:: PDIFF_6 semantic presentation

REAL is non empty V33() V156() V157() V158() V162() set
NAT is V6() V33() V38() V39() V156() V157() V158() V159() V160() V161() V162() Element of K6(REAL)
K6(REAL) is V33() set
REAL * is functional FinSequence-membered FinSequenceSet of REAL
K7(NAT,(REAL *)) is set
K6(K7(NAT,(REAL *))) is set
COMPLEX is non empty V33() V156() V162() set
NAT is V6() V33() V38() V39() V156() V157() V158() V159() V160() V161() V162() set
K6(NAT) is V33() set
K6(NAT) is V33() set
K7(NAT,REAL) is V33() complex-yielding V146() V147() set
K6(K7(NAT,REAL)) is V33() set
K190() is non empty set
K7(K190(),K190()) is set
K7(K7(K190(),K190()),K190()) is set
K6(K7(K7(K190(),K190()),K190())) is set
K7(REAL,K190()) is V33() set
K7(K7(REAL,K190()),K190()) is V33() set
K6(K7(K7(REAL,K190()),K190())) is V33() set
K196() is RLSStruct
the carrier of K196() is set
K6( the carrier of K196()) is set
K200() is Element of K6( the carrier of K196())
K7(K200(),K200()) is set
K7(K7(K200(),K200()),REAL) is complex-yielding V146() V147() set
K6(K7(K7(K200(),K200()),REAL)) is set
K203() is Element of K6( the carrier of K196())
K7(K203(),REAL) is complex-yielding V146() V147() set
K6(K7(K203(),REAL)) is set
RAT is non empty V33() V156() V157() V158() V159() V162() set
INT is non empty V33() V156() V157() V158() V159() V160() V162() set
1 is non empty V6() V10() V11() V12() ext-real positive non negative V33() V38() V120() V155() V156() V157() V158() V159() V160() V161() Element of NAT
K7(1,1) is RAT -valued INT -valued complex-yielding V146() V147() V148() set
K6(K7(1,1)) is set
K7(K7(1,1),1) is RAT -valued INT -valued complex-yielding V146() V147() V148() set
K6(K7(K7(1,1),1)) is set
K7(K7(1,1),REAL) is complex-yielding V146() V147() set
K6(K7(K7(1,1),REAL)) is set
K7(REAL,REAL) is V33() complex-yielding V146() V147() set
K7(K7(REAL,REAL),REAL) is V33() complex-yielding V146() V147() set
K6(K7(K7(REAL,REAL),REAL)) is V33() set
2 is non empty V6() V10() V11() V12() ext-real positive non negative V33() V38() V120() V155() V156() V157() V158() V159() V160() V161() Element of NAT
K7(2,2) is RAT -valued INT -valued complex-yielding V146() V147() V148() set
K7(K7(2,2),REAL) is complex-yielding V146() V147() set
K6(K7(K7(2,2),REAL)) is set
K6(K7(REAL,REAL)) is V33() set
TOP-REAL 2 is non empty V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital TopSpace-like V198() V205() L20()
the carrier of (TOP-REAL 2) is non empty set
REAL 1 is non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered FinSequenceSet of REAL
1 -tuples_on REAL is non empty functional FinSequence-membered FinSequenceSet of REAL
K7((REAL 1),(REAL 1)) is set
K6(K7((REAL 1),(REAL 1))) is set
REAL-NS 1 is non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() V108() V109() V142() L15()
the carrier of (REAL-NS 1) is non empty V2() set
R_NormSpace_of_BoundedLinearOperators ((REAL-NS 1),(REAL-NS 1)) is non empty V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() V109() V142() L15()
BoundedLinearOperators ((REAL-NS 1),(REAL-NS 1)) is non empty Element of K6( the carrier of (R_VectorSpace_of_LinearOperators ((REAL-NS 1),(REAL-NS 1))))
R_VectorSpace_of_LinearOperators ((REAL-NS 1),(REAL-NS 1)) is non empty V71() strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V204() RLSStruct
LinearOperators ((REAL-NS 1),(REAL-NS 1)) is non empty functional Element of K6( the carrier of K279( the carrier of (REAL-NS 1),(REAL-NS 1)))
K279( the carrier of (REAL-NS 1),(REAL-NS 1)) is non empty V71() strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V204() RLSStruct
K123( the carrier of (REAL-NS 1), the carrier of (REAL-NS 1)) is non empty functional M4( the carrier of (REAL-NS 1), the carrier of (REAL-NS 1))
FuncZero ( the carrier of (REAL-NS 1),(REAL-NS 1)) is Relation-like the carrier of (REAL-NS 1) -defined the carrier of (REAL-NS 1) -valued Function-like total V30( the carrier of (REAL-NS 1), the carrier of (REAL-NS 1)) Element of K123( the carrier of (REAL-NS 1), the carrier of (REAL-NS 1))
0. (REAL-NS 1) is V52( REAL-NS 1) Element of the carrier of (REAL-NS 1)
the carrier of (REAL-NS 1) --> (0. (REAL-NS 1)) is non empty Relation-like the carrier of (REAL-NS 1) -defined the carrier of (REAL-NS 1) -valued Function-like total V30( the carrier of (REAL-NS 1), the carrier of (REAL-NS 1)) Element of K6(K7( the carrier of (REAL-NS 1), the carrier of (REAL-NS 1)))
K7( the carrier of (REAL-NS 1), the carrier of (REAL-NS 1)) is set
K6(K7( the carrier of (REAL-NS 1), the carrier of (REAL-NS 1))) is set
FuncAdd ( the carrier of (REAL-NS 1),(REAL-NS 1)) is Relation-like K7(K123( the carrier of (REAL-NS 1), the carrier of (REAL-NS 1)),K123( the carrier of (REAL-NS 1), the carrier of (REAL-NS 1))) -defined K123( the carrier of (REAL-NS 1), the carrier of (REAL-NS 1)) -valued Function-like total V30(K7(K123( the carrier of (REAL-NS 1), the carrier of (REAL-NS 1)),K123( the carrier of (REAL-NS 1), the carrier of (REAL-NS 1))),K123( the carrier of (REAL-NS 1), the carrier of (REAL-NS 1))) Element of K6(K7(K7(K123( the carrier of (REAL-NS 1), the carrier of (REAL-NS 1)),K123( the carrier of (REAL-NS 1), the carrier of (REAL-NS 1))),K123( the carrier of (REAL-NS 1), the carrier of (REAL-NS 1))))
K7(K123( the carrier of (REAL-NS 1), the carrier of (REAL-NS 1)),K123( the carrier of (REAL-NS 1), the carrier of (REAL-NS 1))) is set
K7(K7(K123( the carrier of (REAL-NS 1), the carrier of (REAL-NS 1)),K123( the carrier of (REAL-NS 1), the carrier of (REAL-NS 1))),K123( the carrier of (REAL-NS 1), the carrier of (REAL-NS 1))) is set
K6(K7(K7(K123( the carrier of (REAL-NS 1), the carrier of (REAL-NS 1)),K123( the carrier of (REAL-NS 1), the carrier of (REAL-NS 1))),K123( the carrier of (REAL-NS 1), the carrier of (REAL-NS 1)))) is set
FuncExtMult ( the carrier of (REAL-NS 1),(REAL-NS 1)) is Relation-like K7(REAL,K123( the carrier of (REAL-NS 1), the carrier of (REAL-NS 1))) -defined K123( the carrier of (REAL-NS 1), the carrier of (REAL-NS 1)) -valued Function-like total V30(K7(REAL,K123( the carrier of (REAL-NS 1), the carrier of (REAL-NS 1))),K123( the carrier of (REAL-NS 1), the carrier of (REAL-NS 1))) Element of K6(K7(K7(REAL,K123( the carrier of (REAL-NS 1), the carrier of (REAL-NS 1))),K123( the carrier of (REAL-NS 1), the carrier of (REAL-NS 1))))
K7(REAL,K123( the carrier of (REAL-NS 1), the carrier of (REAL-NS 1))) is V33() set
K7(K7(REAL,K123( the carrier of (REAL-NS 1), the carrier of (REAL-NS 1))),K123( the carrier of (REAL-NS 1), the carrier of (REAL-NS 1))) is V33() set
K6(K7(K7(REAL,K123( the carrier of (REAL-NS 1), the carrier of (REAL-NS 1))),K123( the carrier of (REAL-NS 1), the carrier of (REAL-NS 1)))) is V33() set
RLSStruct(# K123( the carrier of (REAL-NS 1), the carrier of (REAL-NS 1)),(FuncZero ( the carrier of (REAL-NS 1),(REAL-NS 1))),(FuncAdd ( the carrier of (REAL-NS 1),(REAL-NS 1))),(FuncExtMult ( the carrier of (REAL-NS 1),(REAL-NS 1))) #) is strict RLSStruct
the carrier of K279( the carrier of (REAL-NS 1),(REAL-NS 1)) is non empty set
K6( the carrier of K279( the carrier of (REAL-NS 1),(REAL-NS 1))) is set
K199(K279( the carrier of (REAL-NS 1),(REAL-NS 1)),(LinearOperators ((REAL-NS 1),(REAL-NS 1)))) is Element of LinearOperators ((REAL-NS 1),(REAL-NS 1))
K197(K279( the carrier of (REAL-NS 1),(REAL-NS 1)),(LinearOperators ((REAL-NS 1),(REAL-NS 1)))) is Relation-like K7((LinearOperators ((REAL-NS 1),(REAL-NS 1))),(LinearOperators ((REAL-NS 1),(REAL-NS 1)))) -defined LinearOperators ((REAL-NS 1),(REAL-NS 1)) -valued Function-like total V30(K7((LinearOperators ((REAL-NS 1),(REAL-NS 1))),(LinearOperators ((REAL-NS 1),(REAL-NS 1)))), LinearOperators ((REAL-NS 1),(REAL-NS 1))) Element of K6(K7(K7((LinearOperators ((REAL-NS 1),(REAL-NS 1))),(LinearOperators ((REAL-NS 1),(REAL-NS 1)))),(LinearOperators ((REAL-NS 1),(REAL-NS 1)))))
K7((LinearOperators ((REAL-NS 1),(REAL-NS 1))),(LinearOperators ((REAL-NS 1),(REAL-NS 1)))) is set
K7(K7((LinearOperators ((REAL-NS 1),(REAL-NS 1))),(LinearOperators ((REAL-NS 1),(REAL-NS 1)))),(LinearOperators ((REAL-NS 1),(REAL-NS 1)))) is set
K6(K7(K7((LinearOperators ((REAL-NS 1),(REAL-NS 1))),(LinearOperators ((REAL-NS 1),(REAL-NS 1)))),(LinearOperators ((REAL-NS 1),(REAL-NS 1))))) is set
K198(K279( the carrier of (REAL-NS 1),(REAL-NS 1)),(LinearOperators ((REAL-NS 1),(REAL-NS 1)))) is Relation-like K7(REAL,(LinearOperators ((REAL-NS 1),(REAL-NS 1)))) -defined LinearOperators ((REAL-NS 1),(REAL-NS 1)) -valued Function-like total V30(K7(REAL,(LinearOperators ((REAL-NS 1),(REAL-NS 1)))), LinearOperators ((REAL-NS 1),(REAL-NS 1))) Element of K6(K7(K7(REAL,(LinearOperators ((REAL-NS 1),(REAL-NS 1)))),(LinearOperators ((REAL-NS 1),(REAL-NS 1)))))
K7(REAL,(LinearOperators ((REAL-NS 1),(REAL-NS 1)))) is V33() set
K7(K7(REAL,(LinearOperators ((REAL-NS 1),(REAL-NS 1)))),(LinearOperators ((REAL-NS 1),(REAL-NS 1)))) is V33() set
K6(K7(K7(REAL,(LinearOperators ((REAL-NS 1),(REAL-NS 1)))),(LinearOperators ((REAL-NS 1),(REAL-NS 1))))) is V33() set
RLSStruct(# (LinearOperators ((REAL-NS 1),(REAL-NS 1))),K199(K279( the carrier of (REAL-NS 1),(REAL-NS 1)),(LinearOperators ((REAL-NS 1),(REAL-NS 1)))),K197(K279( the carrier of (REAL-NS 1),(REAL-NS 1)),(LinearOperators ((REAL-NS 1),(REAL-NS 1)))),K198(K279( the carrier of (REAL-NS 1),(REAL-NS 1)),(LinearOperators ((REAL-NS 1),(REAL-NS 1)))) #) is V71() strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RLSStruct
the carrier of (R_VectorSpace_of_LinearOperators ((REAL-NS 1),(REAL-NS 1))) is non empty set
K6( the carrier of (R_VectorSpace_of_LinearOperators ((REAL-NS 1),(REAL-NS 1)))) is set
K199((R_VectorSpace_of_LinearOperators ((REAL-NS 1),(REAL-NS 1))),(BoundedLinearOperators ((REAL-NS 1),(REAL-NS 1)))) is Element of BoundedLinearOperators ((REAL-NS 1),(REAL-NS 1))
K197((R_VectorSpace_of_LinearOperators ((REAL-NS 1),(REAL-NS 1))),(BoundedLinearOperators ((REAL-NS 1),(REAL-NS 1)))) is Relation-like K7((BoundedLinearOperators ((REAL-NS 1),(REAL-NS 1))),(BoundedLinearOperators ((REAL-NS 1),(REAL-NS 1)))) -defined BoundedLinearOperators ((REAL-NS 1),(REAL-NS 1)) -valued Function-like total V30(K7((BoundedLinearOperators ((REAL-NS 1),(REAL-NS 1))),(BoundedLinearOperators ((REAL-NS 1),(REAL-NS 1)))), BoundedLinearOperators ((REAL-NS 1),(REAL-NS 1))) Element of K6(K7(K7((BoundedLinearOperators ((REAL-NS 1),(REAL-NS 1))),(BoundedLinearOperators ((REAL-NS 1),(REAL-NS 1)))),(BoundedLinearOperators ((REAL-NS 1),(REAL-NS 1)))))
K7((BoundedLinearOperators ((REAL-NS 1),(REAL-NS 1))),(BoundedLinearOperators ((REAL-NS 1),(REAL-NS 1)))) is set
K7(K7((BoundedLinearOperators ((REAL-NS 1),(REAL-NS 1))),(BoundedLinearOperators ((REAL-NS 1),(REAL-NS 1)))),(BoundedLinearOperators ((REAL-NS 1),(REAL-NS 1)))) is set
K6(K7(K7((BoundedLinearOperators ((REAL-NS 1),(REAL-NS 1))),(BoundedLinearOperators ((REAL-NS 1),(REAL-NS 1)))),(BoundedLinearOperators ((REAL-NS 1),(REAL-NS 1))))) is set
K198((R_VectorSpace_of_LinearOperators ((REAL-NS 1),(REAL-NS 1))),(BoundedLinearOperators ((REAL-NS 1),(REAL-NS 1)))) is Relation-like K7(REAL,(BoundedLinearOperators ((REAL-NS 1),(REAL-NS 1)))) -defined BoundedLinearOperators ((REAL-NS 1),(REAL-NS 1)) -valued Function-like total V30(K7(REAL,(BoundedLinearOperators ((REAL-NS 1),(REAL-NS 1)))), BoundedLinearOperators ((REAL-NS 1),(REAL-NS 1))) Element of K6(K7(K7(REAL,(BoundedLinearOperators ((REAL-NS 1),(REAL-NS 1)))),(BoundedLinearOperators ((REAL-NS 1),(REAL-NS 1)))))
K7(REAL,(BoundedLinearOperators ((REAL-NS 1),(REAL-NS 1)))) is V33() set
K7(K7(REAL,(BoundedLinearOperators ((REAL-NS 1),(REAL-NS 1)))),(BoundedLinearOperators ((REAL-NS 1),(REAL-NS 1)))) is V33() set
K6(K7(K7(REAL,(BoundedLinearOperators ((REAL-NS 1),(REAL-NS 1)))),(BoundedLinearOperators ((REAL-NS 1),(REAL-NS 1))))) is V33() set
BoundedLinearOperatorsNorm ((REAL-NS 1),(REAL-NS 1)) is non empty Relation-like BoundedLinearOperators ((REAL-NS 1),(REAL-NS 1)) -defined REAL -valued Function-like total V30( BoundedLinearOperators ((REAL-NS 1),(REAL-NS 1)), REAL ) complex-yielding V146() V147() Element of K6(K7((BoundedLinearOperators ((REAL-NS 1),(REAL-NS 1))),REAL))
K7((BoundedLinearOperators ((REAL-NS 1),(REAL-NS 1))),REAL) is V33() complex-yielding V146() V147() set
K6(K7((BoundedLinearOperators ((REAL-NS 1),(REAL-NS 1))),REAL)) is V33() set
G15((BoundedLinearOperators ((REAL-NS 1),(REAL-NS 1))),K199((R_VectorSpace_of_LinearOperators ((REAL-NS 1),(REAL-NS 1))),(BoundedLinearOperators ((REAL-NS 1),(REAL-NS 1)))),K197((R_VectorSpace_of_LinearOperators ((REAL-NS 1),(REAL-NS 1))),(BoundedLinearOperators ((REAL-NS 1),(REAL-NS 1)))),K198((R_VectorSpace_of_LinearOperators ((REAL-NS 1),(REAL-NS 1))),(BoundedLinearOperators ((REAL-NS 1),(REAL-NS 1)))),(BoundedLinearOperatorsNorm ((REAL-NS 1),(REAL-NS 1)))) is V108() L15()
the carrier of (R_NormSpace_of_BoundedLinearOperators ((REAL-NS 1),(REAL-NS 1))) is non empty set
K7(COMPLEX,COMPLEX) is V33() complex-yielding set
K6(K7(COMPLEX,COMPLEX)) is V33() set
K7(K7(COMPLEX,COMPLEX),COMPLEX) is V33() complex-yielding set
K6(K7(K7(COMPLEX,COMPLEX),COMPLEX)) is V33() set
K7(RAT,RAT) is RAT -valued V33() complex-yielding V146() V147() set
K6(K7(RAT,RAT)) is V33() set
K7(K7(RAT,RAT),RAT) is RAT -valued V33() complex-yielding V146() V147() set
K6(K7(K7(RAT,RAT),RAT)) is V33() set
K7(INT,INT) is RAT -valued INT -valued V33() complex-yielding V146() V147() set
K6(K7(INT,INT)) is V33() set
K7(K7(INT,INT),INT) is RAT -valued INT -valued V33() complex-yielding V146() V147() set
K6(K7(K7(INT,INT),INT)) is V33() set
K7(NAT,NAT) is RAT -valued INT -valued complex-yielding V146() V147() V148() set
K7(K7(NAT,NAT),NAT) is RAT -valued INT -valued complex-yielding V146() V147() V148() set
K6(K7(K7(NAT,NAT),NAT)) is set
K537() is non empty V73() L8()
the carrier of K537() is non empty set
K542() is non empty L8()
K543() is non empty V73() M15(K542())
K544() is non empty V73() V121() V220() M18(K542(),K543())
K546() is non empty V73() V121() V123() V125() L8()
K547() is non empty V73() V121() V220() M15(K546())
ExtREAL is non empty V157() set
{} is empty V6() functional V38() V40( {} ) V156() V157() V158() V159() V160() V161() V162() complex-functions-membered ext-real-functions-membered real-functions-membered rational-functions-membered integer-functions-membered natural-functions-membered set
{{},1} is non empty set
K271() is V94() L11()
the carrier of K271() is set
the carrier of K271() * is functional FinSequence-membered FinSequenceSet of the carrier of K271()
TOP-REAL 1 is non empty V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital TopSpace-like V198() V205() L20()
the carrier of (TOP-REAL 1) is non empty set
0 is empty V6() V10() V11() V12() ext-real non positive non negative functional V33() V38() V40( {} ) V120() V155() V156() V157() V158() V159() V160() V161() V162() complex-functions-membered ext-real-functions-membered real-functions-membered rational-functions-membered integer-functions-membered natural-functions-membered Element of NAT
- 1 is V11() V12() ext-real non positive Element of REAL
sqrreal is non empty Relation-like REAL -defined REAL -valued Function-like total V30( REAL , REAL ) complex-yielding V146() V147() Element of K6(K7(REAL,REAL))
<*> REAL is Relation-like NAT -defined REAL -valued Function-like FinSequence-like complex-yielding V146() V147() FinSequence of REAL
Sum (<*> REAL) is V11() V12() ext-real Element of REAL
proj (1,1) is non empty Relation-like REAL 1 -defined REAL -valued Function-like total V30( REAL 1, REAL ) complex-yielding V146() V147() Element of K6(K7((REAL 1),REAL))
K7((REAL 1),REAL) is V33() complex-yielding V146() V147() set
K6(K7((REAL 1),REAL)) is V33() set
(proj (1,1)) " is Relation-like Function-like set
m is V6() V10() V11() V12() ext-real non negative V33() V38() V120() V155() V156() V157() V158() V159() V160() V161() Element of NAT
REAL m is non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered FinSequenceSet of REAL
m -tuples_on REAL is non empty functional FinSequence-membered FinSequenceSet of REAL
REAL-NS m is non empty V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() V108() V109() V142() L15()
the carrier of (REAL-NS m) is non empty set
n is V6() V10() V11() V12() ext-real non negative V33() V38() V120() V155() V156() V157() V158() V159() V160() V161() Element of NAT
REAL n is non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered FinSequenceSet of REAL
n -tuples_on REAL is non empty functional FinSequence-membered FinSequenceSet of REAL
K7((REAL m),(REAL n)) is set
K6(K7((REAL m),(REAL n))) is set
REAL-NS n is non empty V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() V108() V109() V142() L15()
the carrier of (REAL-NS n) is non empty set
K7( the carrier of (REAL-NS m), the carrier of (REAL-NS n)) is set
K6(K7( the carrier of (REAL-NS m), the carrier of (REAL-NS n))) is set
f is set
n is non empty V6() V10() V11() V12() ext-real positive non negative V33() V38() V120() V155() V156() V157() V158() V159() V160() V161() Element of NAT
REAL n is non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered FinSequenceSet of REAL
n -tuples_on REAL is non empty functional FinSequence-membered FinSequenceSet of REAL
m is non empty V6() V10() V11() V12() ext-real positive non negative V33() V38() V120() V155() V156() V157() V158() V159() V160() V161() Element of NAT
REAL m is non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered FinSequenceSet of REAL
m -tuples_on REAL is non empty functional FinSequence-membered FinSequenceSet of REAL
K7((REAL n),(REAL m)) is set
K6(K7((REAL n),(REAL m))) is set
REAL-NS n is non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() V108() V109() V142() L15()
the carrier of (REAL-NS n) is non empty V2() set
REAL-NS m is non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() V108() V109() V142() L15()
the carrier of (REAL-NS m) is non empty V2() set
K7( the carrier of (REAL-NS n), the carrier of (REAL-NS m)) is set
K6(K7( the carrier of (REAL-NS n), the carrier of (REAL-NS m))) is set
f is Relation-like REAL n -defined REAL m -valued Function-like complex-functions-valued ext-real-functions-valued real-functions-valued Element of K6(K7((REAL n),(REAL m)))
Z is Relation-like the carrier of (REAL-NS n) -defined the carrier of (REAL-NS m) -valued Function-like Element of K6(K7( the carrier of (REAL-NS n), the carrier of (REAL-NS m)))
g is Relation-like NAT -defined REAL -valued Function-like V40(n) FinSequence-like complex-yielding V146() V147() Element of REAL n
Z0 is Element of the carrier of (REAL-NS n)
c₇ is Relation-like the carrier of (REAL-NS n) -defined the carrier of (REAL-NS m) -valued Function-like Element of K6(K7( the carrier of (REAL-NS n), the carrier of (REAL-NS m)))
x is Element of the carrier of (REAL-NS n)
n is non empty V6() V10() V11() V12() ext-real positive non negative V33() V38() V120() V155() V156() V157() V158() V159() V160() V161() Element of NAT
REAL n is non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered FinSequenceSet of REAL
n -tuples_on REAL is non empty functional FinSequence-membered FinSequenceSet of REAL
m is non empty V6() V10() V11() V12() ext-real positive non negative V33() V38() V120() V155() V156() V157() V158() V159() V160() V161() Element of NAT
REAL m is non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered FinSequenceSet of REAL
m -tuples_on REAL is non empty functional FinSequence-membered FinSequenceSet of REAL
K7((REAL n),(REAL m)) is set
K6(K7((REAL n),(REAL m))) is set
REAL-NS n is non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() V108() V109() V142() L15()
the carrier of (REAL-NS n) is non empty V2() set
REAL-NS m is non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() V108() V109() V142() L15()
the carrier of (REAL-NS m) is non empty V2() set
K7( the carrier of (REAL-NS n), the carrier of (REAL-NS m)) is set
K6(K7( the carrier of (REAL-NS n), the carrier of (REAL-NS m))) is set
f is Relation-like REAL n -defined REAL m -valued Function-like complex-functions-valued ext-real-functions-valued real-functions-valued Element of K6(K7((REAL n),(REAL m)))
Z is Relation-like the carrier of (REAL-NS n) -defined the carrier of (REAL-NS m) -valued Function-like Element of K6(K7( the carrier of (REAL-NS n), the carrier of (REAL-NS m)))
g is Relation-like NAT -defined REAL -valued Function-like V40(n) FinSequence-like complex-yielding V146() V147() Element of REAL n
diff (f,g) is non empty Relation-like REAL n -defined REAL m -valued Function-like total V30( REAL n, REAL m) complex-functions-valued ext-real-functions-valued real-functions-valued Element of K6(K7((REAL n),(REAL m)))
Z0 is Element of the carrier of (REAL-NS n)
diff (Z,Z0) is Relation-like Function-like Element of the carrier of (R_NormSpace_of_BoundedLinearOperators ((REAL-NS n),(REAL-NS m)))
R_NormSpace_of_BoundedLinearOperators ((REAL-NS n),(REAL-NS m)) is non empty V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() V109() V142() L15()
BoundedLinearOperators ((REAL-NS n),(REAL-NS m)) is non empty Element of K6( the carrier of (R_VectorSpace_of_LinearOperators ((REAL-NS n),(REAL-NS m))))
R_VectorSpace_of_LinearOperators ((REAL-NS n),(REAL-NS m)) is non empty V71() strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V204() RLSStruct
LinearOperators ((REAL-NS n),(REAL-NS m)) is non empty functional Element of K6( the carrier of K279( the carrier of (REAL-NS n),(REAL-NS m)))
K279( the carrier of (REAL-NS n),(REAL-NS m)) is non empty V71() strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V204() RLSStruct
K123( the carrier of (REAL-NS n), the carrier of (REAL-NS m)) is non empty functional M4( the carrier of (REAL-NS n), the carrier of (REAL-NS m))
FuncZero ( the carrier of (REAL-NS n),(REAL-NS m)) is Relation-like the carrier of (REAL-NS n) -defined the carrier of (REAL-NS m) -valued Function-like total V30( the carrier of (REAL-NS n), the carrier of (REAL-NS m)) Element of K123( the carrier of (REAL-NS n), the carrier of (REAL-NS m))
0. (REAL-NS m) is V52( REAL-NS m) Element of the carrier of (REAL-NS m)
the carrier of (REAL-NS n) --> (0. (REAL-NS m)) is non empty Relation-like the carrier of (REAL-NS n) -defined the carrier of (REAL-NS m) -valued Function-like total V30( the carrier of (REAL-NS n), the carrier of (REAL-NS m)) Element of K6(K7( the carrier of (REAL-NS n), the carrier of (REAL-NS m)))
FuncAdd ( the carrier of (REAL-NS n),(REAL-NS m)) is Relation-like K7(K123( the carrier of (REAL-NS n), the carrier of (REAL-NS m)),K123( the carrier of (REAL-NS n), the carrier of (REAL-NS m))) -defined K123( the carrier of (REAL-NS n), the carrier of (REAL-NS m)) -valued Function-like total V30(K7(K123( the carrier of (REAL-NS n), the carrier of (REAL-NS m)),K123( the carrier of (REAL-NS n), the carrier of (REAL-NS m))),K123( the carrier of (REAL-NS n), the carrier of (REAL-NS m))) Element of K6(K7(K7(K123( the carrier of (REAL-NS n), the carrier of (REAL-NS m)),K123( the carrier of (REAL-NS n), the carrier of (REAL-NS m))),K123( the carrier of (REAL-NS n), the carrier of (REAL-NS m))))
K7(K123( the carrier of (REAL-NS n), the carrier of (REAL-NS m)),K123( the carrier of (REAL-NS n), the carrier of (REAL-NS m))) is set
K7(K7(K123( the carrier of (REAL-NS n), the carrier of (REAL-NS m)),K123( the carrier of (REAL-NS n), the carrier of (REAL-NS m))),K123( the carrier of (REAL-NS n), the carrier of (REAL-NS m))) is set
K6(K7(K7(K123( the carrier of (REAL-NS n), the carrier of (REAL-NS m)),K123( the carrier of (REAL-NS n), the carrier of (REAL-NS m))),K123( the carrier of (REAL-NS n), the carrier of (REAL-NS m)))) is set
FuncExtMult ( the carrier of (REAL-NS n),(REAL-NS m)) is Relation-like K7(REAL,K123( the carrier of (REAL-NS n), the carrier of (REAL-NS m))) -defined K123( the carrier of (REAL-NS n), the carrier of (REAL-NS m)) -valued Function-like total V30(K7(REAL,K123( the carrier of (REAL-NS n), the carrier of (REAL-NS m))),K123( the carrier of (REAL-NS n), the carrier of (REAL-NS m))) Element of K6(K7(K7(REAL,K123( the carrier of (REAL-NS n), the carrier of (REAL-NS m))),K123( the carrier of (REAL-NS n), the carrier of (REAL-NS m))))
K7(REAL,K123( the carrier of (REAL-NS n), the carrier of (REAL-NS m))) is V33() set
K7(K7(REAL,K123( the carrier of (REAL-NS n), the carrier of (REAL-NS m))),K123( the carrier of (REAL-NS n), the carrier of (REAL-NS m))) is V33() set
K6(K7(K7(REAL,K123( the carrier of (REAL-NS n), the carrier of (REAL-NS m))),K123( the carrier of (REAL-NS n), the carrier of (REAL-NS m)))) is V33() set
RLSStruct(# K123( the carrier of (REAL-NS n), the carrier of (REAL-NS m)),(FuncZero ( the carrier of (REAL-NS n),(REAL-NS m))),(FuncAdd ( the carrier of (REAL-NS n),(REAL-NS m))),(FuncExtMult ( the carrier of (REAL-NS n),(REAL-NS m))) #) is strict RLSStruct
the carrier of K279( the carrier of (REAL-NS n),(REAL-NS m)) is non empty set
K6( the carrier of K279( the carrier of (REAL-NS n),(REAL-NS m))) is set
K199(K279( the carrier of (REAL-NS n),(REAL-NS m)),(LinearOperators ((REAL-NS n),(REAL-NS m)))) is Element of LinearOperators ((REAL-NS n),(REAL-NS m))
K197(K279( the carrier of (REAL-NS n),(REAL-NS m)),(LinearOperators ((REAL-NS n),(REAL-NS m)))) is Relation-like K7((LinearOperators ((REAL-NS n),(REAL-NS m))),(LinearOperators ((REAL-NS n),(REAL-NS m)))) -defined LinearOperators ((REAL-NS n),(REAL-NS m)) -valued Function-like total V30(K7((LinearOperators ((REAL-NS n),(REAL-NS m))),(LinearOperators ((REAL-NS n),(REAL-NS m)))), LinearOperators ((REAL-NS n),(REAL-NS m))) Element of K6(K7(K7((LinearOperators ((REAL-NS n),(REAL-NS m))),(LinearOperators ((REAL-NS n),(REAL-NS m)))),(LinearOperators ((REAL-NS n),(REAL-NS m)))))
K7((LinearOperators ((REAL-NS n),(REAL-NS m))),(LinearOperators ((REAL-NS n),(REAL-NS m)))) is set
K7(K7((LinearOperators ((REAL-NS n),(REAL-NS m))),(LinearOperators ((REAL-NS n),(REAL-NS m)))),(LinearOperators ((REAL-NS n),(REAL-NS m)))) is set
K6(K7(K7((LinearOperators ((REAL-NS n),(REAL-NS m))),(LinearOperators ((REAL-NS n),(REAL-NS m)))),(LinearOperators ((REAL-NS n),(REAL-NS m))))) is set
K198(K279( the carrier of (REAL-NS n),(REAL-NS m)),(LinearOperators ((REAL-NS n),(REAL-NS m)))) is Relation-like K7(REAL,(LinearOperators ((REAL-NS n),(REAL-NS m)))) -defined LinearOperators ((REAL-NS n),(REAL-NS m)) -valued Function-like total V30(K7(REAL,(LinearOperators ((REAL-NS n),(REAL-NS m)))), LinearOperators ((REAL-NS n),(REAL-NS m))) Element of K6(K7(K7(REAL,(LinearOperators ((REAL-NS n),(REAL-NS m)))),(LinearOperators ((REAL-NS n),(REAL-NS m)))))
K7(REAL,(LinearOperators ((REAL-NS n),(REAL-NS m)))) is V33() set
K7(K7(REAL,(LinearOperators ((REAL-NS n),(REAL-NS m)))),(LinearOperators ((REAL-NS n),(REAL-NS m)))) is V33() set
K6(K7(K7(REAL,(LinearOperators ((REAL-NS n),(REAL-NS m)))),(LinearOperators ((REAL-NS n),(REAL-NS m))))) is V33() set
RLSStruct(# (LinearOperators ((REAL-NS n),(REAL-NS m))),K199(K279( the carrier of (REAL-NS n),(REAL-NS m)),(LinearOperators ((REAL-NS n),(REAL-NS m)))),K197(K279( the carrier of (REAL-NS n),(REAL-NS m)),(LinearOperators ((REAL-NS n),(REAL-NS m)))),K198(K279( the carrier of (REAL-NS n),(REAL-NS m)),(LinearOperators ((REAL-NS n),(REAL-NS m)))) #) is V71() strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RLSStruct
the carrier of (R_VectorSpace_of_LinearOperators ((REAL-NS n),(REAL-NS m))) is non empty set
K6( the carrier of (R_VectorSpace_of_LinearOperators ((REAL-NS n),(REAL-NS m)))) is set
K199((R_VectorSpace_of_LinearOperators ((REAL-NS n),(REAL-NS m))),(BoundedLinearOperators ((REAL-NS n),(REAL-NS m)))) is Element of BoundedLinearOperators ((REAL-NS n),(REAL-NS m))
K197((R_VectorSpace_of_LinearOperators ((REAL-NS n),(REAL-NS m))),(BoundedLinearOperators ((REAL-NS n),(REAL-NS m)))) is Relation-like K7((BoundedLinearOperators ((REAL-NS n),(REAL-NS m))),(BoundedLinearOperators ((REAL-NS n),(REAL-NS m)))) -defined BoundedLinearOperators ((REAL-NS n),(REAL-NS m)) -valued Function-like total V30(K7((BoundedLinearOperators ((REAL-NS n),(REAL-NS m))),(BoundedLinearOperators ((REAL-NS n),(REAL-NS m)))), BoundedLinearOperators ((REAL-NS n),(REAL-NS m))) Element of K6(K7(K7((BoundedLinearOperators ((REAL-NS n),(REAL-NS m))),(BoundedLinearOperators ((REAL-NS n),(REAL-NS m)))),(BoundedLinearOperators ((REAL-NS n),(REAL-NS m)))))
K7((BoundedLinearOperators ((REAL-NS n),(REAL-NS m))),(BoundedLinearOperators ((REAL-NS n),(REAL-NS m)))) is set
K7(K7((BoundedLinearOperators ((REAL-NS n),(REAL-NS m))),(BoundedLinearOperators ((REAL-NS n),(REAL-NS m)))),(BoundedLinearOperators ((REAL-NS n),(REAL-NS m)))) is set
K6(K7(K7((BoundedLinearOperators ((REAL-NS n),(REAL-NS m))),(BoundedLinearOperators ((REAL-NS n),(REAL-NS m)))),(BoundedLinearOperators ((REAL-NS n),(REAL-NS m))))) is set
K198((R_VectorSpace_of_LinearOperators ((REAL-NS n),(REAL-NS m))),(BoundedLinearOperators ((REAL-NS n),(REAL-NS m)))) is Relation-like K7(REAL,(BoundedLinearOperators ((REAL-NS n),(REAL-NS m)))) -defined BoundedLinearOperators ((REAL-NS n),(REAL-NS m)) -valued Function-like total V30(K7(REAL,(BoundedLinearOperators ((REAL-NS n),(REAL-NS m)))), BoundedLinearOperators ((REAL-NS n),(REAL-NS m))) Element of K6(K7(K7(REAL,(BoundedLinearOperators ((REAL-NS n),(REAL-NS m)))),(BoundedLinearOperators ((REAL-NS n),(REAL-NS m)))))
K7(REAL,(BoundedLinearOperators ((REAL-NS n),(REAL-NS m)))) is V33() set
K7(K7(REAL,(BoundedLinearOperators ((REAL-NS n),(REAL-NS m)))),(BoundedLinearOperators ((REAL-NS n),(REAL-NS m)))) is V33() set
K6(K7(K7(REAL,(BoundedLinearOperators ((REAL-NS n),(REAL-NS m)))),(BoundedLinearOperators ((REAL-NS n),(REAL-NS m))))) is V33() set
BoundedLinearOperatorsNorm ((REAL-NS n),(REAL-NS m)) is non empty Relation-like BoundedLinearOperators ((REAL-NS n),(REAL-NS m)) -defined REAL -valued Function-like total V30( BoundedLinearOperators ((REAL-NS n),(REAL-NS m)), REAL ) complex-yielding V146() V147() Element of K6(K7((BoundedLinearOperators ((REAL-NS n),(REAL-NS m))),REAL))
K7((BoundedLinearOperators ((REAL-NS n),(REAL-NS m))),REAL) is V33() complex-yielding V146() V147() set
K6(K7((BoundedLinearOperators ((REAL-NS n),(REAL-NS m))),REAL)) is V33() set
G15((BoundedLinearOperators ((REAL-NS n),(REAL-NS m))),K199((R_VectorSpace_of_LinearOperators ((REAL-NS n),(REAL-NS m))),(BoundedLinearOperators ((REAL-NS n),(REAL-NS m)))),K197((R_VectorSpace_of_LinearOperators ((REAL-NS n),(REAL-NS m))),(BoundedLinearOperators ((REAL-NS n),(REAL-NS m)))),K198((R_VectorSpace_of_LinearOperators ((REAL-NS n),(REAL-NS m))),(BoundedLinearOperators ((REAL-NS n),(REAL-NS m)))),(BoundedLinearOperatorsNorm ((REAL-NS n),(REAL-NS m)))) is V108() L15()
the carrier of (R_NormSpace_of_BoundedLinearOperators ((REAL-NS n),(REAL-NS m))) is non empty set
c₇ is Relation-like the carrier of (REAL-NS n) -defined the carrier of (REAL-NS m) -valued Function-like Element of K6(K7( the carrier of (REAL-NS n), the carrier of (REAL-NS m)))
x is Element of the carrier of (REAL-NS n)
diff (c₇,x) is Relation-like Function-like Element of the carrier of (R_NormSpace_of_BoundedLinearOperators ((REAL-NS n),(REAL-NS m)))
m is V6() V10() V11() V12() ext-real non negative V33() V38() V120() V155() V156() V157() V158() V159() V160() V161() Element of NAT
REAL m is non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered FinSequenceSet of REAL
m -tuples_on REAL is non empty functional FinSequence-membered FinSequenceSet of REAL
REAL-NS m is non empty V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() V108() V109() V142() L15()
the carrier of (REAL-NS m) is non empty set
n is V6() V10() V11() V12() ext-real non negative V33() V38() V120() V155() V156() V157() V158() V159() V160() V161() Element of NAT
REAL n is non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered FinSequenceSet of REAL
n -tuples_on REAL is non empty functional FinSequence-membered FinSequenceSet of REAL
K7((REAL m),(REAL n)) is set
K6(K7((REAL m),(REAL n))) is set
REAL-NS n is non empty V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() V108() V109() V142() L15()
the carrier of (REAL-NS n) is non empty set
K7( the carrier of (REAL-NS m), the carrier of (REAL-NS n)) is set
K6(K7( the carrier of (REAL-NS m), the carrier of (REAL-NS n))) is set
f is Relation-like REAL m -defined REAL n -valued Function-like complex-functions-valued ext-real-functions-valued real-functions-valued Element of K6(K7((REAL m),(REAL n)))
Z is Relation-like REAL m -defined REAL n -valued Function-like complex-functions-valued ext-real-functions-valued real-functions-valued Element of K6(K7((REAL m),(REAL n)))
f + Z is Relation-like REAL m -defined REAL n -valued Function-like complex-functions-valued ext-real-functions-valued real-functions-valued Element of K6(K7((REAL m),(REAL n)))
g is Relation-like the carrier of (REAL-NS m) -defined the carrier of (REAL-NS n) -valued Function-like Element of K6(K7( the carrier of (REAL-NS m), the carrier of (REAL-NS n)))
Z0 is Relation-like the carrier of (REAL-NS m) -defined the carrier of (REAL-NS n) -valued Function-like Element of K6(K7( the carrier of (REAL-NS m), the carrier of (REAL-NS n)))
g + Z0 is Relation-like the carrier of (REAL-NS m) -defined the carrier of (REAL-NS n) -valued Function-like Element of K6(K7( the carrier of (REAL-NS m), the carrier of (REAL-NS n)))
dom f is functional complex-functions-membered ext-real-functions-membered real-functions-membered Element of K6((REAL m))
K6((REAL m)) is set
dom Z is functional complex-functions-membered ext-real-functions-membered real-functions-membered Element of K6((REAL m))
(dom f) /\ (dom Z) is functional complex-functions-membered ext-real-functions-membered real-functions-membered Element of K6((REAL m))
c₇ is Relation-like REAL m -defined REAL n -valued Function-like complex-functions-valued ext-real-functions-valued real-functions-valued Element of K6(K7((REAL m),(REAL n)))
dom c₇ is functional complex-functions-membered ext-real-functions-membered real-functions-membered Element of K6((REAL m))
f <++> Z is Relation-like (REAL m) /\ (REAL m) -defined R_PFuncs ((DOMS (REAL n)) /\ (DOMS (REAL n))) -valued Function-like complex-functions-valued ext-real-functions-valued real-functions-valued Element of K6(K7(((REAL m) /\ (REAL m)),(R_PFuncs ((DOMS (REAL n)) /\ (DOMS (REAL n))))))
(REAL m) /\ (REAL m) is set
DOMS (REAL n) is set
(DOMS (REAL n)) /\ (DOMS (REAL n)) is set
R_PFuncs ((DOMS (REAL n)) /\ (DOMS (REAL n))) is functional complex-functions-membered ext-real-functions-membered real-functions-membered set
K7(((REAL m) /\ (REAL m)),(R_PFuncs ((DOMS (REAL n)) /\ (DOMS (REAL n))))) is set
K6(K7(((REAL m) /\ (REAL m)),(R_PFuncs ((DOMS (REAL n)) /\ (DOMS (REAL n)))))) is set
x is set
(g + Z0) . x is set
f . x is Relation-like Function-like complex-yielding V146() V147() set
Z . x is Relation-like Function-like complex-yielding V146() V147() set
(f . x) + (Z . x) is Relation-like Function-like set
dom (g + Z0) is Element of K6( the carrier of (REAL-NS m))
K6( the carrier of (REAL-NS m)) is set
f /. x is V40(n) complex-yielding V146() V147() Element of REAL n
Z /. x is V40(n) complex-yielding V146() V147() Element of REAL n
g /. x is Element of the carrier of (REAL-NS n)
Z0 /. x is Element of the carrier of (REAL-NS n)
c₇ . x is Relation-like Function-like complex-yielding V146() V147() set
(g + Z0) /. x is Element of the carrier of (REAL-NS n)
(g /. x) + (Z0 /. x) is Element of the carrier of (REAL-NS n)
m is V6() V10() V11() V12() ext-real non negative V33() V38() V120() V155() V156() V157() V158() V159() V160() V161() Element of NAT
REAL m is non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered FinSequenceSet of REAL
m -tuples_on REAL is non empty functional FinSequence-membered FinSequenceSet of REAL
REAL-NS m is non empty V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() V108() V109() V142() L15()
the carrier of (REAL-NS m) is non empty set
n is V6() V10() V11() V12() ext-real non negative V33() V38() V120() V155() V156() V157() V158() V159() V160() V161() Element of NAT
REAL n is non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered FinSequenceSet of REAL
n -tuples_on REAL is non empty functional FinSequence-membered FinSequenceSet of REAL
K7((REAL m),(REAL n)) is set
K6(K7((REAL m),(REAL n))) is set
REAL-NS n is non empty V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() V108() V109() V142() L15()
the carrier of (REAL-NS n) is non empty set
K7( the carrier of (REAL-NS m), the carrier of (REAL-NS n)) is set
K6(K7( the carrier of (REAL-NS m), the carrier of (REAL-NS n))) is set
f is Relation-like REAL m -defined REAL n -valued Function-like complex-functions-valued ext-real-functions-valued real-functions-valued Element of K6(K7((REAL m),(REAL n)))
Z is Relation-like REAL m -defined REAL n -valued Function-like complex-functions-valued ext-real-functions-valued real-functions-valued Element of K6(K7((REAL m),(REAL n)))
f - Z is Relation-like REAL m -defined REAL n -valued Function-like complex-functions-valued ext-real-functions-valued real-functions-valued Element of K6(K7((REAL m),(REAL n)))
g is Relation-like the carrier of (REAL-NS m) -defined the carrier of (REAL-NS n) -valued Function-like Element of K6(K7( the carrier of (REAL-NS m), the carrier of (REAL-NS n)))
Z0 is Relation-like the carrier of (REAL-NS m) -defined the carrier of (REAL-NS n) -valued Function-like Element of K6(K7( the carrier of (REAL-NS m), the carrier of (REAL-NS n)))
g - Z0 is Relation-like the carrier of (REAL-NS m) -defined the carrier of (REAL-NS n) -valued Function-like Element of K6(K7( the carrier of (REAL-NS m), the carrier of (REAL-NS n)))
dom f is functional complex-functions-membered ext-real-functions-membered real-functions-membered Element of K6((REAL m))
K6((REAL m)) is set
dom Z is functional complex-functions-membered ext-real-functions-membered real-functions-membered Element of K6((REAL m))
(dom f) /\ (dom Z) is functional complex-functions-membered ext-real-functions-membered real-functions-membered Element of K6((REAL m))
c₇ is Relation-like REAL m -defined REAL n -valued Function-like complex-functions-valued ext-real-functions-valued real-functions-valued Element of K6(K7((REAL m),(REAL n)))
dom c₇ is functional complex-functions-membered ext-real-functions-membered real-functions-membered Element of K6((REAL m))
f <--> Z is Relation-like (REAL m) /\ (REAL m) -defined R_PFuncs ((DOMS (REAL n)) /\ (DOMS (REAL n))) -valued Function-like complex-functions-valued ext-real-functions-valued real-functions-valued Element of K6(K7(((REAL m) /\ (REAL m)),(R_PFuncs ((DOMS (REAL n)) /\ (DOMS (REAL n))))))
(REAL m) /\ (REAL m) is set
DOMS (REAL n) is set
(DOMS (REAL n)) /\ (DOMS (REAL n)) is set
R_PFuncs ((DOMS (REAL n)) /\ (DOMS (REAL n))) is functional complex-functions-membered ext-real-functions-membered real-functions-membered set
K7(((REAL m) /\ (REAL m)),(R_PFuncs ((DOMS (REAL n)) /\ (DOMS (REAL n))))) is set
K6(K7(((REAL m) /\ (REAL m)),(R_PFuncs ((DOMS (REAL n)) /\ (DOMS (REAL n)))))) is set
x is set
(g - Z0) . x is set
f . x is Relation-like Function-like complex-yielding V146() V147() set
Z . x is Relation-like Function-like complex-yielding V146() V147() set
(f . x) - (Z . x) is Relation-like Function-like set
- (Z . x) is Relation-like Function-like complex-yielding set
- 1 is V11() V12() ext-real non positive set
(- 1) (#) (Z . x) is Relation-like Function-like set
(f . x) + (- (Z . x)) is Relation-like Function-like set
dom (g - Z0) is Element of K6( the carrier of (REAL-NS m))
K6( the carrier of (REAL-NS m)) is set
f /. x is V40(n) complex-yielding V146() V147() Element of REAL n
Z /. x is V40(n) complex-yielding V146() V147() Element of REAL n
g /. x is Element of the carrier of (REAL-NS n)
Z0 /. x is Element of the carrier of (REAL-NS n)
c₇ . x is Relation-like Function-like complex-yielding V146() V147() set
(g - Z0) /. x is Element of the carrier of (REAL-NS n)
(g /. x) - (Z0 /. x) is Element of the carrier of (REAL-NS n)
m is V6() V10() V11() V12() ext-real non negative V33() V38() V120() V155() V156() V157() V158() V159() V160() V161() Element of NAT
REAL m is non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered FinSequenceSet of REAL
m -tuples_on REAL is non empty functional FinSequence-membered FinSequenceSet of REAL
REAL-NS m is non empty V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() V108() V109() V142() L15()
the carrier of (REAL-NS m) is non empty set
n is V6() V10() V11() V12() ext-real non negative V33() V38() V120() V155() V156() V157() V158() V159() V160() V161() Element of NAT
REAL n is non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered FinSequenceSet of REAL
n -tuples_on REAL is non empty functional FinSequence-membered FinSequenceSet of REAL
K7((REAL m),(REAL n)) is set
K6(K7((REAL m),(REAL n))) is set
REAL-NS n is non empty V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() V108() V109() V142() L15()
the carrier of (REAL-NS n) is non empty set
K7( the carrier of (REAL-NS m), the carrier of (REAL-NS n)) is set
K6(K7( the carrier of (REAL-NS m), the carrier of (REAL-NS n))) is set
f is Relation-like REAL m -defined REAL n -valued Function-like complex-functions-valued ext-real-functions-valued real-functions-valued Element of K6(K7((REAL m),(REAL n)))
Z is Relation-like the carrier of (REAL-NS m) -defined the carrier of (REAL-NS n) -valued Function-like Element of K6(K7( the carrier of (REAL-NS m), the carrier of (REAL-NS n)))
g is V11() V12() ext-real Element of REAL
g (#) f is Relation-like REAL m -defined REAL n -valued Function-like complex-functions-valued ext-real-functions-valued real-functions-valued Element of K6(K7((REAL m),(REAL n)))
g (#) Z is Relation-like the carrier of (REAL-NS m) -defined the carrier of (REAL-NS n) -valued Function-like Element of K6(K7( the carrier of (REAL-NS m), the carrier of (REAL-NS n)))
dom f is functional complex-functions-membered ext-real-functions-membered real-functions-membered Element of K6((REAL m))
K6((REAL m)) is set
Z0 is Relation-like REAL m -defined REAL n -valued Function-like complex-functions-valued ext-real-functions-valued real-functions-valued Element of K6(K7((REAL m),(REAL n)))
dom Z0 is functional complex-functions-membered ext-real-functions-membered real-functions-membered Element of K6((REAL m))
f [#] g is Relation-like REAL m -defined R_PFuncs (DOMS (REAL n)) -valued Function-like complex-functions-valued ext-real-functions-valued real-functions-valued Element of K6(K7((REAL m),(R_PFuncs (DOMS (REAL n)))))
DOMS (REAL n) is set
R_PFuncs (DOMS (REAL n)) is functional complex-functions-membered ext-real-functions-membered real-functions-membered set
K7((REAL m),(R_PFuncs (DOMS (REAL n)))) is set
K6(K7((REAL m),(R_PFuncs (DOMS (REAL n))))) is set
c₇ is set
Z0 . c₇ is Relation-like Function-like complex-yielding V146() V147() set
f . c₇ is Relation-like Function-like complex-yielding V146() V147() set
g (#) (f . c₇) is Relation-like Function-like set
dom (g (#) Z) is Element of K6( the carrier of (REAL-NS m))
K6( the carrier of (REAL-NS m)) is set
f /. c₇ is V40(n) complex-yielding V146() V147() Element of REAL n
Z /. c₇ is Element of the carrier of (REAL-NS n)
(g (#) Z) /. c₇ is Element of the carrier of (REAL-NS n)
g * (Z /. c₇) is Element of the carrier of (REAL-NS n)
m is V6() V10() V11() V12() ext-real non negative V33() V38() V120() V155() V156() V157() V158() V159() V160() V161() Element of NAT
REAL m is non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered FinSequenceSet of REAL
m -tuples_on REAL is non empty functional FinSequence-membered FinSequenceSet of REAL
REAL-NS m is non empty V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() V108() V109() V142() L15()
n is V6() V10() V11() V12() ext-real non negative V33() V38() V120() V155() V156() V157() V158() V159() V160() V161() Element of NAT
REAL n is non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered FinSequenceSet of REAL
n -tuples_on REAL is non empty functional FinSequence-membered FinSequenceSet of REAL
K7((REAL m),(REAL n)) is set
K6(K7((REAL m),(REAL n))) is set
REAL-NS n is non empty V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() V108() V109() V142() L15()
R_NormSpace_of_BoundedLinearOperators ((REAL-NS m),(REAL-NS n)) is non empty V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() V109() V142() L15()
BoundedLinearOperators ((REAL-NS m),(REAL-NS n)) is non empty Element of K6( the carrier of (R_VectorSpace_of_LinearOperators ((REAL-NS m),(REAL-NS n))))
R_VectorSpace_of_LinearOperators ((REAL-NS m),(REAL-NS n)) is non empty V71() strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V204() RLSStruct
LinearOperators ((REAL-NS m),(REAL-NS n)) is non empty functional Element of K6( the carrier of K279( the carrier of (REAL-NS m),(REAL-NS n)))
the carrier of (REAL-NS m) is non empty set
K279( the carrier of (REAL-NS m),(REAL-NS n)) is non empty V71() strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V204() RLSStruct
the carrier of (REAL-NS n) is non empty set
K123( the carrier of (REAL-NS m), the carrier of (REAL-NS n)) is non empty functional M4( the carrier of (REAL-NS m), the carrier of (REAL-NS n))
FuncZero ( the carrier of (REAL-NS m),(REAL-NS n)) is Relation-like the carrier of (REAL-NS m) -defined the carrier of (REAL-NS n) -valued Function-like total V30( the carrier of (REAL-NS m), the carrier of (REAL-NS n)) Element of K123( the carrier of (REAL-NS m), the carrier of (REAL-NS n))
0. (REAL-NS n) is V52( REAL-NS n) Element of the carrier of (REAL-NS n)
the carrier of (REAL-NS m) --> (0. (REAL-NS n)) is non empty Relation-like the carrier of (REAL-NS m) -defined the carrier of (REAL-NS n) -valued Function-like total V30( the carrier of (REAL-NS m), the carrier of (REAL-NS n)) Element of K6(K7( the carrier of (REAL-NS m), the carrier of (REAL-NS n)))
K7( the carrier of (REAL-NS m), the carrier of (REAL-NS n)) is set
K6(K7( the carrier of (REAL-NS m), the carrier of (REAL-NS n))) is set
FuncAdd ( the carrier of (REAL-NS m),(REAL-NS n)) is Relation-like K7(K123( the carrier of (REAL-NS m), the carrier of (REAL-NS n)),K123( the carrier of (REAL-NS m), the carrier of (REAL-NS n))) -defined K123( the carrier of (REAL-NS m), the carrier of (REAL-NS n)) -valued Function-like total V30(K7(K123( the carrier of (REAL-NS m), the carrier of (REAL-NS n)),K123( the carrier of (REAL-NS m), the carrier of (REAL-NS n))),K123( the carrier of (REAL-NS m), the carrier of (REAL-NS n))) Element of K6(K7(K7(K123( the carrier of (REAL-NS m), the carrier of (REAL-NS n)),K123( the carrier of (REAL-NS m), the carrier of (REAL-NS n))),K123( the carrier of (REAL-NS m), the carrier of (REAL-NS n))))
K7(K123( the carrier of (REAL-NS m), the carrier of (REAL-NS n)),K123( the carrier of (REAL-NS m), the carrier of (REAL-NS n))) is set
K7(K7(K123( the carrier of (REAL-NS m), the carrier of (REAL-NS n)),K123( the carrier of (REAL-NS m), the carrier of (REAL-NS n))),K123( the carrier of (REAL-NS m), the carrier of (REAL-NS n))) is set
K6(K7(K7(K123( the carrier of (REAL-NS m), the carrier of (REAL-NS n)),K123( the carrier of (REAL-NS m), the carrier of (REAL-NS n))),K123( the carrier of (REAL-NS m), the carrier of (REAL-NS n)))) is set
FuncExtMult ( the carrier of (REAL-NS m),(REAL-NS n)) is Relation-like K7(REAL,K123( the carrier of (REAL-NS m), the carrier of (REAL-NS n))) -defined K123( the carrier of (REAL-NS m), the carrier of (REAL-NS n)) -valued Function-like total V30(K7(REAL,K123( the carrier of (REAL-NS m), the carrier of (REAL-NS n))),K123( the carrier of (REAL-NS m), the carrier of (REAL-NS n))) Element of K6(K7(K7(REAL,K123( the carrier of (REAL-NS m), the carrier of (REAL-NS n))),K123( the carrier of (REAL-NS m), the carrier of (REAL-NS n))))
K7(REAL,K123( the carrier of (REAL-NS m), the carrier of (REAL-NS n))) is V33() set
K7(K7(REAL,K123( the carrier of (REAL-NS m), the carrier of (REAL-NS n))),K123( the carrier of (REAL-NS m), the carrier of (REAL-NS n))) is V33() set
K6(K7(K7(REAL,K123( the carrier of (REAL-NS m), the carrier of (REAL-NS n))),K123( the carrier of (REAL-NS m), the carrier of (REAL-NS n)))) is V33() set
RLSStruct(# K123( the carrier of (REAL-NS m), the carrier of (REAL-NS n)),(FuncZero ( the carrier of (REAL-NS m),(REAL-NS n))),(FuncAdd ( the carrier of (REAL-NS m),(REAL-NS n))),(FuncExtMult ( the carrier of (REAL-NS m),(REAL-NS n))) #) is strict RLSStruct
the carrier of K279( the carrier of (REAL-NS m),(REAL-NS n)) is non empty set
K6( the carrier of K279( the carrier of (REAL-NS m),(REAL-NS n))) is set
K199(K279( the carrier of (REAL-NS m),(REAL-NS n)),(LinearOperators ((REAL-NS m),(REAL-NS n)))) is Element of LinearOperators ((REAL-NS m),(REAL-NS n))
K197(K279( the carrier of (REAL-NS m),(REAL-NS n)),(LinearOperators ((REAL-NS m),(REAL-NS n)))) is Relation-like K7((LinearOperators ((REAL-NS m),(REAL-NS n))),(LinearOperators ((REAL-NS m),(REAL-NS n)))) -defined LinearOperators ((REAL-NS m),(REAL-NS n)) -valued Function-like total V30(K7((LinearOperators ((REAL-NS m),(REAL-NS n))),(LinearOperators ((REAL-NS m),(REAL-NS n)))), LinearOperators ((REAL-NS m),(REAL-NS n))) Element of K6(K7(K7((LinearOperators ((REAL-NS m),(REAL-NS n))),(LinearOperators ((REAL-NS m),(REAL-NS n)))),(LinearOperators ((REAL-NS m),(REAL-NS n)))))
K7((LinearOperators ((REAL-NS m),(REAL-NS n))),(LinearOperators ((REAL-NS m),(REAL-NS n)))) is set
K7(K7((LinearOperators ((REAL-NS m),(REAL-NS n))),(LinearOperators ((REAL-NS m),(REAL-NS n)))),(LinearOperators ((REAL-NS m),(REAL-NS n)))) is set
K6(K7(K7((LinearOperators ((REAL-NS m),(REAL-NS n))),(LinearOperators ((REAL-NS m),(REAL-NS n)))),(LinearOperators ((REAL-NS m),(REAL-NS n))))) is set
K198(K279( the carrier of (REAL-NS m),(REAL-NS n)),(LinearOperators ((REAL-NS m),(REAL-NS n)))) is Relation-like K7(REAL,(LinearOperators ((REAL-NS m),(REAL-NS n)))) -defined LinearOperators ((REAL-NS m),(REAL-NS n)) -valued Function-like total V30(K7(REAL,(LinearOperators ((REAL-NS m),(REAL-NS n)))), LinearOperators ((REAL-NS m),(REAL-NS n))) Element of K6(K7(K7(REAL,(LinearOperators ((REAL-NS m),(REAL-NS n)))),(LinearOperators ((REAL-NS m),(REAL-NS n)))))
K7(REAL,(LinearOperators ((REAL-NS m),(REAL-NS n)))) is V33() set
K7(K7(REAL,(LinearOperators ((REAL-NS m),(REAL-NS n)))),(LinearOperators ((REAL-NS m),(REAL-NS n)))) is V33() set
K6(K7(K7(REAL,(LinearOperators ((REAL-NS m),(REAL-NS n)))),(LinearOperators ((REAL-NS m),(REAL-NS n))))) is V33() set
RLSStruct(# (LinearOperators ((REAL-NS m),(REAL-NS n))),K199(K279( the carrier of (REAL-NS m),(REAL-NS n)),(LinearOperators ((REAL-NS m),(REAL-NS n)))),K197(K279( the carrier of (REAL-NS m),(REAL-NS n)),(LinearOperators ((REAL-NS m),(REAL-NS n)))),K198(K279( the carrier of (REAL-NS m),(REAL-NS n)),(LinearOperators ((REAL-NS m),(REAL-NS n)))) #) is V71() strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RLSStruct
the carrier of (R_VectorSpace_of_LinearOperators ((REAL-NS m),(REAL-NS n))) is non empty set
K6( the carrier of (R_VectorSpace_of_LinearOperators ((REAL-NS m),(REAL-NS n)))) is set
K199((R_VectorSpace_of_LinearOperators ((REAL-NS m),(REAL-NS n))),(BoundedLinearOperators ((REAL-NS m),(REAL-NS n)))) is Element of BoundedLinearOperators ((REAL-NS m),(REAL-NS n))
K197((R_VectorSpace_of_LinearOperators ((REAL-NS m),(REAL-NS n))),(BoundedLinearOperators ((REAL-NS m),(REAL-NS n)))) is Relation-like K7((BoundedLinearOperators ((REAL-NS m),(REAL-NS n))),(BoundedLinearOperators ((REAL-NS m),(REAL-NS n)))) -defined BoundedLinearOperators ((REAL-NS m),(REAL-NS n)) -valued Function-like total V30(K7((BoundedLinearOperators ((REAL-NS m),(REAL-NS n))),(BoundedLinearOperators ((REAL-NS m),(REAL-NS n)))), BoundedLinearOperators ((REAL-NS m),(REAL-NS n))) Element of K6(K7(K7((BoundedLinearOperators ((REAL-NS m),(REAL-NS n))),(BoundedLinearOperators ((REAL-NS m),(REAL-NS n)))),(BoundedLinearOperators ((REAL-NS m),(REAL-NS n)))))
K7((BoundedLinearOperators ((REAL-NS m),(REAL-NS n))),(BoundedLinearOperators ((REAL-NS m),(REAL-NS n)))) is set
K7(K7((BoundedLinearOperators ((REAL-NS m),(REAL-NS n))),(BoundedLinearOperators ((REAL-NS m),(REAL-NS n)))),(BoundedLinearOperators ((REAL-NS m),(REAL-NS n)))) is set
K6(K7(K7((BoundedLinearOperators ((REAL-NS m),(REAL-NS n))),(BoundedLinearOperators ((REAL-NS m),(REAL-NS n)))),(BoundedLinearOperators ((REAL-NS m),(REAL-NS n))))) is set
K198((R_VectorSpace_of_LinearOperators ((REAL-NS m),(REAL-NS n))),(BoundedLinearOperators ((REAL-NS m),(REAL-NS n)))) is Relation-like K7(REAL,(BoundedLinearOperators ((REAL-NS m),(REAL-NS n)))) -defined BoundedLinearOperators ((REAL-NS m),(REAL-NS n)) -valued Function-like total V30(K7(REAL,(BoundedLinearOperators ((REAL-NS m),(REAL-NS n)))), BoundedLinearOperators ((REAL-NS m),(REAL-NS n))) Element of K6(K7(K7(REAL,(BoundedLinearOperators ((REAL-NS m),(REAL-NS n)))),(BoundedLinearOperators ((REAL-NS m),(REAL-NS n)))))
K7(REAL,(BoundedLinearOperators ((REAL-NS m),(REAL-NS n)))) is V33() set
K7(K7(REAL,(BoundedLinearOperators ((REAL-NS m),(REAL-NS n)))),(BoundedLinearOperators ((REAL-NS m),(REAL-NS n)))) is V33() set
K6(K7(K7(REAL,(BoundedLinearOperators ((REAL-NS m),(REAL-NS n)))),(BoundedLinearOperators ((REAL-NS m),(REAL-NS n))))) is V33() set
BoundedLinearOperatorsNorm ((REAL-NS m),(REAL-NS n)) is non empty Relation-like BoundedLinearOperators ((REAL-NS m),(REAL-NS n)) -defined REAL -valued Function-like total V30( BoundedLinearOperators ((REAL-NS m),(REAL-NS n)), REAL ) complex-yielding V146() V147() Element of K6(K7((BoundedLinearOperators ((REAL-NS m),(REAL-NS n))),REAL))
K7((BoundedLinearOperators ((REAL-NS m),(REAL-NS n))),REAL) is V33() complex-yielding V146() V147() set
K6(K7((BoundedLinearOperators ((REAL-NS m),(REAL-NS n))),REAL)) is V33() set
G15((BoundedLinearOperators ((REAL-NS m),(REAL-NS n))),K199((R_VectorSpace_of_LinearOperators ((REAL-NS m),(REAL-NS n))),(BoundedLinearOperators ((REAL-NS m),(REAL-NS n)))),K197((R_VectorSpace_of_LinearOperators ((REAL-NS m),(REAL-NS n))),(BoundedLinearOperators ((REAL-NS m),(REAL-NS n)))),K198((R_VectorSpace_of_LinearOperators ((REAL-NS m),(REAL-NS n))),(BoundedLinearOperators ((REAL-NS m),(REAL-NS n)))),(BoundedLinearOperatorsNorm ((REAL-NS m),(REAL-NS n)))) is V108() L15()
the carrier of (R_NormSpace_of_BoundedLinearOperators ((REAL-NS m),(REAL-NS n))) is non empty set
f is non empty Relation-like REAL m -defined REAL n -valued Function-like total V30( REAL m, REAL n) complex-functions-valued ext-real-functions-valued real-functions-valued Element of K6(K7((REAL m),(REAL n)))
Z is non empty Relation-like REAL m -defined REAL n -valued Function-like total V30( REAL m, REAL n) complex-functions-valued ext-real-functions-valued real-functions-valued Element of K6(K7((REAL m),(REAL n)))
f + Z is Relation-like REAL m -defined REAL n -valued Function-like complex-functions-valued ext-real-functions-valued real-functions-valued Element of K6(K7((REAL m),(REAL n)))
g is Relation-like Function-like Element of the carrier of (R_NormSpace_of_BoundedLinearOperators ((REAL-NS m),(REAL-NS n)))
Z0 is Relation-like Function-like Element of the carrier of (R_NormSpace_of_BoundedLinearOperators ((REAL-NS m),(REAL-NS n)))
g + Z0 is Relation-like Function-like Element of the carrier of (R_NormSpace_of_BoundedLinearOperators ((REAL-NS m),(REAL-NS n)))
dom g is set
dom Z0 is set
dom f is functional complex-functions-membered ext-real-functions-membered real-functions-membered Element of K6((REAL m))
K6((REAL m)) is set
dom Z is functional complex-functions-membered ext-real-functions-membered real-functions-membered Element of K6((REAL m))
(dom f) /\ (dom Z) is functional complex-functions-membered ext-real-functions-membered real-functions-membered Element of K6((REAL m))
c₇ is non empty Relation-like REAL m -defined REAL n -valued Function-like total V30( REAL m, REAL n) complex-functions-valued ext-real-functions-valued real-functions-valued Element of K6(K7((REAL m),(REAL n)))
dom c₇ is functional complex-functions-membered ext-real-functions-membered real-functions-membered Element of K6((REAL m))
f <++> Z is Relation-like (REAL m) /\ (REAL m) -defined R_PFuncs ((DOMS (REAL n)) /\ (DOMS (REAL n))) -valued Function-like complex-functions-valued ext-real-functions-valued real-functions-valued Element of K6(K7(((REAL m) /\ (REAL m)),(R_PFuncs ((DOMS (REAL n)) /\ (DOMS (REAL n))))))
