:: PDIFF_7 semantic presentation

REAL is non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() set
NAT is complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below Element of K6(REAL)
K6(REAL) is set
REAL * is non empty functional FinSequence-membered FinSequenceSet of REAL
K7(NAT,(REAL *)) is set
K6(K7(NAT,(REAL *))) is set
COMPLEX is non empty non finite complex-membered V205() set
omega is complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below set
K6(omega) is set
K6(NAT) is set
K7(NAT,REAL) is complex-valued ext-real-valued real-valued set
K6(K7(NAT,REAL)) is 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 set
K7(K7(REAL,K190()),K190()) is set
K6(K7(K7(REAL,K190()),K190())) is 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-valued ext-real-valued real-valued set
K6(K7(K7(K200(),K200()),REAL)) is set
K203() is Element of K6( the carrier of K196())
K7(K203(),REAL) is complex-valued ext-real-valued real-valued set
K6(K7(K203(),REAL)) is set
1 is non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below Element of NAT
INT is non empty non finite complex-membered ext-real-membered real-membered rational-membered integer-membered V205() set
K7(1,1) is RAT -valued INT -valued complex-valued ext-real-valued real-valued natural-valued set
RAT is non empty non finite complex-membered ext-real-membered real-membered rational-membered V205() set
K6(K7(1,1)) is set
K7(K7(1,1),1) is RAT -valued INT -valued complex-valued ext-real-valued real-valued natural-valued set
K6(K7(K7(1,1),1)) is set
K7(K7(1,1),REAL) is complex-valued ext-real-valued real-valued set
K6(K7(K7(1,1),REAL)) is set
K7(REAL,REAL) is complex-valued ext-real-valued real-valued set
K7(K7(REAL,REAL),REAL) is complex-valued ext-real-valued real-valued set
K6(K7(K7(REAL,REAL),REAL)) is set
2 is non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below Element of NAT
K7(2,2) is RAT -valued INT -valued complex-valued ext-real-valued real-valued natural-valued set
K7(K7(2,2),REAL) is complex-valued ext-real-valued real-valued set
K6(K7(K7(2,2),REAL)) is set
K6(K7(REAL,REAL)) is set
TOP-REAL 2 is non empty V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital TopSpace-like V148() V214() V215() L19()
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
{ b₁ where b₁ is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like Element of REAL * : len b₁ = 1 } is set
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() strict RealNormSpace-like V185() NORMSTR
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() RealNormSpace-like V185() NORMSTR
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 V214() RLSStruct
LinearOperators ((REAL-NS 1),(REAL-NS 1)) is non empty functional Element of K6( the carrier of K318( the carrier of (REAL-NS 1),(REAL-NS 1)))
K318( 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 V214() RLSStruct
Funcs ( the carrier of (REAL-NS 1), the carrier of (REAL-NS 1)) is non empty functional FUNCTION_DOMAIN of 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 quasi_total Element of Funcs ( 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)
K214( 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 quasi_total 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((Funcs ( the carrier of (REAL-NS 1), the carrier of (REAL-NS 1))),(Funcs ( the carrier of (REAL-NS 1), the carrier of (REAL-NS 1)))) -defined Funcs ( the carrier of (REAL-NS 1), the carrier of (REAL-NS 1)) -valued Function-like total quasi_total Element of K6(K7(K7((Funcs ( the carrier of (REAL-NS 1), the carrier of (REAL-NS 1))),(Funcs ( the carrier of (REAL-NS 1), the carrier of (REAL-NS 1)))),(Funcs ( the carrier of (REAL-NS 1), the carrier of (REAL-NS 1)))))
K7((Funcs ( the carrier of (REAL-NS 1), the carrier of (REAL-NS 1))),(Funcs ( the carrier of (REAL-NS 1), the carrier of (REAL-NS 1)))) is set
K7(K7((Funcs ( the carrier of (REAL-NS 1), the carrier of (REAL-NS 1))),(Funcs ( the carrier of (REAL-NS 1), the carrier of (REAL-NS 1)))),(Funcs ( the carrier of (REAL-NS 1), the carrier of (REAL-NS 1)))) is set
K6(K7(K7((Funcs ( the carrier of (REAL-NS 1), the carrier of (REAL-NS 1))),(Funcs ( the carrier of (REAL-NS 1), the carrier of (REAL-NS 1)))),(Funcs ( 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,(Funcs ( the carrier of (REAL-NS 1), the carrier of (REAL-NS 1)))) -defined Funcs ( the carrier of (REAL-NS 1), the carrier of (REAL-NS 1)) -valued Function-like total quasi_total Element of K6(K7(K7(REAL,(Funcs ( the carrier of (REAL-NS 1), the carrier of (REAL-NS 1)))),(Funcs ( the carrier of (REAL-NS 1), the carrier of (REAL-NS 1)))))
K7(REAL,(Funcs ( the carrier of (REAL-NS 1), the carrier of (REAL-NS 1)))) is set
K7(K7(REAL,(Funcs ( the carrier of (REAL-NS 1), the carrier of (REAL-NS 1)))),(Funcs ( the carrier of (REAL-NS 1), the carrier of (REAL-NS 1)))) is set
K6(K7(K7(REAL,(Funcs ( the carrier of (REAL-NS 1), the carrier of (REAL-NS 1)))),(Funcs ( the carrier of (REAL-NS 1), the carrier of (REAL-NS 1))))) is set
RLSStruct(# (Funcs ( 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 K318( the carrier of (REAL-NS 1),(REAL-NS 1)) is non empty set
K6( the carrier of K318( the carrier of (REAL-NS 1),(REAL-NS 1))) is set
K199(K318( the carrier of (REAL-NS 1),(REAL-NS 1)),(LinearOperators ((REAL-NS 1),(REAL-NS 1)))) is Relation-like Function-like Element of LinearOperators ((REAL-NS 1),(REAL-NS 1))
K197(K318( 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 quasi_total 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(K318( 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 quasi_total 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 set
K7(K7(REAL,(LinearOperators ((REAL-NS 1),(REAL-NS 1)))),(LinearOperators ((REAL-NS 1),(REAL-NS 1)))) is set
K6(K7(K7(REAL,(LinearOperators ((REAL-NS 1),(REAL-NS 1)))),(LinearOperators ((REAL-NS 1),(REAL-NS 1))))) is set
RLSStruct(# (LinearOperators ((REAL-NS 1),(REAL-NS 1))),K199(K318( the carrier of (REAL-NS 1),(REAL-NS 1)),(LinearOperators ((REAL-NS 1),(REAL-NS 1)))),K197(K318( the carrier of (REAL-NS 1),(REAL-NS 1)),(LinearOperators ((REAL-NS 1),(REAL-NS 1)))),K198(K318( 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 quasi_total 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 quasi_total 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 set
K7(K7(REAL,(BoundedLinearOperators ((REAL-NS 1),(REAL-NS 1)))),(BoundedLinearOperators ((REAL-NS 1),(REAL-NS 1)))) is set
K6(K7(K7(REAL,(BoundedLinearOperators ((REAL-NS 1),(REAL-NS 1)))),(BoundedLinearOperators ((REAL-NS 1),(REAL-NS 1))))) is 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 quasi_total complex-valued ext-real-valued real-valued Element of K6(K7((BoundedLinearOperators ((REAL-NS 1),(REAL-NS 1))),REAL))
K7((BoundedLinearOperators ((REAL-NS 1),(REAL-NS 1))),REAL) is complex-valued ext-real-valued real-valued set
K6(K7((BoundedLinearOperators ((REAL-NS 1),(REAL-NS 1))),REAL)) is set
NORMSTR(# (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 strict NORMSTR
the carrier of (R_NormSpace_of_BoundedLinearOperators ((REAL-NS 1),(REAL-NS 1))) is non empty set
K7(COMPLEX,COMPLEX) is complex-valued set
K6(K7(COMPLEX,COMPLEX)) is set
K7(K7(COMPLEX,COMPLEX),COMPLEX) is complex-valued set
K6(K7(K7(COMPLEX,COMPLEX),COMPLEX)) is set
K7(RAT,RAT) is RAT -valued complex-valued ext-real-valued real-valued set
K6(K7(RAT,RAT)) is set
K7(K7(RAT,RAT),RAT) is RAT -valued complex-valued ext-real-valued real-valued set
K6(K7(K7(RAT,RAT),RAT)) is set
K7(INT,INT) is RAT -valued INT -valued complex-valued ext-real-valued real-valued set
K6(K7(INT,INT)) is set
K7(K7(INT,INT),INT) is RAT -valued INT -valued complex-valued ext-real-valued real-valued set
K6(K7(K7(INT,INT),INT)) is set
K7(NAT,NAT) is RAT -valued INT -valued complex-valued ext-real-valued real-valued natural-valued set
K7(K7(NAT,NAT),NAT) is RAT -valued INT -valued complex-valued ext-real-valued real-valued natural-valued set
K6(K7(K7(NAT,NAT),NAT)) is set
K541() is non empty V73() L8()
the carrier of K541() is non empty set
K546() is non empty V73() V164() V165() V166() V168() V224() V225() V226() V227() V228() V229() L8()
K547() is non empty V73() V166() V168() V227() V228() V229() M17(K546())
K548() is non empty V73() V164() V166() V168() V227() V228() V229() V230() M20(K546(),K547())
K550() is non empty V73() V164() V166() V168() L8()
K551() is non empty V73() V164() V166() V168() V230() M17(K550())
ExtREAL is non empty ext-real-membered V263() set
{} is empty Function-like functional FinSequence-membered complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below V263() complex-functions-membered ext-real-functions-membered real-functions-membered rational-functions-membered integer-functions-membered natural-functions-membered set
the empty Function-like functional FinSequence-membered complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below V263() complex-functions-membered ext-real-functions-membered real-functions-membered rational-functions-membered integer-functions-membered natural-functions-membered set is empty Function-like functional FinSequence-membered complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below V263() 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
K310() is V94() L11()
the carrier of K310() is set
the carrier of K310() * is non empty functional FinSequence-membered FinSequenceSet of the carrier of K310()
K7(COMPLEX,REAL) is complex-valued ext-real-valued real-valued set
K6(K7(COMPLEX,REAL)) is set
3 is non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below Element of NAT
0 is empty ordinal natural V11() real ext-real non positive non negative Function-like functional FinSequence-membered V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below V263() complex-functions-membered ext-real-functions-membered real-functions-membered rational-functions-membered integer-functions-membered natural-functions-membered Element of NAT
sqrreal is non empty Relation-like REAL -defined REAL -valued Function-like total quasi_total complex-valued ext-real-valued real-valued Element of K6(K7(REAL,REAL))
sqrt 1 is V11() real ext-real Element of REAL
proj (1,1) is non empty Relation-like REAL 1 -defined REAL -valued Function-like total quasi_total complex-valued ext-real-valued real-valued Element of K6(K7((REAL 1),REAL))
K7((REAL 1),REAL) is complex-valued ext-real-valued real-valued set
K6(K7((REAL 1),REAL)) is set
(proj (1,1)) " is Relation-like Function-like set
K7(REAL,(REAL 1)) is set
K6(K7(REAL,(REAL 1))) is set
proj1 ((proj (1,1)) ") is set
proj2 ((proj (1,1)) ") is set
<*1*> is non empty V2() Relation-like NAT -defined NAT -valued Function-like one-to-one constant finite 1 -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued natural-valued increasing decreasing non-decreasing non-increasing FinSequence of NAT
[1,1] is V27() set
{1,1} is non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below set
{1} is non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below set
{{1,1},{1}} is non empty set
{[1,1]} is non empty Function-like set
m is ordinal natural V11() real ext-real non negative set
TOP-REAL m is non empty V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital TopSpace-like V148() V214() V215() L19()
the carrier of (TOP-REAL m) is non empty set
f is Relation-like NAT -defined Function-like finite m -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of the carrier of (TOP-REAL m)
i is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
X is Relation-like NAT -defined Function-like finite m -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of the carrier of (TOP-REAL m)
y0 is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
f + X is Relation-like NAT -defined Function-like finite m -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of the carrier of (TOP-REAL m)
f + X is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
i + y0 is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
f + X is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
X is ordinal natural V11() real ext-real non negative set
TOP-REAL X is non empty V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital TopSpace-like V148() V214() V215() L19()
the carrier of (TOP-REAL X) is non empty set
m is V11() real ext-real set
f is V11() real ext-real set
i is Relation-like NAT -defined Function-like finite X -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of the carrier of (TOP-REAL X)
y0 is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
m * i is Relation-like NAT -defined Function-like finite X -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of the carrier of (TOP-REAL X)
m * i is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
f * y0 is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
m is ordinal natural V11() real ext-real non negative set
TOP-REAL m is non empty V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital TopSpace-like V148() V214() V215() L19()
the carrier of (TOP-REAL m) is non empty set
f is Relation-like NAT -defined Function-like finite m -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of the carrier of (TOP-REAL m)
X is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
- f is Relation-like NAT -defined Function-like finite m -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of the carrier of (TOP-REAL m)
- f is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
K38(1) is non empty V11() real ext-real non positive negative set
K38(1) * f is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
- X is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
K38(1) * X is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
- f is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
m is ordinal natural V11() real ext-real non negative set
TOP-REAL m is non empty V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital TopSpace-like V148() V214() V215() L19()
the carrier of (TOP-REAL m) is non empty set
f is Relation-like NAT -defined Function-like finite m -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of the carrier of (TOP-REAL m)
i is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
X is Relation-like NAT -defined Function-like finite m -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of the carrier of (TOP-REAL m)
y0 is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
f - X is Relation-like NAT -defined Function-like finite m -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of the carrier of (TOP-REAL m)
f - X is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
- X is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
K38(1) * X is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
f + (- X) is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
i - y0 is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
- y0 is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
K38(1) * y0 is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
i + (- y0) is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
f - X is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
m is non empty V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() RealNormSpace-like NORMSTR
the carrier of m is non empty set
f is Element of the carrier of m
X is Neighbourhood of f
i is Neighbourhood of f
X /\ i is Element of K6( the carrier of m)
K6( the carrier of m) is set
y0 is Neighbourhood of f
r is V11() real ext-real Element of REAL
{ b₁ where b₁ is Element of the carrier of m : not r <= ||.(b₁ - f).|| } is set
m is ordinal natural V11() real ext-real non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of NAT
f is ordinal natural V11() real ext-real non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of NAT
f -' m is ordinal natural V11() real ext-real non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of NAT
m + (f -' m) is ordinal natural V11() real ext-real non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of NAT
m + f is ordinal natural V11() real ext-real non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of NAT
(m + f) -' m is ordinal natural V11() real ext-real non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of NAT
f - m is V11() real ext-real V163() V198() Element of INT
(m + f) - m is V11() real ext-real V163() V198() Element of INT
m is ordinal natural V11() real ext-real non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of NAT
0* m is Relation-like NAT -defined REAL -valued Function-like finite m -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued 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
{ b₁ where b₁ is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like Element of REAL * : len b₁ = m } is set
m |-> 0 is Relation-like empty-yielding NAT -defined REAL -valued Function-like finite m -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of m -tuples_on REAL
Seg m is finite m -element complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below bounded_above real-bounded Element of K6(NAT)
{ b₁ where b₁ is ordinal natural V11() real ext-real non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of NAT : ( 1 <= b₁ & b₁ <= m ) } is set
K213((Seg m),0) is Relation-like Seg m -defined INT -valued RAT -valued {0} -valued Function-like total quasi_total finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued natural-valued Element of K6(K7((Seg m),{0}))
{0} is non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below set
K7((Seg m),{0}) is INT -valued RAT -valued complex-valued ext-real-valued real-valued natural-valued set
K6(K7((Seg m),{0})) is set
f is ordinal natural V11() real ext-real non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of NAT
0* f is Relation-like NAT -defined REAL -valued Function-like finite f -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL f
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
{ b₁ where b₁ is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like Element of REAL * : len b₁ = f } is set
f |-> 0 is Relation-like empty-yielding NAT -defined REAL -valued Function-like finite f -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of f -tuples_on REAL
Seg f is finite f -element complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below bounded_above real-bounded Element of K6(NAT)
{ b₁ where b₁ is ordinal natural V11() real ext-real non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of NAT : ( 1 <= b₁ & b₁ <= f ) } is set
K213((Seg f),0) is Relation-like Seg f -defined INT -valued RAT -valued {0} -valued Function-like total quasi_total finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued natural-valued Element of K6(K7((Seg f),{0}))
K7((Seg f),{0}) is INT -valued RAT -valued complex-valued ext-real-valued real-valued natural-valued set
K6(K7((Seg f),{0})) is set
(0* f) | m is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
(0* f) | (Seg m) is Relation-like NAT -defined Function-like FinSubsequence-like complex-valued ext-real-valued real-valued set
len (0* m) is ordinal natural V11() real ext-real non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of NAT
f -' m is ordinal natural V11() real ext-real non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of NAT
0* (f -' m) is Relation-like NAT -defined REAL -valued Function-like finite f -' m -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL (f -' m)
REAL (f -' m) is non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered FinSequenceSet of REAL
(f -' m) -tuples_on REAL is non empty functional FinSequence-membered FinSequenceSet of REAL
{ b₁ where b₁ is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like Element of REAL * : len b₁ = f -' m } is set
(f -' m) |-> 0 is Relation-like empty-yielding NAT -defined REAL -valued Function-like finite f -' m -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of (f -' m) -tuples_on REAL
Seg (f -' m) is finite f -' m -element complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below bounded_above real-bounded Element of K6(NAT)
{ b₁ where b₁ is ordinal natural V11() real ext-real non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of NAT : ( 1 <= b₁ & b₁ <= f -' m ) } is set
K213((Seg (f -' m)),0) is Relation-like Seg (f -' m) -defined INT -valued RAT -valued {0} -valued Function-like total quasi_total finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued natural-valued Element of K6(K7((Seg (f -' m)),{0}))
K7((Seg (f -' m)),{0}) is INT -valued RAT -valued complex-valued ext-real-valued real-valued natural-valued set
K6(K7((Seg (f -' m)),{0})) is set
(0* m) ^ (0* (f -' m)) is Relation-like NAT -defined REAL -valued Function-like finite K335(m,(f -' m)) -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
K335(m,(f -' m)) is ordinal natural V11() real ext-real non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of NAT
((0* m) ^ (0* (f -' m))) | (len (0* m)) is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
Seg (len (0* m)) is finite len (0* m) -element complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below bounded_above real-bounded Element of K6(NAT)
{ b₁ where b₁ is ordinal natural V11() real ext-real non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of NAT : ( 1 <= b₁ & b₁ <= len (0* m) ) } is set
((0* m) ^ (0* (f -' m))) | (Seg (len (0* m))) is Relation-like NAT -defined Function-like FinSubsequence-like complex-valued ext-real-valued real-valued set
m + (f -' m) is ordinal natural V11() real ext-real non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of NAT
m + f is ordinal natural V11() real ext-real non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of NAT
(m + f) -' m is ordinal natural V11() real ext-real non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of NAT
(m + f) - m is V11() real ext-real V163() V198() Element of INT
m is ordinal natural V11() real ext-real non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of NAT
m -' 1 is ordinal natural V11() real ext-real non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of NAT
0* (m -' 1) is Relation-like NAT -defined REAL -valued Function-like finite m -' 1 -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL (m -' 1)
REAL (m -' 1) is non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered FinSequenceSet of REAL
(m -' 1) -tuples_on REAL is non empty functional FinSequence-membered FinSequenceSet of REAL
{ b₁ where b₁ is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like Element of REAL * : len b₁ = m -' 1 } is set
(m -' 1) |-> 0 is Relation-like empty-yielding NAT -defined REAL -valued Function-like finite m -' 1 -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of (m -' 1) -tuples_on REAL
Seg (m -' 1) is finite m -' 1 -element complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below bounded_above real-bounded Element of K6(NAT)
{ b₁ where b₁ is ordinal natural V11() real ext-real non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of NAT : ( 1 <= b₁ & b₁ <= m -' 1 ) } is set
K213((Seg (m -' 1)),0) is Relation-like Seg (m -' 1) -defined INT -valued RAT -valued {0} -valued Function-like total quasi_total finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued natural-valued Element of K6(K7((Seg (m -' 1)),{0}))
{0} is non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below set
K7((Seg (m -' 1)),{0}) is INT -valued RAT -valued complex-valued ext-real-valued real-valued natural-valued set
K6(K7((Seg (m -' 1)),{0})) is set
f is ordinal natural V11() real ext-real non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of NAT
0* f is Relation-like NAT -defined REAL -valued Function-like finite f -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL f
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
{ b₁ where b₁ is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like Element of REAL * : len b₁ = f } is set
f |-> 0 is Relation-like empty-yielding NAT -defined REAL -valued Function-like finite f -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of f -tuples_on REAL
Seg f is finite f -element complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below bounded_above real-bounded Element of K6(NAT)
{ b₁ where b₁ is ordinal natural V11() real ext-real non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of NAT : ( 1 <= b₁ & b₁ <= f ) } is set
K213((Seg f),0) is Relation-like Seg f -defined INT -valued RAT -valued {0} -valued Function-like total quasi_total finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued natural-valued Element of K6(K7((Seg f),{0}))
K7((Seg f),{0}) is INT -valued RAT -valued complex-valued ext-real-valued real-valued natural-valued set
K6(K7((Seg f),{0})) is set
(0* f) | (m -' 1) is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
(0* f) | (Seg (m -' 1)) is Relation-like NAT -defined Function-like FinSubsequence-like complex-valued ext-real-valued real-valued set
m - 1 is V11() real ext-real V163() V198() Element of INT
f - 1 is V11() real ext-real V163() V198() Element of INT
m is ordinal natural V11() real ext-real non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of NAT
f is ordinal natural V11() real ext-real non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of NAT
0* f is Relation-like NAT -defined REAL -valued Function-like finite f -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL f
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
{ b₁ where b₁ is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like Element of REAL * : len b₁ = f } is set
f |-> 0 is Relation-like empty-yielding NAT -defined REAL -valued Function-like finite f -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of f -tuples_on REAL
Seg f is finite f -element complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below bounded_above real-bounded Element of K6(NAT)
{ b₁ where b₁ is ordinal natural V11() real ext-real non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of NAT : ( 1 <= b₁ & b₁ <= f ) } is set
K213((Seg f),0) is Relation-like Seg f -defined INT -valued RAT -valued {0} -valued Function-like total quasi_total finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued natural-valued Element of K6(K7((Seg f),{0}))
{0} is non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below set
K7((Seg f),{0}) is INT -valued RAT -valued complex-valued ext-real-valued real-valued natural-valued set
K6(K7((Seg f),{0})) is set
(0* f) /^ m is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
f -' m is ordinal natural V11() real ext-real non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of NAT
0* (f -' m) is Relation-like NAT -defined REAL -valued Function-like finite f -' m -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL (f -' m)
REAL (f -' m) is non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered FinSequenceSet of REAL
(f -' m) -tuples_on REAL is non empty functional FinSequence-membered FinSequenceSet of REAL
{ b₁ where b₁ is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like Element of REAL * : len b₁ = f -' m } is set
(f -' m) |-> 0 is Relation-like empty-yielding NAT -defined REAL -valued Function-like finite f -' m -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of (f -' m) -tuples_on REAL
Seg (f -' m) is finite f -' m -element complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below bounded_above real-bounded Element of K6(NAT)
{ b₁ where b₁ is ordinal natural V11() real ext-real non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of NAT : ( 1 <= b₁ & b₁ <= f -' m ) } is set
K213((Seg (f -' m)),0) is Relation-like Seg (f -' m) -defined INT -valued RAT -valued {0} -valued Function-like total quasi_total finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued natural-valued Element of K6(K7((Seg (f -' m)),{0}))
K7((Seg (f -' m)),{0}) is INT -valued RAT -valued complex-valued ext-real-valued real-valued natural-valued set
K6(K7((Seg (f -' m)),{0})) is set
len ((0* f) /^ m) is ordinal natural V11() real ext-real non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of NAT
len (0* f) is ordinal natural V11() real ext-real non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of NAT
(len (0* f)) -' m is ordinal natural V11() real ext-real non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of NAT
len (0* (f -' m)) is ordinal natural V11() real ext-real non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of NAT
X is ordinal natural V11() real ext-real non negative set
((0* f) /^ m) . X is V11() real ext-real Element of REAL
(0* (f -' m)) . X is V11() real ext-real Element of REAL
dom ((0* f) /^ m) is complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of K6(NAT)
X + m is ordinal natural V11() real ext-real non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of NAT
(0* f) . (X + m) is V11() real ext-real Element of REAL
m is ordinal natural V11() real ext-real non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of NAT
f is ordinal natural V11() real ext-real non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of NAT
0* f is Relation-like NAT -defined REAL -valued Function-like finite f -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL f
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
{ b₁ where b₁ is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like Element of REAL * : len b₁ = f } is set
f |-> 0 is Relation-like empty-yielding NAT -defined REAL -valued Function-like finite f -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of f -tuples_on REAL
Seg f is finite f -element complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below bounded_above real-bounded Element of K6(NAT)
{ b₁ where b₁ is ordinal natural V11() real ext-real non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of NAT : ( 1 <= b₁ & b₁ <= f ) } is set
K213((Seg f),0) is Relation-like Seg f -defined INT -valued RAT -valued {0} -valued Function-like total quasi_total finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued natural-valued Element of K6(K7((Seg f),{0}))
{0} is non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below set
K7((Seg f),{0}) is INT -valued RAT -valued complex-valued ext-real-valued real-valued natural-valued set
K6(K7((Seg f),{0})) is set
(0* f) /^ m is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
f -' m is ordinal natural V11() real ext-real non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of NAT
0* (f -' m) is Relation-like NAT -defined REAL -valued Function-like finite f -' m -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL (f -' m)
REAL (f -' m) is non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered FinSequenceSet of REAL
(f -' m) -tuples_on REAL is non empty functional FinSequence-membered FinSequenceSet of REAL
{ b₁ where b₁ is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like Element of REAL * : len b₁ = f -' m } is set
(f -' m) |-> 0 is Relation-like empty-yielding NAT -defined REAL -valued Function-like finite f -' m -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of (f -' m) -tuples_on REAL
Seg (f -' m) is finite f -' m -element complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below bounded_above real-bounded Element of K6(NAT)
{ b₁ where b₁ is ordinal natural V11() real ext-real non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of NAT : ( 1 <= b₁ & b₁ <= f -' m ) } is set
K213((Seg (f -' m)),0) is Relation-like Seg (f -' m) -defined INT -valued RAT -valued {0} -valued Function-like total quasi_total finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued natural-valued Element of K6(K7((Seg (f -' m)),{0}))
K7((Seg (f -' m)),{0}) is INT -valued RAT -valued complex-valued ext-real-valued real-valued natural-valued set
K6(K7((Seg (f -' m)),{0})) is set
len (0* f) is ordinal natural V11() real ext-real non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of NAT
f - m is V11() real ext-real V163() V198() Element of INT
m is ordinal natural V11() real ext-real non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of NAT
0* m is Relation-like NAT -defined REAL -valued Function-like finite m -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued 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
{ b₁ where b₁ is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like Element of REAL * : len b₁ = m } is set
m |-> 0 is Relation-like empty-yielding NAT -defined REAL -valued Function-like finite m -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of m -tuples_on REAL
Seg m is finite m -element complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below bounded_above real-bounded Element of K6(NAT)
{ b₁ where b₁ is ordinal natural V11() real ext-real non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of NAT : ( 1 <= b₁ & b₁ <= m ) } is set
K213((Seg m),0) is Relation-like Seg m -defined INT -valued RAT -valued {0} -valued Function-like total quasi_total finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued natural-valued Element of K6(K7((Seg m),{0}))
{0} is non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below set
K7((Seg m),{0}) is INT -valued RAT -valued complex-valued ext-real-valued real-valued natural-valued set
K6(K7((Seg m),{0})) is set
f is ordinal natural V11() real ext-real non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of NAT
(0* m) /^ f is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
m -' f is ordinal natural V11() real ext-real non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of NAT
0* (m -' f) is Relation-like NAT -defined REAL -valued Function-like finite m -' f -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL (m -' f)
REAL (m -' f) is non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered FinSequenceSet of REAL
(m -' f) -tuples_on REAL is non empty functional FinSequence-membered FinSequenceSet of REAL
{ b₁ where b₁ is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like Element of REAL * : len b₁ = m -' f } is set
(m -' f) |-> 0 is Relation-like empty-yielding NAT -defined REAL -valued Function-like finite m -' f -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of (m -' f) -tuples_on REAL
Seg (m -' f) is finite m -' f -element complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below bounded_above real-bounded Element of K6(NAT)
{ b₁ where b₁ is ordinal natural V11() real ext-real non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of NAT : ( 1 <= b₁ & b₁ <= m -' f ) } is set
K213((Seg (m -' f)),0) is Relation-like Seg (m -' f) -defined INT -valued RAT -valued {0} -valued Function-like total quasi_total finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued natural-valued Element of K6(K7((Seg (m -' f)),{0}))
K7((Seg (m -' f)),{0}) is INT -valued RAT -valued complex-valued ext-real-valued real-valued natural-valued set
K6(K7((Seg (m -' f)),{0})) is set
m is ordinal natural V11() real ext-real non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of NAT
f is ordinal natural V11() real ext-real non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of NAT
m -' 1 is ordinal natural V11() real ext-real non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of NAT
0* f is Relation-like NAT -defined REAL -valued Function-like finite f -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL f
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
{ b₁ where b₁ is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like Element of REAL * : len b₁ = f } is set
f |-> 0 is Relation-like empty-yielding NAT -defined REAL -valued Function-like finite f -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of f -tuples_on REAL
Seg f is finite f -element complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below bounded_above real-bounded Element of K6(NAT)
{ b₁ where b₁ is ordinal natural V11() real ext-real non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of NAT : ( 1 <= b₁ & b₁ <= f ) } is set
K213((Seg f),0) is Relation-like Seg f -defined INT -valued RAT -valued {0} -valued Function-like total quasi_total finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued natural-valued Element of K6(K7((Seg f),{0}))
{0} is non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below set
K7((Seg f),{0}) is INT -valued RAT -valued complex-valued ext-real-valued real-valued natural-valued set
K6(K7((Seg f),{0})) is set
(0* f) | (m -' 1) is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
Seg (m -' 1) is finite m -' 1 -element complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below bounded_above real-bounded Element of K6(NAT)
{ b₁ where b₁ is ordinal natural V11() real ext-real non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of NAT : ( 1 <= b₁ & b₁ <= m -' 1 ) } is set
(0* f) | (Seg (m -' 1)) is Relation-like NAT -defined Function-like FinSubsequence-like complex-valued ext-real-valued real-valued set
0* (m -' 1) is Relation-like NAT -defined REAL -valued Function-like finite m -' 1 -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL (m -' 1)
REAL (m -' 1) is non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered FinSequenceSet of REAL
(m -' 1) -tuples_on REAL is non empty functional FinSequence-membered FinSequenceSet of REAL
{ b₁ where b₁ is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like Element of REAL * : len b₁ = m -' 1 } is set
(m -' 1) |-> 0 is Relation-like empty-yielding NAT -defined REAL -valued Function-like finite m -' 1 -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of (m -' 1) -tuples_on REAL
K213((Seg (m -' 1)),0) is Relation-like Seg (m -' 1) -defined INT -valued RAT -valued {0} -valued Function-like total quasi_total finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued natural-valued Element of K6(K7((Seg (m -' 1)),{0}))
K7((Seg (m -' 1)),{0}) is INT -valued RAT -valued complex-valued ext-real-valued real-valued natural-valued set
K6(K7((Seg (m -' 1)),{0})) is set
X is ordinal natural V11() real ext-real non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of NAT
i is ordinal natural V11() real ext-real non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of NAT
0* i is Relation-like NAT -defined REAL -valued Function-like finite i -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL i
REAL i is non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered FinSequenceSet of REAL
i -tuples_on REAL is non empty functional FinSequence-membered FinSequenceSet of REAL
{ b₁ where b₁ is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like Element of REAL * : len b₁ = i } is set
i |-> 0 is Relation-like empty-yielding NAT -defined REAL -valued Function-like finite i -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of i -tuples_on REAL
Seg i is finite i -element complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below bounded_above real-bounded Element of K6(NAT)
{ b₁ where b₁ is ordinal natural V11() real ext-real non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of NAT : ( 1 <= b₁ & b₁ <= i ) } is set
K213((Seg i),0) is Relation-like Seg i -defined INT -valued RAT -valued {0} -valued Function-like total quasi_total finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued natural-valued Element of K6(K7((Seg i),{0}))
K7((Seg i),{0}) is INT -valued RAT -valued complex-valued ext-real-valued real-valued natural-valued set
K6(K7((Seg i),{0})) is set
(0* i) /^ X is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
i -' X is ordinal natural V11() real ext-real non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of NAT
0* (i -' X) is Relation-like NAT -defined REAL -valued Function-like finite i -' X -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL (i -' X)
REAL (i -' X) is non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered FinSequenceSet of REAL
(i -' X) -tuples_on REAL is non empty functional FinSequence-membered FinSequenceSet of REAL
{ b₁ where b₁ is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like Element of REAL * : len b₁ = i -' X } is set
(i -' X) |-> 0 is Relation-like empty-yielding NAT -defined REAL -valued Function-like finite i -' X -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of (i -' X) -tuples_on REAL
Seg (i -' X) is finite i -' X -element complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below bounded_above real-bounded Element of K6(NAT)
{ b₁ where b₁ is ordinal natural V11() real ext-real non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of NAT : ( 1 <= b₁ & b₁ <= i -' X ) } is set
K213((Seg (i -' X)),0) is Relation-like Seg (i -' X) -defined INT -valued RAT -valued {0} -valued Function-like total quasi_total finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued natural-valued Element of K6(K7((Seg (i -' X)),{0}))
K7((Seg (i -' X)),{0}) is INT -valued RAT -valued complex-valued ext-real-valued real-valued natural-valued set
K6(K7((Seg (i -' X)),{0})) is set
m is ordinal natural V11() real ext-real non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of NAT
f is ordinal natural V11() real ext-real non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of NAT
REAL-NS f is non empty V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V185() NORMSTR
the carrier of (REAL-NS f) is non empty set
0. (REAL-NS f) is V52( REAL-NS f) Element of the carrier of (REAL-NS f)
reproj (m,(0. (REAL-NS f))) is non empty Relation-like the carrier of (REAL-NS 1) -defined the carrier of (REAL-NS f) -valued Function-like total quasi_total Element of K6(K7( the carrier of (REAL-NS 1), the carrier of (REAL-NS f)))
K7( the carrier of (REAL-NS 1), the carrier of (REAL-NS f)) is set
K6(K7( the carrier of (REAL-NS 1), the carrier of (REAL-NS f))) is set
X is Element of the carrier of (REAL-NS 1)
(reproj (m,(0. (REAL-NS f)))) . X is Element of the carrier of (REAL-NS f)
||.((reproj (m,(0. (REAL-NS f)))) . X).|| is V11() real ext-real Element of REAL
the U8 of (REAL-NS f) is non empty Relation-like the carrier of (REAL-NS f) -defined REAL -valued Function-like total quasi_total complex-valued ext-real-valued real-valued Element of K6(K7( the carrier of (REAL-NS f),REAL))
K7( the carrier of (REAL-NS f),REAL) is complex-valued ext-real-valued real-valued set
K6(K7( the carrier of (REAL-NS f),REAL)) is set
the U8 of (REAL-NS f) . ((reproj (m,(0. (REAL-NS f)))) . X) is V11() real ext-real Element of REAL
||.X.|| is V11() real ext-real Element of REAL
the U8 of (REAL-NS 1) is non empty Relation-like the carrier of (REAL-NS 1) -defined REAL -valued Function-like total quasi_total complex-valued ext-real-valued real-valued Element of K6(K7( the carrier of (REAL-NS 1),REAL))
K7( the carrier of (REAL-NS 1),REAL) is complex-valued ext-real-valued real-valued set
K6(K7( the carrier of (REAL-NS 1),REAL)) is set
the U8 of (REAL-NS 1) . X is V11() real ext-real Element of REAL
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
{ b₁ where b₁ is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like Element of REAL * : len b₁ = f } is set
i is V11() real ext-real Element of REAL
<*i*> is non empty V2() Relation-like NAT -defined REAL -valued Function-like one-to-one constant finite 1 -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued increasing decreasing non-decreasing non-increasing FinSequence of REAL
[1,i] is V27() set
{1,i} is non empty complex-membered ext-real-membered real-membered set
{{1,i},{1}} is non empty set
{[1,i]} is non empty Function-like set
y0 is Relation-like NAT -defined REAL -valued Function-like finite f -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL f
reproj (m,y0) is non empty Relation-like REAL -defined REAL f -valued Function-like total quasi_total complex-functions-valued ext-real-functions-valued real-functions-valued Element of K6(K7(REAL,(REAL f)))
K7(REAL,(REAL f)) is set
K6(K7(REAL,(REAL f))) is set
(reproj (m,y0)) . i is Relation-like NAT -defined Function-like finite f -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL f
Replace (y0,m,i) is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
len y0 is ordinal natural V11() real ext-real non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of NAT
m -' 1 is ordinal natural V11() real ext-real non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of NAT
y0 | (m -' 1) is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
Seg (m -' 1) is finite m -' 1 -element complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below bounded_above real-bounded Element of K6(NAT)
{ b₁ where b₁ is ordinal natural V11() real ext-real non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of NAT : ( 1 <= b₁ & b₁ <= m -' 1 ) } is set
y0 | (Seg (m -' 1)) is Relation-like NAT -defined Function-like FinSubsequence-like complex-valued ext-real-valued real-valued set
(y0 | (m -' 1)) ^ <*i*> is non empty Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
y0 /^ m is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
((y0 | (m -' 1)) ^ <*i*>) ^ (y0 /^ m) is non empty Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
|.(((y0 | (m -' 1)) ^ <*i*>) ^ (y0 /^ m)).| is V11() real ext-real non negative Element of REAL
sqr (((y0 | (m -' 1)) ^ <*i*>) ^ (y0 /^ m)) is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
mlt ((((y0 | (m -' 1)) ^ <*i*>) ^ (y0 /^ m)),(((y0 | (m -' 1)) ^ <*i*>) ^ (y0 /^ m))) is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
Sum (sqr (((y0 | (m -' 1)) ^ <*i*>) ^ (y0 /^ m))) is V11() real ext-real Element of REAL
sqrt (Sum (sqr (((y0 | (m -' 1)) ^ <*i*>) ^ (y0 /^ m)))) is V11() real ext-real Element of REAL
0* f is Relation-like NAT -defined REAL -valued Function-like finite f -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL f
f |-> 0 is Relation-like empty-yielding NAT -defined REAL -valued Function-like finite f -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of f -tuples_on REAL
Seg f is finite f -element complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below bounded_above real-bounded Element of K6(NAT)
{ b₁ where b₁ is ordinal natural V11() real ext-real non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of NAT : ( 1 <= b₁ & b₁ <= f ) } is set
K213((Seg f),0) is Relation-like Seg f -defined INT -valued RAT -valued {0} -valued Function-like total quasi_total finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued natural-valued Element of K6(K7((Seg f),{0}))
{0} is non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below set
K7((Seg f),{0}) is INT -valued RAT -valued complex-valued ext-real-valued real-valued natural-valued set
K6(K7((Seg f),{0})) is set
(0* f) | (m -' 1) is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
(0* f) | (Seg (m -' 1)) is Relation-like NAT -defined Function-like FinSubsequence-like complex-valued ext-real-valued real-valued set
sqr (y0 | (m -' 1)) is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
mlt ((y0 | (m -' 1)),(y0 | (m -' 1))) is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
Sum (sqr (y0 | (m -' 1))) is V11() real ext-real Element of REAL
sqrt (Sum (sqr (y0 | (m -' 1)))) is V11() real ext-real Element of REAL
0* (m -' 1) is Relation-like NAT -defined REAL -valued Function-like finite m -' 1 -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL (m -' 1)
REAL (m -' 1) is non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered FinSequenceSet of REAL
(m -' 1) -tuples_on REAL is non empty functional FinSequence-membered FinSequenceSet of REAL
{ b₁ where b₁ is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like Element of REAL * : len b₁ = m -' 1 } is set
(m -' 1) |-> 0 is Relation-like empty-yielding NAT -defined REAL -valued Function-like finite m -' 1 -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of (m -' 1) -tuples_on REAL
K213((Seg (m -' 1)),0) is Relation-like Seg (m -' 1) -defined INT -valued RAT -valued {0} -valued Function-like total quasi_total finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued natural-valued Element of K6(K7((Seg (m -' 1)),{0}))
K7((Seg (m -' 1)),{0}) is INT -valued RAT -valued complex-valued ext-real-valued real-valued natural-valued set
K6(K7((Seg (m -' 1)),{0})) is set
|.(0* (m -' 1)).| is V11() real ext-real non negative Element of REAL
sqr (0* (m -' 1)) is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
mlt ((0* (m -' 1)),(0* (m -' 1))) is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
Sum (sqr (0* (m -' 1))) is V11() real ext-real Element of REAL
sqrt (Sum (sqr (0* (m -' 1)))) is V11() real ext-real Element of REAL
(0* f) /^ m is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
sqr (y0 /^ m) is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
mlt ((y0 /^ m),(y0 /^ m)) is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
Sum (sqr (y0 /^ m)) is V11() real ext-real Element of REAL
sqrt (Sum (sqr (y0 /^ m))) is V11() real ext-real Element of REAL
f -' m is ordinal natural V11() real ext-real non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of NAT
0* (f -' m) is Relation-like NAT -defined REAL -valued Function-like finite f -' m -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL (f -' m)
REAL (f -' m) is non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered FinSequenceSet of REAL
(f -' m) -tuples_on REAL is non empty functional FinSequence-membered FinSequenceSet of REAL
{ b₁ where b₁ is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like Element of REAL * : len b₁ = f -' m } is set
(f -' m) |-> 0 is Relation-like empty-yielding NAT -defined REAL -valued Function-like finite f -' m -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of (f -' m) -tuples_on REAL
Seg (f -' m) is finite f -' m -element complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below bounded_above real-bounded Element of K6(NAT)
{ b₁ where b₁ is ordinal natural V11() real ext-real non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of NAT : ( 1 <= b₁ & b₁ <= f -' m ) } is set
K213((Seg (f -' m)),0) is Relation-like Seg (f -' m) -defined INT -valued RAT -valued {0} -valued Function-like total quasi_total finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued natural-valued Element of K6(K7((Seg (f -' m)),{0}))
K7((Seg (f -' m)),{0}) is INT -valued RAT -valued complex-valued ext-real-valued real-valued natural-valued set
K6(K7((Seg (f -' m)),{0})) is set
|.(0* (f -' m)).| is V11() real ext-real non negative Element of REAL
sqr (0* (f -' m)) is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
mlt ((0* (f -' m)),(0* (f -' m))) is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
Sum (sqr (0* (f -' m))) is V11() real ext-real Element of REAL
sqrt (Sum (sqr (0* (f -' m)))) is V11() real ext-real Element of REAL
sqr ((y0 | (m -' 1)) ^ <*i*>) is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
mlt (((y0 | (m -' 1)) ^ <*i*>),((y0 | (m -' 1)) ^ <*i*>)) is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
(sqr ((y0 | (m -' 1)) ^ <*i*>)) ^ (sqr (y0 /^ m)) is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
sqr <*i*> is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
mlt (<*i*>,<*i*>) is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
(sqr (y0 | (m -' 1))) ^ (sqr <*i*>) is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
((sqr (y0 | (m -' 1))) ^ (sqr <*i*>)) ^ (sqr (y0 /^ m)) is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
i ^2 is V11() real ext-real Element of REAL
K37(i,i) is V11() real ext-real set
<*(i ^2)*> is non empty V2() Relation-like NAT -defined REAL -valued Function-like one-to-one constant finite 1 -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued increasing decreasing non-decreasing non-increasing FinSequence of REAL
[1,(i ^2)] is V27() set
{1,(i ^2)} is non empty complex-membered ext-real-membered real-membered set
{{1,(i ^2)},{1}} is non empty set
{[1,(i ^2)]} is non empty Function-like set
(sqr (y0 | (m -' 1))) ^ <*(i ^2)*> is non empty Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
((sqr (y0 | (m -' 1))) ^ <*(i ^2)*>) ^ (sqr (y0 /^ m)) is non empty Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
Sum ((sqr (y0 | (m -' 1))) ^ <*(i ^2)*>) is V11() real ext-real Element of REAL
(Sum ((sqr (y0 | (m -' 1))) ^ <*(i ^2)*>)) + (Sum (sqr (y0 /^ m))) is V11() real ext-real Element of REAL
(Sum (sqr (y0 | (m -' 1)))) + (i ^2) is V11() real ext-real Element of REAL
((Sum (sqr (y0 | (m -' 1)))) + (i ^2)) + (Sum (sqr (y0 /^ m))) is V11() real ext-real Element of REAL
|.i.| is V11() real ext-real Element of REAL
(proj (1,1)) . <*i*> is V11() real ext-real Element of REAL
m is ordinal natural V11() real ext-real non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below 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
{ b₁ where b₁ is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like Element of REAL * : len b₁ = m } is set
f is ordinal natural V11() real ext-real non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of NAT
0* m is Relation-like NAT -defined REAL -valued Function-like finite m -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL m
m |-> 0 is Relation-like empty-yielding NAT -defined REAL -valued Function-like finite m -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of m -tuples_on REAL
Seg m is finite m -element complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below bounded_above real-bounded Element of K6(NAT)
{ b₁ where b₁ is ordinal natural V11() real ext-real non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of NAT : ( 1 <= b₁ & b₁ <= m ) } is set
K213((Seg m),0) is Relation-like Seg m -defined INT -valued RAT -valued {0} -valued Function-like total quasi_total finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued natural-valued Element of K6(K7((Seg m),{0}))
{0} is non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below set
K7((Seg m),{0}) is INT -valued RAT -valued complex-valued ext-real-valued real-valued natural-valued set
K6(K7((Seg m),{0})) is set
reproj (f,(0* m)) is non empty Relation-like REAL -defined REAL m -valued Function-like total quasi_total complex-functions-valued ext-real-functions-valued real-functions-valued Element of K6(K7(REAL,(REAL m)))
K7(REAL,(REAL m)) is set
K6(K7(REAL,(REAL m))) is set
proj (f,m) is non empty Relation-like REAL m -defined REAL -valued Function-like total quasi_total complex-valued ext-real-valued real-valued Element of K6(K7((REAL m),REAL))
K7((REAL m),REAL) is complex-valued ext-real-valued real-valued set
K6(K7((REAL m),REAL)) is set
X is Relation-like NAT -defined REAL -valued Function-like finite m -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL m
reproj (f,X) is non empty Relation-like REAL -defined REAL m -valued Function-like total quasi_total complex-functions-valued ext-real-functions-valued real-functions-valued Element of K6(K7(REAL,(REAL m)))
(proj (f,m)) . X is V11() real ext-real Element of REAL
i is V11() real ext-real Element of REAL
(reproj (f,X)) . i is Relation-like NAT -defined Function-like finite m -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL m
((reproj (f,X)) . i) - X is Relation-like NAT -defined REAL -valued Function-like finite m -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL m
- X is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
K38(1) * X is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
((reproj (f,X)) . i) + (- X) is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
i - ((proj (f,m)) . X) is V11() real ext-real Element of REAL
(reproj (f,(0* m))) . (i - ((proj (f,m)) . X)) is Relation-like NAT -defined Function-like finite m -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL m
X - ((reproj (f,X)) . i) is Relation-like NAT -defined REAL -valued Function-like finite m -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL m
- ((reproj (f,X)) . i) is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
K38(1) * ((reproj (f,X)) . i) is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
X + (- ((reproj (f,X)) . i)) is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
((proj (f,m)) . X) - i is V11() real ext-real Element of REAL
(reproj (f,(0* m))) . (((proj (f,m)) . X) - i) is Relation-like NAT -defined Function-like finite m -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL m
y0 is Relation-like NAT -defined Function-like finite m -element FinSequence-like FinSubsequence-like set
dom y0 is complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of K6(NAT)
r is Relation-like NAT -defined Function-like finite m -element FinSequence-like FinSubsequence-like set
dom r is complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of K6(NAT)
s is Relation-like NAT -defined Function-like finite m -element FinSequence-like FinSubsequence-like set
dom s is complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of K6(NAT)
y1 is Relation-like NAT -defined Function-like finite m -element FinSequence-like FinSubsequence-like set
dom y1 is complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of K6(NAT)
dom X is complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of K6(NAT)
dom (0* m) is complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of K6(NAT)
Replace (X,f,i) is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
len X is ordinal natural V11() real ext-real non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of NAT
len (0* m) is ordinal natural V11() real ext-real non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of NAT
len (Replace (X,f,i)) is ordinal natural V11() real ext-real non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of NAT
a is ordinal natural V11() real ext-real non negative set
y0 . a is set
r . a is set
dom (Replace (X,f,i)) is complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of K6(NAT)
(Replace (X,f,i)) /. a is V11() real ext-real Element of REAL
(Replace (X,f,i)) . a is V11() real ext-real Element of REAL
DD is Relation-like NAT -defined REAL -valued Function-like finite m -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of m -tuples_on REAL
DD . a is V11() real ext-real Element of REAL
PP is Relation-like NAT -defined REAL -valued Function-like finite m -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of m -tuples_on REAL
PP . a is V11() real ext-real Element of REAL
(DD . a) - (PP . a) is V11() real ext-real Element of REAL
Replace ((0* m),f,(i - ((proj (f,m)) . X))) is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
(Replace ((0* m),f,(i - ((proj (f,m)) . X)))) /. a is V11() real ext-real Element of REAL
i - (PP . a) is V11() real ext-real Element of REAL
PP /. a is V11() real ext-real Element of REAL
(0* m) /. a is V11() real ext-real Element of REAL
(m |-> 0) . a is empty V11() real ext-real non positive non negative Function-like functional FinSequence-membered complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below V263() complex-functions-membered ext-real-functions-membered real-functions-membered rational-functions-membered integer-functions-membered natural-functions-membered Element of REAL
a is ordinal natural V11() real ext-real non negative set
s . a is set
y1 . a is set
dom (Replace (X,f,i)) is complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of K6(NAT)
(Replace (X,f,i)) /. a is V11() real ext-real Element of REAL
(Replace (X,f,i)) . a is V11() real ext-real Element of REAL
PP is Relation-like NAT -defined REAL -valued Function-like finite m -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of m -tuples_on REAL
PP . a is V11() real ext-real Element of REAL
DD is Relation-like NAT -defined REAL -valued Function-like finite m -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of m -tuples_on REAL
DD . a is V11() real ext-real Element of REAL
(PP . a) - (DD . a) is V11() real ext-real Element of REAL
Replace ((0* m),f,(((proj (f,m)) . X) - i)) is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
(Replace ((0* m),f,(((proj (f,m)) . X) - i))) /. a is V11() real ext-real Element of REAL
(PP . a) - i is V11() real ext-real Element of REAL
PP /. a is V11() real ext-real Element of REAL
(0* m) /. a is V11() real ext-real Element of REAL
(m |-> 0) . a is empty V11() real ext-real non positive non negative Function-like functional FinSequence-membered complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below V263() complex-functions-membered ext-real-functions-membered real-functions-membered rational-functions-membered integer-functions-membered natural-functions-membered Element of REAL
m is ordinal natural V11() real ext-real non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below 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() strict RealNormSpace-like V185() NORMSTR
the carrier of (REAL-NS m) is non empty set
f is ordinal natural V11() real ext-real non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of NAT
0. (REAL-NS m) is V52( REAL-NS m) Element of the carrier of (REAL-NS m)
reproj (f,(0. (REAL-NS m))) is non empty Relation-like the carrier of (REAL-NS 1) -defined the carrier of (REAL-NS m) -valued Function-like total quasi_total Element of K6(K7( the carrier of (REAL-NS 1), the carrier of (REAL-NS m)))
K7( the carrier of (REAL-NS 1), the carrier of (REAL-NS m)) is set
K6(K7( the carrier of (REAL-NS 1), the carrier of (REAL-NS m))) is set
Proj (f,m) is non empty Relation-like the carrier of (REAL-NS m) -defined the carrier of (REAL-NS 1) -valued Function-like total quasi_total 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
X is Element of the carrier of (REAL-NS m)
reproj (f,X) is non empty Relation-like the carrier of (REAL-NS 1) -defined the carrier of (REAL-NS m) -valued Function-like total quasi_total Element of K6(K7( the carrier of (REAL-NS 1), the carrier of (REAL-NS m)))
(Proj (f,m)) . X is Element of the carrier of (REAL-NS 1)
i is Element of the carrier of (REAL-NS 1)
(reproj (f,X)) . i is Element of the carrier of (REAL-NS m)
((reproj (f,X)) . i) - X is Element of the carrier of (REAL-NS m)
i - ((Proj (f,m)) . X) is Element of the carrier of (REAL-NS 1)
(reproj (f,(0. (REAL-NS m)))) . (i - ((Proj (f,m)) . X)) is Element of the carrier of (REAL-NS m)
X - ((reproj (f,X)) . i) is Element of the carrier of (REAL-NS m)
((Proj (f,m)) . X) - i is Element of the carrier of (REAL-NS 1)
(reproj (f,(0. (REAL-NS m)))) . (((Proj (f,m)) . X) - i) is Element of the carrier of (REAL-NS 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
{ b₁ where b₁ is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like Element of REAL * : len b₁ = m } is set
y0 is V11() real ext-real Element of REAL
<*y0*> is non empty V2() Relation-like NAT -defined REAL -valued Function-like one-to-one constant finite 1 -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued increasing decreasing non-decreasing non-increasing FinSequence of REAL
[1,y0] is V27() set
{1,y0} is non empty complex-membered ext-real-membered real-membered set
{{1,y0},{1}} is non empty set
{[1,y0]} is non empty Function-like set
r is Relation-like NAT -defined REAL -valued Function-like finite m -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL m
reproj (f,r) is non empty Relation-like REAL -defined REAL m -valued Function-like total quasi_total complex-functions-valued ext-real-functions-valued real-functions-valued Element of K6(K7(REAL,(REAL m)))
K7(REAL,(REAL m)) is set
K6(K7(REAL,(REAL m))) is set
(reproj (f,r)) . y0 is Relation-like NAT -defined Function-like finite m -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL m
y1 is Relation-like NAT -defined REAL -valued Function-like finite m -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL m
y1 - r is Relation-like NAT -defined REAL -valued Function-like finite m -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL m
- r is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
K38(1) * r is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
y1 + (- r) is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
0* m is Relation-like NAT -defined REAL -valued Function-like finite m -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL m
m |-> 0 is Relation-like empty-yielding NAT -defined REAL -valued Function-like finite m -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of m -tuples_on REAL
Seg m is finite m -element complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below bounded_above real-bounded Element of K6(NAT)
{ b₁ where b₁ is ordinal natural V11() real ext-real non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of NAT : ( 1 <= b₁ & b₁ <= m ) } is set
K213((Seg m),0) is Relation-like Seg m -defined INT -valued RAT -valued {0} -valued Function-like total quasi_total finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued natural-valued Element of K6(K7((Seg m),{0}))
{0} is non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below set
K7((Seg m),{0}) is INT -valued RAT -valued complex-valued ext-real-valued real-valued natural-valued set
K6(K7((Seg m),{0})) is set
reproj (f,(0* m)) is non empty Relation-like REAL -defined REAL m -valued Function-like total quasi_total complex-functions-valued ext-real-functions-valued real-functions-valued Element of K6(K7(REAL,(REAL m)))
proj (f,m) is non empty Relation-like REAL m -defined REAL -valued Function-like total quasi_total complex-valued ext-real-valued real-valued Element of K6(K7((REAL m),REAL))
K7((REAL m),REAL) is complex-valued ext-real-valued real-valued set
K6(K7((REAL m),REAL)) is set
(proj (f,m)) . r is V11() real ext-real Element of REAL
y0 - ((proj (f,m)) . r) is V11() real ext-real Element of REAL
(reproj (f,(0* m))) . (y0 - ((proj (f,m)) . r)) is Relation-like NAT -defined Function-like finite m -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL m
<*((proj (f,m)) . r)*> is non empty V2() Relation-like NAT -defined REAL -valued Function-like one-to-one constant finite 1 -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued increasing decreasing non-decreasing non-increasing FinSequence of REAL
[1,((proj (f,m)) . r)] is V27() set
{1,((proj (f,m)) . r)} is non empty complex-membered ext-real-membered real-membered set
{{1,((proj (f,m)) . r)},{1}} is non empty set
{[1,((proj (f,m)) . r)]} is non empty Function-like set
DD is V11() real ext-real Element of REAL
<*DD*> is non empty V2() Relation-like NAT -defined REAL -valued Function-like one-to-one constant finite 1 -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued increasing decreasing non-decreasing non-increasing FinSequence of REAL
[1,DD] is V27() set
{1,DD} is non empty complex-membered ext-real-membered real-membered set
{{1,DD},{1}} is non empty set
{[1,DD]} is non empty Function-like set
a is Relation-like NAT -defined REAL -valued Function-like finite m -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL m
reproj (f,a) is non empty Relation-like REAL -defined REAL m -valued Function-like total quasi_total complex-functions-valued ext-real-functions-valued real-functions-valued Element of K6(K7(REAL,(REAL m)))
(reproj (f,a)) . DD is Relation-like NAT -defined Function-like finite m -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL m
s is Relation-like NAT -defined REAL -valued Function-like finite 1 -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL 1
PP is Relation-like NAT -defined REAL -valued Function-like finite 1 -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL 1
s - PP is Relation-like NAT -defined REAL -valued Function-like finite 1 -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL 1
- PP is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
K38(1) * PP is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
s + (- PP) is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
<*(y0 - ((proj (f,m)) . r))*> is non empty V2() Relation-like NAT -defined REAL -valued Function-like one-to-one constant finite 1 -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued increasing decreasing non-decreasing non-increasing FinSequence of REAL
[1,(y0 - ((proj (f,m)) . r))] is V27() set
{1,(y0 - ((proj (f,m)) . r))} is non empty complex-membered ext-real-membered real-membered set
{{1,(y0 - ((proj (f,m)) . r))},{1}} is non empty set
{[1,(y0 - ((proj (f,m)) . r))]} is non empty Function-like set
r - y1 is Relation-like NAT -defined REAL -valued Function-like finite m -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL m
- y1 is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
K38(1) * y1 is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
r + (- y1) is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
((proj (f,m)) . r) - y0 is V11() real ext-real Element of REAL
(reproj (f,(0* m))) . (((proj (f,m)) . r) - y0) is Relation-like NAT -defined Function-like finite m -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL m
t is V11() real ext-real Element of REAL
<*t*> is non empty V2() Relation-like NAT -defined REAL -valued Function-like one-to-one constant finite 1 -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued increasing decreasing non-decreasing non-increasing FinSequence of REAL
