X,REAL)):] is non empty set
[:[:(Funcs ( the carrier of X,REAL)),(Funcs ( the carrier of X,REAL)):],(Funcs ( the carrier of X,REAL)):] is non empty set
bool [:[:(Funcs ( the carrier of X,REAL)),(Funcs ( the carrier of X,REAL)):],(Funcs ( the carrier of X,REAL)):] is non empty set
RealFuncExtMult the carrier of X is non empty Relation-like [:REAL,(Funcs ( the carrier of X,REAL)):] -defined Funcs ( the carrier of X,REAL) -valued Function-like total quasi_total Element of bool [:[:REAL,(Funcs ( the carrier of X,REAL)):],(Funcs ( the carrier of X,REAL)):]
[:REAL,(Funcs ( the carrier of X,REAL)):] is non empty set
[:[:REAL,(Funcs ( the carrier of X,REAL)):],(Funcs ( the carrier of X,REAL)):] is non empty set
bool [:[:REAL,(Funcs ( the carrier of X,REAL)):],(Funcs ( the carrier of X,REAL)):] is non empty set
RLSStruct(# (Funcs ( the carrier of X,REAL)),(RealFuncZero the carrier of X),(RealFuncAdd the carrier of X),(RealFuncExtMult the carrier of X) #) is strict RLSStruct
the carrier of (RealVectSpace the carrier of X) is non empty set
bool the carrier of (RealVectSpace the carrier of X) is non empty set
X is set
F is non empty Relation-like the carrier of X -defined REAL -valued Function-like total quasi_total V152() V153() V154() Element of bool [: the carrier of X,REAL:]
G is non empty Element of bool the carrier of X
support F is set
bool G is non empty set
F is non empty Relation-like the carrier of X -defined REAL -valued Function-like total quasi_total V152() V153() V154() Element of bool [: the carrier of X,REAL:]
G is V185() V186() V187() Element of bool REAL
F " G is Element of bool the carrier of X
[#] 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 V11() real ext-real non positive non negative V185() V186() V187() V188() V189() V190() V191() closed bounded_below V268() 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 V11() real ext-real set
{} X is empty proper epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural V11() real ext-real non positive non negative V185() V186() V187() V188() V189() V190() V191() closed bounded_below V268() compact 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 (RealVectSpace the carrier of X)
the carrier of X is non empty set
RealVectSpace the carrier of X is non empty right_complementable strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RLSStruct
Funcs ( the carrier of X,REAL) is non empty functional FUNCTION_DOMAIN of the carrier of X, REAL
RealFuncZero the carrier of X is Relation-like the carrier of X -defined REAL -valued Function-like total quasi_total V152() V153() V154() Element of Funcs ( the carrier of X,REAL)
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 V152() V153() V154() V155() Element of bool [: the carrier of X,NAT:]
[: the carrier of X,NAT:] is non empty RAT -valued INT -valued V152() V153() V154() V155() set
bool [: the carrier of X,NAT:] is non empty set
RealFuncAdd the carrier of X is non empty Relation-like [:(Funcs ( the carrier of X,REAL)),(Funcs ( the carrier of X,REAL)):] -defined Funcs ( the carrier of X,REAL) -valued Function-like total quasi_total Element of bool [:[:(Funcs ( the carrier of X,REAL)),(Funcs ( the carrier of X,REAL)):],(Funcs ( the carrier of X,REAL)):]
[:(Funcs ( the carrier of X,REAL)),(Funcs ( the carrier of X,REAL)):] is non empty set
[:[:(Funcs ( the carrier of X,REAL)),(Funcs ( the carrier of X,REAL)):],(Funcs ( the carrier of X,REAL)):] is non empty set
bool [:[:(Funcs ( the carrier of X,REAL)),(Funcs ( the carrier of X,REAL)):],(Funcs ( the carrier of X,REAL)):] is non empty set
RealFuncExtMult the carrier of X is non empty Relation-like [:REAL,(Funcs ( the carrier of X,REAL)):] -defined Funcs ( the carrier of X,REAL) -valued Function-like total quasi_total Element of bool [:[:REAL,(Funcs ( the carrier of X,REAL)):],(Funcs ( the carrier of X,REAL)):]
[:REAL,(Funcs ( the carrier of X,REAL)):] is non empty set
[:[:REAL,(Funcs ( the carrier of X,REAL)):],(Funcs ( the carrier of X,REAL)):] is non empty set
bool [:[:REAL,(Funcs ( the carrier of X,REAL)):],(Funcs ( the carrier of X,REAL)):] is non empty set
RLSStruct(# (Funcs ( the carrier of X,REAL)),(RealFuncZero the carrier of X),(RealFuncAdd the carrier of X),(RealFuncExtMult the carrier of X) #) is strict RLSStruct
the carrier of (RealVectSpace the carrier of X) is non empty set
bool the carrier of (RealVectSpace the carrier of X) is non empty set
[: the carrier of X,REAL:] is non empty V152() V153() V154() set
bool [: the carrier of X,REAL:] 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 REAL -valued Function-like total quasi_total V152() V153() V154() Element of bool [: the carrier of X,REAL:] : ( b₁ is continuous & 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
RAlgebra the carrier of X is non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative unital associative commutative right-distributive right_unital well-unital left_unital strict vector-associative AlgebraStr
RealFuncMult the carrier of X is non empty Relation-like [:(Funcs ( the carrier of X,REAL)),(Funcs ( the carrier of X,REAL)):] -defined Funcs ( the carrier of X,REAL) -valued Function-like total quasi_total Element of bool [:[:(Funcs ( the carrier of X,REAL)),(Funcs ( the carrier of X,REAL)):],(Funcs ( the carrier of X,REAL)):]
RealFuncUnit the carrier of X is Relation-like the carrier of X -defined REAL -valued Function-like total quasi_total V152() V153() V154() Element of Funcs ( the carrier of X,REAL)
the carrier of X --> 1 is non empty Relation-like the carrier of X -defined NAT -valued RAT -valued INT -valued Function-like total quasi_total V152() V153() V154() V155() Element of bool [: the carrier of X,NAT:]
AlgebraStr(# (Funcs ( the carrier of X,REAL)),(RealFuncMult the carrier of X),(RealFuncAdd the carrier of X),(RealFuncExtMult the carrier of X),(RealFuncUnit the carrier of X),(RealFuncZero the carrier of X) #) is strict AlgebraStr
the carrier of (RAlgebra the carrier of X) is non empty set
bool the carrier of (RAlgebra the carrier of X) is non empty set
X is non empty TopSpace-like TopStruct
the carrier of X is non empty set
RAlgebra the carrier of X is non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative unital associative commutative right-distributive right_unital well-unital left_unital strict vector-associative AlgebraStr
Funcs ( the carrier of X,REAL) is non empty functional FUNCTION_DOMAIN of the carrier of X, REAL
RealFuncMult the carrier of X is non empty Relation-like [:(Funcs ( the carrier of X,REAL)),(Funcs ( the carrier of X,REAL)):] -defined Funcs ( the carrier of X,REAL) -valued Function-like total quasi_total Element of bool [:[:(Funcs ( the carrier of X,REAL)),(Funcs ( the carrier of X,REAL)):],(Funcs ( the carrier of X,REAL)):]
[:(Funcs ( the carrier of X,REAL)),(Funcs ( the carrier of X,REAL)):] is non empty set
[:[:(Funcs ( the carrier of X,REAL)),(Funcs ( the carrier of X,REAL)):],(Funcs ( the carrier of X,REAL)):] is non empty set
bool [:[:(Funcs ( the carrier of X,REAL)),(Funcs ( the carrier of X,REAL)):],(Funcs ( the carrier of X,REAL)):] is non empty set
RealFuncAdd the carrier of X is non empty Relation-like [:(Funcs ( the carrier of X,REAL)),(Funcs ( the carrier of X,REAL)):] -defined Funcs ( the carrier of X,REAL) -valued Function-like total quasi_total Element of bool [:[:(Funcs ( the carrier of X,REAL)),(Funcs ( the carrier of X,REAL)):],(Funcs ( the carrier of X,REAL)):]
RealFuncExtMult the carrier of X is non empty Relation-like [:REAL,(Funcs ( the carrier of X,REAL)):] -defined Funcs ( the carrier of X,REAL) -valued Function-like total quasi_total Element of bool [:[:REAL,(Funcs ( the carrier of X,REAL)):],(Funcs ( the carrier of X,REAL)):]
[:REAL,(Funcs ( the carrier of X,REAL)):] is non empty set
[:[:REAL,(Funcs ( the carrier of X,REAL)):],(Funcs ( the carrier of X,REAL)):] is non empty set
bool [:[:REAL,(Funcs ( the carrier of X,REAL)):],(Funcs ( the carrier of X,REAL)):] is non empty set
RealFuncUnit the carrier of X is Relation-like the carrier of X -defined REAL -valued Function-like total quasi_total V152() V153() V154() Element of Funcs ( the carrier of X,REAL)
the carrier of X --> 1 is non empty Relation-like the carrier of X -defined NAT -valued RAT -valued INT -valued Function-like total quasi_total V152() V153() V154() V155() Element of bool [: the carrier of X,NAT:]
[: the carrier of X,NAT:] is non empty RAT -valued INT -valued V152() V153() V154() V155() set
bool [: the carrier of X,NAT:] is non empty set
RealFuncZero the carrier of X is Relation-like the carrier of X -defined REAL -valued Function-like total quasi_total V152() V153() V154() Element of Funcs ( the carrier of X,REAL)
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 V152() V153() V154() V155() Element of bool [: the carrier of X,NAT:]
AlgebraStr(# (Funcs ( the carrier of X,REAL)),(RealFuncMult the carrier of X),(RealFuncAdd the carrier of X),(RealFuncExtMult the carrier of X),(RealFuncUnit the carrier of X),(RealFuncZero the carrier of X) #) is strict AlgebraStr
the carrier of (RAlgebra the carrier of X) is non empty set
(X) is non empty Element of bool the carrier of (RealVectSpace the carrier of X)
RealVectSpace the carrier of X is non empty right_complementable strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RLSStruct
RLSStruct(# (Funcs ( the carrier of X,REAL)),(RealFuncZero the carrier of X),(RealFuncAdd the carrier of X),(RealFuncExtMult the carrier of X) #) is strict RLSStruct
the carrier of (RealVectSpace the carrier of X) is non empty set
bool the carrier of (RealVectSpace the carrier of X) is non empty set
[: the carrier of X,REAL:] is non empty V152() V153() V154() set
bool [: the carrier of X,REAL:] 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 REAL -valued Function-like total quasi_total V152() V153() V154() Element of bool [: the carrier of X,REAL:] : ( b₁ is continuous & 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 right_complementable Element of the carrier of (RAlgebra the carrier of X)
FB is right_complementable Element of the carrier of (RAlgebra the carrier of X)
G + FB is right_complementable Element of the carrier of (RAlgebra the carrier of X)
the addF of (RAlgebra the carrier of X) is non empty Relation-like [: the carrier of (RAlgebra the carrier of X), the carrier of (RAlgebra the carrier of X):] -defined the carrier of (RAlgebra the carrier of X) -valued Function-like total quasi_total Element of bool [:[: the carrier of (RAlgebra the carrier of X), the carrier of (RAlgebra the carrier of X):], the carrier of (RAlgebra the carrier of X):]
[: the carrier of (RAlgebra the carrier of X), the carrier of (RAlgebra the carrier of X):] is non empty set
[:[: the carrier of (RAlgebra the carrier of X), the carrier of (RAlgebra the carrier of X):], the carrier of (RAlgebra the carrier of X):] is non empty set
bool [:[: the carrier of (RAlgebra the carrier of X), the carrier of (RAlgebra the carrier of X):], the carrier of (RAlgebra the carrier of X):] is non empty set
the addF of (RAlgebra the carrier of X) . (G,FB) is right_complementable Element of the carrier of (RAlgebra 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 (RAlgebra the carrier of X) . [G,FB] is set
GB is non empty Relation-like the carrier of X -defined REAL -valued Function-like total quasi_total V152() V153() V154() Element of bool [: the carrier of X,REAL:]
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 REAL -valued Function-like total quasi_total V152() V153() V154() Element of bool [: the carrier of X,REAL:]
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 add-closed having-inverse additively-closed multiplicatively-closed additively-linearly-closed Element of bool the carrier of (RAlgebra the carrier of X)
bool the carrier of (RAlgebra the carrier of X) is non empty set
{ b₁ where b₁ is non empty Relation-like the carrier of X -defined REAL -valued Function-like total quasi_total V152() V153() V154() continuous Element of bool [: the carrier of X,REAL:] : 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 right_complementable Element of the carrier of (RAlgebra the carrier of X)
the addF of (RAlgebra the carrier of X) . (FB,G) is right_complementable Element of the carrier of (RAlgebra 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 (RAlgebra the carrier of X) . [FB,G] is set
r is non empty Relation-like the carrier of X -defined REAL -valued Function-like total quasi_total V152() V153() V154() continuous Element of bool [: the carrier of X,REAL:]
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 non empty Relation-like the carrier of X -defined REAL -valued Function-like total quasi_total V152() V153() V154() Element of bool [: the carrier of X,REAL:]
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 REAL -valued Function-like total quasi_total V152() V153() V154() Element of Funcs ( the carrier of X,REAL)
dom fvu1 is Element of bool the carrier of X
c₁₉ is set
fvu1 . c₁₉ is V11() real ext-real set
aF . c₁₉ is V11() real ext-real set
GB . c₁₉ is V11() real ext-real set
(aF . c₁₉) + (GB . c₁₉) is V11() real ext-real set
c₂₀ is non empty Element of bool the carrier of X
bool c₂₀ is non empty set
c₂₁ is non empty Element of bool the carrier of X
bool c₂₁ is non empty set
X is non empty TopSpace-like TopStruct
the carrier of X is non empty set
RAlgebra the carrier of X is non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative unital associative commutative right-distributive right_unital well-unital left_unital strict vector-associative AlgebraStr
Funcs ( the carrier of X,REAL) is non empty functional FUNCTION_DOMAIN of the carrier of X, REAL
RealFuncMult the carrier of X is non empty Relation-like [:(Funcs ( the carrier of X,REAL)),(Funcs ( the carrier of X,REAL)):] -defined Funcs ( the carrier of X,REAL) -valued Function-like total quasi_total Element of bool [:[:(Funcs ( the carrier of X,REAL)),(Funcs ( the carrier of X,REAL)):],(Funcs ( the carrier of X,REAL)):]
[:(Funcs ( the carrier of X,REAL)),(Funcs ( the carrier of X,REAL)):] is non empty set
[:[:(Funcs ( the carrier of X,REAL)),(Funcs ( the carrier of X,REAL)):],(Funcs ( the carrier of X,REAL)):] is non empty set
bool [:[:(Funcs ( the carrier of X,REAL)),(Funcs ( the carrier of X,REAL)):],(Funcs ( the carrier of X,REAL)):] is non empty set
RealFuncAdd the carrier of X is non empty Relation-like [:(Funcs ( the carrier of X,REAL)),(Funcs ( the carrier of X,REAL)):] -defined Funcs ( the carrier of X,REAL) -valued Function-like total quasi_total Element of bool [:[:(Funcs ( the carrier of X,REAL)),(Funcs ( the carrier of X,REAL)):],(Funcs ( the carrier of X,REAL)):]
RealFuncExtMult the carrier of X is non empty Relation-like [:REAL,(Funcs ( the carrier of X,REAL)):] -defined Funcs ( the carrier of X,REAL) -valued Function-like total quasi_total Element of bool [:[:REAL,(Funcs ( the carrier of X,REAL)):],(Funcs ( the carrier of X,REAL)):]
[:REAL,(Funcs ( the carrier of X,REAL)):] is non empty set
[:[:REAL,(Funcs ( the carrier of X,REAL)):],(Funcs ( the carrier of X,REAL)):] is non empty set
bool [:[:REAL,(Funcs ( the carrier of X,REAL)):],(Funcs ( the carrier of X,REAL)):] is non empty set
RealFuncUnit the carrier of X is Relation-like the carrier of X -defined REAL -valued Function-like total quasi_total V152() V153() V154() Element of Funcs ( the carrier of X,REAL)
the carrier of X --> 1 is non empty Relation-like the carrier of X -defined NAT -valued RAT -valued INT -valued Function-like total quasi_total V152() V153() V154() V155() Element of bool [: the carrier of X,NAT:]
[: the carrier of X,NAT:] is non empty RAT -valued INT -valued V152() V153() V154() V155() set
bool [: the carrier of X,NAT:] is non empty set
RealFuncZero the carrier of X is Relation-like the carrier of X -defined REAL -valued Function-like total quasi_total V152() V153() V154() Element of Funcs ( the carrier of X,REAL)
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 V152() V153() V154() V155() Element of bool [: the carrier of X,NAT:]
AlgebraStr(# (Funcs ( the carrier of X,REAL)),(RealFuncMult the carrier of X),(RealFuncAdd the carrier of X),(RealFuncExtMult the carrier of X),(RealFuncUnit the carrier of X),(RealFuncZero the carrier of X) #) is strict AlgebraStr
the carrier of (RAlgebra the carrier of X) is non empty set
(X) is non empty Element of bool the carrier of (RealVectSpace the carrier of X)
RealVectSpace the carrier of X is non empty right_complementable strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RLSStruct
RLSStruct(# (Funcs ( the carrier of X,REAL)),(RealFuncZero the carrier of X),(RealFuncAdd the carrier of X),(RealFuncExtMult the carrier of X) #) is strict RLSStruct
the carrier of (RealVectSpace the carrier of X) is non empty set
bool the carrier of (RealVectSpace the carrier of X) is non empty set
[: the carrier of X,REAL:] is non empty V152() V153() V154() set
bool [: the carrier of X,REAL:] 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 REAL -valued Function-like total quasi_total V152() V153() V154() Element of bool [: the carrier of X,REAL:] : ( b₁ is continuous & 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 V11() real ext-real Element of REAL
FB is right_complementable Element of the carrier of (RAlgebra the carrier of X)
G * FB is right_complementable Element of the carrier of (RAlgebra the carrier of X)
the Mult of (RAlgebra the carrier of X) is non empty Relation-like [:REAL, the carrier of (RAlgebra the carrier of X):] -defined the carrier of (RAlgebra the carrier of X) -valued Function-like total quasi_total Element of bool [:[:REAL, the carrier of (RAlgebra the carrier of X):], the carrier of (RAlgebra the carrier of X):]
[:REAL, the carrier of (RAlgebra the carrier of X):] is non empty set
[:[:REAL, the carrier of (RAlgebra the carrier of X):], the carrier of (RAlgebra the carrier of X):] is non empty set
bool [:[:REAL, the carrier of (RAlgebra the carrier of X):], the carrier of (RAlgebra the carrier of X):] is non empty set
the Mult of (RAlgebra the carrier of X) . (G,FB) is set
[G,FB] is set
{G,FB} is non empty set
{G} is non empty V185() V186() V187() set
{{G,FB},{G}} is non empty set
the Mult of (RAlgebra the carrier of X) . [G,FB] is set
GB is non empty Relation-like the carrier of X -defined REAL -valued Function-like total quasi_total V152() V153() V154() Element of bool [: the carrier of X,REAL:]
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 add-closed having-inverse additively-closed multiplicatively-closed additively-linearly-closed Element of bool the carrier of (RAlgebra the carrier of X)
bool the carrier of (RAlgebra the carrier of X) is non empty set
{ b₁ where b₁ is non empty Relation-like the carrier of X -defined REAL -valued Function-like total quasi_total V152() V153() V154() continuous Element of bool [: the carrier of X,REAL:] : 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 REAL -valued Function-like total quasi_total V152() V153() V154() continuous Element of bool [: the carrier of X,REAL:]
dom GB is Element of bool the carrier of X
G (#) GB is non empty Relation-like the carrier of X -defined REAL -valued Function-like total quasi_total V152() V153() V154() Element of bool [: the carrier of X,REAL:]
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
s1 is Element of bool the carrier of X
Cl s1 is Element of bool the carrier of X
uu1 is Element of bool the carrier of X
Cl uu1 is Element of bool the carrier of X
fau1 is Relation-like the carrier of X -defined REAL -valued Function-like total quasi_total V152() V153() V154() Element of Funcs ( the carrier of X,REAL)
dom fau1 is Element of bool the carrier of X
A2 is set
fau1 . A2 is V11() real ext-real set
GB . A2 is V11() real ext-real set
G * (GB . A2) is V11() real ext-real Element of REAL
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