(REAL m) /\ (REAL m) is set
DOMS (REAL n) is set
(DOMS (REAL n)) /\ (DOMS (REAL n)) is set
R_PFuncs ((DOMS (REAL n)) /\ (DOMS (REAL n))) is functional complex-functions-membered ext-real-functions-membered real-functions-membered set
K7(((REAL m) /\ (REAL m)),(R_PFuncs ((DOMS (REAL n)) /\ (DOMS (REAL n))))) is set
K6(K7(((REAL m) /\ (REAL m)),(R_PFuncs ((DOMS (REAL n)) /\ (DOMS (REAL n)))))) is set
x is set
c₇ . x is Relation-like Function-like complex-yielding V146() V147() set
f . x is Relation-like Function-like complex-yielding V146() V147() set
Z . x is Relation-like Function-like complex-yielding V146() V147() set
(f . x) + (Z . x) is Relation-like Function-like set
L is Element of the carrier of (REAL-NS m)
c₇ . L is Relation-like Function-like complex-yielding V146() V147() set
g . L is Element of the carrier of (REAL-NS n)
Z0 . L is Element of the carrier of (REAL-NS n)
(g . L) + (Z0 . L) is Element of the carrier of (REAL-NS n)
L0 is Relation-like NAT -defined REAL -valued Function-like V40(m) FinSequence-like complex-yielding V146() V147() Element of REAL m
f /. L0 is V40(n) complex-yielding V146() V147() Element of REAL n
f . L0 is Relation-like Function-like V40(n) complex-yielding V146() V147() Element of REAL n
Z /. L0 is V40(n) complex-yielding V146() V147() Element of REAL n
Z . L0 is Relation-like Function-like V40(n) complex-yielding V146() V147() Element of REAL n
(f /. L0) + (Z /. L0) is Relation-like NAT -defined REAL -valued Function-like V40(n) FinSequence-like complex-yielding V146() V147() Element of REAL n
m is V6() V10() V11() V12() ext-real non negative V33() V38() V120() V155() V156() V157() V158() V159() V160() V161() Element of NAT
REAL m is non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered FinSequenceSet of REAL
m -tuples_on REAL is non empty functional FinSequence-membered FinSequenceSet of REAL
REAL-NS m is non empty V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() V108() V109() V142() L15()
n is V6() V10() V11() V12() ext-real non negative V33() V38() V120() V155() V156() V157() V158() V159() V160() V161() Element of NAT
REAL n is non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered FinSequenceSet of REAL
n -tuples_on REAL is non empty functional FinSequence-membered FinSequenceSet of REAL
K7((REAL m),(REAL n)) is set
K6(K7((REAL m),(REAL n))) is set
REAL-NS n is non empty V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() V108() V109() V142() L15()
R_NormSpace_of_BoundedLinearOperators ((REAL-NS m),(REAL-NS n)) is non empty V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() V109() V142() L15()
BoundedLinearOperators ((REAL-NS m),(REAL-NS n)) is non empty Element of K6( the carrier of (R_VectorSpace_of_LinearOperators ((REAL-NS m),(REAL-NS n))))
R_VectorSpace_of_LinearOperators ((REAL-NS m),(REAL-NS n)) is non empty V71() strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V204() RLSStruct
LinearOperators ((REAL-NS m),(REAL-NS n)) is non empty functional Element of K6( the carrier of K279( the carrier of (REAL-NS m),(REAL-NS n)))
the carrier of (REAL-NS m) is non empty set
K279( the carrier of (REAL-NS m),(REAL-NS n)) is non empty V71() strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V204() RLSStruct
the carrier of (REAL-NS n) is non empty set
K123( the carrier of (REAL-NS m), the carrier of (REAL-NS n)) is non empty functional M4( the carrier of (REAL-NS m), the carrier of (REAL-NS n))
FuncZero ( the carrier of (REAL-NS m),(REAL-NS n)) is Relation-like the carrier of (REAL-NS m) -defined the carrier of (REAL-NS n) -valued Function-like total V30( the carrier of (REAL-NS m), the carrier of (REAL-NS n)) Element of K123( the carrier of (REAL-NS m), the carrier of (REAL-NS n))
0. (REAL-NS n) is V52( REAL-NS n) Element of the carrier of (REAL-NS n)
the carrier of (REAL-NS m) --> (0. (REAL-NS n)) is non empty Relation-like the carrier of (REAL-NS m) -defined the carrier of (REAL-NS n) -valued Function-like total V30( the carrier of (REAL-NS m), the carrier of (REAL-NS n)) Element of K6(K7( the carrier of (REAL-NS m), the carrier of (REAL-NS n)))
K7( the carrier of (REAL-NS m), the carrier of (REAL-NS n)) is set
K6(K7( the carrier of (REAL-NS m), the carrier of (REAL-NS n))) is set
FuncAdd ( the carrier of (REAL-NS m),(REAL-NS n)) is Relation-like K7(K123( the carrier of (REAL-NS m), the carrier of (REAL-NS n)),K123( the carrier of (REAL-NS m), the carrier of (REAL-NS n))) -defined K123( the carrier of (REAL-NS m), the carrier of (REAL-NS n)) -valued Function-like total V30(K7(K123( the carrier of (REAL-NS m), the carrier of (REAL-NS n)),K123( the carrier of (REAL-NS m), the carrier of (REAL-NS n))),K123( the carrier of (REAL-NS m), the carrier of (REAL-NS n))) Element of K6(K7(K7(K123( the carrier of (REAL-NS m), the carrier of (REAL-NS n)),K123( the carrier of (REAL-NS m), the carrier of (REAL-NS n))),K123( the carrier of (REAL-NS m), the carrier of (REAL-NS n))))
K7(K123( the carrier of (REAL-NS m), the carrier of (REAL-NS n)),K123( the carrier of (REAL-NS m), the carrier of (REAL-NS n))) is set
K7(K7(K123( the carrier of (REAL-NS m), the carrier of (REAL-NS n)),K123( the carrier of (REAL-NS m), the carrier of (REAL-NS n))),K123( the carrier of (REAL-NS m), the carrier of (REAL-NS n))) is set
K6(K7(K7(K123( the carrier of (REAL-NS m), the carrier of (REAL-NS n)),K123( the carrier of (REAL-NS m), the carrier of (REAL-NS n))),K123( the carrier of (REAL-NS m), the carrier of (REAL-NS n)))) is set
FuncExtMult ( the carrier of (REAL-NS m),(REAL-NS n)) is Relation-like K7(REAL,K123( the carrier of (REAL-NS m), the carrier of (REAL-NS n))) -defined K123( the carrier of (REAL-NS m), the carrier of (REAL-NS n)) -valued Function-like total V30(K7(REAL,K123( the carrier of (REAL-NS m), the carrier of (REAL-NS n))),K123( the carrier of (REAL-NS m), the carrier of (REAL-NS n))) Element of K6(K7(K7(REAL,K123( the carrier of (REAL-NS m), the carrier of (REAL-NS n))),K123( the carrier of (REAL-NS m), the carrier of (REAL-NS n))))
K7(REAL,K123( the carrier of (REAL-NS m), the carrier of (REAL-NS n))) is V33() set
K7(K7(REAL,K123( the carrier of (REAL-NS m), the carrier of (REAL-NS n))),K123( the carrier of (REAL-NS m), the carrier of (REAL-NS n))) is V33() set
K6(K7(K7(REAL,K123( the carrier of (REAL-NS m), the carrier of (REAL-NS n))),K123( the carrier of (REAL-NS m), the carrier of (REAL-NS n)))) is V33() set
RLSStruct(# K123( the carrier of (REAL-NS m), the carrier of (REAL-NS n)),(FuncZero ( the carrier of (REAL-NS m),(REAL-NS n))),(FuncAdd ( the carrier of (REAL-NS m),(REAL-NS n))),(FuncExtMult ( the carrier of (REAL-NS m),(REAL-NS n))) #) is strict RLSStruct
the carrier of K279( the carrier of (REAL-NS m),(REAL-NS n)) is non empty set
K6( the carrier of K279( the carrier of (REAL-NS m),(REAL-NS n))) is set
K199(K279( the carrier of (REAL-NS m),(REAL-NS n)),(LinearOperators ((REAL-NS m),(REAL-NS n)))) is Element of LinearOperators ((REAL-NS m),(REAL-NS n))
K197(K279( the carrier of (REAL-NS m),(REAL-NS n)),(LinearOperators ((REAL-NS m),(REAL-NS n)))) is Relation-like K7((LinearOperators ((REAL-NS m),(REAL-NS n))),(LinearOperators ((REAL-NS m),(REAL-NS n)))) -defined LinearOperators ((REAL-NS m),(REAL-NS n)) -valued Function-like total V30(K7((LinearOperators ((REAL-NS m),(REAL-NS n))),(LinearOperators ((REAL-NS m),(REAL-NS n)))), LinearOperators ((REAL-NS m),(REAL-NS n))) Element of K6(K7(K7((LinearOperators ((REAL-NS m),(REAL-NS n))),(LinearOperators ((REAL-NS m),(REAL-NS n)))),(LinearOperators ((REAL-NS m),(REAL-NS n)))))
K7((LinearOperators ((REAL-NS m),(REAL-NS n))),(LinearOperators ((REAL-NS m),(REAL-NS n)))) is set
K7(K7((LinearOperators ((REAL-NS m),(REAL-NS n))),(LinearOperators ((REAL-NS m),(REAL-NS n)))),(LinearOperators ((REAL-NS m),(REAL-NS n)))) is set
K6(K7(K7((LinearOperators ((REAL-NS m),(REAL-NS n))),(LinearOperators ((REAL-NS m),(REAL-NS n)))),(LinearOperators ((REAL-NS m),(REAL-NS n))))) is set
K198(K279( the carrier of (REAL-NS m),(REAL-NS n)),(LinearOperators ((REAL-NS m),(REAL-NS n)))) is Relation-like K7(REAL,(LinearOperators ((REAL-NS m),(REAL-NS n)))) -defined LinearOperators ((REAL-NS m),(REAL-NS n)) -valued Function-like total V30(K7(REAL,(LinearOperators ((REAL-NS m),(REAL-NS n)))), LinearOperators ((REAL-NS m),(REAL-NS n))) Element of K6(K7(K7(REAL,(LinearOperators ((REAL-NS m),(REAL-NS n)))),(LinearOperators ((REAL-NS m),(REAL-NS n)))))
K7(REAL,(LinearOperators ((REAL-NS m),(REAL-NS n)))) is V33() set
K7(K7(REAL,(LinearOperators ((REAL-NS m),(REAL-NS n)))),(LinearOperators ((REAL-NS m),(REAL-NS n)))) is V33() set
K6(K7(K7(REAL,(LinearOperators ((REAL-NS m),(REAL-NS n)))),(LinearOperators ((REAL-NS m),(REAL-NS n))))) is V33() set
RLSStruct(# (LinearOperators ((REAL-NS m),(REAL-NS n))),K199(K279( the carrier of (REAL-NS m),(REAL-NS n)),(LinearOperators ((REAL-NS m),(REAL-NS n)))),K197(K279( the carrier of (REAL-NS m),(REAL-NS n)),(LinearOperators ((REAL-NS m),(REAL-NS n)))),K198(K279( the carrier of (REAL-NS m),(REAL-NS n)),(LinearOperators ((REAL-NS m),(REAL-NS n)))) #) is V71() strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RLSStruct
the carrier of (R_VectorSpace_of_LinearOperators ((REAL-NS m),(REAL-NS n))) is non empty set
K6( the carrier of (R_VectorSpace_of_LinearOperators ((REAL-NS m),(REAL-NS n)))) is set
K199((R_VectorSpace_of_LinearOperators ((REAL-NS m),(REAL-NS n))),(BoundedLinearOperators ((REAL-NS m),(REAL-NS n)))) is Element of BoundedLinearOperators ((REAL-NS m),(REAL-NS n))
K197((R_VectorSpace_of_LinearOperators ((REAL-NS m),(REAL-NS n))),(BoundedLinearOperators ((REAL-NS m),(REAL-NS n)))) is Relation-like K7((BoundedLinearOperators ((REAL-NS m),(REAL-NS n))),(BoundedLinearOperators ((REAL-NS m),(REAL-NS n)))) -defined BoundedLinearOperators ((REAL-NS m),(REAL-NS n)) -valued Function-like total V30(K7((BoundedLinearOperators ((REAL-NS m),(REAL-NS n))),(BoundedLinearOperators ((REAL-NS m),(REAL-NS n)))), BoundedLinearOperators ((REAL-NS m),(REAL-NS n))) Element of K6(K7(K7((BoundedLinearOperators ((REAL-NS m),(REAL-NS n))),(BoundedLinearOperators ((REAL-NS m),(REAL-NS n)))),(BoundedLinearOperators ((REAL-NS m),(REAL-NS n)))))
K7((BoundedLinearOperators ((REAL-NS m),(REAL-NS n))),(BoundedLinearOperators ((REAL-NS m),(REAL-NS n)))) is set
K7(K7((BoundedLinearOperators ((REAL-NS m),(REAL-NS n))),(BoundedLinearOperators ((REAL-NS m),(REAL-NS n)))),(BoundedLinearOperators ((REAL-NS m),(REAL-NS n)))) is set
K6(K7(K7((BoundedLinearOperators ((REAL-NS m),(REAL-NS n))),(BoundedLinearOperators ((REAL-NS m),(REAL-NS n)))),(BoundedLinearOperators ((REAL-NS m),(REAL-NS n))))) is set
K198((R_VectorSpace_of_LinearOperators ((REAL-NS m),(REAL-NS n))),(BoundedLinearOperators ((REAL-NS m),(REAL-NS n)))) is Relation-like K7(REAL,(BoundedLinearOperators ((REAL-NS m),(REAL-NS n)))) -defined BoundedLinearOperators ((REAL-NS m),(REAL-NS n)) -valued Function-like total V30(K7(REAL,(BoundedLinearOperators ((REAL-NS m),(REAL-NS n)))), BoundedLinearOperators ((REAL-NS m),(REAL-NS n))) Element of K6(K7(K7(REAL,(BoundedLinearOperators ((REAL-NS m),(REAL-NS n)))),(BoundedLinearOperators ((REAL-NS m),(REAL-NS n)))))
K7(REAL,(BoundedLinearOperators ((REAL-NS m),(REAL-NS n)))) is V33() set
K7(K7(REAL,(BoundedLinearOperators ((REAL-NS m),(REAL-NS n)))),(BoundedLinearOperators ((REAL-NS m),(REAL-NS n)))) is V33() set
K6(K7(K7(REAL,(BoundedLinearOperators ((REAL-NS m),(REAL-NS n)))),(BoundedLinearOperators ((REAL-NS m),(REAL-NS n))))) is V33() set
BoundedLinearOperatorsNorm ((REAL-NS m),(REAL-NS n)) is non empty Relation-like BoundedLinearOperators ((REAL-NS m),(REAL-NS n)) -defined REAL -valued Function-like total V30( BoundedLinearOperators ((REAL-NS m),(REAL-NS n)), REAL ) complex-yielding V146() V147() Element of K6(K7((BoundedLinearOperators ((REAL-NS m),(REAL-NS n))),REAL))
K7((BoundedLinearOperators ((REAL-NS m),(REAL-NS n))),REAL) is V33() complex-yielding V146() V147() set
K6(K7((BoundedLinearOperators ((REAL-NS m),(REAL-NS n))),REAL)) is V33() set
G15((BoundedLinearOperators ((REAL-NS m),(REAL-NS n))),K199((R_VectorSpace_of_LinearOperators ((REAL-NS m),(REAL-NS n))),(BoundedLinearOperators ((REAL-NS m),(REAL-NS n)))),K197((R_VectorSpace_of_LinearOperators ((REAL-NS m),(REAL-NS n))),(BoundedLinearOperators ((REAL-NS m),(REAL-NS n)))),K198((R_VectorSpace_of_LinearOperators ((REAL-NS m),(REAL-NS n))),(BoundedLinearOperators ((REAL-NS m),(REAL-NS n)))),(BoundedLinearOperatorsNorm ((REAL-NS m),(REAL-NS n)))) is V108() L15()
the carrier of (R_NormSpace_of_BoundedLinearOperators ((REAL-NS m),(REAL-NS n))) is non empty set
f is non empty Relation-like REAL m -defined REAL n -valued Function-like total V30( REAL m, REAL n) complex-functions-valued ext-real-functions-valued real-functions-valued Element of K6(K7((REAL m),(REAL n)))
Z is non empty Relation-like REAL m -defined REAL n -valued Function-like total V30( REAL m, REAL n) complex-functions-valued ext-real-functions-valued real-functions-valued Element of K6(K7((REAL m),(REAL n)))
f - Z is Relation-like REAL m -defined REAL n -valued Function-like complex-functions-valued ext-real-functions-valued real-functions-valued Element of K6(K7((REAL m),(REAL n)))
g is Relation-like Function-like Element of the carrier of (R_NormSpace_of_BoundedLinearOperators ((REAL-NS m),(REAL-NS n)))
Z0 is Relation-like Function-like Element of the carrier of (R_NormSpace_of_BoundedLinearOperators ((REAL-NS m),(REAL-NS n)))
g - Z0 is Relation-like Function-like Element of the carrier of (R_NormSpace_of_BoundedLinearOperators ((REAL-NS m),(REAL-NS n)))
dom g is set
dom Z0 is set
dom f is functional complex-functions-membered ext-real-functions-membered real-functions-membered Element of K6((REAL m))
K6((REAL m)) is set
dom Z is functional complex-functions-membered ext-real-functions-membered real-functions-membered Element of K6((REAL m))
(dom f) /\ (dom Z) is functional complex-functions-membered ext-real-functions-membered real-functions-membered Element of K6((REAL m))
c₇ is non empty Relation-like REAL m -defined REAL n -valued Function-like total V30( REAL m, REAL n) complex-functions-valued ext-real-functions-valued real-functions-valued Element of K6(K7((REAL m),(REAL n)))
dom c₇ is functional complex-functions-membered ext-real-functions-membered real-functions-membered Element of K6((REAL m))
f <--> Z is Relation-like (REAL m) /\ (REAL m) -defined R_PFuncs ((DOMS (REAL n)) /\ (DOMS (REAL n))) -valued Function-like complex-functions-valued ext-real-functions-valued real-functions-valued Element of K6(K7(((REAL m) /\ (REAL m)),(R_PFuncs ((DOMS (REAL n)) /\ (DOMS (REAL n))))))
(REAL m) /\ (REAL m) is set
DOMS (REAL n) is set
(DOMS (REAL n)) /\ (DOMS (REAL n)) is set
R_PFuncs ((DOMS (REAL n)) /\ (DOMS (REAL n))) is functional complex-functions-membered ext-real-functions-membered real-functions-membered set
K7(((REAL m) /\ (REAL m)),(R_PFuncs ((DOMS (REAL n)) /\ (DOMS (REAL n))))) is set
K6(K7(((REAL m) /\ (REAL m)),(R_PFuncs ((DOMS (REAL n)) /\ (DOMS (REAL n)))))) is set
x is set
c₇ . x is Relation-like Function-like complex-yielding V146() V147() set
f . x is Relation-like Function-like complex-yielding V146() V147() set
Z . x is Relation-like Function-like complex-yielding V146() V147() set
(f . x) - (Z . x) is Relation-like Function-like set
- (Z . x) is Relation-like Function-like complex-yielding set
- 1 is V11() V12() ext-real non positive set
(- 1) (#) (Z . x) is Relation-like Function-like set
(f . x) + (- (Z . x)) is Relation-like Function-like set
L is Element of the carrier of (REAL-NS m)
c₇ . L is Relation-like Function-like complex-yielding V146() V147() set
g . L is Element of the carrier of (REAL-NS n)
Z0 . L is Element of the carrier of (REAL-NS n)
(g . L) - (Z0 . L) is Element of the carrier of (REAL-NS n)
L0 is Relation-like NAT -defined REAL -valued Function-like V40(m) FinSequence-like complex-yielding V146() V147() Element of REAL m
f /. L0 is V40(n) complex-yielding V146() V147() Element of REAL n
f . L0 is Relation-like Function-like V40(n) complex-yielding V146() V147() Element of REAL n
Z /. L0 is V40(n) complex-yielding V146() V147() Element of REAL n
Z . L0 is Relation-like Function-like V40(n) complex-yielding V146() V147() Element of REAL n
(f /. L0) - (Z /. L0) is Relation-like NAT -defined REAL -valued Function-like V40(n) FinSequence-like complex-yielding V146() V147() Element of REAL n
m is V6() V10() V11() V12() ext-real non negative V33() V38() V120() V155() V156() V157() V158() V159() V160() V161() Element of NAT
REAL m is non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered FinSequenceSet of REAL
m -tuples_on REAL is non empty functional FinSequence-membered FinSequenceSet of REAL
REAL-NS m is non empty V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() V108() V109() V142() L15()
n is V6() V10() V11() V12() ext-real non negative V33() V38() V120() V155() V156() V157() V158() V159() V160() V161() Element of NAT
REAL n is non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered FinSequenceSet of REAL
n -tuples_on REAL is non empty functional FinSequence-membered FinSequenceSet of REAL
K7((REAL m),(REAL n)) is set
K6(K7((REAL m),(REAL n))) is set
REAL-NS n is non empty V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() V108() V109() V142() L15()
R_NormSpace_of_BoundedLinearOperators ((REAL-NS m),(REAL-NS n)) is non empty V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() V109() V142() L15()
BoundedLinearOperators ((REAL-NS m),(REAL-NS n)) is non empty Element of K6( the carrier of (R_VectorSpace_of_LinearOperators ((REAL-NS m),(REAL-NS n))))
R_VectorSpace_of_LinearOperators ((REAL-NS m),(REAL-NS n)) is non empty V71() strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V204() RLSStruct
LinearOperators ((REAL-NS m),(REAL-NS n)) is non empty functional Element of K6( the carrier of K279( the carrier of (REAL-NS m),(REAL-NS n)))
the carrier of (REAL-NS m) is non empty set
K279( the carrier of (REAL-NS m),(REAL-NS n)) is non empty V71() strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V204() RLSStruct
the carrier of (REAL-NS n) is non empty set
K123( the carrier of (REAL-NS m), the carrier of (REAL-NS n)) is non empty functional M4( the carrier of (REAL-NS m), the carrier of (REAL-NS n))
FuncZero ( the carrier of (REAL-NS m),(REAL-NS n)) is Relation-like the carrier of (REAL-NS m) -defined the carrier of (REAL-NS n) -valued Function-like total V30( the carrier of (REAL-NS m), the carrier of (REAL-NS n)) Element of K123( the carrier of (REAL-NS m), the carrier of (REAL-NS n))
0. (REAL-NS n) is V52( REAL-NS n) Element of the carrier of (REAL-NS n)
the carrier of (REAL-NS m) --> (0. (REAL-NS n)) is non empty Relation-like the carrier of (REAL-NS m) -defined the carrier of (REAL-NS n) -valued Function-like total V30( the carrier of (REAL-NS m), the carrier of (REAL-NS n)) Element of K6(K7( the carrier of (REAL-NS m), the carrier of (REAL-NS n)))
K7( the carrier of (REAL-NS m), the carrier of (REAL-NS n)) is set
K6(K7( the carrier of (REAL-NS m), the carrier of (REAL-NS n))) is set
FuncAdd ( the carrier of (REAL-NS m),(REAL-NS n)) is Relation-like K7(K123( the carrier of (REAL-NS m), the carrier of (REAL-NS n)),K123( the carrier of (REAL-NS m), the carrier of (REAL-NS n))) -defined K123( the carrier of (REAL-NS m), the carrier of (REAL-NS n)) -valued Function-like total V30(K7(K123( the carrier of (REAL-NS m), the carrier of (REAL-NS n)),K123( the carrier of (REAL-NS m), the carrier of (REAL-NS n))),K123( the carrier of (REAL-NS m), the carrier of (REAL-NS n))) Element of K6(K7(K7(K123( the carrier of (REAL-NS m), the carrier of (REAL-NS n)),K123( the carrier of (REAL-NS m), the carrier of (REAL-NS n))),K123( the carrier of (REAL-NS m), the carrier of (REAL-NS n))))
K7(K123( the carrier of (REAL-NS m), the carrier of (REAL-NS n)),K123( the carrier of (REAL-NS m), the carrier of (REAL-NS n))) is set
K7(K7(K123( the carrier of (REAL-NS m), the carrier of (REAL-NS n)),K123( the carrier of (REAL-NS m), the carrier of (REAL-NS n))),K123( the carrier of (REAL-NS m), the carrier of (REAL-NS n))) is set
K6(K7(K7(K123( the carrier of (REAL-NS m), the carrier of (REAL-NS n)),K123( the carrier of (REAL-NS m), the carrier of (REAL-NS n))),K123( the carrier of (REAL-NS m), the carrier of (REAL-NS n)))) is set
FuncExtMult ( the carrier of (REAL-NS m),(REAL-NS n)) is Relation-like K7(REAL,K123( the carrier of (REAL-NS m), the carrier of (REAL-NS n))) -defined K123( the carrier of (REAL-NS m), the carrier of (REAL-NS n)) -valued Function-like total V30(K7(REAL,K123( the carrier of (REAL-NS m), the carrier of (REAL-NS n))),K123( the carrier of (REAL-NS m), the carrier of (REAL-NS n))) Element of K6(K7(K7(REAL,K123( the carrier of (REAL-NS m), the carrier of (REAL-NS n))),K123( the carrier of (REAL-NS m), the carrier of (REAL-NS n))))
K7(REAL,K123( the carrier of (REAL-NS m), the carrier of (REAL-NS n))) is V33() set
K7(K7(REAL,K123( the carrier of (REAL-NS m), the carrier of (REAL-NS n))),K123( the carrier of (REAL-NS m), the carrier of (REAL-NS n))) is V33() set
K6(K7(K7(REAL,K123( the carrier of (REAL-NS m), the carrier of (REAL-NS n))),K123( the carrier of (REAL-NS m), the carrier of (REAL-NS n)))) is V33() set
RLSStruct(# K123( the carrier of (REAL-NS m), the carrier of (REAL-NS n)),(FuncZero ( the carrier of (REAL-NS m),(REAL-NS n))),(FuncAdd ( the carrier of (REAL-NS m),(REAL-NS n))),(FuncExtMult ( the carrier of (REAL-NS m),(REAL-NS n))) #) is strict RLSStruct
the carrier of K279( the carrier of (REAL-NS m),(REAL-NS n)) is non empty set
K6( the carrier of K279( the carrier of (REAL-NS m),(REAL-NS n))) is set
K199(K279( the carrier of (REAL-NS m),(REAL-NS n)),(LinearOperators ((REAL-NS m),(REAL-NS n)))) is Element of LinearOperators ((REAL-NS m),(REAL-NS n))
K197(K279( the carrier of (REAL-NS m),(REAL-NS n)),(LinearOperators ((REAL-NS m),(REAL-NS n)))) is Relation-like K7((LinearOperators ((REAL-NS m),(REAL-NS n))),(LinearOperators ((REAL-NS m),(REAL-NS n)))) -defined LinearOperators ((REAL-NS m),(REAL-NS n)) -valued Function-like total V30(K7((LinearOperators ((REAL-NS m),(REAL-NS n))),(LinearOperators ((REAL-NS m),(REAL-NS n)))), LinearOperators ((REAL-NS m),(REAL-NS n))) Element of K6(K7(K7((LinearOperators ((REAL-NS m),(REAL-NS n))),(LinearOperators ((REAL-NS m),(REAL-NS n)))),(LinearOperators ((REAL-NS m),(REAL-NS n)))))
K7((LinearOperators ((REAL-NS m),(REAL-NS n))),(LinearOperators ((REAL-NS m),(REAL-NS n)))) is set
K7(K7((LinearOperators ((REAL-NS m),(REAL-NS n))),(LinearOperators ((REAL-NS m),(REAL-NS n)))),(LinearOperators ((REAL-NS m),(REAL-NS n)))) is set
K6(K7(K7((LinearOperators ((REAL-NS m),(REAL-NS n))),(LinearOperators ((REAL-NS m),(REAL-NS n)))),(LinearOperators ((REAL-NS m),(REAL-NS n))))) is set
K198(K279( the carrier of (REAL-NS m),(REAL-NS n)),(LinearOperators ((REAL-NS m),(REAL-NS n)))) is Relation-like K7(REAL,(LinearOperators ((REAL-NS m),(REAL-NS n)))) -defined LinearOperators ((REAL-NS m),(REAL-NS n)) -valued Function-like total V30(K7(REAL,(LinearOperators ((REAL-NS m),(REAL-NS n)))), LinearOperators ((REAL-NS m),(REAL-NS n))) Element of K6(K7(K7(REAL,(LinearOperators ((REAL-NS m),(REAL-NS n)))),(LinearOperators ((REAL-NS m),(REAL-NS n)))))
K7(REAL,(LinearOperators ((REAL-NS m),(REAL-NS n)))) is V33() set
K7(K7(REAL,(LinearOperators ((REAL-NS m),(REAL-NS n)))),(LinearOperators ((REAL-NS m),(REAL-NS n)))) is V33() set
K6(K7(K7(REAL,(LinearOperators ((REAL-NS m),(REAL-NS n)))),(LinearOperators ((REAL-NS m),(REAL-NS n))))) is V33() set
RLSStruct(# (LinearOperators ((REAL-NS m),(REAL-NS n))),K199(K279( the carrier of (REAL-NS m),(REAL-NS n)),(LinearOperators ((REAL-NS m),(REAL-NS n)))),K197(K279( the carrier of (REAL-NS m),(REAL-NS n)),(LinearOperators ((REAL-NS m),(REAL-NS n)))),K198(K279( the carrier of (REAL-NS m),(REAL-NS n)),(LinearOperators ((REAL-NS m),(REAL-NS n)))) #) is V71() strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RLSStruct
the carrier of (R_VectorSpace_of_LinearOperators ((REAL-NS m),(REAL-NS n))) is non empty set
K6( the carrier of (R_VectorSpace_of_LinearOperators ((REAL-NS m),(REAL-NS n)))) is set
K199((R_VectorSpace_of_LinearOperators ((REAL-NS m),(REAL-NS n))),(BoundedLinearOperators ((REAL-NS m),(REAL-NS n)))) is Element of BoundedLinearOperators ((REAL-NS m),(REAL-NS n))
K197((R_VectorSpace_of_LinearOperators ((REAL-NS m),(REAL-NS n))),(BoundedLinearOperators ((REAL-NS m),(REAL-NS n)))) is Relation-like K7((BoundedLinearOperators ((REAL-NS m),(REAL-NS n))),(BoundedLinearOperators ((REAL-NS m),(REAL-NS n)))) -defined BoundedLinearOperators ((REAL-NS m),(REAL-NS n)) -valued Function-like total V30(K7((BoundedLinearOperators ((REAL-NS m),(REAL-NS n))),(BoundedLinearOperators ((REAL-NS m),(REAL-NS n)))), BoundedLinearOperators ((REAL-NS m),(REAL-NS n))) Element of K6(K7(K7((BoundedLinearOperators ((REAL-NS m),(REAL-NS n))),(BoundedLinearOperators ((REAL-NS m),(REAL-NS n)))),(BoundedLinearOperators ((REAL-NS m),(REAL-NS n)))))
K7((BoundedLinearOperators ((REAL-NS m),(REAL-NS n))),(BoundedLinearOperators ((REAL-NS m),(REAL-NS n)))) is set
K7(K7((BoundedLinearOperators ((REAL-NS m),(REAL-NS n))),(BoundedLinearOperators ((REAL-NS m),(REAL-NS n)))),(BoundedLinearOperators ((REAL-NS m),(REAL-NS n)))) is set
K6(K7(K7((BoundedLinearOperators ((REAL-NS m),(REAL-NS n))),(BoundedLinearOperators ((REAL-NS m),(REAL-NS n)))),(BoundedLinearOperators ((REAL-NS m),(REAL-NS n))))) is set
K198((R_VectorSpace_of_LinearOperators ((REAL-NS m),(REAL-NS n))),(BoundedLinearOperators ((REAL-NS m),(REAL-NS n)))) is Relation-like K7(REAL,(BoundedLinearOperators ((REAL-NS m),(REAL-NS n)))) -defined BoundedLinearOperators ((REAL-NS m),(REAL-NS n)) -valued Function-like total V30(K7(REAL,(BoundedLinearOperators ((REAL-NS m),(REAL-NS n)))), BoundedLinearOperators ((REAL-NS m),(REAL-NS n))) Element of K6(K7(K7(REAL,(BoundedLinearOperators ((REAL-NS m),(REAL-NS n)))),(BoundedLinearOperators ((REAL-NS m),(REAL-NS n)))))
K7(REAL,(BoundedLinearOperators ((REAL-NS m),(REAL-NS n)))) is V33() set
K7(K7(REAL,(BoundedLinearOperators ((REAL-NS m),(REAL-NS n)))),(BoundedLinearOperators ((REAL-NS m),(REAL-NS n)))) is V33() set
K6(K7(K7(REAL,(BoundedLinearOperators ((REAL-NS m),(REAL-NS n)))),(BoundedLinearOperators ((REAL-NS m),(REAL-NS n))))) is V33() set
BoundedLinearOperatorsNorm ((REAL-NS m),(REAL-NS n)) is non empty Relation-like BoundedLinearOperators ((REAL-NS m),(REAL-NS n)) -defined REAL -valued Function-like total V30( BoundedLinearOperators ((REAL-NS m),(REAL-NS n)), REAL ) complex-yielding V146() V147() Element of K6(K7((BoundedLinearOperators ((REAL-NS m),(REAL-NS n))),REAL))
K7((BoundedLinearOperators ((REAL-NS m),(REAL-NS n))),REAL) is V33() complex-yielding V146() V147() set
K6(K7((BoundedLinearOperators ((REAL-NS m),(REAL-NS n))),REAL)) is V33() set
G15((BoundedLinearOperators ((REAL-NS m),(REAL-NS n))),K199((R_VectorSpace_of_LinearOperators ((REAL-NS m),(REAL-NS n))),(BoundedLinearOperators ((REAL-NS m),(REAL-NS n)))),K197((R_VectorSpace_of_LinearOperators ((REAL-NS m),(REAL-NS n))),(BoundedLinearOperators ((REAL-NS m),(REAL-NS n)))),K198((R_VectorSpace_of_LinearOperators ((REAL-NS m),(REAL-NS n))),(BoundedLinearOperators ((REAL-NS m),(REAL-NS n)))),(BoundedLinearOperatorsNorm ((REAL-NS m),(REAL-NS n)))) is V108() L15()
the carrier of (R_NormSpace_of_BoundedLinearOperators ((REAL-NS m),(REAL-NS n))) is non empty set
f is non empty Relation-like REAL m -defined REAL n -valued Function-like total V30( REAL m, REAL n) complex-functions-valued ext-real-functions-valued real-functions-valued Element of K6(K7((REAL m),(REAL n)))
Z is Relation-like Function-like Element of the carrier of (R_NormSpace_of_BoundedLinearOperators ((REAL-NS m),(REAL-NS n)))
g is V11() V12() ext-real Element of REAL
g (#) f is Relation-like REAL m -defined REAL n -valued Function-like complex-functions-valued ext-real-functions-valued real-functions-valued Element of K6(K7((REAL m),(REAL n)))
g * Z is Relation-like Function-like Element of the carrier of (R_NormSpace_of_BoundedLinearOperators ((REAL-NS m),(REAL-NS n)))
dom f is functional complex-functions-membered ext-real-functions-membered real-functions-membered Element of K6((REAL m))
K6((REAL m)) is set
Z0 is non empty Relation-like REAL m -defined REAL n -valued Function-like total V30( REAL m, REAL n) complex-functions-valued ext-real-functions-valued real-functions-valued Element of K6(K7((REAL m),(REAL n)))
dom Z0 is functional complex-functions-membered ext-real-functions-membered real-functions-membered Element of K6((REAL m))
f [#] g is Relation-like REAL m -defined R_PFuncs (DOMS (REAL n)) -valued Function-like complex-functions-valued ext-real-functions-valued real-functions-valued Element of K6(K7((REAL m),(R_PFuncs (DOMS (REAL n)))))
DOMS (REAL n) is set
R_PFuncs (DOMS (REAL n)) is functional complex-functions-membered ext-real-functions-membered real-functions-membered set
K7((REAL m),(R_PFuncs (DOMS (REAL n)))) is set
K6(K7((REAL m),(R_PFuncs (DOMS (REAL n))))) is set
c₇ is set
Z0 . c₇ is Relation-like Function-like complex-yielding V146() V147() set
f . c₇ is Relation-like Function-like complex-yielding V146() V147() set
g (#) (f . c₇) is Relation-like Function-like set
x is Element of the carrier of (REAL-NS m)
Z0 . x is Relation-like Function-like complex-yielding V146() V147() set
Z . x is Element of the carrier of (REAL-NS n)
g * (Z . x) is Element of the carrier of (REAL-NS n)
L is Relation-like NAT -defined REAL -valued Function-like V40(m) FinSequence-like complex-yielding V146() V147() Element of REAL m
f /. L is V40(n) complex-yielding V146() V147() Element of REAL n
f . L is Relation-like Function-like V40(n) complex-yielding V146() V147() Element of REAL n
g * (f /. L) is Relation-like NAT -defined REAL -valued Function-like V40(n) FinSequence-like complex-yielding V146() V147() Element of REAL n
m is non empty V6() V10() V11() V12() ext-real positive non negative V33() V38() V120() V155() V156() V157() V158() V159() V160() V161() Element of NAT
REAL m is non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered FinSequenceSet of REAL
m -tuples_on REAL is non empty functional FinSequence-membered FinSequenceSet of REAL
n is non empty V6() V10() V11() V12() ext-real positive non negative V33() V38() V120() V155() V156() V157() V158() V159() V160() V161() Element of NAT
REAL n is non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered FinSequenceSet of REAL
n -tuples_on REAL is non empty functional FinSequence-membered FinSequenceSet of REAL
K7((REAL m),(REAL n)) is set
K6(K7((REAL m),(REAL n))) is set
REAL-NS m is non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() V108() V109() V142() L15()
REAL-NS n is non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() V108() V109() V142() L15()
R_NormSpace_of_BoundedLinearOperators ((REAL-NS m),(REAL-NS n)) is non empty V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() V109() V142() L15()
BoundedLinearOperators ((REAL-NS m),(REAL-NS n)) is non empty Element of K6( the carrier of (R_VectorSpace_of_LinearOperators ((REAL-NS m),(REAL-NS n))))
R_VectorSpace_of_LinearOperators ((REAL-NS m),(REAL-NS n)) is non empty V71() strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V204() RLSStruct
LinearOperators ((REAL-NS m),(REAL-NS n)) is non empty functional Element of K6( the carrier of K279( the carrier of (REAL-NS m),(REAL-NS n)))
the carrier of (REAL-NS m) is non empty V2() set
K279( the carrier of (REAL-NS m),(REAL-NS n)) is non empty V71() strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V204() RLSStruct
the carrier of (REAL-NS n) is non empty V2() set
K123( the carrier of (REAL-NS m), the carrier of (REAL-NS n)) is non empty functional M4( the carrier of (REAL-NS m), the carrier of (REAL-NS n))
FuncZero ( the carrier of (REAL-NS m),(REAL-NS n)) is Relation-like the carrier of (REAL-NS m) -defined the carrier of (REAL-NS n) -valued Function-like total V30( the carrier of (REAL-NS m), the carrier of (REAL-NS n)) Element of K123( the carrier of (REAL-NS m), the carrier of (REAL-NS n))
0. (REAL-NS n) is V52( REAL-NS n) Element of the carrier of (REAL-NS n)
the carrier of (REAL-NS m) --> (0. (REAL-NS n)) is non empty Relation-like the carrier of (REAL-NS m) -defined the carrier of (REAL-NS n) -valued Function-like total V30( the carrier of (REAL-NS m), the carrier of (REAL-NS n)) Element of K6(K7( the carrier of (REAL-NS m), the carrier of (REAL-NS n)))
K7( the carrier of (REAL-NS m), the carrier of (REAL-NS n)) is set
K6(K7( the carrier of (REAL-NS m), the carrier of (REAL-NS n))) is set
FuncAdd ( the carrier of (REAL-NS m),(REAL-NS n)) is Relation-like K7(K123( the carrier of (REAL-NS m), the carrier of (REAL-NS n)),K123( the carrier of (REAL-NS m), the carrier of (REAL-NS n))) -defined K123( the carrier of (REAL-NS m), the carrier of (REAL-NS n)) -valued Function-like total V30(K7(K123( the carrier of (REAL-NS m), the carrier of (REAL-NS n)),K123( the carrier of (REAL-NS m), the carrier of (REAL-NS n))),K123( the carrier of (REAL-NS m), the carrier of (REAL-NS n))) Element of K6(K7(K7(K123( the carrier of (REAL-NS m), the carrier of (REAL-NS n)),K123( the carrier of (REAL-NS m), the carrier of (REAL-NS n))),K123( the carrier of (REAL-NS m), the carrier of (REAL-NS n))))
K7(K123( the carrier of (REAL-NS m), the carrier of (REAL-NS n)),K123( the carrier of (REAL-NS m), the carrier of (REAL-NS n))) is set
K7(K7(K123( the carrier of (REAL-NS m), the carrier of (REAL-NS n)),K123( the carrier of (REAL-NS m), the carrier of (REAL-NS n))),K123( the carrier of (REAL-NS m), the carrier of (REAL-NS n))) is set
K6(K7(K7(K123( the carrier of (REAL-NS m), the carrier of (REAL-NS n)),K123( the carrier of (REAL-NS m), the carrier of (REAL-NS n))),K123( the carrier of (REAL-NS m), the carrier of (REAL-NS n)))) is set
FuncExtMult ( the carrier of (REAL-NS m),(REAL-NS n)) is Relation-like K7(REAL,K123( the carrier of (REAL-NS m), the carrier of (REAL-NS n))) -defined K123( the carrier of (REAL-NS m), the carrier of (REAL-NS n)) -valued Function-like total V30(K7(REAL,K123( the carrier of (REAL-NS m), the carrier of (REAL-NS n))),K123( the carrier of (REAL-NS m), the carrier of (REAL-NS n))) Element of K6(K7(K7(REAL,K123( the carrier of (REAL-NS m), the carrier of (REAL-NS n))),K123( the carrier of (REAL-NS m), the carrier of (REAL-NS n))))
K7(REAL,K123( the carrier of (REAL-NS m), the carrier of (REAL-NS n))) is V33() set
K7(K7(REAL,K123( the carrier of (REAL-NS m), the carrier of (REAL-NS n))),K123( the carrier of (REAL-NS m), the carrier of (REAL-NS n))) is V33() set
K6(K7(K7(REAL,K123( the carrier of (REAL-NS m), the carrier of (REAL-NS n))),K123( the carrier of (REAL-NS m), the carrier of (REAL-NS n)))) is V33() set
RLSStruct(# K123( the carrier of (REAL-NS m), the carrier of (REAL-NS n)),(FuncZero ( the carrier of (REAL-NS m),(REAL-NS n))),(FuncAdd ( the carrier of (REAL-NS m),(REAL-NS n))),(FuncExtMult ( the carrier of (REAL-NS m),(REAL-NS n))) #) is strict RLSStruct
the carrier of K279( the carrier of (REAL-NS m),(REAL-NS n)) is non empty set
K6( the carrier of K279( the carrier of (REAL-NS m),(REAL-NS n))) is set
K199(K279( the carrier of (REAL-NS m),(REAL-NS n)),(LinearOperators ((REAL-NS m),(REAL-NS n)))) is Element of LinearOperators ((REAL-NS m),(REAL-NS n))
K197(K279( the carrier of (REAL-NS m),(REAL-NS n)),(LinearOperators ((REAL-NS m),(REAL-NS n)))) is Relation-like K7((LinearOperators ((REAL-NS m),(REAL-NS n))),(LinearOperators ((REAL-NS m),(REAL-NS n)))) -defined LinearOperators ((REAL-NS m),(REAL-NS n)) -valued Function-like total V30(K7((LinearOperators ((REAL-NS m),(REAL-NS n))),(LinearOperators ((REAL-NS m),(REAL-NS n)))), LinearOperators ((REAL-NS m),(REAL-NS n))) Element of K6(K7(K7((LinearOperators ((REAL-NS m),(REAL-NS n))),(LinearOperators ((REAL-NS m),(REAL-NS n)))),(LinearOperators ((REAL-NS m),(REAL-NS n)))))
K7((LinearOperators ((REAL-NS m),(REAL-NS n))),(LinearOperators ((REAL-NS m),(REAL-NS n)))) is set
K7(K7((LinearOperators ((REAL-NS m),(REAL-NS n))),(LinearOperators ((REAL-NS m),(REAL-NS n)))),(LinearOperators ((REAL-NS m),(REAL-NS n)))) is set
K6(K7(K7((LinearOperators ((REAL-NS m),(REAL-NS n))),(LinearOperators ((REAL-NS m),(REAL-NS n)))),(LinearOperators ((REAL-NS m),(REAL-NS n))))) is set
K198(K279( the carrier of (REAL-NS m),(REAL-NS n)),(LinearOperators ((REAL-NS m),(REAL-NS n)))) is Relation-like K7(REAL,(LinearOperators ((REAL-NS m),(REAL-NS n)))) -defined LinearOperators ((REAL-NS m),(REAL-NS n)) -valued Function-like total V30(K7(REAL,(LinearOperators ((REAL-NS m),(REAL-NS n)))), LinearOperators ((REAL-NS m),(REAL-NS n))) Element of K6(K7(K7(REAL,(LinearOperators ((REAL-NS m),(REAL-NS n)))),(LinearOperators ((REAL-NS m),(REAL-NS n)))))
K7(REAL,(LinearOperators ((REAL-NS m),(REAL-NS n)))) is V33() set
K7(K7(REAL,(LinearOperators ((REAL-NS m),(REAL-NS n)))),(LinearOperators ((REAL-NS m),(REAL-NS n)))) is V33() set
K6(K7(K7(REAL,(LinearOperators ((REAL-NS m),(REAL-NS n)))),(LinearOperators ((REAL-NS m),(REAL-NS n))))) is V33() set
RLSStruct(# (LinearOperators ((REAL-NS m),(REAL-NS n))),K199(K279( the carrier of (REAL-NS m),(REAL-NS n)),(LinearOperators ((REAL-NS m),(REAL-NS n)))),K197(K279( the carrier of (REAL-NS m),(REAL-NS n)),(LinearOperators ((REAL-NS m),(REAL-NS n)))),K198(K279( the carrier of (REAL-NS m),(REAL-NS n)),(LinearOperators ((REAL-NS m),(REAL-NS n)))) #) is V71() strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RLSStruct
the carrier of (R_VectorSpace_of_LinearOperators ((REAL-NS m),(REAL-NS n))) is non empty set
K6( the carrier of (R_VectorSpace_of_LinearOperators ((REAL-NS m),(REAL-NS n)))) is set
K199((R_VectorSpace_of_LinearOperators ((REAL-NS m),(REAL-NS n))),(BoundedLinearOperators ((REAL-NS m),(REAL-NS n)))) is Element of BoundedLinearOperators ((REAL-NS m),(REAL-NS n))
K197((R_VectorSpace_of_LinearOperators ((REAL-NS m),(REAL-NS n))),(BoundedLinearOperators ((REAL-NS m),(REAL-NS n)))) is Relation-like K7((BoundedLinearOperators ((REAL-NS m),(REAL-NS n))),(BoundedLinearOperators ((REAL-NS m),(REAL-NS n)))) -defined BoundedLinearOperators ((REAL-NS m),(REAL-NS n)) -valued Function-like total V30(K7((BoundedLinearOperators ((REAL-NS m),(REAL-NS n))),(BoundedLinearOperators ((REAL-NS m),(REAL-NS n)))), BoundedLinearOperators ((REAL-NS m),(REAL-NS n))) Element of K6(K7(K7((BoundedLinearOperators ((REAL-NS m),(REAL-NS n))),(BoundedLinearOperators ((REAL-NS m),(REAL-NS n)))),(BoundedLinearOperators ((REAL-NS m),(REAL-NS n)))))
K7((BoundedLinearOperators ((REAL-NS m),(REAL-NS n))),(BoundedLinearOperators ((REAL-NS m),(REAL-NS n)))) is set
K7(K7((BoundedLinearOperators ((REAL-NS m),(REAL-NS n))),(BoundedLinearOperators ((REAL-NS m),(REAL-NS n)))),(BoundedLinearOperators ((REAL-NS m),(REAL-NS n)))) is set
K6(K7(K7((BoundedLinearOperators ((REAL-NS m),(REAL-NS n))),(BoundedLinearOperators ((REAL-NS m),(REAL-NS n)))),(BoundedLinearOperators ((REAL-NS m),(REAL-NS n))))) is set
K198((R_VectorSpace_of_LinearOperators ((REAL-NS m),(REAL-NS n))),(BoundedLinearOperators ((REAL-NS m),(REAL-NS n)))) is Relation-like K7(REAL,(BoundedLinearOperators ((REAL-NS m),(REAL-NS n)))) -defined BoundedLinearOperators ((REAL-NS m),(REAL-NS n)) -valued Function-like total V30(K7(REAL,(BoundedLinearOperators ((REAL-NS m),(REAL-NS n)))), BoundedLinearOperators ((REAL-NS m),(REAL-NS n))) Element of K6(K7(K7(REAL,(BoundedLinearOperators ((REAL-NS m),(REAL-NS n)))),(BoundedLinearOperators ((REAL-NS m),(REAL-NS n)))))
K7(REAL,(BoundedLinearOperators ((REAL-NS m),(REAL-NS n)))) is V33() set
K7(K7(REAL,(BoundedLinearOperators ((REAL-NS m),(REAL-NS n)))),(BoundedLinearOperators ((REAL-NS m),(REAL-NS n)))) is V33() set
K6(K7(K7(REAL,(BoundedLinearOperators ((REAL-NS m),(REAL-NS n)))),(BoundedLinearOperators ((REAL-NS m),(REAL-NS n))))) is V33() set
BoundedLinearOperatorsNorm ((REAL-NS m),(REAL-NS n)) is non empty Relation-like BoundedLinearOperators ((REAL-NS m),(REAL-NS n)) -defined REAL -valued Function-like total V30( BoundedLinearOperators ((REAL-NS m),(REAL-NS n)), REAL ) complex-yielding V146() V147() Element of K6(K7((BoundedLinearOperators ((REAL-NS m),(REAL-NS n))),REAL))
K7((BoundedLinearOperators ((REAL-NS m),(REAL-NS n))),REAL) is V33() complex-yielding V146() V147() set
K6(K7((BoundedLinearOperators ((REAL-NS m),(REAL-NS n))),REAL)) is V33() set
G15((BoundedLinearOperators ((REAL-NS m),(REAL-NS n))),K199((R_VectorSpace_of_LinearOperators ((REAL-NS m),(REAL-NS n))),(BoundedLinearOperators ((REAL-NS m),(REAL-NS n)))),K197((R_VectorSpace_of_LinearOperators ((REAL-NS m),(REAL-NS n))),(BoundedLinearOperators ((REAL-NS m),(REAL-NS n)))),K198((R_VectorSpace_of_LinearOperators ((REAL-NS m),(REAL-NS n))),(BoundedLinearOperators ((REAL-NS m),(REAL-NS n)))),(BoundedLinearOperatorsNorm ((REAL-NS m),(REAL-NS n)))) is V108() L15()
the carrier of (R_NormSpace_of_BoundedLinearOperators ((REAL-NS m),(REAL-NS n))) is non empty set
f is Relation-like REAL m -defined REAL n -valued Function-like complex-functions-valued ext-real-functions-valued real-functions-valued Element of K6(K7((REAL m),(REAL n)))
Z is Relation-like NAT -defined REAL -valued Function-like V40(m) FinSequence-like complex-yielding V146() V147() Element of REAL m
diff (f,Z) is non empty Relation-like REAL m -defined REAL n -valued Function-like total V30( REAL m, REAL n) complex-functions-valued ext-real-functions-valued real-functions-valued Element of K6(K7((REAL m),(REAL n)))
g is Relation-like the carrier of (REAL-NS m) -defined the carrier of (REAL-NS n) -valued Function-like Element of K6(K7( the carrier of (REAL-NS m), the carrier of (REAL-NS n)))
Z0 is Element of the carrier of (REAL-NS m)
diff (g,Z0) is Relation-like Function-like Element of the carrier of (R_NormSpace_of_BoundedLinearOperators ((REAL-NS m),(REAL-NS n)))
n is V6() V10() V11() V12() ext-real non negative V33() V38() set
REAL n is non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered FinSequenceSet of REAL
n -tuples_on REAL is non empty functional FinSequence-membered FinSequenceSet of REAL
m is V6() V10() V11() V12() ext-real non negative V33() V38() set
REAL m is non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered FinSequenceSet of REAL
m -tuples_on REAL is non empty functional FinSequence-membered FinSequenceSet of REAL
K7((REAL n),(REAL m)) is set
K6(K7((REAL n),(REAL m))) is set
m is V6() V10() V11() V12() ext-real non negative V33() V38() V120() V155() V156() V157() V158() V159() V160() V161() Element of NAT
REAL m is non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered FinSequenceSet of REAL
m -tuples_on REAL is non empty functional FinSequence-membered FinSequenceSet of REAL
0* m is Relation-like NAT -defined REAL -valued Function-like V40(m) FinSequence-like complex-yielding V146() V147() Element of REAL m
m |-> 0 is Relation-like empty-yielding NAT -defined REAL -valued Function-like V40(m) FinSequence-like complex-yielding V146() V147() Element of m -tuples_on REAL
n is V6() V10() V11() V12() ext-real non negative V33() V38() V120() V155() V156() V157() V158() V159() V160() V161() Element of NAT
REAL n is non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered FinSequenceSet of REAL
n -tuples_on REAL is non empty functional FinSequence-membered FinSequenceSet of REAL
K7((REAL m),(REAL n)) is set
K6(K7((REAL m),(REAL n))) is set
0* n is Relation-like NAT -defined REAL -valued Function-like V40(n) FinSequence-like complex-yielding V146() V147() Element of REAL n
n |-> 0 is Relation-like empty-yielding NAT -defined REAL -valued Function-like V40(n) FinSequence-like complex-yielding V146() V147() Element of n -tuples_on REAL
f is non empty Relation-like REAL m -defined REAL n -valued Function-like total V30( REAL m, REAL n) complex-functions-valued ext-real-functions-valued real-functions-valued Element of K6(K7((REAL m),(REAL n)))
f . (0* m) is Relation-like Function-like V40(n) complex-yielding V146() V147() Element of REAL n
(0* m) + (0* m) is Relation-like NAT -defined REAL -valued Function-like V40(m) FinSequence-like complex-yielding V146() V147() Element of REAL m
K497() is Relation-like K7(REAL,REAL) -defined REAL -valued Function-like total V30(K7(REAL,REAL), REAL ) complex-yielding V146() V147() Element of K6(K7(K7(REAL,REAL),REAL))
K497() .: ((0* m),(0* m)) is set
f . ((0* m) + (0* m)) is Relation-like Function-like V40(n) complex-yielding V146() V147() Element of REAL n
(f . (0* m)) + (f . (0* m)) is Relation-like NAT -defined REAL -valued Function-like V40(n) FinSequence-like complex-yielding V146() V147() Element of REAL n
((f . (0* m)) + (f . (0* m))) - (f . (0* m)) is Relation-like NAT -defined REAL -valued Function-like V40(n) FinSequence-like complex-yielding V146() V147() Element of REAL n
- (f . (0* m)) is Relation-like Function-like complex-yielding set
- 1 is V11() V12() ext-real non positive set
(- 1) (#) (f . (0* m)) is Relation-like Function-like set
((f . (0* m)) + (f . (0* m))) + (- (f . (0* m))) is Relation-like Function-like set
m is V6() V10() V11() V12() ext-real non negative V33() V38() V120() V155() V156() V157() V158() V159() V160() V161() Element of NAT
REAL m is non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered FinSequenceSet of REAL
m -tuples_on REAL is non empty functional FinSequence-membered FinSequenceSet of REAL
REAL-NS m is non empty V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() V108() V109() V142() L15()
the carrier of (REAL-NS m) is non empty set
n is V6() V10() V11() V12() ext-real non negative V33() V38() V120() V155() V156() V157() V158() V159() V160() V161() Element of NAT
REAL n is non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered FinSequenceSet of REAL
n -tuples_on REAL is non empty functional FinSequence-membered FinSequenceSet of REAL
K7((REAL m),(REAL n)) is set
K6(K7((REAL m),(REAL n))) is set
REAL-NS n is non empty V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() V108() V109() V142() L15()
the carrier of (REAL-NS n) is non empty set
K7( the carrier of (REAL-NS m), the carrier of (REAL-NS n)) is set
K6(K7( the carrier of (REAL-NS m), the carrier of (REAL-NS n))) is set
f is non empty Relation-like REAL m -defined REAL n -valued Function-like total V30( REAL m, REAL n) complex-functions-valued ext-real-functions-valued real-functions-valued Element of K6(K7((REAL m),(REAL n)))
Z is non empty Relation-like the carrier of (REAL-NS m) -defined the carrier of (REAL-NS n) -valued Function-like total V30( the carrier of (REAL-NS m), the carrier of (REAL-NS n)) Element of K6(K7( the carrier of (REAL-NS m), the carrier of (REAL-NS n)))
g is Element of the carrier of (REAL-NS m)
Z0 is Element of the carrier of (REAL-NS m)
g + Z0 is Element of the carrier of (REAL-NS m)
Z . (g + Z0) is Element of the carrier of (REAL-NS n)
c₇ is Relation-like NAT -defined REAL -valued Function-like V40(m) FinSequence-like complex-yielding V146() V147() Element of REAL m
x is Relation-like NAT -defined REAL -valued Function-like V40(m) FinSequence-like complex-yielding V146() V147() Element of REAL m
c₇ + x is Relation-like NAT -defined REAL -valued Function-like V40(m) FinSequence-like complex-yielding V146() V147() Element of REAL m
K497() is Relation-like K7(REAL,REAL) -defined REAL -valued Function-like total V30(K7(REAL,REAL), REAL ) complex-yielding V146() V147() Element of K6(K7(K7(REAL,REAL),REAL))
K497() .: (c₇,x) is set
f . (c₇ + x) is Relation-like Function-like V40(n) complex-yielding V146() V147() Element of REAL n
f . c₇ is Relation-like Function-like V40(n) complex-yielding V146() V147() Element of REAL n
f . x is Relation-like Function-like V40(n) complex-yielding V146() V147() Element of REAL n
(f . c₇) + (f . x) is Relation-like NAT -defined REAL -valued Function-like V40(n) FinSequence-like complex-yielding V146() V147() Element of REAL n
Z . g is Element of the carrier of (REAL-NS n)
Z . Z0 is Element of the carrier of (REAL-NS n)
(Z . g) + (Z . Z0) is Element of the carrier of (REAL-NS n)
g is Relation-like NAT -defined REAL -valued Function-like V40(m) FinSequence-like complex-yielding V146() V147() Element of REAL m
Z0 is Relation-like NAT -defined REAL -valued Function-like V40(m) FinSequence-like complex-yielding V146() V147() Element of REAL m
g + Z0 is Relation-like NAT -defined REAL -valued Function-like V40(m) FinSequence-like complex-yielding V146() V147() Element of REAL m
K497() .: (g,Z0) is set
f . (g + Z0) is Relation-like Function-like V40(n) complex-yielding V146() V147() Element of REAL n
c₇ is Element of the carrier of (REAL-NS m)
x is Element of the carrier of (REAL-NS m)
c₇ + x is Element of the carrier of (REAL-NS m)
Z . (c₇ + x) is Element of the carrier of (REAL-NS n)
Z . c₇ is Element of the carrier of (REAL-NS n)
Z . x is Element of the carrier of (REAL-NS n)
(Z . c₇) + (Z . x) is Element of the carrier of (REAL-NS n)
f . g is Relation-like Function-like V40(n) complex-yielding V146() V147() Element of REAL n
f . Z0 is Relation-like Function-like V40(n) complex-yielding V146() V147() Element of REAL n
(f . g) + (f . Z0) is Relation-like NAT -defined REAL -valued Function-like V40(n) FinSequence-like complex-yielding V146() V147() Element of REAL n
m is V6() V10() V11() V12() ext-real non negative V33() V38() V120() V155() V156() V157() V158() V159() V160() V161() Element of NAT
REAL m is non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered FinSequenceSet of REAL
m -tuples_on REAL is non empty functional FinSequence-membered FinSequenceSet of REAL
REAL-NS m is non empty V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() V108() V109() V142() L15()
the carrier of (REAL-NS m) is non empty set
n is V6() V10() V11() V12() ext-real non negative V33() V38() V120() V155() V156() V157() V158() V159() V160() V161() Element of NAT
REAL n is non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered FinSequenceSet of REAL
n -tuples_on REAL is non empty functional FinSequence-membered FinSequenceSet of REAL
K7((REAL m),(REAL n)) is set
K6(K7((REAL m),(REAL n))) is set
REAL-NS n is non empty V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() V108() V109() V142() L15()
the carrier of (REAL-NS n) is non empty set
K7( the carrier of (REAL-NS m), the carrier of (REAL-NS n)) is set
K6(K7( the carrier of (REAL-NS m), the carrier of (REAL-NS n))) is set
f is non empty Relation-like REAL m -defined REAL n -valued Function-like total V30( REAL m, REAL n) complex-functions-valued ext-real-functions-valued real-functions-valued Element of K6(K7((REAL m),(REAL n)))
Z is non empty Relation-like the carrier of (REAL-NS m) -defined the carrier of (REAL-NS n) -valued Function-like total V30( the carrier of (REAL-NS m), the carrier of (REAL-NS n)) Element of K6(K7( the carrier of (REAL-NS m), the carrier of (REAL-NS n)))
g is Element of the carrier of (REAL-NS m)
Z0 is V11() V12() ext-real Element of REAL
Z0 * g is Element of the carrier of (REAL-NS m)
Z . (Z0 * g) is Element of the carrier of (REAL-NS n)
c₇ is Relation-like NAT -defined REAL -valued Function-like V40(m) FinSequence-like complex-yielding V146() V147() Element of REAL m
Z0 * c₇ is Relation-like NAT -defined REAL -valued Function-like V40(m) FinSequence-like complex-yielding V146() V147() Element of REAL m
K515(Z0) is non empty Relation-like REAL -defined REAL -valued Function-like total V30( REAL , REAL ) complex-yielding V146() V147() Element of K6(K7(REAL,REAL))
K499() is Relation-like K7(REAL,REAL) -defined REAL -valued Function-like total V30(K7(REAL,REAL), REAL ) complex-yielding V146() V147() Element of K6(K7(K7(REAL,REAL),REAL))
id REAL is non empty Relation-like REAL -defined REAL -valued total V30( REAL , REAL ) complex-yielding V146() V147() Element of K6(K7(REAL,REAL))
K499() [;] (Z0,(id REAL)) is set
c₇ (#) K515(Z0) is Relation-like complex-yielding V146() V147() set
f . (Z0 * c₇) is Relation-like Function-like V40(n) complex-yielding V146() V147() Element of REAL n
f . c₇ is Relation-like Function-like V40(n) complex-yielding V146() V147() Element of REAL n
Z0 * (f . c₇) is Relation-like NAT -defined REAL -valued Function-like V40(n) FinSequence-like complex-yielding V146() V147() Element of REAL n
Z . g is Element of the carrier of (REAL-NS n)
Z0 * (Z . g) is Element of the carrier of (REAL-NS n)
g is Relation-like NAT -defined REAL -valued Function-like V40(m) FinSequence-like complex-yielding V146() V147() Element of REAL m
Z0 is V11() V12() ext-real Element of REAL
Z0 * g is Relation-like NAT -defined REAL -valued Function-like V40(m) FinSequence-like complex-yielding V146() V147() Element of REAL m
K515(Z0) is non empty Relation-like REAL -defined REAL -valued Function-like total V30( REAL , REAL ) complex-yielding V146() V147() Element of K6(K7(REAL,REAL))
K499() [;] (Z0,(id REAL)) is set
g (#) K515(Z0) is Relation-like complex-yielding V146() V147() set
f . (Z0 * g) is Relation-like Function-like V40(n) complex-yielding V146() V147() Element of REAL n
c₇ is Element of the carrier of (REAL-NS m)
Z0 * c₇ is Element of the carrier of (REAL-NS m)
Z . (Z0 * c₇) is Element of the carrier of (REAL-NS n)
Z . c₇ is Element of the carrier of (REAL-NS n)
Z0 * (Z . c₇) is Element of the carrier of (REAL-NS n)
f . g is Relation-like Function-like V40(n) complex-yielding V146() V147() Element of REAL n
Z0 * (f . g) is Relation-like NAT -defined REAL -valued Function-like V40(n) FinSequence-like complex-yielding V146() V147() Element of REAL n
n is V6() V10() V11() V12() ext-real non negative V33() V38() set
REAL n is non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered FinSequenceSet of REAL
n -tuples_on REAL is non empty functional FinSequence-membered FinSequenceSet of REAL
m is V6() V10() V11() V12() ext-real non negative V33() V38() set
REAL m is non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered FinSequenceSet of REAL
m -tuples_on REAL is non empty functional FinSequence-membered FinSequenceSet of REAL
K7((REAL n),(REAL m)) is set
K6(K7((REAL n),(REAL m))) is set
0* m is Relation-like NAT -defined REAL -valued Function-like V40(m) FinSequence-like complex-yielding V146() V147() Element of REAL m
m |-> 0 is Relation-like empty-yielding NAT -defined REAL -valued Function-like V40(m) FinSequence-like complex-yielding V146() V147() Element of m -tuples_on REAL
(REAL n) --> (0* m) is non empty Relation-like REAL n -defined REAL m -valued Function-like total V30( REAL n, REAL m) complex-functions-valued ext-real-functions-valued real-functions-valued Element of K6(K7((REAL n),(REAL m)))
f is non empty Relation-like REAL n -defined REAL m -valued Function-like total V30( REAL n, REAL m) complex-functions-valued ext-real-functions-valued real-functions-valued Element of K6(K7((REAL n),(REAL m)))
Z is V6() V10() V11() V12() ext-real non negative V33() V38() V120() V155() V156() V157() V158() V159() V160() V161() Element of NAT
REAL-NS Z is non empty V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() V108() V109() V142() L15()
0. (REAL-NS Z) is V52( REAL-NS Z) Element of the carrier of (REAL-NS Z)
the carrier of (REAL-NS Z) is non empty set
g is Relation-like NAT -defined REAL -valued Function-like V40(n) FinSequence-like complex-yielding V146() V147() Element of REAL n
Z0 is Relation-like NAT -defined REAL -valued Function-like V40(n) FinSequence-like complex-yielding V146() V147() Element of REAL n
g + Z0 is Relation-like NAT -defined REAL -valued Function-like V40(n) FinSequence-like complex-yielding V146() V147() Element of REAL n
K497() is Relation-like K7(REAL,REAL) -defined REAL -valued Function-like total V30(K7(REAL,REAL), REAL ) complex-yielding V146() V147() Element of K6(K7(K7(REAL,REAL),REAL))
K497() .: (g,Z0) is set
f . (g + Z0) is Relation-like Function-like V40(m) complex-yielding V146() V147() Element of REAL m
(0. (REAL-NS Z)) + (0. (REAL-NS Z)) is Element of the carrier of (REAL-NS Z)
(0* m) + (0* m) is Relation-like NAT -defined REAL -valued Function-like V40(m) FinSequence-like complex-yielding V146() V147() Element of REAL m
K497() .: ((0* m),(0* m)) is set
f . g is Relation-like Function-like V40(m) complex-yielding V146() V147() Element of REAL m