[1,t] is V27() set
{1,t} is non empty complex-membered ext-real-membered real-membered set
{{1,t},{1}} is non empty set
{[1,t]} is non empty Function-like set
tm is Relation-like NAT -defined REAL -valued Function-like finite m -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL m
reproj (f,tm) is non empty Relation-like REAL -defined REAL m -valued Function-like total quasi_total complex-functions-valued ext-real-functions-valued real-functions-valued Element of K6(K7(REAL,(REAL m)))
(reproj (f,tm)) . t is Relation-like NAT -defined Function-like finite m -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL m
PP - s is Relation-like NAT -defined REAL -valued Function-like finite 1 -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL 1
- s is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
K38(1) * s is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
PP + (- s) is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
<*(((proj (f,m)) . r) - y0)*> is non empty V2() Relation-like NAT -defined REAL -valued Function-like one-to-one constant finite 1 -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued increasing decreasing non-decreasing non-increasing FinSequence of REAL
[1,(((proj (f,m)) . r) - y0)] is V27() set
{1,(((proj (f,m)) . r) - y0)} is non empty complex-membered ext-real-membered real-membered set
{{1,(((proj (f,m)) . r) - y0)},{1}} is non empty set
{[1,(((proj (f,m)) . r) - y0)]} is non empty Function-like set
m is ordinal natural V11() real ext-real non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below 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() strict RealNormSpace-like V185() NORMSTR
the carrier of (REAL-NS m) is non empty set
K6( the carrier of (REAL-NS m)) is set
f is Element of the carrier of (REAL-NS m)
X is Element of K6( the carrier of (REAL-NS m))
i is ordinal natural V11() real ext-real non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of NAT
Proj (i,m) is non empty Relation-like the carrier of (REAL-NS m) -defined the carrier of (REAL-NS 1) -valued Function-like total quasi_total 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
(Proj (i,m)) . f is Element of the carrier of (REAL-NS 1)
reproj (i,f) is non empty Relation-like the carrier of (REAL-NS 1) -defined the carrier of (REAL-NS m) -valued Function-like total quasi_total Element of K6(K7( the carrier of (REAL-NS 1), the carrier of (REAL-NS m)))
K7( the carrier of (REAL-NS 1), the carrier of (REAL-NS m)) is set
K6(K7( the carrier of (REAL-NS 1), the carrier of (REAL-NS m))) is set
y0 is V11() real ext-real Element of REAL
{ b₁ where b₁ is Element of the carrier of (REAL-NS m) : not y0 <= ||.(b₁ - f).|| } is set
{ b₁ where b₁ is Element of the carrier of (REAL-NS 1) : not y0 <= ||.(b₁ - ((Proj (i,m)) . f)).|| } is set
s is Neighbourhood of (Proj (i,m)) . f
y1 is Element of the carrier of (REAL-NS 1)
(reproj (i,f)) . y1 is Element of the carrier of (REAL-NS m)
((reproj (i,f)) . y1) - f is Element of the carrier of (REAL-NS m)
||.(((reproj (i,f)) . y1) - f).|| is V11() real ext-real Element of REAL
the U8 of (REAL-NS m) is non empty Relation-like the carrier of (REAL-NS m) -defined REAL -valued Function-like total quasi_total complex-valued ext-real-valued real-valued Element of K6(K7( the carrier of (REAL-NS m),REAL))
K7( the carrier of (REAL-NS m),REAL) is complex-valued ext-real-valued real-valued set
K6(K7( the carrier of (REAL-NS m),REAL)) is set
the U8 of (REAL-NS m) . (((reproj (i,f)) . y1) - f) is V11() real ext-real Element of REAL
0. (REAL-NS m) is V52( REAL-NS m) Element of the carrier of (REAL-NS m)
reproj (i,(0. (REAL-NS m))) is non empty Relation-like the carrier of (REAL-NS 1) -defined the carrier of (REAL-NS m) -valued Function-like total quasi_total Element of K6(K7( the carrier of (REAL-NS 1), the carrier of (REAL-NS m)))
y1 - ((Proj (i,m)) . f) is Element of the carrier of (REAL-NS 1)
(reproj (i,(0. (REAL-NS m)))) . (y1 - ((Proj (i,m)) . f)) is Element of the carrier of (REAL-NS m)
||.((reproj (i,(0. (REAL-NS m)))) . (y1 - ((Proj (i,m)) . f))).|| is V11() real ext-real Element of REAL
the U8 of (REAL-NS m) . ((reproj (i,(0. (REAL-NS m)))) . (y1 - ((Proj (i,m)) . f))) is V11() real ext-real Element of REAL
||.(y1 - ((Proj (i,m)) . f)).|| is V11() real ext-real Element of REAL
the U8 of (REAL-NS 1) is non empty Relation-like the carrier of (REAL-NS 1) -defined REAL -valued Function-like total quasi_total complex-valued ext-real-valued real-valued Element of K6(K7( the carrier of (REAL-NS 1),REAL))
K7( the carrier of (REAL-NS 1),REAL) is complex-valued ext-real-valued real-valued set
K6(K7( the carrier of (REAL-NS 1),REAL)) is set
the U8 of (REAL-NS 1) . (y1 - ((Proj (i,m)) . f)) is V11() real ext-real Element of REAL
PP is Element of the carrier of (REAL-NS 1)
PP - ((Proj (i,m)) . f) is Element of the carrier of (REAL-NS 1)
||.(PP - ((Proj (i,m)) . f)).|| is V11() real ext-real Element of REAL
the U8 of (REAL-NS 1) . (PP - ((Proj (i,m)) . f)) is V11() real ext-real Element of REAL
m is non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below 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() strict RealNormSpace-like V185() NORMSTR
the carrier of (REAL-NS m) is non empty V2() set
f is non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below Element of NAT
REAL-NS f is non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V185() NORMSTR
the carrier of (REAL-NS f) is non empty V2() set
K7( the carrier of (REAL-NS m), the carrier of (REAL-NS f)) is set
K6(K7( the carrier of (REAL-NS m), the carrier of (REAL-NS f))) is set
K6( the carrier of (REAL-NS m)) is set
X is ordinal natural V11() real ext-real non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of NAT
i is Relation-like the carrier of (REAL-NS m) -defined the carrier of (REAL-NS f) -valued Function-like Element of K6(K7( the carrier of (REAL-NS m), the carrier of (REAL-NS f)))
dom i is Element of K6( the carrier of (REAL-NS m))
y0 is Element of K6( the carrier of (REAL-NS m))
R_NormSpace_of_BoundedLinearOperators ((REAL-NS 1),(REAL-NS f)) is non empty V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() RealNormSpace-like V185() NORMSTR
BoundedLinearOperators ((REAL-NS 1),(REAL-NS f)) is non empty Element of K6( the carrier of (R_VectorSpace_of_LinearOperators ((REAL-NS 1),(REAL-NS f))))
R_VectorSpace_of_LinearOperators ((REAL-NS 1),(REAL-NS f)) is non empty V71() strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V214() RLSStruct
LinearOperators ((REAL-NS 1),(REAL-NS f)) is non empty functional Element of K6( the carrier of K318( the carrier of (REAL-NS 1),(REAL-NS f)))
K318( the carrier of (REAL-NS 1),(REAL-NS f)) is non empty V71() strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V214() RLSStruct
Funcs ( the carrier of (REAL-NS 1), the carrier of (REAL-NS f)) is non empty functional FUNCTION_DOMAIN of the carrier of (REAL-NS 1), the carrier of (REAL-NS f)
FuncZero ( the carrier of (REAL-NS 1),(REAL-NS f)) is Relation-like the carrier of (REAL-NS 1) -defined the carrier of (REAL-NS f) -valued Function-like total quasi_total Element of Funcs ( the carrier of (REAL-NS 1), the carrier of (REAL-NS f))
0. (REAL-NS f) is V52( REAL-NS f) Element of the carrier of (REAL-NS f)
K214( the carrier of (REAL-NS f), the carrier of (REAL-NS 1),(0. (REAL-NS f))) is non empty Relation-like the carrier of (REAL-NS 1) -defined the carrier of (REAL-NS f) -valued Function-like total quasi_total Element of K6(K7( the carrier of (REAL-NS 1), the carrier of (REAL-NS f)))
K7( the carrier of (REAL-NS 1), the carrier of (REAL-NS f)) is set
K6(K7( the carrier of (REAL-NS 1), the carrier of (REAL-NS f))) is set
FuncAdd ( the carrier of (REAL-NS 1),(REAL-NS f)) is Relation-like K7((Funcs ( the carrier of (REAL-NS 1), the carrier of (REAL-NS f))),(Funcs ( the carrier of (REAL-NS 1), the carrier of (REAL-NS f)))) -defined Funcs ( the carrier of (REAL-NS 1), the carrier of (REAL-NS f)) -valued Function-like total quasi_total Element of K6(K7(K7((Funcs ( the carrier of (REAL-NS 1), the carrier of (REAL-NS f))),(Funcs ( the carrier of (REAL-NS 1), the carrier of (REAL-NS f)))),(Funcs ( the carrier of (REAL-NS 1), the carrier of (REAL-NS f)))))
K7((Funcs ( the carrier of (REAL-NS 1), the carrier of (REAL-NS f))),(Funcs ( the carrier of (REAL-NS 1), the carrier of (REAL-NS f)))) is set
K7(K7((Funcs ( the carrier of (REAL-NS 1), the carrier of (REAL-NS f))),(Funcs ( the carrier of (REAL-NS 1), the carrier of (REAL-NS f)))),(Funcs ( the carrier of (REAL-NS 1), the carrier of (REAL-NS f)))) is set
K6(K7(K7((Funcs ( the carrier of (REAL-NS 1), the carrier of (REAL-NS f))),(Funcs ( the carrier of (REAL-NS 1), the carrier of (REAL-NS f)))),(Funcs ( the carrier of (REAL-NS 1), the carrier of (REAL-NS f))))) is set
FuncExtMult ( the carrier of (REAL-NS 1),(REAL-NS f)) is Relation-like K7(REAL,(Funcs ( the carrier of (REAL-NS 1), the carrier of (REAL-NS f)))) -defined Funcs ( the carrier of (REAL-NS 1), the carrier of (REAL-NS f)) -valued Function-like total quasi_total Element of K6(K7(K7(REAL,(Funcs ( the carrier of (REAL-NS 1), the carrier of (REAL-NS f)))),(Funcs ( the carrier of (REAL-NS 1), the carrier of (REAL-NS f)))))
K7(REAL,(Funcs ( the carrier of (REAL-NS 1), the carrier of (REAL-NS f)))) is set
K7(K7(REAL,(Funcs ( the carrier of (REAL-NS 1), the carrier of (REAL-NS f)))),(Funcs ( the carrier of (REAL-NS 1), the carrier of (REAL-NS f)))) is set
K6(K7(K7(REAL,(Funcs ( the carrier of (REAL-NS 1), the carrier of (REAL-NS f)))),(Funcs ( the carrier of (REAL-NS 1), the carrier of (REAL-NS f))))) is set
RLSStruct(# (Funcs ( the carrier of (REAL-NS 1), the carrier of (REAL-NS f))),(FuncZero ( the carrier of (REAL-NS 1),(REAL-NS f))),(FuncAdd ( the carrier of (REAL-NS 1),(REAL-NS f))),(FuncExtMult ( the carrier of (REAL-NS 1),(REAL-NS f))) #) is strict RLSStruct
the carrier of K318( the carrier of (REAL-NS 1),(REAL-NS f)) is non empty set
K6( the carrier of K318( the carrier of (REAL-NS 1),(REAL-NS f))) is set
K199(K318( the carrier of (REAL-NS 1),(REAL-NS f)),(LinearOperators ((REAL-NS 1),(REAL-NS f)))) is Relation-like Function-like Element of LinearOperators ((REAL-NS 1),(REAL-NS f))
K197(K318( the carrier of (REAL-NS 1),(REAL-NS f)),(LinearOperators ((REAL-NS 1),(REAL-NS f)))) is Relation-like K7((LinearOperators ((REAL-NS 1),(REAL-NS f))),(LinearOperators ((REAL-NS 1),(REAL-NS f)))) -defined LinearOperators ((REAL-NS 1),(REAL-NS f)) -valued Function-like total quasi_total Element of K6(K7(K7((LinearOperators ((REAL-NS 1),(REAL-NS f))),(LinearOperators ((REAL-NS 1),(REAL-NS f)))),(LinearOperators ((REAL-NS 1),(REAL-NS f)))))
K7((LinearOperators ((REAL-NS 1),(REAL-NS f))),(LinearOperators ((REAL-NS 1),(REAL-NS f)))) is set
K7(K7((LinearOperators ((REAL-NS 1),(REAL-NS f))),(LinearOperators ((REAL-NS 1),(REAL-NS f)))),(LinearOperators ((REAL-NS 1),(REAL-NS f)))) is set
K6(K7(K7((LinearOperators ((REAL-NS 1),(REAL-NS f))),(LinearOperators ((REAL-NS 1),(REAL-NS f)))),(LinearOperators ((REAL-NS 1),(REAL-NS f))))) is set
K198(K318( the carrier of (REAL-NS 1),(REAL-NS f)),(LinearOperators ((REAL-NS 1),(REAL-NS f)))) is Relation-like K7(REAL,(LinearOperators ((REAL-NS 1),(REAL-NS f)))) -defined LinearOperators ((REAL-NS 1),(REAL-NS f)) -valued Function-like total quasi_total Element of K6(K7(K7(REAL,(LinearOperators ((REAL-NS 1),(REAL-NS f)))),(LinearOperators ((REAL-NS 1),(REAL-NS f)))))
K7(REAL,(LinearOperators ((REAL-NS 1),(REAL-NS f)))) is set
K7(K7(REAL,(LinearOperators ((REAL-NS 1),(REAL-NS f)))),(LinearOperators ((REAL-NS 1),(REAL-NS f)))) is set
K6(K7(K7(REAL,(LinearOperators ((REAL-NS 1),(REAL-NS f)))),(LinearOperators ((REAL-NS 1),(REAL-NS f))))) is set
RLSStruct(# (LinearOperators ((REAL-NS 1),(REAL-NS f))),K199(K318( the carrier of (REAL-NS 1),(REAL-NS f)),(LinearOperators ((REAL-NS 1),(REAL-NS f)))),K197(K318( the carrier of (REAL-NS 1),(REAL-NS f)),(LinearOperators ((REAL-NS 1),(REAL-NS f)))),K198(K318( the carrier of (REAL-NS 1),(REAL-NS f)),(LinearOperators ((REAL-NS 1),(REAL-NS f)))) #) 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 f))) is non empty set
K6( the carrier of (R_VectorSpace_of_LinearOperators ((REAL-NS 1),(REAL-NS f)))) is set
K199((R_VectorSpace_of_LinearOperators ((REAL-NS 1),(REAL-NS f))),(BoundedLinearOperators ((REAL-NS 1),(REAL-NS f)))) is Element of BoundedLinearOperators ((REAL-NS 1),(REAL-NS f))
K197((R_VectorSpace_of_LinearOperators ((REAL-NS 1),(REAL-NS f))),(BoundedLinearOperators ((REAL-NS 1),(REAL-NS f)))) is Relation-like K7((BoundedLinearOperators ((REAL-NS 1),(REAL-NS f))),(BoundedLinearOperators ((REAL-NS 1),(REAL-NS f)))) -defined BoundedLinearOperators ((REAL-NS 1),(REAL-NS f)) -valued Function-like total quasi_total Element of K6(K7(K7((BoundedLinearOperators ((REAL-NS 1),(REAL-NS f))),(BoundedLinearOperators ((REAL-NS 1),(REAL-NS f)))),(BoundedLinearOperators ((REAL-NS 1),(REAL-NS f)))))
K7((BoundedLinearOperators ((REAL-NS 1),(REAL-NS f))),(BoundedLinearOperators ((REAL-NS 1),(REAL-NS f)))) is set
K7(K7((BoundedLinearOperators ((REAL-NS 1),(REAL-NS f))),(BoundedLinearOperators ((REAL-NS 1),(REAL-NS f)))),(BoundedLinearOperators ((REAL-NS 1),(REAL-NS f)))) is set
K6(K7(K7((BoundedLinearOperators ((REAL-NS 1),(REAL-NS f))),(BoundedLinearOperators ((REAL-NS 1),(REAL-NS f)))),(BoundedLinearOperators ((REAL-NS 1),(REAL-NS f))))) is set
K198((R_VectorSpace_of_LinearOperators ((REAL-NS 1),(REAL-NS f))),(BoundedLinearOperators ((REAL-NS 1),(REAL-NS f)))) is Relation-like K7(REAL,(BoundedLinearOperators ((REAL-NS 1),(REAL-NS f)))) -defined BoundedLinearOperators ((REAL-NS 1),(REAL-NS f)) -valued Function-like total quasi_total Element of K6(K7(K7(REAL,(BoundedLinearOperators ((REAL-NS 1),(REAL-NS f)))),(BoundedLinearOperators ((REAL-NS 1),(REAL-NS f)))))
K7(REAL,(BoundedLinearOperators ((REAL-NS 1),(REAL-NS f)))) is set
K7(K7(REAL,(BoundedLinearOperators ((REAL-NS 1),(REAL-NS f)))),(BoundedLinearOperators ((REAL-NS 1),(REAL-NS f)))) is set
K6(K7(K7(REAL,(BoundedLinearOperators ((REAL-NS 1),(REAL-NS f)))),(BoundedLinearOperators ((REAL-NS 1),(REAL-NS f))))) is set
BoundedLinearOperatorsNorm ((REAL-NS 1),(REAL-NS f)) is non empty Relation-like BoundedLinearOperators ((REAL-NS 1),(REAL-NS f)) -defined REAL -valued Function-like total quasi_total complex-valued ext-real-valued real-valued Element of K6(K7((BoundedLinearOperators ((REAL-NS 1),(REAL-NS f))),REAL))
K7((BoundedLinearOperators ((REAL-NS 1),(REAL-NS f))),REAL) is complex-valued ext-real-valued real-valued set
K6(K7((BoundedLinearOperators ((REAL-NS 1),(REAL-NS f))),REAL)) is set
NORMSTR(# (BoundedLinearOperators ((REAL-NS 1),(REAL-NS f))),K199((R_VectorSpace_of_LinearOperators ((REAL-NS 1),(REAL-NS f))),(BoundedLinearOperators ((REAL-NS 1),(REAL-NS f)))),K197((R_VectorSpace_of_LinearOperators ((REAL-NS 1),(REAL-NS f))),(BoundedLinearOperators ((REAL-NS 1),(REAL-NS f)))),K198((R_VectorSpace_of_LinearOperators ((REAL-NS 1),(REAL-NS f))),(BoundedLinearOperators ((REAL-NS 1),(REAL-NS f)))),(BoundedLinearOperatorsNorm ((REAL-NS 1),(REAL-NS f))) #) is strict NORMSTR
PP is Element of the carrier of (REAL-NS m)
Proj (X,m) is non empty Relation-like the carrier of (REAL-NS m) -defined the carrier of (REAL-NS 1) -valued Function-like total quasi_total 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
(Proj (X,m)) . PP is Element of the carrier of (REAL-NS 1)
i | y0 is Relation-like the carrier of (REAL-NS m) -defined the carrier of (REAL-NS f) -valued Function-like Element of K6(K7( the carrier of (REAL-NS m), the carrier of (REAL-NS f)))
reproj (X,PP) is non empty Relation-like the carrier of (REAL-NS 1) -defined the carrier of (REAL-NS m) -valued Function-like total quasi_total Element of K6(K7( the carrier of (REAL-NS 1), the carrier of (REAL-NS m)))
K7( the carrier of (REAL-NS 1), the carrier of (REAL-NS m)) is set
K6(K7( the carrier of (REAL-NS 1), the carrier of (REAL-NS m))) is set
(i | y0) * (reproj (X,PP)) is Relation-like the carrier of (REAL-NS 1) -defined the carrier of (REAL-NS f) -valued Function-like Element of K6(K7( the carrier of (REAL-NS 1), the carrier of (REAL-NS f)))
DD is Element of the carrier of (REAL-NS 1)
dom ((i | y0) * (reproj (X,PP))) is Element of K6( the carrier of (REAL-NS 1))
K6( the carrier of (REAL-NS 1)) is set
the carrier of (R_NormSpace_of_BoundedLinearOperators ((REAL-NS 1),(REAL-NS f))) is non empty set
((i | y0) * (reproj (X,PP))) /. DD is Element of the carrier of (REAL-NS f)
a is Neighbourhood of DD
t is Relation-like Function-like Element of the carrier of (R_NormSpace_of_BoundedLinearOperators ((REAL-NS 1),(REAL-NS f)))
tm is Relation-like the carrier of (REAL-NS 1) -defined the carrier of (REAL-NS f) -valued Function-like RestFunc-like Element of K6(K7( the carrier of (REAL-NS 1), the carrier of (REAL-NS f)))
t is Relation-like Function-like Element of the carrier of (R_NormSpace_of_BoundedLinearOperators ((REAL-NS 1),(REAL-NS f)))
tm is Relation-like the carrier of (REAL-NS 1) -defined the carrier of (REAL-NS f) -valued Function-like RestFunc-like Element of K6(K7( the carrier of (REAL-NS 1), the carrier of (REAL-NS f)))
Sw0 is Neighbourhood of DD
w1 is Element of the carrier of (REAL-NS 1)
(reproj (X,PP)) . w1 is Element of the carrier of (REAL-NS m)
(dom i) /\ y0 is Element of K6( the carrier of (REAL-NS m))
dom (i | y0) is Element of K6( the carrier of (REAL-NS m))
a /\ Sw0 is Element of K6( the carrier of (REAL-NS 1))
i * (reproj (X,PP)) is Relation-like the carrier of (REAL-NS 1) -defined the carrier of (REAL-NS f) -valued Function-like Element of K6(K7( the carrier of (REAL-NS 1), the carrier of (REAL-NS f)))
dom (i * (reproj (X,PP))) is Element of K6( the carrier of (REAL-NS 1))
w1 is Neighbourhood of DD
Sw1 is Element of the carrier of (REAL-NS 1)
dom (reproj (X,PP)) is Element of K6( the carrier of (REAL-NS 1))
(reproj (X,PP)) . Sw1 is Element of the carrier of (REAL-NS m)
(reproj (X,PP)) . DD is Element of the carrier of (REAL-NS m)
((i | y0) * (reproj (X,PP))) /. Sw1 is Element of the carrier of (REAL-NS f)
(reproj (X,PP)) /. Sw1 is Element of the carrier of (REAL-NS m)
(i | y0) /. ((reproj (X,PP)) /. Sw1) is Element of the carrier of (REAL-NS f)
i /. ((reproj (X,PP)) /. Sw1) is Element of the carrier of (REAL-NS f)
(i * (reproj (X,PP))) /. Sw1 is Element of the carrier of (REAL-NS f)
(reproj (X,PP)) /. DD is Element of the carrier of (REAL-NS m)
(i | y0) /. ((reproj (X,PP)) /. DD) is Element of the carrier of (REAL-NS f)
i /. ((reproj (X,PP)) /. DD) is Element of the carrier of (REAL-NS f)
(i * (reproj (X,PP))) /. DD is Element of the carrier of (REAL-NS f)
((i * (reproj (X,PP))) /. Sw1) - ((i * (reproj (X,PP))) /. DD) is Element of the carrier of (REAL-NS f)
Sw1 - DD is Element of the carrier of (REAL-NS 1)
t . (Sw1 - DD) is Element of the carrier of (REAL-NS f)
tm /. (Sw1 - DD) is Element of the carrier of (REAL-NS f)
(t . (Sw1 - DD)) + (tm /. (Sw1 - DD)) is Element of the carrier of (REAL-NS f)
i | y0 is Relation-like the carrier of (REAL-NS m) -defined the carrier of (REAL-NS f) -valued Function-like Element of K6(K7( the carrier of (REAL-NS m), the carrier of (REAL-NS f)))
PP is Element of the carrier of (REAL-NS m)
Proj (X,m) is non empty Relation-like the carrier of (REAL-NS m) -defined the carrier of (REAL-NS 1) -valued Function-like total quasi_total 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
(Proj (X,m)) . PP is Element of the carrier of (REAL-NS 1)
reproj (X,PP) is non empty Relation-like the carrier of (REAL-NS 1) -defined the carrier of (REAL-NS m) -valued Function-like total quasi_total Element of K6(K7( the carrier of (REAL-NS 1), the carrier of (REAL-NS m)))
K7( the carrier of (REAL-NS 1), the carrier of (REAL-NS m)) is set
K6(K7( the carrier of (REAL-NS 1), the carrier of (REAL-NS m))) is set
i * (reproj (X,PP)) is Relation-like the carrier of (REAL-NS 1) -defined the carrier of (REAL-NS f) -valued Function-like Element of K6(K7( the carrier of (REAL-NS 1), the carrier of (REAL-NS f)))
DD is Element of the carrier of (REAL-NS 1)
dom (i * (reproj (X,PP))) is Element of K6( the carrier of (REAL-NS 1))
K6( the carrier of (REAL-NS 1)) is set
the carrier of (R_NormSpace_of_BoundedLinearOperators ((REAL-NS 1),(REAL-NS f))) is non empty set
(i * (reproj (X,PP))) /. DD is Element of the carrier of (REAL-NS f)
a is Neighbourhood of DD
t is Neighbourhood of DD
tm is Element of the carrier of (REAL-NS 1)
(reproj (X,PP)) . tm is Element of the carrier of (REAL-NS m)
(dom i) /\ y0 is Element of K6( the carrier of (REAL-NS m))
dom (i | y0) is Element of K6( the carrier of (REAL-NS m))
(i | y0) * (reproj (X,PP)) is Relation-like the carrier of (REAL-NS 1) -defined the carrier of (REAL-NS f) -valued Function-like Element of K6(K7( the carrier of (REAL-NS 1), the carrier of (REAL-NS f)))
dom ((i | y0) * (reproj (X,PP))) is Element of K6( the carrier of (REAL-NS 1))
tm is set
Sw0 is Element of the carrier of (REAL-NS 1)
(reproj (X,PP)) . Sw0 is Element of the carrier of (REAL-NS m)
dom (reproj (X,PP)) is Element of K6( the carrier of (REAL-NS 1))
a /\ t is Element of K6( the carrier of (REAL-NS 1))
tm is Neighbourhood of DD
Sw0 is Relation-like Function-like Element of the carrier of (R_NormSpace_of_BoundedLinearOperators ((REAL-NS 1),(REAL-NS f)))
w1 is Relation-like the carrier of (REAL-NS 1) -defined the carrier of (REAL-NS f) -valued Function-like RestFunc-like Element of K6(K7( the carrier of (REAL-NS 1), the carrier of (REAL-NS f)))
Sw0 is Relation-like Function-like Element of the carrier of (R_NormSpace_of_BoundedLinearOperators ((REAL-NS 1),(REAL-NS f)))
w1 is Relation-like the carrier of (REAL-NS 1) -defined the carrier of (REAL-NS f) -valued Function-like RestFunc-like Element of K6(K7( the carrier of (REAL-NS 1), the carrier of (REAL-NS f)))
Sw1 is Element of the carrier of (REAL-NS 1)
dom (reproj (X,PP)) is Element of K6( the carrier of (REAL-NS 1))
(reproj (X,PP)) . Sw1 is Element of the carrier of (REAL-NS m)
(reproj (X,PP)) . DD is Element of the carrier of (REAL-NS m)
((i | y0) * (reproj (X,PP))) /. Sw1 is Element of the carrier of (REAL-NS f)
(reproj (X,PP)) /. Sw1 is Element of the carrier of (REAL-NS m)
(i | y0) /. ((reproj (X,PP)) /. Sw1) is Element of the carrier of (REAL-NS f)
i /. ((reproj (X,PP)) /. Sw1) is Element of the carrier of (REAL-NS f)
(i * (reproj (X,PP))) /. Sw1 is Element of the carrier of (REAL-NS f)
((i | y0) * (reproj (X,PP))) /. DD is Element of the carrier of (REAL-NS f)
(reproj (X,PP)) /. DD is Element of the carrier of (REAL-NS m)
(i | y0) /. ((reproj (X,PP)) /. DD) is Element of the carrier of (REAL-NS f)
i /. ((reproj (X,PP)) /. DD) is Element of the carrier of (REAL-NS f)
(((i | y0) * (reproj (X,PP))) /. Sw1) - (((i | y0) * (reproj (X,PP))) /. DD) is Element of the carrier of (REAL-NS f)
Sw1 - DD is Element of the carrier of (REAL-NS 1)
Sw0 . (Sw1 - DD) is Element of the carrier of (REAL-NS f)
w1 /. (Sw1 - DD) is Element of the carrier of (REAL-NS f)
(Sw0 . (Sw1 - DD)) + (w1 /. (Sw1 - DD)) is Element of the carrier of (REAL-NS f)
PP is Element of the carrier of (REAL-NS m)
m is non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below 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
{ b₁ where b₁ is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like Element of REAL * : len b₁ = m } is set
f is Relation-like NAT -defined REAL -valued Function-like finite m -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL m
X is V11() real ext-real Element of REAL
i is ordinal natural V11() real ext-real non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of NAT
Replace (f,i,X) is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
len (Replace (f,i,X)) is ordinal natural V11() real ext-real non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of NAT
len f is ordinal natural V11() real ext-real non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of NAT
m is non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below Element of NAT
0* m is Relation-like NAT -defined REAL -valued Function-like finite m -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued 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
{ b₁ where b₁ is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like Element of REAL * : len b₁ = m } is set
m |-> 0 is Relation-like empty-yielding NAT -defined REAL -valued Function-like finite m -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of m -tuples_on REAL
Seg m is non empty finite m -element complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end right_end bounded_below bounded_above real-bounded Element of K6(NAT)
{ b₁ where b₁ is ordinal natural V11() real ext-real non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of NAT : ( 1 <= b₁ & b₁ <= m ) } is set
K213((Seg m),0) is non empty Relation-like Seg m -defined INT -valued RAT -valued {0} -valued Function-like total quasi_total finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued natural-valued Element of K6(K7((Seg m),{0}))
{0} is non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below set
K7((Seg m),{0}) is INT -valued RAT -valued complex-valued ext-real-valued real-valued natural-valued set
K6(K7((Seg m),{0})) is set
f is V11() real ext-real Element of REAL
i is ordinal natural V11() real ext-real non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of NAT
X is ordinal natural V11() real ext-real non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of NAT
Replace ((0* m),X,f) is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
(Replace ((0* m),X,f)) . i is V11() real ext-real Element of REAL
len (0* m) is ordinal natural V11() real ext-real non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of NAT
dom (0* m) is complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of K6(NAT)
(0* m) +* (X,f) is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
((0* m) +* (X,f)) . i is V11() real ext-real Element of REAL
(0* m) . i is V11() real ext-real Element of REAL
m is non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below Element of NAT
0* m is Relation-like NAT -defined REAL -valued Function-like finite m -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued 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
{ b₁ where b₁ is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like Element of REAL * : len b₁ = m } is set
m |-> 0 is Relation-like empty-yielding NAT -defined REAL -valued Function-like finite m -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of m -tuples_on REAL
Seg m is non empty finite m -element complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end right_end bounded_below bounded_above real-bounded Element of K6(NAT)
{ b₁ where b₁ is ordinal natural V11() real ext-real non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of NAT : ( 1 <= b₁ & b₁ <= m ) } is set
K213((Seg m),0) is non empty Relation-like Seg m -defined INT -valued RAT -valued {0} -valued Function-like total quasi_total finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued natural-valued Element of K6(K7((Seg m),{0}))
{0} is non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below set
K7((Seg m),{0}) is INT -valued RAT -valued complex-valued ext-real-valued real-valued natural-valued set
K6(K7((Seg m),{0})) is set
f is V11() real ext-real Element of REAL
X is V11() real ext-real Element of REAL
f + X is V11() real ext-real Element of REAL
i is ordinal natural V11() real ext-real non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of NAT
Replace ((0* m),i,(f + X)) is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
Replace ((0* m),i,f) is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
Replace ((0* m),i,X) is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
(Replace ((0* m),i,f)) + (Replace ((0* m),i,X)) is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of K6(K7(NAT,REAL))
len (Replace ((0* m),i,(f + X))) is ordinal natural V11() real ext-real non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of NAT
len (Replace ((0* m),i,f)) is ordinal natural V11() real ext-real non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of NAT
len (Replace ((0* m),i,X)) is ordinal natural V11() real ext-real non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of NAT
len ((Replace ((0* m),i,f)) + (Replace ((0* m),i,X))) is ordinal natural V11() real ext-real non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of NAT
y0 is ordinal natural V11() real ext-real non negative set
(Replace ((0* m),i,(f + X))) . y0 is V11() real ext-real Element of REAL
((Replace ((0* m),i,f)) + (Replace ((0* m),i,X))) . y0 is V11() real ext-real Element of REAL
dom ((Replace ((0* m),i,f)) + (Replace ((0* m),i,X))) is complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of K6(NAT)
dom (Replace ((0* m),i,f)) is complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of K6(NAT)
dom (Replace ((0* m),i,X)) is complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of K6(NAT)
(dom (Replace ((0* m),i,f))) /\ (dom (Replace ((0* m),i,X))) is complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of K6(NAT)
r is ordinal natural V11() real ext-real non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of NAT
((Replace ((0* m),i,f)) + (Replace ((0* m),i,X))) . r is V11() real ext-real Element of REAL
(Replace ((0* m),i,f)) . r is V11() real ext-real Element of REAL
(Replace ((0* m),i,X)) . r is V11() real ext-real Element of REAL
((Replace ((0* m),i,f)) . r) + ((Replace ((0* m),i,X)) . r) is V11() real ext-real Element of REAL
f + ((Replace ((0* m),i,X)) . r) is V11() real ext-real Element of REAL
0 + ((Replace ((0* m),i,X)) . r) is V11() real ext-real Element of REAL
m is non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below Element of NAT
0* m is Relation-like NAT -defined REAL -valued Function-like finite m -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued 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
{ b₁ where b₁ is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like Element of REAL * : len b₁ = m } is set
m |-> 0 is Relation-like empty-yielding NAT -defined REAL -valued Function-like finite m -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of m -tuples_on REAL
Seg m is non empty finite m -element complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end right_end bounded_below bounded_above real-bounded Element of K6(NAT)
{ b₁ where b₁ is ordinal natural V11() real ext-real non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of NAT : ( 1 <= b₁ & b₁ <= m ) } is set
K213((Seg m),0) is non empty Relation-like Seg m -defined INT -valued RAT -valued {0} -valued Function-like total quasi_total finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued natural-valued Element of K6(K7((Seg m),{0}))
{0} is non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below set
K7((Seg m),{0}) is INT -valued RAT -valued complex-valued ext-real-valued real-valued natural-valued set
K6(K7((Seg m),{0})) is set
X is V11() real ext-real Element of REAL
f is V11() real ext-real Element of REAL
X * f is V11() real ext-real Element of REAL
i is ordinal natural V11() real ext-real non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of NAT
Replace ((0* m),i,(X * f)) is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
Replace ((0* m),i,f) is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
X * (Replace ((0* m),i,f)) is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
len (Replace ((0* m),i,(X * f))) is ordinal natural V11() real ext-real non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of NAT
len (Replace ((0* m),i,f)) is ordinal natural V11() real ext-real non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of NAT
len (X * (Replace ((0* m),i,f))) is ordinal natural V11() real ext-real non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of NAT
y0 is ordinal natural V11() real ext-real non negative set
(Replace ((0* m),i,(X * f))) . y0 is V11() real ext-real Element of REAL
(X * (Replace ((0* m),i,f))) . y0 is V11() real ext-real Element of REAL
r is ordinal natural V11() real ext-real non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of NAT
(Replace ((0* m),i,(X * f))) . r is V11() real ext-real Element of REAL
(Replace ((0* m),i,f)) . r is V11() real ext-real Element of REAL
X * ((Replace ((0* m),i,f)) . r) is V11() real ext-real Element of REAL
r is ordinal natural V11() real ext-real non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of NAT
(Replace ((0* m),i,f)) . r is V11() real ext-real Element of REAL
(Replace ((0* m),i,(X * f))) . r is V11() real ext-real Element of REAL
X * ((Replace ((0* m),i,f)) . r) is V11() real ext-real Element of REAL
r is ordinal natural V11() real ext-real non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of NAT
m is non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below Element of NAT
0* m is Relation-like NAT -defined REAL -valued Function-like finite m -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued 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
{ b₁ where b₁ is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like Element of REAL * : len b₁ = m } is set
m |-> 0 is Relation-like empty-yielding NAT -defined REAL -valued Function-like finite m -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of m -tuples_on REAL
Seg m is non empty finite m -element complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end right_end bounded_below bounded_above real-bounded Element of K6(NAT)
{ b₁ where b₁ is ordinal natural V11() real ext-real non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of NAT : ( 1 <= b₁ & b₁ <= m ) } is set
K213((Seg m),0) is non empty Relation-like Seg m -defined INT -valued RAT -valued {0} -valued Function-like total quasi_total finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued natural-valued Element of K6(K7((Seg m),{0}))
{0} is non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below set
K7((Seg m),{0}) is INT -valued RAT -valued complex-valued ext-real-valued real-valued natural-valued set
K6(K7((Seg m),{0})) is set
f is V11() real ext-real Element of REAL
X is ordinal natural V11() real ext-real non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of NAT
Replace ((0* m),X,f) is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
len (0* m) is ordinal natural V11() real ext-real non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of NAT
proj1 (0* m) is complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below set
(0* m) . X is V11() real ext-real Element of REAL
m is non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below 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
{ b₁ where b₁ is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like Element of REAL * : len b₁ = m } is set
0* m is Relation-like NAT -defined REAL -valued Function-like finite m -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL m
m |-> 0 is Relation-like empty-yielding NAT -defined REAL -valued Function-like finite m -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of m -tuples_on REAL
Seg m is non empty finite m -element complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end right_end bounded_below bounded_above real-bounded Element of K6(NAT)
{ b₁ where b₁ is ordinal natural V11() real ext-real non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of NAT : ( 1 <= b₁ & b₁ <= m ) } is set
K213((Seg m),0) is non empty Relation-like Seg m -defined INT -valued RAT -valued {0} -valued Function-like total quasi_total finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued natural-valued Element of K6(K7((Seg m),{0}))
{0} is non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below set
K7((Seg m),{0}) is INT -valued RAT -valued complex-valued ext-real-valued real-valued natural-valued set
K6(K7((Seg m),{0})) is set
X is V11() real ext-real Element of REAL
f is V11() real ext-real Element of REAL
f - X is V11() real ext-real Element of REAL
X - f is V11() real ext-real Element of REAL
i is Relation-like NAT -defined REAL -valued Function-like finite m -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL m
y0 is ordinal natural V11() real ext-real non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of NAT
proj (y0,m) is non empty Relation-like REAL m -defined REAL -valued Function-like total quasi_total complex-valued ext-real-valued real-valued Element of K6(K7((REAL m),REAL))
K7((REAL m),REAL) is complex-valued ext-real-valued real-valued set
K6(K7((REAL m),REAL)) is set
(proj (y0,m)) . i is V11() real ext-real Element of REAL
Replace (i,y0,f) is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
(Replace (i,y0,f)) - i is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of K6(K7(NAT,REAL))
- i is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
K38(1) * i is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
(Replace (i,y0,f)) + (- i) is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
Replace ((0* m),y0,(f - X)) is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
i - (Replace (i,y0,f)) is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of K6(K7(NAT,REAL))
- (Replace (i,y0,f)) is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
K38(1) * (Replace (i,y0,f)) is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
i + (- (Replace (i,y0,f))) is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
Replace ((0* m),y0,(X - f)) is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
len (Replace ((0* m),y0,(f - X))) is ordinal natural V11() real ext-real non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of NAT
Replace ((0* m),y0,f) is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
len (Replace ((0* m),y0,f)) is ordinal natural V11() real ext-real non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of NAT
- X is V11() real ext-real Element of REAL
Replace ((0* m),y0,(- X)) is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
len (Replace ((0* m),y0,(- X))) is ordinal natural V11() real ext-real non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of NAT
dom (Replace (i,y0,f)) is complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of K6(NAT)
dom i is complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of K6(NAT)
- i is Relation-like NAT -defined REAL -valued Function-like finite m -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL m
dom (- i) is complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of K6(NAT)
- (Replace (i,y0,f)) is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of K6(K7(NAT,REAL))
dom (- (Replace (i,y0,f))) is complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of K6(NAT)
dom ((Replace (i,y0,f)) - i) is complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of K6(NAT)
(dom (Replace (i,y0,f))) /\ (dom (- i)) is complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of K6(NAT)
len (0* m) is ordinal natural V11() real ext-real non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of NAT
len ((Replace (i,y0,f)) - i) is ordinal natural V11() real ext-real non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of NAT
r is ordinal natural V11() real ext-real non negative set
(Replace ((0* m),y0,(f - X))) . r is V11() real ext-real Element of REAL
((Replace (i,y0,f)) - i) . r is V11() real ext-real Element of REAL
s is ordinal natural V11() real ext-real non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of NAT
(- i) . s is V11() real ext-real Element of REAL
- 1 is non empty V11() real ext-real non positive negative V163() V198() Element of INT
i . s is V11() real ext-real Element of REAL
(- 1) * (i . s) is V11() real ext-real Element of REAL
((Replace (i,y0,f)) - i) . s is V11() real ext-real Element of REAL
(Replace (i,y0,f)) . s is V11() real ext-real Element of REAL
- (i . s) is V11() real ext-real Element of REAL
((Replace (i,y0,f)) . s) + (- (i . s)) is V11() real ext-real Element of REAL
((Replace (i,y0,f)) . s) - (i . s) is V11() real ext-real Element of REAL
len i is ordinal natural V11() real ext-real non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of NAT
(Replace (i,y0,f)) . y0 is V11() real ext-real Element of REAL
(Replace ((0* m),y0,f)) + (Replace ((0* m),y0,(- X))) is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of K6(K7(NAT,REAL))
dom ((Replace ((0* m),y0,f)) + (Replace ((0* m),y0,(- X)))) is complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of K6(NAT)
dom (Replace ((0* m),y0,f)) is complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of K6(NAT)
dom (Replace ((0* m),y0,(- X))) is complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of K6(NAT)
(dom (Replace ((0* m),y0,f))) /\ (dom (Replace ((0* m),y0,(- X)))) is complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of K6(NAT)
((Replace ((0* m),y0,f)) + (Replace ((0* m),y0,(- X)))) . s is V11() real ext-real Element of REAL
(Replace ((0* m),y0,f)) . s is V11() real ext-real Element of REAL
(Replace ((0* m),y0,(- X))) . s is V11() real ext-real Element of REAL
((Replace ((0* m),y0,f)) . s) + ((Replace ((0* m),y0,(- X))) . s) is V11() real ext-real Element of REAL
(Replace ((0* m),y0,(f - X))) . s is V11() real ext-real Element of REAL
f + (- X) is V11() real ext-real Element of REAL
Replace ((0* m),y0,(f + (- X))) is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
(Replace ((0* m),y0,(f + (- X)))) . s is V11() real ext-real Element of REAL
f + ((Replace ((0* m),y0,(- X))) . s) is V11() real ext-real Element of REAL
f - ((proj (y0,m)) . i) is V11() real ext-real Element of REAL
(Replace ((0* m),y0,(f - X))) . s is V11() real ext-real Element of REAL
(i . s) - (i . s) is V11() real ext-real Element of REAL
- 1 is non empty V11() real ext-real non positive negative V163() V198() Element of INT
- (Replace ((0* m),y0,(f - X))) is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of K6(K7(NAT,REAL))
K38(1) * (Replace ((0* m),y0,(f - X))) is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
r is V11() real ext-real Element of REAL
r * (f - X) is V11() real ext-real Element of REAL
Replace ((0* m),y0,(r * (f - X))) is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
m is non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below 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
{ b₁ where b₁ is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like Element of REAL * : len b₁ = m } is set
0* m is Relation-like NAT -defined REAL -valued Function-like finite m -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL m
m |-> 0 is Relation-like empty-yielding NAT -defined REAL -valued Function-like finite m -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of m -tuples_on REAL
Seg m is non empty finite m -element complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end right_end bounded_below bounded_above real-bounded Element of K6(NAT)
{ b₁ where b₁ is ordinal natural V11() real ext-real non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of NAT : ( 1 <= b₁ & b₁ <= m ) } is set
K213((Seg m),0) is non empty Relation-like Seg m -defined INT -valued RAT -valued {0} -valued Function-like total quasi_total finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued natural-valued Element of K6(K7((Seg m),{0}))
{0} is non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below set
K7((Seg m),{0}) is INT -valued RAT -valued complex-valued ext-real-valued real-valued natural-valued set
K6(K7((Seg m),{0})) is set
f is V11() real ext-real Element of REAL
X is V11() real ext-real Element of REAL
f + X is V11() real ext-real Element of REAL
i is ordinal natural V11() real ext-real non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of NAT
reproj (i,(0* m)) is non empty Relation-like REAL -defined REAL m -valued Function-like total quasi_total complex-functions-valued ext-real-functions-valued real-functions-valued Element of K6(K7(REAL,(REAL m)))
K7(REAL,(REAL m)) is set
K6(K7(REAL,(REAL m))) is set
(reproj (i,(0* m))) . (f + X) is Relation-like NAT -defined Function-like finite m -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL m
(reproj (i,(0* m))) . f is Relation-like NAT -defined Function-like finite m -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL m
(reproj (i,(0* m))) . X is Relation-like NAT -defined Function-like finite m -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL m
((reproj (i,(0* m))) . f) + ((reproj (i,(0* m))) . X) is Relation-like NAT -defined REAL -valued Function-like finite m -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL m
Replace ((0* m),i,f) is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
Replace ((0* m),i,X) is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
Replace ((0* m),i,(f + X)) is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
m is non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below 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() strict RealNormSpace-like V185() NORMSTR
the carrier of (REAL-NS m) is non empty V2() set
0. (REAL-NS m) is V52( REAL-NS m) Element of the carrier of (REAL-NS m)
f is Element of the carrier of (REAL-NS 1)
X is Element of the carrier of (REAL-NS 1)
f + X is Element of the carrier of (REAL-NS 1)
i is ordinal natural V11() real ext-real non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of NAT
reproj (i,(0. (REAL-NS m))) is non empty Relation-like the carrier of (REAL-NS 1) -defined the carrier of (REAL-NS m) -valued Function-like total quasi_total Element of K6(K7( the carrier of (REAL-NS 1), the carrier of (REAL-NS m)))
K7( the carrier of (REAL-NS 1), the carrier of (REAL-NS m)) is set
K6(K7( the carrier of (REAL-NS 1), the carrier of (REAL-NS m))) is set
(reproj (i,(0. (REAL-NS m)))) . (f + X) is Element of the carrier of (REAL-NS m)
(reproj (i,(0. (REAL-NS m)))) . f is Element of the carrier of (REAL-NS m)
(reproj (i,(0. (REAL-NS m)))) . X is Element of the carrier of (REAL-NS m)
((reproj (i,(0. (REAL-NS m)))) . f) + ((reproj (i,(0. (REAL-NS m)))) . X) is Element of the carrier of (REAL-NS 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
{ b₁ where b₁ is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like Element of REAL * : len b₁ = m } is set
y0 is V11() real ext-real Element of REAL
<*y0*> is non empty V2() Relation-like NAT -defined REAL -valued Function-like one-to-one constant finite 1 -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued increasing decreasing non-decreasing non-increasing FinSequence of REAL
[1,y0] is V27() set
{1,y0} is non empty complex-membered ext-real-membered real-membered set
{{1,y0},{1}} is non empty set
{[1,y0]} is non empty Function-like set
r is Relation-like NAT -defined REAL -valued Function-like finite m -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL m
reproj (i,r) is non empty Relation-like REAL -defined REAL m -valued Function-like total quasi_total complex-functions-valued ext-real-functions-valued real-functions-valued Element of K6(K7(REAL,(REAL m)))
K7(REAL,(REAL m)) is set
K6(K7(REAL,(REAL m))) is set
(reproj (i,r)) . y0 is Relation-like NAT -defined Function-like finite m -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL m
s is V11() real ext-real Element of REAL
<*s*> is non empty V2() Relation-like NAT -defined REAL -valued Function-like one-to-one constant finite 1 -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued increasing decreasing non-decreasing non-increasing FinSequence of REAL
[1,s] is V27() set
{1,s} is non empty complex-membered ext-real-membered real-membered set
{{1,s},{1}} is non empty set
{[1,s]} is non empty Function-like set
y1 is Relation-like NAT -defined REAL -valued Function-like finite m -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL m
reproj (i,y1) is non empty Relation-like REAL -defined REAL m -valued Function-like total quasi_total complex-functions-valued ext-real-functions-valued real-functions-valued Element of K6(K7(REAL,(REAL m)))
(reproj (i,y1)) . s is Relation-like NAT -defined Function-like finite m -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL m
PP is V11() real ext-real Element of REAL
<*PP*> is non empty V2() Relation-like NAT -defined REAL -valued Function-like one-to-one constant finite 1 -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued increasing decreasing non-decreasing non-increasing FinSequence of REAL
[1,PP] is V27() set
{1,PP} is non empty complex-membered ext-real-membered real-membered set
{{1,PP},{1}} is non empty set
{[1,PP]} is non empty Function-like set
DD is Relation-like NAT -defined REAL -valued Function-like finite m -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL m
reproj (i,DD) is non empty Relation-like REAL -defined REAL m -valued Function-like total quasi_total complex-functions-valued ext-real-functions-valued real-functions-valued Element of K6(K7(REAL,(REAL m)))
(reproj (i,DD)) . PP is Relation-like NAT -defined Function-like finite m -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL m
0* m is Relation-like NAT -defined REAL -valued Function-like finite m -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL m
m |-> 0 is Relation-like empty-yielding NAT -defined REAL -valued Function-like finite m -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of m -tuples_on REAL
Seg m is non empty finite m -element complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end right_end bounded_below bounded_above real-bounded Element of K6(NAT)
{ b₁ where b₁ is ordinal natural V11() real ext-real non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of NAT : ( 1 <= b₁ & b₁ <= m ) } is set
K213((Seg m),0) is non empty Relation-like Seg m -defined INT -valued RAT -valued {0} -valued Function-like total quasi_total finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued natural-valued Element of K6(K7((Seg m),{0}))
{0} is non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below set
K7((Seg m),{0}) is INT -valued RAT -valued complex-valued ext-real-valued real-valued natural-valued set
K6(K7((Seg m),{0})) is set
a is Relation-like NAT -defined REAL -valued Function-like finite 1 -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL 1
t is Relation-like NAT -defined REAL -valued Function-like finite 1 -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL 1
a + t is Relation-like NAT -defined REAL -valued Function-like finite 1 -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL 1
y0 + s is V11() real ext-real Element of REAL
<*(y0 + s)*> is non empty V2() Relation-like NAT -defined REAL -valued Function-like one-to-one constant finite 1 -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued increasing decreasing non-decreasing non-increasing FinSequence of REAL
[1,(y0 + s)] is V27() set
{1,(y0 + s)} is non empty complex-membered ext-real-membered real-membered set
{{1,(y0 + s)},{1}} is non empty set
{[1,(y0 + s)]} is non empty Function-like set
reproj (i,(0* m)) is non empty Relation-like REAL -defined REAL m -valued Function-like total quasi_total complex-functions-valued ext-real-functions-valued real-functions-valued Element of K6(K7(REAL,(REAL m)))
(reproj (i,(0* m))) . y0 is Relation-like NAT -defined Function-like finite m -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL m
(reproj (i,(0* m))) . s is Relation-like NAT -defined Function-like finite m -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL m
((reproj (i,(0* m))) . y0) + ((reproj (i,(0* m))) . s) is Relation-like NAT -defined REAL -valued Function-like finite m -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL m
(reproj (i,(0* m))) . (y0 + s) is Relation-like NAT -defined Function-like finite m -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL m
m is non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below 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
{ b₁ where b₁ is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like Element of REAL * : len b₁ = m } is set
0* m is Relation-like NAT -defined REAL -valued Function-like finite m -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL m
m |-> 0 is Relation-like empty-yielding NAT -defined REAL -valued Function-like finite m -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of m -tuples_on REAL
Seg m is non empty finite m -element complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end right_end bounded_below bounded_above real-bounded Element of K6(NAT)
{ b₁ where b₁ is ordinal natural V11() real ext-real non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of NAT : ( 1 <= b₁ & b₁ <= m ) } is set
K213((Seg m),0) is non empty Relation-like Seg m -defined INT -valued RAT -valued {0} -valued Function-like total quasi_total finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued natural-valued Element of K6(K7((Seg m),{0}))
{0} is non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below set
K7((Seg m),{0}) is INT -valued RAT -valued complex-valued ext-real-valued real-valued natural-valued set
K6(K7((Seg m),{0})) is set
X is V11() real ext-real Element of REAL
f is V11() real ext-real Element of REAL
X * f is V11() real ext-real Element of REAL
i is ordinal natural V11() real ext-real non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of NAT
reproj (i,(0* m)) is non empty Relation-like REAL -defined REAL m -valued Function-like total quasi_total complex-functions-valued ext-real-functions-valued real-functions-valued Element of K6(K7(REAL,(REAL m)))
K7(REAL,(REAL m)) is set
K6(K7(REAL,(REAL m))) is set
(reproj (i,(0* m))) . (X * f) is Relation-like NAT -defined Function-like finite m -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL m
(reproj (i,(0* m))) . f is Relation-like NAT -defined Function-like finite m -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL m
X * ((reproj (i,(0* m))) . f) is Relation-like NAT -defined REAL -valued Function-like finite m -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL m
Replace ((0* m),i,(X * f)) is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
Replace ((0* m),i,f) is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
X * (Replace ((0* m),i,f)) is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
m is non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below 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() strict RealNormSpace-like V185() NORMSTR
the carrier of (REAL-NS m) is non empty V2() set
0. (REAL-NS m) is V52( REAL-NS m) Element of the carrier of (REAL-NS m)
f is Element of the carrier of (REAL-NS 1)
X is V11() real ext-real Element of REAL
X * f is Element of the carrier of (REAL-NS 1)
i is ordinal natural V11() real ext-real non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of NAT
reproj (i,(0. (REAL-NS m))) is non empty Relation-like the carrier of (REAL-NS 1) -defined the carrier of (REAL-NS m) -valued Function-like total quasi_total Element of K6(K7( the carrier of (REAL-NS 1), the carrier of (REAL-NS m)))
K7( the carrier of (REAL-NS 1), the carrier of (REAL-NS m)) is set
K6(K7( the carrier of (REAL-NS 1), the carrier of (REAL-NS m))) is set
(reproj (i,(0. (REAL-NS m)))) . (X * f) is Element of the carrier of (REAL-NS m)
(reproj (i,(0. (REAL-NS m)))) . f is Element of the carrier of (REAL-NS m)
X * ((reproj (i,(0. (REAL-NS m)))) . f) is Element of the carrier of (REAL-NS 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
{ b₁ where b₁ is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like Element of REAL * : len b₁ = m } is set
y0 is V11() real ext-real Element of REAL
<*y0*> is non empty V2() Relation-like NAT -defined REAL -valued Function-like one-to-one constant finite 1 -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued increasing decreasing non-decreasing non-increasing FinSequence of REAL
[1,y0] is V27() set
{1,y0} is non empty complex-membered ext-real-membered real-membered set
{{1,y0},{1}} is non empty set
{[1,y0]} is non empty Function-like set
r is Relation-like NAT -defined REAL -valued Function-like finite m -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL m
reproj (i,r) is non empty Relation-like REAL -defined REAL m -valued Function-like total quasi_total complex-functions-valued ext-real-functions-valued real-functions-valued Element of K6(K7(REAL,(REAL m)))
K7(REAL,(REAL m)) is set
K6(K7(REAL,(REAL m))) is set
(reproj (i,r)) . y0 is Relation-like NAT -defined Function-like finite m -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL m
s is V11() real ext-real Element of REAL
<*s*> is non empty V2() Relation-like NAT -defined REAL -valued Function-like one-to-one constant finite 1 -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued increasing decreasing non-decreasing non-increasing FinSequence of REAL
[1,s] is V27() set
{1,s} is non empty complex-membered ext-real-membered real-membered set
{{1,s},{1}} is non empty set
{[1,s]} is non empty Function-like set
y1 is Relation-like NAT -defined REAL -valued Function-like finite m -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL m
reproj (i,y1) is non empty Relation-like REAL -defined REAL m -valued Function-like total quasi_total complex-functions-valued ext-real-functions-valued real-functions-valued Element of K6(K7(REAL,(REAL m)))
(reproj (i,y1)) . s is Relation-like NAT -defined Function-like finite m -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL m
0* m is Relation-like NAT -defined REAL -valued Function-like finite m -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL m
m |-> 0 is Relation-like empty-yielding NAT -defined REAL -valued Function-like finite m -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of m -tuples_on REAL
Seg m is non empty finite m -element complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end right_end bounded_below bounded_above real-bounded Element of K6(NAT)
{ b₁ where b₁ is ordinal natural V11() real ext-real non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of NAT : ( 1 <= b₁ & b₁ <= m ) } is set
K213((Seg m),0) is non empty Relation-like Seg m -defined INT -valued RAT -valued {0} -valued Function-like total quasi_total finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued natural-valued Element of K6(K7((Seg m),{0}))
{0} is non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below set
K7((Seg m),{0}) is INT -valued RAT -valued complex-valued ext-real-valued real-valued natural-valued set
K6(K7((Seg m),{0})) is set
PP is Relation-like NAT -defined REAL -valued Function-like finite 1 -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL 1
X * PP is Relation-like NAT -defined REAL -valued Function-like finite 1 -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL 1
X * y0 is V11() real ext-real Element of REAL
<*(X * y0)*> is non empty V2() Relation-like NAT -defined REAL -valued Function-like one-to-one constant finite 1 -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued increasing decreasing non-decreasing non-increasing FinSequence of REAL
[1,(X * y0)] is V27() set
{1,(X * y0)} is non empty complex-membered ext-real-membered real-membered set
{{1,(X * y0)},{1}} is non empty set
{[1,(X * y0)]} is non empty Function-like set
reproj (i,(0* m)) is non empty Relation-like REAL -defined REAL m -valued Function-like total quasi_total complex-functions-valued ext-real-functions-valued real-functions-valued Element of K6(K7(REAL,(REAL m)))
(reproj (i,(0* m))) . y0 is Relation-like NAT -defined Function-like finite m -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL m
X * ((reproj (i,(0* m))) . y0) is Relation-like NAT -defined REAL -valued Function-like finite m -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL m
(reproj (i,(0* m))) . (X * y0) is Relation-like NAT -defined Function-like finite m -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL m
m is non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below 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
{ b₁ where b₁ is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like Element of REAL * : len b₁ = m } is set
0* m is Relation-like NAT -defined REAL -valued Function-like finite m -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL m
m |-> 0 is Relation-like empty-yielding NAT -defined REAL -valued Function-like finite m -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of m -tuples_on REAL
Seg m is non empty finite m -element complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end right_end bounded_below bounded_above real-bounded Element of K6(NAT)
{ b₁ where b₁ is ordinal natural V11() real ext-real non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of NAT : ( 1 <= b₁ & b₁ <= m ) } is set
K213((Seg m),0) is non empty Relation-like Seg m -defined INT -valued RAT -valued {0} -valued Function-like total quasi_total finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued natural-valued Element of K6(K7((Seg m),{0}))
{0} is non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below set
K7((Seg m),{0}) is INT -valued RAT -valued complex-valued ext-real-valued real-valued natural-valued set
K6(K7((Seg m),{0})) is set
f is V11() real ext-real Element of REAL
X is ordinal natural V11() real ext-real non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of NAT
reproj (X,(0* m)) is non empty Relation-like REAL -defined REAL m -valued Function-like total quasi_total complex-functions-valued ext-real-functions-valued real-functions-valued Element of K6(K7(REAL,(REAL m)))
K7(REAL,(REAL m)) is set
K6(K7(REAL,(REAL m))) is set
(reproj (X,(0* m))) . f is Relation-like NAT -defined Function-like finite m -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL m
Replace ((0* m),X,f) is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
m is non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below 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() strict RealNormSpace-like V185() NORMSTR
the carrier of (REAL-NS m) is non empty V2() set
0. (REAL-NS m) is V52( REAL-NS m) Element of the carrier of (REAL-NS m)
f is Element of the carrier of (REAL-NS 1)
X is ordinal natural V11() real ext-real non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of NAT
reproj (X,(0. (REAL-NS m))) is non empty Relation-like the carrier of (REAL-NS 1) -defined the carrier of (REAL-NS m) -valued Function-like total quasi_total Element of K6(K7( the carrier of (REAL-NS 1), the carrier of (REAL-NS m)))
K7( the carrier of (REAL-NS 1), the carrier of (REAL-NS m)) is set
K6(K7( the carrier of (REAL-NS 1), the carrier of (REAL-NS m))) is set
(reproj (X,(0. (REAL-NS m)))) . f is Element of the carrier of (REAL-NS 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
{ b₁ where b₁ is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like Element of REAL * : len b₁ = m } is set
i is V11() real ext-real Element of REAL
<*i*> is non empty V2() Relation-like NAT -defined REAL -valued Function-like one-to-one constant finite 1 -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued increasing decreasing non-decreasing non-increasing FinSequence of REAL
[1,i] is V27() set
{1,i} is non empty complex-membered ext-real-membered real-membered set
{{1,i},{1}} is non empty set
{[1,i]} is non empty Function-like set
y0 is Relation-like NAT -defined REAL -valued Function-like finite m -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL m
reproj (X,y0) is non empty Relation-like REAL -defined REAL m -valued Function-like total quasi_total complex-functions-valued ext-real-functions-valued real-functions-valued Element of K6(K7(REAL,(REAL m)))
K7(REAL,(REAL m)) is set
K6(K7(REAL,(REAL m))) is set
(reproj (X,y0)) . i is Relation-like NAT -defined Function-like finite m -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL m
0* m is Relation-like NAT -defined REAL -valued Function-like finite m -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL m
m |-> 0 is Relation-like empty-yielding NAT -defined REAL -valued Function-like finite m -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of m -tuples_on REAL
Seg m is non empty finite m -element complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end right_end bounded_below bounded_above real-bounded Element of K6(NAT)
{ b₁ where b₁ is ordinal natural V11() real ext-real non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of NAT : ( 1 <= b₁ & b₁ <= m ) } is set
K213((Seg m),0) is non empty Relation-like Seg m -defined INT -valued RAT -valued {0} -valued Function-like total quasi_total finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued natural-valued Element of K6(K7((Seg m),{0}))
{0} is non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below set
K7((Seg m),{0}) is INT -valued RAT -valued complex-valued ext-real-valued real-valued natural-valued set
K6(K7((Seg m),{0})) is set
0* 1 is Relation-like NAT -defined REAL -valued Function-like finite 1 -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL 1
1 |-> 0 is Relation-like empty-yielding NAT -defined REAL -valued Function-like finite 1 -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of 1 -tuples_on REAL
Seg 1 is non empty finite 1 -element complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end right_end bounded_below bounded_above real-bounded Element of K6(NAT)
{ b₁ where b₁ is ordinal natural V11() real ext-real non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of NAT : ( 1 <= b₁ & b₁ <= 1 ) } is set
K213((Seg 1),0) is non empty Relation-like Seg 1 -defined INT -valued RAT -valued {0} -valued Function-like total quasi_total finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued natural-valued Element of K6(K7((Seg 1),{0}))
K7((Seg 1),{0}) is INT -valued RAT -valued complex-valued ext-real-valued real-valued natural-valued set
K6(K7((Seg 1),{0})) is set
m is non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below 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
{ b₁ where b₁ is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like Element of REAL * : len b₁ = m } is set
0* m is Relation-like NAT -defined REAL -valued Function-like finite m -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL m
m |-> 0 is Relation-like empty-yielding NAT -defined REAL -valued Function-like finite m -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of m -tuples_on REAL
Seg m is non empty finite m -element complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end right_end bounded_below bounded_above real-bounded Element of K6(NAT)
{ b₁ where b₁ is ordinal natural V11() real ext-real non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of NAT : ( 1 <= b₁ & b₁ <= m ) } is set
K213((Seg m),0) is non empty Relation-like Seg m -defined INT -valued RAT -valued {0} -valued Function-like total quasi_total finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued natural-valued Element of K6(K7((Seg m),{0}))
{0} is non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below set
K7((Seg m),{0}) is INT -valued RAT -valued complex-valued ext-real-valued real-valued natural-valued set
K6(K7((Seg m),{0})) is set
X is V11() real ext-real Element of REAL
f is V11() real ext-real Element of REAL
f - X is V11() real ext-real Element of REAL
X - f is V11() real ext-real Element of REAL
i is Relation-like NAT -defined REAL -valued Function-like finite m -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL m
y0 is ordinal natural V11() real ext-real non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of NAT
proj (y0,m) is non empty Relation-like REAL m -defined REAL -valued Function-like total quasi_total complex-valued ext-real-valued real-valued Element of K6(K7((REAL m),REAL))
K7((REAL m),REAL) is complex-valued ext-real-valued real-valued set
K6(K7((REAL m),REAL)) is set
(proj (y0,m)) . i is V11() real ext-real Element of REAL
reproj (y0,i) is non empty Relation-like REAL -defined REAL m -valued Function-like total quasi_total complex-functions-valued ext-real-functions-valued real-functions-valued Element of K6(K7(REAL,(REAL m)))
K7(REAL,(REAL m)) is set
K6(K7(REAL,(REAL m))) is set
(reproj (y0,i)) . f is Relation-like NAT -defined Function-like finite m -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL m
((reproj (y0,i)) . f) - i is Relation-like NAT -defined REAL -valued Function-like finite m -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL m
- i is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
K38(1) * i is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
((reproj (y0,i)) . f) + (- i) is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
reproj (y0,(0* m)) is non empty Relation-like REAL -defined REAL m -valued Function-like total quasi_total complex-functions-valued ext-real-functions-valued real-functions-valued Element of K6(K7(REAL,(REAL m)))