RAlgebra the carrier of X is non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative unital associative commutative right-distributive right_unital well-unital left_unital strict vector-associative AlgebraStr
Funcs ( the carrier of X,REAL) is non empty functional FUNCTION_DOMAIN of the carrier of X, REAL
RealFuncMult the carrier of X is non empty Relation-like [:(Funcs ( the carrier of X,REAL)),(Funcs ( the carrier of X,REAL)):] -defined Funcs ( the carrier of X,REAL) -valued Function-like total quasi_total Element of bool [:[:(Funcs ( the carrier of X,REAL)),(Funcs ( the carrier of X,REAL)):],(Funcs ( the carrier of X,REAL)):]
[:(Funcs ( the carrier of X,REAL)),(Funcs ( the carrier of X,REAL)):] is non empty set
[:[:(Funcs ( the carrier of X,REAL)),(Funcs ( the carrier of X,REAL)):],(Funcs ( the carrier of X,REAL)):] is non empty set
bool [:[:(Funcs ( the carrier of X,REAL)),(Funcs ( the carrier of X,REAL)):],(Funcs ( the carrier of X,REAL)):] is non empty set
RealFuncAdd the carrier of X is non empty Relation-like [:(Funcs ( the carrier of X,REAL)),(Funcs ( the carrier of X,REAL)):] -defined Funcs ( the carrier of X,REAL) -valued Function-like total quasi_total Element of bool [:[:(Funcs ( the carrier of X,REAL)),(Funcs ( the carrier of X,REAL)):],(Funcs ( the carrier of X,REAL)):]
RealFuncExtMult the carrier of X is non empty Relation-like [:REAL,(Funcs ( the carrier of X,REAL)):] -defined Funcs ( the carrier of X,REAL) -valued Function-like total quasi_total Element of bool [:[:REAL,(Funcs ( the carrier of X,REAL)):],(Funcs ( the carrier of X,REAL)):]
[:REAL,(Funcs ( the carrier of X,REAL)):] is non empty set
[:[:REAL,(Funcs ( the carrier of X,REAL)):],(Funcs ( the carrier of X,REAL)):] is non empty set
bool [:[:REAL,(Funcs ( the carrier of X,REAL)):],(Funcs ( the carrier of X,REAL)):] is non empty set
RealFuncUnit the carrier of X is Relation-like the carrier of X -defined REAL -valued Function-like total quasi_total V152() V153() V154() Element of Funcs ( the carrier of X,REAL)
the carrier of X --> 1 is non empty Relation-like the carrier of X -defined NAT -valued RAT -valued INT -valued Function-like total quasi_total V152() V153() V154() V155() Element of bool [: the carrier of X,NAT:]
[: the carrier of X,NAT:] is non empty RAT -valued INT -valued V152() V153() V154() V155() set
bool [: the carrier of X,NAT:] is non empty set
RealFuncZero the carrier of X is Relation-like the carrier of X -defined REAL -valued Function-like total quasi_total V152() V153() V154() Element of Funcs ( the carrier of X,REAL)
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 V152() V153() V154() V155() Element of bool [: the carrier of X,NAT:]
AlgebraStr(# (Funcs ( the carrier of X,REAL)),(RealFuncMult the carrier of X),(RealFuncAdd the carrier of X),(RealFuncExtMult the carrier of X),(RealFuncUnit the carrier of X),(RealFuncZero the carrier of X) #) is strict AlgebraStr
the carrier of (RAlgebra the carrier of X) is non empty set
(X) is non empty Element of bool the carrier of (RealVectSpace the carrier of X)
RealVectSpace the carrier of X is non empty right_complementable strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RLSStruct
RLSStruct(# (Funcs ( the carrier of X,REAL)),(RealFuncZero the carrier of X),(RealFuncAdd the carrier of X),(RealFuncExtMult the carrier of X) #) is strict RLSStruct
the carrier of (RealVectSpace the carrier of X) is non empty set
bool the carrier of (RealVectSpace the carrier of X) is non empty set
[: the carrier of X,REAL:] is non empty V152() V153() V154() set
bool [: the carrier of X,REAL:] 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 REAL -valued Function-like total quasi_total V152() V153() V154() Element of bool [: the carrier of X,REAL:] : ( b₁ is continuous & 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 right_complementable Element of the carrier of (RAlgebra the carrier of X)
- G is right_complementable Element of the carrier of (RAlgebra the carrier of X)
(- 1) * G is right_complementable Element of the carrier of (RAlgebra the carrier of X)
the Mult of (RAlgebra the carrier of X) is non empty Relation-like [:REAL, the carrier of (RAlgebra the carrier of X):] -defined the carrier of (RAlgebra the carrier of X) -valued Function-like total quasi_total Element of bool [:[:REAL, the carrier of (RAlgebra the carrier of X):], the carrier of (RAlgebra the carrier of X):]
[:REAL, the carrier of (RAlgebra the carrier of X):] is non empty set
[:[:REAL, the carrier of (RAlgebra the carrier of X):], the carrier of (RAlgebra the carrier of X):] is non empty set
bool [:[:REAL, the carrier of (RAlgebra the carrier of X):], the carrier of (RAlgebra the carrier of X):] is non empty set
the Mult of (RAlgebra the carrier of X) . ((- 1),G) is set
[(- 1),G] is set
{(- 1),G} is non empty set
{(- 1)} is non empty V185() V186() V187() set
{{(- 1),G},{(- 1)}} is non empty set
the Mult of (RAlgebra the carrier of X) . [(- 1),G] is set
X is non empty TopSpace-like TopStruct
the carrier of X is non empty set
RAlgebra the carrier of X is non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative unital associative commutative right-distributive right_unital well-unital left_unital strict vector-associative AlgebraStr
Funcs ( the carrier of X,REAL) is non empty functional FUNCTION_DOMAIN of the carrier of X, REAL
RealFuncMult the carrier of X is non empty Relation-like [:(Funcs ( the carrier of X,REAL)),(Funcs ( the carrier of X,REAL)):] -defined Funcs ( the carrier of X,REAL) -valued Function-like total quasi_total Element of bool [:[:(Funcs ( the carrier of X,REAL)),(Funcs ( the carrier of X,REAL)):],(Funcs ( the carrier of X,REAL)):]
[:(Funcs ( the carrier of X,REAL)),(Funcs ( the carrier of X,REAL)):] is non empty set
[:[:(Funcs ( the carrier of X,REAL)),(Funcs ( the carrier of X,REAL)):],(Funcs ( the carrier of X,REAL)):] is non empty set
bool [:[:(Funcs ( the carrier of X,REAL)),(Funcs ( the carrier of X,REAL)):],(Funcs ( the carrier of X,REAL)):] is non empty set
RealFuncAdd the carrier of X is non empty Relation-like [:(Funcs ( the carrier of X,REAL)),(Funcs ( the carrier of X,REAL)):] -defined Funcs ( the carrier of X,REAL) -valued Function-like total quasi_total Element of bool [:[:(Funcs ( the carrier of X,REAL)),(Funcs ( the carrier of X,REAL)):],(Funcs ( the carrier of X,REAL)):]
RealFuncExtMult the carrier of X is non empty Relation-like [:REAL,(Funcs ( the carrier of X,REAL)):] -defined Funcs ( the carrier of X,REAL) -valued Function-like total quasi_total Element of bool [:[:REAL,(Funcs ( the carrier of X,REAL)):],(Funcs ( the carrier of X,REAL)):]
[:REAL,(Funcs ( the carrier of X,REAL)):] is non empty set
[:[:REAL,(Funcs ( the carrier of X,REAL)):],(Funcs ( the carrier of X,REAL)):] is non empty set
bool [:[:REAL,(Funcs ( the carrier of X,REAL)):],(Funcs ( the carrier of X,REAL)):] is non empty set
RealFuncUnit the carrier of X is Relation-like the carrier of X -defined REAL -valued Function-like total quasi_total V152() V153() V154() Element of Funcs ( the carrier of X,REAL)
the carrier of X --> 1 is non empty Relation-like the carrier of X -defined NAT -valued RAT -valued INT -valued Function-like total quasi_total V152() V153() V154() V155() Element of bool [: the carrier of X,NAT:]
[: the carrier of X,NAT:] is non empty RAT -valued INT -valued V152() V153() V154() V155() set
bool [: the carrier of X,NAT:] is non empty set
RealFuncZero the carrier of X is Relation-like the carrier of X -defined REAL -valued Function-like total quasi_total V152() V153() V154() Element of Funcs ( the carrier of X,REAL)
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 V152() V153() V154() V155() Element of bool [: the carrier of X,NAT:]
AlgebraStr(# (Funcs ( the carrier of X,REAL)),(RealFuncMult the carrier of X),(RealFuncAdd the carrier of X),(RealFuncExtMult the carrier of X),(RealFuncUnit the carrier of X),(RealFuncZero the carrier of X) #) is strict AlgebraStr
the carrier of (RAlgebra the carrier of X) is non empty set
bool the carrier of (RAlgebra the carrier of X) is non empty set
(X) is non empty Element of bool the carrier of (RealVectSpace the carrier of X)
RealVectSpace the carrier of X is non empty right_complementable strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RLSStruct
RLSStruct(# (Funcs ( the carrier of X,REAL)),(RealFuncZero the carrier of X),(RealFuncAdd the carrier of X),(RealFuncExtMult the carrier of X) #) is strict RLSStruct
the carrier of (RealVectSpace the carrier of X) is non empty set
bool the carrier of (RealVectSpace the carrier of X) is non empty set
[: the carrier of X,REAL:] is non empty V152() V153() V154() set
bool [: the carrier of X,REAL:] 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 REAL -valued Function-like total quasi_total V152() V153() V154() Element of bool [: the carrier of X,REAL:] : ( b₁ is continuous & 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 (RAlgebra the carrier of X)
G is right_complementable Element of the carrier of (RAlgebra the carrier of X)
FB is right_complementable Element of the carrier of (RAlgebra the carrier of X)
G + FB is right_complementable Element of the carrier of (RAlgebra the carrier of X)
the addF of (RAlgebra the carrier of X) is non empty Relation-like [: the carrier of (RAlgebra the carrier of X), the carrier of (RAlgebra the carrier of X):] -defined the carrier of (RAlgebra the carrier of X) -valued Function-like total quasi_total Element of bool [:[: the carrier of (RAlgebra the carrier of X), the carrier of (RAlgebra the carrier of X):], the carrier of (RAlgebra the carrier of X):]
[: the carrier of (RAlgebra the carrier of X), the carrier of (RAlgebra the carrier of X):] is non empty set
[:[: the carrier of (RAlgebra the carrier of X), the carrier of (RAlgebra the carrier of X):], the carrier of (RAlgebra the carrier of X):] is non empty set
bool [:[: the carrier of (RAlgebra the carrier of X), the carrier of (RAlgebra the carrier of X):], the carrier of (RAlgebra the carrier of X):] is non empty set
the addF of (RAlgebra the carrier of X) . (G,FB) is right_complementable Element of the carrier of (RAlgebra 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 (RAlgebra the carrier of X) . [G,FB] is set
G is right_complementable Element of the carrier of (RAlgebra the carrier of X)
- G is right_complementable Element of the carrier of (RAlgebra the carrier of X)
FB is right_complementable Element of the carrier of (RAlgebra the carrier of X)
G is V11() real ext-real Element of REAL
G * FB is right_complementable Element of the carrier of (RAlgebra the carrier of X)
the Mult of (RAlgebra the carrier of X) is non empty Relation-like [:REAL, the carrier of (RAlgebra the carrier of X):] -defined the carrier of (RAlgebra the carrier of X) -valued Function-like total quasi_total Element of bool [:[:REAL, the carrier of (RAlgebra the carrier of X):], the carrier of (RAlgebra the carrier of X):]
[:REAL, the carrier of (RAlgebra the carrier of X):] is non empty set
[:[:REAL, the carrier of (RAlgebra the carrier of X):], the carrier of (RAlgebra the carrier of X):] is non empty set
bool [:[:REAL, the carrier of (RAlgebra the carrier of X):], the carrier of (RAlgebra the carrier of X):] is non empty set
the Mult of (RAlgebra the carrier of X) . (G,FB) is set
[G,FB] is set
{G,FB} is non empty set
{G} is non empty V185() V186() V187() set
{{G,FB},{G}} is non empty set
the Mult of (RAlgebra the carrier of X) . [G,FB] is set
X is non empty TopSpace-like TopStruct
the carrier of X is non empty set
RealVectSpace the carrier of X is non empty right_complementable strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RLSStruct
Funcs ( the carrier of X,REAL) is non empty functional FUNCTION_DOMAIN of the carrier of X, REAL
RealFuncZero the carrier of X is Relation-like the carrier of X -defined REAL -valued Function-like total quasi_total V152() V153() V154() Element of Funcs ( the carrier of X,REAL)
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 V152() V153() V154() V155() Element of bool [: the carrier of X,NAT:]
[: the carrier of X,NAT:] is non empty RAT -valued INT -valued V152() V153() V154() V155() set
bool [: the carrier of X,NAT:] is non empty set
RealFuncAdd the carrier of X is non empty Relation-like [:(Funcs ( the carrier of X,REAL)),(Funcs ( the carrier of X,REAL)):] -defined Funcs ( the carrier of X,REAL) -valued Function-like total quasi_total Element of bool [:[:(Funcs ( the carrier of X,REAL)),(Funcs ( the carrier of X,REAL)):],(Funcs ( the carrier of X,REAL)):]
[:(Funcs ( the carrier of X,REAL)),(Funcs ( the carrier of X,REAL)):] is non empty set
[:[:(Funcs ( the carrier of X,REAL)),(Funcs ( the carrier of X,REAL)):],(Funcs ( the carrier of X,REAL)):] is non empty set
bool [:[:(Funcs ( the carrier of X,REAL)),(Funcs ( the carrier of X,REAL)):],(Funcs ( the carrier of X,REAL)):] is non empty set
RealFuncExtMult the carrier of X is non empty Relation-like [:REAL,(Funcs ( the carrier of X,REAL)):] -defined Funcs ( the carrier of X,REAL) -valued Function-like total quasi_total Element of bool [:[:REAL,(Funcs ( the carrier of X,REAL)):],(Funcs ( the carrier of X,REAL)):]
[:REAL,(Funcs ( the carrier of X,REAL)):] is non empty set
[:[:REAL,(Funcs ( the carrier of X,REAL)):],(Funcs ( the carrier of X,REAL)):] is non empty set
bool [:[:REAL,(Funcs ( the carrier of X,REAL)):],(Funcs ( the carrier of X,REAL)):] is non empty set
RLSStruct(# (Funcs ( the carrier of X,REAL)),(RealFuncZero the carrier of X),(RealFuncAdd the carrier of X),(RealFuncExtMult the carrier of X) #) is strict RLSStruct
(X) is non empty Element of bool the carrier of (RealVectSpace the carrier of X)
the carrier of (RealVectSpace the carrier of X) is non empty set
bool the carrier of (RealVectSpace the carrier of X) is non empty set
[: the carrier of X,REAL:] is non empty V152() V153() V154() set
bool [: the carrier of X,REAL:] 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 REAL -valued Function-like total quasi_total V152() V153() V154() Element of bool [: the carrier of X,REAL:] : ( b₁ is continuous & 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 right_complementable Element of the carrier of (RealVectSpace the carrier of X)
FB is right_complementable Element of the carrier of (RealVectSpace the carrier of X)
G + FB is right_complementable Element of the carrier of (RealVectSpace the carrier of X)
the addF of (RealVectSpace the carrier of X) is non empty Relation-like [: the carrier of (RealVectSpace the carrier of X), the carrier of (RealVectSpace the carrier of X):] -defined the carrier of (RealVectSpace the carrier of X) -valued Function-like total quasi_total Element of bool [:[: the carrier of (RealVectSpace the carrier of X), the carrier of (RealVectSpace the carrier of X):], the carrier of (RealVectSpace the carrier of X):]
[: the carrier of (RealVectSpace the carrier of X), the carrier of (RealVectSpace the carrier of X):] is non empty set
[:[: the carrier of (RealVectSpace the carrier of X), the carrier of (RealVectSpace the carrier of X):], the carrier of (RealVectSpace the carrier of X):] is non empty set
bool [:[: the carrier of (RealVectSpace the carrier of X), the carrier of (RealVectSpace the carrier of X):], the carrier of (RealVectSpace the carrier of X):] is non empty set
the addF of (RealVectSpace the carrier of X) . (G,FB) is right_complementable Element of the carrier of (RealVectSpace 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 (RealVectSpace the carrier of X) . [G,FB] is set
RAlgebra the carrier of X is non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative unital associative commutative right-distributive right_unital well-unital left_unital strict vector-associative AlgebraStr
RealFuncMult the carrier of X is non empty Relation-like [:(Funcs ( the carrier of X,REAL)),(Funcs ( the carrier of X,REAL)):] -defined Funcs ( the carrier of X,REAL) -valued Function-like total quasi_total Element of bool [:[:(Funcs ( the carrier of X,REAL)),(Funcs ( the carrier of X,REAL)):],(Funcs ( the carrier of X,REAL)):]
RealFuncUnit the carrier of X is Relation-like the carrier of X -defined REAL -valued Function-like total quasi_total V152() V153() V154() Element of Funcs ( the carrier of X,REAL)
the carrier of X --> 1 is non empty Relation-like the carrier of X -defined NAT -valued RAT -valued INT -valued Function-like total quasi_total V152() V153() V154() V155() Element of bool [: the carrier of X,NAT:]
AlgebraStr(# (Funcs ( the carrier of X,REAL)),(RealFuncMult the carrier of X),(RealFuncAdd the carrier of X),(RealFuncExtMult the carrier of X),(RealFuncUnit the carrier of X),(RealFuncZero the carrier of X) #) is strict AlgebraStr
the carrier of (RAlgebra the carrier of X) is non empty set
aF is right_complementable Element of the carrier of (RAlgebra the carrier of X)
aG is right_complementable Element of the carrier of (RAlgebra the carrier of X)
aF + aG is right_complementable Element of the carrier of (RAlgebra the carrier of X)
the addF of (RAlgebra the carrier of X) is non empty Relation-like [: the carrier of (RAlgebra the carrier of X), the carrier of (RAlgebra the carrier of X):] -defined the carrier of (RAlgebra the carrier of X) -valued Function-like total quasi_total Element of bool [:[: the carrier of (RAlgebra the carrier of X), the carrier of (RAlgebra the carrier of X):], the carrier of (RAlgebra the carrier of X):]
[: the carrier of (RAlgebra the carrier of X), the carrier of (RAlgebra the carrier of X):] is non empty set
[:[: the carrier of (RAlgebra the carrier of X), the carrier of (RAlgebra the carrier of X):], the carrier of (RAlgebra the carrier of X):] is non empty set
bool [:[: the carrier of (RAlgebra the carrier of X), the carrier of (RAlgebra the carrier of X):], the carrier of (RAlgebra the carrier of X):] is non empty set
the addF of (RAlgebra the carrier of X) . (aF,aG) is right_complementable Element of the carrier of (RAlgebra 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 (RAlgebra the carrier of X) . [aF,aG] is set
G is V11() real ext-real Element of REAL
FB is right_complementable Element of the carrier of (RealVectSpace the carrier of X)
G * FB is right_complementable Element of the carrier of (RealVectSpace the carrier of X)
the Mult of (RealVectSpace the carrier of X) is non empty Relation-like [:REAL, the carrier of (RealVectSpace the carrier of X):] -defined the carrier of (RealVectSpace the carrier of X) -valued Function-like total quasi_total Element of bool [:[:REAL, the carrier of (RealVectSpace the carrier of X):], the carrier of (RealVectSpace the carrier of X):]
[:REAL, the carrier of (RealVectSpace the carrier of X):] is non empty set
[:[:REAL, the carrier of (RealVectSpace the carrier of X):], the carrier of (RealVectSpace the carrier of X):] is non empty set
bool [:[:REAL, the carrier of (RealVectSpace the carrier of X):], the carrier of (RealVectSpace the carrier of X):] is non empty set
the Mult of (RealVectSpace the carrier of X) . (G,FB) is set
[G,FB] is set
{G,FB} is non empty set
{G} is non empty V185() V186() V187() set
{{G,FB},{G}} is non empty set
the Mult of (RealVectSpace the carrier of X) . [G,FB] is set
RAlgebra the carrier of X is non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative unital associative commutative right-distributive right_unital well-unital left_unital strict vector-associative AlgebraStr
RealFuncMult the carrier of X is non empty Relation-like [:(Funcs ( the carrier of X,REAL)),(Funcs ( the carrier of X,REAL)):] -defined Funcs ( the carrier of X,REAL) -valued Function-like total quasi_total Element of bool [:[:(Funcs ( the carrier of X,REAL)),(Funcs ( the carrier of X,REAL)):],(Funcs ( the carrier of X,REAL)):]
RealFuncUnit the carrier of X is Relation-like the carrier of X -defined REAL -valued Function-like total quasi_total V152() V153() V154() Element of Funcs ( the carrier of X,REAL)