(f . g) + (0* m) is Relation-like NAT -defined REAL -valued Function-like V40(m) FinSequence-like complex-yielding V146() V147() Element of REAL m
f . Z0 is Relation-like Function-like V40(m) complex-yielding V146() V147() Element of REAL m
(f . g) + (f . Z0) is Relation-like NAT -defined REAL -valued Function-like V40(m) FinSequence-like complex-yielding V146() V147() Element of REAL m
g is Relation-like NAT -defined REAL -valued Function-like V40(n) FinSequence-like complex-yielding V146() V147() Element of REAL n
Z0 is V11() V12() ext-real Element of REAL
Z0 * g is Relation-like NAT -defined REAL -valued Function-like V40(n) FinSequence-like complex-yielding V146() V147() Element of REAL n
K515(Z0) is non empty Relation-like REAL -defined REAL -valued Function-like total V30( REAL , REAL ) complex-yielding V146() V147() Element of K6(K7(REAL,REAL))
K499() is Relation-like K7(REAL,REAL) -defined REAL -valued Function-like total V30(K7(REAL,REAL), REAL ) complex-yielding V146() V147() Element of K6(K7(K7(REAL,REAL),REAL))
id REAL is non empty Relation-like REAL -defined REAL -valued total V30( REAL , REAL ) complex-yielding V146() V147() Element of K6(K7(REAL,REAL))
K499() [;] (Z0,(id REAL)) is set
g (#) K515(Z0) is Relation-like complex-yielding V146() V147() set
f . (Z0 * g) is Relation-like Function-like V40(m) complex-yielding V146() V147() Element of REAL m
Z0 * (0. (REAL-NS Z)) is Element of the carrier of (REAL-NS Z)
Z0 * (0* m) is Relation-like NAT -defined REAL -valued Function-like V40(m) FinSequence-like complex-yielding V146() V147() Element of REAL m
(0* m) (#) K515(Z0) is Relation-like complex-yielding V146() V147() set
f . g is Relation-like Function-like V40(m) complex-yielding V146() V147() Element of REAL m
Z0 * (f . g) is Relation-like NAT -defined REAL -valued Function-like V40(m) FinSequence-like complex-yielding V146() V147() Element of REAL m
n is V6() V10() V11() V12() ext-real non negative V33() V38() set
REAL n is non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered FinSequenceSet of REAL
n -tuples_on REAL is non empty functional FinSequence-membered FinSequenceSet of REAL
m is V6() V10() V11() V12() ext-real non negative V33() V38() set
REAL m is non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered FinSequenceSet of REAL
m -tuples_on REAL is non empty functional FinSequence-membered FinSequenceSet of REAL
K7((REAL n),(REAL m)) is set
K6(K7((REAL n),(REAL m))) is set
0* m is Relation-like NAT -defined REAL -valued Function-like V40(m) FinSequence-like complex-yielding V146() V147() Element of REAL m
m |-> 0 is Relation-like empty-yielding NAT -defined REAL -valued Function-like V40(m) FinSequence-like complex-yielding V146() V147() Element of m -tuples_on REAL
(REAL n) --> (0* m) is non empty Relation-like REAL n -defined REAL n -defined REAL m -valued REAL m -valued Function-like total total V30( REAL n, REAL m) V30( REAL n, REAL m) complex-functions-valued ext-real-functions-valued real-functions-valued (m,n) (m,n) Element of K6(K7((REAL n),(REAL m)))
m is V6() V10() V11() V12() ext-real non negative V33() V38() V120() V155() V156() V157() V158() V159() V160() V161() Element of NAT
REAL m is non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered FinSequenceSet of REAL
m -tuples_on REAL is non empty functional FinSequence-membered FinSequenceSet of REAL
REAL-NS m is non empty V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() V108() V109() V142() L15()
the carrier of (REAL-NS m) is non empty set
n is V6() V10() V11() V12() ext-real non negative V33() V38() V120() V155() V156() V157() V158() V159() V160() V161() Element of NAT
REAL n is non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered FinSequenceSet of REAL
n -tuples_on REAL is non empty functional FinSequence-membered FinSequenceSet of REAL
K7((REAL m),(REAL n)) is set
K6(K7((REAL m),(REAL n))) is set
REAL-NS n is non empty V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() V108() V109() V142() L15()
the carrier of (REAL-NS n) is non empty set
K7( the carrier of (REAL-NS m), the carrier of (REAL-NS n)) is set
K6(K7( the carrier of (REAL-NS m), the carrier of (REAL-NS n))) is set
f is set
m is V6() V10() V11() V12() ext-real non negative V33() V38() set
REAL m is non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered FinSequenceSet of REAL
m -tuples_on REAL is non empty functional FinSequence-membered FinSequenceSet of REAL
n is V6() V10() V11() V12() ext-real non negative V33() V38() set
REAL n is non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered FinSequenceSet of REAL
n -tuples_on REAL is non empty functional FinSequence-membered FinSequenceSet of REAL
K7((REAL m),(REAL n)) is set
K6(K7((REAL m),(REAL n))) is set
f is non empty Relation-like REAL m -defined REAL n -valued Function-like total V30( REAL m, REAL n) complex-functions-valued ext-real-functions-valued real-functions-valued Element of K6(K7((REAL m),(REAL n)))
Z is Relation-like NAT -defined REAL -valued Function-like V40(m) FinSequence-like complex-yielding V146() V147() Element of REAL m
f . Z is Relation-like Function-like complex-yielding V146() V147() set
f . Z is Relation-like Function-like V40(n) complex-yielding V146() V147() Element of REAL n
m is V6() V10() V11() V12() ext-real non negative V33() V38() set
REAL m is non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered FinSequenceSet of REAL
m -tuples_on REAL is non empty functional FinSequence-membered FinSequenceSet of REAL
n is V6() V10() V11() V12() ext-real non negative V33() V38() set
REAL n is non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered FinSequenceSet of REAL
n -tuples_on REAL is non empty functional FinSequence-membered FinSequenceSet of REAL
K7((REAL m),(REAL n)) is set
K6(K7((REAL m),(REAL n))) is set
m is V6() V10() V11() V12() ext-real non negative V33() V38() V120() V155() V156() V157() V158() V159() V160() V161() Element of NAT
REAL m is non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered FinSequenceSet of REAL
m -tuples_on REAL is non empty functional FinSequence-membered FinSequenceSet of REAL
n is Relation-like NAT -defined REAL m -valued Function-like FinSequence-like complex-functions-valued ext-real-functions-valued real-functions-valued FinSequence of REAL m
len n is V6() V10() V11() V12() ext-real non negative V33() V38() V120() V155() V156() V157() V158() V159() V160() V161() Element of NAT
f is Relation-like NAT -defined REAL m -valued Function-like FinSequence-like complex-functions-valued ext-real-functions-valued real-functions-valued FinSequence of REAL m
len f is V6() V10() V11() V12() ext-real non negative V33() V38() V120() V155() V156() V157() V158() V159() V160() V161() Element of NAT
(len f) + 1 is non empty V6() V10() V11() V12() ext-real positive non negative V33() V38() V120() V155() V156() V157() V158() V159() V160() V161() Element of NAT
n | (len f) is Relation-like NAT -defined REAL m -valued Function-like FinSequence-like complex-functions-valued ext-real-functions-valued real-functions-valued FinSequence of REAL m
Seg (len f) is V156() V157() V158() V159() V160() V161() Element of K6(NAT)
{ b₁ where b₁ is V6() V10() V11() V12() ext-real non negative V33() V38() V120() V155() V156() V157() V158() V159() V160() V161() Element of NAT : ( 1 <= b₁ & b₁ <= len f ) } is set
n | (Seg (len f)) is Relation-like set
n . (len n) is Relation-like Function-like complex-yielding V146() V147() set
Sum n is Relation-like NAT -defined REAL -valued Function-like V40(m) FinSequence-like complex-yielding V146() V147() Element of REAL m
Sum f is Relation-like NAT -defined REAL -valued Function-like V40(m) FinSequence-like complex-yielding V146() V147() Element of REAL m
accum n is Relation-like NAT -defined REAL m -valued Function-like FinSequence-like complex-functions-valued ext-real-functions-valued real-functions-valued FinSequence of REAL m
len (accum n) is V6() V10() V11() V12() ext-real non negative V33() V38() V120() V155() V156() V157() V158() V159() V160() V161() Element of NAT
n . 1 is Relation-like Function-like complex-yielding V146() V147() set
(accum n) . 1 is Relation-like Function-like complex-yielding V146() V147() set
Z is V6() V10() V11() V12() ext-real non negative V33() V38() set
Z + 1 is non empty V6() V10() V11() V12() ext-real positive non negative V33() V38() V120() V155() V156() V157() V158() V159() V160() V161() Element of NAT
(accum n) . (Z + 1) is Relation-like Function-like complex-yielding V146() V147() set
(accum n) /. Z is V40(m) complex-yielding V146() V147() Element of REAL m
n /. (Z + 1) is V40(m) complex-yielding V146() V147() Element of REAL m
((accum n) /. Z) + (n /. (Z + 1)) is Relation-like NAT -defined REAL -valued Function-like V40(m) FinSequence-like complex-yielding V146() V147() Element of REAL m
(accum n) . (len n) is Relation-like Function-like complex-yielding V146() V147() set
Z0 is Relation-like NAT -defined REAL -valued Function-like V40(m) FinSequence-like complex-yielding V146() V147() Element of REAL m
(Sum f) + Z0 is Relation-like NAT -defined REAL -valued Function-like V40(m) FinSequence-like complex-yielding V146() V147() Element of REAL m
K497() is Relation-like K7(REAL,REAL) -defined REAL -valued Function-like total V30(K7(REAL,REAL), REAL ) complex-yielding V146() V147() Element of K6(K7(K7(REAL,REAL),REAL))
K497() .: ((Sum f),Z0) is set
0* m is Relation-like NAT -defined REAL -valued Function-like V40(m) FinSequence-like complex-yielding V146() V147() Element of REAL m
m |-> 0 is Relation-like empty-yielding NAT -defined REAL -valued Function-like V40(m) FinSequence-like complex-yielding V146() V147() Element of m -tuples_on REAL
accum f is Relation-like NAT -defined REAL m -valued Function-like FinSequence-like complex-functions-valued ext-real-functions-valued real-functions-valued FinSequence of REAL m
len (accum f) is V6() V10() V11() V12() ext-real non negative V33() V38() V120() V155() V156() V157() V158() V159() V160() V161() Element of NAT
f . 1 is Relation-like Function-like complex-yielding V146() V147() set
(accum f) . 1 is Relation-like Function-like complex-yielding V146() V147() set
Z0 is V6() V10() V11() V12() ext-real non negative V33() V38() set
Z0 + 1 is non empty V6() V10() V11() V12() ext-real positive non negative V33() V38() V120() V155() V156() V157() V158() V159() V160() V161() Element of NAT
(accum f) . (Z0 + 1) is Relation-like Function-like complex-yielding V146() V147() set
(accum f) /. Z0 is V40(m) complex-yielding V146() V147() Element of REAL m
f /. (Z0 + 1) is V40(m) complex-yielding V146() V147() Element of REAL m
((accum f) /. Z0) + (f /. (Z0 + 1)) is Relation-like NAT -defined REAL -valued Function-like V40(m) FinSequence-like complex-yielding V146() V147() Element of REAL m
(accum f) . (len f) is Relation-like Function-like complex-yielding V146() V147() set
(accum n) | (len f) is Relation-like NAT -defined REAL m -valued Function-like FinSequence-like complex-functions-valued ext-real-functions-valued real-functions-valued FinSequence of REAL m
(accum n) | (Seg (len f)) is Relation-like set
len ((accum n) | (len f)) is V6() V10() V11() V12() ext-real non negative V33() V38() V120() V155() V156() V157() V158() V159() V160() V161() Element of NAT
((accum n) | (len f)) . 0 is Relation-like Function-like complex-yielding V146() V147() set
(accum f) . 0 is Relation-like Function-like complex-yielding V146() V147() set
c₇ is V6() V10() V11() V12() ext-real non negative V33() V38() set
((accum n) | (len f)) . c₇ is Relation-like Function-like complex-yielding V146() V147() set
(accum f) . c₇ is Relation-like Function-like complex-yielding V146() V147() set
c₇ + 1 is non empty V6() V10() V11() V12() ext-real positive non negative V33() V38() V120() V155() V156() V157() V158() V159() V160() V161() Element of NAT
((accum n) | (len f)) . (c₇ + 1) is Relation-like Function-like complex-yielding V146() V147() set
(accum n) . (c₇ + 1) is Relation-like Function-like complex-yielding V146() V147() set
n . (c₇ + 1) is Relation-like Function-like complex-yielding V146() V147() set
n | (Seg (len f)) is Relation-like NAT -defined REAL m -valued Function-like complex-functions-valued ext-real-functions-valued real-functions-valued Element of K6(K7(NAT,(REAL m)))
K7(NAT,(REAL m)) is V33() set
K6(K7(NAT,(REAL m))) is V33() set
(n | (Seg (len f))) . (c₇ + 1) is Relation-like Function-like complex-yielding V146() V147() set
(accum n) /. c₇ is V40(m) complex-yielding V146() V147() Element of REAL m
(accum n) . c₇ is Relation-like Function-like complex-yielding V146() V147() set
(accum f) /. c₇ is V40(m) complex-yielding V146() V147() Element of REAL m
n /. (c₇ + 1) is V40(m) complex-yielding V146() V147() Element of REAL m
f /. (c₇ + 1) is V40(m) complex-yielding V146() V147() Element of REAL m
((accum n) /. c₇) + (n /. (c₇ + 1)) is Relation-like NAT -defined REAL -valued Function-like V40(m) FinSequence-like complex-yielding V146() V147() Element of REAL m
(accum f) . (c₇ + 1) is Relation-like Function-like complex-yielding V146() V147() set
dom ((accum n) | (len f)) is V156() V157() V158() V159() V160() V161() Element of K6(NAT)
dom (accum n) is V156() V157() V158() V159() V160() V161() Element of K6(NAT)
(accum n) . (len f) is Relation-like Function-like complex-yielding V146() V147() set
(accum n) /. (len f) is V40(m) complex-yielding V146() V147() Element of REAL m
((accum n) | (len f)) . (len f) is Relation-like Function-like complex-yielding V146() V147() set
dom n is V156() V157() V158() V159() V160() V161() Element of K6(NAT)
Seg ((len f) + 1) is V156() V157() V158() V159() V160() V161() Element of K6(NAT)
{ b₁ where b₁ is V6() V10() V11() V12() ext-real non negative V33() V38() V120() V155() V156() V157() V158() V159() V160() V161() Element of NAT : ( 1 <= b₁ & b₁ <= (len f) + 1 ) } is set
rng n is functional complex-functions-membered ext-real-functions-membered real-functions-membered Element of K6((REAL m))
K6((REAL m)) is set
c₇ is Relation-like NAT -defined REAL -valued Function-like V40(m) FinSequence-like complex-yielding V146() V147() Element of REAL m
(Sum f) + c₇ is Relation-like NAT -defined REAL -valued Function-like V40(m) FinSequence-like complex-yielding V146() V147() Element of REAL m
K497() is Relation-like K7(REAL,REAL) -defined REAL -valued Function-like total V30(K7(REAL,REAL), REAL ) complex-yielding V146() V147() Element of K6(K7(K7(REAL,REAL),REAL))
K497() .: ((Sum f),c₇) is set
n /. ((len f) + 1) is V40(m) complex-yielding V146() V147() Element of REAL m
((accum n) /. (len f)) + (n /. ((len f) + 1)) is Relation-like NAT -defined REAL -valued Function-like V40(m) FinSequence-like complex-yielding V146() V147() Element of REAL m
m is V6() V10() V11() V12() ext-real non negative V33() V38() V120() V155() V156() V157() V158() V159() V160() V161() Element of NAT
REAL m is non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered FinSequenceSet of REAL
m -tuples_on REAL is non empty functional FinSequence-membered FinSequenceSet of REAL
n is V6() V10() V11() V12() ext-real non negative V33() V38() V120() V155() V156() V157() V158() V159() V160() V161() Element of NAT
REAL n is non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered FinSequenceSet of REAL
n -tuples_on REAL is non empty functional FinSequence-membered FinSequenceSet of REAL
K7((REAL m),(REAL n)) is set
K6(K7((REAL m),(REAL n))) is set
f is non empty Relation-like REAL m -defined REAL n -valued Function-like total V30( REAL m, REAL n) complex-functions-valued ext-real-functions-valued real-functions-valued (n,m) (n,m) Element of K6(K7((REAL m),(REAL n)))
Z is Relation-like NAT -defined REAL m -valued Function-like FinSequence-like complex-functions-valued ext-real-functions-valued real-functions-valued FinSequence of REAL m
len Z is V6() V10() V11() V12() ext-real non negative V33() V38() V120() V155() V156() V157() V158() V159() V160() V161() Element of NAT
dom Z is V156() V157() V158() V159() V160() V161() Element of K6(NAT)
Sum Z is Relation-like NAT -defined REAL -valued Function-like V40(m) FinSequence-like complex-yielding V146() V147() Element of REAL m
(m,n,f,(Sum Z)) is Relation-like NAT -defined REAL -valued Function-like V40(n) FinSequence-like complex-yielding V146() V147() Element of REAL n
g is Relation-like NAT -defined REAL n -valued Function-like FinSequence-like complex-functions-valued ext-real-functions-valued real-functions-valued FinSequence of REAL n
len g is V6() V10() V11() V12() ext-real non negative V33() V38() V120() V155() V156() V157() V158() V159() V160() V161() Element of NAT
Sum g is Relation-like NAT -defined REAL -valued Function-like V40(n) FinSequence-like complex-yielding V146() V147() Element of REAL n
0* m is Relation-like NAT -defined REAL -valued Function-like V40(m) FinSequence-like complex-yielding V146() V147() Element of REAL m
m |-> 0 is Relation-like empty-yielding NAT -defined REAL -valued Function-like V40(m) FinSequence-like complex-yielding V146() V147() Element of m -tuples_on REAL
0* n is Relation-like NAT -defined REAL -valued Function-like V40(n) FinSequence-like complex-yielding V146() V147() Element of REAL n
n |-> 0 is Relation-like empty-yielding NAT -defined REAL -valued Function-like V40(n) FinSequence-like complex-yielding V146() V147() Element of n -tuples_on REAL
Z is V6() V10() V11() V12() ext-real non negative V33() V38() V120() V155() V156() V157() V158() V159() V160() V161() Element of NAT
Z + 1 is non empty V6() V10() V11() V12() ext-real positive non negative V33() V38() V120() V155() V156() V157() V158() V159() V160() V161() Element of NAT
g is Relation-like NAT -defined REAL m -valued Function-like FinSequence-like complex-functions-valued ext-real-functions-valued real-functions-valued FinSequence of REAL m
len g is V6() V10() V11() V12() ext-real non negative V33() V38() V120() V155() V156() V157() V158() V159() V160() V161() Element of NAT
Z0 is Relation-like NAT -defined REAL n -valued Function-like FinSequence-like complex-functions-valued ext-real-functions-valued real-functions-valued FinSequence of REAL n
len Z0 is V6() V10() V11() V12() ext-real non negative V33() V38() V120() V155() V156() V157() V158() V159() V160() V161() Element of NAT
dom g is V156() V157() V158() V159() V160() V161() Element of K6(NAT)
g | Z is Relation-like NAT -defined REAL m -valued Function-like FinSequence-like complex-functions-valued ext-real-functions-valued real-functions-valued FinSequence of REAL m
Seg Z is V156() V157() V158() V159() V160() V161() Element of K6(NAT)
{ b₁ where b₁ is V6() V10() V11() V12() ext-real non negative V33() V38() V120() V155() V156() V157() V158() V159() V160() V161() Element of NAT : ( 1 <= b₁ & b₁ <= Z ) } is set
g | (Seg Z) is Relation-like set
Z0 | Z is Relation-like NAT -defined REAL n -valued Function-like FinSequence-like complex-functions-valued ext-real-functions-valued real-functions-valued FinSequence of REAL n
Z0 | (Seg Z) is Relation-like set
len (g | Z) is V6() V10() V11() V12() ext-real non negative V33() V38() V120() V155() V156() V157() V158() V159() V160() V161() Element of NAT
len (Z0 | Z) is V6() V10() V11() V12() ext-real non negative V33() V38() V120() V155() V156() V157() V158() V159() V160() V161() Element of NAT
dom (g | Z) is V156() V157() V158() V159() V160() V161() Element of K6(NAT)
L is V6() V10() V11() V12() ext-real non negative V33() V38() V120() V155() V156() V157() V158() V159() V160() V161() Element of NAT
(Z0 | Z) . L is Relation-like Function-like complex-yielding V146() V147() set
(g | Z) . L is Relation-like Function-like complex-yielding V146() V147() set
f . ((g | Z) . L) is Relation-like Function-like complex-yielding V146() V147() set
Z0 . L is Relation-like Function-like complex-yielding V146() V147() set
g . L is Relation-like Function-like complex-yielding V146() V147() set
f . (g . L) is Relation-like Function-like complex-yielding V146() V147() set
Sum (Z0 | Z) is Relation-like NAT -defined REAL -valued Function-like V40(n) FinSequence-like complex-yielding V146() V147() Element of REAL n
Sum (g | Z) is Relation-like NAT -defined REAL -valued Function-like V40(m) FinSequence-like complex-yielding V146() V147() Element of REAL m
(m,n,f,(Sum (g | Z))) is Relation-like NAT -defined REAL -valued Function-like V40(n) FinSequence-like complex-yielding V146() V147() Element of REAL n
g . (len g) is Relation-like Function-like complex-yielding V146() V147() set
Sum g is Relation-like NAT -defined REAL -valued Function-like V40(m) FinSequence-like complex-yielding V146() V147() Element of REAL m
L is Relation-like NAT -defined REAL -valued Function-like V40(m) FinSequence-like complex-yielding V146() V147() Element of REAL m
(Sum (g | Z)) + L is Relation-like NAT -defined REAL -valued Function-like V40(m) FinSequence-like complex-yielding V146() V147() Element of REAL m
K497() is Relation-like K7(REAL,REAL) -defined REAL -valued Function-like total V30(K7(REAL,REAL), REAL ) complex-yielding V146() V147() Element of K6(K7(K7(REAL,REAL),REAL))
K497() .: ((Sum (g | Z)),L) is set
Z0 . (len Z0) is Relation-like Function-like complex-yielding V146() V147() set
Sum Z0 is Relation-like NAT -defined REAL -valued Function-like V40(n) FinSequence-like complex-yielding V146() V147() Element of REAL n
L0 is Relation-like NAT -defined REAL -valued Function-like V40(n) FinSequence-like complex-yielding V146() V147() Element of REAL n
(Sum (Z0 | Z)) + L0 is Relation-like NAT -defined REAL -valued Function-like V40(n) FinSequence-like complex-yielding V146() V147() Element of REAL n
K497() .: ((Sum (Z0 | Z)),L0) is set
Seg (Z + 1) is V156() V157() V158() V159() V160() V161() Element of K6(NAT)
{ b₁ where b₁ is V6() V10() V11() V12() ext-real non negative V33() V38() V120() V155() V156() V157() V158() V159() V160() V161() Element of NAT : ( 1 <= b₁ & b₁ <= Z + 1 ) } is set
(m,n,f,L) is Relation-like NAT -defined REAL -valued Function-like V40(n) FinSequence-like complex-yielding V146() V147() Element of REAL n
(m,n,f,(Sum g)) is Relation-like NAT -defined REAL -valued Function-like V40(n) FinSequence-like complex-yielding V146() V147() Element of REAL n
g is Relation-like NAT -defined REAL m -valued Function-like FinSequence-like complex-functions-valued ext-real-functions-valued real-functions-valued FinSequence of REAL m
len g is V6() V10() V11() V12() ext-real non negative V33() V38() V120() V155() V156() V157() V158() V159() V160() V161() Element of NAT
Z0 is Relation-like NAT -defined REAL n -valued Function-like FinSequence-like complex-functions-valued ext-real-functions-valued real-functions-valued FinSequence of REAL n
len Z0 is V6() V10() V11() V12() ext-real non negative V33() V38() V120() V155() V156() V157() V158() V159() V160() V161() Element of NAT
dom g is V156() V157() V158() V159() V160() V161() Element of K6(NAT)
Sum Z0 is Relation-like NAT -defined REAL -valued Function-like V40(n) FinSequence-like complex-yielding V146() V147() Element of REAL n
Sum g is Relation-like NAT -defined REAL -valued Function-like V40(m) FinSequence-like complex-yielding V146() V147() Element of REAL m
(m,n,f,(Sum g)) is Relation-like NAT -defined REAL -valued Function-like V40(n) FinSequence-like complex-yielding V146() V147() Element of REAL n
Z is Relation-like NAT -defined REAL m -valued Function-like FinSequence-like complex-functions-valued ext-real-functions-valued real-functions-valued FinSequence of REAL m
len Z is V6() V10() V11() V12() ext-real non negative V33() V38() V120() V155() V156() V157() V158() V159() V160() V161() Element of NAT
dom Z is V156() V157() V158() V159() V160() V161() Element of K6(NAT)
Sum Z is Relation-like NAT -defined REAL -valued Function-like V40(m) FinSequence-like complex-yielding V146() V147() Element of REAL m
(m,n,f,(Sum Z)) is Relation-like NAT -defined REAL -valued Function-like V40(n) FinSequence-like complex-yielding V146() V147() Element of REAL n
g is Relation-like NAT -defined REAL n -valued Function-like FinSequence-like complex-functions-valued ext-real-functions-valued real-functions-valued FinSequence of REAL n
len g is V6() V10() V11() V12() ext-real non negative V33() V38() V120() V155() V156() V157() V158() V159() V160() V161() Element of NAT
Sum g is Relation-like NAT -defined REAL -valued Function-like V40(n) FinSequence-like complex-yielding V146() V147() Element of REAL n
m is V6() V10() V11() V12() ext-real non negative V33() V38() V120() V155() V156() V157() V158() V159() V160() V161() Element of NAT
REAL m is non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered FinSequenceSet of REAL
m -tuples_on REAL is non empty functional FinSequence-membered FinSequenceSet of REAL
n is Relation-like NAT -defined REAL m -valued Function-like FinSequence-like complex-functions-valued ext-real-functions-valued real-functions-valued FinSequence of REAL m
len n is V6() V10() V11() V12() ext-real non negative V33() V38() V120() V155() V156() V157() V158() V159() V160() V161() Element of NAT
dom n is V156() V157() V158() V159() V160() V161() Element of K6(NAT)
Sum n is Relation-like NAT -defined REAL -valued Function-like V40(m) FinSequence-like complex-yielding V146() V147() Element of REAL m
|.(Sum n).| is V11() V12() ext-real non negative Element of REAL
sqr (Sum n) is Relation-like NAT -defined REAL -valued Function-like FinSequence-like complex-yielding V146() V147() FinSequence of REAL
(Sum n) (#) sqrreal is Relation-like complex-yielding V146() V147() set
(Sum n) (#) (Sum n) is Relation-like Function-like set
K499() is Relation-like K7(REAL,REAL) -defined REAL -valued Function-like total V30(K7(REAL,REAL), REAL ) complex-yielding V146() V147() Element of K6(K7(K7(REAL,REAL),REAL))
K499() .: ((Sum n),(Sum n)) is set
Sum (sqr (Sum n)) is V11() V12() ext-real Element of REAL
sqrt (Sum (sqr (Sum n))) is V11() V12() ext-real Element of REAL
f is Relation-like NAT -defined REAL -valued Function-like FinSequence-like complex-yielding V146() V147() FinSequence of REAL
len f is V6() V10() V11() V12() ext-real non negative V33() V38() V120() V155() V156() V157() V158() V159() V160() V161() Element of NAT
Sum f is V11() V12() ext-real Element of REAL
0* m is Relation-like NAT -defined REAL -valued Function-like V40(m) FinSequence-like complex-yielding V146() V147() Element of REAL m
m |-> 0 is Relation-like empty-yielding NAT -defined REAL -valued Function-like V40(m) FinSequence-like complex-yielding V146() V147() Element of m -tuples_on REAL
n is V6() V10() V11() V12() ext-real non negative V33() V38() V120() V155() V156() V157() V158() V159() V160() V161() Element of NAT
f is Relation-like NAT -defined REAL m -valued Function-like FinSequence-like complex-functions-valued ext-real-functions-valued real-functions-valued FinSequence of REAL m
f | n is Relation-like NAT -defined REAL m -valued Function-like FinSequence-like complex-functions-valued ext-real-functions-valued real-functions-valued FinSequence of REAL m
Seg n is V156() V157() V158() V159() V160() V161() Element of K6(NAT)
{ b₁ where b₁ is V6() V10() V11() V12() ext-real non negative V33() V38() V120() V155() V156() V157() V158() V159() V160() V161() Element of NAT : ( 1 <= b₁ & b₁ <= n ) } is set
f | (Seg n) is Relation-like set
Z is Relation-like NAT -defined REAL -valued Function-like FinSequence-like complex-yielding V146() V147() FinSequence of REAL
Z | n is Relation-like NAT -defined REAL -valued Function-like FinSequence-like complex-yielding V146() V147() FinSequence of REAL
Z | (Seg n) is Relation-like complex-yielding V146() V147() set
n + 1 is non empty V6() V10() V11() V12() ext-real positive non negative V33() V38() V120() V155() V156() V157() V158() V159() V160() V161() Element of NAT
len f is V6() V10() V11() V12() ext-real non negative V33() V38() V120() V155() V156() V157() V158() V159() V160() V161() Element of NAT
len Z is V6() V10() V11() V12() ext-real non negative V33() V38() V120() V155() V156() V157() V158() V159() V160() V161() Element of NAT
dom f is V156() V157() V158() V159() V160() V161() Element of K6(NAT)
dom (f | n) is V156() V157() V158() V159() V160() V161() Element of K6(NAT)
c₇ is V6() V10() V11() V12() ext-real non negative V33() V38() V120() V155() V156() V157() V158() V159() V160() V161() Element of NAT
(f | n) . c₇ is Relation-like Function-like complex-yielding V146() V147() set
(Z | n) . c₇ is V11() V12() ext-real Element of REAL
f . c₇ is Relation-like Function-like complex-yielding V146() V147() set
Z . c₇ is V11() V12() ext-real Element of REAL
x is Relation-like NAT -defined REAL -valued Function-like V40(m) FinSequence-like complex-yielding V146() V147() Element of REAL m
|.x.| is V11() V12() ext-real non negative Element of REAL
sqr x is Relation-like NAT -defined REAL -valued Function-like FinSequence-like complex-yielding V146() V147() FinSequence of REAL
x (#) sqrreal is Relation-like complex-yielding V146() V147() set
x (#) x is Relation-like Function-like set
K499() is Relation-like K7(REAL,REAL) -defined REAL -valued Function-like total V30(K7(REAL,REAL), REAL ) complex-yielding V146() V147() Element of K6(K7(K7(REAL,REAL),REAL))
K499() .: (x,x) is set
Sum (sqr x) is V11() V12() ext-real Element of REAL
sqrt (Sum (sqr x)) is V11() V12() ext-real Element of REAL
Seg (n + 1) is V156() V157() V158() V159() V160() V161() Element of K6(NAT)
{ b₁ where b₁ is V6() V10() V11() V12() ext-real non negative V33() V38() V120() V155() V156() V157() V158() V159() V160() V161() Element of NAT : ( 1 <= b₁ & b₁ <= n + 1 ) } is set
f . (n + 1) is Relation-like Function-like complex-yielding V146() V147() set
Z . (n + 1) is V11() V12() ext-real Element of REAL
c₇ is Relation-like NAT -defined REAL -valued Function-like V40(m) FinSequence-like complex-yielding V146() V147() Element of REAL m
|.c₇.| is V11() V12() ext-real non negative Element of REAL
sqr c₇ is Relation-like NAT -defined REAL -valued Function-like FinSequence-like complex-yielding V146() V147() FinSequence of REAL
c₇ (#) sqrreal is Relation-like complex-yielding V146() V147() set
c₇ (#) c₇ is Relation-like Function-like set
K499() is Relation-like K7(REAL,REAL) -defined REAL -valued Function-like total V30(K7(REAL,REAL), REAL ) complex-yielding V146() V147() Element of K6(K7(K7(REAL,REAL),REAL))
K499() .: (c₇,c₇) is set
Sum (sqr c₇) is V11() V12() ext-real Element of REAL
sqrt (Sum (sqr c₇)) is V11() V12() ext-real Element of REAL
Z /. (n + 1) is V11() V12() ext-real Element of REAL
<*(Z /. (n + 1))*> is Relation-like NAT -defined REAL -valued Function-like FinSequence-like complex-yielding V146() V147() FinSequence of REAL
[1,(Z /. (n + 1))] is set
{[1,(Z /. (n + 1))]} is non empty V2() V40(1) set
(Z | n) ^ <*(Z /. (n + 1))*> is Relation-like NAT -defined REAL -valued Function-like FinSequence-like complex-yielding V146() V147() FinSequence of REAL
<*(Z . (n + 1))*> is Relation-like NAT -defined REAL -valued Function-like FinSequence-like complex-yielding V146() V147() FinSequence of REAL
[1,(Z . (n + 1))] is set
{[1,(Z . (n + 1))]} is non empty V2() V40(1) set
(Z | n) ^ <*(Z . (n + 1))*> is Relation-like NAT -defined REAL -valued Function-like FinSequence-like complex-yielding V146() V147() FinSequence of REAL
Sum Z is V11() V12() ext-real Element of REAL
Sum (Z | n) is V11() V12() ext-real Element of REAL
(Sum (Z | n)) + (Z . (n + 1)) is V11() V12() ext-real Element of REAL
len (f | n) is V6() V10() V11() V12() ext-real non negative V33() V38() V120() V155() V156() V157() V158() V159() V160() V161() Element of NAT
f . (len f) is Relation-like Function-like complex-yielding V146() V147() set
Sum f is Relation-like NAT -defined REAL -valued Function-like V40(m) FinSequence-like complex-yielding V146() V147() Element of REAL m
Sum (f | n) is Relation-like NAT -defined REAL -valued Function-like V40(m) FinSequence-like complex-yielding V146() V147() Element of REAL m
(Sum (f | n)) + c₇ is Relation-like NAT -defined REAL -valued Function-like V40(m) FinSequence-like complex-yielding V146() V147() Element of REAL m
K497() is Relation-like K7(REAL,REAL) -defined REAL -valued Function-like total V30(K7(REAL,REAL), REAL ) complex-yielding V146() V147() Element of K6(K7(K7(REAL,REAL),REAL))
K497() .: ((Sum (f | n)),c₇) is set
|.((Sum (f | n)) + c₇).| is V11() V12() ext-real non negative Element of REAL
sqr ((Sum (f | n)) + c₇) is Relation-like NAT -defined REAL -valued Function-like FinSequence-like complex-yielding V146() V147() FinSequence of REAL
((Sum (f | n)) + c₇) (#) sqrreal is Relation-like complex-yielding V146() V147() set
((Sum (f | n)) + c₇) (#) ((Sum (f | n)) + c₇) is Relation-like Function-like set
K499() .: (((Sum (f | n)) + c₇),((Sum (f | n)) + c₇)) is set
Sum (sqr ((Sum (f | n)) + c₇)) is V11() V12() ext-real Element of REAL
sqrt (Sum (sqr ((Sum (f | n)) + c₇))) is V11() V12() ext-real Element of REAL
|.(Sum (f | n)).| is V11() V12() ext-real non negative Element of REAL
sqr (Sum (f | n)) is Relation-like NAT -defined REAL -valued Function-like FinSequence-like complex-yielding V146() V147() FinSequence of REAL
(Sum (f | n)) (#) sqrreal is Relation-like complex-yielding V146() V147() set
(Sum (f | n)) (#) (Sum (f | n)) is Relation-like Function-like set
K499() .: ((Sum (f | n)),(Sum (f | n))) is set
Sum (sqr (Sum (f | n))) is V11() V12() ext-real Element of REAL
sqrt (Sum (sqr (Sum (f | n)))) is V11() V12() ext-real Element of REAL
|.(Sum (f | n)).| + |.c₇.| is V11() V12() ext-real non negative Element of REAL
len (Z | n) is V6() V10() V11() V12() ext-real non negative V33() V38() V120() V155() V156() V157() V158() V159() V160() V161() Element of NAT
|.(Sum f).| is V11() V12() ext-real non negative Element of REAL
sqr (Sum f) is Relation-like NAT -defined REAL -valued Function-like FinSequence-like complex-yielding V146() V147() FinSequence of REAL
(Sum f) (#) sqrreal is Relation-like complex-yielding V146() V147() set
(Sum f) (#) (Sum f) is Relation-like Function-like set
K499() .: ((Sum f),(Sum f)) is set
Sum (sqr (Sum f)) is V11() V12() ext-real Element of REAL
sqrt (Sum (sqr (Sum f))) is V11() V12() ext-real Element of REAL
x is Relation-like NAT -defined REAL -valued Function-like V40(m) FinSequence-like complex-yielding V146() V147() Element of REAL m
(Sum (f | n)) + x is Relation-like NAT -defined REAL -valued Function-like V40(m) FinSequence-like complex-yielding V146() V147() Element of REAL m
K497() .: ((Sum (f | n)),x) is set
f is Relation-like NAT -defined REAL m -valued Function-like FinSequence-like complex-functions-valued ext-real-functions-valued real-functions-valued FinSequence of REAL m
len f is V6() V10() V11() V12() ext-real non negative V33() V38() V120() V155() V156() V157() V158() V159() V160() V161() Element of NAT
Z is Relation-like NAT -defined REAL -valued Function-like FinSequence-like complex-yielding V146() V147() FinSequence of REAL
len Z is V6() V10() V11() V12() ext-real non negative V33() V38() V120() V155() V156() V157() V158() V159() V160() V161() Element of NAT
dom f is V156() V157() V158() V159() V160() V161() Element of K6(NAT)
Sum f is Relation-like NAT -defined REAL -valued Function-like V40(m) FinSequence-like complex-yielding V146() V147() Element of REAL m
|.(Sum f).| is V11() V12() ext-real non negative Element of REAL
sqr (Sum f) is Relation-like NAT -defined REAL -valued Function-like FinSequence-like complex-yielding V146() V147() FinSequence of REAL
(Sum f) (#) sqrreal is Relation-like complex-yielding V146() V147() set
(Sum f) (#) (Sum f) is Relation-like Function-like set
K499() .: ((Sum f),(Sum f)) is set
Sum (sqr (Sum f)) is V11() V12() ext-real Element of REAL
sqrt (Sum (sqr (Sum f))) is V11() V12() ext-real Element of REAL
Sum Z is V11() V12() ext-real Element of REAL
n is Relation-like NAT -defined REAL m -valued Function-like FinSequence-like complex-functions-valued ext-real-functions-valued real-functions-valued FinSequence of REAL m
len n is V6() V10() V11() V12() ext-real non negative V33() V38() V120() V155() V156() V157() V158() V159() V160() V161() Element of NAT
f is Relation-like NAT -defined REAL -valued Function-like FinSequence-like complex-yielding V146() V147() FinSequence of REAL
len f is V6() V10() V11() V12() ext-real non negative V33() V38() V120() V155() V156() V157() V158() V159() V160() V161() Element of NAT
dom n is V156() V157() V158() V159() V160() V161() Element of K6(NAT)
Sum n is Relation-like NAT -defined REAL -valued Function-like V40(m) FinSequence-like complex-yielding V146() V147() Element of REAL m
|.(Sum n).| is V11() V12() ext-real non negative Element of REAL
sqr (Sum n) is Relation-like NAT -defined REAL -valued Function-like FinSequence-like complex-yielding V146() V147() FinSequence of REAL
(Sum n) (#) sqrreal is Relation-like complex-yielding V146() V147() set
(Sum n) (#) (Sum n) is Relation-like Function-like set
K499() .: ((Sum n),(Sum n)) is set
Sum (sqr (Sum n)) is V11() V12() ext-real Element of REAL
sqrt (Sum (sqr (Sum n))) is V11() V12() ext-real Element of REAL
Sum f is V11() V12() ext-real Element of REAL
m is V6() V10() V11() V12() ext-real non negative V33() V38() set
REAL m is non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered FinSequenceSet of REAL
m -tuples_on REAL is non empty functional FinSequence-membered FinSequenceSet of REAL
n is V6() V10() V11() V12() ext-real non negative V33() V38() set
REAL n is non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered FinSequenceSet of REAL
n -tuples_on REAL is non empty functional FinSequence-membered FinSequenceSet of REAL
K7((REAL m),(REAL n)) is set
K6(K7((REAL m),(REAL n))) is set
f is non empty Relation-like REAL m -defined REAL n -valued Function-like total V30( REAL m, REAL n) complex-functions-valued ext-real-functions-valued real-functions-valued (n,m) (n,m) Element of K6(K7((REAL m),(REAL n)))
Seg m is V156() V157() V158() V159() V160() V161() Element of K6(NAT)
{ b₁ where b₁ is V6() V10() V11() V12() ext-real non negative V33() V38() V120() V155() V156() V157() V158() V159() V160() V161() Element of NAT : ( 1 <= b₁ & b₁ <= m ) } is set
Z is V6() V10() V11() V12() ext-real non negative V33() V38() set
Base_FinSeq (m,Z) is Relation-like NAT -defined REAL -valued Function-like V40(m) FinSequence-like complex-yielding V146() V147() Element of REAL m
(m,n,f,(Base_FinSeq (m,Z))) is Relation-like NAT -defined REAL -valued Function-like V40(n) FinSequence-like complex-yielding V146() V147() Element of REAL n
|.(m,n,f,(Base_FinSeq (m,Z))).| is V11() V12() ext-real non negative Element of REAL
sqr (m,n,f,(Base_FinSeq (m,Z))) is Relation-like NAT -defined REAL -valued Function-like FinSequence-like complex-yielding V146() V147() FinSequence of REAL
(m,n,f,(Base_FinSeq (m,Z))) (#) sqrreal is Relation-like complex-yielding V146() V147() set
(m,n,f,(Base_FinSeq (m,Z))) (#) (m,n,f,(Base_FinSeq (m,Z))) is Relation-like Function-like set
K499() is Relation-like K7(REAL,REAL) -defined REAL -valued Function-like total V30(K7(REAL,REAL), REAL ) complex-yielding V146() V147() Element of K6(K7(K7(REAL,REAL),REAL))
K499() .: ((m,n,f,(Base_FinSeq (m,Z))),(m,n,f,(Base_FinSeq (m,Z)))) is set
Sum (sqr (m,n,f,(Base_FinSeq (m,Z)))) is V11() V12() ext-real Element of REAL
sqrt (Sum (sqr (m,n,f,(Base_FinSeq (m,Z))))) is V11() V12() ext-real Element of REAL
Z is Relation-like NAT -defined REAL -valued Function-like FinSequence-like complex-yielding V146() V147() FinSequence of REAL
dom Z is V156() V157() V158() V159() V160() V161() Element of K6(NAT)
Sum Z is V11() V12() ext-real Element of REAL
(Sum Z) + 1 is V11() V12() ext-real Element of REAL
Z0 is V6() V10() V11() V12() ext-real non negative V33() V38() set
Z . Z0 is V11() V12() ext-real Element of REAL
Base_FinSeq (m,Z0) is Relation-like NAT -defined REAL -valued Function-like V40(m) FinSequence-like complex-yielding V146() V147() Element of REAL m
(m,n,f,(Base_FinSeq (m,Z0))) is Relation-like NAT -defined REAL -valued Function-like V40(n) FinSequence-like complex-yielding V146() V147() Element of REAL n
|.(m,n,f,(Base_FinSeq (m,Z0))).| is V11() V12() ext-real non negative Element of REAL
sqr (m,n,f,(Base_FinSeq (m,Z0))) is Relation-like NAT -defined REAL -valued Function-like FinSequence-like complex-yielding V146() V147() FinSequence of REAL
(m,n,f,(Base_FinSeq (m,Z0))) (#) sqrreal is Relation-like complex-yielding V146() V147() set
(m,n,f,(Base_FinSeq (m,Z0))) (#) (m,n,f,(Base_FinSeq (m,Z0))) is Relation-like Function-like set
K499() is Relation-like K7(REAL,REAL) -defined REAL -valued Function-like total V30(K7(REAL,REAL), REAL ) complex-yielding V146() V147() Element of K6(K7(K7(REAL,REAL),REAL))
K499() .: ((m,n,f,(Base_FinSeq (m,Z0))),(m,n,f,(Base_FinSeq (m,Z0)))) is set
Sum (sqr (m,n,f,(Base_FinSeq (m,Z0)))) is V11() V12() ext-real Element of REAL
sqrt (Sum (sqr (m,n,f,(Base_FinSeq (m,Z0))))) is V11() V12() ext-real Element of REAL
c₇ is Relation-like NAT -defined REAL -valued Function-like V40(m) FinSequence-like complex-yielding V146() V147() Element of REAL m
ProjFinSeq c₇ is Relation-like NAT -defined REAL m -valued Function-like FinSequence-like complex-functions-valued ext-real-functions-valued real-functions-valued FinSequence of REAL m
len (ProjFinSeq c₇) is V6() V10() V11() V12() ext-real non negative V33() V38() V120() V155() V156() V157() V158() V159() V160() V161() Element of NAT
x is V6() V10() V11() V12() ext-real non negative V33() V38() set
dom (ProjFinSeq c₇) is V156() V157() V158() V159() V160() V161() Element of K6(NAT)
(ProjFinSeq c₇) . x is Relation-like Function-like complex-yielding V146() V147() set
L is Relation-like NAT -defined REAL -valued Function-like V40(m) FinSequence-like complex-yielding V146() V147() Element of REAL m
(m,n,f,L) is Relation-like NAT -defined REAL -valued Function-like V40(n) FinSequence-like complex-yielding V146() V147() Element of REAL n
f . ((ProjFinSeq c₇) . x) is Relation-like Function-like complex-yielding V146() V147() set
x is Relation-like NAT -defined REAL n -valued Function-like FinSequence-like complex-functions-valued ext-real-functions-valued real-functions-valued FinSequence of REAL n
dom x is V156() V157() V158() V159() V160() V161() Element of K6(NAT)
Sum (ProjFinSeq c₇) is Relation-like NAT -defined REAL -valued Function-like V40(m) FinSequence-like complex-yielding V146() V147() Element of REAL m
len x is V6() V10() V11() V12() ext-real non negative V33() V38() V120() V155() V156() V157() V158() V159() V160() V161() Element of NAT
L is V6() V10() V11() V12() ext-real non negative V33() V38() V120() V155() V156() V157() V158() V159() V160() V161() Element of NAT
x . L is Relation-like Function-like complex-yielding V146() V147() set
(ProjFinSeq c₇) . L is Relation-like Function-like complex-yielding V146() V147() set
f . ((ProjFinSeq c₇) . L) is Relation-like Function-like complex-yielding V146() V147() set
Sum x is Relation-like NAT -defined REAL -valued Function-like V40(n) FinSequence-like complex-yielding V146() V147() Element of REAL n
(m,n,f,c₇) is Relation-like NAT -defined REAL -valued Function-like V40(n) FinSequence-like complex-yielding V146() V147() Element of REAL n
|.c₇.| is V11() V12() ext-real non negative Element of REAL
sqr c₇ is Relation-like NAT -defined REAL -valued Function-like FinSequence-like complex-yielding V146() V147() FinSequence of REAL
c₇ (#) sqrreal is Relation-like complex-yielding V146() V147() set
c₇ (#) c₇ is Relation-like Function-like set
K499() .: (c₇,c₇) is set
Sum (sqr c₇) is V11() V12() ext-real Element of REAL
sqrt (Sum (sqr c₇)) is V11() V12() ext-real Element of REAL
L0 is V6() V10() V11() V12() ext-real non negative V33() V38() set
x . L0 is Relation-like Function-like complex-yielding V146() V147() set
Base_FinSeq (m,L0) is Relation-like NAT -defined REAL -valued Function-like V40(m) FinSequence-like complex-yielding V146() V147() Element of REAL m
(m,n,f,(Base_FinSeq (m,L0))) is Relation-like NAT -defined REAL -valued Function-like V40(n) FinSequence-like complex-yielding V146() V147() Element of REAL n
|.(m,n,f,(Base_FinSeq (m,L0))).| is V11() V12() ext-real non negative Element of REAL
sqr (m,n,f,(Base_FinSeq (m,L0))) is Relation-like NAT -defined REAL -valued Function-like FinSequence-like complex-yielding V146() V147() FinSequence of REAL
(m,n,f,(Base_FinSeq (m,L0))) (#) sqrreal is Relation-like complex-yielding V146() V147() set
(m,n,f,(Base_FinSeq (m,L0))) (#) (m,n,f,(Base_FinSeq (m,L0))) is Relation-like Function-like set
K499() .: ((m,n,f,(Base_FinSeq (m,L0))),(m,n,f,(Base_FinSeq (m,L0)))) is set
Sum (sqr (m,n,f,(Base_FinSeq (m,L0)))) is V11() V12() ext-real Element of REAL
sqrt (Sum (sqr (m,n,f,(Base_FinSeq (m,L0))))) is V11() V12() ext-real Element of REAL
|.c₇.| * |.(m,n,f,(Base_FinSeq (m,L0))).| is V11() V12() ext-real non negative Element of REAL
(ProjFinSeq c₇) . L0 is Relation-like Function-like complex-yielding V146() V147() set
|(c₇,(Base_FinSeq (m,L0)))| is V11() V12() ext-real Element of REAL
mlt (c₇,(Base_FinSeq (m,L0))) is Relation-like NAT -defined REAL -valued Function-like FinSequence-like complex-yielding V146() V147() FinSequence of REAL
K499() .: (c₇,(Base_FinSeq (m,L0))) is set
Sum (mlt (c₇,(Base_FinSeq (m,L0)))) is V11() V12() ext-real Element of REAL
|(c₇,(Base_FinSeq (m,L0)))| * (Base_FinSeq (m,L0)) is Relation-like NAT -defined REAL -valued Function-like V40(m) FinSequence-like complex-yielding V146() V147() Element of REAL m
K515(|(c₇,(Base_FinSeq (m,L0)))|) is non empty Relation-like REAL -defined REAL -valued Function-like total V30( REAL , REAL ) complex-yielding V146() V147() Element of K6(K7(REAL,REAL))
id REAL is non empty Relation-like REAL -defined REAL -valued total V30( REAL , REAL ) complex-yielding V146() V147() Element of K6(K7(REAL,REAL))
K499() [;] (|(c₇,(Base_FinSeq (m,L0)))|,(id REAL)) is set
(Base_FinSeq (m,L0)) (#) K515(|(c₇,(Base_FinSeq (m,L0)))|) is Relation-like complex-yielding V146() V147() set
R is Relation-like NAT -defined REAL -valued Function-like V40(n) FinSequence-like complex-yielding V146() V147() Element of REAL n
|.R.| is V11() V12() ext-real non negative Element of REAL
sqr R is Relation-like NAT -defined REAL -valued Function-like FinSequence-like complex-yielding V146() V147() FinSequence of REAL
R (#) sqrreal is Relation-like complex-yielding V146() V147() set
R (#) R is Relation-like Function-like set
K499() .: (R,R) is set
Sum (sqr R) is V11() V12() ext-real Element of REAL
sqrt (Sum (sqr R)) is V11() V12() ext-real Element of REAL
f . ((ProjFinSeq c₇) . L0) is Relation-like Function-like complex-yielding V146() V147() set
|(c₇,(Base_FinSeq (m,L0)))| * (m,n,f,(Base_FinSeq (m,L0))) is Relation-like NAT -defined REAL -valued Function-like V40(n) FinSequence-like complex-yielding V146() V147() Element of REAL n
(m,n,f,(Base_FinSeq (m,L0))) (#) K515(|(c₇,(Base_FinSeq (m,L0)))|) is Relation-like complex-yielding V146() V147() set
L is Relation-like NAT -defined REAL -valued Function-like V40(m) FinSequence-like complex-yielding V146() V147() Element of REAL m
L . L0 is V11() V12() ext-real Element of REAL
(L . L0) * (m,n,f,(Base_FinSeq (m,L0))) is Relation-like NAT -defined REAL -valued Function-like V40(n) FinSequence-like complex-yielding V146() V147() Element of REAL n
K515((L . L0)) is non empty Relation-like REAL -defined REAL -valued Function-like total V30( REAL , REAL ) complex-yielding V146() V147() Element of K6(K7(REAL,REAL))
K499() [;] ((L . L0),(id REAL)) is set
(m,n,f,(Base_FinSeq (m,L0))) (#) K515((L . L0)) is Relation-like complex-yielding V146() V147() set
|.(L . L0).| is V11() V12() ext-real Element of REAL
|.(L . L0).| * |.(m,n,f,(Base_FinSeq (m,L0))).| is V11() V12() ext-real Element of REAL
L0 is V6() V10() V11() V12() ext-real non negative V33() V38() set
x . L0 is Relation-like Function-like complex-yielding V146() V147() set
R is Relation-like NAT -defined REAL -valued Function-like V40(n) FinSequence-like complex-yielding V146() V147() Element of REAL n
|.R.| is V11() V12() ext-real non negative Element of REAL
sqr R is Relation-like NAT -defined REAL -valued Function-like FinSequence-like complex-yielding V146() V147() FinSequence of REAL
R (#) sqrreal is Relation-like complex-yielding V146() V147() set
R (#) R is Relation-like Function-like set
K499() .: (R,R) is set
Sum (sqr R) is V11() V12() ext-real Element of REAL
sqrt (Sum (sqr R)) is V11() V12() ext-real Element of REAL
L0 is Relation-like NAT -defined REAL -valued Function-like FinSequence-like complex-yielding V146() V147() FinSequence of REAL
dom L0 is V156() V157() V158() V159() V160() V161() Element of K6(NAT)
len L0 is V6() V10() V11() V12() ext-real non negative V33() V38() V120() V155() V156() V157() V158() V159() V160() V161() Element of NAT
R is V6() V10() V11() V12() ext-real non negative V33() V38() V120() V155() V156() V157() V158() V159() V160() V161() Element of NAT
x . R is Relation-like Function-like complex-yielding V146() V147() set
L0 . R is V11() V12() ext-real Element of REAL
|.(m,n,f,c₇).| is V11() V12() ext-real non negative Element of REAL
sqr (m,n,f,c₇) is Relation-like NAT -defined REAL -valued Function-like FinSequence-like complex-yielding V146() V147() FinSequence of REAL
(m,n,f,c₇) (#) sqrreal is Relation-like complex-yielding V146() V147() set
(m,n,f,c₇) (#) (m,n,f,c₇) is Relation-like Function-like set
K499() .: ((m,n,f,c₇),(m,n,f,c₇)) is set
Sum (sqr (m,n,f,c₇)) is V11() V12() ext-real Element of REAL
sqrt (Sum (sqr (m,n,f,c₇))) is V11() V12() ext-real Element of REAL
Sum L0 is V11() V12() ext-real Element of REAL
R is V6() V10() V11() V12() ext-real non negative V33() V38() set
x . R is Relation-like Function-like complex-yielding V146() V147() set
L0 . R is V11() V12() ext-real Element of REAL
Base_FinSeq (m,R) is Relation-like NAT -defined REAL -valued Function-like V40(m) FinSequence-like complex-yielding V146() V147() Element of REAL m
(m,n,f,(Base_FinSeq (m,R))) is Relation-like NAT -defined REAL -valued Function-like V40(n) FinSequence-like complex-yielding V146() V147() Element of REAL n
|.(m,n,f,(Base_FinSeq (m,R))).| is V11() V12() ext-real non negative Element of REAL
sqr (m,n,f,(Base_FinSeq (m,R))) is Relation-like NAT -defined REAL -valued Function-like FinSequence-like complex-yielding V146() V147() FinSequence of REAL
(m,n,f,(Base_FinSeq (m,R))) (#) sqrreal is Relation-like complex-yielding V146() V147() set
(m,n,f,(Base_FinSeq (m,R))) (#) (m,n,f,(Base_FinSeq (m,R))) is Relation-like Function-like set
K499() .: ((m,n,f,(Base_FinSeq (m,R))),(m,n,f,(Base_FinSeq (m,R)))) is set
Sum (sqr (m,n,f,(Base_FinSeq (m,R)))) is V11() V12() ext-real Element of REAL
sqrt (Sum (sqr (m,n,f,(Base_FinSeq (m,R))))) is V11() V12() ext-real Element of REAL
|.c₇.| * |.(m,n,f,(Base_FinSeq (m,R))).| is V11() V12() ext-real non negative Element of REAL
Z . R is V11() V12() ext-real Element of REAL
|.c₇.| * (Z . R) is V11() V12() ext-real Element of REAL
R is Relation-like NAT -defined REAL -valued Function-like V40(n) FinSequence-like complex-yielding V146() V147() Element of REAL n
|.R.| is V11() V12() ext-real non negative Element of REAL
sqr R is Relation-like NAT -defined REAL -valued Function-like FinSequence-like complex-yielding V146() V147() FinSequence of REAL
R (#) sqrreal is Relation-like complex-yielding V146() V147() set
R (#) R is Relation-like Function-like set
K499() .: (R,R) is set
Sum (sqr R) is V11() V12() ext-real Element of REAL
sqrt (Sum (sqr R)) is V11() V12() ext-real Element of REAL
N is Relation-like NAT -defined REAL -valued Function-like V40(n) FinSequence-like complex-yielding V146() V147() Element of REAL n
|.N.| is V11() V12() ext-real non negative Element of REAL
sqr N is Relation-like NAT -defined REAL -valued Function-like FinSequence-like complex-yielding V146() V147() FinSequence of REAL
N (#) sqrreal is Relation-like complex-yielding V146() V147() set
N (#) N is Relation-like Function-like set
K499() .: (N,N) is set
Sum (sqr N) is V11() V12() ext-real Element of REAL
sqrt (Sum (sqr N)) is V11() V12() ext-real Element of REAL
|.c₇.| * Z is Relation-like NAT -defined REAL -valued Function-like FinSequence-like complex-yielding V146() V147() FinSequence of REAL
K515(|.c₇.|) is non empty Relation-like REAL -defined REAL -valued Function-like total V30( REAL , REAL ) complex-yielding V146() V147() Element of K6(K7(REAL,REAL))
id REAL is non empty Relation-like REAL -defined REAL -valued total V30( REAL , REAL ) complex-yielding V146() V147() Element of K6(K7(REAL,REAL))
K499() [;] (|.c₇.|,(id REAL)) is set
Z (#) K515(|.c₇.|) is Relation-like complex-yielding V146() V147() set
(|.c₇.| * Z) . R is V11() V12() ext-real Element of REAL
len Z is V6() V10() V11() V12() ext-real non negative V33() V38() V120() V155() V156() V157() V158() V159() V160() V161() Element of NAT
Z0 is V6() V10() V11() V12() ext-real non negative V33() V38() set
REAL Z0 is non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered FinSequenceSet of REAL
Z0 -tuples_on REAL is non empty functional FinSequence-membered FinSequenceSet of REAL
R is Relation-like NAT -defined REAL -valued Function-like V40(Z0) FinSequence-like complex-yielding V146() V147() Element of REAL Z0
|.c₇.| * R is Relation-like NAT -defined REAL -valued Function-like V40(Z0) FinSequence-like complex-yielding V146() V147() Element of REAL Z0
R (#) K515(|.c₇.|) is Relation-like complex-yielding V146() V147() set
Sum (|.c₇.| * Z) is V11() V12() ext-real Element of REAL
|.c₇.| * (Sum Z) is V11() V12() ext-real Element of REAL
0 + (Sum Z) is V11() V12() ext-real Element of REAL
1 + (Sum Z) is V11() V12() ext-real Element of REAL
|.c₇.| * (1 + (Sum Z)) is V11() V12() ext-real Element of REAL
((Sum Z) + 1) * |.c₇.| is V11() V12() ext-real Element of REAL
m is V6() V10() V11() V12() ext-real non negative V33() V38() V120() V155() V156() V157() V158() V159() V160() V161() Element of NAT
REAL-NS m is non empty V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() V108() V109() V142() L15()
the carrier of (REAL-NS m) is non empty set
n is V6() V10() V11() V12() ext-real non negative V33() V38() V120() V155() V156() V157() V158() V159() V160() V161() Element of NAT
REAL-NS n is non empty V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() V108() V109() V142() L15()
the carrier of (REAL-NS n) is non empty set
K7( the carrier of (REAL-NS m), the carrier of (REAL-NS n)) is set
K6(K7( the carrier of (REAL-NS m), the carrier of (REAL-NS n))) is set
f is non empty Relation-like the carrier of (REAL-NS m) -defined the carrier of (REAL-NS n) -valued Function-like total V30( the carrier of (REAL-NS m), the carrier of (REAL-NS n)) additive homogeneous Element of K6(K7( the carrier of (REAL-NS m), the carrier of (REAL-NS n)))
REAL m is non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered FinSequenceSet of REAL
m -tuples_on REAL is non empty functional FinSequence-membered FinSequenceSet of REAL
REAL n is non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered FinSequenceSet of REAL
n -tuples_on REAL is non empty functional FinSequence-membered FinSequenceSet of REAL
K7((REAL m),(REAL n)) is set
K6(K7((REAL m),(REAL n))) is set
Z is non empty Relation-like REAL m -defined REAL n -valued Function-like total V30( REAL m, REAL n) complex-functions-valued ext-real-functions-valued real-functions-valued (n,m) (n,m) (m,n) Element of K6(K7((REAL m),(REAL n)))
g is V11() V12() ext-real Element of REAL
Z0 is Element of the carrier of (REAL-NS m)
f . Z0 is Element of the carrier of (REAL-NS n)
||.(f . Z0).|| is V11() V12() ext-real Element of REAL
||.Z0.|| is V11() V12() ext-real Element of REAL
g * ||.Z0.|| is V11() V12() ext-real Element of REAL
c₇ is Relation-like NAT -defined REAL -valued Function-like V40(m) FinSequence-like complex-yielding V146() V147() Element of REAL m
(m,n,Z,c₇) is Relation-like NAT -defined REAL -valued Function-like V40(n) FinSequence-like complex-yielding V146() V147() Element of REAL n
x is Relation-like NAT -defined REAL -valued Function-like V40(n) FinSequence-like complex-yielding V146() V147() Element of REAL n
|.x.| is V11() V12() ext-real non negative Element of REAL
sqr x is Relation-like NAT -defined REAL -valued Function-like FinSequence-like complex-yielding V146() V147() FinSequence of REAL