(reproj (y0,(0* m))) . (f - X) is Relation-like NAT -defined Function-like finite m -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL m
i - ((reproj (y0,i)) . f) is Relation-like NAT -defined REAL -valued Function-like finite m -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL m
- ((reproj (y0,i)) . f) is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
K38(1) * ((reproj (y0,i)) . f) is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
i + (- ((reproj (y0,i)) . f)) is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
(reproj (y0,(0* m))) . (X - f) is Relation-like NAT -defined Function-like finite m -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL m
Replace (i,y0,f) is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
(Replace (i,y0,f)) - i is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of K6(K7(NAT,REAL))
(Replace (i,y0,f)) + (- i) is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
Replace ((0* m),y0,(f - X)) is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
i - (Replace (i,y0,f)) is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of K6(K7(NAT,REAL))
- (Replace (i,y0,f)) is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
K38(1) * (Replace (i,y0,f)) is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
i + (- (Replace (i,y0,f))) is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
Replace ((0* m),y0,(X - f)) is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
m is non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below 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() strict RealNormSpace-like V185() NORMSTR
the carrier of (REAL-NS m) is non empty V2() set
0. (REAL-NS m) is V52( REAL-NS m) Element of the carrier of (REAL-NS m)
X is Element of the carrier of (REAL-NS 1)
f is Element of the carrier of (REAL-NS 1)
f - X is Element of the carrier of (REAL-NS 1)
X - f is Element of the carrier of (REAL-NS 1)
i is ordinal natural V11() real ext-real non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of NAT
Proj (i,m) is non empty Relation-like the carrier of (REAL-NS m) -defined the carrier of (REAL-NS 1) -valued Function-like total quasi_total 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
reproj (i,(0. (REAL-NS m))) is non empty Relation-like the carrier of (REAL-NS 1) -defined the carrier of (REAL-NS m) -valued Function-like total quasi_total Element of K6(K7( the carrier of (REAL-NS 1), the carrier of (REAL-NS m)))
K7( the carrier of (REAL-NS 1), the carrier of (REAL-NS m)) is set
K6(K7( the carrier of (REAL-NS 1), the carrier of (REAL-NS m))) is set
(reproj (i,(0. (REAL-NS m)))) . (f - X) is Element of the carrier of (REAL-NS m)
(reproj (i,(0. (REAL-NS m)))) . (X - f) is Element of the carrier of (REAL-NS m)
y0 is Element of the carrier of (REAL-NS m)
(Proj (i,m)) . y0 is Element of the carrier of (REAL-NS 1)
reproj (i,y0) is non empty Relation-like the carrier of (REAL-NS 1) -defined the carrier of (REAL-NS m) -valued Function-like total quasi_total Element of K6(K7( the carrier of (REAL-NS 1), the carrier of (REAL-NS m)))
(reproj (i,y0)) . f is Element of the carrier of (REAL-NS m)
((reproj (i,y0)) . f) - y0 is Element of the carrier of (REAL-NS m)
y0 - ((reproj (i,y0)) . f) is Element of the carrier of (REAL-NS 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
{ b₁ where b₁ is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like Element of REAL * : len b₁ = m } is set
r is V11() real ext-real Element of REAL
<*r*> is non empty V2() Relation-like NAT -defined REAL -valued Function-like one-to-one constant finite 1 -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued increasing decreasing non-decreasing non-increasing FinSequence of REAL
[1,r] is V27() set
{1,r} is non empty complex-membered ext-real-membered real-membered set
{{1,r},{1}} is non empty set
{[1,r]} is non empty Function-like set
s is Relation-like NAT -defined REAL -valued Function-like finite m -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL m
reproj (i,s) is non empty Relation-like REAL -defined REAL m -valued Function-like total quasi_total complex-functions-valued ext-real-functions-valued real-functions-valued Element of K6(K7(REAL,(REAL m)))
K7(REAL,(REAL m)) is set
K6(K7(REAL,(REAL m))) is set
(reproj (i,s)) . r is Relation-like NAT -defined Function-like finite m -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL m
(reproj (i,y0)) . X is Element of the carrier of (REAL-NS m)
y1 is V11() real ext-real Element of REAL
<*y1*> is non empty V2() Relation-like NAT -defined REAL -valued Function-like one-to-one constant finite 1 -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued increasing decreasing non-decreasing non-increasing FinSequence of REAL
[1,y1] is V27() set
{1,y1} is non empty complex-membered ext-real-membered real-membered set
{{1,y1},{1}} is non empty set
{[1,y1]} is non empty Function-like set
PP is Relation-like NAT -defined REAL -valued Function-like finite m -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL m
reproj (i,PP) is non empty Relation-like REAL -defined REAL m -valued Function-like total quasi_total complex-functions-valued ext-real-functions-valued real-functions-valued Element of K6(K7(REAL,(REAL m)))
(reproj (i,PP)) . y1 is Relation-like NAT -defined Function-like finite m -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL m
DD is V11() real ext-real Element of REAL
<*DD*> is non empty V2() Relation-like NAT -defined REAL -valued Function-like one-to-one constant finite 1 -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued increasing decreasing non-decreasing non-increasing FinSequence of REAL
[1,DD] is V27() set
{1,DD} is non empty complex-membered ext-real-membered real-membered set
{{1,DD},{1}} is non empty set
{[1,DD]} is non empty Function-like set
a is Relation-like NAT -defined REAL -valued Function-like finite m -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL m
reproj (i,a) is non empty Relation-like REAL -defined REAL m -valued Function-like total quasi_total complex-functions-valued ext-real-functions-valued real-functions-valued Element of K6(K7(REAL,(REAL m)))
(reproj (i,a)) . DD is Relation-like NAT -defined Function-like finite m -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL m
t is V11() real ext-real Element of REAL
<*t*> is non empty V2() Relation-like NAT -defined REAL -valued Function-like one-to-one constant finite 1 -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued increasing decreasing non-decreasing non-increasing FinSequence of REAL
[1,t] is V27() set
{1,t} is non empty complex-membered ext-real-membered real-membered set
{{1,t},{1}} is non empty set
{[1,t]} is non empty Function-like set
tm is Relation-like NAT -defined REAL -valued Function-like finite m -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL m
reproj (i,tm) is non empty Relation-like REAL -defined REAL m -valued Function-like total quasi_total complex-functions-valued ext-real-functions-valued real-functions-valued Element of K6(K7(REAL,(REAL m)))
(reproj (i,tm)) . t is Relation-like NAT -defined Function-like finite m -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL m
0* m is Relation-like NAT -defined REAL -valued Function-like finite m -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL m
m |-> 0 is Relation-like empty-yielding NAT -defined REAL -valued Function-like finite m -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of m -tuples_on REAL
Seg m is non empty finite m -element complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end right_end bounded_below bounded_above real-bounded Element of K6(NAT)
{ b₁ where b₁ is ordinal natural V11() real ext-real non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of NAT : ( 1 <= b₁ & b₁ <= m ) } is set
K213((Seg m),0) is non empty Relation-like Seg m -defined INT -valued RAT -valued {0} -valued Function-like total quasi_total finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued natural-valued Element of K6(K7((Seg m),{0}))
{0} is non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below set
K7((Seg m),{0}) is INT -valued RAT -valued complex-valued ext-real-valued real-valued natural-valued set
K6(K7((Seg m),{0})) is set
Sw0 is Relation-like NAT -defined REAL -valued Function-like finite 1 -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL 1
w1 is Relation-like NAT -defined REAL -valued Function-like finite 1 -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL 1
Sw0 - w1 is Relation-like NAT -defined REAL -valued Function-like finite 1 -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL 1
- w1 is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
K38(1) * w1 is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
Sw0 + (- w1) is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
w1 - Sw0 is Relation-like NAT -defined REAL -valued Function-like finite 1 -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL 1
- Sw0 is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
K38(1) * Sw0 is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
w1 + (- Sw0) is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
r - y1 is V11() real ext-real Element of REAL
<*(r - y1)*> is non empty V2() Relation-like NAT -defined REAL -valued Function-like one-to-one constant finite 1 -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued increasing decreasing non-decreasing non-increasing FinSequence of REAL
[1,(r - y1)] is V27() set
{1,(r - y1)} is non empty complex-membered ext-real-membered real-membered set
{{1,(r - y1)},{1}} is non empty set
{[1,(r - y1)]} is non empty Function-like set
y1 - r is V11() real ext-real Element of REAL
<*(y1 - r)*> is non empty V2() Relation-like NAT -defined REAL -valued Function-like one-to-one constant finite 1 -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued increasing decreasing non-decreasing non-increasing FinSequence of REAL
[1,(y1 - r)] is V27() set
{1,(y1 - r)} is non empty complex-membered ext-real-membered real-membered set
{{1,(y1 - r)},{1}} is non empty set
{[1,(y1 - r)]} is non empty Function-like set
reproj (i,(0* m)) is non empty Relation-like REAL -defined REAL m -valued Function-like total quasi_total complex-functions-valued ext-real-functions-valued real-functions-valued Element of K6(K7(REAL,(REAL m)))
(reproj (i,(0* m))) . (r - y1) is Relation-like NAT -defined Function-like finite m -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL m
(reproj (i,(0* m))) . (y1 - r) is Relation-like NAT -defined Function-like finite m -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL m
proj (i,m) is non empty Relation-like REAL m -defined REAL -valued Function-like total quasi_total complex-valued ext-real-valued real-valued Element of K6(K7((REAL m),REAL))
K7((REAL m),REAL) is complex-valued ext-real-valued real-valued set
K6(K7((REAL m),REAL)) is set
(proj (i,m)) . y0 is V11() real ext-real Element of REAL
<*((proj (i,m)) . y0)*> is non empty V2() Relation-like NAT -defined REAL -valued Function-like one-to-one constant finite 1 -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued increasing decreasing non-decreasing non-increasing FinSequence of REAL
[1,((proj (i,m)) . y0)] is V27() set
{1,((proj (i,m)) . y0)} is non empty complex-membered ext-real-membered real-membered set
{{1,((proj (i,m)) . y0)},{1}} is non empty set
{[1,((proj (i,m)) . y0)]} is non empty Function-like set
(proj (i,m)) . s is V11() real ext-real Element of REAL
((reproj (i,s)) . r) - s is Relation-like NAT -defined REAL -valued Function-like finite m -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL m
- s is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
K38(1) * s is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
((reproj (i,s)) . r) + (- s) is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
s - ((reproj (i,s)) . r) is Relation-like NAT -defined REAL -valued Function-like finite m -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL m
- ((reproj (i,s)) . r) is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
K38(1) * ((reproj (i,s)) . r) is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
s + (- ((reproj (i,s)) . r)) is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
m is non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below 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() strict RealNormSpace-like V185() NORMSTR
the carrier of (REAL-NS m) is non empty V2() set
f is non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below Element of NAT
REAL-NS f is non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V185() NORMSTR
the carrier of (REAL-NS f) is non empty V2() set
K7( the carrier of (REAL-NS f), the carrier of (REAL-NS m)) is set
K6(K7( the carrier of (REAL-NS f), the carrier of (REAL-NS m))) is set
0. (REAL-NS f) is V52( REAL-NS f) Element of the carrier of (REAL-NS f)
X is ordinal natural V11() real ext-real non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of NAT
reproj (X,(0. (REAL-NS f))) is non empty Relation-like the carrier of (REAL-NS 1) -defined the carrier of (REAL-NS f) -valued Function-like total quasi_total Element of K6(K7( the carrier of (REAL-NS 1), the carrier of (REAL-NS f)))
K7( the carrier of (REAL-NS 1), the carrier of (REAL-NS f)) is set
K6(K7( the carrier of (REAL-NS 1), the carrier of (REAL-NS f))) is set
i is Relation-like the carrier of (REAL-NS f) -defined the carrier of (REAL-NS m) -valued Function-like Element of K6(K7( the carrier of (REAL-NS f), the carrier of (REAL-NS m)))
y0 is Element of the carrier of (REAL-NS f)
partdiff (i,y0,X) is Relation-like Function-like Element of the carrier of (R_NormSpace_of_BoundedLinearOperators ((REAL-NS 1),(REAL-NS m)))
R_NormSpace_of_BoundedLinearOperators ((REAL-NS 1),(REAL-NS m)) is non empty V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() RealNormSpace-like V185() NORMSTR
BoundedLinearOperators ((REAL-NS 1),(REAL-NS m)) is non empty Element of K6( the carrier of (R_VectorSpace_of_LinearOperators ((REAL-NS 1),(REAL-NS m))))
R_VectorSpace_of_LinearOperators ((REAL-NS 1),(REAL-NS m)) is non empty V71() strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V214() RLSStruct
LinearOperators ((REAL-NS 1),(REAL-NS m)) is non empty functional Element of K6( the carrier of K318( the carrier of (REAL-NS 1),(REAL-NS m)))
K318( the carrier of (REAL-NS 1),(REAL-NS m)) is non empty V71() strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V214() RLSStruct
Funcs ( the carrier of (REAL-NS 1), the carrier of (REAL-NS m)) is non empty functional FUNCTION_DOMAIN of the carrier of (REAL-NS 1), the carrier of (REAL-NS m)
FuncZero ( the carrier of (REAL-NS 1),(REAL-NS m)) is Relation-like the carrier of (REAL-NS 1) -defined the carrier of (REAL-NS m) -valued Function-like total quasi_total Element of Funcs ( the carrier of (REAL-NS 1), the carrier of (REAL-NS m))
0. (REAL-NS m) is V52( REAL-NS m) Element of the carrier of (REAL-NS m)
K214( the carrier of (REAL-NS m), the carrier of (REAL-NS 1),(0. (REAL-NS m))) is non empty Relation-like the carrier of (REAL-NS 1) -defined the carrier of (REAL-NS m) -valued Function-like total quasi_total Element of K6(K7( the carrier of (REAL-NS 1), the carrier of (REAL-NS m)))
K7( the carrier of (REAL-NS 1), the carrier of (REAL-NS m)) is set
K6(K7( the carrier of (REAL-NS 1), the carrier of (REAL-NS m))) is set
FuncAdd ( the carrier of (REAL-NS 1),(REAL-NS m)) is Relation-like K7((Funcs ( the carrier of (REAL-NS 1), the carrier of (REAL-NS m))),(Funcs ( the carrier of (REAL-NS 1), the carrier of (REAL-NS m)))) -defined Funcs ( the carrier of (REAL-NS 1), the carrier of (REAL-NS m)) -valued Function-like total quasi_total Element of K6(K7(K7((Funcs ( the carrier of (REAL-NS 1), the carrier of (REAL-NS m))),(Funcs ( the carrier of (REAL-NS 1), the carrier of (REAL-NS m)))),(Funcs ( the carrier of (REAL-NS 1), the carrier of (REAL-NS m)))))
K7((Funcs ( the carrier of (REAL-NS 1), the carrier of (REAL-NS m))),(Funcs ( the carrier of (REAL-NS 1), the carrier of (REAL-NS m)))) is set
K7(K7((Funcs ( the carrier of (REAL-NS 1), the carrier of (REAL-NS m))),(Funcs ( the carrier of (REAL-NS 1), the carrier of (REAL-NS m)))),(Funcs ( the carrier of (REAL-NS 1), the carrier of (REAL-NS m)))) is set
K6(K7(K7((Funcs ( the carrier of (REAL-NS 1), the carrier of (REAL-NS m))),(Funcs ( the carrier of (REAL-NS 1), the carrier of (REAL-NS m)))),(Funcs ( the carrier of (REAL-NS 1), the carrier of (REAL-NS m))))) is set
FuncExtMult ( the carrier of (REAL-NS 1),(REAL-NS m)) is Relation-like K7(REAL,(Funcs ( the carrier of (REAL-NS 1), the carrier of (REAL-NS m)))) -defined Funcs ( the carrier of (REAL-NS 1), the carrier of (REAL-NS m)) -valued Function-like total quasi_total Element of K6(K7(K7(REAL,(Funcs ( the carrier of (REAL-NS 1), the carrier of (REAL-NS m)))),(Funcs ( the carrier of (REAL-NS 1), the carrier of (REAL-NS m)))))
K7(REAL,(Funcs ( the carrier of (REAL-NS 1), the carrier of (REAL-NS m)))) is set
K7(K7(REAL,(Funcs ( the carrier of (REAL-NS 1), the carrier of (REAL-NS m)))),(Funcs ( the carrier of (REAL-NS 1), the carrier of (REAL-NS m)))) is set
K6(K7(K7(REAL,(Funcs ( the carrier of (REAL-NS 1), the carrier of (REAL-NS m)))),(Funcs ( the carrier of (REAL-NS 1), the carrier of (REAL-NS m))))) is set
RLSStruct(# (Funcs ( the carrier of (REAL-NS 1), the carrier of (REAL-NS m))),(FuncZero ( the carrier of (REAL-NS 1),(REAL-NS m))),(FuncAdd ( the carrier of (REAL-NS 1),(REAL-NS m))),(FuncExtMult ( the carrier of (REAL-NS 1),(REAL-NS m))) #) is strict RLSStruct
the carrier of K318( the carrier of (REAL-NS 1),(REAL-NS m)) is non empty set
K6( the carrier of K318( the carrier of (REAL-NS 1),(REAL-NS m))) is set
K199(K318( the carrier of (REAL-NS 1),(REAL-NS m)),(LinearOperators ((REAL-NS 1),(REAL-NS m)))) is Relation-like Function-like Element of LinearOperators ((REAL-NS 1),(REAL-NS m))
K197(K318( the carrier of (REAL-NS 1),(REAL-NS m)),(LinearOperators ((REAL-NS 1),(REAL-NS m)))) is Relation-like K7((LinearOperators ((REAL-NS 1),(REAL-NS m))),(LinearOperators ((REAL-NS 1),(REAL-NS m)))) -defined LinearOperators ((REAL-NS 1),(REAL-NS m)) -valued Function-like total quasi_total Element of K6(K7(K7((LinearOperators ((REAL-NS 1),(REAL-NS m))),(LinearOperators ((REAL-NS 1),(REAL-NS m)))),(LinearOperators ((REAL-NS 1),(REAL-NS m)))))
K7((LinearOperators ((REAL-NS 1),(REAL-NS m))),(LinearOperators ((REAL-NS 1),(REAL-NS m)))) is set
K7(K7((LinearOperators ((REAL-NS 1),(REAL-NS m))),(LinearOperators ((REAL-NS 1),(REAL-NS m)))),(LinearOperators ((REAL-NS 1),(REAL-NS m)))) is set
K6(K7(K7((LinearOperators ((REAL-NS 1),(REAL-NS m))),(LinearOperators ((REAL-NS 1),(REAL-NS m)))),(LinearOperators ((REAL-NS 1),(REAL-NS m))))) is set
K198(K318( the carrier of (REAL-NS 1),(REAL-NS m)),(LinearOperators ((REAL-NS 1),(REAL-NS m)))) is Relation-like K7(REAL,(LinearOperators ((REAL-NS 1),(REAL-NS m)))) -defined LinearOperators ((REAL-NS 1),(REAL-NS m)) -valued Function-like total quasi_total Element of K6(K7(K7(REAL,(LinearOperators ((REAL-NS 1),(REAL-NS m)))),(LinearOperators ((REAL-NS 1),(REAL-NS m)))))
K7(REAL,(LinearOperators ((REAL-NS 1),(REAL-NS m)))) is set
K7(K7(REAL,(LinearOperators ((REAL-NS 1),(REAL-NS m)))),(LinearOperators ((REAL-NS 1),(REAL-NS m)))) is set
K6(K7(K7(REAL,(LinearOperators ((REAL-NS 1),(REAL-NS m)))),(LinearOperators ((REAL-NS 1),(REAL-NS m))))) is set
RLSStruct(# (LinearOperators ((REAL-NS 1),(REAL-NS m))),K199(K318( the carrier of (REAL-NS 1),(REAL-NS m)),(LinearOperators ((REAL-NS 1),(REAL-NS m)))),K197(K318( the carrier of (REAL-NS 1),(REAL-NS m)),(LinearOperators ((REAL-NS 1),(REAL-NS m)))),K198(K318( the carrier of (REAL-NS 1),(REAL-NS m)),(LinearOperators ((REAL-NS 1),(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 1),(REAL-NS m))) is non empty set
K6( the carrier of (R_VectorSpace_of_LinearOperators ((REAL-NS 1),(REAL-NS m)))) is set
K199((R_VectorSpace_of_LinearOperators ((REAL-NS 1),(REAL-NS m))),(BoundedLinearOperators ((REAL-NS 1),(REAL-NS m)))) is Element of BoundedLinearOperators ((REAL-NS 1),(REAL-NS m))
K197((R_VectorSpace_of_LinearOperators ((REAL-NS 1),(REAL-NS m))),(BoundedLinearOperators ((REAL-NS 1),(REAL-NS m)))) is Relation-like K7((BoundedLinearOperators ((REAL-NS 1),(REAL-NS m))),(BoundedLinearOperators ((REAL-NS 1),(REAL-NS m)))) -defined BoundedLinearOperators ((REAL-NS 1),(REAL-NS m)) -valued Function-like total quasi_total Element of K6(K7(K7((BoundedLinearOperators ((REAL-NS 1),(REAL-NS m))),(BoundedLinearOperators ((REAL-NS 1),(REAL-NS m)))),(BoundedLinearOperators ((REAL-NS 1),(REAL-NS m)))))
K7((BoundedLinearOperators ((REAL-NS 1),(REAL-NS m))),(BoundedLinearOperators ((REAL-NS 1),(REAL-NS m)))) is set
K7(K7((BoundedLinearOperators ((REAL-NS 1),(REAL-NS m))),(BoundedLinearOperators ((REAL-NS 1),(REAL-NS m)))),(BoundedLinearOperators ((REAL-NS 1),(REAL-NS m)))) is set
K6(K7(K7((BoundedLinearOperators ((REAL-NS 1),(REAL-NS m))),(BoundedLinearOperators ((REAL-NS 1),(REAL-NS m)))),(BoundedLinearOperators ((REAL-NS 1),(REAL-NS m))))) is set
K198((R_VectorSpace_of_LinearOperators ((REAL-NS 1),(REAL-NS m))),(BoundedLinearOperators ((REAL-NS 1),(REAL-NS m)))) is Relation-like K7(REAL,(BoundedLinearOperators ((REAL-NS 1),(REAL-NS m)))) -defined BoundedLinearOperators ((REAL-NS 1),(REAL-NS m)) -valued Function-like total quasi_total Element of K6(K7(K7(REAL,(BoundedLinearOperators ((REAL-NS 1),(REAL-NS m)))),(BoundedLinearOperators ((REAL-NS 1),(REAL-NS m)))))
K7(REAL,(BoundedLinearOperators ((REAL-NS 1),(REAL-NS m)))) is set
K7(K7(REAL,(BoundedLinearOperators ((REAL-NS 1),(REAL-NS m)))),(BoundedLinearOperators ((REAL-NS 1),(REAL-NS m)))) is set
K6(K7(K7(REAL,(BoundedLinearOperators ((REAL-NS 1),(REAL-NS m)))),(BoundedLinearOperators ((REAL-NS 1),(REAL-NS m))))) is set
BoundedLinearOperatorsNorm ((REAL-NS 1),(REAL-NS m)) is non empty Relation-like BoundedLinearOperators ((REAL-NS 1),(REAL-NS m)) -defined REAL -valued Function-like total quasi_total complex-valued ext-real-valued real-valued Element of K6(K7((BoundedLinearOperators ((REAL-NS 1),(REAL-NS m))),REAL))
K7((BoundedLinearOperators ((REAL-NS 1),(REAL-NS m))),REAL) is complex-valued ext-real-valued real-valued set
K6(K7((BoundedLinearOperators ((REAL-NS 1),(REAL-NS m))),REAL)) is set
NORMSTR(# (BoundedLinearOperators ((REAL-NS 1),(REAL-NS m))),K199((R_VectorSpace_of_LinearOperators ((REAL-NS 1),(REAL-NS m))),(BoundedLinearOperators ((REAL-NS 1),(REAL-NS m)))),K197((R_VectorSpace_of_LinearOperators ((REAL-NS 1),(REAL-NS m))),(BoundedLinearOperators ((REAL-NS 1),(REAL-NS m)))),K198((R_VectorSpace_of_LinearOperators ((REAL-NS 1),(REAL-NS m))),(BoundedLinearOperators ((REAL-NS 1),(REAL-NS m)))),(BoundedLinearOperatorsNorm ((REAL-NS 1),(REAL-NS m))) #) is strict NORMSTR
the carrier of (R_NormSpace_of_BoundedLinearOperators ((REAL-NS 1),(REAL-NS m))) is non empty set
reproj (X,y0) is non empty Relation-like the carrier of (REAL-NS 1) -defined the carrier of (REAL-NS f) -valued Function-like total quasi_total Element of K6(K7( the carrier of (REAL-NS 1), the carrier of (REAL-NS f)))
i * (reproj (X,y0)) is Relation-like the carrier of (REAL-NS 1) -defined the carrier of (REAL-NS m) -valued Function-like Element of K6(K7( the carrier of (REAL-NS 1), the carrier of (REAL-NS m)))
Proj (X,f) is non empty Relation-like the carrier of (REAL-NS f) -defined the carrier of (REAL-NS 1) -valued Function-like total quasi_total Element of K6(K7( the carrier of (REAL-NS f), the carrier of (REAL-NS 1)))
K7( the carrier of (REAL-NS f), the carrier of (REAL-NS 1)) is set
K6(K7( the carrier of (REAL-NS f), the carrier of (REAL-NS 1))) is set
(Proj (X,f)) . y0 is Element of the carrier of (REAL-NS 1)
diff ((i * (reproj (X,y0))),((Proj (X,f)) . y0)) is Relation-like Function-like Element of the carrier of (R_NormSpace_of_BoundedLinearOperators ((REAL-NS 1),(REAL-NS m)))
diff (i,y0) is Relation-like Function-like Element of the carrier of (R_NormSpace_of_BoundedLinearOperators ((REAL-NS f),(REAL-NS m)))
R_NormSpace_of_BoundedLinearOperators ((REAL-NS f),(REAL-NS m)) is non empty V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() RealNormSpace-like V185() NORMSTR
BoundedLinearOperators ((REAL-NS f),(REAL-NS m)) is non empty Element of K6( the carrier of (R_VectorSpace_of_LinearOperators ((REAL-NS f),(REAL-NS m))))
R_VectorSpace_of_LinearOperators ((REAL-NS f),(REAL-NS m)) is non empty V71() strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V214() RLSStruct
LinearOperators ((REAL-NS f),(REAL-NS m)) is non empty functional Element of K6( the carrier of K318( the carrier of (REAL-NS f),(REAL-NS m)))
K318( the carrier of (REAL-NS f),(REAL-NS m)) is non empty V71() strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V214() RLSStruct
Funcs ( the carrier of (REAL-NS f), the carrier of (REAL-NS m)) is non empty functional FUNCTION_DOMAIN of the carrier of (REAL-NS f), the carrier of (REAL-NS m)
FuncZero ( the carrier of (REAL-NS f),(REAL-NS m)) is Relation-like the carrier of (REAL-NS f) -defined the carrier of (REAL-NS m) -valued Function-like total quasi_total Element of Funcs ( the carrier of (REAL-NS f), the carrier of (REAL-NS m))
K214( the carrier of (REAL-NS m), the carrier of (REAL-NS f),(0. (REAL-NS m))) is non empty Relation-like the carrier of (REAL-NS f) -defined the carrier of (REAL-NS m) -valued Function-like total quasi_total Element of K6(K7( the carrier of (REAL-NS f), the carrier of (REAL-NS m)))
FuncAdd ( the carrier of (REAL-NS f),(REAL-NS m)) is Relation-like K7((Funcs ( the carrier of (REAL-NS f), the carrier of (REAL-NS m))),(Funcs ( the carrier of (REAL-NS f), the carrier of (REAL-NS m)))) -defined Funcs ( the carrier of (REAL-NS f), the carrier of (REAL-NS m)) -valued Function-like total quasi_total Element of K6(K7(K7((Funcs ( the carrier of (REAL-NS f), the carrier of (REAL-NS m))),(Funcs ( the carrier of (REAL-NS f), the carrier of (REAL-NS m)))),(Funcs ( the carrier of (REAL-NS f), the carrier of (REAL-NS m)))))
K7((Funcs ( the carrier of (REAL-NS f), the carrier of (REAL-NS m))),(Funcs ( the carrier of (REAL-NS f), the carrier of (REAL-NS m)))) is set
K7(K7((Funcs ( the carrier of (REAL-NS f), the carrier of (REAL-NS m))),(Funcs ( the carrier of (REAL-NS f), the carrier of (REAL-NS m)))),(Funcs ( the carrier of (REAL-NS f), the carrier of (REAL-NS m)))) is set
K6(K7(K7((Funcs ( the carrier of (REAL-NS f), the carrier of (REAL-NS m))),(Funcs ( the carrier of (REAL-NS f), the carrier of (REAL-NS m)))),(Funcs ( the carrier of (REAL-NS f), the carrier of (REAL-NS m))))) is set
FuncExtMult ( the carrier of (REAL-NS f),(REAL-NS m)) is Relation-like K7(REAL,(Funcs ( the carrier of (REAL-NS f), the carrier of (REAL-NS m)))) -defined Funcs ( the carrier of (REAL-NS f), the carrier of (REAL-NS m)) -valued Function-like total quasi_total Element of K6(K7(K7(REAL,(Funcs ( the carrier of (REAL-NS f), the carrier of (REAL-NS m)))),(Funcs ( the carrier of (REAL-NS f), the carrier of (REAL-NS m)))))
K7(REAL,(Funcs ( the carrier of (REAL-NS f), the carrier of (REAL-NS m)))) is set
K7(K7(REAL,(Funcs ( the carrier of (REAL-NS f), the carrier of (REAL-NS m)))),(Funcs ( the carrier of (REAL-NS f), the carrier of (REAL-NS m)))) is set
K6(K7(K7(REAL,(Funcs ( the carrier of (REAL-NS f), the carrier of (REAL-NS m)))),(Funcs ( the carrier of (REAL-NS f), the carrier of (REAL-NS m))))) is set
RLSStruct(# (Funcs ( the carrier of (REAL-NS f), the carrier of (REAL-NS m))),(FuncZero ( the carrier of (REAL-NS f),(REAL-NS m))),(FuncAdd ( the carrier of (REAL-NS f),(REAL-NS m))),(FuncExtMult ( the carrier of (REAL-NS f),(REAL-NS m))) #) is strict RLSStruct
the carrier of K318( the carrier of (REAL-NS f),(REAL-NS m)) is non empty set
K6( the carrier of K318( the carrier of (REAL-NS f),(REAL-NS m))) is set
K199(K318( the carrier of (REAL-NS f),(REAL-NS m)),(LinearOperators ((REAL-NS f),(REAL-NS m)))) is Relation-like Function-like Element of LinearOperators ((REAL-NS f),(REAL-NS m))
K197(K318( the carrier of (REAL-NS f),(REAL-NS m)),(LinearOperators ((REAL-NS f),(REAL-NS m)))) is Relation-like K7((LinearOperators ((REAL-NS f),(REAL-NS m))),(LinearOperators ((REAL-NS f),(REAL-NS m)))) -defined LinearOperators ((REAL-NS f),(REAL-NS m)) -valued Function-like total quasi_total Element of K6(K7(K7((LinearOperators ((REAL-NS f),(REAL-NS m))),(LinearOperators ((REAL-NS f),(REAL-NS m)))),(LinearOperators ((REAL-NS f),(REAL-NS m)))))
K7((LinearOperators ((REAL-NS f),(REAL-NS m))),(LinearOperators ((REAL-NS f),(REAL-NS m)))) is set
K7(K7((LinearOperators ((REAL-NS f),(REAL-NS m))),(LinearOperators ((REAL-NS f),(REAL-NS m)))),(LinearOperators ((REAL-NS f),(REAL-NS m)))) is set
K6(K7(K7((LinearOperators ((REAL-NS f),(REAL-NS m))),(LinearOperators ((REAL-NS f),(REAL-NS m)))),(LinearOperators ((REAL-NS f),(REAL-NS m))))) is set
K198(K318( the carrier of (REAL-NS f),(REAL-NS m)),(LinearOperators ((REAL-NS f),(REAL-NS m)))) is Relation-like K7(REAL,(LinearOperators ((REAL-NS f),(REAL-NS m)))) -defined LinearOperators ((REAL-NS f),(REAL-NS m)) -valued Function-like total quasi_total Element of K6(K7(K7(REAL,(LinearOperators ((REAL-NS f),(REAL-NS m)))),(LinearOperators ((REAL-NS f),(REAL-NS m)))))
K7(REAL,(LinearOperators ((REAL-NS f),(REAL-NS m)))) is set
K7(K7(REAL,(LinearOperators ((REAL-NS f),(REAL-NS m)))),(LinearOperators ((REAL-NS f),(REAL-NS m)))) is set
K6(K7(K7(REAL,(LinearOperators ((REAL-NS f),(REAL-NS m)))),(LinearOperators ((REAL-NS f),(REAL-NS m))))) is set
RLSStruct(# (LinearOperators ((REAL-NS f),(REAL-NS m))),K199(K318( the carrier of (REAL-NS f),(REAL-NS m)),(LinearOperators ((REAL-NS f),(REAL-NS m)))),K197(K318( the carrier of (REAL-NS f),(REAL-NS m)),(LinearOperators ((REAL-NS f),(REAL-NS m)))),K198(K318( the carrier of (REAL-NS f),(REAL-NS m)),(LinearOperators ((REAL-NS f),(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 f),(REAL-NS m))) is non empty set
K6( the carrier of (R_VectorSpace_of_LinearOperators ((REAL-NS f),(REAL-NS m)))) is set
K199((R_VectorSpace_of_LinearOperators ((REAL-NS f),(REAL-NS m))),(BoundedLinearOperators ((REAL-NS f),(REAL-NS m)))) is Element of BoundedLinearOperators ((REAL-NS f),(REAL-NS m))
K197((R_VectorSpace_of_LinearOperators ((REAL-NS f),(REAL-NS m))),(BoundedLinearOperators ((REAL-NS f),(REAL-NS m)))) is Relation-like K7((BoundedLinearOperators ((REAL-NS f),(REAL-NS m))),(BoundedLinearOperators ((REAL-NS f),(REAL-NS m)))) -defined BoundedLinearOperators ((REAL-NS f),(REAL-NS m)) -valued Function-like total quasi_total Element of K6(K7(K7((BoundedLinearOperators ((REAL-NS f),(REAL-NS m))),(BoundedLinearOperators ((REAL-NS f),(REAL-NS m)))),(BoundedLinearOperators ((REAL-NS f),(REAL-NS m)))))
K7((BoundedLinearOperators ((REAL-NS f),(REAL-NS m))),(BoundedLinearOperators ((REAL-NS f),(REAL-NS m)))) is set
K7(K7((BoundedLinearOperators ((REAL-NS f),(REAL-NS m))),(BoundedLinearOperators ((REAL-NS f),(REAL-NS m)))),(BoundedLinearOperators ((REAL-NS f),(REAL-NS m)))) is set
K6(K7(K7((BoundedLinearOperators ((REAL-NS f),(REAL-NS m))),(BoundedLinearOperators ((REAL-NS f),(REAL-NS m)))),(BoundedLinearOperators ((REAL-NS f),(REAL-NS m))))) is set
K198((R_VectorSpace_of_LinearOperators ((REAL-NS f),(REAL-NS m))),(BoundedLinearOperators ((REAL-NS f),(REAL-NS m)))) is Relation-like K7(REAL,(BoundedLinearOperators ((REAL-NS f),(REAL-NS m)))) -defined BoundedLinearOperators ((REAL-NS f),(REAL-NS m)) -valued Function-like total quasi_total Element of K6(K7(K7(REAL,(BoundedLinearOperators ((REAL-NS f),(REAL-NS m)))),(BoundedLinearOperators ((REAL-NS f),(REAL-NS m)))))
K7(REAL,(BoundedLinearOperators ((REAL-NS f),(REAL-NS m)))) is set
K7(K7(REAL,(BoundedLinearOperators ((REAL-NS f),(REAL-NS m)))),(BoundedLinearOperators ((REAL-NS f),(REAL-NS m)))) is set
K6(K7(K7(REAL,(BoundedLinearOperators ((REAL-NS f),(REAL-NS m)))),(BoundedLinearOperators ((REAL-NS f),(REAL-NS m))))) is set
BoundedLinearOperatorsNorm ((REAL-NS f),(REAL-NS m)) is non empty Relation-like BoundedLinearOperators ((REAL-NS f),(REAL-NS m)) -defined REAL -valued Function-like total quasi_total complex-valued ext-real-valued real-valued Element of K6(K7((BoundedLinearOperators ((REAL-NS f),(REAL-NS m))),REAL))
K7((BoundedLinearOperators ((REAL-NS f),(REAL-NS m))),REAL) is complex-valued ext-real-valued real-valued set
K6(K7((BoundedLinearOperators ((REAL-NS f),(REAL-NS m))),REAL)) is set
NORMSTR(# (BoundedLinearOperators ((REAL-NS f),(REAL-NS m))),K199((R_VectorSpace_of_LinearOperators ((REAL-NS f),(REAL-NS m))),(BoundedLinearOperators ((REAL-NS f),(REAL-NS m)))),K197((R_VectorSpace_of_LinearOperators ((REAL-NS f),(REAL-NS m))),(BoundedLinearOperators ((REAL-NS f),(REAL-NS m)))),K198((R_VectorSpace_of_LinearOperators ((REAL-NS f),(REAL-NS m))),(BoundedLinearOperators ((REAL-NS f),(REAL-NS m)))),(BoundedLinearOperatorsNorm ((REAL-NS f),(REAL-NS m))) #) is strict NORMSTR
the carrier of (R_NormSpace_of_BoundedLinearOperators ((REAL-NS f),(REAL-NS m))) is non empty set
(reproj (X,(0. (REAL-NS f)))) (#) (diff (i,y0)) is Relation-like Function-like set
dom i is Element of K6( the carrier of (REAL-NS f))
K6( the carrier of (REAL-NS f)) is set
i /. y0 is Element of the carrier of (REAL-NS m)
r is Neighbourhood of y0
s is Relation-like the carrier of (REAL-NS f) -defined the carrier of (REAL-NS m) -valued Function-like RestFunc-like Element of K6(K7( the carrier of (REAL-NS f), the carrier of (REAL-NS m)))
s is Relation-like the carrier of (REAL-NS f) -defined the carrier of (REAL-NS m) -valued Function-like RestFunc-like Element of K6(K7( the carrier of (REAL-NS f), the carrier of (REAL-NS m)))
y1 is V11() real ext-real Element of REAL
{ b₁ where b₁ is Element of the carrier of (REAL-NS f) : not y1 <= ||.(b₁ - y0).|| } is set
DD is Element of the carrier of (REAL-NS 1)
{ b₁ where b₁ is Element of the carrier of (REAL-NS 1) : not y1 <= ||.(b₁ - DD).|| } is set
tm is set
Sw0 is Element of the carrier of (REAL-NS 1)
Sw0 - DD is Element of the carrier of (REAL-NS 1)
||.(Sw0 - DD).|| is V11() real ext-real Element of REAL
the U8 of (REAL-NS 1) is non empty Relation-like the carrier of (REAL-NS 1) -defined REAL -valued Function-like total quasi_total complex-valued ext-real-valued real-valued Element of K6(K7( the carrier of (REAL-NS 1),REAL))
K7( the carrier of (REAL-NS 1),REAL) is complex-valued ext-real-valued real-valued set
K6(K7( the carrier of (REAL-NS 1),REAL)) is set
the U8 of (REAL-NS 1) . (Sw0 - DD) is V11() real ext-real Element of REAL
K6( the carrier of (REAL-NS 1)) is set
tm is Neighbourhood of DD
Sw0 is Element of the carrier of (REAL-NS 1)
(reproj (X,y0)) . Sw0 is Element of the carrier of (REAL-NS f)
((reproj (X,y0)) . Sw0) - y0 is Element of the carrier of (REAL-NS f)
Sw0 - DD is Element of the carrier of (REAL-NS 1)
(reproj (X,(0. (REAL-NS f)))) . (Sw0 - DD) is Element of the carrier of (REAL-NS f)
||.(((reproj (X,y0)) . Sw0) - y0).|| is V11() real ext-real Element of REAL
the U8 of (REAL-NS f) is non empty Relation-like the carrier of (REAL-NS f) -defined REAL -valued Function-like total quasi_total complex-valued ext-real-valued real-valued Element of K6(K7( the carrier of (REAL-NS f),REAL))
K7( the carrier of (REAL-NS f),REAL) is complex-valued ext-real-valued real-valued set
K6(K7( the carrier of (REAL-NS f),REAL)) is set
the U8 of (REAL-NS f) . (((reproj (X,y0)) . Sw0) - y0) is V11() real ext-real Element of REAL
w1 is Element of the carrier of (REAL-NS 1)
w1 - DD is Element of the carrier of (REAL-NS 1)
||.(w1 - DD).|| is V11() real ext-real Element of REAL
the U8 of (REAL-NS 1) is non empty Relation-like the carrier of (REAL-NS 1) -defined REAL -valued Function-like total quasi_total complex-valued ext-real-valued real-valued Element of K6(K7( the carrier of (REAL-NS 1),REAL))
K7( the carrier of (REAL-NS 1),REAL) is complex-valued ext-real-valued real-valued set
K6(K7( the carrier of (REAL-NS 1),REAL)) is set
the U8 of (REAL-NS 1) . (w1 - DD) is V11() real ext-real Element of REAL
dom s is Element of K6( the carrier of (REAL-NS f))
s * (reproj (X,(0. (REAL-NS f)))) is Relation-like the carrier of (REAL-NS 1) -defined the carrier of (REAL-NS m) -valued Function-like Element of K6(K7( the carrier of (REAL-NS 1), the carrier of (REAL-NS m)))
dom (reproj (X,(0. (REAL-NS f)))) is Element of K6( the carrier of (REAL-NS 1))
Sw0 is Relation-like the carrier of (REAL-NS 1) -defined the carrier of (REAL-NS m) -valued Function-like Element of K6(K7( the carrier of (REAL-NS 1), the carrier of (REAL-NS m)))
dom Sw0 is Element of K6( the carrier of (REAL-NS 1))
w1 is V11() real ext-real Element of REAL
Sw1 is V11() real ext-real Element of REAL
w2 is Element of the carrier of (REAL-NS 1)
||.w2.|| is V11() real ext-real Element of REAL
the U8 of (REAL-NS 1) is non empty Relation-like the carrier of (REAL-NS 1) -defined REAL -valued Function-like total quasi_total complex-valued ext-real-valued real-valued Element of K6(K7( the carrier of (REAL-NS 1),REAL))
K7( the carrier of (REAL-NS 1),REAL) is complex-valued ext-real-valued real-valued set
K6(K7( the carrier of (REAL-NS 1),REAL)) is set
the U8 of (REAL-NS 1) . w2 is V11() real ext-real Element of REAL
(reproj (X,(0. (REAL-NS f)))) . w2 is Element of the carrier of (REAL-NS f)
||.((reproj (X,(0. (REAL-NS f)))) . w2).|| is V11() real ext-real Element of REAL
the U8 of (REAL-NS f) is non empty Relation-like the carrier of (REAL-NS f) -defined REAL -valued Function-like total quasi_total complex-valued ext-real-valued real-valued Element of K6(K7( the carrier of (REAL-NS f),REAL))
K7( the carrier of (REAL-NS f),REAL) is complex-valued ext-real-valued real-valued set
K6(K7( the carrier of (REAL-NS f),REAL)) is set
the U8 of (REAL-NS f) . ((reproj (X,(0. (REAL-NS f)))) . w2) is V11() real ext-real Element of REAL
s /. ((reproj (X,(0. (REAL-NS f)))) . w2) is Element of the carrier of (REAL-NS m)
s . ((reproj (X,(0. (REAL-NS f)))) . w2) is set
Sw0 . w2 is set
Sw0 /. w2 is Element of the carrier of (REAL-NS m)
||.w2.|| " is V11() real ext-real Element of REAL
||.(Sw0 /. w2).|| is V11() real ext-real Element of REAL
the U8 of (REAL-NS m) is non empty Relation-like the carrier of (REAL-NS m) -defined REAL -valued Function-like total quasi_total complex-valued ext-real-valued real-valued Element of K6(K7( the carrier of (REAL-NS m),REAL))
K7( the carrier of (REAL-NS m),REAL) is complex-valued ext-real-valued real-valued set
K6(K7( the carrier of (REAL-NS m),REAL)) is set
the U8 of (REAL-NS m) . (Sw0 /. w2) is V11() real ext-real Element of REAL
(||.w2.|| ") * ||.(Sw0 /. w2).|| is V11() real ext-real Element of REAL
w2 is Element of the carrier of (REAL-NS 1)
||.w2.|| is V11() real ext-real Element of REAL
the U8 of (REAL-NS 1) . w2 is V11() real ext-real Element of REAL
||.w2.|| " is V11() real ext-real Element of REAL
Sw0 /. w2 is Element of the carrier of (REAL-NS m)
||.(Sw0 /. w2).|| is V11() real ext-real Element of REAL
the U8 of (REAL-NS m) . (Sw0 /. w2) is V11() real ext-real Element of REAL
(||.w2.|| ") * ||.(Sw0 /. w2).|| is V11() real ext-real Element of REAL
Sw1 is non empty Relation-like the carrier of (REAL-NS f) -defined the carrier of (REAL-NS m) -valued Function-like total quasi_total V182( REAL-NS f, REAL-NS m) V183( REAL-NS f, REAL-NS m) Lipschitzian Element of K6(K7( the carrier of (REAL-NS f), the carrier of (REAL-NS m)))
Sw1 * (reproj (X,(0. (REAL-NS f)))) is non empty Relation-like the carrier of (REAL-NS 1) -defined the carrier of (REAL-NS m) -valued Function-like total quasi_total Element of K6(K7( the carrier of (REAL-NS 1), the carrier of (REAL-NS m)))
w2 is non empty Relation-like the carrier of (REAL-NS 1) -defined the carrier of (REAL-NS m) -valued Function-like total quasi_total Element of K6(K7( the carrier of (REAL-NS 1), the carrier of (REAL-NS m)))
ys is Element of the carrier of (REAL-NS 1)
rm is Element of the carrier of (REAL-NS 1)
ys + rm is Element of the carrier of (REAL-NS 1)
w2 . (ys + rm) is Element of the carrier of (REAL-NS m)
(reproj (X,(0. (REAL-NS f)))) . (ys + rm) is Element of the carrier of (REAL-NS f)
Sw1 . ((reproj (X,(0. (REAL-NS f)))) . (ys + rm)) is Element of the carrier of (REAL-NS m)
(reproj (X,(0. (REAL-NS f)))) . ys is Element of the carrier of (REAL-NS f)
(reproj (X,(0. (REAL-NS f)))) . rm is Element of the carrier of (REAL-NS f)
((reproj (X,(0. (REAL-NS f)))) . ys) + ((reproj (X,(0. (REAL-NS f)))) . rm) is Element of the carrier of (REAL-NS f)
Sw1 . (((reproj (X,(0. (REAL-NS f)))) . ys) + ((reproj (X,(0. (REAL-NS f)))) . rm)) is Element of the carrier of (REAL-NS m)
Sw1 . ((reproj (X,(0. (REAL-NS f)))) . ys) is Element of the carrier of (REAL-NS m)
Sw1 . ((reproj (X,(0. (REAL-NS f)))) . rm) is Element of the carrier of (REAL-NS m)
(Sw1 . ((reproj (X,(0. (REAL-NS f)))) . ys)) + (Sw1 . ((reproj (X,(0. (REAL-NS f)))) . rm)) is Element of the carrier of (REAL-NS m)
w2 . ys is Element of the carrier of (REAL-NS m)
(w2 . ys) + (Sw1 . ((reproj (X,(0. (REAL-NS f)))) . rm)) is Element of the carrier of (REAL-NS m)
w2 . rm is Element of the carrier of (REAL-NS m)
(w2 . ys) + (w2 . rm) is Element of the carrier of (REAL-NS m)
ys is Element of the carrier of (REAL-NS 1)
rm is V11() real ext-real Element of REAL
rm * ys is Element of the carrier of (REAL-NS 1)
w2 . (rm * ys) is Element of the carrier of (REAL-NS m)
(reproj (X,(0. (REAL-NS f)))) . (rm * ys) is Element of the carrier of (REAL-NS f)
Sw1 . ((reproj (X,(0. (REAL-NS f)))) . (rm * ys)) is Element of the carrier of (REAL-NS m)
(reproj (X,(0. (REAL-NS f)))) . ys is Element of the carrier of (REAL-NS f)
rm * ((reproj (X,(0. (REAL-NS f)))) . ys) is Element of the carrier of (REAL-NS f)
Sw1 . (rm * ((reproj (X,(0. (REAL-NS f)))) . ys)) is Element of the carrier of (REAL-NS m)
Sw1 . ((reproj (X,(0. (REAL-NS f)))) . ys) is Element of the carrier of (REAL-NS m)
rm * (Sw1 . ((reproj (X,(0. (REAL-NS f)))) . ys)) is Element of the carrier of (REAL-NS m)
w2 . ys is Element of the carrier of (REAL-NS m)
rm * (w2 . ys) is Element of the carrier of (REAL-NS m)
ys is non empty Relation-like the carrier of (REAL-NS 1) -defined the carrier of (REAL-NS m) -valued Function-like total quasi_total V182( REAL-NS 1, REAL-NS m) V183( REAL-NS 1, REAL-NS m) Lipschitzian Element of K6(K7( the carrier of (REAL-NS 1), the carrier of (REAL-NS m)))
rs is set
(reproj (X,y0)) .: tm is Element of K6( the carrier of (REAL-NS f))
i is Element of the carrier of (REAL-NS 1)
(reproj (X,y0)) . i is Element of the carrier of (REAL-NS f)
i is Relation-like the carrier of (REAL-NS f) -defined the carrier of (REAL-NS m) -valued Function-like RestFunc-like Element of K6(K7( the carrier of (REAL-NS f), the carrier of (REAL-NS m)))
dom (reproj (X,y0)) is Element of K6( the carrier of (REAL-NS 1))
dom (i * (reproj (X,y0))) is Element of K6( the carrier of (REAL-NS 1))
(i * (reproj (X,y0))) /. DD is Element of the carrier of (REAL-NS m)
rm is Relation-like Function-like Element of the carrier of (R_NormSpace_of_BoundedLinearOperators ((REAL-NS 1),(REAL-NS m)))
w1 is Relation-like the carrier of (REAL-NS 1) -defined the carrier of (REAL-NS m) -valued Function-like RestFunc-like Element of K6(K7( the carrier of (REAL-NS 1), the carrier of (REAL-NS m)))
rs is Element of the carrier of (REAL-NS 1)
(i * (reproj (X,y0))) /. rs is Element of the carrier of (REAL-NS m)
((i * (reproj (X,y0))) /. rs) - ((i * (reproj (X,y0))) /. DD) is Element of the carrier of (REAL-NS m)
rs - DD is Element of the carrier of (REAL-NS 1)
rm . (rs - DD) is Element of the carrier of (REAL-NS m)
w1 /. (rs - DD) is Element of the carrier of (REAL-NS m)
(rm . (rs - DD)) + (w1 /. (rs - DD)) is Element of the carrier of (REAL-NS m)
(reproj (X,y0)) . rs is Element of the carrier of (REAL-NS f)
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
{ b₁ where b₁ is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like Element of REAL * : len b₁ = f } is set
(reproj (X,y0)) . DD is Element of the carrier of (REAL-NS f)
i is V11() real ext-real Element of REAL
<*i*> is non empty V2() Relation-like NAT -defined REAL -valued Function-like one-to-one constant finite 1 -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued increasing decreasing non-decreasing non-increasing FinSequence of REAL
[1,i] is V27() set
{1,i} is non empty complex-membered ext-real-membered real-membered set
{{1,i},{1}} is non empty set
{[1,i]} is non empty Function-like set
v is Relation-like NAT -defined REAL -valued Function-like finite f -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL f
reproj (X,v) is non empty Relation-like REAL -defined REAL f -valued Function-like total quasi_total complex-functions-valued ext-real-functions-valued real-functions-valued Element of K6(K7(REAL,(REAL f)))
K7(REAL,(REAL f)) is set
K6(K7(REAL,(REAL f))) is set
(reproj (X,v)) . i is Relation-like NAT -defined Function-like finite f -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL f
v . X is V11() real ext-real Element of REAL
proj (X,f) is non empty Relation-like REAL f -defined REAL -valued Function-like total quasi_total complex-valued ext-real-valued real-valued Element of K6(K7((REAL f),REAL))
K7((REAL f),REAL) is complex-valued ext-real-valued real-valued set
K6(K7((REAL f),REAL)) is set
(proj (X,f)) . y0 is V11() real ext-real Element of REAL
<*((proj (X,f)) . y0)*> is non empty V2() Relation-like NAT -defined REAL -valued Function-like one-to-one constant finite 1 -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued increasing decreasing non-decreasing non-increasing FinSequence of REAL
[1,((proj (X,f)) . y0)] is V27() set
{1,((proj (X,f)) . y0)} is non empty complex-membered ext-real-membered real-membered set
{{1,((proj (X,f)) . y0)},{1}} is non empty set
{[1,((proj (X,f)) . y0)]} is non empty Function-like set
(proj (X,f)) . v is V11() real ext-real Element of REAL
(reproj (X,v)) . (v . X) is Relation-like NAT -defined Function-like finite f -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL f
p1 is V11() real ext-real Element of REAL
Replace (v,X,p1) is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
(i * (reproj (X,y0))) . rs is set
i . ((reproj (X,y0)) . rs) is set
i /. ((reproj (X,y0)) . rs) is Element of the carrier of (REAL-NS m)
p0 is Relation-like the carrier of (REAL-NS f) -defined the carrier of (REAL-NS m) -valued Function-like RestFunc-like Element of K6(K7( the carrier of (REAL-NS f), the carrier of (REAL-NS m)))
(i * (reproj (X,y0))) . DD is set
i . ((reproj (X,y0)) . DD) is set
(i /. ((reproj (X,y0)) . rs)) - (i /. y0) is Element of the carrier of (REAL-NS m)
p0 is Relation-like the carrier of (REAL-NS f) -defined the carrier of (REAL-NS m) -valued Function-like RestFunc-like Element of K6(K7( the carrier of (REAL-NS f), the carrier of (REAL-NS m)))
(reproj (X,(0. (REAL-NS f)))) . (rs - DD) is Element of the carrier of (REAL-NS f)
s /. ((reproj (X,(0. (REAL-NS f)))) . (rs - DD)) is Element of the carrier of (REAL-NS m)
s . ((reproj (X,(0. (REAL-NS f)))) . (rs - DD)) is set
w1 . (rs - DD) is set
((reproj (X,y0)) . rs) - y0 is Element of the carrier of (REAL-NS f)
(diff (i,y0)) . (((reproj (X,y0)) . rs) - y0) is Element of the carrier of (REAL-NS m)
s /. (((reproj (X,y0)) . rs) - y0) is Element of the carrier of (REAL-NS m)
((diff (i,y0)) . (((reproj (X,y0)) . rs) - y0)) + (s /. (((reproj (X,y0)) . rs) - y0)) is Element of the carrier of (REAL-NS m)
Sw1 . ((reproj (X,(0. (REAL-NS f)))) . (rs - DD)) is Element of the carrier of (REAL-NS m)
(Sw1 . ((reproj (X,(0. (REAL-NS f)))) . (rs - DD))) + (s /. (((reproj (X,y0)) . rs) - y0)) is Element of the carrier of (REAL-NS m)
(Sw1 . ((reproj (X,(0. (REAL-NS f)))) . (rs - DD))) + (s /. ((reproj (X,(0. (REAL-NS f)))) . (rs - DD))) is Element of the carrier of (REAL-NS m)
<*1*> is non empty V2() Relation-like NAT -defined REAL -valued Function-like one-to-one constant finite 1 -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued increasing decreasing non-decreasing non-increasing FinSequence of REAL
m is non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below 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
{ b₁ where b₁ is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like Element of REAL * : len b₁ = m } is set
f is non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below 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
{ b₁ where b₁ is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like Element of REAL * : len b₁ = f } is set
K7((REAL f),(REAL m)) is set
K6(K7((REAL f),(REAL m))) is set
REAL-NS f is non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V185() NORMSTR
the carrier of (REAL-NS f) is non empty V2() set
0. (REAL-NS f) is V52( REAL-NS f) Element of the carrier of (REAL-NS f)
X is ordinal natural V11() real ext-real non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of NAT
reproj (X,(0. (REAL-NS f))) is non empty Relation-like the carrier of (REAL-NS 1) -defined the carrier of (REAL-NS f) -valued Function-like total quasi_total Element of K6(K7( the carrier of (REAL-NS 1), the carrier of (REAL-NS f)))
K7( the carrier of (REAL-NS 1), the carrier of (REAL-NS f)) is set
K6(K7( the carrier of (REAL-NS 1), the carrier of (REAL-NS f))) is set
i is Relation-like REAL f -defined REAL m -valued Function-like complex-functions-valued ext-real-functions-valued real-functions-valued Element of K6(K7((REAL f),(REAL m)))
y0 is Relation-like NAT -defined REAL -valued Function-like finite f -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL f
partdiff (i,y0,X) is Relation-like NAT -defined REAL -valued Function-like finite m -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL m
diff (i,y0) is non empty Relation-like REAL f -defined REAL m -valued Function-like total quasi_total complex-functions-valued ext-real-functions-valued real-functions-valued Element of K6(K7((REAL f),(REAL m)))
(diff (i,y0)) * (reproj (X,(0. (REAL-NS f)))) is Relation-like the carrier of (REAL-NS 1) -defined REAL m -valued Function-like complex-functions-valued ext-real-functions-valued real-functions-valued Element of K6(K7( the carrier of (REAL-NS 1),(REAL m)))
K7( the carrier of (REAL-NS 1),(REAL m)) is set
K6(K7( the carrier of (REAL-NS 1),(REAL m))) is set
((diff (i,y0)) * (reproj (X,(0. (REAL-NS f))))) . <*1*> is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
REAL-NS m is non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V185() NORMSTR
the carrier of (REAL-NS m) is non empty V2() set
K7( the carrier of (REAL-NS f), the carrier of (REAL-NS m)) is set
K6(K7( the carrier of (REAL-NS f), the carrier of (REAL-NS m))) is set
r is Relation-like the carrier of (REAL-NS f) -defined the carrier of (REAL-NS m) -valued Function-like Element of K6(K7( the carrier of (REAL-NS f), the carrier of (REAL-NS m)))
s is Element of the carrier of (REAL-NS f)
partdiff (r,s,X) is Relation-like Function-like Element of the carrier of (R_NormSpace_of_BoundedLinearOperators ((REAL-NS 1),(REAL-NS m)))
R_NormSpace_of_BoundedLinearOperators ((REAL-NS 1),(REAL-NS m)) is non empty V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() RealNormSpace-like V185() NORMSTR
BoundedLinearOperators ((REAL-NS 1),(REAL-NS m)) is non empty Element of K6( the carrier of (R_VectorSpace_of_LinearOperators ((REAL-NS 1),(REAL-NS m))))
R_VectorSpace_of_LinearOperators ((REAL-NS 1),(REAL-NS m)) is non empty V71() strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V214() RLSStruct
LinearOperators ((REAL-NS 1),(REAL-NS m)) is non empty functional Element of K6( the carrier of K318( the carrier of (REAL-NS 1),(REAL-NS m)))
K318( the carrier of (REAL-NS 1),(REAL-NS m)) is non empty V71() strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V214() RLSStruct
Funcs ( the carrier of (REAL-NS 1), the carrier of (REAL-NS m)) is non empty functional FUNCTION_DOMAIN of the carrier of (REAL-NS 1), the carrier of (REAL-NS m)
FuncZero ( the carrier of (REAL-NS 1),(REAL-NS m)) is Relation-like the carrier of (REAL-NS 1) -defined the carrier of (REAL-NS m) -valued Function-like total quasi_total Element of Funcs ( the carrier of (REAL-NS 1), the carrier of (REAL-NS m))
0. (REAL-NS m) is V52( REAL-NS m) Element of the carrier of (REAL-NS m)
K214( the carrier of (REAL-NS m), the carrier of (REAL-NS 1),(0. (REAL-NS m))) is non empty Relation-like the carrier of (REAL-NS 1) -defined the carrier of (REAL-NS m) -valued Function-like total quasi_total Element of K6(K7( the carrier of (REAL-NS 1), the carrier of (REAL-NS m)))
K7( the carrier of (REAL-NS 1), the carrier of (REAL-NS m)) is set
K6(K7( the carrier of (REAL-NS 1), the carrier of (REAL-NS m))) is set
FuncAdd ( the carrier of (REAL-NS 1),(REAL-NS m)) is Relation-like K7((Funcs ( the carrier of (REAL-NS 1), the carrier of (REAL-NS m))),(Funcs ( the carrier of (REAL-NS 1), the carrier of (REAL-NS m)))) -defined Funcs ( the carrier of (REAL-NS 1), the carrier of (REAL-NS m)) -valued Function-like total quasi_total Element of K6(K7(K7((Funcs ( the carrier of (REAL-NS 1), the carrier of (REAL-NS m))),(Funcs ( the carrier of (REAL-NS 1), the carrier of (REAL-NS m)))),(Funcs ( the carrier of (REAL-NS 1), the carrier of (REAL-NS m)))))
K7((Funcs ( the carrier of (REAL-NS 1), the carrier of (REAL-NS m))),(Funcs ( the carrier of (REAL-NS 1), the carrier of (REAL-NS m)))) is set
K7(K7((Funcs ( the carrier of (REAL-NS 1), the carrier of (REAL-NS m))),(Funcs ( the carrier of (REAL-NS 1), the carrier of (REAL-NS m)))),(Funcs ( the carrier of (REAL-NS 1), the carrier of (REAL-NS m)))) is set
K6(K7(K7((Funcs ( the carrier of (REAL-NS 1), the carrier of (REAL-NS m))),(Funcs ( the carrier of (REAL-NS 1), the carrier of (REAL-NS m)))),(Funcs ( the carrier of (REAL-NS 1), the carrier of (REAL-NS m))))) is set
FuncExtMult ( the carrier of (REAL-NS 1),(REAL-NS m)) is Relation-like K7(REAL,(Funcs ( the carrier of (REAL-NS 1), the carrier of (REAL-NS m)))) -defined Funcs ( the carrier of (REAL-NS 1), the carrier of (REAL-NS m)) -valued Function-like total quasi_total Element of K6(K7(K7(REAL,(Funcs ( the carrier of (REAL-NS 1), the carrier of (REAL-NS m)))),(Funcs ( the carrier of (REAL-NS 1), the carrier of (REAL-NS m)))))
K7(REAL,(Funcs ( the carrier of (REAL-NS 1), the carrier of (REAL-NS m)))) is set
K7(K7(REAL,(Funcs ( the carrier of (REAL-NS 1), the carrier of (REAL-NS m)))),(Funcs ( the carrier of (REAL-NS 1), the carrier of (REAL-NS m)))) is set
K6(K7(K7(REAL,(Funcs ( the carrier of (REAL-NS 1), the carrier of (REAL-NS m)))),(Funcs ( the carrier of (REAL-NS 1), the carrier of (REAL-NS m))))) is set
RLSStruct(# (Funcs ( the carrier of (REAL-NS 1), the carrier of (REAL-NS m))),(FuncZero ( the carrier of (REAL-NS 1),(REAL-NS m))),(FuncAdd ( the carrier of (REAL-NS 1),(REAL-NS m))),(FuncExtMult ( the carrier of (REAL-NS 1),(REAL-NS m))) #) is strict RLSStruct
the carrier of K318( the carrier of (REAL-NS 1),(REAL-NS m)) is non empty set
K6( the carrier of K318( the carrier of (REAL-NS 1),(REAL-NS m))) is set
K199(K318( the carrier of (REAL-NS 1),(REAL-NS m)),(LinearOperators ((REAL-NS 1),(REAL-NS m)))) is Relation-like Function-like Element of LinearOperators ((REAL-NS 1),(REAL-NS m))
K197(K318( the carrier of (REAL-NS 1),(REAL-NS m)),(LinearOperators ((REAL-NS 1),(REAL-NS m)))) is Relation-like K7((LinearOperators ((REAL-NS 1),(REAL-NS m))),(LinearOperators ((REAL-NS 1),(REAL-NS m)))) -defined LinearOperators ((REAL-NS 1),(REAL-NS m)) -valued Function-like total quasi_total Element of K6(K7(K7((LinearOperators ((REAL-NS 1),(REAL-NS m))),(LinearOperators ((REAL-NS 1),(REAL-NS m)))),(LinearOperators ((REAL-NS 1),(REAL-NS m)))))
K7((LinearOperators ((REAL-NS 1),(REAL-NS m))),(LinearOperators ((REAL-NS 1),(REAL-NS m)))) is set
K7(K7((LinearOperators ((REAL-NS 1),(REAL-NS m))),(LinearOperators ((REAL-NS 1),(REAL-NS m)))),(LinearOperators ((REAL-NS 1),(REAL-NS m)))) is set
K6(K7(K7((LinearOperators ((REAL-NS 1),(REAL-NS m))),(LinearOperators ((REAL-NS 1),(REAL-NS m)))),(LinearOperators ((REAL-NS 1),(REAL-NS m))))) is set
K198(K318( the carrier of (REAL-NS 1),(REAL-NS m)),(LinearOperators ((REAL-NS 1),(REAL-NS m)))) is Relation-like K7(REAL,(LinearOperators ((REAL-NS 1),(REAL-NS m)))) -defined LinearOperators ((REAL-NS 1),(REAL-NS m)) -valued Function-like total quasi_total Element of K6(K7(K7(REAL,(LinearOperators ((REAL-NS 1),(REAL-NS m)))),(LinearOperators ((REAL-NS 1),(REAL-NS m)))))
K7(REAL,(LinearOperators ((REAL-NS 1),(REAL-NS m)))) is set
K7(K7(REAL,(LinearOperators ((REAL-NS 1),(REAL-NS m)))),(LinearOperators ((REAL-NS 1),(REAL-NS m)))) is set
K6(K7(K7(REAL,(LinearOperators ((REAL-NS 1),(REAL-NS m)))),(LinearOperators ((REAL-NS 1),(REAL-NS m))))) is set
RLSStruct(# (LinearOperators ((REAL-NS 1),(REAL-NS m))),K199(K318( the carrier of (REAL-NS 1),(REAL-NS m)),(LinearOperators ((REAL-NS 1),(REAL-NS m)))),K197(K318( the carrier of (REAL-NS 1),(REAL-NS m)),(LinearOperators ((REAL-NS 1),(REAL-NS m)))),K198(K318( the carrier of (REAL-NS 1),(REAL-NS m)),(LinearOperators ((REAL-NS 1),(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 1),(REAL-NS m))) is non empty set
K6( the carrier of (R_VectorSpace_of_LinearOperators ((REAL-NS 1),(REAL-NS m)))) is set
K199((R_VectorSpace_of_LinearOperators ((REAL-NS 1),(REAL-NS m))),(BoundedLinearOperators ((REAL-NS 1),(REAL-NS m)))) is Element of BoundedLinearOperators ((REAL-NS 1),(REAL-NS m))
K197((R_VectorSpace_of_LinearOperators ((REAL-NS 1),(REAL-NS m))),(BoundedLinearOperators ((REAL-NS 1),(REAL-NS m)))) is Relation-like K7((BoundedLinearOperators ((REAL-NS 1),(REAL-NS m))),(BoundedLinearOperators ((REAL-NS 1),(REAL-NS m)))) -defined BoundedLinearOperators ((REAL-NS 1),(REAL-NS m)) -valued Function-like total quasi_total Element of K6(K7(K7((BoundedLinearOperators ((REAL-NS 1),(REAL-NS m))),(BoundedLinearOperators ((REAL-NS 1),(REAL-NS m)))),(BoundedLinearOperators ((REAL-NS 1),(REAL-NS m)))))
K7((BoundedLinearOperators ((REAL-NS 1),(REAL-NS m))),(BoundedLinearOperators ((REAL-NS 1),(REAL-NS m)))) is set
K7(K7((BoundedLinearOperators ((REAL-NS 1),(REAL-NS m))),(BoundedLinearOperators ((REAL-NS 1),(REAL-NS m)))),(BoundedLinearOperators ((REAL-NS 1),(REAL-NS m)))) is set
K6(K7(K7((BoundedLinearOperators ((REAL-NS 1),(REAL-NS m))),(BoundedLinearOperators ((REAL-NS 1),(REAL-NS m)))),(BoundedLinearOperators ((REAL-NS 1),(REAL-NS m))))) is set
K198((R_VectorSpace_of_LinearOperators ((REAL-NS 1),(REAL-NS m))),(BoundedLinearOperators ((REAL-NS 1),(REAL-NS m)))) is Relation-like K7(REAL,(BoundedLinearOperators ((REAL-NS 1),(REAL-NS m)))) -defined BoundedLinearOperators ((REAL-NS 1),(REAL-NS m)) -valued Function-like total quasi_total Element of K6(K7(K7(REAL,(BoundedLinearOperators ((REAL-NS 1),(REAL-NS m)))),(BoundedLinearOperators ((REAL-NS 1),(REAL-NS m)))))
K7(REAL,(BoundedLinearOperators ((REAL-NS 1),(REAL-NS m)))) is set
K7(K7(REAL,(BoundedLinearOperators ((REAL-NS 1),(REAL-NS m)))),(BoundedLinearOperators ((REAL-NS 1),(REAL-NS m)))) is set
K6(K7(K7(REAL,(BoundedLinearOperators ((REAL-NS 1),(REAL-NS m)))),(BoundedLinearOperators ((REAL-NS 1),(REAL-NS m))))) is set
BoundedLinearOperatorsNorm ((REAL-NS 1),(REAL-NS m)) is non empty Relation-like BoundedLinearOperators ((REAL-NS 1),(REAL-NS m)) -defined REAL -valued Function-like total quasi_total complex-valued ext-real-valued real-valued Element of K6(K7((BoundedLinearOperators ((REAL-NS 1),(REAL-NS m))),REAL))
K7((BoundedLinearOperators ((REAL-NS 1),(REAL-NS m))),REAL) is complex-valued ext-real-valued real-valued set
K6(K7((BoundedLinearOperators ((REAL-NS 1),(REAL-NS m))),REAL)) is set
NORMSTR(# (BoundedLinearOperators ((REAL-NS 1),(REAL-NS m))),K199((R_VectorSpace_of_LinearOperators ((REAL-NS 1),(REAL-NS m))),(BoundedLinearOperators ((REAL-NS 1),(REAL-NS m)))),K197((R_VectorSpace_of_LinearOperators ((REAL-NS 1),(REAL-NS m))),(BoundedLinearOperators ((REAL-NS 1),(REAL-NS m)))),K198((R_VectorSpace_of_LinearOperators ((REAL-NS 1),(REAL-NS m))),(BoundedLinearOperators ((REAL-NS 1),(REAL-NS m)))),(BoundedLinearOperatorsNorm ((REAL-NS 1),(REAL-NS m))) #) is strict NORMSTR
the carrier of (R_NormSpace_of_BoundedLinearOperators ((REAL-NS 1),(REAL-NS m))) is non empty set
reproj (X,s) is non empty Relation-like the carrier of (REAL-NS 1) -defined the carrier of (REAL-NS f) -valued Function-like total quasi_total Element of K6(K7( the carrier of (REAL-NS 1), the carrier of (REAL-NS f)))
r * (reproj (X,s)) is Relation-like the carrier of (REAL-NS 1) -defined the carrier of (REAL-NS m) -valued Function-like Element of K6(K7( the carrier of (REAL-NS 1), the carrier of (REAL-NS m)))
Proj (X,f) is non empty Relation-like the carrier of (REAL-NS f) -defined the carrier of (REAL-NS 1) -valued Function-like total quasi_total Element of K6(K7( the carrier of (REAL-NS f), the carrier of (REAL-NS 1)))
K7( the carrier of (REAL-NS f), the carrier of (REAL-NS 1)) is set
K6(K7( the carrier of (REAL-NS f), the carrier of (REAL-NS 1))) is set
(Proj (X,f)) . s is Element of the carrier of (REAL-NS 1)
diff ((r * (reproj (X,s))),((Proj (X,f)) . s)) is Relation-like Function-like Element of the carrier of (R_NormSpace_of_BoundedLinearOperators ((REAL-NS 1),(REAL-NS m)))
diff (r,s) is Relation-like Function-like Element of the carrier of (R_NormSpace_of_BoundedLinearOperators ((REAL-NS f),(REAL-NS m)))
R_NormSpace_of_BoundedLinearOperators ((REAL-NS f),(REAL-NS m)) is non empty V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() RealNormSpace-like V185() NORMSTR
BoundedLinearOperators ((REAL-NS f),(REAL-NS m)) is non empty Element of K6( the carrier of (R_VectorSpace_of_LinearOperators ((REAL-NS f),(REAL-NS m))))
R_VectorSpace_of_LinearOperators ((REAL-NS f),(REAL-NS m)) is non empty V71() strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V214() RLSStruct
LinearOperators ((REAL-NS f),(REAL-NS m)) is non empty functional Element of K6( the carrier of K318( the carrier of (REAL-NS f),(REAL-NS m)))
K318( the carrier of (REAL-NS f),(REAL-NS m)) is non empty V71() strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V214() RLSStruct
Funcs ( the carrier of (REAL-NS f), the carrier of (REAL-NS m)) is non empty functional FUNCTION_DOMAIN of the carrier of (REAL-NS f), the carrier of (REAL-NS m)
FuncZero ( the carrier of (REAL-NS f),(REAL-NS m)) is Relation-like the carrier of (REAL-NS f) -defined the carrier of (REAL-NS m) -valued Function-like total quasi_total Element of Funcs ( the carrier of (REAL-NS f), the carrier of (REAL-NS m))
K214( the carrier of (REAL-NS m), the carrier of (REAL-NS f),(0. (REAL-NS m))) is non empty Relation-like the carrier of (REAL-NS f) -defined the carrier of (REAL-NS m) -valued Function-like total quasi_total Element of K6(K7( the carrier of (REAL-NS f), the carrier of (REAL-NS m)))
FuncAdd ( the carrier of (REAL-NS f),(REAL-NS m)) is Relation-like K7((Funcs ( the carrier of (REAL-NS f), the carrier of (REAL-NS m))),(Funcs ( the carrier of (REAL-NS f), the carrier of (REAL-NS m)))) -defined Funcs ( the carrier of (REAL-NS f), the carrier of (REAL-NS m)) -valued Function-like total quasi_total Element of K6(K7(K7((Funcs ( the carrier of (REAL-NS f), the carrier of (REAL-NS m))),(Funcs ( the carrier of (REAL-NS f), the carrier of (REAL-NS m)))),(Funcs ( the carrier of (REAL-NS f), the carrier of (REAL-NS m)))))
K7((Funcs ( the carrier of (REAL-NS f), the carrier of (REAL-NS m))),(Funcs ( the carrier of (REAL-NS f), the carrier of (REAL-NS m)))) is set
K7(K7((Funcs ( the carrier of (REAL-NS f), the carrier of (REAL-NS m))),(Funcs ( the carrier of (REAL-NS f), the carrier of (REAL-NS m)))),(Funcs ( the carrier of (REAL-NS f), the carrier of (REAL-NS m)))) is set
K6(K7(K7((Funcs ( the carrier of (REAL-NS f), the carrier of (REAL-NS m))),(Funcs ( the carrier of (REAL-NS f), the carrier of (REAL-NS m)))),(Funcs ( the carrier of (REAL-NS f), the carrier of (REAL-NS m))))) is set
FuncExtMult ( the carrier of (REAL-NS f),(REAL-NS m)) is Relation-like K7(REAL,(Funcs ( the carrier of (REAL-NS f), the carrier of (REAL-NS m)))) -defined Funcs ( the carrier of (REAL-NS f), the carrier of (REAL-NS m)) -valued Function-like total quasi_total Element of K6(K7(K7(REAL,(Funcs ( the carrier of (REAL-NS f), the carrier of (REAL-NS m)))),(Funcs ( the carrier of (REAL-NS f), the carrier of (REAL-NS m)))))
K7(REAL,(Funcs ( the carrier of (REAL-NS f), the carrier of (REAL-NS m)))) is set
K7(K7(REAL,(Funcs ( the carrier of (REAL-NS f), the carrier of (REAL-NS m)))),(Funcs ( the carrier of (REAL-NS f), the carrier of (REAL-NS m)))) is set
K6(K7(K7(REAL,(Funcs ( the carrier of (REAL-NS f), the carrier of (REAL-NS m)))),(Funcs ( the carrier of (REAL-NS f), the carrier of (REAL-NS m))))) is set
RLSStruct(# (Funcs ( the carrier of (REAL-NS f), the carrier of (REAL-NS m))),(FuncZero ( the carrier of (REAL-NS f),(REAL-NS m))),(FuncAdd ( the carrier of (REAL-NS f),(REAL-NS m))),(FuncExtMult ( the carrier of (REAL-NS f),(REAL-NS m))) #) is strict RLSStruct
the carrier of K318( the carrier of (REAL-NS f),(REAL-NS m)) is non empty set
K6( the carrier of K318( the carrier of (REAL-NS f),(REAL-NS m))) is set
K199(K318( the carrier of (REAL-NS f),(REAL-NS m)),(LinearOperators ((REAL-NS f),(REAL-NS m)))) is Relation-like Function-like Element of LinearOperators ((REAL-NS f),(REAL-NS m))
K197(K318( the carrier of (REAL-NS f),(REAL-NS m)),(LinearOperators ((REAL-NS f),(REAL-NS m)))) is Relation-like K7((LinearOperators ((REAL-NS f),(REAL-NS m))),(LinearOperators ((REAL-NS f),(REAL-NS m)))) -defined LinearOperators ((REAL-NS f),(REAL-NS m)) -valued Function-like total quasi_total Element of K6(K7(K7((LinearOperators ((REAL-NS f),(REAL-NS m))),(LinearOperators ((REAL-NS f),(REAL-NS m)))),(LinearOperators ((REAL-NS f),(REAL-NS m)))))
K7((LinearOperators ((REAL-NS f),(REAL-NS m))),(LinearOperators ((REAL-NS f),(REAL-NS m)))) is set
K7(K7((LinearOperators ((REAL-NS f),(REAL-NS m))),(LinearOperators ((REAL-NS f),(REAL-NS m)))),(LinearOperators ((REAL-NS f),(REAL-NS m)))) is set
K6(K7(K7((LinearOperators ((REAL-NS f),(REAL-NS m))),(LinearOperators ((REAL-NS f),(REAL-NS m)))),(LinearOperators ((REAL-NS f),(REAL-NS m))))) is set
K198(K318( the carrier of (REAL-NS f),(REAL-NS m)),(LinearOperators ((REAL-NS f),(REAL-NS m)))) is Relation-like K7(REAL,(LinearOperators ((REAL-NS f),(REAL-NS m)))) -defined LinearOperators ((REAL-NS f),(REAL-NS m)) -valued Function-like total quasi_total Element of K6(K7(K7(REAL,(LinearOperators ((REAL-NS f),(REAL-NS m)))),(LinearOperators ((REAL-NS f),(REAL-NS m)))))
K7(REAL,(LinearOperators ((REAL-NS f),(REAL-NS m)))) is set
K7(K7(REAL,(LinearOperators ((REAL-NS f),(REAL-NS m)))),(LinearOperators ((REAL-NS f),(REAL-NS m)))) is set
K6(K7(K7(REAL,(LinearOperators ((REAL-NS f),(REAL-NS m)))),(LinearOperators ((REAL-NS f),(REAL-NS m))))) is set
RLSStruct(# (LinearOperators ((REAL-NS f),(REAL-NS m))),K199(K318( the carrier of (REAL-NS f),(REAL-NS m)),(LinearOperators ((REAL-NS f),(REAL-NS m)))),K197(K318( the carrier of (REAL-NS f),(REAL-NS m)),(LinearOperators ((REAL-NS f),(REAL-NS m)))),K198(K318( the carrier of (REAL-NS f),(REAL-NS m)),(LinearOperators ((REAL-NS f),(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 f),(REAL-NS m))) is non empty set
K6( the carrier of (R_VectorSpace_of_LinearOperators ((REAL-NS f),(REAL-NS m)))) is set
K199((R_VectorSpace_of_LinearOperators ((REAL-NS f),(REAL-NS m))),(BoundedLinearOperators ((REAL-NS f),(REAL-NS m)))) is Element of BoundedLinearOperators ((REAL-NS f),(REAL-NS m))
K197((R_VectorSpace_of_LinearOperators ((REAL-NS f),(REAL-NS m))),(BoundedLinearOperators ((REAL-NS f),(REAL-NS m)))) is Relation-like K7((BoundedLinearOperators ((REAL-NS f),(REAL-NS m))),(BoundedLinearOperators ((REAL-NS f),(REAL-NS m)))) -defined BoundedLinearOperators ((REAL-NS f),(REAL-NS m)) -valued Function-like total quasi_total Element of K6(K7(K7((BoundedLinearOperators ((REAL-NS f),(REAL-NS m))),(BoundedLinearOperators ((REAL-NS f),(REAL-NS m)))),(BoundedLinearOperators ((REAL-NS f),(REAL-NS m)))))
K7((BoundedLinearOperators ((REAL-NS f),(REAL-NS m))),(BoundedLinearOperators ((REAL-NS f),(REAL-NS m)))) is set
K7(K7((BoundedLinearOperators ((REAL-NS f),(REAL-NS m))),(BoundedLinearOperators ((REAL-NS f),(REAL-NS m)))),(BoundedLinearOperators ((REAL-NS f),(REAL-NS m)))) is set
K6(K7(K7((BoundedLinearOperators ((REAL-NS f),(REAL-NS m))),(BoundedLinearOperators ((REAL-NS f),(REAL-NS m)))),(BoundedLinearOperators ((REAL-NS f),(REAL-NS m))))) is set
K198((R_VectorSpace_of_LinearOperators ((REAL-NS f),(REAL-NS m))),(BoundedLinearOperators ((REAL-NS f),(REAL-NS m)))) is Relation-like K7(REAL,(BoundedLinearOperators ((REAL-NS f),(REAL-NS m)))) -defined BoundedLinearOperators ((REAL-NS f),(REAL-NS m)) -valued Function-like total quasi_total Element of K6(K7(K7(REAL,(BoundedLinearOperators ((REAL-NS f),(REAL-NS m)))),(BoundedLinearOperators ((REAL-NS f),(REAL-NS m)))))