the carrier of X --> 1 is non empty Relation-like the carrier of X -defined NAT -valued RAT -valued INT -valued Function-like total quasi_total V152() V153() V154() V155() Element of bool [: the carrier of X,NAT:]
AlgebraStr(# (Funcs ( the carrier of X,REAL)),(RealFuncMult the carrier of X),(RealFuncAdd the carrier of X),(RealFuncExtMult the carrier of X),(RealFuncUnit the carrier of X),(RealFuncZero the carrier of X) #) is strict AlgebraStr
the carrier of (RAlgebra the carrier of X) is non empty set
aFB is right_complementable Element of the carrier of (RAlgebra the carrier of X)
G * aFB is right_complementable Element of the carrier of (RAlgebra the carrier of X)
the Mult of (RAlgebra the carrier of X) is non empty Relation-like [:REAL, the carrier of (RAlgebra the carrier of X):] -defined the carrier of (RAlgebra the carrier of X) -valued Function-like total quasi_total Element of bool [:[:REAL, the carrier of (RAlgebra the carrier of X):], the carrier of (RAlgebra the carrier of X):]
[:REAL, the carrier of (RAlgebra the carrier of X):] is non empty set
[:[:REAL, the carrier of (RAlgebra the carrier of X):], the carrier of (RAlgebra the carrier of X):] is non empty set
bool [:[:REAL, the carrier of (RAlgebra the carrier of X):], the carrier of (RAlgebra the carrier of X):] is non empty set
the Mult of (RAlgebra the carrier of X) . (G,aFB) is set
[G,aFB] is set
{G,aFB} is non empty set
{{G,aFB},{G}} is non empty set
the Mult of (RAlgebra 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 (RealVectSpace the carrier of X)
the carrier of X is non empty set
RealVectSpace the carrier of X is non empty right_complementable strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RLSStruct
Funcs ( the carrier of X,REAL) is non empty functional FUNCTION_DOMAIN of the carrier of X, REAL
RealFuncZero the carrier of X is Relation-like the carrier of X -defined REAL -valued Function-like total quasi_total V152() V153() V154() Element of Funcs ( the carrier of X,REAL)
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 V152() V153() V154() V155() Element of bool [: the carrier of X,NAT:]
[: the carrier of X,NAT:] is non empty RAT -valued INT -valued V152() V153() V154() V155() set
bool [: the carrier of X,NAT:] is non empty set
RealFuncAdd the carrier of X is non empty Relation-like [:(Funcs ( the carrier of X,REAL)),(Funcs ( the carrier of X,REAL)):] -defined Funcs ( the carrier of X,REAL) -valued Function-like total quasi_total Element of bool [:[:(Funcs ( the carrier of X,REAL)),(Funcs ( the carrier of X,REAL)):],(Funcs ( the carrier of X,REAL)):]
[:(Funcs ( the carrier of X,REAL)),(Funcs ( the carrier of X,REAL)):] is non empty set
[:[:(Funcs ( the carrier of X,REAL)),(Funcs ( the carrier of X,REAL)):],(Funcs ( the carrier of X,REAL)):] is non empty set
bool [:[:(Funcs ( the carrier of X,REAL)),(Funcs ( the carrier of X,REAL)):],(Funcs ( the carrier of X,REAL)):] is non empty set
RealFuncExtMult the carrier of X is non empty Relation-like [:REAL,(Funcs ( the carrier of X,REAL)):] -defined Funcs ( the carrier of X,REAL) -valued Function-like total quasi_total Element of bool [:[:REAL,(Funcs ( the carrier of X,REAL)):],(Funcs ( the carrier of X,REAL)):]
[:REAL,(Funcs ( the carrier of X,REAL)):] is non empty set
[:[:REAL,(Funcs ( the carrier of X,REAL)):],(Funcs ( the carrier of X,REAL)):] is non empty set
bool [:[:REAL,(Funcs ( the carrier of X,REAL)):],(Funcs ( the carrier of X,REAL)):] is non empty set
RLSStruct(# (Funcs ( the carrier of X,REAL)),(RealFuncZero the carrier of X),(RealFuncAdd the carrier of X),(RealFuncExtMult the carrier of X) #) is strict RLSStruct
the carrier of (RealVectSpace the carrier of X) is non empty set
bool the carrier of (RealVectSpace the carrier of X) is non empty set
[: the carrier of X,REAL:] is non empty V152() V153() V154() set
bool [: the carrier of X,REAL:] 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 REAL -valued Function-like total quasi_total V152() V153() V154() Element of bool [: the carrier of X,REAL:] : ( b₁ is continuous & 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 TopSpace-like TopStruct
(X) is non empty linearly-closed Element of bool the carrier of (RealVectSpace the carrier of X)
the carrier of X is non empty set
RealVectSpace the carrier of X is non empty right_complementable strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RLSStruct
Funcs ( the carrier of X,REAL) is non empty functional FUNCTION_DOMAIN of the carrier of X, REAL
RealFuncZero the carrier of X is Relation-like the carrier of X -defined REAL -valued Function-like total quasi_total V152() V153() V154() Element of Funcs ( the carrier of X,REAL)
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 V152() V153() V154() V155() Element of bool [: the carrier of X,NAT:]
[: the carrier of X,NAT:] is non empty RAT -valued INT -valued V152() V153() V154() V155() set
bool [: the carrier of X,NAT:] is non empty set
RealFuncAdd the carrier of X is non empty Relation-like [:(Funcs ( the carrier of X,REAL)),(Funcs ( the carrier of X,REAL)):] -defined Funcs ( the carrier of X,REAL) -valued Function-like total quasi_total Element of bool [:[:(Funcs ( the carrier of X,REAL)),(Funcs ( the carrier of X,REAL)):],(Funcs ( the carrier of X,REAL)):]
[:(Funcs ( the carrier of X,REAL)),(Funcs ( the carrier of X,REAL)):] is non empty set
[:[:(Funcs ( the carrier of X,REAL)),(Funcs ( the carrier of X,REAL)):],(Funcs ( the carrier of X,REAL)):] is non empty set
bool [:[:(Funcs ( the carrier of X,REAL)),(Funcs ( the carrier of X,REAL)):],(Funcs ( the carrier of X,REAL)):] is non empty set
RealFuncExtMult the carrier of X is non empty Relation-like [:REAL,(Funcs ( the carrier of X,REAL)):] -defined Funcs ( the carrier of X,REAL) -valued Function-like total quasi_total Element of bool [:[:REAL,(Funcs ( the carrier of X,REAL)):],(Funcs ( the carrier of X,REAL)):]
[:REAL,(Funcs ( the carrier of X,REAL)):] is non empty set
[:[:REAL,(Funcs ( the carrier of X,REAL)):],(Funcs ( the carrier of X,REAL)):] is non empty set
bool [:[:REAL,(Funcs ( the carrier of X,REAL)):],(Funcs ( the carrier of X,REAL)):] is non empty set
RLSStruct(# (Funcs ( the carrier of X,REAL)),(RealFuncZero the carrier of X),(RealFuncAdd the carrier of X),(RealFuncExtMult the carrier of X) #) is strict RLSStruct
the carrier of (RealVectSpace the carrier of X) is non empty set
bool the carrier of (RealVectSpace the carrier of X) is non empty set
[: the carrier of X,REAL:] is non empty V152() V153() V154() set
bool [: the carrier of X,REAL:] 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 REAL -valued Function-like total quasi_total V152() V153() V154() Element of bool [: the carrier of X,REAL:] : ( b₁ is continuous & 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),(RealVectSpace the carrier of X)) is Element of (X)
Add_ ((X),(RealVectSpace 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),(RealVectSpace the carrier of X)) is non empty Relation-like [:REAL,(X):] -defined (X) -valued Function-like total quasi_total Element of bool [:[:REAL,(X):],(X):]
[:REAL,(X):] is non empty set
[:[:REAL,(X):],(X):] is non empty set
bool [:[:REAL,(X):],(X):] is non empty set
RLSStruct(# (X),(Zero_ ((X),(RealVectSpace the carrier of X))),(Add_ ((X),(RealVectSpace the carrier of X))),(Mult_ ((X),(RealVectSpace the carrier of X))) #) is strict RLSStruct
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 RLSStruct
(X) is non empty linearly-closed Element of bool the carrier of (RealVectSpace the carrier of X)
the carrier of X is non empty set
RealVectSpace the carrier of X is non empty right_complementable strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RLSStruct
Funcs ( the carrier of X,REAL) is non empty functional FUNCTION_DOMAIN of the carrier of X, REAL
RealFuncZero the carrier of X is Relation-like the carrier of X -defined REAL -valued Function-like total quasi_total V152() V153() V154() Element of Funcs ( the carrier of X,REAL)
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 V152() V153() V154() V155() Element of bool [: the carrier of X,NAT:]
[: the carrier of X,NAT:] is non empty RAT -valued INT -valued V152() V153() V154() V155() set
bool [: the carrier of X,NAT:] is non empty set
RealFuncAdd the carrier of X is non empty Relation-like [:(Funcs ( the carrier of X,REAL)),(Funcs ( the carrier of X,REAL)):] -defined Funcs ( the carrier of X,REAL) -valued Function-like total quasi_total Element of bool [:[:(Funcs ( the carrier of X,REAL)),(Funcs ( the carrier of X,REAL)):],(Funcs ( the carrier of X,REAL)):]
[:(Funcs ( the carrier of X,REAL)),(Funcs ( the carrier of X,REAL)):] is non empty set
[:[:(Funcs ( the carrier of X,REAL)),(Funcs ( the carrier of X,REAL)):],(Funcs ( the carrier of X,REAL)):] is non empty set
bool [:[:(Funcs ( the carrier of X,REAL)),(Funcs ( the carrier of X,REAL)):],(Funcs ( the carrier of X,REAL)):] is non empty set
RealFuncExtMult the carrier of X is non empty Relation-like [:REAL,(Funcs ( the carrier of X,REAL)):] -defined Funcs ( the carrier of X,REAL) -valued Function-like total quasi_total Element of bool [:[:REAL,(Funcs ( the carrier of X,REAL)):],(Funcs ( the carrier of X,REAL)):]
[:REAL,(Funcs ( the carrier of X,REAL)):] is non empty set
[:[:REAL,(Funcs ( the carrier of X,REAL)):],(Funcs ( the carrier of X,REAL)):] is non empty set
bool [:[:REAL,(Funcs ( the carrier of X,REAL)):],(Funcs ( the carrier of X,REAL)):] is non empty set
RLSStruct(# (Funcs ( the carrier of X,REAL)),(RealFuncZero the carrier of X),(RealFuncAdd the carrier of X),(RealFuncExtMult the carrier of X) #) is strict RLSStruct
the carrier of (RealVectSpace the carrier of X) is non empty set
bool the carrier of (RealVectSpace the carrier of X) is non empty set
[: the carrier of X,REAL:] is non empty V152() V153() V154() set
bool [: the carrier of X,REAL:] 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 REAL -valued Function-like total quasi_total V152() V153() V154() Element of bool [: the carrier of X,REAL:] : ( b₁ is continuous & 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),(RealVectSpace the carrier of X)) is Element of (X)
Add_ ((X),(RealVectSpace 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),(RealVectSpace the carrier of X)) is non empty Relation-like [:REAL,(X):] -defined (X) -valued Function-like total quasi_total Element of bool [:[:REAL,(X):],(X):]
[:REAL,(X):] is non empty set
[:[:REAL,(X):],(X):] is non empty set
bool [:[:REAL,(X):],(X):] is non empty set
RLSStruct(# (X),(Zero_ ((X),(RealVectSpace the carrier of X))),(Add_ ((X),(RealVectSpace the carrier of X))),(Mult_ ((X),(RealVectSpace the carrier of X))) #) is strict RLSStruct
X is non empty TopSpace-like TopStruct
(X) is non empty linearly-closed Element of bool the carrier of (RealVectSpace the carrier of X)
the carrier of X is non empty set
RealVectSpace the carrier of X is non empty right_complementable strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RLSStruct
Funcs ( the carrier of X,REAL) is non empty functional FUNCTION_DOMAIN of the carrier of X, REAL
RealFuncZero the carrier of X is Relation-like the carrier of X -defined REAL -valued Function-like total quasi_total V152() V153() V154() Element of Funcs ( the carrier of X,REAL)
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 V152() V153() V154() V155() Element of bool [: the carrier of X,NAT:]
[: the carrier of X,NAT:] is non empty RAT -valued INT -valued V152() V153() V154() V155() set
bool [: the carrier of X,NAT:] is non empty set
RealFuncAdd the carrier of X is non empty Relation-like [:(Funcs ( the carrier of X,REAL)),(Funcs ( the carrier of X,REAL)):] -defined Funcs ( the carrier of X,REAL) -valued Function-like total quasi_total Element of bool [:[:(Funcs ( the carrier of X,REAL)),(Funcs ( the carrier of X,REAL)):],(Funcs ( the carrier of X,REAL)):]
[:(Funcs ( the carrier of X,REAL)),(Funcs ( the carrier of X,REAL)):] is non empty set
[:[:(Funcs ( the carrier of X,REAL)),(Funcs ( the carrier of X,REAL)):],(Funcs ( the carrier of X,REAL)):] is non empty set
bool [:[:(Funcs ( the carrier of X,REAL)),(Funcs ( the carrier of X,REAL)):],(Funcs ( the carrier of X,REAL)):] is non empty set
RealFuncExtMult the carrier of X is non empty Relation-like [:REAL,(Funcs ( the carrier of X,REAL)):] -defined Funcs ( the carrier of X,REAL) -valued Function-like total quasi_total Element of bool [:[:REAL,(Funcs ( the carrier of X,REAL)):],(Funcs ( the carrier of X,REAL)):]
[:REAL,(Funcs ( the carrier of X,REAL)):] is non empty set
[:[:REAL,(Funcs ( the carrier of X,REAL)):],(Funcs ( the carrier of X,REAL)):] is non empty set
bool [:[:REAL,(Funcs ( the carrier of X,REAL)):],(Funcs ( the carrier of X,REAL)):] is non empty set
RLSStruct(# (Funcs ( the carrier of X,REAL)),(RealFuncZero the carrier of X),(RealFuncAdd the carrier of X),(RealFuncExtMult the carrier of X) #) is strict RLSStruct
the carrier of (RealVectSpace the carrier of X) is non empty set
bool the carrier of (RealVectSpace the carrier of X) is non empty set
[: the carrier of X,REAL:] is non empty V152() V153() V154() set
bool [: the carrier of X,REAL:] 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 REAL -valued Function-like total quasi_total V152() V153() V154() Element of bool [: the carrier of X,REAL:] : ( b₁ is continuous & 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
BoundedFunctions the carrier of X is non empty add-closed having-inverse additively-closed multiplicatively-closed additively-linearly-closed Element of bool the carrier of (RAlgebra the carrier of X)
RAlgebra the carrier of X is non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative unital associative commutative right-distributive right_unital well-unital left_unital strict vector-associative AlgebraStr
RealFuncMult the carrier of X is non empty Relation-like [:(Funcs ( the carrier of X,REAL)),(Funcs ( the carrier of X,REAL)):] -defined Funcs ( the carrier of X,REAL) -valued Function-like total quasi_total Element of bool [:[:(Funcs ( the carrier of X,REAL)),(Funcs ( the carrier of X,REAL)):],(Funcs ( the carrier of X,REAL)):]
RealFuncUnit the carrier of X is Relation-like the carrier of X -defined REAL -valued Function-like total quasi_total V152() V153() V154() Element of Funcs ( the carrier of X,REAL)
the carrier of X --> 1 is non empty Relation-like the carrier of X -defined NAT -valued RAT -valued INT -valued Function-like total quasi_total V152() V153() V154() V155() Element of bool [: the carrier of X,NAT:]
AlgebraStr(# (Funcs ( the carrier of X,REAL)),(RealFuncMult the carrier of X),(RealFuncAdd the carrier of X),(RealFuncExtMult the carrier of X),(RealFuncUnit the carrier of X),(RealFuncZero the carrier of X) #) is strict AlgebraStr
the carrier of (RAlgebra the carrier of X) is non empty set
bool the carrier of (RAlgebra the carrier of X) is non empty set
{ b₁ where b₁ is non empty Relation-like the carrier of X -defined REAL -valued Function-like total quasi_total V152() V153() V154() Element of bool [: the carrier of X,REAL:] : b₁ | the carrier of X is bounded } is set
X is set
F is non empty Relation-like the carrier of X -defined REAL -valued Function-like total quasi_total V152() V153() V154() Element of bool [: the carrier of X,REAL:]
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 .: G is non empty V185() V186() V187() Element of bool REAL
aFB is non empty Element of bool the carrier of X
bool aFB is non empty set
GB is non empty V185() V186() V187() bounded_below bounded_above real-bounded Element of bool REAL
inf GB is V11() real ext-real set
sup GB is V11() real ext-real set
[.(inf GB),(sup GB).] is compact closed V185() V186() V187() V268() Element of bool REAL
{ b₁ where b₁ is V11() real ext-real Element of REAL : ( inf GB <= b₁ & b₁ <= sup GB ) } is set
aFB is V11() real ext-real set
abs aFB is V11() real ext-real Element of REAL
aF is V11() real ext-real set
abs aF is V11() real ext-real Element of REAL
(abs aFB) + (abs aF) is V11() real ext-real Element of REAL
((abs aFB) + (abs aF)) + 1 is V11() real ext-real Element of REAL
aG is V11() real ext-real set
- aG is V11() real ext-real set
r is Element of G
F . r is V11() real ext-real Element of REAL
[.aFB,aF.] is compact closed V185() V186() V187() V268() Element of bool REAL
{ b₁ where b₁ is V11() real ext-real Element of REAL : ( aFB <= b₁ & b₁ <= aF ) } is set
s1 is V11() real ext-real Element of REAL
- (abs aFB) is V11() real ext-real Element of REAL
(- (abs aFB)) - (abs aF) is V11() real ext-real Element of REAL
- (abs aF) is V11() real ext-real set
(- (abs aFB)) + (- (abs aF)) is V11() real ext-real set
aFB - 0 is V11() real ext-real Element of REAL
- 0 is empty epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural V11() real ext-real non positive non negative V185() V186() V187() V188() V189() V190() V191() bounded_below V268() set
aFB + (- 0) is V11() real ext-real set
((- (abs aFB)) - (abs aF)) - 1 is V11() real ext-real Element of REAL
- 1 is V11() real ext-real non positive set
((- (abs aFB)) - (abs aF)) + (- 1) is V11() real ext-real set
aF + 0 is V11() real ext-real Element of REAL
(abs aF) + (abs aFB) is V11() real ext-real Element of REAL
r is V11() real ext-real set
- r is V11() real ext-real set
s1 is Element of the carrier of X
F . s1 is V11() 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
(- 1) * 0 is empty epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural V11() real ext-real non positive non negative V185() V186() V187() V188() V189() V190() V191() bounded_below V268() Element of REAL
(- 1) * r is V11() real ext-real Element of REAL
the carrier of X \ G is Element of bool the carrier of X
s1 is V11() real ext-real set
- s1 is V11() real ext-real set
the carrier of X /\ (dom F) is Element of bool the carrier of X
uu1 is set
F . uu1 is V11() real ext-real set
F | the carrier of X is Relation-like the carrier of X -defined REAL -valued Function-like V152() V153() V154() Element of bool [: the carrier of X,REAL:]
uu1 is set
F . uu1 is V11() real ext-real set
the carrier of X /\ the carrier of X is set
uu1 is non empty Element of bool the carrier of X
bool uu1 is non empty set
X is non empty TopSpace-like TopStruct
the carrier of X is non empty set
BoundedFunctions the carrier of X is non empty add-closed having-inverse additively-closed multiplicatively-closed additively-linearly-closed Element of bool the carrier of (RAlgebra the carrier of X)
RAlgebra the carrier of X is non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative unital associative commutative right-distributive right_unital well-unital left_unital strict vector-associative AlgebraStr
Funcs ( the carrier of X,REAL) is non empty functional FUNCTION_DOMAIN of the carrier of X, REAL
RealFuncMult the carrier of X is non empty Relation-like [:(Funcs ( the carrier of X,REAL)),(Funcs ( the carrier of X,REAL)):] -defined Funcs ( the carrier of X,REAL) -valued Function-like total quasi_total Element of bool [:[:(Funcs ( the carrier of X,REAL)),(Funcs ( the carrier of X,REAL)):],(Funcs ( the carrier of X,REAL)):]
[:(Funcs ( the carrier of X,REAL)),(Funcs ( the carrier of X,REAL)):] is non empty set
[:[:(Funcs ( the carrier of X,REAL)),(Funcs ( the carrier of X,REAL)):],(Funcs ( the carrier of X,REAL)):] is non empty set
bool [:[:(Funcs ( the carrier of X,REAL)),(Funcs ( the carrier of X,REAL)):],(Funcs ( the carrier of X,REAL)):] is non empty set
RealFuncAdd the carrier of X is non empty Relation-like [:(Funcs ( the carrier of X,REAL)),(Funcs ( the carrier of X,REAL)):] -defined Funcs ( the carrier of X,REAL) -valued Function-like total quasi_total Element of bool [:[:(Funcs ( the carrier of X,REAL)),(Funcs ( the carrier of X,REAL)):],(Funcs ( the carrier of X,REAL)):]