x (#) sqrreal is Relation-like complex-yielding V146() V147() set
x (#) x is Relation-like Function-like set
K499() is Relation-like K7(REAL,REAL) -defined REAL -valued Function-like total V30(K7(REAL,REAL), REAL ) complex-yielding V146() V147() Element of K6(K7(K7(REAL,REAL),REAL))
K499() .: (x,x) is set
Sum (sqr x) is V11() V12() ext-real Element of REAL
sqrt (Sum (sqr x)) is V11() V12() ext-real Element of REAL
|.c₇.| is V11() V12() ext-real non negative Element of REAL
sqr c₇ is Relation-like NAT -defined REAL -valued Function-like FinSequence-like complex-yielding V146() V147() FinSequence of REAL
c₇ (#) sqrreal is Relation-like complex-yielding V146() V147() set
c₇ (#) c₇ is Relation-like Function-like set
K499() .: (c₇,c₇) is set
Sum (sqr c₇) is V11() V12() ext-real Element of REAL
sqrt (Sum (sqr c₇)) is V11() V12() ext-real Element of REAL
g * |.c₇.| is V11() V12() ext-real Element of REAL
m is non empty V6() V10() V11() V12() ext-real positive non negative V33() V38() V120() V155() V156() V157() V158() V159() V160() V161() Element of NAT
REAL m is non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered FinSequenceSet of REAL
m -tuples_on REAL is non empty functional FinSequence-membered FinSequenceSet of REAL
n is non empty V6() V10() V11() V12() ext-real positive non negative V33() V38() V120() V155() V156() V157() V158() V159() V160() V161() Element of NAT
REAL n is non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered FinSequenceSet of REAL
n -tuples_on REAL is non empty functional FinSequence-membered FinSequenceSet of REAL
K7((REAL m),(REAL n)) is set
K6(K7((REAL m),(REAL n))) is set
REAL-NS m is non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() V108() V109() V142() L15()
the carrier of (REAL-NS m) is non empty V2() set
REAL-NS n is non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() V108() V109() V142() L15()
the carrier of (REAL-NS n) is non empty V2() set
K7( the carrier of (REAL-NS m), the carrier of (REAL-NS n)) is set
K6(K7( the carrier of (REAL-NS m), the carrier of (REAL-NS n))) is set
f is Relation-like REAL m -defined REAL n -valued Function-like complex-functions-valued ext-real-functions-valued real-functions-valued Element of K6(K7((REAL m),(REAL n)))
Z is Relation-like NAT -defined REAL -valued Function-like V40(m) FinSequence-like complex-yielding V146() V147() Element of REAL m
diff (f,Z) is non empty Relation-like REAL m -defined REAL n -valued Function-like total V30( REAL m, REAL n) complex-functions-valued ext-real-functions-valued real-functions-valued Element of K6(K7((REAL m),(REAL n)))
R_NormSpace_of_BoundedLinearOperators ((REAL-NS m),(REAL-NS n)) is non empty V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() V109() V142() L15()
BoundedLinearOperators ((REAL-NS m),(REAL-NS n)) is non empty Element of K6( the carrier of (R_VectorSpace_of_LinearOperators ((REAL-NS m),(REAL-NS n))))
R_VectorSpace_of_LinearOperators ((REAL-NS m),(REAL-NS n)) is non empty V71() strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V204() RLSStruct
LinearOperators ((REAL-NS m),(REAL-NS n)) is non empty functional Element of K6( the carrier of K279( the carrier of (REAL-NS m),(REAL-NS n)))
K279( the carrier of (REAL-NS m),(REAL-NS n)) is non empty V71() strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V204() RLSStruct
K123( the carrier of (REAL-NS m), the carrier of (REAL-NS n)) is non empty functional M4( the carrier of (REAL-NS m), the carrier of (REAL-NS n))
FuncZero ( the carrier of (REAL-NS m),(REAL-NS n)) is Relation-like the carrier of (REAL-NS m) -defined the carrier of (REAL-NS n) -valued Function-like total V30( the carrier of (REAL-NS m), the carrier of (REAL-NS n)) Element of K123( the carrier of (REAL-NS m), the carrier of (REAL-NS n))
0. (REAL-NS n) is V52( REAL-NS n) Element of the carrier of (REAL-NS n)
the carrier of (REAL-NS m) --> (0. (REAL-NS n)) is non empty Relation-like the carrier of (REAL-NS m) -defined the carrier of (REAL-NS n) -valued Function-like total V30( the carrier of (REAL-NS m), the carrier of (REAL-NS n)) Element of K6(K7( the carrier of (REAL-NS m), the carrier of (REAL-NS n)))
FuncAdd ( the carrier of (REAL-NS m),(REAL-NS n)) is Relation-like K7(K123( the carrier of (REAL-NS m), the carrier of (REAL-NS n)),K123( the carrier of (REAL-NS m), the carrier of (REAL-NS n))) -defined K123( the carrier of (REAL-NS m), the carrier of (REAL-NS n)) -valued Function-like total V30(K7(K123( the carrier of (REAL-NS m), the carrier of (REAL-NS n)),K123( the carrier of (REAL-NS m), the carrier of (REAL-NS n))),K123( the carrier of (REAL-NS m), the carrier of (REAL-NS n))) Element of K6(K7(K7(K123( the carrier of (REAL-NS m), the carrier of (REAL-NS n)),K123( the carrier of (REAL-NS m), the carrier of (REAL-NS n))),K123( the carrier of (REAL-NS m), the carrier of (REAL-NS n))))
K7(K123( the carrier of (REAL-NS m), the carrier of (REAL-NS n)),K123( the carrier of (REAL-NS m), the carrier of (REAL-NS n))) is set
K7(K7(K123( the carrier of (REAL-NS m), the carrier of (REAL-NS n)),K123( the carrier of (REAL-NS m), the carrier of (REAL-NS n))),K123( the carrier of (REAL-NS m), the carrier of (REAL-NS n))) is set
K6(K7(K7(K123( the carrier of (REAL-NS m), the carrier of (REAL-NS n)),K123( the carrier of (REAL-NS m), the carrier of (REAL-NS n))),K123( the carrier of (REAL-NS m), the carrier of (REAL-NS n)))) is set
FuncExtMult ( the carrier of (REAL-NS m),(REAL-NS n)) is Relation-like K7(REAL,K123( the carrier of (REAL-NS m), the carrier of (REAL-NS n))) -defined K123( the carrier of (REAL-NS m), the carrier of (REAL-NS n)) -valued Function-like total V30(K7(REAL,K123( the carrier of (REAL-NS m), the carrier of (REAL-NS n))),K123( the carrier of (REAL-NS m), the carrier of (REAL-NS n))) Element of K6(K7(K7(REAL,K123( the carrier of (REAL-NS m), the carrier of (REAL-NS n))),K123( the carrier of (REAL-NS m), the carrier of (REAL-NS n))))
K7(REAL,K123( the carrier of (REAL-NS m), the carrier of (REAL-NS n))) is V33() set
K7(K7(REAL,K123( the carrier of (REAL-NS m), the carrier of (REAL-NS n))),K123( the carrier of (REAL-NS m), the carrier of (REAL-NS n))) is V33() set
K6(K7(K7(REAL,K123( the carrier of (REAL-NS m), the carrier of (REAL-NS n))),K123( the carrier of (REAL-NS m), the carrier of (REAL-NS n)))) is V33() set
RLSStruct(# K123( the carrier of (REAL-NS m), the carrier of (REAL-NS n)),(FuncZero ( the carrier of (REAL-NS m),(REAL-NS n))),(FuncAdd ( the carrier of (REAL-NS m),(REAL-NS n))),(FuncExtMult ( the carrier of (REAL-NS m),(REAL-NS n))) #) is strict RLSStruct
the carrier of K279( the carrier of (REAL-NS m),(REAL-NS n)) is non empty set
K6( the carrier of K279( the carrier of (REAL-NS m),(REAL-NS n))) is set
K199(K279( the carrier of (REAL-NS m),(REAL-NS n)),(LinearOperators ((REAL-NS m),(REAL-NS n)))) is Element of LinearOperators ((REAL-NS m),(REAL-NS n))
K197(K279( the carrier of (REAL-NS m),(REAL-NS n)),(LinearOperators ((REAL-NS m),(REAL-NS n)))) is Relation-like K7((LinearOperators ((REAL-NS m),(REAL-NS n))),(LinearOperators ((REAL-NS m),(REAL-NS n)))) -defined LinearOperators ((REAL-NS m),(REAL-NS n)) -valued Function-like total V30(K7((LinearOperators ((REAL-NS m),(REAL-NS n))),(LinearOperators ((REAL-NS m),(REAL-NS n)))), LinearOperators ((REAL-NS m),(REAL-NS n))) Element of K6(K7(K7((LinearOperators ((REAL-NS m),(REAL-NS n))),(LinearOperators ((REAL-NS m),(REAL-NS n)))),(LinearOperators ((REAL-NS m),(REAL-NS n)))))
K7((LinearOperators ((REAL-NS m),(REAL-NS n))),(LinearOperators ((REAL-NS m),(REAL-NS n)))) is set
K7(K7((LinearOperators ((REAL-NS m),(REAL-NS n))),(LinearOperators ((REAL-NS m),(REAL-NS n)))),(LinearOperators ((REAL-NS m),(REAL-NS n)))) is set
K6(K7(K7((LinearOperators ((REAL-NS m),(REAL-NS n))),(LinearOperators ((REAL-NS m),(REAL-NS n)))),(LinearOperators ((REAL-NS m),(REAL-NS n))))) is set
K198(K279( the carrier of (REAL-NS m),(REAL-NS n)),(LinearOperators ((REAL-NS m),(REAL-NS n)))) is Relation-like K7(REAL,(LinearOperators ((REAL-NS m),(REAL-NS n)))) -defined LinearOperators ((REAL-NS m),(REAL-NS n)) -valued Function-like total V30(K7(REAL,(LinearOperators ((REAL-NS m),(REAL-NS n)))), LinearOperators ((REAL-NS m),(REAL-NS n))) Element of K6(K7(K7(REAL,(LinearOperators ((REAL-NS m),(REAL-NS n)))),(LinearOperators ((REAL-NS m),(REAL-NS n)))))
K7(REAL,(LinearOperators ((REAL-NS m),(REAL-NS n)))) is V33() set
K7(K7(REAL,(LinearOperators ((REAL-NS m),(REAL-NS n)))),(LinearOperators ((REAL-NS m),(REAL-NS n)))) is V33() set
K6(K7(K7(REAL,(LinearOperators ((REAL-NS m),(REAL-NS n)))),(LinearOperators ((REAL-NS m),(REAL-NS n))))) is V33() set
RLSStruct(# (LinearOperators ((REAL-NS m),(REAL-NS n))),K199(K279( the carrier of (REAL-NS m),(REAL-NS n)),(LinearOperators ((REAL-NS m),(REAL-NS n)))),K197(K279( the carrier of (REAL-NS m),(REAL-NS n)),(LinearOperators ((REAL-NS m),(REAL-NS n)))),K198(K279( the carrier of (REAL-NS m),(REAL-NS n)),(LinearOperators ((REAL-NS m),(REAL-NS n)))) #) is V71() strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RLSStruct
the carrier of (R_VectorSpace_of_LinearOperators ((REAL-NS m),(REAL-NS n))) is non empty set
K6( the carrier of (R_VectorSpace_of_LinearOperators ((REAL-NS m),(REAL-NS n)))) is set
K199((R_VectorSpace_of_LinearOperators ((REAL-NS m),(REAL-NS n))),(BoundedLinearOperators ((REAL-NS m),(REAL-NS n)))) is Element of BoundedLinearOperators ((REAL-NS m),(REAL-NS n))
K197((R_VectorSpace_of_LinearOperators ((REAL-NS m),(REAL-NS n))),(BoundedLinearOperators ((REAL-NS m),(REAL-NS n)))) is Relation-like K7((BoundedLinearOperators ((REAL-NS m),(REAL-NS n))),(BoundedLinearOperators ((REAL-NS m),(REAL-NS n)))) -defined BoundedLinearOperators ((REAL-NS m),(REAL-NS n)) -valued Function-like total V30(K7((BoundedLinearOperators ((REAL-NS m),(REAL-NS n))),(BoundedLinearOperators ((REAL-NS m),(REAL-NS n)))), BoundedLinearOperators ((REAL-NS m),(REAL-NS n))) Element of K6(K7(K7((BoundedLinearOperators ((REAL-NS m),(REAL-NS n))),(BoundedLinearOperators ((REAL-NS m),(REAL-NS n)))),(BoundedLinearOperators ((REAL-NS m),(REAL-NS n)))))
K7((BoundedLinearOperators ((REAL-NS m),(REAL-NS n))),(BoundedLinearOperators ((REAL-NS m),(REAL-NS n)))) is set
K7(K7((BoundedLinearOperators ((REAL-NS m),(REAL-NS n))),(BoundedLinearOperators ((REAL-NS m),(REAL-NS n)))),(BoundedLinearOperators ((REAL-NS m),(REAL-NS n)))) is set
K6(K7(K7((BoundedLinearOperators ((REAL-NS m),(REAL-NS n))),(BoundedLinearOperators ((REAL-NS m),(REAL-NS n)))),(BoundedLinearOperators ((REAL-NS m),(REAL-NS n))))) is set
K198((R_VectorSpace_of_LinearOperators ((REAL-NS m),(REAL-NS n))),(BoundedLinearOperators ((REAL-NS m),(REAL-NS n)))) is Relation-like K7(REAL,(BoundedLinearOperators ((REAL-NS m),(REAL-NS n)))) -defined BoundedLinearOperators ((REAL-NS m),(REAL-NS n)) -valued Function-like total V30(K7(REAL,(BoundedLinearOperators ((REAL-NS m),(REAL-NS n)))), BoundedLinearOperators ((REAL-NS m),(REAL-NS n))) Element of K6(K7(K7(REAL,(BoundedLinearOperators ((REAL-NS m),(REAL-NS n)))),(BoundedLinearOperators ((REAL-NS m),(REAL-NS n)))))
K7(REAL,(BoundedLinearOperators ((REAL-NS m),(REAL-NS n)))) is V33() set
K7(K7(REAL,(BoundedLinearOperators ((REAL-NS m),(REAL-NS n)))),(BoundedLinearOperators ((REAL-NS m),(REAL-NS n)))) is V33() set
K6(K7(K7(REAL,(BoundedLinearOperators ((REAL-NS m),(REAL-NS n)))),(BoundedLinearOperators ((REAL-NS m),(REAL-NS n))))) is V33() set
BoundedLinearOperatorsNorm ((REAL-NS m),(REAL-NS n)) is non empty Relation-like BoundedLinearOperators ((REAL-NS m),(REAL-NS n)) -defined REAL -valued Function-like total V30( BoundedLinearOperators ((REAL-NS m),(REAL-NS n)), REAL ) complex-yielding V146() V147() Element of K6(K7((BoundedLinearOperators ((REAL-NS m),(REAL-NS n))),REAL))
K7((BoundedLinearOperators ((REAL-NS m),(REAL-NS n))),REAL) is V33() complex-yielding V146() V147() set
K6(K7((BoundedLinearOperators ((REAL-NS m),(REAL-NS n))),REAL)) is V33() set
G15((BoundedLinearOperators ((REAL-NS m),(REAL-NS n))),K199((R_VectorSpace_of_LinearOperators ((REAL-NS m),(REAL-NS n))),(BoundedLinearOperators ((REAL-NS m),(REAL-NS n)))),K197((R_VectorSpace_of_LinearOperators ((REAL-NS m),(REAL-NS n))),(BoundedLinearOperators ((REAL-NS m),(REAL-NS n)))),K198((R_VectorSpace_of_LinearOperators ((REAL-NS m),(REAL-NS n))),(BoundedLinearOperators ((REAL-NS m),(REAL-NS n)))),(BoundedLinearOperatorsNorm ((REAL-NS m),(REAL-NS n)))) is V108() L15()
the carrier of (R_NormSpace_of_BoundedLinearOperators ((REAL-NS m),(REAL-NS n))) is non empty set
m is non empty V6() V10() V11() V12() ext-real positive non negative V33() V38() V120() V155() V156() V157() V158() V159() V160() V161() Element of NAT
REAL m is non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered FinSequenceSet of REAL
m -tuples_on REAL is non empty functional FinSequence-membered FinSequenceSet of REAL
n is non empty V6() V10() V11() V12() ext-real positive non negative V33() V38() V120() V155() V156() V157() V158() V159() V160() V161() Element of NAT
REAL n is non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered FinSequenceSet of REAL
n -tuples_on REAL is non empty functional FinSequence-membered FinSequenceSet of REAL
K7((REAL m),(REAL n)) is set
K6(K7((REAL m),(REAL n))) is set
f is Relation-like REAL m -defined REAL n -valued Function-like complex-functions-valued ext-real-functions-valued real-functions-valued Element of K6(K7((REAL m),(REAL n)))
Z is Relation-like NAT -defined REAL -valued Function-like V40(m) FinSequence-like complex-yielding V146() V147() Element of REAL m
diff (f,Z) is non empty Relation-like REAL m -defined REAL n -valued Function-like total V30( REAL m, REAL n) complex-functions-valued ext-real-functions-valued real-functions-valued Element of K6(K7((REAL m),(REAL n)))
REAL-NS m is non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() V108() V109() V142() L15()
the carrier of (REAL-NS m) is non empty V2() set
REAL-NS n is non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() V108() V109() V142() L15()
the carrier of (REAL-NS n) is non empty V2() set
K7( the carrier of (REAL-NS m), the carrier of (REAL-NS n)) is set
K6(K7( the carrier of (REAL-NS m), the carrier of (REAL-NS n))) is set
n is non empty V6() V10() V11() V12() ext-real positive non negative V33() V38() V120() V155() V156() V157() V158() V159() V160() V161() Element of NAT
REAL n is non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered FinSequenceSet of REAL
n -tuples_on REAL is non empty functional FinSequence-membered FinSequenceSet of REAL
m is non empty V6() V10() V11() V12() ext-real positive non negative V33() V38() V120() V155() V156() V157() V158() V159() V160() V161() Element of NAT
REAL m is non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered FinSequenceSet of REAL
m -tuples_on REAL is non empty functional FinSequence-membered FinSequenceSet of REAL
K7((REAL n),(REAL m)) is set
K6(K7((REAL n),(REAL m))) is set
f is Relation-like REAL n -defined REAL m -valued Function-like complex-functions-valued ext-real-functions-valued real-functions-valued Element of K6(K7((REAL n),(REAL m)))
Z is Relation-like REAL n -defined REAL m -valued Function-like complex-functions-valued ext-real-functions-valued real-functions-valued Element of K6(K7((REAL n),(REAL m)))
f + Z is Relation-like REAL n -defined REAL m -valued Function-like complex-functions-valued ext-real-functions-valued real-functions-valued Element of K6(K7((REAL n),(REAL m)))
g is Relation-like NAT -defined REAL -valued Function-like V40(n) FinSequence-like complex-yielding V146() V147() Element of REAL n
diff ((f + Z),g) is non empty Relation-like REAL n -defined REAL m -valued Function-like total V30( REAL n, REAL m) complex-functions-valued ext-real-functions-valued real-functions-valued Element of K6(K7((REAL n),(REAL m)))
diff (f,g) is non empty Relation-like REAL n -defined REAL m -valued Function-like total V30( REAL n, REAL m) complex-functions-valued ext-real-functions-valued real-functions-valued Element of K6(K7((REAL n),(REAL m)))
diff (Z,g) is non empty Relation-like REAL n -defined REAL m -valued Function-like total V30( REAL n, REAL m) complex-functions-valued ext-real-functions-valued real-functions-valued Element of K6(K7((REAL n),(REAL m)))
(diff (f,g)) + (diff (Z,g)) is Relation-like REAL n -defined REAL m -valued Function-like complex-functions-valued ext-real-functions-valued real-functions-valued Element of K6(K7((REAL n),(REAL m)))
REAL-NS n is non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() V108() V109() V142() L15()
the carrier of (REAL-NS n) is non empty V2() set
REAL-NS m is non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() V108() V109() V142() L15()
the carrier of (REAL-NS m) is non empty V2() set
K7( the carrier of (REAL-NS n), the carrier of (REAL-NS m)) is set
K6(K7( the carrier of (REAL-NS n), the carrier of (REAL-NS m))) is set
Z0 is Relation-like the carrier of (REAL-NS n) -defined the carrier of (REAL-NS m) -valued Function-like Element of K6(K7( the carrier of (REAL-NS n), the carrier of (REAL-NS m)))
x is Element of the carrier of (REAL-NS n)
c₇ is Relation-like the carrier of (REAL-NS n) -defined the carrier of (REAL-NS m) -valued Function-like Element of K6(K7( the carrier of (REAL-NS n), the carrier of (REAL-NS m)))
Z0 + c₇ is Relation-like the carrier of (REAL-NS n) -defined the carrier of (REAL-NS m) -valued Function-like Element of K6(K7( the carrier of (REAL-NS n), the carrier of (REAL-NS m)))
diff ((Z0 + c₇),x) is Relation-like Function-like Element of the carrier of (R_NormSpace_of_BoundedLinearOperators ((REAL-NS n),(REAL-NS m)))
R_NormSpace_of_BoundedLinearOperators ((REAL-NS n),(REAL-NS m)) is non empty V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() V109() V142() L15()
BoundedLinearOperators ((REAL-NS n),(REAL-NS m)) is non empty Element of K6( the carrier of (R_VectorSpace_of_LinearOperators ((REAL-NS n),(REAL-NS m))))
R_VectorSpace_of_LinearOperators ((REAL-NS n),(REAL-NS m)) is non empty V71() strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V204() RLSStruct
LinearOperators ((REAL-NS n),(REAL-NS m)) is non empty functional Element of K6( the carrier of K279( the carrier of (REAL-NS n),(REAL-NS m)))
K279( the carrier of (REAL-NS n),(REAL-NS m)) is non empty V71() strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V204() RLSStruct
K123( the carrier of (REAL-NS n), the carrier of (REAL-NS m)) is non empty functional M4( the carrier of (REAL-NS n), the carrier of (REAL-NS m))
FuncZero ( the carrier of (REAL-NS n),(REAL-NS m)) is Relation-like the carrier of (REAL-NS n) -defined the carrier of (REAL-NS m) -valued Function-like total V30( the carrier of (REAL-NS n), the carrier of (REAL-NS m)) Element of K123( the carrier of (REAL-NS n), the carrier of (REAL-NS m))
0. (REAL-NS m) is V52( REAL-NS m) Element of the carrier of (REAL-NS m)
the carrier of (REAL-NS n) --> (0. (REAL-NS m)) is non empty Relation-like the carrier of (REAL-NS n) -defined the carrier of (REAL-NS m) -valued Function-like total V30( the carrier of (REAL-NS n), the carrier of (REAL-NS m)) Element of K6(K7( the carrier of (REAL-NS n), the carrier of (REAL-NS m)))
FuncAdd ( the carrier of (REAL-NS n),(REAL-NS m)) is Relation-like K7(K123( the carrier of (REAL-NS n), the carrier of (REAL-NS m)),K123( the carrier of (REAL-NS n), the carrier of (REAL-NS m))) -defined K123( the carrier of (REAL-NS n), the carrier of (REAL-NS m)) -valued Function-like total V30(K7(K123( the carrier of (REAL-NS n), the carrier of (REAL-NS m)),K123( the carrier of (REAL-NS n), the carrier of (REAL-NS m))),K123( the carrier of (REAL-NS n), the carrier of (REAL-NS m))) Element of K6(K7(K7(K123( the carrier of (REAL-NS n), the carrier of (REAL-NS m)),K123( the carrier of (REAL-NS n), the carrier of (REAL-NS m))),K123( the carrier of (REAL-NS n), the carrier of (REAL-NS m))))
K7(K123( the carrier of (REAL-NS n), the carrier of (REAL-NS m)),K123( the carrier of (REAL-NS n), the carrier of (REAL-NS m))) is set
K7(K7(K123( the carrier of (REAL-NS n), the carrier of (REAL-NS m)),K123( the carrier of (REAL-NS n), the carrier of (REAL-NS m))),K123( the carrier of (REAL-NS n), the carrier of (REAL-NS m))) is set
K6(K7(K7(K123( the carrier of (REAL-NS n), the carrier of (REAL-NS m)),K123( the carrier of (REAL-NS n), the carrier of (REAL-NS m))),K123( the carrier of (REAL-NS n), the carrier of (REAL-NS m)))) is set
FuncExtMult ( the carrier of (REAL-NS n),(REAL-NS m)) is Relation-like K7(REAL,K123( the carrier of (REAL-NS n), the carrier of (REAL-NS m))) -defined K123( the carrier of (REAL-NS n), the carrier of (REAL-NS m)) -valued Function-like total V30(K7(REAL,K123( the carrier of (REAL-NS n), the carrier of (REAL-NS m))),K123( the carrier of (REAL-NS n), the carrier of (REAL-NS m))) Element of K6(K7(K7(REAL,K123( the carrier of (REAL-NS n), the carrier of (REAL-NS m))),K123( the carrier of (REAL-NS n), the carrier of (REAL-NS m))))
K7(REAL,K123( the carrier of (REAL-NS n), the carrier of (REAL-NS m))) is V33() set
K7(K7(REAL,K123( the carrier of (REAL-NS n), the carrier of (REAL-NS m))),K123( the carrier of (REAL-NS n), the carrier of (REAL-NS m))) is V33() set
K6(K7(K7(REAL,K123( the carrier of (REAL-NS n), the carrier of (REAL-NS m))),K123( the carrier of (REAL-NS n), the carrier of (REAL-NS m)))) is V33() set
RLSStruct(# K123( the carrier of (REAL-NS n), the carrier of (REAL-NS m)),(FuncZero ( the carrier of (REAL-NS n),(REAL-NS m))),(FuncAdd ( the carrier of (REAL-NS n),(REAL-NS m))),(FuncExtMult ( the carrier of (REAL-NS n),(REAL-NS m))) #) is strict RLSStruct
the carrier of K279( the carrier of (REAL-NS n),(REAL-NS m)) is non empty set
K6( the carrier of K279( the carrier of (REAL-NS n),(REAL-NS m))) is set
K199(K279( the carrier of (REAL-NS n),(REAL-NS m)),(LinearOperators ((REAL-NS n),(REAL-NS m)))) is Element of LinearOperators ((REAL-NS n),(REAL-NS m))
K197(K279( the carrier of (REAL-NS n),(REAL-NS m)),(LinearOperators ((REAL-NS n),(REAL-NS m)))) is Relation-like K7((LinearOperators ((REAL-NS n),(REAL-NS m))),(LinearOperators ((REAL-NS n),(REAL-NS m)))) -defined LinearOperators ((REAL-NS n),(REAL-NS m)) -valued Function-like total V30(K7((LinearOperators ((REAL-NS n),(REAL-NS m))),(LinearOperators ((REAL-NS n),(REAL-NS m)))), LinearOperators ((REAL-NS n),(REAL-NS m))) Element of K6(K7(K7((LinearOperators ((REAL-NS n),(REAL-NS m))),(LinearOperators ((REAL-NS n),(REAL-NS m)))),(LinearOperators ((REAL-NS n),(REAL-NS m)))))
K7((LinearOperators ((REAL-NS n),(REAL-NS m))),(LinearOperators ((REAL-NS n),(REAL-NS m)))) is set
K7(K7((LinearOperators ((REAL-NS n),(REAL-NS m))),(LinearOperators ((REAL-NS n),(REAL-NS m)))),(LinearOperators ((REAL-NS n),(REAL-NS m)))) is set
K6(K7(K7((LinearOperators ((REAL-NS n),(REAL-NS m))),(LinearOperators ((REAL-NS n),(REAL-NS m)))),(LinearOperators ((REAL-NS n),(REAL-NS m))))) is set
K198(K279( the carrier of (REAL-NS n),(REAL-NS m)),(LinearOperators ((REAL-NS n),(REAL-NS m)))) is Relation-like K7(REAL,(LinearOperators ((REAL-NS n),(REAL-NS m)))) -defined LinearOperators ((REAL-NS n),(REAL-NS m)) -valued Function-like total V30(K7(REAL,(LinearOperators ((REAL-NS n),(REAL-NS m)))), LinearOperators ((REAL-NS n),(REAL-NS m))) Element of K6(K7(K7(REAL,(LinearOperators ((REAL-NS n),(REAL-NS m)))),(LinearOperators ((REAL-NS n),(REAL-NS m)))))
K7(REAL,(LinearOperators ((REAL-NS n),(REAL-NS m)))) is V33() set
K7(K7(REAL,(LinearOperators ((REAL-NS n),(REAL-NS m)))),(LinearOperators ((REAL-NS n),(REAL-NS m)))) is V33() set
K6(K7(K7(REAL,(LinearOperators ((REAL-NS n),(REAL-NS m)))),(LinearOperators ((REAL-NS n),(REAL-NS m))))) is V33() set
RLSStruct(# (LinearOperators ((REAL-NS n),(REAL-NS m))),K199(K279( the carrier of (REAL-NS n),(REAL-NS m)),(LinearOperators ((REAL-NS n),(REAL-NS m)))),K197(K279( the carrier of (REAL-NS n),(REAL-NS m)),(LinearOperators ((REAL-NS n),(REAL-NS m)))),K198(K279( the carrier of (REAL-NS n),(REAL-NS m)),(LinearOperators ((REAL-NS n),(REAL-NS m)))) #) is V71() strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RLSStruct
the carrier of (R_VectorSpace_of_LinearOperators ((REAL-NS n),(REAL-NS m))) is non empty set
K6( the carrier of (R_VectorSpace_of_LinearOperators ((REAL-NS n),(REAL-NS m)))) is set
K199((R_VectorSpace_of_LinearOperators ((REAL-NS n),(REAL-NS m))),(BoundedLinearOperators ((REAL-NS n),(REAL-NS m)))) is Element of BoundedLinearOperators ((REAL-NS n),(REAL-NS m))
K197((R_VectorSpace_of_LinearOperators ((REAL-NS n),(REAL-NS m))),(BoundedLinearOperators ((REAL-NS n),(REAL-NS m)))) is Relation-like K7((BoundedLinearOperators ((REAL-NS n),(REAL-NS m))),(BoundedLinearOperators ((REAL-NS n),(REAL-NS m)))) -defined BoundedLinearOperators ((REAL-NS n),(REAL-NS m)) -valued Function-like total V30(K7((BoundedLinearOperators ((REAL-NS n),(REAL-NS m))),(BoundedLinearOperators ((REAL-NS n),(REAL-NS m)))), BoundedLinearOperators ((REAL-NS n),(REAL-NS m))) Element of K6(K7(K7((BoundedLinearOperators ((REAL-NS n),(REAL-NS m))),(BoundedLinearOperators ((REAL-NS n),(REAL-NS m)))),(BoundedLinearOperators ((REAL-NS n),(REAL-NS m)))))
K7((BoundedLinearOperators ((REAL-NS n),(REAL-NS m))),(BoundedLinearOperators ((REAL-NS n),(REAL-NS m)))) is set
K7(K7((BoundedLinearOperators ((REAL-NS n),(REAL-NS m))),(BoundedLinearOperators ((REAL-NS n),(REAL-NS m)))),(BoundedLinearOperators ((REAL-NS n),(REAL-NS m)))) is set
K6(K7(K7((BoundedLinearOperators ((REAL-NS n),(REAL-NS m))),(BoundedLinearOperators ((REAL-NS n),(REAL-NS m)))),(BoundedLinearOperators ((REAL-NS n),(REAL-NS m))))) is set
K198((R_VectorSpace_of_LinearOperators ((REAL-NS n),(REAL-NS m))),(BoundedLinearOperators ((REAL-NS n),(REAL-NS m)))) is Relation-like K7(REAL,(BoundedLinearOperators ((REAL-NS n),(REAL-NS m)))) -defined BoundedLinearOperators ((REAL-NS n),(REAL-NS m)) -valued Function-like total V30(K7(REAL,(BoundedLinearOperators ((REAL-NS n),(REAL-NS m)))), BoundedLinearOperators ((REAL-NS n),(REAL-NS m))) Element of K6(K7(K7(REAL,(BoundedLinearOperators ((REAL-NS n),(REAL-NS m)))),(BoundedLinearOperators ((REAL-NS n),(REAL-NS m)))))
K7(REAL,(BoundedLinearOperators ((REAL-NS n),(REAL-NS m)))) is V33() set
K7(K7(REAL,(BoundedLinearOperators ((REAL-NS n),(REAL-NS m)))),(BoundedLinearOperators ((REAL-NS n),(REAL-NS m)))) is V33() set
K6(K7(K7(REAL,(BoundedLinearOperators ((REAL-NS n),(REAL-NS m)))),(BoundedLinearOperators ((REAL-NS n),(REAL-NS m))))) is V33() set
BoundedLinearOperatorsNorm ((REAL-NS n),(REAL-NS m)) is non empty Relation-like BoundedLinearOperators ((REAL-NS n),(REAL-NS m)) -defined REAL -valued Function-like total V30( BoundedLinearOperators ((REAL-NS n),(REAL-NS m)), REAL ) complex-yielding V146() V147() Element of K6(K7((BoundedLinearOperators ((REAL-NS n),(REAL-NS m))),REAL))
K7((BoundedLinearOperators ((REAL-NS n),(REAL-NS m))),REAL) is V33() complex-yielding V146() V147() set
K6(K7((BoundedLinearOperators ((REAL-NS n),(REAL-NS m))),REAL)) is V33() set
G15((BoundedLinearOperators ((REAL-NS n),(REAL-NS m))),K199((R_VectorSpace_of_LinearOperators ((REAL-NS n),(REAL-NS m))),(BoundedLinearOperators ((REAL-NS n),(REAL-NS m)))),K197((R_VectorSpace_of_LinearOperators ((REAL-NS n),(REAL-NS m))),(BoundedLinearOperators ((REAL-NS n),(REAL-NS m)))),K198((R_VectorSpace_of_LinearOperators ((REAL-NS n),(REAL-NS m))),(BoundedLinearOperators ((REAL-NS n),(REAL-NS m)))),(BoundedLinearOperatorsNorm ((REAL-NS n),(REAL-NS m)))) is V108() L15()
the carrier of (R_NormSpace_of_BoundedLinearOperators ((REAL-NS n),(REAL-NS m))) is non empty set
diff (Z0,x) is Relation-like Function-like Element of the carrier of (R_NormSpace_of_BoundedLinearOperators ((REAL-NS n),(REAL-NS m)))
diff (c₇,x) is Relation-like Function-like Element of the carrier of (R_NormSpace_of_BoundedLinearOperators ((REAL-NS n),(REAL-NS m)))
(diff (Z0,x)) + (diff (c₇,x)) is Relation-like Function-like Element of the carrier of (R_NormSpace_of_BoundedLinearOperators ((REAL-NS n),(REAL-NS m)))
n is non empty V6() V10() V11() V12() ext-real positive non negative V33() V38() V120() V155() V156() V157() V158() V159() V160() V161() Element of NAT
REAL n is non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered FinSequenceSet of REAL
n -tuples_on REAL is non empty functional FinSequence-membered FinSequenceSet of REAL
m is non empty V6() V10() V11() V12() ext-real positive non negative V33() V38() V120() V155() V156() V157() V158() V159() V160() V161() Element of NAT
REAL m is non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered FinSequenceSet of REAL
m -tuples_on REAL is non empty functional FinSequence-membered FinSequenceSet of REAL
K7((REAL n),(REAL m)) is set
K6(K7((REAL n),(REAL m))) is set
f is Relation-like REAL n -defined REAL m -valued Function-like complex-functions-valued ext-real-functions-valued real-functions-valued Element of K6(K7((REAL n),(REAL m)))
Z is Relation-like REAL n -defined REAL m -valued Function-like complex-functions-valued ext-real-functions-valued real-functions-valued Element of K6(K7((REAL n),(REAL m)))
f - Z is Relation-like REAL n -defined REAL m -valued Function-like complex-functions-valued ext-real-functions-valued real-functions-valued Element of K6(K7((REAL n),(REAL m)))
g is Relation-like NAT -defined REAL -valued Function-like V40(n) FinSequence-like complex-yielding V146() V147() Element of REAL n
diff ((f - Z),g) is non empty Relation-like REAL n -defined REAL m -valued Function-like total V30( REAL n, REAL m) complex-functions-valued ext-real-functions-valued real-functions-valued Element of K6(K7((REAL n),(REAL m)))
diff (f,g) is non empty Relation-like REAL n -defined REAL m -valued Function-like total V30( REAL n, REAL m) complex-functions-valued ext-real-functions-valued real-functions-valued Element of K6(K7((REAL n),(REAL m)))
diff (Z,g) is non empty Relation-like REAL n -defined REAL m -valued Function-like total V30( REAL n, REAL m) complex-functions-valued ext-real-functions-valued real-functions-valued Element of K6(K7((REAL n),(REAL m)))
(diff (f,g)) - (diff (Z,g)) is Relation-like REAL n -defined REAL m -valued Function-like complex-functions-valued ext-real-functions-valued real-functions-valued Element of K6(K7((REAL n),(REAL m)))
REAL-NS n is non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() V108() V109() V142() L15()
the carrier of (REAL-NS n) is non empty V2() set
REAL-NS m is non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() V108() V109() V142() L15()
the carrier of (REAL-NS m) is non empty V2() set
K7( the carrier of (REAL-NS n), the carrier of (REAL-NS m)) is set
K6(K7( the carrier of (REAL-NS n), the carrier of (REAL-NS m))) is set
Z0 is Relation-like the carrier of (REAL-NS n) -defined the carrier of (REAL-NS m) -valued Function-like Element of K6(K7( the carrier of (REAL-NS n), the carrier of (REAL-NS m)))
x is Element of the carrier of (REAL-NS n)
c₇ is Relation-like the carrier of (REAL-NS n) -defined the carrier of (REAL-NS m) -valued Function-like Element of K6(K7( the carrier of (REAL-NS n), the carrier of (REAL-NS m)))
Z0 - c₇ is Relation-like the carrier of (REAL-NS n) -defined the carrier of (REAL-NS m) -valued Function-like Element of K6(K7( the carrier of (REAL-NS n), the carrier of (REAL-NS m)))
diff ((Z0 - c₇),x) is Relation-like Function-like Element of the carrier of (R_NormSpace_of_BoundedLinearOperators ((REAL-NS n),(REAL-NS m)))
R_NormSpace_of_BoundedLinearOperators ((REAL-NS n),(REAL-NS m)) is non empty V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() V109() V142() L15()
BoundedLinearOperators ((REAL-NS n),(REAL-NS m)) is non empty Element of K6( the carrier of (R_VectorSpace_of_LinearOperators ((REAL-NS n),(REAL-NS m))))
R_VectorSpace_of_LinearOperators ((REAL-NS n),(REAL-NS m)) is non empty V71() strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V204() RLSStruct
LinearOperators ((REAL-NS n),(REAL-NS m)) is non empty functional Element of K6( the carrier of K279( the carrier of (REAL-NS n),(REAL-NS m)))
K279( the carrier of (REAL-NS n),(REAL-NS m)) is non empty V71() strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V204() RLSStruct
K123( the carrier of (REAL-NS n), the carrier of (REAL-NS m)) is non empty functional M4( the carrier of (REAL-NS n), the carrier of (REAL-NS m))
FuncZero ( the carrier of (REAL-NS n),(REAL-NS m)) is Relation-like the carrier of (REAL-NS n) -defined the carrier of (REAL-NS m) -valued Function-like total V30( the carrier of (REAL-NS n), the carrier of (REAL-NS m)) Element of K123( the carrier of (REAL-NS n), the carrier of (REAL-NS m))
0. (REAL-NS m) is V52( REAL-NS m) Element of the carrier of (REAL-NS m)
the carrier of (REAL-NS n) --> (0. (REAL-NS m)) is non empty Relation-like the carrier of (REAL-NS n) -defined the carrier of (REAL-NS m) -valued Function-like total V30( the carrier of (REAL-NS n), the carrier of (REAL-NS m)) Element of K6(K7( the carrier of (REAL-NS n), the carrier of (REAL-NS m)))
FuncAdd ( the carrier of (REAL-NS n),(REAL-NS m)) is Relation-like K7(K123( the carrier of (REAL-NS n), the carrier of (REAL-NS m)),K123( the carrier of (REAL-NS n), the carrier of (REAL-NS m))) -defined K123( the carrier of (REAL-NS n), the carrier of (REAL-NS m)) -valued Function-like total V30(K7(K123( the carrier of (REAL-NS n), the carrier of (REAL-NS m)),K123( the carrier of (REAL-NS n), the carrier of (REAL-NS m))),K123( the carrier of (REAL-NS n), the carrier of (REAL-NS m))) Element of K6(K7(K7(K123( the carrier of (REAL-NS n), the carrier of (REAL-NS m)),K123( the carrier of (REAL-NS n), the carrier of (REAL-NS m))),K123( the carrier of (REAL-NS n), the carrier of (REAL-NS m))))
K7(K123( the carrier of (REAL-NS n), the carrier of (REAL-NS m)),K123( the carrier of (REAL-NS n), the carrier of (REAL-NS m))) is set
K7(K7(K123( the carrier of (REAL-NS n), the carrier of (REAL-NS m)),K123( the carrier of (REAL-NS n), the carrier of (REAL-NS m))),K123( the carrier of (REAL-NS n), the carrier of (REAL-NS m))) is set
K6(K7(K7(K123( the carrier of (REAL-NS n), the carrier of (REAL-NS m)),K123( the carrier of (REAL-NS n), the carrier of (REAL-NS m))),K123( the carrier of (REAL-NS n), the carrier of (REAL-NS m)))) is set
FuncExtMult ( the carrier of (REAL-NS n),(REAL-NS m)) is Relation-like K7(REAL,K123( the carrier of (REAL-NS n), the carrier of (REAL-NS m))) -defined K123( the carrier of (REAL-NS n), the carrier of (REAL-NS m)) -valued Function-like total V30(K7(REAL,K123( the carrier of (REAL-NS n), the carrier of (REAL-NS m))),K123( the carrier of (REAL-NS n), the carrier of (REAL-NS m))) Element of K6(K7(K7(REAL,K123( the carrier of (REAL-NS n), the carrier of (REAL-NS m))),K123( the carrier of (REAL-NS n), the carrier of (REAL-NS m))))
K7(REAL,K123( the carrier of (REAL-NS n), the carrier of (REAL-NS m))) is V33() set
K7(K7(REAL,K123( the carrier of (REAL-NS n), the carrier of (REAL-NS m))),K123( the carrier of (REAL-NS n), the carrier of (REAL-NS m))) is V33() set
K6(K7(K7(REAL,K123( the carrier of (REAL-NS n), the carrier of (REAL-NS m))),K123( the carrier of (REAL-NS n), the carrier of (REAL-NS m)))) is V33() set
RLSStruct(# K123( the carrier of (REAL-NS n), the carrier of (REAL-NS m)),(FuncZero ( the carrier of (REAL-NS n),(REAL-NS m))),(FuncAdd ( the carrier of (REAL-NS n),(REAL-NS m))),(FuncExtMult ( the carrier of (REAL-NS n),(REAL-NS m))) #) is strict RLSStruct
the carrier of K279( the carrier of (REAL-NS n),(REAL-NS m)) is non empty set
K6( the carrier of K279( the carrier of (REAL-NS n),(REAL-NS m))) is set
K199(K279( the carrier of (REAL-NS n),(REAL-NS m)),(LinearOperators ((REAL-NS n),(REAL-NS m)))) is Element of LinearOperators ((REAL-NS n),(REAL-NS m))
K197(K279( the carrier of (REAL-NS n),(REAL-NS m)),(LinearOperators ((REAL-NS n),(REAL-NS m)))) is Relation-like K7((LinearOperators ((REAL-NS n),(REAL-NS m))),(LinearOperators ((REAL-NS n),(REAL-NS m)))) -defined LinearOperators ((REAL-NS n),(REAL-NS m)) -valued Function-like total V30(K7((LinearOperators ((REAL-NS n),(REAL-NS m))),(LinearOperators ((REAL-NS n),(REAL-NS m)))), LinearOperators ((REAL-NS n),(REAL-NS m))) Element of K6(K7(K7((LinearOperators ((REAL-NS n),(REAL-NS m))),(LinearOperators ((REAL-NS n),(REAL-NS m)))),(LinearOperators ((REAL-NS n),(REAL-NS m)))))
K7((LinearOperators ((REAL-NS n),(REAL-NS m))),(LinearOperators ((REAL-NS n),(REAL-NS m)))) is set
K7(K7((LinearOperators ((REAL-NS n),(REAL-NS m))),(LinearOperators ((REAL-NS n),(REAL-NS m)))),(LinearOperators ((REAL-NS n),(REAL-NS m)))) is set
K6(K7(K7((LinearOperators ((REAL-NS n),(REAL-NS m))),(LinearOperators ((REAL-NS n),(REAL-NS m)))),(LinearOperators ((REAL-NS n),(REAL-NS m))))) is set
K198(K279( the carrier of (REAL-NS n),(REAL-NS m)),(LinearOperators ((REAL-NS n),(REAL-NS m)))) is Relation-like K7(REAL,(LinearOperators ((REAL-NS n),(REAL-NS m)))) -defined LinearOperators ((REAL-NS n),(REAL-NS m)) -valued Function-like total V30(K7(REAL,(LinearOperators ((REAL-NS n),(REAL-NS m)))), LinearOperators ((REAL-NS n),(REAL-NS m))) Element of K6(K7(K7(REAL,(LinearOperators ((REAL-NS n),(REAL-NS m)))),(LinearOperators ((REAL-NS n),(REAL-NS m)))))
K7(REAL,(LinearOperators ((REAL-NS n),(REAL-NS m)))) is V33() set
K7(K7(REAL,(LinearOperators ((REAL-NS n),(REAL-NS m)))),(LinearOperators ((REAL-NS n),(REAL-NS m)))) is V33() set
K6(K7(K7(REAL,(LinearOperators ((REAL-NS n),(REAL-NS m)))),(LinearOperators ((REAL-NS n),(REAL-NS m))))) is V33() set
RLSStruct(# (LinearOperators ((REAL-NS n),(REAL-NS m))),K199(K279( the carrier of (REAL-NS n),(REAL-NS m)),(LinearOperators ((REAL-NS n),(REAL-NS m)))),K197(K279( the carrier of (REAL-NS n),(REAL-NS m)),(LinearOperators ((REAL-NS n),(REAL-NS m)))),K198(K279( the carrier of (REAL-NS n),(REAL-NS m)),(LinearOperators ((REAL-NS n),(REAL-NS m)))) #) is V71() strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RLSStruct
the carrier of (R_VectorSpace_of_LinearOperators ((REAL-NS n),(REAL-NS m))) is non empty set
K6( the carrier of (R_VectorSpace_of_LinearOperators ((REAL-NS n),(REAL-NS m)))) is set
K199((R_VectorSpace_of_LinearOperators ((REAL-NS n),(REAL-NS m))),(BoundedLinearOperators ((REAL-NS n),(REAL-NS m)))) is Element of BoundedLinearOperators ((REAL-NS n),(REAL-NS m))
K197((R_VectorSpace_of_LinearOperators ((REAL-NS n),(REAL-NS m))),(BoundedLinearOperators ((REAL-NS n),(REAL-NS m)))) is Relation-like K7((BoundedLinearOperators ((REAL-NS n),(REAL-NS m))),(BoundedLinearOperators ((REAL-NS n),(REAL-NS m)))) -defined BoundedLinearOperators ((REAL-NS n),(REAL-NS m)) -valued Function-like total V30(K7((BoundedLinearOperators ((REAL-NS n),(REAL-NS m))),(BoundedLinearOperators ((REAL-NS n),(REAL-NS m)))), BoundedLinearOperators ((REAL-NS n),(REAL-NS m))) Element of K6(K7(K7((BoundedLinearOperators ((REAL-NS n),(REAL-NS m))),(BoundedLinearOperators ((REAL-NS n),(REAL-NS m)))),(BoundedLinearOperators ((REAL-NS n),(REAL-NS m)))))
K7((BoundedLinearOperators ((REAL-NS n),(REAL-NS m))),(BoundedLinearOperators ((REAL-NS n),(REAL-NS m)))) is set
K7(K7((BoundedLinearOperators ((REAL-NS n),(REAL-NS m))),(BoundedLinearOperators ((REAL-NS n),(REAL-NS m)))),(BoundedLinearOperators ((REAL-NS n),(REAL-NS m)))) is set
K6(K7(K7((BoundedLinearOperators ((REAL-NS n),(REAL-NS m))),(BoundedLinearOperators ((REAL-NS n),(REAL-NS m)))),(BoundedLinearOperators ((REAL-NS n),(REAL-NS m))))) is set
K198((R_VectorSpace_of_LinearOperators ((REAL-NS n),(REAL-NS m))),(BoundedLinearOperators ((REAL-NS n),(REAL-NS m)))) is Relation-like K7(REAL,(BoundedLinearOperators ((REAL-NS n),(REAL-NS m)))) -defined BoundedLinearOperators ((REAL-NS n),(REAL-NS m)) -valued Function-like total V30(K7(REAL,(BoundedLinearOperators ((REAL-NS n),(REAL-NS m)))), BoundedLinearOperators ((REAL-NS n),(REAL-NS m))) Element of K6(K7(K7(REAL,(BoundedLinearOperators ((REAL-NS n),(REAL-NS m)))),(BoundedLinearOperators ((REAL-NS n),(REAL-NS m)))))
K7(REAL,(BoundedLinearOperators ((REAL-NS n),(REAL-NS m)))) is V33() set
K7(K7(REAL,(BoundedLinearOperators ((REAL-NS n),(REAL-NS m)))),(BoundedLinearOperators ((REAL-NS n),(REAL-NS m)))) is V33() set
K6(K7(K7(REAL,(BoundedLinearOperators ((REAL-NS n),(REAL-NS m)))),(BoundedLinearOperators ((REAL-NS n),(REAL-NS m))))) is V33() set
BoundedLinearOperatorsNorm ((REAL-NS n),(REAL-NS m)) is non empty Relation-like BoundedLinearOperators ((REAL-NS n),(REAL-NS m)) -defined REAL -valued Function-like total V30( BoundedLinearOperators ((REAL-NS n),(REAL-NS m)), REAL ) complex-yielding V146() V147() Element of K6(K7((BoundedLinearOperators ((REAL-NS n),(REAL-NS m))),REAL))
K7((BoundedLinearOperators ((REAL-NS n),(REAL-NS m))),REAL) is V33() complex-yielding V146() V147() set
K6(K7((BoundedLinearOperators ((REAL-NS n),(REAL-NS m))),REAL)) is V33() set
G15((BoundedLinearOperators ((REAL-NS n),(REAL-NS m))),K199((R_VectorSpace_of_LinearOperators ((REAL-NS n),(REAL-NS m))),(BoundedLinearOperators ((REAL-NS n),(REAL-NS m)))),K197((R_VectorSpace_of_LinearOperators ((REAL-NS n),(REAL-NS m))),(BoundedLinearOperators ((REAL-NS n),(REAL-NS m)))),K198((R_VectorSpace_of_LinearOperators ((REAL-NS n),(REAL-NS m))),(BoundedLinearOperators ((REAL-NS n),(REAL-NS m)))),(BoundedLinearOperatorsNorm ((REAL-NS n),(REAL-NS m)))) is V108() L15()
the carrier of (R_NormSpace_of_BoundedLinearOperators ((REAL-NS n),(REAL-NS m))) is non empty set
diff (Z0,x) is Relation-like Function-like Element of the carrier of (R_NormSpace_of_BoundedLinearOperators ((REAL-NS n),(REAL-NS m)))
diff (c₇,x) is Relation-like Function-like Element of the carrier of (R_NormSpace_of_BoundedLinearOperators ((REAL-NS n),(REAL-NS m)))
(diff (Z0,x)) - (diff (c₇,x)) is Relation-like Function-like Element of the carrier of (R_NormSpace_of_BoundedLinearOperators ((REAL-NS n),(REAL-NS m)))
n is non empty V6() V10() V11() V12() ext-real positive non negative V33() V38() V120() V155() V156() V157() V158() V159() V160() V161() Element of NAT
REAL n is non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered FinSequenceSet of REAL
n -tuples_on REAL is non empty functional FinSequence-membered FinSequenceSet of REAL
m is non empty V6() V10() V11() V12() ext-real positive non negative V33() V38() V120() V155() V156() V157() V158() V159() V160() V161() Element of NAT
REAL m is non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered FinSequenceSet of REAL
m -tuples_on REAL is non empty functional FinSequence-membered FinSequenceSet of REAL
K7((REAL n),(REAL m)) is set
K6(K7((REAL n),(REAL m))) is set
f is Relation-like REAL n -defined REAL m -valued Function-like complex-functions-valued ext-real-functions-valued real-functions-valued Element of K6(K7((REAL n),(REAL m)))
Z is Relation-like NAT -defined REAL -valued Function-like V40(n) FinSequence-like complex-yielding V146() V147() Element of REAL n
diff (f,Z) is non empty Relation-like REAL n -defined REAL m -valued Function-like total V30( REAL n, REAL m) complex-functions-valued ext-real-functions-valued real-functions-valued Element of K6(K7((REAL n),(REAL m)))
g is V11() V12() ext-real Element of REAL
g (#) f is Relation-like REAL n -defined REAL m -valued Function-like complex-functions-valued ext-real-functions-valued real-functions-valued Element of K6(K7((REAL n),(REAL m)))
diff ((g (#) f),Z) is non empty Relation-like REAL n -defined REAL m -valued Function-like total V30( REAL n, REAL m) complex-functions-valued ext-real-functions-valued real-functions-valued Element of K6(K7((REAL n),(REAL m)))
g (#) (diff (f,Z)) is Relation-like REAL n -defined REAL m -valued Function-like complex-functions-valued ext-real-functions-valued real-functions-valued Element of K6(K7((REAL n),(REAL m)))
REAL-NS n is non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() V108() V109() V142() L15()
the carrier of (REAL-NS n) is non empty V2() set
REAL-NS m is non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() V108() V109() V142() L15()
the carrier of (REAL-NS m) is non empty V2() set
K7( the carrier of (REAL-NS n), the carrier of (REAL-NS m)) is set
K6(K7( the carrier of (REAL-NS n), the carrier of (REAL-NS m))) is set
Z0 is Relation-like the carrier of (REAL-NS n) -defined the carrier of (REAL-NS m) -valued Function-like Element of K6(K7( the carrier of (REAL-NS n), the carrier of (REAL-NS m)))
c₇ is Element of the carrier of (REAL-NS n)
g (#) Z0 is Relation-like the carrier of (REAL-NS n) -defined the carrier of (REAL-NS m) -valued Function-like Element of K6(K7( the carrier of (REAL-NS n), the carrier of (REAL-NS m)))
diff ((g (#) Z0),c₇) is Relation-like Function-like Element of the carrier of (R_NormSpace_of_BoundedLinearOperators ((REAL-NS n),(REAL-NS m)))
R_NormSpace_of_BoundedLinearOperators ((REAL-NS n),(REAL-NS m)) is non empty V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() V109() V142() L15()
BoundedLinearOperators ((REAL-NS n),(REAL-NS m)) is non empty Element of K6( the carrier of (R_VectorSpace_of_LinearOperators ((REAL-NS n),(REAL-NS m))))
R_VectorSpace_of_LinearOperators ((REAL-NS n),(REAL-NS m)) is non empty V71() strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V204() RLSStruct
LinearOperators ((REAL-NS n),(REAL-NS m)) is non empty functional Element of K6( the carrier of K279( the carrier of (REAL-NS n),(REAL-NS m)))
K279( the carrier of (REAL-NS n),(REAL-NS m)) is non empty V71() strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V204() RLSStruct
K123( the carrier of (REAL-NS n), the carrier of (REAL-NS m)) is non empty functional M4( the carrier of (REAL-NS n), the carrier of (REAL-NS m))
FuncZero ( the carrier of (REAL-NS n),(REAL-NS m)) is Relation-like the carrier of (REAL-NS n) -defined the carrier of (REAL-NS m) -valued Function-like total V30( the carrier of (REAL-NS n), the carrier of (REAL-NS m)) Element of K123( the carrier of (REAL-NS n), the carrier of (REAL-NS m))
0. (REAL-NS m) is V52( REAL-NS m) Element of the carrier of (REAL-NS m)
the carrier of (REAL-NS n) --> (0. (REAL-NS m)) is non empty Relation-like the carrier of (REAL-NS n) -defined the carrier of (REAL-NS m) -valued Function-like total V30( the carrier of (REAL-NS n), the carrier of (REAL-NS m)) Element of K6(K7( the carrier of (REAL-NS n), the carrier of (REAL-NS m)))
FuncAdd ( the carrier of (REAL-NS n),(REAL-NS m)) is Relation-like K7(K123( the carrier of (REAL-NS n), the carrier of (REAL-NS m)),K123( the carrier of (REAL-NS n), the carrier of (REAL-NS m))) -defined K123( the carrier of (REAL-NS n), the carrier of (REAL-NS m)) -valued Function-like total V30(K7(K123( the carrier of (REAL-NS n), the carrier of (REAL-NS m)),K123( the carrier of (REAL-NS n), the carrier of (REAL-NS m))),K123( the carrier of (REAL-NS n), the carrier of (REAL-NS m))) Element of K6(K7(K7(K123( the carrier of (REAL-NS n), the carrier of (REAL-NS m)),K123( the carrier of (REAL-NS n), the carrier of (REAL-NS m))),K123( the carrier of (REAL-NS n), the carrier of (REAL-NS m))))
K7(K123( the carrier of (REAL-NS n), the carrier of (REAL-NS m)),K123( the carrier of (REAL-NS n), the carrier of (REAL-NS m))) is set
K7(K7(K123( the carrier of (REAL-NS n), the carrier of (REAL-NS m)),K123( the carrier of (REAL-NS n), the carrier of (REAL-NS m))),K123( the carrier of (REAL-NS n), the carrier of (REAL-NS m))) is set
K6(K7(K7(K123( the carrier of (REAL-NS n), the carrier of (REAL-NS m)),K123( the carrier of (REAL-NS n), the carrier of (REAL-NS m))),K123( the carrier of (REAL-NS n), the carrier of (REAL-NS m)))) is set
FuncExtMult ( the carrier of (REAL-NS n),(REAL-NS m)) is Relation-like K7(REAL,K123( the carrier of (REAL-NS n), the carrier of (REAL-NS m))) -defined K123( the carrier of (REAL-NS n), the carrier of (REAL-NS m)) -valued Function-like total V30(K7(REAL,K123( the carrier of (REAL-NS n), the carrier of (REAL-NS m))),K123( the carrier of (REAL-NS n), the carrier of (REAL-NS m))) Element of K6(K7(K7(REAL,K123( the carrier of (REAL-NS n), the carrier of (REAL-NS m))),K123( the carrier of (REAL-NS n), the carrier of (REAL-NS m))))
K7(REAL,K123( the carrier of (REAL-NS n), the carrier of (REAL-NS m))) is V33() set
K7(K7(REAL,K123( the carrier of (REAL-NS n), the carrier of (REAL-NS m))),K123( the carrier of (REAL-NS n), the carrier of (REAL-NS m))) is V33() set
K6(K7(K7(REAL,K123( the carrier of (REAL-NS n), the carrier of (REAL-NS m))),K123( the carrier of (REAL-NS n), the carrier of (REAL-NS m)))) is V33() set
RLSStruct(# K123( the carrier of (REAL-NS n), the carrier of (REAL-NS m)),(FuncZero ( the carrier of (REAL-NS n),(REAL-NS m))),(FuncAdd ( the carrier of (REAL-NS n),(REAL-NS m))),(FuncExtMult ( the carrier of (REAL-NS n),(REAL-NS m))) #) is strict RLSStruct
the carrier of K279( the carrier of (REAL-NS n),(REAL-NS m)) is non empty set
K6( the carrier of K279( the carrier of (REAL-NS n),(REAL-NS m))) is set
K199(K279( the carrier of (REAL-NS n),(REAL-NS m)),(LinearOperators ((REAL-NS n),(REAL-NS m)))) is Element of LinearOperators ((REAL-NS n),(REAL-NS m))
K197(K279( the carrier of (REAL-NS n),(REAL-NS m)),(LinearOperators ((REAL-NS n),(REAL-NS m)))) is Relation-like K7((LinearOperators ((REAL-NS n),(REAL-NS m))),(LinearOperators ((REAL-NS n),(REAL-NS m)))) -defined LinearOperators ((REAL-NS n),(REAL-NS m)) -valued Function-like total V30(K7((LinearOperators ((REAL-NS n),(REAL-NS m))),(LinearOperators ((REAL-NS n),(REAL-NS m)))), LinearOperators ((REAL-NS n),(REAL-NS m))) Element of K6(K7(K7((LinearOperators ((REAL-NS n),(REAL-NS m))),(LinearOperators ((REAL-NS n),(REAL-NS m)))),(LinearOperators ((REAL-NS n),(REAL-NS m)))))
K7((LinearOperators ((REAL-NS n),(REAL-NS m))),(LinearOperators ((REAL-NS n),(REAL-NS m)))) is set
K7(K7((LinearOperators ((REAL-NS n),(REAL-NS m))),(LinearOperators ((REAL-NS n),(REAL-NS m)))),(LinearOperators ((REAL-NS n),(REAL-NS m)))) is set
K6(K7(K7((LinearOperators ((REAL-NS n),(REAL-NS m))),(LinearOperators ((REAL-NS n),(REAL-NS m)))),(LinearOperators ((REAL-NS n),(REAL-NS m))))) is set
K198(K279( the carrier of (REAL-NS n),(REAL-NS m)),(LinearOperators ((REAL-NS n),(REAL-NS m)))) is Relation-like K7(REAL,(LinearOperators ((REAL-NS n),(REAL-NS m)))) -defined LinearOperators ((REAL-NS n),(REAL-NS m)) -valued Function-like total V30(K7(REAL,(LinearOperators ((REAL-NS n),(REAL-NS m)))), LinearOperators ((REAL-NS n),(REAL-NS m))) Element of K6(K7(K7(REAL,(LinearOperators ((REAL-NS n),(REAL-NS m)))),(LinearOperators ((REAL-NS n),(REAL-NS m)))))
K7(REAL,(LinearOperators ((REAL-NS n),(REAL-NS m)))) is V33() set
K7(K7(REAL,(LinearOperators ((REAL-NS n),(REAL-NS m)))),(LinearOperators ((REAL-NS n),(REAL-NS m)))) is V33() set
K6(K7(K7(REAL,(LinearOperators ((REAL-NS n),(REAL-NS m)))),(LinearOperators ((REAL-NS n),(REAL-NS m))))) is V33() set
RLSStruct(# (LinearOperators ((REAL-NS n),(REAL-NS m))),K199(K279( the carrier of (REAL-NS n),(REAL-NS m)),(LinearOperators ((REAL-NS n),(REAL-NS m)))),K197(K279( the carrier of (REAL-NS n),(REAL-NS m)),(LinearOperators ((REAL-NS n),(REAL-NS m)))),K198(K279( the carrier of (REAL-NS n),(REAL-NS m)),(LinearOperators ((REAL-NS n),(REAL-NS m)))) #) is V71() strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RLSStruct
the carrier of (R_VectorSpace_of_LinearOperators ((REAL-NS n),(REAL-NS m))) is non empty set
K6( the carrier of (R_VectorSpace_of_LinearOperators ((REAL-NS n),(REAL-NS m)))) is set
K199((R_VectorSpace_of_LinearOperators ((REAL-NS n),(REAL-NS m))),(BoundedLinearOperators ((REAL-NS n),(REAL-NS m)))) is Element of BoundedLinearOperators ((REAL-NS n),(REAL-NS m))
K197((R_VectorSpace_of_LinearOperators ((REAL-NS n),(REAL-NS m))),(BoundedLinearOperators ((REAL-NS n),(REAL-NS m)))) is Relation-like K7((BoundedLinearOperators ((REAL-NS n),(REAL-NS m))),(BoundedLinearOperators ((REAL-NS n),(REAL-NS m)))) -defined BoundedLinearOperators ((REAL-NS n),(REAL-NS m)) -valued Function-like total V30(K7((BoundedLinearOperators ((REAL-NS n),(REAL-NS m))),(BoundedLinearOperators ((REAL-NS n),(REAL-NS m)))), BoundedLinearOperators ((REAL-NS n),(REAL-NS m))) Element of K6(K7(K7((BoundedLinearOperators ((REAL-NS n),(REAL-NS m))),(BoundedLinearOperators ((REAL-NS n),(REAL-NS m)))),(BoundedLinearOperators ((REAL-NS n),(REAL-NS m)))))
K7((BoundedLinearOperators ((REAL-NS n),(REAL-NS m))),(BoundedLinearOperators ((REAL-NS n),(REAL-NS m)))) is set
K7(K7((BoundedLinearOperators ((REAL-NS n),(REAL-NS m))),(BoundedLinearOperators ((REAL-NS n),(REAL-NS m)))),(BoundedLinearOperators ((REAL-NS n),(REAL-NS m)))) is set
K6(K7(K7((BoundedLinearOperators ((REAL-NS n),(REAL-NS m))),(BoundedLinearOperators ((REAL-NS n),(REAL-NS m)))),(BoundedLinearOperators ((REAL-NS n),(REAL-NS m))))) is set
K198((R_VectorSpace_of_LinearOperators ((REAL-NS n),(REAL-NS m))),(BoundedLinearOperators ((REAL-NS n),(REAL-NS m)))) is Relation-like K7(REAL,(BoundedLinearOperators ((REAL-NS n),(REAL-NS m)))) -defined BoundedLinearOperators ((REAL-NS n),(REAL-NS m)) -valued Function-like total V30(K7(REAL,(BoundedLinearOperators ((REAL-NS n),(REAL-NS m)))), BoundedLinearOperators ((REAL-NS n),(REAL-NS m))) Element of K6(K7(K7(REAL,(BoundedLinearOperators ((REAL-NS n),(REAL-NS m)))),(BoundedLinearOperators ((REAL-NS n),(REAL-NS m)))))
K7(REAL,(BoundedLinearOperators ((REAL-NS n),(REAL-NS m)))) is V33() set
K7(K7(REAL,(BoundedLinearOperators ((REAL-NS n),(REAL-NS m)))),(BoundedLinearOperators ((REAL-NS n),(REAL-NS m)))) is V33() set
K6(K7(K7(REAL,(BoundedLinearOperators ((REAL-NS n),(REAL-NS m)))),(BoundedLinearOperators ((REAL-NS n),(REAL-NS m))))) is V33() set
BoundedLinearOperatorsNorm ((REAL-NS n),(REAL-NS m)) is non empty Relation-like BoundedLinearOperators ((REAL-NS n),(REAL-NS m)) -defined REAL -valued Function-like total V30( BoundedLinearOperators ((REAL-NS n),(REAL-NS m)), REAL ) complex-yielding V146() V147() Element of K6(K7((BoundedLinearOperators ((REAL-NS n),(REAL-NS m))),REAL))
K7((BoundedLinearOperators ((REAL-NS n),(REAL-NS m))),REAL) is V33() complex-yielding V146() V147() set
K6(K7((BoundedLinearOperators ((REAL-NS n),(REAL-NS m))),REAL)) is V33() set
G15((BoundedLinearOperators ((REAL-NS n),(REAL-NS m))),K199((R_VectorSpace_of_LinearOperators ((REAL-NS n),(REAL-NS m))),(BoundedLinearOperators ((REAL-NS n),(REAL-NS m)))),K197((R_VectorSpace_of_LinearOperators ((REAL-NS n),(REAL-NS m))),(BoundedLinearOperators ((REAL-NS n),(REAL-NS m)))),K198((R_VectorSpace_of_LinearOperators ((REAL-NS n),(REAL-NS m))),(BoundedLinearOperators ((REAL-NS n),(REAL-NS m)))),(BoundedLinearOperatorsNorm ((REAL-NS n),(REAL-NS m)))) is V108() L15()
the carrier of (R_NormSpace_of_BoundedLinearOperators ((REAL-NS n),(REAL-NS m))) is non empty set
diff (Z0,c₇) is Relation-like Function-like Element of the carrier of (R_NormSpace_of_BoundedLinearOperators ((REAL-NS n),(REAL-NS m)))
g * (diff (Z0,c₇)) is Relation-like Function-like Element of the carrier of (R_NormSpace_of_BoundedLinearOperators ((REAL-NS n),(REAL-NS m)))
m is V6() V10() V11() V12() ext-real non negative V33() V38() V120() V155() V156() V157() V158() V159() V160() V161() Element of NAT
REAL m is non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered FinSequenceSet of REAL
m -tuples_on REAL is non empty functional FinSequence-membered FinSequenceSet of REAL
REAL-NS m is non empty V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() V108() V109() V142() L15()
the carrier of (REAL-NS m) is non empty set
n is Relation-like NAT -defined REAL -valued Function-like V40(m) FinSequence-like complex-yielding V146() V147() Element of REAL m
f is Element of the carrier of (REAL-NS m)
Z is V11() V12() ext-real Element of REAL
{ b₁ where b₁ is Relation-like NAT -defined REAL -valued Function-like V40(m) FinSequence-like complex-yielding V146() V147() Element of REAL m : not Z <= |.(b₁ - n).| } is set
{ b₁ where b₁ is Element of the carrier of (REAL-NS m) : not Z <= ||.(b₁ - f).|| } is set
g is set
Z0 is Relation-like NAT -defined REAL -valued Function-like V40(m) FinSequence-like complex-yielding V146() V147() Element of REAL m
Z0 - n is Relation-like NAT -defined REAL -valued Function-like V40(m) FinSequence-like complex-yielding V146() V147() Element of REAL m
K498() is Relation-like K7(REAL,REAL) -defined REAL -valued Function-like total V30(K7(REAL,REAL), REAL ) complex-yielding V146() V147() Element of K6(K7(K7(REAL,REAL),REAL))
K497() is Relation-like K7(REAL,REAL) -defined REAL -valued Function-like total V30(K7(REAL,REAL), REAL ) complex-yielding V146() V147() Element of K6(K7(K7(REAL,REAL),REAL))