K7(REAL,(BoundedLinearOperators ((REAL-NS f),(REAL-NS m)))) is set
K7(K7(REAL,(BoundedLinearOperators ((REAL-NS f),(REAL-NS m)))),(BoundedLinearOperators ((REAL-NS f),(REAL-NS m)))) is set
K6(K7(K7(REAL,(BoundedLinearOperators ((REAL-NS f),(REAL-NS m)))),(BoundedLinearOperators ((REAL-NS f),(REAL-NS m))))) is set
BoundedLinearOperatorsNorm ((REAL-NS f),(REAL-NS m)) is non empty Relation-like BoundedLinearOperators ((REAL-NS f),(REAL-NS m)) -defined REAL -valued Function-like total quasi_total complex-valued ext-real-valued real-valued Element of K6(K7((BoundedLinearOperators ((REAL-NS f),(REAL-NS m))),REAL))
K7((BoundedLinearOperators ((REAL-NS f),(REAL-NS m))),REAL) is complex-valued ext-real-valued real-valued set
K6(K7((BoundedLinearOperators ((REAL-NS f),(REAL-NS m))),REAL)) is set
NORMSTR(# (BoundedLinearOperators ((REAL-NS f),(REAL-NS m))),K199((R_VectorSpace_of_LinearOperators ((REAL-NS f),(REAL-NS m))),(BoundedLinearOperators ((REAL-NS f),(REAL-NS m)))),K197((R_VectorSpace_of_LinearOperators ((REAL-NS f),(REAL-NS m))),(BoundedLinearOperators ((REAL-NS f),(REAL-NS m)))),K198((R_VectorSpace_of_LinearOperators ((REAL-NS f),(REAL-NS m))),(BoundedLinearOperators ((REAL-NS f),(REAL-NS m)))),(BoundedLinearOperatorsNorm ((REAL-NS f),(REAL-NS m))) #) is strict NORMSTR
the carrier of (R_NormSpace_of_BoundedLinearOperators ((REAL-NS f),(REAL-NS m))) is non empty set
(reproj (X,(0. (REAL-NS f)))) (#) (diff (r,s)) is Relation-like Function-like set
y1 is Relation-like the carrier of (REAL-NS f) -defined the carrier of (REAL-NS m) -valued Function-like Element of K6(K7( the carrier of (REAL-NS f), the carrier of (REAL-NS m)))
PP is Element of the carrier of (REAL-NS f)
partdiff (y1,PP,X) is Relation-like Function-like Element of the carrier of (R_NormSpace_of_BoundedLinearOperators ((REAL-NS 1),(REAL-NS m)))
reproj (X,PP) is non empty Relation-like the carrier of (REAL-NS 1) -defined the carrier of (REAL-NS f) -valued Function-like total quasi_total Element of K6(K7( the carrier of (REAL-NS 1), the carrier of (REAL-NS f)))
y1 * (reproj (X,PP)) is Relation-like the carrier of (REAL-NS 1) -defined the carrier of (REAL-NS m) -valued Function-like Element of K6(K7( the carrier of (REAL-NS 1), the carrier of (REAL-NS m)))
(Proj (X,f)) . PP is Element of the carrier of (REAL-NS 1)
diff ((y1 * (reproj (X,PP))),((Proj (X,f)) . PP)) is Relation-like Function-like Element of the carrier of (R_NormSpace_of_BoundedLinearOperators ((REAL-NS 1),(REAL-NS m)))
(partdiff (y1,PP,X)) . <*1*> is set
DD is Relation-like the carrier of (REAL-NS f) -defined the carrier of (REAL-NS m) -valued Function-like Element of K6(K7( the carrier of (REAL-NS f), the carrier of (REAL-NS m)))
a is Element of the carrier of (REAL-NS f)
diff (DD,a) is Relation-like Function-like Element of the carrier of (R_NormSpace_of_BoundedLinearOperators ((REAL-NS f),(REAL-NS m)))
m is non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below 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
{ b₁ where b₁ is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like Element of REAL * : len b₁ = m } is set
K7((REAL m),REAL) is complex-valued ext-real-valued real-valued set
K6(K7((REAL m),REAL)) is set
m is non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below 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
{ b₁ where b₁ is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like Element of REAL * : len b₁ = m } is set
K7((REAL m),REAL) is complex-valued ext-real-valued real-valued set
K6(K7((REAL m),REAL)) is set
f is Relation-like REAL m -defined REAL -valued Function-like complex-valued ext-real-valued real-valued Element of K6(K7((REAL m),REAL))
<>* f is Relation-like REAL m -defined REAL 1 -valued Function-like complex-functions-valued ext-real-functions-valued real-functions-valued Element of K6(K7((REAL m),(REAL 1)))
K7((REAL m),(REAL 1)) is set
K6(K7((REAL m),(REAL 1))) is set
f (#) ((proj (1,1)) ") is Relation-like Function-like set
X is Relation-like NAT -defined REAL -valued Function-like finite m -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL m
diff ((<>* f),X) is non empty Relation-like REAL m -defined REAL 1 -valued Function-like total quasi_total complex-functions-valued ext-real-functions-valued real-functions-valued Element of K6(K7((REAL m),(REAL 1)))
(proj (1,1)) * (diff ((<>* f),X)) is non empty Relation-like REAL m -defined REAL -valued Function-like total quasi_total complex-valued ext-real-valued real-valued Element of K6(K7((REAL m),REAL))
m is non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below 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
{ b₁ where b₁ is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like Element of REAL * : len b₁ = m } is set
K7((REAL m),REAL) is complex-valued ext-real-valued real-valued set
K6(K7((REAL m),REAL)) is set
0* m is Relation-like NAT -defined REAL -valued Function-like finite m -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL m
m |-> 0 is Relation-like empty-yielding NAT -defined REAL -valued Function-like finite m -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of m -tuples_on REAL
Seg m is non empty finite m -element complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end right_end bounded_below bounded_above real-bounded Element of K6(NAT)
{ b₁ where b₁ is ordinal natural V11() real ext-real non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of NAT : ( 1 <= b₁ & b₁ <= m ) } is set
K213((Seg m),0) is non empty Relation-like Seg m -defined INT -valued RAT -valued {0} -valued Function-like total quasi_total finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued natural-valued Element of K6(K7((Seg m),{0}))
{0} is non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below set
K7((Seg m),{0}) is INT -valued RAT -valued complex-valued ext-real-valued real-valued natural-valued set
K6(K7((Seg m),{0})) is set
f is ordinal natural V11() real ext-real non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of NAT
proj (f,m) is non empty Relation-like REAL m -defined REAL -valued Function-like total quasi_total complex-valued ext-real-valued real-valued Element of K6(K7((REAL m),REAL))
reproj (f,(0* m)) is non empty Relation-like REAL -defined REAL m -valued Function-like total quasi_total complex-functions-valued ext-real-functions-valued real-functions-valued Element of K6(K7(REAL,(REAL m)))
K7(REAL,(REAL m)) is set
K6(K7(REAL,(REAL m))) is set
(reproj (f,(0* m))) . 1 is Relation-like NAT -defined Function-like finite m -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL m
X is Relation-like REAL m -defined REAL -valued Function-like complex-valued ext-real-valued real-valued Element of K6(K7((REAL m),REAL))
i is Relation-like NAT -defined REAL -valued Function-like finite m -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL m
partdiff (X,i,f) is V11() real ext-real Element of REAL
reproj (f,i) is non empty Relation-like REAL -defined REAL m -valued Function-like total quasi_total complex-functions-valued ext-real-functions-valued real-functions-valued Element of K6(K7(REAL,(REAL m)))
X * (reproj (f,i)) is Relation-like REAL -defined REAL -valued Function-like complex-valued ext-real-valued real-valued Element of K6(K7(REAL,REAL))
(proj (f,m)) . i is V11() real ext-real Element of REAL
diff ((X * (reproj (f,i))),((proj (f,m)) . i)) is V11() real ext-real Element of REAL
(m,X,i) is non empty Relation-like REAL m -defined REAL -valued Function-like total quasi_total complex-valued ext-real-valued real-valued Element of K6(K7((REAL m),REAL))
<>* X is Relation-like REAL m -defined REAL 1 -valued Function-like complex-functions-valued ext-real-functions-valued real-functions-valued Element of K6(K7((REAL m),(REAL 1)))
K7((REAL m),(REAL 1)) is set
K6(K7((REAL m),(REAL 1))) is set
X (#) ((proj (1,1)) ") is Relation-like Function-like set
diff ((<>* X),i) is non empty Relation-like REAL m -defined REAL 1 -valued Function-like total quasi_total complex-functions-valued ext-real-functions-valued real-functions-valued Element of K6(K7((REAL m),(REAL 1)))
(proj (1,1)) * (diff ((<>* X),i)) is non empty Relation-like REAL m -defined REAL -valued Function-like total quasi_total complex-valued ext-real-valued real-valued Element of K6(K7((REAL m),REAL))
(m,X,i) . ((reproj (f,(0* m))) . 1) is V11() real ext-real Element of REAL
partdiff ((<>* X),i,f) is Relation-like NAT -defined REAL -valued Function-like finite 1 -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL 1
REAL-NS m is non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V185() NORMSTR
the carrier of (REAL-NS m) is non empty V2() set
0. (REAL-NS m) is V52( REAL-NS m) Element of the carrier of (REAL-NS m)
reproj (f,(0. (REAL-NS m))) is non empty Relation-like the carrier of (REAL-NS 1) -defined the carrier of (REAL-NS m) -valued Function-like total quasi_total Element of K6(K7( the carrier of (REAL-NS 1), the carrier of (REAL-NS m)))
K7( the carrier of (REAL-NS 1), the carrier of (REAL-NS m)) is set
K6(K7( the carrier of (REAL-NS 1), the carrier of (REAL-NS m))) is set
(diff ((<>* X),i)) * (reproj (f,(0. (REAL-NS m)))) is Relation-like the carrier of (REAL-NS 1) -defined REAL 1 -valued Function-like complex-functions-valued ext-real-functions-valued real-functions-valued Element of K6(K7( the carrier of (REAL-NS 1),(REAL 1)))
K7( the carrier of (REAL-NS 1),(REAL 1)) is set
K6(K7( the carrier of (REAL-NS 1),(REAL 1))) is set
((diff ((<>* X),i)) * (reproj (f,(0. (REAL-NS m))))) . <*1*> is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued 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
y0 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)))
r is Element of the carrier of (REAL-NS m)
<*(partdiff (X,i,f))*> is non empty V2() Relation-like NAT -defined REAL -valued Function-like one-to-one constant finite 1 -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued increasing decreasing non-decreasing non-increasing FinSequence of REAL
[1,(partdiff (X,i,f))] is V27() set
{1,(partdiff (X,i,f))} is non empty complex-membered ext-real-membered real-membered set
{{1,(partdiff (X,i,f))},{1}} is non empty set
{[1,(partdiff (X,i,f))]} is non empty Function-like set
y0 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)))
r is Element of the carrier of (REAL-NS m)
partdiff (y0,r,f) is Relation-like Function-like Element of the carrier of (R_NormSpace_of_BoundedLinearOperators ((REAL-NS 1),(REAL-NS 1)))
reproj (f,r) is non empty Relation-like the carrier of (REAL-NS 1) -defined the carrier of (REAL-NS m) -valued Function-like total quasi_total Element of K6(K7( the carrier of (REAL-NS 1), the carrier of (REAL-NS m)))
y0 * (reproj (f,r)) is Relation-like the carrier of (REAL-NS 1) -defined the carrier of (REAL-NS 1) -valued Function-like Element of K6(K7( the carrier of (REAL-NS 1), the carrier of (REAL-NS 1)))
Proj (f,m) is non empty Relation-like the carrier of (REAL-NS m) -defined the carrier of (REAL-NS 1) -valued Function-like total quasi_total Element of K6(K7( the carrier of (REAL-NS m), the carrier of (REAL-NS 1)))
(Proj (f,m)) . r is Element of the carrier of (REAL-NS 1)
diff ((y0 * (reproj (f,r))),((Proj (f,m)) . r)) is Relation-like Function-like Element of the carrier of (R_NormSpace_of_BoundedLinearOperators ((REAL-NS 1),(REAL-NS 1)))
(partdiff (y0,r,f)) . <*1*> is set
s 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)))
y1 is Element of the carrier of (REAL-NS m)
(reproj (f,(0. (REAL-NS m)))) . <*1*> is set
(diff ((<>* X),i)) . ((reproj (f,(0. (REAL-NS m)))) . <*1*>) is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
y0 is V11() real ext-real Element of REAL
<*y0*> is non empty V2() Relation-like NAT -defined REAL -valued Function-like one-to-one constant finite 1 -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued increasing decreasing non-decreasing non-increasing FinSequence of REAL
[1,y0] is V27() set
{1,y0} is non empty complex-membered ext-real-membered real-membered set
{{1,y0},{1}} is non empty set
{[1,y0]} is non empty Function-like set
r is Relation-like NAT -defined REAL -valued Function-like finite m -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL m
reproj (f,r) is non empty Relation-like REAL -defined REAL m -valued Function-like total quasi_total complex-functions-valued ext-real-functions-valued real-functions-valued Element of K6(K7(REAL,(REAL m)))
(reproj (f,r)) . y0 is Relation-like NAT -defined Function-like finite m -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL m
(diff ((<>* X),i)) . ((reproj (f,(0* m))) . 1) is Relation-like NAT -defined REAL -valued Function-like finite 1 -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL 1
(proj (1,1)) . ((diff ((<>* X),i)) . ((reproj (f,(0* m))) . 1)) is V11() real ext-real Element of REAL
m is non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below 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
{ b₁ where b₁ is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like Element of REAL * : len b₁ = m } is set
i is Relation-like NAT -defined REAL m -valued Function-like finite FinSequence-like FinSubsequence-like complex-functions-valued ext-real-functions-valued real-functions-valued FinSequence of REAL m
len i is ordinal natural V11() real ext-real non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of NAT
f is Relation-like NAT -defined REAL m -valued Function-like finite FinSequence-like FinSubsequence-like complex-functions-valued ext-real-functions-valued real-functions-valued FinSequence of REAL m
len f is ordinal natural V11() real ext-real non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of NAT
X is Relation-like NAT -defined REAL m -valued Function-like finite FinSequence-like FinSubsequence-like complex-functions-valued ext-real-functions-valued real-functions-valued FinSequence of REAL m
len X is ordinal natural V11() real ext-real non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of NAT
dom i is complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of K6(NAT)
Sum i is Relation-like NAT -defined REAL -valued Function-like finite m -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL m
Sum f is Relation-like NAT -defined REAL -valued Function-like finite m -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL m
Sum X is Relation-like NAT -defined REAL -valued Function-like finite m -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL m
(Sum f) + (Sum X) is Relation-like NAT -defined REAL -valued Function-like finite m -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL m
0* m is Relation-like NAT -defined REAL -valued Function-like finite m -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL m
m |-> 0 is Relation-like empty-yielding NAT -defined REAL -valued Function-like finite m -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of m -tuples_on REAL
Seg m is non empty finite m -element complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end right_end bounded_below bounded_above real-bounded Element of K6(NAT)
{ b₁ where b₁ is ordinal natural V11() real ext-real non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of NAT : ( 1 <= b₁ & b₁ <= m ) } is set
K213((Seg m),0) is non empty Relation-like Seg m -defined INT -valued RAT -valued {0} -valued Function-like total quasi_total finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued natural-valued Element of K6(K7((Seg m),{0}))
{0} is non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below set
K7((Seg m),{0}) is INT -valued RAT -valued complex-valued ext-real-valued real-valued natural-valued set
K6(K7((Seg m),{0})) is set
TOP-REAL m is non empty V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital TopSpace-like V148() V214() V215() L19()
0. (TOP-REAL m) is Relation-like NAT -defined Function-like finite m -element FinSequence-like FinSubsequence-like V52( TOP-REAL m) complex-valued ext-real-valued real-valued Element of the carrier of (TOP-REAL m)
the carrier of (TOP-REAL m) is non empty set
(0. (TOP-REAL m)) + (0. (TOP-REAL m)) is Relation-like NAT -defined Function-like finite m -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of the carrier of (TOP-REAL m)
(0. (TOP-REAL m)) + (0. (TOP-REAL m)) is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
(0. (TOP-REAL m)) + (0. (TOP-REAL m)) is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
f is ordinal natural V11() real ext-real non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of NAT
f + 1 is non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below Element of NAT
y0 is Relation-like NAT -defined REAL m -valued Function-like finite FinSequence-like FinSubsequence-like complex-functions-valued ext-real-functions-valued real-functions-valued FinSequence of REAL m
len y0 is ordinal natural V11() real ext-real non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of NAT
X is Relation-like NAT -defined REAL m -valued Function-like finite FinSequence-like FinSubsequence-like complex-functions-valued ext-real-functions-valued real-functions-valued FinSequence of REAL m
len X is ordinal natural V11() real ext-real non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of NAT
i is Relation-like NAT -defined REAL m -valued Function-like finite FinSequence-like FinSubsequence-like complex-functions-valued ext-real-functions-valued real-functions-valued FinSequence of REAL m
len i is ordinal natural V11() real ext-real non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of NAT
dom y0 is complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of K6(NAT)
X | f is Relation-like NAT -defined REAL m -valued Function-like finite FinSequence-like FinSubsequence-like complex-functions-valued ext-real-functions-valued real-functions-valued FinSequence of REAL m
Seg f is finite f -element complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below bounded_above real-bounded Element of K6(NAT)
{ b₁ where b₁ is ordinal natural V11() real ext-real non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of NAT : ( 1 <= b₁ & b₁ <= f ) } is set
X | (Seg f) is Relation-like NAT -defined Function-like FinSubsequence-like set
i | f is Relation-like NAT -defined REAL m -valued Function-like finite FinSequence-like FinSubsequence-like complex-functions-valued ext-real-functions-valued real-functions-valued FinSequence of REAL m
i | (Seg f) is Relation-like NAT -defined Function-like FinSubsequence-like set
y0 | f is Relation-like NAT -defined REAL m -valued Function-like finite FinSequence-like FinSubsequence-like complex-functions-valued ext-real-functions-valued real-functions-valued FinSequence of REAL m
y0 | (Seg f) is Relation-like NAT -defined Function-like FinSubsequence-like set
dom X is complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of K6(NAT)
dom i is complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of K6(NAT)
len (X | f) is ordinal natural V11() real ext-real non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of NAT
len (i | f) is ordinal natural V11() real ext-real non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of NAT
len (y0 | f) is ordinal natural V11() real ext-real non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of NAT
dom (y0 | f) is complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of K6(NAT)
PP is ordinal natural V11() real ext-real non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of NAT
(y0 | f) /. PP is Relation-like NAT -defined Function-like finite m -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL m
(X | f) /. PP is Relation-like NAT -defined Function-like finite m -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL m
(i | f) /. PP is Relation-like NAT -defined Function-like finite m -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL m
((X | f) /. PP) + ((i | f) /. PP) is Relation-like NAT -defined REAL -valued Function-like finite m -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL m
i | (Seg f) is Relation-like NAT -defined REAL m -valued Function-like FinSubsequence-like complex-functions-valued ext-real-functions-valued real-functions-valued Element of K6(K7(NAT,(REAL m)))
K7(NAT,(REAL m)) is set
K6(K7(NAT,(REAL m))) is set
dom (i | (Seg f)) is complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of K6(NAT)
X | (Seg f) is Relation-like NAT -defined REAL m -valued Function-like FinSubsequence-like complex-functions-valued ext-real-functions-valued real-functions-valued Element of K6(K7(NAT,(REAL m)))
dom (X | (Seg f)) is complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of K6(NAT)
y0 | (Seg f) is Relation-like NAT -defined REAL m -valued Function-like FinSubsequence-like complex-functions-valued ext-real-functions-valued real-functions-valued Element of K6(K7(NAT,(REAL m)))
dom (y0 | (Seg f)) is complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of K6(NAT)
y0 /. PP is Relation-like NAT -defined Function-like finite m -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL m
X /. PP is Relation-like NAT -defined Function-like finite m -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL m
i /. PP is Relation-like NAT -defined Function-like finite m -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL m
(X /. PP) + (i /. PP) is Relation-like NAT -defined REAL -valued Function-like finite m -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL m
X . PP is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
(X | (Seg f)) . PP is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
(X | (Seg f)) /. PP is Relation-like NAT -defined Function-like finite m -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL m
i . PP is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
(i | (Seg f)) . PP is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
(i | (Seg f)) /. PP is Relation-like NAT -defined Function-like finite m -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL m
(y0 | (Seg f)) . PP is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
y0 . PP is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
Sum (y0 | f) is Relation-like NAT -defined REAL -valued Function-like finite m -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL m
Sum (X | f) is Relation-like NAT -defined REAL -valued Function-like finite m -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL m
Sum (i | f) is Relation-like NAT -defined REAL -valued Function-like finite m -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL m
(Sum (X | f)) + (Sum (i | f)) is Relation-like NAT -defined REAL -valued Function-like finite m -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL m
X . (len X) is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
Sum X is Relation-like NAT -defined REAL -valued Function-like finite m -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL m
PP is Relation-like NAT -defined REAL -valued Function-like finite m -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL m
(Sum (X | f)) + PP is Relation-like NAT -defined REAL -valued Function-like finite m -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL m
i . (len i) is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
Sum i is Relation-like NAT -defined REAL -valued Function-like finite m -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL m
DD is Relation-like NAT -defined REAL -valued Function-like finite m -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL m
(Sum (i | f)) + DD is Relation-like NAT -defined REAL -valued Function-like finite m -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL m
y0 . (len y0) is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
Sum y0 is Relation-like NAT -defined REAL -valued Function-like finite m -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL m
a is Relation-like NAT -defined REAL -valued Function-like finite m -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL m
(Sum (y0 | f)) + a is Relation-like NAT -defined REAL -valued Function-like finite m -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL m
Seg (f + 1) is non empty finite f + 1 -element complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end right_end bounded_below bounded_above real-bounded Element of K6(NAT)
{ b₁ where b₁ is ordinal natural V11() real ext-real non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of NAT : ( 1 <= b₁ & b₁ <= f + 1 ) } is set
y0 /. (len y0) is Relation-like NAT -defined Function-like finite m -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL m
X /. (len X) is Relation-like NAT -defined Function-like finite m -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL m
i /. (len i) is Relation-like NAT -defined Function-like finite m -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL m
(X /. (len X)) + (i /. (len i)) is Relation-like NAT -defined REAL -valued Function-like finite m -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL m
t is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
tm is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
t + tm is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
Sw0 is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
(t + tm) + Sw0 is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
w1 is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
((t + tm) + Sw0) + w1 is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
t + Sw0 is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
(t + Sw0) + tm is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
((t + Sw0) + tm) + w1 is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
(Sum X) + (Sum i) is Relation-like NAT -defined REAL -valued Function-like finite m -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL m
y0 is Relation-like NAT -defined REAL m -valued Function-like finite FinSequence-like FinSubsequence-like complex-functions-valued ext-real-functions-valued real-functions-valued FinSequence of REAL m
len y0 is ordinal natural V11() real ext-real non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of NAT
X is Relation-like NAT -defined REAL m -valued Function-like finite FinSequence-like FinSubsequence-like complex-functions-valued ext-real-functions-valued real-functions-valued FinSequence of REAL m
len X is ordinal natural V11() real ext-real non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of NAT
i is Relation-like NAT -defined REAL m -valued Function-like finite FinSequence-like FinSubsequence-like complex-functions-valued ext-real-functions-valued real-functions-valued FinSequence of REAL m
len i is ordinal natural V11() real ext-real non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of NAT
dom y0 is complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of K6(NAT)
Sum y0 is Relation-like NAT -defined REAL -valued Function-like finite m -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL m
Sum X is Relation-like NAT -defined REAL -valued Function-like finite m -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL m
Sum i is Relation-like NAT -defined REAL -valued Function-like finite m -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL m
(Sum X) + (Sum i) is Relation-like NAT -defined REAL -valued Function-like finite m -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL m
f is Relation-like NAT -defined REAL m -valued Function-like finite FinSequence-like FinSubsequence-like complex-functions-valued ext-real-functions-valued real-functions-valued FinSequence of REAL m
dom f is complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of K6(NAT)
X is Relation-like NAT -defined REAL m -valued Function-like finite FinSequence-like FinSubsequence-like complex-functions-valued ext-real-functions-valued real-functions-valued FinSequence of REAL m
dom X is complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of K6(NAT)
i is Relation-like NAT -defined REAL m -valued Function-like finite FinSequence-like FinSubsequence-like complex-functions-valued ext-real-functions-valued real-functions-valued FinSequence of REAL m
f + X 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 set
K6(K7(NAT,(REAL m))) is set
Sum i is Relation-like NAT -defined REAL -valued Function-like finite m -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL m
Sum f is Relation-like NAT -defined REAL -valued Function-like finite m -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL m
Sum X is Relation-like NAT -defined REAL -valued Function-like finite m -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL m
(Sum f) + (Sum X) is Relation-like NAT -defined REAL -valued Function-like finite m -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL m
len X is ordinal natural V11() real ext-real non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of NAT
len f is ordinal natural V11() real ext-real non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of NAT
f <++> X is Relation-like NAT /\ NAT -defined R_PFuncs ((DOMS (REAL m)) /\ (DOMS (REAL m))) -valued Function-like complex-functions-valued ext-real-functions-valued real-functions-valued Element of K6(K7((NAT /\ NAT),(R_PFuncs ((DOMS (REAL m)) /\ (DOMS (REAL m))))))
NAT /\ NAT is complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below set
DOMS (REAL m) is set
(DOMS (REAL m)) /\ (DOMS (REAL m)) is set
R_PFuncs ((DOMS (REAL m)) /\ (DOMS (REAL m))) is functional complex-functions-membered ext-real-functions-membered real-functions-membered set
K7((NAT /\ NAT),(R_PFuncs ((DOMS (REAL m)) /\ (DOMS (REAL m))))) is set
K6(K7((NAT /\ NAT),(R_PFuncs ((DOMS (REAL m)) /\ (DOMS (REAL m)))))) is set
dom i is complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of K6(NAT)
(dom f) /\ (dom X) is complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of K6(NAT)
len i is ordinal natural V11() real ext-real non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of NAT
y0 is ordinal natural V11() real ext-real non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of NAT
i /. y0 is Relation-like NAT -defined Function-like finite m -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL m
f /. y0 is Relation-like NAT -defined Function-like finite m -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL m
X /. y0 is Relation-like NAT -defined Function-like finite m -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL m
(f /. y0) + (X /. y0) is Relation-like NAT -defined REAL -valued Function-like finite m -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL m
m is non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below 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
{ b₁ where b₁ is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like Element of REAL * : len b₁ = m } is set
f is V11() real ext-real Element of REAL
X is Relation-like NAT -defined REAL m -valued Function-like finite FinSequence-like FinSubsequence-like complex-functions-valued ext-real-functions-valued real-functions-valued FinSequence of REAL m
len X is ordinal natural V11() real ext-real non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of NAT
i is Relation-like NAT -defined REAL m -valued Function-like finite FinSequence-like FinSubsequence-like complex-functions-valued ext-real-functions-valued real-functions-valued FinSequence of REAL m
len i is ordinal natural V11() real ext-real non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of NAT
dom X is complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of K6(NAT)
Sum i is Relation-like NAT -defined REAL -valued Function-like finite m -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL m
Sum X is Relation-like NAT -defined REAL -valued Function-like finite m -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL m
f * (Sum X) is Relation-like NAT -defined REAL -valued Function-like finite m -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL m
0* m is Relation-like NAT -defined REAL -valued Function-like finite m -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL m
m |-> 0 is Relation-like empty-yielding NAT -defined REAL -valued Function-like finite m -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of m -tuples_on REAL
Seg m is non empty finite m -element complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end right_end bounded_below bounded_above real-bounded Element of K6(NAT)
{ b₁ where b₁ is ordinal natural V11() real ext-real non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of NAT : ( 1 <= b₁ & b₁ <= m ) } is set
K213((Seg m),0) is non empty Relation-like Seg m -defined INT -valued RAT -valued {0} -valued Function-like total quasi_total finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued natural-valued Element of K6(K7((Seg m),{0}))
{0} is non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below set
K7((Seg m),{0}) is INT -valued RAT -valued complex-valued ext-real-valued real-valued natural-valued set
K6(K7((Seg m),{0})) is set
TOP-REAL m is non empty V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital TopSpace-like V148() V214() V215() L19()
0. (TOP-REAL m) is Relation-like NAT -defined Function-like finite m -element FinSequence-like FinSubsequence-like V52( TOP-REAL m) complex-valued ext-real-valued real-valued Element of the carrier of (TOP-REAL m)
the carrier of (TOP-REAL m) is non empty set
y0 is V11() real ext-real set
y0 * (0. (TOP-REAL m)) is Relation-like NAT -defined Function-like finite m -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of the carrier of (TOP-REAL m)
y0 * (0. (TOP-REAL m)) is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
X is ordinal natural V11() real ext-real non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of NAT
X + 1 is non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below Element of NAT
i is Relation-like NAT -defined REAL m -valued Function-like finite FinSequence-like FinSubsequence-like complex-functions-valued ext-real-functions-valued real-functions-valued FinSequence of REAL m
len i is ordinal natural V11() real ext-real non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of NAT
y0 is Relation-like NAT -defined REAL m -valued Function-like finite FinSequence-like FinSubsequence-like complex-functions-valued ext-real-functions-valued real-functions-valued FinSequence of REAL m
len y0 is ordinal natural V11() real ext-real non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of NAT
dom i is complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of K6(NAT)
dom y0 is complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of K6(NAT)
i | X is Relation-like NAT -defined REAL m -valued Function-like finite FinSequence-like FinSubsequence-like complex-functions-valued ext-real-functions-valued real-functions-valued FinSequence of REAL m
Seg X is finite X -element complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below bounded_above real-bounded Element of K6(NAT)
{ b₁ where b₁ is ordinal natural V11() real ext-real non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of NAT : ( 1 <= b₁ & b₁ <= X ) } is set
i | (Seg X) is Relation-like NAT -defined Function-like FinSubsequence-like set
y0 | X is Relation-like NAT -defined REAL m -valued Function-like finite FinSequence-like FinSubsequence-like complex-functions-valued ext-real-functions-valued real-functions-valued FinSequence of REAL m
y0 | (Seg X) is Relation-like NAT -defined Function-like FinSubsequence-like set
len (i | X) is ordinal natural V11() real ext-real non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of NAT
len (y0 | X) is ordinal natural V11() real ext-real non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of NAT
dom (i | X) is complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of K6(NAT)
y1 is ordinal natural V11() real ext-real non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of NAT
(y0 | X) /. y1 is Relation-like NAT -defined Function-like finite m -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL m
(i | X) /. y1 is Relation-like NAT -defined Function-like finite m -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL m
f * ((i | X) /. y1) is Relation-like NAT -defined REAL -valued Function-like finite m -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL m
y0 | (Seg X) is Relation-like NAT -defined REAL m -valued Function-like FinSubsequence-like complex-functions-valued ext-real-functions-valued real-functions-valued Element of K6(K7(NAT,(REAL m)))
K7(NAT,(REAL m)) is set
K6(K7(NAT,(REAL m))) is set
dom (y0 | (Seg X)) is complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of K6(NAT)
i /. y1 is Relation-like NAT -defined Function-like finite m -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL m
i . y1 is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
i | (Seg X) is Relation-like NAT -defined REAL m -valued Function-like FinSubsequence-like complex-functions-valued ext-real-functions-valued real-functions-valued Element of K6(K7(NAT,(REAL m)))
(i | (Seg X)) . y1 is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
(y0 | (Seg X)) . y1 is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
y0 . y1 is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
y0 /. y1 is Relation-like NAT -defined Function-like finite m -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL m
Sum (y0 | X) is Relation-like NAT -defined REAL -valued Function-like finite m -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL m
Sum (i | X) is Relation-like NAT -defined REAL -valued Function-like finite m -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL m
f * (Sum (i | X)) is Relation-like NAT -defined REAL -valued Function-like finite m -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL m
i . (len i) is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
Sum i is Relation-like NAT -defined REAL -valued Function-like finite m -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL m
y1 is Relation-like NAT -defined REAL -valued Function-like finite m -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL m
(Sum (i | X)) + y1 is Relation-like NAT -defined REAL -valued Function-like finite m -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL m
y0 . (len y0) is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
Sum y0 is Relation-like NAT -defined REAL -valued Function-like finite m -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL m
PP is Relation-like NAT -defined REAL -valued Function-like finite m -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL m
(Sum (y0 | X)) + PP is Relation-like NAT -defined REAL -valued Function-like finite m -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL m
Seg (X + 1) is non empty finite X + 1 -element complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end right_end bounded_below bounded_above real-bounded Element of K6(NAT)
{ b₁ where b₁ is ordinal natural V11() real ext-real non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of NAT : ( 1 <= b₁ & b₁ <= X + 1 ) } is set
y0 /. (len y0) is Relation-like NAT -defined Function-like finite m -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL m
i /. (len y0) is Relation-like NAT -defined Function-like finite m -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL m
f * (i /. (len y0)) is Relation-like NAT -defined REAL -valued Function-like finite m -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL m
f * y1 is Relation-like NAT -defined REAL -valued Function-like finite m -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL m
f * (Sum i) is Relation-like NAT -defined REAL -valued Function-like finite m -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL m
i is Relation-like NAT -defined REAL m -valued Function-like finite FinSequence-like FinSubsequence-like complex-functions-valued ext-real-functions-valued real-functions-valued FinSequence of REAL m
len i is ordinal natural V11() real ext-real non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of NAT
y0 is Relation-like NAT -defined REAL m -valued Function-like finite FinSequence-like FinSubsequence-like complex-functions-valued ext-real-functions-valued real-functions-valued FinSequence of REAL m
len y0 is ordinal natural V11() real ext-real non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of NAT
dom i is complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of K6(NAT)
Sum y0 is Relation-like NAT -defined REAL -valued Function-like finite m -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL m
Sum i is Relation-like NAT -defined REAL -valued Function-like finite m -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL m
f * (Sum i) is Relation-like NAT -defined REAL -valued Function-like finite m -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL m
i is Relation-like NAT -defined REAL m -valued Function-like finite FinSequence-like FinSubsequence-like complex-functions-valued ext-real-functions-valued real-functions-valued FinSequence of REAL m
X is Relation-like NAT -defined REAL m -valued Function-like finite FinSequence-like FinSubsequence-like complex-functions-valued ext-real-functions-valued real-functions-valued FinSequence of REAL m
f (#) X 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 set
K6(K7(NAT,(REAL m))) is set
Sum i is Relation-like NAT -defined REAL -valued Function-like finite m -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL m
Sum X is Relation-like NAT -defined REAL -valued Function-like finite m -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL m
f * (Sum X) is Relation-like NAT -defined REAL -valued Function-like finite m -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL m
X [#] f is Relation-like NAT -defined R_PFuncs (DOMS (REAL m)) -valued Function-like complex-functions-valued ext-real-functions-valued real-functions-valued Element of K6(K7(NAT,(R_PFuncs (DOMS (REAL m)))))
DOMS (REAL m) is set
R_PFuncs (DOMS (REAL m)) is functional complex-functions-membered ext-real-functions-membered real-functions-membered set
K7(NAT,(R_PFuncs (DOMS (REAL m)))) is set
K6(K7(NAT,(R_PFuncs (DOMS (REAL m))))) is set
dom i is complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of K6(NAT)
dom X is complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of K6(NAT)
len X is ordinal natural V11() real ext-real non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of NAT
len i is ordinal natural V11() real ext-real non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of NAT
y0 is ordinal natural V11() real ext-real non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of NAT
i /. y0 is Relation-like NAT -defined Function-like finite m -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL m
X /. y0 is Relation-like NAT -defined Function-like finite m -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL m
f * (X /. y0) is Relation-like NAT -defined REAL -valued Function-like finite m -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL m
m is non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below 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
{ b₁ where b₁ is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like Element of REAL * : len b₁ = m } is set
f is non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below 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
{ b₁ where b₁ is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like Element of REAL * : len b₁ = f } is set
K7((REAL m),(REAL f)) is set
K6(K7((REAL m),(REAL f))) is set
Seg m is non empty finite m -element complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end right_end bounded_below bounded_above real-bounded Element of K6(NAT)
{ b₁ where b₁ is ordinal natural V11() real ext-real non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of NAT : ( 1 <= b₁ & b₁ <= m ) } is set
X is Relation-like REAL m -defined REAL f -valued Function-like complex-functions-valued ext-real-functions-valued real-functions-valued Element of K6(K7((REAL m),(REAL f)))
i is Relation-like NAT -defined REAL -valued Function-like finite m -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL m
y0 is Relation-like NAT -defined REAL -valued Function-like finite m -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL m
r is ordinal natural V11() real ext-real non negative set
s is ordinal natural V11() real ext-real non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of NAT
partdiff (X,i,s) is Relation-like NAT -defined REAL -valued Function-like finite f -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL f
proj (s,m) is non empty Relation-like REAL m -defined REAL -valued Function-like total quasi_total complex-valued ext-real-valued real-valued Element of K6(K7((REAL m),REAL))
K7((REAL m),REAL) is complex-valued ext-real-valued real-valued set
K6(K7((REAL m),REAL)) is set
(proj (s,m)) . y0 is V11() real ext-real Element of REAL
((proj (s,m)) . y0) * (partdiff (X,i,s)) is Relation-like NAT -defined REAL -valued Function-like finite f -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL f
r is Relation-like NAT -defined REAL f -valued Function-like finite FinSequence-like FinSubsequence-like complex-functions-valued ext-real-functions-valued real-functions-valued FinSequence of REAL f
dom r is complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of K6(NAT)
s is ordinal natural V11() real ext-real non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of NAT
r . s is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
partdiff (X,i,s) is Relation-like NAT -defined REAL -valued Function-like finite f -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL f
proj (s,m) is non empty Relation-like REAL m -defined REAL -valued Function-like total quasi_total complex-valued ext-real-valued real-valued Element of K6(K7((REAL m),REAL))
K7((REAL m),REAL) is complex-valued ext-real-valued real-valued set
K6(K7((REAL m),REAL)) is set
(proj (s,m)) . y0 is V11() real ext-real Element of REAL
((proj (s,m)) . y0) * (partdiff (X,i,s)) is Relation-like NAT -defined REAL -valued Function-like finite f -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL f
y1 is ordinal natural V11() real ext-real non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of NAT
partdiff (X,i,y1) is Relation-like NAT -defined REAL -valued Function-like finite f -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL f
proj (y1,m) is non empty Relation-like REAL m -defined REAL -valued Function-like total quasi_total complex-valued ext-real-valued real-valued Element of K6(K7((REAL m),REAL))
(proj (y1,m)) . y0 is V11() real ext-real Element of REAL
((proj (y1,m)) . y0) * (partdiff (X,i,y1)) is Relation-like NAT -defined REAL -valued Function-like finite f -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL f
Sum r is Relation-like NAT -defined REAL -valued Function-like finite f -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL f
s is Relation-like NAT -defined REAL -valued Function-like finite f -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL f
y1 is Relation-like NAT -defined REAL f -valued Function-like finite FinSequence-like FinSubsequence-like complex-functions-valued ext-real-functions-valued real-functions-valued FinSequence of REAL f
dom y1 is complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of K6(NAT)
Sum y1 is Relation-like NAT -defined REAL -valued Function-like finite f -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL f
y0 is non empty Relation-like REAL m -defined REAL f -valued Function-like total quasi_total complex-functions-valued ext-real-functions-valued real-functions-valued Element of K6(K7((REAL m),(REAL f)))
r is Relation-like NAT -defined REAL -valued Function-like finite m -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL m
s is Relation-like NAT -defined REAL -valued Function-like finite m -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL m
r + s is Relation-like NAT -defined REAL -valued Function-like finite m -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL m
y0 . (r + s) is Relation-like NAT -defined REAL -valued Function-like finite f -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL f
y0 . r is Relation-like NAT -defined REAL -valued Function-like finite f -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL f
y0 . s is Relation-like NAT -defined REAL -valued Function-like finite f -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL f
(y0 . r) + (y0 . s) is Relation-like NAT -defined REAL -valued Function-like finite f -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL f
y1 is Relation-like NAT -defined REAL f -valued Function-like finite FinSequence-like FinSubsequence-like complex-functions-valued ext-real-functions-valued real-functions-valued FinSequence of REAL f
dom y1 is complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of K6(NAT)
Sum y1 is Relation-like NAT -defined REAL -valued Function-like finite f -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL f
PP is Relation-like NAT -defined REAL f -valued Function-like finite FinSequence-like FinSubsequence-like complex-functions-valued ext-real-functions-valued real-functions-valued FinSequence of REAL f
dom PP is complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of K6(NAT)
Sum PP is Relation-like NAT -defined REAL -valued Function-like finite f -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL f
DD is Relation-like NAT -defined REAL f -valued Function-like finite FinSequence-like FinSubsequence-like complex-functions-valued ext-real-functions-valued real-functions-valued FinSequence of REAL f
dom DD is complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of K6(NAT)
Sum DD is Relation-like NAT -defined REAL -valued Function-like finite f -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL f
y1 + PP is Relation-like NAT -defined REAL f -valued Function-like complex-functions-valued ext-real-functions-valued real-functions-valued Element of K6(K7(NAT,(REAL f)))
K7(NAT,(REAL f)) is set
K6(K7(NAT,(REAL f))) is set
y1 <++> PP is Relation-like NAT /\ NAT -defined R_PFuncs ((DOMS (REAL f)) /\ (DOMS (REAL f))) -valued Function-like complex-functions-valued ext-real-functions-valued real-functions-valued Element of K6(K7((NAT /\ NAT),(R_PFuncs ((DOMS (REAL f)) /\ (DOMS (REAL f))))))
NAT /\ NAT is complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below set
DOMS (REAL f) is set
(DOMS (REAL f)) /\ (DOMS (REAL f)) is set
R_PFuncs ((DOMS (REAL f)) /\ (DOMS (REAL f))) is functional complex-functions-membered ext-real-functions-membered real-functions-membered set
K7((NAT /\ NAT),(R_PFuncs ((DOMS (REAL f)) /\ (DOMS (REAL f))))) is set
K6(K7((NAT /\ NAT),(R_PFuncs ((DOMS (REAL f)) /\ (DOMS (REAL f)))))) is set
(dom y1) /\ (dom PP) is complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of K6(NAT)
a is set
t is ordinal natural V11() real ext-real non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of NAT
y1 . t is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
partdiff (X,i,t) is Relation-like NAT -defined REAL -valued Function-like finite f -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL f
proj (t,m) is non empty Relation-like REAL m -defined REAL -valued Function-like total quasi_total complex-valued ext-real-valued real-valued Element of K6(K7((REAL m),REAL))
K7((REAL m),REAL) is complex-valued ext-real-valued real-valued set
K6(K7((REAL m),REAL)) is set
(proj (t,m)) . r is V11() real ext-real Element of REAL
((proj (t,m)) . r) * (partdiff (X,i,t)) is Relation-like NAT -defined REAL -valued Function-like finite f -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL f
PP . t is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
(proj (t,m)) . s is V11() real ext-real Element of REAL
((proj (t,m)) . s) * (partdiff (X,i,t)) is Relation-like NAT -defined REAL -valued Function-like finite f -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL f
DD . a is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
(proj (t,m)) . (r + s) is V11() real ext-real Element of REAL
((proj (t,m)) . (r + s)) * (partdiff (X,i,t)) is Relation-like NAT -defined REAL -valued Function-like finite f -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL f
(r + s) . t is V11() real ext-real Element of REAL
((r + s) . t) * (partdiff (X,i,t)) is Relation-like NAT -defined REAL -valued Function-like finite f -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL f
r . t is V11() real ext-real Element of REAL
s . t is V11() real ext-real Element of REAL
(r . t) + (s . t) is V11() real ext-real Element of REAL
((r . t) + (s . t)) * (partdiff (X,i,t)) is Relation-like NAT -defined REAL -valued Function-like finite f -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL f
((proj (t,m)) . r) + (s . t) is V11() real ext-real Element of REAL
(((proj (t,m)) . r) + (s . t)) * (partdiff (X,i,t)) is Relation-like NAT -defined REAL -valued Function-like finite f -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL f
((proj (t,m)) . r) + ((proj (t,m)) . s) is V11() real ext-real Element of REAL
(((proj (t,m)) . r) + ((proj (t,m)) . s)) * (partdiff (X,i,t)) is Relation-like NAT -defined REAL -valued Function-like finite f -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL f
y1 . a is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
PP . a is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
(y1 . a) + (PP . a) is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
r is Relation-like NAT -defined REAL -valued Function-like finite m -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL m
y0 . r is Relation-like NAT -defined REAL -valued Function-like finite f -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL f
s is V11() real ext-real Element of REAL
s * r is Relation-like NAT -defined REAL -valued Function-like finite m -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL m
y0 . (s * r) is Relation-like NAT -defined REAL -valued Function-like finite f -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL f
s * (y0 . r) is Relation-like NAT -defined REAL -valued Function-like finite f -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL f
y1 is Relation-like NAT -defined REAL f -valued Function-like finite FinSequence-like FinSubsequence-like complex-functions-valued ext-real-functions-valued real-functions-valued FinSequence of REAL f
dom y1 is complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of K6(NAT)
Sum y1 is Relation-like NAT -defined REAL -valued Function-like finite f -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL f
PP is Relation-like NAT -defined REAL f -valued Function-like finite FinSequence-like FinSubsequence-like complex-functions-valued ext-real-functions-valued real-functions-valued FinSequence of REAL f
dom PP is complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of K6(NAT)
Sum PP is Relation-like NAT -defined REAL -valued Function-like finite f -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL f
s (#) y1 is Relation-like NAT -defined REAL f -valued Function-like complex-functions-valued ext-real-functions-valued real-functions-valued Element of K6(K7(NAT,(REAL f)))
K7(NAT,(REAL f)) is set
K6(K7(NAT,(REAL f))) is set
y1 [#] s is Relation-like NAT -defined R_PFuncs (DOMS (REAL f)) -valued Function-like complex-functions-valued ext-real-functions-valued real-functions-valued Element of K6(K7(NAT,(R_PFuncs (DOMS (REAL f)))))
DOMS (REAL f) is set
R_PFuncs (DOMS (REAL f)) is functional complex-functions-membered ext-real-functions-membered real-functions-membered set
K7(NAT,(R_PFuncs (DOMS (REAL f)))) is set
K6(K7(NAT,(R_PFuncs (DOMS (REAL f))))) is set
DD is set
a is ordinal natural V11() real ext-real non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of NAT
y1 . a is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
partdiff (X,i,a) is Relation-like NAT -defined REAL -valued Function-like finite f -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL f
proj (a,m) is non empty Relation-like REAL m -defined REAL -valued Function-like total quasi_total complex-valued ext-real-valued real-valued Element of K6(K7((REAL m),REAL))
K7((REAL m),REAL) is complex-valued ext-real-valued real-valued set
K6(K7((REAL m),REAL)) is set
(proj (a,m)) . r is V11() real ext-real Element of REAL
((proj (a,m)) . r) * (partdiff (X,i,a)) is Relation-like NAT -defined REAL -valued Function-like finite f -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL f
PP . DD is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
(proj (a,m)) . (s * r) is V11() real ext-real Element of REAL
((proj (a,m)) . (s * r)) * (partdiff (X,i,a)) is Relation-like NAT -defined REAL -valued Function-like finite f -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL f
(s * r) . a is V11() real ext-real Element of REAL
((s * r) . a) * (partdiff (X,i,a)) is Relation-like NAT -defined REAL -valued Function-like finite f -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL f
r . a is V11() real ext-real Element of REAL
s * (r . a) is V11() real ext-real Element of REAL
(s * (r . a)) * (partdiff (X,i,a)) is Relation-like NAT -defined REAL -valued Function-like finite f -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL f
s * ((proj (a,m)) . r) is V11() real ext-real Element of REAL
(s * ((proj (a,m)) . r)) * (partdiff (X,i,a)) is Relation-like NAT -defined REAL -valued Function-like finite f -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL f
y1 . DD is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
s * (y1 . DD) is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
r is non empty Relation-like REAL m -defined REAL f -valued Function-like total quasi_total V255(f,m) V256(f,m) Lipschitzian complex-functions-valued ext-real-functions-valued real-functions-valued Element of K6(K7((REAL m),(REAL f)))
s is Relation-like NAT -defined REAL -valued Function-like finite m -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL m
r . s is Relation-like NAT -defined REAL -valued Function-like finite f -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL f
m is non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below 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
{ b₁ where b₁ is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like Element of REAL * : len b₁ = m } is set
f is Relation-like NAT -defined REAL m -valued Function-like finite FinSequence-like FinSubsequence-like complex-functions-valued ext-real-functions-valued real-functions-valued FinSequence of REAL m
len f is ordinal natural V11() real ext-real non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of NAT
X is Relation-like NAT -defined REAL m -valued Function-like finite FinSequence-like FinSubsequence-like complex-functions-valued ext-real-functions-valued real-functions-valued FinSequence of REAL m
len X is ordinal natural V11() real ext-real non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of NAT
(len X) + 1 is non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below Element of NAT
dom X is complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of K6(NAT)
f /. 1 is Relation-like NAT -defined Function-like finite m -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL m
f /. (len f) is Relation-like NAT -defined Function-like finite m -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL m
(f /. 1) - (f /. (len f)) is Relation-like NAT -defined REAL -valued Function-like finite m -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL m
- (f /. (len f)) is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
K38(1) * (f /. (len f)) is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
(f /. 1) + (- (f /. (len f))) is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
Sum X is Relation-like NAT -defined REAL -valued Function-like finite m -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL m
0* m is Relation-like NAT -defined REAL -valued Function-like finite m -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL m
m |-> 0 is Relation-like empty-yielding NAT -defined REAL -valued Function-like finite m -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of m -tuples_on REAL
Seg m is non empty finite m -element complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end right_end bounded_below bounded_above real-bounded Element of K6(NAT)
{ b₁ where b₁ is ordinal natural V11() real ext-real non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of NAT : ( 1 <= b₁ & b₁ <= m ) } is set
K213((Seg m),0) is non empty Relation-like Seg m -defined INT -valued RAT -valued {0} -valued Function-like total quasi_total finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued natural-valued Element of K6(K7((Seg m),{0}))
{0} is non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below set
K7((Seg m),{0}) is INT -valued RAT -valued complex-valued ext-real-valued real-valued natural-valued set
K6(K7((Seg m),{0})) is set
accum X is Relation-like NAT -defined REAL m -valued Function-like finite FinSequence-like FinSubsequence-like complex-functions-valued ext-real-functions-valued real-functions-valued FinSequence of REAL m
(accum X) . (len X) is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
(accum X) . 1 is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
1 + 1 is non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below Element of NAT
f /. (1 + 1) is Relation-like NAT -defined Function-like finite m -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL m
(f /. 1) - (f /. (1 + 1)) is Relation-like NAT -defined REAL -valued Function-like finite m -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL m
- (f /. (1 + 1)) is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
K38(1) * (f /. (1 + 1)) is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
(f /. 1) + (- (f /. (1 + 1))) is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
Seg (len X) is finite len X -element complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below bounded_above real-bounded Element of K6(NAT)
{ b₁ where b₁ is ordinal natural V11() real ext-real non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of NAT : ( 1 <= b₁ & b₁ <= len X ) } is set
X . 1 is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
X /. 1 is Relation-like NAT -defined Function-like finite m -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL m
i is ordinal natural V11() real ext-real non negative set
(accum X) . i is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
i + 1 is non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below Element of NAT
f /. (i + 1) is Relation-like NAT -defined Function-like finite m -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL m
(f /. 1) - (f /. (i + 1)) is Relation-like NAT -defined REAL -valued Function-like finite m -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL m
- (f /. (i + 1)) is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
K38(1) * (f /. (i + 1)) is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
(f /. 1) + (- (f /. (i + 1))) is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
(accum X) . (i + 1) is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
(i + 1) + 1 is non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below Element of NAT
f /. ((i + 1) + 1) is Relation-like NAT -defined Function-like finite m -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL m
(f /. 1) - (f /. ((i + 1) + 1)) is Relation-like NAT -defined REAL -valued Function-like finite m -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL m
- (f /. ((i + 1) + 1)) is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
K38(1) * (f /. ((i + 1) + 1)) is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
(f /. 1) + (- (f /. ((i + 1) + 1))) is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
len (accum X) is ordinal natural V11() real ext-real non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of NAT
dom (accum X) is complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of K6(NAT)
(accum X) /. i is Relation-like NAT -defined Function-like finite m -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL m
X /. (i + 1) is Relation-like NAT -defined Function-like finite m -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL m
((accum X) /. i) + (X /. (i + 1)) is Relation-like NAT -defined REAL -valued Function-like finite m -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL m
(f /. (i + 1)) - (f /. ((i + 1) + 1)) is Relation-like NAT -defined REAL -valued Function-like finite m -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL m
(f /. (i + 1)) + (- (f /. ((i + 1) + 1))) is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
((accum X) /. i) + ((f /. (i + 1)) - (f /. ((i + 1) + 1))) is Relation-like NAT -defined REAL -valued Function-like finite m -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL m
TOP-REAL m is non empty V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital TopSpace-like V148() V214() V215() L19()
the carrier of (TOP-REAL m) is non empty set
i + 2 is non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below Element of NAT
f /. (i + 2) is Relation-like NAT -defined Function-like finite m -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL m
s is Relation-like NAT -defined Function-like finite m -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of the carrier of (TOP-REAL m)
y0 is Relation-like NAT -defined Function-like finite m -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of the carrier of (TOP-REAL m)
r is Relation-like NAT -defined Function-like finite m -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of the carrier of (TOP-REAL m)
y0 - r is Relation-like NAT -defined Function-like finite m -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of the carrier of (TOP-REAL m)
y0 - r is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
- r is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
K38(1) * r is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
y0 + (- r) is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
y0 - r is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
s + (y0 - r) is Relation-like NAT -defined Function-like finite m -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of the carrier of (TOP-REAL m)
s + (y0 - r) is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
s + (y0 - r) is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
s + y0 is Relation-like NAT -defined Function-like finite m -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of the carrier of (TOP-REAL m)
s + y0 is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
s + y0 is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
(s + y0) - r is Relation-like NAT -defined Function-like finite m -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of the carrier of (TOP-REAL m)
(s + y0) - r is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
(s + y0) + (- r) is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
(s + y0) - r is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
m is non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below 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
{ b₁ where b₁ is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like Element of REAL * : len b₁ = m } is set
f is Relation-like NAT -defined REAL m -valued Function-like finite FinSequence-like FinSubsequence-like complex-functions-valued ext-real-functions-valued real-functions-valued FinSequence of REAL m
len f is ordinal natural V11() real ext-real non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of NAT
i is Relation-like NAT -defined REAL m -valued Function-like finite FinSequence-like FinSubsequence-like complex-functions-valued ext-real-functions-valued real-functions-valued FinSequence of REAL m
len i is ordinal natural V11() real ext-real non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of NAT
X is Relation-like NAT -defined REAL m -valued Function-like finite FinSequence-like FinSubsequence-like complex-functions-valued ext-real-functions-valued real-functions-valued FinSequence of REAL m
len X is ordinal natural V11() real ext-real non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of NAT
dom i is complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of K6(NAT)
Sum i is Relation-like NAT -defined REAL -valued Function-like finite m -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL m
Sum f is Relation-like NAT -defined REAL -valued Function-like finite m -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL m
Sum X is Relation-like NAT -defined REAL -valued Function-like finite m -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL m
(Sum f) - (Sum X) is Relation-like NAT -defined REAL -valued Function-like finite m -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL m
- (Sum X) is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