RealFuncExtMult the carrier of X is non empty Relation-like [:REAL,(Funcs ( the carrier of X,REAL)):] -defined Funcs ( the carrier of X,REAL) -valued Function-like total quasi_total Element of bool [:[:REAL,(Funcs ( the carrier of X,REAL)):],(Funcs ( the carrier of X,REAL)):]
[:REAL,(Funcs ( the carrier of X,REAL)):] is non empty set
[:[:REAL,(Funcs ( the carrier of X,REAL)):],(Funcs ( the carrier of X,REAL)):] is non empty set
bool [:[:REAL,(Funcs ( the carrier of X,REAL)):],(Funcs ( the carrier of X,REAL)):] is non empty set
RealFuncUnit the carrier of X is Relation-like the carrier of X -defined REAL -valued Function-like total quasi_total V152() V153() V154() Element of Funcs ( the carrier of X,REAL)
the carrier of X --> 1 is non empty Relation-like the carrier of X -defined NAT -valued RAT -valued INT -valued Function-like total quasi_total V152() V153() V154() V155() Element of bool [: the carrier of X,NAT:]
[: the carrier of X,NAT:] is non empty RAT -valued INT -valued V152() V153() V154() V155() set
bool [: the carrier of X,NAT:] is non empty set
RealFuncZero the carrier of X is Relation-like the carrier of X -defined REAL -valued Function-like total quasi_total V152() V153() V154() Element of Funcs ( the carrier of X,REAL)
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 V152() V153() V154() V155() Element of bool [: the carrier of X,NAT:]
AlgebraStr(# (Funcs ( the carrier of X,REAL)),(RealFuncMult the carrier of X),(RealFuncAdd the carrier of X),(RealFuncExtMult the carrier of X),(RealFuncUnit the carrier of X),(RealFuncZero the carrier of X) #) is strict AlgebraStr
the carrier of (RAlgebra the carrier of X) is non empty set
bool the carrier of (RAlgebra the carrier of X) is non empty set
[: the carrier of X,REAL:] is non empty V152() V153() V154() set
bool [: the carrier of X,REAL:] is non empty set
{ b₁ where b₁ is non empty Relation-like the carrier of X -defined REAL -valued Function-like total quasi_total V152() V153() V154() Element of bool [: the carrier of X,REAL:] : b₁ | the carrier of X is bounded } is set
BoundedFunctionsNorm the carrier of X is non empty Relation-like BoundedFunctions the carrier of X -defined REAL -valued Function-like total quasi_total V152() V153() V154() Element of bool [:(BoundedFunctions the carrier of X),REAL:]
[:(BoundedFunctions the carrier of X),REAL:] is non empty V152() V153() V154() set
bool [:(BoundedFunctions the carrier of X),REAL:] is non empty set
(X) is non empty linearly-closed Element of bool the carrier of (RealVectSpace the carrier of X)
RealVectSpace the carrier of X is non empty right_complementable strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RLSStruct
RLSStruct(# (Funcs ( the carrier of X,REAL)),(RealFuncZero the carrier of X),(RealFuncAdd the carrier of X),(RealFuncExtMult the carrier of X) #) is strict RLSStruct
the carrier of (RealVectSpace the carrier of X) is non empty set
bool the carrier of (RealVectSpace 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 REAL -valued Function-like total quasi_total V152() V153() V154() Element of bool [: the carrier of X,REAL:] : ( b₁ is continuous & 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
(BoundedFunctionsNorm the carrier of X) | (X) is Relation-like BoundedFunctions the carrier of X -defined REAL -valued Function-like V152() V153() V154() Element of bool [:(BoundedFunctions the carrier of X),REAL:]
[:(X),REAL:] is non empty V152() V153() V154() 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 (RealVectSpace the carrier of X)
the carrier of X is non empty set
RealVectSpace the carrier of X is non empty right_complementable strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RLSStruct
Funcs ( the carrier of X,REAL) is non empty functional FUNCTION_DOMAIN of the carrier of X, REAL
RealFuncZero the carrier of X is Relation-like the carrier of X -defined REAL -valued Function-like total quasi_total V152() V153() V154() Element of Funcs ( the carrier of X,REAL)
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 V152() V153() V154() V155() Element of bool [: the carrier of X,NAT:]
[: the carrier of X,NAT:] is non empty RAT -valued INT -valued V152() V153() V154() V155() set
bool [: the carrier of X,NAT:] is non empty set
RealFuncAdd the carrier of X is non empty Relation-like [:(Funcs ( the carrier of X,REAL)),(Funcs ( the carrier of X,REAL)):] -defined Funcs ( the carrier of X,REAL) -valued Function-like total quasi_total Element of bool [:[:(Funcs ( the carrier of X,REAL)),(Funcs ( the carrier of X,REAL)):],(Funcs ( the carrier of X,REAL)):]
[:(Funcs ( the carrier of X,REAL)),(Funcs ( the carrier of X,REAL)):] is non empty set
[:[:(Funcs ( the carrier of X,REAL)),(Funcs ( the carrier of X,REAL)):],(Funcs ( the carrier of X,REAL)):] is non empty set
bool [:[:(Funcs ( the carrier of X,REAL)),(Funcs ( the carrier of X,REAL)):],(Funcs ( the carrier of X,REAL)):] is non empty set
RealFuncExtMult the carrier of X is non empty Relation-like [:REAL,(Funcs ( the carrier of X,REAL)):] -defined Funcs ( the carrier of X,REAL) -valued Function-like total quasi_total Element of bool [:[:REAL,(Funcs ( the carrier of X,REAL)):],(Funcs ( the carrier of X,REAL)):]
[:REAL,(Funcs ( the carrier of X,REAL)):] is non empty set
[:[:REAL,(Funcs ( the carrier of X,REAL)):],(Funcs ( the carrier of X,REAL)):] is non empty set
bool [:[:REAL,(Funcs ( the carrier of X,REAL)):],(Funcs ( the carrier of X,REAL)):] is non empty set
RLSStruct(# (Funcs ( the carrier of X,REAL)),(RealFuncZero the carrier of X),(RealFuncAdd the carrier of X),(RealFuncExtMult the carrier of X) #) is strict RLSStruct
the carrier of (RealVectSpace the carrier of X) is non empty set
bool the carrier of (RealVectSpace the carrier of X) is non empty set
[: the carrier of X,REAL:] is non empty V152() V153() V154() set
bool [: the carrier of X,REAL:] 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 REAL -valued Function-like total quasi_total V152() V153() V154() Element of bool [: the carrier of X,REAL:] : ( b₁ is continuous & 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),(RealVectSpace the carrier of X)) is Element of (X)
Add_ ((X),(RealVectSpace 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),(RealVectSpace the carrier of X)) is non empty Relation-like [:REAL,(X):] -defined (X) -valued Function-like total quasi_total Element of bool [:[:REAL,(X):],(X):]
[:REAL,(X):] is non empty set
[:[:REAL,(X):],(X):] is non empty set
bool [:[:REAL,(X):],(X):] is non empty set
(X) is non empty Relation-like (X) -defined REAL -valued Function-like total quasi_total V152() V153() V154() Element of bool [:(X),REAL:]
[:(X),REAL:] is non empty V152() V153() V154() set
bool [:(X),REAL:] is non empty set
BoundedFunctions the carrier of X is non empty add-closed having-inverse additively-closed multiplicatively-closed additively-linearly-closed Element of bool the carrier of (RAlgebra the carrier of X)
RAlgebra the carrier of X is non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative unital associative commutative right-distributive right_unital well-unital left_unital strict vector-associative AlgebraStr
RealFuncMult the carrier of X is non empty Relation-like [:(Funcs ( the carrier of X,REAL)),(Funcs ( the carrier of X,REAL)):] -defined Funcs ( the carrier of X,REAL) -valued Function-like total quasi_total Element of bool [:[:(Funcs ( the carrier of X,REAL)),(Funcs ( the carrier of X,REAL)):],(Funcs ( the carrier of X,REAL)):]
RealFuncUnit the carrier of X is Relation-like the carrier of X -defined REAL -valued Function-like total quasi_total V152() V153() V154() Element of Funcs ( the carrier of X,REAL)
the carrier of X --> 1 is non empty Relation-like the carrier of X -defined NAT -valued RAT -valued INT -valued Function-like total quasi_total V152() V153() V154() V155() Element of bool [: the carrier of X,NAT:]
AlgebraStr(# (Funcs ( the carrier of X,REAL)),(RealFuncMult the carrier of X),(RealFuncAdd the carrier of X),(RealFuncExtMult the carrier of X),(RealFuncUnit the carrier of X),(RealFuncZero the carrier of X) #) is strict AlgebraStr
the carrier of (RAlgebra the carrier of X) is non empty set
bool the carrier of (RAlgebra the carrier of X) is non empty set
{ b₁ where b₁ is non empty Relation-like the carrier of X -defined REAL -valued Function-like total quasi_total V152() V153() V154() Element of bool [: the carrier of X,REAL:] : b₁ | the carrier of X is bounded } is set
BoundedFunctionsNorm the carrier of X is non empty Relation-like BoundedFunctions the carrier of X -defined REAL -valued Function-like total quasi_total V152() V153() V154() Element of bool [:(BoundedFunctions the carrier of X),REAL:]
[:(BoundedFunctions the carrier of X),REAL:] is non empty V152() V153() V154() set
bool [:(BoundedFunctions the carrier of X),REAL:] is non empty set
(BoundedFunctionsNorm the carrier of X) | (X) is Relation-like BoundedFunctions the carrier of X -defined REAL -valued Function-like V152() V153() V154() Element of bool [:(BoundedFunctions the carrier of X),REAL:]
NORMSTR(# (X),(Zero_ ((X),(RealVectSpace the carrier of X))),(Add_ ((X),(RealVectSpace the carrier of X))),(Mult_ ((X),(RealVectSpace the carrier of X))),(X) #) is strict NORMSTR
X is non empty TopSpace-like TopStruct
(X) is non empty NORMSTR
(X) is non empty linearly-closed Element of bool the carrier of (RealVectSpace the carrier of X)
the carrier of X is non empty set
RealVectSpace the carrier of X is non empty right_complementable strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RLSStruct
Funcs ( the carrier of X,REAL) is non empty functional FUNCTION_DOMAIN of the carrier of X, REAL
RealFuncZero the carrier of X is Relation-like the carrier of X -defined REAL -valued Function-like total quasi_total V152() V153() V154() Element of Funcs ( the carrier of X,REAL)
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 V152() V153() V154() V155() Element of bool [: the carrier of X,NAT:]
[: the carrier of X,NAT:] is non empty RAT -valued INT -valued V152() V153() V154() V155() set
bool [: the carrier of X,NAT:] is non empty set
RealFuncAdd the carrier of X is non empty Relation-like [:(Funcs ( the carrier of X,REAL)),(Funcs ( the carrier of X,REAL)):] -defined Funcs ( the carrier of X,REAL) -valued Function-like total quasi_total Element of bool [:[:(Funcs ( the carrier of X,REAL)),(Funcs ( the carrier of X,REAL)):],(Funcs ( the carrier of X,REAL)):]
[:(Funcs ( the carrier of X,REAL)),(Funcs ( the carrier of X,REAL)):] is non empty set
[:[:(Funcs ( the carrier of X,REAL)),(Funcs ( the carrier of X,REAL)):],(Funcs ( the carrier of X,REAL)):] is non empty set
bool [:[:(Funcs ( the carrier of X,REAL)),(Funcs ( the carrier of X,REAL)):],(Funcs ( the carrier of X,REAL)):] is non empty set
RealFuncExtMult the carrier of X is non empty Relation-like [:REAL,(Funcs ( the carrier of X,REAL)):] -defined Funcs ( the carrier of X,REAL) -valued Function-like total quasi_total Element of bool [:[:REAL,(Funcs ( the carrier of X,REAL)):],(Funcs ( the carrier of X,REAL)):]
[:REAL,(Funcs ( the carrier of X,REAL)):] is non empty set
[:[:REAL,(Funcs ( the carrier of X,REAL)):],(Funcs ( the carrier of X,REAL)):] is non empty set
bool [:[:REAL,(Funcs ( the carrier of X,REAL)):],(Funcs ( the carrier of X,REAL)):] is non empty set
RLSStruct(# (Funcs ( the carrier of X,REAL)),(RealFuncZero the carrier of X),(RealFuncAdd the carrier of X),(RealFuncExtMult the carrier of X) #) is strict RLSStruct
the carrier of (RealVectSpace the carrier of X) is non empty set
bool the carrier of (RealVectSpace the carrier of X) is non empty set
[: the carrier of X,REAL:] is non empty V152() V153() V154() set
bool [: the carrier of X,REAL:] 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 REAL -valued Function-like total quasi_total V152() V153() V154() Element of bool [: the carrier of X,REAL:] : ( b₁ is continuous & 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),(RealVectSpace the carrier of X)) is Element of (X)
Add_ ((X),(RealVectSpace 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),(RealVectSpace the carrier of X)) is non empty Relation-like [:REAL,(X):] -defined (X) -valued Function-like total quasi_total Element of bool [:[:REAL,(X):],(X):]
[:REAL,(X):] is non empty set
[:[:REAL,(X):],(X):] is non empty set
bool [:[:REAL,(X):],(X):] is non empty set
(X) is non empty Relation-like (X) -defined REAL -valued Function-like total quasi_total V152() V153() V154() Element of bool [:(X),REAL:]
[:(X),REAL:] is non empty V152() V153() V154() set
bool [:(X),REAL:] is non empty set
BoundedFunctions the carrier of X is non empty add-closed having-inverse additively-closed multiplicatively-closed additively-linearly-closed Element of bool the carrier of (RAlgebra the carrier of X)
RAlgebra the carrier of X is non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative unital associative commutative right-distributive right_unital well-unital left_unital strict vector-associative AlgebraStr
RealFuncMult the carrier of X is non empty Relation-like [:(Funcs ( the carrier of X,REAL)),(Funcs ( the carrier of X,REAL)):] -defined Funcs ( the carrier of X,REAL) -valued Function-like total quasi_total Element of bool [:[:(Funcs ( the carrier of X,REAL)),(Funcs ( the carrier of X,REAL)):],(Funcs ( the carrier of X,REAL)):]
RealFuncUnit the carrier of X is Relation-like the carrier of X -defined REAL -valued Function-like total quasi_total V152() V153() V154() Element of Funcs ( the carrier of X,REAL)
the carrier of X --> 1 is non empty Relation-like the carrier of X -defined NAT -valued RAT -valued INT -valued Function-like total quasi_total V152() V153() V154() V155() Element of bool [: the carrier of X,NAT:]
AlgebraStr(# (Funcs ( the carrier of X,REAL)),(RealFuncMult the carrier of X),(RealFuncAdd the carrier of X),(RealFuncExtMult the carrier of X),(RealFuncUnit the carrier of X),(RealFuncZero the carrier of X) #) is strict AlgebraStr
the carrier of (RAlgebra the carrier of X) is non empty set
bool the carrier of (RAlgebra the carrier of X) is non empty set
{ b₁ where b₁ is non empty Relation-like the carrier of X -defined REAL -valued Function-like total quasi_total V152() V153() V154() Element of bool [: the carrier of X,REAL:] : b₁ | the carrier of X is bounded } is set
BoundedFunctionsNorm the carrier of X is non empty Relation-like BoundedFunctions the carrier of X -defined REAL -valued Function-like total quasi_total V152() V153() V154() Element of bool [:(BoundedFunctions the carrier of X),REAL:]
[:(BoundedFunctions the carrier of X),REAL:] is non empty V152() V153() V154() set
bool [:(BoundedFunctions the carrier of X),REAL:] is non empty set
(BoundedFunctionsNorm the carrier of X) | (X) is Relation-like BoundedFunctions the carrier of X -defined REAL -valued Function-like V152() V153() V154() Element of bool [:(BoundedFunctions the carrier of X),REAL:]
NORMSTR(# (X),(Zero_ ((X),(RealVectSpace the carrier of X))),(Add_ ((X),(RealVectSpace the carrier of X))),(Mult_ ((X),(RealVectSpace the carrier of X))),(X) #) is strict NORMSTR
X is non empty TopSpace-like TopStruct
(X) is non empty linearly-closed Element of bool the carrier of (RealVectSpace the carrier of X)
the carrier of X is non empty set
RealVectSpace the carrier of X is non empty right_complementable strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RLSStruct
Funcs ( the carrier of X,REAL) is non empty functional FUNCTION_DOMAIN of the carrier of X, REAL
RealFuncZero the carrier of X is Relation-like the carrier of X -defined REAL -valued Function-like total quasi_total V152() V153() V154() Element of Funcs ( the carrier of X,REAL)
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 V152() V153() V154() V155() Element of bool [: the carrier of X,NAT:]
[: the carrier of X,NAT:] is non empty RAT -valued INT -valued V152() V153() V154() V155() set
bool [: the carrier of X,NAT:] is non empty set
RealFuncAdd the carrier of X is non empty Relation-like [:(Funcs ( the carrier of X,REAL)),(Funcs ( the carrier of X,REAL)):] -defined Funcs ( the carrier of X,REAL) -valued Function-like total quasi_total Element of bool [:[:(Funcs ( the carrier of X,REAL)),(Funcs ( the carrier of X,REAL)):],(Funcs ( the carrier of X,REAL)):]
[:(Funcs ( the carrier of X,REAL)),(Funcs ( the carrier of X,REAL)):] is non empty set
[:[:(Funcs ( the carrier of X,REAL)),(Funcs ( the carrier of X,REAL)):],(Funcs ( the carrier of X,REAL)):] is non empty set
bool [:[:(Funcs ( the carrier of X,REAL)),(Funcs ( the carrier of X,REAL)):],(Funcs ( the carrier of X,REAL)):] is non empty set
RealFuncExtMult the carrier of X is non empty Relation-like [:REAL,(Funcs ( the carrier of X,REAL)):] -defined Funcs ( the carrier of X,REAL) -valued Function-like total quasi_total Element of bool [:[:REAL,(Funcs ( the carrier of X,REAL)):],(Funcs ( the carrier of X,REAL)):]
[:REAL,(Funcs ( the carrier of X,REAL)):] is non empty set
[:[:REAL,(Funcs ( the carrier of X,REAL)):],(Funcs ( the carrier of X,REAL)):] is non empty set
bool [:[:REAL,(Funcs ( the carrier of X,REAL)):],(Funcs ( the carrier of X,REAL)):] is non empty set
RLSStruct(# (Funcs ( the carrier of X,REAL)),(RealFuncZero the carrier of X),(RealFuncAdd the carrier of X),(RealFuncExtMult the carrier of X) #) is strict RLSStruct
the carrier of (RealVectSpace the carrier of X) is non empty set
bool the carrier of (RealVectSpace the carrier of X) is non empty set
[: the carrier of X,REAL:] is non empty V152() V153() V154() set
bool [: the carrier of X,REAL:] 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 REAL -valued Function-like total quasi_total V152() V153() V154() Element of bool [: the carrier of X,REAL:] : ( b₁ is continuous & 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 add-closed having-inverse additively-closed multiplicatively-closed additively-linearly-closed Element of bool the carrier of (RAlgebra the carrier of X)
RAlgebra the carrier of X is non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative unital associative commutative right-distributive right_unital well-unital left_unital strict vector-associative AlgebraStr
RealFuncMult the carrier of X is non empty Relation-like [:(Funcs ( the carrier of X,REAL)),(Funcs ( the carrier of X,REAL)):] -defined Funcs ( the carrier of X,REAL) -valued Function-like total quasi_total Element of bool [:[:(Funcs ( the carrier of X,REAL)),(Funcs ( the carrier of X,REAL)):],(Funcs ( the carrier of X,REAL)):]
RealFuncUnit the carrier of X is Relation-like the carrier of X -defined REAL -valued Function-like total quasi_total V152() V153() V154() Element of Funcs ( the carrier of X,REAL)
the carrier of X --> 1 is non empty Relation-like the carrier of X -defined NAT -valued RAT -valued INT -valued Function-like total quasi_total V152() V153() V154() V155() Element of bool [: the carrier of X,NAT:]
AlgebraStr(# (Funcs ( the carrier of X,REAL)),(RealFuncMult the carrier of X),(RealFuncAdd the carrier of X),(RealFuncExtMult the carrier of X),(RealFuncUnit the carrier of X),(RealFuncZero the carrier of X) #) is strict AlgebraStr
the carrier of (RAlgebra the carrier of X) is non empty set
bool the carrier of (RAlgebra the carrier of X) is non empty set
{ b₁ where b₁ is non empty Relation-like the carrier of X -defined REAL -valued Function-like total quasi_total V152() V153() V154() continuous Element of bool [: the carrier of X,REAL:] : verum } is set
X is set
F is non empty Relation-like the carrier of X -defined REAL -valued Function-like total quasi_total V152() V153() V154() Element of bool [: the carrier of X,REAL:]
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 RLSStruct
(X) is non empty linearly-closed Element of bool the carrier of (RealVectSpace the carrier of X)
the carrier of X is non empty set
RealVectSpace the carrier of X is non empty right_complementable strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RLSStruct