id REAL is non empty Relation-like REAL -defined REAL -valued total V30( REAL , REAL ) complex-yielding V146() V147() Element of K6(K7(REAL,REAL))
K495() is non empty Relation-like REAL -defined REAL -valued Function-like total V30( REAL , REAL ) complex-yielding V146() V147() Element of K6(K7(REAL,REAL))
K231(REAL,K497(),(id REAL),K495()) is Relation-like K7(REAL,REAL) -defined REAL -valued Function-like total V30(K7(REAL,REAL), REAL ) complex-yielding V146() V147() Element of K6(K7(K7(REAL,REAL),REAL))
K498() .: (Z0,n) is set
- n is Relation-like Function-like complex-yielding set
n (#) K495() is Relation-like complex-yielding V146() V147() set
- 1 is V11() V12() ext-real non positive set
(- 1) (#) n is Relation-like Function-like set
K515((- 1)) is non empty Relation-like REAL -defined REAL -valued Function-like total V30( REAL , REAL ) complex-yielding V146() V147() Element of K6(K7(REAL,REAL))
K499() is Relation-like K7(REAL,REAL) -defined REAL -valued Function-like total V30(K7(REAL,REAL), REAL ) complex-yielding V146() V147() Element of K6(K7(K7(REAL,REAL),REAL))
K499() [;] ((- 1),(id REAL)) is set
n (#) K515((- 1)) is Relation-like complex-yielding V146() V147() set
Z0 + (- n) is Relation-like Function-like set
|.(Z0 - n).| is V11() V12() ext-real non negative Element of REAL
sqr (Z0 - n) is Relation-like NAT -defined REAL -valued Function-like FinSequence-like complex-yielding V146() V147() FinSequence of REAL
(Z0 - n) (#) sqrreal is Relation-like complex-yielding V146() V147() set
(Z0 - n) (#) (Z0 - n) is Relation-like Function-like set
K499() .: ((Z0 - n),(Z0 - n)) is set
Sum (sqr (Z0 - n)) is V11() V12() ext-real Element of REAL
sqrt (Sum (sqr (Z0 - n))) is V11() V12() ext-real Element of REAL
c₇ is Element of the carrier of (REAL-NS m)
c₇ - f is Element of the carrier of (REAL-NS m)
||.(c₇ - f).|| is V11() V12() ext-real Element of REAL
g is set
Z0 is Element of the carrier of (REAL-NS m)
Z0 - f is Element of the carrier of (REAL-NS m)
||.(Z0 - f).|| is V11() V12() ext-real Element of REAL
c₇ is Relation-like NAT -defined REAL -valued Function-like V40(m) FinSequence-like complex-yielding V146() V147() Element of REAL m
c₇ - n is Relation-like NAT -defined REAL -valued Function-like V40(m) FinSequence-like complex-yielding V146() V147() Element of REAL m
K498() .: (c₇,n) is set
c₇ + (- n) is Relation-like Function-like set
|.(c₇ - n).| is V11() V12() ext-real non negative Element of REAL
sqr (c₇ - n) is Relation-like NAT -defined REAL -valued Function-like FinSequence-like complex-yielding V146() V147() FinSequence of REAL
(c₇ - n) (#) sqrreal is Relation-like complex-yielding V146() V147() set
(c₇ - n) (#) (c₇ - n) is Relation-like Function-like set
K499() .: ((c₇ - n),(c₇ - n)) is set
Sum (sqr (c₇ - n)) is V11() V12() ext-real Element of REAL
sqrt (Sum (sqr (c₇ - n))) is V11() V12() ext-real Element of REAL
m is non empty V6() V10() V11() V12() ext-real positive non negative V33() V38() V120() V155() V156() V157() V158() V159() V160() V161() Element of NAT
REAL m is non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered FinSequenceSet of REAL
m -tuples_on REAL is non empty functional FinSequence-membered FinSequenceSet of REAL
n is non empty V6() V10() V11() V12() ext-real positive non negative V33() V38() V120() V155() V156() V157() V158() V159() V160() V161() Element of NAT
REAL n is non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered FinSequenceSet of REAL
n -tuples_on REAL is non empty functional FinSequence-membered FinSequenceSet of REAL
K7((REAL m),(REAL n)) is set
K6(K7((REAL m),(REAL n))) is set
0* m is Relation-like NAT -defined REAL -valued Function-like V40(m) FinSequence-like complex-yielding V146() V147() Element of REAL m
m |-> 0 is Relation-like empty-yielding NAT -defined REAL -valued Function-like V40(m) FinSequence-like complex-yielding V146() V147() Element of m -tuples_on REAL
f is Relation-like REAL m -defined REAL n -valued Function-like complex-functions-valued ext-real-functions-valued real-functions-valued Element of K6(K7((REAL m),(REAL n)))
dom f is functional complex-functions-membered ext-real-functions-membered real-functions-membered Element of K6((REAL m))
K6((REAL m)) is set
Z is Relation-like NAT -defined REAL -valued Function-like V40(m) FinSequence-like complex-yielding V146() V147() Element of REAL m
f /. Z is V40(n) complex-yielding V146() V147() Element of REAL n
diff (f,Z) is non empty Relation-like REAL m -defined REAL n -valued Function-like total V30( REAL m, REAL n) complex-functions-valued ext-real-functions-valued real-functions-valued Element of K6(K7((REAL m),(REAL n)))
g is non empty Relation-like REAL m -defined REAL n -valued Function-like total V30( REAL m, REAL n) complex-functions-valued ext-real-functions-valued real-functions-valued Element of K6(K7((REAL m),(REAL n)))
Z0 is non empty Relation-like REAL m -defined REAL n -valued Function-like total V30( REAL m, REAL n) complex-functions-valued ext-real-functions-valued real-functions-valued Element of K6(K7((REAL m),(REAL n)))
REAL-NS m is non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() V108() V109() V142() L15()
the carrier of (REAL-NS m) is non empty V2() set
REAL-NS n is non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() V108() V109() V142() L15()
the carrier of (REAL-NS n) is non empty V2() set
K7( the carrier of (REAL-NS m), the carrier of (REAL-NS n)) is set
K6(K7( the carrier of (REAL-NS m), the carrier of (REAL-NS n))) is set
L is V11() V12() ext-real Element of REAL
{ b₁ where b₁ is Relation-like NAT -defined REAL -valued Function-like V40(m) FinSequence-like complex-yielding V146() V147() Element of REAL m : not L <= |.(b₁ - Z).| } is set
L is V11() V12() ext-real Element of REAL
{ b₁ where b₁ is Relation-like NAT -defined REAL -valued Function-like V40(m) FinSequence-like complex-yielding V146() V147() Element of REAL m : not L <= |.(b₁ - Z).| } is set
L0 is V11() V12() ext-real Element of REAL
{ b₁ where b₁ is Relation-like NAT -defined REAL -valued Function-like V40(m) FinSequence-like complex-yielding V146() V147() Element of REAL m : not L0 <= |.(b₁ - Z).| } is set
R_NormSpace_of_BoundedLinearOperators ((REAL-NS m),(REAL-NS n)) is non empty V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() V109() V142() L15()
BoundedLinearOperators ((REAL-NS m),(REAL-NS n)) is non empty Element of K6( the carrier of (R_VectorSpace_of_LinearOperators ((REAL-NS m),(REAL-NS n))))
R_VectorSpace_of_LinearOperators ((REAL-NS m),(REAL-NS n)) is non empty V71() strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V204() RLSStruct
LinearOperators ((REAL-NS m),(REAL-NS n)) is non empty functional Element of K6( the carrier of K279( the carrier of (REAL-NS m),(REAL-NS n)))
K279( the carrier of (REAL-NS m),(REAL-NS n)) is non empty V71() strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V204() RLSStruct
K123( the carrier of (REAL-NS m), the carrier of (REAL-NS n)) is non empty functional M4( the carrier of (REAL-NS m), the carrier of (REAL-NS n))
FuncZero ( the carrier of (REAL-NS m),(REAL-NS n)) is Relation-like the carrier of (REAL-NS m) -defined the carrier of (REAL-NS n) -valued Function-like total V30( the carrier of (REAL-NS m), the carrier of (REAL-NS n)) Element of K123( the carrier of (REAL-NS m), the carrier of (REAL-NS n))
0. (REAL-NS n) is V52( REAL-NS n) Element of the carrier of (REAL-NS n)
the carrier of (REAL-NS m) --> (0. (REAL-NS n)) is non empty Relation-like the carrier of (REAL-NS m) -defined the carrier of (REAL-NS n) -valued Function-like total V30( the carrier of (REAL-NS m), the carrier of (REAL-NS n)) Element of K6(K7( the carrier of (REAL-NS m), the carrier of (REAL-NS n)))
FuncAdd ( the carrier of (REAL-NS m),(REAL-NS n)) is Relation-like K7(K123( the carrier of (REAL-NS m), the carrier of (REAL-NS n)),K123( the carrier of (REAL-NS m), the carrier of (REAL-NS n))) -defined K123( the carrier of (REAL-NS m), the carrier of (REAL-NS n)) -valued Function-like total V30(K7(K123( the carrier of (REAL-NS m), the carrier of (REAL-NS n)),K123( the carrier of (REAL-NS m), the carrier of (REAL-NS n))),K123( the carrier of (REAL-NS m), the carrier of (REAL-NS n))) Element of K6(K7(K7(K123( the carrier of (REAL-NS m), the carrier of (REAL-NS n)),K123( the carrier of (REAL-NS m), the carrier of (REAL-NS n))),K123( the carrier of (REAL-NS m), the carrier of (REAL-NS n))))
K7(K123( the carrier of (REAL-NS m), the carrier of (REAL-NS n)),K123( the carrier of (REAL-NS m), the carrier of (REAL-NS n))) is set
K7(K7(K123( the carrier of (REAL-NS m), the carrier of (REAL-NS n)),K123( the carrier of (REAL-NS m), the carrier of (REAL-NS n))),K123( the carrier of (REAL-NS m), the carrier of (REAL-NS n))) is set
K6(K7(K7(K123( the carrier of (REAL-NS m), the carrier of (REAL-NS n)),K123( the carrier of (REAL-NS m), the carrier of (REAL-NS n))),K123( the carrier of (REAL-NS m), the carrier of (REAL-NS n)))) is set
FuncExtMult ( the carrier of (REAL-NS m),(REAL-NS n)) is Relation-like K7(REAL,K123( the carrier of (REAL-NS m), the carrier of (REAL-NS n))) -defined K123( the carrier of (REAL-NS m), the carrier of (REAL-NS n)) -valued Function-like total V30(K7(REAL,K123( the carrier of (REAL-NS m), the carrier of (REAL-NS n))),K123( the carrier of (REAL-NS m), the carrier of (REAL-NS n))) Element of K6(K7(K7(REAL,K123( the carrier of (REAL-NS m), the carrier of (REAL-NS n))),K123( the carrier of (REAL-NS m), the carrier of (REAL-NS n))))
K7(REAL,K123( the carrier of (REAL-NS m), the carrier of (REAL-NS n))) is V33() set
K7(K7(REAL,K123( the carrier of (REAL-NS m), the carrier of (REAL-NS n))),K123( the carrier of (REAL-NS m), the carrier of (REAL-NS n))) is V33() set
K6(K7(K7(REAL,K123( the carrier of (REAL-NS m), the carrier of (REAL-NS n))),K123( the carrier of (REAL-NS m), the carrier of (REAL-NS n)))) is V33() set
RLSStruct(# K123( the carrier of (REAL-NS m), the carrier of (REAL-NS n)),(FuncZero ( the carrier of (REAL-NS m),(REAL-NS n))),(FuncAdd ( the carrier of (REAL-NS m),(REAL-NS n))),(FuncExtMult ( the carrier of (REAL-NS m),(REAL-NS n))) #) is strict RLSStruct
the carrier of K279( the carrier of (REAL-NS m),(REAL-NS n)) is non empty set
K6( the carrier of K279( the carrier of (REAL-NS m),(REAL-NS n))) is set
K199(K279( the carrier of (REAL-NS m),(REAL-NS n)),(LinearOperators ((REAL-NS m),(REAL-NS n)))) is Element of LinearOperators ((REAL-NS m),(REAL-NS n))
K197(K279( the carrier of (REAL-NS m),(REAL-NS n)),(LinearOperators ((REAL-NS m),(REAL-NS n)))) is Relation-like K7((LinearOperators ((REAL-NS m),(REAL-NS n))),(LinearOperators ((REAL-NS m),(REAL-NS n)))) -defined LinearOperators ((REAL-NS m),(REAL-NS n)) -valued Function-like total V30(K7((LinearOperators ((REAL-NS m),(REAL-NS n))),(LinearOperators ((REAL-NS m),(REAL-NS n)))), LinearOperators ((REAL-NS m),(REAL-NS n))) Element of K6(K7(K7((LinearOperators ((REAL-NS m),(REAL-NS n))),(LinearOperators ((REAL-NS m),(REAL-NS n)))),(LinearOperators ((REAL-NS m),(REAL-NS n)))))
K7((LinearOperators ((REAL-NS m),(REAL-NS n))),(LinearOperators ((REAL-NS m),(REAL-NS n)))) is set
K7(K7((LinearOperators ((REAL-NS m),(REAL-NS n))),(LinearOperators ((REAL-NS m),(REAL-NS n)))),(LinearOperators ((REAL-NS m),(REAL-NS n)))) is set
K6(K7(K7((LinearOperators ((REAL-NS m),(REAL-NS n))),(LinearOperators ((REAL-NS m),(REAL-NS n)))),(LinearOperators ((REAL-NS m),(REAL-NS n))))) is set
K198(K279( the carrier of (REAL-NS m),(REAL-NS n)),(LinearOperators ((REAL-NS m),(REAL-NS n)))) is Relation-like K7(REAL,(LinearOperators ((REAL-NS m),(REAL-NS n)))) -defined LinearOperators ((REAL-NS m),(REAL-NS n)) -valued Function-like total V30(K7(REAL,(LinearOperators ((REAL-NS m),(REAL-NS n)))), LinearOperators ((REAL-NS m),(REAL-NS n))) Element of K6(K7(K7(REAL,(LinearOperators ((REAL-NS m),(REAL-NS n)))),(LinearOperators ((REAL-NS m),(REAL-NS n)))))
K7(REAL,(LinearOperators ((REAL-NS m),(REAL-NS n)))) is V33() set
K7(K7(REAL,(LinearOperators ((REAL-NS m),(REAL-NS n)))),(LinearOperators ((REAL-NS m),(REAL-NS n)))) is V33() set
K6(K7(K7(REAL,(LinearOperators ((REAL-NS m),(REAL-NS n)))),(LinearOperators ((REAL-NS m),(REAL-NS n))))) is V33() set
RLSStruct(# (LinearOperators ((REAL-NS m),(REAL-NS n))),K199(K279( the carrier of (REAL-NS m),(REAL-NS n)),(LinearOperators ((REAL-NS m),(REAL-NS n)))),K197(K279( the carrier of (REAL-NS m),(REAL-NS n)),(LinearOperators ((REAL-NS m),(REAL-NS n)))),K198(K279( the carrier of (REAL-NS m),(REAL-NS n)),(LinearOperators ((REAL-NS m),(REAL-NS n)))) #) is V71() strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RLSStruct
the carrier of (R_VectorSpace_of_LinearOperators ((REAL-NS m),(REAL-NS n))) is non empty set
K6( the carrier of (R_VectorSpace_of_LinearOperators ((REAL-NS m),(REAL-NS n)))) is set
K199((R_VectorSpace_of_LinearOperators ((REAL-NS m),(REAL-NS n))),(BoundedLinearOperators ((REAL-NS m),(REAL-NS n)))) is Element of BoundedLinearOperators ((REAL-NS m),(REAL-NS n))
K197((R_VectorSpace_of_LinearOperators ((REAL-NS m),(REAL-NS n))),(BoundedLinearOperators ((REAL-NS m),(REAL-NS n)))) is Relation-like K7((BoundedLinearOperators ((REAL-NS m),(REAL-NS n))),(BoundedLinearOperators ((REAL-NS m),(REAL-NS n)))) -defined BoundedLinearOperators ((REAL-NS m),(REAL-NS n)) -valued Function-like total V30(K7((BoundedLinearOperators ((REAL-NS m),(REAL-NS n))),(BoundedLinearOperators ((REAL-NS m),(REAL-NS n)))), BoundedLinearOperators ((REAL-NS m),(REAL-NS n))) Element of K6(K7(K7((BoundedLinearOperators ((REAL-NS m),(REAL-NS n))),(BoundedLinearOperators ((REAL-NS m),(REAL-NS n)))),(BoundedLinearOperators ((REAL-NS m),(REAL-NS n)))))
K7((BoundedLinearOperators ((REAL-NS m),(REAL-NS n))),(BoundedLinearOperators ((REAL-NS m),(REAL-NS n)))) is set
K7(K7((BoundedLinearOperators ((REAL-NS m),(REAL-NS n))),(BoundedLinearOperators ((REAL-NS m),(REAL-NS n)))),(BoundedLinearOperators ((REAL-NS m),(REAL-NS n)))) is set
K6(K7(K7((BoundedLinearOperators ((REAL-NS m),(REAL-NS n))),(BoundedLinearOperators ((REAL-NS m),(REAL-NS n)))),(BoundedLinearOperators ((REAL-NS m),(REAL-NS n))))) is set
K198((R_VectorSpace_of_LinearOperators ((REAL-NS m),(REAL-NS n))),(BoundedLinearOperators ((REAL-NS m),(REAL-NS n)))) is Relation-like K7(REAL,(BoundedLinearOperators ((REAL-NS m),(REAL-NS n)))) -defined BoundedLinearOperators ((REAL-NS m),(REAL-NS n)) -valued Function-like total V30(K7(REAL,(BoundedLinearOperators ((REAL-NS m),(REAL-NS n)))), BoundedLinearOperators ((REAL-NS m),(REAL-NS n))) Element of K6(K7(K7(REAL,(BoundedLinearOperators ((REAL-NS m),(REAL-NS n)))),(BoundedLinearOperators ((REAL-NS m),(REAL-NS n)))))
K7(REAL,(BoundedLinearOperators ((REAL-NS m),(REAL-NS n)))) is V33() set
K7(K7(REAL,(BoundedLinearOperators ((REAL-NS m),(REAL-NS n)))),(BoundedLinearOperators ((REAL-NS m),(REAL-NS n)))) is V33() set
K6(K7(K7(REAL,(BoundedLinearOperators ((REAL-NS m),(REAL-NS n)))),(BoundedLinearOperators ((REAL-NS m),(REAL-NS n))))) is V33() set
BoundedLinearOperatorsNorm ((REAL-NS m),(REAL-NS n)) is non empty Relation-like BoundedLinearOperators ((REAL-NS m),(REAL-NS n)) -defined REAL -valued Function-like total V30( BoundedLinearOperators ((REAL-NS m),(REAL-NS n)), REAL ) complex-yielding V146() V147() Element of K6(K7((BoundedLinearOperators ((REAL-NS m),(REAL-NS n))),REAL))
K7((BoundedLinearOperators ((REAL-NS m),(REAL-NS n))),REAL) is V33() complex-yielding V146() V147() set
K6(K7((BoundedLinearOperators ((REAL-NS m),(REAL-NS n))),REAL)) is V33() set
G15((BoundedLinearOperators ((REAL-NS m),(REAL-NS n))),K199((R_VectorSpace_of_LinearOperators ((REAL-NS m),(REAL-NS n))),(BoundedLinearOperators ((REAL-NS m),(REAL-NS n)))),K197((R_VectorSpace_of_LinearOperators ((REAL-NS m),(REAL-NS n))),(BoundedLinearOperators ((REAL-NS m),(REAL-NS n)))),K198((R_VectorSpace_of_LinearOperators ((REAL-NS m),(REAL-NS n))),(BoundedLinearOperators ((REAL-NS m),(REAL-NS n)))),(BoundedLinearOperatorsNorm ((REAL-NS m),(REAL-NS n)))) is V108() L15()
the carrier of (R_NormSpace_of_BoundedLinearOperators ((REAL-NS m),(REAL-NS n))) is non empty set
dom Z0 is functional complex-functions-membered ext-real-functions-membered real-functions-membered Element of K6((REAL m))
R is V11() V12() ext-real Element of REAL
N is V11() V12() ext-real Element of REAL
N is V11() V12() ext-real Element of REAL
0. (REAL-NS m) is V52( REAL-NS m) Element of the carrier of (REAL-NS m)
R is non empty Relation-like the carrier of (REAL-NS m) -defined the carrier of (REAL-NS n) -valued Function-like total V30( the carrier of (REAL-NS m), the carrier of (REAL-NS n)) Element of K6(K7( the carrier of (REAL-NS m), the carrier of (REAL-NS n)))
x is Element of the carrier of (REAL-NS m)
||.x.|| is V11() V12() ext-real Element of REAL
||.x.|| " is V11() V12() ext-real Element of REAL
R /. x is Element of the carrier of (REAL-NS n)
R . x is Element of the carrier of (REAL-NS n)
||.(R /. x).|| is V11() V12() ext-real Element of REAL
(||.x.|| ") * ||.(R /. x).|| is V11() V12() ext-real Element of REAL
y is Relation-like NAT -defined REAL -valued Function-like V40(m) FinSequence-like complex-yielding V146() V147() Element of REAL m
(m,n,Z0,y) is Relation-like NAT -defined REAL -valued Function-like V40(n) FinSequence-like complex-yielding V146() V147() Element of REAL n
|.y.| is V11() V12() ext-real non negative Element of REAL
sqr y is Relation-like NAT -defined REAL -valued Function-like FinSequence-like complex-yielding V146() V147() FinSequence of REAL
y (#) sqrreal is Relation-like complex-yielding V146() V147() set
y (#) y is Relation-like Function-like set
K499() is Relation-like K7(REAL,REAL) -defined REAL -valued Function-like total V30(K7(REAL,REAL), REAL ) complex-yielding V146() V147() Element of K6(K7(K7(REAL,REAL),REAL))
K499() .: (y,y) is set
Sum (sqr y) is V11() V12() ext-real Element of REAL
sqrt (Sum (sqr y)) is V11() V12() ext-real Element of REAL
|.y.| " is V11() V12() ext-real non negative Element of REAL
Rz is Relation-like NAT -defined REAL -valued Function-like V40(n) FinSequence-like complex-yielding V146() V147() Element of REAL n
|.Rz.| is V11() V12() ext-real non negative Element of REAL
sqr Rz is Relation-like NAT -defined REAL -valued Function-like FinSequence-like complex-yielding V146() V147() FinSequence of REAL
Rz (#) sqrreal is Relation-like complex-yielding V146() V147() set
Rz (#) Rz is Relation-like Function-like set
K499() .: (Rz,Rz) is set
Sum (sqr Rz) is V11() V12() ext-real Element of REAL
sqrt (Sum (sqr Rz)) is V11() V12() ext-real Element of REAL
(|.y.| ") * |.Rz.| is V11() V12() ext-real non negative Element of REAL
(||.x.|| ") * |.Rz.| is V11() V12() ext-real Element of REAL
x is Element of the carrier of (REAL-NS m)
{ b₁ where b₁ is Element of the carrier of (REAL-NS m) : not L <= ||.(b₁ - x).|| } is set
N is Neighbourhood of x
c₇ is Relation-like the carrier of (REAL-NS m) -defined the carrier of (REAL-NS n) -valued Function-like Element of K6(K7( the carrier of (REAL-NS m), the carrier of (REAL-NS n)))
dom c₇ is Element of K6( the carrier of (REAL-NS m))
K6( the carrier of (REAL-NS m)) is set
c₇ /. x is Element of the carrier of (REAL-NS n)
L0 is Relation-like Function-like Element of the carrier of (R_NormSpace_of_BoundedLinearOperators ((REAL-NS m),(REAL-NS n)))
R is Relation-like the carrier of (REAL-NS m) -defined the carrier of (REAL-NS n) -valued Function-like RestFunc-like Element of K6(K7( the carrier of (REAL-NS m), the carrier of (REAL-NS n)))
x is Element of the carrier of (REAL-NS m)
c₇ /. x is Element of the carrier of (REAL-NS n)
(c₇ /. x) - (c₇ /. x) is Element of the carrier of (REAL-NS n)
x - x is Element of the carrier of (REAL-NS m)
L0 . (x - x) is Element of the carrier of (REAL-NS n)
R /. (x - x) is Element of the carrier of (REAL-NS n)
(L0 . (x - x)) + (R /. (x - x)) is Element of the carrier of (REAL-NS n)
y is Relation-like NAT -defined REAL -valued Function-like V40(m) FinSequence-like complex-yielding V146() V147() Element of REAL m
f /. y is V40(n) complex-yielding V146() V147() Element of REAL n
(f /. y) - (f /. Z) is Relation-like NAT -defined REAL -valued Function-like V40(n) FinSequence-like complex-yielding V146() V147() Element of REAL n
y - Z is Relation-like NAT -defined REAL -valued Function-like V40(m) FinSequence-like complex-yielding V146() V147() Element of REAL m
K498() is Relation-like K7(REAL,REAL) -defined REAL -valued Function-like total V30(K7(REAL,REAL), REAL ) complex-yielding V146() V147() Element of K6(K7(K7(REAL,REAL),REAL))
K497() is Relation-like K7(REAL,REAL) -defined REAL -valued Function-like total V30(K7(REAL,REAL), REAL ) complex-yielding V146() V147() Element of K6(K7(K7(REAL,REAL),REAL))
id REAL is non empty Relation-like REAL -defined REAL -valued total V30( REAL , REAL ) complex-yielding V146() V147() Element of K6(K7(REAL,REAL))
K495() is non empty Relation-like REAL -defined REAL -valued Function-like total V30( REAL , REAL ) complex-yielding V146() V147() Element of K6(K7(REAL,REAL))
K231(REAL,K497(),(id REAL),K495()) is Relation-like K7(REAL,REAL) -defined REAL -valued Function-like total V30(K7(REAL,REAL), REAL ) complex-yielding V146() V147() Element of K6(K7(K7(REAL,REAL),REAL))
K498() .: (y,Z) is set
- Z is Relation-like Function-like complex-yielding set
Z (#) K495() is Relation-like complex-yielding V146() V147() set
- 1 is V11() V12() ext-real non positive set
(- 1) (#) Z is Relation-like Function-like set
K515((- 1)) is non empty Relation-like REAL -defined REAL -valued Function-like total V30( REAL , REAL ) complex-yielding V146() V147() Element of K6(K7(REAL,REAL))
K499() is Relation-like K7(REAL,REAL) -defined REAL -valued Function-like total V30(K7(REAL,REAL), REAL ) complex-yielding V146() V147() Element of K6(K7(K7(REAL,REAL),REAL))
K499() [;] ((- 1),(id REAL)) is set
Z (#) K515((- 1)) is Relation-like complex-yielding V146() V147() set
y + (- Z) is Relation-like Function-like set
(m,n,g,(y - Z)) is Relation-like NAT -defined REAL -valued Function-like V40(n) FinSequence-like complex-yielding V146() V147() Element of REAL n
(m,n,Z0,(y - Z)) is Relation-like NAT -defined REAL -valued Function-like V40(n) FinSequence-like complex-yielding V146() V147() Element of REAL n
(m,n,g,(y - Z)) + (m,n,Z0,(y - Z)) is Relation-like NAT -defined REAL -valued Function-like V40(n) FinSequence-like complex-yielding V146() V147() Element of REAL n
K497() .: ((m,n,g,(y - Z)),(m,n,Z0,(y - Z))) is set
Rz is Relation-like NAT -defined REAL -valued Function-like V40(m) FinSequence-like complex-yielding V146() V147() Element of REAL m
Rz - Z is Relation-like NAT -defined REAL -valued Function-like V40(m) FinSequence-like complex-yielding V146() V147() Element of REAL m
K498() .: (Rz,Z) is set
Rz + (- Z) is Relation-like Function-like set
|.(Rz - Z).| is V11() V12() ext-real non negative Element of REAL
sqr (Rz - Z) is Relation-like NAT -defined REAL -valued Function-like FinSequence-like complex-yielding V146() V147() FinSequence of REAL
(Rz - Z) (#) sqrreal is Relation-like complex-yielding V146() V147() set
(Rz - Z) (#) (Rz - Z) is Relation-like Function-like set
K499() .: ((Rz - Z),(Rz - Z)) is set
Sum (sqr (Rz - Z)) is V11() V12() ext-real Element of REAL
sqrt (Sum (sqr (Rz - Z))) is V11() V12() ext-real Element of REAL
f . y is Relation-like Function-like complex-yielding V146() V147() set
f . Z is Relation-like Function-like complex-yielding V146() V147() set
diff (c₇,x) is Relation-like Function-like Element of the carrier of (R_NormSpace_of_BoundedLinearOperators ((REAL-NS m),(REAL-NS n)))
m is non empty V6() V10() V11() V12() ext-real positive non negative V33() V38() V120() V155() V156() V157() V158() V159() V160() V161() Element of NAT
REAL m is non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered FinSequenceSet of REAL
m -tuples_on REAL is non empty functional FinSequence-membered FinSequenceSet of REAL
n is non empty V6() V10() V11() V12() ext-real positive non negative V33() V38() V120() V155() V156() V157() V158() V159() V160() V161() Element of NAT
REAL n is non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered FinSequenceSet of REAL
n -tuples_on REAL is non empty functional FinSequence-membered FinSequenceSet of REAL
K7((REAL m),(REAL n)) is set
K6(K7((REAL m),(REAL n))) is set
0* m is Relation-like NAT -defined REAL -valued Function-like V40(m) FinSequence-like complex-yielding V146() V147() Element of REAL m
m |-> 0 is Relation-like empty-yielding NAT -defined REAL -valued Function-like V40(m) FinSequence-like complex-yielding V146() V147() Element of m -tuples_on REAL
f is Relation-like REAL m -defined REAL n -valued Function-like complex-functions-valued ext-real-functions-valued real-functions-valued Element of K6(K7((REAL m),(REAL n)))
dom f is functional complex-functions-membered ext-real-functions-membered real-functions-membered Element of K6((REAL m))
K6((REAL m)) is set
Z is Relation-like NAT -defined REAL -valued Function-like V40(m) FinSequence-like complex-yielding V146() V147() Element of REAL m
f /. Z is V40(n) complex-yielding V146() V147() Element of REAL n
REAL-NS m is non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() V108() V109() V142() L15()
the carrier of (REAL-NS m) is non empty V2() set
REAL-NS n is non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() V108() V109() V142() L15()
the carrier of (REAL-NS n) is non empty V2() set
K7( the carrier of (REAL-NS m), the carrier of (REAL-NS n)) is set
K6(K7( the carrier of (REAL-NS m), the carrier of (REAL-NS n))) is set
g is Relation-like the carrier of (REAL-NS m) -defined the carrier of (REAL-NS n) -valued Function-like Element of K6(K7( the carrier of (REAL-NS m), the carrier of (REAL-NS n)))
Z0 is Element of the carrier of (REAL-NS m)
dom g is Element of K6( the carrier of (REAL-NS m))
K6( the carrier of (REAL-NS m)) is set
R_NormSpace_of_BoundedLinearOperators ((REAL-NS m),(REAL-NS n)) is non empty V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() V109() V142() L15()
BoundedLinearOperators ((REAL-NS m),(REAL-NS n)) is non empty Element of K6( the carrier of (R_VectorSpace_of_LinearOperators ((REAL-NS m),(REAL-NS n))))
R_VectorSpace_of_LinearOperators ((REAL-NS m),(REAL-NS n)) is non empty V71() strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V204() RLSStruct
LinearOperators ((REAL-NS m),(REAL-NS n)) is non empty functional Element of K6( the carrier of K279( the carrier of (REAL-NS m),(REAL-NS n)))
K279( the carrier of (REAL-NS m),(REAL-NS n)) is non empty V71() strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V204() RLSStruct
K123( the carrier of (REAL-NS m), the carrier of (REAL-NS n)) is non empty functional M4( the carrier of (REAL-NS m), the carrier of (REAL-NS n))
FuncZero ( the carrier of (REAL-NS m),(REAL-NS n)) is Relation-like the carrier of (REAL-NS m) -defined the carrier of (REAL-NS n) -valued Function-like total V30( the carrier of (REAL-NS m), the carrier of (REAL-NS n)) Element of K123( the carrier of (REAL-NS m), the carrier of (REAL-NS n))
0. (REAL-NS n) is V52( REAL-NS n) Element of the carrier of (REAL-NS n)
the carrier of (REAL-NS m) --> (0. (REAL-NS n)) is non empty Relation-like the carrier of (REAL-NS m) -defined the carrier of (REAL-NS n) -valued Function-like total V30( the carrier of (REAL-NS m), the carrier of (REAL-NS n)) Element of K6(K7( the carrier of (REAL-NS m), the carrier of (REAL-NS n)))
FuncAdd ( the carrier of (REAL-NS m),(REAL-NS n)) is Relation-like K7(K123( the carrier of (REAL-NS m), the carrier of (REAL-NS n)),K123( the carrier of (REAL-NS m), the carrier of (REAL-NS n))) -defined K123( the carrier of (REAL-NS m), the carrier of (REAL-NS n)) -valued Function-like total V30(K7(K123( the carrier of (REAL-NS m), the carrier of (REAL-NS n)),K123( the carrier of (REAL-NS m), the carrier of (REAL-NS n))),K123( the carrier of (REAL-NS m), the carrier of (REAL-NS n))) Element of K6(K7(K7(K123( the carrier of (REAL-NS m), the carrier of (REAL-NS n)),K123( the carrier of (REAL-NS m), the carrier of (REAL-NS n))),K123( the carrier of (REAL-NS m), the carrier of (REAL-NS n))))
K7(K123( the carrier of (REAL-NS m), the carrier of (REAL-NS n)),K123( the carrier of (REAL-NS m), the carrier of (REAL-NS n))) is set
K7(K7(K123( the carrier of (REAL-NS m), the carrier of (REAL-NS n)),K123( the carrier of (REAL-NS m), the carrier of (REAL-NS n))),K123( the carrier of (REAL-NS m), the carrier of (REAL-NS n))) is set
K6(K7(K7(K123( the carrier of (REAL-NS m), the carrier of (REAL-NS n)),K123( the carrier of (REAL-NS m), the carrier of (REAL-NS n))),K123( the carrier of (REAL-NS m), the carrier of (REAL-NS n)))) is set
FuncExtMult ( the carrier of (REAL-NS m),(REAL-NS n)) is Relation-like K7(REAL,K123( the carrier of (REAL-NS m), the carrier of (REAL-NS n))) -defined K123( the carrier of (REAL-NS m), the carrier of (REAL-NS n)) -valued Function-like total V30(K7(REAL,K123( the carrier of (REAL-NS m), the carrier of (REAL-NS n))),K123( the carrier of (REAL-NS m), the carrier of (REAL-NS n))) Element of K6(K7(K7(REAL,K123( the carrier of (REAL-NS m), the carrier of (REAL-NS n))),K123( the carrier of (REAL-NS m), the carrier of (REAL-NS n))))
K7(REAL,K123( the carrier of (REAL-NS m), the carrier of (REAL-NS n))) is V33() set
K7(K7(REAL,K123( the carrier of (REAL-NS m), the carrier of (REAL-NS n))),K123( the carrier of (REAL-NS m), the carrier of (REAL-NS n))) is V33() set
K6(K7(K7(REAL,K123( the carrier of (REAL-NS m), the carrier of (REAL-NS n))),K123( the carrier of (REAL-NS m), the carrier of (REAL-NS n)))) is V33() set
RLSStruct(# K123( the carrier of (REAL-NS m), the carrier of (REAL-NS n)),(FuncZero ( the carrier of (REAL-NS m),(REAL-NS n))),(FuncAdd ( the carrier of (REAL-NS m),(REAL-NS n))),(FuncExtMult ( the carrier of (REAL-NS m),(REAL-NS n))) #) is strict RLSStruct
the carrier of K279( the carrier of (REAL-NS m),(REAL-NS n)) is non empty set
K6( the carrier of K279( the carrier of (REAL-NS m),(REAL-NS n))) is set
K199(K279( the carrier of (REAL-NS m),(REAL-NS n)),(LinearOperators ((REAL-NS m),(REAL-NS n)))) is Element of LinearOperators ((REAL-NS m),(REAL-NS n))
K197(K279( the carrier of (REAL-NS m),(REAL-NS n)),(LinearOperators ((REAL-NS m),(REAL-NS n)))) is Relation-like K7((LinearOperators ((REAL-NS m),(REAL-NS n))),(LinearOperators ((REAL-NS m),(REAL-NS n)))) -defined LinearOperators ((REAL-NS m),(REAL-NS n)) -valued Function-like total V30(K7((LinearOperators ((REAL-NS m),(REAL-NS n))),(LinearOperators ((REAL-NS m),(REAL-NS n)))), LinearOperators ((REAL-NS m),(REAL-NS n))) Element of K6(K7(K7((LinearOperators ((REAL-NS m),(REAL-NS n))),(LinearOperators ((REAL-NS m),(REAL-NS n)))),(LinearOperators ((REAL-NS m),(REAL-NS n)))))
K7((LinearOperators ((REAL-NS m),(REAL-NS n))),(LinearOperators ((REAL-NS m),(REAL-NS n)))) is set
K7(K7((LinearOperators ((REAL-NS m),(REAL-NS n))),(LinearOperators ((REAL-NS m),(REAL-NS n)))),(LinearOperators ((REAL-NS m),(REAL-NS n)))) is set
K6(K7(K7((LinearOperators ((REAL-NS m),(REAL-NS n))),(LinearOperators ((REAL-NS m),(REAL-NS n)))),(LinearOperators ((REAL-NS m),(REAL-NS n))))) is set
K198(K279( the carrier of (REAL-NS m),(REAL-NS n)),(LinearOperators ((REAL-NS m),(REAL-NS n)))) is Relation-like K7(REAL,(LinearOperators ((REAL-NS m),(REAL-NS n)))) -defined LinearOperators ((REAL-NS m),(REAL-NS n)) -valued Function-like total V30(K7(REAL,(LinearOperators ((REAL-NS m),(REAL-NS n)))), LinearOperators ((REAL-NS m),(REAL-NS n))) Element of K6(K7(K7(REAL,(LinearOperators ((REAL-NS m),(REAL-NS n)))),(LinearOperators ((REAL-NS m),(REAL-NS n)))))
K7(REAL,(LinearOperators ((REAL-NS m),(REAL-NS n)))) is V33() set
K7(K7(REAL,(LinearOperators ((REAL-NS m),(REAL-NS n)))),(LinearOperators ((REAL-NS m),(REAL-NS n)))) is V33() set
K6(K7(K7(REAL,(LinearOperators ((REAL-NS m),(REAL-NS n)))),(LinearOperators ((REAL-NS m),(REAL-NS n))))) is V33() set
RLSStruct(# (LinearOperators ((REAL-NS m),(REAL-NS n))),K199(K279( the carrier of (REAL-NS m),(REAL-NS n)),(LinearOperators ((REAL-NS m),(REAL-NS n)))),K197(K279( the carrier of (REAL-NS m),(REAL-NS n)),(LinearOperators ((REAL-NS m),(REAL-NS n)))),K198(K279( the carrier of (REAL-NS m),(REAL-NS n)),(LinearOperators ((REAL-NS m),(REAL-NS n)))) #) is V71() strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RLSStruct
the carrier of (R_VectorSpace_of_LinearOperators ((REAL-NS m),(REAL-NS n))) is non empty set
K6( the carrier of (R_VectorSpace_of_LinearOperators ((REAL-NS m),(REAL-NS n)))) is set
K199((R_VectorSpace_of_LinearOperators ((REAL-NS m),(REAL-NS n))),(BoundedLinearOperators ((REAL-NS m),(REAL-NS n)))) is Element of BoundedLinearOperators ((REAL-NS m),(REAL-NS n))
K197((R_VectorSpace_of_LinearOperators ((REAL-NS m),(REAL-NS n))),(BoundedLinearOperators ((REAL-NS m),(REAL-NS n)))) is Relation-like K7((BoundedLinearOperators ((REAL-NS m),(REAL-NS n))),(BoundedLinearOperators ((REAL-NS m),(REAL-NS n)))) -defined BoundedLinearOperators ((REAL-NS m),(REAL-NS n)) -valued Function-like total V30(K7((BoundedLinearOperators ((REAL-NS m),(REAL-NS n))),(BoundedLinearOperators ((REAL-NS m),(REAL-NS n)))), BoundedLinearOperators ((REAL-NS m),(REAL-NS n))) Element of K6(K7(K7((BoundedLinearOperators ((REAL-NS m),(REAL-NS n))),(BoundedLinearOperators ((REAL-NS m),(REAL-NS n)))),(BoundedLinearOperators ((REAL-NS m),(REAL-NS n)))))
K7((BoundedLinearOperators ((REAL-NS m),(REAL-NS n))),(BoundedLinearOperators ((REAL-NS m),(REAL-NS n)))) is set
K7(K7((BoundedLinearOperators ((REAL-NS m),(REAL-NS n))),(BoundedLinearOperators ((REAL-NS m),(REAL-NS n)))),(BoundedLinearOperators ((REAL-NS m),(REAL-NS n)))) is set
K6(K7(K7((BoundedLinearOperators ((REAL-NS m),(REAL-NS n))),(BoundedLinearOperators ((REAL-NS m),(REAL-NS n)))),(BoundedLinearOperators ((REAL-NS m),(REAL-NS n))))) is set
K198((R_VectorSpace_of_LinearOperators ((REAL-NS m),(REAL-NS n))),(BoundedLinearOperators ((REAL-NS m),(REAL-NS n)))) is Relation-like K7(REAL,(BoundedLinearOperators ((REAL-NS m),(REAL-NS n)))) -defined BoundedLinearOperators ((REAL-NS m),(REAL-NS n)) -valued Function-like total V30(K7(REAL,(BoundedLinearOperators ((REAL-NS m),(REAL-NS n)))), BoundedLinearOperators ((REAL-NS m),(REAL-NS n))) Element of K6(K7(K7(REAL,(BoundedLinearOperators ((REAL-NS m),(REAL-NS n)))),(BoundedLinearOperators ((REAL-NS m),(REAL-NS n)))))
K7(REAL,(BoundedLinearOperators ((REAL-NS m),(REAL-NS n)))) is V33() set
K7(K7(REAL,(BoundedLinearOperators ((REAL-NS m),(REAL-NS n)))),(BoundedLinearOperators ((REAL-NS m),(REAL-NS n)))) is V33() set
K6(K7(K7(REAL,(BoundedLinearOperators ((REAL-NS m),(REAL-NS n)))),(BoundedLinearOperators ((REAL-NS m),(REAL-NS n))))) is V33() set
BoundedLinearOperatorsNorm ((REAL-NS m),(REAL-NS n)) is non empty Relation-like BoundedLinearOperators ((REAL-NS m),(REAL-NS n)) -defined REAL -valued Function-like total V30( BoundedLinearOperators ((REAL-NS m),(REAL-NS n)), REAL ) complex-yielding V146() V147() Element of K6(K7((BoundedLinearOperators ((REAL-NS m),(REAL-NS n))),REAL))
K7((BoundedLinearOperators ((REAL-NS m),(REAL-NS n))),REAL) is V33() complex-yielding V146() V147() set
K6(K7((BoundedLinearOperators ((REAL-NS m),(REAL-NS n))),REAL)) is V33() set
G15((BoundedLinearOperators ((REAL-NS m),(REAL-NS n))),K199((R_VectorSpace_of_LinearOperators ((REAL-NS m),(REAL-NS n))),(BoundedLinearOperators ((REAL-NS m),(REAL-NS n)))),K197((R_VectorSpace_of_LinearOperators ((REAL-NS m),(REAL-NS n))),(BoundedLinearOperators ((REAL-NS m),(REAL-NS n)))),K198((R_VectorSpace_of_LinearOperators ((REAL-NS m),(REAL-NS n))),(BoundedLinearOperators ((REAL-NS m),(REAL-NS n)))),(BoundedLinearOperatorsNorm ((REAL-NS m),(REAL-NS n)))) is V108() L15()
the carrier of (R_NormSpace_of_BoundedLinearOperators ((REAL-NS m),(REAL-NS n))) is non empty set
g /. Z0 is Element of the carrier of (REAL-NS n)
c₇ is Neighbourhood of Z0
x is Relation-like Function-like Element of the carrier of (R_NormSpace_of_BoundedLinearOperators ((REAL-NS m),(REAL-NS n)))
L is Relation-like the carrier of (REAL-NS m) -defined the carrier of (REAL-NS n) -valued Function-like RestFunc-like Element of K6(K7( the carrier of (REAL-NS m), the carrier of (REAL-NS n)))
x is Relation-like Function-like Element of the carrier of (R_NormSpace_of_BoundedLinearOperators ((REAL-NS m),(REAL-NS n)))
L is Relation-like the carrier of (REAL-NS m) -defined the carrier of (REAL-NS n) -valued Function-like RestFunc-like Element of K6(K7( the carrier of (REAL-NS m), the carrier of (REAL-NS n)))
L0 is V11() V12() ext-real Element of REAL
{ b₁ where b₁ is Element of the carrier of (REAL-NS m) : not L0 <= ||.(b₁ - Z0).|| } is set
R is V11() V12() ext-real Element of REAL
{ b₁ where b₁ is Element of the carrier of (REAL-NS m) : not R <= ||.(b₁ - Z0).|| } is set
R is Relation-like Function-like Element of the carrier of (R_NormSpace_of_BoundedLinearOperators ((REAL-NS m),(REAL-NS n)))
N is Relation-like the carrier of (REAL-NS m) -defined the carrier of (REAL-NS n) -valued Function-like RestFunc-like Element of K6(K7( the carrier of (REAL-NS m), the carrier of (REAL-NS n)))
{ b₁ where b₁ is Relation-like NAT -defined REAL -valued Function-like V40(m) FinSequence-like complex-yielding V146() V147() Element of REAL m : not R <= |.(b₁ - Z).| } is set
dom L is Element of K6( the carrier of (REAL-NS m))
R is non empty Relation-like REAL m -defined REAL n -valued Function-like total V30( REAL m, REAL n) complex-functions-valued ext-real-functions-valued real-functions-valued Element of K6(K7((REAL m),(REAL n)))
N is non empty Relation-like REAL m -defined REAL n -valued Function-like total V30( REAL m, REAL n) complex-functions-valued ext-real-functions-valued real-functions-valued Element of K6(K7((REAL m),(REAL n)))
N is non empty Relation-like REAL m -defined REAL n -valued Function-like total V30( REAL m, REAL n) complex-functions-valued ext-real-functions-valued real-functions-valued Element of K6(K7((REAL m),(REAL n)))
x is non empty Relation-like REAL m -defined REAL n -valued Function-like total V30( REAL m, REAL n) complex-functions-valued ext-real-functions-valued real-functions-valued Element of K6(K7((REAL m),(REAL n)))
y is V11() V12() ext-real Element of REAL
0. (REAL-NS m) is V52( REAL-NS m) Element of the carrier of (REAL-NS m)
Rz is V11() V12() ext-real Element of REAL
R0 is Relation-like NAT -defined REAL -valued Function-like V40(m) FinSequence-like complex-yielding V146() V147() Element of REAL m
|.R0.| is V11() V12() ext-real non negative Element of REAL
sqr R0 is Relation-like NAT -defined REAL -valued Function-like FinSequence-like complex-yielding V146() V147() FinSequence of REAL
R0 (#) sqrreal is Relation-like complex-yielding V146() V147() set
R0 (#) R0 is Relation-like Function-like set
K499() is Relation-like K7(REAL,REAL) -defined REAL -valued Function-like total V30(K7(REAL,REAL), REAL ) complex-yielding V146() V147() Element of K6(K7(K7(REAL,REAL),REAL))
K499() .: (R0,R0) is set
Sum (sqr R0) is V11() V12() ext-real Element of REAL
sqrt (Sum (sqr R0)) is V11() V12() ext-real Element of REAL
(m,n,x,R0) is Relation-like NAT -defined REAL -valued Function-like V40(n) FinSequence-like complex-yielding V146() V147() Element of REAL n
|.R0.| " is V11() V12() ext-real non negative Element of REAL
SRz is Relation-like NAT -defined REAL -valued Function-like V40(n) FinSequence-like complex-yielding V146() V147() Element of REAL n
|.SRz.| is V11() V12() ext-real non negative Element of REAL
sqr SRz is Relation-like NAT -defined REAL -valued Function-like FinSequence-like complex-yielding V146() V147() FinSequence of REAL
SRz (#) sqrreal is Relation-like complex-yielding V146() V147() set
SRz (#) SRz is Relation-like Function-like set
K499() .: (SRz,SRz) is set
Sum (sqr SRz) is V11() V12() ext-real Element of REAL
sqrt (Sum (sqr SRz)) is V11() V12() ext-real Element of REAL
(|.R0.| ") * |.SRz.| is V11() V12() ext-real non negative Element of REAL
c₂₀ is Element of the carrier of (REAL-NS m)
||.c₂₀.|| is V11() V12() ext-real Element of REAL
||.c₂₀.|| " is V11() V12() ext-real Element of REAL
L /. c₂₀ is Element of the carrier of (REAL-NS n)
||.(L /. c₂₀).|| is V11() V12() ext-real Element of REAL
(||.c₂₀.|| ") * ||.(L /. c₂₀).|| is V11() V12() ext-real Element of REAL
(|.R0.| ") * ||.(L /. c₂₀).|| is V11() V12() ext-real Element of REAL
y is Relation-like NAT -defined REAL -valued Function-like V40(m) FinSequence-like complex-yielding V146() V147() Element of REAL m
y - Z is Relation-like NAT -defined REAL -valued Function-like V40(m) FinSequence-like complex-yielding V146() V147() Element of REAL m
K498() is Relation-like K7(REAL,REAL) -defined REAL -valued Function-like total V30(K7(REAL,REAL), REAL ) complex-yielding V146() V147() Element of K6(K7(K7(REAL,REAL),REAL))
K497() is Relation-like K7(REAL,REAL) -defined REAL -valued Function-like total V30(K7(REAL,REAL), REAL ) complex-yielding V146() V147() Element of K6(K7(K7(REAL,REAL),REAL))
id REAL is non empty Relation-like REAL -defined REAL -valued total V30( REAL , REAL ) complex-yielding V146() V147() Element of K6(K7(REAL,REAL))
K495() is non empty Relation-like REAL -defined REAL -valued Function-like total V30( REAL , REAL ) complex-yielding V146() V147() Element of K6(K7(REAL,REAL))
K231(REAL,K497(),(id REAL),K495()) is Relation-like K7(REAL,REAL) -defined REAL -valued Function-like total V30(K7(REAL,REAL), REAL ) complex-yielding V146() V147() Element of K6(K7(K7(REAL,REAL),REAL))
K498() .: (y,Z) is set
- Z is Relation-like Function-like complex-yielding set
Z (#) K495() is Relation-like complex-yielding V146() V147() set
- 1 is V11() V12() ext-real non positive set
(- 1) (#) Z is Relation-like Function-like set
K515((- 1)) is non empty Relation-like REAL -defined REAL -valued Function-like total V30( REAL , REAL ) complex-yielding V146() V147() Element of K6(K7(REAL,REAL))
K499() is Relation-like K7(REAL,REAL) -defined REAL -valued Function-like total V30(K7(REAL,REAL), REAL ) complex-yielding V146() V147() Element of K6(K7(K7(REAL,REAL),REAL))
K499() [;] ((- 1),(id REAL)) is set
Z (#) K515((- 1)) is Relation-like complex-yielding V146() V147() set
y + (- Z) is Relation-like Function-like set
|.(y - Z).| is V11() V12() ext-real non negative Element of REAL
sqr (y - Z) is Relation-like NAT -defined REAL -valued Function-like FinSequence-like complex-yielding V146() V147() FinSequence of REAL
(y - Z) (#) sqrreal is Relation-like complex-yielding V146() V147() set
(y - Z) (#) (y - Z) is Relation-like Function-like set
K499() .: ((y - Z),(y - Z)) is set
Sum (sqr (y - Z)) is V11() V12() ext-real Element of REAL
sqrt (Sum (sqr (y - Z))) is V11() V12() ext-real Element of REAL
f /. y is V40(n) complex-yielding V146() V147() Element of REAL n
(f /. y) - (f /. Z) is Relation-like NAT -defined REAL -valued Function-like V40(n) FinSequence-like complex-yielding V146() V147() Element of REAL n
(m,n,N,(y - Z)) is Relation-like NAT -defined REAL -valued Function-like V40(n) FinSequence-like complex-yielding V146() V147() Element of REAL n
(m,n,x,(y - Z)) is Relation-like NAT -defined REAL -valued Function-like V40(n) FinSequence-like complex-yielding V146() V147() Element of REAL n
(m,n,N,(y - Z)) + (m,n,x,(y - Z)) is Relation-like NAT -defined REAL -valued Function-like V40(n) FinSequence-like complex-yielding V146() V147() Element of REAL n
K497() .: ((m,n,N,(y - Z)),(m,n,x,(y - Z))) is set
Rz is Element of the carrier of (REAL-NS m)
g /. Rz is Element of the carrier of (REAL-NS n)
(g /. Rz) - (g /. Z0) is Element of the carrier of (REAL-NS n)
Rz - Z0 is Element of the carrier of (REAL-NS m)
x . (Rz - Z0) is Element of the carrier of (REAL-NS n)
L /. (Rz - Z0) is Element of the carrier of (REAL-NS n)
(x . (Rz - Z0)) + (L /. (Rz - Z0)) is Element of the carrier of (REAL-NS n)
f . y is Relation-like Function-like complex-yielding V146() V147() set
f . Z is Relation-like Function-like complex-yielding V146() V147() set
R0 is Relation-like Function-like Element of the carrier of (R_NormSpace_of_BoundedLinearOperators ((REAL-NS m),(REAL-NS n)))
SRz is Relation-like the carrier of (REAL-NS m) -defined the carrier of (REAL-NS n) -valued Function-like RestFunc-like Element of K6(K7( the carrier of (REAL-NS m), the carrier of (REAL-NS n)))
R0 is Relation-like Function-like Element of the carrier of (R_NormSpace_of_BoundedLinearOperators ((REAL-NS m),(REAL-NS n)))
SRz is Relation-like the carrier of (REAL-NS m) -defined the carrier of (REAL-NS n) -valued Function-like RestFunc-like Element of K6(K7( the carrier of (REAL-NS m), the carrier of (REAL-NS n)))
c₇ is V11() V12() ext-real Element of REAL
{ b₁ where b₁ is Relation-like NAT -defined REAL -valued Function-like V40(m) FinSequence-like complex-yielding V146() V147() Element of REAL m : not c₇ <= |.(b₁ - Z).| } is set
x is non empty Relation-like REAL m -defined REAL n -valued Function-like total V30( REAL m, REAL n) complex-functions-valued ext-real-functions-valued real-functions-valued Element of K6(K7((REAL m),(REAL n)))
L is non empty Relation-like REAL m -defined REAL n -valued Function-like total V30( REAL m, REAL n) complex-functions-valued ext-real-functions-valued real-functions-valued Element of K6(K7((REAL m),(REAL n)))
m is V6() V10() V11() V12() ext-real non negative V33() V38() V120() V155() V156() V157() V158() V159() V160() V161() Element of NAT
REAL-NS m is non empty V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() V108() V109() V142() L15()
the carrier of (REAL-NS m) is non empty set
n is V6() V10() V11() V12() ext-real non negative V33() V38() V120() V155() V156() V157() V158() V159() V160() V161() Element of NAT
Proj (n,m) is non empty Relation-like the carrier of (REAL-NS m) -defined the carrier of (REAL-NS 1) -valued Function-like total V30( the carrier of (REAL-NS m), the carrier of (REAL-NS 1)) Element of K6(K7( the carrier of (REAL-NS m), the carrier of (REAL-NS 1)))
K7( the carrier of (REAL-NS m), the carrier of (REAL-NS 1)) is set
K6(K7( the carrier of (REAL-NS m), the carrier of (REAL-NS 1))) is set
f is Element of the carrier of (REAL-NS m)
Z is Element of the carrier of (REAL-NS m)
f + Z is Element of the carrier of (REAL-NS m)
(Proj (n,m)) . (f + Z) is Element of the carrier of (REAL-NS 1)
(Proj (n,m)) . f is Element of the carrier of (REAL-NS 1)
(Proj (n,m)) . Z is Element of the carrier of (REAL-NS 1)
((Proj (n,m)) . f) + ((Proj (n,m)) . Z) is Element of the carrier of (REAL-NS 1)
REAL m is non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered FinSequenceSet of REAL
m -tuples_on REAL is non empty functional FinSequence-membered FinSequenceSet of REAL
g is Relation-like NAT -defined REAL -valued Function-like V40(m) FinSequence-like complex-yielding V146() V147() Element of REAL m
g . n is V11() V12() ext-real Element of REAL
Z0 is Relation-like NAT -defined REAL -valued Function-like V40(m) FinSequence-like complex-yielding V146() V147() Element of REAL m
Z0 . n is V11() V12() ext-real Element of REAL
proj (n,m) is non empty Relation-like REAL m -defined REAL -valued Function-like total V30( REAL m, REAL ) complex-yielding V146() V147() Element of K6(K7((REAL m),REAL))
K7((REAL m),REAL) is V33() complex-yielding V146() V147() set
K6(K7((REAL m),REAL)) is V33() set
(proj (n,m)) . f is V11() V12() ext-real Element of REAL
<*((proj (n,m)) . f)*> is Relation-like NAT -defined REAL -valued Function-like FinSequence-like complex-yielding V146() V147() FinSequence of REAL
[1,((proj (n,m)) . f)] is set
{[1,((proj (n,m)) . f)]} is non empty V2() V40(1) set
(proj (n,m)) . Z is V11() V12() ext-real Element of REAL
<*((proj (n,m)) . Z)*> is Relation-like NAT -defined REAL -valued Function-like FinSequence-like complex-yielding V146() V147() FinSequence of REAL
[1,((proj (n,m)) . Z)] is set
{[1,((proj (n,m)) . Z)]} is non empty V2() V40(1) set
c₇ is V11() V12() ext-real Element of REAL
<*c₇*> is Relation-like NAT -defined REAL -valued Function-like FinSequence-like complex-yielding V146() V147() FinSequence of REAL
[1,c₇] is set
{[1,c₇]} is non empty V2() V40(1) set
x is V11() V12() ext-real Element of REAL
<*x*> is Relation-like NAT -defined REAL -valued Function-like FinSequence-like complex-yielding V146() V147() FinSequence of REAL
[1,x] is set
{[1,x]} is non empty V2() V40(1) set
(proj (n,m)) . (f + Z) is V11() V12() ext-real Element of REAL
<*((proj (n,m)) . (f + Z))*> is Relation-like NAT -defined REAL -valued Function-like FinSequence-like complex-yielding V146() V147() FinSequence of REAL
[1,((proj (n,m)) . (f + Z))] is set
{[1,((proj (n,m)) . (f + Z))]} is non empty V2() V40(1) set
g + Z0 is Relation-like NAT -defined REAL -valued Function-like V40(m) FinSequence-like complex-yielding V146() V147() Element of REAL m
K497() is Relation-like K7(REAL,REAL) -defined REAL -valued Function-like total V30(K7(REAL,REAL), REAL ) complex-yielding V146() V147() Element of K6(K7(K7(REAL,REAL),REAL))
K497() .: (g,Z0) is set
(proj (n,m)) . (g + Z0) is V11() V12() ext-real Element of REAL
<*((proj (n,m)) . (g + Z0))*> is Relation-like NAT -defined REAL -valued Function-like FinSequence-like complex-yielding V146() V147() FinSequence of REAL
[1,((proj (n,m)) . (g + Z0))] is set
{[1,((proj (n,m)) . (g + Z0))]} is non empty V2() V40(1) set
(g + Z0) . n is V11() V12() ext-real Element of REAL
<*((g + Z0) . n)*> is Relation-like NAT -defined REAL -valued Function-like FinSequence-like complex-yielding V146() V147() FinSequence of REAL
[1,((g + Z0) . n)] is set
{[1,((g + Z0) . n)]} is non empty V2() V40(1) set
(g . n) + (Z0 . n) is V11() V12() ext-real Element of REAL
<*((g . n) + (Z0 . n))*> is Relation-like NAT -defined REAL -valued Function-like FinSequence-like complex-yielding V146() V147() FinSequence of REAL
[1,((g . n) + (Z0 . n))] is set
{[1,((g . n) + (Z0 . n))]} is non empty V2() V40(1) set
<*c₇*> + <*x*> is Relation-like NAT -defined REAL -valued Function-like FinSequence-like complex-yielding V146() V147() FinSequence of REAL
K497() .: (<*c₇*>,<*x*>) is set
m is V6() V10() V11() V12() ext-real non negative V33() V38() V120() V155() V156() V157() V158() V159() V160() V161() Element of NAT
REAL-NS m is non empty V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() V108() V109() V142() L15()
the carrier of (REAL-NS m) is non empty set
n is V6() V10() V11() V12() ext-real non negative V33() V38() V120() V155() V156() V157() V158() V159() V160() V161() Element of NAT
Proj (n,m) is non empty Relation-like the carrier of (REAL-NS m) -defined the carrier of (REAL-NS 1) -valued Function-like total V30( the carrier of (REAL-NS m), the carrier of (REAL-NS 1)) Element of K6(K7( the carrier of (REAL-NS m), the carrier of (REAL-NS 1)))
K7( the carrier of (REAL-NS m), the carrier of (REAL-NS 1)) is set
K6(K7( the carrier of (REAL-NS m), the carrier of (REAL-NS 1))) is set
f is Element of the carrier of (REAL-NS m)
(Proj (n,m)) . f is Element of the carrier of (REAL-NS 1)
Z is V11() V12() ext-real Element of REAL
Z * f is Element of the carrier of (REAL-NS m)
(Proj (n,m)) . (Z * f) is Element of the carrier of (REAL-NS 1)
Z * ((Proj (n,m)) . f) is Element of the carrier of (REAL-NS 1)
REAL m is non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered FinSequenceSet of REAL
m -tuples_on REAL is non empty functional FinSequence-membered FinSequenceSet of REAL
g is Relation-like NAT -defined REAL -valued Function-like V40(m) FinSequence-like complex-yielding V146() V147() Element of REAL m
g . n is V11() V12() ext-real Element of REAL
proj (n,m) is non empty Relation-like REAL m -defined REAL -valued Function-like total V30( REAL m, REAL ) complex-yielding V146() V147() Element of K6(K7((REAL m),REAL))
K7((REAL m),REAL) is V33() complex-yielding V146() V147() set
K6(K7((REAL m),REAL)) is V33() set
(proj (n,m)) . f is V11() V12() ext-real Element of REAL
<*((proj (n,m)) . f)*> is Relation-like NAT -defined REAL -valued Function-like FinSequence-like complex-yielding V146() V147() FinSequence of REAL
[1,((proj (n,m)) . f)] is set
{[1,((proj (n,m)) . f)]} is non empty V2() V40(1) set
Z0 is V11() V12() ext-real Element of REAL
<*Z0*> is Relation-like NAT -defined REAL -valued Function-like FinSequence-like complex-yielding V146() V147() FinSequence of REAL
[1,Z0] is set
{[1,Z0]} is non empty V2() V40(1) set
(proj (n,m)) . (Z * f) is V11() V12() ext-real Element of REAL
<*((proj (n,m)) . (Z * f))*> is Relation-like NAT -defined REAL -valued Function-like FinSequence-like complex-yielding V146() V147() FinSequence of REAL
[1,((proj (n,m)) . (Z * f))] is set
{[1,((proj (n,m)) . (Z * f))]} is non empty V2() V40(1) set
Z * g is Relation-like NAT -defined REAL -valued Function-like V40(m) FinSequence-like complex-yielding V146() V147() Element of REAL m
K515(Z) is non empty Relation-like REAL -defined REAL -valued Function-like total V30( REAL , REAL ) complex-yielding V146() V147() Element of K6(K7(REAL,REAL))
K499() is Relation-like K7(REAL,REAL) -defined REAL -valued Function-like total V30(K7(REAL,REAL), REAL ) complex-yielding V146() V147() Element of K6(K7(K7(REAL,REAL),REAL))
id REAL is non empty Relation-like REAL -defined REAL -valued total V30( REAL , REAL ) complex-yielding V146() V147() Element of K6(K7(REAL,REAL))
K499() [;] (Z,(id REAL)) is set
g (#) K515(Z) is Relation-like complex-yielding V146() V147() set
(proj (n,m)) . (Z * g) is V11() V12() ext-real Element of REAL
<*((proj (n,m)) . (Z * g))*> is Relation-like NAT -defined REAL -valued Function-like FinSequence-like complex-yielding V146() V147() FinSequence of REAL
[1,((proj (n,m)) . (Z * g))] is set
{[1,((proj (n,m)) . (Z * g))]} is non empty V2() V40(1) set
(Z * g) . n is V11() V12() ext-real Element of REAL
<*((Z * g) . n)*> is Relation-like NAT -defined REAL -valued Function-like FinSequence-like complex-yielding V146() V147() FinSequence of REAL
[1,((Z * g) . n)] is set
{[1,((Z * g) . n)]} is non empty V2() V40(1) set
Z * (g . n) is V11() V12() ext-real Element of REAL
<*(Z * (g . n))*> is Relation-like NAT -defined REAL -valued Function-like FinSequence-like complex-yielding V146() V147() FinSequence of REAL
[1,(Z * (g . n))] is set
{[1,(Z * (g . n))]} is non empty V2() V40(1) set
Z * <*Z0*> is Relation-like NAT -defined REAL -valued Function-like FinSequence-like complex-yielding V146() V147() FinSequence of REAL
<*Z0*> (#) K515(Z) is Relation-like complex-yielding V146() V147() set
m is non empty V6() V10() V11() V12() ext-real positive non negative V33() V38() V120() V155() V156() V157() V158() V159() V160() V161() Element of NAT
REAL-NS m is non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() V108() V109() V142() L15()
the carrier of (REAL-NS m) is non empty V2() set
n is non empty V6() V10() V11() V12() ext-real positive non negative V33() V38() V120() V155() V156() V157() V158() V159() V160() V161() Element of NAT
REAL-NS n is non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() V108() V109() V142() L15()
the carrier of (REAL-NS n) is non empty V2() set
K7( the carrier of (REAL-NS m), the carrier of (REAL-NS n)) is set
K6(K7( the carrier of (REAL-NS m), the carrier of (REAL-NS n))) is set
f is Relation-like the carrier of (REAL-NS m) -defined the carrier of (REAL-NS n) -valued Function-like Element of K6(K7( the carrier of (REAL-NS m), the carrier of (REAL-NS n)))
Z is Element of the carrier of (REAL-NS m)
diff (f,Z) is Relation-like Function-like Element of the carrier of (R_NormSpace_of_BoundedLinearOperators ((REAL-NS m),(REAL-NS n)))
R_NormSpace_of_BoundedLinearOperators ((REAL-NS m),(REAL-NS n)) is non empty V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() V109() V142() L15()
BoundedLinearOperators ((REAL-NS m),(REAL-NS n)) is non empty Element of K6( the carrier of (R_VectorSpace_of_LinearOperators ((REAL-NS m),(REAL-NS n))))
R_VectorSpace_of_LinearOperators ((REAL-NS m),(REAL-NS n)) is non empty V71() strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V204() RLSStruct
LinearOperators ((REAL-NS m),(REAL-NS n)) is non empty functional Element of K6( the carrier of K279( the carrier of (REAL-NS m),(REAL-NS n)))
K279( the carrier of (REAL-NS m),(REAL-NS n)) is non empty V71() strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V204() RLSStruct
K123( the carrier of (REAL-NS m), the carrier of (REAL-NS n)) is non empty functional M4( the carrier of (REAL-NS m), the carrier of (REAL-NS n))