K38(1) * (Sum X) is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
(Sum f) + (- (Sum X)) is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
Seg (len i) is finite len i -element complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below bounded_above real-bounded Element of K6(NAT)
{ b₁ where b₁ is ordinal natural V11() real ext-real non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of NAT : ( 1 <= b₁ & b₁ <= len i ) } is set
dom X is complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of K6(NAT)
Seg (len X) is finite len X -element complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below bounded_above real-bounded Element of K6(NAT)
{ b₁ where b₁ is ordinal natural V11() real ext-real non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of NAT : ( 1 <= b₁ & b₁ <= len X ) } is set
dom f is complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of K6(NAT)
Seg (len f) is finite len f -element complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below bounded_above real-bounded Element of K6(NAT)
{ b₁ where b₁ is ordinal natural V11() real ext-real non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of NAT : ( 1 <= b₁ & b₁ <= len f ) } is set
y0 is ordinal natural V11() real ext-real non negative set
f /. y0 is Relation-like NAT -defined Function-like finite m -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL m
i /. y0 is Relation-like NAT -defined Function-like finite m -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL m
X /. y0 is Relation-like NAT -defined Function-like finite m -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL m
(i /. y0) + (X /. y0) is Relation-like NAT -defined REAL -valued Function-like finite m -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL m
TOP-REAL m is non empty V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital TopSpace-like V148() V214() V215() L19()
the carrier of (TOP-REAL m) is non empty set
r is Relation-like NAT -defined Function-like finite m -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of the carrier of (TOP-REAL m)
s is Relation-like NAT -defined Function-like finite m -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of the carrier of (TOP-REAL m)
y1 is Relation-like NAT -defined Function-like finite m -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of the carrier of (TOP-REAL m)
s - y1 is Relation-like NAT -defined Function-like finite m -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of the carrier of (TOP-REAL m)
s - y1 is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
- y1 is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
K38(1) * y1 is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
s + (- y1) is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
s - y1 is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
r + y1 is Relation-like NAT -defined Function-like finite m -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of the carrier of (TOP-REAL m)
r + y1 is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
r + y1 is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
i + X 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 set
K6(K7(NAT,(REAL m))) is set
i <++> X is Relation-like NAT /\ NAT -defined R_PFuncs ((DOMS (REAL m)) /\ (DOMS (REAL m))) -valued Function-like complex-functions-valued ext-real-functions-valued real-functions-valued Element of K6(K7((NAT /\ NAT),(R_PFuncs ((DOMS (REAL m)) /\ (DOMS (REAL m))))))
NAT /\ NAT is complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below set
DOMS (REAL m) is set
(DOMS (REAL m)) /\ (DOMS (REAL m)) is set
R_PFuncs ((DOMS (REAL m)) /\ (DOMS (REAL m))) is functional complex-functions-membered ext-real-functions-membered real-functions-membered set
K7((NAT /\ NAT),(R_PFuncs ((DOMS (REAL m)) /\ (DOMS (REAL m))))) is set
K6(K7((NAT /\ NAT),(R_PFuncs ((DOMS (REAL m)) /\ (DOMS (REAL m)))))) is set
dom (i + X) is complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of K6(NAT)
(dom i) /\ (dom X) is complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of K6(NAT)
TOP-REAL m is non empty V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital TopSpace-like V148() V214() V215() L19()
the carrier of (TOP-REAL m) is non empty set
y1 is ordinal natural V11() real ext-real non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of NAT
f . y1 is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
(i + X) . y1 is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
f /. y1 is Relation-like NAT -defined Function-like finite m -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL m
i /. y1 is Relation-like NAT -defined Function-like finite m -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL m
X /. y1 is Relation-like NAT -defined Function-like finite m -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL m
(i /. y1) + (X /. y1) is Relation-like NAT -defined REAL -valued Function-like finite m -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL m
(i + X) /. y1 is Relation-like NAT -defined Function-like finite m -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL m
r is Relation-like NAT -defined Function-like finite m -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of the carrier of (TOP-REAL m)
y0 is Relation-like NAT -defined Function-like finite m -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of the carrier of (TOP-REAL m)
s is Relation-like NAT -defined Function-like finite m -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of the carrier of (TOP-REAL m)
y0 + s is Relation-like NAT -defined Function-like finite m -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of the carrier of (TOP-REAL m)
y0 + s is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
y0 + s is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
r - s is Relation-like NAT -defined Function-like finite m -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of the carrier of (TOP-REAL m)
r - s is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
- s is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
K38(1) * s is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
r + (- s) is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
r - s is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
m is non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below 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
{ b₁ where b₁ is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like Element of REAL * : len b₁ = m } is set
f is non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below 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
{ b₁ where b₁ is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like Element of REAL * : len b₁ = f } is set
K7((REAL m),(REAL f)) is set
K6(K7((REAL m),(REAL f))) is set
m + 1 is non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below Element of NAT
X is Relation-like REAL m -defined REAL f -valued Function-like complex-functions-valued ext-real-functions-valued real-functions-valued Element of K6(K7((REAL m),(REAL f)))
y0 is Relation-like NAT -defined REAL -valued Function-like finite m -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL m
i is Relation-like NAT -defined REAL -valued Function-like finite m -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL m
|.y0.| is V11() real ext-real non negative Element of REAL
sqr y0 is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
mlt (y0,y0) is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
Sum (sqr y0) is V11() real ext-real Element of REAL
sqrt (Sum (sqr y0)) is V11() real ext-real Element of REAL
i + y0 is Relation-like NAT -defined REAL -valued Function-like finite m -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL m
X /. (i + y0) is Relation-like NAT -defined Function-like finite f -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL f
X /. i is Relation-like NAT -defined Function-like finite f -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL f
(X /. (i + y0)) - (X /. i) is Relation-like NAT -defined REAL -valued Function-like finite f -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL f
- (X /. i) is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
K38(1) * (X /. i) is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
(X /. (i + y0)) + (- (X /. i)) is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
len y0 is ordinal natural V11() real ext-real non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of NAT
Seg (m + 1) is non empty finite m + 1 -element complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end right_end bounded_below bounded_above real-bounded Element of K6(NAT)
{ b₁ where b₁ is ordinal natural V11() real ext-real non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of NAT : ( 1 <= b₁ & b₁ <= m + 1 ) } is set
r is ordinal natural V11() real ext-real non negative set
(m + 1) -' r is ordinal natural V11() real ext-real non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of NAT
y0 | ((m + 1) -' r) is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
Seg ((m + 1) -' r) is finite (m + 1) -' r -element complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below bounded_above real-bounded Element of K6(NAT)
{ b₁ where b₁ is ordinal natural V11() real ext-real non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of NAT : ( 1 <= b₁ & b₁ <= (m + 1) -' r ) } is set
y0 | (Seg ((m + 1) -' r)) is Relation-like NAT -defined Function-like FinSubsequence-like complex-valued ext-real-valued real-valued set
r -' 1 is ordinal natural V11() real ext-real non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of NAT
0* (r -' 1) is Relation-like NAT -defined REAL -valued Function-like finite r -' 1 -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL (r -' 1)
REAL (r -' 1) is non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered FinSequenceSet of REAL
(r -' 1) -tuples_on REAL is non empty functional FinSequence-membered FinSequenceSet of REAL
{ b₁ where b₁ is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like Element of REAL * : len b₁ = r -' 1 } is set
(r -' 1) |-> 0 is Relation-like empty-yielding NAT -defined REAL -valued Function-like finite r -' 1 -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of (r -' 1) -tuples_on REAL
Seg (r -' 1) is finite r -' 1 -element complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below bounded_above real-bounded Element of K6(NAT)
{ b₁ where b₁ is ordinal natural V11() real ext-real non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of NAT : ( 1 <= b₁ & b₁ <= r -' 1 ) } is set
K213((Seg (r -' 1)),0) is Relation-like Seg (r -' 1) -defined INT -valued RAT -valued {0} -valued Function-like total quasi_total finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued natural-valued Element of K6(K7((Seg (r -' 1)),{0}))
{0} is non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below set
K7((Seg (r -' 1)),{0}) is INT -valued RAT -valued complex-valued ext-real-valued real-valued natural-valued set
K6(K7((Seg (r -' 1)),{0})) is set
(y0 | ((m + 1) -' r)) ^ (0* (r -' 1)) is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
r - 1 is V11() real ext-real Element of REAL
(m + 1) - r is V11() real ext-real Element of REAL
m + r is non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below Element of NAT
len (y0 | ((m + 1) -' r)) is ordinal natural V11() real ext-real non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of NAT
len (0* (r -' 1)) is ordinal natural V11() real ext-real non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of NAT
len ((y0 | ((m + 1) -' r)) ^ (0* (r -' 1))) is ordinal natural V11() real ext-real non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of NAT
((m + 1) -' r) + (r -' 1) is ordinal natural V11() real ext-real non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of NAT
(((m + 1) -' r) + (r -' 1)) -tuples_on REAL is non empty functional FinSequence-membered FinSequenceSet of REAL
{ b₁ where b₁ is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like Element of REAL * : len b₁ = ((m + 1) -' r) + (r -' 1) } is set
r is Relation-like NAT -defined REAL m -valued Function-like finite FinSequence-like FinSubsequence-like complex-functions-valued ext-real-functions-valued real-functions-valued FinSequence of REAL m
dom r is complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of K6(NAT)
s is ordinal natural V11() real ext-real non negative set
r /. s is Relation-like NAT -defined Function-like finite m -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL m
r . s is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
(m + 1) -' s is ordinal natural V11() real ext-real non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of NAT
y0 | ((m + 1) -' s) is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
Seg ((m + 1) -' s) is finite (m + 1) -' s -element complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below bounded_above real-bounded Element of K6(NAT)
{ b₁ where b₁ is ordinal natural V11() real ext-real non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of NAT : ( 1 <= b₁ & b₁ <= (m + 1) -' s ) } is set
y0 | (Seg ((m + 1) -' s)) is Relation-like NAT -defined Function-like FinSubsequence-like complex-valued ext-real-valued real-valued set
s -' 1 is ordinal natural V11() real ext-real non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of NAT
0* (s -' 1) is Relation-like NAT -defined REAL -valued Function-like finite s -' 1 -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL (s -' 1)
REAL (s -' 1) is non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered FinSequenceSet of REAL
(s -' 1) -tuples_on REAL is non empty functional FinSequence-membered FinSequenceSet of REAL
{ b₁ where b₁ is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like Element of REAL * : len b₁ = s -' 1 } is set
(s -' 1) |-> 0 is Relation-like empty-yielding NAT -defined REAL -valued Function-like finite s -' 1 -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of (s -' 1) -tuples_on REAL
Seg (s -' 1) is finite s -' 1 -element complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below bounded_above real-bounded Element of K6(NAT)
{ b₁ where b₁ is ordinal natural V11() real ext-real non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of NAT : ( 1 <= b₁ & b₁ <= s -' 1 ) } is set
K213((Seg (s -' 1)),0) is Relation-like Seg (s -' 1) -defined INT -valued RAT -valued {0} -valued Function-like total quasi_total finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued natural-valued Element of K6(K7((Seg (s -' 1)),{0}))
{0} is non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below set
K7((Seg (s -' 1)),{0}) is INT -valued RAT -valued complex-valued ext-real-valued real-valued natural-valued set
K6(K7((Seg (s -' 1)),{0})) is set
(y0 | ((m + 1) -' s)) ^ (0* (s -' 1)) is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
s is Relation-like NAT -defined REAL f -valued Function-like finite FinSequence-like FinSubsequence-like complex-functions-valued ext-real-functions-valued real-functions-valued FinSequence of REAL f
len s is ordinal natural V11() real ext-real non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of NAT
dom s is complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of K6(NAT)
y1 is ordinal natural V11() real ext-real non negative set
s /. y1 is Relation-like NAT -defined Function-like finite f -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL f
s . y1 is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
r /. y1 is Relation-like NAT -defined Function-like finite m -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL m
i + (r /. y1) is Relation-like NAT -defined REAL -valued Function-like finite m -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL m
X /. (i + (r /. y1)) is Relation-like NAT -defined Function-like finite f -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL f
y1 is Relation-like NAT -defined REAL f -valued Function-like finite FinSequence-like FinSubsequence-like complex-functions-valued ext-real-functions-valued real-functions-valued FinSequence of REAL f
len y1 is ordinal natural V11() real ext-real non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of NAT
dom y1 is complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of K6(NAT)
PP is ordinal natural V11() real ext-real non negative set
y1 /. PP is Relation-like NAT -defined Function-like finite f -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL f
y1 . PP is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
s /. PP is Relation-like NAT -defined Function-like finite f -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL f
PP + 1 is non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below Element of NAT
s /. (PP + 1) is Relation-like NAT -defined Function-like finite f -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL f
(s /. PP) - (s /. (PP + 1)) is Relation-like NAT -defined REAL -valued Function-like finite f -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL f
- (s /. (PP + 1)) is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
K38(1) * (s /. (PP + 1)) is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
(s /. PP) + (- (s /. (PP + 1))) is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
(m + 1) -' 1 is ordinal natural V11() real ext-real non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of NAT
(m + 1) - 1 is V11() real ext-real V163() V198() Element of INT
r /. 1 is Relation-like NAT -defined Function-like finite m -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL m
y0 | ((m + 1) -' 1) is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
Seg ((m + 1) -' 1) is finite (m + 1) -' 1 -element complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below bounded_above real-bounded Element of K6(NAT)
{ b₁ where b₁ is ordinal natural V11() real ext-real non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of NAT : ( 1 <= b₁ & b₁ <= (m + 1) -' 1 ) } is set
y0 | (Seg ((m + 1) -' 1)) is Relation-like NAT -defined Function-like FinSubsequence-like complex-valued ext-real-valued real-valued set
1 -' 1 is ordinal natural V11() real ext-real non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of NAT
0* (1 -' 1) is Relation-like NAT -defined REAL -valued Function-like finite 1 -' 1 -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL (1 -' 1)
REAL (1 -' 1) is non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered FinSequenceSet of REAL
(1 -' 1) -tuples_on REAL is non empty functional FinSequence-membered FinSequenceSet of REAL
{ b₁ where b₁ is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like Element of REAL * : len b₁ = 1 -' 1 } is set
(1 -' 1) |-> 0 is Relation-like empty-yielding NAT -defined REAL -valued Function-like finite 1 -' 1 -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of (1 -' 1) -tuples_on REAL
Seg (1 -' 1) is finite 1 -' 1 -element complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below bounded_above real-bounded Element of K6(NAT)
{ b₁ where b₁ is ordinal natural V11() real ext-real non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of NAT : ( 1 <= b₁ & b₁ <= 1 -' 1 ) } is set
K213((Seg (1 -' 1)),0) is Relation-like Seg (1 -' 1) -defined INT -valued RAT -valued {0} -valued Function-like total quasi_total finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued natural-valued Element of K6(K7((Seg (1 -' 1)),{0}))
K7((Seg (1 -' 1)),{0}) is INT -valued RAT -valued complex-valued ext-real-valued real-valued natural-valued set
K6(K7((Seg (1 -' 1)),{0})) is set
(y0 | ((m + 1) -' 1)) ^ (0* (1 -' 1)) is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
y0 | m is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
Seg m is non empty finite m -element complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end right_end bounded_below bounded_above real-bounded Element of K6(NAT)
{ b₁ where b₁ is ordinal natural V11() real ext-real non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of NAT : ( 1 <= b₁ & b₁ <= m ) } is set
y0 | (Seg m) is Relation-like NAT -defined Function-like FinSubsequence-like complex-valued ext-real-valued real-valued set
0* 0 is Relation-like NAT -defined REAL -valued Function-like finite 0 -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL 0
REAL 0 is non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered FinSequenceSet of REAL
0 -tuples_on REAL is non empty functional FinSequence-membered FinSequenceSet of REAL
{ b₁ where b₁ is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like Element of REAL * : len b₁ = 0 } is set
0 |-> 0 is Relation-like empty-yielding NAT -defined REAL -valued Function-like finite 0 -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of 0 -tuples_on REAL
Seg 0 is empty Function-like functional finite 0 -element FinSequence-membered complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below bounded_above real-bounded V263() complex-functions-membered ext-real-functions-membered real-functions-membered rational-functions-membered integer-functions-membered natural-functions-membered Element of K6(NAT)
{ b₁ where b₁ is ordinal natural V11() real ext-real non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of NAT : ( 1 <= b₁ & b₁ <= 0 ) } is set
K213((Seg 0),0) is empty Relation-like Seg 0 -defined INT -valued RAT -valued {0} -valued Function-like one-to-one constant functional total quasi_total finite FinSequence-like FinSubsequence-like FinSequence-membered complex-valued ext-real-valued real-valued natural-valued complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below bounded_above real-bounded V263() complex-functions-membered ext-real-functions-membered real-functions-membered rational-functions-membered integer-functions-membered natural-functions-membered complex-functions-valued ext-real-functions-valued real-functions-valued rational-functions-valued integer-functions-valued natural-functions-valued Element of K6(K7((Seg 0),{0}))
K7((Seg 0),{0}) is INT -valued RAT -valued complex-valued ext-real-valued real-valued natural-valued set
K6(K7((Seg 0),{0})) is set
(y0 | m) ^ (0* 0) is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
y0 ^ (0* 0) is Relation-like NAT -defined REAL -valued Function-like finite K335(m,0) -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
K335(m,0) is non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below Element of NAT
(m + 1) -' (len s) is ordinal natural V11() real ext-real non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of NAT
(m + 1) - (len s) is V11() real ext-real V163() V198() Element of INT
(len s) -' 1 is ordinal natural V11() real ext-real non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of NAT
(len s) - 1 is V11() real ext-real V163() V198() Element of INT
r /. (len s) is Relation-like NAT -defined Function-like finite m -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL m
y0 | 0 is empty Relation-like NAT -defined REAL -valued RAT -valued Function-like one-to-one constant functional finite FinSequence-like FinSubsequence-like FinSequence-membered complex-valued ext-real-valued real-valued natural-valued complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below bounded_above real-bounded V263() complex-functions-membered ext-real-functions-membered real-functions-membered rational-functions-membered integer-functions-membered natural-functions-membered complex-functions-valued ext-real-functions-valued real-functions-valued rational-functions-valued integer-functions-valued natural-functions-valued FinSequence of REAL
y0 | (Seg 0) is Relation-like NAT -defined Function-like FinSubsequence-like complex-valued ext-real-valued real-valued set
0* ((len s) -' 1) is Relation-like NAT -defined REAL -valued Function-like finite (len s) -' 1 -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL ((len s) -' 1)
REAL ((len s) -' 1) is non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered FinSequenceSet of REAL
((len s) -' 1) -tuples_on REAL is non empty functional FinSequence-membered FinSequenceSet of REAL
{ b₁ where b₁ is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like Element of REAL * : len b₁ = (len s) -' 1 } is set
((len s) -' 1) |-> 0 is Relation-like empty-yielding NAT -defined REAL -valued Function-like finite (len s) -' 1 -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of ((len s) -' 1) -tuples_on REAL
Seg ((len s) -' 1) is finite (len s) -' 1 -element complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below bounded_above real-bounded Element of K6(NAT)
{ b₁ where b₁ is ordinal natural V11() real ext-real non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of NAT : ( 1 <= b₁ & b₁ <= (len s) -' 1 ) } is set
K213((Seg ((len s) -' 1)),0) is Relation-like Seg ((len s) -' 1) -defined INT -valued RAT -valued {0} -valued Function-like total quasi_total finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued natural-valued Element of K6(K7((Seg ((len s) -' 1)),{0}))
K7((Seg ((len s) -' 1)),{0}) is INT -valued RAT -valued complex-valued ext-real-valued real-valued natural-valued set
K6(K7((Seg ((len s) -' 1)),{0})) is set
(y0 | 0) ^ (0* ((len s) -' 1)) is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
0* m is Relation-like NAT -defined REAL -valued Function-like finite m -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL m
m |-> 0 is Relation-like empty-yielding NAT -defined REAL -valued Function-like finite m -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of m -tuples_on REAL
K213((Seg m),0) is non empty Relation-like Seg m -defined INT -valued RAT -valued {0} -valued Function-like total quasi_total finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued natural-valued Element of K6(K7((Seg m),{0}))
K7((Seg m),{0}) is INT -valued RAT -valued complex-valued ext-real-valued real-valued natural-valued set
K6(K7((Seg m),{0})) is set
s /. 1 is Relation-like NAT -defined Function-like finite f -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL f
s /. (len s) is Relation-like NAT -defined Function-like finite f -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL f
i + (r /. (len s)) is Relation-like NAT -defined REAL -valued Function-like finite m -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL m
X /. (i + (r /. (len s))) is Relation-like NAT -defined Function-like finite f -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL f
len r is ordinal natural V11() real ext-real non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of NAT
Sum y1 is Relation-like NAT -defined REAL -valued Function-like finite f -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL f
PP is ordinal natural V11() real ext-real non negative set
s /. PP is Relation-like NAT -defined Function-like finite f -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL f
r /. PP is Relation-like NAT -defined Function-like finite m -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL m
i + (r /. PP) is Relation-like NAT -defined REAL -valued Function-like finite m -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL m
X /. (i + (r /. PP)) is Relation-like NAT -defined Function-like finite f -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL f
PP + 1 is non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below Element of NAT
s /. (PP + 1) is Relation-like NAT -defined Function-like finite f -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL f
r /. (PP + 1) is Relation-like NAT -defined Function-like finite m -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL m
i + (r /. (PP + 1)) is Relation-like NAT -defined REAL -valued Function-like finite m -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL m
X /. (i + (r /. (PP + 1))) is Relation-like NAT -defined Function-like finite f -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL f
y1 /. PP is Relation-like NAT -defined Function-like finite f -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL f
(X /. (i + (r /. PP))) - (X /. (i + (r /. (PP + 1)))) is Relation-like NAT -defined REAL -valued Function-like finite f -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL f
- (X /. (i + (r /. (PP + 1)))) is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
K38(1) * (X /. (i + (r /. (PP + 1)))) is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
(X /. (i + (r /. PP))) + (- (X /. (i + (r /. (PP + 1))))) is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
PP is ordinal natural V11() real ext-real non negative set
r /. PP is Relation-like NAT -defined Function-like finite m -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL m
DD is Relation-like NAT -defined REAL -valued Function-like finite m -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL m
|.DD.| is V11() real ext-real non negative Element of REAL
sqr DD is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
mlt (DD,DD) is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
Sum (sqr DD) is V11() real ext-real Element of REAL
sqrt (Sum (sqr DD)) is V11() real ext-real Element of REAL
(m + 1) -' PP is ordinal natural V11() real ext-real non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of NAT
y0 | ((m + 1) -' PP) is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
Seg ((m + 1) -' PP) is finite (m + 1) -' PP -element complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below bounded_above real-bounded Element of K6(NAT)
{ b₁ where b₁ is ordinal natural V11() real ext-real non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of NAT : ( 1 <= b₁ & b₁ <= (m + 1) -' PP ) } is set
y0 | (Seg ((m + 1) -' PP)) is Relation-like NAT -defined Function-like FinSubsequence-like complex-valued ext-real-valued real-valued set
PP -' 1 is ordinal natural V11() real ext-real non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of NAT
0* (PP -' 1) is Relation-like NAT -defined REAL -valued Function-like finite PP -' 1 -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL (PP -' 1)
REAL (PP -' 1) is non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered FinSequenceSet of REAL
(PP -' 1) -tuples_on REAL is non empty functional FinSequence-membered FinSequenceSet of REAL
{ b₁ where b₁ is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like Element of REAL * : len b₁ = PP -' 1 } is set
(PP -' 1) |-> 0 is Relation-like empty-yielding NAT -defined REAL -valued Function-like finite PP -' 1 -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of (PP -' 1) -tuples_on REAL
Seg (PP -' 1) is finite PP -' 1 -element complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below bounded_above real-bounded Element of K6(NAT)
{ b₁ where b₁ is ordinal natural V11() real ext-real non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of NAT : ( 1 <= b₁ & b₁ <= PP -' 1 ) } is set
K213((Seg (PP -' 1)),0) is Relation-like Seg (PP -' 1) -defined INT -valued RAT -valued {0} -valued Function-like total quasi_total finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued natural-valued Element of K6(K7((Seg (PP -' 1)),{0}))
K7((Seg (PP -' 1)),{0}) is INT -valued RAT -valued complex-valued ext-real-valued real-valued natural-valued set
K6(K7((Seg (PP -' 1)),{0})) is set
(y0 | ((m + 1) -' PP)) ^ (0* (PP -' 1)) is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
sqr DD is Relation-like NAT -defined REAL -valued Function-like finite m -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of m -tuples_on REAL
sqr y0 is Relation-like NAT -defined REAL -valued Function-like finite m -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of m -tuples_on REAL
a is ordinal natural V11() real ext-real non negative set
(sqr DD) . a is V11() real ext-real Element of REAL
(sqr y0) . a is V11() real ext-real Element of REAL
len DD is ordinal natural V11() real ext-real non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of NAT
dom ((y0 | ((m + 1) -' PP)) ^ (0* (PP -' 1))) is complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of K6(NAT)
dom (y0 | ((m + 1) -' PP)) is complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of K6(NAT)
len (y0 | ((m + 1) -' PP)) is ordinal natural V11() real ext-real non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of NAT
Seg (len (y0 | ((m + 1) -' PP))) is finite len (y0 | ((m + 1) -' PP)) -element complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below bounded_above real-bounded Element of K6(NAT)
{ b₁ where b₁ is ordinal natural V11() real ext-real non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of NAT : ( 1 <= b₁ & b₁ <= len (y0 | ((m + 1) -' PP)) ) } is set
sqrreal * DD is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
(sqrreal * DD) . a is V11() real ext-real Element of REAL
((y0 | ((m + 1) -' PP)) ^ (0* (PP -' 1))) . a is V11() real ext-real Element of REAL
sqrreal . (((y0 | ((m + 1) -' PP)) ^ (0* (PP -' 1))) . a) is V11() real ext-real Element of REAL
(y0 | ((m + 1) -' PP)) . a is V11() real ext-real Element of REAL
sqrreal . ((y0 | ((m + 1) -' PP)) . a) is V11() real ext-real Element of REAL
y0 . a is V11() real ext-real Element of REAL
sqrreal . (y0 . a) is V11() real ext-real Element of REAL
(y0 . a) ^2 is V11() real ext-real Element of REAL
K37((y0 . a),(y0 . a)) is V11() real ext-real set
dom (0* (PP -' 1)) is complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of K6(NAT)
len (y0 | ((m + 1) -' PP)) is ordinal natural V11() real ext-real non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of NAT
t is ordinal natural V11() real ext-real non negative set
(len (y0 | ((m + 1) -' PP))) + t is ordinal natural V11() real ext-real non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of NAT
t is ordinal natural V11() real ext-real non negative set
(len (y0 | ((m + 1) -' PP))) + t is ordinal natural V11() real ext-real non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of NAT
sqrreal * DD is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
(sqrreal * DD) . a is V11() real ext-real Element of REAL
((y0 | ((m + 1) -' PP)) ^ (0* (PP -' 1))) . a is V11() real ext-real Element of REAL
sqrreal . (((y0 | ((m + 1) -' PP)) ^ (0* (PP -' 1))) . a) is V11() real ext-real Element of REAL
(0* (PP -' 1)) . t is V11() real ext-real Element of REAL
sqrreal . ((0* (PP -' 1)) . t) is V11() real ext-real Element of REAL
((0* (PP -' 1)) . t) ^2 is V11() real ext-real Element of REAL
K37(((0* (PP -' 1)) . t),((0* (PP -' 1)) . t)) is V11() real ext-real set
y0 . a is V11() real ext-real Element of REAL
(y0 . a) ^2 is V11() real ext-real Element of REAL
K37((y0 . a),(y0 . a)) is V11() real ext-real set
(y0 . a) * (y0 . a) is V11() real ext-real Element of REAL
dom (y0 | ((m + 1) -' PP)) is complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of K6(NAT)
dom (0* (PP -' 1)) is complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of K6(NAT)
len (y0 | ((m + 1) -' PP)) is ordinal natural V11() real ext-real non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of NAT
Sum (sqr DD) is V11() real ext-real Element of REAL
PP is ordinal natural V11() real ext-real non negative set
r /. PP is Relation-like NAT -defined Function-like finite m -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL m
(m + 1) -' PP is ordinal natural V11() real ext-real non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of NAT
y0 | ((m + 1) -' PP) is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
Seg ((m + 1) -' PP) is finite (m + 1) -' PP -element complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below bounded_above real-bounded Element of K6(NAT)
{ b₁ where b₁ is ordinal natural V11() real ext-real non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of NAT : ( 1 <= b₁ & b₁ <= (m + 1) -' PP ) } is set
y0 | (Seg ((m + 1) -' PP)) is Relation-like NAT -defined Function-like FinSubsequence-like complex-valued ext-real-valued real-valued set
PP -' 1 is ordinal natural V11() real ext-real non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of NAT
0* (PP -' 1) is Relation-like NAT -defined REAL -valued Function-like finite PP -' 1 -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL (PP -' 1)
REAL (PP -' 1) is non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered FinSequenceSet of REAL
(PP -' 1) -tuples_on REAL is non empty functional FinSequence-membered FinSequenceSet of REAL
{ b₁ where b₁ is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like Element of REAL * : len b₁ = PP -' 1 } is set
(PP -' 1) |-> 0 is Relation-like empty-yielding NAT -defined REAL -valued Function-like finite PP -' 1 -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of (PP -' 1) -tuples_on REAL
Seg (PP -' 1) is finite PP -' 1 -element complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below bounded_above real-bounded Element of K6(NAT)
{ b₁ where b₁ is ordinal natural V11() real ext-real non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of NAT : ( 1 <= b₁ & b₁ <= PP -' 1 ) } is set
K213((Seg (PP -' 1)),0) is Relation-like Seg (PP -' 1) -defined INT -valued RAT -valued {0} -valued Function-like total quasi_total finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued natural-valued Element of K6(K7((Seg (PP -' 1)),{0}))
K7((Seg (PP -' 1)),{0}) is INT -valued RAT -valued complex-valued ext-real-valued real-valued natural-valued set
K6(K7((Seg (PP -' 1)),{0})) is set
(y0 | ((m + 1) -' PP)) ^ (0* (PP -' 1)) is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
DD is ordinal natural V11() real ext-real non negative set
y1 /. DD is Relation-like NAT -defined Function-like finite f -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL f
r /. DD is Relation-like NAT -defined Function-like finite m -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL m
i + (r /. DD) is Relation-like NAT -defined REAL -valued Function-like finite m -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL m
X /. (i + (r /. DD)) is Relation-like NAT -defined Function-like finite f -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL f
DD + 1 is non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below Element of NAT
r /. (DD + 1) is Relation-like NAT -defined Function-like finite m -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL m
i + (r /. (DD + 1)) is Relation-like NAT -defined REAL -valued Function-like finite m -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL m
X /. (i + (r /. (DD + 1))) is Relation-like NAT -defined Function-like finite f -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL f
(X /. (i + (r /. DD))) - (X /. (i + (r /. (DD + 1)))) is Relation-like NAT -defined REAL -valued Function-like finite f -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL f
- (X /. (i + (r /. (DD + 1)))) is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
K38(1) * (X /. (i + (r /. (DD + 1)))) is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
(X /. (i + (r /. DD))) + (- (X /. (i + (r /. (DD + 1))))) is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
a is ordinal natural V11() real ext-real non negative set
r /. a is Relation-like NAT -defined Function-like finite m -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL m
t is Relation-like NAT -defined REAL -valued Function-like finite m -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL m
|.t.| is V11() real ext-real non negative Element of REAL
sqr t is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
mlt (t,t) is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
Sum (sqr t) is V11() real ext-real Element of REAL
sqrt (Sum (sqr t)) is V11() real ext-real Element of REAL
m is non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below 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
{ b₁ where b₁ is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like Element of REAL * : len b₁ = m } is set
K7((REAL m),(REAL 1)) is set
K6(K7((REAL m),(REAL 1))) is set
K7((REAL m),REAL) is complex-valued ext-real-valued real-valued set
K6(K7((REAL m),REAL)) is set
f is Relation-like REAL m -defined REAL 1 -valued Function-like complex-functions-valued ext-real-functions-valued real-functions-valued Element of K6(K7((REAL m),(REAL 1)))
dom f is functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered Element of K6((REAL m))
K6((REAL m)) is set
X is set
f /. X is Relation-like NAT -defined Function-like finite 1 -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL 1
(proj (1,1)) . (f /. X) is V11() real ext-real Element of REAL
X is Relation-like REAL m -defined REAL -valued Function-like complex-valued ext-real-valued real-valued Element of K6(K7((REAL m),REAL))
dom X is functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered Element of K6((REAL m))
<>* X is Relation-like REAL m -defined REAL 1 -valued Function-like complex-functions-valued ext-real-functions-valued real-functions-valued Element of K6(K7((REAL m),(REAL 1)))
X (#) ((proj (1,1)) ") is Relation-like Function-like set
i is set
i is set
rng X is complex-membered ext-real-membered real-membered Element of K6(REAL)
i is non empty Relation-like REAL -defined REAL 1 -valued Function-like total quasi_total complex-functions-valued ext-real-functions-valued real-functions-valued Element of K6(K7(REAL,(REAL 1)))
proj1 (X (#) ((proj (1,1)) ")) is set
dom (<>* X) is functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered Element of K6((REAL m))
i is Relation-like NAT -defined REAL -valued Function-like finite m -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL m
(<>* X) . i is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
f . i is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
X . i is V11() real ext-real Element of REAL
((proj (1,1)) ") . (X . i) is set
f /. i is Relation-like NAT -defined Function-like finite 1 -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL 1
(proj (1,1)) . (f /. i) is V11() real ext-real Element of REAL
((proj (1,1)) ") . ((proj (1,1)) . (f /. i)) is set
y0 is V11() real ext-real Element of REAL
<*y0*> is non empty V2() Relation-like NAT -defined REAL -valued Function-like one-to-one constant finite 1 -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued increasing decreasing non-decreasing non-increasing FinSequence of REAL
[1,y0] is V27() set
{1,y0} is non empty complex-membered ext-real-membered real-membered set
{{1,y0},{1}} is non empty set
{[1,y0]} is non empty Function-like set
m is non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below 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
{ b₁ where b₁ is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like Element of REAL * : len b₁ = m } is set
f is non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below 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
{ b₁ where b₁ is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like Element of REAL * : len b₁ = f } is set
K7((REAL m),(REAL f)) is set
K6(K7((REAL m),(REAL f))) 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() strict RealNormSpace-like V185() NORMSTR
the carrier of (REAL-NS m) 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() strict RealNormSpace-like V185() NORMSTR
the carrier of (REAL-NS f) is non empty V2() set
K7( the carrier of (REAL-NS m), the carrier of (REAL-NS f)) is set
K6(K7( the carrier of (REAL-NS m), the carrier of (REAL-NS f))) is set
X is Relation-like REAL m -defined REAL f -valued Function-like complex-functions-valued ext-real-functions-valued real-functions-valued Element of K6(K7((REAL m),(REAL f)))
dom X is functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered Element of K6((REAL m))
K6((REAL m)) is set
i is Relation-like the carrier of (REAL-NS m) -defined the carrier of (REAL-NS f) -valued Function-like Element of K6(K7( the carrier of (REAL-NS m), the carrier of (REAL-NS f)))
y0 is Relation-like NAT -defined REAL -valued Function-like finite m -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL m
X /. y0 is Relation-like NAT -defined Function-like finite f -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL f
r is Element of the carrier of (REAL-NS m)
i /. r is Element of the carrier of (REAL-NS f)
i . r is set
m is non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below 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
{ b₁ where b₁ is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like Element of REAL * : len b₁ = m } is set
K6((REAL m)) is set
m is non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below 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
{ b₁ where b₁ is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like Element of REAL * : len b₁ = m } is set
K6((REAL m)) is set
f is functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered Element of K6((REAL m))
REAL-NS m is non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V185() NORMSTR
the carrier of (REAL-NS m) is non empty V2() set
K6( the carrier of (REAL-NS m)) is set
X is Element of K6( the carrier of (REAL-NS m))
i is Relation-like NAT -defined REAL -valued Function-like finite m -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL m
TopSpaceNorm (REAL-NS m) is non empty TopSpace-like TopStruct
MetricSpaceNorm (REAL-NS m) is non empty V137() V138() V139() V140() L18()
distance_by_norm_of (REAL-NS m) is Relation-like K7( the carrier of (REAL-NS m), the carrier of (REAL-NS m)) -defined REAL -valued Function-like total quasi_total complex-valued ext-real-valued real-valued Element of K6(K7(K7( the carrier of (REAL-NS m), the carrier of (REAL-NS m)),REAL))
K7( the carrier of (REAL-NS m), the carrier of (REAL-NS m)) is set
K7(K7( the carrier of (REAL-NS m), the carrier of (REAL-NS m)),REAL) is complex-valued ext-real-valued real-valued set
K6(K7(K7( the carrier of (REAL-NS m), the carrier of (REAL-NS m)),REAL)) is set
G18( the carrier of (REAL-NS m),(distance_by_norm_of (REAL-NS m))) is V132() L18()
K515((MetricSpaceNorm (REAL-NS m))) is TopStruct
the carrier of (MetricSpaceNorm (REAL-NS m)) is non empty set
K514((MetricSpaceNorm (REAL-NS m))) is Element of K6(K6( the carrier of (MetricSpaceNorm (REAL-NS m))))
K6( the carrier of (MetricSpaceNorm (REAL-NS m))) is set
K6(K6( the carrier of (MetricSpaceNorm (REAL-NS m)))) is set
TopStruct(# the carrier of (MetricSpaceNorm (REAL-NS m)),K514((MetricSpaceNorm (REAL-NS m))) #) is strict TopStruct
the carrier of (TopSpaceNorm (REAL-NS m)) is non empty set
K6( the carrier of (TopSpaceNorm (REAL-NS m))) is set
y0 is Element of K6( the carrier of (TopSpaceNorm (REAL-NS m)))
r is Element of the carrier of (REAL-NS m)
s is V11() real ext-real Element of REAL
{ b₁ where b₁ is Element of the carrier of (REAL-NS m) : not s <= ||.(r - b₁).|| } is set
y1 is V11() real ext-real Element of REAL
{ b₁ where b₁ is Relation-like NAT -defined REAL -valued Function-like finite m -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL m : not y1 <= |.(b₁ - i).| } is set
PP is set
DD is Relation-like NAT -defined REAL -valued Function-like finite m -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL m
DD - i is Relation-like NAT -defined REAL -valued Function-like finite m -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL m
- i is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
K38(1) * i is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
DD + (- i) is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
|.(DD - i).| is V11() real ext-real non negative Element of REAL
sqr (DD - i) is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
mlt ((DD - i),(DD - i)) is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
Sum (sqr (DD - i)) is V11() real ext-real Element of REAL
sqrt (Sum (sqr (DD - i))) is V11() real ext-real Element of REAL
a is Element of the carrier of (REAL-NS m)
a - r is Element of the carrier of (REAL-NS m)
||.(a - r).|| is V11() real ext-real Element of REAL
the U8 of (REAL-NS m) is non empty Relation-like the carrier of (REAL-NS m) -defined REAL -valued Function-like total quasi_total complex-valued ext-real-valued real-valued Element of K6(K7( the carrier of (REAL-NS m),REAL))
K7( the carrier of (REAL-NS m),REAL) is complex-valued ext-real-valued real-valued set
K6(K7( the carrier of (REAL-NS m),REAL)) is set
the U8 of (REAL-NS m) . (a - r) is V11() real ext-real Element of REAL
r - a is Element of the carrier of (REAL-NS m)
||.(r - a).|| is V11() real ext-real Element of REAL
the U8 of (REAL-NS m) . (r - a) is V11() real ext-real Element of REAL
{ b₁ where b₁ is Element of the carrier of (REAL-NS m) : not y1 <= ||.(r - b₁).|| } is set
X is Element of K6( the carrier of (REAL-NS m))
i is Element of K6( the carrier of (TopSpaceNorm (REAL-NS m)))
y0 is Element of the carrier of (REAL-NS m)
r is Relation-like NAT -defined REAL -valued Function-like finite m -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL m
s is V11() real ext-real Element of REAL
{ b₁ where b₁ is Relation-like NAT -defined REAL -valued Function-like finite m -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL m : not s <= |.(b₁ - r).| } is set
{ b₁ where b₁ is Element of the carrier of (REAL-NS m) : not s <= ||.(y0 - b₁).|| } is set
y1 is set
PP is Element of the carrier of (REAL-NS m)
y0 - PP is Element of the carrier of (REAL-NS m)
||.(y0 - PP).|| is V11() real ext-real Element of REAL
the U8 of (REAL-NS m) is non empty Relation-like the carrier of (REAL-NS m) -defined REAL -valued Function-like total quasi_total complex-valued ext-real-valued real-valued Element of K6(K7( the carrier of (REAL-NS m),REAL))
K7( the carrier of (REAL-NS m),REAL) is complex-valued ext-real-valued real-valued set
K6(K7( the carrier of (REAL-NS m),REAL)) is set
the U8 of (REAL-NS m) . (y0 - PP) is V11() real ext-real Element of REAL
DD is Relation-like NAT -defined REAL -valued Function-like finite m -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL m
DD - r is Relation-like NAT -defined REAL -valued Function-like finite m -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL m
- r is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
K38(1) * r is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
DD + (- r) is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
|.(DD - r).| is V11() real ext-real non negative Element of REAL
sqr (DD - r) is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
mlt ((DD - r),(DD - r)) is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
Sum (sqr (DD - r)) is V11() real ext-real Element of REAL
sqrt (Sum (sqr (DD - r))) is V11() real ext-real Element of REAL
PP - y0 is Element of the carrier of (REAL-NS m)
||.(PP - y0).|| is V11() real ext-real Element of REAL
the U8 of (REAL-NS m) . (PP - y0) is V11() real ext-real Element of REAL
m is non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below 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
{ b₁ where b₁ is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like Element of REAL * : len b₁ = m } is set
f is non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below 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
{ b₁ where b₁ is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like Element of REAL * : len b₁ = f } is set
K7((REAL m),(REAL f)) is set
K6(K7((REAL m),(REAL f))) is set
m is ordinal natural V11() real ext-real non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of NAT
f is set
X is non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below Element of NAT
REAL X is non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered FinSequenceSet of REAL
X -tuples_on REAL is non empty functional FinSequence-membered FinSequenceSet of REAL
{ b₁ where b₁ is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like Element of REAL * : len b₁ = X } is set
i is non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below Element of NAT
REAL i is non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered FinSequenceSet of REAL
i -tuples_on REAL is non empty functional FinSequence-membered FinSequenceSet of REAL
{ b₁ where b₁ is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like Element of REAL * : len b₁ = i } is set
K7((REAL X),(REAL i)) is set
K6(K7((REAL X),(REAL i))) is set
K6((REAL X)) is set
y0 is Relation-like REAL X -defined REAL i -valued Function-like complex-functions-valued ext-real-functions-valued real-functions-valued Element of K6(K7((REAL X),(REAL i)))
dom y0 is functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered Element of K6((REAL X))
m is non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below 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
{ b₁ where b₁ is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like Element of REAL * : len b₁ = m } is set
f is non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below 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
{ b₁ where b₁ is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like Element of REAL * : len b₁ = f } is set
K7((REAL m),(REAL f)) is set
K6(K7((REAL m),(REAL f))) 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() strict RealNormSpace-like V185() NORMSTR
the carrier of (REAL-NS m) 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() strict RealNormSpace-like V185() NORMSTR
the carrier of (REAL-NS f) is non empty V2() set
K7( the carrier of (REAL-NS m), the carrier of (REAL-NS f)) is set
K6(K7( the carrier of (REAL-NS m), the carrier of (REAL-NS f))) is set
X is ordinal natural V11() real ext-real non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of NAT
i is Relation-like REAL m -defined REAL f -valued Function-like complex-functions-valued ext-real-functions-valued real-functions-valued Element of K6(K7((REAL m),(REAL f)))
y0 is Relation-like the carrier of (REAL-NS m) -defined the carrier of (REAL-NS f) -valued Function-like Element of K6(K7( the carrier of (REAL-NS m), the carrier of (REAL-NS f)))
r is set
dom i is functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered Element of K6((REAL m))
K6((REAL m)) is set
i | r is Relation-like REAL m -defined REAL f -valued Function-like complex-functions-valued ext-real-functions-valued real-functions-valued Element of K6(K7((REAL m),(REAL f)))
s is Relation-like NAT -defined REAL -valued Function-like finite m -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL m
s is Element of the carrier of (REAL-NS m)
y1 is Relation-like NAT -defined REAL -valued Function-like finite m -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL m
y0 | r is Relation-like the carrier of (REAL-NS m) -defined the carrier of (REAL-NS f) -valued Function-like Element of K6(K7( the carrier of (REAL-NS m), the carrier of (REAL-NS f)))
PP is Relation-like the carrier of (REAL-NS m) -defined the carrier of (REAL-NS f) -valued Function-like Element of K6(K7( the carrier of (REAL-NS m), the carrier of (REAL-NS f)))
DD is Element of the carrier of (REAL-NS m)
dom y0 is Element of K6( the carrier of (REAL-NS m))
K6( the carrier of (REAL-NS m)) is set
s is Element of the carrier of (REAL-NS m)
s is Relation-like NAT -defined REAL -valued Function-like finite m -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL m
y1 is Element of the carrier of (REAL-NS m)
m is non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below 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
{ b₁ where b₁ is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like Element of REAL * : len b₁ = m } is set
f is non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below 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
{ b₁ where b₁ is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like Element of REAL * : len b₁ = f } is set
K7((REAL m),(REAL f)) is set
K6(K7((REAL m),(REAL f))) is set
K6((REAL m)) is set
X is ordinal natural V11() real ext-real non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of NAT
i is Relation-like REAL m -defined REAL f -valued Function-like complex-functions-valued ext-real-functions-valued real-functions-valued Element of K6(K7((REAL m),(REAL f)))
dom i is functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered Element of K6((REAL m))
y0 is functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered Element of K6((REAL m))
REAL-NS m is non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V185() NORMSTR
the carrier of (REAL-NS m) is non empty V2() set
K6( the carrier of (REAL-NS m)) is set
r is Element of K6( the carrier of (REAL-NS m))
REAL-NS f is non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V185() NORMSTR
the carrier of (REAL-NS f) is non empty V2() set
K7( the carrier of (REAL-NS m), the carrier of (REAL-NS f)) is set
K6(K7( the carrier of (REAL-NS m), the carrier of (REAL-NS f))) is set
s is Relation-like the carrier of (REAL-NS m) -defined the carrier of (REAL-NS f) -valued Function-like Element of K6(K7( the carrier of (REAL-NS m), the carrier of (REAL-NS f)))
y1 is Relation-like NAT -defined REAL -valued Function-like finite m -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL m
PP is Element of the carrier of (REAL-NS m)
y1 is Element of the carrier of (REAL-NS m)
PP is Relation-like NAT -defined REAL -valued Function-like finite m -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL m
DD is Relation-like the carrier of (REAL-NS m) -defined the carrier of (REAL-NS f) -valued Function-like Element of K6(K7( the carrier of (REAL-NS m), the carrier of (REAL-NS f)))
a is Element of the carrier of (REAL-NS m)
m is non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below 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
{ b₁ where b₁ is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like Element of REAL * : len b₁ = m } is set
f is non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below 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
{ b₁ where b₁ is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like Element of REAL * : len b₁ = f } is set
K7((REAL m),(REAL f)) is set
K6(K7((REAL m),(REAL f))) is set
X is ordinal natural V11() real ext-real non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of NAT
i is Relation-like REAL m -defined REAL f -valued Function-like complex-functions-valued ext-real-functions-valued real-functions-valued Element of K6(K7((REAL m),(REAL f)))
y0 is set
r is Relation-like REAL m -defined REAL f -valued Function-like complex-functions-valued ext-real-functions-valued real-functions-valued Element of K6(K7((REAL m),(REAL f)))
dom r is functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered Element of K6((REAL m))
K6((REAL m)) is set
s is set
s is set
s is Relation-like NAT -defined REAL -valued Function-like finite m -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL m
r . s is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
partdiff (i,s,X) is Relation-like NAT -defined REAL -valued Function-like finite f -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL f
r /. s is Relation-like NAT -defined Function-like finite f -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL f
r is Relation-like REAL m -defined REAL f -valued Function-like complex-functions-valued ext-real-functions-valued real-functions-valued Element of K6(K7((REAL m),(REAL f)))
dom r is functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered Element of K6((REAL m))
K6((REAL m)) is set
s is Relation-like REAL m -defined REAL f -valued Function-like complex-functions-valued ext-real-functions-valued real-functions-valued Element of K6(K7((REAL m),(REAL f)))
dom s is functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered Element of K6((REAL m))
y1 is Relation-like NAT -defined REAL -valued Function-like finite m -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL m
r /. y1 is Relation-like NAT -defined Function-like finite f -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL f
partdiff (i,y1,X) is Relation-like NAT -defined REAL -valued Function-like finite f -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL f
s /. y1 is Relation-like NAT -defined Function-like finite f -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL f
m is non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below 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
{ b₁ where b₁ is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like Element of REAL * : len b₁ = m } is set
f is non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below 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
{ b₁ where b₁ is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like Element of REAL * : len b₁ = f } is set
K7((REAL m),(REAL f)) is set
K6(K7((REAL m),(REAL f))) is set
m is non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below 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
{ b₁ where b₁ is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like Element of REAL * : len b₁ = m } is set
f is non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below 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
{ b₁ where b₁ is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like Element of REAL * : len b₁ = f } is set
K7((REAL m),(REAL f)) is set
K6(K7((REAL m),(REAL f))) 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() strict RealNormSpace-like V185() NORMSTR
the carrier of (REAL-NS m) 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() strict RealNormSpace-like V185() NORMSTR
the carrier of (REAL-NS f) is non empty V2() set
K7( the carrier of (REAL-NS m), the carrier of (REAL-NS f)) is set
K6(K7( the carrier of (REAL-NS m), the carrier of (REAL-NS f))) is set
X is Relation-like REAL m -defined REAL f -valued Function-like complex-functions-valued ext-real-functions-valued real-functions-valued Element of K6(K7((REAL m),(REAL f)))
i is Relation-like the carrier of (REAL-NS m) -defined the carrier of (REAL-NS f) -valued Function-like Element of K6(K7( the carrier of (REAL-NS m), the carrier of (REAL-NS f)))
y0 is Relation-like NAT -defined REAL -valued Function-like finite m -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL m
r is Element of the carrier of (REAL-NS m)
s is Element of the carrier of (REAL-NS m)
y1 is Relation-like the carrier of (REAL-NS m) -defined the carrier of (REAL-NS f) -valued Function-like Element of K6(K7( the carrier of (REAL-NS m), the carrier of (REAL-NS f)))
m is non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below 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
{ b₁ where b₁ is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like Element of REAL * : len b₁ = m } is set
f is non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below 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
{ b₁ where b₁ is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like Element of REAL * : len b₁ = f } is set
K7((REAL m),(REAL f)) is set
K6(K7((REAL m),(REAL f))) is set
X is Relation-like REAL m -defined REAL f -valued Function-like complex-functions-valued ext-real-functions-valued real-functions-valued Element of K6(K7((REAL m),(REAL f)))
dom X is functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered Element of K6((REAL m))
K6((REAL m)) is set
i is Relation-like NAT -defined REAL -valued Function-like finite m -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL m
X /. i is Relation-like NAT -defined Function-like finite f -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL f
REAL-NS m is non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V185() NORMSTR
the carrier of (REAL-NS m) 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() strict RealNormSpace-like V185() NORMSTR
the carrier of (REAL-NS f) is non empty V2() set
K7( the carrier of (REAL-NS m), the carrier of (REAL-NS f)) is set
K6(K7( the carrier of (REAL-NS m), the carrier of (REAL-NS f))) is set
y0 is Relation-like the carrier of (REAL-NS m) -defined the carrier of (REAL-NS f) -valued Function-like Element of K6(K7( the carrier of (REAL-NS m), the carrier of (REAL-NS f)))
r is Element of the carrier of (REAL-NS m)
dom y0 is Element of K6( the carrier of (REAL-NS m))
K6( the carrier of (REAL-NS m)) is set
y0 /. r is Element of the carrier of (REAL-NS f)
s is V11() real ext-real Element of REAL
s is V11() real ext-real Element of REAL
y1 is V11() real ext-real Element of REAL
PP is Relation-like NAT -defined REAL -valued Function-like finite m -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL m
PP - i is Relation-like NAT -defined REAL -valued Function-like finite m -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL m
- i is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
K38(1) * i is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
PP + (- i) is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
|.(PP - i).| is V11() real ext-real non negative Element of REAL
sqr (PP - i) is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
mlt ((PP - i),(PP - i)) is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
Sum (sqr (PP - i)) is V11() real ext-real Element of REAL
sqrt (Sum (sqr (PP - i))) is V11() real ext-real Element of REAL
DD is Element of the carrier of (REAL-NS m)
DD - r is Element of the carrier of (REAL-NS m)
||.(DD - r).|| is V11() real ext-real Element of REAL
the U8 of (REAL-NS m) is non empty Relation-like the carrier of (REAL-NS m) -defined REAL -valued Function-like total quasi_total complex-valued ext-real-valued real-valued Element of K6(K7( the carrier of (REAL-NS m),REAL))
K7( the carrier of (REAL-NS m),REAL) is complex-valued ext-real-valued real-valued set
K6(K7( the carrier of (REAL-NS m),REAL)) is set
the U8 of (REAL-NS m) . (DD - r) is V11() real ext-real Element of REAL
y0 /. DD is Element of the carrier of (REAL-NS f)
(y0 /. DD) - (y0 /. r) is Element of the carrier of (REAL-NS f)
||.((y0 /. DD) - (y0 /. r)).|| is V11() real ext-real Element of REAL
the U8 of (REAL-NS f) is non empty Relation-like the carrier of (REAL-NS f) -defined REAL -valued Function-like total quasi_total complex-valued ext-real-valued real-valued Element of K6(K7( the carrier of (REAL-NS f),REAL))
K7( the carrier of (REAL-NS f),REAL) is complex-valued ext-real-valued real-valued set
K6(K7( the carrier of (REAL-NS f),REAL)) is set
the U8 of (REAL-NS f) . ((y0 /. DD) - (y0 /. r)) is V11() real ext-real Element of REAL
X /. PP is Relation-like NAT -defined Function-like finite f -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL f
(X /. PP) - (X /. i) is Relation-like NAT -defined REAL -valued Function-like finite f -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL f
- (X /. i) is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
K38(1) * (X /. i) is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
(X /. PP) + (- (X /. i)) is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
|.((X /. PP) - (X /. i)).| is V11() real ext-real non negative Element of REAL
sqr ((X /. PP) - (X /. i)) is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
mlt (((X /. PP) - (X /. i)),((X /. PP) - (X /. i))) is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
Sum (sqr ((X /. PP) - (X /. i))) is V11() real ext-real Element of REAL
sqrt (Sum (sqr ((X /. PP) - (X /. i)))) is V11() real ext-real Element of REAL
s is Element of the carrier of (REAL-NS m)
y0 /. s is Element of the carrier of (REAL-NS f)
y1 is V11() real ext-real Element of REAL
PP is V11() real ext-real Element of REAL
DD is Element of the carrier of (REAL-NS m)
DD - s is Element of the carrier of (REAL-NS m)
||.(DD - s).|| is V11() real ext-real Element of REAL
the U8 of (REAL-NS m) is non empty Relation-like the carrier of (REAL-NS m) -defined REAL -valued Function-like total quasi_total complex-valued ext-real-valued real-valued Element of K6(K7( the carrier of (REAL-NS m),REAL))
K7( the carrier of (REAL-NS m),REAL) is complex-valued ext-real-valued real-valued set
K6(K7( the carrier of (REAL-NS m),REAL)) is set
the U8 of (REAL-NS m) . (DD - s) is V11() real ext-real Element of REAL
a is Relation-like NAT -defined REAL -valued Function-like finite m -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL m
a - i is Relation-like NAT -defined REAL -valued Function-like finite m -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL m
- i is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
K38(1) * i is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
a + (- i) is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
|.(a - i).| is V11() real ext-real non negative Element of REAL
sqr (a - i) is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
mlt ((a - i),(a - i)) is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
Sum (sqr (a - i)) is V11() real ext-real Element of REAL
sqrt (Sum (sqr (a - i))) is V11() real ext-real Element of REAL
X /. a is Relation-like NAT -defined Function-like finite f -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL f