Funcs ( the carrier of X,REAL) is non empty functional FUNCTION_DOMAIN of the carrier of X, REAL
RealFuncZero the carrier of X is Relation-like the carrier of X -defined REAL -valued Function-like total quasi_total V152() V153() V154() Element of Funcs ( the carrier of X,REAL)
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 V152() V153() V154() V155() Element of bool [: the carrier of X,NAT:]
[: the carrier of X,NAT:] is non empty RAT -valued INT -valued V152() V153() V154() V155() set
bool [: the carrier of X,NAT:] is non empty set
RealFuncAdd the carrier of X is non empty Relation-like [:(Funcs ( the carrier of X,REAL)),(Funcs ( the carrier of X,REAL)):] -defined Funcs ( the carrier of X,REAL) -valued Function-like total quasi_total Element of bool [:[:(Funcs ( the carrier of X,REAL)),(Funcs ( the carrier of X,REAL)):],(Funcs ( the carrier of X,REAL)):]
[:(Funcs ( the carrier of X,REAL)),(Funcs ( the carrier of X,REAL)):] is non empty set
[:[:(Funcs ( the carrier of X,REAL)),(Funcs ( the carrier of X,REAL)):],(Funcs ( the carrier of X,REAL)):] is non empty set
bool [:[:(Funcs ( the carrier of X,REAL)),(Funcs ( the carrier of X,REAL)):],(Funcs ( the carrier of X,REAL)):] is non empty set
RealFuncExtMult the carrier of X is non empty Relation-like [:REAL,(Funcs ( the carrier of X,REAL)):] -defined Funcs ( the carrier of X,REAL) -valued Function-like total quasi_total Element of bool [:[:REAL,(Funcs ( the carrier of X,REAL)):],(Funcs ( the carrier of X,REAL)):]
[:REAL,(Funcs ( the carrier of X,REAL)):] is non empty set
[:[:REAL,(Funcs ( the carrier of X,REAL)):],(Funcs ( the carrier of X,REAL)):] is non empty set
bool [:[:REAL,(Funcs ( the carrier of X,REAL)):],(Funcs ( the carrier of X,REAL)):] is non empty set
RLSStruct(# (Funcs ( the carrier of X,REAL)),(RealFuncZero the carrier of X),(RealFuncAdd the carrier of X),(RealFuncExtMult the carrier of X) #) is strict RLSStruct
the carrier of (RealVectSpace the carrier of X) is non empty set
bool the carrier of (RealVectSpace the carrier of X) is non empty set
[: the carrier of X,REAL:] is non empty V152() V153() V154() set
bool [: the carrier of X,REAL:] 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 REAL -valued Function-like total quasi_total V152() V153() V154() Element of bool [: the carrier of X,REAL:] : ( b₁ is continuous & 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),(RealVectSpace the carrier of X)) is Element of (X)
Add_ ((X),(RealVectSpace 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),(RealVectSpace the carrier of X)) is non empty Relation-like [:REAL,(X):] -defined (X) -valued Function-like total quasi_total Element of bool [:[:REAL,(X):],(X):]
[:REAL,(X):] is non empty set
[:[:REAL,(X):],(X):] is non empty set
bool [:[:REAL,(X):],(X):] is non empty set
RLSStruct(# (X),(Zero_ ((X),(RealVectSpace the carrier of X))),(Add_ ((X),(RealVectSpace the carrier of X))),(Mult_ ((X),(RealVectSpace the carrier of X))) #) is strict RLSStruct
0. (X) is V49((X)) right_complementable Element of the carrier of (X)
the carrier of (X) is non empty set
the ZeroF of (X) is right_complementable Element of the carrier of (X)
(X,0) is non empty Relation-like the carrier of X -defined REAL -valued Function-like total quasi_total V152() V153() V154() continuous Element of bool [: the carrier of X,REAL:]
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 V152() V153() V154() V155() Element of bool [: the carrier of X,{0}:]
{0} is non empty V185() V186() V187() V188() V189() V190() left_end bounded_below set
[: the carrier of X,{0}:] is non empty RAT -valued INT -valued V152() V153() V154() V155() set
bool [: the carrier of X,{0}:] is non empty set
0. (RealVectSpace the carrier of X) is V49( RealVectSpace the carrier of X) right_complementable Element of the carrier of (RealVectSpace the carrier of X)
the ZeroF of (RealVectSpace the carrier of X) is right_complementable Element of the carrier of (RealVectSpace the carrier of X)
X is non empty TopSpace-like TopStruct
(X) is non empty strict NORMSTR
(X) is non empty linearly-closed Element of bool the carrier of (RealVectSpace the carrier of X)
the carrier of X is non empty set
RealVectSpace the carrier of X is non empty right_complementable strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RLSStruct
Funcs ( the carrier of X,REAL) is non empty functional FUNCTION_DOMAIN of the carrier of X, REAL
RealFuncZero the carrier of X is Relation-like the carrier of X -defined REAL -valued Function-like total quasi_total V152() V153() V154() Element of Funcs ( the carrier of X,REAL)
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 V152() V153() V154() V155() Element of bool [: the carrier of X,NAT:]
[: the carrier of X,NAT:] is non empty RAT -valued INT -valued V152() V153() V154() V155() set
bool [: the carrier of X,NAT:] is non empty set
RealFuncAdd the carrier of X is non empty Relation-like [:(Funcs ( the carrier of X,REAL)),(Funcs ( the carrier of X,REAL)):] -defined Funcs ( the carrier of X,REAL) -valued Function-like total quasi_total Element of bool [:[:(Funcs ( the carrier of X,REAL)),(Funcs ( the carrier of X,REAL)):],(Funcs ( the carrier of X,REAL)):]
[:(Funcs ( the carrier of X,REAL)),(Funcs ( the carrier of X,REAL)):] is non empty set
[:[:(Funcs ( the carrier of X,REAL)),(Funcs ( the carrier of X,REAL)):],(Funcs ( the carrier of X,REAL)):] is non empty set
bool [:[:(Funcs ( the carrier of X,REAL)),(Funcs ( the carrier of X,REAL)):],(Funcs ( the carrier of X,REAL)):] is non empty set
RealFuncExtMult the carrier of X is non empty Relation-like [:REAL,(Funcs ( the carrier of X,REAL)):] -defined Funcs ( the carrier of X,REAL) -valued Function-like total quasi_total Element of bool [:[:REAL,(Funcs ( the carrier of X,REAL)):],(Funcs ( the carrier of X,REAL)):]
[:REAL,(Funcs ( the carrier of X,REAL)):] is non empty set
[:[:REAL,(Funcs ( the carrier of X,REAL)):],(Funcs ( the carrier of X,REAL)):] is non empty set
bool [:[:REAL,(Funcs ( the carrier of X,REAL)):],(Funcs ( the carrier of X,REAL)):] is non empty set
RLSStruct(# (Funcs ( the carrier of X,REAL)),(RealFuncZero the carrier of X),(RealFuncAdd the carrier of X),(RealFuncExtMult the carrier of X) #) is strict RLSStruct
the carrier of (RealVectSpace the carrier of X) is non empty set
bool the carrier of (RealVectSpace the carrier of X) is non empty set
[: the carrier of X,REAL:] is non empty V152() V153() V154() set
bool [: the carrier of X,REAL:] 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 REAL -valued Function-like total quasi_total V152() V153() V154() Element of bool [: the carrier of X,REAL:] : ( b₁ is continuous & 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),(RealVectSpace the carrier of X)) is Element of (X)
Add_ ((X),(RealVectSpace 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),(RealVectSpace the carrier of X)) is non empty Relation-like [:REAL,(X):] -defined (X) -valued Function-like total quasi_total Element of bool [:[:REAL,(X):],(X):]
[:REAL,(X):] is non empty set
[:[:REAL,(X):],(X):] is non empty set
bool [:[:REAL,(X):],(X):] is non empty set
(X) is non empty Relation-like (X) -defined REAL -valued Function-like total quasi_total V152() V153() V154() Element of bool [:(X),REAL:]
[:(X),REAL:] is non empty V152() V153() V154() set
bool [:(X),REAL:] is non empty set
BoundedFunctions the carrier of X is non empty add-closed having-inverse additively-closed multiplicatively-closed additively-linearly-closed Element of bool the carrier of (RAlgebra the carrier of X)
RAlgebra the carrier of X is non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative unital associative commutative right-distributive right_unital well-unital left_unital strict vector-associative AlgebraStr
RealFuncMult the carrier of X is non empty Relation-like [:(Funcs ( the carrier of X,REAL)),(Funcs ( the carrier of X,REAL)):] -defined Funcs ( the carrier of X,REAL) -valued Function-like total quasi_total Element of bool [:[:(Funcs ( the carrier of X,REAL)),(Funcs ( the carrier of X,REAL)):],(Funcs ( the carrier of X,REAL)):]
RealFuncUnit the carrier of X is Relation-like the carrier of X -defined REAL -valued Function-like total quasi_total V152() V153() V154() Element of Funcs ( the carrier of X,REAL)
the carrier of X --> 1 is non empty Relation-like the carrier of X -defined NAT -valued RAT -valued INT -valued Function-like total quasi_total V152() V153() V154() V155() Element of bool [: the carrier of X,NAT:]
AlgebraStr(# (Funcs ( the carrier of X,REAL)),(RealFuncMult the carrier of X),(RealFuncAdd the carrier of X),(RealFuncExtMult the carrier of X),(RealFuncUnit the carrier of X),(RealFuncZero the carrier of X) #) is strict AlgebraStr
the carrier of (RAlgebra the carrier of X) is non empty set
bool the carrier of (RAlgebra the carrier of X) is non empty set
{ b₁ where b₁ is non empty Relation-like the carrier of X -defined REAL -valued Function-like total quasi_total V152() V153() V154() Element of bool [: the carrier of X,REAL:] : b₁ | the carrier of X is bounded } is set
BoundedFunctionsNorm the carrier of X is non empty Relation-like BoundedFunctions the carrier of X -defined REAL -valued Function-like total quasi_total V152() V153() V154() Element of bool [:(BoundedFunctions the carrier of X),REAL:]
[:(BoundedFunctions the carrier of X),REAL:] is non empty V152() V153() V154() set
bool [:(BoundedFunctions the carrier of X),REAL:] is non empty set
(BoundedFunctionsNorm the carrier of X) | (X) is Relation-like BoundedFunctions the carrier of X -defined REAL -valued Function-like V152() V153() V154() Element of bool [:(BoundedFunctions the carrier of X),REAL:]
NORMSTR(# (X),(Zero_ ((X),(RealVectSpace the carrier of X))),(Add_ ((X),(RealVectSpace the carrier of X))),(Mult_ ((X),(RealVectSpace the carrier of X))),(X) #) is strict NORMSTR
0. (X) is V49((X)) 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 REAL -valued Function-like total quasi_total V152() V153() V154() continuous Element of bool [: the carrier of X,REAL:]
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 V152() V153() V154() V155() Element of bool [: the carrier of X,{0}:]
{0} is non empty V185() V186() V187() V188() V189() V190() left_end bounded_below set
[: the carrier of X,{0}:] is non empty RAT -valued INT -valued V152() V153() V154() V155() 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 RLSStruct
RLSStruct(# (X),(Zero_ ((X),(RealVectSpace the carrier of X))),(Add_ ((X),(RealVectSpace the carrier of X))),(Mult_ ((X),(RealVectSpace the carrier of X))) #) is strict RLSStruct
0. (X) is V49((X)) right_complementable Element of the carrier of (X)
the carrier of (X) is non empty set
the ZeroF of (X) is right_complementable Element of the carrier of (X)
X is non empty TopSpace-like TopStruct
(X) is non empty strict NORMSTR
(X) is non empty linearly-closed Element of bool the carrier of (RealVectSpace the carrier of X)
the carrier of X is non empty set
RealVectSpace the carrier of X is non empty right_complementable strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RLSStruct
Funcs ( the carrier of X,REAL) is non empty functional FUNCTION_DOMAIN of the carrier of X, REAL
RealFuncZero the carrier of X is Relation-like the carrier of X -defined REAL -valued Function-like total quasi_total V152() V153() V154() Element of Funcs ( the carrier of X,REAL)
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 V152() V153() V154() V155() Element of bool [: the carrier of X,NAT:]
[: the carrier of X,NAT:] is non empty RAT -valued INT -valued V152() V153() V154() V155() set
bool [: the carrier of X,NAT:] is non empty set
RealFuncAdd the carrier of X is non empty Relation-like [:(Funcs ( the carrier of X,REAL)),(Funcs ( the carrier of X,REAL)):] -defined Funcs ( the carrier of X,REAL) -valued Function-like total quasi_total Element of bool [:[:(Funcs ( the carrier of X,REAL)),(Funcs ( the carrier of X,REAL)):],(Funcs ( the carrier of X,REAL)):]
[:(Funcs ( the carrier of X,REAL)),(Funcs ( the carrier of X,REAL)):] is non empty set
[:[:(Funcs ( the carrier of X,REAL)),(Funcs ( the carrier of X,REAL)):],(Funcs ( the carrier of X,REAL)):] is non empty set
bool [:[:(Funcs ( the carrier of X,REAL)),(Funcs ( the carrier of X,REAL)):],(Funcs ( the carrier of X,REAL)):] is non empty set
RealFuncExtMult the carrier of X is non empty Relation-like [:REAL,(Funcs ( the carrier of X,REAL)):] -defined Funcs ( the carrier of X,REAL) -valued Function-like total quasi_total Element of bool [:[:REAL,(Funcs ( the carrier of X,REAL)):],(Funcs ( the carrier of X,REAL)):]
[:REAL,(Funcs ( the carrier of X,REAL)):] is non empty set
[:[:REAL,(Funcs ( the carrier of X,REAL)):],(Funcs ( the carrier of X,REAL)):] is non empty set
bool [:[:REAL,(Funcs ( the carrier of X,REAL)):],(Funcs ( the carrier of X,REAL)):] is non empty set
RLSStruct(# (Funcs ( the carrier of X,REAL)),(RealFuncZero the carrier of X),(RealFuncAdd the carrier of X),(RealFuncExtMult the carrier of X) #) is strict RLSStruct
the carrier of (RealVectSpace the carrier of X) is non empty set
bool the carrier of (RealVectSpace the carrier of X) is non empty set
[: the carrier of X,REAL:] is non empty V152() V153() V154() set
bool [: the carrier of X,REAL:] 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 REAL -valued Function-like total quasi_total V152() V153() V154() Element of bool [: the carrier of X,REAL:] : ( b₁ is continuous & 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),(RealVectSpace the carrier of X)) is Element of (X)
Add_ ((X),(RealVectSpace 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),(RealVectSpace the carrier of X)) is non empty Relation-like [:REAL,(X):] -defined (X) -valued Function-like total quasi_total Element of bool [:[:REAL,(X):],(X):]
[:REAL,(X):] is non empty set
[:[:REAL,(X):],(X):] is non empty set
bool [:[:REAL,(X):],(X):] is non empty set
(X) is non empty Relation-like (X) -defined REAL -valued Function-like total quasi_total V152() V153() V154() Element of bool [:(X),REAL:]
[:(X),REAL:] is non empty V152() V153() V154() set
bool [:(X),REAL:] is non empty set
BoundedFunctions the carrier of X is non empty add-closed having-inverse additively-closed multiplicatively-closed additively-linearly-closed Element of bool the carrier of (RAlgebra the carrier of X)
RAlgebra the carrier of X is non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative unital associative commutative right-distributive right_unital well-unital left_unital strict vector-associative AlgebraStr
RealFuncMult the carrier of X is non empty Relation-like [:(Funcs ( the carrier of X,REAL)),(Funcs ( the carrier of X,REAL)):] -defined Funcs ( the carrier of X,REAL) -valued Function-like total quasi_total Element of bool [:[:(Funcs ( the carrier of X,REAL)),(Funcs ( the carrier of X,REAL)):],(Funcs ( the carrier of X,REAL)):]
RealFuncUnit the carrier of X is Relation-like the carrier of X -defined REAL -valued Function-like total quasi_total V152() V153() V154() Element of Funcs ( the carrier of X,REAL)
the carrier of X --> 1 is non empty Relation-like the carrier of X -defined NAT -valued RAT -valued INT -valued Function-like total quasi_total V152() V153() V154() V155() Element of bool [: the carrier of X,NAT:]
AlgebraStr(# (Funcs ( the carrier of X,REAL)),(RealFuncMult the carrier of X),(RealFuncAdd the carrier of X),(RealFuncExtMult the carrier of X),(RealFuncUnit the carrier of X),(RealFuncZero the carrier of X) #) is strict AlgebraStr
the carrier of (RAlgebra the carrier of X) is non empty set
bool the carrier of (RAlgebra the carrier of X) is non empty set
{ b₁ where b₁ is non empty Relation-like the carrier of X -defined REAL -valued Function-like total quasi_total V152() V153() V154() Element of bool [: the carrier of X,REAL:] : b₁ | the carrier of X is bounded } is set
BoundedFunctionsNorm the carrier of X is non empty Relation-like BoundedFunctions the carrier of X -defined REAL -valued Function-like total quasi_total V152() V153() V154() Element of bool [:(BoundedFunctions the carrier of X),REAL:]
[:(BoundedFunctions the carrier of X),REAL:] is non empty V152() V153() V154() set
bool [:(BoundedFunctions the carrier of X),REAL:] is non empty set
(BoundedFunctionsNorm the carrier of X) | (X) is Relation-like BoundedFunctions the carrier of X -defined REAL -valued Function-like V152() V153() V154() Element of bool [:(BoundedFunctions the carrier of X),REAL:]
NORMSTR(# (X),(Zero_ ((X),(RealVectSpace the carrier of X))),(Add_ ((X),(RealVectSpace the carrier of X))),(Mult_ ((X),(RealVectSpace the carrier of X))),(X) #) is strict NORMSTR
the carrier of (X) is non empty set
R_Normed_Algebra_of_BoundedFunctions the carrier of X is non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital discerning reflexive RealNormSpace-like unital complete Normed_AlgebraStr
mult_ ((BoundedFunctions the carrier of X),(RAlgebra the carrier of X)) is non empty Relation-like [:(BoundedFunctions the carrier of X),(BoundedFunctions the carrier of X):] -defined BoundedFunctions the carrier of X -valued Function-like total quasi_total Element of bool [:[:(BoundedFunctions the carrier of X),(BoundedFunctions the carrier of X):],(BoundedFunctions the carrier of X):]
[:(BoundedFunctions the carrier of X),(BoundedFunctions the carrier of X):] is non empty set
[:[:(BoundedFunctions the carrier of X),(BoundedFunctions the carrier of X):],(BoundedFunctions the carrier of X):] is non empty set
bool [:[:(BoundedFunctions the carrier of X),(BoundedFunctions the carrier of X):],(BoundedFunctions the carrier of X):] is non empty set
Add_ ((BoundedFunctions the carrier of X),(RAlgebra the carrier of X)) is non empty Relation-like [:(BoundedFunctions the carrier of X),(BoundedFunctions the carrier of X):] -defined BoundedFunctions the carrier of X -valued Function-like total quasi_total Element of bool [:[:(BoundedFunctions the carrier of X),(BoundedFunctions the carrier of X):],(BoundedFunctions the carrier of X):]
Mult_ ((BoundedFunctions the carrier of X),(RAlgebra the carrier of X)) is non empty Relation-like [:REAL,(BoundedFunctions the carrier of X):] -defined BoundedFunctions the carrier of X -valued Function-like total quasi_total Element of bool [:[:REAL,(BoundedFunctions the carrier of X):],(BoundedFunctions the carrier of X):]
[:REAL,(BoundedFunctions the carrier of X):] is non empty set
[:[:REAL,(BoundedFunctions the carrier of X):],(BoundedFunctions the carrier of X):] is non empty set
bool [:[:REAL,(BoundedFunctions the carrier of X):],(BoundedFunctions the carrier of X):] is non empty set
One_ ((BoundedFunctions the carrier of X),(RAlgebra the carrier of X)) is Element of BoundedFunctions the carrier of X
Zero_ ((BoundedFunctions the carrier of X),(RAlgebra the carrier of X)) is Element of BoundedFunctions the carrier of X
Normed_AlgebraStr(# (BoundedFunctions the carrier of X),(mult_ ((BoundedFunctions the carrier of X),(RAlgebra the carrier of X))),(Add_ ((BoundedFunctions the carrier of X),(RAlgebra the carrier of X))),(Mult_ ((BoundedFunctions the carrier of X),(RAlgebra the carrier of X))),(One_ ((BoundedFunctions the carrier of X),(RAlgebra the carrier of X))),(Zero_ ((BoundedFunctions the carrier of X),(RAlgebra the carrier of X))),(BoundedFunctionsNorm the carrier of X) #) is strict Normed_AlgebraStr
the carrier of (R_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 right_complementable Element of the carrier of (R_Normed_Algebra_of_BoundedFunctions the carrier of X)
FB is right_complementable Element of the carrier of (R_Normed_Algebra_of_BoundedFunctions the carrier of X)
G + FB is right_complementable Element of the carrier of (R_Normed_Algebra_of_BoundedFunctions the carrier of X)