FuncZero ( the carrier of (REAL-NS m),(REAL-NS n)) is Relation-like the carrier of (REAL-NS m) -defined the carrier of (REAL-NS n) -valued Function-like total V30( the carrier of (REAL-NS m), the carrier of (REAL-NS n)) Element of K123( the carrier of (REAL-NS m), the carrier of (REAL-NS n))
0. (REAL-NS n) is V52( REAL-NS n) Element of the carrier of (REAL-NS n)
the carrier of (REAL-NS m) --> (0. (REAL-NS n)) is non empty Relation-like the carrier of (REAL-NS m) -defined the carrier of (REAL-NS n) -valued Function-like total V30( the carrier of (REAL-NS m), the carrier of (REAL-NS n)) Element of K6(K7( the carrier of (REAL-NS m), the carrier of (REAL-NS n)))
FuncAdd ( the carrier of (REAL-NS m),(REAL-NS n)) is Relation-like K7(K123( the carrier of (REAL-NS m), the carrier of (REAL-NS n)),K123( the carrier of (REAL-NS m), the carrier of (REAL-NS n))) -defined K123( the carrier of (REAL-NS m), the carrier of (REAL-NS n)) -valued Function-like total V30(K7(K123( the carrier of (REAL-NS m), the carrier of (REAL-NS n)),K123( the carrier of (REAL-NS m), the carrier of (REAL-NS n))),K123( the carrier of (REAL-NS m), the carrier of (REAL-NS n))) Element of K6(K7(K7(K123( the carrier of (REAL-NS m), the carrier of (REAL-NS n)),K123( the carrier of (REAL-NS m), the carrier of (REAL-NS n))),K123( the carrier of (REAL-NS m), the carrier of (REAL-NS n))))
K7(K123( the carrier of (REAL-NS m), the carrier of (REAL-NS n)),K123( the carrier of (REAL-NS m), the carrier of (REAL-NS n))) is set
K7(K7(K123( the carrier of (REAL-NS m), the carrier of (REAL-NS n)),K123( the carrier of (REAL-NS m), the carrier of (REAL-NS n))),K123( the carrier of (REAL-NS m), the carrier of (REAL-NS n))) is set
K6(K7(K7(K123( the carrier of (REAL-NS m), the carrier of (REAL-NS n)),K123( the carrier of (REAL-NS m), the carrier of (REAL-NS n))),K123( the carrier of (REAL-NS m), the carrier of (REAL-NS n)))) is set
FuncExtMult ( the carrier of (REAL-NS m),(REAL-NS n)) is Relation-like K7(REAL,K123( the carrier of (REAL-NS m), the carrier of (REAL-NS n))) -defined K123( the carrier of (REAL-NS m), the carrier of (REAL-NS n)) -valued Function-like total V30(K7(REAL,K123( the carrier of (REAL-NS m), the carrier of (REAL-NS n))),K123( the carrier of (REAL-NS m), the carrier of (REAL-NS n))) Element of K6(K7(K7(REAL,K123( the carrier of (REAL-NS m), the carrier of (REAL-NS n))),K123( the carrier of (REAL-NS m), the carrier of (REAL-NS n))))
K7(REAL,K123( the carrier of (REAL-NS m), the carrier of (REAL-NS n))) is V33() set
K7(K7(REAL,K123( the carrier of (REAL-NS m), the carrier of (REAL-NS n))),K123( the carrier of (REAL-NS m), the carrier of (REAL-NS n))) is V33() set
K6(K7(K7(REAL,K123( the carrier of (REAL-NS m), the carrier of (REAL-NS n))),K123( the carrier of (REAL-NS m), the carrier of (REAL-NS n)))) is V33() set
RLSStruct(# K123( the carrier of (REAL-NS m), the carrier of (REAL-NS n)),(FuncZero ( the carrier of (REAL-NS m),(REAL-NS n))),(FuncAdd ( the carrier of (REAL-NS m),(REAL-NS n))),(FuncExtMult ( the carrier of (REAL-NS m),(REAL-NS n))) #) is strict RLSStruct
the carrier of K279( the carrier of (REAL-NS m),(REAL-NS n)) is non empty set
K6( the carrier of K279( the carrier of (REAL-NS m),(REAL-NS n))) is set
K199(K279( the carrier of (REAL-NS m),(REAL-NS n)),(LinearOperators ((REAL-NS m),(REAL-NS n)))) is Element of LinearOperators ((REAL-NS m),(REAL-NS n))
K197(K279( the carrier of (REAL-NS m),(REAL-NS n)),(LinearOperators ((REAL-NS m),(REAL-NS n)))) is Relation-like K7((LinearOperators ((REAL-NS m),(REAL-NS n))),(LinearOperators ((REAL-NS m),(REAL-NS n)))) -defined LinearOperators ((REAL-NS m),(REAL-NS n)) -valued Function-like total V30(K7((LinearOperators ((REAL-NS m),(REAL-NS n))),(LinearOperators ((REAL-NS m),(REAL-NS n)))), LinearOperators ((REAL-NS m),(REAL-NS n))) Element of K6(K7(K7((LinearOperators ((REAL-NS m),(REAL-NS n))),(LinearOperators ((REAL-NS m),(REAL-NS n)))),(LinearOperators ((REAL-NS m),(REAL-NS n)))))
K7((LinearOperators ((REAL-NS m),(REAL-NS n))),(LinearOperators ((REAL-NS m),(REAL-NS n)))) is set
K7(K7((LinearOperators ((REAL-NS m),(REAL-NS n))),(LinearOperators ((REAL-NS m),(REAL-NS n)))),(LinearOperators ((REAL-NS m),(REAL-NS n)))) is set
K6(K7(K7((LinearOperators ((REAL-NS m),(REAL-NS n))),(LinearOperators ((REAL-NS m),(REAL-NS n)))),(LinearOperators ((REAL-NS m),(REAL-NS n))))) is set
K198(K279( the carrier of (REAL-NS m),(REAL-NS n)),(LinearOperators ((REAL-NS m),(REAL-NS n)))) is Relation-like K7(REAL,(LinearOperators ((REAL-NS m),(REAL-NS n)))) -defined LinearOperators ((REAL-NS m),(REAL-NS n)) -valued Function-like total V30(K7(REAL,(LinearOperators ((REAL-NS m),(REAL-NS n)))), LinearOperators ((REAL-NS m),(REAL-NS n))) Element of K6(K7(K7(REAL,(LinearOperators ((REAL-NS m),(REAL-NS n)))),(LinearOperators ((REAL-NS m),(REAL-NS n)))))
K7(REAL,(LinearOperators ((REAL-NS m),(REAL-NS n)))) is V33() set
K7(K7(REAL,(LinearOperators ((REAL-NS m),(REAL-NS n)))),(LinearOperators ((REAL-NS m),(REAL-NS n)))) is V33() set
K6(K7(K7(REAL,(LinearOperators ((REAL-NS m),(REAL-NS n)))),(LinearOperators ((REAL-NS m),(REAL-NS n))))) is V33() set
RLSStruct(# (LinearOperators ((REAL-NS m),(REAL-NS n))),K199(K279( the carrier of (REAL-NS m),(REAL-NS n)),(LinearOperators ((REAL-NS m),(REAL-NS n)))),K197(K279( the carrier of (REAL-NS m),(REAL-NS n)),(LinearOperators ((REAL-NS m),(REAL-NS n)))),K198(K279( the carrier of (REAL-NS m),(REAL-NS n)),(LinearOperators ((REAL-NS m),(REAL-NS n)))) #) is V71() strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RLSStruct
the carrier of (R_VectorSpace_of_LinearOperators ((REAL-NS m),(REAL-NS n))) is non empty set
K6( the carrier of (R_VectorSpace_of_LinearOperators ((REAL-NS m),(REAL-NS n)))) is set
K199((R_VectorSpace_of_LinearOperators ((REAL-NS m),(REAL-NS n))),(BoundedLinearOperators ((REAL-NS m),(REAL-NS n)))) is Element of BoundedLinearOperators ((REAL-NS m),(REAL-NS n))
K197((R_VectorSpace_of_LinearOperators ((REAL-NS m),(REAL-NS n))),(BoundedLinearOperators ((REAL-NS m),(REAL-NS n)))) is Relation-like K7((BoundedLinearOperators ((REAL-NS m),(REAL-NS n))),(BoundedLinearOperators ((REAL-NS m),(REAL-NS n)))) -defined BoundedLinearOperators ((REAL-NS m),(REAL-NS n)) -valued Function-like total V30(K7((BoundedLinearOperators ((REAL-NS m),(REAL-NS n))),(BoundedLinearOperators ((REAL-NS m),(REAL-NS n)))), BoundedLinearOperators ((REAL-NS m),(REAL-NS n))) Element of K6(K7(K7((BoundedLinearOperators ((REAL-NS m),(REAL-NS n))),(BoundedLinearOperators ((REAL-NS m),(REAL-NS n)))),(BoundedLinearOperators ((REAL-NS m),(REAL-NS n)))))
K7((BoundedLinearOperators ((REAL-NS m),(REAL-NS n))),(BoundedLinearOperators ((REAL-NS m),(REAL-NS n)))) is set
K7(K7((BoundedLinearOperators ((REAL-NS m),(REAL-NS n))),(BoundedLinearOperators ((REAL-NS m),(REAL-NS n)))),(BoundedLinearOperators ((REAL-NS m),(REAL-NS n)))) is set
K6(K7(K7((BoundedLinearOperators ((REAL-NS m),(REAL-NS n))),(BoundedLinearOperators ((REAL-NS m),(REAL-NS n)))),(BoundedLinearOperators ((REAL-NS m),(REAL-NS n))))) is set
K198((R_VectorSpace_of_LinearOperators ((REAL-NS m),(REAL-NS n))),(BoundedLinearOperators ((REAL-NS m),(REAL-NS n)))) is Relation-like K7(REAL,(BoundedLinearOperators ((REAL-NS m),(REAL-NS n)))) -defined BoundedLinearOperators ((REAL-NS m),(REAL-NS n)) -valued Function-like total V30(K7(REAL,(BoundedLinearOperators ((REAL-NS m),(REAL-NS n)))), BoundedLinearOperators ((REAL-NS m),(REAL-NS n))) Element of K6(K7(K7(REAL,(BoundedLinearOperators ((REAL-NS m),(REAL-NS n)))),(BoundedLinearOperators ((REAL-NS m),(REAL-NS n)))))
K7(REAL,(BoundedLinearOperators ((REAL-NS m),(REAL-NS n)))) is V33() set
K7(K7(REAL,(BoundedLinearOperators ((REAL-NS m),(REAL-NS n)))),(BoundedLinearOperators ((REAL-NS m),(REAL-NS n)))) is V33() set
K6(K7(K7(REAL,(BoundedLinearOperators ((REAL-NS m),(REAL-NS n)))),(BoundedLinearOperators ((REAL-NS m),(REAL-NS n))))) is V33() set
BoundedLinearOperatorsNorm ((REAL-NS m),(REAL-NS n)) is non empty Relation-like BoundedLinearOperators ((REAL-NS m),(REAL-NS n)) -defined REAL -valued Function-like total V30( BoundedLinearOperators ((REAL-NS m),(REAL-NS n)), REAL ) complex-yielding V146() V147() Element of K6(K7((BoundedLinearOperators ((REAL-NS m),(REAL-NS n))),REAL))
K7((BoundedLinearOperators ((REAL-NS m),(REAL-NS n))),REAL) is V33() complex-yielding V146() V147() set
K6(K7((BoundedLinearOperators ((REAL-NS m),(REAL-NS n))),REAL)) is V33() set
G15((BoundedLinearOperators ((REAL-NS m),(REAL-NS n))),K199((R_VectorSpace_of_LinearOperators ((REAL-NS m),(REAL-NS n))),(BoundedLinearOperators ((REAL-NS m),(REAL-NS n)))),K197((R_VectorSpace_of_LinearOperators ((REAL-NS m),(REAL-NS n))),(BoundedLinearOperators ((REAL-NS m),(REAL-NS n)))),K198((R_VectorSpace_of_LinearOperators ((REAL-NS m),(REAL-NS n))),(BoundedLinearOperators ((REAL-NS m),(REAL-NS n)))),(BoundedLinearOperatorsNorm ((REAL-NS m),(REAL-NS n)))) is V108() L15()
the carrier of (R_NormSpace_of_BoundedLinearOperators ((REAL-NS m),(REAL-NS n))) is non empty set
g is V6() V10() V11() V12() ext-real non negative V33() V38() V120() V155() V156() V157() V158() V159() V160() V161() Element of NAT
Proj (g,n) is non empty Relation-like the carrier of (REAL-NS n) -defined the carrier of (REAL-NS 1) -valued Function-like total V30( the carrier of (REAL-NS n), the carrier of (REAL-NS 1)) Element of K6(K7( the carrier of (REAL-NS n), the carrier of (REAL-NS 1)))
K7( the carrier of (REAL-NS n), the carrier of (REAL-NS 1)) is set
K6(K7( the carrier of (REAL-NS n), the carrier of (REAL-NS 1))) is set
(Proj (g,n)) * f is Relation-like the carrier of (REAL-NS m) -defined the carrier of (REAL-NS 1) -valued Function-like Element of K6(K7( the carrier of (REAL-NS m), the carrier of (REAL-NS 1)))
K7( the carrier of (REAL-NS m), the carrier of (REAL-NS 1)) is set
K6(K7( the carrier of (REAL-NS m), the carrier of (REAL-NS 1))) is set
(diff (f,Z)) (#) (Proj (g,n)) is Relation-like set
diff (((Proj (g,n)) * f),Z) is Relation-like Function-like Element of the carrier of (R_NormSpace_of_BoundedLinearOperators ((REAL-NS m),(REAL-NS 1)))
R_NormSpace_of_BoundedLinearOperators ((REAL-NS m),(REAL-NS 1)) is non empty V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() V109() V142() L15()
BoundedLinearOperators ((REAL-NS m),(REAL-NS 1)) is non empty Element of K6( the carrier of (R_VectorSpace_of_LinearOperators ((REAL-NS m),(REAL-NS 1))))
R_VectorSpace_of_LinearOperators ((REAL-NS m),(REAL-NS 1)) is non empty V71() strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V204() RLSStruct
LinearOperators ((REAL-NS m),(REAL-NS 1)) is non empty functional Element of K6( the carrier of K279( the carrier of (REAL-NS m),(REAL-NS 1)))
K279( the carrier of (REAL-NS m),(REAL-NS 1)) is non empty V71() strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V204() RLSStruct
K123( the carrier of (REAL-NS m), the carrier of (REAL-NS 1)) is non empty functional M4( the carrier of (REAL-NS m), the carrier of (REAL-NS 1))
FuncZero ( the carrier of (REAL-NS m),(REAL-NS 1)) is Relation-like the carrier of (REAL-NS m) -defined the carrier of (REAL-NS 1) -valued Function-like total V30( the carrier of (REAL-NS m), the carrier of (REAL-NS 1)) Element of K123( the carrier of (REAL-NS m), the carrier of (REAL-NS 1))
the carrier of (REAL-NS m) --> (0. (REAL-NS 1)) is non empty Relation-like the carrier of (REAL-NS m) -defined the carrier of (REAL-NS 1) -valued Function-like total V30( the carrier of (REAL-NS m), the carrier of (REAL-NS 1)) Element of K6(K7( the carrier of (REAL-NS m), the carrier of (REAL-NS 1)))
FuncAdd ( the carrier of (REAL-NS m),(REAL-NS 1)) is Relation-like K7(K123( the carrier of (REAL-NS m), the carrier of (REAL-NS 1)),K123( the carrier of (REAL-NS m), the carrier of (REAL-NS 1))) -defined K123( the carrier of (REAL-NS m), the carrier of (REAL-NS 1)) -valued Function-like total V30(K7(K123( the carrier of (REAL-NS m), the carrier of (REAL-NS 1)),K123( the carrier of (REAL-NS m), the carrier of (REAL-NS 1))),K123( the carrier of (REAL-NS m), the carrier of (REAL-NS 1))) Element of K6(K7(K7(K123( the carrier of (REAL-NS m), the carrier of (REAL-NS 1)),K123( the carrier of (REAL-NS m), the carrier of (REAL-NS 1))),K123( the carrier of (REAL-NS m), the carrier of (REAL-NS 1))))
K7(K123( the carrier of (REAL-NS m), the carrier of (REAL-NS 1)),K123( the carrier of (REAL-NS m), the carrier of (REAL-NS 1))) is set
K7(K7(K123( the carrier of (REAL-NS m), the carrier of (REAL-NS 1)),K123( the carrier of (REAL-NS m), the carrier of (REAL-NS 1))),K123( the carrier of (REAL-NS m), the carrier of (REAL-NS 1))) is set
K6(K7(K7(K123( the carrier of (REAL-NS m), the carrier of (REAL-NS 1)),K123( the carrier of (REAL-NS m), the carrier of (REAL-NS 1))),K123( the carrier of (REAL-NS m), the carrier of (REAL-NS 1)))) is set
FuncExtMult ( the carrier of (REAL-NS m),(REAL-NS 1)) is Relation-like K7(REAL,K123( the carrier of (REAL-NS m), the carrier of (REAL-NS 1))) -defined K123( the carrier of (REAL-NS m), the carrier of (REAL-NS 1)) -valued Function-like total V30(K7(REAL,K123( the carrier of (REAL-NS m), the carrier of (REAL-NS 1))),K123( the carrier of (REAL-NS m), the carrier of (REAL-NS 1))) Element of K6(K7(K7(REAL,K123( the carrier of (REAL-NS m), the carrier of (REAL-NS 1))),K123( the carrier of (REAL-NS m), the carrier of (REAL-NS 1))))
K7(REAL,K123( the carrier of (REAL-NS m), the carrier of (REAL-NS 1))) is V33() set
K7(K7(REAL,K123( the carrier of (REAL-NS m), the carrier of (REAL-NS 1))),K123( the carrier of (REAL-NS m), the carrier of (REAL-NS 1))) is V33() set
K6(K7(K7(REAL,K123( the carrier of (REAL-NS m), the carrier of (REAL-NS 1))),K123( the carrier of (REAL-NS m), the carrier of (REAL-NS 1)))) is V33() set
RLSStruct(# K123( the carrier of (REAL-NS m), the carrier of (REAL-NS 1)),(FuncZero ( the carrier of (REAL-NS m),(REAL-NS 1))),(FuncAdd ( the carrier of (REAL-NS m),(REAL-NS 1))),(FuncExtMult ( the carrier of (REAL-NS m),(REAL-NS 1))) #) is strict RLSStruct
the carrier of K279( the carrier of (REAL-NS m),(REAL-NS 1)) is non empty set
K6( the carrier of K279( the carrier of (REAL-NS m),(REAL-NS 1))) is set
K199(K279( the carrier of (REAL-NS m),(REAL-NS 1)),(LinearOperators ((REAL-NS m),(REAL-NS 1)))) is Element of LinearOperators ((REAL-NS m),(REAL-NS 1))
K197(K279( the carrier of (REAL-NS m),(REAL-NS 1)),(LinearOperators ((REAL-NS m),(REAL-NS 1)))) is Relation-like K7((LinearOperators ((REAL-NS m),(REAL-NS 1))),(LinearOperators ((REAL-NS m),(REAL-NS 1)))) -defined LinearOperators ((REAL-NS m),(REAL-NS 1)) -valued Function-like total V30(K7((LinearOperators ((REAL-NS m),(REAL-NS 1))),(LinearOperators ((REAL-NS m),(REAL-NS 1)))), LinearOperators ((REAL-NS m),(REAL-NS 1))) Element of K6(K7(K7((LinearOperators ((REAL-NS m),(REAL-NS 1))),(LinearOperators ((REAL-NS m),(REAL-NS 1)))),(LinearOperators ((REAL-NS m),(REAL-NS 1)))))
K7((LinearOperators ((REAL-NS m),(REAL-NS 1))),(LinearOperators ((REAL-NS m),(REAL-NS 1)))) is set
K7(K7((LinearOperators ((REAL-NS m),(REAL-NS 1))),(LinearOperators ((REAL-NS m),(REAL-NS 1)))),(LinearOperators ((REAL-NS m),(REAL-NS 1)))) is set
K6(K7(K7((LinearOperators ((REAL-NS m),(REAL-NS 1))),(LinearOperators ((REAL-NS m),(REAL-NS 1)))),(LinearOperators ((REAL-NS m),(REAL-NS 1))))) is set
K198(K279( the carrier of (REAL-NS m),(REAL-NS 1)),(LinearOperators ((REAL-NS m),(REAL-NS 1)))) is Relation-like K7(REAL,(LinearOperators ((REAL-NS m),(REAL-NS 1)))) -defined LinearOperators ((REAL-NS m),(REAL-NS 1)) -valued Function-like total V30(K7(REAL,(LinearOperators ((REAL-NS m),(REAL-NS 1)))), LinearOperators ((REAL-NS m),(REAL-NS 1))) Element of K6(K7(K7(REAL,(LinearOperators ((REAL-NS m),(REAL-NS 1)))),(LinearOperators ((REAL-NS m),(REAL-NS 1)))))
K7(REAL,(LinearOperators ((REAL-NS m),(REAL-NS 1)))) is V33() set
K7(K7(REAL,(LinearOperators ((REAL-NS m),(REAL-NS 1)))),(LinearOperators ((REAL-NS m),(REAL-NS 1)))) is V33() set
K6(K7(K7(REAL,(LinearOperators ((REAL-NS m),(REAL-NS 1)))),(LinearOperators ((REAL-NS m),(REAL-NS 1))))) is V33() set
RLSStruct(# (LinearOperators ((REAL-NS m),(REAL-NS 1))),K199(K279( the carrier of (REAL-NS m),(REAL-NS 1)),(LinearOperators ((REAL-NS m),(REAL-NS 1)))),K197(K279( the carrier of (REAL-NS m),(REAL-NS 1)),(LinearOperators ((REAL-NS m),(REAL-NS 1)))),K198(K279( the carrier of (REAL-NS m),(REAL-NS 1)),(LinearOperators ((REAL-NS m),(REAL-NS 1)))) #) is V71() strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RLSStruct
the carrier of (R_VectorSpace_of_LinearOperators ((REAL-NS m),(REAL-NS 1))) is non empty set
K6( the carrier of (R_VectorSpace_of_LinearOperators ((REAL-NS m),(REAL-NS 1)))) is set
K199((R_VectorSpace_of_LinearOperators ((REAL-NS m),(REAL-NS 1))),(BoundedLinearOperators ((REAL-NS m),(REAL-NS 1)))) is Element of BoundedLinearOperators ((REAL-NS m),(REAL-NS 1))
K197((R_VectorSpace_of_LinearOperators ((REAL-NS m),(REAL-NS 1))),(BoundedLinearOperators ((REAL-NS m),(REAL-NS 1)))) is Relation-like K7((BoundedLinearOperators ((REAL-NS m),(REAL-NS 1))),(BoundedLinearOperators ((REAL-NS m),(REAL-NS 1)))) -defined BoundedLinearOperators ((REAL-NS m),(REAL-NS 1)) -valued Function-like total V30(K7((BoundedLinearOperators ((REAL-NS m),(REAL-NS 1))),(BoundedLinearOperators ((REAL-NS m),(REAL-NS 1)))), BoundedLinearOperators ((REAL-NS m),(REAL-NS 1))) Element of K6(K7(K7((BoundedLinearOperators ((REAL-NS m),(REAL-NS 1))),(BoundedLinearOperators ((REAL-NS m),(REAL-NS 1)))),(BoundedLinearOperators ((REAL-NS m),(REAL-NS 1)))))
K7((BoundedLinearOperators ((REAL-NS m),(REAL-NS 1))),(BoundedLinearOperators ((REAL-NS m),(REAL-NS 1)))) is set
K7(K7((BoundedLinearOperators ((REAL-NS m),(REAL-NS 1))),(BoundedLinearOperators ((REAL-NS m),(REAL-NS 1)))),(BoundedLinearOperators ((REAL-NS m),(REAL-NS 1)))) is set
K6(K7(K7((BoundedLinearOperators ((REAL-NS m),(REAL-NS 1))),(BoundedLinearOperators ((REAL-NS m),(REAL-NS 1)))),(BoundedLinearOperators ((REAL-NS m),(REAL-NS 1))))) is set
K198((R_VectorSpace_of_LinearOperators ((REAL-NS m),(REAL-NS 1))),(BoundedLinearOperators ((REAL-NS m),(REAL-NS 1)))) is Relation-like K7(REAL,(BoundedLinearOperators ((REAL-NS m),(REAL-NS 1)))) -defined BoundedLinearOperators ((REAL-NS m),(REAL-NS 1)) -valued Function-like total V30(K7(REAL,(BoundedLinearOperators ((REAL-NS m),(REAL-NS 1)))), BoundedLinearOperators ((REAL-NS m),(REAL-NS 1))) Element of K6(K7(K7(REAL,(BoundedLinearOperators ((REAL-NS m),(REAL-NS 1)))),(BoundedLinearOperators ((REAL-NS m),(REAL-NS 1)))))
K7(REAL,(BoundedLinearOperators ((REAL-NS m),(REAL-NS 1)))) is V33() set
K7(K7(REAL,(BoundedLinearOperators ((REAL-NS m),(REAL-NS 1)))),(BoundedLinearOperators ((REAL-NS m),(REAL-NS 1)))) is V33() set
K6(K7(K7(REAL,(BoundedLinearOperators ((REAL-NS m),(REAL-NS 1)))),(BoundedLinearOperators ((REAL-NS m),(REAL-NS 1))))) is V33() set
BoundedLinearOperatorsNorm ((REAL-NS m),(REAL-NS 1)) is non empty Relation-like BoundedLinearOperators ((REAL-NS m),(REAL-NS 1)) -defined REAL -valued Function-like total V30( BoundedLinearOperators ((REAL-NS m),(REAL-NS 1)), REAL ) complex-yielding V146() V147() Element of K6(K7((BoundedLinearOperators ((REAL-NS m),(REAL-NS 1))),REAL))
K7((BoundedLinearOperators ((REAL-NS m),(REAL-NS 1))),REAL) is V33() complex-yielding V146() V147() set
K6(K7((BoundedLinearOperators ((REAL-NS m),(REAL-NS 1))),REAL)) is V33() set
G15((BoundedLinearOperators ((REAL-NS m),(REAL-NS 1))),K199((R_VectorSpace_of_LinearOperators ((REAL-NS m),(REAL-NS 1))),(BoundedLinearOperators ((REAL-NS m),(REAL-NS 1)))),K197((R_VectorSpace_of_LinearOperators ((REAL-NS m),(REAL-NS 1))),(BoundedLinearOperators ((REAL-NS m),(REAL-NS 1)))),K198((R_VectorSpace_of_LinearOperators ((REAL-NS m),(REAL-NS 1))),(BoundedLinearOperators ((REAL-NS m),(REAL-NS 1)))),(BoundedLinearOperatorsNorm ((REAL-NS m),(REAL-NS 1)))) is V108() L15()
the carrier of (R_NormSpace_of_BoundedLinearOperators ((REAL-NS m),(REAL-NS 1))) is non empty set
dom f is Element of K6( the carrier of (REAL-NS m))
K6( the carrier of (REAL-NS m)) is set
f /. Z is Element of the carrier of (REAL-NS n)
Z0 is Neighbourhood of Z
c₇ is Relation-like the carrier of (REAL-NS m) -defined the carrier of (REAL-NS n) -valued Function-like RestFunc-like Element of K6(K7( the carrier of (REAL-NS m), the carrier of (REAL-NS n)))
c₇ is Relation-like the carrier of (REAL-NS m) -defined the carrier of (REAL-NS n) -valued Function-like RestFunc-like Element of K6(K7( the carrier of (REAL-NS m), the carrier of (REAL-NS n)))
L is non empty Relation-like the carrier of (REAL-NS m) -defined the carrier of (REAL-NS n) -valued Function-like total V30( the carrier of (REAL-NS m), the carrier of (REAL-NS n)) additive homogeneous Lipschitzian Element of K6(K7( the carrier of (REAL-NS m), the carrier of (REAL-NS n)))
L0 is non empty Relation-like the carrier of (REAL-NS n) -defined the carrier of (REAL-NS 1) -valued Function-like total V30( the carrier of (REAL-NS n), the carrier of (REAL-NS 1)) Element of K6(K7( the carrier of (REAL-NS n), the carrier of (REAL-NS 1)))
L0 * L is non empty Relation-like the carrier of (REAL-NS m) -defined the carrier of (REAL-NS 1) -valued Function-like total V30( the carrier of (REAL-NS m), the carrier of (REAL-NS 1)) Element of K6(K7( the carrier of (REAL-NS m), the carrier of (REAL-NS 1)))
R is non empty Relation-like the carrier of (REAL-NS m) -defined the carrier of (REAL-NS 1) -valued Function-like total V30( the carrier of (REAL-NS m), the carrier of (REAL-NS 1)) Element of K6(K7( the carrier of (REAL-NS m), the carrier of (REAL-NS 1)))
R is Element of the carrier of (REAL-NS m)
N is Element of the carrier of (REAL-NS m)
R + N is Element of the carrier of (REAL-NS m)
R . (R + N) is Element of the carrier of (REAL-NS 1)
R . R is Element of the carrier of (REAL-NS 1)
R . N is Element of the carrier of (REAL-NS 1)
(R . R) + (R . N) is Element of the carrier of (REAL-NS 1)
dom R is Element of K6( the carrier of (REAL-NS m))
L . (R + N) is Element of the carrier of (REAL-NS n)
L . R is Element of the carrier of (REAL-NS n)
L . N is Element of the carrier of (REAL-NS n)
(L . R) + (L . N) is Element of the carrier of (REAL-NS n)
(Proj (g,n)) . (L . (R + N)) is Element of the carrier of (REAL-NS 1)
(Proj (g,n)) . (L . R) is Element of the carrier of (REAL-NS 1)
(Proj (g,n)) . (L . N) is Element of the carrier of (REAL-NS 1)
((Proj (g,n)) . (L . R)) + ((Proj (g,n)) . (L . N)) is Element of the carrier of (REAL-NS 1)
((Proj (g,n)) . (L . R)) + (R . N) is Element of the carrier of (REAL-NS 1)
R is Element of the carrier of (REAL-NS m)
R . R is Element of the carrier of (REAL-NS 1)
N is V11() V12() ext-real Element of REAL
N * R is Element of the carrier of (REAL-NS m)
R . (N * R) is Element of the carrier of (REAL-NS 1)
N * (R . R) is Element of the carrier of (REAL-NS 1)
dom R is Element of K6( the carrier of (REAL-NS m))
L . (N * R) is Element of the carrier of (REAL-NS n)
L . R is Element of the carrier of (REAL-NS n)
N * (L . R) is Element of the carrier of (REAL-NS n)
(Proj (g,n)) . (L . (N * R)) is Element of the carrier of (REAL-NS 1)
(Proj (g,n)) . (L . R) is Element of the carrier of (REAL-NS 1)
N * ((Proj (g,n)) . (L . R)) is Element of the carrier of (REAL-NS 1)
R is non empty Relation-like the carrier of (REAL-NS m) -defined the carrier of (REAL-NS 1) -valued Function-like total V30( the carrier of (REAL-NS m), the carrier of (REAL-NS 1)) additive homogeneous Lipschitzian Element of K6(K7( the carrier of (REAL-NS m), the carrier of (REAL-NS 1)))
N is non empty Relation-like the carrier of (REAL-NS m) -defined the carrier of (REAL-NS n) -valued Function-like total V30( the carrier of (REAL-NS m), the carrier of (REAL-NS n)) Element of K6(K7( the carrier of (REAL-NS m), the carrier of (REAL-NS n)))
(Proj (g,n)) * N is non empty Relation-like the carrier of (REAL-NS m) -defined the carrier of (REAL-NS 1) -valued Function-like total V30( the carrier of (REAL-NS m), the carrier of (REAL-NS 1)) Element of K6(K7( the carrier of (REAL-NS m), the carrier of (REAL-NS 1)))
(Proj (g,n)) * c₇ is Relation-like the carrier of (REAL-NS m) -defined the carrier of (REAL-NS 1) -valued Function-like Element of K6(K7( the carrier of (REAL-NS m), the carrier of (REAL-NS 1)))
dom c₇ is Element of K6( the carrier of (REAL-NS m))
0. (REAL-NS m) is V52( REAL-NS m) Element of the carrier of (REAL-NS m)
x is Relation-like the carrier of (REAL-NS m) -defined the carrier of (REAL-NS 1) -valued Function-like Element of K6(K7( the carrier of (REAL-NS m), the carrier of (REAL-NS 1)))
y is V11() V12() ext-real Element of REAL
Rz is V11() V12() ext-real Element of REAL
R0 is Element of the carrier of (REAL-NS m)
||.R0.|| is V11() V12() ext-real Element of REAL
||.R0.|| " is V11() V12() ext-real Element of REAL
x /. R0 is Element of the carrier of (REAL-NS 1)
||.(x /. R0).|| is V11() V12() ext-real Element of REAL
(||.R0.|| ") * ||.(x /. R0).|| is V11() V12() ext-real Element of REAL
c₇ /. R0 is Element of the carrier of (REAL-NS n)
c₇ . R0 is set
Seg n is V156() V157() V158() V159() V160() V161() Element of K6(NAT)
{ b₁ where b₁ is V6() V10() V11() V12() ext-real non negative V33() V38() V120() V155() V156() V157() V158() V159() V160() V161() Element of NAT : ( 1 <= b₁ & b₁ <= n ) } is set
REAL n is non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered FinSequenceSet of REAL
n -tuples_on REAL is non empty functional FinSequence-membered FinSequenceSet of REAL
SRz is Element of the carrier of (REAL-NS n)
c₂₀ is Relation-like NAT -defined REAL -valued Function-like V40(n) FinSequence-like complex-yielding V146() V147() Element of REAL n
c₂₀ . g is V11() V12() ext-real Element of REAL
dom (Proj (g,n)) is Element of K6( the carrier of (REAL-NS n))
K6( the carrier of (REAL-NS n)) is set
dom ((Proj (g,n)) * c₇) is Element of K6( the carrier of (REAL-NS m))
((Proj (g,n)) * c₇) . R0 is set
(Proj (g,n)) . (c₇ . R0) is set
proj (g,n) is non empty Relation-like REAL n -defined REAL -valued Function-like total V30( REAL n, REAL ) complex-yielding V146() V147() Element of K6(K7((REAL n),REAL))
K7((REAL n),REAL) is V33() complex-yielding V146() V147() set
K6(K7((REAL n),REAL)) is V33() set
(proj (g,n)) . c₂₀ is V11() V12() ext-real Element of REAL
<*((proj (g,n)) . c₂₀)*> is Relation-like NAT -defined REAL -valued Function-like FinSequence-like complex-yielding V146() V147() FinSequence of REAL
[1,((proj (g,n)) . c₂₀)] is set
{[1,((proj (g,n)) . c₂₀)]} is non empty V2() V40(1) set
i is V11() V12() ext-real Element of REAL
<*i*> is Relation-like NAT -defined REAL -valued Function-like FinSequence-like complex-yielding V146() V147() FinSequence of REAL
[1,i] is set
{[1,i]} is non empty V2() V40(1) set
abs i is V11() V12() ext-real Element of REAL
||.(c₇ /. R0).|| is V11() V12() ext-real Element of REAL
(||.R0.|| ") * (abs i) is V11() V12() ext-real Element of REAL
(||.R0.|| ") * ||.(c₇ /. R0).|| is V11() V12() ext-real Element of REAL
rng ((Proj (g,n)) * c₇) is Element of K6( the carrier of (REAL-NS 1))
K6( the carrier of (REAL-NS 1)) is set
Rzi is Element of the carrier of (REAL-NS 1)
||.Rzi.|| is V11() V12() ext-real Element of REAL
|.<*i*>.| is V11() V12() ext-real non negative Element of REAL
sqr <*i*> is Relation-like NAT -defined REAL -valued Function-like FinSequence-like complex-yielding V146() V147() FinSequence of REAL
<*i*> (#) sqrreal is Relation-like complex-yielding V146() V147() set
<*i*> (#) <*i*> is Relation-like Function-like set
K499() is Relation-like K7(REAL,REAL) -defined REAL -valued Function-like total V30(K7(REAL,REAL), REAL ) complex-yielding V146() V147() Element of K6(K7(K7(REAL,REAL),REAL))
K499() .: (<*i*>,<*i*>) is set
Sum (sqr <*i*>) is V11() V12() ext-real Element of REAL
sqrt (Sum (sqr <*i*>)) is V11() V12() ext-real Element of REAL
(||.R0.|| ") * ||.Rzi.|| is V11() V12() ext-real Element of REAL
dom (Proj (g,n)) is Element of K6( the carrier of (REAL-NS n))
K6( the carrier of (REAL-NS n)) is set
rng f is Element of K6( the carrier of (REAL-NS n))
dom ((Proj (g,n)) * f) is Element of K6( the carrier of (REAL-NS m))
REAL n is non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered FinSequenceSet of REAL
n -tuples_on REAL is non empty functional FinSequence-membered FinSequenceSet of REAL
((Proj (g,n)) * f) /. Z is Element of the carrier of (REAL-NS 1)
N is Relation-like Function-like Element of the carrier of (R_NormSpace_of_BoundedLinearOperators ((REAL-NS m),(REAL-NS 1)))
x is Relation-like the carrier of (REAL-NS m) -defined the carrier of (REAL-NS 1) -valued Function-like RestFunc-like Element of K6(K7( the carrier of (REAL-NS m), the carrier of (REAL-NS 1)))
Rz is Element of the carrier of (REAL-NS m)
((Proj (g,n)) * f) /. Rz is Element of the carrier of (REAL-NS 1)
(((Proj (g,n)) * f) /. Rz) - (((Proj (g,n)) * f) /. Z) is Element of the carrier of (REAL-NS 1)
Rz - Z is Element of the carrier of (REAL-NS m)
N . (Rz - Z) is Element of the carrier of (REAL-NS 1)
x /. (Rz - Z) is Element of the carrier of (REAL-NS 1)
(N . (Rz - Z)) + (x /. (Rz - Z)) is Element of the carrier of (REAL-NS 1)
f /. Rz is Element of the carrier of (REAL-NS n)
(f /. Rz) - (f /. Z) is Element of the carrier of (REAL-NS n)
(diff (f,Z)) . (Rz - Z) is Element of the carrier of (REAL-NS n)
c₇ /. (Rz - Z) is Element of the carrier of (REAL-NS n)
((diff (f,Z)) . (Rz - Z)) + (c₇ /. (Rz - Z)) is Element of the carrier of (REAL-NS n)
((Proj (g,n)) * f) . Rz is set
((Proj (g,n)) * f) . Z is set
R0 is Relation-like the carrier of (REAL-NS m) -defined the carrier of (REAL-NS n) -valued Function-like RestFunc-like Element of K6(K7( the carrier of (REAL-NS m), the carrier of (REAL-NS n)))
f . Rz is set
f . Z is set
R0 is Relation-like the carrier of (REAL-NS m) -defined the carrier of (REAL-NS n) -valued Function-like RestFunc-like Element of K6(K7( the carrier of (REAL-NS m), the carrier of (REAL-NS n)))
rng ((Proj (g,n)) * f) is Element of K6( the carrier of (REAL-NS 1))
K6( the carrier of (REAL-NS 1)) is set
SRz is Relation-like the carrier of (REAL-NS m) -defined the carrier of (REAL-NS n) -valued Function-like RestFunc-like Element of K6(K7( the carrier of (REAL-NS m), the carrier of (REAL-NS n)))
c₂₀ is Relation-like the carrier of (REAL-NS m) -defined the carrier of (REAL-NS n) -valued Function-like RestFunc-like Element of K6(K7( the carrier of (REAL-NS m), the carrier of (REAL-NS n)))
i is Relation-like the carrier of (REAL-NS m) -defined the carrier of (REAL-NS n) -valued Function-like RestFunc-like Element of K6(K7( the carrier of (REAL-NS m), the carrier of (REAL-NS n)))
Rzi is Relation-like the carrier of (REAL-NS m) -defined the carrier of (REAL-NS n) -valued Function-like RestFunc-like Element of K6(K7( the carrier of (REAL-NS m), the carrier of (REAL-NS n)))
(Proj (g,n)) . (f . Rz) is set
i is Relation-like NAT -defined REAL -valued Function-like V40(n) FinSequence-like complex-yielding V146() V147() Element of REAL n
rng (Proj (g,n)) is Element of K6( the carrier of (REAL-NS 1))
(Proj (g,n)) . (f . Z) is set
Rzi is Relation-like NAT -defined REAL -valued Function-like V40(n) FinSequence-like complex-yielding V146() V147() Element of REAL n
dom x is Element of K6( the carrier of (REAL-NS m))
x . (Rz - Z) is set
Ldiff is Relation-like NAT -defined REAL -valued Function-like V40(n) FinSequence-like complex-yielding V146() V147() Element of REAL n
(Proj (g,n)) . Ldiff is set
dom c₇ is Element of K6( the carrier of (REAL-NS m))
c₇ . (Rz - Z) is set
rng c₇ is Element of K6( the carrier of (REAL-NS n))
Rdiff is Relation-like NAT -defined REAL -valued Function-like V40(n) FinSequence-like complex-yielding V146() V147() Element of REAL n
(Proj (g,n)) . Rdiff is set
dom N is set
Ldxx0 is Element of the carrier of (REAL-NS 1)
Rdxx0 is Element of the carrier of (REAL-NS 1)
Ldxx0 + Rdxx0 is Element of the carrier of (REAL-NS 1)
ProjLdiff is non empty V2() Relation-like NAT -defined REAL -valued Function-like V40(1) FinSequence-like complex-yielding V146() V147() V149() V150() V151() V152() Element of REAL 1
ProjRdiff is non empty V2() Relation-like NAT -defined REAL -valued Function-like V40(1) FinSequence-like complex-yielding V146() V147() V149() V150() V151() V152() Element of REAL 1
ProjLdiff + ProjRdiff is non empty V2() Relation-like NAT -defined REAL -valued Function-like V40(1) FinSequence-like complex-yielding V146() V147() V149() V150() V151() V152() Element of REAL 1
K497() is Relation-like K7(REAL,REAL) -defined REAL -valued Function-like total V30(K7(REAL,REAL), REAL ) complex-yielding V146() V147() Element of K6(K7(K7(REAL,REAL),REAL))
K497() .: (ProjLdiff,ProjRdiff) is set
proj (g,n) is non empty Relation-like REAL n -defined REAL -valued Function-like total V30( REAL n, REAL ) complex-yielding V146() V147() Element of K6(K7((REAL n),REAL))
K7((REAL n),REAL) is V33() complex-yielding V146() V147() set
K6(K7((REAL n),REAL)) is V33() set
(proj (g,n)) . Ldiff is V11() V12() ext-real Element of REAL
<*((proj (g,n)) . Ldiff)*> is Relation-like NAT -defined REAL -valued Function-like FinSequence-like complex-yielding V146() V147() FinSequence of REAL
[1,((proj (g,n)) . Ldiff)] is set
{[1,((proj (g,n)) . Ldiff)]} is non empty V2() V40(1) set
Ldiff . g is V11() V12() ext-real Element of REAL
<*(Ldiff . g)*> is Relation-like NAT -defined REAL -valued Function-like FinSequence-like complex-yielding V146() V147() FinSequence of REAL
[1,(Ldiff . g)] is set
{[1,(Ldiff . g)]} is non empty V2() V40(1) set
(proj (g,n)) . Rdiff is V11() V12() ext-real Element of REAL
<*((proj (g,n)) . Rdiff)*> is Relation-like NAT -defined REAL -valued Function-like FinSequence-like complex-yielding V146() V147() FinSequence of REAL
[1,((proj (g,n)) . Rdiff)] is set
{[1,((proj (g,n)) . Rdiff)]} is non empty V2() V40(1) set
Rdiff . g is V11() V12() ext-real Element of REAL
<*(Rdiff . g)*> is Relation-like NAT -defined REAL -valued Function-like FinSequence-like complex-yielding V146() V147() FinSequence of REAL
[1,(Rdiff . g)] is set
{[1,(Rdiff . g)]} is non empty V2() V40(1) set
R0 is non empty V2() Relation-like NAT -defined REAL -valued Function-like V40(1) FinSequence-like complex-yielding V146() V147() V149() V150() V151() V152() Element of REAL 1
SRz is non empty V2() Relation-like NAT -defined REAL -valued Function-like V40(1) FinSequence-like complex-yielding V146() V147() V149() V150() V151() V152() Element of REAL 1
c₂₀ is non empty V2() Relation-like NAT -defined REAL -valued Function-like V40(1) FinSequence-like complex-yielding V146() V147() V149() V150() V151() V152() Element of REAL 1
SRz - c₂₀ is non empty V2() Relation-like NAT -defined REAL -valued Function-like V40(1) FinSequence-like complex-yielding V146() V147() V149() V150() V151() V152() Element of REAL 1
K498() is Relation-like K7(REAL,REAL) -defined REAL -valued Function-like total V30(K7(REAL,REAL), REAL ) complex-yielding V146() V147() Element of K6(K7(K7(REAL,REAL),REAL))
id REAL is non empty Relation-like REAL -defined REAL -valued total V30( REAL , REAL ) complex-yielding V146() V147() Element of K6(K7(REAL,REAL))
K495() is non empty Relation-like REAL -defined REAL -valued Function-like total V30( REAL , REAL ) complex-yielding V146() V147() Element of K6(K7(REAL,REAL))
K231(REAL,K497(),(id REAL),K495()) is Relation-like K7(REAL,REAL) -defined REAL -valued Function-like total V30(K7(REAL,REAL), REAL ) complex-yielding V146() V147() Element of K6(K7(K7(REAL,REAL),REAL))
K498() .: (SRz,c₂₀) is set
- c₂₀ is Relation-like Function-like complex-yielding set
c₂₀ (#) K495() is Relation-like complex-yielding V146() V147() set
- 1 is V11() V12() ext-real non positive set
(- 1) (#) c₂₀ is Relation-like Function-like set
K515((- 1)) is non empty Relation-like REAL -defined REAL -valued Function-like total V30( REAL , REAL ) complex-yielding V146() V147() Element of K6(K7(REAL,REAL))
K499() is Relation-like K7(REAL,REAL) -defined REAL -valued Function-like total V30(K7(REAL,REAL), REAL ) complex-yielding V146() V147() Element of K6(K7(K7(REAL,REAL),REAL))
K499() [;] ((- 1),(id REAL)) is set
c₂₀ (#) K515((- 1)) is Relation-like complex-yielding V146() V147() set
SRz + (- c₂₀) is Relation-like Function-like set
ProjGx is non empty V2() Relation-like NAT -defined REAL -valued Function-like V40(1) FinSequence-like complex-yielding V146() V147() V149() V150() V151() V152() Element of REAL 1
ProjGx - c₂₀ is non empty V2() Relation-like NAT -defined REAL -valued Function-like V40(1) FinSequence-like complex-yielding V146() V147() V149() V150() V151() V152() Element of REAL 1
K498() .: (ProjGx,c₂₀) is set
ProjGx + (- c₂₀) is Relation-like Function-like set
ProjGx0 is non empty V2() Relation-like NAT -defined REAL -valued Function-like V40(1) FinSequence-like complex-yielding V146() V147() V149() V150() V151() V152() Element of REAL 1
ProjGx - ProjGx0 is non empty V2() Relation-like NAT -defined REAL -valued Function-like V40(1) FinSequence-like complex-yielding V146() V147() V149() V150() V151() V152() Element of REAL 1
K498() .: (ProjGx,ProjGx0) is set
- ProjGx0 is Relation-like Function-like complex-yielding set
ProjGx0 (#) K495() is Relation-like complex-yielding V146() V147() set
(- 1) (#) ProjGx0 is Relation-like Function-like set
ProjGx0 (#) K515((- 1)) is Relation-like complex-yielding V146() V147() set
ProjGx + (- ProjGx0) is Relation-like Function-like set
Gx1 is Element of the carrier of (REAL-NS n)
(proj (g,n)) . Gx1 is V11() V12() ext-real Element of REAL
<*((proj (g,n)) . Gx1)*> is Relation-like NAT -defined REAL -valued Function-like FinSequence-like complex-yielding V146() V147() FinSequence of REAL
[1,((proj (g,n)) . Gx1)] is set
{[1,((proj (g,n)) . Gx1)]} is non empty V2() V40(1) set
<*((proj (g,n)) . Gx1)*> - ProjGx0 is Relation-like NAT -defined REAL -valued Function-like FinSequence-like complex-yielding V146() V147() FinSequence of REAL
K498() .: (<*((proj (g,n)) . Gx1)*>,ProjGx0) is set
<*((proj (g,n)) . Gx1)*> + (- ProjGx0) is Relation-like Function-like set
Gx01 is Element of the carrier of (REAL-NS n)
(proj (g,n)) . Gx01 is V11() V12() ext-real Element of REAL
<*((proj (g,n)) . Gx01)*> is Relation-like NAT -defined REAL -valued Function-like FinSequence-like complex-yielding V146() V147() FinSequence of REAL
[1,((proj (g,n)) . Gx01)] is set
{[1,((proj (g,n)) . Gx01)]} is non empty V2() V40(1) set
<*((proj (g,n)) . Gx1)*> - <*((proj (g,n)) . Gx01)*> is Relation-like NAT -defined REAL -valued Function-like FinSequence-like complex-yielding V146() V147() FinSequence of REAL
K498() .: (<*((proj (g,n)) . Gx1)*>,<*((proj (g,n)) . Gx01)*>) is set
- <*((proj (g,n)) . Gx01)*> is Relation-like Function-like complex-yielding set
<*((proj (g,n)) . Gx01)*> (#) K495() is Relation-like complex-yielding V146() V147() set
(- 1) (#) <*((proj (g,n)) . Gx01)*> is Relation-like Function-like set
<*((proj (g,n)) . Gx01)*> (#) K515((- 1)) is Relation-like complex-yielding V146() V147() set
<*((proj (g,n)) . Gx1)*> + (- <*((proj (g,n)) . Gx01)*>) is Relation-like Function-like set
i . g is V11() V12() ext-real Element of REAL
<*(i . g)*> is Relation-like NAT -defined REAL -valued Function-like FinSequence-like complex-yielding V146() V147() FinSequence of REAL
[1,(i . g)] is set
{[1,(i . g)]} is non empty V2() V40(1) set
<*(i . g)*> - <*((proj (g,n)) . Gx01)*> is Relation-like NAT -defined REAL -valued Function-like FinSequence-like complex-yielding V146() V147() FinSequence of REAL
K498() .: (<*(i . g)*>,<*((proj (g,n)) . Gx01)*>) is set
<*(i . g)*> + (- <*((proj (g,n)) . Gx01)*>) is Relation-like Function-like set
Rzi . g is V11() V12() ext-real Element of REAL
<*(Rzi . g)*> is Relation-like NAT -defined REAL -valued Function-like FinSequence-like complex-yielding V146() V147() FinSequence of REAL
[1,(Rzi . g)] is set
{[1,(Rzi . g)]} is non empty V2() V40(1) set
<*(i . g)*> - <*(Rzi . g)*> is Relation-like NAT -defined REAL -valued Function-like FinSequence-like complex-yielding V146() V147() FinSequence of REAL
K498() .: (<*(i . g)*>,<*(Rzi . g)*>) is set
- <*(Rzi . g)*> is Relation-like Function-like complex-yielding set
<*(Rzi . g)*> (#) K495() is Relation-like complex-yielding V146() V147() set
(- 1) (#) <*(Rzi . g)*> is Relation-like Function-like set
<*(Rzi . g)*> (#) K515((- 1)) is Relation-like complex-yielding V146() V147() set
<*(i . g)*> + (- <*(Rzi . g)*>) is Relation-like Function-like set
(i . g) - (Rzi . g) is V11() V12() ext-real Element of REAL
- (Rzi . g) is V11() V12() ext-real set
(i . g) + (- (Rzi . g)) is V11() V12() ext-real set
<*((i . g) - (Rzi . g))*> is Relation-like NAT -defined REAL -valued Function-like FinSequence-like complex-yielding V146() V147() FinSequence of REAL
[1,((i . g) - (Rzi . g))] is set
{[1,((i . g) - (Rzi . g))]} is non empty V2() V40(1) set
Gsx is Relation-like NAT -defined REAL -valued Function-like V40(n) FinSequence-like complex-yielding V146() V147() Element of REAL n
Gsx0 is Relation-like NAT -defined REAL -valued Function-like V40(n) FinSequence-like complex-yielding V146() V147() Element of REAL n
Gsx - Gsx0 is Relation-like NAT -defined REAL -valued Function-like V40(n) FinSequence-like complex-yielding V146() V147() Element of REAL n
K498() .: (Gsx,Gsx0) is set
- Gsx0 is Relation-like Function-like complex-yielding set
Gsx0 (#) K495() is Relation-like complex-yielding V146() V147() set
(- 1) (#) Gsx0 is Relation-like Function-like set
Gsx0 (#) K515((- 1)) is Relation-like complex-yielding V146() V147() set
Gsx + (- Gsx0) is Relation-like Function-like set
(Gsx - Gsx0) . g is V11() V12() ext-real Element of REAL
<*((Gsx - Gsx0) . g)*> is Relation-like NAT -defined REAL -valued Function-like FinSequence-like complex-yielding V146() V147() FinSequence of REAL
[1,((Gsx - Gsx0) . g)] is set
{[1,((Gsx - Gsx0) . g)]} is non empty V2() V40(1) set
diffGR is Relation-like NAT -defined REAL -valued Function-like V40(n) FinSequence-like complex-yielding V146() V147() Element of REAL n
diffGR . g is V11() V12() ext-real Element of REAL
<*(diffGR . g)*> is Relation-like NAT -defined REAL -valued Function-like FinSequence-like complex-yielding V146() V147() FinSequence of REAL
[1,(diffGR . g)] is set
{[1,(diffGR . g)]} is non empty V2() V40(1) set
Rsdiff is Relation-like NAT -defined REAL -valued Function-like V40(n) FinSequence-like complex-yielding V146() V147() Element of REAL n
Ldiff + Rsdiff is Relation-like NAT -defined REAL -valued Function-like V40(n) FinSequence-like complex-yielding V146() V147() Element of REAL n
K497() .: (Ldiff,Rsdiff) is set
(Ldiff + Rsdiff) . g is V11() V12() ext-real Element of REAL
<*((Ldiff + Rsdiff) . g)*> is Relation-like NAT -defined REAL -valued Function-like FinSequence-like complex-yielding V146() V147() FinSequence of REAL
[1,((Ldiff + Rsdiff) . g)] is set
{[1,((Ldiff + Rsdiff) . g)]} is non empty V2() V40(1) set
Rsdiff . g is V11() V12() ext-real Element of REAL
(Ldiff . g) + (Rsdiff . g) is V11() V12() ext-real Element of REAL
<*((Ldiff . g) + (Rsdiff . g))*> is Relation-like NAT -defined REAL -valued Function-like FinSequence-like complex-yielding V146() V147() FinSequence of REAL
[1,((Ldiff . g) + (Rsdiff . g))] is set
{[1,((Ldiff . g) + (Rsdiff . g))]} is non empty V2() V40(1) set
c₄₂ is Relation-like the carrier of (REAL-NS m) -defined the carrier of (REAL-NS n) -valued Function-like RestFunc-like Element of K6(K7( the carrier of (REAL-NS m), the carrier of (REAL-NS n)))
c₄₂ is Relation-like the carrier of (REAL-NS m) -defined the carrier of (REAL-NS n) -valued Function-like RestFunc-like Element of K6(K7( the carrier of (REAL-NS m), the carrier of (REAL-NS n)))
Rz is Relation-like the carrier of (REAL-NS m) -defined the carrier of (REAL-NS n) -valued Function-like RestFunc-like Element of K6(K7( the carrier of (REAL-NS m), the carrier of (REAL-NS n)))
Rz is Relation-like the carrier of (REAL-NS m) -defined the carrier of (REAL-NS n) -valued Function-like RestFunc-like Element of K6(K7( the carrier of (REAL-NS m), the carrier of (REAL-NS n)))
m is non empty V6() V10() V11() V12() ext-real positive non negative V33() V38() V120() V155() V156() V157() V158() V159() V160() V161() Element of NAT
REAL-NS m is non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() V108() V109() V142() L15()
the carrier of (REAL-NS m) is non empty V2() set
n is non empty V6() V10() V11() V12() ext-real positive non negative V33() V38() V120() V155() V156() V157() V158() V159() V160() V161() Element of NAT
REAL-NS n is non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() V108() V109() V142() L15()
the carrier of (REAL-NS n) is non empty V2() set
K7( the carrier of (REAL-NS m), the carrier of (REAL-NS n)) is set
K6(K7( the carrier of (REAL-NS m), the carrier of (REAL-NS n))) is set
f is Relation-like the carrier of (REAL-NS m) -defined the carrier of (REAL-NS n) -valued Function-like Element of K6(K7( the carrier of (REAL-NS m), the carrier of (REAL-NS n)))
Z is Element of the carrier of (REAL-NS m)
R_NormSpace_of_BoundedLinearOperators ((REAL-NS m),(REAL-NS n)) is non empty V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() V109() V142() L15()
BoundedLinearOperators ((REAL-NS m),(REAL-NS n)) is non empty Element of K6( the carrier of (R_VectorSpace_of_LinearOperators ((REAL-NS m),(REAL-NS n))))
R_VectorSpace_of_LinearOperators ((REAL-NS m),(REAL-NS n)) is non empty V71() strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V204() RLSStruct
LinearOperators ((REAL-NS m),(REAL-NS n)) is non empty functional Element of K6( the carrier of K279( the carrier of (REAL-NS m),(REAL-NS n)))
K279( the carrier of (REAL-NS m),(REAL-NS n)) is non empty V71() strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V204() RLSStruct
K123( the carrier of (REAL-NS m), the carrier of (REAL-NS n)) is non empty functional M4( the carrier of (REAL-NS m), the carrier of (REAL-NS n))
FuncZero ( the carrier of (REAL-NS m),(REAL-NS n)) is Relation-like the carrier of (REAL-NS m) -defined the carrier of (REAL-NS n) -valued Function-like total V30( the carrier of (REAL-NS m), the carrier of (REAL-NS n)) Element of K123( the carrier of (REAL-NS m), the carrier of (REAL-NS n))
0. (REAL-NS n) is V52( REAL-NS n) Element of the carrier of (REAL-NS n)
the carrier of (REAL-NS m) --> (0. (REAL-NS n)) is non empty Relation-like the carrier of (REAL-NS m) -defined the carrier of (REAL-NS n) -valued Function-like total V30( the carrier of (REAL-NS m), the carrier of (REAL-NS n)) Element of K6(K7( the carrier of (REAL-NS m), the carrier of (REAL-NS n)))
FuncAdd ( the carrier of (REAL-NS m),(REAL-NS n)) is Relation-like K7(K123( the carrier of (REAL-NS m), the carrier of (REAL-NS n)),K123( the carrier of (REAL-NS m), the carrier of (REAL-NS n))) -defined K123( the carrier of (REAL-NS m), the carrier of (REAL-NS n)) -valued Function-like total V30(K7(K123( the carrier of (REAL-NS m), the carrier of (REAL-NS n)),K123( the carrier of (REAL-NS m), the carrier of (REAL-NS n))),K123( the carrier of (REAL-NS m), the carrier of (REAL-NS n))) Element of K6(K7(K7(K123( the carrier of (REAL-NS m), the carrier of (REAL-NS n)),K123( the carrier of (REAL-NS m), the carrier of (REAL-NS n))),K123( the carrier of (REAL-NS m), the carrier of (REAL-NS n))))
K7(K123( the carrier of (REAL-NS m), the carrier of (REAL-NS n)),K123( the carrier of (REAL-NS m), the carrier of (REAL-NS n))) is set
K7(K7(K123( the carrier of (REAL-NS m), the carrier of (REAL-NS n)),K123( the carrier of (REAL-NS m), the carrier of (REAL-NS n))),K123( the carrier of (REAL-NS m), the carrier of (REAL-NS n))) is set
K6(K7(K7(K123( the carrier of (REAL-NS m), the carrier of (REAL-NS n)),K123( the carrier of (REAL-NS m), the carrier of (REAL-NS n))),K123( the carrier of (REAL-NS m), the carrier of (REAL-NS n)))) is set
FuncExtMult ( the carrier of (REAL-NS m),(REAL-NS n)) is Relation-like K7(REAL,K123( the carrier of (REAL-NS m), the carrier of (REAL-NS n))) -defined K123( the carrier of (REAL-NS m), the carrier of (REAL-NS n)) -valued Function-like total V30(K7(REAL,K123( the carrier of (REAL-NS m), the carrier of (REAL-NS n))),K123( the carrier of (REAL-NS m), the carrier of (REAL-NS n))) Element of K6(K7(K7(REAL,K123( the carrier of (REAL-NS m), the carrier of (REAL-NS n))),K123( the carrier of (REAL-NS m), the carrier of (REAL-NS n))))
K7(REAL,K123( the carrier of (REAL-NS m), the carrier of (REAL-NS n))) is V33() set
K7(K7(REAL,K123( the carrier of (REAL-NS m), the carrier of (REAL-NS n))),K123( the carrier of (REAL-NS m), the carrier of (REAL-NS n))) is V33() set
K6(K7(K7(REAL,K123( the carrier of (REAL-NS m), the carrier of (REAL-NS n))),K123( the carrier of (REAL-NS m), the carrier of (REAL-NS n)))) is V33() set
RLSStruct(# K123( the carrier of (REAL-NS m), the carrier of (REAL-NS n)),(FuncZero ( the carrier of (REAL-NS m),(REAL-NS n))),(FuncAdd ( the carrier of (REAL-NS m),(REAL-NS n))),(FuncExtMult ( the carrier of (REAL-NS m),(REAL-NS n))) #) is strict RLSStruct
the carrier of K279( the carrier of (REAL-NS m),(REAL-NS n)) is non empty set
K6( the carrier of K279( the carrier of (REAL-NS m),(REAL-NS n))) is set
K199(K279( the carrier of (REAL-NS m),(REAL-NS n)),(LinearOperators ((REAL-NS m),(REAL-NS n)))) is Element of LinearOperators ((REAL-NS m),(REAL-NS n))
K197(K279( the carrier of (REAL-NS m),(REAL-NS n)),(LinearOperators ((REAL-NS m),(REAL-NS n)))) is Relation-like K7((LinearOperators ((REAL-NS m),(REAL-NS n))),(LinearOperators ((REAL-NS m),(REAL-NS n)))) -defined LinearOperators ((REAL-NS m),(REAL-NS n)) -valued Function-like total V30(K7((LinearOperators ((REAL-NS m),(REAL-NS n))),(LinearOperators ((REAL-NS m),(REAL-NS n)))), LinearOperators ((REAL-NS m),(REAL-NS n))) Element of K6(K7(K7((LinearOperators ((REAL-NS m),(REAL-NS n))),(LinearOperators ((REAL-NS m),(REAL-NS n)))),(LinearOperators ((REAL-NS m),(REAL-NS n)))))
K7((LinearOperators ((REAL-NS m),(REAL-NS n))),(LinearOperators ((REAL-NS m),(REAL-NS n)))) is set
K7(K7((LinearOperators ((REAL-NS m),(REAL-NS n))),(LinearOperators ((REAL-NS m),(REAL-NS n)))),(LinearOperators ((REAL-NS m),(REAL-NS n)))) is set
K6(K7(K7((LinearOperators ((REAL-NS m),(REAL-NS n))),(LinearOperators ((REAL-NS m),(REAL-NS n)))),(LinearOperators ((REAL-NS m),(REAL-NS n))))) is set
K198(K279( the carrier of (REAL-NS m),(REAL-NS n)),(LinearOperators ((REAL-NS m),(REAL-NS n)))) is Relation-like K7(REAL,(LinearOperators ((REAL-NS m),(REAL-NS n)))) -defined LinearOperators ((REAL-NS m),(REAL-NS n)) -valued Function-like total V30(K7(REAL,(LinearOperators ((REAL-NS m),(REAL-NS n)))), LinearOperators ((REAL-NS m),(REAL-NS n))) Element of K6(K7(K7(REAL,(LinearOperators ((REAL-NS m),(REAL-NS n)))),(LinearOperators ((REAL-NS m),(REAL-NS n)))))
K7(REAL,(LinearOperators ((REAL-NS m),(REAL-NS n)))) is V33() set
K7(K7(REAL,(LinearOperators ((REAL-NS m),(REAL-NS n)))),(LinearOperators ((REAL-NS m),(REAL-NS n)))) is V33() set
K6(K7(K7(REAL,(LinearOperators ((REAL-NS m),(REAL-NS n)))),(LinearOperators ((REAL-NS m),(REAL-NS n))))) is V33() set
RLSStruct(# (LinearOperators ((REAL-NS m),(REAL-NS n))),K199(K279( the carrier of (REAL-NS m),(REAL-NS n)),(LinearOperators ((REAL-NS m),(REAL-NS n)))),K197(K279( the carrier of (REAL-NS m),(REAL-NS n)),(LinearOperators ((REAL-NS m),(REAL-NS n)))),K198(K279( the carrier of (REAL-NS m),(REAL-NS n)),(LinearOperators ((REAL-NS m),(REAL-NS n)))) #) is V71() strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RLSStruct
the carrier of (R_VectorSpace_of_LinearOperators ((REAL-NS m),(REAL-NS n))) is non empty set
K6( the carrier of (R_VectorSpace_of_LinearOperators ((REAL-NS m),(REAL-NS n)))) is set
K199((R_VectorSpace_of_LinearOperators ((REAL-NS m),(REAL-NS n))),(BoundedLinearOperators ((REAL-NS m),(REAL-NS n)))) is Element of BoundedLinearOperators ((REAL-NS m),(REAL-NS n))
K197((R_VectorSpace_of_LinearOperators ((REAL-NS m),(REAL-NS n))),(BoundedLinearOperators ((REAL-NS m),(REAL-NS n)))) is Relation-like K7((BoundedLinearOperators ((REAL-NS m),(REAL-NS n))),(BoundedLinearOperators ((REAL-NS m),(REAL-NS n)))) -defined BoundedLinearOperators ((REAL-NS m),(REAL-NS n)) -valued Function-like total V30(K7((BoundedLinearOperators ((REAL-NS m),(REAL-NS n))),(BoundedLinearOperators ((REAL-NS m),(REAL-NS n)))), BoundedLinearOperators ((REAL-NS m),(REAL-NS n))) Element of K6(K7(K7((BoundedLinearOperators ((REAL-NS m),(REAL-NS n))),(BoundedLinearOperators ((REAL-NS m),(REAL-NS n)))),(BoundedLinearOperators ((REAL-NS m),(REAL-NS n)))))
K7((BoundedLinearOperators ((REAL-NS m),(REAL-NS n))),(BoundedLinearOperators ((REAL-NS m),(REAL-NS n)))) is set
K7(K7((BoundedLinearOperators ((REAL-NS m),(REAL-NS n))),(BoundedLinearOperators ((REAL-NS m),(REAL-NS n)))),(BoundedLinearOperators ((REAL-NS m),(REAL-NS n)))) is set
K6(K7(K7((BoundedLinearOperators ((REAL-NS m),(REAL-NS n))),(BoundedLinearOperators ((REAL-NS m),(REAL-NS n)))),(BoundedLinearOperators ((REAL-NS m),(REAL-NS n))))) is set
K198((R_VectorSpace_of_LinearOperators ((REAL-NS m),(REAL-NS n))),(BoundedLinearOperators ((REAL-NS m),(REAL-NS n)))) is Relation-like K7(REAL,(BoundedLinearOperators ((REAL-NS m),(REAL-NS n)))) -defined BoundedLinearOperators ((REAL-NS m),(REAL-NS n)) -valued Function-like total V30(K7(REAL,(BoundedLinearOperators ((REAL-NS m),(REAL-NS n)))), BoundedLinearOperators ((REAL-NS m),(REAL-NS n))) Element of K6(K7(K7(REAL,(BoundedLinearOperators ((REAL-NS m),(REAL-NS n)))),(BoundedLinearOperators ((REAL-NS m),(REAL-NS n)))))
K7(REAL,(BoundedLinearOperators ((REAL-NS m),(REAL-NS n)))) is V33() set
K7(K7(REAL,(BoundedLinearOperators ((REAL-NS m),(REAL-NS n)))),(BoundedLinearOperators ((REAL-NS m),(REAL-NS n)))) is V33() set
K6(K7(K7(REAL,(BoundedLinearOperators ((REAL-NS m),(REAL-NS n)))),(BoundedLinearOperators ((REAL-NS m),(REAL-NS n))))) is V33() set
BoundedLinearOperatorsNorm ((REAL-NS m),(REAL-NS n)) is non empty Relation-like BoundedLinearOperators ((REAL-NS m),(REAL-NS n)) -defined REAL -valued Function-like total V30( BoundedLinearOperators ((REAL-NS m),(REAL-NS n)), REAL ) complex-yielding V146() V147() Element of K6(K7((BoundedLinearOperators ((REAL-NS m),(REAL-NS n))),REAL))
K7((BoundedLinearOperators ((REAL-NS m),(REAL-NS n))),REAL) is V33() complex-yielding V146() V147() set
K6(K7((BoundedLinearOperators ((REAL-NS m),(REAL-NS n))),REAL)) is V33() set
G15((BoundedLinearOperators ((REAL-NS m),(REAL-NS n))),K199((R_VectorSpace_of_LinearOperators ((REAL-NS m),(REAL-NS n))),(BoundedLinearOperators ((REAL-NS m),(REAL-NS n)))),K197((R_VectorSpace_of_LinearOperators ((REAL-NS m),(REAL-NS n))),(BoundedLinearOperators ((REAL-NS m),(REAL-NS n)))),K198((R_VectorSpace_of_LinearOperators ((REAL-NS m),(REAL-NS n))),(BoundedLinearOperators ((REAL-NS m),(REAL-NS n)))),(BoundedLinearOperatorsNorm ((REAL-NS m),(REAL-NS n)))) is V108() L15()
{ b₁ where b₁ is Element of the carrier of (REAL-NS m) : not a₂ <= ||.(b₁ - Z).|| } is set
K7( the carrier of (REAL-NS m), the carrier of (REAL-NS 1)) is set
K6(K7( the carrier of (REAL-NS m), the carrier of (REAL-NS 1))) is set
Seg n is V156() V157() V158() V159() V160() V161() Element of K6(NAT)
{ b₁ where b₁ is V6() V10() V11() V12() ext-real non negative V33() V38() V120() V155() V156() V157() V158() V159() V160() V161() Element of NAT : ( 1 <= b₁ & b₁ <= n ) } is set
Z0 is V6() V10() V11() V12() ext-real non negative V33() V38() V120() V155() V156() V157() V158() V159() V160() V161() Element of NAT
Proj (Z0,n) is non empty Relation-like the carrier of (REAL-NS n) -defined the carrier of (REAL-NS 1) -valued Function-like total V30( the carrier of (REAL-NS n), the carrier of (REAL-NS 1)) Element of K6(K7( the carrier of (REAL-NS n), the carrier of (REAL-NS 1)))
K7( the carrier of (REAL-NS n), the carrier of (REAL-NS 1)) is set
K6(K7( the carrier of (REAL-NS n), the carrier of (REAL-NS 1))) is set
(Proj (Z0,n)) * f is Relation-like the carrier of (REAL-NS m) -defined the carrier of (REAL-NS 1) -valued Function-like Element of K6(K7( the carrier of (REAL-NS m), the carrier of (REAL-NS 1)))
dom ((Proj (Z0,n)) * f) is Element of K6( the carrier of (REAL-NS m))
K6( the carrier of (REAL-NS m)) is set
((Proj (Z0,n)) * f) /. Z is Element of the carrier of (REAL-NS 1)
diff (((Proj (Z0,n)) * f),Z) is Relation-like Function-like Element of the carrier of (R_NormSpace_of_BoundedLinearOperators ((REAL-NS m),(REAL-NS 1)))