(X /. a) - (X /. i) is Relation-like NAT -defined REAL -valued Function-like finite f -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL f
- (X /. i) is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
K38(1) * (X /. i) is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
(X /. a) + (- (X /. i)) is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
|.((X /. a) - (X /. i)).| is V11() real ext-real non negative Element of REAL
sqr ((X /. a) - (X /. i)) is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
mlt (((X /. a) - (X /. i)),((X /. a) - (X /. i))) is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
Sum (sqr ((X /. a) - (X /. i))) is V11() real ext-real Element of REAL
sqrt (Sum (sqr ((X /. a) - (X /. i)))) is V11() real ext-real Element of REAL
y0 /. DD is Element of the carrier of (REAL-NS f)
(y0 /. DD) - (y0 /. s) is Element of the carrier of (REAL-NS f)
||.((y0 /. DD) - (y0 /. s)).|| is V11() real ext-real Element of REAL
the U8 of (REAL-NS f) is non empty Relation-like the carrier of (REAL-NS f) -defined REAL -valued Function-like total quasi_total complex-valued ext-real-valued real-valued Element of K6(K7( the carrier of (REAL-NS f),REAL))
K7( the carrier of (REAL-NS f),REAL) is complex-valued ext-real-valued real-valued set
K6(K7( the carrier of (REAL-NS f),REAL)) is set
the U8 of (REAL-NS f) . ((y0 /. DD) - (y0 /. s)) is V11() real ext-real Element of REAL
m is non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below 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
{ b₁ where b₁ is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like Element of REAL * : len b₁ = m } is set
f is non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below 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
{ b₁ where b₁ is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like Element of REAL * : len b₁ = f } is set
K7((REAL m),(REAL f)) is set
K6(K7((REAL m),(REAL f))) is set
m is non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below 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
{ b₁ where b₁ is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like Element of REAL * : len b₁ = m } is set
f is non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below 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
{ b₁ where b₁ is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like Element of REAL * : len b₁ = f } is set
K7((REAL m),(REAL f)) is set
K6(K7((REAL m),(REAL f))) 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() strict RealNormSpace-like V185() NORMSTR
the carrier of (REAL-NS m) 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() strict RealNormSpace-like V185() NORMSTR
the carrier of (REAL-NS f) is non empty V2() set
K7( the carrier of (REAL-NS m), the carrier of (REAL-NS f)) is set
K6(K7( the carrier of (REAL-NS m), the carrier of (REAL-NS f))) is set
X is Relation-like REAL m -defined REAL f -valued Function-like complex-functions-valued ext-real-functions-valued real-functions-valued Element of K6(K7((REAL m),(REAL f)))
i is Relation-like the carrier of (REAL-NS m) -defined the carrier of (REAL-NS f) -valued Function-like Element of K6(K7( the carrier of (REAL-NS m), the carrier of (REAL-NS f)))
y0 is set
dom X is functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered Element of K6((REAL m))
K6((REAL m)) is set
X | y0 is Relation-like REAL m -defined REAL f -valued Function-like complex-functions-valued ext-real-functions-valued real-functions-valued Element of K6(K7((REAL m),(REAL f)))
r is Relation-like NAT -defined REAL -valued Function-like finite m -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL m
r is Element of the carrier of (REAL-NS m)
s is Relation-like NAT -defined REAL -valued Function-like finite m -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL m
i | y0 is Relation-like the carrier of (REAL-NS m) -defined the carrier of (REAL-NS f) -valued Function-like Element of K6(K7( the carrier of (REAL-NS m), the carrier of (REAL-NS f)))
dom i 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 is Relation-like NAT -defined REAL -valued Function-like finite m -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL m
s is Element of the carrier of (REAL-NS m)
m is non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below 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
{ b₁ where b₁ is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like Element of REAL * : len b₁ = m } is set
f is non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below 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
{ b₁ where b₁ is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like Element of REAL * : len b₁ = f } is set
K7((REAL m),(REAL f)) is set
K6(K7((REAL m),(REAL f))) is set
X is Relation-like REAL m -defined REAL f -valued Function-like complex-functions-valued ext-real-functions-valued real-functions-valued Element of K6(K7((REAL m),(REAL f)))
dom X is functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered Element of K6((REAL m))
K6((REAL m)) is set
i 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() strict RealNormSpace-like V185() NORMSTR
the carrier of (REAL-NS m) 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() strict RealNormSpace-like V185() NORMSTR
the carrier of (REAL-NS f) is non empty V2() set
K7( the carrier of (REAL-NS m), the carrier of (REAL-NS f)) is set
K6(K7( the carrier of (REAL-NS m), the carrier of (REAL-NS f))) is set
y0 is Relation-like the carrier of (REAL-NS m) -defined the carrier of (REAL-NS f) -valued Function-like Element of K6(K7( the carrier of (REAL-NS m), the carrier of (REAL-NS f)))
r is Relation-like NAT -defined REAL -valued Function-like finite m -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL m
X /. r is Relation-like NAT -defined Function-like finite f -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL f
s is V11() real ext-real Element of REAL
y1 is Element of the carrier of (REAL-NS m)
y0 /. y1 is Element of the carrier of (REAL-NS f)
PP is V11() real ext-real Element of REAL
DD is Relation-like NAT -defined REAL -valued Function-like finite m -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL m
DD - r is Relation-like NAT -defined REAL -valued Function-like finite m -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL m
- r is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
K38(1) * r is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
DD + (- r) is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
|.(DD - r).| is V11() real ext-real non negative Element of REAL
sqr (DD - r) is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
mlt ((DD - r),(DD - r)) is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
Sum (sqr (DD - r)) is V11() real ext-real Element of REAL
sqrt (Sum (sqr (DD - r))) is V11() real ext-real Element of REAL
a is Element of the carrier of (REAL-NS m)
a - y1 is Element of the carrier of (REAL-NS m)
||.(a - y1).|| is V11() real ext-real Element of REAL
the U8 of (REAL-NS m) is non empty Relation-like the carrier of (REAL-NS m) -defined REAL -valued Function-like total quasi_total complex-valued ext-real-valued real-valued Element of K6(K7( the carrier of (REAL-NS m),REAL))
K7( the carrier of (REAL-NS m),REAL) is complex-valued ext-real-valued real-valued set
K6(K7( the carrier of (REAL-NS m),REAL)) is set
the U8 of (REAL-NS m) . (a - y1) is V11() real ext-real Element of REAL
y0 /. a is Element of the carrier of (REAL-NS f)
(y0 /. a) - (y0 /. y1) is Element of the carrier of (REAL-NS f)
||.((y0 /. a) - (y0 /. y1)).|| is V11() real ext-real Element of REAL
the U8 of (REAL-NS f) is non empty Relation-like the carrier of (REAL-NS f) -defined REAL -valued Function-like total quasi_total complex-valued ext-real-valued real-valued Element of K6(K7( the carrier of (REAL-NS f),REAL))
K7( the carrier of (REAL-NS f),REAL) is complex-valued ext-real-valued real-valued set
K6(K7( the carrier of (REAL-NS f),REAL)) is set
the U8 of (REAL-NS f) . ((y0 /. a) - (y0 /. y1)) is V11() real ext-real Element of REAL
X /. DD is Relation-like NAT -defined Function-like finite f -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL f
(X /. DD) - (X /. r) is Relation-like NAT -defined REAL -valued Function-like finite f -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL f
- (X /. r) is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
K38(1) * (X /. r) is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
(X /. DD) + (- (X /. r)) is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
|.((X /. DD) - (X /. r)).| is V11() real ext-real non negative Element of REAL
sqr ((X /. DD) - (X /. r)) is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
mlt (((X /. DD) - (X /. r)),((X /. DD) - (X /. r))) is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
Sum (sqr ((X /. DD) - (X /. r))) is V11() real ext-real Element of REAL
sqrt (Sum (sqr ((X /. DD) - (X /. r)))) is V11() real ext-real Element of REAL
r is Element of the carrier of (REAL-NS m)
y0 /. r is Element of the carrier of (REAL-NS f)
s is V11() real ext-real Element of REAL
y1 is Relation-like NAT -defined REAL -valued Function-like finite m -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL m
X /. y1 is Relation-like NAT -defined Function-like finite f -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL f
PP is V11() real ext-real Element of REAL
DD is Element of the carrier of (REAL-NS m)
DD - r is Element of the carrier of (REAL-NS m)
||.(DD - r).|| is V11() real ext-real Element of REAL
the U8 of (REAL-NS m) is non empty Relation-like the carrier of (REAL-NS m) -defined REAL -valued Function-like total quasi_total complex-valued ext-real-valued real-valued Element of K6(K7( the carrier of (REAL-NS m),REAL))
K7( the carrier of (REAL-NS m),REAL) is complex-valued ext-real-valued real-valued set
K6(K7( the carrier of (REAL-NS m),REAL)) is set
the U8 of (REAL-NS m) . (DD - r) is V11() real ext-real Element of REAL
a is Relation-like NAT -defined REAL -valued Function-like finite m -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL m
a - y1 is Relation-like NAT -defined REAL -valued Function-like finite m -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL m
- y1 is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
K38(1) * y1 is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
a + (- y1) is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
|.(a - y1).| is V11() real ext-real non negative Element of REAL
sqr (a - y1) is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
mlt ((a - y1),(a - y1)) is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
Sum (sqr (a - y1)) is V11() real ext-real Element of REAL
sqrt (Sum (sqr (a - y1))) is V11() real ext-real Element of REAL
X /. a is Relation-like NAT -defined Function-like finite f -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL f
(X /. a) - (X /. y1) is Relation-like NAT -defined REAL -valued Function-like finite f -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL f
- (X /. y1) is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
K38(1) * (X /. y1) is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
(X /. a) + (- (X /. y1)) is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
|.((X /. a) - (X /. y1)).| is V11() real ext-real non negative Element of REAL
sqr ((X /. a) - (X /. y1)) is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
mlt (((X /. a) - (X /. y1)),((X /. a) - (X /. y1))) is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
Sum (sqr ((X /. a) - (X /. y1))) is V11() real ext-real Element of REAL
sqrt (Sum (sqr ((X /. a) - (X /. y1)))) is V11() real ext-real Element of REAL
y0 /. DD is Element of the carrier of (REAL-NS f)
(y0 /. DD) - (y0 /. r) is Element of the carrier of (REAL-NS f)
||.((y0 /. DD) - (y0 /. r)).|| is V11() real ext-real Element of REAL
the U8 of (REAL-NS f) is non empty Relation-like the carrier of (REAL-NS f) -defined REAL -valued Function-like total quasi_total complex-valued ext-real-valued real-valued Element of K6(K7( the carrier of (REAL-NS f),REAL))
K7( the carrier of (REAL-NS f),REAL) is complex-valued ext-real-valued real-valued set
K6(K7( the carrier of (REAL-NS f),REAL)) is set
the U8 of (REAL-NS f) . ((y0 /. DD) - (y0 /. r)) is V11() real ext-real Element of REAL
m is non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below 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
{ b₁ where b₁ is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like Element of REAL * : len b₁ = m } is set
X is Relation-like NAT -defined REAL -valued Function-like finite m -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL m
f is Relation-like NAT -defined REAL -valued Function-like finite m -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL m
i is ordinal natural V11() real ext-real non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of NAT
reproj (i,f) is non empty Relation-like REAL -defined REAL m -valued Function-like total quasi_total complex-functions-valued ext-real-functions-valued real-functions-valued Element of K6(K7(REAL,(REAL m)))
K7(REAL,(REAL m)) is set
K6(K7(REAL,(REAL m))) is set
proj (i,m) is non empty Relation-like REAL m -defined REAL -valued Function-like total quasi_total complex-valued ext-real-valued real-valued Element of K6(K7((REAL m),REAL))
K7((REAL m),REAL) is complex-valued ext-real-valued real-valued set
K6(K7((REAL m),REAL)) is set
(proj (i,m)) . X is V11() real ext-real Element of REAL
y0 is V11() real ext-real Element of REAL
(reproj (i,f)) . y0 is Relation-like NAT -defined Function-like finite m -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL m
Replace (f,i,y0) is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
len f is ordinal natural V11() real ext-real non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of NAT
len X is ordinal natural V11() real ext-real non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of NAT
dom X is complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of K6(NAT)
X /. i is V11() real ext-real Element of REAL
X . i is V11() real ext-real Element of REAL
m is non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below 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
{ b₁ where b₁ is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like Element of REAL * : len b₁ = m } is set
K7((REAL m),REAL) is complex-valued ext-real-valued real-valued set
K6(K7((REAL m),REAL)) is set
i is Relation-like NAT -defined REAL -valued Function-like finite m -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL m
X is Relation-like NAT -defined REAL -valued Function-like finite m -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL m
y0 is ordinal natural V11() real ext-real non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of NAT
reproj (y0,X) is non empty Relation-like REAL -defined REAL m -valued Function-like total quasi_total complex-functions-valued ext-real-functions-valued real-functions-valued Element of K6(K7(REAL,(REAL m)))
K7(REAL,(REAL m)) is set
K6(K7(REAL,(REAL m))) is set
reproj (y0,i) is non empty Relation-like REAL -defined REAL m -valued Function-like total quasi_total complex-functions-valued ext-real-functions-valued real-functions-valued Element of K6(K7(REAL,(REAL m)))
r is V11() real ext-real Element of REAL
(reproj (y0,X)) . r is Relation-like NAT -defined Function-like finite m -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL m
Replace (X,y0,r) is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
len X is ordinal natural V11() real ext-real non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of NAT
len i is ordinal natural V11() real ext-real non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of NAT
y0 -' 1 is ordinal natural V11() real ext-real non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of NAT
X | (y0 -' 1) is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
Seg (y0 -' 1) is finite y0 -' 1 -element complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below bounded_above real-bounded Element of K6(NAT)
{ b₁ where b₁ is ordinal natural V11() real ext-real non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of NAT : ( 1 <= b₁ & b₁ <= y0 -' 1 ) } is set
X | (Seg (y0 -' 1)) is Relation-like NAT -defined Function-like FinSubsequence-like complex-valued ext-real-valued real-valued set
<*r*> is non empty V2() Relation-like NAT -defined REAL -valued Function-like one-to-one constant finite 1 -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued increasing decreasing non-decreasing non-increasing FinSequence of REAL
[1,r] is V27() set
{1,r} is non empty complex-membered ext-real-membered real-membered set
{{1,r},{1}} is non empty set
{[1,r]} is non empty Function-like set
(X | (y0 -' 1)) ^ <*r*> is non empty Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
X /^ y0 is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
((X | (y0 -' 1)) ^ <*r*>) ^ (X /^ y0) is non empty Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
dom (reproj (y0,X)) is complex-membered ext-real-membered real-membered Element of K6(REAL)
dom (reproj (y0,i)) is complex-membered ext-real-membered real-membered Element of K6(REAL)
s is set
(reproj (y0,X)) . s is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
(reproj (y0,i)) . s is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
y1 is V11() real ext-real Element of REAL
Replace (X,y0,y1) is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
<*y1*> is non empty V2() Relation-like NAT -defined REAL -valued Function-like one-to-one constant finite 1 -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued increasing decreasing non-decreasing non-increasing FinSequence of REAL
[1,y1] is V27() set
{1,y1} is non empty complex-membered ext-real-membered real-membered set
{{1,y1},{1}} is non empty set
{[1,y1]} is non empty Function-like set
(X | (y0 -' 1)) ^ <*y1*> is non empty Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
((X | (y0 -' 1)) ^ <*y1*>) ^ (X /^ y0) is non empty Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
Replace (i,y0,y1) is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
i | (y0 -' 1) is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
i | (Seg (y0 -' 1)) is Relation-like NAT -defined Function-like FinSubsequence-like complex-valued ext-real-valued real-valued set
(i | (y0 -' 1)) ^ <*y1*> is non empty Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
i /^ y0 is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
((i | (y0 -' 1)) ^ <*y1*>) ^ (i /^ y0) is non empty Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
dom (i | (y0 -' 1)) is complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of K6(NAT)
dom (X | (y0 -' 1)) is complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of K6(NAT)
PP is ordinal natural V11() real ext-real non negative set
(i | (y0 -' 1)) /. PP is V11() real ext-real Element of REAL
(X | (y0 -' 1)) /. PP is V11() real ext-real Element of REAL
len (X | (y0 -' 1)) is ordinal natural V11() real ext-real non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of NAT
Seg (len (X | (y0 -' 1))) is finite len (X | (y0 -' 1)) -element complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below bounded_above real-bounded Element of K6(NAT)
{ b₁ where b₁ is ordinal natural V11() real ext-real non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of NAT : ( 1 <= b₁ & b₁ <= len (X | (y0 -' 1)) ) } is set
(i | (y0 -' 1)) . PP is V11() real ext-real Element of REAL
(((X | (y0 -' 1)) ^ <*r*>) ^ (X /^ y0)) . PP is V11() real ext-real Element of REAL
<*r*> ^ (X /^ y0) is non empty Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
(X | (y0 -' 1)) ^ (<*r*> ^ (X /^ y0)) is non empty Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
((X | (y0 -' 1)) ^ (<*r*> ^ (X /^ y0))) . PP is V11() real ext-real Element of REAL
(X | (y0 -' 1)) . PP is V11() real ext-real Element of REAL
len (i /^ y0) is ordinal natural V11() real ext-real non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of NAT
(len i) -' y0 is ordinal natural V11() real ext-real non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of NAT
len (X /^ y0) is ordinal natural V11() real ext-real non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of NAT
(len X) -' y0 is ordinal natural V11() real ext-real non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of NAT
PP is ordinal natural V11() real ext-real non negative set
(i /^ y0) . PP is V11() real ext-real Element of REAL
(X /^ y0) . PP is V11() real ext-real Element of REAL
dom (i /^ y0) is complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of K6(NAT)
dom (X /^ y0) is complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of K6(NAT)
len (X | (y0 -' 1)) is ordinal natural V11() real ext-real non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of NAT
len <*r*> is ordinal natural V11() real ext-real non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of NAT
y0 - 1 is V11() real ext-real V163() V198() Element of INT
len ((X | (y0 -' 1)) ^ <*r*>) is ordinal natural V11() real ext-real non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of NAT
(y0 - 1) + 1 is V11() real ext-real V163() V198() Element of INT
y0 + PP is ordinal natural V11() real ext-real non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of NAT
i . (y0 + PP) is V11() real ext-real Element of REAL
m is non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below 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
{ b₁ where b₁ is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like Element of REAL * : len b₁ = m } is set
K7((REAL m),REAL) is complex-valued ext-real-valued real-valued set
K6(K7((REAL m),REAL)) is set
f is Relation-like REAL m -defined REAL -valued Function-like complex-valued ext-real-valued real-valued Element of K6(K7((REAL m),REAL))
X is Relation-like REAL -defined REAL -valued Function-like complex-valued ext-real-valued real-valued Element of K6(K7(REAL,REAL))
y0 is Relation-like NAT -defined REAL -valued Function-like finite m -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL m
i is Relation-like NAT -defined REAL -valued Function-like finite m -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL m
r is ordinal natural V11() real ext-real non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of NAT
reproj (r,i) is non empty Relation-like REAL -defined REAL m -valued Function-like total quasi_total complex-functions-valued ext-real-functions-valued real-functions-valued Element of K6(K7(REAL,(REAL m)))
K7(REAL,(REAL m)) is set
K6(K7(REAL,(REAL m))) is set
f * (reproj (r,i)) is Relation-like REAL -defined REAL -valued Function-like complex-valued ext-real-valued real-valued Element of K6(K7(REAL,REAL))
partdiff (f,y0,r) is V11() real ext-real Element of REAL
reproj (r,y0) is non empty Relation-like REAL -defined REAL m -valued Function-like total quasi_total complex-functions-valued ext-real-functions-valued real-functions-valued Element of K6(K7(REAL,(REAL m)))
f * (reproj (r,y0)) is Relation-like REAL -defined REAL -valued Function-like complex-valued ext-real-valued real-valued Element of K6(K7(REAL,REAL))
proj (r,m) is non empty Relation-like REAL m -defined REAL -valued Function-like total quasi_total complex-valued ext-real-valued real-valued Element of K6(K7((REAL m),REAL))
(proj (r,m)) . y0 is V11() real ext-real Element of REAL
diff ((f * (reproj (r,y0))),((proj (r,m)) . y0)) is V11() real ext-real Element of REAL
s is V11() real ext-real Element of REAL
(reproj (r,i)) . s is Relation-like NAT -defined Function-like finite m -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL m
diff (X,s) is V11() real ext-real Element of REAL
m is non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below 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
{ b₁ where b₁ is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like Element of REAL * : len b₁ = m } is set
K7((REAL m),REAL) is complex-valued ext-real-valued real-valued set
K6(K7((REAL m),REAL)) is set
f is Relation-like REAL m -defined REAL -valued Function-like complex-valued ext-real-valued real-valued Element of K6(K7((REAL m),REAL))
dom f is functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered Element of K6((REAL m))
K6((REAL m)) is set
i is V11() real ext-real Element of REAL
X is V11() real ext-real Element of REAL
[.X,i.] is complex-membered ext-real-membered real-membered closed V263() Element of K6(REAL)
{ b₁ where b₁ is V11() real ext-real Element of REAL : ( X <= b₁ & b₁ <= i ) } is set
].X,i.[ is complex-membered ext-real-membered real-membered open non left_end non right_end V263() Element of K6(REAL)
{ b₁ where b₁ is V11() real ext-real Element of REAL : ( not b₁ <= X & not i <= b₁ ) } is set
i - X is V11() real ext-real Element of REAL
y0 is Relation-like NAT -defined REAL -valued Function-like finite m -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL m
r is ordinal natural V11() real ext-real non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of NAT
reproj (r,y0) is non empty Relation-like REAL -defined REAL m -valued Function-like total quasi_total complex-functions-valued ext-real-functions-valued real-functions-valued Element of K6(K7(REAL,(REAL m)))
K7(REAL,(REAL m)) is set
K6(K7(REAL,(REAL m))) is set
(reproj (r,y0)) . i is Relation-like NAT -defined Function-like finite m -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL m
f /. ((reproj (r,y0)) . i) is V11() real ext-real Element of REAL
(reproj (r,y0)) . X is Relation-like NAT -defined Function-like finite m -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL m
f /. ((reproj (r,y0)) . X) is V11() real ext-real Element of REAL
(f /. ((reproj (r,y0)) . i)) - (f /. ((reproj (r,y0)) . X)) is V11() real ext-real Element of REAL
f * (reproj (r,y0)) is Relation-like REAL -defined REAL -valued Function-like complex-valued ext-real-valued real-valued Element of K6(K7(REAL,REAL))
y1 is set
dom (reproj (r,y0)) is complex-membered ext-real-membered real-membered Element of K6(REAL)
PP is V11() real ext-real Element of REAL
(reproj (r,y0)) . PP is Relation-like NAT -defined Function-like finite m -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL m
dom (f * (reproj (r,y0))) is complex-membered ext-real-membered real-membered Element of K6(REAL)
y1 is V11() real ext-real Element of REAL
(reproj (r,y0)) . y1 is Relation-like NAT -defined Function-like finite m -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL m
proj (r,m) is non empty Relation-like REAL m -defined REAL -valued Function-like total quasi_total complex-valued ext-real-valued real-valued Element of K6(K7((REAL m),REAL))
(proj (r,m)) . ((reproj (r,y0)) . y1) is V11() real ext-real Element of REAL
reproj (r,((reproj (r,y0)) . y1)) is non empty Relation-like REAL -defined REAL m -valued Function-like total quasi_total complex-functions-valued ext-real-functions-valued real-functions-valued Element of K6(K7(REAL,(REAL m)))
f * (reproj (r,((reproj (r,y0)) . y1))) is Relation-like REAL -defined REAL -valued Function-like complex-valued ext-real-valued real-valued Element of K6(K7(REAL,REAL))
y1 is set
PP is V11() real ext-real Element of REAL
y1 is V11() real ext-real Element of REAL
y1 is V11() real ext-real set
PP is V11() real ext-real set
(f * (reproj (r,y0))) . y1 is V11() real ext-real Element of REAL
DD is V11() real ext-real set
a is V11() real ext-real set
t is V11() real ext-real set
t - y1 is V11() real ext-real Element of REAL
abs (t - y1) is V11() real ext-real Element of REAL
(f * (reproj (r,y0))) . t is V11() real ext-real Element of REAL
((f * (reproj (r,y0))) . t) - ((f * (reproj (r,y0))) . y1) is V11() real ext-real Element of REAL
abs (((f * (reproj (r,y0))) . t) - ((f * (reproj (r,y0))) . y1)) is V11() real ext-real Element of REAL
(f * (reproj (r,y0))) | [.X,i.] is Relation-like REAL -defined REAL -valued Function-like complex-valued ext-real-valued real-valued Element of K6(K7(REAL,REAL))
(f * (reproj (r,y0))) . i is V11() real ext-real Element of REAL
(f * (reproj (r,y0))) . X is V11() real ext-real Element of REAL
((f * (reproj (r,y0))) . i) - ((f * (reproj (r,y0))) . X) is V11() real ext-real Element of REAL
(((f * (reproj (r,y0))) . i) - ((f * (reproj (r,y0))) . X)) / (i - X) is V11() real ext-real Element of REAL
y1 is V11() real ext-real Element of REAL
diff ((f * (reproj (r,y0))),y1) is V11() real ext-real Element of REAL
(diff ((f * (reproj (r,y0))),y1)) * (i - X) is V11() real ext-real Element of REAL
f . ((reproj (r,y0)) . X) is V11() real ext-real Element of REAL
f . ((reproj (r,y0)) . i) is V11() real ext-real Element of REAL
(reproj (r,y0)) . y1 is Relation-like NAT -defined Function-like finite m -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL m
PP is Relation-like NAT -defined REAL -valued Function-like finite m -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL m
partdiff (f,PP,r) is V11() real ext-real Element of REAL
reproj (r,PP) is non empty Relation-like REAL -defined REAL m -valued Function-like total quasi_total complex-functions-valued ext-real-functions-valued real-functions-valued Element of K6(K7(REAL,(REAL m)))
f * (reproj (r,PP)) is Relation-like REAL -defined REAL -valued Function-like complex-valued ext-real-valued real-valued Element of K6(K7(REAL,REAL))
(proj (r,m)) . PP is V11() real ext-real Element of REAL
diff ((f * (reproj (r,PP))),((proj (r,m)) . PP)) is V11() real ext-real Element of REAL
m is non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below 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
{ b₁ where b₁ is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like Element of REAL * : len b₁ = m } is set
K7((REAL m),REAL) is complex-valued ext-real-valued real-valued set
K6(K7((REAL m),REAL)) is set
f is Relation-like REAL m -defined REAL -valued Function-like complex-valued ext-real-valued real-valued Element of K6(K7((REAL m),REAL))
dom f is functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered Element of K6((REAL m))
K6((REAL m)) is set
X is V11() real ext-real Element of REAL
i is V11() real ext-real Element of REAL
[.X,i.] is complex-membered ext-real-membered real-membered closed V263() Element of K6(REAL)
{ b₁ where b₁ is V11() real ext-real Element of REAL : ( X <= b₁ & b₁ <= i ) } is set
i - X is V11() real ext-real Element of REAL
y0 is Relation-like NAT -defined REAL -valued Function-like finite m -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL m
r is ordinal natural V11() real ext-real non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of NAT
reproj (r,y0) is non empty Relation-like REAL -defined REAL m -valued Function-like total quasi_total complex-functions-valued ext-real-functions-valued real-functions-valued Element of K6(K7(REAL,(REAL m)))
K7(REAL,(REAL m)) is set
K6(K7(REAL,(REAL m))) is set
(reproj (r,y0)) . i is Relation-like NAT -defined Function-like finite m -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL m
f /. ((reproj (r,y0)) . i) is V11() real ext-real Element of REAL
(reproj (r,y0)) . X is Relation-like NAT -defined Function-like finite m -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL m
f /. ((reproj (r,y0)) . X) is V11() real ext-real Element of REAL
(f /. ((reproj (r,y0)) . i)) - (f /. ((reproj (r,y0)) . X)) is V11() real ext-real Element of REAL
s is Relation-like NAT -defined REAL -valued Function-like finite m -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL m
partdiff (f,s,r) is V11() real ext-real Element of REAL
reproj (r,s) is non empty Relation-like REAL -defined REAL m -valued Function-like total quasi_total complex-functions-valued ext-real-functions-valued real-functions-valued Element of K6(K7(REAL,(REAL m)))
f * (reproj (r,s)) is Relation-like REAL -defined REAL -valued Function-like complex-valued ext-real-valued real-valued Element of K6(K7(REAL,REAL))
proj (r,m) is non empty Relation-like REAL m -defined REAL -valued Function-like total quasi_total complex-valued ext-real-valued real-valued Element of K6(K7((REAL m),REAL))
(proj (r,m)) . s is V11() real ext-real Element of REAL
diff ((f * (reproj (r,s))),((proj (r,m)) . s)) is V11() real ext-real Element of REAL
(i - X) * (partdiff (f,s,r)) is V11() real ext-real Element of REAL
].X,i.[ is complex-membered ext-real-membered real-membered open non left_end non right_end V263() Element of K6(REAL)
{ b₁ where b₁ is V11() real ext-real Element of REAL : ( not b₁ <= X & not i <= b₁ ) } is set
s is V11() real ext-real Element of REAL
y1 is Relation-like NAT -defined REAL -valued Function-like finite m -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL m
(reproj (r,y0)) . s is Relation-like NAT -defined Function-like finite m -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL m
partdiff (f,y1,r) is V11() real ext-real Element of REAL
reproj (r,y1) is non empty Relation-like REAL -defined REAL m -valued Function-like total quasi_total complex-functions-valued ext-real-functions-valued real-functions-valued Element of K6(K7(REAL,(REAL m)))
f * (reproj (r,y1)) is Relation-like REAL -defined REAL -valued Function-like complex-valued ext-real-valued real-valued Element of K6(K7(REAL,REAL))
proj (r,m) is non empty Relation-like REAL m -defined REAL -valued Function-like total quasi_total complex-valued ext-real-valued real-valued Element of K6(K7((REAL m),REAL))
(proj (r,m)) . y1 is V11() real ext-real Element of REAL
diff ((f * (reproj (r,y1))),((proj (r,m)) . y1)) is V11() real ext-real Element of REAL
(i - X) * (partdiff (f,y1,r)) is V11() real ext-real Element of REAL
m is non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below 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
{ b₁ where b₁ is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like Element of REAL * : len b₁ = m } is set
X is Relation-like NAT -defined REAL -valued Function-like finite m -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL m
f is Relation-like NAT -defined REAL -valued Function-like finite m -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL m
X - f is Relation-like NAT -defined REAL -valued Function-like finite m -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL m
- f is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
K38(1) * f is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
X + (- f) is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
|.(X - f).| is V11() real ext-real non negative Element of REAL
sqr (X - f) is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
mlt ((X - f),(X - f)) is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
Sum (sqr (X - f)) is V11() real ext-real Element of REAL
sqrt (Sum (sqr (X - f))) is V11() real ext-real Element of REAL
i is Relation-like NAT -defined REAL -valued Function-like finite m -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL m
i - f is Relation-like NAT -defined REAL -valued Function-like finite m -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL m
i + (- f) is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
|.(i - f).| is V11() real ext-real non negative Element of REAL
sqr (i - f) is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
mlt ((i - f),(i - f)) is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
Sum (sqr (i - f)) is V11() real ext-real Element of REAL
sqrt (Sum (sqr (i - f))) is V11() real ext-real Element of REAL
y0 is Relation-like NAT -defined REAL -valued Function-like finite m -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL m
y0 - f is Relation-like NAT -defined REAL -valued Function-like finite m -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL m
y0 + (- f) is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
|.(y0 - f).| is V11() real ext-real non negative Element of REAL
sqr (y0 - f) is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
mlt ((y0 - f),(y0 - f)) is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
Sum (sqr (y0 - f)) is V11() real ext-real Element of REAL
sqrt (Sum (sqr (y0 - f))) is V11() real ext-real Element of REAL
r is ordinal natural V11() real ext-real non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of NAT
proj (r,m) is non empty Relation-like REAL m -defined REAL -valued Function-like total quasi_total complex-valued ext-real-valued real-valued Element of K6(K7((REAL m),REAL))
K7((REAL m),REAL) is complex-valued ext-real-valued real-valued set
K6(K7((REAL m),REAL)) is set
(proj (r,m)) . X is V11() real ext-real Element of REAL
reproj (r,X) is non empty Relation-like REAL -defined REAL m -valued Function-like total quasi_total complex-functions-valued ext-real-functions-valued real-functions-valued Element of K6(K7(REAL,(REAL m)))
K7(REAL,(REAL m)) is set
K6(K7(REAL,(REAL m))) is set
s is V11() real ext-real Element of REAL
y1 is V11() real ext-real Element of REAL
PP is V11() real ext-real Element of REAL
(reproj (r,X)) . PP is Relation-like NAT -defined Function-like finite m -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL m
DD is V11() real ext-real Element of REAL
[.y1,PP.] is complex-membered ext-real-membered real-membered closed V263() Element of K6(REAL)
{ b₁ where b₁ is V11() real ext-real Element of REAL : ( y1 <= b₁ & b₁ <= PP ) } is set
(reproj (r,X)) . DD is Relation-like NAT -defined Function-like finite m -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL m
sqr (X - f) is Relation-like NAT -defined REAL -valued Function-like finite m -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of m -tuples_on REAL
Sum (sqr (X - f)) is V11() real ext-real Element of REAL
|.(X - f).| ^2 is V11() real ext-real Element of REAL
K37(|.(X - f).|,|.(X - f).|) is V11() real ext-real non negative set
sqr (y0 - f) is Relation-like NAT -defined REAL -valued Function-like finite m -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of m -tuples_on REAL
Sum (sqr (y0 - f)) is V11() real ext-real Element of REAL
|.(y0 - f).| ^2 is V11() real ext-real Element of REAL
K37(|.(y0 - f).|,|.(y0 - f).|) is V11() real ext-real non negative set
sqr (i - f) is Relation-like NAT -defined REAL -valued Function-like finite m -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of m -tuples_on REAL
Sum (sqr (i - f)) is V11() real ext-real Element of REAL
|.(i - f).| ^2 is V11() real ext-real Element of REAL
K37(|.(i - f).|,|.(i - f).|) is V11() real ext-real non negative set
(proj (r,m)) . i is V11() real ext-real Element of REAL
(proj (r,m)) . y0 is V11() real ext-real Element of REAL
Seg m is non empty finite m -element complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end right_end bounded_below bounded_above real-bounded Element of K6(NAT)
{ b₁ where b₁ is ordinal natural V11() real ext-real non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of NAT : ( 1 <= b₁ & b₁ <= m ) } is set
dom (X - f) is complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of K6(NAT)
dom (y0 - f) is complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of K6(NAT)
dom (i - f) is complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of K6(NAT)
w1 is ordinal natural V11() real ext-real non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of NAT
(sqr (X - f)) . w1 is V11() real ext-real Element of REAL
(sqr (y0 - f)) . w1 is V11() real ext-real Element of REAL
dom X is complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of K6(NAT)
len X is ordinal natural V11() real ext-real non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of NAT
Seg (len X) is finite len X -element complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below bounded_above real-bounded Element of K6(NAT)
{ b₁ where b₁ is ordinal natural V11() real ext-real non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of NAT : ( 1 <= b₁ & b₁ <= len X ) } is set
Replace (X,r,DD) is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
len (Replace (X,r,DD)) is ordinal natural V11() real ext-real non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of NAT
Seg (len (Replace (X,r,DD))) is finite len (Replace (X,r,DD)) -element complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below bounded_above real-bounded Element of K6(NAT)
{ b₁ where b₁ is ordinal natural V11() real ext-real non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of NAT : ( 1 <= b₁ & b₁ <= len (Replace (X,r,DD)) ) } is set
dom (Replace (X,r,DD)) is complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of K6(NAT)
sqrreal * (X - f) is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
sqrreal * (y0 - f) is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
(X - f) . w1 is V11() real ext-real Element of REAL
sqrreal . ((X - f) . w1) is V11() real ext-real Element of REAL
(y0 - f) . w1 is V11() real ext-real Element of REAL
sqrreal . ((y0 - f) . w1) is V11() real ext-real Element of REAL
((X - f) . w1) ^2 is V11() real ext-real Element of REAL
K37(((X - f) . w1),((X - f) . w1)) is V11() real ext-real set
((y0 - f) . w1) ^2 is V11() real ext-real Element of REAL
K37(((y0 - f) . w1),((y0 - f) . w1)) is V11() real ext-real set
X . w1 is V11() real ext-real Element of REAL
f . w1 is V11() real ext-real Element of REAL
(X . w1) - (f . w1) is V11() real ext-real Element of REAL
((X . w1) - (f . w1)) ^2 is V11() real ext-real Element of REAL
K37(((X . w1) - (f . w1)),((X . w1) - (f . w1))) is V11() real ext-real set
y0 . w1 is V11() real ext-real Element of REAL
(y0 . w1) - (f . w1) is V11() real ext-real Element of REAL
((y0 . w1) - (f . w1)) ^2 is V11() real ext-real Element of REAL
K37(((y0 . w1) - (f . w1)),((y0 . w1) - (f . w1))) is V11() real ext-real set
(Replace (X,r,DD)) . w1 is V11() real ext-real Element of REAL
(Replace (X,r,DD)) /. w1 is V11() real ext-real Element of REAL
X /. w1 is V11() real ext-real Element of REAL
w1 is ordinal natural V11() real ext-real non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of NAT
(sqr (i - f)) . w1 is V11() real ext-real Element of REAL
(sqr (y0 - f)) . w1 is V11() real ext-real Element of REAL
dom i is complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of K6(NAT)
len i is ordinal natural V11() real ext-real non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of NAT
Seg (len i) is finite len i -element complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below bounded_above real-bounded Element of K6(NAT)
{ b₁ where b₁ is ordinal natural V11() real ext-real non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of NAT : ( 1 <= b₁ & b₁ <= len i ) } is set
Replace (i,r,DD) is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
len (Replace (i,r,DD)) is ordinal natural V11() real ext-real non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of NAT
Seg (len (Replace (i,r,DD))) is finite len (Replace (i,r,DD)) -element complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below bounded_above real-bounded Element of K6(NAT)
{ b₁ where b₁ is ordinal natural V11() real ext-real non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of NAT : ( 1 <= b₁ & b₁ <= len (Replace (i,r,DD)) ) } is set
dom (Replace (i,r,DD)) is complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of K6(NAT)
sqrreal * (i - f) is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
sqrreal * (y0 - f) is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
(i - f) . w1 is V11() real ext-real Element of REAL
sqrreal . ((i - f) . w1) is V11() real ext-real Element of REAL
(y0 - f) . w1 is V11() real ext-real Element of REAL
sqrreal . ((y0 - f) . w1) is V11() real ext-real Element of REAL
((i - f) . w1) ^2 is V11() real ext-real Element of REAL
K37(((i - f) . w1),((i - f) . w1)) is V11() real ext-real set
((y0 - f) . w1) ^2 is V11() real ext-real Element of REAL
K37(((y0 - f) . w1),((y0 - f) . w1)) is V11() real ext-real set
i . w1 is V11() real ext-real Element of REAL
f . w1 is V11() real ext-real Element of REAL
(i . w1) - (f . w1) is V11() real ext-real Element of REAL
((i . w1) - (f . w1)) ^2 is V11() real ext-real Element of REAL
K37(((i . w1) - (f . w1)),((i . w1) - (f . w1))) is V11() real ext-real set
y0 . w1 is V11() real ext-real Element of REAL
(y0 . w1) - (f . w1) is V11() real ext-real Element of REAL
((y0 . w1) - (f . w1)) ^2 is V11() real ext-real Element of REAL
K37(((y0 . w1) - (f . w1)),((y0 . w1) - (f . w1))) is V11() real ext-real set
reproj (r,i) is non empty Relation-like REAL -defined REAL m -valued Function-like total quasi_total complex-functions-valued ext-real-functions-valued real-functions-valued Element of K6(K7(REAL,(REAL m)))
(reproj (r,i)) . DD is Relation-like NAT -defined Function-like finite m -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL m
((reproj (r,i)) . DD) . w1 is V11() real ext-real Element of REAL
(Replace (i,r,DD)) . w1 is V11() real ext-real Element of REAL
(Replace (i,r,DD)) /. w1 is V11() real ext-real Element of REAL
i /. w1 is V11() real ext-real Element of REAL
(sqr (y0 - f)) . r is V11() real ext-real Element of REAL
(sqr (X - f)) . r is V11() real ext-real Element of REAL
len (sqr (y0 - f)) is ordinal natural V11() real ext-real non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of NAT
len (sqr (X - f)) is ordinal natural V11() real ext-real non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of NAT
dom (sqr (y0 - f)) is complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of K6(NAT)
w1 is ordinal natural V11() real ext-real non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of NAT
(sqr (y0 - f)) . w1 is V11() real ext-real Element of REAL
(sqr (X - f)) . w1 is V11() real ext-real Element of REAL
(sqr (i - f)) . r is V11() real ext-real Element of REAL
len (sqr (i - f)) is ordinal natural V11() real ext-real non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of NAT
w1 is ordinal natural V11() real ext-real non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of NAT
(sqr (y0 - f)) . w1 is V11() real ext-real Element of REAL
(sqr (i - f)) . w1 is V11() real ext-real Element of REAL
sqrreal * (X - f) is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
sqrreal * (y0 - f) is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
sqrreal * (i - f) is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
(X - f) . r is V11() real ext-real Element of REAL
sqrreal . ((X - f) . r) is V11() real ext-real Element of REAL
(y0 - f) . r is V11() real ext-real Element of REAL
sqrreal . ((y0 - f) . r) is V11() real ext-real Element of REAL
(i - f) . r is V11() real ext-real Element of REAL
sqrreal . ((i - f) . r) is V11() real ext-real Element of REAL
((X - f) . r) ^2 is V11() real ext-real Element of REAL
K37(((X - f) . r),((X - f) . r)) is V11() real ext-real set
((y0 - f) . r) ^2 is V11() real ext-real Element of REAL
K37(((y0 - f) . r),((y0 - f) . r)) is V11() real ext-real set
((i - f) . r) ^2 is V11() real ext-real Element of REAL
K37(((i - f) . r),((i - f) . r)) is V11() real ext-real set
X . r is V11() real ext-real Element of REAL
y0 . r is V11() real ext-real Element of REAL
i . r is V11() real ext-real Element of REAL
f . r is V11() real ext-real Element of REAL
y1 - (f . r) is V11() real ext-real Element of REAL
(y1 - (f . r)) ^2 is V11() real ext-real Element of REAL
K37((y1 - (f . r)),(y1 - (f . r))) is V11() real ext-real set
DD - (f . r) is V11() real ext-real Element of REAL
(DD - (f . r)) ^2 is V11() real ext-real Element of REAL
K37((DD - (f . r)),(DD - (f . r))) is V11() real ext-real set
PP - (f . r) is V11() real ext-real Element of REAL
(PP - (f . r)) ^2 is V11() real ext-real Element of REAL
K37((PP - (f . r)),(PP - (f . r))) is V11() real ext-real set
(f . r) - y1 is V11() real ext-real Element of REAL
(f . r) - DD is V11() real ext-real Element of REAL
((f . r) - y1) ^2 is V11() real ext-real Element of REAL
K37(((f . r) - y1),((f . r) - y1)) is V11() real ext-real set
((f . r) - DD) ^2 is V11() real ext-real Element of REAL
K37(((f . r) - DD),((f . r) - DD)) is V11() real ext-real set
(f . r) - DD is V11() real ext-real Element of REAL
(f . r) - y1 is V11() real ext-real Element of REAL
((f . r) - y1) ^2 is V11() real ext-real Element of REAL
K37(((f . r) - y1),((f . r) - y1)) is V11() real ext-real set
((f . r) - DD) ^2 is V11() real ext-real Element of REAL
K37(((f . r) - DD),((f . r) - DD)) is V11() real ext-real set
m is non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below 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
{ b₁ where b₁ is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like Element of REAL * : len b₁ = m } is set
K7((REAL m),REAL) is complex-valued ext-real-valued real-valued set
K6(K7((REAL m),REAL)) is set
K6((REAL m)) is set
f is Relation-like REAL m -defined REAL -valued Function-like complex-valued ext-real-valued real-valued Element of K6(K7((REAL m),REAL))
dom f is functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered Element of K6((REAL m))
X is functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered Element of K6((REAL m))
i is Relation-like NAT -defined REAL -valued Function-like finite m -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL m
y0 is Relation-like NAT -defined REAL -valued Function-like finite m -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL m
y0 - i is Relation-like NAT -defined REAL -valued Function-like finite m -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL m
- i is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
K38(1) * i is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
y0 + (- i) is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
|.(y0 - i).| is V11() real ext-real non negative Element of REAL
sqr (y0 - i) is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
mlt ((y0 - i),(y0 - i)) is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
Sum (sqr (y0 - i)) is V11() real ext-real Element of REAL
sqrt (Sum (sqr (y0 - i))) is V11() real ext-real Element of REAL
r is Relation-like NAT -defined REAL -valued Function-like finite m -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL m
r - i is Relation-like NAT -defined REAL -valued Function-like finite m -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL m
r + (- i) is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
|.(r - i).| is V11() real ext-real non negative Element of REAL
sqr (r - i) is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
mlt ((r - i),(r - i)) is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
Sum (sqr (r - i)) is V11() real ext-real Element of REAL
sqrt (Sum (sqr (r - i))) is V11() real ext-real Element of REAL
f /. r is V11() real ext-real Element of REAL
f /. y0 is V11() real ext-real Element of REAL
(f /. r) - (f /. y0) is V11() real ext-real Element of REAL
s is ordinal natural V11() real ext-real non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of NAT
reproj (s,y0) is non empty Relation-like REAL -defined REAL m -valued Function-like total quasi_total complex-functions-valued ext-real-functions-valued real-functions-valued Element of K6(K7(REAL,(REAL m)))
K7(REAL,(REAL m)) is set
K6(K7(REAL,(REAL m))) is set
proj (s,m) is non empty Relation-like REAL m -defined REAL -valued Function-like total quasi_total complex-valued ext-real-valued real-valued Element of K6(K7((REAL m),REAL))
(proj (s,m)) . y0 is V11() real ext-real Element of REAL
y1 is V11() real ext-real Element of REAL
PP is V11() real ext-real Element of REAL
(reproj (s,y0)) . PP is Relation-like NAT -defined Function-like finite m -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL m
DD is V11() real ext-real Element of REAL
PP - DD is V11() real ext-real Element of REAL
(proj (s,m)) . r is V11() real ext-real Element of REAL
(reproj (s,y0)) . DD is Relation-like NAT -defined Function-like finite m -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL m
y0 . s is V11() real ext-real Element of REAL
(reproj (s,y0)) . (y0 . s) is Relation-like NAT -defined Function-like finite m -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL m
Replace (y0,s,(y0 . s)) is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
[.DD,PP.] is complex-membered ext-real-membered real-membered closed V263() Element of K6(REAL)
{ b₁ where b₁ is V11() real ext-real Element of REAL : ( DD <= b₁ & b₁ <= PP ) } is set
a is V11() real ext-real Element of REAL
(reproj (s,y0)) . a is Relation-like NAT -defined Function-like finite m -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL m
((reproj (s,y0)) . a) - i is Relation-like NAT -defined REAL -valued Function-like finite m -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL m
((reproj (s,y0)) . a) + (- i) is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
|.(((reproj (s,y0)) . a) - i).| is V11() real ext-real non negative Element of REAL
sqr (((reproj (s,y0)) . a) - i) is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
mlt ((((reproj (s,y0)) . a) - i),(((reproj (s,y0)) . a) - i)) is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
Sum (sqr (((reproj (s,y0)) . a) - i)) is V11() real ext-real Element of REAL
sqrt (Sum (sqr (((reproj (s,y0)) . a) - i))) is V11() real ext-real Element of REAL
a is V11() real ext-real Element of REAL
(reproj (s,y0)) . a is Relation-like NAT -defined Function-like finite m -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL m
a is V11() real ext-real Element of REAL
(reproj (s,y0)) . a is Relation-like NAT -defined Function-like finite m -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL m
f /. ((reproj (s,y0)) . PP) is V11() real ext-real Element of REAL
f /. ((reproj (s,y0)) . DD) is V11() real ext-real Element of REAL
(f /. ((reproj (s,y0)) . PP)) - (f /. ((reproj (s,y0)) . DD)) is V11() real ext-real Element of REAL
a is V11() real ext-real Element of REAL
t is Relation-like NAT -defined REAL -valued Function-like finite m -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL m
(reproj (s,y0)) . a is Relation-like NAT -defined Function-like finite m -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL m
partdiff (f,t,s) is V11() real ext-real Element of REAL
reproj (s,t) is non empty Relation-like REAL -defined REAL m -valued Function-like total quasi_total complex-functions-valued ext-real-functions-valued real-functions-valued Element of K6(K7(REAL,(REAL m)))
f * (reproj (s,t)) is Relation-like REAL -defined REAL -valued Function-like complex-valued ext-real-valued real-valued Element of K6(K7(REAL,REAL))
(proj (s,m)) . t is V11() real ext-real Element of REAL
diff ((f * (reproj (s,t))),((proj (s,m)) . t)) is V11() real ext-real Element of REAL
(PP - DD) * (partdiff (f,t,s)) is V11() real ext-real Element of REAL
t - i is Relation-like NAT -defined REAL -valued Function-like finite m -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL m
t + (- i) is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
|.(t - i).| is V11() real ext-real non negative Element of REAL
sqr (t - i) is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
mlt ((t - i),(t - i)) is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
Sum (sqr (t - i)) is V11() real ext-real Element of REAL
sqrt (Sum (sqr (t - i))) is V11() real ext-real Element of REAL
dom y0 is complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of K6(NAT)
Seg m is non empty finite m -element complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end right_end bounded_below bounded_above real-bounded Element of K6(NAT)
{ b₁ where b₁ is ordinal natural V11() real ext-real non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of NAT : ( 1 <= b₁ & b₁ <= m ) } is set
len y0 is ordinal natural V11() real ext-real non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of NAT
Replace (y0,s,PP) is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
len (Replace (y0,s,PP)) is ordinal natural V11() real ext-real non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of NAT
dom (Replace (y0,s,PP)) is complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of K6(NAT)
r . s is V11() real ext-real Element of REAL
(Replace (y0,s,PP)) /. s is V11() real ext-real Element of REAL
reproj (s,r) is non empty Relation-like REAL -defined REAL m -valued Function-like total quasi_total complex-functions-valued ext-real-functions-valued real-functions-valued Element of K6(K7(REAL,(REAL m)))
(reproj (s,r)) . DD is Relation-like NAT -defined Function-like finite m -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL m
[.PP,DD.] is complex-membered ext-real-membered real-membered closed V263() Element of K6(REAL)
{ b₁ where b₁ is V11() real ext-real Element of REAL : ( PP <= b₁ & b₁ <= DD ) } is set
a is V11() real ext-real Element of REAL
(reproj (s,r)) . a is Relation-like NAT -defined Function-like finite m -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL m
((reproj (s,r)) . a) - i is Relation-like NAT -defined REAL -valued Function-like finite m -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL m
((reproj (s,r)) . a) + (- i) is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
|.(((reproj (s,r)) . a) - i).| is V11() real ext-real non negative Element of REAL
sqr (((reproj (s,r)) . a) - i) is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
mlt ((((reproj (s,r)) . a) - i),(((reproj (s,r)) . a) - i)) is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
Sum (sqr (((reproj (s,r)) . a) - i)) is V11() real ext-real Element of REAL
sqrt (Sum (sqr (((reproj (s,r)) . a) - i))) is V11() real ext-real Element of REAL
a is V11() real ext-real Element of REAL
(reproj (s,r)) . a is Relation-like NAT -defined Function-like finite m -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL m
a is V11() real ext-real Element of REAL
(reproj (s,r)) . a is Relation-like NAT -defined Function-like finite m -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL m
f /. ((reproj (s,r)) . DD) is V11() real ext-real Element of REAL
(reproj (s,r)) . PP is Relation-like NAT -defined Function-like finite m -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL m
f /. ((reproj (s,r)) . PP) is V11() real ext-real Element of REAL
(f /. ((reproj (s,r)) . DD)) - (f /. ((reproj (s,r)) . PP)) is V11() real ext-real Element of REAL
DD - PP is V11() real ext-real Element of REAL
a is V11() real ext-real Element of REAL
t is Relation-like NAT -defined REAL -valued Function-like finite m -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL m
(reproj (s,r)) . a is Relation-like NAT -defined Function-like finite m -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL m
partdiff (f,t,s) is V11() real ext-real Element of REAL
reproj (s,t) is non empty Relation-like REAL -defined REAL m -valued Function-like total quasi_total complex-functions-valued ext-real-functions-valued real-functions-valued Element of K6(K7(REAL,(REAL m)))
f * (reproj (s,t)) is Relation-like REAL -defined REAL -valued Function-like complex-valued ext-real-valued real-valued Element of K6(K7(REAL,REAL))
(proj (s,m)) . t is V11() real ext-real Element of REAL
diff ((f * (reproj (s,t))),((proj (s,m)) . t)) is V11() real ext-real Element of REAL
(DD - PP) * (partdiff (f,t,s)) is V11() real ext-real Element of REAL
t - i is Relation-like NAT -defined REAL -valued Function-like finite m -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL m
t + (- i) is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
|.(t - i).| is V11() real ext-real non negative Element of REAL
sqr (t - i) is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
mlt ((t - i),(t - i)) is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
Sum (sqr (t - i)) is V11() real ext-real Element of REAL
sqrt (Sum (sqr (t - i))) is V11() real ext-real Element of REAL
Replace (r,s,(r . s)) is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
(f /. y0) - (f /. r) is V11() real ext-real Element of REAL
(PP - DD) * (partdiff (f,t,s)) is V11() real ext-real Element of REAL
m is non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below 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
{ b₁ where b₁ is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like Element of REAL * : len b₁ = m } is set
m + 1 is non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below Element of NAT
f is Relation-like NAT -defined REAL m -valued Function-like finite FinSequence-like FinSubsequence-like complex-functions-valued ext-real-functions-valued real-functions-valued FinSequence of REAL m
len f is ordinal natural V11() real ext-real non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of NAT
dom f is complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of K6(NAT)
X is Relation-like NAT -defined REAL -valued Function-like finite m -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL m
i is Relation-like NAT -defined REAL -valued Function-like finite m -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL m
i + X is Relation-like NAT -defined REAL -valued Function-like finite m -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL m
y0 is ordinal natural V11() real ext-real non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of NAT
f /. y0 is Relation-like NAT -defined Function-like finite m -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL m
i + (f /. y0) is Relation-like NAT -defined REAL -valued Function-like finite m -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL m
(m + 1) -' y0 is ordinal natural V11() real ext-real non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of NAT
y0 + 1 is non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below Element of NAT
f /. (y0 + 1) is Relation-like NAT -defined Function-like finite m -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL m
i + (f /. (y0 + 1)) is Relation-like NAT -defined REAL -valued Function-like finite m -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL m
reproj (((m + 1) -' y0),(i + (f /. (y0 + 1)))) is non empty Relation-like REAL -defined REAL m -valued Function-like total quasi_total complex-functions-valued ext-real-functions-valued real-functions-valued Element of K6(K7(REAL,(REAL m)))
K7(REAL,(REAL m)) is set
K6(K7(REAL,(REAL m))) is set
proj (((m + 1) -' y0),m) is non empty Relation-like REAL m -defined REAL -valued Function-like total quasi_total complex-valued ext-real-valued real-valued Element of K6(K7((REAL m),REAL))
K7((REAL m),REAL) is complex-valued ext-real-valued real-valued set
K6(K7((REAL m),REAL)) is set
(proj (((m + 1) -' y0),m)) . (i + X) is V11() real ext-real Element of REAL
(reproj (((m + 1) -' y0),(i + (f /. (y0 + 1))))) . ((proj (((m + 1) -' y0),m)) . (i + X)) is Relation-like NAT -defined Function-like finite m -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL m
m - y0 is V11() real ext-real V163() V198() Element of INT
Seg m is non empty finite m -element complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end right_end bounded_below bounded_above real-bounded Element of K6(NAT)
{ b₁ where b₁ is ordinal natural V11() real ext-real non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of NAT : ( 1 <= b₁ & b₁ <= m ) } is set
Seg (m + 1) is non empty finite m + 1 -element complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end right_end bounded_below bounded_above real-bounded Element of K6(NAT)
{ b₁ where b₁ is ordinal natural V11() real ext-real non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of NAT : ( 1 <= b₁ & b₁ <= m + 1 ) } is set
X | ((m + 1) -' y0) is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
Seg ((m + 1) -' y0) is finite (m + 1) -' y0 -element complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below bounded_above real-bounded Element of K6(NAT)
{ b₁ where b₁ is ordinal natural V11() real ext-real non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of NAT : ( 1 <= b₁ & b₁ <= (m + 1) -' y0 ) } is set
X | (Seg ((m + 1) -' y0)) is Relation-like NAT -defined Function-like FinSubsequence-like complex-valued ext-real-valued real-valued set
y0 -' 1 is ordinal natural V11() real ext-real non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of NAT
0* (y0 -' 1) is Relation-like NAT -defined REAL -valued Function-like finite y0 -' 1 -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL (y0 -' 1)
REAL (y0 -' 1) is non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered FinSequenceSet of REAL
(y0 -' 1) -tuples_on REAL is non empty functional FinSequence-membered FinSequenceSet of REAL
{ b₁ where b₁ is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like Element of REAL * : len b₁ = y0 -' 1 } is set
(y0 -' 1) |-> 0 is Relation-like empty-yielding NAT -defined REAL -valued Function-like finite y0 -' 1 -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of (y0 -' 1) -tuples_on REAL
Seg (y0 -' 1) is finite y0 -' 1 -element complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below bounded_above real-bounded Element of K6(NAT)
{ b₁ where b₁ is ordinal natural V11() real ext-real non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of NAT : ( 1 <= b₁ & b₁ <= y0 -' 1 ) } is set
K213((Seg (y0 -' 1)),0) is Relation-like Seg (y0 -' 1) -defined INT -valued RAT -valued {0} -valued Function-like total quasi_total finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued natural-valued Element of K6(K7((Seg (y0 -' 1)),{0}))
{0} is non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below set
K7((Seg (y0 -' 1)),{0}) is INT -valued RAT -valued complex-valued ext-real-valued real-valued natural-valued set
K6(K7((Seg (y0 -' 1)),{0})) is set
(X | ((m + 1) -' y0)) ^ (0* (y0 -' 1)) is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
(m + 1) -' (y0 + 1) is ordinal natural V11() real ext-real non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of NAT
X | ((m + 1) -' (y0 + 1)) is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
Seg ((m + 1) -' (y0 + 1)) is finite (m + 1) -' (y0 + 1) -element complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below bounded_above real-bounded Element of K6(NAT)
{ b₁ where b₁ is ordinal natural V11() real ext-real non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of NAT : ( 1 <= b₁ & b₁ <= (m + 1) -' (y0 + 1) ) } is set
X | (Seg ((m + 1) -' (y0 + 1))) is Relation-like NAT -defined Function-like FinSubsequence-like complex-valued ext-real-valued real-valued set
(y0 + 1) -' 1 is ordinal natural V11() real ext-real non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of NAT
0* ((y0 + 1) -' 1) is Relation-like NAT -defined REAL -valued Function-like finite (y0 + 1) -' 1 -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL ((y0 + 1) -' 1)
REAL ((y0 + 1) -' 1) is non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered FinSequenceSet of REAL
((y0 + 1) -' 1) -tuples_on REAL is non empty functional FinSequence-membered FinSequenceSet of REAL
{ b₁ where b₁ is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like Element of REAL * : len b₁ = (y0 + 1) -' 1 } is set
((y0 + 1) -' 1) |-> 0 is Relation-like empty-yielding NAT -defined REAL -valued Function-like finite (y0 + 1) -' 1 -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of ((y0 + 1) -' 1) -tuples_on REAL
Seg ((y0 + 1) -' 1) is finite (y0 + 1) -' 1 -element complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below bounded_above real-bounded Element of K6(NAT)
{ b₁ where b₁ is ordinal natural V11() real ext-real non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of NAT : ( 1 <= b₁ & b₁ <= (y0 + 1) -' 1 ) } is set
K213((Seg ((y0 + 1) -' 1)),0) is Relation-like Seg ((y0 + 1) -' 1) -defined INT -valued RAT -valued {0} -valued Function-like total quasi_total finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued natural-valued Element of K6(K7((Seg ((y0 + 1) -' 1)),{0}))
K7((Seg ((y0 + 1) -' 1)),{0}) is INT -valued RAT -valued complex-valued ext-real-valued real-valued natural-valued set
K6(K7((Seg ((y0 + 1) -' 1)),{0})) is set
(X | ((m + 1) -' (y0 + 1))) ^ (0* ((y0 + 1) -' 1)) is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
m -' y0 is ordinal natural V11() real ext-real non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of NAT
(m -' y0) + 1 is non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below Element of NAT
1 - y0 is V11() real ext-real V163() V198() Element of INT
(m + 1) - y0 is V11() real ext-real V163() V198() Element of INT
m + (1 - y0) is V11() real ext-real V163() V198() Element of INT
dom (i + X) is complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of K6(NAT)
dom X is complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of K6(NAT)
dom i is complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of K6(NAT)
len X is ordinal natural V11() real ext-real non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of NAT
len (X | ((m + 1) -' y0)) is ordinal natural V11() real ext-real non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of NAT
(m + 1) - (y0 + 1) is V11() real ext-real V163() V198() Element of INT