the addF of (R_Normed_Algebra_of_BoundedFunctions the carrier of X) is non empty Relation-like [: the carrier of (R_Normed_Algebra_of_BoundedFunctions the carrier of X), the carrier of (R_Normed_Algebra_of_BoundedFunctions the carrier of X):] -defined the carrier of (R_Normed_Algebra_of_BoundedFunctions the carrier of X) -valued Function-like total quasi_total Element of bool [:[: the carrier of (R_Normed_Algebra_of_BoundedFunctions the carrier of X), the carrier of (R_Normed_Algebra_of_BoundedFunctions the carrier of X):], the carrier of (R_Normed_Algebra_of_BoundedFunctions the carrier of X):]
[: the carrier of (R_Normed_Algebra_of_BoundedFunctions the carrier of X), the carrier of (R_Normed_Algebra_of_BoundedFunctions the carrier of X):] is non empty set
[:[: the carrier of (R_Normed_Algebra_of_BoundedFunctions the carrier of X), the carrier of (R_Normed_Algebra_of_BoundedFunctions the carrier of X):], the carrier of (R_Normed_Algebra_of_BoundedFunctions the carrier of X):] is non empty set
bool [:[: the carrier of (R_Normed_Algebra_of_BoundedFunctions the carrier of X), the carrier of (R_Normed_Algebra_of_BoundedFunctions the carrier of X):], the carrier of (R_Normed_Algebra_of_BoundedFunctions the carrier of X):] is non empty set
the addF of (R_Normed_Algebra_of_BoundedFunctions the carrier of X) . (G,FB) is right_complementable Element of the carrier of (R_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 (R_Normed_Algebra_of_BoundedFunctions the carrier of X) . [G,FB] is set
the addF of (RealVectSpace the carrier of X) is non empty Relation-like [: the carrier of (RealVectSpace the carrier of X), the carrier of (RealVectSpace the carrier of X):] -defined the carrier of (RealVectSpace the carrier of X) -valued Function-like total quasi_total Element of bool [:[: the carrier of (RealVectSpace the carrier of X), the carrier of (RealVectSpace the carrier of X):], the carrier of (RealVectSpace the carrier of X):]
[: the carrier of (RealVectSpace the carrier of X), the carrier of (RealVectSpace the carrier of X):] is non empty set
[:[: the carrier of (RealVectSpace the carrier of X), the carrier of (RealVectSpace the carrier of X):], the carrier of (RealVectSpace the carrier of X):] is non empty set
bool [:[: the carrier of (RealVectSpace the carrier of X), the carrier of (RealVectSpace the carrier of X):], the carrier of (RealVectSpace the carrier of X):] is non empty set
the addF of (RealVectSpace the carrier of X) || (X) is Relation-like Function-like set
the addF of (RealVectSpace 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 (RealVectSpace the carrier of X) || (X)) . [X,F] is set
the addF of (RAlgebra the carrier of X) is non empty Relation-like [: the carrier of (RAlgebra the carrier of X), the carrier of (RAlgebra the carrier of X):] -defined the carrier of (RAlgebra the carrier of X) -valued Function-like total quasi_total Element of bool [:[: the carrier of (RAlgebra the carrier of X), the carrier of (RAlgebra the carrier of X):], the carrier of (RAlgebra the carrier of X):]
[: the carrier of (RAlgebra the carrier of X), the carrier of (RAlgebra the carrier of X):] is non empty set
[:[: the carrier of (RAlgebra the carrier of X), the carrier of (RAlgebra the carrier of X):], the carrier of (RAlgebra the carrier of X):] is non empty set
bool [:[: the carrier of (RAlgebra the carrier of X), the carrier of (RAlgebra the carrier of X):], the carrier of (RAlgebra the carrier of X):] is non empty set
the addF of (RAlgebra the carrier of X) . [X,F] is set
the addF of (RAlgebra the carrier of X) || (BoundedFunctions the carrier of X) is Relation-like Function-like set
the addF of (RAlgebra the carrier of X) | [:(BoundedFunctions the carrier of X),(BoundedFunctions the carrier of X):] is Relation-like set
[G,FB] is Element of [: the carrier of (R_Normed_Algebra_of_BoundedFunctions the carrier of X), the carrier of (R_Normed_Algebra_of_BoundedFunctions the carrier of X):]
( the addF of (RAlgebra the carrier of X) || (BoundedFunctions the carrier of X)) . [G,FB] is set
X is non empty TopSpace-like TopStruct
(X) is non empty strict NORMSTR
(X) is non empty linearly-closed Element of bool the carrier of (RealVectSpace the carrier of X)
the carrier of X is non empty set
RealVectSpace the carrier of X is non empty right_complementable strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RLSStruct
Funcs ( the carrier of X,REAL) is non empty functional FUNCTION_DOMAIN of the carrier of X, REAL
RealFuncZero the carrier of X is Relation-like the carrier of X -defined REAL -valued Function-like total quasi_total V152() V153() V154() Element of Funcs ( the carrier of X,REAL)
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 V152() V153() V154() V155() Element of bool [: the carrier of X,NAT:]
[: the carrier of X,NAT:] is non empty RAT -valued INT -valued V152() V153() V154() V155() set
bool [: the carrier of X,NAT:] is non empty set
RealFuncAdd the carrier of X is non empty Relation-like [:(Funcs ( the carrier of X,REAL)),(Funcs ( the carrier of X,REAL)):] -defined Funcs ( the carrier of X,REAL) -valued Function-like total quasi_total Element of bool [:[:(Funcs ( the carrier of X,REAL)),(Funcs ( the carrier of X,REAL)):],(Funcs ( the carrier of X,REAL)):]
[:(Funcs ( the carrier of X,REAL)),(Funcs ( the carrier of X,REAL)):] is non empty set
[:[:(Funcs ( the carrier of X,REAL)),(Funcs ( the carrier of X,REAL)):],(Funcs ( the carrier of X,REAL)):] is non empty set
bool [:[:(Funcs ( the carrier of X,REAL)),(Funcs ( the carrier of X,REAL)):],(Funcs ( the carrier of X,REAL)):] is non empty set
RealFuncExtMult the carrier of X is non empty Relation-like [:REAL,(Funcs ( the carrier of X,REAL)):] -defined Funcs ( the carrier of X,REAL) -valued Function-like total quasi_total Element of bool [:[:REAL,(Funcs ( the carrier of X,REAL)):],(Funcs ( the carrier of X,REAL)):]
[:REAL,(Funcs ( the carrier of X,REAL)):] is non empty set
[:[:REAL,(Funcs ( the carrier of X,REAL)):],(Funcs ( the carrier of X,REAL)):] is non empty set
bool [:[:REAL,(Funcs ( the carrier of X,REAL)):],(Funcs ( the carrier of X,REAL)):] is non empty set
RLSStruct(# (Funcs ( the carrier of X,REAL)),(RealFuncZero the carrier of X),(RealFuncAdd the carrier of X),(RealFuncExtMult the carrier of X) #) is strict RLSStruct
the carrier of (RealVectSpace the carrier of X) is non empty set
bool the carrier of (RealVectSpace the carrier of X) is non empty set
[: the carrier of X,REAL:] is non empty V152() V153() V154() set
bool [: the carrier of X,REAL:] 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 REAL -valued Function-like total quasi_total V152() V153() V154() Element of bool [: the carrier of X,REAL:] : ( b₁ is continuous & 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),(RealVectSpace the carrier of X)) is Element of (X)
Add_ ((X),(RealVectSpace 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),(RealVectSpace the carrier of X)) is non empty Relation-like [:REAL,(X):] -defined (X) -valued Function-like total quasi_total Element of bool [:[:REAL,(X):],(X):]
[:REAL,(X):] is non empty set
[:[:REAL,(X):],(X):] is non empty set
bool [:[:REAL,(X):],(X):] is non empty set
(X) is non empty Relation-like (X) -defined REAL -valued Function-like total quasi_total V152() V153() V154() Element of bool [:(X),REAL:]
[:(X),REAL:] is non empty V152() V153() V154() set
bool [:(X),REAL:] is non empty set
BoundedFunctions the carrier of X is non empty add-closed having-inverse additively-closed multiplicatively-closed additively-linearly-closed Element of bool the carrier of (RAlgebra the carrier of X)
RAlgebra the carrier of X is non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative unital associative commutative right-distributive right_unital well-unital left_unital strict vector-associative AlgebraStr
RealFuncMult the carrier of X is non empty Relation-like [:(Funcs ( the carrier of X,REAL)),(Funcs ( the carrier of X,REAL)):] -defined Funcs ( the carrier of X,REAL) -valued Function-like total quasi_total Element of bool [:[:(Funcs ( the carrier of X,REAL)),(Funcs ( the carrier of X,REAL)):],(Funcs ( the carrier of X,REAL)):]
RealFuncUnit the carrier of X is Relation-like the carrier of X -defined REAL -valued Function-like total quasi_total V152() V153() V154() Element of Funcs ( the carrier of X,REAL)
the carrier of X --> 1 is non empty Relation-like the carrier of X -defined NAT -valued RAT -valued INT -valued Function-like total quasi_total V152() V153() V154() V155() Element of bool [: the carrier of X,NAT:]
AlgebraStr(# (Funcs ( the carrier of X,REAL)),(RealFuncMult the carrier of X),(RealFuncAdd the carrier of X),(RealFuncExtMult the carrier of X),(RealFuncUnit the carrier of X),(RealFuncZero the carrier of X) #) is strict AlgebraStr
the carrier of (RAlgebra the carrier of X) is non empty set
bool the carrier of (RAlgebra the carrier of X) is non empty set
{ b₁ where b₁ is non empty Relation-like the carrier of X -defined REAL -valued Function-like total quasi_total V152() V153() V154() Element of bool [: the carrier of X,REAL:] : b₁ | the carrier of X is bounded } is set
BoundedFunctionsNorm the carrier of X is non empty Relation-like BoundedFunctions the carrier of X -defined REAL -valued Function-like total quasi_total V152() V153() V154() Element of bool [:(BoundedFunctions the carrier of X),REAL:]
[:(BoundedFunctions the carrier of X),REAL:] is non empty V152() V153() V154() set
bool [:(BoundedFunctions the carrier of X),REAL:] is non empty set
(BoundedFunctionsNorm the carrier of X) | (X) is Relation-like BoundedFunctions the carrier of X -defined REAL -valued Function-like V152() V153() V154() Element of bool [:(BoundedFunctions the carrier of X),REAL:]
NORMSTR(# (X),(Zero_ ((X),(RealVectSpace the carrier of X))),(Add_ ((X),(RealVectSpace the carrier of X))),(Mult_ ((X),(RealVectSpace the carrier of X))),(X) #) is strict NORMSTR
the carrier of (X) is non empty set
R_Normed_Algebra_of_BoundedFunctions the carrier of X is non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital discerning reflexive RealNormSpace-like unital complete Normed_AlgebraStr
mult_ ((BoundedFunctions the carrier of X),(RAlgebra the carrier of X)) is non empty Relation-like [:(BoundedFunctions the carrier of X),(BoundedFunctions the carrier of X):] -defined BoundedFunctions the carrier of X -valued Function-like total quasi_total Element of bool [:[:(BoundedFunctions the carrier of X),(BoundedFunctions the carrier of X):],(BoundedFunctions the carrier of X):]
[:(BoundedFunctions the carrier of X),(BoundedFunctions the carrier of X):] is non empty set
[:[:(BoundedFunctions the carrier of X),(BoundedFunctions the carrier of X):],(BoundedFunctions the carrier of X):] is non empty set
bool [:[:(BoundedFunctions the carrier of X),(BoundedFunctions the carrier of X):],(BoundedFunctions the carrier of X):] is non empty set
Add_ ((BoundedFunctions the carrier of X),(RAlgebra the carrier of X)) is non empty Relation-like [:(BoundedFunctions the carrier of X),(BoundedFunctions the carrier of X):] -defined BoundedFunctions the carrier of X -valued Function-like total quasi_total Element of bool [:[:(BoundedFunctions the carrier of X),(BoundedFunctions the carrier of X):],(BoundedFunctions the carrier of X):]
Mult_ ((BoundedFunctions the carrier of X),(RAlgebra the carrier of X)) is non empty Relation-like [:REAL,(BoundedFunctions the carrier of X):] -defined BoundedFunctions the carrier of X -valued Function-like total quasi_total Element of bool [:[:REAL,(BoundedFunctions the carrier of X):],(BoundedFunctions the carrier of X):]
[:REAL,(BoundedFunctions the carrier of X):] is non empty set
[:[:REAL,(BoundedFunctions the carrier of X):],(BoundedFunctions the carrier of X):] is non empty set
bool [:[:REAL,(BoundedFunctions the carrier of X):],(BoundedFunctions the carrier of X):] is non empty set
One_ ((BoundedFunctions the carrier of X),(RAlgebra the carrier of X)) is Element of BoundedFunctions the carrier of X
Zero_ ((BoundedFunctions the carrier of X),(RAlgebra the carrier of X)) is Element of BoundedFunctions the carrier of X
Normed_AlgebraStr(# (BoundedFunctions the carrier of X),(mult_ ((BoundedFunctions the carrier of X),(RAlgebra the carrier of X))),(Add_ ((BoundedFunctions the carrier of X),(RAlgebra the carrier of X))),(Mult_ ((BoundedFunctions the carrier of X),(RAlgebra the carrier of X))),(One_ ((BoundedFunctions the carrier of X),(RAlgebra the carrier of X))),(Zero_ ((BoundedFunctions the carrier of X),(RAlgebra the carrier of X))),(BoundedFunctionsNorm the carrier of X) #) is strict Normed_AlgebraStr
the carrier of (R_Normed_Algebra_of_BoundedFunctions the carrier of X) is non empty set
X is V11() real ext-real Element of REAL
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 [:REAL, the carrier of (X):] -defined the carrier of (X) -valued Function-like total quasi_total Element of bool [:[:REAL, the carrier of (X):], the carrier of (X):]
[:REAL, the carrier of (X):] is non empty set
[:[:REAL, the carrier of (X):], the carrier of (X):] is non empty set
bool [:[:REAL, the carrier of (X):], the carrier of (X):] is non empty set
the Mult of (X) . (X,F) is set
[X,F] is set
{X,F} is non empty set
{X} is non empty V185() V186() V187() set
{{X,F},{X}} is non empty set
the Mult of (X) . [X,F] is set
G is right_complementable Element of the carrier of (R_Normed_Algebra_of_BoundedFunctions the carrier of X)
X * G is right_complementable Element of the carrier of (R_Normed_Algebra_of_BoundedFunctions the carrier of X)
the Mult of (R_Normed_Algebra_of_BoundedFunctions the carrier of X) is non empty Relation-like [:REAL, the carrier of (R_Normed_Algebra_of_BoundedFunctions the carrier of X):] -defined the carrier of (R_Normed_Algebra_of_BoundedFunctions the carrier of X) -valued Function-like total quasi_total Element of bool [:[:REAL, the carrier of (R_Normed_Algebra_of_BoundedFunctions the carrier of X):], the carrier of (R_Normed_Algebra_of_BoundedFunctions the carrier of X):]
[:REAL, the carrier of (R_Normed_Algebra_of_BoundedFunctions the carrier of X):] is non empty set
[:[:REAL, the carrier of (R_Normed_Algebra_of_BoundedFunctions the carrier of X):], the carrier of (R_Normed_Algebra_of_BoundedFunctions the carrier of X):] is non empty set
bool [:[:REAL, the carrier of (R_Normed_Algebra_of_BoundedFunctions the carrier of X):], the carrier of (R_Normed_Algebra_of_BoundedFunctions the carrier of X):] is non empty set
the Mult of (R_Normed_Algebra_of_BoundedFunctions the carrier of X) . (X,G) is set
[X,G] is set
{X,G} is non empty set
{{X,G},{X}} is non empty set
the Mult of (R_Normed_Algebra_of_BoundedFunctions the carrier of X) . [X,G] is set
[:REAL, the carrier of (RAlgebra the carrier of X):] is non empty set
the Mult of (RAlgebra the carrier of X) is non empty Relation-like [:REAL, the carrier of (RAlgebra the carrier of X):] -defined the carrier of (RAlgebra the carrier of X) -valued Function-like total quasi_total Element of bool [:[:REAL, the carrier of (RAlgebra the carrier of X):], the carrier of (RAlgebra the carrier of X):]
[:[:REAL, the carrier of (RAlgebra the carrier of X):], the carrier of (RAlgebra the carrier of X):] is non empty set
bool [:[:REAL, the carrier of (RAlgebra the carrier of X):], the carrier of (RAlgebra the carrier of X):] is non empty set
the Mult of (RAlgebra the carrier of X) | [:REAL,(X):] is Relation-like [:REAL, the carrier of (RAlgebra the carrier of X):] -defined the carrier of (RAlgebra the carrier of X) -valued Function-like Element of bool [:[:REAL, the carrier of (RAlgebra the carrier of X):], the carrier of (RAlgebra the carrier of X):]
[X,F] is Element of [:REAL, the carrier of (X):]
( the Mult of (RAlgebra the carrier of X) | [:REAL,(X):]) . [X,F] is set
the Mult of (RAlgebra the carrier of X) . [X,F] is set
the Mult of (RAlgebra the carrier of X) | [:REAL,(BoundedFunctions the carrier of X):] is Relation-like [:REAL, the carrier of (RAlgebra the carrier of X):] -defined the carrier of (RAlgebra the carrier of X) -valued Function-like Element of bool [:[:REAL, the carrier of (RAlgebra the carrier of X):], the carrier of (RAlgebra the carrier of X):]
[X,G] is Element of [:REAL, the carrier of (R_Normed_Algebra_of_BoundedFunctions the carrier of X):]
( the Mult of (RAlgebra the carrier of X) | [:REAL,(BoundedFunctions the carrier of X):]) . [X,G] is set
X is V11() real ext-real Element of REAL
abs X is V11() real ext-real Element of REAL
X is non empty TopSpace-like TopStruct
(X) is non empty strict NORMSTR
(X) is non empty linearly-closed Element of bool the carrier of (RealVectSpace the carrier of X)
the carrier of X is non empty set
RealVectSpace the carrier of X is non empty right_complementable strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RLSStruct
Funcs ( the carrier of X,REAL) is non empty functional FUNCTION_DOMAIN of the carrier of X, REAL
RealFuncZero the carrier of X is Relation-like the carrier of X -defined REAL -valued Function-like total quasi_total V152() V153() V154() Element of Funcs ( the carrier of X,REAL)
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 V152() V153() V154() V155() Element of bool [: the carrier of X,NAT:]
[: the carrier of X,NAT:] is non empty RAT -valued INT -valued V152() V153() V154() V155() set
bool [: the carrier of X,NAT:] is non empty set
RealFuncAdd the carrier of X is non empty Relation-like [:(Funcs ( the carrier of X,REAL)),(Funcs ( the carrier of X,REAL)):] -defined Funcs ( the carrier of X,REAL) -valued Function-like total quasi_total Element of bool [:[:(Funcs ( the carrier of X,REAL)),(Funcs ( the carrier of X,REAL)):],(Funcs ( the carrier of X,REAL)):]
[:(Funcs ( the carrier of X,REAL)),(Funcs ( the carrier of X,REAL)):] is non empty set
[:[:(Funcs ( the carrier of X,REAL)),(Funcs ( the carrier of X,REAL)):],(Funcs ( the carrier of X,REAL)):] is non empty set
bool [:[:(Funcs ( the carrier of X,REAL)),(Funcs ( the carrier of X,REAL)):],(Funcs ( the carrier of X,REAL)):] is non empty set
RealFuncExtMult the carrier of X is non empty Relation-like [:REAL,(Funcs ( the carrier of X,REAL)):] -defined Funcs ( the carrier of X,REAL) -valued Function-like total quasi_total Element of bool [:[:REAL,(Funcs ( the carrier of X,REAL)):],(Funcs ( the carrier of X,REAL)):]
[:REAL,(Funcs ( the carrier of X,REAL)):] is non empty set
[:[:REAL,(Funcs ( the carrier of X,REAL)):],(Funcs ( the carrier of X,REAL)):] is non empty set
bool [:[:REAL,(Funcs ( the carrier of X,REAL)):],(Funcs ( the carrier of X,REAL)):] is non empty set
RLSStruct(# (Funcs ( the carrier of X,REAL)),(RealFuncZero the carrier of X),(RealFuncAdd the carrier of X),(RealFuncExtMult the carrier of X) #) is strict RLSStruct
the carrier of (RealVectSpace the carrier of X) is non empty set
bool the carrier of (RealVectSpace the carrier of X) is non empty set
[: the carrier of X,REAL:] is non empty V152() V153() V154() set
bool [: the carrier of X,REAL:] 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 REAL -valued Function-like total quasi_total V152() V153() V154() Element of bool [: the carrier of X,REAL:] : ( b₁ is continuous & 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),(RealVectSpace the carrier of X)) is Element of (X)
Add_ ((X),(RealVectSpace 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),(RealVectSpace the carrier of X)) is non empty Relation-like [:REAL,(X):] -defined (X) -valued Function-like total quasi_total Element of bool [:[:REAL,(X):],(X):]
[:REAL,(X):] is non empty set
[:[:REAL,(X):],(X):] is non empty set
bool [:[:REAL,(X):],(X):] is non empty set
(X) is non empty Relation-like (X) -defined REAL -valued Function-like total quasi_total V152() V153() V154() Element of bool [:(X),REAL:]
[:(X),REAL:] is non empty V152() V153() V154() set
bool [:(X),REAL:] is non empty set
BoundedFunctions the carrier of X is non empty add-closed having-inverse additively-closed multiplicatively-closed additively-linearly-closed Element of bool the carrier of (RAlgebra the carrier of X)