R_NormSpace_of_BoundedLinearOperators ((REAL-NS m),(REAL-NS 1)) is non empty V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() V109() V142() L15()
BoundedLinearOperators ((REAL-NS m),(REAL-NS 1)) is non empty Element of K6( the carrier of (R_VectorSpace_of_LinearOperators ((REAL-NS m),(REAL-NS 1))))
R_VectorSpace_of_LinearOperators ((REAL-NS m),(REAL-NS 1)) is non empty V71() strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V204() RLSStruct
LinearOperators ((REAL-NS m),(REAL-NS 1)) is non empty functional Element of K6( the carrier of K279( the carrier of (REAL-NS m),(REAL-NS 1)))
K279( the carrier of (REAL-NS m),(REAL-NS 1)) is non empty V71() strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V204() RLSStruct
K123( the carrier of (REAL-NS m), the carrier of (REAL-NS 1)) is non empty functional M4( the carrier of (REAL-NS m), the carrier of (REAL-NS 1))
FuncZero ( the carrier of (REAL-NS m),(REAL-NS 1)) is Relation-like the carrier of (REAL-NS m) -defined the carrier of (REAL-NS 1) -valued Function-like total V30( the carrier of (REAL-NS m), the carrier of (REAL-NS 1)) Element of K123( the carrier of (REAL-NS m), the carrier of (REAL-NS 1))
the carrier of (REAL-NS m) --> (0. (REAL-NS 1)) is non empty Relation-like the carrier of (REAL-NS m) -defined the carrier of (REAL-NS 1) -valued Function-like total V30( the carrier of (REAL-NS m), the carrier of (REAL-NS 1)) Element of K6(K7( the carrier of (REAL-NS m), the carrier of (REAL-NS 1)))
FuncAdd ( the carrier of (REAL-NS m),(REAL-NS 1)) is Relation-like K7(K123( the carrier of (REAL-NS m), the carrier of (REAL-NS 1)),K123( the carrier of (REAL-NS m), the carrier of (REAL-NS 1))) -defined K123( the carrier of (REAL-NS m), the carrier of (REAL-NS 1)) -valued Function-like total V30(K7(K123( the carrier of (REAL-NS m), the carrier of (REAL-NS 1)),K123( the carrier of (REAL-NS m), the carrier of (REAL-NS 1))),K123( the carrier of (REAL-NS m), the carrier of (REAL-NS 1))) Element of K6(K7(K7(K123( the carrier of (REAL-NS m), the carrier of (REAL-NS 1)),K123( the carrier of (REAL-NS m), the carrier of (REAL-NS 1))),K123( the carrier of (REAL-NS m), the carrier of (REAL-NS 1))))
K7(K123( the carrier of (REAL-NS m), the carrier of (REAL-NS 1)),K123( the carrier of (REAL-NS m), the carrier of (REAL-NS 1))) is set
K7(K7(K123( the carrier of (REAL-NS m), the carrier of (REAL-NS 1)),K123( the carrier of (REAL-NS m), the carrier of (REAL-NS 1))),K123( the carrier of (REAL-NS m), the carrier of (REAL-NS 1))) is set
K6(K7(K7(K123( the carrier of (REAL-NS m), the carrier of (REAL-NS 1)),K123( the carrier of (REAL-NS m), the carrier of (REAL-NS 1))),K123( the carrier of (REAL-NS m), the carrier of (REAL-NS 1)))) is set
FuncExtMult ( the carrier of (REAL-NS m),(REAL-NS 1)) is Relation-like K7(REAL,K123( the carrier of (REAL-NS m), the carrier of (REAL-NS 1))) -defined K123( the carrier of (REAL-NS m), the carrier of (REAL-NS 1)) -valued Function-like total V30(K7(REAL,K123( the carrier of (REAL-NS m), the carrier of (REAL-NS 1))),K123( the carrier of (REAL-NS m), the carrier of (REAL-NS 1))) Element of K6(K7(K7(REAL,K123( the carrier of (REAL-NS m), the carrier of (REAL-NS 1))),K123( the carrier of (REAL-NS m), the carrier of (REAL-NS 1))))
K7(REAL,K123( the carrier of (REAL-NS m), the carrier of (REAL-NS 1))) is V33() set
K7(K7(REAL,K123( the carrier of (REAL-NS m), the carrier of (REAL-NS 1))),K123( the carrier of (REAL-NS m), the carrier of (REAL-NS 1))) is V33() set
K6(K7(K7(REAL,K123( the carrier of (REAL-NS m), the carrier of (REAL-NS 1))),K123( the carrier of (REAL-NS m), the carrier of (REAL-NS 1)))) is V33() set
RLSStruct(# K123( the carrier of (REAL-NS m), the carrier of (REAL-NS 1)),(FuncZero ( the carrier of (REAL-NS m),(REAL-NS 1))),(FuncAdd ( the carrier of (REAL-NS m),(REAL-NS 1))),(FuncExtMult ( the carrier of (REAL-NS m),(REAL-NS 1))) #) is strict RLSStruct
the carrier of K279( the carrier of (REAL-NS m),(REAL-NS 1)) is non empty set
K6( the carrier of K279( the carrier of (REAL-NS m),(REAL-NS 1))) is set
K199(K279( the carrier of (REAL-NS m),(REAL-NS 1)),(LinearOperators ((REAL-NS m),(REAL-NS 1)))) is Element of LinearOperators ((REAL-NS m),(REAL-NS 1))
K197(K279( the carrier of (REAL-NS m),(REAL-NS 1)),(LinearOperators ((REAL-NS m),(REAL-NS 1)))) is Relation-like K7((LinearOperators ((REAL-NS m),(REAL-NS 1))),(LinearOperators ((REAL-NS m),(REAL-NS 1)))) -defined LinearOperators ((REAL-NS m),(REAL-NS 1)) -valued Function-like total V30(K7((LinearOperators ((REAL-NS m),(REAL-NS 1))),(LinearOperators ((REAL-NS m),(REAL-NS 1)))), LinearOperators ((REAL-NS m),(REAL-NS 1))) Element of K6(K7(K7((LinearOperators ((REAL-NS m),(REAL-NS 1))),(LinearOperators ((REAL-NS m),(REAL-NS 1)))),(LinearOperators ((REAL-NS m),(REAL-NS 1)))))
K7((LinearOperators ((REAL-NS m),(REAL-NS 1))),(LinearOperators ((REAL-NS m),(REAL-NS 1)))) is set
K7(K7((LinearOperators ((REAL-NS m),(REAL-NS 1))),(LinearOperators ((REAL-NS m),(REAL-NS 1)))),(LinearOperators ((REAL-NS m),(REAL-NS 1)))) is set
K6(K7(K7((LinearOperators ((REAL-NS m),(REAL-NS 1))),(LinearOperators ((REAL-NS m),(REAL-NS 1)))),(LinearOperators ((REAL-NS m),(REAL-NS 1))))) is set
K198(K279( the carrier of (REAL-NS m),(REAL-NS 1)),(LinearOperators ((REAL-NS m),(REAL-NS 1)))) is Relation-like K7(REAL,(LinearOperators ((REAL-NS m),(REAL-NS 1)))) -defined LinearOperators ((REAL-NS m),(REAL-NS 1)) -valued Function-like total V30(K7(REAL,(LinearOperators ((REAL-NS m),(REAL-NS 1)))), LinearOperators ((REAL-NS m),(REAL-NS 1))) Element of K6(K7(K7(REAL,(LinearOperators ((REAL-NS m),(REAL-NS 1)))),(LinearOperators ((REAL-NS m),(REAL-NS 1)))))
K7(REAL,(LinearOperators ((REAL-NS m),(REAL-NS 1)))) is V33() set
K7(K7(REAL,(LinearOperators ((REAL-NS m),(REAL-NS 1)))),(LinearOperators ((REAL-NS m),(REAL-NS 1)))) is V33() set
K6(K7(K7(REAL,(LinearOperators ((REAL-NS m),(REAL-NS 1)))),(LinearOperators ((REAL-NS m),(REAL-NS 1))))) is V33() set
RLSStruct(# (LinearOperators ((REAL-NS m),(REAL-NS 1))),K199(K279( the carrier of (REAL-NS m),(REAL-NS 1)),(LinearOperators ((REAL-NS m),(REAL-NS 1)))),K197(K279( the carrier of (REAL-NS m),(REAL-NS 1)),(LinearOperators ((REAL-NS m),(REAL-NS 1)))),K198(K279( the carrier of (REAL-NS m),(REAL-NS 1)),(LinearOperators ((REAL-NS m),(REAL-NS 1)))) #) is V71() strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RLSStruct
the carrier of (R_VectorSpace_of_LinearOperators ((REAL-NS m),(REAL-NS 1))) is non empty set
K6( the carrier of (R_VectorSpace_of_LinearOperators ((REAL-NS m),(REAL-NS 1)))) is set
K199((R_VectorSpace_of_LinearOperators ((REAL-NS m),(REAL-NS 1))),(BoundedLinearOperators ((REAL-NS m),(REAL-NS 1)))) is Element of BoundedLinearOperators ((REAL-NS m),(REAL-NS 1))
K197((R_VectorSpace_of_LinearOperators ((REAL-NS m),(REAL-NS 1))),(BoundedLinearOperators ((REAL-NS m),(REAL-NS 1)))) is Relation-like K7((BoundedLinearOperators ((REAL-NS m),(REAL-NS 1))),(BoundedLinearOperators ((REAL-NS m),(REAL-NS 1)))) -defined BoundedLinearOperators ((REAL-NS m),(REAL-NS 1)) -valued Function-like total V30(K7((BoundedLinearOperators ((REAL-NS m),(REAL-NS 1))),(BoundedLinearOperators ((REAL-NS m),(REAL-NS 1)))), BoundedLinearOperators ((REAL-NS m),(REAL-NS 1))) Element of K6(K7(K7((BoundedLinearOperators ((REAL-NS m),(REAL-NS 1))),(BoundedLinearOperators ((REAL-NS m),(REAL-NS 1)))),(BoundedLinearOperators ((REAL-NS m),(REAL-NS 1)))))
K7((BoundedLinearOperators ((REAL-NS m),(REAL-NS 1))),(BoundedLinearOperators ((REAL-NS m),(REAL-NS 1)))) is set
K7(K7((BoundedLinearOperators ((REAL-NS m),(REAL-NS 1))),(BoundedLinearOperators ((REAL-NS m),(REAL-NS 1)))),(BoundedLinearOperators ((REAL-NS m),(REAL-NS 1)))) is set
K6(K7(K7((BoundedLinearOperators ((REAL-NS m),(REAL-NS 1))),(BoundedLinearOperators ((REAL-NS m),(REAL-NS 1)))),(BoundedLinearOperators ((REAL-NS m),(REAL-NS 1))))) is set
K198((R_VectorSpace_of_LinearOperators ((REAL-NS m),(REAL-NS 1))),(BoundedLinearOperators ((REAL-NS m),(REAL-NS 1)))) is Relation-like K7(REAL,(BoundedLinearOperators ((REAL-NS m),(REAL-NS 1)))) -defined BoundedLinearOperators ((REAL-NS m),(REAL-NS 1)) -valued Function-like total V30(K7(REAL,(BoundedLinearOperators ((REAL-NS m),(REAL-NS 1)))), BoundedLinearOperators ((REAL-NS m),(REAL-NS 1))) Element of K6(K7(K7(REAL,(BoundedLinearOperators ((REAL-NS m),(REAL-NS 1)))),(BoundedLinearOperators ((REAL-NS m),(REAL-NS 1)))))
K7(REAL,(BoundedLinearOperators ((REAL-NS m),(REAL-NS 1)))) is V33() set
K7(K7(REAL,(BoundedLinearOperators ((REAL-NS m),(REAL-NS 1)))),(BoundedLinearOperators ((REAL-NS m),(REAL-NS 1)))) is V33() set
K6(K7(K7(REAL,(BoundedLinearOperators ((REAL-NS m),(REAL-NS 1)))),(BoundedLinearOperators ((REAL-NS m),(REAL-NS 1))))) is V33() set
BoundedLinearOperatorsNorm ((REAL-NS m),(REAL-NS 1)) is non empty Relation-like BoundedLinearOperators ((REAL-NS m),(REAL-NS 1)) -defined REAL -valued Function-like total V30( BoundedLinearOperators ((REAL-NS m),(REAL-NS 1)), REAL ) complex-yielding V146() V147() Element of K6(K7((BoundedLinearOperators ((REAL-NS m),(REAL-NS 1))),REAL))
K7((BoundedLinearOperators ((REAL-NS m),(REAL-NS 1))),REAL) is V33() complex-yielding V146() V147() set
K6(K7((BoundedLinearOperators ((REAL-NS m),(REAL-NS 1))),REAL)) is V33() set
G15((BoundedLinearOperators ((REAL-NS m),(REAL-NS 1))),K199((R_VectorSpace_of_LinearOperators ((REAL-NS m),(REAL-NS 1))),(BoundedLinearOperators ((REAL-NS m),(REAL-NS 1)))),K197((R_VectorSpace_of_LinearOperators ((REAL-NS m),(REAL-NS 1))),(BoundedLinearOperators ((REAL-NS m),(REAL-NS 1)))),K198((R_VectorSpace_of_LinearOperators ((REAL-NS m),(REAL-NS 1))),(BoundedLinearOperators ((REAL-NS m),(REAL-NS 1)))),(BoundedLinearOperatorsNorm ((REAL-NS m),(REAL-NS 1)))) is V108() L15()
the carrier of (R_NormSpace_of_BoundedLinearOperators ((REAL-NS m),(REAL-NS 1))) is non empty set
c₇ is Neighbourhood of Z
x is V11() V12() ext-real Element of REAL
{ b₁ where b₁ is Element of the carrier of (REAL-NS m) : not x <= ||.(b₁ - Z).|| } is set
L is V11() V12() ext-real Element of REAL
{ b₁ where b₁ is Element of the carrier of (REAL-NS m) : not L <= ||.(b₁ - Z).|| } is set
L0 is Relation-like the carrier of (REAL-NS m) -defined the carrier of (REAL-NS 1) -valued Function-like RestFunc-like Element of K6(K7( the carrier of (REAL-NS m), the carrier of (REAL-NS 1)))
L0 is Relation-like the carrier of (REAL-NS m) -defined the carrier of (REAL-NS 1) -valued Function-like RestFunc-like Element of K6(K7( the carrier of (REAL-NS m), the carrier of (REAL-NS 1)))
L0 is Relation-like the carrier of (REAL-NS m) -defined the carrier of (REAL-NS 1) -valued Function-like RestFunc-like Element of K6(K7( the carrier of (REAL-NS m), the carrier of (REAL-NS 1)))
R is Element of the carrier of (REAL-NS m)
((Proj (Z0,n)) * f) /. R is Element of the carrier of (REAL-NS 1)
(((Proj (Z0,n)) * f) /. R) - (((Proj (Z0,n)) * f) /. Z) is Element of the carrier of (REAL-NS 1)
R - Z is Element of the carrier of (REAL-NS m)
(diff (((Proj (Z0,n)) * f),Z)) . (R - Z) is Element of the carrier of (REAL-NS 1)
L0 /. (R - Z) is Element of the carrier of (REAL-NS 1)
((diff (((Proj (Z0,n)) * f),Z)) . (R - Z)) + (L0 /. (R - Z)) is Element of the carrier of (REAL-NS 1)
Z0 is Relation-like NAT -defined REAL -valued Function-like FinSequence-like complex-yielding V146() V147() FinSequence of REAL
dom Z0 is V156() V157() V158() V159() V160() V161() Element of K6(NAT)
min Z0 is V11() V12() ext-real Element of REAL
min_p Z0 is V6() V10() V11() V12() ext-real non negative V33() V38() V120() V155() V156() V157() V158() V159() V160() V161() Element of NAT
Z0 . (min_p Z0) is V11() V12() ext-real Element of REAL
len Z0 is V6() V10() V11() V12() ext-real non negative V33() V38() V120() V155() V156() V157() V158() V159() V160() V161() Element of NAT
Z0 /. (min_p Z0) is V11() V12() ext-real Element of REAL
REAL n is non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered FinSequenceSet of REAL
n -tuples_on REAL is non empty functional FinSequence-membered FinSequenceSet of REAL
x is Element of the carrier of (REAL-NS m)
L is V6() V10() V11() V12() ext-real non negative V33() V38() V120() V155() V156() V157() V158() V159() V160() V161() Element of NAT
Proj (L,n) is non empty Relation-like the carrier of (REAL-NS n) -defined the carrier of (REAL-NS 1) -valued Function-like total V30( the carrier of (REAL-NS n), the carrier of (REAL-NS 1)) Element of K6(K7( the carrier of (REAL-NS n), the carrier of (REAL-NS 1)))
K7( the carrier of (REAL-NS n), the carrier of (REAL-NS 1)) is set
K6(K7( the carrier of (REAL-NS n), the carrier of (REAL-NS 1))) is set
(Proj (L,n)) * f is Relation-like the carrier of (REAL-NS m) -defined the carrier of (REAL-NS 1) -valued Function-like Element of K6(K7( the carrier of (REAL-NS m), the carrier of (REAL-NS 1)))
diff (((Proj (L,n)) * f),Z) is Relation-like Function-like Element of the carrier of (R_NormSpace_of_BoundedLinearOperators ((REAL-NS m),(REAL-NS 1)))
R_NormSpace_of_BoundedLinearOperators ((REAL-NS m),(REAL-NS 1)) is non empty V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() V109() V142() L15()
BoundedLinearOperators ((REAL-NS m),(REAL-NS 1)) is non empty Element of K6( the carrier of (R_VectorSpace_of_LinearOperators ((REAL-NS m),(REAL-NS 1))))
R_VectorSpace_of_LinearOperators ((REAL-NS m),(REAL-NS 1)) is non empty V71() strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V204() RLSStruct
LinearOperators ((REAL-NS m),(REAL-NS 1)) is non empty functional Element of K6( the carrier of K279( the carrier of (REAL-NS m),(REAL-NS 1)))
K279( the carrier of (REAL-NS m),(REAL-NS 1)) is non empty V71() strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V204() RLSStruct
K123( the carrier of (REAL-NS m), the carrier of (REAL-NS 1)) is non empty functional M4( the carrier of (REAL-NS m), the carrier of (REAL-NS 1))
FuncZero ( the carrier of (REAL-NS m),(REAL-NS 1)) is Relation-like the carrier of (REAL-NS m) -defined the carrier of (REAL-NS 1) -valued Function-like total V30( the carrier of (REAL-NS m), the carrier of (REAL-NS 1)) Element of K123( the carrier of (REAL-NS m), the carrier of (REAL-NS 1))
the carrier of (REAL-NS m) --> (0. (REAL-NS 1)) is non empty Relation-like the carrier of (REAL-NS m) -defined the carrier of (REAL-NS 1) -valued Function-like total V30( the carrier of (REAL-NS m), the carrier of (REAL-NS 1)) Element of K6(K7( the carrier of (REAL-NS m), the carrier of (REAL-NS 1)))
FuncAdd ( the carrier of (REAL-NS m),(REAL-NS 1)) is Relation-like K7(K123( the carrier of (REAL-NS m), the carrier of (REAL-NS 1)),K123( the carrier of (REAL-NS m), the carrier of (REAL-NS 1))) -defined K123( the carrier of (REAL-NS m), the carrier of (REAL-NS 1)) -valued Function-like total V30(K7(K123( the carrier of (REAL-NS m), the carrier of (REAL-NS 1)),K123( the carrier of (REAL-NS m), the carrier of (REAL-NS 1))),K123( the carrier of (REAL-NS m), the carrier of (REAL-NS 1))) Element of K6(K7(K7(K123( the carrier of (REAL-NS m), the carrier of (REAL-NS 1)),K123( the carrier of (REAL-NS m), the carrier of (REAL-NS 1))),K123( the carrier of (REAL-NS m), the carrier of (REAL-NS 1))))
K7(K123( the carrier of (REAL-NS m), the carrier of (REAL-NS 1)),K123( the carrier of (REAL-NS m), the carrier of (REAL-NS 1))) is set
K7(K7(K123( the carrier of (REAL-NS m), the carrier of (REAL-NS 1)),K123( the carrier of (REAL-NS m), the carrier of (REAL-NS 1))),K123( the carrier of (REAL-NS m), the carrier of (REAL-NS 1))) is set
K6(K7(K7(K123( the carrier of (REAL-NS m), the carrier of (REAL-NS 1)),K123( the carrier of (REAL-NS m), the carrier of (REAL-NS 1))),K123( the carrier of (REAL-NS m), the carrier of (REAL-NS 1)))) is set
FuncExtMult ( the carrier of (REAL-NS m),(REAL-NS 1)) is Relation-like K7(REAL,K123( the carrier of (REAL-NS m), the carrier of (REAL-NS 1))) -defined K123( the carrier of (REAL-NS m), the carrier of (REAL-NS 1)) -valued Function-like total V30(K7(REAL,K123( the carrier of (REAL-NS m), the carrier of (REAL-NS 1))),K123( the carrier of (REAL-NS m), the carrier of (REAL-NS 1))) Element of K6(K7(K7(REAL,K123( the carrier of (REAL-NS m), the carrier of (REAL-NS 1))),K123( the carrier of (REAL-NS m), the carrier of (REAL-NS 1))))
K7(REAL,K123( the carrier of (REAL-NS m), the carrier of (REAL-NS 1))) is V33() set
K7(K7(REAL,K123( the carrier of (REAL-NS m), the carrier of (REAL-NS 1))),K123( the carrier of (REAL-NS m), the carrier of (REAL-NS 1))) is V33() set
K6(K7(K7(REAL,K123( the carrier of (REAL-NS m), the carrier of (REAL-NS 1))),K123( the carrier of (REAL-NS m), the carrier of (REAL-NS 1)))) is V33() set
RLSStruct(# K123( the carrier of (REAL-NS m), the carrier of (REAL-NS 1)),(FuncZero ( the carrier of (REAL-NS m),(REAL-NS 1))),(FuncAdd ( the carrier of (REAL-NS m),(REAL-NS 1))),(FuncExtMult ( the carrier of (REAL-NS m),(REAL-NS 1))) #) is strict RLSStruct
the carrier of K279( the carrier of (REAL-NS m),(REAL-NS 1)) is non empty set
K6( the carrier of K279( the carrier of (REAL-NS m),(REAL-NS 1))) is set
K199(K279( the carrier of (REAL-NS m),(REAL-NS 1)),(LinearOperators ((REAL-NS m),(REAL-NS 1)))) is Element of LinearOperators ((REAL-NS m),(REAL-NS 1))
K197(K279( the carrier of (REAL-NS m),(REAL-NS 1)),(LinearOperators ((REAL-NS m),(REAL-NS 1)))) is Relation-like K7((LinearOperators ((REAL-NS m),(REAL-NS 1))),(LinearOperators ((REAL-NS m),(REAL-NS 1)))) -defined LinearOperators ((REAL-NS m),(REAL-NS 1)) -valued Function-like total V30(K7((LinearOperators ((REAL-NS m),(REAL-NS 1))),(LinearOperators ((REAL-NS m),(REAL-NS 1)))), LinearOperators ((REAL-NS m),(REAL-NS 1))) Element of K6(K7(K7((LinearOperators ((REAL-NS m),(REAL-NS 1))),(LinearOperators ((REAL-NS m),(REAL-NS 1)))),(LinearOperators ((REAL-NS m),(REAL-NS 1)))))
K7((LinearOperators ((REAL-NS m),(REAL-NS 1))),(LinearOperators ((REAL-NS m),(REAL-NS 1)))) is set
K7(K7((LinearOperators ((REAL-NS m),(REAL-NS 1))),(LinearOperators ((REAL-NS m),(REAL-NS 1)))),(LinearOperators ((REAL-NS m),(REAL-NS 1)))) is set
K6(K7(K7((LinearOperators ((REAL-NS m),(REAL-NS 1))),(LinearOperators ((REAL-NS m),(REAL-NS 1)))),(LinearOperators ((REAL-NS m),(REAL-NS 1))))) is set
K198(K279( the carrier of (REAL-NS m),(REAL-NS 1)),(LinearOperators ((REAL-NS m),(REAL-NS 1)))) is Relation-like K7(REAL,(LinearOperators ((REAL-NS m),(REAL-NS 1)))) -defined LinearOperators ((REAL-NS m),(REAL-NS 1)) -valued Function-like total V30(K7(REAL,(LinearOperators ((REAL-NS m),(REAL-NS 1)))), LinearOperators ((REAL-NS m),(REAL-NS 1))) Element of K6(K7(K7(REAL,(LinearOperators ((REAL-NS m),(REAL-NS 1)))),(LinearOperators ((REAL-NS m),(REAL-NS 1)))))
K7(REAL,(LinearOperators ((REAL-NS m),(REAL-NS 1)))) is V33() set
K7(K7(REAL,(LinearOperators ((REAL-NS m),(REAL-NS 1)))),(LinearOperators ((REAL-NS m),(REAL-NS 1)))) is V33() set
K6(K7(K7(REAL,(LinearOperators ((REAL-NS m),(REAL-NS 1)))),(LinearOperators ((REAL-NS m),(REAL-NS 1))))) is V33() set
RLSStruct(# (LinearOperators ((REAL-NS m),(REAL-NS 1))),K199(K279( the carrier of (REAL-NS m),(REAL-NS 1)),(LinearOperators ((REAL-NS m),(REAL-NS 1)))),K197(K279( the carrier of (REAL-NS m),(REAL-NS 1)),(LinearOperators ((REAL-NS m),(REAL-NS 1)))),K198(K279( the carrier of (REAL-NS m),(REAL-NS 1)),(LinearOperators ((REAL-NS m),(REAL-NS 1)))) #) is V71() strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RLSStruct
the carrier of (R_VectorSpace_of_LinearOperators ((REAL-NS m),(REAL-NS 1))) is non empty set
K6( the carrier of (R_VectorSpace_of_LinearOperators ((REAL-NS m),(REAL-NS 1)))) is set
K199((R_VectorSpace_of_LinearOperators ((REAL-NS m),(REAL-NS 1))),(BoundedLinearOperators ((REAL-NS m),(REAL-NS 1)))) is Element of BoundedLinearOperators ((REAL-NS m),(REAL-NS 1))
K197((R_VectorSpace_of_LinearOperators ((REAL-NS m),(REAL-NS 1))),(BoundedLinearOperators ((REAL-NS m),(REAL-NS 1)))) is Relation-like K7((BoundedLinearOperators ((REAL-NS m),(REAL-NS 1))),(BoundedLinearOperators ((REAL-NS m),(REAL-NS 1)))) -defined BoundedLinearOperators ((REAL-NS m),(REAL-NS 1)) -valued Function-like total V30(K7((BoundedLinearOperators ((REAL-NS m),(REAL-NS 1))),(BoundedLinearOperators ((REAL-NS m),(REAL-NS 1)))), BoundedLinearOperators ((REAL-NS m),(REAL-NS 1))) Element of K6(K7(K7((BoundedLinearOperators ((REAL-NS m),(REAL-NS 1))),(BoundedLinearOperators ((REAL-NS m),(REAL-NS 1)))),(BoundedLinearOperators ((REAL-NS m),(REAL-NS 1)))))
K7((BoundedLinearOperators ((REAL-NS m),(REAL-NS 1))),(BoundedLinearOperators ((REAL-NS m),(REAL-NS 1)))) is set
K7(K7((BoundedLinearOperators ((REAL-NS m),(REAL-NS 1))),(BoundedLinearOperators ((REAL-NS m),(REAL-NS 1)))),(BoundedLinearOperators ((REAL-NS m),(REAL-NS 1)))) is set
K6(K7(K7((BoundedLinearOperators ((REAL-NS m),(REAL-NS 1))),(BoundedLinearOperators ((REAL-NS m),(REAL-NS 1)))),(BoundedLinearOperators ((REAL-NS m),(REAL-NS 1))))) is set
K198((R_VectorSpace_of_LinearOperators ((REAL-NS m),(REAL-NS 1))),(BoundedLinearOperators ((REAL-NS m),(REAL-NS 1)))) is Relation-like K7(REAL,(BoundedLinearOperators ((REAL-NS m),(REAL-NS 1)))) -defined BoundedLinearOperators ((REAL-NS m),(REAL-NS 1)) -valued Function-like total V30(K7(REAL,(BoundedLinearOperators ((REAL-NS m),(REAL-NS 1)))), BoundedLinearOperators ((REAL-NS m),(REAL-NS 1))) Element of K6(K7(K7(REAL,(BoundedLinearOperators ((REAL-NS m),(REAL-NS 1)))),(BoundedLinearOperators ((REAL-NS m),(REAL-NS 1)))))
K7(REAL,(BoundedLinearOperators ((REAL-NS m),(REAL-NS 1)))) is V33() set
K7(K7(REAL,(BoundedLinearOperators ((REAL-NS m),(REAL-NS 1)))),(BoundedLinearOperators ((REAL-NS m),(REAL-NS 1)))) is V33() set
K6(K7(K7(REAL,(BoundedLinearOperators ((REAL-NS m),(REAL-NS 1)))),(BoundedLinearOperators ((REAL-NS m),(REAL-NS 1))))) is V33() set
BoundedLinearOperatorsNorm ((REAL-NS m),(REAL-NS 1)) is non empty Relation-like BoundedLinearOperators ((REAL-NS m),(REAL-NS 1)) -defined REAL -valued Function-like total V30( BoundedLinearOperators ((REAL-NS m),(REAL-NS 1)), REAL ) complex-yielding V146() V147() Element of K6(K7((BoundedLinearOperators ((REAL-NS m),(REAL-NS 1))),REAL))
K7((BoundedLinearOperators ((REAL-NS m),(REAL-NS 1))),REAL) is V33() complex-yielding V146() V147() set
K6(K7((BoundedLinearOperators ((REAL-NS m),(REAL-NS 1))),REAL)) is V33() set
G15((BoundedLinearOperators ((REAL-NS m),(REAL-NS 1))),K199((R_VectorSpace_of_LinearOperators ((REAL-NS m),(REAL-NS 1))),(BoundedLinearOperators ((REAL-NS m),(REAL-NS 1)))),K197((R_VectorSpace_of_LinearOperators ((REAL-NS m),(REAL-NS 1))),(BoundedLinearOperators ((REAL-NS m),(REAL-NS 1)))),K198((R_VectorSpace_of_LinearOperators ((REAL-NS m),(REAL-NS 1))),(BoundedLinearOperators ((REAL-NS m),(REAL-NS 1)))),(BoundedLinearOperatorsNorm ((REAL-NS m),(REAL-NS 1)))) is V108() L15()
the carrier of (R_NormSpace_of_BoundedLinearOperators ((REAL-NS m),(REAL-NS 1))) is non empty set
(diff (((Proj (L,n)) * f),Z)) . x is Element of the carrier of (REAL-NS 1)
(proj (1,1)) . ((diff (((Proj (L,n)) * f),Z)) . x) is V11() V12() ext-real Element of REAL
L is Relation-like NAT -defined REAL -valued Function-like FinSequence-like complex-yielding V146() V147() FinSequence of REAL
dom L is V156() V157() V158() V159() V160() V161() Element of K6(NAT)
len L is V6() V10() V11() V12() ext-real non negative V33() V38() V120() V155() V156() V157() V158() V159() V160() V161() Element of NAT
R is V6() V10() V11() V12() ext-real non negative V33() V38() V120() V155() V156() V157() V158() V159() V160() V161() Element of NAT
L0 is Relation-like NAT -defined REAL -valued Function-like V40(n) FinSequence-like complex-yielding V146() V147() Element of REAL n
L0 /. R is V11() V12() ext-real Element of REAL
Proj (R,n) is non empty Relation-like the carrier of (REAL-NS n) -defined the carrier of (REAL-NS 1) -valued Function-like total V30( the carrier of (REAL-NS n), the carrier of (REAL-NS 1)) Element of K6(K7( the carrier of (REAL-NS n), the carrier of (REAL-NS 1)))
K7( the carrier of (REAL-NS n), the carrier of (REAL-NS 1)) is set
K6(K7( the carrier of (REAL-NS n), the carrier of (REAL-NS 1))) is set
(Proj (R,n)) * f is Relation-like the carrier of (REAL-NS m) -defined the carrier of (REAL-NS 1) -valued Function-like Element of K6(K7( the carrier of (REAL-NS m), the carrier of (REAL-NS 1)))
diff (((Proj (R,n)) * f),Z) is Relation-like Function-like Element of the carrier of (R_NormSpace_of_BoundedLinearOperators ((REAL-NS m),(REAL-NS 1)))
R_NormSpace_of_BoundedLinearOperators ((REAL-NS m),(REAL-NS 1)) is non empty V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() V109() V142() L15()
BoundedLinearOperators ((REAL-NS m),(REAL-NS 1)) is non empty Element of K6( the carrier of (R_VectorSpace_of_LinearOperators ((REAL-NS m),(REAL-NS 1))))
R_VectorSpace_of_LinearOperators ((REAL-NS m),(REAL-NS 1)) is non empty V71() strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V204() RLSStruct
LinearOperators ((REAL-NS m),(REAL-NS 1)) is non empty functional Element of K6( the carrier of K279( the carrier of (REAL-NS m),(REAL-NS 1)))
K279( the carrier of (REAL-NS m),(REAL-NS 1)) is non empty V71() strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V204() RLSStruct
K123( the carrier of (REAL-NS m), the carrier of (REAL-NS 1)) is non empty functional M4( the carrier of (REAL-NS m), the carrier of (REAL-NS 1))
FuncZero ( the carrier of (REAL-NS m),(REAL-NS 1)) is Relation-like the carrier of (REAL-NS m) -defined the carrier of (REAL-NS 1) -valued Function-like total V30( the carrier of (REAL-NS m), the carrier of (REAL-NS 1)) Element of K123( the carrier of (REAL-NS m), the carrier of (REAL-NS 1))
the carrier of (REAL-NS m) --> (0. (REAL-NS 1)) is non empty Relation-like the carrier of (REAL-NS m) -defined the carrier of (REAL-NS 1) -valued Function-like total V30( the carrier of (REAL-NS m), the carrier of (REAL-NS 1)) Element of K6(K7( the carrier of (REAL-NS m), the carrier of (REAL-NS 1)))
FuncAdd ( the carrier of (REAL-NS m),(REAL-NS 1)) is Relation-like K7(K123( the carrier of (REAL-NS m), the carrier of (REAL-NS 1)),K123( the carrier of (REAL-NS m), the carrier of (REAL-NS 1))) -defined K123( the carrier of (REAL-NS m), the carrier of (REAL-NS 1)) -valued Function-like total V30(K7(K123( the carrier of (REAL-NS m), the carrier of (REAL-NS 1)),K123( the carrier of (REAL-NS m), the carrier of (REAL-NS 1))),K123( the carrier of (REAL-NS m), the carrier of (REAL-NS 1))) Element of K6(K7(K7(K123( the carrier of (REAL-NS m), the carrier of (REAL-NS 1)),K123( the carrier of (REAL-NS m), the carrier of (REAL-NS 1))),K123( the carrier of (REAL-NS m), the carrier of (REAL-NS 1))))
K7(K123( the carrier of (REAL-NS m), the carrier of (REAL-NS 1)),K123( the carrier of (REAL-NS m), the carrier of (REAL-NS 1))) is set
K7(K7(K123( the carrier of (REAL-NS m), the carrier of (REAL-NS 1)),K123( the carrier of (REAL-NS m), the carrier of (REAL-NS 1))),K123( the carrier of (REAL-NS m), the carrier of (REAL-NS 1))) is set
K6(K7(K7(K123( the carrier of (REAL-NS m), the carrier of (REAL-NS 1)),K123( the carrier of (REAL-NS m), the carrier of (REAL-NS 1))),K123( the carrier of (REAL-NS m), the carrier of (REAL-NS 1)))) is set
FuncExtMult ( the carrier of (REAL-NS m),(REAL-NS 1)) is Relation-like K7(REAL,K123( the carrier of (REAL-NS m), the carrier of (REAL-NS 1))) -defined K123( the carrier of (REAL-NS m), the carrier of (REAL-NS 1)) -valued Function-like total V30(K7(REAL,K123( the carrier of (REAL-NS m), the carrier of (REAL-NS 1))),K123( the carrier of (REAL-NS m), the carrier of (REAL-NS 1))) Element of K6(K7(K7(REAL,K123( the carrier of (REAL-NS m), the carrier of (REAL-NS 1))),K123( the carrier of (REAL-NS m), the carrier of (REAL-NS 1))))
K7(REAL,K123( the carrier of (REAL-NS m), the carrier of (REAL-NS 1))) is V33() set
K7(K7(REAL,K123( the carrier of (REAL-NS m), the carrier of (REAL-NS 1))),K123( the carrier of (REAL-NS m), the carrier of (REAL-NS 1))) is V33() set
K6(K7(K7(REAL,K123( the carrier of (REAL-NS m), the carrier of (REAL-NS 1))),K123( the carrier of (REAL-NS m), the carrier of (REAL-NS 1)))) is V33() set
RLSStruct(# K123( the carrier of (REAL-NS m), the carrier of (REAL-NS 1)),(FuncZero ( the carrier of (REAL-NS m),(REAL-NS 1))),(FuncAdd ( the carrier of (REAL-NS m),(REAL-NS 1))),(FuncExtMult ( the carrier of (REAL-NS m),(REAL-NS 1))) #) is strict RLSStruct
the carrier of K279( the carrier of (REAL-NS m),(REAL-NS 1)) is non empty set
K6( the carrier of K279( the carrier of (REAL-NS m),(REAL-NS 1))) is set
K199(K279( the carrier of (REAL-NS m),(REAL-NS 1)),(LinearOperators ((REAL-NS m),(REAL-NS 1)))) is Element of LinearOperators ((REAL-NS m),(REAL-NS 1))
K197(K279( the carrier of (REAL-NS m),(REAL-NS 1)),(LinearOperators ((REAL-NS m),(REAL-NS 1)))) is Relation-like K7((LinearOperators ((REAL-NS m),(REAL-NS 1))),(LinearOperators ((REAL-NS m),(REAL-NS 1)))) -defined LinearOperators ((REAL-NS m),(REAL-NS 1)) -valued Function-like total V30(K7((LinearOperators ((REAL-NS m),(REAL-NS 1))),(LinearOperators ((REAL-NS m),(REAL-NS 1)))), LinearOperators ((REAL-NS m),(REAL-NS 1))) Element of K6(K7(K7((LinearOperators ((REAL-NS m),(REAL-NS 1))),(LinearOperators ((REAL-NS m),(REAL-NS 1)))),(LinearOperators ((REAL-NS m),(REAL-NS 1)))))
K7((LinearOperators ((REAL-NS m),(REAL-NS 1))),(LinearOperators ((REAL-NS m),(REAL-NS 1)))) is set
K7(K7((LinearOperators ((REAL-NS m),(REAL-NS 1))),(LinearOperators ((REAL-NS m),(REAL-NS 1)))),(LinearOperators ((REAL-NS m),(REAL-NS 1)))) is set
K6(K7(K7((LinearOperators ((REAL-NS m),(REAL-NS 1))),(LinearOperators ((REAL-NS m),(REAL-NS 1)))),(LinearOperators ((REAL-NS m),(REAL-NS 1))))) is set
K198(K279( the carrier of (REAL-NS m),(REAL-NS 1)),(LinearOperators ((REAL-NS m),(REAL-NS 1)))) is Relation-like K7(REAL,(LinearOperators ((REAL-NS m),(REAL-NS 1)))) -defined LinearOperators ((REAL-NS m),(REAL-NS 1)) -valued Function-like total V30(K7(REAL,(LinearOperators ((REAL-NS m),(REAL-NS 1)))), LinearOperators ((REAL-NS m),(REAL-NS 1))) Element of K6(K7(K7(REAL,(LinearOperators ((REAL-NS m),(REAL-NS 1)))),(LinearOperators ((REAL-NS m),(REAL-NS 1)))))
K7(REAL,(LinearOperators ((REAL-NS m),(REAL-NS 1)))) is V33() set
K7(K7(REAL,(LinearOperators ((REAL-NS m),(REAL-NS 1)))),(LinearOperators ((REAL-NS m),(REAL-NS 1)))) is V33() set
K6(K7(K7(REAL,(LinearOperators ((REAL-NS m),(REAL-NS 1)))),(LinearOperators ((REAL-NS m),(REAL-NS 1))))) is V33() set
RLSStruct(# (LinearOperators ((REAL-NS m),(REAL-NS 1))),K199(K279( the carrier of (REAL-NS m),(REAL-NS 1)),(LinearOperators ((REAL-NS m),(REAL-NS 1)))),K197(K279( the carrier of (REAL-NS m),(REAL-NS 1)),(LinearOperators ((REAL-NS m),(REAL-NS 1)))),K198(K279( the carrier of (REAL-NS m),(REAL-NS 1)),(LinearOperators ((REAL-NS m),(REAL-NS 1)))) #) is V71() strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RLSStruct
the carrier of (R_VectorSpace_of_LinearOperators ((REAL-NS m),(REAL-NS 1))) is non empty set
K6( the carrier of (R_VectorSpace_of_LinearOperators ((REAL-NS m),(REAL-NS 1)))) is set
K199((R_VectorSpace_of_LinearOperators ((REAL-NS m),(REAL-NS 1))),(BoundedLinearOperators ((REAL-NS m),(REAL-NS 1)))) is Element of BoundedLinearOperators ((REAL-NS m),(REAL-NS 1))
K197((R_VectorSpace_of_LinearOperators ((REAL-NS m),(REAL-NS 1))),(BoundedLinearOperators ((REAL-NS m),(REAL-NS 1)))) is Relation-like K7((BoundedLinearOperators ((REAL-NS m),(REAL-NS 1))),(BoundedLinearOperators ((REAL-NS m),(REAL-NS 1)))) -defined BoundedLinearOperators ((REAL-NS m),(REAL-NS 1)) -valued Function-like total V30(K7((BoundedLinearOperators ((REAL-NS m),(REAL-NS 1))),(BoundedLinearOperators ((REAL-NS m),(REAL-NS 1)))), BoundedLinearOperators ((REAL-NS m),(REAL-NS 1))) Element of K6(K7(K7((BoundedLinearOperators ((REAL-NS m),(REAL-NS 1))),(BoundedLinearOperators ((REAL-NS m),(REAL-NS 1)))),(BoundedLinearOperators ((REAL-NS m),(REAL-NS 1)))))
K7((BoundedLinearOperators ((REAL-NS m),(REAL-NS 1))),(BoundedLinearOperators ((REAL-NS m),(REAL-NS 1)))) is set
K7(K7((BoundedLinearOperators ((REAL-NS m),(REAL-NS 1))),(BoundedLinearOperators ((REAL-NS m),(REAL-NS 1)))),(BoundedLinearOperators ((REAL-NS m),(REAL-NS 1)))) is set
K6(K7(K7((BoundedLinearOperators ((REAL-NS m),(REAL-NS 1))),(BoundedLinearOperators ((REAL-NS m),(REAL-NS 1)))),(BoundedLinearOperators ((REAL-NS m),(REAL-NS 1))))) is set
K198((R_VectorSpace_of_LinearOperators ((REAL-NS m),(REAL-NS 1))),(BoundedLinearOperators ((REAL-NS m),(REAL-NS 1)))) is Relation-like K7(REAL,(BoundedLinearOperators ((REAL-NS m),(REAL-NS 1)))) -defined BoundedLinearOperators ((REAL-NS m),(REAL-NS 1)) -valued Function-like total V30(K7(REAL,(BoundedLinearOperators ((REAL-NS m),(REAL-NS 1)))), BoundedLinearOperators ((REAL-NS m),(REAL-NS 1))) Element of K6(K7(K7(REAL,(BoundedLinearOperators ((REAL-NS m),(REAL-NS 1)))),(BoundedLinearOperators ((REAL-NS m),(REAL-NS 1)))))
K7(REAL,(BoundedLinearOperators ((REAL-NS m),(REAL-NS 1)))) is V33() set
K7(K7(REAL,(BoundedLinearOperators ((REAL-NS m),(REAL-NS 1)))),(BoundedLinearOperators ((REAL-NS m),(REAL-NS 1)))) is V33() set
K6(K7(K7(REAL,(BoundedLinearOperators ((REAL-NS m),(REAL-NS 1)))),(BoundedLinearOperators ((REAL-NS m),(REAL-NS 1))))) is V33() set
BoundedLinearOperatorsNorm ((REAL-NS m),(REAL-NS 1)) is non empty Relation-like BoundedLinearOperators ((REAL-NS m),(REAL-NS 1)) -defined REAL -valued Function-like total V30( BoundedLinearOperators ((REAL-NS m),(REAL-NS 1)), REAL ) complex-yielding V146() V147() Element of K6(K7((BoundedLinearOperators ((REAL-NS m),(REAL-NS 1))),REAL))
K7((BoundedLinearOperators ((REAL-NS m),(REAL-NS 1))),REAL) is V33() complex-yielding V146() V147() set
K6(K7((BoundedLinearOperators ((REAL-NS m),(REAL-NS 1))),REAL)) is V33() set
G15((BoundedLinearOperators ((REAL-NS m),(REAL-NS 1))),K199((R_VectorSpace_of_LinearOperators ((REAL-NS m),(REAL-NS 1))),(BoundedLinearOperators ((REAL-NS m),(REAL-NS 1)))),K197((R_VectorSpace_of_LinearOperators ((REAL-NS m),(REAL-NS 1))),(BoundedLinearOperators ((REAL-NS m),(REAL-NS 1)))),K198((R_VectorSpace_of_LinearOperators ((REAL-NS m),(REAL-NS 1))),(BoundedLinearOperators ((REAL-NS m),(REAL-NS 1)))),(BoundedLinearOperatorsNorm ((REAL-NS m),(REAL-NS 1)))) is V108() L15()
the carrier of (R_NormSpace_of_BoundedLinearOperators ((REAL-NS m),(REAL-NS 1))) is non empty set
(diff (((Proj (R,n)) * f),Z)) . x is Element of the carrier of (REAL-NS 1)
(proj (1,1)) . ((diff (((Proj (R,n)) * f),Z)) . x) is V11() V12() ext-real Element of REAL
L0 . R is V11() V12() ext-real Element of REAL
R is Element of the carrier of (REAL-NS n)
R is Relation-like NAT -defined REAL -valued Function-like V40(n) FinSequence-like complex-yielding V146() V147() Element of REAL n
x is non empty Relation-like the carrier of (REAL-NS m) -defined the carrier of (REAL-NS n) -valued Function-like total V30( the carrier of (REAL-NS m), the carrier of (REAL-NS n)) Element of K6(K7( the carrier of (REAL-NS m), the carrier of (REAL-NS n)))
L is Element of the carrier of (REAL-NS m)
L0 is Element of the carrier of (REAL-NS m)
L + L0 is Element of the carrier of (REAL-NS m)
x . (L + L0) is Element of the carrier of (REAL-NS n)
x . L is Element of the carrier of (REAL-NS n)
x . L0 is Element of the carrier of (REAL-NS n)
(x . L) + (x . L0) is Element of the carrier of (REAL-NS n)
R is Relation-like NAT -defined REAL -valued Function-like V40(n) FinSequence-like complex-yielding V146() V147() Element of REAL n
R is Relation-like NAT -defined REAL -valued Function-like V40(n) FinSequence-like complex-yielding V146() V147() Element of REAL n
N is Relation-like NAT -defined REAL -valued Function-like V40(n) FinSequence-like complex-yielding V146() V147() Element of REAL n
dom N is V40(n) V156() V157() V158() V159() V160() V161() Element of K6(NAT)
R + R is Relation-like NAT -defined REAL -valued Function-like V40(n) FinSequence-like complex-yielding V146() V147() Element of REAL n
K497() is Relation-like K7(REAL,REAL) -defined REAL -valued Function-like total V30(K7(REAL,REAL), REAL ) complex-yielding V146() V147() Element of K6(K7(K7(REAL,REAL),REAL))
K497() .: (R,R) is set
dom (R + R) is V40(n) V156() V157() V158() V159() V160() V161() Element of K6(NAT)
N is V6() V10() V11() V12() ext-real non negative V33() V38() set
N . N is V11() V12() ext-real Element of REAL
(R + R) . N is V11() V12() ext-real Element of REAL
x is V6() V10() V11() V12() ext-real non negative V33() V38() V120() V155() V156() V157() V158() V159() V160() V161() Element of NAT
N . x is V11() V12() ext-real Element of REAL
Proj (x,n) is non empty Relation-like the carrier of (REAL-NS n) -defined the carrier of (REAL-NS 1) -valued Function-like total V30( the carrier of (REAL-NS n), the carrier of (REAL-NS 1)) Element of K6(K7( the carrier of (REAL-NS n), the carrier of (REAL-NS 1)))
K7( the carrier of (REAL-NS n), the carrier of (REAL-NS 1)) is set
K6(K7( the carrier of (REAL-NS n), the carrier of (REAL-NS 1))) is set
(Proj (x,n)) * f is Relation-like the carrier of (REAL-NS m) -defined the carrier of (REAL-NS 1) -valued Function-like Element of K6(K7( the carrier of (REAL-NS m), the carrier of (REAL-NS 1)))
diff (((Proj (x,n)) * f),Z) is Relation-like Function-like Element of the carrier of (R_NormSpace_of_BoundedLinearOperators ((REAL-NS m),(REAL-NS 1)))
R_NormSpace_of_BoundedLinearOperators ((REAL-NS m),(REAL-NS 1)) is non empty V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() V109() V142() L15()
BoundedLinearOperators ((REAL-NS m),(REAL-NS 1)) is non empty Element of K6( the carrier of (R_VectorSpace_of_LinearOperators ((REAL-NS m),(REAL-NS 1))))
R_VectorSpace_of_LinearOperators ((REAL-NS m),(REAL-NS 1)) is non empty V71() strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V204() RLSStruct
LinearOperators ((REAL-NS m),(REAL-NS 1)) is non empty functional Element of K6( the carrier of K279( the carrier of (REAL-NS m),(REAL-NS 1)))
K279( the carrier of (REAL-NS m),(REAL-NS 1)) is non empty V71() strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V204() RLSStruct
K123( the carrier of (REAL-NS m), the carrier of (REAL-NS 1)) is non empty functional M4( the carrier of (REAL-NS m), the carrier of (REAL-NS 1))
FuncZero ( the carrier of (REAL-NS m),(REAL-NS 1)) is Relation-like the carrier of (REAL-NS m) -defined the carrier of (REAL-NS 1) -valued Function-like total V30( the carrier of (REAL-NS m), the carrier of (REAL-NS 1)) Element of K123( the carrier of (REAL-NS m), the carrier of (REAL-NS 1))
the carrier of (REAL-NS m) --> (0. (REAL-NS 1)) is non empty Relation-like the carrier of (REAL-NS m) -defined the carrier of (REAL-NS 1) -valued Function-like total V30( the carrier of (REAL-NS m), the carrier of (REAL-NS 1)) Element of K6(K7( the carrier of (REAL-NS m), the carrier of (REAL-NS 1)))
FuncAdd ( the carrier of (REAL-NS m),(REAL-NS 1)) is Relation-like K7(K123( the carrier of (REAL-NS m), the carrier of (REAL-NS 1)),K123( the carrier of (REAL-NS m), the carrier of (REAL-NS 1))) -defined K123( the carrier of (REAL-NS m), the carrier of (REAL-NS 1)) -valued Function-like total V30(K7(K123( the carrier of (REAL-NS m), the carrier of (REAL-NS 1)),K123( the carrier of (REAL-NS m), the carrier of (REAL-NS 1))),K123( the carrier of (REAL-NS m), the carrier of (REAL-NS 1))) Element of K6(K7(K7(K123( the carrier of (REAL-NS m), the carrier of (REAL-NS 1)),K123( the carrier of (REAL-NS m), the carrier of (REAL-NS 1))),K123( the carrier of (REAL-NS m), the carrier of (REAL-NS 1))))
K7(K123( the carrier of (REAL-NS m), the carrier of (REAL-NS 1)),K123( the carrier of (REAL-NS m), the carrier of (REAL-NS 1))) is set
K7(K7(K123( the carrier of (REAL-NS m), the carrier of (REAL-NS 1)),K123( the carrier of (REAL-NS m), the carrier of (REAL-NS 1))),K123( the carrier of (REAL-NS m), the carrier of (REAL-NS 1))) is set
K6(K7(K7(K123( the carrier of (REAL-NS m), the carrier of (REAL-NS 1)),K123( the carrier of (REAL-NS m), the carrier of (REAL-NS 1))),K123( the carrier of (REAL-NS m), the carrier of (REAL-NS 1)))) is set
FuncExtMult ( the carrier of (REAL-NS m),(REAL-NS 1)) is Relation-like K7(REAL,K123( the carrier of (REAL-NS m), the carrier of (REAL-NS 1))) -defined K123( the carrier of (REAL-NS m), the carrier of (REAL-NS 1)) -valued Function-like total V30(K7(REAL,K123( the carrier of (REAL-NS m), the carrier of (REAL-NS 1))),K123( the carrier of (REAL-NS m), the carrier of (REAL-NS 1))) Element of K6(K7(K7(REAL,K123( the carrier of (REAL-NS m), the carrier of (REAL-NS 1))),K123( the carrier of (REAL-NS m), the carrier of (REAL-NS 1))))
K7(REAL,K123( the carrier of (REAL-NS m), the carrier of (REAL-NS 1))) is V33() set
K7(K7(REAL,K123( the carrier of (REAL-NS m), the carrier of (REAL-NS 1))),K123( the carrier of (REAL-NS m), the carrier of (REAL-NS 1))) is V33() set
K6(K7(K7(REAL,K123( the carrier of (REAL-NS m), the carrier of (REAL-NS 1))),K123( the carrier of (REAL-NS m), the carrier of (REAL-NS 1)))) is V33() set
RLSStruct(# K123( the carrier of (REAL-NS m), the carrier of (REAL-NS 1)),(FuncZero ( the carrier of (REAL-NS m),(REAL-NS 1))),(FuncAdd ( the carrier of (REAL-NS m),(REAL-NS 1))),(FuncExtMult ( the carrier of (REAL-NS m),(REAL-NS 1))) #) is strict RLSStruct
the carrier of K279( the carrier of (REAL-NS m),(REAL-NS 1)) is non empty set
K6( the carrier of K279( the carrier of (REAL-NS m),(REAL-NS 1))) is set
K199(K279( the carrier of (REAL-NS m),(REAL-NS 1)),(LinearOperators ((REAL-NS m),(REAL-NS 1)))) is Element of LinearOperators ((REAL-NS m),(REAL-NS 1))
K197(K279( the carrier of (REAL-NS m),(REAL-NS 1)),(LinearOperators ((REAL-NS m),(REAL-NS 1)))) is Relation-like K7((LinearOperators ((REAL-NS m),(REAL-NS 1))),(LinearOperators ((REAL-NS m),(REAL-NS 1)))) -defined LinearOperators ((REAL-NS m),(REAL-NS 1)) -valued Function-like total V30(K7((LinearOperators ((REAL-NS m),(REAL-NS 1))),(LinearOperators ((REAL-NS m),(REAL-NS 1)))), LinearOperators ((REAL-NS m),(REAL-NS 1))) Element of K6(K7(K7((LinearOperators ((REAL-NS m),(REAL-NS 1))),(LinearOperators ((REAL-NS m),(REAL-NS 1)))),(LinearOperators ((REAL-NS m),(REAL-NS 1)))))
K7((LinearOperators ((REAL-NS m),(REAL-NS 1))),(LinearOperators ((REAL-NS m),(REAL-NS 1)))) is set
K7(K7((LinearOperators ((REAL-NS m),(REAL-NS 1))),(LinearOperators ((REAL-NS m),(REAL-NS 1)))),(LinearOperators ((REAL-NS m),(REAL-NS 1)))) is set
K6(K7(K7((LinearOperators ((REAL-NS m),(REAL-NS 1))),(LinearOperators ((REAL-NS m),(REAL-NS 1)))),(LinearOperators ((REAL-NS m),(REAL-NS 1))))) is set
K198(K279( the carrier of (REAL-NS m),(REAL-NS 1)),(LinearOperators ((REAL-NS m),(REAL-NS 1)))) is Relation-like K7(REAL,(LinearOperators ((REAL-NS m),(REAL-NS 1)))) -defined LinearOperators ((REAL-NS m),(REAL-NS 1)) -valued Function-like total V30(K7(REAL,(LinearOperators ((REAL-NS m),(REAL-NS 1)))), LinearOperators ((REAL-NS m),(REAL-NS 1))) Element of K6(K7(K7(REAL,(LinearOperators ((REAL-NS m),(REAL-NS 1)))),(LinearOperators ((REAL-NS m),(REAL-NS 1)))))
K7(REAL,(LinearOperators ((REAL-NS m),(REAL-NS 1)))) is V33() set
K7(K7(REAL,(LinearOperators ((REAL-NS m),(REAL-NS 1)))),(LinearOperators ((REAL-NS m),(REAL-NS 1)))) is V33() set
K6(K7(K7(REAL,(LinearOperators ((REAL-NS m),(REAL-NS 1)))),(LinearOperators ((REAL-NS m),(REAL-NS 1))))) is V33() set
RLSStruct(# (LinearOperators ((REAL-NS m),(REAL-NS 1))),K199(K279( the carrier of (REAL-NS m),(REAL-NS 1)),(LinearOperators ((REAL-NS m),(REAL-NS 1)))),K197(K279( the carrier of (REAL-NS m),(REAL-NS 1)),(LinearOperators ((REAL-NS m),(REAL-NS 1)))),K198(K279( the carrier of (REAL-NS m),(REAL-NS 1)),(LinearOperators ((REAL-NS m),(REAL-NS 1)))) #) is V71() strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RLSStruct
the carrier of (R_VectorSpace_of_LinearOperators ((REAL-NS m),(REAL-NS 1))) is non empty set
K6( the carrier of (R_VectorSpace_of_LinearOperators ((REAL-NS m),(REAL-NS 1)))) is set
K199((R_VectorSpace_of_LinearOperators ((REAL-NS m),(REAL-NS 1))),(BoundedLinearOperators ((REAL-NS m),(REAL-NS 1)))) is Element of BoundedLinearOperators ((REAL-NS m),(REAL-NS 1))
K197((R_VectorSpace_of_LinearOperators ((REAL-NS m),(REAL-NS 1))),(BoundedLinearOperators ((REAL-NS m),(REAL-NS 1)))) is Relation-like K7((BoundedLinearOperators ((REAL-NS m),(REAL-NS 1))),(BoundedLinearOperators ((REAL-NS m),(REAL-NS 1)))) -defined BoundedLinearOperators ((REAL-NS m),(REAL-NS 1)) -valued Function-like total V30(K7((BoundedLinearOperators ((REAL-NS m),(REAL-NS 1))),(BoundedLinearOperators ((REAL-NS m),(REAL-NS 1)))), BoundedLinearOperators ((REAL-NS m),(REAL-NS 1))) Element of K6(K7(K7((BoundedLinearOperators ((REAL-NS m),(REAL-NS 1))),(BoundedLinearOperators ((REAL-NS m),(REAL-NS 1)))),(BoundedLinearOperators ((REAL-NS m),(REAL-NS 1)))))
K7((BoundedLinearOperators ((REAL-NS m),(REAL-NS 1))),(BoundedLinearOperators ((REAL-NS m),(REAL-NS 1)))) is set
K7(K7((BoundedLinearOperators ((REAL-NS m),(REAL-NS 1))),(BoundedLinearOperators ((REAL-NS m),(REAL-NS 1)))),(BoundedLinearOperators ((REAL-NS m),(REAL-NS 1)))) is set
K6(K7(K7((BoundedLinearOperators ((REAL-NS m),(REAL-NS 1))),(BoundedLinearOperators ((REAL-NS m),(REAL-NS 1)))),(BoundedLinearOperators ((REAL-NS m),(REAL-NS 1))))) is set
K198((R_VectorSpace_of_LinearOperators ((REAL-NS m),(REAL-NS 1))),(BoundedLinearOperators ((REAL-NS m),(REAL-NS 1)))) is Relation-like K7(REAL,(BoundedLinearOperators ((REAL-NS m),(REAL-NS 1)))) -defined BoundedLinearOperators ((REAL-NS m),(REAL-NS 1)) -valued Function-like total V30(K7(REAL,(BoundedLinearOperators ((REAL-NS m),(REAL-NS 1)))), BoundedLinearOperators ((REAL-NS m),(REAL-NS 1))) Element of K6(K7(K7(REAL,(BoundedLinearOperators ((REAL-NS m),(REAL-NS 1)))),(BoundedLinearOperators ((REAL-NS m),(REAL-NS 1)))))
K7(REAL,(BoundedLinearOperators ((REAL-NS m),(REAL-NS 1)))) is V33() set
K7(K7(REAL,(BoundedLinearOperators ((REAL-NS m),(REAL-NS 1)))),(BoundedLinearOperators ((REAL-NS m),(REAL-NS 1)))) is V33() set
K6(K7(K7(REAL,(BoundedLinearOperators ((REAL-NS m),(REAL-NS 1)))),(BoundedLinearOperators ((REAL-NS m),(REAL-NS 1))))) is V33() set
BoundedLinearOperatorsNorm ((REAL-NS m),(REAL-NS 1)) is non empty Relation-like BoundedLinearOperators ((REAL-NS m),(REAL-NS 1)) -defined REAL -valued Function-like total V30( BoundedLinearOperators ((REAL-NS m),(REAL-NS 1)), REAL ) complex-yielding V146() V147() Element of K6(K7((BoundedLinearOperators ((REAL-NS m),(REAL-NS 1))),REAL))
K7((BoundedLinearOperators ((REAL-NS m),(REAL-NS 1))),REAL) is V33() complex-yielding V146() V147() set
K6(K7((BoundedLinearOperators ((REAL-NS m),(REAL-NS 1))),REAL)) is V33() set
G15((BoundedLinearOperators ((REAL-NS m),(REAL-NS 1))),K199((R_VectorSpace_of_LinearOperators ((REAL-NS m),(REAL-NS 1))),(BoundedLinearOperators ((REAL-NS m),(REAL-NS 1)))),K197((R_VectorSpace_of_LinearOperators ((REAL-NS m),(REAL-NS 1))),(BoundedLinearOperators ((REAL-NS m),(REAL-NS 1)))),K198((R_VectorSpace_of_LinearOperators ((REAL-NS m),(REAL-NS 1))),(BoundedLinearOperators ((REAL-NS m),(REAL-NS 1)))),(BoundedLinearOperatorsNorm ((REAL-NS m),(REAL-NS 1)))) is V108() L15()
the carrier of (R_NormSpace_of_BoundedLinearOperators ((REAL-NS m),(REAL-NS 1))) is non empty set
(diff (((Proj (x,n)) * f),Z)) . (L + L0) is Element of the carrier of (REAL-NS 1)
(proj (1,1)) . ((diff (((Proj (x,n)) * f),Z)) . (L + L0)) is V11() V12() ext-real Element of REAL
R . x is V11() V12() ext-real Element of REAL
(diff (((Proj (x,n)) * f),Z)) . L is Element of the carrier of (REAL-NS 1)
(proj (1,1)) . ((diff (((Proj (x,n)) * f),Z)) . L) is V11() V12() ext-real Element of REAL
R . x is V11() V12() ext-real Element of REAL
(diff (((Proj (x,n)) * f),Z)) . L0 is Element of the carrier of (REAL-NS 1)
(proj (1,1)) . ((diff (((Proj (x,n)) * f),Z)) . L0) is V11() V12() ext-real Element of REAL
((diff (((Proj (x,n)) * f),Z)) . L) + ((diff (((Proj (x,n)) * f),Z)) . L0) is Element of the carrier of (REAL-NS 1)
y is non empty V2() Relation-like NAT -defined REAL -valued Function-like V40(1) FinSequence-like complex-yielding V146() V147() V149() V150() V151() V152() Element of REAL 1
Rz is non empty V2() Relation-like NAT -defined REAL -valued Function-like V40(1) FinSequence-like complex-yielding V146() V147() V149() V150() V151() V152() Element of REAL 1
R0 is non empty V2() Relation-like NAT -defined REAL -valued Function-like V40(1) FinSequence-like complex-yielding V146() V147() V149() V150() V151() V152() Element of REAL 1
Rz + R0 is non empty V2() Relation-like NAT -defined REAL -valued Function-like V40(1) FinSequence-like complex-yielding V146() V147() V149() V150() V151() V152() Element of REAL 1
K497() .: (Rz,R0) is set
(Rz + R0) . 1 is V11() V12() ext-real Element of REAL
Rz . 1 is V11() V12() ext-real Element of REAL
R0 . 1 is V11() V12() ext-real Element of REAL
(Rz . 1) + (R0 . 1) is V11() V12() ext-real Element of REAL
R . N is V11() V12() ext-real Element of REAL
(R . N) + (R0 . 1) is V11() V12() ext-real Element of REAL
R . N is V11() V12() ext-real Element of REAL
(R . N) + (R . N) is V11() V12() ext-real Element of REAL
L is Element of the carrier of (REAL-NS m)
x . L is Element of the carrier of (REAL-NS n)
L0 is V11() V12() ext-real Element of REAL
L0 * L is Element of the carrier of (REAL-NS m)
x . (L0 * L) is Element of the carrier of (REAL-NS n)
L0 * (x . L) is Element of the carrier of (REAL-NS n)
R is Relation-like NAT -defined REAL -valued Function-like V40(n) FinSequence-like complex-yielding V146() V147() Element of REAL n
R is Relation-like NAT -defined REAL -valued Function-like V40(n) FinSequence-like complex-yielding V146() V147() Element of REAL n
L0 * R is Relation-like NAT -defined REAL -valued Function-like V40(n) FinSequence-like complex-yielding V146() V147() Element of REAL n
K515(L0) is non empty Relation-like REAL -defined REAL -valued Function-like total V30( REAL , REAL ) complex-yielding V146() V147() Element of K6(K7(REAL,REAL))
K499() is Relation-like K7(REAL,REAL) -defined REAL -valued Function-like total V30(K7(REAL,REAL), REAL ) complex-yielding V146() V147() Element of K6(K7(K7(REAL,REAL),REAL))
id REAL is non empty Relation-like REAL -defined REAL -valued total V30( REAL , REAL ) complex-yielding V146() V147() Element of K6(K7(REAL,REAL))
K499() [;] (L0,(id REAL)) is set
R (#) K515(L0) is Relation-like complex-yielding V146() V147() set
dom (L0 * R) is V40(n) V156() V157() V158() V159() V160() V161() Element of K6(NAT)
dom R is V40(n) V156() V157() V158() V159() V160() V161() Element of K6(NAT)
N is V6() V10() V11() V12() ext-real non negative V33() V38() set
(L0 * R) . N is V11() V12() ext-real Element of REAL
R . N is V11() V12() ext-real Element of REAL
N is V6() V10() V11() V12() ext-real non negative V33() V38() V120() V155() V156() V157() V158() V159() V160() V161() Element of NAT
R . N is V11() V12() ext-real Element of REAL
Proj (N,n) is non empty Relation-like the carrier of (REAL-NS n) -defined the carrier of (REAL-NS 1) -valued Function-like total V30( the carrier of (REAL-NS n), the carrier of (REAL-NS 1)) Element of K6(K7( the carrier of (REAL-NS n), the carrier of (REAL-NS 1)))
K7( the carrier of (REAL-NS n), the carrier of (REAL-NS 1)) is set
K6(K7( the carrier of (REAL-NS n), the carrier of (REAL-NS 1))) is set
(Proj (N,n)) * f is Relation-like the carrier of (REAL-NS m) -defined the carrier of (REAL-NS 1) -valued Function-like Element of K6(K7( the carrier of (REAL-NS m), the carrier of (REAL-NS 1)))
diff (((Proj (N,n)) * f),Z) is Relation-like Function-like Element of the carrier of (R_NormSpace_of_BoundedLinearOperators ((REAL-NS m),(REAL-NS 1)))
R_NormSpace_of_BoundedLinearOperators ((REAL-NS m),(REAL-NS 1)) is non empty V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() V109() V142() L15()
BoundedLinearOperators ((REAL-NS m),(REAL-NS 1)) is non empty Element of K6( the carrier of (R_VectorSpace_of_LinearOperators ((REAL-NS m),(REAL-NS 1))))
R_VectorSpace_of_LinearOperators ((REAL-NS m),(REAL-NS 1)) is non empty V71() strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V204() RLSStruct
LinearOperators ((REAL-NS m),(REAL-NS 1)) is non empty functional Element of K6( the carrier of K279( the carrier of (REAL-NS m),(REAL-NS 1)))
K279( the carrier of (REAL-NS m),(REAL-NS 1)) is non empty V71() strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V204() RLSStruct
K123( the carrier of (REAL-NS m), the carrier of (REAL-NS 1)) is non empty functional M4( the carrier of (REAL-NS m), the carrier of (REAL-NS 1))
FuncZero ( the carrier of (REAL-NS m),(REAL-NS 1)) is Relation-like the carrier of (REAL-NS m) -defined the carrier of (REAL-NS 1) -valued Function-like total V30( the carrier of (REAL-NS m), the carrier of (REAL-NS 1)) Element of K123( the carrier of (REAL-NS m), the carrier of (REAL-NS 1))
the carrier of (REAL-NS m) --> (0. (REAL-NS 1)) is non empty Relation-like the carrier of (REAL-NS m) -defined the carrier of (REAL-NS 1) -valued Function-like total V30( the carrier of (REAL-NS m), the carrier of (REAL-NS 1)) Element of K6(K7( the carrier of (REAL-NS m), the carrier of (REAL-NS 1)))
FuncAdd ( the carrier of (REAL-NS m),(REAL-NS 1)) is Relation-like K7(K123( the carrier of (REAL-NS m), the carrier of (REAL-NS 1)),K123( the carrier of (REAL-NS m), the carrier of (REAL-NS 1))) -defined K123( the carrier of (REAL-NS m), the carrier of (REAL-NS 1)) -valued Function-like total V30(K7(K123( the carrier of (REAL-NS m), the carrier of (REAL-NS 1)),K123( the carrier of (REAL-NS m), the carrier of (REAL-NS 1))),K123( the carrier of (REAL-NS m), the carrier of (REAL-NS 1))) Element of K6(K7(K7(K123( the carrier of (REAL-NS m), the carrier of (REAL-NS 1)),K123( the carrier of (REAL-NS m), the carrier of (REAL-NS 1))),K123( the carrier of (REAL-NS m), the carrier of (REAL-NS 1))))
K7(K123( the carrier of (REAL-NS m), the carrier of (REAL-NS 1)),K123( the carrier of (REAL-NS m), the carrier of (REAL-NS 1))) is set
K7(K7(K123( the carrier of (REAL-NS m), the carrier of (REAL-NS 1)),K123( the carrier of (REAL-NS m), the carrier of (REAL-NS 1))),K123( the carrier of (REAL-NS m), the carrier of (REAL-NS 1))) is set
K6(K7(K7(K123( the carrier of (REAL-NS m), the carrier of (REAL-NS 1)),K123( the carrier of (REAL-NS m), the carrier of (REAL-NS 1))),K123( the carrier of (REAL-NS m), the carrier of (REAL-NS 1)))) is set
FuncExtMult ( the carrier of (REAL-NS m),(REAL-NS 1)) is Relation-like K7(REAL,K123( the carrier of (REAL-NS m), the carrier of (REAL-NS 1))) -defined K123( the carrier of (REAL-NS m), the carrier of (REAL-NS 1)) -valued Function-like total V30(K7(REAL,K123( the carrier of (REAL-NS m), the carrier of (REAL-NS 1))),K123( the carrier of (REAL-NS m), the carrier of (REAL-NS 1))) Element of K6(K7(K7(REAL,K123( the carrier of (REAL-NS m), the carrier of (REAL-NS 1))),K123( the carrier of (REAL-NS m), the carrier of (REAL-NS 1))))
K7(REAL,K123( the carrier of (REAL-NS m), the carrier of (REAL-NS 1))) is V33() set
K7(K7(REAL,K123( the carrier of (REAL-NS m), the carrier of (REAL-NS 1))),K123( the carrier of (REAL-NS m), the carrier of (REAL-NS 1))) is V33() set
K6(K7(K7(REAL,K123( the carrier of (REAL-NS m), the carrier of (REAL-NS 1))),K123( the carrier of (REAL-NS m), the carrier of (REAL-NS 1)))) is V33() set
RLSStruct(# K123( the carrier of (REAL-NS m), the carrier of (REAL-NS 1)),(FuncZero ( the carrier of (REAL-NS m),(REAL-NS 1))),(FuncAdd ( the carrier of (REAL-NS m),(REAL-NS 1))),(FuncExtMult ( the carrier of (REAL-NS m),(REAL-NS 1))) #) is strict RLSStruct
the carrier of K279( the carrier of (REAL-NS m),(REAL-NS 1)) is non empty set
K6( the carrier of K279( the carrier of (REAL-NS m),(REAL-NS 1))) is set
K199(K279( the carrier of (REAL-NS m),(REAL-NS 1)),(LinearOperators ((REAL-NS m),(REAL-NS 1)))) is Element of LinearOperators ((REAL-NS m),(REAL-NS 1))
K197(K279( the carrier of (REAL-NS m),(REAL-NS 1)),(LinearOperators ((REAL-NS m),(REAL-NS 1)))) is Relation-like K7((LinearOperators ((REAL-NS m),(REAL-NS 1))),(LinearOperators ((REAL-NS m),(REAL-NS 1)))) -defined LinearOperators ((REAL-NS m),(REAL-NS 1)) -valued Function-like total V30(K7((LinearOperators ((REAL-NS m),(REAL-NS 1))),(LinearOperators ((REAL-NS m),(REAL-NS 1)))), LinearOperators ((REAL-NS m),(REAL-NS 1))) Element of K6(K7(K7((LinearOperators ((REAL-NS m),(REAL-NS 1))),(LinearOperators ((REAL-NS m),(REAL-NS 1)))),(LinearOperators ((REAL-NS m),(REAL-NS 1)))))
K7((LinearOperators ((REAL-NS m),(REAL-NS 1))),(LinearOperators ((REAL-NS m),(REAL-NS 1)))) is set
K7(K7((LinearOperators ((REAL-NS m),(REAL-NS 1))),(LinearOperators ((REAL-NS m),(REAL-NS 1)))),(LinearOperators ((REAL-NS m),(REAL-NS 1)))) is set