len (X | ((m + 1) -' (y0 + 1))) is ordinal natural V11() real ext-real non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of NAT
(i + X) . ((m + 1) -' y0) is V11() real ext-real Element of REAL
i . ((m + 1) -' y0) is V11() real ext-real Element of REAL
X . ((m + 1) -' y0) is V11() real ext-real Element of REAL
(i . ((m + 1) -' y0)) + (X . ((m + 1) -' y0)) is V11() real ext-real Element of REAL
X /. ((m + 1) -' y0) is V11() real ext-real Element of REAL
(i . ((m + 1) -' y0)) + (X /. ((m + 1) -' y0)) is V11() real ext-real Element of REAL
i /. ((m + 1) -' y0) is V11() real ext-real Element of REAL
(i /. ((m + 1) -' y0)) + (X /. ((m + 1) -' y0)) is V11() real ext-real Element of REAL
Replace ((i + (f /. (y0 + 1))),((m + 1) -' y0),((i /. ((m + 1) -' y0)) + (X /. ((m + 1) -' y0)))) is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
(reproj (((m + 1) -' y0),(i + (f /. (y0 + 1))))) /. ((proj (((m + 1) -' y0),m)) . (i + X)) is Relation-like NAT -defined Function-like finite m -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL m
y1 is Relation-like NAT -defined REAL -valued Function-like finite m -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL m
i + y1 is Relation-like NAT -defined REAL -valued Function-like finite m -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL m
len (i + y1) is ordinal natural V11() real ext-real non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of NAT
s is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
len s is ordinal natural V11() real ext-real non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of NAT
DD is ordinal natural V11() real ext-real non negative set
len (i + (f /. y0)) is ordinal natural V11() real ext-real non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of NAT
Seg (len (i + (f /. y0))) is finite len (i + (f /. y0)) -element complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below bounded_above real-bounded Element of K6(NAT)
{ b₁ where b₁ is ordinal natural V11() real ext-real non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of NAT : ( 1 <= b₁ & b₁ <= len (i + (f /. y0)) ) } is set
len (i + (f /. (y0 + 1))) is ordinal natural V11() real ext-real non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of NAT
Seg (len (i + (f /. (y0 + 1)))) is finite len (i + (f /. (y0 + 1))) -element complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below bounded_above real-bounded Element of K6(NAT)
{ b₁ where b₁ is ordinal natural V11() real ext-real non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of NAT : ( 1 <= b₁ & b₁ <= len (i + (f /. (y0 + 1))) ) } is set
len i is ordinal natural V11() real ext-real non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of NAT
Seg (len i) is finite len i -element complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below bounded_above real-bounded Element of K6(NAT)
{ b₁ where b₁ is ordinal natural V11() real ext-real non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of NAT : ( 1 <= b₁ & b₁ <= len i ) } is set
Seg (len X) is finite len X -element complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below bounded_above real-bounded Element of K6(NAT)
{ b₁ where b₁ is ordinal natural V11() real ext-real non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of NAT : ( 1 <= b₁ & b₁ <= len X ) } is set
dom (i + (f /. y0)) is complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of K6(NAT)
dom (i + (f /. (y0 + 1))) is complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of K6(NAT)
dom s is complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of K6(NAT)
(i + (f /. y0)) . DD is V11() real ext-real Element of REAL
i . DD is V11() real ext-real Element of REAL
y1 . DD is V11() real ext-real Element of REAL
(i . DD) + (y1 . DD) is V11() real ext-real Element of REAL
(i + (f /. (y0 + 1))) . DD is V11() real ext-real Element of REAL
PP is Relation-like NAT -defined REAL -valued Function-like finite m -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL m
PP . DD is V11() real ext-real Element of REAL
(i . DD) + (PP . DD) is V11() real ext-real Element of REAL
X | (Seg ((m + 1) -' y0)) is Relation-like NAT -defined REAL -valued Function-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of K6(K7(NAT,REAL))
(X | (Seg ((m + 1) -' y0))) . DD is V11() real ext-real Element of REAL
X . DD is V11() real ext-real Element of REAL
r is ordinal natural V11() real ext-real non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of NAT
r + 1 is non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below Element of NAT
Seg r is finite r -element complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below bounded_above real-bounded Element of K6(NAT)
{ b₁ where b₁ is ordinal natural V11() real ext-real non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of NAT : ( 1 <= b₁ & b₁ <= r ) } is set
X | (Seg r) is Relation-like NAT -defined REAL -valued Function-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of K6(K7(NAT,REAL))
(X | (Seg r)) . DD is V11() real ext-real Element of REAL
s /. DD is V11() real ext-real Element of REAL
(i + (f /. (y0 + 1))) /. DD is V11() real ext-real Element of REAL
s . DD is V11() real ext-real Element of REAL
X | (Seg ((m + 1) -' y0)) is Relation-like NAT -defined REAL -valued Function-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of K6(K7(NAT,REAL))
(X | (Seg ((m + 1) -' y0))) . DD is V11() real ext-real Element of REAL
X . DD is V11() real ext-real Element of REAL
s /. DD is V11() real ext-real Element of REAL
i /. DD is V11() real ext-real Element of REAL
X /. DD is V11() real ext-real Element of REAL
(i /. DD) + (X /. DD) is V11() real ext-real Element of REAL
s . DD is V11() real ext-real Element of REAL
(i /. DD) + (X . DD) is V11() real ext-real Element of REAL
DD - ((m + 1) -' y0) is V11() real ext-real Element of REAL
len y1 is ordinal natural V11() real ext-real non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of NAT
a is ordinal natural V11() real ext-real non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of NAT
(0* (y0 -' 1)) . a is V11() real ext-real Element of REAL
(m - y0) + 1 is V11() real ext-real V163() V198() Element of INT
(m - y0) + 0 is V11() real ext-real Element of REAL
DD - (m -' y0) is V11() real ext-real Element of REAL
len PP is ordinal natural V11() real ext-real non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of NAT
(0* ((y0 + 1) -' 1)) . (DD - (m -' y0)) is V11() real ext-real Element of REAL
s /. DD is V11() real ext-real Element of REAL
(i + (f /. (y0 + 1))) /. DD is V11() real ext-real Element of REAL
s . DD is V11() real ext-real Element of REAL
m is non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below 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
{ b₁ where b₁ is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like Element of REAL * : len b₁ = m } is set
K7((REAL m),(REAL 1)) is set
K6(K7((REAL m),(REAL 1))) is set
K6((REAL m)) is set
Seg m is non empty finite m -element complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end right_end bounded_below bounded_above real-bounded Element of K6(NAT)
{ b₁ where b₁ is ordinal natural V11() real ext-real non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of NAT : ( 1 <= b₁ & b₁ <= m ) } is set
f is Relation-like REAL m -defined REAL 1 -valued Function-like complex-functions-valued ext-real-functions-valued real-functions-valued Element of K6(K7((REAL m),(REAL 1)))
X is functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered Element of K6((REAL m))
i is Relation-like NAT -defined REAL -valued Function-like finite m -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL m
diff (f,i) is non empty Relation-like REAL m -defined REAL 1 -valued Function-like total quasi_total complex-functions-valued ext-real-functions-valued real-functions-valued Element of K6(K7((REAL m),(REAL 1)))
y0 is non empty Relation-like REAL m -defined REAL 1 -valued Function-like total quasi_total V255(1,m) V256(1,m) Lipschitzian complex-functions-valued ext-real-functions-valued real-functions-valued Element of K6(K7((REAL m),(REAL 1)))
r is V11() real ext-real Element of REAL
{ b₁ where b₁ is Relation-like NAT -defined REAL -valued Function-like finite m -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL m : not r <= |.(b₁ - i).| } is set
dom f is functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered Element of K6((REAL m))
f /. i is Relation-like NAT -defined Function-like finite 1 -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL 1
y1 is non empty Relation-like REAL m -defined REAL 1 -valued Function-like total quasi_total complex-functions-valued ext-real-functions-valued real-functions-valued Element of K6(K7((REAL m),(REAL 1)))
K7((REAL m),REAL) is complex-valued ext-real-valued real-valued set
K6(K7((REAL m),REAL)) is set
PP is Relation-like REAL m -defined REAL -valued Function-like complex-valued ext-real-valued real-valued Element of K6(K7((REAL m),REAL))
<>* PP is Relation-like REAL m -defined REAL 1 -valued Function-like complex-functions-valued ext-real-functions-valued real-functions-valued Element of K6(K7((REAL m),(REAL 1)))
PP (#) ((proj (1,1)) ") is Relation-like Function-like set
DD is V11() real ext-real Element of REAL
DD / 2 is V11() real ext-real Element of REAL
(DD / 2) / m is V11() real ext-real Element of REAL
tm is ordinal natural V11() real ext-real non negative set
Sw0 is ordinal natural V11() real ext-real non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of NAT
(m,1,Sw0,f,X) is Relation-like REAL m -defined REAL 1 -valued Function-like complex-functions-valued ext-real-functions-valued real-functions-valued Element of K6(K7((REAL m),(REAL 1)))
(m,1,Sw0,f,X) /. i is Relation-like NAT -defined Function-like finite 1 -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL 1
w1 is V11() real ext-real Element of REAL
partdiff (f,i,Sw0) is Relation-like NAT -defined REAL -valued Function-like finite 1 -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL 1
Sw1 is Relation-like NAT -defined REAL -valued Function-like finite m -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL m
Sw1 - i is Relation-like NAT -defined REAL -valued Function-like finite m -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL m
- i is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
K38(1) * i is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
Sw1 + (- i) is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
|.(Sw1 - i).| is V11() real ext-real non negative Element of REAL
sqr (Sw1 - i) is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
mlt ((Sw1 - i),(Sw1 - i)) is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
Sum (sqr (Sw1 - i)) is V11() real ext-real Element of REAL
sqrt (Sum (sqr (Sw1 - i))) is V11() real ext-real Element of REAL
partdiff (f,Sw1,Sw0) is Relation-like NAT -defined REAL -valued Function-like finite 1 -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL 1
(partdiff (f,Sw1,Sw0)) - (partdiff (f,i,Sw0)) is Relation-like NAT -defined REAL -valued Function-like finite 1 -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL 1
- (partdiff (f,i,Sw0)) is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
K38(1) * (partdiff (f,i,Sw0)) is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
(partdiff (f,Sw1,Sw0)) + (- (partdiff (f,i,Sw0))) is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
|.((partdiff (f,Sw1,Sw0)) - (partdiff (f,i,Sw0))).| is V11() real ext-real non negative Element of REAL
sqr ((partdiff (f,Sw1,Sw0)) - (partdiff (f,i,Sw0))) is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
mlt (((partdiff (f,Sw1,Sw0)) - (partdiff (f,i,Sw0))),((partdiff (f,Sw1,Sw0)) - (partdiff (f,i,Sw0)))) is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
Sum (sqr ((partdiff (f,Sw1,Sw0)) - (partdiff (f,i,Sw0)))) is V11() real ext-real Element of REAL
sqrt (Sum (sqr ((partdiff (f,Sw1,Sw0)) - (partdiff (f,i,Sw0))))) is V11() real ext-real Element of REAL
(m,1,Sw0,f,X) /. Sw1 is Relation-like NAT -defined Function-like finite 1 -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL 1
((m,1,Sw0,f,X) /. Sw1) - ((m,1,Sw0,f,X) /. i) is Relation-like NAT -defined REAL -valued Function-like finite 1 -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL 1
- ((m,1,Sw0,f,X) /. i) is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
K38(1) * ((m,1,Sw0,f,X) /. i) is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
((m,1,Sw0,f,X) /. Sw1) + (- ((m,1,Sw0,f,X) /. i)) is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
|.(((m,1,Sw0,f,X) /. Sw1) - ((m,1,Sw0,f,X) /. i)).| is V11() real ext-real non negative Element of REAL
sqr (((m,1,Sw0,f,X) /. Sw1) - ((m,1,Sw0,f,X) /. i)) is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
mlt ((((m,1,Sw0,f,X) /. Sw1) - ((m,1,Sw0,f,X) /. i)),(((m,1,Sw0,f,X) /. Sw1) - ((m,1,Sw0,f,X) /. i))) is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
Sum (sqr (((m,1,Sw0,f,X) /. Sw1) - ((m,1,Sw0,f,X) /. i))) is V11() real ext-real Element of REAL
sqrt (Sum (sqr (((m,1,Sw0,f,X) /. Sw1) - ((m,1,Sw0,f,X) /. i)))) is V11() real ext-real Element of REAL
tm is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
dom tm is complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of K6(NAT)
rng tm is complex-membered ext-real-membered real-membered Element of K6(REAL)
Sw0 is non empty ext-real-membered set
w1 is non empty ext-real-membered left_end right_end set
inf w1 is ext-real set
Sw1 is V11() real ext-real Element of REAL
min (r,Sw1) is V11() real ext-real Element of REAL
ys is set
tm . ys is V11() real ext-real Element of REAL
rm is ordinal natural V11() real ext-real non negative set
tm . rm is V11() real ext-real Element of REAL
rs is Relation-like NAT -defined REAL -valued Function-like finite m -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL m
rs - i is Relation-like NAT -defined REAL -valued Function-like finite m -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL m
- i is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
K38(1) * i is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
rs + (- i) is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
|.(rs - i).| is V11() real ext-real non negative Element of REAL
sqr (rs - i) is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
mlt ((rs - i),(rs - i)) is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
Sum (sqr (rs - i)) is V11() real ext-real Element of REAL
sqrt (Sum (sqr (rs - i))) is V11() real ext-real Element of REAL
rs is Relation-like NAT -defined REAL -valued Function-like finite m -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL m
rs - i is Relation-like NAT -defined REAL -valued Function-like finite m -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL m
rs + (- i) is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
|.(rs - i).| is V11() real ext-real non negative Element of REAL
sqr (rs - i) is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
mlt ((rs - i),(rs - i)) is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
Sum (sqr (rs - i)) is V11() real ext-real Element of REAL
sqrt (Sum (sqr (rs - i))) is V11() real ext-real Element of REAL
i is ordinal natural V11() real ext-real non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of NAT
v is ordinal natural V11() real ext-real non negative set
tm . v is V11() real ext-real Element of REAL
p1 is ordinal natural V11() real ext-real non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of NAT
partdiff (f,i,p1) is Relation-like NAT -defined REAL -valued Function-like finite 1 -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL 1
partdiff (f,rs,i) is Relation-like NAT -defined REAL -valued Function-like finite 1 -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL 1
partdiff (f,i,i) is Relation-like NAT -defined REAL -valued Function-like finite 1 -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL 1
(partdiff (f,rs,i)) - (partdiff (f,i,i)) is Relation-like NAT -defined REAL -valued Function-like finite 1 -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL 1
- (partdiff (f,i,i)) is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
K38(1) * (partdiff (f,i,i)) is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
(partdiff (f,rs,i)) + (- (partdiff (f,i,i))) is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
|.((partdiff (f,rs,i)) - (partdiff (f,i,i))).| is V11() real ext-real non negative Element of REAL
sqr ((partdiff (f,rs,i)) - (partdiff (f,i,i))) is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
mlt (((partdiff (f,rs,i)) - (partdiff (f,i,i))),((partdiff (f,rs,i)) - (partdiff (f,i,i)))) is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
Sum (sqr ((partdiff (f,rs,i)) - (partdiff (f,i,i)))) is V11() real ext-real Element of REAL
sqrt (Sum (sqr ((partdiff (f,rs,i)) - (partdiff (f,i,i))))) is V11() real ext-real Element of REAL
rs is V11() real ext-real Element of REAL
i is ordinal natural V11() real ext-real non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of NAT
partdiff (f,i,i) is Relation-like NAT -defined REAL -valued Function-like finite 1 -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL 1
0* m is Relation-like NAT -defined REAL -valued Function-like finite m -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL m
m |-> 0 is Relation-like empty-yielding NAT -defined REAL -valued Function-like finite m -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of m -tuples_on REAL
K213((Seg m),0) is non empty Relation-like Seg m -defined INT -valued RAT -valued {0} -valued Function-like total quasi_total finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued natural-valued Element of K6(K7((Seg m),{0}))
{0} is non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below set
K7((Seg m),{0}) is INT -valued RAT -valued complex-valued ext-real-valued real-valued natural-valued set
K6(K7((Seg m),{0})) is set
i is Relation-like NAT -defined REAL -valued Function-like finite m -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL m
|.i.| is V11() real ext-real non negative Element of REAL
sqr i is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
mlt (i,i) is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
Sum (sqr i) is V11() real ext-real Element of REAL
sqrt (Sum (sqr i)) is V11() real ext-real Element of REAL
y1 . i is Relation-like NAT -defined REAL -valued Function-like finite 1 -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL 1
|.i.| " is V11() real ext-real non negative Element of REAL
v is Relation-like NAT -defined REAL -valued Function-like finite 1 -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL 1
|.v.| is V11() real ext-real non negative Element of REAL
sqr v is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
mlt (v,v) is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
Sum (sqr v) is V11() real ext-real Element of REAL
sqrt (Sum (sqr v)) is V11() real ext-real Element of REAL
(|.i.| ") * |.v.| is V11() real ext-real non negative Element of REAL
m + 1 is non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below Element of NAT
i + i is Relation-like NAT -defined REAL -valued Function-like finite m -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL m
f /. (i + i) is Relation-like NAT -defined Function-like finite 1 -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL 1
(f /. (i + i)) - (f /. i) is Relation-like NAT -defined REAL -valued Function-like finite 1 -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL 1
- (f /. i) is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
K38(1) * (f /. i) is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
(f /. (i + i)) + (- (f /. i)) is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
p1 is Relation-like NAT -defined REAL m -valued Function-like finite FinSequence-like FinSubsequence-like complex-functions-valued ext-real-functions-valued real-functions-valued FinSequence of REAL m
len p1 is ordinal natural V11() real ext-real non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of NAT
p0 is Relation-like NAT -defined REAL 1 -valued Function-like finite FinSequence-like FinSubsequence-like complex-functions-valued ext-real-functions-valued real-functions-valued FinSequence of REAL 1
len p0 is ordinal natural V11() real ext-real non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of NAT
dom p1 is complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of K6(NAT)
dom p0 is complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of K6(NAT)
Sum p0 is Relation-like NAT -defined REAL -valued Function-like finite 1 -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL 1
y1 /. i is Relation-like NAT -defined Function-like finite 1 -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL 1
y1 . i is Relation-like NAT -defined Function-like finite 1 -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL 1
y0 . i is Relation-like NAT -defined REAL -valued Function-like finite 1 -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL 1
(Sum p0) - (y0 . i) is Relation-like NAT -defined REAL -valued Function-like finite 1 -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL 1
- (y0 . i) is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
K38(1) * (y0 . i) is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
(Sum p0) + (- (y0 . i)) is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
P1 is Relation-like NAT -defined REAL 1 -valued Function-like finite FinSequence-like FinSubsequence-like complex-functions-valued ext-real-functions-valued real-functions-valued FinSequence of REAL 1
dom P1 is complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of K6(NAT)
Sum P1 is Relation-like NAT -defined REAL -valued Function-like finite 1 -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL 1
idseq m is Relation-like NAT -defined Function-like finite m -element FinSequence-like FinSubsequence-like set
id (Seg m) is non empty Relation-like Seg m -defined INT -valued RAT -valued Seg m -valued Function-like one-to-one total quasi_total complex-valued ext-real-valued real-valued natural-valued increasing non-decreasing Element of K6(K7((Seg m),(Seg m)))
K7((Seg m),(Seg m)) is INT -valued RAT -valued complex-valued ext-real-valued real-valued natural-valued set
K6(K7((Seg m),(Seg m))) is set
dom (idseq m) is complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of K6(NAT)
proj2 (idseq m) is set
Rev (idseq m) is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like set
dom (Rev (idseq m)) is complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of K6(NAT)
proj2 (Rev (idseq m)) is set
P0 is non empty Relation-like Seg m -defined Seg m -valued Function-like total quasi_total complex-valued ext-real-valued real-valued natural-valued Element of K6(K7((Seg m),(Seg m)))
K7((dom P1),(dom P1)) is INT -valued RAT -valued complex-valued ext-real-valued real-valued natural-valued set
K6(K7((dom P1),(dom P1))) is set
len (idseq m) is ordinal natural V11() real ext-real non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of NAT
len P1 is ordinal natural V11() real ext-real non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of NAT
PP is Relation-like dom P1 -defined dom P1 -valued Function-like one-to-one total quasi_total onto bijective complex-valued ext-real-valued real-valued natural-valued Element of K6(K7((dom P1),(dom P1)))
P1 (*) PP is Relation-like NAT -defined REAL 1 -valued Function-like finite FinSequence-like FinSubsequence-like complex-functions-valued ext-real-functions-valued real-functions-valued FinSequence of REAL 1
P1 * PP is Relation-like dom P1 -defined REAL 1 -valued Function-like complex-functions-valued ext-real-functions-valued real-functions-valued Element of K6(K7((dom P1),(REAL 1)))
K7((dom P1),(REAL 1)) is set
K6(K7((dom P1),(REAL 1))) is set
dom (P1 (*) PP) is complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of K6(NAT)
dom PP is complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of K6((dom P1))
K6((dom P1)) is set
Rev P1 is Relation-like NAT -defined REAL 1 -valued Function-like finite FinSequence-like FinSubsequence-like complex-functions-valued ext-real-functions-valued real-functions-valued FinSequence of REAL 1
dom (Rev P1) is complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of K6(NAT)
pt is ordinal natural V11() real ext-real non negative set
m - pt is V11() real ext-real Element of REAL
pt1 is ordinal natural V11() real ext-real non negative set
0 + 1 is non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below Element of NAT
(m - pt) + 1 is V11() real ext-real Element of REAL
1 - 1 is V11() real ext-real V163() V198() Element of INT
pt - 1 is V11() real ext-real Element of REAL
m - (pt - 1) is V11() real ext-real Element of REAL
pt1 + 1 is non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below Element of NAT
(Rev P1) . pt is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
(len (idseq m)) - pt is V11() real ext-real Element of REAL
((len (idseq m)) - pt) + 1 is V11() real ext-real Element of REAL
P1 . (((len (idseq m)) - pt) + 1) is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
(idseq m) . (((len (idseq m)) - pt) + 1) is set
P1 . ((idseq m) . (((len (idseq m)) - pt) + 1)) is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
(Rev (idseq m)) . pt is set
P1 . ((Rev (idseq m)) . pt) is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
(P1 (*) PP) . pt is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
Sum (Rev P1) is Relation-like NAT -defined REAL -valued Function-like finite 1 -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL 1
pt is Relation-like NAT -defined REAL 1 -valued Function-like finite FinSequence-like FinSubsequence-like complex-functions-valued ext-real-functions-valued real-functions-valued FinSequence of REAL 1
len pt is ordinal natural V11() real ext-real non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of NAT
dom pt is complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of K6(NAT)
pt1 is ordinal natural V11() real ext-real non negative set
pt /. pt1 is Relation-like NAT -defined Function-like finite 1 -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL 1
pt . pt1 is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
p0 /. pt1 is Relation-like NAT -defined Function-like finite 1 -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL 1
(Rev P1) /. pt1 is Relation-like NAT -defined Function-like finite 1 -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL 1
(p0 /. pt1) - ((Rev P1) /. pt1) is Relation-like NAT -defined REAL -valued Function-like finite 1 -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL 1
- ((Rev P1) /. pt1) is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
K38(1) * ((Rev P1) /. pt1) is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
(p0 /. pt1) + (- ((Rev P1) /. pt1)) is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
len (Rev P1) is ordinal natural V11() real ext-real non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of NAT
Sum pt is Relation-like NAT -defined REAL -valued Function-like finite 1 -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL 1
|.i.| * ((DD / 2) / m) is V11() real ext-real Element of REAL
pt1 is ordinal natural V11() real ext-real non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of NAT
pt . pt1 is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
p0 /. pt1 is Relation-like NAT -defined Function-like finite 1 -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL 1
p1 /. pt1 is Relation-like NAT -defined Function-like finite m -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL m
i + (p1 /. pt1) is Relation-like NAT -defined REAL -valued Function-like finite m -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL m
f /. (i + (p1 /. pt1)) is Relation-like NAT -defined Function-like finite 1 -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL 1
pt1 + 1 is non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below Element of NAT
p1 /. (pt1 + 1) is Relation-like NAT -defined Function-like finite m -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL m
i + (p1 /. (pt1 + 1)) is Relation-like NAT -defined REAL -valued Function-like finite m -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL m
f /. (i + (p1 /. (pt1 + 1))) is Relation-like NAT -defined Function-like finite 1 -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL 1
(f /. (i + (p1 /. pt1))) - (f /. (i + (p1 /. (pt1 + 1)))) is Relation-like NAT -defined REAL -valued Function-like finite 1 -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL 1
- (f /. (i + (p1 /. (pt1 + 1)))) is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
K38(1) * (f /. (i + (p1 /. (pt1 + 1)))) is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
(f /. (i + (p1 /. pt1))) + (- (f /. (i + (p1 /. (pt1 + 1))))) is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
m + pt1 is non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below Element of NAT
(m + 1) - pt1 is V11() real ext-real V163() V198() Element of INT
(m + 1) -' pt1 is ordinal natural V11() real ext-real non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of NAT
k is Relation-like NAT -defined REAL -valued Function-like finite m -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL m
P1 /. ((m + 1) -' pt1) is Relation-like NAT -defined Function-like finite 1 -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL 1
P1 . ((m + 1) -' pt1) is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
partdiff (f,i,((m + 1) -' pt1)) is Relation-like NAT -defined REAL -valued Function-like finite 1 -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL 1
proj (((m + 1) -' pt1),m) is non empty Relation-like REAL m -defined REAL -valued Function-like total quasi_total complex-valued ext-real-valued real-valued Element of K6(K7((REAL m),REAL))
(proj (((m + 1) -' pt1),m)) . i is V11() real ext-real Element of REAL
((proj (((m + 1) -' pt1),m)) . i) * (partdiff (f,i,((m + 1) -' pt1))) is Relation-like NAT -defined REAL -valued Function-like finite 1 -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL 1
k is ordinal natural V11() real ext-real non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of NAT
Seg (m + 1) is non empty finite m + 1 -element complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end right_end bounded_below bounded_above real-bounded Element of K6(NAT)
{ b₁ where b₁ is ordinal natural V11() real ext-real non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of NAT : ( 1 <= b₁ & b₁ <= m + 1 ) } is set
p1 /. k is Relation-like NAT -defined Function-like finite m -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL m
i + (p1 /. k) is Relation-like NAT -defined REAL -valued Function-like finite m -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL m
(i + (p1 /. k)) - i is Relation-like NAT -defined REAL -valued Function-like finite m -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL m
(i + (p1 /. k)) + (- i) is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
j is Relation-like NAT -defined REAL -valued Function-like finite m -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL m
|.j.| is V11() real ext-real non negative Element of REAL
sqr j is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
mlt (j,j) is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
Sum (sqr j) is V11() real ext-real Element of REAL
sqrt (Sum (sqr j)) is V11() real ext-real Element of REAL
|.((i + (p1 /. k)) - i).| is V11() real ext-real non negative Element of REAL
sqr ((i + (p1 /. k)) - i) is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
mlt (((i + (p1 /. k)) - i),((i + (p1 /. k)) - i)) is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
Sum (sqr ((i + (p1 /. k)) - i)) is V11() real ext-real Element of REAL
sqrt (Sum (sqr ((i + (p1 /. k)) - i))) is V11() real ext-real Element of REAL
rng PP is complex-membered ext-real-membered real-membered Element of K6(REAL)
k is non empty Relation-like REAL -defined REAL 1 -valued Function-like total quasi_total complex-functions-valued ext-real-functions-valued real-functions-valued Element of K6(K7(REAL,(REAL 1)))
dom PP is functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered Element of K6((REAL m))
(i + (p1 /. pt1)) - i is Relation-like NAT -defined REAL -valued Function-like finite m -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL m
(i + (p1 /. pt1)) + (- i) is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
|.((i + (p1 /. pt1)) - i).| is V11() real ext-real non negative Element of REAL
sqr ((i + (p1 /. pt1)) - i) is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
mlt (((i + (p1 /. pt1)) - i),((i + (p1 /. pt1)) - i)) is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
Sum (sqr ((i + (p1 /. pt1)) - i)) is V11() real ext-real Element of REAL
sqrt (Sum (sqr ((i + (p1 /. pt1)) - i))) is V11() real ext-real Element of REAL
(<>* PP) . (i + (p1 /. pt1)) is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
PP . (i + (p1 /. pt1)) is V11() real ext-real Element of REAL
((proj (1,1)) ") . (PP . (i + (p1 /. pt1))) is set
<*(PP . (i + (p1 /. pt1)))*> is non empty V2() Relation-like NAT -defined REAL -valued Function-like one-to-one constant finite 1 -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued increasing decreasing non-decreasing non-increasing FinSequence of REAL
[1,(PP . (i + (p1 /. pt1)))] is V27() set
{1,(PP . (i + (p1 /. pt1)))} is non empty complex-membered ext-real-membered real-membered set
{{1,(PP . (i + (p1 /. pt1)))},{1}} is non empty set
{[1,(PP . (i + (p1 /. pt1)))]} is non empty Function-like set
PP /. (i + (p1 /. pt1)) is V11() real ext-real Element of REAL
<*(PP /. (i + (p1 /. pt1)))*> is non empty V2() Relation-like NAT -defined REAL -valued Function-like one-to-one constant finite 1 -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued increasing decreasing non-decreasing non-increasing FinSequence of REAL
[1,(PP /. (i + (p1 /. pt1)))] is V27() set
{1,(PP /. (i + (p1 /. pt1)))} is non empty complex-membered ext-real-membered real-membered set
{{1,(PP /. (i + (p1 /. pt1)))},{1}} is non empty set
{[1,(PP /. (i + (p1 /. pt1)))]} is non empty Function-like set
(i + (p1 /. (pt1 + 1))) - i is Relation-like NAT -defined REAL -valued Function-like finite m -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL m
(i + (p1 /. (pt1 + 1))) + (- i) is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
|.((i + (p1 /. (pt1 + 1))) - i).| is V11() real ext-real non negative Element of REAL
sqr ((i + (p1 /. (pt1 + 1))) - i) is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
mlt (((i + (p1 /. (pt1 + 1))) - i),((i + (p1 /. (pt1 + 1))) - i)) is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
Sum (sqr ((i + (p1 /. (pt1 + 1))) - i)) is V11() real ext-real Element of REAL
sqrt (Sum (sqr ((i + (p1 /. (pt1 + 1))) - i))) is V11() real ext-real Element of REAL
(<>* PP) . (i + (p1 /. (pt1 + 1))) is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
PP . (i + (p1 /. (pt1 + 1))) is V11() real ext-real Element of REAL
((proj (1,1)) ") . (PP . (i + (p1 /. (pt1 + 1)))) is set
<*(PP . (i + (p1 /. (pt1 + 1))))*> is non empty V2() Relation-like NAT -defined REAL -valued Function-like one-to-one constant finite 1 -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued increasing decreasing non-decreasing non-increasing FinSequence of REAL
[1,(PP . (i + (p1 /. (pt1 + 1))))] is V27() set
{1,(PP . (i + (p1 /. (pt1 + 1))))} is non empty complex-membered ext-real-membered real-membered set
{{1,(PP . (i + (p1 /. (pt1 + 1))))},{1}} is non empty set
{[1,(PP . (i + (p1 /. (pt1 + 1))))]} is non empty Function-like set
PP /. (i + (p1 /. (pt1 + 1))) is V11() real ext-real Element of REAL
<*(PP /. (i + (p1 /. (pt1 + 1))))*> is non empty V2() Relation-like NAT -defined REAL -valued Function-like one-to-one constant finite 1 -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued increasing decreasing non-decreasing non-increasing FinSequence of REAL
[1,(PP /. (i + (p1 /. (pt1 + 1))))] is V27() set
{1,(PP /. (i + (p1 /. (pt1 + 1))))} is non empty complex-membered ext-real-membered real-membered set
{{1,(PP /. (i + (p1 /. (pt1 + 1))))},{1}} is non empty set
{[1,(PP /. (i + (p1 /. (pt1 + 1))))]} is non empty Function-like set
partdiff (PP,i,((m + 1) -' pt1)) is V11() real ext-real Element of REAL
reproj (((m + 1) -' pt1),i) is non empty Relation-like REAL -defined REAL m -valued Function-like total quasi_total complex-functions-valued ext-real-functions-valued real-functions-valued Element of K6(K7(REAL,(REAL m)))
K7(REAL,(REAL m)) is set
K6(K7(REAL,(REAL m))) is set
PP * (reproj (((m + 1) -' pt1),i)) is Relation-like REAL -defined REAL -valued Function-like complex-valued ext-real-valued real-valued Element of K6(K7(REAL,REAL))
(proj (((m + 1) -' pt1),m)) . i is V11() real ext-real Element of REAL
diff ((PP * (reproj (((m + 1) -' pt1),i))),((proj (((m + 1) -' pt1),m)) . i)) is V11() real ext-real Element of REAL
<*(partdiff (PP,i,((m + 1) -' pt1)))*> is non empty V2() Relation-like NAT -defined REAL -valued Function-like one-to-one constant finite 1 -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued increasing decreasing non-decreasing non-increasing FinSequence of REAL
[1,(partdiff (PP,i,((m + 1) -' pt1)))] is V27() set
{1,(partdiff (PP,i,((m + 1) -' pt1)))} is non empty complex-membered ext-real-membered real-membered set
{{1,(partdiff (PP,i,((m + 1) -' pt1)))},{1}} is non empty set
{[1,(partdiff (PP,i,((m + 1) -' pt1)))]} is non empty Function-like set
((proj (((m + 1) -' pt1),m)) . i) * (partdiff (PP,i,((m + 1) -' pt1))) is V11() real ext-real Element of REAL
<*(((proj (((m + 1) -' pt1),m)) . i) * (partdiff (PP,i,((m + 1) -' pt1))))*> is non empty V2() Relation-like NAT -defined REAL -valued Function-like one-to-one constant finite 1 -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued increasing decreasing non-decreasing non-increasing FinSequence of REAL
[1,(((proj (((m + 1) -' pt1),m)) . i) * (partdiff (PP,i,((m + 1) -' pt1))))] is V27() set
{1,(((proj (((m + 1) -' pt1),m)) . i) * (partdiff (PP,i,((m + 1) -' pt1))))} is non empty complex-membered ext-real-membered real-membered set
{{1,(((proj (((m + 1) -' pt1),m)) . i) * (partdiff (PP,i,((m + 1) -' pt1))))},{1}} is non empty set
{[1,(((proj (((m + 1) -' pt1),m)) . i) * (partdiff (PP,i,((m + 1) -' pt1))))]} is non empty Function-like set
(PP /. (i + (p1 /. pt1))) - (PP /. (i + (p1 /. (pt1 + 1)))) is V11() real ext-real Element of REAL
<*((PP /. (i + (p1 /. pt1))) - (PP /. (i + (p1 /. (pt1 + 1)))))*> is non empty V2() Relation-like NAT -defined REAL -valued Function-like one-to-one constant finite 1 -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued increasing decreasing non-decreasing non-increasing FinSequence of REAL
[1,((PP /. (i + (p1 /. pt1))) - (PP /. (i + (p1 /. (pt1 + 1)))))] is V27() set
{1,((PP /. (i + (p1 /. pt1))) - (PP /. (i + (p1 /. (pt1 + 1)))))} is non empty complex-membered ext-real-membered real-membered set
{{1,((PP /. (i + (p1 /. pt1))) - (PP /. (i + (p1 /. (pt1 + 1)))))},{1}} is non empty set
{[1,((PP /. (i + (p1 /. pt1))) - (PP /. (i + (p1 /. (pt1 + 1)))))]} is non empty Function-like set
k is Relation-like NAT -defined REAL -valued Function-like finite m -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL m
reproj (((m + 1) -' pt1),(i + (p1 /. (pt1 + 1)))) is non empty Relation-like REAL -defined REAL m -valued Function-like total quasi_total complex-functions-valued ext-real-functions-valued real-functions-valued Element of K6(K7(REAL,(REAL m)))
(proj (((m + 1) -' pt1),m)) . (i + i) is V11() real ext-real Element of REAL
(reproj (((m + 1) -' pt1),(i + (p1 /. (pt1 + 1))))) . ((proj (((m + 1) -' pt1),m)) . (i + i)) is Relation-like NAT -defined Function-like finite m -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL m
(m + 1) -' (pt1 + 1) is ordinal natural V11() real ext-real non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of NAT
(m + 1) - (pt1 + 1) is V11() real ext-real V163() V198() Element of INT
m - pt1 is V11() real ext-real V163() V198() Element of INT
m -' pt1 is ordinal natural V11() real ext-real non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of NAT
(pt1 + 1) -' 1 is ordinal natural V11() real ext-real non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of NAT
(pt1 + 1) - 1 is V11() real ext-real V163() V198() Element of INT
(proj (((m + 1) -' pt1),m)) . (i + (p1 /. (pt1 + 1))) is V11() real ext-real Element of REAL
((proj (((m + 1) -' pt1),m)) . (i + i)) - ((proj (((m + 1) -' pt1),m)) . (i + (p1 /. (pt1 + 1)))) is V11() real ext-real Element of REAL
k is Relation-like NAT -defined REAL -valued Function-like finite m -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL m
k - i is Relation-like NAT -defined REAL -valued Function-like finite m -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL m
k + (- i) is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
|.(k - i).| is V11() real ext-real non negative Element of REAL
sqr (k - i) is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
mlt ((k - i),(k - i)) is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
Sum (sqr (k - i)) is V11() real ext-real Element of REAL
sqrt (Sum (sqr (k - i))) is V11() real ext-real Element of REAL
partdiff (PP,k,((m + 1) -' pt1)) is V11() real ext-real Element of REAL
reproj (((m + 1) -' pt1),k) is non empty Relation-like REAL -defined REAL m -valued Function-like total quasi_total complex-functions-valued ext-real-functions-valued real-functions-valued Element of K6(K7(REAL,(REAL m)))
PP * (reproj (((m + 1) -' pt1),k)) is Relation-like REAL -defined REAL -valued Function-like complex-valued ext-real-valued real-valued Element of K6(K7(REAL,REAL))
(proj (((m + 1) -' pt1),m)) . k is V11() real ext-real Element of REAL
diff ((PP * (reproj (((m + 1) -' pt1),k))),((proj (((m + 1) -' pt1),m)) . k)) is V11() real ext-real Element of REAL
(((proj (((m + 1) -' pt1),m)) . (i + i)) - ((proj (((m + 1) -' pt1),m)) . (i + (p1 /. (pt1 + 1))))) * (partdiff (PP,k,((m + 1) -' pt1))) is V11() real ext-real Element of REAL
partdiff (f,k,((m + 1) -' pt1)) is Relation-like NAT -defined REAL -valued Function-like finite 1 -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL 1
(partdiff (f,k,((m + 1) -' pt1))) - (partdiff (f,i,((m + 1) -' pt1))) is Relation-like NAT -defined REAL -valued Function-like finite 1 -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL 1
- (partdiff (f,i,((m + 1) -' pt1))) is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
K38(1) * (partdiff (f,i,((m + 1) -' pt1))) is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
(partdiff (f,k,((m + 1) -' pt1))) + (- (partdiff (f,i,((m + 1) -' pt1)))) is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
|.((partdiff (f,k,((m + 1) -' pt1))) - (partdiff (f,i,((m + 1) -' pt1)))).| is V11() real ext-real non negative Element of REAL
sqr ((partdiff (f,k,((m + 1) -' pt1))) - (partdiff (f,i,((m + 1) -' pt1)))) is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
mlt (((partdiff (f,k,((m + 1) -' pt1))) - (partdiff (f,i,((m + 1) -' pt1)))),((partdiff (f,k,((m + 1) -' pt1))) - (partdiff (f,i,((m + 1) -' pt1))))) is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
Sum (sqr ((partdiff (f,k,((m + 1) -' pt1))) - (partdiff (f,i,((m + 1) -' pt1))))) is V11() real ext-real Element of REAL
sqrt (Sum (sqr ((partdiff (f,k,((m + 1) -' pt1))) - (partdiff (f,i,((m + 1) -' pt1)))))) is V11() real ext-real Element of REAL
<*(partdiff (PP,k,((m + 1) -' pt1)))*> is non empty V2() Relation-like NAT -defined REAL -valued Function-like one-to-one constant finite 1 -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued increasing decreasing non-decreasing non-increasing FinSequence of REAL
[1,(partdiff (PP,k,((m + 1) -' pt1)))] is V27() set
{1,(partdiff (PP,k,((m + 1) -' pt1)))} is non empty complex-membered ext-real-membered real-membered set
{{1,(partdiff (PP,k,((m + 1) -' pt1)))},{1}} is non empty set
{[1,(partdiff (PP,k,((m + 1) -' pt1)))]} is non empty Function-like set
len i is ordinal natural V11() real ext-real non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of NAT
len i is ordinal natural V11() real ext-real non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of NAT
hj1 is Relation-like NAT -defined REAL -valued Function-like finite m -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL m
len hj1 is ordinal natural V11() real ext-real non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of NAT
dom i is complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of K6(NAT)
dom i is complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of K6(NAT)
dom hj1 is complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of K6(NAT)
(dom i) /\ (dom i) is complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of K6(NAT)
(dom i) /\ (dom hj1) is complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of K6(NAT)
dom (i + i) is complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of K6(NAT)
i + hj1 is Relation-like NAT -defined REAL -valued Function-like finite m -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL m
dom (i + hj1) is complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of K6(NAT)
i | (m -' pt1) is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
Seg (m -' pt1) is finite m -' pt1 -element complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below bounded_above real-bounded Element of K6(NAT)
{ b₁ where b₁ is ordinal natural V11() real ext-real non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of NAT : ( 1 <= b₁ & b₁ <= m -' pt1 ) } is set
i | (Seg (m -' pt1)) is Relation-like NAT -defined Function-like FinSubsequence-like complex-valued ext-real-valued real-valued set
0* pt1 is Relation-like NAT -defined REAL -valued Function-like finite pt1 -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL pt1
REAL pt1 is non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered FinSequenceSet of REAL
pt1 -tuples_on REAL is non empty functional FinSequence-membered FinSequenceSet of REAL
{ b₁ where b₁ is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like Element of REAL * : len b₁ = pt1 } is set
pt1 |-> 0 is Relation-like empty-yielding NAT -defined REAL -valued Function-like finite pt1 -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of pt1 -tuples_on REAL
Seg pt1 is finite pt1 -element complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below bounded_above real-bounded Element of K6(NAT)
{ b₁ where b₁ is ordinal natural V11() real ext-real non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of NAT : ( 1 <= b₁ & b₁ <= pt1 ) } is set
K213((Seg pt1),0) is Relation-like Seg pt1 -defined INT -valued RAT -valued {0} -valued Function-like total quasi_total finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued natural-valued Element of K6(K7((Seg pt1),{0}))
K7((Seg pt1),{0}) is INT -valued RAT -valued complex-valued ext-real-valued real-valued natural-valued set
K6(K7((Seg pt1),{0})) is set
(i | (m -' pt1)) ^ (0* pt1) is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
len (i | (m -' pt1)) is ordinal natural V11() real ext-real non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of NAT
(m -' pt1) + 1 is non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below Element of NAT
hj1 . ((m + 1) -' pt1) is V11() real ext-real Element of REAL
((m + 1) -' pt1) - (m -' pt1) is V11() real ext-real V163() V198() Element of INT
(0* pt1) . (((m + 1) -' pt1) - (m -' pt1)) is V11() real ext-real Element of REAL
(i + i) . ((m + 1) -' pt1) is V11() real ext-real Element of REAL
((i + i) . ((m + 1) -' pt1)) - ((proj (((m + 1) -' pt1),m)) . (i + (p1 /. (pt1 + 1)))) is V11() real ext-real Element of REAL
(i + (p1 /. (pt1 + 1))) . ((m + 1) -' pt1) is V11() real ext-real Element of REAL
((i + i) . ((m + 1) -' pt1)) - ((i + (p1 /. (pt1 + 1))) . ((m + 1) -' pt1)) is V11() real ext-real Element of REAL
i . ((m + 1) -' pt1) is V11() real ext-real Element of REAL
i . ((m + 1) -' pt1) is V11() real ext-real Element of REAL
(i . ((m + 1) -' pt1)) + (i . ((m + 1) -' pt1)) is V11() real ext-real Element of REAL
((i . ((m + 1) -' pt1)) + (i . ((m + 1) -' pt1))) - ((i + (p1 /. (pt1 + 1))) . ((m + 1) -' pt1)) is V11() real ext-real Element of REAL
(i . ((m + 1) -' pt1)) + 0 is V11() real ext-real Element of REAL
((i . ((m + 1) -' pt1)) + (i . ((m + 1) -' pt1))) - ((i . ((m + 1) -' pt1)) + 0) is V11() real ext-real Element of REAL
pt /. pt1 is Relation-like NAT -defined Function-like finite 1 -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL 1
gwj is Relation-like NAT -defined REAL -valued Function-like finite 1 -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL 1
|.gwj.| is V11() real ext-real non negative Element of REAL
sqr gwj is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
mlt (gwj,gwj) is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
Sum (sqr gwj) is V11() real ext-real Element of REAL
sqrt (Sum (sqr gwj)) is V11() real ext-real Element of REAL
Seg (len (Rev P1)) is finite len (Rev P1) -element complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below bounded_above real-bounded Element of K6(NAT)
{ b₁ where b₁ is ordinal natural V11() real ext-real non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of NAT : ( 1 <= b₁ & b₁ <= len (Rev P1) ) } is set
(Rev P1) /. pt1 is Relation-like NAT -defined Function-like finite 1 -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL 1
(Rev P1) . pt1 is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
(m - pt1) + 1 is V11() real ext-real V163() V198() Element of INT
P1 . ((m - pt1) + 1) is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
(p0 /. pt1) - (P1 /. ((m + 1) -' pt1)) is Relation-like NAT -defined REAL -valued Function-like finite 1 -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL 1
- (P1 /. ((m + 1) -' pt1)) is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
K38(1) * (P1 /. ((m + 1) -' pt1)) is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
(p0 /. pt1) + (- (P1 /. ((m + 1) -' pt1))) is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
((proj (((m + 1) -' pt1),m)) . i) * (partdiff (PP,k,((m + 1) -' pt1))) is V11() real ext-real Element of REAL
(((proj (((m + 1) -' pt1),m)) . i) * (partdiff (PP,k,((m + 1) -' pt1)))) - (((proj (((m + 1) -' pt1),m)) . i) * (partdiff (PP,i,((m + 1) -' pt1)))) is V11() real ext-real Element of REAL
<*((((proj (((m + 1) -' pt1),m)) . i) * (partdiff (PP,k,((m + 1) -' pt1)))) - (((proj (((m + 1) -' pt1),m)) . i) * (partdiff (PP,i,((m + 1) -' pt1)))))*> is non empty V2() Relation-like NAT -defined REAL -valued Function-like one-to-one constant finite 1 -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued increasing decreasing non-decreasing non-increasing FinSequence of REAL
[1,((((proj (((m + 1) -' pt1),m)) . i) * (partdiff (PP,k,((m + 1) -' pt1)))) - (((proj (((m + 1) -' pt1),m)) . i) * (partdiff (PP,i,((m + 1) -' pt1)))))] is V27() set
{1,((((proj (((m + 1) -' pt1),m)) . i) * (partdiff (PP,k,((m + 1) -' pt1)))) - (((proj (((m + 1) -' pt1),m)) . i) * (partdiff (PP,i,((m + 1) -' pt1)))))} is non empty complex-membered ext-real-membered real-membered set
{{1,((((proj (((m + 1) -' pt1),m)) . i) * (partdiff (PP,k,((m + 1) -' pt1)))) - (((proj (((m + 1) -' pt1),m)) . i) * (partdiff (PP,i,((m + 1) -' pt1)))))},{1}} is non empty set
{[1,((((proj (((m + 1) -' pt1),m)) . i) * (partdiff (PP,k,((m + 1) -' pt1)))) - (((proj (((m + 1) -' pt1),m)) . i) * (partdiff (PP,i,((m + 1) -' pt1)))))]} is non empty Function-like set
(partdiff (PP,k,((m + 1) -' pt1))) - (partdiff (PP,i,((m + 1) -' pt1))) is V11() real ext-real Element of REAL
((proj (((m + 1) -' pt1),m)) . i) * ((partdiff (PP,k,((m + 1) -' pt1))) - (partdiff (PP,i,((m + 1) -' pt1)))) is V11() real ext-real Element of REAL
<*(((proj (((m + 1) -' pt1),m)) . i) * ((partdiff (PP,k,((m + 1) -' pt1))) - (partdiff (PP,i,((m + 1) -' pt1)))))*> is non empty V2() Relation-like NAT -defined REAL -valued Function-like one-to-one constant finite 1 -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued increasing decreasing non-decreasing non-increasing FinSequence of REAL
[1,(((proj (((m + 1) -' pt1),m)) . i) * ((partdiff (PP,k,((m + 1) -' pt1))) - (partdiff (PP,i,((m + 1) -' pt1)))))] is V27() set
{1,(((proj (((m + 1) -' pt1),m)) . i) * ((partdiff (PP,k,((m + 1) -' pt1))) - (partdiff (PP,i,((m + 1) -' pt1)))))} is non empty complex-membered ext-real-membered real-membered set
{{1,(((proj (((m + 1) -' pt1),m)) . i) * ((partdiff (PP,k,((m + 1) -' pt1))) - (partdiff (PP,i,((m + 1) -' pt1)))))},{1}} is non empty set
{[1,(((proj (((m + 1) -' pt1),m)) . i) * ((partdiff (PP,k,((m + 1) -' pt1))) - (partdiff (PP,i,((m + 1) -' pt1)))))]} is non empty Function-like set
<*((partdiff (PP,k,((m + 1) -' pt1))) - (partdiff (PP,i,((m + 1) -' pt1))))*> is non empty V2() Relation-like NAT -defined REAL -valued Function-like one-to-one constant finite 1 -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued increasing decreasing non-decreasing non-increasing FinSequence of REAL
[1,((partdiff (PP,k,((m + 1) -' pt1))) - (partdiff (PP,i,((m + 1) -' pt1))))] is V27() set
{1,((partdiff (PP,k,((m + 1) -' pt1))) - (partdiff (PP,i,((m + 1) -' pt1))))} is non empty complex-membered ext-real-membered real-membered set
{{1,((partdiff (PP,k,((m + 1) -' pt1))) - (partdiff (PP,i,((m + 1) -' pt1))))},{1}} is non empty set
{[1,((partdiff (PP,k,((m + 1) -' pt1))) - (partdiff (PP,i,((m + 1) -' pt1))))]} is non empty Function-like set
((proj (((m + 1) -' pt1),m)) . i) * <*((partdiff (PP,k,((m + 1) -' pt1))) - (partdiff (PP,i,((m + 1) -' pt1))))*> is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
((proj (((m + 1) -' pt1),m)) . i) * ((partdiff (f,k,((m + 1) -' pt1))) - (partdiff (f,i,((m + 1) -' pt1)))) is Relation-like NAT -defined REAL -valued Function-like finite 1 -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL 1
abs ((proj (((m + 1) -' pt1),m)) . i) is V11() real ext-real Element of REAL
(abs ((proj (((m + 1) -' pt1),m)) . i)) * |.((partdiff (f,k,((m + 1) -' pt1))) - (partdiff (f,i,((m + 1) -' pt1)))).| is V11() real ext-real Element of REAL
(abs ((proj (((m + 1) -' pt1),m)) . i)) * ((DD / 2) / m) is V11() real ext-real Element of REAL
abs (i . ((m + 1) -' pt1)) is V11() real ext-real Element of REAL
pt1 is ordinal natural V11() real ext-real non negative set
pt . pt1 is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
k is Relation-like NAT -defined REAL -valued Function-like finite 1 -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL 1
|.k.| is V11() real ext-real non negative Element of REAL
sqr k is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
mlt (k,k) is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
Sum (sqr k) is V11() real ext-real Element of REAL
sqrt (Sum (sqr k)) is V11() real ext-real Element of REAL
pt1 is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
dom pt1 is complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of K6(NAT)
len pt1 is ordinal natural V11() real ext-real non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of NAT
k is ordinal natural V11() real ext-real non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of NAT
pt . k is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
pt1 . k is V11() real ext-real Element of REAL
|.(Sum pt).| is V11() real ext-real non negative Element of REAL
sqr (Sum pt) is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
mlt ((Sum pt),(Sum pt)) is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
Sum (sqr (Sum pt)) is V11() real ext-real Element of REAL
sqrt (Sum (sqr (Sum pt))) is V11() real ext-real Element of REAL
k is Relation-like NAT -defined REAL -valued Function-like finite m -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL m
Sum k is V11() real ext-real Element of REAL
m |-> (|.i.| * ((DD / 2) / m)) is Relation-like NAT -defined REAL -valued Function-like finite m -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of m -tuples_on REAL
K213((Seg m),(|.i.| * ((DD / 2) / m))) is non empty Relation-like Seg m -defined {(|.i.| * ((DD / 2) / m))} -valued Function-like total quasi_total finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of K6(K7((Seg m),{(|.i.| * ((DD / 2) / m))}))
{(|.i.| * ((DD / 2) / m))} is non empty complex-membered ext-real-membered real-membered set
K7((Seg m),{(|.i.| * ((DD / 2) / m))}) is complex-valued ext-real-valued real-valued set
K6(K7((Seg m),{(|.i.| * ((DD / 2) / m))})) is set
j is ordinal natural V11() real ext-real non negative set
k . j is V11() real ext-real Element of REAL
(m |-> (|.i.| * ((DD / 2) / m))) . j is V11() real ext-real Element of REAL
pt . j is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
mj is Relation-like NAT -defined REAL -valued Function-like finite 1 -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL 1
|.mj.| is V11() real ext-real non negative Element of REAL
sqr mj is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
mlt (mj,mj) is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
Sum (sqr mj) is V11() real ext-real Element of REAL
sqrt (Sum (sqr mj)) is V11() real ext-real Element of REAL
hj1 is Relation-like NAT -defined REAL -valued Function-like finite 1 -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL 1
|.hj1.| is V11() real ext-real non negative Element of REAL
sqr hj1 is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
mlt (hj1,hj1) is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
Sum (sqr hj1) is V11() real ext-real Element of REAL
sqrt (Sum (sqr hj1)) is V11() real ext-real Element of REAL
Sum (m |-> (|.i.| * ((DD / 2) / m))) is V11() real ext-real Element of REAL
m * (|.i.| * ((DD / 2) / m)) is V11() real ext-real Element of REAL
|.v.| * (|.i.| ") is V11() real ext-real non negative Element of REAL
m * |.i.| is V11() real ext-real non negative Element of REAL
(m * |.i.|) * ((DD / 2) / m) is V11() real ext-real Element of REAL
((m * |.i.|) * ((DD / 2) / m)) * (|.i.| ") is V11() real ext-real Element of REAL
((DD / 2) / m) * |.i.| is V11() real ext-real Element of REAL
(((DD / 2) / m) * |.i.|) * (|.i.| ") is V11() real ext-real Element of REAL
m * ((((DD / 2) / m) * |.i.|) * (|.i.| ")) is V11() real ext-real Element of REAL
m * ((DD / 2) / m) is V11() real ext-real Element of REAL
DD is Relation-like NAT -defined REAL -valued Function-like finite m -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL m
DD - i is Relation-like NAT -defined REAL -valued Function-like finite m -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL m
DD + (- i) is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
|.(DD - i).| is V11() real ext-real non negative Element of REAL
sqr (DD - i) is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
mlt ((DD - i),(DD - i)) is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
Sum (sqr (DD - i)) is V11() real ext-real Element of REAL
sqrt (Sum (sqr (DD - i))) is V11() real ext-real Element of REAL
f /. DD is Relation-like NAT -defined Function-like finite 1 -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL 1
(f /. DD) - (f /. i) is Relation-like NAT -defined REAL -valued Function-like finite 1 -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL 1
- (f /. i) is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
K38(1) * (f /. i) is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
(f /. DD) + (- (f /. i)) is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
y0 . (DD - i) is Relation-like NAT -defined REAL -valued Function-like finite 1 -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL 1
y1 . (DD - i) is Relation-like NAT -defined REAL -valued Function-like finite 1 -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL 1
(y0 . (DD - i)) + (y1 . (DD - i)) is Relation-like NAT -defined REAL -valued Function-like finite 1 -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL 1
i + (DD - i) is Relation-like NAT -defined REAL -valued Function-like finite m -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL m
f /. (i + (DD - i)) is Relation-like NAT -defined Function-like finite 1 -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL 1
(f /. (i + (DD - i))) - (f /. i) is Relation-like NAT -defined REAL -valued Function-like finite 1 -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL 1
(f /. (i + (DD - i))) + (- (f /. i)) is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
((f /. (i + (DD - i))) - (f /. i)) - (y0 . (DD - i)) is Relation-like NAT -defined REAL -valued Function-like finite 1 -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL 1
- (y0 . (DD - i)) is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
K38(1) * (y0 . (DD - i)) is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
((f /. (i + (DD - i))) - (f /. i)) + (- (y0 . (DD - i))) is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
(y0 . (DD - i)) - (y0 . (DD - i)) is Relation-like NAT -defined REAL -valued Function-like finite 1 -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL 1
(y0 . (DD - i)) + (- (y0 . (DD - i))) is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
((f /. (i + (DD - i))) - (f /. i)) - ((y0 . (DD - i)) - (y0 . (DD - i))) is Relation-like NAT -defined REAL -valued Function-like finite 1 -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL 1
- ((y0 . (DD - i)) - (y0 . (DD - i))) is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
K38(1) * ((y0 . (DD - i)) - (y0 . (DD - i))) is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
((f /. (i + (DD - i))) - (f /. i)) + (- ((y0 . (DD - i)) - (y0 . (DD - i)))) is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
0* 1 is Relation-like NAT -defined REAL -valued Function-like finite 1 -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL 1
1 |-> 0 is Relation-like empty-yielding NAT -defined REAL -valued Function-like finite 1 -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of 1 -tuples_on REAL
Seg 1 is non empty finite 1 -element complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end right_end bounded_below bounded_above real-bounded Element of K6(NAT)
{ b₁ where b₁ is ordinal natural V11() real ext-real non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of NAT : ( 1 <= b₁ & b₁ <= 1 ) } is set
K213((Seg 1),0) is non empty Relation-like Seg 1 -defined INT -valued RAT -valued {0} -valued Function-like total quasi_total finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued natural-valued Element of K6(K7((Seg 1),{0}))
K7((Seg 1),{0}) is INT -valued RAT -valued complex-valued ext-real-valued real-valued natural-valued set
K6(K7((Seg 1),{0})) is set
((f /. (i + (DD - i))) - (f /. i)) - (0* 1) is Relation-like NAT -defined REAL -valued Function-like finite 1 -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL 1
- (0* 1) is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
K38(1) * (0* 1) is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
((f /. (i + (DD - i))) - (f /. i)) + (- (0* 1)) is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
i + DD is Relation-like NAT -defined REAL -valued Function-like finite m -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL m
(i + DD) - i is Relation-like NAT -defined REAL -valued Function-like finite m -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL m
(i + DD) + (- i) is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
f /. ((i + DD) - i) is Relation-like NAT -defined Function-like finite 1 -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL 1
(f /. ((i + DD) - i)) - (f /. i) is Relation-like NAT -defined REAL -valued Function-like finite 1 -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL 1
(f /. ((i + DD) - i)) + (- (f /. i)) is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
i - i is Relation-like NAT -defined REAL -valued Function-like finite m -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL m
i + (- i) is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
DD + (i - i) is Relation-like NAT -defined REAL -valued Function-like finite m -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL m
f /. (DD + (i - i)) is Relation-like NAT -defined Function-like finite 1 -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL 1
(f /. (DD + (i - i))) - (f /. i) is Relation-like NAT -defined REAL -valued Function-like finite 1 -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL 1
(f /. (DD + (i - i))) + (- (f /. i)) is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
DD + (0* m) is Relation-like NAT -defined REAL -valued Function-like finite m -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL m
f /. (DD + (0* m)) is Relation-like NAT -defined Function-like finite 1 -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL 1
(f /. (DD + (0* m))) - (f /. i) is Relation-like NAT -defined REAL -valued Function-like finite 1 -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL 1
(f /. (DD + (0* m))) + (- (f /. i)) is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
DD is Relation-like NAT -defined REAL -valued Function-like finite m -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL m
(diff (f,i)) . DD is Relation-like NAT -defined REAL -valued Function-like finite 1 -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL 1
m is non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below 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() strict RealNormSpace-like V185() NORMSTR
the carrier of (REAL-NS m) is non empty V2() 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
K6( the carrier of (REAL-NS m)) is set
Seg m is non empty finite m -element complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end right_end bounded_below bounded_above real-bounded Element of K6(NAT)
{ b₁ where b₁ is ordinal natural V11() real ext-real non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of NAT : ( 1 <= b₁ & b₁ <= m ) } is set
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)))
X is Element of K6( the carrier of (REAL-NS m))
i is Element of the carrier of (REAL-NS m)
diff (f,i) 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() RealNormSpace-like V185() NORMSTR
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 V214() RLSStruct
LinearOperators ((REAL-NS m),(REAL-NS 1)) is non empty functional Element of K6( the carrier of K318( the carrier of (REAL-NS m),(REAL-NS 1)))
K318( 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 V214() RLSStruct
Funcs ( the carrier of (REAL-NS m), the carrier of (REAL-NS 1)) is non empty functional FUNCTION_DOMAIN of 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 quasi_total Element of Funcs ( the carrier of (REAL-NS m), the carrier of (REAL-NS 1))
K214( 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 quasi_total 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((Funcs ( the carrier of (REAL-NS m), the carrier of (REAL-NS 1))),(Funcs ( the carrier of (REAL-NS m), the carrier of (REAL-NS 1)))) -defined Funcs ( the carrier of (REAL-NS m), the carrier of (REAL-NS 1)) -valued Function-like total quasi_total Element of K6(K7(K7((Funcs ( the carrier of (REAL-NS m), the carrier of (REAL-NS 1))),(Funcs ( the carrier of (REAL-NS m), the carrier of (REAL-NS 1)))),(Funcs ( the carrier of (REAL-NS m), the carrier of (REAL-NS 1)))))
K7((Funcs ( the carrier of (REAL-NS m), the carrier of (REAL-NS 1))),(Funcs ( the carrier of (REAL-NS m), the carrier of (REAL-NS 1)))) is set
K7(K7((Funcs ( the carrier of (REAL-NS m), the carrier of (REAL-NS 1))),(Funcs ( the carrier of (REAL-NS m), the carrier of (REAL-NS 1)))),(Funcs ( the carrier of (REAL-NS m), the carrier of (REAL-NS 1)))) is set
K6(K7(K7((Funcs ( the carrier of (REAL-NS m), the carrier of (REAL-NS 1))),(Funcs ( the carrier of (REAL-NS m), the carrier of (REAL-NS 1)))),(Funcs ( 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,(Funcs ( the carrier of (REAL-NS m), the carrier of (REAL-NS 1)))) -defined Funcs ( the carrier of (REAL-NS m), the carrier of (REAL-NS 1)) -valued Function-like total quasi_total Element of K6(K7(K7(REAL,(Funcs ( the carrier of (REAL-NS m), the carrier of (REAL-NS 1)))),(Funcs ( the carrier of (REAL-NS m), the carrier of (REAL-NS 1)))))
K7(REAL,(Funcs ( the carrier of (REAL-NS m), the carrier of (REAL-NS 1)))) is set
K7(K7(REAL,(Funcs ( the carrier of (REAL-NS m), the carrier of (REAL-NS 1)))),(Funcs ( the carrier of (REAL-NS m), the carrier of (REAL-NS 1)))) is set
K6(K7(K7(REAL,(Funcs ( the carrier of (REAL-NS m), the carrier of (REAL-NS 1)))),(Funcs ( the carrier of (REAL-NS m), the carrier of (REAL-NS 1))))) is set
RLSStruct(# (Funcs ( 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 K318( the carrier of (REAL-NS m),(REAL-NS 1)) is non empty set