RAlgebra the carrier of X is non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative unital associative commutative right-distributive right_unital well-unital left_unital strict vector-associative AlgebraStr
RealFuncMult the carrier of X is non empty Relation-like [:(Funcs ( the carrier of X,REAL)),(Funcs ( the carrier of X,REAL)):] -defined Funcs ( the carrier of X,REAL) -valued Function-like total quasi_total Element of bool [:[:(Funcs ( the carrier of X,REAL)),(Funcs ( the carrier of X,REAL)):],(Funcs ( the carrier of X,REAL)):]
RealFuncUnit the carrier of X is Relation-like the carrier of X -defined REAL -valued Function-like total quasi_total V152() V153() V154() Element of Funcs ( the carrier of X,REAL)
the carrier of X --> 1 is non empty Relation-like the carrier of X -defined NAT -valued RAT -valued INT -valued Function-like total quasi_total V152() V153() V154() V155() Element of bool [: the carrier of X,NAT:]
AlgebraStr(# (Funcs ( the carrier of X,REAL)),(RealFuncMult the carrier of X),(RealFuncAdd the carrier of X),(RealFuncExtMult the carrier of X),(RealFuncUnit the carrier of X),(RealFuncZero the carrier of X) #) is strict AlgebraStr
the carrier of (RAlgebra the carrier of X) is non empty set
bool the carrier of (RAlgebra the carrier of X) is non empty set
{ b₁ where b₁ is non empty Relation-like the carrier of X -defined REAL -valued Function-like total quasi_total V152() V153() V154() Element of bool [: the carrier of X,REAL:] : b₁ | the carrier of X is bounded } is set
BoundedFunctionsNorm the carrier of X is non empty Relation-like BoundedFunctions the carrier of X -defined REAL -valued Function-like total quasi_total V152() V153() V154() Element of bool [:(BoundedFunctions the carrier of X),REAL:]
[:(BoundedFunctions the carrier of X),REAL:] is non empty V152() V153() V154() set
bool [:(BoundedFunctions the carrier of X),REAL:] is non empty set
(BoundedFunctionsNorm the carrier of X) | (X) is Relation-like BoundedFunctions the carrier of X -defined REAL -valued Function-like V152() V153() V154() Element of bool [:(BoundedFunctions the carrier of X),REAL:]
NORMSTR(# (X),(Zero_ ((X),(RealVectSpace the carrier of X))),(Add_ ((X),(RealVectSpace the carrier of X))),(Mult_ ((X),(RealVectSpace the carrier of X))),(X) #) is strict NORMSTR
the carrier of (X) is non empty set
0. (X) is V49((X)) 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 V11() real ext-real Element of REAL
the U8 of (X) is non empty Relation-like the carrier of (X) -defined REAL -valued Function-like total quasi_total V152() V153() V154() Element of bool [: the carrier of (X),REAL:]
[: the carrier of (X),REAL:] is non empty V152() V153() V154() set
bool [: the carrier of (X),REAL:] is non empty set
the U8 of (X) . F is V11() real ext-real Element of REAL
X * F is Element of the carrier of (X)
the Mult of (X) is non empty Relation-like [:REAL, the carrier of (X):] -defined the carrier of (X) -valued Function-like total quasi_total Element of bool [:[:REAL, the carrier of (X):], the carrier of (X):]
[:REAL, the carrier of (X):] is non empty set
[:[:REAL, the carrier of (X):], the carrier of (X):] is non empty set
bool [:[:REAL, the carrier of (X):], the carrier of (X):] is non empty set
the Mult of (X) . (X,F) is set
[X,F] is set
{X,F} is non empty set
{X} is non empty V185() V186() V187() set
{{X,F},{X}} is non empty set
the Mult of (X) . [X,F] is set
||.(X * F).|| is V11() real ext-real Element of REAL
the U8 of (X) . (X * F) is V11() real ext-real Element of REAL
(abs X) * ||.F.|| is V11() 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 V11() real ext-real Element of REAL
the U8 of (X) . (F + G) is V11() real ext-real Element of REAL
||.G.|| is V11() real ext-real Element of REAL
the U8 of (X) . G is V11() real ext-real Element of REAL
||.F.|| + ||.G.|| is V11() real ext-real Element of REAL
R_Normed_Algebra_of_BoundedFunctions the carrier of X is non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital discerning reflexive RealNormSpace-like unital complete Normed_AlgebraStr
mult_ ((BoundedFunctions the carrier of X),(RAlgebra the carrier of X)) is non empty Relation-like [:(BoundedFunctions the carrier of X),(BoundedFunctions the carrier of X):] -defined BoundedFunctions the carrier of X -valued Function-like total quasi_total Element of bool [:[:(BoundedFunctions the carrier of X),(BoundedFunctions the carrier of X):],(BoundedFunctions the carrier of X):]
[:(BoundedFunctions the carrier of X),(BoundedFunctions the carrier of X):] is non empty set
[:[:(BoundedFunctions the carrier of X),(BoundedFunctions the carrier of X):],(BoundedFunctions the carrier of X):] is non empty set
bool [:[:(BoundedFunctions the carrier of X),(BoundedFunctions the carrier of X):],(BoundedFunctions the carrier of X):] is non empty set
Add_ ((BoundedFunctions the carrier of X),(RAlgebra the carrier of X)) is non empty Relation-like [:(BoundedFunctions the carrier of X),(BoundedFunctions the carrier of X):] -defined BoundedFunctions the carrier of X -valued Function-like total quasi_total Element of bool [:[:(BoundedFunctions the carrier of X),(BoundedFunctions the carrier of X):],(BoundedFunctions the carrier of X):]
Mult_ ((BoundedFunctions the carrier of X),(RAlgebra the carrier of X)) is non empty Relation-like [:REAL,(BoundedFunctions the carrier of X):] -defined BoundedFunctions the carrier of X -valued Function-like total quasi_total Element of bool [:[:REAL,(BoundedFunctions the carrier of X):],(BoundedFunctions the carrier of X):]
[:REAL,(BoundedFunctions the carrier of X):] is non empty set
[:[:REAL,(BoundedFunctions the carrier of X):],(BoundedFunctions the carrier of X):] is non empty set
bool [:[:REAL,(BoundedFunctions the carrier of X):],(BoundedFunctions the carrier of X):] is non empty set
One_ ((BoundedFunctions the carrier of X),(RAlgebra the carrier of X)) is Element of BoundedFunctions the carrier of X
Zero_ ((BoundedFunctions the carrier of X),(RAlgebra the carrier of X)) is Element of BoundedFunctions the carrier of X
Normed_AlgebraStr(# (BoundedFunctions the carrier of X),(mult_ ((BoundedFunctions the carrier of X),(RAlgebra the carrier of X))),(Add_ ((BoundedFunctions the carrier of X),(RAlgebra the carrier of X))),(Mult_ ((BoundedFunctions the carrier of X),(RAlgebra the carrier of X))),(One_ ((BoundedFunctions the carrier of X),(RAlgebra the carrier of X))),(Zero_ ((BoundedFunctions the carrier of X),(RAlgebra the carrier of X))),(BoundedFunctionsNorm the carrier of X) #) is strict Normed_AlgebraStr
the carrier of (R_Normed_Algebra_of_BoundedFunctions the carrier of X) is non empty set
(X,0) is non empty Relation-like the carrier of X -defined REAL -valued Function-like total quasi_total V152() V153() V154() continuous Element of bool [: the carrier of X,REAL:]
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 V152() V153() V154() V155() Element of bool [: the carrier of X,{0}:]
{0} is non empty V185() V186() V187() V188() V189() V190() left_end bounded_below set
[: the carrier of X,{0}:] is non empty RAT -valued INT -valued V152() V153() V154() V155() set
bool [: the carrier of X,{0}:] is non empty set
FB is right_complementable Element of the carrier of (R_Normed_Algebra_of_BoundedFunctions the carrier of X)
||.FB.|| is V11() real ext-real Element of REAL
the U8 of (R_Normed_Algebra_of_BoundedFunctions the carrier of X) is non empty Relation-like the carrier of (R_Normed_Algebra_of_BoundedFunctions the carrier of X) -defined REAL -valued Function-like total quasi_total V152() V153() V154() Element of bool [: the carrier of (R_Normed_Algebra_of_BoundedFunctions the carrier of X),REAL:]
[: the carrier of (R_Normed_Algebra_of_BoundedFunctions the carrier of X),REAL:] is non empty V152() V153() V154() set
bool [: the carrier of (R_Normed_Algebra_of_BoundedFunctions the carrier of X),REAL:] is non empty set
the U8 of (R_Normed_Algebra_of_BoundedFunctions the carrier of X) . FB is V11() real ext-real Element of REAL
0. (R_Normed_Algebra_of_BoundedFunctions the carrier of X) is V49( R_Normed_Algebra_of_BoundedFunctions the carrier of X) right_complementable Element of the carrier of (R_Normed_Algebra_of_BoundedFunctions the carrier of X)
the ZeroF of (R_Normed_Algebra_of_BoundedFunctions the carrier of X) is right_complementable Element of the carrier of (R_Normed_Algebra_of_BoundedFunctions the carrier of X)
R_Normed_Algebra_of_BoundedFunctions the carrier of X is non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital discerning reflexive RealNormSpace-like unital complete Normed_AlgebraStr
mult_ ((BoundedFunctions the carrier of X),(RAlgebra the carrier of X)) is non empty Relation-like [:(BoundedFunctions the carrier of X),(BoundedFunctions the carrier of X):] -defined BoundedFunctions the carrier of X -valued Function-like total quasi_total Element of bool [:[:(BoundedFunctions the carrier of X),(BoundedFunctions the carrier of X):],(BoundedFunctions the carrier of X):]
[:(BoundedFunctions the carrier of X),(BoundedFunctions the carrier of X):] is non empty set
[:[:(BoundedFunctions the carrier of X),(BoundedFunctions the carrier of X):],(BoundedFunctions the carrier of X):] is non empty set
bool [:[:(BoundedFunctions the carrier of X),(BoundedFunctions the carrier of X):],(BoundedFunctions the carrier of X):] is non empty set
Add_ ((BoundedFunctions the carrier of X),(RAlgebra the carrier of X)) is non empty Relation-like [:(BoundedFunctions the carrier of X),(BoundedFunctions the carrier of X):] -defined BoundedFunctions the carrier of X -valued Function-like total quasi_total Element of bool [:[:(BoundedFunctions the carrier of X),(BoundedFunctions the carrier of X):],(BoundedFunctions the carrier of X):]
Mult_ ((BoundedFunctions the carrier of X),(RAlgebra the carrier of X)) is non empty Relation-like [:REAL,(BoundedFunctions the carrier of X):] -defined BoundedFunctions the carrier of X -valued Function-like total quasi_total Element of bool [:[:REAL,(BoundedFunctions the carrier of X):],(BoundedFunctions the carrier of X):]
[:REAL,(BoundedFunctions the carrier of X):] is non empty set
[:[:REAL,(BoundedFunctions the carrier of X):],(BoundedFunctions the carrier of X):] is non empty set
bool [:[:REAL,(BoundedFunctions the carrier of X):],(BoundedFunctions the carrier of X):] is non empty set
One_ ((BoundedFunctions the carrier of X),(RAlgebra the carrier of X)) is Element of BoundedFunctions the carrier of X
Zero_ ((BoundedFunctions the carrier of X),(RAlgebra the carrier of X)) is Element of BoundedFunctions the carrier of X
Normed_AlgebraStr(# (BoundedFunctions the carrier of X),(mult_ ((BoundedFunctions the carrier of X),(RAlgebra the carrier of X))),(Add_ ((BoundedFunctions the carrier of X),(RAlgebra the carrier of X))),(Mult_ ((BoundedFunctions the carrier of X),(RAlgebra the carrier of X))),(One_ ((BoundedFunctions the carrier of X),(RAlgebra the carrier of X))),(Zero_ ((BoundedFunctions the carrier of X),(RAlgebra the carrier of X))),(BoundedFunctionsNorm the carrier of X) #) is strict Normed_AlgebraStr
the carrier of (R_Normed_Algebra_of_BoundedFunctions the carrier of X) is non empty set
FB is right_complementable Element of the carrier of (R_Normed_Algebra_of_BoundedFunctions the carrier of X)
||.FB.|| is V11() real ext-real Element of REAL
the U8 of (R_Normed_Algebra_of_BoundedFunctions the carrier of X) is non empty Relation-like the carrier of (R_Normed_Algebra_of_BoundedFunctions the carrier of X) -defined REAL -valued Function-like total quasi_total V152() V153() V154() Element of bool [: the carrier of (R_Normed_Algebra_of_BoundedFunctions the carrier of X),REAL:]
[: the carrier of (R_Normed_Algebra_of_BoundedFunctions the carrier of X),REAL:] is non empty V152() V153() V154() set
bool [: the carrier of (R_Normed_Algebra_of_BoundedFunctions the carrier of X),REAL:] is non empty set
the U8 of (R_Normed_Algebra_of_BoundedFunctions the carrier of X) . FB is V11() real ext-real Element of REAL
X * FB is right_complementable Element of the carrier of (R_Normed_Algebra_of_BoundedFunctions the carrier of X)
the Mult of (R_Normed_Algebra_of_BoundedFunctions the carrier of X) is non empty Relation-like [:REAL, the carrier of (R_Normed_Algebra_of_BoundedFunctions the carrier of X):] -defined the carrier of (R_Normed_Algebra_of_BoundedFunctions the carrier of X) -valued Function-like total quasi_total Element of bool [:[:REAL, the carrier of (R_Normed_Algebra_of_BoundedFunctions the carrier of X):], the carrier of (R_Normed_Algebra_of_BoundedFunctions the carrier of X):]
[:REAL, the carrier of (R_Normed_Algebra_of_BoundedFunctions the carrier of X):] is non empty set
[:[:REAL, the carrier of (R_Normed_Algebra_of_BoundedFunctions the carrier of X):], the carrier of (R_Normed_Algebra_of_BoundedFunctions the carrier of X):] is non empty set
bool [:[:REAL, the carrier of (R_Normed_Algebra_of_BoundedFunctions the carrier of X):], the carrier of (R_Normed_Algebra_of_BoundedFunctions the carrier of X):] is non empty set
the Mult of (R_Normed_Algebra_of_BoundedFunctions the carrier of X) . (X,FB) is set
[X,FB] is set
{X,FB} is non empty set
{{X,FB},{X}} is non empty set
the Mult of (R_Normed_Algebra_of_BoundedFunctions the carrier of X) . [X,FB] is set
GB is right_complementable Element of the carrier of (R_Normed_Algebra_of_BoundedFunctions the carrier of X)
||.GB.|| is V11() real ext-real Element of REAL
the U8 of (R_Normed_Algebra_of_BoundedFunctions the carrier of X) . GB is V11() real ext-real Element of REAL
aFB is Element of the carrier of (X)
||.aFB.|| is V11() real ext-real Element of REAL
the U8 of (X) . aFB is V11() real ext-real Element of REAL
R_Normed_Algebra_of_BoundedFunctions the carrier of X is non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital discerning reflexive RealNormSpace-like unital complete Normed_AlgebraStr
mult_ ((BoundedFunctions the carrier of X),(RAlgebra the carrier of X)) is non empty Relation-like [:(BoundedFunctions the carrier of X),(BoundedFunctions the carrier of X):] -defined BoundedFunctions the carrier of X -valued Function-like total quasi_total Element of bool [:[:(BoundedFunctions the carrier of X),(BoundedFunctions the carrier of X):],(BoundedFunctions the carrier of X):]
[:(BoundedFunctions the carrier of X),(BoundedFunctions the carrier of X):] is non empty set
[:[:(BoundedFunctions the carrier of X),(BoundedFunctions the carrier of X):],(BoundedFunctions the carrier of X):] is non empty set
bool [:[:(BoundedFunctions the carrier of X),(BoundedFunctions the carrier of X):],(BoundedFunctions the carrier of X):] is non empty set
Add_ ((BoundedFunctions the carrier of X),(RAlgebra the carrier of X)) is non empty Relation-like [:(BoundedFunctions the carrier of X),(BoundedFunctions the carrier of X):] -defined BoundedFunctions the carrier of X -valued Function-like total quasi_total Element of bool [:[:(BoundedFunctions the carrier of X),(BoundedFunctions the carrier of X):],(BoundedFunctions the carrier of X):]
Mult_ ((BoundedFunctions the carrier of X),(RAlgebra the carrier of X)) is non empty Relation-like [:REAL,(BoundedFunctions the carrier of X):] -defined BoundedFunctions the carrier of X -valued Function-like total quasi_total Element of bool [:[:REAL,(BoundedFunctions the carrier of X):],(BoundedFunctions the carrier of X):]
[:REAL,(BoundedFunctions the carrier of X):] is non empty set
[:[:REAL,(BoundedFunctions the carrier of X):],(BoundedFunctions the carrier of X):] is non empty set
bool [:[:REAL,(BoundedFunctions the carrier of X):],(BoundedFunctions the carrier of X):] is non empty set
One_ ((BoundedFunctions the carrier of X),(RAlgebra the carrier of X)) is Element of BoundedFunctions the carrier of X
Zero_ ((BoundedFunctions the carrier of X),(RAlgebra the carrier of X)) is Element of BoundedFunctions the carrier of X
Normed_AlgebraStr(# (BoundedFunctions the carrier of X),(mult_ ((BoundedFunctions the carrier of X),(RAlgebra the carrier of X))),(Add_ ((BoundedFunctions the carrier of X),(RAlgebra the carrier of X))),(Mult_ ((BoundedFunctions the carrier of X),(RAlgebra the carrier of X))),(One_ ((BoundedFunctions the carrier of X),(RAlgebra the carrier of X))),(Zero_ ((BoundedFunctions the carrier of X),(RAlgebra the carrier of X))),(BoundedFunctionsNorm the carrier of X) #) is strict Normed_AlgebraStr
the carrier of (R_Normed_Algebra_of_BoundedFunctions the carrier of X) is non empty set
FB is right_complementable Element of the carrier of (R_Normed_Algebra_of_BoundedFunctions the carrier of X)
||.FB.|| is V11() real ext-real Element of REAL
the U8 of (R_Normed_Algebra_of_BoundedFunctions the carrier of X) is non empty Relation-like the carrier of (R_Normed_Algebra_of_BoundedFunctions the carrier of X) -defined REAL -valued Function-like total quasi_total V152() V153() V154() Element of bool [: the carrier of (R_Normed_Algebra_of_BoundedFunctions the carrier of X),REAL:]
[: the carrier of (R_Normed_Algebra_of_BoundedFunctions the carrier of X),REAL:] is non empty V152() V153() V154() set
bool [: the carrier of (R_Normed_Algebra_of_BoundedFunctions the carrier of X),REAL:] is non empty set
the U8 of (R_Normed_Algebra_of_BoundedFunctions the carrier of X) . FB is V11() real ext-real Element of REAL
GB is right_complementable Element of the carrier of (R_Normed_Algebra_of_BoundedFunctions the carrier of X)
||.GB.|| is V11() real ext-real Element of REAL
the U8 of (R_Normed_Algebra_of_BoundedFunctions the carrier of X) . GB is V11() real ext-real Element of REAL
FB + GB is right_complementable Element of the carrier of (R_Normed_Algebra_of_BoundedFunctions the carrier of X)
the addF of (R_Normed_Algebra_of_BoundedFunctions the carrier of X) is non empty Relation-like [: the carrier of (R_Normed_Algebra_of_BoundedFunctions the carrier of X), the carrier of (R_Normed_Algebra_of_BoundedFunctions the carrier of X):] -defined the carrier of (R_Normed_Algebra_of_BoundedFunctions the carrier of X) -valued Function-like total quasi_total Element of bool [:[: the carrier of (R_Normed_Algebra_of_BoundedFunctions the carrier of X), the carrier of (R_Normed_Algebra_of_BoundedFunctions the carrier of X):], the carrier of (R_Normed_Algebra_of_BoundedFunctions the carrier of X):]
[: the carrier of (R_Normed_Algebra_of_BoundedFunctions the carrier of X), the carrier of (R_Normed_Algebra_of_BoundedFunctions the carrier of X):] is non empty set
[:[: the carrier of (R_Normed_Algebra_of_BoundedFunctions the carrier of X), the carrier of (R_Normed_Algebra_of_BoundedFunctions the carrier of X):], the carrier of (R_Normed_Algebra_of_BoundedFunctions the carrier of X):] is non empty set
bool [:[: the carrier of (R_Normed_Algebra_of_BoundedFunctions the carrier of X), the carrier of (R_Normed_Algebra_of_BoundedFunctions the carrier of X):], the carrier of (R_Normed_Algebra_of_BoundedFunctions the carrier of X):] is non empty set
the addF of (R_Normed_Algebra_of_BoundedFunctions the carrier of X) . (FB,GB) is right_complementable Element of the carrier of (R_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 (R_Normed_Algebra_of_BoundedFunctions the carrier of X) . [FB,GB] is set
||.(FB + GB).|| is V11() real ext-real Element of REAL
the U8 of (R_Normed_Algebra_of_BoundedFunctions the carrier of X) . (FB + GB) is V11() real ext-real Element of REAL
X is non empty TopSpace-like TopStruct
(X) is non empty strict NORMSTR
(X) is non empty linearly-closed Element of bool the carrier of (RealVectSpace the carrier of X)
the carrier of X is non empty set
RealVectSpace the carrier of X is non empty right_complementable strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RLSStruct
Funcs ( the carrier of X,REAL) is non empty functional FUNCTION_DOMAIN of the carrier of X, REAL
RealFuncZero the carrier of X is Relation-like the carrier of X -defined REAL -valued Function-like total quasi_total V152() V153() V154() Element of Funcs ( the carrier of X,REAL)