K6(K7(K7((LinearOperators ((REAL-NS m),(REAL-NS 1))),(LinearOperators ((REAL-NS m),(REAL-NS 1)))),(LinearOperators ((REAL-NS m),(REAL-NS 1))))) is set
K198(K279( the carrier of (REAL-NS m),(REAL-NS 1)),(LinearOperators ((REAL-NS m),(REAL-NS 1)))) is Relation-like K7(REAL,(LinearOperators ((REAL-NS m),(REAL-NS 1)))) -defined LinearOperators ((REAL-NS m),(REAL-NS 1)) -valued Function-like total V30(K7(REAL,(LinearOperators ((REAL-NS m),(REAL-NS 1)))), LinearOperators ((REAL-NS m),(REAL-NS 1))) Element of K6(K7(K7(REAL,(LinearOperators ((REAL-NS m),(REAL-NS 1)))),(LinearOperators ((REAL-NS m),(REAL-NS 1)))))
K7(REAL,(LinearOperators ((REAL-NS m),(REAL-NS 1)))) is V33() set
K7(K7(REAL,(LinearOperators ((REAL-NS m),(REAL-NS 1)))),(LinearOperators ((REAL-NS m),(REAL-NS 1)))) is V33() set
K6(K7(K7(REAL,(LinearOperators ((REAL-NS m),(REAL-NS 1)))),(LinearOperators ((REAL-NS m),(REAL-NS 1))))) is V33() set
RLSStruct(# (LinearOperators ((REAL-NS m),(REAL-NS 1))),K199(K279( the carrier of (REAL-NS m),(REAL-NS 1)),(LinearOperators ((REAL-NS m),(REAL-NS 1)))),K197(K279( the carrier of (REAL-NS m),(REAL-NS 1)),(LinearOperators ((REAL-NS m),(REAL-NS 1)))),K198(K279( the carrier of (REAL-NS m),(REAL-NS 1)),(LinearOperators ((REAL-NS m),(REAL-NS 1)))) #) is V71() strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RLSStruct
the carrier of (R_VectorSpace_of_LinearOperators ((REAL-NS m),(REAL-NS 1))) is non empty set
K6( the carrier of (R_VectorSpace_of_LinearOperators ((REAL-NS m),(REAL-NS 1)))) is set
K199((R_VectorSpace_of_LinearOperators ((REAL-NS m),(REAL-NS 1))),(BoundedLinearOperators ((REAL-NS m),(REAL-NS 1)))) is Element of BoundedLinearOperators ((REAL-NS m),(REAL-NS 1))
K197((R_VectorSpace_of_LinearOperators ((REAL-NS m),(REAL-NS 1))),(BoundedLinearOperators ((REAL-NS m),(REAL-NS 1)))) is Relation-like K7((BoundedLinearOperators ((REAL-NS m),(REAL-NS 1))),(BoundedLinearOperators ((REAL-NS m),(REAL-NS 1)))) -defined BoundedLinearOperators ((REAL-NS m),(REAL-NS 1)) -valued Function-like total V30(K7((BoundedLinearOperators ((REAL-NS m),(REAL-NS 1))),(BoundedLinearOperators ((REAL-NS m),(REAL-NS 1)))), BoundedLinearOperators ((REAL-NS m),(REAL-NS 1))) Element of K6(K7(K7((BoundedLinearOperators ((REAL-NS m),(REAL-NS 1))),(BoundedLinearOperators ((REAL-NS m),(REAL-NS 1)))),(BoundedLinearOperators ((REAL-NS m),(REAL-NS 1)))))
K7((BoundedLinearOperators ((REAL-NS m),(REAL-NS 1))),(BoundedLinearOperators ((REAL-NS m),(REAL-NS 1)))) is set
K7(K7((BoundedLinearOperators ((REAL-NS m),(REAL-NS 1))),(BoundedLinearOperators ((REAL-NS m),(REAL-NS 1)))),(BoundedLinearOperators ((REAL-NS m),(REAL-NS 1)))) is set
K6(K7(K7((BoundedLinearOperators ((REAL-NS m),(REAL-NS 1))),(BoundedLinearOperators ((REAL-NS m),(REAL-NS 1)))),(BoundedLinearOperators ((REAL-NS m),(REAL-NS 1))))) is set
K198((R_VectorSpace_of_LinearOperators ((REAL-NS m),(REAL-NS 1))),(BoundedLinearOperators ((REAL-NS m),(REAL-NS 1)))) is Relation-like K7(REAL,(BoundedLinearOperators ((REAL-NS m),(REAL-NS 1)))) -defined BoundedLinearOperators ((REAL-NS m),(REAL-NS 1)) -valued Function-like total V30(K7(REAL,(BoundedLinearOperators ((REAL-NS m),(REAL-NS 1)))), BoundedLinearOperators ((REAL-NS m),(REAL-NS 1))) Element of K6(K7(K7(REAL,(BoundedLinearOperators ((REAL-NS m),(REAL-NS 1)))),(BoundedLinearOperators ((REAL-NS m),(REAL-NS 1)))))
K7(REAL,(BoundedLinearOperators ((REAL-NS m),(REAL-NS 1)))) is V33() set
K7(K7(REAL,(BoundedLinearOperators ((REAL-NS m),(REAL-NS 1)))),(BoundedLinearOperators ((REAL-NS m),(REAL-NS 1)))) is V33() set
K6(K7(K7(REAL,(BoundedLinearOperators ((REAL-NS m),(REAL-NS 1)))),(BoundedLinearOperators ((REAL-NS m),(REAL-NS 1))))) is V33() set
BoundedLinearOperatorsNorm ((REAL-NS m),(REAL-NS 1)) is non empty Relation-like BoundedLinearOperators ((REAL-NS m),(REAL-NS 1)) -defined REAL -valued Function-like total V30( BoundedLinearOperators ((REAL-NS m),(REAL-NS 1)), REAL ) complex-yielding V146() V147() Element of K6(K7((BoundedLinearOperators ((REAL-NS m),(REAL-NS 1))),REAL))
K7((BoundedLinearOperators ((REAL-NS m),(REAL-NS 1))),REAL) is V33() complex-yielding V146() V147() set
K6(K7((BoundedLinearOperators ((REAL-NS m),(REAL-NS 1))),REAL)) is V33() set
G15((BoundedLinearOperators ((REAL-NS m),(REAL-NS 1))),K199((R_VectorSpace_of_LinearOperators ((REAL-NS m),(REAL-NS 1))),(BoundedLinearOperators ((REAL-NS m),(REAL-NS 1)))),K197((R_VectorSpace_of_LinearOperators ((REAL-NS m),(REAL-NS 1))),(BoundedLinearOperators ((REAL-NS m),(REAL-NS 1)))),K198((R_VectorSpace_of_LinearOperators ((REAL-NS m),(REAL-NS 1))),(BoundedLinearOperators ((REAL-NS m),(REAL-NS 1)))),(BoundedLinearOperatorsNorm ((REAL-NS m),(REAL-NS 1)))) is V108() L15()
the carrier of (R_NormSpace_of_BoundedLinearOperators ((REAL-NS m),(REAL-NS 1))) is non empty set
(diff (((Proj (N,n)) * f),Z)) . L is Element of the carrier of (REAL-NS 1)
(proj (1,1)) . ((diff (((Proj (N,n)) * f),Z)) . L) is V11() V12() ext-real Element of REAL
R . N is V11() V12() ext-real Element of REAL
(diff (((Proj (N,n)) * f),Z)) . (L0 * L) is Element of the carrier of (REAL-NS 1)
(proj (1,1)) . ((diff (((Proj (N,n)) * f),Z)) . (L0 * L)) is V11() V12() ext-real Element of REAL
L0 * ((diff (((Proj (N,n)) * f),Z)) . L) is Element of the carrier of (REAL-NS 1)
x is non empty V2() Relation-like NAT -defined REAL -valued Function-like V40(1) FinSequence-like complex-yielding V146() V147() V149() V150() V151() V152() Element of REAL 1
y is non empty V2() Relation-like NAT -defined REAL -valued Function-like V40(1) FinSequence-like complex-yielding V146() V147() V149() V150() V151() V152() Element of REAL 1
L0 * y is non empty V2() Relation-like NAT -defined REAL -valued Function-like V40(1) FinSequence-like complex-yielding V146() V147() V149() V150() V151() V152() Element of REAL 1
y (#) K515(L0) is Relation-like complex-yielding V146() V147() set
(L0 * y) . 1 is V11() V12() ext-real Element of REAL
y . 1 is V11() V12() ext-real Element of REAL
L0 * (y . 1) is V11() V12() ext-real Element of REAL
R . N is V11() V12() ext-real Element of REAL
L0 * (R . N) is V11() V12() ext-real Element of REAL
the carrier of (R_NormSpace_of_BoundedLinearOperators ((REAL-NS m),(REAL-NS n))) is non empty set
L is non empty Relation-like the carrier of (REAL-NS m) -defined the carrier of (REAL-NS n) -valued Function-like total V30( the carrier of (REAL-NS m), the carrier of (REAL-NS n)) additive homogeneous Lipschitzian Element of K6(K7( the carrier of (REAL-NS m), the carrier of (REAL-NS n)))
f /. Z is Element of the carrier of (REAL-NS n)
R is non empty Relation-like the carrier of (REAL-NS m) -defined the carrier of (REAL-NS n) -valued Function-like total V30( the carrier of (REAL-NS m), the carrier of (REAL-NS n)) Element of K6(K7( the carrier of (REAL-NS m), the carrier of (REAL-NS n)))
dom f is Element of K6( the carrier of (REAL-NS m))
K6( the carrier of (REAL-NS m)) is set
R is Element of the carrier of (REAL-NS m)
R + Z is Element of the carrier of (REAL-NS m)
N is V6() V10() V11() V12() ext-real non negative V33() V38() V120() V155() V156() V157() V158() V159() V160() V161() Element of NAT
Proj (N,n) is non empty Relation-like the carrier of (REAL-NS n) -defined the carrier of (REAL-NS 1) -valued Function-like total V30( the carrier of (REAL-NS n), the carrier of (REAL-NS 1)) Element of K6(K7( the carrier of (REAL-NS n), the carrier of (REAL-NS 1)))
K7( the carrier of (REAL-NS n), the carrier of (REAL-NS 1)) is set
K6(K7( the carrier of (REAL-NS n), the carrier of (REAL-NS 1))) is set
(Proj (N,n)) * R is non empty Relation-like the carrier of (REAL-NS m) -defined the carrier of (REAL-NS 1) -valued Function-like total V30( the carrier of (REAL-NS m), the carrier of (REAL-NS 1)) Element of K6(K7( the carrier of (REAL-NS m), the carrier of (REAL-NS 1)))
((Proj (N,n)) * R) . R is Element of the carrier of (REAL-NS 1)
(Proj (N,n)) * f is Relation-like the carrier of (REAL-NS m) -defined the carrier of (REAL-NS 1) -valued Function-like Element of K6(K7( the carrier of (REAL-NS m), the carrier of (REAL-NS 1)))
((Proj (N,n)) * f) /. (R + Z) is Element of the carrier of (REAL-NS 1)
((Proj (N,n)) * f) /. Z is Element of the carrier of (REAL-NS 1)
(((Proj (N,n)) * f) /. (R + Z)) - (((Proj (N,n)) * f) /. Z) is Element of the carrier of (REAL-NS 1)
(Proj (N,n)) * L is non empty Relation-like the carrier of (REAL-NS m) -defined the carrier of (REAL-NS 1) -valued Function-like total V30( the carrier of (REAL-NS m), the carrier of (REAL-NS 1)) Element of K6(K7( the carrier of (REAL-NS m), the carrier of (REAL-NS 1)))
((Proj (N,n)) * L) . R is Element of the carrier of (REAL-NS 1)
((((Proj (N,n)) * f) /. (R + Z)) - (((Proj (N,n)) * f) /. Z)) - (((Proj (N,n)) * L) . R) is Element of the carrier of (REAL-NS 1)
Z - Z is Element of the carrier of (REAL-NS m)
0. (REAL-NS m) is V52( REAL-NS m) Element of the carrier of (REAL-NS m)
0* m is Relation-like NAT -defined REAL -valued Function-like V40(m) FinSequence-like complex-yielding V146() V147() Element of REAL m
REAL m is non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered FinSequenceSet of REAL
m -tuples_on REAL is non empty functional FinSequence-membered FinSequenceSet of REAL
m |-> 0 is Relation-like empty-yielding NAT -defined REAL -valued Function-like V40(m) FinSequence-like complex-yielding V146() V147() Element of m -tuples_on REAL
||.(Z - Z).|| is V11() V12() ext-real Element of REAL
|.(0* m).| is V11() V12() ext-real non negative Element of REAL
sqr (0* m) is Relation-like NAT -defined REAL -valued Function-like FinSequence-like complex-yielding V146() V147() FinSequence of REAL
(0* m) (#) sqrreal is Relation-like complex-yielding V146() V147() set
(0* m) (#) (0* m) is Relation-like Function-like set
K499() is Relation-like K7(REAL,REAL) -defined REAL -valued Function-like total V30(K7(REAL,REAL), REAL ) complex-yielding V146() V147() Element of K6(K7(K7(REAL,REAL),REAL))
K499() .: ((0* m),(0* m)) is set
Sum (sqr (0* m)) is V11() V12() ext-real Element of REAL
sqrt (Sum (sqr (0* m))) is V11() V12() ext-real Element of REAL
Z0 /. N is V11() V12() ext-real Element of REAL
{ b₁ where b₁ is Element of the carrier of (REAL-NS m) : not Z0 /. N <= ||.(b₁ - Z).|| } is set
dom ((Proj (N,n)) * f) is Element of K6( the carrier of (REAL-NS m))
f /. (R + Z) is Element of the carrier of (REAL-NS n)
L . R is Element of the carrier of (REAL-NS n)
x is Relation-like NAT -defined REAL -valued Function-like V40(n) FinSequence-like complex-yielding V146() V147() Element of REAL n
- x is Relation-like NAT -defined REAL -valued Function-like V40(n) FinSequence-like complex-yielding V146() V147() Element of REAL n
K495() is non empty Relation-like REAL -defined REAL -valued Function-like total V30( REAL , REAL ) complex-yielding V146() V147() Element of K6(K7(REAL,REAL))
x (#) K495() is Relation-like complex-yielding V146() V147() set
- 1 is V11() V12() ext-real non positive set
(- 1) (#) x is Relation-like Function-like set
K515((- 1)) is non empty Relation-like REAL -defined REAL -valued Function-like total V30( REAL , REAL ) complex-yielding V146() V147() Element of K6(K7(REAL,REAL))
id REAL is non empty Relation-like REAL -defined REAL -valued total V30( REAL , REAL ) complex-yielding V146() V147() Element of K6(K7(REAL,REAL))
K499() [;] ((- 1),(id REAL)) is set
x (#) K515((- 1)) is Relation-like complex-yielding V146() V147() set
(- 1) * (f /. Z) is Element of the carrier of (REAL-NS n)
y is Relation-like NAT -defined REAL -valued Function-like V40(n) FinSequence-like complex-yielding V146() V147() Element of REAL n
- y is Relation-like NAT -defined REAL -valued Function-like V40(n) FinSequence-like complex-yielding V146() V147() Element of REAL n
y (#) K495() is Relation-like complex-yielding V146() V147() set
(- 1) (#) y is Relation-like Function-like set
y (#) K515((- 1)) is Relation-like complex-yielding V146() V147() set
(- 1) * (L . R) is Element of the carrier of (REAL-NS n)
N is Relation-like NAT -defined REAL -valued Function-like V40(n) FinSequence-like complex-yielding V146() V147() Element of REAL n
N + (- x) is Relation-like NAT -defined REAL -valued Function-like V40(n) FinSequence-like complex-yielding V146() V147() Element of REAL n
K497() is Relation-like K7(REAL,REAL) -defined REAL -valued Function-like total V30(K7(REAL,REAL), REAL ) complex-yielding V146() V147() Element of K6(K7(K7(REAL,REAL),REAL))
K497() .: (N,(- x)) is set
(f /. (R + Z)) + ((- 1) * (f /. Z)) is Element of the carrier of (REAL-NS n)
(N + (- x)) + (- y) is Relation-like NAT -defined REAL -valued Function-like V40(n) FinSequence-like complex-yielding V146() V147() Element of REAL n
K497() .: ((N + (- x)),(- y)) is set
((f /. (R + Z)) + ((- 1) * (f /. Z))) + ((- 1) * (L . R)) is Element of the carrier of (REAL-NS n)
N - x is Relation-like NAT -defined REAL -valued Function-like V40(n) FinSequence-like complex-yielding V146() V147() Element of REAL n
K498() is Relation-like K7(REAL,REAL) -defined REAL -valued Function-like total V30(K7(REAL,REAL), REAL ) complex-yielding V146() V147() Element of K6(K7(K7(REAL,REAL),REAL))
K231(REAL,K497(),(id REAL),K495()) is Relation-like K7(REAL,REAL) -defined REAL -valued Function-like total V30(K7(REAL,REAL), REAL ) complex-yielding V146() V147() Element of K6(K7(K7(REAL,REAL),REAL))
K498() .: (N,x) is set
- x is Relation-like Function-like complex-yielding set
N + (- x) is Relation-like Function-like set
(f /. (R + Z)) - (f /. Z) is Element of the carrier of (REAL-NS n)
((f /. (R + Z)) - (f /. Z)) - (L . R) is Element of the carrier of (REAL-NS n)
(N - x) - y is Relation-like NAT -defined REAL -valued Function-like V40(n) FinSequence-like complex-yielding V146() V147() Element of REAL n
K498() .: ((N - x),y) is set
- y is Relation-like Function-like complex-yielding set
(N - x) + (- y) is Relation-like Function-like set
R . R is Element of the carrier of (REAL-NS n)
(Proj (N,n)) . (R . R) is Element of the carrier of (REAL-NS 1)
(Proj (N,n)) . (((f /. (R + Z)) - (f /. Z)) - (L . R)) is Element of the carrier of (REAL-NS 1)
(Proj (N,n)) . ((f /. (R + Z)) + ((- 1) * (f /. Z))) is Element of the carrier of (REAL-NS 1)
(Proj (N,n)) . ((- 1) * (L . R)) is Element of the carrier of (REAL-NS 1)
((Proj (N,n)) . ((f /. (R + Z)) + ((- 1) * (f /. Z)))) + ((Proj (N,n)) . ((- 1) * (L . R))) is Element of the carrier of (REAL-NS 1)
(Proj (N,n)) . (f /. (R + Z)) is Element of the carrier of (REAL-NS 1)
(Proj (N,n)) . ((- 1) * (f /. Z)) is Element of the carrier of (REAL-NS 1)
((Proj (N,n)) . (f /. (R + Z))) + ((Proj (N,n)) . ((- 1) * (f /. Z))) is Element of the carrier of (REAL-NS 1)
(((Proj (N,n)) . (f /. (R + Z))) + ((Proj (N,n)) . ((- 1) * (f /. Z)))) + ((Proj (N,n)) . ((- 1) * (L . R))) is Element of the carrier of (REAL-NS 1)
(Proj (N,n)) . (f /. Z) is Element of the carrier of (REAL-NS 1)
(- 1) * ((Proj (N,n)) . (f /. Z)) is Element of the carrier of (REAL-NS 1)
Rz is Element of the carrier of (REAL-NS n)
(- 1) * Rz is Element of the carrier of (REAL-NS n)
(Proj (N,n)) . ((- 1) * Rz) is Element of the carrier of (REAL-NS 1)
(Proj (N,n)) . Rz is Element of the carrier of (REAL-NS 1)
(- 1) * ((Proj (N,n)) . Rz) is Element of the carrier of (REAL-NS 1)
- ((Proj (N,n)) . (f /. Z)) is Element of the carrier of (REAL-NS 1)
((Proj (N,n)) . (f /. (R + Z))) + (- ((Proj (N,n)) . (f /. Z))) is Element of the carrier of (REAL-NS 1)
(Proj (N,n)) . (L . R) is Element of the carrier of (REAL-NS 1)
(- 1) * ((Proj (N,n)) . (L . R)) is Element of the carrier of (REAL-NS 1)
(((Proj (N,n)) . (f /. (R + Z))) + (- ((Proj (N,n)) . (f /. Z)))) + ((- 1) * ((Proj (N,n)) . (L . R))) is Element of the carrier of (REAL-NS 1)
- ((Proj (N,n)) . (L . R)) is Element of the carrier of (REAL-NS 1)
(((Proj (N,n)) . (f /. (R + Z))) + (- ((Proj (N,n)) . (f /. Z)))) + (- ((Proj (N,n)) . (L . R))) is Element of the carrier of (REAL-NS 1)
((Proj (N,n)) . (f /. (R + Z))) - ((Proj (N,n)) . (f /. Z)) is Element of the carrier of (REAL-NS 1)
(((Proj (N,n)) . (f /. (R + Z))) - ((Proj (N,n)) . (f /. Z))) + (- ((Proj (N,n)) . (L . R))) is Element of the carrier of (REAL-NS 1)
(((Proj (N,n)) . (f /. (R + Z))) - ((Proj (N,n)) . (f /. Z))) - ((Proj (N,n)) . (L . R)) is Element of the carrier of (REAL-NS 1)
dom (Proj (N,n)) is Element of K6( the carrier of (REAL-NS n))
K6( the carrier of (REAL-NS n)) is set
(Proj (N,n)) /. (f /. (R + Z)) is Element of the carrier of (REAL-NS 1)
(Proj (N,n)) /. (f /. Z) is Element of the carrier of (REAL-NS 1)
0. (REAL-NS m) is V52( REAL-NS m) Element of the carrier of (REAL-NS m)
R is V11() V12() ext-real Element of REAL
sqrt n is V11() V12() ext-real set
(sqrt n) " is V11() V12() ext-real set
((sqrt n) ") * R is V11() V12() ext-real Element of REAL
N is V6() V10() V11() V12() ext-real non negative V33() V38() V120() V155() V156() V157() V158() V159() V160() V161() Element of NAT
Proj (N,n) is non empty Relation-like the carrier of (REAL-NS n) -defined the carrier of (REAL-NS 1) -valued Function-like total V30( the carrier of (REAL-NS n), the carrier of (REAL-NS 1)) Element of K6(K7( the carrier of (REAL-NS n), the carrier of (REAL-NS 1)))
K7( the carrier of (REAL-NS n), the carrier of (REAL-NS 1)) is set
K6(K7( the carrier of (REAL-NS n), the carrier of (REAL-NS 1))) is set
(Proj (N,n)) * R is non empty Relation-like the carrier of (REAL-NS m) -defined the carrier of (REAL-NS 1) -valued Function-like total V30( the carrier of (REAL-NS m), the carrier of (REAL-NS 1)) Element of K6(K7( the carrier of (REAL-NS m), the carrier of (REAL-NS 1)))
(Proj (N,n)) * f is Relation-like the carrier of (REAL-NS m) -defined the carrier of (REAL-NS 1) -valued Function-like Element of K6(K7( the carrier of (REAL-NS m), the carrier of (REAL-NS 1)))
dom ((Proj (N,n)) * f) is Element of K6( the carrier of (REAL-NS m))
((Proj (N,n)) * f) /. Z is Element of the carrier of (REAL-NS 1)
diff (((Proj (N,n)) * f),Z) is Relation-like Function-like Element of the carrier of (R_NormSpace_of_BoundedLinearOperators ((REAL-NS m),(REAL-NS 1)))
R_NormSpace_of_BoundedLinearOperators ((REAL-NS m),(REAL-NS 1)) is non empty V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() V109() V142() L15()
BoundedLinearOperators ((REAL-NS m),(REAL-NS 1)) is non empty Element of K6( the carrier of (R_VectorSpace_of_LinearOperators ((REAL-NS m),(REAL-NS 1))))
R_VectorSpace_of_LinearOperators ((REAL-NS m),(REAL-NS 1)) is non empty V71() strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V204() RLSStruct
LinearOperators ((REAL-NS m),(REAL-NS 1)) is non empty functional Element of K6( the carrier of K279( the carrier of (REAL-NS m),(REAL-NS 1)))
K279( the carrier of (REAL-NS m),(REAL-NS 1)) is non empty V71() strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V204() RLSStruct
K123( the carrier of (REAL-NS m), the carrier of (REAL-NS 1)) is non empty functional M4( the carrier of (REAL-NS m), the carrier of (REAL-NS 1))
FuncZero ( the carrier of (REAL-NS m),(REAL-NS 1)) is Relation-like the carrier of (REAL-NS m) -defined the carrier of (REAL-NS 1) -valued Function-like total V30( the carrier of (REAL-NS m), the carrier of (REAL-NS 1)) Element of K123( the carrier of (REAL-NS m), the carrier of (REAL-NS 1))
the carrier of (REAL-NS m) --> (0. (REAL-NS 1)) is non empty Relation-like the carrier of (REAL-NS m) -defined the carrier of (REAL-NS 1) -valued Function-like total V30( the carrier of (REAL-NS m), the carrier of (REAL-NS 1)) Element of K6(K7( the carrier of (REAL-NS m), the carrier of (REAL-NS 1)))
FuncAdd ( the carrier of (REAL-NS m),(REAL-NS 1)) is Relation-like K7(K123( the carrier of (REAL-NS m), the carrier of (REAL-NS 1)),K123( the carrier of (REAL-NS m), the carrier of (REAL-NS 1))) -defined K123( the carrier of (REAL-NS m), the carrier of (REAL-NS 1)) -valued Function-like total V30(K7(K123( the carrier of (REAL-NS m), the carrier of (REAL-NS 1)),K123( the carrier of (REAL-NS m), the carrier of (REAL-NS 1))),K123( the carrier of (REAL-NS m), the carrier of (REAL-NS 1))) Element of K6(K7(K7(K123( the carrier of (REAL-NS m), the carrier of (REAL-NS 1)),K123( the carrier of (REAL-NS m), the carrier of (REAL-NS 1))),K123( the carrier of (REAL-NS m), the carrier of (REAL-NS 1))))
K7(K123( the carrier of (REAL-NS m), the carrier of (REAL-NS 1)),K123( the carrier of (REAL-NS m), the carrier of (REAL-NS 1))) is set
K7(K7(K123( the carrier of (REAL-NS m), the carrier of (REAL-NS 1)),K123( the carrier of (REAL-NS m), the carrier of (REAL-NS 1))),K123( the carrier of (REAL-NS m), the carrier of (REAL-NS 1))) is set
K6(K7(K7(K123( the carrier of (REAL-NS m), the carrier of (REAL-NS 1)),K123( the carrier of (REAL-NS m), the carrier of (REAL-NS 1))),K123( the carrier of (REAL-NS m), the carrier of (REAL-NS 1)))) is set
FuncExtMult ( the carrier of (REAL-NS m),(REAL-NS 1)) is Relation-like K7(REAL,K123( the carrier of (REAL-NS m), the carrier of (REAL-NS 1))) -defined K123( the carrier of (REAL-NS m), the carrier of (REAL-NS 1)) -valued Function-like total V30(K7(REAL,K123( the carrier of (REAL-NS m), the carrier of (REAL-NS 1))),K123( the carrier of (REAL-NS m), the carrier of (REAL-NS 1))) Element of K6(K7(K7(REAL,K123( the carrier of (REAL-NS m), the carrier of (REAL-NS 1))),K123( the carrier of (REAL-NS m), the carrier of (REAL-NS 1))))
K7(REAL,K123( the carrier of (REAL-NS m), the carrier of (REAL-NS 1))) is V33() set
K7(K7(REAL,K123( the carrier of (REAL-NS m), the carrier of (REAL-NS 1))),K123( the carrier of (REAL-NS m), the carrier of (REAL-NS 1))) is V33() set
K6(K7(K7(REAL,K123( the carrier of (REAL-NS m), the carrier of (REAL-NS 1))),K123( the carrier of (REAL-NS m), the carrier of (REAL-NS 1)))) is V33() set
RLSStruct(# K123( the carrier of (REAL-NS m), the carrier of (REAL-NS 1)),(FuncZero ( the carrier of (REAL-NS m),(REAL-NS 1))),(FuncAdd ( the carrier of (REAL-NS m),(REAL-NS 1))),(FuncExtMult ( the carrier of (REAL-NS m),(REAL-NS 1))) #) is strict RLSStruct
the carrier of K279( the carrier of (REAL-NS m),(REAL-NS 1)) is non empty set
K6( the carrier of K279( the carrier of (REAL-NS m),(REAL-NS 1))) is set
K199(K279( the carrier of (REAL-NS m),(REAL-NS 1)),(LinearOperators ((REAL-NS m),(REAL-NS 1)))) is Element of LinearOperators ((REAL-NS m),(REAL-NS 1))
K197(K279( the carrier of (REAL-NS m),(REAL-NS 1)),(LinearOperators ((REAL-NS m),(REAL-NS 1)))) is Relation-like K7((LinearOperators ((REAL-NS m),(REAL-NS 1))),(LinearOperators ((REAL-NS m),(REAL-NS 1)))) -defined LinearOperators ((REAL-NS m),(REAL-NS 1)) -valued Function-like total V30(K7((LinearOperators ((REAL-NS m),(REAL-NS 1))),(LinearOperators ((REAL-NS m),(REAL-NS 1)))), LinearOperators ((REAL-NS m),(REAL-NS 1))) Element of K6(K7(K7((LinearOperators ((REAL-NS m),(REAL-NS 1))),(LinearOperators ((REAL-NS m),(REAL-NS 1)))),(LinearOperators ((REAL-NS m),(REAL-NS 1)))))
K7((LinearOperators ((REAL-NS m),(REAL-NS 1))),(LinearOperators ((REAL-NS m),(REAL-NS 1)))) is set
K7(K7((LinearOperators ((REAL-NS m),(REAL-NS 1))),(LinearOperators ((REAL-NS m),(REAL-NS 1)))),(LinearOperators ((REAL-NS m),(REAL-NS 1)))) is set
K6(K7(K7((LinearOperators ((REAL-NS m),(REAL-NS 1))),(LinearOperators ((REAL-NS m),(REAL-NS 1)))),(LinearOperators ((REAL-NS m),(REAL-NS 1))))) is set
K198(K279( the carrier of (REAL-NS m),(REAL-NS 1)),(LinearOperators ((REAL-NS m),(REAL-NS 1)))) is Relation-like K7(REAL,(LinearOperators ((REAL-NS m),(REAL-NS 1)))) -defined LinearOperators ((REAL-NS m),(REAL-NS 1)) -valued Function-like total V30(K7(REAL,(LinearOperators ((REAL-NS m),(REAL-NS 1)))), LinearOperators ((REAL-NS m),(REAL-NS 1))) Element of K6(K7(K7(REAL,(LinearOperators ((REAL-NS m),(REAL-NS 1)))),(LinearOperators ((REAL-NS m),(REAL-NS 1)))))
K7(REAL,(LinearOperators ((REAL-NS m),(REAL-NS 1)))) is V33() set
K7(K7(REAL,(LinearOperators ((REAL-NS m),(REAL-NS 1)))),(LinearOperators ((REAL-NS m),(REAL-NS 1)))) is V33() set
K6(K7(K7(REAL,(LinearOperators ((REAL-NS m),(REAL-NS 1)))),(LinearOperators ((REAL-NS m),(REAL-NS 1))))) is V33() set
RLSStruct(# (LinearOperators ((REAL-NS m),(REAL-NS 1))),K199(K279( the carrier of (REAL-NS m),(REAL-NS 1)),(LinearOperators ((REAL-NS m),(REAL-NS 1)))),K197(K279( the carrier of (REAL-NS m),(REAL-NS 1)),(LinearOperators ((REAL-NS m),(REAL-NS 1)))),K198(K279( the carrier of (REAL-NS m),(REAL-NS 1)),(LinearOperators ((REAL-NS m),(REAL-NS 1)))) #) is V71() strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RLSStruct
the carrier of (R_VectorSpace_of_LinearOperators ((REAL-NS m),(REAL-NS 1))) is non empty set
K6( the carrier of (R_VectorSpace_of_LinearOperators ((REAL-NS m),(REAL-NS 1)))) is set
K199((R_VectorSpace_of_LinearOperators ((REAL-NS m),(REAL-NS 1))),(BoundedLinearOperators ((REAL-NS m),(REAL-NS 1)))) is Element of BoundedLinearOperators ((REAL-NS m),(REAL-NS 1))
K197((R_VectorSpace_of_LinearOperators ((REAL-NS m),(REAL-NS 1))),(BoundedLinearOperators ((REAL-NS m),(REAL-NS 1)))) is Relation-like K7((BoundedLinearOperators ((REAL-NS m),(REAL-NS 1))),(BoundedLinearOperators ((REAL-NS m),(REAL-NS 1)))) -defined BoundedLinearOperators ((REAL-NS m),(REAL-NS 1)) -valued Function-like total V30(K7((BoundedLinearOperators ((REAL-NS m),(REAL-NS 1))),(BoundedLinearOperators ((REAL-NS m),(REAL-NS 1)))), BoundedLinearOperators ((REAL-NS m),(REAL-NS 1))) Element of K6(K7(K7((BoundedLinearOperators ((REAL-NS m),(REAL-NS 1))),(BoundedLinearOperators ((REAL-NS m),(REAL-NS 1)))),(BoundedLinearOperators ((REAL-NS m),(REAL-NS 1)))))
K7((BoundedLinearOperators ((REAL-NS m),(REAL-NS 1))),(BoundedLinearOperators ((REAL-NS m),(REAL-NS 1)))) is set
K7(K7((BoundedLinearOperators ((REAL-NS m),(REAL-NS 1))),(BoundedLinearOperators ((REAL-NS m),(REAL-NS 1)))),(BoundedLinearOperators ((REAL-NS m),(REAL-NS 1)))) is set
K6(K7(K7((BoundedLinearOperators ((REAL-NS m),(REAL-NS 1))),(BoundedLinearOperators ((REAL-NS m),(REAL-NS 1)))),(BoundedLinearOperators ((REAL-NS m),(REAL-NS 1))))) is set
K198((R_VectorSpace_of_LinearOperators ((REAL-NS m),(REAL-NS 1))),(BoundedLinearOperators ((REAL-NS m),(REAL-NS 1)))) is Relation-like K7(REAL,(BoundedLinearOperators ((REAL-NS m),(REAL-NS 1)))) -defined BoundedLinearOperators ((REAL-NS m),(REAL-NS 1)) -valued Function-like total V30(K7(REAL,(BoundedLinearOperators ((REAL-NS m),(REAL-NS 1)))), BoundedLinearOperators ((REAL-NS m),(REAL-NS 1))) Element of K6(K7(K7(REAL,(BoundedLinearOperators ((REAL-NS m),(REAL-NS 1)))),(BoundedLinearOperators ((REAL-NS m),(REAL-NS 1)))))
K7(REAL,(BoundedLinearOperators ((REAL-NS m),(REAL-NS 1)))) is V33() set
K7(K7(REAL,(BoundedLinearOperators ((REAL-NS m),(REAL-NS 1)))),(BoundedLinearOperators ((REAL-NS m),(REAL-NS 1)))) is V33() set
K6(K7(K7(REAL,(BoundedLinearOperators ((REAL-NS m),(REAL-NS 1)))),(BoundedLinearOperators ((REAL-NS m),(REAL-NS 1))))) is V33() set
BoundedLinearOperatorsNorm ((REAL-NS m),(REAL-NS 1)) is non empty Relation-like BoundedLinearOperators ((REAL-NS m),(REAL-NS 1)) -defined REAL -valued Function-like total V30( BoundedLinearOperators ((REAL-NS m),(REAL-NS 1)), REAL ) complex-yielding V146() V147() Element of K6(K7((BoundedLinearOperators ((REAL-NS m),(REAL-NS 1))),REAL))
K7((BoundedLinearOperators ((REAL-NS m),(REAL-NS 1))),REAL) is V33() complex-yielding V146() V147() set
K6(K7((BoundedLinearOperators ((REAL-NS m),(REAL-NS 1))),REAL)) is V33() set
G15((BoundedLinearOperators ((REAL-NS m),(REAL-NS 1))),K199((R_VectorSpace_of_LinearOperators ((REAL-NS m),(REAL-NS 1))),(BoundedLinearOperators ((REAL-NS m),(REAL-NS 1)))),K197((R_VectorSpace_of_LinearOperators ((REAL-NS m),(REAL-NS 1))),(BoundedLinearOperators ((REAL-NS m),(REAL-NS 1)))),K198((R_VectorSpace_of_LinearOperators ((REAL-NS m),(REAL-NS 1))),(BoundedLinearOperators ((REAL-NS m),(REAL-NS 1)))),(BoundedLinearOperatorsNorm ((REAL-NS m),(REAL-NS 1)))) is V108() L15()
the carrier of (R_NormSpace_of_BoundedLinearOperators ((REAL-NS m),(REAL-NS 1))) is non empty set
x is Neighbourhood of Z
y is V11() V12() ext-real Element of REAL
{ b₁ where b₁ is Element of the carrier of (REAL-NS m) : not y <= ||.(b₁ - Z).|| } is set
Rz is Relation-like the carrier of (REAL-NS m) -defined the carrier of (REAL-NS 1) -valued Function-like RestFunc-like Element of K6(K7( the carrier of (REAL-NS m), the carrier of (REAL-NS 1)))
Rz is Relation-like the carrier of (REAL-NS m) -defined the carrier of (REAL-NS 1) -valued Function-like RestFunc-like Element of K6(K7( the carrier of (REAL-NS m), the carrier of (REAL-NS 1)))
R0 is V11() V12() ext-real Element of REAL
min (y,R0) is V11() V12() ext-real Element of REAL
c₂₀ is Element of the carrier of (REAL-NS m)
||.c₂₀.|| is V11() V12() ext-real Element of REAL
||.c₂₀.|| " is V11() V12() ext-real Element of REAL
((Proj (N,n)) * R) /. c₂₀ is Element of the carrier of (REAL-NS 1)
((Proj (N,n)) * R) . c₂₀ is Element of the carrier of (REAL-NS 1)
||.(((Proj (N,n)) * R) /. c₂₀).|| is V11() V12() ext-real Element of REAL
(||.c₂₀.|| ") * ||.(((Proj (N,n)) * R) /. c₂₀).|| is V11() V12() ext-real Element of REAL
Rz /. c₂₀ is Element of the carrier of (REAL-NS 1)
||.(Rz /. c₂₀).|| is V11() V12() ext-real Element of REAL
(||.c₂₀.|| ") * ||.(Rz /. c₂₀).|| is V11() V12() ext-real Element of REAL
c₂₀ + Z is Element of the carrier of (REAL-NS m)
(c₂₀ + Z) - Z is Element of the carrier of (REAL-NS m)
i is Relation-like the carrier of (REAL-NS m) -defined the carrier of (REAL-NS 1) -valued Function-like RestFunc-like Element of K6(K7( the carrier of (REAL-NS m), the carrier of (REAL-NS 1)))
((Proj (N,n)) * f) /. (c₂₀ + Z) is Element of the carrier of (REAL-NS 1)
(((Proj (N,n)) * f) /. (c₂₀ + Z)) - (((Proj (N,n)) * f) /. Z) is Element of the carrier of (REAL-NS 1)
(diff (((Proj (N,n)) * f),Z)) . ((c₂₀ + Z) - Z) is Element of the carrier of (REAL-NS 1)
Rz /. ((c₂₀ + Z) - Z) is Element of the carrier of (REAL-NS 1)
((diff (((Proj (N,n)) * f),Z)) . ((c₂₀ + Z) - Z)) + (Rz /. ((c₂₀ + Z) - Z)) is Element of the carrier of (REAL-NS 1)
(diff (((Proj (N,n)) * f),Z)) . c₂₀ is Element of the carrier of (REAL-NS 1)
((((Proj (N,n)) * f) /. (c₂₀ + Z)) - (((Proj (N,n)) * f) /. Z)) - ((diff (((Proj (N,n)) * f),Z)) . c₂₀) is Element of the carrier of (REAL-NS 1)
(Proj (N,n)) * L is non empty Relation-like the carrier of (REAL-NS m) -defined the carrier of (REAL-NS 1) -valued Function-like total V30( the carrier of (REAL-NS m), the carrier of (REAL-NS 1)) Element of K6(K7( the carrier of (REAL-NS m), the carrier of (REAL-NS 1)))
((Proj (N,n)) * L) . c₂₀ is Element of the carrier of (REAL-NS 1)
((((Proj (N,n)) * f) /. (c₂₀ + Z)) - (((Proj (N,n)) * f) /. Z)) - (((Proj (N,n)) * L) . c₂₀) is Element of the carrier of (REAL-NS 1)
L . c₂₀ is Element of the carrier of (REAL-NS n)
i is Relation-like NAT -defined REAL -valued Function-like V40(n) FinSequence-like complex-yielding V146() V147() Element of REAL n
i . N is V11() V12() ext-real Element of REAL
(proj (1,1)) . ((diff (((Proj (N,n)) * f),Z)) . c₂₀) is V11() V12() ext-real Element of REAL
Rzi is V11() V12() ext-real Element of REAL
<*Rzi*> is Relation-like NAT -defined REAL -valued Function-like FinSequence-like complex-yielding V146() V147() FinSequence of REAL
[1,Rzi] is set
{[1,Rzi]} is non empty V2() V40(1) set
dom L is Element of K6( the carrier of (REAL-NS m))
(Proj (N,n)) . (L . c₂₀) is Element of the carrier of (REAL-NS 1)
proj (N,n) is non empty Relation-like REAL n -defined REAL -valued Function-like total V30( REAL n, REAL ) complex-yielding V146() V147() Element of K6(K7((REAL n),REAL))
K7((REAL n),REAL) is V33() complex-yielding V146() V147() set
K6(K7((REAL n),REAL)) is V33() set
y1 is Element of the carrier of (REAL-NS n)
(proj (N,n)) . y1 is V11() V12() ext-real Element of REAL
<*((proj (N,n)) . y1)*> is Relation-like NAT -defined REAL -valued Function-like FinSequence-like complex-yielding V146() V147() FinSequence of REAL
[1,((proj (N,n)) . y1)] is set
{[1,((proj (N,n)) . y1)]} is non empty V2() V40(1) set
<*((proj (1,1)) . ((diff (((Proj (N,n)) * f),Z)) . c₂₀))*> is Relation-like NAT -defined REAL -valued Function-like FinSequence-like complex-yielding V146() V147() FinSequence of REAL
[1,((proj (1,1)) . ((diff (((Proj (N,n)) * f),Z)) . c₂₀))] is set
{[1,((proj (1,1)) . ((diff (((Proj (N,n)) * f),Z)) . c₂₀))]} is non empty V2() V40(1) set
c₂₀ is Element of the carrier of (REAL-NS m)
||.c₂₀.|| is V11() V12() ext-real Element of REAL
||.c₂₀.|| " is V11() V12() ext-real Element of REAL
((Proj (N,n)) * R) /. c₂₀ is Element of the carrier of (REAL-NS 1)
((Proj (N,n)) * R) . c₂₀ is Element of the carrier of (REAL-NS 1)
||.(((Proj (N,n)) * R) /. c₂₀).|| is V11() V12() ext-real Element of REAL
(||.c₂₀.|| ") * ||.(((Proj (N,n)) * R) /. c₂₀).|| is V11() V12() ext-real Element of REAL
N is Relation-like NAT -defined REAL -valued Function-like FinSequence-like complex-yielding V146() V147() FinSequence of REAL
dom N is V156() V157() V158() V159() V160() V161() Element of K6(NAT)
min N is V11() V12() ext-real Element of REAL
min_p N is V6() V10() V11() V12() ext-real non negative V33() V38() V120() V155() V156() V157() V158() V159() V160() V161() Element of NAT
N . (min_p N) is V11() V12() ext-real Element of REAL
len N is V6() V10() V11() V12() ext-real non negative V33() V38() V120() V155() V156() V157() V158() V159() V160() V161() Element of NAT
N /. (min_p N) is V11() V12() ext-real Element of REAL
y is Element of the carrier of (REAL-NS m)
||.y.|| is V11() V12() ext-real Element of REAL
||.y.|| " is V11() V12() ext-real Element of REAL
R /. y is Element of the carrier of (REAL-NS n)
R . y is Element of the carrier of (REAL-NS n)
||.(R /. y).|| is V11() V12() ext-real Element of REAL
(||.y.|| ") * ||.(R /. y).|| is V11() V12() ext-real Element of REAL
||.y.|| * (((sqrt n) ") * R) is V11() V12() ext-real Element of REAL
(||.y.|| * (((sqrt n) ") * R)) ^2 is V11() V12() ext-real Element of REAL
(||.y.|| * (((sqrt n) ") * R)) * (||.y.|| * (((sqrt n) ") * R)) is V11() V12() ext-real set
n |-> ((||.y.|| * (((sqrt n) ") * R)) ^2) is Relation-like NAT -defined REAL -valued Function-like V40(n) FinSequence-like complex-yielding V146() V147() Element of n -tuples_on REAL
Rz is Relation-like NAT -defined REAL -valued Function-like V40(n) FinSequence-like complex-yielding V146() V147() Element of REAL n
sqr Rz is Relation-like NAT -defined REAL -valued Function-like V40(n) FinSequence-like complex-yielding V146() V147() Element of n -tuples_on REAL
Rz (#) sqrreal is Relation-like complex-yielding V146() V147() set
Rz (#) Rz is Relation-like Function-like set
K499() is Relation-like K7(REAL,REAL) -defined REAL -valued Function-like total V30(K7(REAL,REAL), REAL ) complex-yielding V146() V147() Element of K6(K7(K7(REAL,REAL),REAL))
K499() .: (Rz,Rz) is set
R0 is Relation-like NAT -defined REAL -valued Function-like V40(n) FinSequence-like complex-yielding V146() V147() Element of n -tuples_on REAL
SRz is Relation-like NAT -defined REAL -valued Function-like V40(n) FinSequence-like complex-yielding V146() V147() Element of n -tuples_on REAL
c₂₀ is V6() V10() V11() V12() ext-real non negative V33() V38() set
R0 . c₂₀ is V11() V12() ext-real Element of REAL
SRz . c₂₀ is V11() V12() ext-real Element of REAL
i is V6() V10() V11() V12() ext-real non negative V33() V38() V120() V155() V156() V157() V158() V159() V160() V161() Element of NAT
dom Rz is V40(n) V156() V157() V158() V159() V160() V161() Element of K6(NAT)
(sqr Rz) . i is V11() V12() ext-real Element of REAL
Rz . i is V11() V12() ext-real Element of REAL
sqrreal . (Rz . i) is V11() V12() ext-real Element of REAL
(Rz . i) ^2 is V11() V12() ext-real Element of REAL
(Rz . i) * (Rz . i) is V11() V12() ext-real set
N . i is V11() V12() ext-real Element of REAL
N /. i is V11() V12() ext-real Element of REAL
Proj (i,n) is non empty Relation-like the carrier of (REAL-NS n) -defined the carrier of (REAL-NS 1) -valued Function-like total V30( the carrier of (REAL-NS n), the carrier of (REAL-NS 1)) Element of K6(K7( the carrier of (REAL-NS n), the carrier of (REAL-NS 1)))
K7( the carrier of (REAL-NS n), the carrier of (REAL-NS 1)) is set
K6(K7( the carrier of (REAL-NS n), the carrier of (REAL-NS 1))) is set
(Proj (i,n)) * R is non empty Relation-like the carrier of (REAL-NS m) -defined the carrier of (REAL-NS 1) -valued Function-like total V30( the carrier of (REAL-NS m), the carrier of (REAL-NS 1)) Element of K6(K7( the carrier of (REAL-NS m), the carrier of (REAL-NS 1)))
((Proj (i,n)) * R) /. y is Element of the carrier of (REAL-NS 1)
((Proj (i,n)) * R) . y is Element of the carrier of (REAL-NS 1)
||.(((Proj (i,n)) * R) /. y).|| is V11() V12() ext-real Element of REAL
(||.y.|| ") * ||.(((Proj (i,n)) * R) /. y).|| is V11() V12() ext-real Element of REAL
(((sqrt n) ") * R) / (||.y.|| ") is V11() V12() ext-real Element of REAL
(||.y.|| ") " is V11() V12() ext-real set
(((sqrt n) ") * R) * ((||.y.|| ") ") is V11() V12() ext-real set
<*(Rz . i)*> is Relation-like NAT -defined REAL -valued Function-like FinSequence-like complex-yielding V146() V147() FinSequence of REAL
[1,(Rz . i)] is set
{[1,(Rz . i)]} is non empty V2() V40(1) set
(Proj (i,n)) . (R . y) is Element of the carrier of (REAL-NS 1)
proj (i,n) is non empty Relation-like REAL n -defined REAL -valued Function-like total V30( REAL n, REAL ) complex-yielding V146() V147() Element of K6(K7((REAL n),REAL))
K7((REAL n),REAL) is V33() complex-yielding V146() V147() set
K6(K7((REAL n),REAL)) is V33() set
(proj (i,n)) . Rz is V11() V12() ext-real Element of REAL
<*((proj (i,n)) . Rz)*> is Relation-like NAT -defined REAL -valued Function-like FinSequence-like complex-yielding V146() V147() FinSequence of REAL
[1,((proj (i,n)) . Rz)] is set
{[1,((proj (i,n)) . Rz)]} is non empty V2() V40(1) set
||.(((Proj (i,n)) * R) . y).|| is V11() V12() ext-real Element of REAL
Rzi is non empty V2() Relation-like NAT -defined REAL -valued Function-like V40(1) FinSequence-like complex-yielding V146() V147() V149() V150() V151() V152() Element of REAL 1
|.Rzi.| is V11() V12() ext-real non negative Element of REAL
sqr Rzi is Relation-like NAT -defined REAL -valued Function-like FinSequence-like complex-yielding V146() V147() FinSequence of REAL
Rzi (#) sqrreal is Relation-like complex-yielding V146() V147() set
Rzi (#) Rzi is Relation-like Function-like set
K499() .: (Rzi,Rzi) is set
Sum (sqr Rzi) is V11() V12() ext-real Element of REAL
sqrt (Sum (sqr Rzi)) is V11() V12() ext-real Element of REAL
|.(Rz . i).| is V11() V12() ext-real Element of REAL
|.(Rz . i).| ^2 is V11() V12() ext-real Element of REAL
|.(Rz . i).| * |.(Rz . i).| is V11() V12() ext-real set
Rz . c₂₀ is V11() V12() ext-real Element of REAL
(Rz . c₂₀) ^2 is V11() V12() ext-real Element of REAL
(Rz . c₂₀) * (Rz . c₂₀) is V11() V12() ext-real set
dom (sqr Rz) is V40(n) V156() V157() V158() V159() V160() V161() Element of K6(NAT)
c₂₀ is V6() V10() V11() V12() ext-real non negative V33() V38() set
(sqr Rz) . c₂₀ is V11() V12() ext-real Element of REAL
sqrreal * Rz is Relation-like NAT -defined REAL -valued Function-like complex-yielding V146() V147() Element of K6(K7(NAT,REAL))
dom (sqrreal * Rz) is V156() V157() V158() V159() V160() V161() Element of K6(NAT)
dom Rz is V40(n) V156() V157() V158() V159() V160() V161() Element of K6(NAT)
Rz . c₂₀ is V11() V12() ext-real Element of REAL
sqrreal . (Rz . c₂₀) is V11() V12() ext-real Element of REAL
(Rz . c₂₀) ^2 is V11() V12() ext-real Element of REAL
(Rz . c₂₀) * (Rz . c₂₀) is V11() V12() ext-real set
R0 . 1 is V11() V12() ext-real Element of REAL
SRz . 1 is V11() V12() ext-real Element of REAL
c₂₀ is V6() V10() V11() V12() ext-real non negative V33() V38() set
SRz . c₂₀ is V11() V12() ext-real Element of REAL
R0 . c₂₀ is V11() V12() ext-real Element of REAL
Sum R0 is V11() V12() ext-real Element of REAL
Sum SRz is V11() V12() ext-real Element of REAL
c₂₀ is V6() V10() V11() V12() ext-real non negative V33() V38() set
R0 . c₂₀ is V11() V12() ext-real Element of REAL
SRz . c₂₀ is V11() V12() ext-real Element of REAL
n * ((||.y.|| * (((sqrt n) ") * R)) ^2) is V11() V12() ext-real Element of REAL
Sum (sqr Rz) is V11() V12() ext-real Element of REAL
sqrt (n * ((||.y.|| * (((sqrt n) ") * R)) ^2)) is V11() V12() ext-real Element of REAL
|.Rz.| is V11() V12() ext-real non negative Element of REAL
sqr Rz is Relation-like NAT -defined REAL -valued Function-like FinSequence-like complex-yielding V146() V147() FinSequence of REAL
Sum (sqr Rz) is V11() V12() ext-real Element of REAL
sqrt (Sum (sqr Rz)) is V11() V12() ext-real Element of REAL
sqrt ((||.y.|| * (((sqrt n) ") * R)) ^2) is V11() V12() ext-real Element of REAL
(sqrt n) * (sqrt ((||.y.|| * (((sqrt n) ") * R)) ^2)) is V11() V12() ext-real Element of REAL
|.(||.y.|| * (((sqrt n) ") * R)).| is V11() V12() ext-real Element of REAL
(sqrt n) * |.(||.y.|| * (((sqrt n) ") * R)).| is V11() V12() ext-real Element of REAL
(sqrt n) * (||.y.|| * (((sqrt n) ") * R)) is V11() V12() ext-real Element of REAL
(sqrt n) * (((sqrt n) ") * R) is V11() V12() ext-real Element of REAL
((sqrt n) * (((sqrt n) ") * R)) * ||.y.|| is V11() V12() ext-real Element of REAL
|.Rz.| / ||.y.|| is V11() V12() ext-real Element of REAL
||.y.|| " is V11() V12() ext-real set
|.Rz.| * (||.y.|| ") is V11() V12() ext-real set
(sqrt n) * ((sqrt n) ") is V11() V12() ext-real set
((sqrt n) * ((sqrt n) ")) * R is V11() V12() ext-real Element of REAL
1 * R is V11() V12() ext-real Element of REAL
y is Element of the carrier of (REAL-NS m)
||.y.|| is V11() V12() ext-real Element of REAL
||.y.|| " is V11() V12() ext-real Element of REAL
R /. y is Element of the carrier of (REAL-NS n)
R . y is Element of the carrier of (REAL-NS n)
||.(R /. y).|| is V11() V12() ext-real Element of REAL
(||.y.|| ") * ||.(R /. y).|| is V11() V12() ext-real Element of REAL
{ b₁ where b₁ is Element of the carrier of (REAL-NS m) : not min Z0 <= ||.(b₁ - Z).|| } is set
x is set
N is Neighbourhood of Z
y is Element of the carrier of (REAL-NS m)
y - Z is Element of the carrier of (REAL-NS m)
||.(y - Z).|| is V11() V12() ext-real Element of REAL
Z0 /. 1 is V11() V12() ext-real Element of REAL
{ b₁ where b₁ is Element of the carrier of (REAL-NS m) : not Z0 /. 1 <= ||.(b₁ - Z).|| } is set
Proj (1,n) is non empty Relation-like the carrier of (REAL-NS n) -defined the carrier of (REAL-NS 1) -valued Function-like total V30( the carrier of (REAL-NS n), the carrier of (REAL-NS 1)) Element of K6(K7( the carrier of (REAL-NS n), the carrier of (REAL-NS 1)))
K7( the carrier of (REAL-NS n), the carrier of (REAL-NS 1)) is set
K6(K7( the carrier of (REAL-NS n), the carrier of (REAL-NS 1))) is set
(Proj (1,n)) * f is Relation-like the carrier of (REAL-NS m) -defined the carrier of (REAL-NS 1) -valued Function-like Element of K6(K7( the carrier of (REAL-NS m), the carrier of (REAL-NS 1)))
dom ((Proj (1,n)) * f) is Element of K6( the carrier of (REAL-NS m))
Z0 . 1 is V11() V12() ext-real Element of REAL
L0 is Relation-like Function-like Element of the carrier of (R_NormSpace_of_BoundedLinearOperators ((REAL-NS m),(REAL-NS n)))
R is Relation-like the carrier of (REAL-NS m) -defined the carrier of (REAL-NS n) -valued Function-like RestFunc-like Element of K6(K7( the carrier of (REAL-NS m), the carrier of (REAL-NS n)))
x is Element of the carrier of (REAL-NS m)
dom R is Element of K6( the carrier of (REAL-NS m))
x - Z is Element of the carrier of (REAL-NS m)
R . (x - Z) is set
(x - Z) + Z is Element of the carrier of (REAL-NS m)
f /. ((x - Z) + Z) is Element of the carrier of (REAL-NS n)
(f /. ((x - Z) + Z)) - (f /. Z) is Element of the carrier of (REAL-NS n)
L . (x - Z) is Element of the carrier of (REAL-NS n)
((f /. ((x - Z) + Z)) - (f /. Z)) - (L . (x - Z)) is Element of the carrier of (REAL-NS n)
Z - Z is Element of the carrier of (REAL-NS m)
x - (Z - Z) is Element of the carrier of (REAL-NS m)
f /. (x - (Z - Z)) is Element of the carrier of (REAL-NS n)
(f /. (x - (Z - Z))) - (f /. Z) is Element of the carrier of (REAL-NS n)
L0 . (x - Z) is Element of the carrier of (REAL-NS n)
((f /. (x - (Z - Z))) - (f /. Z)) - (L0 . (x - Z)) is Element of the carrier of (REAL-NS n)
- Z is Element of the carrier of (REAL-NS m)
Z + (- Z) is Element of the carrier of (REAL-NS m)
x - (Z + (- Z)) is Element of the carrier of (REAL-NS m)
f /. (x - (Z + (- Z))) is Element of the carrier of (REAL-NS n)
(f /. (x - (Z + (- Z)))) - (f /. Z) is Element of the carrier of (REAL-NS n)
((f /. (x - (Z + (- Z)))) - (f /. Z)) - (L0 . (x - Z)) is Element of the carrier of (REAL-NS n)
x - (0. (REAL-NS m)) is Element of the carrier of (REAL-NS m)
f /. (x - (0. (REAL-NS m))) is Element of the carrier of (REAL-NS n)
(f /. (x - (0. (REAL-NS m)))) - (f /. Z) is Element of the carrier of (REAL-NS n)
((f /. (x - (0. (REAL-NS m)))) - (f /. Z)) - (L0 . (x - Z)) is Element of the carrier of (REAL-NS n)
f /. x is Element of the carrier of (REAL-NS n)
(f /. x) - (f /. Z) is Element of the carrier of (REAL-NS n)
((f /. x) - (f /. Z)) - (L0 . (x - Z)) is Element of the carrier of (REAL-NS n)
- (L0 . (x - Z)) is Element of the carrier of (REAL-NS n)
((f /. x) - (f /. Z)) + (- (L0 . (x - Z))) is Element of the carrier of (REAL-NS n)
R /. (x - Z) is Element of the carrier of (REAL-NS n)
(L0 . (x - Z)) + (R /. (x - Z)) is Element of the carrier of (REAL-NS n)
(- (L0 . (x - Z))) + (L0 . (x - Z)) is Element of the carrier of (REAL-NS n)
((f /. x) - (f /. Z)) + ((- (L0 . (x - Z))) + (L0 . (x - Z))) is Element of the carrier of (REAL-NS n)
((f /. x) - (f /. Z)) + (0. (REAL-NS n)) is Element of the carrier of (REAL-NS n)
x is Relation-like Function-like Element of the carrier of (R_NormSpace_of_BoundedLinearOperators ((REAL-NS m),(REAL-NS n)))
y is Relation-like the carrier of (REAL-NS m) -defined the carrier of (REAL-NS n) -valued Function-like RestFunc-like Element of K6(K7( the carrier of (REAL-NS m), the carrier of (REAL-NS n)))
Z0 is V6() V10() V11() V12() ext-real non negative V33() V38() V120() V155() V156() V157() V158() V159() V160() V161() Element of NAT
Proj (Z0,n) is non empty Relation-like the carrier of (REAL-NS n) -defined the carrier of (REAL-NS 1) -valued Function-like total V30( the carrier of (REAL-NS n), the carrier of (REAL-NS 1)) Element of K6(K7( the carrier of (REAL-NS n), the carrier of (REAL-NS 1)))
K7( the carrier of (REAL-NS n), the carrier of (REAL-NS 1)) is set
K6(K7( the carrier of (REAL-NS n), the carrier of (REAL-NS 1))) is set
(Proj (Z0,n)) * f is Relation-like the carrier of (REAL-NS m) -defined the carrier of (REAL-NS 1) -valued Function-like Element of K6(K7( the carrier of (REAL-NS m), the carrier of (REAL-NS 1)))
K7( the carrier of (REAL-NS m), the carrier of (REAL-NS 1)) is set
K6(K7( the carrier of (REAL-NS m), the carrier of (REAL-NS 1))) is set
c₇ is V6() V10() V11() V12() ext-real non negative V33() V38() V120() V155() V156() V157() V158() V159() V160() V161() Element of NAT
Proj (c₇,n) is non empty Relation-like the carrier of (REAL-NS n) -defined the carrier of (REAL-NS 1) -valued Function-like total V30( the carrier of (REAL-NS n), the carrier of (REAL-NS 1)) Element of K6(K7( the carrier of (REAL-NS n), the carrier of (REAL-NS 1)))
(Proj (c₇,n)) * f is Relation-like the carrier of (REAL-NS m) -defined the carrier of (REAL-NS 1) -valued Function-like Element of K6(K7( the carrier of (REAL-NS m), the carrier of (REAL-NS 1)))
f is non empty V6() V10() V11() V12() ext-real positive non negative V33() V38() V120() V155() V156() V157() V158() V159() V160() V161() Element of NAT
REAL f is non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered FinSequenceSet of REAL
f -tuples_on REAL is non empty functional FinSequence-membered FinSequenceSet of REAL
n is non empty V6() V10() V11() V12() ext-real positive non negative V33() V38() V120() V155() V156() V157() V158() V159() V160() V161() Element of NAT
REAL n is non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered FinSequenceSet of REAL
n -tuples_on REAL is non empty functional FinSequence-membered FinSequenceSet of REAL
K7((REAL f),(REAL n)) is set
K6(K7((REAL f),(REAL n))) is set
m is set
n is non empty V6() V10() V11() V12() ext-real positive non negative V33() V38() V120() V155() V156() V157() V158() V159() V160() V161() Element of NAT
REAL n is non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered FinSequenceSet of REAL
n -tuples_on REAL is non empty functional FinSequence-membered FinSequenceSet of REAL
f is non empty V6() V10() V11() V12() ext-real positive non negative V33() V38() V120() V155() V156() V157() V158() V159() V160() V161() Element of NAT
REAL f is non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered FinSequenceSet of REAL
f -tuples_on REAL is non empty functional FinSequence-membered FinSequenceSet of REAL
K7((REAL n),(REAL f)) is set
K6(K7((REAL n),(REAL f))) is set
REAL-NS n is non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() V108() V109() V142() L15()
the carrier of (REAL-NS n) is non empty V2() set
REAL-NS f is non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() V108() V109() V142() L15()
the carrier of (REAL-NS f) is non empty V2() set
K7( the carrier of (REAL-NS n), the carrier of (REAL-NS f)) is set
K6(K7( the carrier of (REAL-NS n), the carrier of (REAL-NS f))) is set
Z is Relation-like REAL n -defined REAL f -valued Function-like complex-functions-valued ext-real-functions-valued real-functions-valued Element of K6(K7((REAL n),(REAL f)))
g is Relation-like the carrier of (REAL-NS n) -defined the carrier of (REAL-NS f) -valued Function-like Element of K6(K7( the carrier of (REAL-NS n), the carrier of (REAL-NS f)))
dom Z is functional complex-functions-membered ext-real-functions-membered real-functions-membered Element of K6((REAL n))
K6((REAL n)) is set
Z | m is Relation-like REAL n -defined REAL f -valued Function-like complex-functions-valued ext-real-functions-valued real-functions-valued Element of K6(K7((REAL n),(REAL f)))
Z0 is Relation-like NAT -defined REAL -valued Function-like V40(n) FinSequence-like complex-yielding V146() V147() Element of REAL n
Z0 is Element of the carrier of (REAL-NS n)
c₇ is Relation-like NAT -defined REAL -valued Function-like V40(n) FinSequence-like complex-yielding V146() V147() Element of REAL n
g | m is Relation-like the carrier of (REAL-NS n) -defined the carrier of (REAL-NS f) -valued Function-like Element of K6(K7( the carrier of (REAL-NS n), the carrier of (REAL-NS f)))
x is Relation-like the carrier of (REAL-NS n) -defined the carrier of (REAL-NS f) -valued Function-like Element of K6(K7( the carrier of (REAL-NS n), the carrier of (REAL-NS f)))
L is Element of the carrier of (REAL-NS n)
dom g is Element of K6( the carrier of (REAL-NS n))
K6( the carrier of (REAL-NS n)) is set
Z0 is Element of the carrier of (REAL-NS n)
Z0 is Relation-like NAT -defined REAL -valued Function-like V40(n) FinSequence-like complex-yielding V146() V147() Element of REAL n
c₇ is Element of the carrier of (REAL-NS n)
m is set
n is non empty V6() V10() V11() V12() ext-real positive non negative V33() V38() V120() V155() V156() V157() V158() V159() V160() V161() Element of NAT
REAL n is non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered FinSequenceSet of REAL
n -tuples_on REAL is non empty functional FinSequence-membered FinSequenceSet of REAL
f is non empty V6() V10() V11() V12() ext-real positive non negative V33() V38() V120() V155() V156() V157() V158() V159() V160() V161() Element of NAT
REAL f is non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered FinSequenceSet of REAL
f -tuples_on REAL is non empty functional FinSequence-membered FinSequenceSet of REAL
K7((REAL n),(REAL f)) is set
K6(K7((REAL n),(REAL f))) is set
K6((REAL n)) is set
Z is Relation-like REAL n -defined REAL f -valued Function-like complex-functions-valued ext-real-functions-valued real-functions-valued Element of K6(K7((REAL n),(REAL f)))
dom Z is functional complex-functions-membered ext-real-functions-membered real-functions-membered Element of K6((REAL n))
m is non empty V6() V10() V11() V12() ext-real positive non negative V33() V38() V120() V155() V156() V157() V158() V159() V160() V161() Element of NAT
REAL m is non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered FinSequenceSet of REAL
m -tuples_on REAL is non empty functional FinSequence-membered FinSequenceSet of REAL
n is non empty V6() V10() V11() V12() ext-real positive non negative V33() V38() V120() V155() V156() V157() V158() V159() V160() V161() Element of NAT
REAL n is non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered FinSequenceSet of REAL
n -tuples_on REAL is non empty functional FinSequence-membered FinSequenceSet of REAL
K7((REAL m),(REAL n)) is set
K6(K7((REAL m),(REAL n))) is set
K6((REAL m)) is set
REAL-NS m is non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() V108() V109() V142() L15()
the carrier of (REAL-NS m) is non empty V2() set
K6( the carrier of (REAL-NS m)) is set
f is Relation-like REAL m -defined REAL n -valued Function-like complex-functions-valued ext-real-functions-valued real-functions-valued Element of K6(K7((REAL m),(REAL n)))
dom f is functional complex-functions-membered ext-real-functions-membered real-functions-membered Element of K6((REAL m))
Z is functional complex-functions-membered ext-real-functions-membered real-functions-membered Element of K6((REAL m))
g is Element of K6( the carrier of (REAL-NS m))
g is Element of K6( the carrier of (REAL-NS m))
REAL-NS n is non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() V108() V109() V142() L15()
the carrier of (REAL-NS n) is non empty V2() set
K7( the carrier of (REAL-NS m), the carrier of (REAL-NS n)) is set
K6(K7( the carrier of (REAL-NS m), the carrier of (REAL-NS n))) is set
Z0 is Relation-like the carrier of (REAL-NS m) -defined the carrier of (REAL-NS n) -valued Function-like Element of K6(K7( the carrier of (REAL-NS m), the carrier of (REAL-NS n)))
dom Z0 is Element of K6( the carrier of (REAL-NS m))
c₇ is Element of the carrier of (REAL-NS m)
c₇ is Relation-like NAT -defined REAL -valued Function-like V40(m) FinSequence-like complex-yielding V146() V147() Element of REAL m
x is Element of the carrier of (REAL-NS m)
L is Element of the carrier of (REAL-NS m)
c₇ is Element of the carrier of (REAL-NS m)
x is Relation-like NAT -defined REAL -valued Function-like V40(m) FinSequence-like complex-yielding V146() V147() Element of REAL m
L is Relation-like the carrier of (REAL-NS m) -defined the carrier of (REAL-NS n) -valued Function-like Element of K6(K7( the carrier of (REAL-NS m), the carrier of (REAL-NS n)))
L0 is Element of the carrier of (REAL-NS m)
c₇ is Element of the carrier of (REAL-NS m)
m is non empty V6() V10() V11() V12() ext-real positive non negative V33() V38() V120() V155() V156() V157() V158() V159() V160() V161() Element of NAT
REAL m is non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered FinSequenceSet of REAL
m -tuples_on REAL is non empty functional FinSequence-membered FinSequenceSet of REAL
n is non empty V6() V10() V11() V12() ext-real positive non negative V33() V38() V120() V155() V156() V157() V158() V159() V160() V161() Element of NAT
REAL n is non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered FinSequenceSet of REAL
n -tuples_on REAL is non empty functional FinSequence-membered FinSequenceSet of REAL
K7((REAL m),(REAL n)) is set
K6(K7((REAL m),(REAL n))) is set
K6((REAL m)) is set
REAL-NS m is non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() V108() V109() V142() L15()
the carrier of (REAL-NS m) is non empty V2() set
K6( the carrier of (REAL-NS m)) is set
f is Relation-like REAL m -defined REAL n -valued Function-like complex-functions-valued ext-real-functions-valued real-functions-valued Element of K6(K7((REAL m),(REAL n)))
Z is functional complex-functions-membered ext-real-functions-membered real-functions-membered Element of K6((REAL m))
REAL-NS n is non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() V108() V109() V142() L15()
the carrier of (REAL-NS n) is non empty V2() set
K7( the carrier of (REAL-NS m), the carrier of (REAL-NS n)) is set
K6(K7( the carrier of (REAL-NS m), the carrier of (REAL-NS n))) is set
Z0 is Element of K6( the carrier of (REAL-NS m))
g is Relation-like the carrier of (REAL-NS m) -defined the carrier of (REAL-NS n) -valued Function-like Element of K6(K7( the carrier of (REAL-NS m), the carrier of (REAL-NS n)))