K6( the carrier of K318( the carrier of (REAL-NS m),(REAL-NS 1))) is set
K199(K318( the carrier of (REAL-NS m),(REAL-NS 1)),(LinearOperators ((REAL-NS m),(REAL-NS 1)))) is Relation-like Function-like Element of LinearOperators ((REAL-NS m),(REAL-NS 1))
K197(K318( 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 quasi_total 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(K318( 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 quasi_total 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 set
K7(K7(REAL,(LinearOperators ((REAL-NS m),(REAL-NS 1)))),(LinearOperators ((REAL-NS m),(REAL-NS 1)))) is set
K6(K7(K7(REAL,(LinearOperators ((REAL-NS m),(REAL-NS 1)))),(LinearOperators ((REAL-NS m),(REAL-NS 1))))) is set
RLSStruct(# (LinearOperators ((REAL-NS m),(REAL-NS 1))),K199(K318( the carrier of (REAL-NS m),(REAL-NS 1)),(LinearOperators ((REAL-NS m),(REAL-NS 1)))),K197(K318( the carrier of (REAL-NS m),(REAL-NS 1)),(LinearOperators ((REAL-NS m),(REAL-NS 1)))),K198(K318( 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 quasi_total 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 quasi_total 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 set
K7(K7(REAL,(BoundedLinearOperators ((REAL-NS m),(REAL-NS 1)))),(BoundedLinearOperators ((REAL-NS m),(REAL-NS 1)))) is set
K6(K7(K7(REAL,(BoundedLinearOperators ((REAL-NS m),(REAL-NS 1)))),(BoundedLinearOperators ((REAL-NS m),(REAL-NS 1))))) is 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 quasi_total complex-valued ext-real-valued real-valued Element of K6(K7((BoundedLinearOperators ((REAL-NS m),(REAL-NS 1))),REAL))
K7((BoundedLinearOperators ((REAL-NS m),(REAL-NS 1))),REAL) is complex-valued ext-real-valued real-valued set
K6(K7((BoundedLinearOperators ((REAL-NS m),(REAL-NS 1))),REAL)) is set
NORMSTR(# (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 strict NORMSTR
the carrier of (R_NormSpace_of_BoundedLinearOperators ((REAL-NS m),(REAL-NS 1))) is non empty 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
{ b₁ where b₁ is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like Element of REAL * : len b₁ = m } is set
y0 is Relation-like NAT -defined REAL -valued Function-like finite 1 -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL 1
K7((REAL m),(REAL 1)) is set
K6(K7((REAL m),(REAL 1))) is set
K6((REAL m)) is set
y1 is functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered Element of K6((REAL m))
DD is ordinal natural V11() real ext-real non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of NAT
s is Relation-like REAL m -defined REAL 1 -valued Function-like complex-functions-valued ext-real-functions-valued real-functions-valued Element of K6(K7((REAL m),(REAL 1)))
f `partial| (X,DD) is Relation-like the carrier of (REAL-NS m) -defined the carrier of (R_NormSpace_of_BoundedLinearOperators ((REAL-NS 1),(REAL-NS 1))) -valued Function-like Element of K6(K7( the carrier of (REAL-NS m), the carrier of (R_NormSpace_of_BoundedLinearOperators ((REAL-NS 1),(REAL-NS 1)))))
K7( the carrier of (REAL-NS m), the carrier of (R_NormSpace_of_BoundedLinearOperators ((REAL-NS 1),(REAL-NS 1)))) is set
K6(K7( the carrier of (REAL-NS m), the carrier of (R_NormSpace_of_BoundedLinearOperators ((REAL-NS 1),(REAL-NS 1))))) is set
(m,1,DD,s,y1) is Relation-like REAL m -defined REAL 1 -valued Function-like complex-functions-valued ext-real-functions-valued real-functions-valued Element of K6(K7((REAL m),(REAL 1)))
dom (m,1,DD,s,y1) is functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered Element of K6((REAL m))
a is Relation-like NAT -defined REAL -valued Function-like finite m -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL m
(m,1,DD,s,y1) /. a is Relation-like NAT -defined Function-like finite 1 -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL 1
partdiff (s,a,DD) is Relation-like NAT -defined REAL -valued Function-like finite 1 -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL 1
r is Element of the carrier of (REAL-NS 1)
a is Relation-like NAT -defined REAL -valued Function-like finite m -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL m
(m,1,DD,s,y1) /. a is Relation-like NAT -defined Function-like finite 1 -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL 1
partdiff (s,a,DD) is Relation-like NAT -defined REAL -valued Function-like finite 1 -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL 1
t 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)))
tm is Element of the carrier of (REAL-NS m)
partdiff (t,tm,DD) is Relation-like Function-like Element of the carrier of (R_NormSpace_of_BoundedLinearOperators ((REAL-NS 1),(REAL-NS 1)))
reproj (DD,tm) is non empty Relation-like the carrier of (REAL-NS 1) -defined the carrier of (REAL-NS m) -valued Function-like total quasi_total Element of K6(K7( the carrier of (REAL-NS 1), the carrier of (REAL-NS m)))
K7( the carrier of (REAL-NS 1), the carrier of (REAL-NS m)) is set
K6(K7( the carrier of (REAL-NS 1), the carrier of (REAL-NS m))) is set
t * (reproj (DD,tm)) is Relation-like the carrier of (REAL-NS 1) -defined the carrier of (REAL-NS 1) -valued Function-like Element of K6(K7( the carrier of (REAL-NS 1), the carrier of (REAL-NS 1)))
Proj (DD,m) is non empty Relation-like the carrier of (REAL-NS m) -defined the carrier of (REAL-NS 1) -valued Function-like total quasi_total Element of K6(K7( the carrier of (REAL-NS m), the carrier of (REAL-NS 1)))
(Proj (DD,m)) . tm is Element of the carrier of (REAL-NS 1)
diff ((t * (reproj (DD,tm))),((Proj (DD,m)) . tm)) is Relation-like Function-like Element of the carrier of (R_NormSpace_of_BoundedLinearOperators ((REAL-NS 1),(REAL-NS 1)))
(partdiff (t,tm,DD)) . <*1*> is set
partdiff (f,tm,DD) is Relation-like Function-like Element of the carrier of (R_NormSpace_of_BoundedLinearOperators ((REAL-NS 1),(REAL-NS 1)))
f * (reproj (DD,tm)) is Relation-like the carrier of (REAL-NS 1) -defined the carrier of (REAL-NS 1) -valued Function-like Element of K6(K7( the carrier of (REAL-NS 1), the carrier of (REAL-NS 1)))
diff ((f * (reproj (DD,tm))),((Proj (DD,m)) . tm)) is Relation-like Function-like Element of the carrier of (R_NormSpace_of_BoundedLinearOperators ((REAL-NS 1),(REAL-NS 1)))
(partdiff (f,tm,DD)) . r is Element of the carrier of (REAL-NS 1)
a is Relation-like NAT -defined REAL -valued Function-like finite m -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL m
(m,1,DD,s,y1) /. a is Relation-like NAT -defined Function-like finite 1 -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL 1
t is V11() real ext-real Element of REAL
tm is Element of the carrier of (REAL-NS m)
(f `partial| (X,DD)) /. tm is Relation-like Function-like Element of the carrier of (R_NormSpace_of_BoundedLinearOperators ((REAL-NS 1),(REAL-NS 1)))
Sw0 is V11() real ext-real Element of REAL
w1 is Relation-like NAT -defined REAL -valued Function-like finite m -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL m
w1 - a is Relation-like NAT -defined REAL -valued Function-like finite m -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL m
- a is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
K38(1) * a is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
w1 + (- a) is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
|.(w1 - a).| is V11() real ext-real non negative Element of REAL
sqr (w1 - a) is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
mlt ((w1 - a),(w1 - a)) is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
Sum (sqr (w1 - a)) is V11() real ext-real Element of REAL
sqrt (Sum (sqr (w1 - a))) is V11() real ext-real Element of REAL
(m,1,DD,s,y1) /. w1 is Relation-like NAT -defined Function-like finite 1 -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL 1
((m,1,DD,s,y1) /. w1) - ((m,1,DD,s,y1) /. a) is Relation-like NAT -defined REAL -valued Function-like finite 1 -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL 1
- ((m,1,DD,s,y1) /. a) is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
K38(1) * ((m,1,DD,s,y1) /. a) is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
((m,1,DD,s,y1) /. w1) + (- ((m,1,DD,s,y1) /. a)) is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
|.(((m,1,DD,s,y1) /. w1) - ((m,1,DD,s,y1) /. a)).| is V11() real ext-real non negative Element of REAL
sqr (((m,1,DD,s,y1) /. w1) - ((m,1,DD,s,y1) /. a)) is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
mlt ((((m,1,DD,s,y1) /. w1) - ((m,1,DD,s,y1) /. a)),(((m,1,DD,s,y1) /. w1) - ((m,1,DD,s,y1) /. a))) is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
Sum (sqr (((m,1,DD,s,y1) /. w1) - ((m,1,DD,s,y1) /. a))) is V11() real ext-real Element of REAL
sqrt (Sum (sqr (((m,1,DD,s,y1) /. w1) - ((m,1,DD,s,y1) /. a)))) is V11() real ext-real Element of REAL
Sw1 is Element of the carrier of (REAL-NS m)
Sw1 - tm is Element of the carrier of (REAL-NS m)
||.(Sw1 - tm).|| is V11() real ext-real Element of REAL
the U8 of (REAL-NS m) is non empty Relation-like the carrier of (REAL-NS m) -defined REAL -valued Function-like total quasi_total complex-valued ext-real-valued real-valued Element of K6(K7( the carrier of (REAL-NS m),REAL))
K7( the carrier of (REAL-NS m),REAL) is complex-valued ext-real-valued real-valued set
K6(K7( the carrier of (REAL-NS m),REAL)) is set
the U8 of (REAL-NS m) . (Sw1 - tm) is V11() real ext-real Element of REAL
(f `partial| (X,DD)) /. Sw1 is Relation-like Function-like Element of the carrier of (R_NormSpace_of_BoundedLinearOperators ((REAL-NS 1),(REAL-NS 1)))
((f `partial| (X,DD)) /. Sw1) - ((f `partial| (X,DD)) /. tm) is Relation-like Function-like Element of the carrier of (R_NormSpace_of_BoundedLinearOperators ((REAL-NS 1),(REAL-NS 1)))
||.(((f `partial| (X,DD)) /. Sw1) - ((f `partial| (X,DD)) /. tm)).|| is V11() real ext-real Element of REAL
the U8 of (R_NormSpace_of_BoundedLinearOperators ((REAL-NS 1),(REAL-NS 1))) is non empty Relation-like the carrier of (R_NormSpace_of_BoundedLinearOperators ((REAL-NS 1),(REAL-NS 1))) -defined REAL -valued Function-like total quasi_total complex-valued ext-real-valued real-valued Element of K6(K7( the carrier of (R_NormSpace_of_BoundedLinearOperators ((REAL-NS 1),(REAL-NS 1))),REAL))
K7( the carrier of (R_NormSpace_of_BoundedLinearOperators ((REAL-NS 1),(REAL-NS 1))),REAL) is complex-valued ext-real-valued real-valued set
K6(K7( the carrier of (R_NormSpace_of_BoundedLinearOperators ((REAL-NS 1),(REAL-NS 1))),REAL)) is set
the U8 of (R_NormSpace_of_BoundedLinearOperators ((REAL-NS 1),(REAL-NS 1))) . (((f `partial| (X,DD)) /. Sw1) - ((f `partial| (X,DD)) /. tm)) is V11() real ext-real Element of REAL
partdiff (f,Sw1,DD) is Relation-like Function-like Element of the carrier of (R_NormSpace_of_BoundedLinearOperators ((REAL-NS 1),(REAL-NS 1)))
reproj (DD,Sw1) is non empty Relation-like the carrier of (REAL-NS 1) -defined the carrier of (REAL-NS m) -valued Function-like total quasi_total Element of K6(K7( the carrier of (REAL-NS 1), the carrier of (REAL-NS m)))
K7( the carrier of (REAL-NS 1), the carrier of (REAL-NS m)) is set
K6(K7( the carrier of (REAL-NS 1), the carrier of (REAL-NS m))) is set
f * (reproj (DD,Sw1)) is Relation-like the carrier of (REAL-NS 1) -defined the carrier of (REAL-NS 1) -valued Function-like Element of K6(K7( the carrier of (REAL-NS 1), the carrier of (REAL-NS 1)))
Proj (DD,m) is non empty Relation-like the carrier of (REAL-NS m) -defined the carrier of (REAL-NS 1) -valued Function-like total quasi_total Element of K6(K7( the carrier of (REAL-NS m), the carrier of (REAL-NS 1)))
(Proj (DD,m)) . Sw1 is Element of the carrier of (REAL-NS 1)
diff ((f * (reproj (DD,Sw1))),((Proj (DD,m)) . Sw1)) is Relation-like Function-like Element of the carrier of (R_NormSpace_of_BoundedLinearOperators ((REAL-NS 1),(REAL-NS 1)))
partdiff (f,tm,DD) is Relation-like Function-like Element of the carrier of (R_NormSpace_of_BoundedLinearOperators ((REAL-NS 1),(REAL-NS 1)))
reproj (DD,tm) is non empty Relation-like the carrier of (REAL-NS 1) -defined the carrier of (REAL-NS m) -valued Function-like total quasi_total Element of K6(K7( the carrier of (REAL-NS 1), the carrier of (REAL-NS m)))
f * (reproj (DD,tm)) is Relation-like the carrier of (REAL-NS 1) -defined the carrier of (REAL-NS 1) -valued Function-like Element of K6(K7( the carrier of (REAL-NS 1), the carrier of (REAL-NS 1)))
(Proj (DD,m)) . tm is Element of the carrier of (REAL-NS 1)
diff ((f * (reproj (DD,tm))),((Proj (DD,m)) . tm)) is Relation-like Function-like Element of the carrier of (R_NormSpace_of_BoundedLinearOperators ((REAL-NS 1),(REAL-NS 1)))
(partdiff (f,Sw1,DD)) - (partdiff (f,tm,DD)) is Relation-like Function-like Element of the carrier of (R_NormSpace_of_BoundedLinearOperators ((REAL-NS 1),(REAL-NS 1)))
||.((partdiff (f,Sw1,DD)) - (partdiff (f,tm,DD))).|| is V11() real ext-real Element of REAL
the U8 of (R_NormSpace_of_BoundedLinearOperators ((REAL-NS 1),(REAL-NS 1))) . ((partdiff (f,Sw1,DD)) - (partdiff (f,tm,DD))) is V11() real ext-real Element of REAL
(partdiff (f,Sw1,DD)) . r is Element of the carrier of (REAL-NS 1)
(partdiff (f,tm,DD)) . r is Element of the carrier of (REAL-NS 1)
((partdiff (f,Sw1,DD)) . r) - ((partdiff (f,tm,DD)) . r) is Element of the carrier of (REAL-NS 1)
w2 is non empty Relation-like the carrier of (REAL-NS 1) -defined the carrier of (REAL-NS 1) -valued Function-like total quasi_total V182( REAL-NS 1, REAL-NS 1) V183( REAL-NS 1, REAL-NS 1) Lipschitzian Element of K6(K7( the carrier of (REAL-NS 1), the carrier of (REAL-NS 1)))
w2 . r is Element of the carrier of (REAL-NS 1)
||.(((partdiff (f,Sw1,DD)) . r) - ((partdiff (f,tm,DD)) . r)).|| is V11() real ext-real Element of REAL
the U8 of (REAL-NS 1) is non empty Relation-like the carrier of (REAL-NS 1) -defined REAL -valued Function-like total quasi_total complex-valued ext-real-valued real-valued Element of K6(K7( the carrier of (REAL-NS 1),REAL))
K7( the carrier of (REAL-NS 1),REAL) is complex-valued ext-real-valued real-valued set
K6(K7( the carrier of (REAL-NS 1),REAL)) is set
the U8 of (REAL-NS 1) . (((partdiff (f,Sw1,DD)) . r) - ((partdiff (f,tm,DD)) . r)) is V11() real ext-real Element of REAL
||.r.|| is V11() real ext-real Element of REAL
the U8 of (REAL-NS 1) . r is V11() real ext-real Element of REAL
||.((partdiff (f,Sw1,DD)) - (partdiff (f,tm,DD))).|| * ||.r.|| is V11() real ext-real Element of REAL
|.y0.| is V11() real ext-real non negative Element of REAL
sqr y0 is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
mlt (y0,y0) is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
Sum (sqr y0) is V11() real ext-real Element of REAL
sqrt (Sum (sqr y0)) is V11() real ext-real Element of REAL
abs 1 is V11() real ext-real V198() Element of REAL
ys is Element of the carrier of (REAL-NS m)
partdiff (f,ys,DD) is Relation-like Function-like Element of the carrier of (R_NormSpace_of_BoundedLinearOperators ((REAL-NS 1),(REAL-NS 1)))
reproj (DD,ys) is non empty Relation-like the carrier of (REAL-NS 1) -defined the carrier of (REAL-NS m) -valued Function-like total quasi_total Element of K6(K7( the carrier of (REAL-NS 1), the carrier of (REAL-NS m)))
f * (reproj (DD,ys)) is Relation-like the carrier of (REAL-NS 1) -defined the carrier of (REAL-NS 1) -valued Function-like Element of K6(K7( the carrier of (REAL-NS 1), the carrier of (REAL-NS 1)))
(Proj (DD,m)) . ys is Element of the carrier of (REAL-NS 1)
diff ((f * (reproj (DD,ys))),((Proj (DD,m)) . ys)) is Relation-like Function-like Element of the carrier of (R_NormSpace_of_BoundedLinearOperators ((REAL-NS 1),(REAL-NS 1)))
(partdiff (f,ys,DD)) . r is Element of the carrier of (REAL-NS 1)
rm is Element of the carrier of (REAL-NS m)
partdiff (f,rm,DD) is Relation-like Function-like Element of the carrier of (R_NormSpace_of_BoundedLinearOperators ((REAL-NS 1),(REAL-NS 1)))
reproj (DD,rm) is non empty Relation-like the carrier of (REAL-NS 1) -defined the carrier of (REAL-NS m) -valued Function-like total quasi_total Element of K6(K7( the carrier of (REAL-NS 1), the carrier of (REAL-NS m)))
f * (reproj (DD,rm)) is Relation-like the carrier of (REAL-NS 1) -defined the carrier of (REAL-NS 1) -valued Function-like Element of K6(K7( the carrier of (REAL-NS 1), the carrier of (REAL-NS 1)))
(Proj (DD,m)) . rm is Element of the carrier of (REAL-NS 1)
diff ((f * (reproj (DD,rm))),((Proj (DD,m)) . rm)) is Relation-like Function-like Element of the carrier of (R_NormSpace_of_BoundedLinearOperators ((REAL-NS 1),(REAL-NS 1)))
(partdiff (f,rm,DD)) . r is Element of the carrier of (REAL-NS 1)
PP is Relation-like NAT -defined REAL -valued Function-like finite m -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL m
diff (s,PP) is non empty Relation-like REAL m -defined REAL 1 -valued Function-like total quasi_total complex-functions-valued ext-real-functions-valued real-functions-valued Element of K6(K7((REAL m),(REAL 1)))
DD is Relation-like NAT -defined REAL -valued Function-like finite m -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL m
(diff (s,PP)) . DD is Relation-like NAT -defined REAL -valued Function-like finite 1 -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL 1
DD 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)))
a is Element of the carrier of (REAL-NS m)
DD is Element of the carrier of (REAL-NS m)
a is Relation-like NAT -defined REAL -valued Function-like finite m -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL m
(diff (s,PP)) . a is Relation-like NAT -defined REAL -valued Function-like finite 1 -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL 1
t is Relation-like NAT -defined REAL 1 -valued Function-like finite FinSequence-like FinSubsequence-like complex-functions-valued ext-real-functions-valued real-functions-valued FinSequence of REAL 1
dom t is complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of K6(NAT)
Sum t is Relation-like NAT -defined REAL -valued Function-like finite 1 -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL 1
tm is Relation-like NAT -defined REAL 1 -valued Function-like finite FinSequence-like FinSubsequence-like complex-functions-valued ext-real-functions-valued real-functions-valued FinSequence of REAL 1
dom tm is complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of K6(NAT)
Sw0 is ordinal natural V11() real ext-real non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of NAT
tm . Sw0 is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
partdiff (f,i,Sw0) is Relation-like Function-like Element of the carrier of (R_NormSpace_of_BoundedLinearOperators ((REAL-NS 1),(REAL-NS 1)))
reproj (Sw0,i) is non empty Relation-like the carrier of (REAL-NS 1) -defined the carrier of (REAL-NS m) -valued Function-like total quasi_total Element of K6(K7( the carrier of (REAL-NS 1), the carrier of (REAL-NS m)))
K7( the carrier of (REAL-NS 1), the carrier of (REAL-NS m)) is set
K6(K7( the carrier of (REAL-NS 1), the carrier of (REAL-NS m))) is set
f * (reproj (Sw0,i)) is Relation-like the carrier of (REAL-NS 1) -defined the carrier of (REAL-NS 1) -valued Function-like Element of K6(K7( the carrier of (REAL-NS 1), the carrier of (REAL-NS 1)))
Proj (Sw0,m) is non empty Relation-like the carrier of (REAL-NS m) -defined the carrier of (REAL-NS 1) -valued Function-like total quasi_total Element of K6(K7( the carrier of (REAL-NS m), the carrier of (REAL-NS 1)))
(Proj (Sw0,m)) . i is Element of the carrier of (REAL-NS 1)
diff ((f * (reproj (Sw0,i))),((Proj (Sw0,m)) . i)) is Relation-like Function-like Element of the carrier of (R_NormSpace_of_BoundedLinearOperators ((REAL-NS 1),(REAL-NS 1)))
proj (Sw0,m) is non empty Relation-like REAL m -defined REAL -valued Function-like total quasi_total complex-valued ext-real-valued real-valued Element of K6(K7((REAL m),REAL))
K7((REAL m),REAL) is complex-valued ext-real-valued real-valued set
K6(K7((REAL m),REAL)) is set
(proj (Sw0,m)) . DD is V11() real ext-real Element of REAL
<*((proj (Sw0,m)) . DD)*> is non empty V2() Relation-like NAT -defined REAL -valued Function-like one-to-one constant finite 1 -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued increasing decreasing non-decreasing non-increasing FinSequence of REAL
[1,((proj (Sw0,m)) . DD)] is V27() set
{1,((proj (Sw0,m)) . DD)} is non empty complex-membered ext-real-membered real-membered set
{{1,((proj (Sw0,m)) . DD)},{1}} is non empty set
{[1,((proj (Sw0,m)) . DD)]} is non empty Function-like set
(partdiff (f,i,Sw0)) . <*((proj (Sw0,m)) . DD)*> is set
partdiff (s,PP,Sw0) is Relation-like NAT -defined REAL -valued Function-like finite 1 -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL 1
((proj (Sw0,m)) . DD) * r is Element of the carrier of (REAL-NS 1)
(proj (Sw0,m)) . a is V11() real ext-real Element of REAL
((proj (Sw0,m)) . a) * y0 is Relation-like NAT -defined REAL -valued Function-like finite 1 -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL 1
((proj (Sw0,m)) . a) * 1 is V11() real ext-real Element of REAL
<*(((proj (Sw0,m)) . a) * 1)*> is non empty V2() Relation-like NAT -defined REAL -valued Function-like one-to-one constant finite 1 -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued increasing decreasing non-decreasing non-increasing FinSequence of REAL
[1,(((proj (Sw0,m)) . a) * 1)] is V27() set
{1,(((proj (Sw0,m)) . a) * 1)} is non empty complex-membered ext-real-membered real-membered set
{{1,(((proj (Sw0,m)) . a) * 1)},{1}} is non empty set
{[1,(((proj (Sw0,m)) . a) * 1)]} is non empty Function-like set
((proj (Sw0,m)) . a) * (partdiff (s,PP,Sw0)) is Relation-like NAT -defined REAL -valued Function-like finite 1 -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL 1
w1 is non empty Relation-like the carrier of (REAL-NS 1) -defined the carrier of (REAL-NS 1) -valued Function-like total quasi_total V182( REAL-NS 1, REAL-NS 1) V183( REAL-NS 1, REAL-NS 1) Lipschitzian Element of K6(K7( the carrier of (REAL-NS 1), the carrier of (REAL-NS 1)))
w1 . r is Element of the carrier of (REAL-NS 1)
((proj (Sw0,m)) . a) * (w1 . r) is Element of the carrier of (REAL-NS 1)
Sw1 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)))
w2 is Element of the carrier of (REAL-NS m)
partdiff (Sw1,w2,Sw0) is Relation-like Function-like Element of the carrier of (R_NormSpace_of_BoundedLinearOperators ((REAL-NS 1),(REAL-NS 1)))
reproj (Sw0,w2) is non empty Relation-like the carrier of (REAL-NS 1) -defined the carrier of (REAL-NS m) -valued Function-like total quasi_total Element of K6(K7( the carrier of (REAL-NS 1), the carrier of (REAL-NS m)))
Sw1 * (reproj (Sw0,w2)) is Relation-like the carrier of (REAL-NS 1) -defined the carrier of (REAL-NS 1) -valued Function-like Element of K6(K7( the carrier of (REAL-NS 1), the carrier of (REAL-NS 1)))
(Proj (Sw0,m)) . w2 is Element of the carrier of (REAL-NS 1)
diff ((Sw1 * (reproj (Sw0,w2))),((Proj (Sw0,m)) . w2)) is Relation-like Function-like Element of the carrier of (R_NormSpace_of_BoundedLinearOperators ((REAL-NS 1),(REAL-NS 1)))
(partdiff (Sw1,w2,Sw0)) . <*1*> is set
(diff (f,i)) . DD is Element of the carrier of (REAL-NS 1)
Sum tm is Relation-like NAT -defined REAL -valued Function-like finite 1 -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL 1
Sw0 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)))
w1 is Element of the carrier of (REAL-NS m)
diff (Sw0,w1) is Relation-like Function-like Element of the carrier of (R_NormSpace_of_BoundedLinearOperators ((REAL-NS m),(REAL-NS 1)))
DD is Element of the carrier of (REAL-NS m)
(diff (f,i)) . DD is Element of the carrier of (REAL-NS 1)
m is non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below 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() strict RealNormSpace-like V185() NORMSTR
the carrier of (REAL-NS m) is non empty V2() 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
K6( the carrier of (REAL-NS m)) is set
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() RealNormSpace-like V185() NORMSTR
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 V214() RLSStruct
LinearOperators ((REAL-NS m),(REAL-NS 1)) is non empty functional Element of K6( the carrier of K318( the carrier of (REAL-NS m),(REAL-NS 1)))
K318( 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 V214() RLSStruct
Funcs ( the carrier of (REAL-NS m), the carrier of (REAL-NS 1)) is non empty functional FUNCTION_DOMAIN of 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 quasi_total Element of Funcs ( the carrier of (REAL-NS m), the carrier of (REAL-NS 1))
K214( 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 quasi_total 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((Funcs ( the carrier of (REAL-NS m), the carrier of (REAL-NS 1))),(Funcs ( the carrier of (REAL-NS m), the carrier of (REAL-NS 1)))) -defined Funcs ( the carrier of (REAL-NS m), the carrier of (REAL-NS 1)) -valued Function-like total quasi_total Element of K6(K7(K7((Funcs ( the carrier of (REAL-NS m), the carrier of (REAL-NS 1))),(Funcs ( the carrier of (REAL-NS m), the carrier of (REAL-NS 1)))),(Funcs ( the carrier of (REAL-NS m), the carrier of (REAL-NS 1)))))
K7((Funcs ( the carrier of (REAL-NS m), the carrier of (REAL-NS 1))),(Funcs ( the carrier of (REAL-NS m), the carrier of (REAL-NS 1)))) is set
K7(K7((Funcs ( the carrier of (REAL-NS m), the carrier of (REAL-NS 1))),(Funcs ( the carrier of (REAL-NS m), the carrier of (REAL-NS 1)))),(Funcs ( the carrier of (REAL-NS m), the carrier of (REAL-NS 1)))) is set
K6(K7(K7((Funcs ( the carrier of (REAL-NS m), the carrier of (REAL-NS 1))),(Funcs ( the carrier of (REAL-NS m), the carrier of (REAL-NS 1)))),(Funcs ( 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,(Funcs ( the carrier of (REAL-NS m), the carrier of (REAL-NS 1)))) -defined Funcs ( the carrier of (REAL-NS m), the carrier of (REAL-NS 1)) -valued Function-like total quasi_total Element of K6(K7(K7(REAL,(Funcs ( the carrier of (REAL-NS m), the carrier of (REAL-NS 1)))),(Funcs ( the carrier of (REAL-NS m), the carrier of (REAL-NS 1)))))
K7(REAL,(Funcs ( the carrier of (REAL-NS m), the carrier of (REAL-NS 1)))) is set
K7(K7(REAL,(Funcs ( the carrier of (REAL-NS m), the carrier of (REAL-NS 1)))),(Funcs ( the carrier of (REAL-NS m), the carrier of (REAL-NS 1)))) is set
K6(K7(K7(REAL,(Funcs ( the carrier of (REAL-NS m), the carrier of (REAL-NS 1)))),(Funcs ( the carrier of (REAL-NS m), the carrier of (REAL-NS 1))))) is set
RLSStruct(# (Funcs ( 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 K318( the carrier of (REAL-NS m),(REAL-NS 1)) is non empty set
K6( the carrier of K318( the carrier of (REAL-NS m),(REAL-NS 1))) is set
K199(K318( the carrier of (REAL-NS m),(REAL-NS 1)),(LinearOperators ((REAL-NS m),(REAL-NS 1)))) is Relation-like Function-like Element of LinearOperators ((REAL-NS m),(REAL-NS 1))
K197(K318( 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 quasi_total 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(K318( 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 quasi_total 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 set
K7(K7(REAL,(LinearOperators ((REAL-NS m),(REAL-NS 1)))),(LinearOperators ((REAL-NS m),(REAL-NS 1)))) is set
K6(K7(K7(REAL,(LinearOperators ((REAL-NS m),(REAL-NS 1)))),(LinearOperators ((REAL-NS m),(REAL-NS 1))))) is set
RLSStruct(# (LinearOperators ((REAL-NS m),(REAL-NS 1))),K199(K318( the carrier of (REAL-NS m),(REAL-NS 1)),(LinearOperators ((REAL-NS m),(REAL-NS 1)))),K197(K318( the carrier of (REAL-NS m),(REAL-NS 1)),(LinearOperators ((REAL-NS m),(REAL-NS 1)))),K198(K318( 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 quasi_total 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 quasi_total 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 set
K7(K7(REAL,(BoundedLinearOperators ((REAL-NS m),(REAL-NS 1)))),(BoundedLinearOperators ((REAL-NS m),(REAL-NS 1)))) is set
K6(K7(K7(REAL,(BoundedLinearOperators ((REAL-NS m),(REAL-NS 1)))),(BoundedLinearOperators ((REAL-NS m),(REAL-NS 1))))) is 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 quasi_total complex-valued ext-real-valued real-valued Element of K6(K7((BoundedLinearOperators ((REAL-NS m),(REAL-NS 1))),REAL))
K7((BoundedLinearOperators ((REAL-NS m),(REAL-NS 1))),REAL) is complex-valued ext-real-valued real-valued set
K6(K7((BoundedLinearOperators ((REAL-NS m),(REAL-NS 1))),REAL)) is set
NORMSTR(# (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 strict NORMSTR
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)))
X is Element of K6( the carrier of (REAL-NS m))
f `| X is Relation-like the carrier of (REAL-NS m) -defined the carrier of (R_NormSpace_of_BoundedLinearOperators ((REAL-NS m),(REAL-NS 1))) -valued Function-like Element of K6(K7( the carrier of (REAL-NS m), the carrier of (R_NormSpace_of_BoundedLinearOperators ((REAL-NS m),(REAL-NS 1)))))
the carrier of (R_NormSpace_of_BoundedLinearOperators ((REAL-NS m),(REAL-NS 1))) is non empty set
K7( the carrier of (REAL-NS m), the carrier of (R_NormSpace_of_BoundedLinearOperators ((REAL-NS m),(REAL-NS 1)))) is set
K6(K7( the carrier of (REAL-NS m), the carrier of (R_NormSpace_of_BoundedLinearOperators ((REAL-NS m),(REAL-NS 1))))) is set
i is ordinal natural V11() real ext-real non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of NAT
f `partial| (X,i) is Relation-like the carrier of (REAL-NS m) -defined the carrier of (R_NormSpace_of_BoundedLinearOperators ((REAL-NS 1),(REAL-NS 1))) -valued Function-like Element of K6(K7( the carrier of (REAL-NS m), the carrier of (R_NormSpace_of_BoundedLinearOperators ((REAL-NS 1),(REAL-NS 1)))))
K7( the carrier of (REAL-NS m), the carrier of (R_NormSpace_of_BoundedLinearOperators ((REAL-NS 1),(REAL-NS 1)))) is set
K6(K7( the carrier of (REAL-NS m), the carrier of (R_NormSpace_of_BoundedLinearOperators ((REAL-NS 1),(REAL-NS 1))))) is set
dom (f `partial| (X,i)) is Element of K6( the carrier of (REAL-NS m))
y0 is Element of the carrier of (REAL-NS m)
r is V11() real ext-real Element of REAL
(f `partial| (X,i)) /. y0 is Relation-like Function-like Element of the carrier of (R_NormSpace_of_BoundedLinearOperators ((REAL-NS 1),(REAL-NS 1)))
dom f is Element of K6( the carrier of (REAL-NS m))
i is Element of the carrier of (REAL-NS m)
dom (f `| X) is Element of K6( the carrier of (REAL-NS m))
i is Element of the carrier of (REAL-NS m)
(f `| X) /. i is Relation-like Function-like Element of the carrier of (R_NormSpace_of_BoundedLinearOperators ((REAL-NS m),(REAL-NS 1)))
y0 is V11() real ext-real Element of REAL
2 * m is non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below Element of NAT
y0 / (2 * m) is V11() real ext-real Element of REAL
r is ordinal natural V11() real ext-real non negative set
Seg m is non empty finite m -element complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end right_end bounded_below bounded_above real-bounded Element of K6(NAT)
{ b<sub>1</sub> where b<sub>1</sub> is ordinal natural V11() real ext-real non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of NAT : ( 1 <= b<sub>1</sub> & b<sub>1</sub> <= m ) } is set
s is ordinal natural V11() real ext-real non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of NAT
f `partial| (X,s) is Relation-like the carrier of (REAL-NS m) -defined the carrier of (R_NormSpace_of_BoundedLinearOperators ((REAL-NS 1),(REAL-NS 1))) -valued Function-like Element of K6(K7( the carrier of (REAL-NS m), the carrier of (R_NormSpace_of_BoundedLinearOperators ((REAL-NS 1),(REAL-NS 1)))))
(f `partial| (X,s)) /. i is Relation-like Function-like Element of the carrier of (R_NormSpace_of_BoundedLinearOperators ((REAL-NS 1),(REAL-NS 1)))
y1 is V11() real ext-real Element of REAL
PP is V11() real ext-real Element of REAL
DD is V11() real ext-real Element of REAL
a is ordinal natural V11() real ext-real non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of NAT
t is Element of the carrier of (REAL-NS m)
t - i is Element of the carrier of (REAL-NS m)
||.(t - i).|| is V11() real ext-real Element of REAL
the U8 of (REAL-NS m) is non empty Relation-like the carrier of (REAL-NS m) -defined REAL -valued Function-like total quasi_total complex-valued ext-real-valued real-valued Element of K6(K7( the carrier of (REAL-NS m),REAL))
K7( the carrier of (REAL-NS m),REAL) is complex-valued ext-real-valued real-valued set
K6(K7( the carrier of (REAL-NS m),REAL)) is set
the U8 of (REAL-NS m) . (t - i) is V11() real ext-real Element of REAL
f `partial| (X,a) is Relation-like the carrier of (REAL-NS m) -defined the carrier of (R_NormSpace_of_BoundedLinearOperators ((REAL-NS 1),(REAL-NS 1))) -valued Function-like Element of K6(K7( the carrier of (REAL-NS m), the carrier of (R_NormSpace_of_BoundedLinearOperators ((REAL-NS 1),(REAL-NS 1)))))
(f `partial| (X,a)) /. t is Relation-like Function-like Element of the carrier of (R_NormSpace_of_BoundedLinearOperators ((REAL-NS 1),(REAL-NS 1)))
(f `partial| (X,a)) /. i is Relation-like Function-like Element of the carrier of (R_NormSpace_of_BoundedLinearOperators ((REAL-NS 1),(REAL-NS 1)))
((f `partial| (X,a)) /. t) - ((f `partial| (X,a)) /. i) is Relation-like Function-like Element of the carrier of (R_NormSpace_of_BoundedLinearOperators ((REAL-NS 1),(REAL-NS 1)))
||.(((f `partial| (X,a)) /. t) - ((f `partial| (X,a)) /. i)).|| is V11() real ext-real Element of REAL
the U8 of (R_NormSpace_of_BoundedLinearOperators ((REAL-NS 1),(REAL-NS 1))) is non empty Relation-like the carrier of (R_NormSpace_of_BoundedLinearOperators ((REAL-NS 1),(REAL-NS 1))) -defined REAL -valued Function-like total quasi_total complex-valued ext-real-valued real-valued Element of K6(K7( the carrier of (R_NormSpace_of_BoundedLinearOperators ((REAL-NS 1),(REAL-NS 1))),REAL))
K7( the carrier of (R_NormSpace_of_BoundedLinearOperators ((REAL-NS 1),(REAL-NS 1))),REAL) is complex-valued ext-real-valued real-valued set
K6(K7( the carrier of (R_NormSpace_of_BoundedLinearOperators ((REAL-NS 1),(REAL-NS 1))),REAL)) is set
the U8 of (R_NormSpace_of_BoundedLinearOperators ((REAL-NS 1),(REAL-NS 1))) . (((f `partial| (X,a)) /. t) - ((f `partial| (X,a)) /. i)) is V11() real ext-real Element of REAL
r is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
dom r is complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of K6(NAT)
min r is V11() real ext-real Element of REAL
s is V11() real ext-real Element of REAL
len r is ordinal natural V11() real ext-real non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of NAT
min_p r is ordinal natural V11() real ext-real non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of NAT
r . (min_p r) is V11() real ext-real Element of REAL
y1 is Element of the carrier of (REAL-NS m)
y1 - i is Element of the carrier of (REAL-NS m)
||.(y1 - i).|| is V11() real ext-real Element of REAL
the U8 of (REAL-NS m) is non empty Relation-like the carrier of (REAL-NS m) -defined REAL -valued Function-like total quasi_total complex-valued ext-real-valued real-valued Element of K6(K7( the carrier of (REAL-NS m),REAL))
K7( the carrier of (REAL-NS m),REAL) is complex-valued ext-real-valued real-valued set
K6(K7( the carrier of (REAL-NS m),REAL)) is set
the U8 of (REAL-NS m) . (y1 - i) is V11() real ext-real Element of REAL
(f `| X) /. y1 is Relation-like Function-like Element of the carrier of (R_NormSpace_of_BoundedLinearOperators ((REAL-NS m),(REAL-NS 1)))
((f `| X) /. y1) - ((f `| X) /. i) is Relation-like Function-like Element of the carrier of (R_NormSpace_of_BoundedLinearOperators ((REAL-NS m),(REAL-NS 1)))
||.(((f `| X) /. y1) - ((f `| X) /. i)).|| is V11() real ext-real Element of REAL
the U8 of (R_NormSpace_of_BoundedLinearOperators ((REAL-NS m),(REAL-NS 1))) is non empty Relation-like the carrier of (R_NormSpace_of_BoundedLinearOperators ((REAL-NS m),(REAL-NS 1))) -defined REAL -valued Function-like total quasi_total complex-valued ext-real-valued real-valued Element of K6(K7( the carrier of (R_NormSpace_of_BoundedLinearOperators ((REAL-NS m),(REAL-NS 1))),REAL))
K7( the carrier of (R_NormSpace_of_BoundedLinearOperators ((REAL-NS m),(REAL-NS 1))),REAL) is complex-valued ext-real-valued real-valued set
K6(K7( the carrier of (R_NormSpace_of_BoundedLinearOperators ((REAL-NS m),(REAL-NS 1))),REAL)) is set
the U8 of (R_NormSpace_of_BoundedLinearOperators ((REAL-NS m),(REAL-NS 1))) . (((f `| X) /. y1) - ((f `| X) /. i)) is V11() real ext-real Element of REAL
diff (f,y1) is Relation-like Function-like Element of the carrier of (R_NormSpace_of_BoundedLinearOperators ((REAL-NS m),(REAL-NS 1)))
diff (f,i) is Relation-like Function-like Element of the carrier of (R_NormSpace_of_BoundedLinearOperators ((REAL-NS m),(REAL-NS 1)))
(diff (f,y1)) - (diff (f,i)) is Relation-like Function-like Element of the carrier of (R_NormSpace_of_BoundedLinearOperators ((REAL-NS m),(REAL-NS 1)))
PP is non empty Relation-like the carrier of (REAL-NS m) -defined the carrier of (REAL-NS 1) -valued Function-like total quasi_total V182( REAL-NS m, REAL-NS 1) V183( REAL-NS m, REAL-NS 1) Lipschitzian Element of K6(K7( the carrier of (REAL-NS m), the carrier of (REAL-NS 1)))
PreNorms PP is non empty complex-membered ext-real-membered real-membered Element of K6(REAL)
{ ||.(PP . b₁).|| where b₁ is Element of the carrier of (REAL-NS m) : ||.b₁.|| <= 1 } is set
upper_bound (PreNorms PP) is V11() real ext-real Element of REAL
||.((diff (f,y1)) - (diff (f,i))).|| is V11() real ext-real Element of REAL
the U8 of (R_NormSpace_of_BoundedLinearOperators ((REAL-NS m),(REAL-NS 1))) . ((diff (f,y1)) - (diff (f,i))) is V11() real ext-real Element of REAL
DD is V11() real ext-real set
a is Element of the carrier of (REAL-NS m)
PP . a is Element of the carrier of (REAL-NS 1)
||.(PP . a).|| is V11() real ext-real Element of REAL
the U8 of (REAL-NS 1) is non empty Relation-like the carrier of (REAL-NS 1) -defined REAL -valued Function-like total quasi_total complex-valued ext-real-valued real-valued Element of K6(K7( the carrier of (REAL-NS 1),REAL))
K7( the carrier of (REAL-NS 1),REAL) is complex-valued ext-real-valued real-valued set
K6(K7( the carrier of (REAL-NS 1),REAL)) is set
the U8 of (REAL-NS 1) . (PP . a) is V11() real ext-real Element of REAL
||.a.|| is V11() real ext-real Element of REAL
the U8 of (REAL-NS m) . a is V11() real ext-real Element of REAL
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
{ b₁ where b₁ is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like Element of REAL * : len b₁ = m } is set
(diff (f,i)) . a is Element of the carrier of (REAL-NS 1)
tm is Relation-like NAT -defined REAL 1 -valued Function-like finite FinSequence-like FinSubsequence-like complex-functions-valued ext-real-functions-valued real-functions-valued FinSequence of REAL 1
dom tm is complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of K6(NAT)
Sum tm is Relation-like NAT -defined REAL -valued Function-like finite 1 -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL 1
(diff (f,y1)) . a is Element of the carrier of (REAL-NS 1)
w1 is Relation-like NAT -defined REAL 1 -valued Function-like finite FinSequence-like FinSubsequence-like complex-functions-valued ext-real-functions-valued real-functions-valued FinSequence of REAL 1
dom w1 is complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of K6(NAT)
Sum w1 is Relation-like NAT -defined REAL -valued Function-like finite 1 -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL 1
w2 is Relation-like NAT -defined REAL 1 -valued Function-like finite FinSequence-like FinSubsequence-like complex-functions-valued ext-real-functions-valued real-functions-valued FinSequence of REAL 1
len w2 is ordinal natural V11() real ext-real non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of NAT
dom w2 is complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of K6(NAT)
len w1 is ordinal natural V11() real ext-real non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of NAT
len tm is ordinal natural V11() real ext-real non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of NAT
ys is ordinal natural V11() real ext-real non negative set
w2 . ys is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
w1 /. ys is Relation-like NAT -defined Function-like finite 1 -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL 1
tm /. ys is Relation-like NAT -defined Function-like finite 1 -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL 1
(w1 /. ys) - (tm /. ys) is Relation-like NAT -defined REAL -valued Function-like finite 1 -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL 1
- (tm /. ys) is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
K38(1) * (tm /. ys) is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
(w1 /. ys) + (- (tm /. ys)) is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
w2 /. ys is Relation-like NAT -defined Function-like finite 1 -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL 1
Sum w2 is Relation-like NAT -defined REAL -valued Function-like finite 1 -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL 1
(Sum w1) - (Sum tm) is Relation-like NAT -defined REAL -valued Function-like finite 1 -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL 1
- (Sum tm) is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
K38(1) * (Sum tm) is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
(Sum w1) + (- (Sum tm)) is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
Sw1 is Element of the carrier of (REAL-NS 1)
Sw0 is Element of the carrier of (REAL-NS 1)
Sw1 - Sw0 is Element of the carrier of (REAL-NS 1)
|.(Sum w2).| is V11() real ext-real non negative Element of REAL
sqr (Sum w2) is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
mlt ((Sum w2),(Sum w2)) is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
Sum (sqr (Sum w2)) is V11() real ext-real Element of REAL
sqrt (Sum (sqr (Sum w2))) is V11() real ext-real Element of REAL
ys is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
len ys is ordinal natural V11() real ext-real non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of NAT
dom ys is complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of K6(NAT)
rm is ordinal natural V11() real ext-real non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of NAT
w2 /. rm is Relation-like NAT -defined Function-like finite 1 -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL 1
rs is Relation-like NAT -defined REAL -valued Function-like finite 1 -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL 1
i is Relation-like NAT -defined REAL -valued Function-like finite 1 -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL 1
w2 . rm is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
ys . rm is V11() real ext-real Element of REAL
|.i.| is V11() real ext-real non negative Element of REAL
sqr i is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
mlt (i,i) is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
Sum (sqr i) is V11() real ext-real Element of REAL
sqrt (Sum (sqr i)) is V11() real ext-real Element of REAL
Sum ys is V11() real ext-real Element of REAL
rm is V11() real ext-real Element of REAL
rs is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
len rs is ordinal natural V11() real ext-real non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of NAT
dom rs is complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of K6(NAT)
rng rs is complex-membered ext-real-membered real-membered Element of K6(REAL)
{rm} is non empty complex-membered ext-real-membered real-membered Element of K6(REAL)
i is set
v is set
rs . v is V11() real ext-real Element of REAL
p1 is ordinal natural V11() real ext-real non negative set
rs . p1 is V11() real ext-real Element of REAL
i is set
rs . 1 is V11() real ext-real Element of REAL
m |-> (y0 / (2 * m)) is Relation-like NAT -defined REAL -valued Function-like finite m -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of m -tuples_on REAL
K213((Seg m),(y0 / (2 * m))) is non empty Relation-like Seg m -defined {(y0 / (2 * m))} -valued Function-like total quasi_total finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of K6(K7((Seg m),{(y0 / (2 * m))}))
{(y0 / (2 * m))} is non empty complex-membered ext-real-membered real-membered set
K7((Seg m),{(y0 / (2 * m))}) is complex-valued ext-real-valued real-valued set
K6(K7((Seg m),{(y0 / (2 * m))})) is set
Sum rs is V11() real ext-real Element of REAL
m * (y0 / (2 * m)) is V11() real ext-real Element of REAL
y0 / 2 is V11() real ext-real Element of REAL
(y0 / 2) / m is V11() real ext-real Element of REAL
m * ((y0 / 2) / m) is V11() real ext-real Element of REAL
i is ordinal natural V11() real ext-real non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of NAT
w2 . i is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
ys . i is V11() real ext-real Element of REAL
v is Relation-like NAT -defined REAL -valued Function-like finite 1 -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL 1
|.v.| is V11() real ext-real non negative Element of REAL
sqr v is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
mlt (v,v) is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
Sum (sqr v) is V11() real ext-real Element of REAL
sqrt (Sum (sqr v)) is V11() real ext-real Element of REAL
partdiff (f,y1,i) is Relation-like Function-like Element of the carrier of (R_NormSpace_of_BoundedLinearOperators ((REAL-NS 1),(REAL-NS 1)))
reproj (i,y1) is non empty Relation-like the carrier of (REAL-NS 1) -defined the carrier of (REAL-NS m) -valued Function-like total quasi_total Element of K6(K7( the carrier of (REAL-NS 1), the carrier of (REAL-NS m)))
K7( the carrier of (REAL-NS 1), the carrier of (REAL-NS m)) is set
K6(K7( the carrier of (REAL-NS 1), the carrier of (REAL-NS m))) is set
f * (reproj (i,y1)) is Relation-like the carrier of (REAL-NS 1) -defined the carrier of (REAL-NS 1) -valued Function-like Element of K6(K7( the carrier of (REAL-NS 1), the carrier of (REAL-NS 1)))
Proj (i,m) is non empty Relation-like the carrier of (REAL-NS m) -defined the carrier of (REAL-NS 1) -valued Function-like total quasi_total Element of K6(K7( the carrier of (REAL-NS m), the carrier of (REAL-NS 1)))
(Proj (i,m)) . y1 is Element of the carrier of (REAL-NS 1)
diff ((f * (reproj (i,y1))),((Proj (i,m)) . y1)) is Relation-like Function-like Element of the carrier of (R_NormSpace_of_BoundedLinearOperators ((REAL-NS 1),(REAL-NS 1)))
partdiff (f,i,i) is Relation-like Function-like Element of the carrier of (R_NormSpace_of_BoundedLinearOperators ((REAL-NS 1),(REAL-NS 1)))
reproj (i,i) is non empty Relation-like the carrier of (REAL-NS 1) -defined the carrier of (REAL-NS m) -valued Function-like total quasi_total Element of K6(K7( the carrier of (REAL-NS 1), the carrier of (REAL-NS m)))
f * (reproj (i,i)) is Relation-like the carrier of (REAL-NS 1) -defined the carrier of (REAL-NS 1) -valued Function-like Element of K6(K7( the carrier of (REAL-NS 1), the carrier of (REAL-NS 1)))
(Proj (i,m)) . i is Element of the carrier of (REAL-NS 1)
diff ((f * (reproj (i,i))),((Proj (i,m)) . i)) is Relation-like Function-like Element of the carrier of (R_NormSpace_of_BoundedLinearOperators ((REAL-NS 1),(REAL-NS 1)))
p1 is non empty Relation-like the carrier of (REAL-NS 1) -defined the carrier of (REAL-NS 1) -valued Function-like total quasi_total V182( REAL-NS 1, REAL-NS 1) V183( REAL-NS 1, REAL-NS 1) Lipschitzian Element of K6(K7( the carrier of (REAL-NS 1), the carrier of (REAL-NS 1)))
dom p1 is Element of K6( the carrier of (REAL-NS 1))
K6( the carrier of (REAL-NS 1)) is set
rng p1 is Element of K6( the carrier of (REAL-NS 1))
proj (i,m) is non empty Relation-like REAL m -defined REAL -valued Function-like total quasi_total complex-valued ext-real-valued real-valued Element of K6(K7((REAL m),REAL))
K7((REAL m),REAL) is complex-valued ext-real-valued real-valued set
K6(K7((REAL m),REAL)) is set
(proj (i,m)) . a is V11() real ext-real Element of REAL
<*((proj (i,m)) . a)*> is non empty V2() Relation-like NAT -defined REAL -valued Function-like one-to-one constant finite 1 -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued increasing decreasing non-decreasing non-increasing FinSequence of REAL
[1,((proj (i,m)) . a)] is V27() set
{1,((proj (i,m)) . a)} is non empty complex-membered ext-real-membered real-membered set
{{1,((proj (i,m)) . a)},{1}} is non empty set
{[1,((proj (i,m)) . a)]} is non empty Function-like set
p1 . <*((proj (i,m)) . a)*> is set
p0 is non empty Relation-like the carrier of (REAL-NS 1) -defined the carrier of (REAL-NS 1) -valued Function-like total quasi_total V182( REAL-NS 1, REAL-NS 1) V183( REAL-NS 1, REAL-NS 1) Lipschitzian Element of K6(K7( the carrier of (REAL-NS 1), the carrier of (REAL-NS 1)))
dom p0 is Element of K6( the carrier of (REAL-NS 1))
rng p0 is Element of K6( the carrier of (REAL-NS 1))
p0 . <*((proj (i,m)) . a)*> is set
w1 /. i is Relation-like NAT -defined Function-like finite 1 -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL 1
w1 . i is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
P1 is Element of the carrier of (REAL-NS 1)
tm /. i is Relation-like NAT -defined Function-like finite 1 -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL 1
tm . i is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
P0 is Element of the carrier of (REAL-NS 1)
(w1 /. i) - (tm /. i) is Relation-like NAT -defined REAL -valued Function-like finite 1 -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL 1
- (tm /. i) is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
K38(1) * (tm /. i) is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
(w1 /. i) + (- (tm /. i)) is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
P1 - P0 is Element of the carrier of (REAL-NS 1)
r . i is V11() real ext-real Element of REAL
f `partial| (X,i) is Relation-like the carrier of (REAL-NS m) -defined the carrier of (R_NormSpace_of_BoundedLinearOperators ((REAL-NS 1),(REAL-NS 1))) -valued Function-like Element of K6(K7( the carrier of (REAL-NS m), the carrier of (R_NormSpace_of_BoundedLinearOperators ((REAL-NS 1),(REAL-NS 1)))))
(f `partial| (X,i)) /. y1 is Relation-like Function-like Element of the carrier of (R_NormSpace_of_BoundedLinearOperators ((REAL-NS 1),(REAL-NS 1)))
(f `partial| (X,i)) /. i is Relation-like Function-like Element of the carrier of (R_NormSpace_of_BoundedLinearOperators ((REAL-NS 1),(REAL-NS 1)))
((f `partial| (X,i)) /. y1) - ((f `partial| (X,i)) /. i) is Relation-like Function-like Element of the carrier of (R_NormSpace_of_BoundedLinearOperators ((REAL-NS 1),(REAL-NS 1)))
||.(((f `partial| (X,i)) /. y1) - ((f `partial| (X,i)) /. i)).|| is V11() real ext-real Element of REAL
the U8 of (R_NormSpace_of_BoundedLinearOperators ((REAL-NS 1),(REAL-NS 1))) is non empty Relation-like the carrier of (R_NormSpace_of_BoundedLinearOperators ((REAL-NS 1),(REAL-NS 1))) -defined REAL -valued Function-like total quasi_total complex-valued ext-real-valued real-valued Element of K6(K7( the carrier of (R_NormSpace_of_BoundedLinearOperators ((REAL-NS 1),(REAL-NS 1))),REAL))
K7( the carrier of (R_NormSpace_of_BoundedLinearOperators ((REAL-NS 1),(REAL-NS 1))),REAL) is complex-valued ext-real-valued real-valued set
K6(K7( the carrier of (R_NormSpace_of_BoundedLinearOperators ((REAL-NS 1),(REAL-NS 1))),REAL)) is set
the U8 of (R_NormSpace_of_BoundedLinearOperators ((REAL-NS 1),(REAL-NS 1))) . (((f `partial| (X,i)) /. y1) - ((f `partial| (X,i)) /. i)) is V11() real ext-real Element of REAL
(partdiff (f,y1,i)) - ((f `partial| (X,i)) /. i) is Relation-like Function-like Element of the carrier of (R_NormSpace_of_BoundedLinearOperators ((REAL-NS 1),(REAL-NS 1)))
||.((partdiff (f,y1,i)) - ((f `partial| (X,i)) /. i)).|| is V11() real ext-real Element of REAL
the U8 of (R_NormSpace_of_BoundedLinearOperators ((REAL-NS 1),(REAL-NS 1))) . ((partdiff (f,y1,i)) - ((f `partial| (X,i)) /. i)) is V11() real ext-real Element of REAL
(partdiff (f,y1,i)) - (partdiff (f,i,i)) is Relation-like Function-like Element of the carrier of (R_NormSpace_of_BoundedLinearOperators ((REAL-NS 1),(REAL-NS 1)))
||.((partdiff (f,y1,i)) - (partdiff (f,i,i))).|| is V11() real ext-real Element of REAL
the U8 of (R_NormSpace_of_BoundedLinearOperators ((REAL-NS 1),(REAL-NS 1))) . ((partdiff (f,y1,i)) - (partdiff (f,i,i))) is V11() real ext-real Element of REAL
PP is non empty Relation-like the carrier of (REAL-NS 1) -defined the carrier of (REAL-NS 1) -valued Function-like total quasi_total V182( REAL-NS 1, REAL-NS 1) V183( REAL-NS 1, REAL-NS 1) Lipschitzian Element of K6(K7( the carrier of (REAL-NS 1), the carrier of (REAL-NS 1)))
PreNorms PP is non empty complex-membered ext-real-membered real-membered Element of K6(REAL)
{ ||.(PP . b₁).|| where b₁ is Element of the carrier of (REAL-NS 1) : ||.b₁.|| <= 1 } is set
upper_bound (PreNorms PP) is V11() real ext-real Element of REAL
pt is Element of the carrier of (REAL-NS 1)
PP . pt is Element of the carrier of (REAL-NS 1)
||.pt.|| is V11() real ext-real Element of REAL
the U8 of (REAL-NS 1) . pt is V11() real ext-real Element of REAL
pt1 is Relation-like NAT -defined REAL -valued Function-like finite 1 -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL 1
|.pt1.| is V11() real ext-real non negative Element of REAL
sqr pt1 is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
mlt (pt1,pt1) is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
Sum (sqr pt1) is V11() real ext-real Element of REAL
sqrt (Sum (sqr pt1)) is V11() real ext-real Element of REAL
((proj (i,m)) . a) ^2 is V11() real ext-real Element of REAL
K37(((proj (i,m)) . a),((proj (i,m)) . a)) is V11() real ext-real set
<*(((proj (i,m)) . a) ^2)*> is non empty V2() Relation-like NAT -defined REAL -valued Function-like one-to-one constant finite 1 -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued increasing decreasing non-decreasing non-increasing FinSequence of REAL
[1,(((proj (i,m)) . a) ^2)] is V27() set
{1,(((proj (i,m)) . a) ^2)} is non empty complex-membered ext-real-membered real-membered set
{{1,(((proj (i,m)) . a) ^2)},{1}} is non empty set
{[1,(((proj (i,m)) . a) ^2)]} is non empty Function-like set
Sum <*(((proj (i,m)) . a) ^2)*> is V11() real ext-real Element of REAL
sqrt (Sum <*(((proj (i,m)) . a) ^2)*>) is V11() real ext-real Element of REAL
sqrt (((proj (i,m)) . a) ^2) is V11() real ext-real Element of REAL
t is Relation-like NAT -defined REAL -valued Function-like finite m -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL m
t . i is V11() real ext-real Element of REAL
(t . i) ^2 is V11() real ext-real Element of REAL
K37((t . i),(t . i)) is V11() real ext-real set
|.t.| is V11() real ext-real non negative Element of REAL
sqr t is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
mlt (t,t) is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
Sum (sqr t) is V11() real ext-real Element of REAL
sqrt (Sum (sqr t)) is V11() real ext-real Element of REAL
sqr t is Relation-like NAT -defined REAL -valued Function-like finite m -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of m -tuples_on REAL
Sum (sqr t) is V11() real ext-real Element of REAL
len t is ordinal natural V11() real ext-real non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of NAT
dom t is complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of K6(NAT)
dom (sqr t) is complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of K6(NAT)
k is ordinal natural V11() real ext-real non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of NAT
(sqr t) . k is V11() real ext-real Element of REAL
t . k is V11() real ext-real Element of REAL
(t . k) ^2 is V11() real ext-real Element of REAL
K37((t . k),(t . k)) is V11() real ext-real set
(sqr t) . i is V11() real ext-real Element of REAL
||.(PP . pt).|| is V11() real ext-real Element of REAL
the U8 of (REAL-NS 1) . (PP . pt) is V11() real ext-real Element of REAL
||.(P1 - P0).|| is V11() real ext-real Element of REAL
the U8 of (REAL-NS 1) . (P1 - P0) is V11() real ext-real Element of REAL
rs . i is V11() real ext-real Element of REAL
(diff (f,y1)) - ((f `| X) /. i) is Relation-like Function-like Element of the carrier of (R_NormSpace_of_BoundedLinearOperators ((REAL-NS m),(REAL-NS 1)))
||.((diff (f,y1)) - ((f `| X) /. i)).|| is V11() real ext-real Element of REAL
the U8 of (R_NormSpace_of_BoundedLinearOperators ((REAL-NS m),(REAL-NS 1))) . ((diff (f,y1)) - ((f `| X) /. i)) is V11() real ext-real Element of REAL
i is Element of the carrier of (REAL-NS m)
i is ordinal natural V11() real ext-real non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of NAT
f `partial| (X,i) is Relation-like the carrier of (REAL-NS m) -defined the carrier of (R_NormSpace_of_BoundedLinearOperators ((REAL-NS 1),(REAL-NS 1))) -valued Function-like Element of K6(K7( the carrier of (REAL-NS m), the carrier of (R_NormSpace_of_BoundedLinearOperators ((REAL-NS 1),(REAL-NS 1)))))
y0 is Element of the carrier of (REAL-NS m)
partdiff (f,y0,i) is Relation-like Function-like Element of the carrier of (R_NormSpace_of_BoundedLinearOperators ((REAL-NS 1),(REAL-NS 1)))
reproj (i,y0) is non empty Relation-like the carrier of (REAL-NS 1) -defined the carrier of (REAL-NS m) -valued Function-like total quasi_total Element of K6(K7( the carrier of (REAL-NS 1), the carrier of (REAL-NS m)))
K7( the carrier of (REAL-NS 1), the carrier of (REAL-NS m)) is set
K6(K7( the carrier of (REAL-NS 1), the carrier of (REAL-NS m))) is set
f * (reproj (i,y0)) is Relation-like the carrier of (REAL-NS 1) -defined the carrier of (REAL-NS 1) -valued Function-like Element of K6(K7( the carrier of (REAL-NS 1), the carrier of (REAL-NS 1)))
Proj (i,m) is non empty Relation-like the carrier of (REAL-NS m) -defined the carrier of (REAL-NS 1) -valued Function-like total quasi_total Element of K6(K7( the carrier of (REAL-NS m), the carrier of (REAL-NS 1)))
(Proj (i,m)) . y0 is Element of the carrier of (REAL-NS 1)
diff ((f * (reproj (i,y0))),((Proj (i,m)) . y0)) is Relation-like Function-like Element of the carrier of (R_NormSpace_of_BoundedLinearOperators ((REAL-NS 1),(REAL-NS 1)))
0. (REAL-NS m) is V52( REAL-NS m) Element of the carrier of (REAL-NS m)
reproj (i,(0. (REAL-NS m))) is non empty Relation-like the carrier of (REAL-NS 1) -defined the carrier of (REAL-NS m) -valued Function-like total quasi_total Element of K6(K7( the carrier of (REAL-NS 1), the carrier of (REAL-NS m)))
diff (f,y0) is Relation-like Function-like Element of the carrier of (R_NormSpace_of_BoundedLinearOperators ((REAL-NS m),(REAL-NS 1)))
(reproj (i,(0. (REAL-NS m)))) (#) (diff (f,y0)) is Relation-like Function-like set
y0 is Element of the carrier of (REAL-NS m)
dom (f `partial| (X,i)) is Element of K6( the carrier of (REAL-NS m))
y0 is Element of the carrier of (REAL-NS m)
(f `partial| (X,i)) /. y0 is Relation-like Function-like Element of the carrier of (R_NormSpace_of_BoundedLinearOperators ((REAL-NS 1),(REAL-NS 1)))
r is V11() real ext-real Element of REAL
(f `| X) /. y0 is Relation-like Function-like Element of the carrier of (R_NormSpace_of_BoundedLinearOperators ((REAL-NS m),(REAL-NS 1)))
s is V11() real ext-real Element of REAL
y1 is Element of the carrier of (REAL-NS m)
y1 - y0 is Element of the carrier of (REAL-NS m)
||.(y1 - y0).|| is V11() real ext-real Element of REAL
the U8 of (REAL-NS m) is non empty Relation-like the carrier of (REAL-NS m) -defined REAL -valued Function-like total quasi_total complex-valued ext-real-valued real-valued Element of K6(K7( the carrier of (REAL-NS m),REAL))
K7( the carrier of (REAL-NS m),REAL) is complex-valued ext-real-valued real-valued set
K6(K7( the carrier of (REAL-NS m),REAL)) is set
the U8 of (REAL-NS m) . (y1 - y0) is V11() real ext-real Element of REAL
(f `partial| (X,i)) /. y1 is Relation-like Function-like Element of the carrier of (R_NormSpace_of_BoundedLinearOperators ((REAL-NS 1),(REAL-NS 1)))
((f `partial| (X,i)) /. y1) - ((f `partial| (X,i)) /. y0) is Relation-like Function-like Element of the carrier of (R_NormSpace_of_BoundedLinearOperators ((REAL-NS 1),(REAL-NS 1)))
||.(((f `partial| (X,i)) /. y1) - ((f `partial| (X,i)) /. y0)).|| is V11() real ext-real Element of REAL
the U8 of (R_NormSpace_of_BoundedLinearOperators ((REAL-NS 1),(REAL-NS 1))) is non empty Relation-like the carrier of (R_NormSpace_of_BoundedLinearOperators ((REAL-NS 1),(REAL-NS 1))) -defined REAL -valued Function-like total quasi_total complex-valued ext-real-valued real-valued Element of K6(K7( the carrier of (R_NormSpace_of_BoundedLinearOperators ((REAL-NS 1),(REAL-NS 1))),REAL))
K7( the carrier of (R_NormSpace_of_BoundedLinearOperators ((REAL-NS 1),(REAL-NS 1))),REAL) is complex-valued ext-real-valued real-valued set
K6(K7( the carrier of (R_NormSpace_of_BoundedLinearOperators ((REAL-NS 1),(REAL-NS 1))),REAL)) is set
the U8 of (R_NormSpace_of_BoundedLinearOperators ((REAL-NS 1),(REAL-NS 1))) . (((f `partial| (X,i)) /. y1) - ((f `partial| (X,i)) /. y0)) is V11() real ext-real Element of REAL
(f `| X) /. y1 is Relation-like Function-like Element of the carrier of (R_NormSpace_of_BoundedLinearOperators ((REAL-NS m),(REAL-NS 1)))
((f `| X) /. y1) - ((f `| X) /. y0) is Relation-like Function-like Element of the carrier of (R_NormSpace_of_BoundedLinearOperators ((REAL-NS m),(REAL-NS 1)))
||.(((f `| X) /. y1) - ((f `| X) /. y0)).|| is V11() real ext-real Element of REAL
the U8 of (R_NormSpace_of_BoundedLinearOperators ((REAL-NS m),(REAL-NS 1))) is non empty Relation-like the carrier of (R_NormSpace_of_BoundedLinearOperators ((REAL-NS m),(REAL-NS 1))) -defined REAL -valued Function-like total quasi_total complex-valued ext-real-valued real-valued Element of K6(K7( the carrier of (R_NormSpace_of_BoundedLinearOperators ((REAL-NS m),(REAL-NS 1))),REAL))
K7( the carrier of (R_NormSpace_of_BoundedLinearOperators ((REAL-NS m),(REAL-NS 1))),REAL) is complex-valued ext-real-valued real-valued set
K6(K7( the carrier of (R_NormSpace_of_BoundedLinearOperators ((REAL-NS m),(REAL-NS 1))),REAL)) is set
the U8 of (R_NormSpace_of_BoundedLinearOperators ((REAL-NS m),(REAL-NS 1))) . (((f `| X) /. y1) - ((f `| X) /. y0)) is V11() real ext-real Element of REAL
diff (f,y1) is Relation-like Function-like Element of the carrier of (R_NormSpace_of_BoundedLinearOperators ((REAL-NS m),(REAL-NS 1)))
(diff (f,y1)) - ((f `| X) /. y0) is Relation-like Function-like Element of the carrier of (R_NormSpace_of_BoundedLinearOperators ((REAL-NS m),(REAL-NS 1)))
||.((diff (f,y1)) - ((f `| X) /. y0)).|| is V11() real ext-real Element of REAL
the U8 of (R_NormSpace_of_BoundedLinearOperators ((REAL-NS m),(REAL-NS 1))) . ((diff (f,y1)) - ((f `| X) /. y0)) is V11() real ext-real Element of REAL
diff (f,y0) is Relation-like Function-like Element of the carrier of (R_NormSpace_of_BoundedLinearOperators ((REAL-NS m),(REAL-NS 1)))
(diff (f,y1)) - (diff (f,y0)) is Relation-like Function-like Element of the carrier of (R_NormSpace_of_BoundedLinearOperators ((REAL-NS m),(REAL-NS 1)))
||.((diff (f,y1)) - (diff (f,y0))).|| is V11() real ext-real Element of REAL
the U8 of (R_NormSpace_of_BoundedLinearOperators ((REAL-NS m),(REAL-NS 1))) . ((diff (f,y1)) - (diff (f,y0))) is V11() real ext-real Element of REAL
partdiff (f,y1,i) is Relation-like Function-like Element of the carrier of (R_NormSpace_of_BoundedLinearOperators ((REAL-NS 1),(REAL-NS 1)))
reproj (i,y1) is non empty Relation-like the carrier of (REAL-NS 1) -defined the carrier of (REAL-NS m) -valued Function-like total quasi_total Element of K6(K7( the carrier of (REAL-NS 1), the carrier of (REAL-NS m)))
f * (reproj (i,y1)) is Relation-like the carrier of (REAL-NS 1) -defined the carrier of (REAL-NS 1) -valued Function-like Element of K6(K7( the carrier of (REAL-NS 1), the carrier of (REAL-NS 1)))
(Proj (i,m)) . y1 is Element of the carrier of (REAL-NS 1)
diff ((f * (reproj (i,y1))),((Proj (i,m)) . y1)) is Relation-like Function-like Element of the carrier of (R_NormSpace_of_BoundedLinearOperators ((REAL-NS 1),(REAL-NS 1)))
(reproj (i,(0. (REAL-NS m)))) (#) (diff (f,y1)) is Relation-like Function-like set
partdiff (f,y0,i) is Relation-like Function-like Element of the carrier of (R_NormSpace_of_BoundedLinearOperators ((REAL-NS 1),(REAL-NS 1)))
reproj (i,y0) is non empty Relation-like the carrier of (REAL-NS 1) -defined the carrier of (REAL-NS m) -valued Function-like total quasi_total Element of K6(K7( the carrier of (REAL-NS 1), the carrier of (REAL-NS m)))
f * (reproj (i,y0)) is Relation-like the carrier of (REAL-NS 1) -defined the carrier of (REAL-NS 1) -valued Function-like Element of K6(K7( the carrier of (REAL-NS 1), the carrier of (REAL-NS 1)))
(Proj (i,m)) . y0 is Element of the carrier of (REAL-NS 1)
diff ((f * (reproj (i,y0))),((Proj (i,m)) . y0)) is Relation-like Function-like Element of the carrier of (R_NormSpace_of_BoundedLinearOperators ((REAL-NS 1),(REAL-NS 1)))
(reproj (i,(0. (REAL-NS m)))) (#) (diff (f,y0)) is Relation-like Function-like set
(partdiff (f,y1,i)) - (partdiff (f,y0,i)) is Relation-like Function-like Element of the carrier of (R_NormSpace_of_BoundedLinearOperators ((REAL-NS 1),(REAL-NS 1)))
PP is non empty Relation-like the carrier of (REAL-NS 1) -defined the carrier of (REAL-NS 1) -valued Function-like total quasi_total V182( REAL-NS 1, REAL-NS 1) V183( REAL-NS 1, REAL-NS 1) Lipschitzian Element of K6(K7( the carrier of (REAL-NS 1), the carrier of (REAL-NS 1)))
PreNorms PP is non empty complex-membered ext-real-membered real-membered Element of K6(REAL)
{ ||.(PP . b₁).|| where b₁ is Element of the carrier of (REAL-NS 1) : ||.b₁.|| <= 1 } is set
upper_bound (PreNorms PP) is V11() real ext-real Element of REAL
||.((partdiff (f,y1,i)) - (partdiff (f,y0,i))).|| is V11() real ext-real Element of REAL
the U8 of (R_NormSpace_of_BoundedLinearOperators ((REAL-NS 1),(REAL-NS 1))) . ((partdiff (f,y1,i)) - (partdiff (f,y0,i))) is V11() real ext-real Element of REAL
DD is non empty Relation-like the carrier of (REAL-NS m) -defined the carrier of (REAL-NS 1) -valued Function-like total quasi_total V182( REAL-NS m, REAL-NS 1) V183( REAL-NS m, REAL-NS 1) Lipschitzian Element of K6(K7( the carrier of (REAL-NS m), the carrier of (REAL-NS 1)))
PreNorms DD is non empty complex-membered ext-real-membered real-membered Element of K6(REAL)
{ ||.(DD . b₁).|| where b₁ is Element of the carrier of (REAL-NS m) : ||.b₁.|| <= 1 } is set
upper_bound (PreNorms DD) is V11() real ext-real Element of REAL
a is set
t is Element of the carrier of (REAL-NS 1)
PP . t is Element of the carrier of (REAL-NS 1)
||.(PP . t).|| is V11() real ext-real Element of REAL
the U8 of (REAL-NS 1) is non empty Relation-like the carrier of (REAL-NS 1) -defined REAL -valued Function-like total quasi_total complex-valued ext-real-valued real-valued Element of K6(K7( the carrier of (REAL-NS 1),REAL))
K7( the carrier of (REAL-NS 1),REAL) is complex-valued ext-real-valued real-valued set
K6(K7( the carrier of (REAL-NS 1),REAL)) is set
the U8 of (REAL-NS 1) . (PP . t) is V11() real ext-real Element of REAL
||.t.|| is V11() real ext-real Element of REAL
the U8 of (REAL-NS 1) . t is V11() real ext-real Element of REAL
dom (reproj (i,(0. (REAL-NS m)))) is Element of K6( the carrier of (REAL-NS 1))
K6( the carrier of (REAL-NS 1)) is set
(reproj (i,(0. (REAL-NS m)))) . t is Element of the carrier of (REAL-NS m)
||.((reproj (i,(0. (REAL-NS m)))) . t).|| is V11() real ext-real Element of REAL
the U8 of (REAL-NS m) . ((reproj (i,(0. (REAL-NS m)))) . t) is V11() real ext-real Element of REAL
(partdiff (f,y1,i)) . t is Element of the carrier of (REAL-NS 1)
(diff (f,y1)) . ((reproj (i,(0. (REAL-NS m)))) . t) is Element of the carrier of (REAL-NS 1)
(partdiff (f,y0,i)) . t is Element of the carrier of (REAL-NS 1)
(diff (f,y0)) . ((reproj (i,(0. (REAL-NS m)))) . t) is Element of the carrier of (REAL-NS 1)
((diff (f,y1)) . ((reproj (i,(0. (REAL-NS m)))) . t)) - ((diff (f,y0)) . ((reproj (i,(0. (REAL-NS m)))) . t)) is Element of the carrier of (REAL-NS 1)
||.(((diff (f,y1)) . ((reproj (i,(0. (REAL-NS m)))) . t)) - ((diff (f,y0)) . ((reproj (i,(0. (REAL-NS m)))) . t))).|| is V11() real ext-real Element of REAL
the U8 of (REAL-NS 1) . (((diff (f,y1)) . ((reproj (i,(0. (REAL-NS m)))) . t)) - ((diff (f,y0)) . ((reproj (i,(0. (REAL-NS m)))) . t))) is V11() real ext-real Element of REAL
DD . ((reproj (i,(0. (REAL-NS m)))) . t) is Element of the carrier of (REAL-NS 1)
||.(DD . ((reproj (i,(0. (REAL-NS m)))) . t)).|| is V11() real ext-real Element of REAL
the U8 of (REAL-NS 1) . (DD . ((reproj (i,(0. (REAL-NS m)))) . t)) is V11() real ext-real Element of REAL
(partdiff (f,y1,i)) - ((f `partial| (X,i)) /. y0) is Relation-like Function-like Element of the carrier of (R_NormSpace_of_BoundedLinearOperators ((REAL-NS 1),(REAL-NS 1)))
||.((partdiff (f,y1,i)) - ((f `partial| (X,i)) /. y0)).|| is V11() real ext-real Element of REAL
the U8 of (R_NormSpace_of_BoundedLinearOperators ((REAL-NS 1),(REAL-NS 1))) . ((partdiff (f,y1,i)) - ((f `partial| (X,i)) /. y0)) is V11() real ext-real Element of REAL