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 V152() V153() V154() V155() Element of bool [: the carrier of X,NAT:]
[: the carrier of X,NAT:] is non empty RAT -valued INT -valued V152() V153() V154() V155() set
bool [: the carrier of X,NAT:] is non empty set
RealFuncAdd the carrier of X is non empty Relation-like [:(Funcs ( the carrier of X,REAL)),(Funcs ( the carrier of X,REAL)):] -defined Funcs ( the carrier of X,REAL) -valued Function-like total quasi_total Element of bool [:[:(Funcs ( the carrier of X,REAL)),(Funcs ( the carrier of X,REAL)):],(Funcs ( the carrier of X,REAL)):]
[:(Funcs ( the carrier of X,REAL)),(Funcs ( the carrier of X,REAL)):] is non empty set
[:[:(Funcs ( the carrier of X,REAL)),(Funcs ( the carrier of X,REAL)):],(Funcs ( the carrier of X,REAL)):] is non empty set
bool [:[:(Funcs ( the carrier of X,REAL)),(Funcs ( the carrier of X,REAL)):],(Funcs ( the carrier of X,REAL)):] is non empty set
RealFuncExtMult the carrier of X is non empty Relation-like [:REAL,(Funcs ( the carrier of X,REAL)):] -defined Funcs ( the carrier of X,REAL) -valued Function-like total quasi_total Element of bool [:[:REAL,(Funcs ( the carrier of X,REAL)):],(Funcs ( the carrier of X,REAL)):]
[:REAL,(Funcs ( the carrier of X,REAL)):] is non empty set
[:[:REAL,(Funcs ( the carrier of X,REAL)):],(Funcs ( the carrier of X,REAL)):] is non empty set
bool [:[:REAL,(Funcs ( the carrier of X,REAL)):],(Funcs ( the carrier of X,REAL)):] is non empty set
RLSStruct(# (Funcs ( the carrier of X,REAL)),(RealFuncZero the carrier of X),(RealFuncAdd the carrier of X),(RealFuncExtMult the carrier of X) #) is strict RLSStruct
the carrier of (RealVectSpace the carrier of X) is non empty set
bool the carrier of (RealVectSpace the carrier of X) is non empty set
[: the carrier of X,REAL:] is non empty V152() V153() V154() set
bool [: the carrier of X,REAL:] 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 REAL -valued Function-like total quasi_total V152() V153() V154() Element of bool [: the carrier of X,REAL:] : ( b₁ is continuous & 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),(RealVectSpace the carrier of X)) is Element of (X)
Add_ ((X),(RealVectSpace 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),(RealVectSpace the carrier of X)) is non empty Relation-like [:REAL,(X):] -defined (X) -valued Function-like total quasi_total Element of bool [:[:REAL,(X):],(X):]
[:REAL,(X):] is non empty set
[:[:REAL,(X):],(X):] is non empty set
bool [:[:REAL,(X):],(X):] is non empty set
(X) is non empty Relation-like (X) -defined REAL -valued Function-like total quasi_total V152() V153() V154() Element of bool [:(X),REAL:]
[:(X),REAL:] is non empty V152() V153() V154() set
bool [:(X),REAL:] is non empty set
BoundedFunctions the carrier of X is non empty add-closed having-inverse additively-closed multiplicatively-closed additively-linearly-closed Element of bool the carrier of (RAlgebra the carrier of X)
RAlgebra the carrier of X is non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative unital associative commutative right-distributive right_unital well-unital left_unital strict vector-associative AlgebraStr
RealFuncMult the carrier of X is non empty Relation-like [:(Funcs ( the carrier of X,REAL)),(Funcs ( the carrier of X,REAL)):] -defined Funcs ( the carrier of X,REAL) -valued Function-like total quasi_total Element of bool [:[:(Funcs ( the carrier of X,REAL)),(Funcs ( the carrier of X,REAL)):],(Funcs ( the carrier of X,REAL)):]
RealFuncUnit the carrier of X is Relation-like the carrier of X -defined REAL -valued Function-like total quasi_total V152() V153() V154() Element of Funcs ( the carrier of X,REAL)
the carrier of X --> 1 is non empty Relation-like the carrier of X -defined NAT -valued RAT -valued INT -valued Function-like total quasi_total V152() V153() V154() V155() Element of bool [: the carrier of X,NAT:]
AlgebraStr(# (Funcs ( the carrier of X,REAL)),(RealFuncMult the carrier of X),(RealFuncAdd the carrier of X),(RealFuncExtMult the carrier of X),(RealFuncUnit the carrier of X),(RealFuncZero the carrier of X) #) is strict AlgebraStr
the carrier of (RAlgebra the carrier of X) is non empty set
bool the carrier of (RAlgebra the carrier of X) is non empty set
{ b₁ where b₁ is non empty Relation-like the carrier of X -defined REAL -valued Function-like total quasi_total V152() V153() V154() Element of bool [: the carrier of X,REAL:] : b₁ | the carrier of X is bounded } is set
BoundedFunctionsNorm the carrier of X is non empty Relation-like BoundedFunctions the carrier of X -defined REAL -valued Function-like total quasi_total V152() V153() V154() Element of bool [:(BoundedFunctions the carrier of X),REAL:]
[:(BoundedFunctions the carrier of X),REAL:] is non empty V152() V153() V154() set
bool [:(BoundedFunctions the carrier of X),REAL:] is non empty set
(BoundedFunctionsNorm the carrier of X) | (X) is Relation-like BoundedFunctions the carrier of X -defined REAL -valued Function-like V152() V153() V154() Element of bool [:(BoundedFunctions the carrier of X),REAL:]
NORMSTR(# (X),(Zero_ ((X),(RealVectSpace the carrier of X))),(Add_ ((X),(RealVectSpace the carrier of X))),(Mult_ ((X),(RealVectSpace the carrier of X))),(X) #) is strict NORMSTR
the carrier of (X) is non empty set
X is Element of the carrier of (X)
F is V11() real ext-real Element of REAL
F * X is Element of the carrier of (X)
the Mult of (X) is non empty Relation-like [:REAL, the carrier of (X):] -defined the carrier of (X) -valued Function-like total quasi_total Element of bool [:[:REAL, the carrier of (X):], the carrier of (X):]
[:REAL, the carrier of (X):] is non empty set
[:[:REAL, the carrier of (X):], the carrier of (X):] is non empty set
bool [:[:REAL, the carrier of (X):], the carrier of (X):] is non empty set
the Mult of (X) . (F,X) is set
[F,X] is set
{F,X} is non empty set
{F} is non empty V185() V186() V187() set
{{F,X},{F}} is non empty set
the Mult of (X) . [F,X] is set
||.(F * X).|| is V11() real ext-real Element of REAL
the U8 of (X) is non empty Relation-like the carrier of (X) -defined REAL -valued Function-like total quasi_total V152() V153() V154() Element of bool [: the carrier of (X),REAL:]
[: the carrier of (X),REAL:] is non empty V152() V153() V154() set
bool [: the carrier of (X),REAL:] is non empty set
the U8 of (X) . (F * X) is V11() real ext-real Element of REAL
abs F is V11() real ext-real Element of REAL
||.X.|| is V11() real ext-real Element of REAL
the U8 of (X) . X is V11() real ext-real Element of REAL
(abs F) * ||.X.|| is V11() 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 V11() real ext-real Element of REAL
the U8 of (X) . (G + FB) is V11() real ext-real Element of REAL
||.G.|| is V11() real ext-real Element of REAL
the U8 of (X) . G is V11() real ext-real Element of REAL
||.FB.|| is V11() real ext-real Element of REAL
the U8 of (X) . FB is V11() real ext-real Element of REAL
||.G.|| + ||.FB.|| is V11() real ext-real Element of REAL
X is non empty TopSpace-like TopStruct
(X) is non empty strict NORMSTR
(X) is non empty linearly-closed Element of bool the carrier of (RealVectSpace the carrier of X)
the carrier of X is non empty set
RealVectSpace the carrier of X is non empty right_complementable strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RLSStruct
Funcs ( the carrier of X,REAL) is non empty functional FUNCTION_DOMAIN of the carrier of X, REAL
RealFuncZero the carrier of X is Relation-like the carrier of X -defined REAL -valued Function-like total quasi_total V152() V153() V154() Element of Funcs ( the carrier of X,REAL)
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 V152() V153() V154() V155() Element of bool [: the carrier of X,NAT:]
[: the carrier of X,NAT:] is non empty RAT -valued INT -valued V152() V153() V154() V155() set
bool [: the carrier of X,NAT:] is non empty set
RealFuncAdd the carrier of X is non empty Relation-like [:(Funcs ( the carrier of X,REAL)),(Funcs ( the carrier of X,REAL)):] -defined Funcs ( the carrier of X,REAL) -valued Function-like total quasi_total Element of bool [:[:(Funcs ( the carrier of X,REAL)),(Funcs ( the carrier of X,REAL)):],(Funcs ( the carrier of X,REAL)):]
[:(Funcs ( the carrier of X,REAL)),(Funcs ( the carrier of X,REAL)):] is non empty set
[:[:(Funcs ( the carrier of X,REAL)),(Funcs ( the carrier of X,REAL)):],(Funcs ( the carrier of X,REAL)):] is non empty set
bool [:[:(Funcs ( the carrier of X,REAL)),(Funcs ( the carrier of X,REAL)):],(Funcs ( the carrier of X,REAL)):] is non empty set
RealFuncExtMult the carrier of X is non empty Relation-like [:REAL,(Funcs ( the carrier of X,REAL)):] -defined Funcs ( the carrier of X,REAL) -valued Function-like total quasi_total Element of bool [:[:REAL,(Funcs ( the carrier of X,REAL)):],(Funcs ( the carrier of X,REAL)):]
[:REAL,(Funcs ( the carrier of X,REAL)):] is non empty set
[:[:REAL,(Funcs ( the carrier of X,REAL)):],(Funcs ( the carrier of X,REAL)):] is non empty set
bool [:[:REAL,(Funcs ( the carrier of X,REAL)):],(Funcs ( the carrier of X,REAL)):] is non empty set
RLSStruct(# (Funcs ( the carrier of X,REAL)),(RealFuncZero the carrier of X),(RealFuncAdd the carrier of X),(RealFuncExtMult the carrier of X) #) is strict RLSStruct
the carrier of (RealVectSpace the carrier of X) is non empty set
bool the carrier of (RealVectSpace the carrier of X) is non empty set
[: the carrier of X,REAL:] is non empty V152() V153() V154() set
bool [: the carrier of X,REAL:] 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 REAL -valued Function-like total quasi_total V152() V153() V154() Element of bool [: the carrier of X,REAL:] : ( b₁ is continuous & 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),(RealVectSpace the carrier of X)) is Element of (X)
Add_ ((X),(RealVectSpace 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),(RealVectSpace the carrier of X)) is non empty Relation-like [:REAL,(X):] -defined (X) -valued Function-like total quasi_total Element of bool [:[:REAL,(X):],(X):]
[:REAL,(X):] is non empty set
[:[:REAL,(X):],(X):] is non empty set
bool [:[:REAL,(X):],(X):] is non empty set
(X) is non empty Relation-like (X) -defined REAL -valued Function-like total quasi_total V152() V153() V154() Element of bool [:(X),REAL:]
[:(X),REAL:] is non empty V152() V153() V154() set
bool [:(X),REAL:] is non empty set
BoundedFunctions the carrier of X is non empty add-closed having-inverse additively-closed multiplicatively-closed additively-linearly-closed Element of bool the carrier of (RAlgebra the carrier of X)
RAlgebra the carrier of X is non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative unital associative commutative right-distributive right_unital well-unital left_unital strict vector-associative AlgebraStr
RealFuncMult the carrier of X is non empty Relation-like [:(Funcs ( the carrier of X,REAL)),(Funcs ( the carrier of X,REAL)):] -defined Funcs ( the carrier of X,REAL) -valued Function-like total quasi_total Element of bool [:[:(Funcs ( the carrier of X,REAL)),(Funcs ( the carrier of X,REAL)):],(Funcs ( the carrier of X,REAL)):]
RealFuncUnit the carrier of X is Relation-like the carrier of X -defined REAL -valued Function-like total quasi_total V152() V153() V154() Element of Funcs ( the carrier of X,REAL)
the carrier of X --> 1 is non empty Relation-like the carrier of X -defined NAT -valued RAT -valued INT -valued Function-like total quasi_total V152() V153() V154() V155() Element of bool [: the carrier of X,NAT:]
AlgebraStr(# (Funcs ( the carrier of X,REAL)),(RealFuncMult the carrier of X),(RealFuncAdd the carrier of X),(RealFuncExtMult the carrier of X),(RealFuncUnit the carrier of X),(RealFuncZero the carrier of X) #) is strict AlgebraStr
the carrier of (RAlgebra the carrier of X) is non empty set
bool the carrier of (RAlgebra the carrier of X) is non empty set
{ b₁ where b₁ is non empty Relation-like the carrier of X -defined REAL -valued Function-like total quasi_total V152() V153() V154() Element of bool [: the carrier of X,REAL:] : b₁ | the carrier of X is bounded } is set
BoundedFunctionsNorm the carrier of X is non empty Relation-like BoundedFunctions the carrier of X -defined REAL -valued Function-like total quasi_total V152() V153() V154() Element of bool [:(BoundedFunctions the carrier of X),REAL:]
[:(BoundedFunctions the carrier of X),REAL:] is non empty V152() V153() V154() set
bool [:(BoundedFunctions the carrier of X),REAL:] is non empty set
(BoundedFunctionsNorm the carrier of X) | (X) is Relation-like BoundedFunctions the carrier of X -defined REAL -valued Function-like V152() V153() V154() Element of bool [:(BoundedFunctions the carrier of X),REAL:]
NORMSTR(# (X),(Zero_ ((X),(RealVectSpace the carrier of X))),(Add_ ((X),(RealVectSpace the carrier of X))),(Mult_ ((X),(RealVectSpace the carrier of X))),(X) #) is strict NORMSTR
(X) is non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RLSStruct
RLSStruct(# (X),(Zero_ ((X),(RealVectSpace the carrier of X))),(Add_ ((X),(RealVectSpace the carrier of X))),(Mult_ ((X),(RealVectSpace the carrier of X))) #) is strict RLSStruct
the carrier of (X) is non empty set
0. (X) is V49((X)) 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 V11() real ext-real Element of REAL
the U8 of (X) is non empty Relation-like the carrier of (X) -defined REAL -valued Function-like total quasi_total V152() V153() V154() Element of bool [: the carrier of (X),REAL:]
[: the carrier of (X),REAL:] is non empty V152() V153() V154() set
bool [: the carrier of (X),REAL:] is non empty set
the U8 of (X) . X is V11() real ext-real Element of REAL
||.(0. (X)).|| is V11() real ext-real Element of REAL
the U8 of (X) is non empty Relation-like the carrier of (X) -defined REAL -valued Function-like total quasi_total V152() V153() V154() Element of bool [: the carrier of (X),REAL:]
[: the carrier of (X),REAL:] is non empty V152() V153() V154() set
bool [: the carrier of (X),REAL:] is non empty set
the U8 of (X) . (0. (X)) is V11() real ext-real Element of REAL
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 discerning reflexive strict RealNormSpace-like NORMSTR
(X) is non empty linearly-closed Element of bool the carrier of (RealVectSpace the carrier of X)
the carrier of X is non empty set
RealVectSpace the carrier of X is non empty right_complementable strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RLSStruct
Funcs ( the carrier of X,REAL) is non empty functional FUNCTION_DOMAIN of the carrier of X, REAL
RealFuncZero the carrier of X is Relation-like the carrier of X -defined REAL -valued Function-like total quasi_total V152() V153() V154() Element of Funcs ( the carrier of X,REAL)
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 V152() V153() V154() V155() Element of bool [: the carrier of X,NAT:]
[: the carrier of X,NAT:] is non empty RAT -valued INT -valued V152() V153() V154() V155() set
bool [: the carrier of X,NAT:] is non empty set
RealFuncAdd the carrier of X is non empty Relation-like [:(Funcs ( the carrier of X,REAL)),(Funcs ( the carrier of X,REAL)):] -defined Funcs ( the carrier of X,REAL) -valued Function-like total quasi_total Element of bool [:[:(Funcs ( the carrier of X,REAL)),(Funcs ( the carrier of X,REAL)):],(Funcs ( the carrier of X,REAL)):]
[:(Funcs ( the carrier of X,REAL)),(Funcs ( the carrier of X,REAL)):] is non empty set
[:[:(Funcs ( the carrier of X,REAL)),(Funcs ( the carrier of X,REAL)):],(Funcs ( the carrier of X,REAL)):] is non empty set
bool [:[:(Funcs ( the carrier of X,REAL)),(Funcs ( the carrier of X,REAL)):],(Funcs ( the carrier of X,REAL)):] is non empty set
RealFuncExtMult the carrier of X is non empty Relation-like [:REAL,(Funcs ( the carrier of X,REAL)):] -defined Funcs ( the carrier of X,REAL) -valued Function-like total quasi_total Element of bool [:[:REAL,(Funcs ( the carrier of X,REAL)):],(Funcs ( the carrier of X,REAL)):]
[:REAL,(Funcs ( the carrier of X,REAL)):] is non empty set
[:[:REAL,(Funcs ( the carrier of X,REAL)):],(Funcs ( the carrier of X,REAL)):] is non empty set
bool [:[:REAL,(Funcs ( the carrier of X,REAL)):],(Funcs ( the carrier of X,REAL)):] is non empty set
RLSStruct(# (Funcs ( the carrier of X,REAL)),(RealFuncZero the carrier of X),(RealFuncAdd the carrier of X),(RealFuncExtMult the carrier of X) #) is strict RLSStruct
the carrier of (RealVectSpace the carrier of X) is non empty set
bool the carrier of (RealVectSpace the carrier of X) is non empty set
[: the carrier of X,REAL:] is non empty V152() V153() V154() set
bool [: the carrier of X,REAL:] 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 REAL -valued Function-like total quasi_total V152() V153() V154() Element of bool [: the carrier of X,REAL:] : ( b₁ is continuous & 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),(RealVectSpace the carrier of X)) is Element of (X)
Add_ ((X),(RealVectSpace 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),(RealVectSpace the carrier of X)) is non empty Relation-like [:REAL,(X):] -defined (X) -valued Function-like total quasi_total Element of bool [:[:REAL,(X):],(X):]
[:REAL,(X):] is non empty set
[:[:REAL,(X):],(X):] is non empty set
bool [:[:REAL,(X):],(X):] is non empty set
(X) is non empty Relation-like (X) -defined REAL -valued Function-like total quasi_total V152() V153() V154() Element of bool [:(X),REAL:]
[:(X),REAL:] is non empty V152() V153() V154() set
bool [:(X),REAL:] is non empty set
BoundedFunctions the carrier of X is non empty add-closed having-inverse additively-closed multiplicatively-closed additively-linearly-closed Element of bool the carrier of (RAlgebra the carrier of X)
RAlgebra the carrier of X is non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative unital associative commutative right-distributive right_unital well-unital left_unital strict vector-associative AlgebraStr
RealFuncMult the carrier of X is non empty Relation-like [:(Funcs ( the carrier of X,REAL)),(Funcs ( the carrier of X,REAL)):] -defined Funcs ( the carrier of X,REAL) -valued Function-like total quasi_total Element of bool [:[:(Funcs ( the carrier of X,REAL)),(Funcs ( the carrier of X,REAL)):],(Funcs ( the carrier of X,REAL)):]
RealFuncUnit the carrier of X is Relation-like the carrier of X -defined REAL -valued Function-like total quasi_total V152() V153() V154() Element of Funcs ( the carrier of X,REAL)
the carrier of X --> 1 is non empty Relation-like the carrier of X -defined NAT -valued RAT -valued INT -valued Function-like total quasi_total V152() V153() V154() V155() Element of bool [: the carrier of X,NAT:]
AlgebraStr(# (Funcs ( the carrier of X,REAL)),(RealFuncMult the carrier of X),(RealFuncAdd the carrier of X),(RealFuncExtMult the carrier of X),(RealFuncUnit the carrier of X),(RealFuncZero the carrier of X) #) is strict AlgebraStr
the carrier of (RAlgebra the carrier of X) is non empty set
bool the carrier of (RAlgebra the carrier of X) is non empty set
{ b₁ where b₁ is non empty Relation-like the carrier of X -defined REAL -valued Function-like total quasi_total V152() V153() V154() Element of bool [: the carrier of X,REAL:] : b₁ | the carrier of X is bounded } is set
BoundedFunctionsNorm the carrier of X is non empty Relation-like BoundedFunctions the carrier of X -defined REAL -valued Function-like total quasi_total V152() V153() V154() Element of bool [:(BoundedFunctions the carrier of X),REAL:]
[:(BoundedFunctions the carrier of X),REAL:] is non empty V152() V153() V154() set
bool [:(BoundedFunctions the carrier of X),REAL:] is non empty set
(BoundedFunctionsNorm the carrier of X) | (X) is Relation-like BoundedFunctions the carrier of X -defined REAL -valued Function-like V152() V153() V154() Element of bool [:(BoundedFunctions the carrier of X),REAL:]
NORMSTR(# (X),(Zero_ ((X),(RealVectSpace the carrier of X))),(Add_ ((X),(RealVectSpace the carrier of X))),(Mult_ ((X),(RealVectSpace the carrier of X))),(X) #) is strict NORMSTR